An Optimization Approach for Real-Time Identification Of

AN OPTIMIZATION APPROACH FOR ONLINE IDENTIFICATION

OF HARMONIC RESONANCE DUE TO

PENDING VOLT/VAR

OPERATION

Kerry D. McBee, PE

All Rights Reserve

A dissertation submitted to the Faculty and the Board of Trustees of the Colorado School of Mines in partial fulfillment of the requirements for the degree of Doctor of Philosophy (Electrical Engineering).

Golden, Colorado

Date ______

Signed:______Kerry D. McBee

Signed:______Dr. Marcelo G. Simões Dissertation Advisor

Golden, Colorado

Date ______

Signed:______Dr. Randy Haupt Head of Electrical Engineering and Computer Science Department

ABSTRACT

The emphasis on creating a more efficient distribution system has led many utility companies to employ dynamic voltage and VAr compensation (Volt/VAr) applications that reduce energy demand, generation, and losses associated with the transmission and distribution of energy. To achieve these benefits, Volt/VAr applications rely upon algorithms to control voltage support equipment, such as transformer load tap changers (LTCs), voltage regulators, and capacitor banks. The majority of these algorithms utilize metaheuristic programming methods to determine the Volt/VAr scheme that produces the most energy efficient operating conditions. It has been well documented that the interaction between capacitor bank reactance and the inductive reactance of a distribution system can produce parallel harmonic resonance that can damage utility and customer equipment. The Volt/VAr controlling algorithms that account for harmonics do so in an indirect manner that can mask harmonic resonance conditions. Unlike previous research endeavors, the primary focus of the method described within this dissertation is to identify Volt/VAr schemes that prevent harmonic resonance due to capacitor bank operation. Instead of a metaheuristic approach, the harmonic resonance identification (HRI) algorithm relies upon constrained mixed integer nonlinear programming (MINLP), which is more suited for analyzing impedance characteristics created by the energized states of a system of capacitor banks. Utilizing a numerical approach improves the accuracy of identifying harmonic resonance conditions, while also reducing the complexity of the process by exclusively relying upon the system’s admittance characteristics. The novel harmonic resonance identification method is applicable to distribution systems that are dynamically reconfigured, which can result in a number of unknown harmonic resonance producing conditions, a feature unavailable with existing controlling algorithms. The ability to identify all harmonic resonance producing configurations based upon a required compensation level also provides a utility company with a means to determine if voltage support shall be purchased from an outside source. Documentation within this dissertation describes the engineering and mathematical theories that support the MINLP dependent harmonic resonance identification algorithm.

iii

Dedicated to: My Mother and Stepfather

TABLE OF CONTENTS

ABSTRACT ...... iii

TABLE OF CONTENTS...... v

LIST OF TABLES ...... viii

LIST OF FIGURES ...... xxv

LIST OF ACRONYMS ...... xxx

LIST OF SYMBOLS ...... xxxi

CHAPTER 1 LITERARY REVIEW AND RESEARCH OBJECTIVE ...... 1

1.1 Chapter Overview ...... 1

1.2 Literary Review and State of the Art ...... 1

1.3 Motivation ...... 6

1.4 Research Objective ...... 12

1.5 Dissertation Overview ...... 14

CHAPTER 2 HARMONIC RESONANCE IDENTIFICATION PROCEDURE ...... 15

2.1 Harmonic Resonance Identification Overview ...... 15

2.2 Variable Identification ...... 17

2.3 Objective Function Formulation ...... 19

2.4 Constraint Formulation ...... 24

2.5 Optimization Formulation Summary ...... 24

2.6 Objective Function Characteristics ...... 25

2.7 HRI Algorithm Applicability...... 27

v 2.7.1 Algorithm Limitations ...... 27

2.7.2 Radial and Loop Configuration ...... 29

2.7.3 Scope of System Models ...... 29

2.7.4 Determining Frequencies of Concern ...... 30

2.7.5 Metering Customer Harmonic Impedance ...... 31

2.7.6 Single-Phase and Three-Phase Representation ...... 32

CHAPTER 3 TESTING THE HRI ALGORITHM ...... 34

3.1 Testing Overview ...... 34

3.2 Test Models ...... 34

3.3 Harmonic Resonance Identification Algorithm Convergence Analysis ...... 47

3.4 Harmonic Resonance Identification Algorithm Accuracy Analysis ...... 50

3.5 Accuracy Analysis Summary ...... 61

CHAPTER 4 PROGRAMMING INSTRUCTIONS ...... 62

4.1 Programming Description and Requirements ...... 62

4.2 Analysis Parameters and Field Monitoring ...... 64

4.3 Pseudo Code for Analysis Parameters and Field Monitoring Function ...... 71

4.4 Driving Impedance Function ...... 71

4.5 Pseudo Code for Driving Impedance Derivation Function ...... 82

4.6 Objective Function Development Function ...... 83

4.7 Pseudo Code for Objective Function Development Function ...... 85

4.8 The Optimization Function...... 85

4.9 Pseudo Code for Optimization Function ...... 91

4.10 The Results Reporting Function ...... 91

vi REFERENCES CITED ...... 93

APPENDIX A CAPACITOR BANK EFFECTS ON ADJACENT FEEDERS ...... 101

APPENDIX B RESISTANCE IMPACT ANALYSIS ...... 139

B.1 Elimination of Resistive Component ...... 140

B.2 Frequency Scan Comparison ...... 140

B.3 Analysis of X/R Ratio ...... 183

APPENDIX C AMPL CODE FOR MODELS 1, 2, AND 3 ...... 189

APPENDIX D TEST RESULTS FOR ACCURACY AND CONVERGENCE ANALYSIS ...... 203

vii

LIST OF TABLES

Table 3.1 List of loads, homes, and transformers ...... 38

Table 3.2 Construction attributes for Feeder 1358 ...... 41

Table 3.4 Stiffness comparison ratios for locations of concern ...... 45

Table 3.5 List of models utilized to evaluate the HRI algorithm ...... 47

Table 3.6 Node threshold limits for Expanded IEEE 13 Node Test Feeder Model...... 52

Table 3.7 Node threshold limits for Feeder 1358 Model ...... 52

Table 3.8 Node threshold limits for F1358 connected to F1444 Model ...... 53

Table 3.9 Harmonic Resonance Identification Accuracy Evaluation at 110% (l = 6) and 105% (l = 12) ...... 54

Table 3.10 Harmonic Resonance Identification Accuracy Evaluation at 120% (l = 6) and 110% (l = 12) ...... 55

Table 3.11 HRI optimization results for evaluating the Expanded IEEE 13 Node Test Feeder for three-Capacitor

Bank compensation (2400 kVAr total) ...... 60

Table 3.12 HRI optimization results for evaluating Feeder 1358 for two-Capacitor Bank compensation (1600

kVAr total) ...... 60

Table A-1 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Caps 1 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 1

(300 kVAr) and Cap 2 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 2 (300 kVAr)

energized ...... 108

Table A-2 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Caps 1 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 1

(300 kVAr) and Cap 2 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 2 (300 kVAr)

energized ...... 108

Table A-3 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Caps 1 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 1

(300 kVAr) and Cap 2 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 2 (300 kVAr)

energized ...... 109

viii Table A-4 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized,

Feeder D – no caps energized ...... 109

Table A-5 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized,

Feeder D – no caps energized ...... 110

Table A-6 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized,

Feeder D – no caps energized ...... 110

Table A-7 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr)

energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300

kVAr) and Cap 3 (300 kVAr) energized...... 111

Table A-8 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr)

energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300

kVAr) and Cap 3 (300 kVAr) energized...... 111

Table A-9 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr)

energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300

kVAr) and Cap 3 (300 kVAr) energized...... 112

Table A-10 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1

(300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized ...... 112

Table A-11 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1

(300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized ...... 113

ix Table A-12 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1

(300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized ...... 113

Table A-13 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder

C – no caps energized, Feeder D – no caps energized ...... 114

Table A-14 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder

C – no caps energized, Feeder D – no caps energized ...... 114

Table A-15 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder

C – no caps energized, Feeder D – no caps...... 115

Table A-16 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap

3 (300 kVAr) energized, Feeder C – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D –

Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized...... 115

Table A-17 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap

3 (300 kVAr) energized, Feeder C – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D –

Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized...... 116

Table A-18 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap

3 (300 kVAr) energized, Feeder C – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D –

Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized...... 116

Table A-19 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap

3 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 2 (300 kVAr) and Cap 3 (300

kVAr) energized ...... 117

x Table A-20 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap

3 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300

kVAr) energized ...... 117

Table A-21 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap

3 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300

kVAr) energized ...... 118

Table A-22 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr)

energized, Feeder C – Cap 3 (300 kVAr) energized, Feeder D – Cap 3 (300 kVAr) energized ...... 118

Table A-23 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr)

energized, Feeder C – Cap 3 (300 kVAr) energized, Feeder D – Cap 3 (300 kVAr) energized ...... 119

Table A-24 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr)

energized, Feeder C – Cap 3 (300 kVAr) energized, Feeder D – Cap 3 (300 kVAr) energized ...... 119

Table A-25 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized ...... 120

Table A-26 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized ...... 120

Table A-27 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized ...... 121

Table A-28 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap

xi 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D –

Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized...... 121

Table A-29 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap

3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D –

Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized...... 122

Table A-30 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap

3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D –

Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized...... 122

Table A-31 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) energized ...... 123

Table A-32 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) energized ...... 123

Table A-33 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) energized ...... 124

Table A-34 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – Cap 2 ( 300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300

kVAr) energized ...... 124

Table A-35 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – Cap 2 ( 300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300

kVAr) energized ...... 125

xii Table A-36 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV,

Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr)

energized, Feeder C – Cap 2 ( 300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300

kVAr) energized ...... 125

Table A-37 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV –

Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no

caps energized, Feeder C – no caps energized, Feeder D – no caps energized ...... 126

Table A-38 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV –

Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no

caps energized, Feeder C – no caps energized, Feeder D – no caps energized ...... 126

Table A-39 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV –

Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no

caps energized, Feeder C – no caps energized, Feeder D – no caps energized ...... 127

Table A-40 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV –

Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap

1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1(300 kVAr)

energized ...... 127

Table A-41 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV –

Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap

1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr)

energized ...... 128

Table A-42 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV –

Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap

1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr)

energized ...... 128

Table A-43 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400

xiii kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D –no caps

energized ...... 129

Table A-44 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400

kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D –no caps

energized ...... 130

Table A-45 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400

kVAr) energized, Feeder B – Caps 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder C – Caps 2

(400 kVAr) and Cap 4 (400 kVAr) energized, Feeder D – Caps 2 (400 kVAr) and Cap 4 (400 kVAr)

energized ...... 131

Table A-46 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400

kVAr) energized, Feeder B – Caps 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder C – Caps 2

(400 kVAr) and Cap 4 (400 kVAr) energized, Feeder D – Caps 2 (400 kVAr) and Cap 4 (400 kVAr)

energized ...... 132

Table A-47 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400

kVAr) energized, Feeder B – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr)

energized, Feeder C – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) energized,

Feeder D – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) ...... 133

Table A-48 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400

kVAr) energized, Feeder B – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr)

energized, Feeder C – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) energized,

Feeder D – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) energized ...... 134

xiv Table A-49 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 2 (300 kVAr) and Cap 4 (300 kVAr) energized, Feeder

B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized ...... 135

Table A-50 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 2 (300 kVAr) and Cap 4 (300 kVAr) energized, Feeder

B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized ...... 136

Table A-51 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder

B – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized, Feeder C – Caps 4 (400 kVAr) and Cap 5

(400 kVAr) energized, Feeder D – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized ...... 137

Table A-52 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder

15 kV – Underground version, Feeder A – Cap 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder

B – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized, Feeder C – Caps 4 (400 kVAr) and Cap 5

(400 kVAr) energized, Feeder D – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized ...... 138

Table B-1 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr ...... 144

Table B-2 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr ...... 144

Table B-3 Node driving impedance in ohms (harmonic frequencies 13 – 17) on Expanded IEEE 13 Node Test

Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr ...... 145

Table B-4 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

with Cap 1, 2, and 6 energized, each rated at 300 kVAr ...... 145

Table B-5 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr ...... 146

Table B-6 Node driving reactance in ohms (harmonic frequencies 13 – 17) on Expanded IEEE 13 Node Test

Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr ...... 146

Table B-7 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr ...... 147

xv Table B-8 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr ...... 147

Table B-9 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr ...... 148

Table B-10 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr ...... 148

Table B-11 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr ...... 149

Table B-12 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr ...... 149

Table B-13 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr ...... 150

Table B-14 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr ...... 150

Table B-15 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr .... 151

Table B-16 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr .... 151

Table B-17 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr .... 152

Table B-18 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr .... 152

Table B-19 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr...... 153

Table B-20 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr .... 153

Table B-21 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr .... 154

xvi Table B-22 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr .... 154

Table B-23 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr .... 155

Table B-24 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr .... 155

Table B-25 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr .... 156

Table B-26 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr .... 156

Table B-27 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr...... 157

Table B-28 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr .... 157

Table B-29 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr .... 158

Table B-30 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr .... 158

Table B-31 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr ...... 159

Table B-32 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr ...... 159

Table B-33 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr ...... 160

Table B-34 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr ...... 160

Table B-35 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr ...... 161

xvii Table B-36 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr ...... 161

Table B-37 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr ...... 162

Table B-38 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr ...... 162

Table B-39 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr ...... 163

Table B-40 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr ...... 163

Table B-41 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr ...... 164

Table B-42 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr ...... 164

Table B-43 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr ...... 165

Table B-44 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr ...... 165

Table B-45 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr ...... 166

Table B-46 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr ...... 166

Table B-47 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr ...... 167

Table B-48 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr ...... 167

Table B-49 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr ...... 168

xviii Table B-50 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr ...... 168

Table B-51 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr ...... 169

Table B-52 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr ...... 169

Table B-53 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr ...... 170

Table B-54 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr ...... 170

Table B-55 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr ...... 171

Table B-56 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr ...... 171

Table B-57 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr ...... 172

Table B-58 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr ...... 172

Table B-59 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr ...... 173

Table B-60 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr ...... 173

Table B-61 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr ...... 174

Table B-62 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr ...... 174

Table B-63 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr ...... 175

xix Table B-64 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr ...... 175

Table B-65 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr ...... 176

Table B-66 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr ...... 176

Table B-67 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr ...... 177

Table B-68 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr ...... 177

Table B-69 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr ...... 178

Table B-70 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr ...... 178

Table B-71 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr...... 179

Table B-72 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr...... 179

Table B-73 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr...... 180

Table B-74 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr...... 180

Table B-75 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder

(Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr ...... 181

Table B-76 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr...... 181

Table B-77 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr...... 182

xx Table B-78 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test

Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr...... 182

Table B-79 Typical X/R Ratio of 5 kV Underground AL Cables ...... 184

Table B-80 Typical X/R Ratio of 5 kV Underground CU Cables ...... 185

Table B-81 Typical X/R Ratio of 15 kV Underground AL Cables ...... 186

Table B-82 Typical X/R Ratio of 15 kV Underground CU Cables ...... 187

Table B-83 Typical X/R Ratio of Vertical Overhead Conductors...... 188

Table D-1 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 800 kVAr, Compensation = 1

Cap ...... 204

Table D-2 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 800 kVAr, Compensation = 2

Caps ...... 204

Table D-3 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 800 kVAr, Compensation = 3

Caps ...... 204

Table D-4 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 800 kVAr, Compensation = 4

Caps ...... 206

Table D-5 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 600 kVAr, Compensation = 1

Caps ...... 207

Table D-6 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 600 kVAr, Compensation = 3

Caps ...... 208

Table D-7 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 600 kVAr, Compensation = 4

Caps ...... 209

Table D-8 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 600 kVAr, Compensation = 5

Caps ...... 210

Table D-9 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 1

Cap ...... 210

Table D-10 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 2

Caps ...... 211

xxi Table D-11 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 3

Caps ...... 212

Table D-12 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 4

Caps ...... 213

Table D-13 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 5

Caps ...... 214

Table D-14 HRI Algorithm results for Feeder 1358, Cap = 800 kVAr, Compensation = 1 Cap ...... 215

Table D-15 HRI Algorithm results for Feeder 1358, Cap = 800 kVAr, Compensation = 2 Caps ...... 216

Table D-16 HRI Algorithm results for Feeder 1358, Cap = 800 kVAr, Compensation = 3 Caps ...... 217

Table D-17 HRI Algorithm results for Feeder 1358, Cap = 800 kVAr, Compensation = 4 Caps ...... 218

Table D-18 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 2

Caps ...... 219

Table D-19 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 3

Caps, Optimizations 1 - 25 ...... 220

Table D-20 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 3

Caps, Optimizations 26 - 30 ...... 221

Table D-21 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 4

Caps, Optimizations 1 – 25 ...... 222

Table D-22 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 4

Caps, Optimizations 26 - 50 ...... 223

Table D-23 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 4

Caps, Optimizations 51 - 67 ...... 224

Table D-24 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 2

Caps, 6 locations of concern ...... 225

Table D-25 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 3

Caps, 3 locations of concern, Optimizations 1 – 25 ...... 226

Table D-26 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 2

Caps, 3 locations of concern, Optimizations 26 - 52 ...... 227

xxii Table D-27 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 4

Caps, 6 locations of concern, Optimizations 1 – 25 ...... 228

Table D-28 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 4

Caps, 6 locations of concern, Optimizations 26 – 50 ...... 229

Table D-29 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 4

Caps, 6 locations of concern, Optimizations 51 – 69 ...... 230

Table D-30 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 5

Caps, 6 locations of concern, Optimizations 1 - 25 ...... 231

Table D-31 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 5

Caps, 6 locations of concern, Optimizations 26 - 50 ...... 232

Table D-32 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 5

Caps, 6 locations of concern, Optimizations 51 - 56 ...... 233

Table D-33 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 2 Caps, 12 locations of concern, Phase A Reactances ...... 234

Table D-34 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 2 Caps, 12 locations of concern, Phase B Reactances ...... 235

Table D-35 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 3 Caps, 12 locations of concern, Phase A Reactances, Optimizations

1 - 25 ...... 236

Table D-36 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 3 Caps, 12 locations of concern, Phase A Reactances, Optimizations

26 – 50...... 237

Table D-37 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 3 Caps, 12 locations of concern, Phase B Reactances, Optimizations

1 – 25 ...... 238

Table D-38 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 3 Caps, 12 locations of concern, Phase B Reactances, Optimizations

26 - 50 ...... 239

xxiii Table D-39 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase A Reactances, Optimizations

1 - 25 ...... 240

Table D-40 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase A Reactances, Optimizations

26 – 50...... 241

Table D-41 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase A Reactances, Optimizations

51 – 64...... 242

Table D-42 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase B Reactances, Optimizations

1 – 25 ...... 243

Table D-43 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase B Reactances, Optimizations

26 - 50 ...... 244

Table D-44 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500

kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase B Reactances, Optimizations

51 – 64...... 245

xxiv

LIST OF FIGURES

Figure 1.1 Graph comparing acceptable results of harmonic power flow simulation results ...... 4

Figure 1.2 Example of a frequency scan illustrating driving impedance at a given bus on a distribution feeder ..... 4

Figure 1.3 Harmonic spectrum illustrating increased harmonic distortion created by harmonic resonance induced

by Volt/VAr operation ...... 9

Figure 1.4 Histogram illustrating the THDV of one hundred and thirty-four 13.2 kV feeders ...... 10

Figure 1.5 Example of an eight-feeder distribution system with reconfiguring capabilities through the use of

twenty-two switches ...... 11

Figure 1.6 Configuration #1 for example reconfigurable feeder ...... 11

Figure 1.7 Configuration #2 for example reconfigurable distribution system...... 12

Figure 2.1 Flowchart of Harmonic Resonance Identification algorithm ...... 16

Figure 2.2 Block diagram illustrating Harmonic Resonance Identification Program interfaces ...... 17

Figure 2.3 Graph illustrating harmonic frequency properties of driving reactance during harmonic resonance .... 20

Figure 2.4 Frequency scan of squared driving reactance for capacitor bank energized state #1 ...... 22

Figure 2.5 Frequency scan of squared driving reactance for capacitor bank energized state #2 ...... 22

Figure 2.6 Frequency scan of squared driving reactance for capacitor bank energized state #3 ...... 22

Figure 2.7 Example two-bus two-capacitor bank system ...... 26

Figure 2.8 View #1 of three-dimensional graph of Bus #1 driving point impedance magnitude~Zd~for example

two-bus two-capacitor bank system ...... 27

Figure 2.9 View #2 of three-dimensional graph of Bus #1 driving point impedance magnitude~Zd~for example

two-bus two-capacitor bank system ...... 27

Figure 3.1 IEEE 13 Node Test Feeder ...... 36

Figure 3.2 Expanded IEEE 13 Node Test Feeder with the addition of six capacitor banks ...... 37

Figure 3.3 Diagram of secondary distribution utilized in Expanded IEEE 13 Node Test Feeder ...... 39

Figure 3.4 One-line diagram of the mainline of Feeder F1358 ...... 40

xxv Figure 3.5 One-line diagram of the combined mainlines of Feeders F1358 and F1444 ...... 42

Figure 3.6 One-line diagram of feeder with split mainline distribution ...... 44

Figure 3.7 HRI algorithm results for analyzing Feeder 1358 model (five-capacitor bank system) for three

energized banks, each rated at 800 kVAr ...... 56

Figure 3.8 HRI algorithm results for analyzing the Expanded IEEE 13 Node Test Feeder model (six-capacitor

bank system) for three energized banks, each rated at 600 kVAr ...... 56

Figure 3.9 HRI algorithm results for analyzing the Feeder 1358 connected to Feeder 1444 model (eight-capacitor

bank system) for three energized banks, each rated at 1200 kVAr ...... 57

Figure 3.10 Illustration of the reactive impedance regions associated with parallel harmonic resonance ...... 58

Figure 4.1 Flowchart illustrating the functions that are required to implement the HRI algorithm ...... 63

Figure 4.2 Flowchart of the Analysis Parameters and Field Monitoring Function ...... 64

Figure 4.3 Action block illustrating the program’s requirement to store default parameters ...... 66

Figure 4.4 Illustration of six-feeder system with forty-four switches ...... 67

Figure 4.5 Input block illustrating the interaction with external software to retrieve information regarding

switching configuration and the required level of compensation...... 67

Figure 4.6 Action block illustrating user override capabilities ...... 68

Figure 4.7 Action block illustrating the interpretation of harmonic current readings ...... 69

Figure 4.8 Action block illustrating the input requirement associated with customer harmonic impedance ...... 69

Figure 4.9 Graphical representation of customers with smart meters that collect harmonic impedance information

...... 70

Figure 4.10 Graphical representation of laterals with power quality meters that retrieve harmonic impedance

information ...... 70

Figure 4.11 Flowchart of Driving Impedance Derivation Function...... 72

Figure 4.12 Action block describing the network admittance matrix development requirement ...... 73

Figure 4.13 One-line diagram of a simple two-bus system ...... 74

Figure 4.14 Representation of Node 9 of the sample distribution system ...... 76

Figure 4.15 Action block illustrating admittance matrix simplification requirements ...... 77

Figure 4.16 Admittance matrix with only diagonals represented ...... 77

xxvi Figure 4.17 Admittance matrix illustrating the lack of connections between the circuit being analyzed and buses 2,

4, and 5 ...... 77

Figure 4.18 Elimination of rows and columns that represent buses that are not connected to the circuit being

analyzed ...... 77

Figure 4.19 Simplified network admittance matrix ...... 78

Figure 4.20 Action block illustrating capacitor bank representation ...... 78

Figure 4.21 Two-bus two-capacitor bank system ...... 79

Figure 4.22 Action block illustrating the capacitor bank counting requirements ...... 79

Figure 4.23 Action block illustrating the development of frequency dependent impedance matrices ...... 80

Figure 4.24 Action block illustrating the function’s requirement for extracting the driving reactance expressions

associated with points of concern ...... 81

Figure 4.25 Action block illustrating the simplification of driving reactance expressions ...... 82

Figure 4.26 Diagram illustrating the format of variable interaction terms within the denominator of the impedance

functions ...... 82

Figure 4.27 Flowchart of Objective Function Development Function ...... 84

Figure 4.28 Action block illustrating the function’s requirement for developing the objective function ...... 84

Figure 4.29 Illustration of the function’s requirement for calculating harmonic threshold limit ...... 85

Figure 4.30 Flowchart of Optimization Function ...... 86

Figure 4.31 Action block illustrating the function’s counter feature ...... 87

Figure 4.32 Decision block to determine if meeting the compensation requirement is feasible ...... 88

Figure 4.33 Action block illustrating the function’s ability to adjust the required compensation level ...... 88

Figure 4.34 Action block illustrating the function’s ability to optimize the objective function ...... 89

Figure 4.35 Action block illustrating the requirement for storing solutions identified through maximizing the

objective function ...... 89

Figure 4.36 Decision block illustrating the function’s ability to determine if maximizing the objective function

identified harmonic resonance conditions...... 90

Figure 4.37 Action block illustrating function’s ability to adjust constraints according to the previously identified

conditions ...... 91

xxvii Figure 4.38 Flowchart of the Results Reporting Function ...... 92

Figure A-1 4 kV model utilized in sensitivity study ...... 102

Figure A-2 15 kV model utilized in sensitivity study ...... 103

Figure A-3 Scenario #1 - Frequency scan of driving reactance at Cap 1 location on Feeder A with all underground

construction, 15 kV rating, and 400 kVAr banks ...... 104

Figure A-4 Scenario #1 - Frequency scan of driving reactance at Cap 2 location on Feeder A with all underground

construction, 15 kV rating, and 400 kVAr banks ...... 105

Figure A-5 Scenario #2 - Frequency scan of driving reactance at Cap 1 location on Feeder A with standard IEEE

13 Node Feeder construction, 4 kV rating, and 300 kVAr banks ...... 105

Figure A-6 Scenario #3 - Frequency scan of driving reactance at Cap 1 location on Feeder A with standard IEEE

13 Node Feeder construction, 4 kV rating, and 300 kVAr banks ...... 106

Figure A-7 Scenario #4 - Frequency scan of driving reactance at Cap 1 location on Feeder A with all underground

construction, 15 kV rating, and 400 kVAr banks ...... 106

Figure A-8 Scenario #4 - Frequency scan of driving reactance at Cap 2 location on Feeder A with all underground

construction, 15 kV rating, and 400 kVAr banks ...... 107

Figure B-1 Three-dimensional frequency scans comparing the actual value and absolute value of the reactance

during harmonic resonance conditions ...... 141

Figure B-2 Frequency scan comparison of the squared driving impedance and driving reactance at Cap 1 location

on Feeder A with overhead construction, 4 kV rating, and Caps 1, 3, and 5 (300 kVAr) energized on all

feeders ...... 141

Figure B-3 Frequency scan comparison of the squared driving impedance and driving reactance at Cap 1 location

on Feeder A with overhead construction, 15 kV rating, and Caps 1, 2, 3, 4, and 5 (300 kVAr) energized

on all feeders ...... 142

Figure B-4 Frequency scan comparison of the squared driving impedance and driving reactance at Cap 1 location

on Feeder A with underground construction, 15 kV rating, and Caps 1, 2, 3, 4, and 5 (500 kVAr)

energized on all feeders ...... 142

xxviii Figure B-5 Frequency scan comparison of the squared driving impedance and driving reactance at Cap 3 location

on Feeder A with underground construction, 15 kV rating, and Caps 1, 2, 3, 4, and 5 (500 kVAr)

energized on all feeders ...... 143

xxix

LIST OF ACRONYMS

BONMIN - Basic Open-source Nonlinear Mixed Integer Program

CNT - Counter in Optimization Function

COUENNE - Convex Over and Under Envelope for Nonlinear Estimation

EV - Electric vehicle

HRI - Harmonic Resonance Identification

LTC - Substation transformer load tap changer

MINLP - Constrained Mixed Integer Nonlinear Programming

MINLP BB - Mixed Integer Nonlinear Programming solver that utilized a branch and bound

approach

PCC - Point of common coupling

RMA - Resonance Mode Analysis

SCR - Short circuit ratio

THDv - Voltage total harmonic distortion

Volt/VAr - Dynamic voltage and VAr

xxx

LIST OF SYMBOLS

A - Constant ranging between 10.2 and 11.2, which is utilized in the customer harmonic impedance function

B - Constant ranging between 1.5 and 1.7, which is utilized in the customer harmonic impedance function

C - An array of real numbers that represent the harmonic reactance of specific customers or the aggregate equivalent of a number of customers

DW h - Driving reactance as a function of W for the h harmonic frequency and location l in units l of ohms

hj, - Driving point impedance as a function of W for the h harmonic frequency and location l DWl in units of ohms e - Notation representing a specific row or column of matrix

el - Notation representing row or column dimensions of matrix corresponding to specific l

F - Harmonic resonance threshold factor

Fi - Farad rating of capacitor bank i fW - Objective function utilized in harmonic resonance identification algorithm in units of ohm2 h - Multiple of fundamental frequency, which is also referred to as harmonic frequency within the document

H - Set of harmonic frequencies analyzed within the system

h Hl - Per-unit increase in reactance for location of concern l at the h harmonic frequency

Ih - Customer current demand at the h harmonic frequency in units of amps

ISC-D - Three-phase short circuit current of location of concern with highest impedance path to the fundamental source

ISC-U - Three-phase short circuit current of location of concern with lowest impedance path to the fundamental source

'Ih - Fluctuation in amps at the h harmonic frequency

k - Number of switchable capacitor banks installed on feeder being analyzed

xxxi M - Base value of fW that represents non-harmonic resonance conditions in units of ohms2

L - The set of locations of concern l analyzed within the system l - Locations of concern

ln - Total quantity of locations of concern analyzed within the objective function

P - Maximum number of capacitor bank switching configurations based upon required compensation

Q - General form of required compensation

QCAP - Required compensation in units of the number of capacitor bank

Qi - VAr rating of capacitor bank i

QNEW - New compensation constraint after relaxing QOLD by U

QOLD - Initial compensation constraint utilized by the HRI algorithm to support Volt/VAr operation

QVAR - Required compensation in units of VArs

R - Resistance in units of ohms

RL - Resistance of load in units of ohms for the example two-bus two-capacitor bank system

S - Array consisting of binary integers that represent the energized state of system switches

Sd - Energized state of switch d (1 = closed, 0 = open)

SR - Stiffness comparison ratio, which compares the three-phase short circuit current at the fundamental frequency for two locations of concern

T - Array consisting of real numbers that represent the threshold limits for monitoring specific h frequencies

W - Real number representing harmonic resonance threshold of fW in units of ohm2

TI - Maximum number of interaction terms in a driving reactance expression

U - Compensation adjustment setting in units of VArs or number of capacitor banks

Vh - Voltage at the h harmonic frequency

xxxii Vi - Voltage across capacitor bank i at the fundamental frequency

VPU - Per unit voltage

ΔVh - Fluctuation in voltage at the h harmonic frequency

>@W - Capacitor bank energized state vector of n-dimensions

wi - Variable representing energized state of capacitor bank i

ω - Angular frequency in rad/sec

X - Reactance in units of ohms

X - Reactance in ohms of capacitor bank i Ci

h Xl - Driving reactance without capacitive contribution for location l at the h harmonic frequency in units of ohms

XL - Reactance of load in two-bus two-capacitor bank example system in units of loads

Xn - Reactance of distribution line in two-bus two-capacitor bank example system in units of ohms

Xp - Expected peak reactance value during the occurrence of harmonic resonance in units of ohms

XT - Reactance of substation transformer in two-bus two-capacitor bank example system in units of ohms

>@Y - Network admittance matrix

¬¼ªºY3Bus - Network admittance matrix for example three-bus system

YC - Admittance of capacitor bank in units of siemens

Yw - Admittance of capacitor bank i as function of w in units of siemens Cii i

ªº YW h - An n x n network admittance matrix that is a function of W where only reactance values ¬¼ nn, at the h harmonic frequency are represented in the real plane

ªºYWjh, - An n x n network admittance matrix that is a function of W where both resistance and ¬¼nn, reactance values at the h harmonic frequency are represented

Z - Impedance in units of ohms

xxxiii >@Z - Network impedance matrix

h ZC - Customer harmonic impedance in units of ohms at the h harmonic frequency

Zd - Driving impedance in units of ohms

ZWh - Diagonal entry of network impedance matrix at the h harmonic frequency with only eell, reactance values represented in real plane and where el corresponds to the desired l

ZWjh, - Diagonal entry of network impedance matrix at the h harmonic frequency where el eell, corresponds to the desired l

Zh - Harmonic impedance at the h harmonic frequency in units of ohms

Zi - Impedance of capacitor bank i in units of ohms

ªºZWh - Impedance network matrix at the h harmonic frequency with only reactance values ¬¼nn, represented in the real plane as a function of W with n rows and n columns

ªºZWjh, - Complex impedance network matrix as a function of W for the h harmonic frequency ¬¼nn, with n rows and n columns

xxxiv

CHAPTER 1

LITERARY REVIEW AND RESEARCH OBJECTIVE

1.1 Chapter Overview

The primary goal of the research described within this document was to establish a novel approach for identifying parallel harmonic resonance conditions pending the operation of capacitor banks that are controlled by a

Volt/VAr application. Within this chapter, a literary review is performed to identify historical and currently accepted methods regarding Volt/VAr controlling algorithms and harmonic resonance identification. Each identified method is evaluated for implementation on an electric distribution system that operates dynamically in a harmonic rich environment.

The chapter also includes a section that summarizes the motivation for developing an algorithm that can identify harmonic resonance due to capacitor bank operation. Included within this section is an evaluation of existing levels of harmonic distortion on a typical distribution system. A sample Volt/VAr application that experienced harmonic resonance during normal operating conditions is also presented.

The chapter concludes with a list of scientific contributions that are achieved by the development of the novel harmonic resonance identification method. The hardware and software utilized to test the approach is also described within this section.

1.2 Literary Review and State of the Art

Existing harmonics distortion levels, the increased implementation of nonlinear devices, and the dynamic operation of feeders suggest that an online tool is needed to identify harmonic resonance pending Volt/VAr capacitor bank operation. Such a tool should have the following functions:

1. The ability to identify harmonic resonance on any feeder that is reconfigured due to efficiency

or reliability.

2. The ability to identify harmonic resonance without relying upon user interaction to evaluate

the results, as performed with the typical frequency scan approach.

3. The ability to identify harmonic resonance from driving impedance evaluations instead of

measuring resulting voltages, which can mask harmonic resonance conditions.

This section investigates existing Volt/VAr controlling algorithms for these described qualities to clarify the novelty of such a tool or method. Because the tool is essentially a harmonic resonance identification application directed towards Volt/VAr operation, other methods for identifying harmonic resonance are also evaluated in this section with respect to the previously identified key functions.

The primary goal of existing Volt/VAr controlling algorithms is to determine the appropriate capacitor bank state (energized or de-energized), voltage regulator setting, and/or LTC settings that produce operating conditions that minimize system losses, operating costs, and energy demand while regulating voltage [1] – [6]. To determine the optimal control scheme given a specific load, researchers have mainly applied metaheuristic programming methods, but have recently started investigating the use of numerical programming methods such as

MINLP and sequential convex programming.

Studies have shown that determining the optimal energized state of a system of capacitor banks regarding energy efficiency is a non-convex nonlinear problem, which hinders the convergence to a global extreme [7]. The complexity of the problem led to the employment of metaheuristic programming approaches such as particle swarm optimization, genetic algorithms, and the tabu search method, which have proven to converge to near global extremes when minimizing for system losses, generation, customer energy usage, and their corresponding costs [7] –

[11]. The use of these methods to specifically solve the capacitor bank energized state problem derives from their implementation in capacitor bank allocation methods, which consist of determining the optimal capacitor bank quantity, rating, and locations for energy efficiency [12] – [21].

Recently, researchers have utilized mathematical programming techniques to solve the problem of optimal

Volt/VAr operation. A sequential convex programming approach was utilized in [22] to find suboptimal solutions.

The authors of [23] developed a two-level evaluation technique with both levels relying upon MINLP.

Unfortunately, neither of the controlling algorithms described in [22] or [23] account for harmonic distortion, thereby suggesting that neither is suited for implementation in a harmonic rich environment.

Similar to the previous research endeavors into operational efficiency, the research efforts performed in [7]

– [11], [22], [23] do not account for the resulting harmonic distortion caused by harmonic resonance or the aggregate sum of customer harmonic demand. The only exception to this omission is the work performed in [11], where authors utilized harmonic power flow analysis within the optimization evaluation in the same manner as the authors of [12] - [21] evaluated the effects of harmonic distortion upon capacitor bank allocation. Harmonic distortion is evaluated through the utilization of harmonic voltage constraints, which are applied to voltages that are derived from harmonic power flow simulations. These constraints prevent capacitor bank solutions that will result in harmonic voltages above a predetermined limit. Although the harmonic power flow method is only applied to one of the developed controlling algorithms, the literature suggests that this approach is the most common method for accounting for harmonic distortion in capacitor bank optimization problems [9], [14] - [21], [23] – [27].

The metaheuristic approaches that rely upon harmonic power flow simulations can result in solutions that mask harmonic resonance even if measured data is utilized. These approaches rely upon comparisons to voltage constraints and do not examine the rate of change of the voltage if the harmonic demand were to fluctuate. Figure 1-

1 illustrates the relationship between the harmonic demand at a given location on a distribution feeder and the corresponding harmonic voltage. The difference between harmonic resonance and non-harmonic resonance conditions is the amount of driving point impedance associated with each (i.e., non-harmonic resonance => 1 Ω, harmonic resonance => 5 – 10Ω depending upon customer demand). Location A represents a harmonic voltage of

2.65 VPU for a harmonic demand of 2 amps during harmonic resonance conditions, whereas point B represents a harmonic voltage of 0.50 VPU for non-harmonic resonance conditions for the same harmonic demand. Both resulting voltages reflect acceptable configurations by the harmonic power flow approach because each are below 3%, which is the IEEE 519 threshold limit for a single harmonic voltage on a 69 kV system and below[28]. However, an increase in 1 amp of harmonic current at the specific frequency would result in a harmonic voltage of 4.21 VPU for the harmonic resonance conditions and only 0.59 VPU for the non-resonant conditions. The novel approach described in this document focuses on identifying differences between non-harmonic resonance impedance values and those that occur during harmonic resonance; therefore, the approach described in this paper would identify point

A as a harmonic resonance configuration.

One of the most utilized approaches for identifying harmonic resonance on a distribution system is performing a frequency scan, which consists of graphing the driving point impedance at a single bus or node on the electric system as a function of frequency [29] - [38]. Engineers identify harmonic resonance visually by pinpointing the frequencies that have a significantly high impedance, such as 540 Hz (i.e. h = 9) in the frequency scan illustrated in Figure 1.2. A disadvantage associated with relying upon frequency scans is that they only correspond to the frequency response at a single location for a single capacitor bank combination. A system consisting of k capacitor banks with two operating conditions (energized and de-energized) would require the human interpretation of 2k frequency scans per sensitive location to evaluate all possible harmonic resonance producing conditions. For a single conductor, single load, and single capacitor bank system, scans can be developed manually with ease. However, for systems consisting of multiple capacitor banks, the task becomes more challenging.

Figure 1.1 Graph comparing acceptable results of harmonic power flow simulation results

Figure 1.2 Example of a frequency scan illustrating driving impedance at a given bus on a distribution feeder

Methods described in [39] and [40] identify harmonic resonance by extrapolating driving point impedance from transients created by a single capacitor bank operation. The driving point impedance approximations are developed through remote monitoring of capacitor attributes, thereby eliminating the need to build a system model that includes all system components. This method essentially simplifies the typical frequency scan approach by minimizing system modeling requirements.

Unfortunately, impedance fluctuations that occur on the circuit during the period between capacitor bank operations make this method ill-suited for identifying harmonic resonance pending Volt/VAr operation. Even a staggered capacitor bank operating approach that relies upon a single operation to gauge the effects of pending capacitor bank operations would not be applicable since most customer loads increase or decrease gradually, resulting in only a single capacitor bank adjustment. Another limitation of this approach is that it can only evaluate the occurrence of harmonic resonance at capacitor bank locations, thereby preventing the identification of the harmonic resonance effects for sensitive customers that can be located throughout the circuit.

The most common approach for identifying harmonic resonance since 2005 has consisted of performing harmonic Resonance Mode Analyses (RMA), which also incorporates the use of frequency scans to identify harmonic resonance frequencies [41] - [48]. Unlike the traditional frequency scanning approach, RMA converts the nodal admittance matrix to a modal admittance matrix that represents the topology of the system instead of specific nodes. Evaluating the topology of a power system allows engineers to perform a complete harmonic analysis to identify the devices and configurations that are influencing the harmonic resonance. The application of an RMA approach can answer the following questions:

x Does harmonic resonance originate from a single bus/node?

x What devices are responsible for creating the resonance?

x How far does the resonance propagate throughout the system?

x Where is harmonic resonance more easily observed?

RMA consists of developing a modal network admittance matrix from the standard nodal admittance matrix decomposed into its eigenvalue matrix, left eigenvector matrix, and right eigenvector matrix. The eigenvalue

matrix (also referred to as an admittance modal matrix) is inverted to create a modal impedance matrix, which can be applied to frequency scans. During harmonic resonance, the diagonal values of the eigenvalue matrix approach zero, which allows for a mathematical interpretation of the occurrence of harmonic resonance.

Although RMA is a robust analysis tool in identifying the cause of harmonic resonance, it is not conducive for determining the effects of pending Volt/VAr operations. One of the key advantages of RMA is that it masks nodes that are not the source of the resonance, which is ideal for analyzing the cause of harmonic resonance.

However, it is a disadvantage when identifying nodes that are only affected by the distortion. From the point of view of harmonic sensitive customers, it does not matter whether the resonance originates at their location or another; therefore, an RMA approach may leave some customers exposed to the effects of harmonic resonance.

In summary, the literary review revealed that existing Volt/VAr algorithms do not possess the capability of directly identifying harmonic resonance pending capacitor bank operation on a dynamic feeder. There are several techniques that can identify the occurrence of harmonic resonance on a distribution system with multiple capacitor banks. However, most of the approaches rely upon performing frequency scans that require user interface.

Therefore, none of the existing harmonic resonance identification methods are adequate for employment on a

Volt/VAr application that operates in a harmonic rich environment on a dynamic system. A summary of disadvantages of the reviewed methods include:

x Harmonic power flow methods do not identify harmonic resonance directly.

x Frequency scans are limited to single bus/node and require user interpretation.

x The capacitor bank transient method relies upon the operation of a capacitor bank and frequency scans.

x RMA can identify harmonic resonance mathematically; however, it only identifies the equipment and

topology responsible for harmonic resonance, thereby leaving some customers exposed to harmonic

effects.

1.3 Motivation

The 21st century global emphasis to improve the efficiency of energy generation, transmission, distribution, and utilization has led to the implementation of a number of smart grid concepts on electric distribution systems, one being a two-way communication system that allows a utility company to meter numerous points on the system for

use in system evaluation and operation. One application that is greatly benefiting from the employment of an extensive communication system is a Volt/VAr application [49] – [55]. The implementation costs that were once offset only by the improved operational efficiency are now divided into a number of smart grid functionalities. The reduced cost of implementation has led to the increased installations of Volt/VAr applications along with increased efforts by researchers to find the most efficient controlling algorithms for the application.

The benefits of Volt/VAr applications were documented as early as 1947, but most of the research into designing and controlling did not begin until 1983 [1] – [3], [56]. The application consists of regulating the feeder voltage by controlling the operation of the LTCs and/or voltage support devices located throughout the distribution feeder, such as capacitor banks. Regulating the voltage to a small envelope has proven to reduce distribution losses, energy demand, and generation while also improving the power factor at the substation [49] – [52]. Although the operational benefits were identified a number of years ago, utilities failed to implement Volt/VAr applications system-wide because of the costs associated with installing and operating the required communication system [53],

[55].

During the early stages of Volt/VAr research, the effects of harmonic distortion were ignored. This neglect was most likely due to the low level of nonlinear device implementation and the corresponding low levels of harmonic distortion during this period. Most of the harmonic current propagating throughout a distribution system was due to commercial and industrial customers, unlike today where residential customers are also potential sources.

This lack of concern for harmonic distortion was evident in that none of the capacitor bank research endeavors performed during this period accounted for harmonic distortion [1] - [3], [57].

Although neglected in early Volt/VAr research endeavors, engineers and researchers have well documented the occurrence of parallel harmonic resonance on the distribution system due to capacitor bank operation [58] – [68].

Harmonic distortion effects on utility and customer owned equipment include degradation of insulation properties and the malfunction of electronic equipment that relies upon waveform zero crossings. Harmonic distortion on the distribution system results in the delivery of energy to electrical devices at frequencies that are higher than the fundamental, thereby resulting in the generation of heat at said frequencies. The increased thermal environment degrades the equipment’s insulation properties and can significantly reduce the life of equipment, depending upon the magnitude and duration of the harmonic distortion [29] – [31]. Besides capacitor banks, other equipment

affected by harmonic distortion includes liquid immersed distribution transformers, dry-type commercially owned transformers, customer owned motors, and underground cables [28], [29], and [69].

Because of these accelerated aging effects, high harmonic distortion on the distribution system can offset the operational benefits of a Volt/VAr application. As illustrated in Figure 1.3, a Volt/VAr controlling algorithm can initiate the energizing of a capacitor bank that results in harmonic resonance. The spectrum analysis illustrated in this figure was retrieved with an Outram PM7000 power quality meter that was temporarily installed on a 13.2 kV feeder located in Boulder, Colorado. The controlling algorithm of this Volt/VAr application was developed by

Current Group and only accounts for harmonic distortion by measuring total harmonic voltage distortion (THDV) after capacitor bank operation and comparing the values to a preset threshold limit. The disadvantage of its control methodology is that it can result in cycling between capacitor banks while searching for configurations that do not violate the preset THDV threshold limits. The inclusion of this example is not to suggest that any percentage of

Volt/VAr algorithms in operation today utilize this approach. Its inclusion is only to indicate that there are algorithms currently being employed today that can result in harmonic resonance through normal operation.

Based upon the growing implementation of nonlinear devices in the form of desktop PCs, laptops, compact fluorescent lighting, electric vehicle (EV) chargers, and renewable energy inverters, harmonic distortion on the electric distribution system may continue to increase, thereby increasing the importance of identifying and preventing the occurrence of harmonic resonance due to capacitor bank operation. The occurrence of harmonic resonance is only damaging to customer and utility owned equipment if there is a sufficient amount of harmonic current on the system to induce excessive harmonic voltages. To gain a further understanding of the amount of voltage harmonic distortion on today’s distribution system, as part of the research described within this dissertation, a study was performed to assess the level of THDV on one hundred and thirty-four 13.2 kV distribution feeders located throughout Colorado. Utilizing Outram Ranger PM7000 power quality recorders, measurements were acquired over a two-year period at varying locations on the sample feeders. The penetration of EVs with respect to the number of residential customers was less than 1%, while the percentage of renewable energy generation on the monitored circuits was less than 0.1% of the peak energy demanded.

Figure 1.3 Harmonic spectrum illustrating increased harmonic distortion created by harmonic resonance induced by Volt/VAr operation

The monitoring revealed that the average THDV was 4.73%, which is just below the 5% harmonic voltage limit set forth by IEEE 519 for distortion on a system below 69 kV [28]. Of the one hundred and thirty-four feeders monitored, 45.52% had average THDV values in excess of 5%. Although the monitoring revealed that some feeders possessed voltage distortion above the IEEE 519 threshold limit, the utility company did not experience any adverse effects or record any complaints related to harmonic distortion on any of the monitored feeders. The lack of complaints was not surprising considering the long-term effects of harmonic distortion that can go unnoticed.

Figure 1.4 illustrates a histogram of the THDV monitoring results.

The results of the monitoring study suggests that even today many distribution feeders may be experiencing voltage distortion that exceeds IEEE 519 limits even if customers or utilities are not witnessing the destructive

impact of harmonics. Based upon this information, it is recommended that utility engineers evaluate the level of harmonic distortion on their system prior to implementing new capacitor banks.

Figure 1.4 Histogram illustrating the THDV of one hundred and thirty-four 13.2 kV feeders

Identifying harmonic resonance conditions on a fixed electric distribution circuit is not difficult for a utility company to evaluate. However, smart grid philosophies have initiated the employment of, and research endeavors into, the dynamic operation of distribution circuits based upon fault isolation and operational efficiency [70] – [76].

For an electric distribution system with these dynamic capabilities, the number of possible harmonic resonance configurations increases exponentially. The typical number of switchable capacitor banks utilized for a Volt/VAr application can range from two to six. Considering only two energized states, “switched in” and “switched out”, there are 2k possible system configurations, where k is the number of switchable capacitor banks. Evaluation of all the possible harmonic resonance producing conditions becomes time consuming, and maybe impossible, when considering changes in customer nonlinear device usage, which can fluctuate hourly, daily, and seasonally. Figure

1.5 illustrates an example of a distribution system consisting of eight feeders, all of which can be reconfigured by a central controlling algorithm. Figures 1.6 and 1.7 illustrate two possible switching scenarios that result in multiple driving point impedances for the same location. For this type of dynamic operation, utility engineers require a means to identify all, if any, harmonic resonance producing capacitor bank configurations so that Volt/VAr efficiency can be maintained. If harmonic resonance cannot be avoided in delivering a specific level of

compensation, then utilities require a means to identify how much compensation should be acquired from other sources, such as customers who own inverters, if not limited by IEEE 1547 [77].

Figure 1.5 Example of an eight-feeder distribution system with reconfiguring capabilities through the use of twentytwo switches

Figure 1.6 Configuration #1 for example reconfigurable feeder

Figure 1.7 Configuration #2 for example reconfigurable distribution system

1.4 Research Objective

The research presented in this document consists of a harmonic resonance identification procedure that is applicable to pending Volt/VAr operation on a dynamic system consisting of multiple locations that are sensitive to harmonic distortion. The approach differs from existing Volt/VAr controlling algorithms and harmonic resonance identification methods in that it:

1. Relies upon a mathematical evaluation instead of visual interpretation, thus allowing for automatic

identification of harmonic resonance conditions without user interaction.

2. Can simultaneously evaluate the driving point impedance of multiple locations, phases, and frequencies;

thereby reducing the amount of time associated with identifying harmonic resonance conditions.

3. Is applicable on both radial and loop distribution systems that can be routinely reconfigured based on fault

isolation and operational efficiency.

The algorithm was tested on an expanded version of the IEEE 13 Node Test feeder that was augmented to include additional capacitor banks, distribution transformers, and customer impedances. The parameters of two real circuits located in Boulder, Colorado were also utilized to develop two additional test models. Utilizing these three models, the Volt/VAr harmonic resonance identification approach was evaluated for accuracy and convergence.

Model variances within the test environment included VAr compensation levels, system stiffness, harmonic resonance severity, and limitations regarding the size of the system that can be analyzed. Software and hardware utilized to evaluate the algorithm include:

x Mathematica – Utilized to perform symbolic math, which is required to develop the driving reactance

expressions that comprise the algorithm’s objective function [78]. Installed on a Toshiba, 64 bit, c640,

Satellite series laptop.

x AMPL – Algebraic modeling language utilized to solve high complexity problems [79].

x MINLP BB – Optimization software that utilizes mixed integer nonlinear programing branch and bound

algorithms and nonlinear relaxations for the bounding step [80]. Accessed through NEOS Server located

at http://www.neos-server.org/neos/solvers). Interfaced with AMPL modeling language.

x COUENNE - Optimization software that utilizes mixed integer nonlinear programing branch and bound

algorithms that rely upon constraint reformulation to determine the linear outer-approximation [81].

Accessed through NEOS Server located at http://www.neos-server.org/neos/solvers). Interfaced with

AMPL modeling language.

x BONMIN – Open-source optimization software that utilizes mixed integer nonlinear programing branch

and bound algorithms that rely upon reformulation of the constraints to form outer-approximations [82].

Accessed through NEOS Server located at http://www.neos-server.org/neos/solvers). Interfaced with

AMPL modeling language.

1.5 Dissertation Overview

CHAPTER 1 describes the evolution of Volt/VAr controlling algorithms, including state of the art methods. Besides Volt/VAr research, it also includes a description of currently accepted approaches for identifying harmonic resonance on an electric distribution feeder. All of these methods are evaluated for utilization on distribution feeders that operate dynamically in harmonic rich environments.

CHAPTER 2 defines the structure of the Harmonic Resonance Identification algorithm presented within this dissertation. This chapter provides a full description of the optimization approach that is utilized specifically to identify capacitor bank switching states that produce the highest driving point impedances. A thorough description of the objective function, constraints, and corresponding variables is included. The chapter also provides information regarding the algorithm’s limitations, harmonic metering requirements, system configuration considerations, and system modeling requirements.

The Harmonic Resonance Identification algorithm is analyzed for convergence and accuracy within

CHAPTER 3. A detailed description of the modeling parameters required to validate a convergence and accuracy analysis is provided. Once developed, test models based upon the parameters are utilized to evaluate optimization solver requirements that enhance convergence and accuracy. The optimization solvers tested include MINLP BB,

BONMIN, and COUENNE. The optimization solvers were interfaced utilizing AMPL modeling software. The

AMPL code for each model is included in APPENDIX C. The complete results of the accuracy analysis are included in APPENDIX D.

CHAPTER 4 is comprised of an instruction manual on how to build a computer program that implements the Harmonic Resonance Identification algorithm. Each function that is required to implement the algorithm is described graphically and with pseudo code.

APPENDIX A describes the results of a study that was performed to evaluate the influence that capacitor bank operation has on the occurrence of harmonic resonance on an adjacent feeder. An expanded version of the

IEEE 13 Node Test Feeder was utilized as a model within the study.

APPENDIX B describes the results of a study that evaluates the influence that resistance has upon driving point impedance as calculated from the mainline of a distribution feeder. The expanded version of the IEEE 13

Node Test Feeder utilized in APPENDIX A was also employed within this study.

CHAPTER 2

HARMONIC RESONANCE IDENTIFICATION PROCEDURE

2.1 Harmonic Resonance Identification Overview

The goal of the online Harmonic Resonance Identification (HRI) algorithm described in this dissertation is to identify the pending capacitor bank operations of a Volt/VAr application that will create harmonic resonance conditions. The algorithm is employable as a stand-alone analysis tool, or as a harmonic resonance filtering tool for existing Volt/VAr algorithms that seek the optimal capacitor bank configuration regarding energy efficiency. The algorithm shall have the ability to:

1. Simultaneously evaluate the occurrence of harmonic resonance at multiple points of concern on the

feeder, which may include multiple nodes, frequencies, and/or phases.

2. Determine if satisfying the required VAr compensation with capacitor banks will result in harmonic

resonance at a minimum of one point of concern on the feeder being analyzed.

3. Identify all energized capacitor bank combinations that result in harmonic resonance.

4. Identify how much capacitive compensation can be supported by the Volt/VAr application and how

much VAr support shall be supplied from other sources, if harmonic resonance cannot be avoided

satisfying the required compensation level, the algorithm will.

The HRI algorithm is dependent upon field monitoring, user defined parameters, and an optimization feature that is responsible for identifying the capacitor bank configurations that produce the highest driving point impedance values. First, the algorithm employs these features to assess the possibility of the occurrence of harmonic resonance and, upon identification, extends its search to identify all harmonic resonance producing conditions. Figure 2.1 illustrates a general flowchart of the actions performed by the HRI algorithm.

Identify current switching configuration of system and required VAr compensation level

Maximize objective function to identify capacitor bank configuration that corresponds to the maximum driving point impedance

Does harmonic resonance exist?

Yes

Maximize objective function repeatedly until all harmonic resonance producing capacitor bank configurations are identified

Yes Can VAr compensation be provided without inducing harmonic resonance?

Relax VAr compensation constraint to identify capacitor bank configuration that does not produce harmonic resonance

Identify the amount of compensation required Stop from non-capacitor bank sources

Figure 2.1 Flowchart of Harmonic Resonance Identification algorithm

The optimization function of the HRI algorithm is the feature that identifies harmonic resonance.

Monitoring information, algorithm parameters, and user supplied information all support the optimization function in the harmonic resonance identification process, as illustrated in Figure 2.2. These supporting functions are generic and are customized based upon the user’s electric system topography, equipment, efficiency goals, and customer harmonic demand. Chapter 4 includes detailed instructions on how to implement these supporting functions along with the optimization function. The remainder of this chapter defines the optimization function and the applicability of the HRI algorithm.

User and Volt/Var System Monitoring Software Interface

Harmonic Resonance Identification Program

Optimization Function

Program Parameters

Figure 2.2 Block diagram illustrating Harmonic Resonance Identification Program interfaces

2.2 Variable Identification

The HRI algorithm is directed towards identifying the energized states of a system of capacitor banks so that harmonic resonance does not occur on the corresponding electric feeder. Information required by the algorithm to determine if harmonic resonance will occur includes the following:

1. Current configuration of the electric system

2. Required level of compensation to be supported by the Volt/VAr application

3. Harmonic impedance of customers

17 4. Harmonic frequencies to be analyzed

5. Locations of concern on the feeder that shall be evaluated

6. Energized state of each capacitor bank

Items 1 through 5 are categorized as information that is specific to the existing demographics of the system being analyzed. The energized state of each capacitor bank is the only item that can vary based upon the prevention of harmonic resonance. Therefore, the variable wi is implemented to represent the energized state of the ith capacitor bank on the feeder being evaluated. The variable is binary in that only two states are considered,

“energized” and “de-energized”, which are represented by 1 and 0 respectively. The number of variables is equivalent to the number of switchable capacitor banks on the feeder being analyzed. The variables of the system are defined in Equation (2.1), where W is an n-dimensional vector called the energized state vector.

ªºw1 «»w W «»2 » (2.1) >@«»» «»» ¬¼wn

wi binary i

Capacitor bank energized state variables wi are applied to the network admittance matrix, which is utilized to identify the driving point impedances at given system locations, phases, and harmonic frequencies. Within the network admittance matrix, the capacitor bank admittance contribution (YC) becomes a function of its switching state (wi), as illustrated in Equations (2.2) and (2.3). Let H be the set of harmonic frequencies (h) analyzed within the problem. The binary characteristic of wi allows the network admittance matrix to adjust according to the state of each capacitor bank, which allows for the development of driving point impedance expressions that are functions of all capacitor bank energized states.

h Yj (2.2) C X Ci

h Yw j w (2.3) CiX i Ci

2.3 Objective Function Formulation

A common method for identifying harmonic resonance at a given location is by determining its driving point impedance. A frequency scan is developed by graphing the driving point impedance with respect to h. To narrow the search space for the HRI algorithm, h is restricted to frequencies that align with those of the harmonic currents propagating through the feeder. Although there are a number of methods for calculating driving point impedance directly, for the purpose of applying the HRI algorithm, the most applicable method is through inverting an n  n network admittance matrix to produce the network impedance matrix. The diagonals of the network impedance matrix correspond to the driving point impedances for given buses or nodes on the circuit. The occurrence of harmonic resonance is dependent upon the energized state of each capacitor bank connected to the system; therefore, the network admittance matrix, and the corresponding network impedance matrix are both functions of W, as illustrated in Equation (2.4). Hence, for a given point of concern, which is defined as a location

jh, of concern (l ) on the feeder at a specific h, the driving point impedance ( DWl ) is a function of the capacitor bank state vector W. The driving point impedance is derived from the diagonal entry (el) of the network impedance matrix that corresponds to l, as defined by Equation (2.5). Let L be the set of locations of concern numbered 1 through ln that are analyzed on the circuit.

1 ªºªºZWjh,, YW jh (2.4) ¬¼¬¼nn,, nn

DWZjh,, jh W (2.5) lee ll,

The driving point impedance is comprised of both resistive and reactive components, as represented by the j superscript. However, the frequency in which harmonic resonance occurs is a function of the inductive and capacitive properties of the system, whereas resistance only dampens the magnitude of the resulting impedance at the resonance frequency [83]. This effect is validated in a study performed in Appendix A to define the system modeling requirements to implement the HRI algorithm. The goal of the HRI algorithm is to identify the occurrence of harmonic resonance and not the resulting magnitude; therefore, to simplify the implementation of the algorithm, the resistive component of the impedance is removed without affecting the identification process. Removing the

resistive component results in a driving point impedance that is completely comprised of reactance, which transitions between the positive and negative properties during harmonic resonance, as illustrated in Figure 2.3.

Figure 2.3 Graph illustrating harmonic frequency properties of driving reactance during harmonic resonance

Without the resistive component, the driving point impedance, which is referred from here on as the driving reactance, can be restricted to a single plane. Removing the complex characteristic of the reactive impedance simplifies employment in an optimization solver. Equation (2.6) illustrates the driving reactance expression

h ( DWl ), which is defined without the complex j superscript, and how it relates to its corresponding network impedance matrix that is also represented in the real plane. The driving reactance expressions are developed by inverting a network admittance matrix that only possesses reactive values in the real plane as well, as illustrated in

Equation (2.7). The network admittance matrix shall include all admittance paths between l’s and harmonic sources, which shall include industrial, commercial, and residential contributors. By utilizing the adjugate and determinant to invert the network admittance matrix, driving reactance expressions are created that are conducive to simplification within the objective function.

DWhh Z W (2.6) lee ll,

adjªºYWh 1 ¬¼ ªºªºZWhh YW nn, (2.7) ¬¼¬¼nn,, nn det ªºYWh ¬¼nn,

The polarity transition is eliminated by squaring the driving reactance function, which only exaggerates the resulting value, but does not change the frequency in which harmonic resonance occurs. Maximizing the squared driving reactance expression that corresponds to a specific h at location l as a function of W allows for the identification of the capacitor bank switching states that result in the highest driving reactance value.

The goal of maximizing the objective function is to identify capacitor bank operations that result in harmonic resonance at multiple locations, frequencies, and/or phases of concern on the feeder. The locations can represent customers who are sensitive to harmonic distortion, the head and tail end of a feeder, or different phases of an unbalanced feeder. These different areas of concern require a multi-objective optimization problem that seeks the capacitor bank scheme that produces the maximum amount of driving reactance associated with any of these areas.

In accordance with [84], individual objective functions consisting of the same variables can be linearly combined to avoid conflict with one another. Therefore, to account for the occurrence of harmonic resonance conditions at any of the points of concern, the squared driving reactance expressions are summed to create a single objective function.

The resulting objective function is illustrated in Equation (2.8).

l n 2 fW ªº Dh W (2.8) ¦¬¼l l 1

Consider Figures 2.4 through 2.6, which illustrate frequency scans of the squared driving reactance of a single location of concern for three different capacitor bank switching schemes. If the user is concerned about the

rd th th 3 , 5 , and 7 harmonic frequencies for this single location (ln = 1), then the objective function would take the form illustrated in Equation (2.9). For this system, energized state #2 is the most favorable regarding harmonic resonance with an objective function value of 3.4 ohms2. Energized states #1 and #3 would be categorized as harmonic resonance conditions by the HRI algorithm. It should be noted that harmonic resonance occurs at the 9th harmonic frequency for energized state #2; however, this occurrence does not align with the frequencies that correspond to the harmonic current flowing through the circuit. Recall that high harmonic voltage distortion only occurs during harmonic resonance conditions if there is a sufficient amount of harmonic current propagating through the system at the resonant frequency.

222 fW ªºªºªº D357 W D W D W ¬¼¬¼¬¼111 (2.9)

Figure 2.4 Frequency scan of squared driving reactance for capacitor bank energized state #1

Figure 2.5 Frequency scan of squared driving reactance for capacitor bank energized state #2

Figure 2.6 Frequency scan of squared driving reactance for capacitor bank energized state #3

The identification of a harmonic resonance producing condition is determined by comparing the maximized value of fW to a threshold limit (W ) that represents the occurrence of one or more points of concern experiencing significant increases in reactive impedance. The threshold limit is determined by linearly increasing the base value of fW by a multiplier (F ) that is determined by identifying unacceptable levels of reactive impedance for each point of concern. The objective function’s base value (M ) is determined by setting all wi to zero in each driving reactance expression. The comparison is illustrated in Equations (2.10) through (2.13).

W FM (2.10)

Mf 0 (2.11)

W d fW Harmonic Resonance (2.12)

W ! fW Non-harmonic Resonance (2.13)

h By identifying the per-unit unacceptable increases in reactive impedance (Hl ) for each point of concern, the most conservative value for F is determined utilizing Equation (2.14). This definition, which relies upon the

h base driving reactance value of the point of concern (Xl ) as defined by Equation (2.15), assumes a violation of W can be achieved by the occurrence of harmonic resonance at a single point of concern.

22 hh h XHll X l M F (2.14) M

hh XDll 0 (2.15)

If large differences exist between the base reactance values of different points of concern, the point of concern with the larger value can dominate the value of the objective function, which can mask harmonic resonance at other points. For these conditions, each reactance expression utilized within the objective function can be

normalized to its base value. The resulting objective function, which is defined in Equation (2.16), weighs all driving reactance expression equally.

2 ln ªºDWh fW «»l (2.16) ¦ h l 1 ¬¼«»Xl

2.4 Constraint Formulation

The objective function will have a single constraint representing the amount of compensation (QVAR) required by the Volt/VAr application. As illustrated in Equation (2.10), QVAR must be equivalent to the aggregate sum of VArs (Qi) provided by each energized capacitor bank. If all capacitor banks are of the same rating, Equation

(2.17) can be simplified to Equation (2.18), where the compensation is expressed in the number of capacitor banks required (QCAP).

k (2.17) ¦ Qwi i Q VAR i 1

k (2.18) ¦wQiCAP i 1

Custom constraints that reflect specific capacitor bank operational limitations can also be developed. These types of constraints can be as simple as preventing two banks from operating in unison or limiting specific VAr compensation requirements to specific areas or branches of a circuit.

2.5 Optimization Formulation Summary

Maximize fW

s.t.

l n 2 fW ªº Dh W (2.8) ¦¬¼l l 1

DWhh Z W (2.7) lee ll,

adjªºYWh 1 ¬¼ ªºªºZWhh YW nn, (2.6) ¬¼¬¼nn,, nn det ªºYWh ¬¼nn,

k (2.17) ¦ Qwi i Q VAR i 1

k (2.18) ¦wQiCAP i 1

2.6 Objective Function Characteristics

The objective function possesses specific characteristics that are unrelated to the size of the feeder being analyzed. The function represents a problem that requires binary solutions; therefore, an integer programming method is applicable. The number of binary variables within the objective function is equivalent to the number of switchable capacitor banks installed on the feeder being analyzed. These variables, which represent the energized state of each capacitor bank, are constrained by the amount of VAr compensation required to be supplied by the

Volt/VAr application. The objective function is comprised of driving reactance expressions, which possess both maximum and minimum extremes during the occurrence of harmonic resonance, as illustrated in the frequency scan of Figure 2.3. To limit the results to a single maximum extreme, each driving reactance expression is squared, which converts the objective function into a second degree polynomial. Although the number of variables increases with the number of switchable capacitor banks, the objective function is limited to a second degree polynomial

structure if driving reactance expressions are developed through inverting the network admittance matrix as illustrated in Equation (2.6). Considering the integer variables and the nonlinear structure of the objective function, a constrained Mixed Integer Nonlinear Programming (MINLP) approach is conducive to solving the harmonic resonance identification problem.

A numerical programming approach is selected over a metaheuristic approach because of the nature of the problem and the characteristics of the occurrence of harmonic resonance. Metaheuristic approaches are ideal for identifying near global extremes regarding electric system efficiency, in which a single solution is desired. Previous research endeavors have concluded that the non-convex nature of the system efficiency problem is not suited for identifying minimum or maximum global extremes with a numerical programming approach [7]. However, the nature of this problem is not to identify the most efficient operating condition, but to identify ALL capacitor bank switching conditions that can result in harmonic resonance. Therefore, the emphasis is to identify all local extrema and not just the global extreme.

Although an electric feeder may be supported by a number of capacitor banks, harmonic resonance may only be induced by several of the banks. Harmonic resonance can occur for any energized configuration that includes these “resonant” banks. Consider the illustration in Figure 2.7 of an example two-bus two-capacitor bank system. Figures 2.8 and 2.9 illustrate how the occurrence of harmonic resonance is directly related to the MVAr rating of capacitor bank #1, and how capacitor bank #2 only dampens the magnitude of the resulting impedance.

Therefore, harmonic resonance will occur for any energized capacitor bank configuration that includes this

h “resonant” bank. For a feeder with multiple capacitor banks, each driving reactance expression ( DWl ) may have a number of local extremes associated with each resonant capacitor bank. Therefore, an optimization approach that can identify all local extremes corresponding to each resonant capacitor bank is required.

Figure 2.7 Example two-bus two-capacitor bank system

Figure 2.8 View #1 of three-dimensional graph of Bus #1 driving point impedance magnitude~Zd~for example two- bus two-capacitor bank system

Figure 2.9 View #2 of three-dimensional graph of Bus #1 driving point impedance magnitude~Zd~for example two- bus two-capacitor bank system

2.7 HRI Algorithm Applicability

2.7.1 Limitations Regarding System Size

The method utilized by the HRI algorithm to invert the network admittance matrices is the only limiting factor regarding the number of nodes utilized to model a subject feeder. To enhance simplification of the objective

function, matrix inversion is performed utilizing the determinant and adjugate of the network admittance matrix, which may possess hundreds of nodes and up to six variables that represent capacitor bank energized states. Within a software program that employs the HRI algorithm, this inversion process is performed by implementing a programming function or by establishing an interface with a mathematical software program that possesses symbolic mathematical capabilities, as described in Section 4.4. Through this matrix inversion process, driving reactance expressions that are utilized within the objective function are developed. The resulting driving reactance expressions can possess up to TI interaction terms, as defined by Equation (2.12), each possessing up to k variables.

Therefore, the size of each driving reactance expression is dependent upon the number of switchable capacitor banks analyzed and not the number of nodes utilized to model the system. For example, a feeder with six capacitor banks can result in a driving reactance expression that possesses sixty-four interaction terms in both the numerator and denominator. Ultimately, the maximum number of nodes that can be analyzed with the HRI algorithm is only limited by the capabilities of the mathematical tool employed to create driving reactance expressions symbolically from the frequency and variable dependent network admittance matrices.

k TI 22u (2.12)

The maximum number of locations of concern (ln) that can be analyzed with the HRI algorithm is currently unknown. A single l is defined as a node on the subject feeder that is analyzed for the effects of harmonic resonance. Each location of concern is frequency dependent; therefore, a single node on the feeder that is analyzed for the 3rd, 5th, and 7th harmonic frequencies would equal three points of concern. For accuracy, which is detailed in

Section 3.4, the objective function should be comprised of driving reactance expressions that represent points of concern with varying harmonic resonance characteristics. Identifying the commonality of harmonic resonance characteristics between two locations of concern is performed by evaluating their Stiffness Comparison Ratio (Sr).

Applications of the ratio are detailed on page 42 of Section 3.2, while a definition of values as they correspond to harmonic resonance characteristics is detailed in paragraph two on page 58 of Section 3.4.

The accuracy analysis performed in Section 3.4 revealed that the typical low impedance design of distribution feeders can result in nodes located throughout the feeder that possess similar, if not identical harmonic

resonance characteristics. By limiting the objective function to driving reactance expressions that represent nodes with varying harmonic resonance characteristics, the number of points of concerns experiencing harmonic resonance simultaneously may be as low as the number of harmonic frequencies analyzed. Based upon the research performed in [86] - [88], voltage harmonic distortion on an electrical distribution system is typically caused by harmonic currents at the 3rd, 5th, and 7th harmonic frequencies due to phase cancellation. Utilizing MINLP BB, the HRI algorithm tested successfully with up to five points of concern experiencing harmonic resonance simultaneously. It was concluded that the algorithm, which also tested successfully on an eight switchable capacitor bank system with twelve points of concern, should converge accurately for the number of locations and points of concern identified on a typical distribution feeder if objective function limitations are maintained.

2.7.2 Radial and Loop Configuration

The HRI algorithm is applicable to both radial and loop configured feeders with single and multiple energy sources. The key for application of the HRI algorithm for any of these conditions is the inclusion of all impedances or admittances associated with conductors, cables, and connected equipment that affect driving point impedances at the designated locations of concern. If unknown, the harmonic impedance associated with harmonic sources, whether from a distributed generating unit, transmission system, or connected customer, shall be monitored in accordance to Section 2.7.5. If the impedance behind a given node is considered infinite, then the node shall be grounded in the network admittance matrix in accordance with [29].

2.7.3 Scope of System Models

The HRI algorithm determines the occurrence of harmonic resonance by identifying the capacitor bank configurations that result in the maximum driving reactance. Because the algorithm is dependent upon system reactance, it is imperative that all impedance or admittance values associated with the feeder under evaluation be included in the network admittance matrix. Harmonic resonance conditions on the subject feeder can be affected by the impedance of the substation transformer, upstream transmission system, feeders connected to the same substation transformer, and the operation of capacitor banks on these adjacent feeders. Influences of these items are dependent upon the impedance associated with substation construction, substation equipment, feeder construction, capacitor bank ratings, and capacitor bank locations.

As part of this dissertation, a study was performed to evaluate the influence that capacitor bank operations have on the occurrence of harmonic resonance on feeders that are connected to the same substation transformer.

The results, which are described in Appendix B, suggest that adjacent capacitor bank operations can dampen the magnitude of resonance, but may not affect the frequency in which it occurs. The impact on driving point impedance, and thereby harmonic resonance characteristics, is a function of the amount of impedance associated with the electrical connection between the location of concern and the location of the switching capacitor bank. The more impedance associated with the connection (as compared to the total value of the driving point impedance), the less of an impact it will have on harmonic resonance characteristics.

It is delegated to the utility company employing the HRI algorithm to determine the extent of the admittance matrix based upon system construction attributes.

2.7.4 Determining Frequencies of Concern

There are two options for establishing the harmonic frequencies that are analyzed with the HRI algorithm.

The first approach consists of defining default frequencies that will always be analyzed regardless of the system configuration and harmonic currents produced by customers. Although simplistic in its application, the approach can result in analyzing frequencies that have no impact on the system. The second approach for establishing harmonic frequencies of concerns is beneficial for an electric system that is dynamically switched routinely and, therefore, has varying harmonic currents propagating through it. For this dynamic approach, the short circuit ratio

(SCR) can be utilized to establish which frequencies should be analyzed.

The harmonic current limitations in IEEE 519 are based upon approximations utilizing the SCR as calculated at the customer’s point of common coupling (PCC) [28]. The document was prepared by IEEE Working

Group on Power System Harmonics of the Transmission and Distribution Committee of the IEEE Power

Engineering Society and the Harmonic and Reactive Compensation Subcommittee of the Industrial Power

Conversion Committee of the IEEE Industry Applications Society. The working group assumed that the SCR at the

PCC is an indicator of how the customer’s harmonic current will affect the harmonic voltage on the system. A SCR below 20 at the PCC represents a customer who can greatly influence the harmonic voltage on the feeder, whereas a

SCR greater than 1000 signifies a customer who is one of many and does not greatly influence the harmonic voltage.

On a feeder that is reconfigured due to fault isolation or operational efficiency, the SCR of customers can fluctuate;

hence, their effect on the voltage of said feeder can also fluctuate. Although this may not apply to residential customers because of their low demand at the fundamental frequency, it does apply to industrial and commercial customers who can inject significant amounts of harmonic current into the system.

Consider a large industrial customer who does not affect the harmonic voltage for residential customers located a mile away under the current switching configuration. As described by general circuit analysis, harmonic currents propagate through the circuit based upon the system impedances and their connections throughout the system. A change in system configuration can result in changes to these connections and impedances, which can thereby change the propagation of harmonic currents. Under these new impedance conditions, it is possible that the industrial facility now affects the harmonic voltage of customers located a mile away.

There are two options for utilizing SCR’s to evaluate harmonic current propagation. The first consists of monitoring all commercial and industrial customers and calculating their SCR based upon the new circuit configuration. This will give a utility company an indication of which customers can influence the harmonic voltage distortion and at which frequencies. The second approach consists of calculating the SCR for locations on the system that are remotely monitored and serve a number of customers. This is similar to the approach utilized in

IEEE 519 except that the individual customer rating is replaced with the aggregate demand of customers downstream of the meter, which can be retrieved through historical data. Calculating the SCR at metering locations will indicate how the measured harmonic currents at these locations will affect the voltage through the feeder. Due to legacy system construction, design standards, and equipment differences, it is delegated to the utility company that is employing the HRI algorithm to evaluate the impact that specific SCR’s will have on specific circuits.

2.7.5 Metering Customer Harmonic Impedance

Implementation of the HRI algorithms requires the retrieval of information regarding customer harmonic impedance. The driving reactance expressions are developed from the inversion of frequency specific network admittance matrices, which account for all admittances that impact harmonic resonance on the subject feeder. The harmonic impedance of customers is retrieved through field monitoring, either at the individual customer or remote locations throughout the system. If an invasive method is employed to calculate the harmonic impedance as described in [89], the Gibbs Phenomenon must be taken into account due to the discontinuous signal that is created during the measuring application.

The Gibbs Phenomenon describes the overshoot or undershoot associated with monitoring a discontinuous signal with equipment that relies upon Fourier series expansion to derive at harmonic quantities [90] – [91]. The method described in [89] for determining harmonic impedance consists of measuring changes in harmonic voltage and current at a given node of the system, while also varying the system impedance upstream of the location. The approach is described mathematically in Equation (2.13). Any device and/or algorithm calculating harmonic impedance in this manner must account for the overshoots and undershoots created by the system impedance change.

'Vh Zh h H (2.13) 'Ih

A non-invasive approach for measuring harmonic impedance is the preferred method of capturing the information related to customers. Several non-invasive methods are described in detail in [92] – [97].

Although harmonic resonance is characterized by abrupt changes in driving reactance as a function of frequency, the Gibbs Phenomenon does not impact any other feature of the HRI algorithm. The identification of harmonic resonance is dependent upon calculating a specific driving reactance value for a “fixed” frequency through the use of the network admittance matrix. Fourier series analysis, which fails to converge uniformly at discontinuities, is not utilized in the identification process.

2.7.6 Single-Phase and Three-Phase Representation

The HRI algorithm is applicable to both single-phase and three-phase harmonic resonance evaluations. For balanced systems, where the capacitor banks are connected in a grounded wye configuration, a single-phase representation is sufficient. However, if the feeder is unbalanced or any of the capacitor banks analyzed are connected in an ungrounded wye or delta configuration, the network admittance matrix must consist of a three-phase representation of the circuit.

Three-phase representation is accounted for in the objective function by establishing points of concern for each phase. As illustrated in Equation (2.8), the HRI algorithm objective function is comprised of driving reactance expressions associated with different points of concern on the feeder. For three-phase representation, the driving

reactance associated with each phase that corresponds to a location of concern on the system is included within the objective function.

CHAPTER 3

TESTING THE HRI ALGORITHM

3.1 Testing Overview

This chapter describes the analysis performed utilizing electric feeder models to test the HRI algorithm for convergence and accuracy in identifying harmonic resonance conditions due to capacitor bank operations. Each feeder model possessed impedance attributes that represented the frequency response characteristics typical of distribution feeders in operation today. The Distribution System Analysis Subcommittee of the IEEE PES Power

System Analysis, Computing, and Economics Committee developed a number of test feeders that possess realistic configurations that are comprised of system components such as transformers, voltage regulators, capacitor banks, customer demand, underground cables, and overhead conductors. These models provide a means for researchers to compare and evaluate new power system analysis tools [98]. Unfortunately, the goal of the HRI algorithm corresponds to how capacitor bank operations can influence the harmonic resonance characteristics of a distribution feeder, which is a novel application unrelated to the typical employment of the test feeders. Therefore, the validity of utilizing a standard IEEE test feeder to evaluate the HRI algorithm was thoroughly investigated.

Testing the HRI algorithm for convergence and accuracy requires a model that possesses realistic driving point impedance frequency response characteristics during capacitor bank operations. Engineers and researchers have well documented parallel harmonic resonance, which is characterized by a nonlinear increase in inductive reactance, followed by the rapid transition to capacitive reactance at the resonant frequency, as illustrated in Figure

2.3 in Chapter 2. Therefore, a model that is utilized to test the HRI algorithm must have impedance characteristics and, therefore, a harmonic resonance characteristic that is representative of the distortion that occurs on today’s electric distribution circuits.

3.2 Test Models

Three test models were developed based upon the frequency response criteria. The first model is an expanded version of the IEEE 13 Node Test feeder, which was adjusted to include distribution laterals, transformers, secondary conductors, and customer harmonic impedance. To obtain realistic driving point impedance values, all

impedances found on a distribution system that can affect harmonic resonance were accounted for in the model. To validate the results achieved with the expanded IEEE 13 Test Feeder, the algorithm was applied to two models that were built from attributes of two 13.2 kV distribution feeders located in Boulder, Colorado. The first circuit model, which is referred to from here on as Feeder F1358, serves approximately 2200 customers. By connecting Feeder

F1358 to an adjacent circuit, which is referred to from here on as Feeder F1444, a third model was developed for testing. This third model represents emergency conditions that exist when a feeder is extended due to fault isolation and/or outage restoration. Each model consists of only the single phase representation of a distribution system, which is the typical approach utilized in performing frequency scans [29], [30]. Variations in system stiffness were accounted for in the selection of the locations of concern on each model.

The IEEE 13 Node Test Feeder, which is illustrated in Figure 3.1, was selected as a base model because of its unbalanced loading and construction topography, which includes both overhead and underground construction

[99]. Several of the test feeder attributes were expanded upon to create a model more suitable for testing the HRI algorithm.

The first adjustment corresponds to the MVA rating of the feeder and the number of capacitor banks required to achieve near unity power factor during operation. The test feeder is rated at 4 kV and only delivers approximately 1.2 MVA per phase. Capacitor banks intended to provide VAr support are typically sized in increments of 100 kVAr per phase. Applying these typical ratings to the test feeder would result in the requirement of two capacitor banks to maintain the circuit’s power factor above 0.95 lagging. This small number of capacitor banks is not representative of the maximum number of capacitor banks that can be installed on a typical distribution feeder, which can be as high as six. This high number of capacitor banks typically occurs only on distribution systems rated 15 kV and above. Due to the lower admittance values associated with capacitor banks at higher voltage ratings, system rated at 15 kV result in driving reactance expressions that are more complex than those developed at higher voltage ratings.

The number of capacitor banks was increased by doubling the test feeder in physical size and customer demand. By connecting Node 680 of the standard IEEE 13 Node Test Feeder to Node 632 of a duplicate feeder, the circuit was increased to a size requiring a minimum of four capacitor banks to maintain sufficient power factor.

Because only a single phase representation was required for testing, the upstream feeder was modeled after the IEEE

Test Feeder’s A Phase loading, while the downstream portion of the feeder was modeled after the B Phase loading.

By increasing the loading on the circuit, the standard construction would result in voltage drops along the circuit’s mainline that could possibly violate acceptable operating voltage levels as defined by ANSI Std. C84.1 [100].

Therefore, the voltage on the circuit was increased to 7.62 kV line to neutral (13.2 kV rating). Testing at this voltage also corresponds to the most complex driving reactance expressions, which increases the validity of the accuracy and convergence testing results. To account for the increased voltage rating, underground cables utilized in the standard IEEE 13 Node Test Feeder were replaced with their 15 kV equivalent, which possessed higher inductance. The resulting test feeder, which is referred to from here on as the Expanded IEEE 13 Node Test Feeder, delivers approximately 2.6 MVA per phase and requires a minimum of four capacitor banks to maintain voltage and power factor. To evaluate multiple capacitor bank options, a total of six switchable capacitor banks were installed on the Expanded IEEE 13 Node Test feeder, which is illustrated in Figure 3.2.

Figure 3.1 IEEE 13 Node Test Feeder

The standard IEEE test feeder possesses two capacitor banks: a three-phase 600 kVAr unit installed at node

675 and a single-phase 100 kVAr unit installed at node 611. However, this design does not result in harmonic resonance conditions that are suitable for testing the HRI algorithm. To induce harmonic resonance conditions, the standard banks were removed and capacitor banks with adjustable ratings were added to nodes 646, 632, 634, 611,

671, and 675, as illustrated in Figure 3.2. The capacitor bank sizes were varied during the analysis so that multiple levels of harmonic resonance severity could be evaluated.

Because harmonic resonance is based upon the interaction of inductive reactance and capacitive reactance within the circuit, all components that affect these attributes throughout the circuit must be accounted for within the test model. To ensure model accuracy, distribution transformers, secondary conductors, and customer impedances, which were neglected in the original IEEE 13 Node Test feeder, were added to the expanded version created for testing the HRI algorithm [99]. The influence that customer impedance has upon distribution system harmonic resonance characteristics is defined in the research performed by the author in [101].

Figure 3.2 Expanded IEEE 13 Node Test Feeder with the addition of six capacitor banks

The secondary distribution system was built based upon the load information provided by the IEEE 13

Node Test Feeder description [99]. Customers were assumed to have a demand of 5 kVA. Secondary conductors between the distribution transformers and customers were assumed to be 100 feet of 350 kcmil triplex underground cable that was manufactured by Southwire. Table 3.1 lists the total demand, number of customers, and the number of transformers connected to each node and phase.

Table 3.1 List of loads, homes, and transformers per node per phase for expanded IEEE 13 Node Test Feeder Load Homes Xfmr Node Phase (kVA) (No.) (No.) A 194 39 4 634 B 150 30 3 C 150 30 3 A 0 0 0 645 B 211 42 4 C 0 0 0 A 0 0 0 646 B 265 53 5 C 0 0 0 A 154 31 3 652 B 0 0 0 C 0 0 0 A 443 89 9 671 B 443 89 9 C 443 89 9 A 521 104 10 675 B 91 18 2 C 359 72 7 A 0 0 0 692 B 0 0 0 C 227 45 5 A 0 0 0 611 B 0 0 0 C 188 38 4

The distribution transformers utilized in the model were based upon an ABB 50 kVA, high efficiency transformer. Each transformer served ten customers who were connected in parallel similar to the four- home representation illustrated in Figure 3.3. The specific transformer model was based upon information provided in

[29] and [102] and the specifications were provided by ABB. The no-load losses of the ABB 50 kVA transformer were 98 watts, which led to a core resistance of 588 ohms referred to the secondary side.

Research performed by the author identified Equation (3.1) as the relationship between the harmonic

h impedance (ZC ) of an end-user and the amount of harmonic current that the end-user injects back into the distribution system (Ih) [101]. The harmonic impedance function, which relies upon the constants A (10.2 – 11.2) and B (1.5 – 1.7), is applicable to the 3rd, 5th, and 7th harmonic frequencies. For the expanded IEEE 13 Node Test

Feeder model, the inductive reactance for each customer was assumed to be 0.57 ohms for all frequencies, which corresponds to the harmonic impedance associated with 15% harmonic current injection at the rated fundamental demand (41.7 amps).

Figure 3.3 Diagram of secondary distribution utilized in Expanded IEEE 13 Node Test Feeder

h B ZAICh u (3.1)

The original 5 MVA substation transformer utilized with the IEEE 13 Node Test Feeder was replaced with a 10 MVA transformer to conform to the feeder expansion. The impedance of the transformer remained at 8%.

A second distribution feeder model was developed from design attributes of a 13.2 kV feeder located in

Boulder, Colorado. Cases of harmonic resonance have been well documented on this feeder, which possesses five switchable capacitor banks [86]. Figure 3.4 illustrates a one-line diagram of the mainline of Feeder F1358. The construction attributes of the circuit’s mainline, which include both overhead and underground construction, are listed in Table 3.2. The feeder is rated at 17 MVA and draws its power from a 40 MVA power transformer that serves two other feeders. The impedance of the transformer is 8% with an X/R ratio of 20, which corresponds to typical values [103].

Figure 3.4 One-line diagram of the mainline of Feeder F1358

A third model was developed by connecting Feeder F1358 to an adjacent circuit, Feeder F1444. Table 3.3 lists the construction attributes of the feeder, which consists of both overhead conductors and underground cables.

The combined circuits deliver 34 MVA to over 3500 customers. The voltage is supported by eight switchable

capacitor banks located throughout the system, as illustrated in Figure 3.5. This model represents extreme operating conditions that can occur during dynamic system operation.

Table 3.2 Construction attributes for Feeder 1358 Connection Length [Node] – [Node] Size [ft] Construction *1 – 3 1000 kcmil CU 5172 Underground 2 – 3 500 kcmil AL 1296 Underground 3 – 4 #2 ASCR 501 Vertical Overhead 3 – 7 795 ASCR 6405 Vertical Overhead 5 – 6 1000 kcmil AL 568 Underground 6 – 7 1000 kcmil AL 748 Underground 7 – 8 1000 kcmil AL 1593 Underground 7 – 10 795 ASCR 983 Vertical Overhead 9 – 10 #2 ASCR 1820 Vertical Overhead 10 - 11 795 ASCR 1495 Vertical Overhead 10 - 13 795 ASCR 1152 Vertical Overhead 11 - 12 795 ASCR 3201 Vertical Overhead * The connection also includes the impedance of a three phase 28 MVA power transformer

Locations of concern on each model were determined based upon the stiffness between nodes. The HRI algorithm can be called upon to evaluate the driving reactance at a number of locations on a distribution feeder.

Evaluating system nodes that share the same harmonic resonance characteristics (i.e., resonance occurs at the same frequency and results in the same impedance increase) due to their location on the system is not necessary. The difference in harmonic resonance characteristics between locations is a function of the stiffness of the connection between the two possible nodes. For a stiff connection regarding harmonic resonance, the impedance between nodes is not sufficient to produce two separate harmonic resonance characteristics. For a weak connection, the inductive reactance between locations is sufficient to produce harmonic resonance at varying frequencies. The amount of impedance required to change the harmonic resonance characteristics is also dependent upon the ratings and locations of capacitor banks in reference to the nodes of concern. To analyze the effects of both stiff and weak connections, the locations of concern for each model were selected to represent connections with different stiffness attributes.

Figure 3.5 One-line diagram of the combined mainlines of Feeders F1358 and F1444

Comparing frequency scans for all possible capacitor bank energized state combinations is the most accurate means of measuring the connection stiffness between two nodes regarding harmonic resonance characteristics. Although this approach is sufficient for evaluating the HRI algorithm, it is not an applicable approach for engineers implementing the HRI algorithm who are identifying locations of concern to be analyzed.

Therefore, a method of comparing the harmonic resonance characteristics of two separate nodes that is also applicable to implementation within the HRI algorithm was developed. The approach relies upon the calculated fault current at each node and is referred to from here on as the stiffness comparison ratio.

The stiffness comparison ratio (SR) is based upon the assumption that nodes experiencing different fault currents, which is a function of system impedance, will also possess different harmonic resonance characteristics.

Evaluating the connection strength consists of comparing the magnitude of short circuit current at the fundamental frequency for each location with all switchable capacitor banks de-energized. The node that experiences the highest fault current, which typically occurs at the node with the smallest impedance path to the source, is utilized to

normalize the fault current at the adjacent node, as illustrated in Equation (3.2). The SR is only intended to provide insight into impedance variations as seen from two different nodes, and does not truly measure the amount of impedance between said locations. For instance, a SR = 1 indicates that two nodes experience the same magnitude of fault current, which can occur between two locations near one another or between two locations on opposite sides of a branched feeder. However, it can be inferred that a value other than 1 indicates that the two nodes do not share identical harmonic resonance characteristics, which is the goal of this analysis and a requirement of the objective function. For the purpose of evaluating the connection between nodes, an SR = 1 indicates a stiff connection, which represents two nodes that have identical harmonic resonance characteristics.

ISC D SR (3.2) ISC U

Because utilizing the stiffness comparison ratio is only a guideline to assist in identifying nodes with similar harmonic resonance characteristics, its accuracy relies upon specific application conditions. Utilization is limited to nodes that are served by the same mainline distribution conductor/cable, as illustrated with Nodes A and B in Figure 3.6. This guideline assumes that two nodes that are served by two separate mainline distribution conductor/cables, as illustrated in with Nodes A and D in Figure 3.6, will have different harmonic resonance characteristics regardless of their fault current.

In the US, mainline distribution conductor/cables are typically sized equal to or greater than 350 kcmil.

Resistance is the largest difference between mainline and sub-distribution (lateral) conductors and cables. The inductance, which affects the occurrence of harmonic resonance, differs little between cable/conductor sizes. For example, a larger sized mainline conductor of 795 ASCR that is constructed vertically possesses an inductive reactance of 0.4921 ohms/mile, whereas a #4 ASCR conductor, which is one of the smaller sizes in the sub- distribution class, possesses a reactance of 0.7351 ohms/miles for the same type of construction. Although the lateral conductor reactance is 33% greater, its route may only extend 1000 feet whereas mainline distribution conductors/cables can extend multiple miles.

Figure 3.6 One-line diagram of feeder with split mainline distribution

Another stipulation is associated with nodes located on distribution laterals that are supported by a switchable capacitor bank(s). These nodes shall not be included in a SR evaluation with nodes located on other laterals due to their proximity to the VAr support device. Nodes in this category, exemplified as Node C in Figure

3.6, will most likely have different harmonic resonance characteristics and should therefore, be included in the analysis regardless.

Nodes connected to the secondary side of distribution transformers, which possess sufficient inductive reactance to create individual harmonic resonance characteristics, are not applicable to the stiffness comparison ratio evaluations with nodes connected to the primary windings. As with nodes that are served by different mainline conductors/cables, nodes on the secondary system should possess different harmonic resonance characteristics than those located on the primary system. However, the SR is applicable to nodes connected to the same secondary distribution system.

Applying this overall methodology to the feeder illustrated in Figure 3.6 results in two sets of stiffness comparison ratio evaluations. The first evaluation consists of Nodes A and B, while the second consists of Nodes E and D. Regardless of the fault current, Nodes C would be included within the objective function.

For the expanded IEEE 13 Node Test Feeder, Nodes 692A and 684B were selected as locations of concern.

The SR between these two nodes was 0.524, which reflects that a fault occurring at Node 684B would experience a current that was 52.4% of the fault current that would occur at Node 692A, given the same fault impedance.

Nodes 4 and 11 were designated as locations of concern on Feeder F1358. The connection between these nodes is weaker than the connection between Nodes 692A and 684B, with a comparison ratio of 0.289.

The harmonic resonance effects at Nodes 4 and 15 were analyzed on the combined model of Feeders F1358 and F1444. The comparison stiffness ratio between these two nodes was 0.190, which represents a weaker connection than the tie between the locations of concern on Feeder F1358. The stiffness comparison ratio for each pair of nodes is listed in Table 3.4, along with the corresponding three-phase fault current at each location.

Table 3.3 Stiffness comparison ratios for locations of concern

Node ISC (amps) SR 694A 11,995 0.524 670B 6,288 4 19,885 0.289 11 5,748 4 19,885 0.190 15 3,789

Each location was analyzed for the 3rd, 5th, and 7th harmonic frequencies, which correspond to the information provided in [86] - [88] that identify these frequencies as the most common on an urban distribution system. For the combined feeder, each location of concern was analyzed for the A-phase and B-phase effects. To simulate an imbalance between locations of concern, 1200 kVAr rated capacitor banks were utilized on A-phase, while 1500 kVAr rated capacitor banks were utilized on B-phase.

Considering the locations of concern designated for each model along with their frequency dependency, objective functions were developed to analyze six and twelve driving reactance expressions or points of interest.

The HRI algorithm must converge to accurate solutions for non-harmonic resonance conditions, low harmonic resonance conditions, mild harmonic resonance conditions, and high harmonic resonance conditions. Each category is defined as follows:

Level 0 “Non-harmonic resonance” conditions occur when the driving point impedance at all points of

concern are unaffected by capacitor bank operation.

Level 1 “Low harmonic resonance” conditions occur when the driving point impedance at a single

location of concern and specific frequency is high due to harmonic resonance.

Level 2 “Medium harmonic resonance” conditions occur when the driving point impedances at two

locations of concern for any frequency is high due to harmonic resonance.

Level 3 “High harmonic resonance” conditions occur when the driving point impedances at more than

two locations of concern for any frequency are high due to harmonic resonance.

These ratings are based upon the influence that the occurrence of harmonic resonance has on the feeder being evaluated. The assumption is that none of the locations of concern are connected by a tie that possesses a SR that suggests identical harmonic resonance characteristics. Therefore, all nodes evaluated within the objective function represent parts of the feeder with different harmonic resonance characteristics. The “Low harmonic resonance” condition is an indication that only a small section of the distribution circuit is affected by harmonic resonance. “Medium harmonic resonance” indicates that more than one location on the distribution system has abnormally high impedance due to resonance. A “High harmonic resonance” condition is an indication that three or more points of concern on the circuit are affected by harmonic resonance. Testing the HRI algorithm for each of these conditions is paramount to evaluating the convergence and accuracy of the novel harmonic resonance identification approach. To account for each of these harmonic resonance conditions in the test environment, the ratings of capacitor banks utilized within each test model were adjusted to increase or reduce harmonic resonance effects at specific frequencies.

Including the three different distribution circuits, designated nodes of concerns, and varying capacitor bank ratings, the HRI algorithm was tested on seven different model configurations. The number of capacitor banks, compensation requirements, and points of concern for each model are listed in Table 3.5.

Table 3.4 List of models utilized to evaluate the HRI algorithm

Cap Points of Model Available QCAP Concern 6 1 6 6 2 6 Expanded IEEE 13 Node Test Feeder 6 3 6 Caps = 200 kVAr 6 4 6 6 5 6 6 1 6 Expanded IEEE 13 Node Test Feeder 6 3 6 Caps = 600 kVAr 6 4 6 6 5 6 6 1 6 Expanded IEEE 13 Node Test Feeder 6 2 6 Caps = 800 kVAr 6 3 6 6 4 6 5 1 6 Feeder 1358 5 2 6 Caps = 800 kVAr 5 3 6 5 4 6 8 2 6 Feeder 1358 connected to Feeder 1444 8 3 6 Caps = 1800 kVAr 8 4 6 8 2 6 Feeder 1358 connected to Feeder 1444 8 3 6 Caps = 1200 kVAr 8 4 6 8 5 6 Feeder 1358 connected to Feeder 1444 8 2 12 Caps = 1200 kVAr (Phase A) and 1500 kVAr 8 3 12 (Phase B) 8 4 12

3.3 Harmonic Resonance Identification Algorithm Convergence Analysis

Evaluating the convergence of the harmonic resonance identification problem through the convexity of the objective function, which is comprised of a number of driving reactance expressions, is nearly impossible due to the characteristics associated with an electric distribution feeder and its customers. Although a single driving reactance

expression possesses pseudo-convex characteristics during harmonic resonance conditions, only a percentage of cases produce an objective function that is convex.

The objective function is comprised of a number of driving reactance expressions that represent different points of concern on a distribution feeder. The utility company establishes these points based upon their electric distribution system and specific customer requirements. Therefore, these points of concern can represent multiple phases and frequencies at multiple locations on an electric feeder. When considering the characteristics of an electric system that is dynamically reconfigured, which results in varying possibilities for variable coefficients and variable interaction terms within the objective function, it is nearly impossible to determine the convexity of all objective functions that can be developed by employing the HRI algorithm. However, applying the HRI algorithm to the frequency dependent models developed previously, one can gain insight into the convergence and accuracy of the problem on a typical distribution feeder. The analysis also allows for the identification of algorithms that are employable by optimization solvers to enhance the convergence and accuracy of the harmonic resonance identification problem.

Software utilized to maximize and minimize an objective function is referred to as an optimization

“solver”. The solver can be interfaced with modeling software, such as AMPL or GAMS, or integrated into a modeling system [79], [104], [105]. Because of the complexity of solving MINLPs and their many applications, there are a number of different types of solvers that utilize varying algorithms to achieve good computational performance [105]. Any solver employed to identify harmonic resonance conditions must have the capability of maximizing a non-convex objective function that is comprised of up to six integer variables, a minimum of six driving reactance expressions, and one hundred and twenty-eight interaction terms (possessing up to six integer variables each) per driving reactance expression, which is the resulting characteristics of a typical six-capacitor bank system with six points of concern. Each driving reactance expression should possess different harmonic resonance characteristics, which results in an objective function that is comprised of a number of pseudo-convex expressions with varying frequency responses. Because the objective function is constrained by the required amount of VAr compensation, there is a possibility that an MINLP solver can encounter an infeasible solution if underestimators are inappropriate. From a stability standpoint, the goal was to find a solver that converged at local extrema without encountering these feasibility problems. Three solvers were tested utilizing the frequency dependent models developed in the previous section.

Convex Over and Under ENvelope for Nonlinear Estimations (COUENNE) is a MINLP solver that can identify the global extreme for both convex and non-convex problems [105]. The solver relies upon a linear outer- approximation that is generated from a reformulation. It utilizes a convexification technique to compute linear underestimators of a non-convex function. This convexification is implemented by branching on continuous variables, a process referred to as spatial branching. Unfortunately, the HRI objective function does not possess continuous variables, which hinders the convexification process.

Attempts to utilize this solver were unsuccessful. The solver converged at the optimal solution for W; however, it failed to provide correct values for fW that were verified with Mathematica. The HRI algorithm utilizes the value of fW to determine if the solutions represent harmonic resonance or non-harmonic resonance conditions. Therefore, implementation of this solver will require additional tasks to be performed by the HRI program or external software.

The Basic Open-Source Nonlinear Mixed Integer Programming (BONMIN) solver was employed to analyze the model built from the expanded IEEE 13 Node Test Feeder. The solver only ensures optimal global solutions for convex functions. It implements an outer-approximation algorithm, which can encounter difficulties in solving non-convex problems if relaxations are infeasible due to constraints [105]. When utilized to maximize the objective function derived from the expanded IEEE 13 Node Test Feeder, the solver failed to converge to solutions that were representative of local extrema. The solver produced solutions of 1 for all variables and stated that the problem was infeasible. Due to the failed attempt, no further tests were performed with BONMIN.

The third solver tested was Mixed Integer Nonlinear Programming Branch and Bound (MINLP BB).

Instead of an outer-approximation algorithm, MINLP BB utilizes nonlinear relaxations for the bounding step.

Although the solver does not ensure global solutions for non-convex functions, the goal of the HRI algorithm is to identify all extrema.

MINLP BB was very successful in maximizing the objective function developed from the expanded IEEE

13 Node Test Feeder with 200 kVAr capacitor banks. Upon successfully testing the solver with this model, it was applied to all models and compensation requirements listed in Table 3.5. Maximizing the objective function for a single compensation level under specific constraint conditions was considered a single operation. Constraints were developed from the solutions and the objective function was maximized again to identify the next highest driving

reactance condition, which was considered a second operation. Recall that the HRI algorithm must successfully identify all harmonic resonance producing conditions for a given compensation requirement, not just the solution that represents a global extreme. During testing, MINLP BB was called upon to maximize the objective function

610 times for 7 different models under 27 different compensation constraint conditions. For the different cases,

MINLP BB converged to accurate values for fW during all 610 optimizations. Considering that the models possess attributes that are representative of the typical impedance response characteristics of a distribution feeder under harmonic resonance conditions, it is reasonable to assume that MINLP BB will converge to an accurate solution for most cases encountered on a typical distribution system.

The results align with the information provided in [105], which states that MINLP branch and bound solvers that utilize nonlinear programing relaxations can be successful in finding good solutions for non-convex functions. MINLP BB falls into this category of solver. The other two solvers tested were COUENNE and

BONMIN, both of which utilize outer-approximation algorithms to develop relaxations of the objective function and constraints. Although COUENNE ensures global solutions for non-convex functions, it is speculated that the lack of a continuous variable within the HRI problem prevents the convexification of the objective function, which results in inaccurate values for fW .

3.4 Harmonic Resonance Identification Algorithm Accuracy Analysis

The HRI algorithm was applied to all test cases listed in Table 3.5 utilizing MINLP BB. As defined in

Section 2.3, a harmonic resonance threshold factor (F ) is utilized to distinguish between fW values that represent harmonic resonance conditions from those that did not. The actual harmonic threshold (W ) for the objective function, which is defined in Equation (2.10), is developed through the use of F and the base value (M ) of the objective function, which is defined in Equation (2.11). Comparing the maximized value of fW to W , as illustrated in Equations (2.12) and (2.13), determines if the identified capacitor bank energized states will produce harmonic resonance at any of the points of concern.

W FM (2.10)

Mf 0 (2.11)

W d fW Harmonic Resonance (2.12)

W ! fW Non-harmonic Resonance (2.13)

Prior to applying the HRI algorithm, a utility company is required to identify the relationship between W and the harmonic voltage distortion effects on their system. One attribute utilized to determine this relationship is the amount of resistance within the system as seen from the locations of concern. As analyzed in Appendix B, resistance dampens the driving point impedance during the occurrence of harmonic resonance. A system with high resistance may require an F value greater than 2 that will mask dampened values. Consideration should also be given to the amount of harmonic distortion produced during non-harmonic resonance conditions, which may require the prevention of any increase in driving reactance due to harmonic resonance. For this condition, an F value ranging between 1 and 2 may be required in order to avoid minor impedance increases due to resonance. An inappropriate F can result in the mislabeling of harmonic resonance conditions for a given system. For the accuracy analysis performed on the models listed in Table 3.5, F values of 1.20, 1.10, and 1.05 (twelve points of concern for evaluation only) were utilized, which resulted in three separate harmonic resonance evaluations.

Assuming that increases in fW are caused by a single point of concern experiencing harmonic

h resonance, the relationship between F and the per unit increase in driving reactance (Hl ) for a single point of concern is defined in Equation (3.3). For the accuracy evaluation, harmonic resonance of a single point of concern

h was determined by identifying driving reactance values that exceeded Hl , which was evaluated for three separate harmonic threshold factors (F = 1.20, F = 1.10, and F = 1.05). The individual per unitized harmonic threshold limit for each point of concern is listed in Tables 3.6, 3.7, and 3.8. For implementation of the HRI algorithm, engineers

h shall utilize Hl values in conjunction with Equation (2.14) to determine F threshold values.

2 h ¬¼ªºFfu 00 f Xl H h (3.3) l h Xl

The HRI algorithm was applied to each model listed in Table 3.5 under the three harmonic resonance threshold conditions with the goal of identifying all harmonic resonance conditions associated with each level of

VAr compensation. The test models were developed to represent the general characteristics of feeders experiencing harmonic resonance; therefore, the results of the analysis give insight into how the algorithm will respond when applied to typical distribution circuits. Accuracy was measured by the number of times the HRI algorithm could successfully distinguish between harmonic and non-harmonic resonance conditions. Another factor considered was the algorithm’s ability to identify all harmonic resonance producing conditions prior to identifying a single non- harmonic resonance condition. Any harmonic resonance condition identified after the identification of a single non- harmonic resonance condition is mislabeled by the HRI algorithm.

Table 3.5 Node threshold limits for Expanded IEEE 13 Node Test Feeder Model

Bus and Frequency F = 1.10 F = 1.20 Node 692A at the 3rd Harmonic Frequency 1.28 1.52 Node 684B at the 3rd Harmonic Frequency 1.83 2.38 Node 692A at the 5th Harmonic Frequency 1.18 1.33 Node 684B at the 5th Harmonic Frequency 1.48 1.84 Node 692A at the 7th Harmonic Frequency 1.13 1.26 Node 684B at the 7th Harmonic Frequency 1.33 1.61

Table 3.6 Node threshold limits for Feeder 1358 Model

Bus and Frequency F = 1.10 F = 1.20 Node 4 at the 3rd Harmonic Frequency 2.64 3.59 Node 11 at the 3rd Harmonic Frequency 5.31 7.45 Node 4 at the 5th Harmonic Frequency 1.86 2.43 Node 11 at the 5th Harmonic Frequency 1.16 1.30 Node 4 at the 7th Harmonic Frequency 1.50 1.87 Node 11 at the 7th Harmonic Frequency 1.08 1.16

Table 3.7 Node threshold limits for F1358 connected to F1444 Model

F = 1.10 (6 points) F = 1.20 (6 points) Bus and Frequency F = 1.05 (12 points) F = 1.10 (12 points) Node 4 at the 3rd Harmonic Frequency 1.63 2.08 Node 15 at the 3rd Harmonic Frequency 1.25 1.46 Node 4 at the 5th Harmonic Frequency 1.48 1.83 Node 15 at the 5th Harmonic Frequency 1.18 1.33 Node 4 at the 7th Harmonic Frequency 1.40 1.71 Node 15 at the 7th Harmonic Frequency 1.14 1.27

The results revealed that the HRI algorithm was 100% accurate in identifying the possibility of initiating harmonic resonance conditions by energizing capacitor banks to satisfy a given compensation level. Although

MINLP BB only ensures heuristic global solutions for non-convex problems, the algorithm successfully identified the maximum driving reactance conditions for 22 of the 27 cases (81.4%). Once the occurrence of harmonic resonance was identified, the algorithm successfully classified 1174 of the 1220 cases (96.23%) as either harmonic resonance or non-harmonic resonance conditions. Tables 3.9 and 3.10 summarize the results of the accuracy analysis, while Appendix D contains the detailed results in Tables D-1 through D-42. Three independent conditions were responsible for the 3.77% of cases that were misclassified. Of the 1220 cases, 0.41% were misclassified due to optimization error related to maximizing the objective function repeatedly. The occurrence of multiple points of concern experiencing driving reactance increases that were sufficient to increase fW above W, accounted for

1.97% of the misclassifications. Although the HRI algorithm identified these as harmonic resonance producing conditions, they were classified as errors because a single point of concern “did not” exceed its corresponding

h unacceptable level (Hl ). The third condition, which accounted for 1.39% of the classifications, was due to points of concern experiencing reactance values below their base value during the recovery from a resonance condition.

Failure to identify all harmonic resonance conditions due to maximizing error only accounted for five misclassifications, which is a failure rate of 0.41%. This failure rate is directly related to the algorithm’s ability to identify all harmonic resonance conditions prior to identifying a single non-harmonic resonance condition.

Although this type of error was witnessed during numerous evaluations under extreme harmonic resonance conditions, it was nearly nonexistent for values of the objective function that were below F = 5.

Table 3.8 Harmonic Resonance Identification Accuracy Evaluation at 110% (l = 6) and 105% (l = 12)

HRI No. Algorithm Caps Success Rate Max on Points of [HR Ident/No. No. of Harmonic Model Feeder QCAP Concern HR Cond.] Optimizations Conditions 6 1 6 100% 5 2 6 2 6 78.6% 14 3 Expanded IEEE 13 Node Test Feeder, Caps = 200 kVAr 6 3 6 100% 18 3 6 4 6 92.9% 14 3 6 5 6 100% 5 3 6 1 6 100% 5 2 Expanded IEEE 13 Node Test 6 3 6 73.7% 19 3 Feeder, Caps = 600 kVAr 6 4 6 100% 12 3 6 5 6 100% 5 3 6 1 6 100% 5 3 Expanded IEEE 13 Node Test 6 2 6 85.7% 14 3 Feeder, Caps = 800 kVAr 6 3 6 88.9% 18 3 6 4 6 100% 14 3 5 1 6 50.1% 4 2 Feeder 1358, 5 2 6 100% 6 2 Caps = 800 kVAr 5 3 6 100% 9 2 5 4 6 100% 4 3

Feeder 1358 tied to Feeder 8 2 6 78.6% 14 2 1444 , 8 3 6 100% 30 2 Caps = 1800 kVAr 8 4 6 100% 67 3 8 2 6 100% 18 1 Feeder 1358 tied to Feeder 8 3 6 100% 52 2 1444, Caps = 1200 kVAr 8 4 6 100% 69 2 8 5 6 100% 56 2

Feeder 1358 tied to Feeder 8 2 12 100% 19 2 1444, Caps = 1200 kVAr (Ph- 8 3 12 100% 50 3 A) and 1500 kVAr (Ph-B) 8 4 12 100% 64 3 Total 96.4% 610

Table 3.9 Harmonic Resonance Identification Accuracy Evaluation at 120% (l = 6) and 110% (l = 12)

HRI No. Algorithm Caps Success Rate Max on Points of [HR Ident/No. No. of Harmonic Model Feeder QCAP Concern HR Cond.] Optimizations Conditions 6 1 6 100% 5 2 6 2 6 85.7% 14 2 Expanded IEEE 13 Node Test Feeder, Caps = 200 kVAr 6 3 6 100% 18 2 6 4 6 78.6% 14 2 6 5 6 100% 5 2 6 1 6 100% 5 2 Expanded IEEE 13 Node Test 6 3 6 79% 19 3 Feeder, Caps = 600 kVAr 6 4 6 91.7% 12 3 6 5 6 100% 5 3 6 1 6 100% 5 2 Expanded IEEE 13 Node Test 6 2 6 92.9% 14 3 Feeder, Caps = 800 kVAr 6 3 6 88.9% 18 3 6 4 6 100% 14 3 5 1 6 75.0% 4 1 Feeder 1358, 5 2 6 100% 6 2 Caps = 800 kVAr 5 3 6 100% 9 2 5 4 6 100% 4 2

Feeder 1358 tied to Feeder 8 2 6 85.7% 14 2 1444 , 8 3 6 100% 30 2 Caps = 1800 kVAr 8 4 6 100% 67 3 8 2 6 100% 18 1 Feeder 1358 tied to Feeder 8 3 6 100% 52 1 1444, Caps = 1200 kVAr 8 4 6 100% 69 1 8 5 6 100% 56 2

Feeder 1358 tied to Feeder 8 2 12 100% 19 2 1444, Caps = 1200 kVAr (Ph- 8 3 12 98.2% 50 3 A) and 1500 kVAr (Ph-B) 8 4 12 98.4% 64 3 Total 96.0% 610

The results also indicate that accuracy associated with the success of repeatedly maximizing an objective function is directly related to the number of switchable capacitor banks being analyzed. Increasing the number of banks adds to the number of conditions that can result in harmonic resonance, thereby increasing the complexity of the evaluation. Also, because the addition of a capacitor bank adds more interaction terms to the objective function, the non-convexity of the function increases as more capacitor banks are evaluated. Although only eight capacitor banks were evaluated within the accuracy analysis, which is more than the typical number utilized on a distribution feeder, the results revealed that the congruency of identifying the next maximum value of fW is directly related to the number of capacitor banks being evaluated. Figures 3.7 through 3.9 illustrate results of applying the HRI algorithm to a five-capacitor bank system (Feeder 1358), six-capacitor bank system (Expanded IEEE 13 Node Test

Feeder), and an eight-capacitor bank system, while each were experiencing level 3 harmonic resonance.

180 160 140 120 100 80 ohms^2 60 40 20 0 13579 Optimization Number f(W) Harm Resonance Threshold

Figure 3.7 HRI algorithm results for analyzing Feeder 1358 model (five-capacitor bank system) for three energized banks, each rated at 800 kVAr

1400 1200 1000 800 600 ohms^2 400 200 0 2 4 6 8 10 12 14 16 18

Optimization Number

f(W) Harm Resonance Threshold

Figure 3.8 HRI algorithm results for analyzing the Expanded IEEE 13 Node Test Feeder model (six-capacitor bank system) for three energized banks, each rated at 600 kVAr

300 250 200 150 ohms^2 100 50 0 2 7 12 17 22 27 32 37 42 47 52 57 62 67

Optimization Number

f(W) Harm Resonance Threshold

Figure 3.9 HRI algorithm results for analyzing the Feeder 1358 connected to Feeder 1444 model (eight-capacitor bank system) for three energized banks, each rated at 1200 kVAr

The assumption that only a single point of concern will raise the objective function beyond acceptable threshold limits resulted in misclassifications during 1.97% of the evaluations. Prior to reaching the harmonic resonance frequency, driving point impedance begins to increase as illustrated in Region A of Figure 3.10. Region

A defines the inductive region where reactance values can be 300% greater than those experienced during non- harmonic resonance conditions. Although multiple nodes may not possess identical harmonic resonance characteristics, they may still possess magnitude increases at multiple harmonic frequencies prior to experiencing the significant increase that occurs at their resonant frequency, as illustrated in Region C. Consider three points of concern that experience harmonic resonance at the 5th, 6th, and 7th harmonic frequencies respectively. Although the harmonic resonance frequencies do not coincide, each node may experience increases in reactance impedance at the

3rd harmonic frequency that is higher than its base value. During testing, the HRI algorithm identified these conditions as harmonic resonance producing because the value of fW was above W. These types of occurrences can be eliminated by setting F to a value that only represents extremely high reactance values, such as those illustrated in Region C, which is the region mostly associated with harmonic resonance. Considering that these sub- harmonic resonance values still represent the possibility of increased harmonic voltage, it is possible that W violation due to these occurrences can be viewed as impactful or as warning conditions.

Sub-base reactance values that reduced the overall value of fW , which accounted for 1.39% of the classifications, fell into Region B of Figure 3.10. Region B is classified as the reactance recovery zone, where the

driving reactance is approaching its frequency dependent base value. Within this region, the absolute value of the driving reactance can be lower than its base value, which lowers the value of the objective function. This dampening effect is increased as more points of concern experience harmonic resonance simultaneously. Therefore, to reduce the errors associated with this occurrence, it is recommended that the objective function be limited to points of concern that do not possess identical harmonic resonance characteristics. Identifying points of concern with identical harmonic resonance characteristics is performed by evaluating their Sr as described previously in

Section 3.2.

Figure 3.10 Illustration of the reactive impedance regions associated with parallel harmonic resonance

Although only two nodes of concerns were utilized in the evaluation, the results provide insight into HRI algorithm limitations regarding the number of points of concern that can be implemented into the evaluation without introducing errors into the results. The evaluation indicated that accuracy in identifying all harmonic resonance producing conditions is related to the number of points of concern experiencing harmonic resonance simultaneously and not the total number of nodes evaluated within the objective function. Errors associated with Region B are increased as more driving reactance expressions experience harmonic resonance simultaneously. During testing, the

HRI algorithm successfully identified harmonic resonance conditions with up to five points of concern experiencing harmonic resonance.

A point of concern is defined as the frequency dependent characteristic of a specific location of concern.

Therefore, monitoring two nodes of concern at the 5th and 7th harmonic frequencies is defined as four points of concern. If the objective function is comprised of locations of concern that are determined based upon the stiffness

of connections, which minimizes the duplication of nodes with identical harmonic resonance characteristics, the number of points of concern experiencing harmonic resonance simultaneously is minimized.

A comparison of harmonic resonance characteristics of nodes with varying Sr revealed that a value of 0.30 is required to produce nodes with different harmonic resonance characteristics. Table 3.11 illustrates how Nodes

692A and 684B, which were connected by a tie with an Sr of 0.524 on the Expanded IEEE 13 Test Feeder, possessed similar harmonic resonance characteristics at the 3rd, 5th, and 7th harmonic frequencies. Table 3.12 illustrates how

Nodes 4 and 11 on Feeder F1358, which were connected by a tie of 0.289, experienced harmonic resonance conditions that were more diverse than those experienced by Nodes 692A and 684B

These results reflect the design goal of developing a distribution feeder, which is to provide little impedance to current flowing downstream to utility customers [106]. This design characteristic of a feeder minimizes voltage drops and I2R losses that occur from the substation to the tail end of the circuit. In fact, capacitor bank operations of a Volt/VAr application respond to system deficiencies that are created by varying customer demand and not by the system configuration. Therefore, it should not be shocking that a distribution feeder has similar harmonic resonance characteristics throughout its primary system. Of course, this may not hold true for harmonic frequencies greater than the 7th, where feeder stiffness is weakened by the increased inductance. Overall, these results suggest that for a stiff system being analyzed for resonance below the 7th harmonic frequency, the number of locations of concern “may” be as low as one, if the only desire is to monitor harmonic distortion on the primary. A weak system regarding harmonic resonance characteristics may only require two or three locations of concerns to identify all harmonic resonance conditions on the circuit.

Nodes located on the primary side of distribution transformers are joined in the objective function by locations of concern that are located on secondary systems that serve customers directly. Multiple nodes on the same secondary system shall undergo the same stiffness comparison ratio evaluation as those located on the primary distribution system. However, because distribution transformers can account for up to 90% of the upstream impedance as seen from a customer, it is most likely that only one location of concern is required per secondary system [30]. If base reactance values between l’s located on the primary and secondary systems are significantly different, the objective function format illustrated in Equation (2.16) shall be applied in lieu of the format illustrated in Equation (2.8).

Table 3.10 HRI optimization results for evaluating the Expanded IEEE 13 Node Test Feeder for three-Capacitor Bank compensation (2400 kVAr total)

3rd 5th 7th Objective (ohms) (ohms) (ohms) Caps Optimization Function Node Node Node Node Node Node Required No. Value 692A 684B 692A 684B 692A 684B 2 1 250,050 6.5217 3.0050 14.6040 5.6663 485.870 122.325 2 2 4,708 5.9917 3.0475 11.9946 6.1127 -59.915 -30.105 2 3 1,811 7.0246 2.9763 20.3862 5.9100 -36.5141 -1.9687 2 4 1,238 6.1945 3.0325 11.8954 5.4887 29.4366 12.0326 2 5 694 5.9917 2.7299 10.7937 4.1037 21.8757 5.8230 2 6 631 5.6853 3.0706 9.5495 5.7616 17.2608 12.7497 2 7 432 5.3654 3.1286 8.2497 6.2193 14.3548 9.0579 2 8 332 5.4960 2.7604 8.5010 4.2499 13.1022 6.0308 2 9 214 5.1874 2.8065 7.4098 4.4838 8.2542 6.1093 2 10 190 6.0512 2.8187 11.0967 4.4850 -1.7013 -0.4953 2 11 163 4.9809 2.8538 6.6668 4.6740 7.0109 3.7991 2 12 140 5.5515 2.8514 8.8593 4.6622 -1.2574 -0.6399 2 13 119 5.2395 2.9008 7.6348 4.9489 -1.0281 -0.8451 2 14 108 5.0286 2.9515 6.8409 5.1845 -1.1091 -1.2534 Base Reactance Values 4.8744 2.6099 6.3760 3.7197 9.3422 5.8684

Table 3.11 HRI optimization results for evaluating Feeder 1358 for two-Capacitor Bank compensation (1600 kVAr total)

3rd 5th 7th Objective (ohms) (ohms) (ohms) Caps Optimization Function Required No. Value Node 4 Node 11 Node 4 Node 11 Node 4 Node 11 3 1 169 0.5100 0.2900 0.8901 3.4700 1.5400 12.3207 3 2 110 0.5138 0.3060 0.9098 3.3270 1.4980 9.6993 3 3 109 0.5116 0.3007 0.8990 3.3243 1.4634 9.6663 3 4 45 0.5181 0.2580 0.9329 2.8293 1.5473 5.6564 3 5 45 0.5158 0.2528 0.9215 2.8270 1.5105 5.6399 3 6 38 0.5161 0.2495 0.9234 2.7044 1.5173 5.0437 3 7 39 0.5222 0.2702 0.9532 2.7342 1.5930 5.0779 3 8 38 0.5181 0.2580 0.9329 2.8293 1.5047 5.0306 3 9 33 0.5225 0.2668 0.9548 2.6134 1.5975 4.5288 Base Reactance Values 0.5343 0.2496 0.8303 2.2098 1.1664 3.1184

The accuracy evaluation results also indicate that it is possible that a high number of points of concern can drown out the effects of a single point of concern experiencing excessive impedance due to harmonic resonance.

This drowning effect is a function of the number of points of concern, the base impedance associated with each, and the expected peak driving reactance (Xp) of a single point of concern. As a guideline, the non-harmonic resonance form of the objective function should not be larger than the Xp, as illustrated in Equation (3.4).

fX 0 p (3.4)

Although not observed during testing, there is an error associated with driving reactance values that occur at the zero crossing that separates the inductive reactance zone from the capacitive reactance zone. This zero crossing is associated with pure parallel harmonic resonance, where capacitive and inductive reactances cancel each other, leaving only the resistive component that has been omitted from the objective function. Although the reactance is zero, the actual impedance is much higher due to the omitted resistance. For the frequency scans performed in Appendix B, the zero crossing region, which is defined by ohm values below non-harmonic resonance conditions, only comprised 1.36% of the frequencies that were affected by harmonic resonance. Therefore, it is concluded that the percentage of cases where this effect results in errors should be minimal.

3.5 Accuracy Analysis Summary

In conclusion, testing the HRI algorithm revealed specific conditions that correspond to the accuracy of the approach. The conditions include the following:

x Accurate in identifying the occurrence of harmonic resonance based exclusively upon a specific

level of capacitive compensation,

x Accurate in identifying harmonic resonance on feeders that possess up to eight switchable

capacitor banks, although stability in identifying the next worst harmonic resonance condition by

maximizing the objective function decreases as more capacitor banks are evaluated,

x Accurate in identifying the occurrence of harmonic resonance when up to five points of concern

are experiencing harmonic resonance simultaneously,

x Accuracy errors are minimized by limiting the objective function to driving reactance expressions

that do not possess identical harmonic resonance characteristics.

CHAPTER 4

PROGRAMMING INSTRUCTIONS

4.1 Programming Description and Requirements

A computer program that employs the HRI algorithm must possess five characteristics that can each be performed by individual functions within the program, as illustrated in Figure 4.1. A general description of each function is listed below:

Analysis Parameters and Monitoring – Stores algorithm parameters related to locations and harmonic frequencies

of concern. It retrieves information from system smart devices and software

to identify the current feeder configuration, required VAr compensation,

and customer harmonic impedance. It allows a user to override preset

parameters and settings established through field monitoring.

Driving Impedance Derivation – Identifies driving impedances throughout the system as a function of the

switching states of all capacitor banks connected to the subject feeder.

Objective Function Development – Utilizes driving reactance functions to develop an objective function that is

employed to identify the capacitor bank switching states that produce the

highest driving reactance.

Optimization – Utilizes programming structure or an interface to an optimization solver to

maximize the objective function repeatedly until all capacitor bank

switching configurations that result in harmonic resonance are identified.

Results Reporting – Summarizes the results of utilizing the aforementioned functions to

determine if harmonic resonance exists and under what conditions.

The remainder of this section describes each function in detail, which includes software requirements,

“inputs” from functions, “outputs” to functions, actions to be performed within the function, and the governing equations as required. It should be noted that descriptions are intended to provide an overall structure for implementing the HRI algorithm; and functions can be augmented or combined depending upon the programming structure and language utilized by the programmer. To enhance this flexibility, pseudo code is also provided for each function.

START

Analysis Parameters & Field Monitoring

Driving Impedance Derivation

Objective Function Development

Optimization

Results Reporting

STOP

Figure 4.1 Flowchart illustrating the functions that are required to implement the HRI algorithm

4.2 Analysis Parameters and Field Monitoring

The Analysis Parameters and Field Monitoring function shall be structured so that it is responsible for storing and retrieving HRI algorithm parameters. As illustrated in Figure 4.2, the function is also responsible for transmitting this information to other functions.

START

Input L, U, F, T, User Defined (H- optional)

Store algorithm default values

External Input Software S, Q

Interpret harmonic STOP frequencies of concern from metered values (optional)

Driving Output Impedance H, S, L, Field Derivation Input C C Monitoring function

Allow user to adjust values Output Optimization S, H, Q, L, U, and F Q, F, U function

Figure 4.2 Flowchart of the Analysis Parameters and Field Monitoring Function

The goals of the Analysis Parameters and Field Monitoring function are as follows:

1. Storage of algorithm parameters

2. Determination of the harmonic frequencies analyzed within the HRI program (two options)

3. Retrieval of customer harmonic impedance through metering

4. Retrieval of required compensation level

5. Retrieval of system configuration based upon current status of tie switches located throughout the

electric system of concern

The function shall be structured to store default program values, as illustrated in Figure 4.3. The default values include the following:

U - The integer variable defines the flexibility of satisfying the required compensation level. Once it is

determined that all capacitor bank configurations satisfying the required compensation level will result in

harmonic resonance, the compensation constraint is adjusted by the value of U , which allows for the

identification of the level of support that can be provided without producing harmonic resonance. Units of

this variable shall match the units of the compensation level.

L- A set of integers that define all locations on the electric system designated as locations of concern (l) that

will be analyzed with the HRI algorithm. An electric system can be divided into nodes that represent

specific buses or changes in system attributes, such as voltage. Figure 4-4 illustrates an example of an

electric distribution system consisting of eighteen nodes, six feeders, and four varying customer

impedances. The L array will consist of the node identifiers that coincide with l’s for the purpose of the

HRI algorithm. For example, if nodes 2, 4, 5, and 7 are considered l’s, L would be comprised of (2, 4, 5,

7).

H- A set of integers that define all of the harmonic frequencies that are to be analyzed with the HRI algorithm.

For example, if the user wants to establish that the 3rd, 5th, and 7th harmonic frequencies are always to be

analyzed, H will consist of (3, 5, 7). There are two options in determining the values of this array. The

first method consists of utilizing predefined values established by the user. However, the array can also be

defined through remote monitoring, in which case the H array is defined with the T array that is described

further in this section. The maximum size for the H array is equal to the maximum number of frequencies

that can be analyzed during a single analysis.

F- The integer variable defines the threshold between harmonic resonance conditions and non-harmonic

resonance conditions as a multiple of the objective function’s base value (M ), which represents driving

impedances without any capacitive contributions. For example, an F of 1.5 defines any objective function

value that is 150% greater than the base value as the occurrence of harmonic resonance.

T - An array consisting of real numbers that define monitoring threshold limits to indicate when a specific

frequency warrants inclusion into the HRI analysis. Consider the user defined threshold limits of 5%, 15%,

and 20% for the 3rd, 5th, and 7th harmonic frequencies respectively. In this case, the T array is comprised of

(0, 0, 0.05, 0, 0.15, 0, 0.20).

Store algorithm default values

Figure 4.3 Action block illustrating the program’s requirement to store default parameters

The function shall be structured to retrieve the required level of compensation as illustrated in Figure 4.5.

The required compensation level is provided in VArs (QVAR) or in the number of capacitor banks (QCAP). The general form of compensation is defined by Q, which is unitless and is utilized to describe compensation in general.

The required compensation level is determined from an algorithm that is external to the HRI algorithm.

Figure 4.4 Illustration of six-feeder system with forty-four switches

External Input Software S, Q

Figure 4.5 Input block illustrating the interaction with external software to retrieve information regarding switching configuration and the required level of compensation

The function shall be structured to retrieve the current configuration of the feeder being analyzed as illustrated in Figure 4.5. The switching configuration is defined by information provided in the S array, which is defined below.

S- An array of integers representing values that define the status of switches located throughout the

distribution system being analyzed with the HRI algorithm. The size of the array is defined by the number

of switches that can be connected to the feeder of concern. Values within the array are binary in that a 1

represents a closed state and 0 represents an open state. Figure 4-6 illustrates an example of an abbreviated

distribution system that possesses forty-four switches designated S1 – S44. If S2, S11, and S13 were the

only switches that were closed with respect to Feeder B, then array S with respect to Feeder B would be as

follows:

S = [0, 1,0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

0, 0, 0]

The function shall be structured to allow the user, as illustrated in Figure 4.6, to adjust any of the settings established through default parameters, field monitoring, or interfaces with other programs.

Allow user to adjust values S, H, Q, L, U, and F

Figure 4.6 Action block illustrating user override capabilities

The function shall be structured to interpret harmonic current measurements from metered locations on the circuit to determine which harmonic frequencies should be analyzed, as illustrated in Figure 4.7. This option is utilized in lieu of having predetermined harmonic frequencies of concern. Harmonic current measurements can be

acquired from substation equipment, power quality devices located on the circuit, synchrophasors, and/or revenue class smart meters. The utility company must utilize system configurations, customer load dynamics, available monitoring, and/or historical harmonic information to determine how the measured harmonic demand will correlate to the threshold limits stored within the T array.

Interpret harmonic frequencies of concern from metered values (optional)

Figure 4.7 Action block illustrating the interpretation of harmonic current readings

The function shall be structured to retrieve harmonic impedance information associated with customers

h connected to the feeder to be analyzed, as illustrated in Figure 4.8. Harmonic impedance (ZC ) information pertaining to customers is collected in the C array, which is frequency dependent and defined below.

C - An array of real numbers that represent the metered impedance values at the h frequency for individual

customers or the equivalent of multiple customers.

Field Input C Monitoring

Figure 4.8 Action block illustrating the input requirement associated with customer harmonic impedance

There are two options for retrieving metered harmonic impedance information, each of which is accounted for differently in the Driving Impedance Derivation function. The harmonic impedances associated with the utility company’s conductors/cables are known and fixed, leaving only the harmonic impedances associated with customer

equipment as the only unknown harmonic impedance in the system. A utility company can monitor customer harmonic impedances with revenue class smart meters that possess this capability, as illustrated in Figure 4.9. A second option is to utilize a single device to measure the equivalent harmonic impedance of a number of customers as illustrated in Figure 4.10.

Figure 4.9 Graphical representation of customers with smart meters that collect harmonic impedance information

Figure 4.10 Graphical representation of laterals with power quality meters that retrieve harmonic impedance information

Each monitoring device utilized for retrieving metered harmonic impedance information must have the capability of measuring this attribute with an invasive method or noninvasive method, which is the preferred method. A noninvasive method can measure harmonic impedance without disturbing the distribution system. The use of an invasive method requires augmentation of the distribution system’s impedance, while measuring the resulting harmonic voltage and current at the metering location, as defined in Equation (2.14). The impedance augmentation can be performed by operating capacitor banks with sufficient VAr ratings to change the system

impedance seen from the metering location. More information regarding the different non-invasive methods is described in [92] – [97], while the invasive method of calculating harmonic impedance is described in [89].

4.3 Pseudo Code for Analysis Parameters and Field Monitoring Function

Use variables: Q, U, F

Define array: S[1-Sn] where Sn is the total number of switches on the circuit to be

analyzed

Define array: L[1-Ln] where Ln is the total number of locations to be analyzed

Defined array: H [1 – Hn] where Hn is total number of harmonic frequencies to be analyzed

Define array: C[1-Ln] where Cn is the total number of harmonic impedances associated with customers

Defined array: T [1 – Tn] where Tn is total number of harmonic frequencies to be analyzed

OPTIONAL – use only with metered harmonic parameters

Define flexibility (U) of satisfying compensation constraint Q

Accept Q, F, H, S, L, C

Change Q, U, L, and H according to user’s commands

Output H, S, L, C to “Driving Impedance Function”

Output Q, F, U to “Optimization Function”

End program

4.4 Driving Impedance Function

The goal of the Driving Impedance function is to develop expressions for the driving reactance associated with each node of concern with respect to capacitor bank switching status and harmonic frequencies of concern. As illustrated in Figure 4.11, the function is required to develop frequency specific network admittance matrices and network impedance matrices based upon the existing switching configuration of the circuit being analyzed. Once the frequency specific network impedance matrices are developed, the function identifies which driving reactance expressions are to be utilized in the Objective Function Development function.

The function shall possess a structure that can construct a network admittance matrix, as illustrated in

Figure 4.12. The admittance matrix is developed by utilizing information regarding the architecture of the electric

system, the current switching configuration, monitored field values, and the harmonic frequencies of concern.

Developing a function of this nature requires an understanding of the mathematics and engineering that govern the development of the network admittance matrix.

Start

Analysis Parameters Input & H, S, L, C Field Monitoring

Stop Develop network admittance matrices based upon SandH Objective Output Function ZD Development Function Eliminate rows and columns not connected

Output Optimization k Function Add capacitor banks to simplified network admittance matrices

Determine the number of Eliminate interactive terms that do not cap banks within the admittance matrix impact solutions

Develop network impedance matrices Extract driving from the network reactance expression admittance matrices

Figure 4.11 Flowchart of Driving Impedance Derivation Function

Develop network admittance matrices based upon S and H

Figure 4.12 Action block describing the network admittance matrix development requirement

Two of the most common matrix formulations in power system modeling are the network admittance matrix and the network impedance matrix [28]. Kirchhoff’s current law is the governing principle behind admittance matrix development, whereas the impedance matrix is based upon Kirchhoff’s voltage law. The different underlying principles lend themselves to different approaches for the development of each matrix. Due to the ease of the approach, the impedance matrix is typically built from the admittance matrix through inversion. In this case, the impedance matrix is developed with the relationship illustrated in Equation (4.1). An admittance matrix representing a three node system is illustrated in Equation (4.2). Characteristics of the admittance matrix include:

x The diagonal elements in position Ym,m are the sum of all admittances connected to node m

x Off diagonal elements in position Ym,p are the negative value of the positive admittance between node m

and node p

x The ground bus is not represented by any nodes within the matrix

1 >@ZY >@ (4.1)

ªºYYY1,1 1,2 1,3 «» (4.2) ¬¼ªºYYYY3Bus «» 2,1 2,2 2,3 «» ¬¼YYY3,1 3,2 3,3

Consider the two-bus system illustrated in Figure 4.13, where the substation transformer and line impedances are represented by their reactive (XT and Xn) values as a function of harmonic frequency (h). The load is represented by its resistive (RL) and reactive components (XL). The imaginary unit j is utilized instead of i to conform to traditional power system analysis guidelines. During application of the HRI algorithm, the impedances of the substation transformer and distribution lines are known values. The harmonic impedance of customers is dependent upon their device usage. Hence, for implementation of the HRI algorithm, only the load impedances at different frequencies are unknown. As described previously, customer harmonic impedance is acquired through the utilization of smart meters and/or power quality devices that are installed on the distribution system.

Admittance (Y ) is the inverse of impedance (Z ), which is generically represented by its resistive (R) and reactive component (X ) as a function of h, as illustrated in Equation (4.3). The matrix in Equation (4.4) is developed by applying the previously identified admittance matrix guidelines. Each harmonic frequency analyzed with the HRI algorithm will have a corresponding frequency dependent admittance matrix.

Figure 4.13 One-line diagram of a simple two-bus system

11 Y (4.3) ZRjXh

ªº11 1 jj j «»hX Xh Xh ªºY «»Tn n (4.4) ¬¼2Bus «»111 «»jj ¬¼XhnLLn R jXh Xh

A sensitivity analysis was performed to evaluate the importance of considering resistance in the identification of harmonic resonance. The study consisted of evaluating the expanded form of the IEEE 13 Node

Test Feeder and the X/R ratio of typical impedances for conductors and underground cables. The analysis revealed that the resistive component of the impedance can be ignored without affecting the HRI algorithm results for distribution systems that are configured in a manner similar to the IEEE 13 Node Test Feeder. The results can be found in Appendix B of this document. Therefore, the network admittance matrix can be simplified to a singular plane format, as illustrated in Equation (4.5). It should be noted that an admittance matrix will be developed for each value within H, which consists of Hn frequencies to be analyzed.

ªº11 1 «»hX X h X h ªºY «»Tn n (4.5) ¬¼2Bus «»111 «» ¬¼XhnLn Xh Xh

Applying values that will be known through field monitoring, algorithm parameters, and design characteristics, the resulting network admittance matrix will take the form illustrated in Equation (4.6).

ªº0.3251 0.2215 ¬¼ªºY2Bus «» (4.6) ¬¼0.2215 0.3428

Along with including all impedance paths between the locations of concern and harmonic sources, the network admittance matrix utilized for the HRI algorithm shall include nodes for all possible configurations, as illustrated in Figure 4.4. All branches that can be connected or disconnected from the circuit analyzed must be represented by a switch. Array S will possess the status of all switches utilized to define the switching configuration with respect to the feeder that is to be analyzed with the HRI algorithm. To incorporate the switching configuration into the admittance matrix, the S array value is multiplied by the admittance of the branch that is connected to its corresponding switch. Consider node 9 from the example distribution system presented in Figure 4.4, which is connected to three admittances branches through switches 20, 21, and 22, as illustrated in Figure 4.14. Assuming

that each impedance is represented by its switch’s identification variable, the network admittance matrix diagonal value for node 9 is defined by Equation (4.7).

Figure 4.14 Representation of Node 9 of the sample distribution system

11 1 YS9,9 20 S 21 S 22 (4.7) ZZZSZ20 21 S 22

The values of array S are binary and only identify the state of each switch, such as 1 representing a closed state, while 0 represents an open state. Assuming that the 20th and 22nd values of array S indicate a closed state, while the 21st value indicates an open state, the resulting diagonal would be as illustrated in Equation (4.8).

11 Y9,9 (4.8) ZZSS20 22

Applying the switching status values to develop the network admittance matrix will result in a matrix that possesses diagonals with 0 values, which represent buses or nodes that are not connected to the feeder that is being analyzed. The function shall be structured to simplify the matrix to only include attributes that represent the current system configuration with respect to the circuit being analyzed, as illustrated in Figure 4.15. To simplify the creation of the network impedance matrix from the admittance matrix, the rows and columns associated with diagonals equaling zero must be eliminated, thereby reducing the size of the admittance matrix. Figures 4.16 through 4.19 illustrate the general approach for reducing the matrix.

Eliminate rows and columns not connected

Figure 4.15 Action block illustrating admittance matrix simplification requirements

ªY1,1 º « Y » « 2,2 » « Y3,3 » >@Y « » « Y4,4 » « Y » « 5,5 » ¬« Y6,6 ¼»

Figure 4.16 Admittance matrix with only diagonals represented

ªY1,1 º « » « 0 » « Y3,3 » >@Y « » « 0 » « 0 » « » ¬« Y6,6 ¼»

Figure 4.17 Admittance matrix illustrating the lack of connections between the circuit being analyzed and buses 2, 4, and 5

Figure 4.18 Elimination of rows and columns that represent buses that are not connected to the circuit being analyzed

ªY1,1 º « » >@Y « Y3,3 » « » ¬ Y6,6 ¼

Figure 4.19 Simplified network admittance matrix

As illustrated in Figure 4.20, capacitor banks are added to the network admittance matrix after it has been reduced to only reflect attributes of the feeder being analyzed. The admittance of a capacitor bank is defined by

Equation 2.2. The switching status of the ith capacitor bank is represented by wi, as illustrated in Equation (2.3).

Unlike array S values, which are known quantities, wi are unknown quantities that are utilized as variables within the objective function that is developed in the Objective Function Development function. The energized state vector W of n-dimensions is illustrated in Equation 2.1.

Add capacitor banks to simplified network admittance matrices

Figure 4.20 Action block illustrating capacitor bank representation

h Yj (2.2) C X Ci

h Yw j w (2.3) CiX i Ci

ªºw1 «»w W «»2 » (2.1) >@«»» «»» ¬¼wn

Consider adding two capacitor banks (#1 and #2) to the simple two-bus system, which results in Figure

4.21. Assuming the frequency dependent admittance of each capacitor bank is 0.12 S, the resulting network admittance would be as illustrated in Equation (4.9).

ªº0.3251 0.12w1 0.2215 ¬¼ªºY2Bus «» (4.9) ¬¼0.2215 0.3428 0.12w2

Figure 4.21 Two-bus two-capacitor bank system

As illustrated in Figure 4.22, the function shall be structured to count the number of capacitor banks that are represented in the new admittance matrix. The task of developing a means to perform this action is delegated to the program designer, who shall find the most efficient means regarding the programming language utilized.

Determine the number of cap banks within the admittance matrix

Figure 4.22 Action block illustrating the capacitor bank counting requirements

The function shall be structured to develop network impedance matrices as functions of W, as illustrated in

Figure 4.23. Although several methods have been developed for creating an impedance matrix directly from system

attributes, evaluation of these methods for utilization within the HRI algorithm proved that the simplest means is through admittance matrix inversion [107] – [112].

Develop network impedance matrices from the network admittance matrices

Figure 4.23 Action block illustrating the development of frequency dependent impedance matrices

The method in which the admittance matrix is inverted is delegated to the programmer, who shall find the most accurate and efficient means regarding the programming language. It has been determined that utilization of the adjugate and determinant to invert the matrix is conducive to providing driving reactance expressions with common denominators and variable interaction terms that are easily eliminated due to their low impact on the final solution. Having common denominators in driving reactance expression simplifies the development of the objective function, which is described in further detail within the Objective Function Development function description. The elimination of interaction terms is described in further detail later in this section. The square matrix characteristic of the admittance matrix and the method in which it is developed provide a nonzero determinant, which makes it a suitable inversion technique for the HRI algorithm.

One option for inverting the W dependent admittance matrix is interfacing the program with mathematical software such as MatLab or Mathmatica, which both utilize the determinant and adjugate in symbolic form to invert a matrix. A second option is to develop a function within the program that is capable of inverting a matrix that varies in size and possesses multiple variables. Limitations in inverting the W dependent admittance matrix directly correspond to limitations in the size of the system and the number of capacitor banks that can be analyzed with the

HRI program.

The function shall be structured to extract the driving reactance expressions for locations of concern from the impedance matrices, as illustrated by the action block in Figure 4.24. The L array stores l information that is utilized to identify which diagonals should be extracted from the impedance matrices. It is delegated to the programmer to develop a method of extracting these diagonal elements. An L array consisting of (3, 5, and 19) would correspond to driving reactances associated with nodes 3, 5, and 19.

Extract driving reactance expression

Figure 4.24 Action block illustrating the function’s requirement for extracting the driving reactance expressions associated with points of concern

The function shall be structured to simplify the driving reaction expressions, as illustrated in Figure 4.25.

Driving reactance expressions, which possess the capacitor bank energized state variables for each bank in the system, are simplified by eliminating variable interaction terms that have little impact on the resulting impedance value. The development of the determinant provides impedance functions that possess denominators with IT interaction terms. The number of terms is dependent upon the number of variables (capacitor bank states) within the admittance matrix. The nonlinear relationship between the number of capacitor banks in the system and the number of interaction terms in the denominator of the driving reactance expression is illustrated in Figure 4.26 and defined in Equation (4.10), where k represents the number of switchable capacitor banks in the system. Fortunately, the capacitor bank admittance values is less than one for the most commonly utilized banks (100 kVAr – 400 kVAr) on systems below 25 kV; therefore, many interaction terms can be eliminated without impacting the accuracy of the

HRI algorithm. Consider the two-bus two-capacitor bank example where the admittance matrix’s diagonals consist of adding 0.12w1 or 0.12w2 from the original bus admittance. Multiplication of these terms during the formation of the determinant will result in an interaction term of 0.0144 w1 w2, which is only 12% of the value with a single variable coefficient. During the convergence and accuracy analysis performed in Chapter 3, it was discovered that interaction terms with more than three variables could be eliminated without affecting the results. The programmer shall perform a study regarding their subject system to determine if and which interaction terms can be eliminated from inclusion within the final driving reactance expressions.

k TI 2 (4.10)

Eliminate interactive terms that do not impact solutions

Figure 4.25 Action block illustrating the simplification of driving reactance expressions

Figure 4.26 Diagram illustrating the format of variable interaction terms within the denominator of the impedance functions

4.5 Pseudo Code for Driving Impedance Derivation Function

Use integer variable k

Define array: S [1-Sn] where Sn is the total number of switches on the circuit to be analyzed

Define array: L [1-Ln] where Ln is the total number of locations to be analyzed

Define array: H [1 – Hn] where Hn is the total number of harmonic frequencies to be analyzed,

Define array: C[1-Ln] where Cn is the total number of harmonic impedances associated with customers to be

analyzed

Accept S, L, H, C

Apply S to develop admittance matrix

Eliminate rows and columns with diagonal elements that equal zero

Apply H to develop frequency specific admittance matrices

Add capacitor bank to admittance matrices with status variable wi

Count total number of capacitor banks represented in each admittance matrix and assign to k

Invert admittance matrices to develop impedance matrices

Extract diagonal elements of each impedance matrix corresponding to values of L

Simplify driving impedance values by eliminating interaction terms that do not impact impedance calculations

by more than 1%

Output matrix diagonal elements (driving reactance expression) to “Objective Function Development” function

Output k to “Optimization” Function

End program

4.6 Objective Function Development Function

The Objective Function Development function shall be structured to utilize the driving reactance expressions developed in the Driving Impedance function to derive an objective function. As illustrated in Figure

4.27, the function also establishes a base value that represents non-harmonic resonance conditions.

The function shall be structured so that an objective function is derived from the driving reactance expressions received from the Driving Impedance function, as defined by the action box illustrated in Figure 4.28.

The objective function is comprised of summing the square of each individual driving reactance expression that represents the l’s at all frequencies of concern. Consider the two-bus two-capacitor bank system previously developed. If buses 1 and 2 were considered l’s and the frequencies of concern were the 3rd, 5th, and 7th harmonic frequencies, then the objective function illustrated in Equation (2.8) would take the form illustrated in Equation

(4.11). Once developed, the W dependent function is set to O.

l n 2 fW ªº Dh W (2.8) ¦¬¼l l 1

222222 fW ªºªºªºªºªºªº D335577 W D W D W D W D W D W ¬¼¬¼¬¼¬¼¬¼¬¼121212 (4.11)

Start

Driving Input Z Impedance D Derivation

Develop objective function

Determine objective function base value

Output Optimization O and M Function

Stop

Figure 4.27 Flowchart of Objective Function Development Function

Develop objective function

Figure 4.28 Action block illustrating the function’s requirement for developing the objective function

The function shall be structured to determine the objective function value that represents non-harmonic resonance conditions, as illustrated in Figure 4.29 and defined in Equation (4.12).

l n 2 MD ªºh 0 ¦¬¼l (4.12) l 1

Determine objective function base value

Figure 4.29 Illustration of the function’s requirement for calculating harmonic threshold limit

4.7 Pseudo Code for Objective Function Development Function

Use variables: O, M,

Define variables for the assignment of driving reactance expressions, A1 thru AN where N is equal to the

number of locations times the number of frequencies analyzed

Accept C1 thru CN where N is equal to the number of locations times the number of frequencies analyzed

Set (C1 through CN) equal to (A1 through AN)

Square each A expression

Sum squared A expressions and set to O

Set all variables to zero in O, calculate value, set to M

Output O and M to “Optimization” function

End program

4.8 The Optimization Function

The goal of the Optimization function is to verify the feasibility of finding a solution that meets the required compensation level, maximize the objective function, interpret the results, and repeat until non-harmonic resonance conditions are identified. Figure 4.30 illustrates the function’s requirements.

Start

Analysis Input Parameters & Q, U, F Field Monitoring

Driving Impedance Input Derivation k Function

Objective Input Function M and O Development Function

Increase iteration value by 1

Is Q constraint Adjust Q feasible? requirements

YES

Maximize objective function

Record solutions

YES Does object function value Apply solutions represent the occurrence to constraints of harmonic resonance?

Solution Output Evaluation recorded Stop Function solutions

Figure 4.30 Flowchart of Optimization Function

The function shall have a structure that counts the number of times the objective function is maximized for a specific compensation level Q. The programmer shall determine the most efficient means of implementing this feature, which is illustrated in the action block in Figure 4.31.

Increase iteration value by 1

Figure 4.31 Action block illustrating the function’s counter feature

The counter values are utilized to evaluate the feasibility of satisfying the compensation Q constraint, which represents the number of capacitor banks (QCAP) or VArs (QVAR) required for adequate compensation. This action, illustrated in the decision block of Figure 4.32, provides the function’s ability to identify when all capacitor bank configurations for a given Q have been analyzed. The counter value (CNT ) cannot be greater than the maximum number of possible capacitor bank combinations (P ), as illustrated in Equations (4.13) and (4.14).

Consider the two-bus two-capacitor bank system previously utilized. If QCAPS was equal to one, there would only be two possible capacitor bank switching scenarios that could satisfy this requirement, each represented by a single capacitor bank engaged. Upon identifying that these two configurations that will result in harmonic resonance, both

CNT and P would be equal to 2. Since non-harmonic resonance conditions have not been found, the function will attempt to maximize the objective function again; however, this time the CNT would equal 3, which would initiate the function to look for another solution that is near, but not exactly equal to, the desired QCAP.

CNT ! P Q is infeasible (4.13)

CNT d P Q is feasible (4.14)

Is Q constraint feasible?

YES

Figure 4.32 Decision block to determine if meeting the compensation requirement is feasible

The function shall be structured to adjust Q, as illustrated in Figure 4.33, when all possible capacitor bank switching scenarios have been analyzed. This feature gives the function the ability to search for additional compensation solutions when it determines that the desired Q cannot be satisfied while preventing the occurrence of harmonic resonance. Compensation requirements are adjusted in accordance to Equation (4.15), where U is defined in the Analysis Parameters and Field Monitoring function. The units of U shall match the units of the required compensation, which can take the form of QCAP or QVAR.

QQUNEW OLD r (4.15)

Adjust Q requirements

Figure 4.33 Action block illustrating the function’s ability to adjust the required compensation level

The function shall be structured to maximize the developed objective function, as illustrated in Figure 4.34.

There are two options for maximizing the objective function. The first consists of programming MINLP BB

algorithms into the program itself. The second option for maximizing the objective function is to interface the HRI program with an internal or external solver that has MINLP BB capabilities. For this approach, the key is providing a format that allows transmission of the objective function and constraints from the HRI program to the solver.

Although identifying an interfacing means may be difficult, it may prove simpler than implementing the MINLP BB algorithms directly into the HRI program.

YES

Maximize objective function

Figure 4.34 Action block illustrating the function’s ability to optimize the objective function

The function shall be structured so the solutions found through maximizing the objective function are stored as illustrated in Figure 4.35.

Record solutions

Figure 4.35 Action block illustrating the requirement for storing solutions identified through maximizing the objective function

The function shall be structured so that it can distinguish between harmonic resonance conditions and non- harmonic resonance conditions, as illustrated in the decision block in Figure 4.36. The value of the objective function is compared to a magnified version of M, which is the non-harmonic resonance value of the objective function. The non-harmonic resonance objective function value is increased by a buffer (F) that allows for fluctuations in the impedance that do not represent harmonic resonance conditions. For example, an F = 1.25 would result in a comparison value of 1.25M. An objective function value greater than 1.25M indicates that undesirable

harmonic resonance condition exists. Solution constraints are set or adjusted and the optimizing algorithm is rerun, if harmonic resonance conditions exist. The approach is described mathematically in Equations (4.16) and (4.17).

OFMd Non harmonic resonance conditions (4.16)

OFM! Harmonic resonance conditions (4.17)

Does object function value represent the occurrence of harmonic resonance?

Figure 4.36 Decision block illustrating the function’s ability to determine if maximizing the objective function identified harmonic resonance conditions

The function shall be structured so that solutions previously identified through maximizing the objective function are eliminated from the search space when the objective function is maximized again. As illustrated in

Figure 4.37, constraints are developed from previous solutions to prevent solution duplication. Constraints are developed by restricting the total sum of previous solutions to values that are less than the required compensation.

Consider a four capacitor bank system that has a QCAP. The first solution identifies w1 and w2 as energized, or as both having a value of 1. The constraints would take the form as illustrated in Equation (4.18), where a repeat of the solution would violate the constraint. If QCAP was utilized instead of QVAR, the constraint would take the form illustrated in Equation (4.19).

Qw11 Qw 2 2 QVAR (4.18)

ww12 QCAP (4.19)

Apply solutions to constraints

Figure 4.37 Action block illustrating function’s ability to adjust constraints according to the previously identified conditions

4.9 Pseudo Code for Optimization Function

Use variables: O, M, Q, F, U, k, P

Accept O, M, Q, F, U, k

Increase counter by 1

Utilize k to calculate P

Evaluate the feasibility of satisfying Q

If not feasible, Adjust Q by U

Else, send objective function and Q to optimization solver or function

Record solutions

Evaluate if harmonic resonance condition exists

If yes, add solution constraint and go to counter

Else, output data to “Results Reporting” Function

End program

4.10 The Results Reporting Function

The goal of the Results Reporting function is to summarize the analyses that were performed utilizing the previous four functions. Figure 4.38 illustrates the flowchart for the function. Information that the function shall provide to the user, but is not limited to, includes:

x Analysis parameters such as H, S, U, T, L, and desired Q,

x Is Q achieved without harmonic resonance,

x Capacitor bank configurations resulting in harmonic resonance,

x Level of compensation achieved without harmonic resonance,

x Required compensation from outside source, such as a customer inverter.

Start

Input Optimization solution Function results

Summarize results from harmonic analysis

output User or Volt/Var software summary

Stop

Figure 4.38 Flowchart of the Results Reporting Function

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100

APPENDIX A CAPACITOR BANK EFFECTS ON ADJACENT FEEDERS

101

The objective function within the HRI algorithm must account for all system components that affect harmonic resonance, which includes the impedance of feeders connected to the same substation transformer and the capacitor bank operations that may occur on them. To evaluate how capacitor bank switching operations may affect harmonic resonance on an adjacent feeder, a sensitivity study was performed utilizing the 4kV and 15kV models developed from the IEEE 13 Node Test Feeder that is described in detail in Chapter 3.

Two distribution system models were developed for the 4 kV and 15 kV single feeder models. Each system contained four feeders, referred to from here on as Feeders A, B, C, and D. Because the analysis only requires a single phase representation, each feeder is modeled after a specific phase of the IEEE 13 Node Test Feeder, which is unbalanced by design. Feeders A and D were based upon the attributes of phase A of the Test Feeder, while Feeders

B and C were constructed from phases B and C respectively. Figure A-1 illustrates the 4 kV four-feeder test system.

The 15 kV distribution system was developed by attaching a duplicate version of the IEEE 13 Node Test Feeder, as described in detail in Chapter 3. The B phase description of the test feeder was added to the tail end of Feeders A and D, while the C phase and A phase descriptions were added to the tail end of Feeders B and C. To account for the different voltage ratings, 15 kV cables were utilized in lieu of the 5 kV cables implemented in the 4 kV system.

Although the size of the primary cables did not change, the amount of reactance associated with each cable was changed to account for the additional insulation. The 15 kV model is illustrated in Figure A-2.

Figure A-1 4 kV model utilized in sensitivity study

102

Figure A-2 15 kV model utilized in sensitivity study

To validate the results acquired from analyzing these two nodes, third and fourth models were developed by converting the standard IEEE 13 Node Test Feeder configuration to one that was all underground, which represents the construction of many distribution systems in service today.

Capacitor banks were varied between 200 kVAr and 400 kVAr, which produced harmonic resonance conditions between the 5th and 12th harmonic frequencies for all models.

A 40 MVA substation transformer was utilized on the 4 kV model, while a 60 MVA transformer was utilized on the 15 kV model. Both transformers possessed an impedance of 10% with an X/R ratio equivalent to 25 in accordance with [101].

Considering the different types of construction and voltage classes, there were a total of four cases evaluated, which are listed below:

x 4 kV , Expanded IEEE 13 Node Test Feeder Configuration and conductor/cable type

x 4 kV, Expanded IEEE 13 Node Test Feeder Configuration, All underground cables

x 15 kV , Expanded IEEE 13 Node Test Feeder Configuration and conductor/cable type

x 15 kV, Expanded IEEE 13 Node Test Feeder Configuration, All underground cables

103

The results indicate that capacitor bank operations on adjacent feeders can have an effect on driving impedance harmonic resonance characteristics. The effects can include a dampening of the impedance, as illustrated in Figures A-3 and A-4. Operation of capacitor banks can also shift the resonance frequency to a slightly lower or higher frequency, as illustrated in Figures A-5 and A-6. However, for the models and simulations observed during the sensitivity study, the overall impact to impedance magnitude was minimal. The frequency shift and/or impedance dampening affect would not result in false harmonic resonance identification because the bandwidth of the impedance that is affected by resonance can span ± 2 harmonic frequencies, as illustrated in Figures A-3 through

A-8. The harmonic resonance identification algorithm seeks the capacitor bank configurations resulting in the greatest driving reactance values at specific frequencies and not the exact magnitude of the impedance; therefore, the dampened and possibly shifted impedance will still be higher than the impedance during non-resonance conditions.

Frequency scan information for the complete analysis is listed in Tables A-1 through A-52.

In summary, for implementation of the HRI algorithm, it is possible to limit capacitor bank model requirements to the feeder being analyzed. However, the onus is placed upon the utility company to determine the extent to which adjacent feeder components influence the magnitude and frequency of harmonic resonance.

Although adjacent feeder effects were minimal on the developed test feeders, the effects were based upon systems that possessed typical power transformer impedance values and feeders with stiff impedance characteristics.

Variations in these attributes, the number of adjacent feeders, or substation bus construction can affect the influence of adjacent feeder capacitor bank operation.

70 60 50 40

- ohms - ohms 30 ~

Z 20 ~ 10 0 1 6 11 16

Harmonic Frequency

FEEDER A: CAP 2, CAP 4 FEEDER B: NONE FEEDER C: NONE FEEDER D: NONE

FEEDER A: CAP 2, CAP 4 FEEDER B: CAP 2 FEEDER C: CAP 2 FEEDER D: CAP 2

Figure A-3 Scenario #1 - Frequency scan of driving reactance at Cap 1 location on Feeder A with all underground construction, 15 kV rating, and 400 kVAr banks

104

70 60 50 40 - ohms - ohms

~ 30 Z

~ 20 10 0 0 2 4 6 8 10121416

Harmonic Frequency

FEEDER A: CAP 2, CAP 4 FEEDER B: NONE FEEDER C: NONE FEEDER D: NONE

FEEDER A: CAP 2, CAP 4 FEEDER B: CAP 2 FEEDER C: CAP 2 FEEDER D: CAP 2

Figure A-4 Scenario #1 - Frequency scan of driving reactance at Cap 2 location on Feeder A with all underground construction, 15 kV rating, and 400 kVAr banks

70 60 50 40

- ohms - ohms 30 ~

Z 20 ~ 10 0 0246810121416

Harmonic Frequency

FEEDER A: CAP 1, CAP 3 FEEDER B: NONE FEEDER C: NONE FEEDER D: NONE

FEEDER A: CAP 1, CAP 3 FEEDER B: CAP 2, CAP 3 FEEDER C: CAP 2, CAP 3 FEEDER D: CAP 2, CAP 3

FEEDER A: CAP 1, CAP 3 FEEDER B: CAP 2, CAP 3 FEEDER C: NONE FEEDER D: CAP 2, CAP 3

FEEDER A: CAP 1, CAP 3 FEEDER B: CAP 3 FEEDER C: CAP 3 FEEDER D: CAP 3

Figure A-5 Scenario #2 - Frequency scan of driving reactance at Cap 1 location on Feeder A with standard IEEE 13 Node Feeder construction, 4 kV rating, and 300 kVAr banks

105

70 60 50

- ohms 40 ~

Z 30 ~ 20 10 0 0246810121416

Harmonic Fequency

FEEDER A: CAP 2 FEEDER B: CAP 3 FEEDER C: CAP 1, CAP 2 FEEDER D: CAP 1, CAP 2 FEEDER A: CAP 2 FEEDER B: NONE FEEDER C: NONE FEEDER D: NONE FEEDER A: CAP 2 FEEDER B: CAP 1, CAP 3 FEEDER C: CAP 1, CAP 3 FEEDER D: CAP 1, CAP 3 FEEDER A: CAP 2 FEEDER B: CAP 1 FEEDER C: CAP 1 FEEDER D: CAP 1

Figure A-6 Scenario #3 - Frequency scan of driving reactance at Cap 1 location on Feeder A with standard IEEE 13 Node Feeder construction, 4 kV rating, and 300 kVAr banks

70 60 50 - ohms - ohms

~ 40 Z

~ 30 20 10 0 0 2 4 6 8 10 12 14 16 Harmonic Frequency

FEEDER A: CAP 1, CAP 3, CAP 5 FEEDER B: NONE FEEDER C: NONE FEEDER D: NONE

FEEDER A: CAP 1, CAP 3, CAP 5 FEEDER B: CAP 2, CAP 4 FEEDER C: CAP 2, CAP 4 FEEDER D: CAP 2, CAP 4

FEEDER A: CAP 1, CAP 3, CAP 5 FEEDER B: CAP 2, CAP 3, CAP 5 FEEDER C: CAP 2, CAP 3, CAP 5 FEEDER D: CAP 2, CAP 3, CAP 5

Figure A-7 Scenario #4 - Frequency scan of driving reactance at Cap 1 location on Feeder A with all underground construction, 15 kV rating, and 400 kVAr banks

106

70 60 50 40 - ohms - ohms

~ 30 Z

~ 20 10 0 0246810121416 Harmonic Frequency

FEEDER A: CAP 1, CAP 3, CAP 5 FEEDER B: NONE FEEDER C: NONE FEEDER D: NONE

FEEDER A: CAP 1, CAP 3, CAP 5 FEEDER B: CAP 2, CAP 4 FEEDER C: CAP 2, CAP 4 FEEDER D: CAP 2, CAP 4 FEEDER A: CAP 1, CAP 3, CAP 5 FEEDER B: CAP 2, CAP 3, CAP 5 FEEDER C: CAP 2, CAP 3, CAP 5 FEEDER D: CAP 2, CAP 3, CAP 5

Figure A-8 Scenario #4 - Frequency scan of driving reactance at Cap 2 location on Feeder A with all underground construction, 15 kV rating, and 400 kVAr banks

107

Table A-1 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Caps 1 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 2 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 2 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0304+0.0969j 0.0346+0.1969j 0.0440+0.3234j 0.0865+0.5440j 1.6781+0.7377j 0.3331-0.1963j 645 0.0846+0.2538j 0.0924+0.5237j 0.1128+0.8529j 0.1911+1.3719j 2.5993+2.0277j 1.2701+1.1456j 632 0.1518+0.3603j 0.1595+0.7368j 0.1796+1.1721j 0.2565+1.7951j 2.6323+2.5562j 1.3215+1.7973j 634 0.4732+0.9780j 0.4654+3.06299j 0.4806+6.1302j 0.5432+10.1174j 1.9675+14.8367j 1.1514+193319j 611 0.1680+0.4835j 0.1977+1.0552j 0.2733+1.8514j 0.5127+3.2928j 5.3852+5.8863j 5.9380+7.9139j 684 0.2336+0.6801j 0.2633+1.4408j 0.3389+2.4297j 0.5784+4.0640j 5.4508+6.8503j 6.0036+9.0706j 671 0.2376+0.5596j 0.2666+1.11985j 0.3409+2.0662j 0.5766+3.5679j 5.3984+6.2191j 5.9448+8.3028j 675 0.4226+0.6350j 0.4512+1.3433j 0.5228+2.2776j 0.7514+3.8431j 5.5159+6.5674j 6.0504+8.7057j 680 0.2170+0.5160j 0.2442+1.1096j 0.3141+1.9253j 0.5384+3.36901j 5.2563+5.9687j 5.7884+7.9774j

Table A-2 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Caps 1 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 2 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 2 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.0790+0.1856j 0.0340+0.3603j 0.0339+0.5162j 0.0458+0.6889j 0.0987+0.9843j 1.3039+2.251j 645 1.0161-1.6700j 0.1388+0.1318j 0.08424+0.6838j 0.0717+1.0380j 0.0824+1.3610j 0.4545+1.9540j 632 1.0743-0.8688j 0.2074+1.0194j 0.1514+1.6760j 0.1409+2.1377j 0.1514+2.5688j 0.5192+3.2675j 634 0.9835+23.5066j 0.5960+30.4841j 0.6096+37.2281j 0.6446+44.2411j 0.688+51.5642j 0.8550+59.2671j 611 7.5828-18.1878j 0.6092-7.0312j 0.1825-4.3978j 0.0832-3.2645j 0.0512-2.6084j 0.1551-2.0772j 684 7.6484-16.8383j 0.6746-5.4890j 0.2481-2.6628j 0.1489-1.3366j 0.1168-0.4877j 0.2207+0.2362j 671 7.6004-17.4639j 0.6838-6.3477j 0.2586-3.6639j 0.1593-2.465j 0.1269-1.7388j 0.2294-1.1360j 675 7.7834-16.7216j 0.8969-5.6637j 0.4670-2.9310j 0.3649-1.6673j 0.3308-0.08712j 0.4308-0.1971j 680 7.5450117.3335j 0.7010-6.4946j 0.2707-3.8970j 0.1677-2.7547j 0.1329-2.0758j 0.2314-1.5177j

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Table A-3 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Caps 1 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 2 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 2 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.1837-0.1204j 0.0564+0.3920j 0.0512+0.6209j 0.08644+.8306j 645 0.1056+1.39801j 0.0655+1.7284j 0.0642+1.9648j 0.0777+2.1939j 632 0.1745+2.82961j 0.1348+3.2685j 0.1335+3.6147j 0.1469+3.95371j 634 0.7729+66.862j 0.8006+74.9903j 0.8397+83.3062j 0.8820+91.81184j 611 0.329-2.0319j 0.0144-1.7706j 0.0096-1.5870j 0.0077-1.4425j 684 0.0985+0.4743j 0.0801+0.9284j 0.0752+1.3047j 0.0733+1.6421j 671 0.1082-1.0145j 0.897-0.6791j 0.0848-0.4206j 0.0828-0.2008j 675 0.3104+0.0002j 0.2915+0.4095j 0.2861+.07426j 0.2838+1.0374j 680 0.1119-1.4325j 0.0928-1.1370j 0.0873-.0917j 0.0848-0.7350j

Table A-4 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 1 2 3 4 5 6 0.0296+0.0960j 650 0.0307+0.1885j 0.0321+0.2866j 0.354+0.3956j 0.0481+0.5388j 0.2380+0.9197j 645 0.080+0.2531j 0.0893+0.5166j 0.1013+0.8190j 0.1316+1.2156j 0.2454+1.9139j 1.9419+4.719j 632 0.1512+0.3596j 0.1564+0.7298j 0.1693+1.1389j 0.1982+1.6411j 0.3099+2.4413j 1.9805+5.3212j 634 0.4727+0.9773j 0.4629+3.0573j 0.4722+6.1063j 0.5039+10.0173j 0.5918+14.7877j 1.5382+21.2409j 611 0.1675+0.4869j 0.1952+1.0499j .2613+1.8215j 0.4313+3.1113j 1.0650+6.0691j 10.4786+20.6778j 684 0.2332+0.6796j 0.2608+1.4355j 0.3274+2.3999j 0.4970+3.8824j 1.1306+7.0331j 10.5443+218346j 671 0.2371+0.5591j 0.2642+1.1933j 0.3296+2.0341j 0.4964+3.3880j 1.1207+6.3941j 10.4261+20.946j 675 0.4222+0.6345j 0.4489+1.3381j 0.5119+2.2474j 0.6730+3.6653j 1.2810+6.7208j 10.4408+21.2383j 680 0.2166+0.5156j 0.2420+1.1044j 0.3034+1.8961j 0.4612+3.1920j 1.0590+6.1115j 10.1133+20.4066j

109

Table A-5 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.0979+0.2761j 0.0372+0.5324j 0.0307+0.6614j 0.0289+0.7659j 0.0282+0.8617j 0.0279+0.9532j 645 0.6752-1.5716j 0.1413+0.1320j 0.0846+.07023j 0.0690+1.0547j 0.0629+1.3286j 0.0599+1.5654j 632 0.7329-0.7720j 0.2099+1.0196j 0.1537+1.6943j 0.1382+2.1514j 0.1321+2.5368j 01292+2.8830j 634 0.8166+23.5574j 0.5972+30.4842j 0.6107+37.2359j 0.6436+44.2476j 0.6812+2.5368 0.7202+59.1395j 611 3.4135-15.0495j 0.4696-6.5237j 0.1589-4.2764j 0.0737-3.2374j 0.0403-2.6329j 0.0244-2.2343j 684 3.4791-13.7000j 0.5352-4.9814j 0.2245-2.5414j 0.1393-1.3095j 0.1059-0.5123j 0.0900+0.0791j 671 3.4692-14.3606j 0.5452-5.8453j 0.2351-3.5437j 0.1498-2.4383j 0.1161-1.7631j 0.1000-1.2917j 675 3.6835-13.6634j 0.7582-5.1667j 0.4434-2.8121j 0.3554-1.6408j 0.3202-0.8953j 0.3030-0.3514j 680 3.4766-14.3087j 0.5628-6.0022j 0.2472-3.7791j 0.1583-2.7285j 0.1224-2.0997j 0.1048-1.6708j

Table A-6 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 13 14 15 16 650 0.0277+1.0426j 0.0277+1.0426j 0.0275+1.2175j 0.0275+1.3039j 645 0.0583+1.7818j 0.0583+1.7818j 0.0568+2.1809j 0.0564+2.3706j 632 0.1276+3.209j 0.1276+3.2091j 0.1262+3.8286j 0.1258+4.1286j 634 0.7598+66.9833j 0.7598+66.9833j 0.8385+83.660j 0.8771+91.8644j 611 0.0158-1.9497j 0.0158-1.9497j 0.0077-1.5668j 0.0056-1.4309j 684 0.0814+0.5565j 0.0814+0.5565j 0.0733+1.3249j 0.0712+1.6537j 671 0.0913-0.0808j 0.0913-0.9331j 0.0829-0.4007j 0.0808-0.1893j 675 0.0951-1.3527j 0.2935+0.0808j 0.2842+0.7623j 0.2817+1.0488j 680 0.0951-1.35271j 0.0951-1.35271j 0.0853-0.8975j 0.0827-0.7237j

110

Table A-7 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0306+0.0972j 0.0356+0.2003j 0.0503+0.3413j 0.1535+0.6621j 0.3495-0.3407j 0.1119+0.2928j 645 0.0848+0.2541j 0.0936+0.5266j 0.1187+0.8693j 0.2665+1.4956j 0.5840+0.6268j 1.0030+2.8678j 632 0.1520+0.3606j 0.1606+0.7397j 0.1855+1.1883j 0.3305+1.9169j 0.6451+1.1736j 1.0565+3.4950j 634 0.4734+0.9782j 0.4663+3.0652j 0.4849+6.1418j 0.5925+10.1964j 0.7804+14.0315j 1.0214+20.2564j 611 0.1681+0.4875j 0.1986+1.0573j 0.2792+1.8659j 0.6108+3.4344j 1.4708+3.6549j 5.9515+14.4072j 684 0.2337+0.6803j 0.2642+1.4429j 0.3449+2.4442j 0.6764+4.2055j 1.5364+4.6188j 6.0171+15.5639j 671 0.2376+0.5598j 0.2675+1.2007j 0.3468+2.0780j 0.6734+3.7082j 1.5258+4.0043j 5.9508+14.73202j 675 0.4227+0.6352j 0.4521+1.3455j 0.5285+2.2910j 0.8464+3.9833j 1.6930+4.3573j 6.0349+15.0698j 680 0.2171+0.5163j 0.2451+1.1117j 0.3197+1.9394j 0.6321+3.5075j 1.4726+3.7700j 5.7579+14.2849j

Table A-8 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.0830+0.2423j 0.0603+0.5423j 0.1369+0.9598j 2.6422+2.7612j 0.2028-0.6062j 0.2028-0.6062j 645 0.8003-1.6348j 0.1415+0.1316j 0.1037+0.7383j 0.6508+1.451j 0.0989+0.9337j 0.0989+0.9337j 632 0.8613-0.8378j 0.2101+1.0193j 0.1725+1.7299j 0.7127+2.5467j 0.1679+2.1465j 0.1679+2.1465j 634 0.8776+23.5271j 0.5972+30.484j 0.6189+37.2508j 0.8719+44.39841j 0.6921+51.4009j 0.6921+51.4098j 611 4.7820-16.6228j 0.5287-6.4746j 0.2049-4.0106j 0.9287-2.3984j 0.1009-2.9098j 0.1009-2.9081j 684 4.8477-15.2733j 0.5944-4.9323j 0.2706-2.2755j 0.9943-0.4706j 0.1666-0.7875j 0.1666-0.7875j 671 4.8258-15.9171j 0.6037-5.7965j 0.2806-3.2805j 0.9958-1.6070j 0.1763-2.0355j 0.1763-2.0355j 675 5.0315-15.1997j 0.8159-5.1183j 0.4877-2.514j 1.1910-0.8159j 0.3804-1.1649j 0.3804-1.1649j 680 4.8152-15.8293j 0.6200-5.9542j 0.2908-3.5206j 0.9857-1.9099j 0.1824-2.3669j 0.1824-2.3669j

111

Table A-9 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.0330+0.2114j 0.0308+0.3618j 0.0406+0.4949j 0.0917+0.6581j 645 0.0571+1.5079j 0.0565+1.7179j 0.0601+1.9192j 0.0794+2.1295j 632 0.1265+2.9382j 0.1259+3.2582j 0.1295+3.5695j 0.1486+3.8899j 634 0.7582+66.8972j 0.7979+74.9873j 0.8384+83.2937j 0.8822+91.8017j 611 0.0204-2.0089j 0.0133-1.772j 0.0095-1.5913j 0.0080-1.4467j 684 0.0861+0.4973j 0.0789+0.9269j 0.0751+1.3005j 0.0737+1.6378j 671 0.0959-0.9918j 0.0886-0.6806j 0.0847-0.4249j 0.0831-1.0333j 675 0.2982+0.0226j 0.2903+0.4808j 0.2860+0.7384j 0.2841+1.0333j 680 0.0997-1.14103j 0.0917-1.1385j 0.0872-0.9212j 0.0851-0.7391j

Table A-10 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0302+0.0967j 0.0332+0.1950j 0.0391+0.3117j 0.05324+0.4707j 0.1099+0.7667j 1.9806+1.6888j 645 0.0845+0.2537j 0.0915+0.5221j 0.1081+0.8421j 0.1534+1.2950j 0.3574+2.2328j 8.0605+5.6111j 632 0.1516+0.3602j 0.1586+0.7352j 0.1750+1.1615j 0.2194+1.7193j 0.4199+2.7556j 8.0121+6.2065j 634 0.4731+0.9778j 0.4647+3.0616j 0.4772+6.1226j 0.5184+10.0682j 0.6598+14.9736j 4.8223+21.6726j 611 0.1679+0.4872j 0.1970+1.0540j 0.2687+1.8419j 0.4627+3.2040j 1.3233+6.6347j 33.5151+20.4205j 684 0.2335+0.6800j 0.2626+1.4396j 0.3343+2.4203j 0.5283+3.9751j 1.3890+7.5986j 33.5808+21.5772j 671 0.2374+0.5595j 0.2659+1.1973j 0.3364+2.0543j 0.5271+3.4799j 1.3757+6.9544j 33.2353+20.7175j 675 0.4225+0.6349j 0.4505+1.3421j 0.5184+2.2674j 0.7030+3.7564j 1.5308+7.2766j 33.0201+21.1004j 680 0.2169+0.5160j 0.2436+1.1084j 0.3098+1.9160j 0.4906+3.2823j 1.3054+6.6628j 32.4921+20.3114j

112

Table A-11 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.1981+.3569j 1.8032+2.158j 0.2121-0.7583j 0.0530-0.1491j 0.0319+0.0791j 0.0262+0.2282j 645 0.3949-1.1649j 0.1672+0.1077j 0.0814+0.5184j 0.0638+0.8587j 0.0581+1.1187j 0.0558+1.3449j 632 0.4610-0.3710j 0.2355+0.9957j 0.1506+1.5123j 0.1131+1.9605j 0.1275+2.3294j 0.1259+2.6650j 634 0.6816+23.759j 0.6087+30.473j 0.6081+37.158j 0.604+44.1715j 0.6783+51.4769j 0.7178+59.0640j 611 1.6924-7.9908j 4.2424-0.2253j 0.5162-5.4418j 0.1236-3.5676j 0.557-2.7813j 0.0309-2.3154j 684 1.7580-6.6413j 4.3080+1.3170j 0.5818-3.7067j 0.1892-1.6393j 0.1213-0.6607j 0.0965-0.0050j 671 1.7581-7.3732j 4.254+0.3943j 0.5898-4.6973j 0.1994-2.7647j 0.1314-1.9100j 0.1065-1.3720j 675 1.9644-6.7521j 4.4328+1.0211j 0.7976-3.9532j 0.4053-1.9639j 0.3356-1.0407j 0.3096-0.4309j 680 1.7638-7.4613j 4.1965+0.1360j 0.5996-4.9097j 0.2081-3.0486j 0.1379-2.2439j 0.1115-1.7497j

Table A-12 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 2 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.0278+0.3474j 0.0379+0.4628j 0.0792+0.6050j 0.3917+0.7618j 645 0.0559+1.5527j 0.0592+1.7531j 0.0743+1.9590j 0.1915+2.1677j 632 0.1253+2.9826j 0.1286+3.2929j 0.1436+3.6089j 0.2596+3.9278j 634 0.7580+66.9113j 0.7988+74.9976j 0.8424+83.3046j 0.9115+91.8112j 611 0.0194-1.9993j 0.0133-1.7672j 0.0106-1.5875j 0.0153-1.4438j 684 0.0850+0.5069j 0.0789+0.9318j 0.0762+1.3042j 0.0809+1.6407j 671 0.0948-0.9823j 0.0886-0.6758j 0.0858-0.4211j 0.0903-0.2021j 675 0.2971+0.0321j 0.2904+0.4128j 0.2871+0.7421j 0.2912+1.0362j 680 0.0987-1.4009j 0.0917-1.1338j 0.0822-0.9175j 0.0922-0.7362j

113

Table A-13 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0302+0.096j 0.0351+0.1904j 0.0503+0.2947j 0.1058+0.4150j 0.3154+0.3685j 0.0560+0.3627j 645 0.0896+0.235j 0.1290+0.5329j 0.2641+0.8897j 0.7578+1.3841j 2.6149+0.3903j 0.3074-0.2263j 632 0.1563+0.360j 0.1955+0.7460j 0.3283+1.2087j 0.8144+1.8080j 2.6464+0.9429j 0.3737+0.4437j 634 0.4772+0.978j 0.4941+3.0702j 0.5879+6.1556j 0.9076+10.1214j 1.9679+13.8777j 0.6255+18.6043j 611 0.1712+0.4874j 0.2276+1.0652j 0.4333+1.9212j 1.4262+3.5768j 8.5788+2.8274j 2.7968-3.6948j 684 0.2460+0.6905j 0.3345+1.5533j 0.6543+2.9422j 2.27772+5.9166j 15.5746+5.6104j 6.3942-7.6280j 671 0.2407+0.5597j 0.2962+1.2085j 0.4991+2.1332j 1.4805+3.8506j 8.5646+3.1948j 2.8480-3.1948j 675 0.4257+0.6351j 0.4803+1.3536j 0.6788+2.34671j 1.6443+4.1289j 8.6638+3.58871j 3.0330-2.6875j 680 0.2201+.5162j 0.2730+1.199j 0.4684+1.9951j 1.4224+3.6539j 83826+3.023j 2.8152-3.3200j

Table A-14 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.0688+0.5377j 0.1380+0.6356j 0.2014+0.6715j 0.2231+0.6807j 0.2119+0.7006j 0.1898+0.7392j 645 0.4235+0.7535j 1.0382+1.0464j 1.6001+0.7886j 1.7909+0.2944j 1.6900-.1037j 1.4930-0.3359j 632 0.4877+1.5216j 1.0946+1.9228j 1.6503+1.7804j 1.8399+1.4039j 1.7411+1.1218j 1.5471+1.0035j 634 0.7096+24.7028j 0.2936-1.0655j 1.2461+37.2624j 1.3080+43.9422j 1.2611+51.0266j 1.1932+58.4911j 611 0.7953-2.0076j 1.5010-4.1797j 0.1132-0.4445j 0.0650+0.0017j 0.0646+0.3299j 0.0732+0.5869j 684 2.5721-5.3203j 0.3662-0.4410j 0.8981-3.5207j 0.5004-2.9933j 0.2714-2.555j 0.1512-2.2039j 671 0.8640-1.4497j 0.5653+0.1828j 0.1870+0.2505j 0.1390+0.7689j 0.1384+1.1706j 0.1467+1.5018j 675 1.0616-0.8912j 0.5633+0.1828j 0.3859+0.9439j 0.3373+1.5342j 0.3360+2.0091j 0.3438+2.4142j 680 0.8578-1.6537j 0.3643-0.7003j 0.1836-0.0562j 0.1368+0.4186j 0.1352+0.7792j 0.1428+1.0706j

114

Table A-15 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps Harmonic Frequency Node 13 14 15 16 650 0.1679+0.7908j 0.1494+0.8499j 0.1344+0.9129j 0.1223+0.9779j 645 1.2987-0.4522j 1.1343-0.5032j 1.008-0.5197j 0.8931-0.5182j 632 1.3555+0.9997j 1.1933+1.0605j 1.0615+1.1555j 0.9551+1.2684j 634 1.1398+66.275j 1.1060+74.3286j 1.0890+82.6160j 1.0843+91.1108j 611 0.0799+0.8033j 0.0838+0.9956j 0.0855+1.1726j 0.858+1.3392j 684 0.0885-1.93535j 0.0545-1.7263j 0.0352-1.5602j 0.0237-1.4253j 671 0.1533+1.7927j 0.1570+2.0599j 0.1587+2.3119j 0.1589+2.5535j 675 0.3500+2.7795j 0.3535+3.1213j 0.3550+3.4480j 0.3551+3.7644j 680 0.1488+1.3325j 0.1522+1.5510j 0.1535+1.7647j 0.1536+1.9682j

Table A-16 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0311+0.0971j 0.0403+0.2008j 0.0845+0.3547j 0.7393+0.0887j 0.1166+0.2005j 0.0398+0.2759j 645 0.0900+0.2545j 0.1346+0.5416j 0.3152+0.9446j 1.3708+0.4842j 1.9631+0.9665j 0.3172-0.2219j 632 0.1570+0.3611j 0.2009+0.7547j 0.3785+1.2628j 1.4192+0.9229j 2.0035+1.5098j 0.3834+0.4481j 634 0.4778+0.9788j 0.4984+3.0772j 0.6244+6.1943j 1.2955+9.5388j 1.5905+14.2200j 0.6307+18.6066j 611 0.1717+0.4879j 0.2320+1.0718j 0.4828+1.9698j 2.1493+2.2829j 7.0080+4.9463j 3.0492-4.0897j 684 0.2456+0.6911j 0.3394+1.5604j 0.7155+2.9995j 3.2418+4.0509j 12.9568+9.4987j 6.9472-8.6287j 671 0.2412+0.5602j 0.3005+1.2151j 0.5481+2.1814j 2.1986+2.5708j 7.0064+5.2906j 3.0983-3.5855j 675 0.4262+0.6356j 0.4845+1.3601j 0.7268+2.3948j 2.3624+2.8665j 7.1118+5.6559j 3.2823-3.0733j 680 0.2206+0.5167j 0.2771+1.1263j 0.5128+2.043j 2.1374+2.4050j 6.8399+5.0689j 3.0630-3.7020j

115

Table A-17 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.0475+0.4153j 0.0923+0.5072j 0.1398+0.5572j 0.1632+0.5830j 0.1623+0.6125j 0.1522+0.6573j 645 0.3948+0.7325j 0.9598+1.0388j 1.5201+0.8392j 1.7578+0.3704j 1.6928-0.0428j 1.5077-0.2978j 632 0.4594+1.5008j 1.0172+1.9153j 1.5712+1.8303j 1.8071+1.4789j 1.7439+1.1820j 1.5616+10411j 634 0.6953+24.6927j 0.9743+30.8896j 1.2129+37.2841j 1.2956+43.9719j 1.2624+51.0486j 1.1984+58.5039j 611 0.7663-2.0735j 0.2805-1.0719j 0.1114-0.4443j 0.0650+0.0016j 0.0648+0.3303j 0.0735+0.5875j 684 2.4377-5.563j 1.3944-4.2133j 0.8533-3.5006j 0.4902-2.9754j 0.2710-2.5436j 0.1522-2.2006j 671 0.8354-1.5153j 0.353-0.4474j 0.1852+0.2507j 0.1390+0.7689j 0.1385+1.1710j 0.1470+1.5024j 675 1.0335-0.9562j 0.5524+0.1766j 0.3842+0.9411j 0.3373+1.5342j 0.3361+2.0095j 0.3441+2.4148j 680 0.8300-1.7182j 0.3516-0.7066j 0.1838-0.0560j 0.1368+0.4186j 0.1354+0.7794j 0.1431+1.0712j

Table A-18 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.1453+0.7210j 0.1706+0.8317j 0.3129+0.5148j 0.0864+0.7466j 645 1.3115-0.4320j 1.1319-0.4972j 1.0185-0.4483j 0.9167-0.5014j 632 1.3682+1.0197j 1.1909+1.0665j 1.0790+1.2262j 0.9786+1.2850j 634 1.1439+66.2814j 1.1054+74.3304j 1.0941+82.635j 1.0905+91.1151j 611 0.0803+0.8038j 0.0837+0.9958j 0.0863+1.1757j 0.0870+1.3400j 684 0.0891-1.9342j 0.0544-1.7261j 0.0356-1.5582j 0.0240-1.4249j 671 0.1537+1.7933j 0.1570+2.0601j 0.1595+2.3149j 0.1601+2.5543j 675 0.3504+27800j 0.3535+3.1215j 0.3558+3.4510j 0.3563+3.7652j 680 0.1492+1.3230j 0.1521+1.5512j 0.1543+1.7676j 0.1547+1.9690j

116

Table A-19 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0309+0.0971j 0.0395+0.2003j 0.0800+0.3513j 0.7509+0.1482j 0.1255+0.2209j 0.419+0.2826j 645 0.0898+0.2545j 0.1338+0.5414j 0.3095+0.9421j 1.4297+0.5556j 2.0347+0.9468j 0.3164-0.2222j 632 0.1569+0.3610j 0.202+0.7543j 0.3729+1.2603j 1.4771+0.9932j 2.0741+1.4906j 0.3826+0.4478j 634 0.477+0.9788j 0.4978+3.0769j 0.6203+6.1926j 1.3338+9.5843j 1.6323+14.2080j 0.6303+18.6064j 611 0.1716+0.4876j 0.2313+1.0715j 0.4774+1.9678j 2.2378+2.3726j 7.2080+4.843j 3.0330-4.0560j 684 0.2455+0.6911j 0.3387+1.5601j 0.7089+2.9972j 3.3712+4.1739j 13.3035+9.2994j 6.9130-8.5441j 671 0.2411+0.5602j 0.2999+1.2148j 0.5427+2.1793j 2.2860+2.6597j 7.2045+5.1887j 3.0822-3.35521j 675 0.4261+0.6356j 0.4839+1.3598j 0.7216+2.3927j 2.4484+2.9550j 7.3084+5.5559j 3.2663-3.0404j 680 0.2205+0.5167j 0.2765+1.1260j 0.5106+2.0409j 2.2224+2.4929j 7.0350+4.9702j 3.0471-3.6693j

Table A-20 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.0502+0.4239j 0.0964+0.5159j 0.1448+0.5649j 0.1679+0.5897j 0.1662+0.6186j 0.1553+0.6629j 645 0.3971+0.7338j 0.9654+1.0388j 1.5255+0.8352j 1.7598+0.3648j 1.6923-0.0472j 1.5065-0.3005j 632 0.4616+1.5021j 1.0227+1.9156j 1.5766+1.8264j 18091+1.4773j 1.7434+1.1777j 1.5604+1.0385j 634 0.6964+24.6933j 0.9768+30.8895j 1.2151+37.2824j 1.2964+43.9698j 1.2622+51.0470j 1.1980+58.5030j 611 0.7690-2.0690j 0.2815-1.0716j 0.1115-0.4443j 0.0650+0.0016j 0.0648+0.3303j 0.0735+0.5875j 684 2.4496-5.5467j 1.04022-4.2117j 0.8564-3.5023j 0.4908-2.9767j 0.2710-2.5442j 0.1521-2.2008j 671 0.8380-1.5108j 0.3542-0.4470j 0.1853+0.2506j 0.1390+0.7689j 0.1385+1.1710j 0.1470+1.5023j 675 1.0361-0.9518j 0.5534+0.1768j 0.3843+0.944j 0.3373+1.5342j 0.3361+2.0095j 0.3441+2.4148j 680 0.8326-1.7138j 0.3525-0.7062j 0.1840-0.0561j 0.1368+0.4186j 0.1353+0.7795j 0.1430+1.0711j

117

Table A-21 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 2 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 01472+0.7258j 0.1693+0.8330j 0.2922+0.5382j 0.0896+0.7643j 645 1.3105-0.4334j 1.1320-0.4976j 1.0188-0.4535j 0.9149-05027j 632 1.3672+1.0183j 1.1910+1.0661j 1.0793+1.2210j 09767+1.2837j 634 1.1436+66.2809j 1.1054+74.3302j 1.0942+82.6342j 1.0900+91.1147j 611 0.0803+0.8038j 0.0837+0.9958j 0.0863+1.1754j 0.0869+1.3399j 684 0.0890-1.9343j 0.0544-1.7261j 0.0356-1.5583j 0.0241-1.4249j 671 0.1536+1.7932j 0.1570+2.0601j 0.1595+2.3147j 01600+2.5543j 675 03504+2.7800j 0.3535+3.1215j 0.3558+3.4507j 0.3562+3.7652j 680 0.1492+1.3230j 0.1521+1.5512j 0.1543+1.7674j 0.1547+1.9689j

Table A-22 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 3 (300 kVAr) energized, Feeder D – Cap 3 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0306+0.0966j 0.0373+0.1951j 0.0603+0.316j 0.1762+0.4991j 0.5614+0.0443j 0.0584+0.2018j 645 0.896+0.2541j 0.1314+0.5370j 0.2800+.0910j 0.9137+1.44558j 1.6615-0.5805j 0.3233-0.2149j 632 0.1566+0.3607j 0.1978+0.7499j 0.3489+1.2288j 0.9678+1.8690j 1.7082-0.0147j 0.3894+0.4550j 634 0.4775+0.9784j 04959+3.0734j 0.5993+6.170j 1.0082+10.1600j 1.4023+13.3164j 0.6340+18.610j 611 01414+04874j 0.2295+1.0657j 0.4488+1.939j 1.6450+3.6418j 5.1403+0.5477j 3.3906-4.3087j 684 02433+0.6908j 0.3366+1.5565j 0.6736+2.9638j 2.5900+5.9986j 9.2864+1.8405j 7.7443-9.2149j 671 0.2409+0.559j 0.2981+1.2115j 0.5145+2.1512j 1.6970+3.9153j 5.1633+0.9330j 3.4366-3.8019j 675 0.4260+0.6353j 0.4821+1.3565j 0.6938+2.3674j 1.8582+4.1942j 5.3077+1.3341j 3.6180-3.2863j 680 0.2203+0.5165j 0.2784+1.1228j 0.4822+2.0130j 1.6342+3.7197j 5.0615+0.7819j 3.3961-3.9123j

118

Table A-23 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 3 (300 kVAr) energized, Feeder D – Cap 3 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.0485+0.3890j 0.0914+0.4981j 0.1397+0.5553j 0.1632+0.5831j 0.1608+0.6111j 0.1472+0.6503j 645 0.3898+0.7269j 09548+1.0370j 1.5188+0.8394j 1.7578+0.3703j 1.6932-0.0416j 1.5098-0.2945j 632 0.4544+1.4953j 1.0122+1.9135j 1.569+1.8305j 1.8072+1.4789j 1.7443+1.1832j 1.5637+1.0444j 634 0.6928+24.609j 09720+30.888j 1.2123+37.2842j 1.2956+43.9719j 1.2626+51.0490j 1.1991+58.505j 611 0.7632-2.0885j 0.2798-1.0726j 0.1114-0.4443j 0.0650+0.0016j 0.0648+0.3303j 0.0735+0.5876j 684 2.4206-5.6192j 1.3880-4.2173j 0.8526-3.5006j 0.4902-2.9754j 0.2710-2.5434j 0.1523-2.2003j 671 0.8323-1.5301j 0.3525-0.4480j 0.1852+0.2507j 0.1390+0.7689j 0.1385+1.1710j 0.1470+1.5024j 675 1.0305-0.9709j 0.5518+0.1758j 0.3842+0.9441j 0.3373+1.5342j 0.3361+2.0095j 0.3441+24149j 680 0.8271-1.7328j 0.3509-0.7072j 0.1838-0.0561j 0.1368+0.4168j 0.1354+0.7794j 0.1431+10712j

Table A-24 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 3 (300 kVAr) energized, Feeder C – Cap 3 (300 kVAr) energized, Feeder D – Cap 3 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.1321+0.6988j 0.1185+0.7528j 0.1072+0.8100j 0.0980+0.8686j 645 1.3175-0.4260j 1.1512-0.4866j 1.0151-0.5088j 09046-0.5109j 632 1.3741+1.0256j 1.2100+1.0770j 1.0757+1.1663j 0.9668+1.2756j 634 1.1458+66.2832j 1.1111+74.3334j 1.0929+82.6189j 1.0874+91.1126j 611 0.0805+0.8039j 0.0844+0.9962j 0.0861+1.1731j 0.0864+1.3395j 684 0.0894-1.9338j 0.0551-1.7256j 0.0356-1.5599j 002339-1.4251j 671 0.1538+1.7934j 0.1577+2.0605j 01593+2.3123j 0.1595+2.5539j 675 0.3506+2.7802j 0.3541+3.1218j 0.3556+3.4484j 0.3557+3.7648j 680 01493+1.3231j 0.1528+1.5516j 0.1541+1.7551j 0.1542+1.9685j

119

Table A-25 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0308+0.0968j 0.0381+0.1970j 0.0611+0.3206j 01561+0.4899j 0.4762+0.3514j 0.1141+0.4936j 645 0.0898+0.2543j 0.1323+0.5385j 0.2816+0.8144j 0.8801+1.4481j 2.6364-0.1502j 0.2903-0.2297j 632 0.1563+0.3608j 0.1986+0.7515j 0.3458+1.2330j 09347+1.8712j 2.6682+0.4104j 0.3569+0.4409j 634 0.4776+0.9786j 0.4966+3.0746j 0.6007+6.1731j 0.9866+10.1617j 1.9763+13.5611j 0.6163+18.6026j 611 0.1716+0.4877j 0.2301+1.0694j 04508+1.9433j 1.5997+3.6489j 83025+1.2109j 2.5458-2.9772j 684 0.2454+0.6909j 0.3373+1.5578j 0.6761+2.9684j 2.5262+6.0108j 14.9536+2.7629j 5.8948-5.8419j 671 0.2410+0.5600j 0.2987+1.2127j 0.5164+2.1551j 1.6522+3.9223j 8.2932+1.5939j 2.5986-2.4846j 675 0.4261+0.6355j 0.4827+1.3577j 0.6957+2.3685j 1.8138+4.2008j 8.4026+2.0029j 2.7834-1.9855j 680 0.2204+0.5166j 02754+1.1239j 0.4851+20168j 1.5901+3.7257j 8.1272+1.4515j 2.5666-2.6247j

Table A-26 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.4353+1.1626j 2.2518-0.4986j 0.2929-0.5861j 0.0604-0.1431j 0.0329+0.0765j 0.0372+0.2135j 645 0.6231+0.8095j 0.9624-0.3387j 0.6806+0.7862j 1.3697+0.7320j 1.6171+0.2810j 1.5473-0.0938j 632 0.6845+1.5770j 1.0207+0.555j 0.7422+1.7775j 1.4235+1.8360j 1.6689+1.502j 1.6007+1.2429j 634 0.8081+24.7289j 0.9661+30.2653j 0.8609+37.2672j 1.1470+44.1147j 1.2368+51.1659j 1.2127+58.5723j 611 1.0888-1.7505j 0.3708-1.3097j 0.0974-0.4524j 0.0652+0.0016j 0.0648+0.3324j 00745+0.5908j 684 3.8021-4.2701j 1.8051-61225j 0.4266-3.5897j 0.3842-2.8965j 0.2565-2.4997j 0.1542-2.1830j 671 1.1543-1.1501j 0.4429-0.6827j 0.1714+0.2426j 0.1391+0.7688j 0.1386+1.1731j 0.1480+1.5057j 675 1.3480-0.5936j 0.6419-0.0563j 0.3705+0.9361j 0.3374+1.5341j 0.3362+2.0116j 0.3451+2.418j 680 1.1412-1.3583j 0.4405-09371j 0.1703-0.0640j 0.1369+0.4185j 0.1354+0.7816j 0.1440+1.0744j

120

Table A-27 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.0466+0.3145j 0.0601+0.4072j 0.0966+0.5149j 0.3142+0.6240j 645 1.3756-0.3115j 1.2009-0.4216j 1.0472-0.4708j 0.9080-0.4706j 632 1.4315+1.1390j 1.2592+1.1413j 1.1074+1.2040j 0.9699+1.3155j 634 1.1646+66.3190j 1.1260+74.3523j 1.1020+82629j 1.0883+91.1231j 611 0.0822+0.8069j 0.0863+0.9984j 0.0875+1.1747j 0.0866+1.3415j 684 0.0922-1.9272j 0.0569-1.7230j 0.0364-1.5588j 0.0240-1.4243j 671 0.1556+1.7963j 0.1569+2.0627j 0.1607+2.3139j 0.1597+2.5558j 675 0.3523+2.7831j 0.3560+3.1240j 0.3570+3.4500j 0.3559+3.7667j 680 0.1510+1.3260j 0.1546+1.5538j 0.1555+1.7667j 0.1544+1.9704j

Table A-28 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0313+0.0973j 0.0407+0.2023j 0.0744+0.3508j 0.3639+0.6317j 0.2672-0.1723 0.0554+0.1994j 645 0.0901+0.2547j 0.1352+0.5431j 0.3067+0.9422j 1.2624+1.4871j 0.8028+0.2878j 0.3237-0.2150j 632 0.1572+0.3612j 0.2015+0.7560j 0.3701+1.2604j 1.3111+1.9101j 0.8610+0.8398j 0.3898+0.4549j 634 0.4779+0.9789j 0.4988+3.0782j 0.6183+6.1927 1.2326+10.1842j 0.9059+13.8313j 0.6343+18.6102j 611 0.1718+0.4880j 0.2324+1.0728j 0.4748+1.9680j 2.1229+3.6558j 3.1393+3.6635j 3.3872-4.3285j 684 0.2457+0.6912j 0.3398+1.5614j 0.7058+2.9975j 3.2674+5.9966j 5.9852+7.5333j 7.7329+9.2627j 671 0.2413+0.5603j 0.3009+1.2161j 0.5402+2.1795j 2.1701+3.9300j 3.1777+4.0151j 3.4332-3.8216j 675 0.4263+0.6357j 0.4849+1.3610j 0.719+2.3929j 2.3263+4.2118j 3.3281+4.3756j 3.6148-3.3057j 680 0.2207+0.5169j 0.2775+1.1273j 0.5081+2.0410j 2.0980+3.7376j 3.0930+3.7918j 3.3929-3.9317j

121

Table A-29 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.658+0.4234j 0.1750+0.6395j 0.4816+0.8720j 1.5485+0.5794j 0.3400-0.5316j 0.0561-0.0142j 645 0.4002+0.7306j 1.0494+1.0279j 1.7317+0.5745j 1.1820-0.4497j 1.2867+0.5253j 1.5325+0.0087j 632 0.4647+1.4989j 1.1056+1.9046j 1.7803+1.5690j 1.2387+0.6683j 1.3422+1.7433j 1.580+1.3442j 634 0.6979+24.6917j 1.0148+30.884j 1.2999+37.1719j 1.0672+43.6568j 1.1188+51.2559j 1.2082+58.6069j 611 0.7777-2.0715j 0.2967-1.0680j 0.1173-0.4470j 0.0652+43.6568j 0.632+0.3344j 0.0744+0.5925j 684 2.4821-5.5592j 1.5219-4.2019j 0.9814-3.6208j 0.3701-3.1941j 0.2077-2.4701j 0.1522-2.1744j 671 0.8464-1.5132j 0.3693-0.4434j 0.1910+0.2480j 0.1391+0.7691j 0.1370+1.1751j 0.1479+1.507j 675 1.0446-0.9541j 0.5683+0.1804j 0.3900+0.9415j 0.3374+1.5344j 0.3346+2.0136j 0.3450+2.4198j 680 0.8411-1.7161j 0.3673-0.7026j 0.1896-0.0587j 0.1370+0.4189j 0.1338+0.7836j 0.1440+1.0761j

Table A-30 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.0387+0.1978j 0.0449+0.3270j 0.0585+0.4329j 0.1025+0.5589j 645 1.3875-0.2752j 1.2128-0.4067j 1.0605-0.4641j 0.9321-0.4838j 632 1.4433+1.1748j 1.2709+1.1560j 1.1206+1.2106j 0.9937+1.3024j 634 1.1685+66.3303j 1.1296+74.3567j 1.1057+82.6311j 1.0945+91.1196j 611 0.0826+0.8078j 0.0867+0.9989j 0.0881+1.1749j 0.0877+1.3408j 684 0.0927-1.9252j 0.0573-1.7224j 0.0368-1.5586j 0.0245-1.4245j 671 0.1559+1.7973j 0.1599+2.0632j 0.1613+2.3142j 0.1609+2.5552j 675 0.1514+1.3269j 0.1550+1.5542j 0.1561+1.7669j 0.1555+1.9698j 680 0.1514+1.3269j 0.1550+1.5542j 0.1561+1.7669j 0.1555+1.9698j

122

Table A-31 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0307+0.0968j 0.0375+0.1966j 0.0594+0.3190j 0.1502+0.4854j 0.4654+0.3579j 0.1071+0.4847j 645 0.0897+0.2543j 0.1318+0.5382j 0.2798+0.9130j 0.8688+1.4464j 2.6525-0.1113j 0.2917-0.2298j 632 0.1567+0.3608j 0.1981+0.7512j 0.3438+1.2317j 0.9236+1.8694j 2.6840+0.4488j 0.3582+0.4403j 634 0.4775+0.9786j 0.4962+3.0744j 0.5992+6.17721j 0.9793+10.1607j 1.9861+13.5837j 0.6170+18.6025j 611 0.1715+0.4877j 0.2297+1.0692j 0.4488+1.9422j 1.5842+3.6479j 8.3747+1.3161j 2.5509-3.0369j 684 0.2453+0.6909j 0.3369+1.5576j 0.6736+2.9670j 2.5042+6.0101j 15.0904+2.9437j 5.8978-5.9881j 671 0.2410+0.5600j 0.2983+1.2125j 0.5144+2.1539j 1.6368+3.9212j 8.3646+1.6981j 2.6037-2.5437j 675 0.4260+0.6355j 0.4825+1.3575j 0.6988+2.3674j 1.7986+4.1997j 8.4728+2.1064j 2.7887-2.0440j 680 0.2204+0.5166j 0.2750+1.1237j 0.4832+2.0157j 1.5751+3.7245j 8.1965+1.5542j 2.5719-2.6827j

Table A-32 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.3735+1.1119j 2.4221-0.1662j 0.3182-0.6126j 0.0618-0.1451j 0.0322+0.0761j 0.0337+0.2073j 645 0.6003+0.8114j 1.1783-0.3465j 0.6601+0.7691j 1.3680+0.7320j 1.6173+0.2814j 1.5488-0.0910j 632 0.6620+1.5788j 1.2338+0.5481j 0.7219+1.7606j 1.4218+1.8360j 1.6691+1.5025j 1.6021+1.2457j 634 0.7969+24.7300j 1.0639+30.2603j 0.8521+37.2602j 1.1463+44.1147j 1.2369+51.1661j 1.2132+58.5733j 611 1.0469-1.72251j 0.4089-1.2969j 0.0972-0.4528j 0.652+0.0016j 0.0648+0.3324j 0.0745+0.5909j 684 3.6346-4.3317j 2.1067-6.0694j 0.4173-3.6000j 0.3838-2.89651j 0.2565-2.4997j 0.1543-2.1828j 671 1.1128-1.1700j 0.4803-0.6701j 0.1712+0.2422j 0.1391+0.7688j 0.1386+1.1731j 0.1480+1.5057j 675 1.3070-0.6134j 0.6789-0.0437j 0.3703+0.9357j 0.3374+1.5341j 0.3362+2.0116j 0.3451+2.4181j 680 1.1006-1.3780j 0.4772-0.9245j 0.1701-0.0664j 0.1370+0.4185j 0.1354+0.7816j 0.1441+1.0744j

123

Table A-33 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – no caps energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.0377+0.2972j 0.0396+0.3688j 0.0400+0.4312j 0.0396+0.4883j 645 1.3798-0.3067j 1.2093-0.4159j 1.0634-0.4656j 0.9438-0.4834j 632 1.4357+1.1437j 1.2675+1.1469j 1.1234+1.2091j 1.0053+1.3029j 634 1.1659+66.3204j 1.1285+74.354j 1.1065+82.6307j 1.0976+91.1197j 611 0.0823+0.8070j 0.0866+0.9986j 0.0882+1.1749j 0.0883+1.3408j 684 0.0924-1.9270j 0.0572-1.7228j 0.369-1.5586j 0.0247-1.4245j 671 0.1557+1.7965j 0.1598+2.0629j 0.1614+2.3141j 0.1614+2.5552j 675 0.3524+2.7832j 0.3563+3.1242j 0.3577+3.4502j 0.3567+3.7661j 680 0.1511+1.3261j 0.1549+1.5539j 0.1562+1.7669j 0.1560+1.9698j

Table A-34 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 2 ( 300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0310+0.0971j 0.0390+0.1994j 0.0674+0.3340j 0.2271+0.5562j 0.5129-0.0709j 0.0723+0.2938j 645 0.0899+0.2545j 0.1333+0.5406j 0.2919+0.9269j 1.0237+1.4853j 1.2573-0.4760j 0.3131-0.2199j 632 0.1569+0.3610j 0.1997+0.7535j 0.3556+1.2454j 1.0760+1.9080j 1.3097+0.0878j 0.3794+0.4500j 634 0.4777+0.9788j 0.4974+3.0763j 0.6078+6.1819j 1.0792+10.184j 1.1633+13.3807j 0.6286+18.6077j 611 0.1716+04876j 0.2310+1.0709j 0.4606+1.9545j 1.7989+3.6821j 4.007+1.1121j 3.1103-3.9023j 684 0.2455+0.6911j 0..383+15595j 0.6882+2.9817j 2.8099+6.0486j 7.3199+2.9339j 7.1252-8.1819j 671 0.2411+0.5602j 0.2995+1.2142j 0.5216+23.1662j 1.8493+3.9555j 4.0404+1.4905j 3.1587-3.399j 675 0.4261+0.6356j 0.4836+1.3559j 0.7052+2.3796j 2.0887+4.2348j 4.1937+1.8805j 3.3413-2.8894j 680 0.2205+0.5167j 0.2762+1.1255j 0.4944+2.0278j 1.7832+3.7599j 3.9563+1.3209j 3.1212-3.5195j

124

Table A-35 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 2 ( 300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.2334+0.6481j 0.4100+0.8891j 1.7801-0.0960j 0.1831-0.3333j 0.0824+0.1319j 0.0940+0.0196j 645 0.4794+0.7436j 1.2377+0.9660j 0.9613-0.3206j 1.2061+0.7380j 1.6055+0.2378j 1.5162-0.0077j 632 0.5428+1.5118j 1.2915+1.843j 1.0201+0.6846j 1.2619+1.8419j 1.6574+14593j 1.5699+1.3281j 634 0.7369+24.6975j 1.0997+30.8547j 0.9715+36.8018j 1.0835+44.1180j 1.2324+51.150j 1.2027+58.6015j 611 0.9036-1.9677j 0.333-10.0664j 0.117-0.4689j 0.0652+0.0016j 0.0647+0.3321j 0.0741+0.5923j 684 3.0001-5.2077j 1.8013-4.232j 0.6522-4.1367j 0.3435-2.8995j 0.2554-2.5059j 0.150-2.1759j 671 0.9712-1.1410j 0.4055-0.4418j 0.1856+0.2262j 0.1391+0.7688j 0.1384+1.1728j 0.1477+1.5071j 675 1.1675-0.8519j 0.6024+0.1821j 0.3846+0.9199j 0.3374+1.5341j 0.3360+2.0113j 0.3447+2.4195j 680 0.9628-1.6146j 0.4028-0.7009j 0.1843-0.0800j 0.1370+0.4185j 0.1353+0.7814j 0.1437+1.0758j

Table A-36 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 2 ( 300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.0372+0.2365j 0.0376+0.3356j 0.0382+0.4088j 0.0381+0.4713j 645 1.3848-0.2876j 1.2134-0.4093j 1.0659-0.4628j 0.9454-0.4820j 632 1.4406+1.1626j 12715+1.153j 1.1260+1.2119j 1.0070+1.3042j 634 1.1676+66.3264j 1.1297+74.3559j 1.1072+82.6314j 1.0980+91.1200j 611 0.0852+0.8075j 00867+0.9988j 0.0833+1.1750j 0.0884+1.3409j 684 0.0926-1.9259j 0.0573-1.7225j 0.0369-1.5585j 0.0247-14245j 671 0.1558+1.7969j 0.1599+2.0631j 0.1615+2.3142j 0.1615+2.5552j 675 0.3525+2.7837j 0.3564+3.1244j 0.3578+3.4503j 0.3577+3.7661j 680 0.1513+1.3266j 0.1550+1.5542j 0.1563+1.7570j 0.1561+1.9699j

125

Table A-37 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV – Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0419+0.0476j 0.0493+0.0747j 0.0524+0.1048j 0.0557+0.1374j 0.0605+01723j 0.0683+0.2097j 645 0.0810+0.0780j 0.0912+0.1350j 0.0993+0.1968j 0.1112+0.2641j 0.1303+0.3380j 0.1608+0.4204j 632 0.2452+0.1015j 0.2564+0.1774j 0.268+0.2589j 0.2768+0.3462j 0.2960+0.4403j 0.3265+05430j 634 0.5623+0.7518j 0.5433+2.6163j 0.5367+5.4718j 0.5441+9.1713j 0.5668+13.598j 0.6006+18.658j 611 0.1514+0.1390j 0.1662+0.2535j 0.1800+0.3781j 0.2007+0.5159j 0.2335+0.6720j 0.2865+08538j 684 0.1968+0.1708j 0.2141+0.3189j 0.2326+0.4807j 0.2608+0.6621j 0.3053+0.8710j 0.3767+1.1195j 671 0.2130+0.1512j 0.2281+0.2765j 0.2419+0.4122j 0.2626+0.5611j 02953+0.7284j 03482+0.9213j 675 0.4721+0.1925j 0.4894+0.3460j 0.5037+0.5123j 0.5243+0.6926j 0.5567+0.8915j 0.6091+1.1162j 680 0.2225+0.1619j 0.2393+0.2911j 0.2543+0.4315j 0.2737+0.5855j 0.3059+0.7579j 0.3579+0.9558j

Table A-38 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV – Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.0809+0.2503j 0.1017+0.2942j 0.1370+0.3406j 0.1982+0.3828j 0.2998+0.3936j 0.4131+03045j 645 0.2100+0.5137j 0.2904+0.6192j 0.4258+0.7334j 0.6599+0.8311j 1.0466+0.8094j 1.4772+0.4115j 632 0.3756+0.6565j 0.4558+0.7824j 0.5910+0.9170j 08246+1.0353j 1.2107+1.0347j 1.6412+06586j 634 0.6470+24.272j 0.7084+30.373j 0.7910+36.9032j 0.9066+43.803j 1.0668+50.998j 1.2235+58.3732j 611 0.3740+1.0717j 05244+1.3388j 0.7958+1.6658 1.3114+2.0296j 2.2845+2.2325j 3.6773+1.5777j 684 0.4950+1.4244j 0.6997+1.8086j 1.0745+2.295j 1.8017+2.8654j 3.2177+3.2592j 5.3574+2.4556j 671 0.4355+1.1503j 0.5855+1.4285j 08564+1.7666j 1.3711+2.1415j 2.3427+2.3561j 3.7340+1.7145j 675 0.6958+1.3770j 0.8446+1.6870j 1.1138+2.0571j 1.6257+2.4646j 2.5932+2.7135j 3.9809+2.1106j 680 0.4438+1.1896j 0.5915+1.4721j 0.8586+1.8148j 1.3667+2.1946j 2.3275+2.4174j 3.7065+1.7944j

126

Table A-39 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV – Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 13 14 15 16 650 0.3915+0.1329j 0.2466+0.719j 0.1377+0.1170j 0.831+0.1867j 645 1.3957-0.3059j 0.8466-0.5999j 0.4342-0.4914j 0.2279-0.2900j 632 1.5609-0.0372j 1.0133-0.3110j 0.6016-0.1828j 0.3955+0.0385j 634 1.2002+65.944j 1.0465+73.9212j 0.9552+82.2442j 0.9329+90.7901j 611 4.1340-0.2774j 3.1236-1.5532j 2.0776-1.7858j 1.4169-1.6439j 684 6.2907-0.2165j 5.0158-2.2371j 3.5581-2.7470j 2.6161-2.6671j 671 4.1913-0.1265j 3.1830-1.3897j 2.1387-1.6112j 1.4788-1.4585j 675 4.4414+0.3107j 3.4404-0.9176j 2.4008-1.1083j 1.7432-0.9252j 680 4.1655-0.0183j 3.1725-1.2645j 2.1401-1.4802j 1.4869-1.3243j

Table A-40 Impedance (ohms) at harmonic frequencies 1 – 6 for Expanded IEEE 13 Node Test Feeder 4 kV – Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1(300 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 650 0.0422+0.0476j 0.0503+0.0751j 0.0548+0.1065j 0.0603+0.1418j 0.0688+0.1817j 0.0828+0.2275j 645 0.0813+0.0780j 0.0922+0.1354j 0.1017+0.1984j 0.1161+0.2683j 0.1397+0.3472j 0.1790+04381j 632 0.2455+0.1015j 0.2574+0.1778j 0.2672+0.2605j 0.2817+0.3505j 0.3054+0.4495j 0.346+0.5606j 634 0.5626+0.7518j 0.5441+2.6166j 0.5384+5.4729j 0.5483+9.1741j 0.5724+13.604j 0.6106+18.6680j 611 0.1517+01390j 0.1671+0.2539j 0.1853+0.3796j 0.2056+0.5201j 0.2453+0.6814j 0.3069+0.8726j 684 0.1970+0.1708j 0.2151+0.3193j 0.2350+0.4823j 0.2659+0.6663j 0.3160+0.8807j 0.3992+1.1393j 671 0.2133+0.1512j 0.2290+0.2769j 0.2442+0.4137j 0.2675+0.5653j 0.3053+0.7377j 0.3685+0.9400j 675 0.4724+0.1925j 0.4904+0.3464j 0.506+0.5139j 0.5291+0.6968j 0.5666+0.9009j 0.6293+1.1350j 680 0.2228+0.1620j 0.2402+0.2915j 0.2556+0.4330j 0.2785+0.5897j 0.3157+0.7672j 0.3778+0.9745j

127

Table A-41 Impedance (ohms) at harmonic frequencies 7 – 12 for Expanded IEEE 13 Node Test Feeder 4 kV – Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 7 8 9 10 11 12 650 0.1065+0.2812j 0.1492+0.3442j 0.2296+0.4127j 0.3814+0.4549j 05939+0.3536j 0.6216+0.0719j 645 0.2457+0.5447j 0.3634+0.6680j 0.5801+0.7927j 0.9746+0.8262j 1.4885+0.4657j 1.4754-0.3249j 632 0.4111+0.6875j 0.5285+0.8315j 0.7447+0.9765j 1.1386+1.0309j 1.6522+0.6924j 1.6404-0.0763j 634 0.6649+24.2876j 0.7421+30.3954 0.8566+36.9274j 1.0300+43.7999j 1.2262+50.8702j 1.219+58.1231j 611 0.4172+1.1062j 0.6201+1.3958j 1.0177+1.7342j 1.8102+1.9777j 3.0309+1.5028j 3.4277-0.0773j 684 0.5441+1.4619j 0.8122+1.8717j 1.3445+2.3694j 2.4296+2.7781j 4.1738+2.2486j 4.9018+0.1332j 671 0.4785+1.1848j 0.6810+1.4854j 1.0779+1.8351j 1.8692+2.0901j 3.0886+1.6280j 3.4859+0.0617j 675 0.7384+1.4115j 0.9395+1.7442j 1.3341+2.1262j 2.1218+2.4151j 3.3380+1.9908j 3.7384+0.4643j 680 0.4861+1.2240j 0.6856+1.5292j 1.0772+1.8837j 1.8595+2.1461j 3.0679+1.7005j 3.4672+0.1592j

Table A-42 Impedance (ohms) at harmonic frequencies 13 – 16 for Expanded IEEE 13 Node Test Feeder 4 kV – Underground version, Feeder A – Cap 1 (300 kVAr) and Cap 3 (300 kVAr) energized, Feeder B – Cap 1 (300 kVAr) energized, Feeder C – Cap 1 (300 kVAr) energized, Feeder D – Cap 1 (300 kVAr) energized Harmonic Frequency Node 13 14 15 16 650 0.4238-0.0570j 0.2650-0.0088j 0.1844+0.0966j 0.1579+0.2292j 645 0.9132-0.6751j 0.4896-0.5978j 0.2690-0.4265j 0.1614-0.2575j 632 1.0798-0.4063j 0.6570-0.3093j 0.4366-0.1182j 0.3290+0.0709j 634 1.0455+65.8293j 0.9405+73.9232j 0.9094+82.2628j 0.9156+90.7988j 611 2.5332-1.0637j 1.7064-1.1457j 1.2394-0.9475j 1.0096-0.7025j 684 3.8143-1.2850j 2.7465-1.4471j 2.16373-1.1735j 1.9515-0.7863j 671 2.5933-0.9125j 1.7678-0.9836j 1.3013-0.7746j 1.0716-0.5187j 675 2.8522-0.4761j 2.0302-0.5166j 1.5651-0.2771j 1.3358+0.0097j 680 2.5880-0.8010j 1.7717-0.8674j 1.3096-0.6554j 10816-0.3961j

128

Table A-43 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D –no caps energized Harmonic Frequency Node 1 2 3 4 5 6 7 8 650A 0.06 0.12 0.18 0.25 0.32 0.40 0.51 0.58 645A 0.16 0.31 0.48 0.68 0.92 1.25 1.91 3.76 632A 0.26 0.40 0.58 0.79 1.05 1.41 2.10 3.93 634A 2.35 6.89 13.33 20.93 29.34 38.63 47.83 57.30 611A 0.28 0.57 0.90 1.30 1.85 2.78 5.05 15.17 684A 0.35 0.72 1.16 1.73 2.56 4.06 7.95 26.68 671A 0.31 0.60 0.94 1.35 1.91 2.85 5.14 15.22 675A 0.50 0.76 1.10 1.53 2.12 3.09 5.42 15.45 680A 0.32 0.61 0.95 1.36 1.93 2.87 5.13 15.07 632B 0.41 0.84 1.34 2.00 2.96 4.70 9.29 31.75 645B 0.45 0.88 1.39 2.06 3.04 4.79 9.36 31.59 646B 0.47 0.91 1.43 2.11 3.09 4.85 9.42 31.56 633B 0.49 0.91 1.42 2.1 3.08 4.84 9.46 31.89 634B 2.51 7.16 13.66 21.34 29.85 39.06 49.08 58.05 671B 0.54 1.11 1.79 2.70 4.08 6.63 13.47 47.67 675B 0.66 1.34 2.14 3.17 4.67 7.33 14.28 47.85 680B 0.57 1.14 1.83 2.74 4.13 6.66 13.44 47.17

129

Table A-44 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D –no caps energized Harmonic Frequency Node 9 10 11 12 13 14 15 16 650A 0.38 0.52 0.65 0.69 0.76 0.81 0.84 0.83 645A 0.19 0.92 1.30 1.77 2.14 2.46 2.70 2.82 632A 0.44 1.17 1.57 2.07 2.44 2.76 3.00 3.09 634A 67.33 77.66 88.09 98.67 109.34 120.10 130.93 141.83 611A 3.70 1.15 0.47 0.63 1.23 1.80 2.41 3.13 684A 7.81 3.53 2.49 0.77 0.16 1.05 2.14 3.61 671A 3.62 1.05 0.37 0.78 1.38 1.97 2.59 3.32 675A 3.44 0.88 0.42 1.23 1.85 2.47 3.13 3.89 680A 3.53 0.98 0.30 0.85 1.46 2.05 2.67 3.39 632B 9.64 4.69 2.85 1.88 1.20 0.65 0.14 0.42 645B 9.44 4.49 2.66 1.66 0.97 0.42 0.21 0.72 646B 9.34 4.38 2.56 1.54 0.85 0.31 0.36 0.91 633B 9.52 4.52 2.68 1.67 0.98 0.44 0.28 0.78 634B 66.16 77.05 77.25 98.63 109.12 119.94 130.84 141.81 671B 15.11 7.76 3.34 3.72 2.86 2.32 2.03 2.07 675B 14.15 6.64 2.28 2.44 1.62 1.36 1.62 2.24 680B 14.80 7.51 3.16 3.50 2.65 2.13 1.90 2.04

130

Table A-45 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400 kVAr) energized, Feeder B – Caps 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder C – Caps 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder D – Caps 2 (400 kVAr) and Cap 4 (400 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 7 8 650A 0.06 0.12 0.19 0.26 0.35 0.49 0.82 0.66 645A 0.16 0.31 0.49 0.69 0.96 1.37 2.38 2.37 632A 0.26 0.41 0.59 0.81 1.09 1.53 2.58 2.39 634A 2.35 6.89 13.33 20.93 29.35 38.37 47.89 57.05 611A 0.28 0.57 0.90 1.31 1.88 2.90 5.73 7.12 684A 0.35 0.72 1.16 1.74 2.60 4.19 8.86 11.85 671A 0.31 0.60 0.94 1.36 1.95 2.97 5.82 7.12 675A 0.50 0.76 1.10 1.54 2.15 3.21 6.11 7.19 680A 0.32 0.61 0.95 1.37 1.96 2.98 5.81 7.03 632B 0.41 0.84 1.34 2.01 3.00 4.48 10.24 13.59 645B 0.45 0.88 1.40 2.07 3.07 4.92 10.31 13.50 646B 0.47 0.91 1.43 211 3.12 4.98 10.37 13.47 633B 0.49 0.91 1.43 2.10 3.11 4.98 10.42 13.66 634B 2.51 7.16 13.66 21.34 29.86 39.09 49.19 56.44 671B 0.54 1.11 1.79 2.71 4.11 6.77 14.63 19.40 675B 0.66 1.34 2.15 3.18 4.70 7.48 15.41 19.11 680B 0.57 1.14 1.83 2.75 4.16 6.81 14.59 19.18

131

Table A-46 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400 kVAr) energized, Feeder B – Caps 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder C – Caps 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder D – Caps 2 (400 kVAr) and Cap 4 (400 kVAr) energized Harmonic Frequency Node 9 10 11 12 13 14 15 16 650A 0.12 0.16 0.32 0.42 0.51 0.57 0.63 0.65 645A 0.38 0.79 1.21 1.58 1.92 2.26 2.54 2.73 632A 0.61 1.03 1.48 1.87 2.24 2.57 2.86 3.01 634A 67.36 77.64 88.09 98.66 109.33 120.1 130.94 141.84 611A 2.93 1.21 0.12 0.63 1.21 1.79 2.43 3.19 684A 6.94 3.83 1.88 0.78 0.16 1.05 2.16 3.66 671A 2.84 1.11 0.16 0.77 1.36 1.96 2.60 3.37 675A 2.61 0.93 0.59 1.22 1.83 2.46 3.15 3.94 680A 2.75 1.03 0.21 0.85 1.44 2.04 2.68 3.44 632B 7.22 5.22 3.00 1.91 1.21 0.64 0.14 0.42 645B 7.03 5.02 2.78 1.69 0.98 0.42 0.21 0.72 646B 6.93 4.90 2.67 1.57 0.85 0.31 0.36 0.91 633B 7.08 5.05 2.80 1.70 0.98 0.44 0.28 0.78 634B 56.23 76.98 87.68 98.37 109.12 119.94 130.64 141.81 671B 8.51 9.00 547 3.85 2.87 2.22 1.87 1.92 675B 7.60 7.86 422 2.52 1.54 1.18 1.49 2.21 680B 8.28 8.72 523 3.62 2.65 2.02 1.74 1.90

132

Table A-47 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400 kVAr) energized, Feeder B – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) energized, Feeder C – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) energized, Feeder D – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) Harmonic Frequency Node 1 2 3 4 5 6 7 8 650A 0.06 0.12 0.19 0.27 0.39 0.67 0.81 0.14 645A 0.16 0.31 0.49 0.70 0.99 1.57 0.81 2.56 632A 0.26 0.41 0.59 0.82 1.13 1.74 0.95 2.79 634A 2.35 6.89 13.33 20.93 29.36 38.41 47.48 57.52 611A 0.28 0.57 0.90 1.31 1.91 3.10 2.15 12.62 684A 0.35 0.72 1.16 1.74 2.63 4.43 3.79 22.94 671A 0.31 0.60 0.94 1.36 1.97 3.17 2.25 12.71 675A 0.50 0.76 1.10 1.54 2.18 3.42 2.59 13.03 680A 0.32 0.61 1.10 1.38 1.99 3.18 2.27 12.62 632B 0.41 0.84 0.95 2.01 3.02 5.07 4.49 27.89 645B 0.45 0.88 1.35 2.08 3.10 5.15 4.91 27.84 646B 0.47 0.91 1.40 2.12 3.15 5.21 4.99 28.08 633B 0.49 0.91 1.43 2.11 3.14 5.21 4.99 60.19 634B 2.51 7.16 13.66 21.34 29.87 39.13 48.26 43.19 671B 0.54 1.11 1.79 2.71 4.14 7.02 7.73 43.87 675B 0.66 1.34 2.15 3.19 4.73 7.72 8.51 42.80 680B 0.57 1.14 1.83 2.76 4.18 7.05 7.78 43.25

133

Table A-48 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 1 (300 kVAr), Cap 3 (300 kVAr), and Cap 5 (400 kVAr) energized, Feeder B – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) energized, Feeder C – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) energized, Feeder D – Caps 2 (400 kVAr), Cap 3 (400 kVAr), and Cap 5 (400 kVAr) energized Harmonic Frequency Node 9 10 11 12 13 14 15 16 650A 0.25 0.36 0.46 0.55 0.65 0.65 1.17 0.18 645A 0.19 0.87 1.29 1.67 2.04 2.04 2.78 2.28 632A 0.43 1.11 1.56 1.96 2.35 2.35 3.04 2.61 634A 67.33 77.65 88.09 98.66 109.33 109.33 130.91 141.86 611A 4.10 1.17 0.12 0.63 1.22 1.22 2.31 3.21 684A 8.83 3.65 1.86 0.78 0.16 0.16 2.09 3.71 671A 4.02 1.08 0.16 0.77 1.37 1.37 2.49 3.40 675A 3.83 0.90 0.59 1.22 1.84 1.84 3.04 3.96 680A 3.92 1.00 0.21 0.85 1.46 1.46 2.58 3.48 632B 11.04 4.91 2.94 1.90 1.20 1.20 0.14 0.43 645B 10.83 4.71 2.73 1.68 0.97 0.97 0.21 0.73 646B 10.72 4.60 2.61 1.55 0.84 0.84 0.36 0.90 633B 10.91 4.74 2.74 1.69 0.97 0.97 0.28 0.78 634B 65.99 77.02 87.69 98.37 109.12 109.12 130.84 141.81 671B 17.67 8.27 5.29 3.79 2.86 2.86 2.37 1.46 675B 16.70 7.14 4.05 2.47 1.58 1.58 1.76 1.99 680B 17.33 8.01 5.05 3.55 2.64 2.64 2.22 1.46

134

Table A-49 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 2 (300 kVAr) and Cap 4 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 1 2 3 4 5 6 7 8 650A 0.06 0.12 0.18 0.24 0.31 0.38 0.47 0.60 645A 0.16 0.31 0.47 0.65 0.85 1.10 1.47 2.7 632A 026 0.40 0.57 0.76 0.97 1.24 1.63 2.44 634A 2.35 6.89 13.32 20.92 29.33 38.62 47.78 57.62 611A 0.28 0.57 0.90 1.30 1.83 2.63 4.06 7.75 684A 0.35 0.71 1.13 1.64 2.32 3.34 5.19 10.05 671A 0.31 0.60 0.94 1.35 1.89 2.70 4.14 7.84 675A 0.50 0.76 1.10 1.52 2.09 2.91 4.38 8.10 680A 0.32 0.61 0.95 1.36 1.91 2.71 4.14 7.81 632B 0.41 0.83. 1.32 1.94 2.77 4.05 6.39 12.60 645B 0.45 0.87 1.38 2.00 2.84 4.13 6.48 12.66 646B 0.47 0.90 1.41 2.03 2.90 4.19 6.54 12.71 633B 0.49 0.90 1.41 2.03 2.88 4.18 6.54 12.77 634B 2.51 7.16 13.65 21.32 29.81 38.93 48.63 59.14 671B 0.54 1.07 1.67 2.35 3.21 4.42 6.47 11.53 675B 0.66 1.31 2.02 2.82 3.80 5.13 7.29 12.47 680B 0.56 1.10 1.70 2.40 3.27 4.48 6.52 11.54

135

Table A-50 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 2 (300 kVAr) and Cap 4 (300 kVAr) energized, Feeder B – no caps energized, Feeder C – no caps energized, Feeder D – no caps energized Harmonic Frequency Node 9 10 11 12 13 14 15 16 650A 1.48 0.31 0.49 0.58 0.66 0.73 0.80 0.86 645A 9.54 0.91 0.31 0.75 1.04 1.27 1.48 1.68 632A 9.78 0.79 0.58 1.01 1.32 1.56 1.79 2.02 634A 68.31 77.46 88.00 98.60 109.29 120.08 130.94 141.88 611A 46.95 10.14 4.40 2.56 1.56 0.87 0.30 0.28 684A 62.15 13.87 6.34 4.01 2.84 2.12 1.62 1.24 671A 47.00 10.02 4.28 2.42 1.42 0.71 0.17 0.47 675A 47.20 9.70 3.93 2.06 1.06 0.46 0.54 1.05 680A 46.54 9.84 4.15 2.31 1.31 0.62 0.14 0.58 632B 79.47 18.12 8.47 5.48 3.97 3.01 2.30 1.71 645B 79.01 17.80 8.21 5.22 3.69 2.72 2.00 1.41 646B 78.89 17.65 8.07 5.07 3.55 2.57 1.84 1.23 633B 79.61 17.92 8.25 5.24 3.71 2.73 2.00 1.40 634B 75.44 75.56 87.12 98.06 108.90 119.78 130.71 141.69 671B 63.71 12.12 4.35 1.79 0.41 0.58 1.33 2.00 675B 64.56 10.96 3.06 0.42 1.15 2.22 3.10 3.89 680B 63.08 11.80 4.10 1.56 0.24 0.84 1.60 2.27

136

Table A-51 Impedance magnitude (ohms) at harmonic frequencies 1 – 8 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder B – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized, Feeder C – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized, Feeder D – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized Harmonic Frequency Node 1 2 3 4 5 6 7 8 650A 0.06 0.12 0.19 0.27 0.38 0.66 0.33 0.24 645A 0.16 0.31 0.48 0.67 0.92 1.39 0.67 1.65 632A 0.26 0.40 0.57 0.78 1.04 1.53 0.87 1.83 634A 2.35 6.89 13.33 20.93 29.34 38.38 47.62 57.53 611A 0.28 0.57 0.90 1.32 1.89 2.93 2.85 6.40 684A 0.35 0.72 1.11 1.65 2.37 3.64 3.89 8.47 671A 0.31 0.60 0.94 1.37 1.95 3.00 2.94 6.48 675A 0.50 0.76 1.10 1.54 2.14 3.22 3.21 6.75 680A 0.32 0.61 0.96 1.38 1.96 3.01 2.96 6.47 632B 0.41 0.83 1.33 1.95 2.82 4.35 5.00 10.78 645B 0.45 0.87 1.38 2.01 2.89 4.43 5.10 10.85 646B 0.47 0.90 1.42 2.06 2.94 4.49 5.17 10.91 633B 0.49 0.91 1.41 2.04 2.93 4.49 5.16 10.94 634B 2.51 7.16 13.66 21.32 29.82 38.99 48.38 58.87 671B 0.54 1.07 1.67 2..36 3.25 5.36 5.37 10.10 675B 0.66 1.31 2.02 2.83 3.84 5.31 6.19 11.04 680B 0.56 1.10 1.71 2.41 3.31 4.71 5.44 10.13

137

Table A-52 Impedance magnitude (ohms) at harmonic frequencies 9 – 16 for Expanded IEEE 13 Node Test Feeder 15 kV – Underground version, Feeder A – Cap 2 (400 kVAr) and Cap 4 (400 kVAr) energized, Feeder B – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized, Feeder C – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized, Feeder D – Caps 4 (400 kVAr) and Cap 5 (400 kVAr) energized Harmonic Frequency Node 9 10 11 12 13 14 15 16 650A 0.62 0.26 0.41 0.50 0.60 0.70 0.86 1.25 645A 5.40 0.95 0.30 0.73 1.02 1.26 1.50 1.86 632A 5.63 0.84 0.57 1.00 1.30 1.55 1.82 2.20 634A 67.96 77.46 88.00 98.60 109.29 120.07 130.95 141.89 611A 28.70 11.30 4.55 2.59 1.57 0.87 0.30 0.28 684A 38.51 15.62 6.61 4.09 2.87 2.13 1.161 1.21 671A 28.77 11.18 4.42 2.45 1.43 0.72 0.17 0.47 675A 29.03 10.86 4.08 2.09 1.06 0.46 0.54 1.05 680A 28.52 10.99 8.91 2.34 1.32 0.62 0.14 0.58 632B 49.76 20.59 8.64 5.64 4.03 3.02 2.27 1.55 645B 49.51 20.25 8.50 5.38 3.77 2.74 1.98 1.25 646B 49.51 20.09 8.69 5.23 3.62 2.59 1.81 1.08 633B 49.93 20.39 8.65 5.40 3.77 2.74 1.97 1.24 634B 73.13 75.39 87.10 98.04 108.90 119.78 130.71 141.70 671B 40.69 14.06 4.69 1.92 0.47 0.56 1.35 2.12 675B 41.67 12.91 3.40 0.54 1.10 2.20 3.12 4.01 680B 40.35 13.72 4.44 1.69 0.28 0.82 1.62 2.40

138

APPENDIX B RESISTANCE IMPACT ANALYSIS

139

B.1 Elimination of Resistive Component

Elimination of the resistive component can reduce the number of interaction terms in the driving impedance functions by 50%, while also eliminating the need to perform the analysis in both the real and imaginary planes. A single driving impedance function can consist of one-hundred and twenty-eight interaction terms for a six capacitor bank system. To determine if the resistive component can be eliminated without influencing the accuracy of the

HRI algorithm, two approaches were taken to examine the influence that resistance has on feeder mainline driving impedance.

1. Utilizing the 5 kV and 15 kV distribution models developed in Appendix A, frequency scans of augmented

driving impedances with and without the resistive component were compared to one another.

2. The X/R ratio as a function of harmonic frequency was evaluated for overhead conductors and

underground cables rated 5 kV and 15 kV to identify the dominant impedance characteristic on a

distribution feeder.

B.2 Frequency Scan Comparison

Frequency scans were performed on the 5 kV and 15 kV models developed in Appendix A. Capacitor banks were varied between 300 kVAr and 1000 kVAr per phase to create harmonic resonance conditions below the

21st harmonic frequency. The goal was to compare the magnitude and resonance frequency of driving impedances with and without their resistive components. Performing a frequency scan exclusively with the reactance component would result in graphs that possessed both positive and negative properties, as illustrated in Figure B-1.

These negative and positive properties hinder an optimization approach from identifying harmonic resonance since the condition is characterized by both minimum and maximum extremes. Therefore, if reactance is the only impedance attribute utilized within the objection function, it must be squared so that maximizing will identify all conditions associated with harmonic resonance.

140 40 20 0 ohms ohms -20 1 2 3 4 5 6 7 8 9 10 11 12 13 -40 14 15 16 17 18 19 20

Harmonic Frequency

X lXl

Figure B-1 Three-dimensional frequency scans comparing the actual value and absolute value of the reactance during harmonic resonance conditions

The results indicated that the resistive component dampens the magnitude of the impedance, but does not change the frequency in which resonance occurs. The magnitude difference between the squared impedance and the squared reactance varied between a multiple of 1.2 and 5.5 of the dampened resistive value, which occurred during the peak of the harmonic resonance period. It should be noted that even though the actual impedance was less than the purely reactance impedance, the magnitude with the resistive component was still between 6 and 34 times greater than the impedance during non-resonance conditions. These results suggest that utilizing reactance exclusively in an objective function is sufficient for identifying harmonic resonance conditions. Figures B-2 through B-5 illustrate several of the frequency scans. The results of the analysis are listed in Tables B-1 through 78.

400 350 300 250 200 150 (ohsm)^2 (ohsm)^2 100 50 0 024681012141618 Harmonic Frequency

Z X

Figure B-2 Frequency scan comparison of the squared driving impedance and driving reactance at Cap 1 location on Feeder A with overhead construction, 4 kV rating, and Caps 1, 3, and 5 (300 kVAr) energized on all feeders

141

70 60 50 40 30 ohms^2 ohms^2 20 10 0 0 2 4 6 8 1012141618

Harmonic Frequency Z X

Figure B-3 Frequency scan comparison of the squared driving impedance and driving reactance at Cap 1 location on Feeder A with overhead construction, 15 kV rating, and Caps 1, 2, 3, 4, and 5 (300 kVAr) energized on all feeders

70 60 50 40

ohms^2 30 20 10 0 024681012141618

Harmonic Frequency

Z X

Figure B-4 Frequency scan comparison of the squared driving impedance and driving reactance at Cap 1 location on Feeder A with underground construction, 15 kV rating, and Caps 1, 2, 3, 4, and 5 (500 kVAr) energized on all feeders

142

70 60 50 40

ohms^2 ohms^2 30 20 10 0 0 2 4 6 8 10 12 14 16 18

Harmonic Frequency Z X

Figure B-5 Frequency scan comparison of the squared driving impedance and driving reactance at Cap 3 location on Feeder A with underground construction, 15 kV rating, and Caps 1, 2, 3, 4, and 5 (500 kVAr) energized on all feeders

143

Table B-1 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0176+0.0520j 0.0188+0.1014j 0.0194+0.1517j 0.0220+0.2035j 0.0208+0.2575j 0.0218+0.3145j 645 0.0307+0.1233j 0.0317+02420j 0.0326+0.3621j 0.0340+0.4865j 0.0361+0.6184j 0.0393+0.7614j 632 0.0553+0.2610j 0.0546+0.5044j 0.05520.7420j 0.0574+0.9812j 0.0617+1.2292j 0.0685+1.4948j 634 0.0813+0.3046j 0.799+0.5894j 0.0799+0.8665j 0.0815+1.1431j 0.0852+1.4266j 0.0914+1.7254j 611 0.0679+0.3308j 0.0670+0.6401j 0.0681+0.9441j 0.07163+1.2538j 0.0779+1.5804j 00881+1.9375j 684 00804+0.3998j 0.0791+0.7733j 0.0807+1.1419j 0.0858+1.5213j 0.0949+1.9278j 0.1092+2.3811j 671 0.2775+0.6016j 0.2723+1.1696j 0.2704+1.7268j 0.2720+2.2886j 0.2775+2.8706j 0.2880+3.4914j 675 0.0931+0.4695j 0.0918+0.9101j 0.0945+1.3488j 0.1017+1.8067j 0.1142+2.3062j 0.1339+2.8749j 680 0.1186+0.5123j 0.1161+0.9920j 0.1177+1.4664j 0.1238+1.9563j 0.1352+2.4835j 0.1535+3.0746j

Table B-2 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0230+0.3758j 0.0252+0.4431j 0.0281+0.5191j 0.0328+0.6087j 0.0413+0.7216j 0.0602+0.8823j 645 0.0440+0.9208j 0.0508+1.1041j 0.0616+1.3237j 0.0796+1.6017j 0.1142+1.9845j 01963+2.5898j 632 0.0789+1.7897j 0.0950+2.1315j 0.1210+2.5496j 0.1660+3.0991j 0.2546+3.8975j 0.4694+5.2504j 634 0.101+2.0505j 0.1164+2.41489j 0.1410+2.8585j 0.1839+3.4215j 0.2683+4.21922j 0.4727+5.5414j 611 0.1037+2.3435j 0.1279+2.8267j 0.1673+3.4343j 02365+4.2551j 0.7341+5.4789j 0.7121+7.6015j 684 01312+2.9082j 0.1655+3.5504j 0.2215+4.3776j 0.3205+5.5204j 0.5185+7.2593j 10080+10.3258j 671 0.3056+4.1760j 0.3345+4.9624j 0.3830+5.9145j 0.4703+7.1518j 0.6472+8.9299j 1.0879+11.9163j 675 0.1641+3.5512j 0.2112+4.3940j 0.2885+5.5029j 0.4257+7.0649j 0.7016+9.4804j 1.3877+13.7945j 680 01820+3.7661j 0.2268+4.6142j 0.3004+5.7135j 0.4311+7.2409j 0.6940+9.5762j 1.3469+13.7094j

144

Table B-3 Node driving impedance in ohms (harmonic frequencies 13 – 17) on Expanded IEEE 13 Node Test Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr Harmonic Frequency Node 13 14 15 16 17 650 0.1255+1.11779j 0.8994+2.1870j 0.4549-0.3989j 0.0905+0.3985j 0.0553+0.7067j 645 0.4944+3.8414j 4.1951+8.6038j 2.2114-4.2636j 0.3775-0.6640j 0.1815+0.5368j 632 1.2597+8.2719j 11.0962+205251j 5.7310-14.0093j 0.8731-4.9668j 0.3352-2.4251j 634 1.2236+8.4431j 10.5524+20.0845j 5.4442-12.5954j 0.8478-3.9874j 0.3401-1.5431j 611 1.9691+12.4306j 17.8183+32.2859j 9.3817-23.8025j 1.4395-92714j 0.5466-5.2522j 684 2.8420+17.3894j 26.1307+46.6818j 13.9354-36.1463j 2.1568-14.8096j 0.8224-8.9449j 671 2.7486+18.5461j 23.9347+45.4700j 12.9398-29.8731j 2.1699-10.2581j 0.9942-4.7349j 675 3.9752+23.8219j 37.0832+65.6685j 20.0411-52.6371j 3.1412-22.2849j 1.2133-13.9585j 680 3.8056+23.2538j 35.2051+62.9243j 18.9729-491141j 2.9818-20.3001j 1.1609-12.3671j

Table B-4 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.06 0.1049 0.1542 0.2055 0.2592 0.3160 645 0.13 0.2466 0.3654 0.4892 0.6207 0.7637 632 0.27 0.5110 0.7466 0.9849 1.2327 1.4983 634 0.33 0.6035 0.8760 1.1502 1.4329 1.7312 611 0.35 0.6478 0.9495 1.2583 1.5846 1.9419 684 0.42 0.7820 1.1481 1.5266 1.9329 2.3866 671 0.81 1.2729 1.7947 2.3392 2.9112 3.5258 675 0.49 0.9200 1.3559 1.8128 2.3123 2.8816 680 0.55 1.0092 1.4781 1.9657 2.4920 3.0832

145

Table B-5 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.3773 0.4446 0.5207 0.6106 0.7243 0.8876 645 0.9231 1.1068 1.327 1.6064 1.9926 2.6087 632 1.7936 2.1363 2.5562 3.1093 3.9166 5.2983 634 2.0563 2.4252 2.8663 3.4326 4.2387 5.5882 611 2.3486 2.8331 3.4434 4.2697 5.5074 7.6749 684 2.9146 3.5588 4.3897 5.5405 7.2991 10.4302 671 4.2065 4.9912 5.9438 7.1858 8.9795 12.0219 675 3.5592 4.4047 5.5188 7.0917 9.5343 13.9385 680 3.7757 4.6261 5.7300 7.2677 9.6287 13.8474

Table B-6 Node driving reactance in ohms (harmonic frequencies 13 – 17) on Expanded IEEE 13 Node Test Feeder with Cap 1, 2, and 6 energized, each rated at 300 kVAr Harmonic Frequency Node 13 14 15 16 17 650 1.1968 2.6886 -0.5602 0.3908 0.7059 645 3.9222 10.9787 -5.0798 -0.708 0.5282 632 8.4849 26.8416 -16.1559 -5.0825 -2.4502 634 8.6467 26.0776 -14.6237 -4.0954 -1.5657 611 12.7652 424363 -27.3231 -9.4653 -5.2954 684 17.8733 61.5716 -41.3787 -151014 -9.0108 671 18.9923 59.0035 -34.6431 -10.5135 -4.7841 675 24.4993 86.8007 -60.1326 -22.7099 -14.0554 680 23.8987 82.9720 -56.2275 -20.6991 -12.457

146

Table B-7 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0187+0.0268j 0.0205+0.0502j 0.0213+0.0736j 0.0218+0.0975j 0.0223+0.1219j 0.0228+0.1471j 645 0.0330+0.0644j 0.0346+0.1245j 0.0355+0.1849j 0.0363+0.2465j 0.0373+0.3100j 0.0388+0.3763j 632 0.0610+0.1384j 0.0619+02699j 0.0630+0.4023j 0.0652+0.5391j 0.0687+0.6841j 0.0739+0.8417j 634 0.1415+01775j 0.1411+0.3465j 0.1401+0.5152j 0.1418+0.6875j 0.1440+0.8667j 0.1479+1.0572j 611 0.0751+0.1755j 0.0757+0.3429j 0.0769+0.5112j 0.0795+0.6852j 0.0839+0.8698j 0.0905+1.0705j 684 0.0890+0.2121j 0.0892+0.4152j 0.0906+0.6190j 0.0937+0.8300j 0.0992+1.0544j 0.1076+1.2997j 671 0.2189+0.2749j 02179+05387j 0.2180+0.8025j 0.2199+1.0725j 0.2241+1.3548j 0.2311+1.6568j 675 0.1029+02493j 0.1024+0.4865j 0.1030+0.7224j 0.1055+0.9631j 0.1103+1.2147j 0.1179+1.4841j 680 0.1826+0.2883j 0.1799+0.5618j 0.1784+0.8324j 0.1789+1.1060j 0.1817+1.3887j 0.1872+1.6871j

Table B-8 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0234+0.1733j 0.0243+0.2007j 0.0254+0.2298j 0.0269+0.2612j 0.0291+0.2960j 0.0324+0.3360j 645 0.0408+0.4466j 0.0436+0.5226j 0.0475+0.6065j 0.0532+0.7020j 0.0618+0.8151j 0.0758+0.9563j 632 0.0815+1.0173j 0.0922+1.2187j 0.1077+1.4569j 0.1307+1.7488j 0.1662+2.1226j 0.2249+2.6282j 634 0.1539+1.2643j 0.1629+1.4952j 0.1763+1.7604j 0.1966+2.0761j 0.2285+2.4688j 0.2820+2.9859j 611 0.1002+1.2949j 0.1140+1.5528j 0.1339+1.8588j 0.1653+2.2354j 0.2094+2.7195j 0.2857+3.3771j 684 0.1197+1.5753j 0.1371+1.8940j 0.1623+2.2744j 0.1997+2.7453j 0.2578+3.3540j 0.3545+4.1851j 671 0.248+1.9876j 0.2574+2.3596j 0.2805+2.7909j 0.3151+3.3093j 0.3696+3.9607j 0.4608+4.82671j 675 0.1292+1.7801j 0.1456+2.1146j 0.1693+2.5050j 0.2045+2.9775j 0.2593+3.5757j 0.3500+4.3772j 680 0.1961+2.0097j 0.2097+2.3678j 0.2303+2.7779j 0.2615+3.2650j 0.3107+3.8703j 0.3934+4.6673j

147

Table B-9 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0379+0.3849j 0.0489+0.4509j 0.0774+0.5576j 0.2116+0.8024j 1.5504-0.0029j 0.1253-0.0593j 645 0.1009+1.14651j 0.1535+1.4340j 0.2985+1.9579j 1.0248+3.3034j 8.9771-1.1741j 0.7271-1.9380j 632 0.3324+3.3648j 0.5626+4.5606j 1.2121+6.8760j 4.5456+13.1053j 42.3655-7.6224j 3.5136-11.9759j 634 0.3808+3.7212j 0.5942+4.8908j 1.2004+7.1204j 4.3320+13.0602j 40.1713-6.3795j 3.4398-10.5911j 611 0.4259+4.3395j 0.7273+5.9085j 1.5806+8.9583j 5.9752+17.1917j 56.0623-10.1553j 4.6708-16.0304j 684 0.5322+5.4063j 0.9150+74038j 2.0005+11.2961j 7.6031+21.1391j 71.6772-13.0502j 5.9976-20.6597j 671 0.6295+6.0698j 0.9946+8.06351j 2.0349+11.8900j 7.4267+22.1391j 69.4488-11.3644j 5.9712-18.8103j 675 0.5164+5.5359j 0.8732+7.4060j 1.8804+11.0128j 7.0527+20.6940j 65.7672-11.4594j 5.4874-18.2226 680 0.5465+5.8018j 0.8772+7.6090j 1.8174+11.060j 6.6763+20.2703j 62.3147-9.9065j 5.2989-16.4706j

Table B-10 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 19 20 650 0.0438+0.1740j 0.0293+0.2812j 645 0.2049-0.7226j 0.0909-0.2291j 632 0.9676-6.4540j 0.4393-4.3105j 634 1.0130-5.3411j 0.5060-3.2799j 611 1.2883-8.7532j 0.5835-5.9429j 684 1.6610-11.3684j 0.7555-7.7808j 671 1.7574-9.7775j 0.8736-6.2511j 675 1.5042-9.6469j 0.6831-6.3167j 680 1.5300-8.3386j 0.7421-5.1578j

148

Table B-11 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.04 0.0585 0.0797 0.1023 0.1230 0.1507 645 0.08 0.1355 0.1928 0.2527 0.3122 0.3810 632 0.17 0.2859 0.4136 0.5482 0.6921 0.8492 634 0.37 0.4450 0.5807 0.7365 0.9060 1.0903 611 0.21 0.3615 0.5244 0.6958 0.8790 1.0793 684 0.25 0.4363 0.6339 0.8420 1.065 1.3098 671 0.59 0.6943 0.9061 1.1503 1.4176 1.7099 675 0.29 0.5100 0.7387 0.9761 1.226 1.4947 680 0.50 0.6660 0.9010 1.1572 1.4296 1.7216

Table B-12 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.1765 0.2036 0.2326 0.2640 0.2989 0.3393 645 0.4510 0.5268 0.6109 0.7068 0.8207 0.9634 632 1.0248 1.2267 1.4660 1.7599 2.1372 2.6495 634 1.2933 1.5215 1.7852 2.1007 2.4949 3.0166 611 1.3038 1.5623 1.8698 2.2489 2.7375 3.4039 684 1.5856 1.9052 2.2874 2.7615 3.760 4.2181 671 2.0342 2.4021 2.8312 3.3494 4.0035 4.8777 675 1.7907 2.1261 2.5180 2.9936 3.5972 4.4090 680 2.0401 2.3958 2.8049 3.2927 3.9011 4.7090

149

Table B-13 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.3889 0.4569 0.5707 0.8746 -7.9926 -0.089 645 1.1572 1.4539 2.0125 3.6682 -47.8093 -2.1368 632 3.4007 4.6354 7.1023 14.7374 -227.999 -12.9561 634 3.7639 4.9680 7.3365 14.5899 -214.8143 -11.5145 611 4.3852 6.0050 9.2534 19.3377 -301.7768 -17.3341 684 5.4632 7.5248 11.6687 24.5494 -385.8786 -22.3326 671 6.1413 8.1939 12.2588 24.7665 -371.7994 -20.4191 675 5.5902 7.5202 11.3610 23.2210 -353.4815 -19.7412 680 5.8594 7.7203 11.3860 22.6328 -333.4006 -17.9065

Table B-14 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 19 20 650 0.1713 0.2812 645 -0.7478 -0.2357 632 -6.5841 -4.3476 634 -5.456 -3.3066 611 -8.927 -5.9926 684 -11.5919 -7.8447 671 -9.9799 -6.2992 675 -9.8472 -6.3732 680 -8.5202 -5.2027

150

Table B-15 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0187+0.0269j 0.0207+0.0503j 0.0217+0.0743j 0.0226+0.0992j 0.0238+0.1257j 0.0254+0.1545j 645 0.0332+0.0645j 0.0354+0.1258j 0.0373+0.1895j 0.0399+0.2581j 0.0439+0.3348j 0.0505+0.4252j 632 0.0616+0.1389j 0.0642+0.2746j 0.0686+0.4189j 0.0763+0.5812j 0.0893+0.7753j 0.1122+1.0245j 634 0.1421+0.1780j 0.1434+0.3512j 0.1464+0.5318j 0.1525+0.7291j 0.1638+0.9565j 0.1843+1.2366j 611 0.0760+0.1763j 0.0792+0.3504j 0.0853+0.5375j 0.0962+0.7515j 0.1149+1.0128j 0.1477+1.3560j 684 0.0901+0.2134j 0.0935+0.4243j 0.1007+0.6510j 0.1138+0.9108j 0.1365+1.2289j 0.1765+1.6485j 671 0.2199+0.2759j 0.2221+0.5477j 0.2279+0.8343j 0.2395+1.1527j 0.2603+1.5277j 0.2979+2.0016j 675 0.1042+0.2505j 0.1074+0.4975j 0.1149+0.7611j 0.1290+1.0603j 0.1535+1.4226j 0.1970+1.8950j 680 0.1838+0.2895j 0.1848+0.5728j 0.1900+0.8708j 0.2015+1.2021j 0.2229+1.5934j 0.2622+2.0903j

Table B-16 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0282+0.1875j 0.0336+0.2287j 0.0479+0.2905j 0.1270+0.4328j 0.4006-0.2000j 0.0375+0.0926j 645 0.0625+0.5404j 0.0880+0.7060j 0.1616+1.0020j 0.6101+1.8253j 2.5747-2.1377 0.1946-0.5478j 632 0.1554+1.3753j 0.2514+1.9383j 0.5408+3.0572j 2.3851+6.4595j 11.1624-10.4842j 0.8314-4.0841j 634 0.2240+1.6145j 0.3138+2.1968j 0.5878+3.3159j 2.3521+6.6449j 10.8963-9.62871j 0.8988-3.4616j 611 0.2102+1.8504j 0.3499+2.6610j 0.7746+4.3015j 3.5046+9.3585j 16.7144-15.9041j 1.2546-6.4564j 684 0.2530+2.2555j 0.4246+3.2544j 0.9478+5.2826j 4.3222+11.5517j 20.7540-19.7499j 1.5659-8.0757j 671 0.3707+2.6587j 0.5354+3.6992j 1.0414+5.7478j 4.3255+11.9545j 20.4950-18.5698j 1.6834-7.1384j 675 0.2801+2.5710j 0.4663+3.6727j 1.0341+5.8918j 4.6951+12.7131j 22.5253-21.2272j 1.7065-8.5404j 680 0.3388+2.7832j 0.5132+3.8867j 1.0508+6.0695j 4.5530+12.7064j 21.9078-19.9273j 1.7623-7.7398j

151

Table B-17 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0233+0.1883j 0.0203+0.2434j 0.0205+0.2869j 0.0213+0.3266j 0.0255+0.3652j 0.0238+0.4046j 645 0.0704-0.0620j 0.0446+0.1702j 0.0369+0.3264j 0.0348+0.4529j 0.0352+0.5673j 0.0371+0.6786j 632 0.2554-2.2090j 0.1239-1.4244j 0.0760-0.9794j 0.0543-0.6809j 0.0433-0.4570j 0.0376-0.2747j 634 0.3345-1.6184j 0.2039-0.8257j 0.1555-0.3609j 0.1329-0.0377j 0.1211+0.2130j 0.1146+0.4230j 611 0.3824-3.7002j 0.1805-2.5709j 0.1053-1.9514j 0.0700-1.5540j 0.0509-1.2728j 0.0396-1.0595j 684 0.4788-4.6675j 0.2261-3.2754j 0.1317-2.5153j 0.0871-2.0308j 0.0629-1.6905j 0.0486-1.4347j 671 0.6095-3.7450j 0.3580-2.3225j 0.2632-1.5182j 0.2180-.09838j 0.1931-0.5907j 0.1782-0.2805j 675 0.5273-4.8195j 0.2537-3.2841j 0.1520-2.4322j 0.1045-1.8765j 0.0794-1.4741j 0.0653-1.1593j 680 0.6077-4.1176j 0.3366-2.6007j 0.2345-1.7431j 0.1860-1.1717j 0.1597-0.7486j 0.1444-0.4107j

Table B-18 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 19 20 650 0.0255+0.4457j 0.0274+0.4895j 645 0.401+0.7925j 0.0442+0.9134j 632 0.0349-0.1163j 0.0342+0.0290j 634 0.1112+0.6093j 0.1097+0.7323j 611 0.0326-0.8890j 0.0280-0.7468j 684 0.0397-1.2322j 0.0339-1.0651j 671 0.1686-0.0227j 0.1693+0.2002j 675 0.0572-0.8974j 0.0531-0.6678j 680 0.1353-0.1245j 0.1302+0.1298j

152

Table B-19 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.04 0.0588 0.0806 0.1044 0.1302 0.1587 645 0.08 0.1372 0.1981 0.2656 0.3415 0.4323 632 0.17 0.2916 0.4318 0.5928 0.7871 1.0385 634 0.37 0.4506 0.5987 0.7803 0.9992 1.2755 611 0.21 0.3703 0.5528 0.7655 1.0275 1.3741 684 0.25 0.4470 0.6684 0.9268 1.246 1.6697 671 0.59 0.7048 0.9404 1.2344 1.5964 2.0649 675 0.30 0.5228 0.7804 1.0779 1.4412 1.9180 680 0.50 0.6787 0.9421 1.2573 1.6407 2.1356

Table B-20 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.1918 0.2339 0.2993 0.4793 -0.595 0.0896 645 0.5488 0.7189 1.0328 2.0602 -4.7548 0.579 632 1.3952 1.9747 3.1617 7.3885 -21.8846 -4.2315 634 1.6544 2.2487 3.4288 7.5390 -20.6423 -3.5902 611 1.8771 2.7115 4.4512 10.7243 -32.9791 -6.6802 684 2.287 3.3151 5.4651 13.2347 -40.9479 -8.3544 671 2.725 3.7881 5.9492 13.6066 -39.3238 -7.3886 675 2.605 3.7385 6.0895 14.5379 -44.2239 -8.8419 680 2.833 3.9634 6.2663 14.4468 -42.159 -8.0169

153

Table B-21 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.1889 0.2444 0.2881 0.3278 0.3665 0.4059 645 -0.0656 0.1700 0.3271 0.4539 0.5686 0.6801 632 -2.2304 -1.4302 -0.9814 -0.6815 -0.457 -0.2744 634 -1.6258 -0.8192 -0.3517 -0.028 0.2225 0.4321 611 -3.7332 -2.5802 -1.9548 -1.5555 -1.2734 -1.0597 684 -4.7085 -3.2868 -2.5194 -2.0325 -1.6912 -1.4349 671 -3.7635 -2.3137 -1.5039 -0.9684 -0.5754 -0.2657 675 -4.8633 -3.2959 -2.4361 -1.8778 -1.4743 -1.1589 680 -4.1458 -2.5992 -1.7352 -1.1622 -0.7389 -0.4011

Table B-22 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 300 kVAr Harmonic Frequency Node 19 20 650 0.4471 0.4910 645 0.7941 0.9153 632 -0.1159 0.0297 634 0.6181 0.7908 611 -0.889 -07468 684 -1.2322 -1.0649 671 -0.0086 0.2137 675 -0.8967 -0.6668 680 -0.1152 0.1388

154

Table B-23 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0188+0.0269j 0.0209+0.0506j 0.0224+0.0753j 0.0241+0.1020j 0.0269+0.1325j 0.0331+0.1717j 645 0.0335+0.0647j 0.0364+0.1274j 0.0399+0.1954j 0.0457+0.2745j 0.0573+0.3760j 0.0860+0.5307j 632 0.0623+0.1396j 0.0675+0.2810j 0.0772+0.4429j 0.0963+0.6487j 0.1370+0.9468j 0.2449+1.4696j 634 0.1429+0.1789j 0.14666+0.3575j 0.1548+0.5555j 0.1718+0.7958j 0.2096+1.1254j 0.3116+1.6736j 611 0.0770+0.1773j 0.8038+0.3596j 0.0974+0.5723j 0.1245+0.8500j 0.1830+1.2641j 0.3392+2.0114j 684 0.0914+0.2146j 0.0991+0.4358j 0.1154+0.6946j 0.1485+1.0338j 0.2200+1.5424j 0.4120+2.4655j 671 0.2212+0.2771j 0.2276+0.5592j 0.2424+0.8777j 0.2734+1.2748j 0.3418+1.8384j 0.5276+2.8096j 675 0.1057+0.2519j 0.1136+0.5104j 0.1312+0.8097j 0.1671+1.19671j 0.2450+1.7692j 0.4543+2.7949j 680 0.1853+0.2909j 0.1909+0.5856j 0.2058+0.9189j 0.2382+1.3369j 0.3107+1.9348j 0.5089+2.9738j

Table B-24 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0562+0.385j 0.4467+0.2803j 0.0370+0.0614j 0.0204+0.1522j 0.0196+0.2011j 0.0206+0.2410j 645 0.2061+0.8642j 2.6362+1.2233j 0.1949-0.4543j 0.0556-0.0017j 0.0365+0.1914j 0.0328+0.3230j 632 0.7231+2.7662j 11.1510+4.4551j 0.8458-3.2576j 0.1902-1.5165j 0.0839-0.8965j 0.0510-0.5615j 634 0.7705+2.9764j 10.9119+4.7091j 0.9186-2.8108j 0.2717-1.0821j 0.1650-0.4225j 0.1311-0.0803j 611 1.0391+3.9092j 16.5088+6.4665j 1.2686-5.1022j 0.2810-2.5604j 0.1176-1.6825j 0.0650-1.2318j 684 1.2766+4.8204j 20.5037+8.0406j 1.5900-6.3979j 0.3543-3.2483j 0.1485-2.1652j 0.0819-1.6133j 671 1.3706+5.1913j 20.2760+8.4806j 1.7128-5.7207j 0.4872-2.5644j 0.2811-1.4383j 0.2136-0.8346j 675 1.3973+5.3858j 22.3952+8.9359j 1.7483-6.8263j 0.3972-3.3642j 0.1725-2.1559j 0.1005-1.5246j 680 1.4137+5.5394j 21.8344+9.1357j 1.8142-6.2690j 0.4818-2.8682j 0.2569-1.6563j 0.1834-1.0057j

155

Table B-25 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0221+0.2790j 0.0239+0.3179j 0.0261+0.3591j 0.0289+0.4052j 0.0325+0.4572j 0.0375+0.5182j 645 0.0335+0.4356j 0.0363+0.5447j 0.0408+0.6590j 0.0472+07855j 0.0562+09323j 0.0694+1.1103j 632 0.0380-0.3365j 0.0327-0.1624j 0.0312-0.0128j 0.322+0.1274j 0.0356+0.2689j 0.0418+0.4223j 634 0.1170+0.1743 0.1108+0.3788j 0.1083+0.5589j 0.1084+0.7289j 0.1008+0.8993j 0.1158+1.0799j 611 0.0425-0.9503j 0.0312-0.7520j 0.0251-0.6002j 0.216-0.4761j 0.0198-0.3690j 0.0192-0.2718j 684 0.0532-1.2721j 0.0387-1.0348j 0.0308-0.8556j 0.0264-0.7113j 0.0243-0.5883j 0.0240-0.4777j 671 0.1840-0.4383j 0.1688-0.1444j 0.1603+0.0291j 0.1553+0.240j 0.1527+0.4745j 0.1517+0.6426j 675 0.0702-1.1193j 0.0561-0.8221j 0.0497-0.5815j 0.0480-0.3698j 0.0501-0.1685j 0.0562+0.0379j 680 0.1518-0.5754j 0.1362-0.2510j 0.1286+0.0177j 0.1256+0.2574j 0.1264+0.4858j 0.1309+0.7176j

Table B-26 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 19 20 650 0.0447+0.5927j 0.0559+0.6888j 645 0.0896+1.3369j 0.1231+1.6435j 632 0.0524+0.6012j 0.0710+0.8273j 634 0.1248+1.2835j 0.1411+1.5300j 611 0.0197-0.1790j 0.0215-0.0850j 684 0.0254-0.3718j 0.0294-0.2626j 671 0.1526+0.8056j 0.1559+0.9712j 675 0.0678+0.2669j 0.0891+0.5448j 680 0.1406+0.9694j 0.1591+1.2656j

156

Table B-27 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.04 0.0593 0.0819 0.1077 0.138 0.1783 645 0.08 0.1393 0.2049 0.2834 0.3862 0.547 632 0.17 0.2993 0.4583 0.6651 0.9693 1.5152 634 0.38 0.4582 0.6247 0.8514 1.1779 1.7417 611 0.21 0.3813 0.5910 0.8706 1.2937 2.0742 684 0.26 0.4606 0.7160 1.0577 1.5774 2.5411 671 0.59 0.7184 0.9875 1.3638 1.924 2.9247 675 0.30 0.5380 0.8333 1.2228 1.8073 2.877 680 0.50 0.6936 0.9942 1.3998 1.9997 3.073

Table B-28 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.2535 1.4536 0.0576 0.1553 0.2025 0.2426 645 0.9217 8.293 -0.4945 -0.0039 0.192 0.3243 632 2.9712 34.4655 -3.4496 -1.5325 -0.8996 -0.5621 634 3.1912 33.8357 -2.9789 -1.0804 -0.4301 -0.0670 611 4.2038 50.9031 -5.3915 -2.585 -1.6877 -1.2333 684 5.1813 63.216 -6.7595 -3.2788 -2.1715 -1.615 671 5.5755 62.6774 -6.046 -2.5664 -1.4192 -0.8133 675 5.7785 69.1773 -7.2209 -3.3969 -2.1621 -1.5258 680 5.9287 67.583 -6.6341 -2.8818 -1.6461 -0.9924

157

Table B-29 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.2807 0.3199 0.3613 0.4073 0.4595 0.5209 645 0.4372 0.5466 0.6612 0.7883 0.9357 1.1148 632 -0.3363 -0.1619 -0.012 0.1284 0.2703 0.4243 634 0.1871 0.3909 0.5703 0.7398 0.9098 1.0903 611 -0.9507 -0.7521 -0.600 -0.4759 -0.3687 -0.2715 684 -1.2725 -1.0346 -0.8552 -0.7108 -0.5877 -0.4768 671 -0.4176 -.1249 0.1104 0.3112 0.4908 0.6582 675 -1.119 -0.8212 -0.5802 -0.3681 -0.1664 0.0407 680 -0.562 -0.2382 0.0209 0.2691 0.4971 0.7289

Table B-30 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Underground construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 19 20 650 0.5962 0.6936 645 1.3434 1.6536 632 06041 0.8321 634 1.2942 1.5419 611 -0.1785 -0.0843 684 -0.3707 -0.2609 671 0.8206 0.9860 675 0.2709 0.5512 680 0.9812 1.2789

158

Table B-31 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0178+0.0523j 0.0196+0.1035j 0.0217+0.1598j 0.0261+0.2285j 0.0438+0.3411j 1.0460+0.5145j 645 0.0314+0.1247j 0.0349+0.2536j 0.0417+0.4067j 0.0598+0.6245j 0.1425+1.0862j 5.7489+2.1003j 632 0.0579+0.2670j 0.0661+0.5539j 0.0880+0.9302j 0.1515+1.5584j 0.4645+3.1711j 23.3812+7.3594j 634 0.0840+0.3105j 0.0913+0.6385j 0.1120+1.0526j 0.1733+1.7118j 0.4776+3.33326j 22.8729+7.4893i 611 0.0717+0.3393j 0.0831+0.7110j 0.1143+1.2152j 0.2058+2.0920j 0.6622+4.4281j 34.6468+10.6952j 684 0.0848+0.4100j 0.0983+0.8590j 0.1361+1.7402j 0.2478+2.5399j 0.8075+5.4085j 42.6926+13.1662j 671 0.2818+0.6118j 0.2908+1.2541j 0.3228+2.0489j 0.4244+3.2815j 0.9495+6.2426j 41.1816+14.0670j 675 0.0978+0.4803j 0.1119+1.0003j 0.1528+1.6950j 0.2740+2.8859j 0.8804+6.0239j 46.1824+14.4401j 680 0.1323+0.5231j 0.1399+1.0816j 0.1747+1.8088j 0.2917+0.30195j 0.8814+6.1325j 45.1529+14.4414j

Table B-32 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0282+0.1596j 0.0195+0.2934j 0.0200+0.3826j 0.0221+0.4677j 0.0251+0.5569j 0.0295+0.6667j 645 0.1127-0.3168j 0.0421+0.2253j 0.0342+0.5004j 0.0356+0.7291j 0.0411+0.9647j 0.0510+1.2390j 632 0.4296-3.2349j 0.1059-1.4077j 0.0531-0.7333j 0.0378-0.3354j 0.0388-0.0339j 0.0354+0.2390j 634 0.4485-2.8425j 0.1306-1.0150j 0.783-0.3150j 0.0630+0.1137j 0.0587+0.4470j 0.0600+0.7506j 611 0.6417-5.2092j 0.1513-2.5767j 0.0680-1.6653j 0.0490-1.1808j 0.0294-0.8636j 0.0241-0.6262j 684 0.7949-6.4968j 0.1871-3.2659j 0.0832-2.1565j 0.0493-1.5738j 0.0350-1.1974j 0.0287-0.9189j 671 0.9826-4.8064j 0.3843-1.5081j 0.2791-0.2493j 0.2432+0.5016j 0.2271-1.0520j 0.2191+1.5063j 675 0.8645-6.7706j 0.2091-3.2275j 0.0986-1.9707j 0.0643-1.2716j 0.0521-0.7786j 0.0502-0.3653j 680 0.8763-6.2855j 0.2326-2.7808j 0.1234-1.5112j 0.0891-0.7879j 0.0767-0.2675j 0.0744+0.1729j

159

Table B-33 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0364+0.7998j 0.04830.9792j 0.0729+1.2524j 0.1441+1.76961j 0.6161+3.4100j 3.6471-5.8520j 645 0.0680+1.5911j 0.0988+2.0890j 0.1703+2.8934j 0.3875+4.5155j 1.9186+9.9714j 12.725-21.7352j 632 0.0423+0.5257j 0.0578+0.8742j 0.0943+1.3775j 0.2090+2.3154j 1.0267+5.3183j 6.9005-11.6994j 634 0.0663+1.0652j 0.0809+1.4364j 0.1124+1.9513j 0.2234+2.8719j 0.9912+5.7236j 6.4735-10.1677j 611 0.0220-0.4291j 0.0222-0.2490j 0.0254-0.0647j 0.0375+0.1613j 0.1225+0.6081j 0.7545-1.02731j 684 0.0267-0.6877j 0.0288-0.4709j 0.0380-0.2305j 0.0712+0.1225j 0.3145+1.0746j 2.0439-3.9271j 671 0.2156+1.9125j 0.2159+2.3018j 0.228+2.7087j 0.2511+3.2135j 0.4664+4.2509j 2.0161-0.0277j 675 0.0564+0.0385j 0.0740+0.5006j 0.1129+1.1390j 0.2580+2.2962j 1.2561+5.9505j 8.3364-14.6979j 680 0.0799+0.6012j 0.0964+1.0819j 0.1377+1.7258j 0.2695+2.8538j 1.2067+6.3169j 7.8149-12.9720j

Table B-34 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 19 20 650 0.1953-0.8763j 0.0820+0.04961j 645 0.6811-5.1441j 0.2387-2.3221j 632 0.3843-2.7139j 0.1437-1.1295j 634 0.3828-1.7124j 0.1589-0.1910j 611 0.0695+0.0778j 0.0520+0.4578j 684 0.1154-1.2185j 0.0426-0.7243j 671 0.2903+2.5710j 0.2242+3.1929j 675 0.4538-3.7754j 0.1644-1.8476j 680 0.4469-2.7014j 0.1776-0.8576j

160

Table B-35 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.06 0.1072 0.1628 0.2315 0.3475 3.980 645 0.13 0.259 0.4116 0.6310 1.108 21.3208 632 0.28 0.5656 0.9394 1.5745 3.2441 85.685 634 0.34 0.6547 1.0665 1.7310 3.4063 84.016 611 0.36 0.7216 1.2269 2.1138 4.5325 126.777 684 0.43 0.8711 1.4838 2.5659 5.5355 156.991 671 0.82 1.3605 2.1232 3.3501 6.394 151.303 675 050 1.0138 1.7099 2.9142 6.161 169.135 680 0.56 1.1022 1.8280 3.0502 6.268 165.582

Table B-36 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.1589 0.2942 0.3855 0.4686 05607 0.6680 645 -0.3267 0.2252 -0.5011 0.7302 0.9661 1.2412 632 -3.2819 -1.4118 -0.734 -0.3353 -0.0334 0.2398 634 -2.8862 -1.017 -0.3139 0.1154 0.4489 0.7527 611 -5.2806 -2.853 -1.6667 -1.1811 -0.8636 -0.6261 684 -6.5851 -3.274 -2.1583 -1.5742 -1.1974 -0.9186 671 -4.8634 -1.4916 -0.2297 0.5200 1.069 1.5219 675 -6.8653 -3.2358 -1.9722 -1.2716 -0.7779 -0.3642 680 -6.3757 -2.786 -1.5107 -0.7863 -0.2655 0.1752

161

Table B-37 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.8014 0.9817 1.257 1.7827 3.5341 -7.8939 645 1.594 2.094 2.9042 4.5515 10.3624 -28.8847 632 0.5269 0.8765 1.3857 2.3337 5.5248 -15.5674 634 1.0676 1.4397 1.9573 2.8901 5.9187 -13.7808 611 -0.4289 -0.2486 -0.0641 0.1633 0.6291 -1.4370 684 -0.6873 -0.4701 0.2287 0.1282 1.137 -5.0714 671 1.927 2.3155 2.7223 3.2298 4.3168 -1.0412 675 0.0402 0.5037 1.1458 2.3192 6.2044 -19.3752 680 0.6041 1.086 1.7333 2.8765 6.5565 -17.3413

Table B-38 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 2, 3, 4, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 19 20 650 -0.891 0.0481 645 -5.218 -2.3319 632 -2.7505 -1.134 634 -1.7456 -0.1944 611 0.0745 0.4585 684 -1.2295 -0.7259 671 2.5706 3.2004 675 -3.8197 -1.8533 680 -2.7416 -0.8621

162

Table B-39 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0176+0.0521j 0.0190+0.1019j 0.0199+0.1536j 0.0210+0.2085j 0.0288+0.2685j 0.0259+0.3379j 645 0.0309+0.1236j 0.0323+0.2443j 0.0341+0.3702j 0.0373+0.5074j 0.0431+0.6657j 0.0542+0.8647j 632 0.0562+0.2632j 0.0586+0.5216j 0.0642+0.8008j 0.0795+1.1283j 0.0980+1.5482j 0.1426+2.1485j 634 0.0823+0.3068j 0.0837+0.6064j 0.0887+0.9246j 0.0995+1.2881j 0.1204+1.7397j 0.1630+2.3644j 611 0.0691+0.3334j 0.0714+0.6605j 0.0787+1.0140j 0.0936+1.4295j 0.1218+1.9648j 0.1791+2.7348j 684 0.0817+0.4028j 0.0843+0.7977j 0.0934+1.2256j 0.1120+1.7327j 0.1477+2.3927j 0.2199+3.3521j 671 0.2787+0.6046j 0.2774+1.1937j 0.2823+1.8089j 0.2964+2.4946j 0.3263+3.3208j 0.3898+4.4256j 675 0.0940+0.4716j 0.0955+0.9266j 0.1034+1.4054j 0.1208+1.9151j 0.1544+2.633j 0.2224+3.5842j 680 0.1195+0.5145j 0.1197+1.0084j 0.1264+1.5224j 0.1432+2.0990j 0.1741+2.8043j 0.2391+3.7681j

Table B-40 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0323+0.4261j 0.0517+0.5661j 0.2218+0.9875i 0.2464-0.2768j 0.0376+0.2606j 0.0252+0.4273j 645 0.0800+1.153j 0.1661+1.6990j 0.9925+3.7022j 1.2842-3.0170j 0.1429-0.4990j 0.0605+0.1109j 632 0.2489+3.1478j 0.6149+5.3062j 4.2523+14.0650j 5.880-16.5486j 0.61814-5.5760j 0.2202-3.1717j 634 0.2651+3.3733j 0.6176+5.4998j 4.1246+13.9935j 5.7011-15.4963j 0.6228-4.8703j 0.2381-2.5098j 611 0.3167+4.0262j 0.7934+6.8370j 5.5555+18.3187j 7.7456-21.9252j 0.8144-7.5636j 0.2859-4.4568j 684 0.3937+4.9759j 0.9945+8.5334j 7.0359+23.1211j 9.8752-28.0446j 1.0440-9.8043j 0.3698-5.8585j 671 0.5470+6.1526j 1.1019+9.6963j 6.7345+23.6220j 9.5511-24.2712j 1.1976-6.9604j 0.5508-3.0641j 675 0.3852+5.1375j 0.9449+8.4484j 6.4905+21.7865j 8.9154-24.7552j 0.9308-8.0482j 0.3285-4.3876j 680 0.3956+5.3113j 0.9346+8.5488j 6.2814+21.4895j 8.6302-23.3977j 0.9240-7.2421j 0.3421-3.6720j

163

Table B-41 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0237+0.5425j 0.0245+0.6463j 0.0265+0.7509j 0.0293+0.8630j 0.0330+0.9878j 0.0380+1.1314j 645 0.0404+0.4291j 0.0337+0.6563j 0.0317+0.8465j 0.0319+1.0232j 0.0339+1.1998j 0.0376+1.3868j 632 0.1162-2.0920j 0.0759-1.4449j 0.0575-0.9852j 0.0490-0.6155j 0.0463+1.1998j 0.0480+0.0361j 634 0.1374-1.4278j 0.0984-0.7642j 0.0805-0.2824j 0.0723+0.1113j 0.0463-0.2860j 0.0711+0.8083j 611 0.1456-3.1003j 0.0893-2.3261j 0.0614-1.8161j 0.0456-1.4481j 0.0696+0.4644j 0.0294-0.9380j 684 0.1905-4.1311j 0.1196-3.1364j 0.0855-2.4675j 0.0676-1.9664j 0.0358-1.1655j 0.0553-1.1907j 671 0.3769-1.2538j .3066-0.1340j 0.2719+0.6772j 0.2530+1.3281j 0.0586-1.5560j 0.2380+2.4076j 675 0.1711-2.7509j 0.1096-1.7839j 0.0807-1.1162j 0.0661-0.6037jj 0.2427+1.8909j 0.0574+0.2046j 680 0.1899-2.0554j 0.1305-1.0859j 0.1025-0.4062j 0.0884+0.1223j 0.0818+0.5667j 0.0800+0.9658j

Table B-42 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 19 20 650 0.0448+1.3020j 0.0548+1.5122j 645 0.0435+1.5960j 0.0532+1.8444j 632 0.0543+0.3801j 0.0674+0.7822j 634 0.0771+1.1702j 0.0894+1.5834j 611 0.0249-0.7484j 0.0218-0.5834j 684 0.0572-0.8366j 0.0653-0.4589j 671 0.2381+2.9093j 0.2438+3.4266j 675 0.060j+0.5708j 0.0683+0.9504j 680 0.0826+1.3478j 0.0903+1.73961j

164

Table B-43 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.06 0.1055 0.1526 0.2106 0.2705 0.3399 645 0.13 0.2491 0.3737 0.5104 0.6688 0.8686 632 0.28 0.5288 0.8065 1.1339 1.5555 2.1587 634 0.34 0.6212 0.9352 1.2972 1.7491 2.3766 611 0.5 0.6689 1.0207 1.4362 1.973 2.7475 684 0.42 .8073 1.2333 1.7406 2.402 3.3676 671 0.82 1.2978 1.8781 2.5473 3.365 4.4683 675 0.49 0.9372 1.4137 1.9597 2.643 3.5664 680 0.55 1.0262 1.5352 2.1103 2.816 3.7865

Table B-44 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.4287 0.571 1.0532 -0.3558 0.2592 0.427 645 1.1596 1.718 4.0078 -3.4537 -0.511 0.109 632 3.1689 5.381 15.3876 -18.5703 -5.636 -3.180 634 3.3955 5.576 15.2692 -17.445 -4.926 -2.512 611 4.0529 6.931 20.0471 24.5895 -7.642 -4.479 684 5.009 8.651 25.3095 -31.4397 -9.906 -5.877 671 6.206 9.8265 25.6673 -27.4611 -7.037 -3.066 675 5.169 8.5628 23.8019 -27.8169 -8.137 -4.402 680 5.343 8.6602 23.4024 -26.3499 -7.322 -3.686

165

Table B-45 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.5431 0.647 0.7517 0.8639 0.9888 1.1326 645 0.4294 0.6565 0.8469 1.0238 1.2006 1.3878 632 -2.0955 -1.4433 -0.9857 -0.6155 -0.2857 0.0363 634 -1.4299 -.07643 -0.2818 0.1132 0.4657 0.8098 611 -3.1053 -2.3282 -1.8171 -1.4487 -1.1658 -0.9382 684 -4.1373 -3.1389 -2.4686 -1.9668 -1.5559 -1.1903 671 -1.2451 -0.1230 0.6885 1.3392 1.9016 2.418 675 -2.7558 -1.7857 -1.1167 -0.6037 -0.1766 0.2053 680 -2.0588 -1.0864 -0.4057 0.1234 0.5681 0.9676

Table B-46 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 5 energized, each rated at 500 kVAr Harmonic Frequency Node 19 20 650 1.3035 1.5142 645 1.5973 1.8461 632 0.3812 0.7841 634 1.1721 1.586 611 -0.7458 -0.5855 684 -0.8357 -0.4574 671 2.9195 3.4369 675 0.5719 0.9522 680 1.349 1.7421

166

Table B-47 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0176+0.0521j 0.0190+0.1019j 0.0199+0.1536j 0.2100+0.2086j 0.0231+0.2696j 0.0271+0.3426j 645 0.0310+0.1239j 0.0329+0.2468j 0.0357+0.3790j 0.0405+0.5300j 0.0495+0.7162j 0.0684+0.9736j 632 0.0563+0.2635j 0.0590+0.5239j 0.0651+0.8090j 0.0795+1.1508j 0.1068+1.6054j 0.1672+2.2970j 634 0.0824+0.3074j 0.0842+0.6087j 0.0902+0.9327j 0.1030+1.3103j 0.1289+1.7958j 0.1859+2.5097j 611 0.0694+0.3337j 0.0720+0.6677j 0.0802+1.0217j 0.0972+1.4514j 0.1314+2.0231j 0.2077+2.8958j 684 0.0818+0.4031j 0.0848+0.7996j 0.0947+1.2328j 0.1158+1.7538j 0.1580+2.4520j 0.2528+3.5266j 671 0.2789+0.6049j 0.2778+1.1956j 0.2837+1.8159j 0.2999+2.5151j 0.3360+3.3782j 0.4206+4.5936j 675 0.0948+0.4735j 0.0985+0.9413j 0.1110+1.4568j 0.1372+2.0843j 0.1898+2.9362j 0.3076+4.2622j 680 0.1203+0.5163j 0.1227+1.0230j 0.1338+1.5732j 0.1583+2.2298 0.2084+3.1018j 0.3215+4.4309j

Table B-48 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0375+0.4449j 0.0887+0.6619j 1.7930-0.3825j 0.0512+0.1201j 0.0247+0.3212j 0.0213+0.4332j 645 0.1208+1.4072j 0.3988+2.5310j 10.7341-3.6041j 0.2754-1.0627j 0.0929-0.0462j 0.0629+0.4422j 632 0.3412+3.5943j 1.2988+7.322j 38.1576-14.5216j 0.9404-6.105j 0.2594-2.9155j 0.1290-1.866j 634 0.3545+3.8083j 1.2791+7.4564j 36.9241-13.610j 0.9363-5.4421j 0.2768-2.3178j 0.1503-1.0902j 611 0.4290+4.5424j 1.6549+9.3094j 49.060-18.7789j 1.2072-8.0503j 0.3243-4.0155j 0.1509-2.5247j 684 0.5281+5.5690j 2.0566+11.5146j 61.4028-23.5810j 1.5139-7.6033j 0.4063-5.1977j 0.1883-3.3477j 671 0.6727+6.7199j 2.0956+11.5146j 58.2056-20.2089j 1.6489-7.6033j 05872-2.6723j 0.3720-0.7410j 675 0.6500+6.8033j 2.0953+12.5309j 76.8254-29.5701j 1.9109-12.9465j 0.5212-6.6330j 0.2499-4.2735j 680 0.6515+6.9339j 2.4900+14.1635j 74.3156-28.1234j 1.8759-12.030j 0.5301-5.8845j 0.2669-3.5630j

167

Table B-49 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0217+0.5272j 0.0239+0.6222j 0.0282+0.7310j 0.0367+0.8724j 0.0572+1.0919j 0.1343+1.5629j 645 0.589+0.8241j 0.0664+1.2121j 0.0585+1.6827j 0.1282+2.3481j 0.2364+3.4802j 0.6642+6.1160j 632 0.0872-0.9660j 0.0738-0.4234j 0.0764+0.0773j 0.0972+0.6473j 0.1609+1.4596j 0.4215+3.1599j 634 0.1095-0.3554j 0.0963+0.2051j 0.0984+0.7221j 0.1179+1.2973j 0.1776+2.1026j 0.4217+3.7290j 611 0.0900-1.7209j 0.0624-1.1974j 0.0483-0.8125j 0.0413-0.5011j 0.0404-0.2213j 0.0548+0.0861j 684 0.1119-2.3543j 0.0778-1.7073j 0.0613-1.2258j 0.0557-0.8199j 0.0632-0.4139j 0.1179+0.1596j 671 0.2991+0.3801j 0.2643+1.1719j 0.2467+1.8048j 0.2394+2.3629j 0.2443+2.9153j 0.2911+3.6114j 675 0.1588-2.9544j 0.1244-2.0203j 0.1194-1.2117j 0.1441-0.3433j 0.2339+0.8760j 06211+3.4229j 680 0.1783-2.2492j 0.1466-1.3102j 0.1394-1.2117j 0.1619+0.3676j 0.2460+1.5548j 0.6084+3.9748j

Table B-50 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr Harmonic Frequency Node 19 20 650 1.5025+4.0224j 0.3927-1.2755j 645 8.6436+20.7063j 2.4205-11.1494j 632 5.2504+12.0192j 1.4347-7.0166j 634 4.9243+12.010j 1.3532-5.6565j 611 0.3760+0.8472j 0.1153-0.2546j 684 1.2382+2.3893j 0.3617-1.9651j 671 1.2785+5.7692j 0.5071+2.1010j 675 7.9705+16.9355j 2.2210-12.3802j 680 7.4611+16.5980j 2.0793-10.6355j

168

Table B-51 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.06 0.1055 0.1562 0.2108 0.2716 0.3448 645 0.13 0.2517 0.3828 0.5334 0.7200 0.9790 632 0.28 0.5312 0.8149 1.1569 1.6132 2.3103 634 0.54 0.6236 0.9435 1.3198 1.8006 2.5248 611 0.35 0.6712 1.0286 1.4585 2.0324 2.9121 684 0.42 0.8094 1.2407 1.7621 2.4631 3.5463 671 0.82 1.2998 1.8853 2.5683 3.4238 4.6452 675 0.49 0.9524 1.4659 2.0941 2.9494 4.2861 680 0.55 1.0413 1.5868 2.2426 3.1171 4.4559

Table B-52 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.4484 0.6771 -4.881 0.1156 0.3212 0.4336 645 1.4192 2.6041 -30.682 -1.0966 -0.497 0.4421 632 3.6293 7.5654 -110.8095 -6.2312 -2.931 -1.6906 634 3.8439 7.6925 -106.8095 -5.5625 -2.331 -1.0926 611 4.5862 9.6195 -142.7938 -8.2135 -4.035 -2.5303 684 5.6229 11.8999 -178.6989 -10.4316 -5.223 -3.3546 671 6.7923 12.9049 -166.6752 -7.7785 -2.679 -0.7318 675 6.8693 14.7114 -223.6376 -13.2036 -6.6646 -4.2817 680 6.9991 14.6255 -215.7716 -12.2771 -5.9135 -3.5695

169

Table B-53 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.5278 0.623 0.723 0.874 1.0954 1.5772 645 0.825 1.2139 1.6857 2.351 3.4957 6.1899 632 -0.967 -0.4232 0.0787 0.647 1.4689 3.2051 634 -0.3552 0.2065 0.7245 1.3015 2.113 3.7724 611 -1.723 -1.1983 -0.8129 -0.5011 -0.2207 0.0892 684 -2.3568 -1.7083 -1.226 -0.8194 -0.4119 0.1699 671 0.392 1.1841 1.8168 2.3746 2.9276 3.6303 675 -2.9569 -2.0203 -1.2099 -0.3384 0.8898 3.4907 680 -2.2503 -1.309 -0.4937 0.3733 1.5688 4.0394

Table B-54 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2, 3, and 6 energized, each rated at 500 kVAr Harmonic Frequency Node 19 20 650 4.6613 -1.352 645 24.4002 -11.632 632 14.2574 -7.3001 634 14.0996 -5.9187 611 0.9973 -0.2728 684 2.9097 -2.034 671 6.2372 2.0488 675 20.340 -12.825 680 19.767 -11.046

170

Table B-55 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr Harmonic Frequency ‘Node 1 2 3 4 5 6 650 0.0176+0.0521j 0.0190+0.1021j 0.0200+0.1541j 0.0212+0.2095j 0.0230+0.2702j 0.0256+0.3390j 645 0.0308+0.1235j 0.0321+0.2436j 0.0337+0.3677j 0.0364+0.5006j 0.0407+0.6491j 0.0479+0.8241j 632 0.0560+0.2626j 0.0574+0.5170j 0.0618+0.7848j 0.0703+1.0863j 0.0853+1.4494j 0.1112+1.9189j 634 0.820+0.3061j 0.0827+0.6018j 0.0863+0.9087j 0.0941+1.2467j 0.1081+1.6427j 0.1326+2.1399j 611 0.0684+0.3317j 0.0688+0.6475j 0.0722+0.9695j 0.799+1.3171j 0.0937+1.7159j 0.1177+2.2064j 684 0.0805+0.3999j 0.0796+0.7744j 0.0820+1.1462j 0.0888+1.5345j 0.104+1.9627j 0.1237+2.4673j 671 0.2776+0.6017j 0.2728+1.1707j 0.2717+1.7310j 0.2748+2.3014j 0.2837+2.9044j 0.3015+3.5744j 675 0.0928+0.4687j 0.0908+0.9038j 0.0925+1.3284j 0.0985+1.7611j 0.1103+2.222j 0.1313+2.7506j 680 0.1183+0.5115j 0.1151+0.9857j 0.1157+1.4462j 0.1207+1.9113j 0.1315+2.4029j 0.1512+2.95301j

Table B-56 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0299+0.4213j 0.0380+0.5282j 0.0571+0.6896j 0.13333+1.0295j 2.2464+2.5488j 0.1765-0.3289j 645 0.610+1.0460j 0.0875+1.1601j 0.1562+1.8917j 0.4526+3.1687j 9.3392+9.6295j 0.7524-3.0659j 632 0.1590+2.5792j 0.2560+3.6170j 0.5248+5.5527j 1.7051+10.5809j 38.2970+37.4318j 3.1926-15.8424j 634 0.1782+2.8195j 0.2740+3.8604j 0.5288+5.7618j 1.6608+10.6280j 36.7504+36.4168j 3.0883-14.6208j 611 0.1619+2.8635j 0.2537+3.8526j 0.4952+5.6336j 1.5561+10.1449j 34.0063+33.8314j 2.7841-13.0918j 684 0.1645+3.1146j 0.2485+4.0498j 0.4670+5.6765j 1.4160+9.6854j 29.9721+30.4057j 2.4026-10.5770j 671 0.3364+4.3734j 0.4111+5.4369j 0.6092+7.1407j 1.4802+11.0663j 27.9958+30.5924j 2.4484-7.4241j 675 0.1697+3.4008j 0.2482+4.3052j 0.4500+5.8259j 1.3151+9.4838j 27.0070+28.0146j 2.1267-8.5923j 680 0.1877+3.6197j 0.2628+4.5282j 0.4526+6.0258j 1.2860+9.5711j 25.9232+27.3746j 2.0629-7.6907j

171

Table B-57 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0497+0.2229j 0.0351+0.4638j 0.0341+0.6478j 0.0376+0.8269j 0.0445+1.0248j 0.0552+1.2622j 645 0.1615-0.9996j 0.0744-0.3027j 0.0480+0.0831j 0.0380+0.3551j 0.0345+0.5786j 0.0348+0.7845j 632 0.6650-7.6103j 0.2759-5.0540j 0.1500-3.8088j 0.0946-3.0663j 0.0658-2.5681j 0.0494-2.2057j 634 0.6617-6.6834j 0.2887-4.1888j 0.1650-2.9518j 0.1149-2.1972j 0.0873-1.6772j 0.0715-1.2878j 611 0.5741-5.7377j 0.2388-3.4110j 0.1320-2.2454j 0.0857-1.5254j 0.0621-1.0224j 0.0489-0.6403j 684 0.4952-4.0611j 0.2077-1.9644j 0.1187-0.8889j 0.0812-0.2063j 0.0628+0.2841j 0.0531+0.6667j 671 0.6544-1.1884j 0.3837+0.9373j 0.2978+2.1109j 0.2610+2.9167j 0.2424+3.5141j 0.2323+4.0633j 675 0.4376-2.6900j 0.1902-0.7602j 0.1147+0.2506j 0.0841+0.9064j 0.0699+1.3867j 0.0631+1.7699j 680 0.4437-1.9973j 0.2067-0.1144j 0.1346+0.8859j 0.1055+1.5448j 0.9022+2.0351j 0.0860+2.4295j

Table B-58 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr Harmonic Frequency Node 19 20 650 0.0722+1.5665j 0.1005+1.9837j 645 0.0384+0.9937j 0.0464+1.2281j 632 0.0396-1.9246j 0.0342-1.6929j 634 0.0621-0.9769j 0.0569-0.7140j 611 0.0414-0.3313j 0.0375-0.0670j 684 0.481+0.9835j 0.0460+1.2591j 671 0.2267+4.5217j 0.2241+4.9392j 675 0.0603+2.0911j 0.0611+2.3729j 680 0.0836+2.7641j 0.0837+3.0600j

172

Table B-59 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.06 0.1056 0.1567 0.2117 0.2721 0.3409 645 0.13 0.2483 0.3712 0.5036 0.652 0.8272 632 0.28 0.524 0.7902 1.0914 1.4549 1.9252 634 0.33 0.6164 0.9190 1.2553 1.6509 2.149 611 0.35 0.6555 0.9754 1.3225 1.7216 2.2134 684 0.42 0.7832 1.1527 1.5402 1.9686 2.4744 671 0.81 1.2741 1.7992 2.3524 2.9457 3.6101 675 0.49 0.9136 1.3354 1.7671 2.2302 2.7578 680 0.55 1.0028 1.4579 1.9207 2.4115 2.9622

Table B-60 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.4235 0.5311 0.6949 1.0502 4.8542 -0.3598 645 1.0501 1.3666 1.9068 3.2427 19.2607 -3.2118 632 2.5897 3.6367 5.6046 10.8631 76.9612 -16.4716 634 2.8316 3.8809 5.8129 10.8997 74.322 -15.2221 611 2.8728 3.8713 5.6817 10.4009 68.9207 -13.6375 684 3.1246 4.0676 5.7209 9.9235 61.3193 -11.045 671 4.4074 5.4739 7.1989 11.2922 59.2741 -7.8471 675 3.4104 4.3224 5.8677 9.6965 55.8553 -9.0033 680 3.6311 4.5462 6.0673 9.7762 54.0718 -8.0829

173

Table B-61 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.2211 0.4641 0.6486 0.828 1.0263 1.2643 645 -1.0116 -0.3053 0.0824 0.355 0.5789 0.7851 632 -7.665 -5.0675 -3.8137 -3.0684 -2.5691 -2.2062 634 -6.7343 -4.2005 -2.9554 -2.1985 -1.6773 -1.2874 611 -5.7838 -3.4219 -2.2491 -1.5269 -1.023 -0.6405 684 -4.0993 -1.9730 -0.8916 -2.072 0.284 0.6669 671 -1.2101 0.9423 2.1204 2.9271 3.5517 4.0733 675 -2.7223 -0.7669 0.2488 0.9062 1.388 1.7706 680 -2.0269 -0.1197 0.8853 1.5456 2.0364 2.431

Table B-62 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1 and 3 energized, each rated at 800 kVAr Harmonic Frequency Node 19 20 650 1.5696 1.9885 645 0.9946 1.2297 632 -1.9248 -1.6928 634 -0.9763 -0.7131 611 -0.3312 -0.667 684 0.984 1.2597 671 4.5314 4.9485 675 2.0919 2.3739 680 2.7657 3.0618

174

Table B-63 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0176+0.0521j 0.0190+0.1021j 0.022+0.1541j 0.0212+0.2095j 0.0230+0.2702j 0.0256+0.3390j 645 0.0308+0.1235j 0.0321+0.2436j 0.0337+0.3677j 0.0364+0.5006j 0.0407+0.6491j 0.0479+0.8241j 632 0.0560+0.2626j 0.0574+0.5170j 0.0618+0.7848j 0.0703+1.0863j 0.0853+1.4494j 0.1112+1.9189j 634 0.0820+0.3061j 0.0827+0.6018j 0.0863+0.9087j 0.0941+1.2467j 0.1081+1.6427j 0.1326+2.1399j 611 0.0684+0.3317j 0.0688+0.6475j 0.0722+0.9695j 0.0799+1.3171j 0.0937+1.7159j 0.117+2.2064j 684 0.0805+0.3999j 0.0796+0.7744j 0.820+1.1462j 0.0888+1.5345j 0.1014+1.9627j 0.1237+2.4673j 671 0.2776+0.6017j 0.2728+1.1707j 0.2717+1.7310j 0.2748+2.3014j 0.2837+2.9044j 0.3015+3.5744j 675 0.0928+0.4687j 0..0908+0.9038j 0.0925+1.3284j 0.0985+1.7611j 0.1103+2.2240j 0.1313+2.7506j 680 0.1183+0.5115j 0.1151+0.9857j 0.1157+1.4462j 0.1207+1.9113j 0.1315+2.4029j 0.1512+0.2953j

Table B-64 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0299+0.4213j 0.0380+0.5282j 0.0571+0.6896j 0.1333+1.0294j 2.2464+2.5488j 0.1765-0.3289j 645 0.0610+1.0460j 0.0875+1.3601j 0.1562+1.8917j 0.4526+3.1687j 9.3392+9.6295j 0.7524-3.06591j 632 0.1590+2.5792j 0.2590+3.6170j 0.5248+5.5527j 1.7051+10.5809j 38.2970+37.4318j 3.1926-15.8424j 634 0.1782+2.8195j 0.2740+3.8604j 0.5288+5.7618j 1.6608+10.6280j 36.7504+36.4168j 3.0883-14.6208j 611 0.1619+2.8635j 0.2537+3.8526j 0.4952+5.6336j 1.5561+10.1449j 34.0063+33.8314j 3.0883-14.6208j 684 0.1645+3.1146j 0.2485+4.0498j 0.4670+5.6765j 1.4150+9.6925j 29.9721+30.4057j 2.7841-13.0918j 671 0.3364+4.3734j 0.4111+5.4369j 0.6092+7.1407j 1.4802+11.0663j 27.9958+30.5924j 2.4026-10.5770j 675 0.1697+3.4008j 0.2482+4.3052j 0.4500+5.8259j 1.3151+9.4838j 27.007+28.0146j 2.1267-8.5923j 680 0.1877+3.6197j 0.2628+4.5282j 0.4562+6.0258j 1.2860+9.5711j 25.9232+27.3746j 2.0629-7.6907j

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Table B-65 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0497+0.2229j 0.0351+0.4683j 0.0341+0.6478j 0.0376+0.8269j 0.0445+1.0248j 0.0552+1.2622j 645 0.1615-0.9996j 0.0744-0.3027j 0.0480+0.0831j 0.0380+0.3551j 0.0345+0.5786j 0.0348+0.7845j 632 0.6650-7.6103j 0.2759-5.0540j 0.1500-3.8088j 0.0946-3.0663j 0.0658-2.5681j 0.0494-2.20571j 634 0.6615-6.6834j 0.2887-4.1888j 0.1680-2.9518j 0.1149-2.1972j 0.873-1.6772j 0.0715-1.2878j 611 0.5741-5.7377j 0.2388-3.4110j 0.1320-2.2454j 0.0857-1.5254j 0.0621-1.0224j 0.0489-0.6403j 684 0.4925-4.0611j 0.2077-1.9644j 0.1187-0.8889j 0.0812-0.2063j 0.0628+0.2841j 0.0531+0.6667j 671 0.6544-1.1884j 0.3837+0.9373j 0.2978+2.1109j 0.2610+2.9167j 0.2424+3.5414j 0.2323+4.0633j 675 0.4376-2.6900j 0.1902-0.7602j 0.1147+0.2506j 0.0841+0.9064j 0.0699+1.38761j 0.0631+1.7699j 680 0.4437-1.9973j 0.2067-0.1144j 0.1346+0.8859j 0.1055+1.5448j 0.0922+2.0351j 0.0860+2.4295j

Table B-66 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 19 20 650 0.0722+1.5665j 0.1005+1.9837j 645 0.0384+0.9937j 0.0464+1.2281j 632 0.0396-1.9246j 0.0342-1.6929j 634 0.0621-0.9769j 0.0569-0.7140j 611 0.0414-0.3313j 0.0375-0.0670j 684 0.0481+0.9835j 0.0460+1.2591j 671 0.2267+4.5217j 0.2241+4.9392j 675 0.0603+2.0911j 0.0601+2.3729j 680 0.0836+2.7641j 0.0837+3.0600j

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Table B-67 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.06 0.1056 0.1567 0.2117 0.2721 0.3409 645 0.13 0.2483 0.3712 0.5036 0.6520 0.8272 632 0.28 0.5240 0.7902 1.0914 1.4549 1.9259 634 0.33 0.6124 0.9190 1.2553 1.6509 2.149 611 0.35 0.6555 0.9754 1.3225 1.7216 2.2134 684 0.42 0.7832 1.1527 1.5402 1.9686 2.4744 671 0.81 1.2741 1.7992 2.3524 2.9457 3.6101 675 0.49 0.9136 1.3354 1.7671 2.2302 2.7578 680 0.55 1.0028 1.4579 1.9207 2.4115 2.9622

Table B-68 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.4235 0.5311 0.6949 1.0502 4.8542 -0.3598 645 1.0501 1.3666 1.9068 3.2427 19.2607 -3.2118 632 2.5897 3.6367 5.6046 10.8631 76.9612 -16.4716 634 2.8316 3.8809 5.8129 10.8997 74.322 -15.2221 611 2.8738 3.8713 5.6812 10.4009 68.9207 -13.6373 684 3.1246 4.0676 5.7209 9.9235 61.3193 -11.045 671 4.407 5.4739 7.1989 11.2922 59.2741 -7.8471 675 3.4107 4.3224 5.8677 9.6965 55.8553 -9.0033 80 3.6311 4.5462 6.0673 9.7762 54.0718 -8.0829

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Table B-69 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.2211 0.4641 0.6486 0.828 1.0263 1.2643 645 -1.0116 -0.3053 0.0824 0.355 0.5789 0.7851 632 -7.665 -5.0675 -3.8137 -3.0684 -2.5691 -2.2062 634 -6.7343 -4.2005 -2.9554 -2.1982 -1.6773 -1.2874 611 -5.7838 -3.4219 -2.2491 -1.5269 -1.023 -0.6405 684 -4.0993 -1.9730 -0.8916 -0.2072 0.284 0.6669 671 -1.2101 0.9423 2.1204 2.9271 3.5517 4.0733 675 -2.7223 -0.7669 0.2488 0.9062 1.388 1.7706 680 -2.0269 -0.1197 0.8853 1.5456 2.0364 2.4310

Table B-70 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 2 and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 19 20 650 1.5696 1.9885 645 0.9946 1.2297 632 -1.9248 -1.6928 634 -0.9763 -0.7131 611 -0.3312 -0.0667 684 0.984 1.2597 671 4.5314 4.9485 675 2.0919 2.3739 680 2.7658 3.0618

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Table B-71 Node driving impedance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.0178+0.0523j 0.0197+0.1037j 0.0220+0.1609j 0.0276+0.2333j 0.0609+0.3773j 0.1366-0.0947j 645 0.0314+0.1246j 0.0349+0.2531j 0.0422+0.4074j 0.0646+0.6378j 0.2222+1.257j 0.7454-1.5050j 632 0.0584+0.2681j 0.0684+0.5636j 0.0962+0.9733j 0.1877+1.7338j 0.8750+4.301j 3.5589-8.37637j 634 0.0845+0.3116j 0.0936+0.6482j 0.1200+1.0952j 0.2087+1.8847j 0.8787+4.4423j 3.5172-8.2982j 611 0.0722+00.3406j 0.0859+0.7228j 0.1246+1.2691j 0.2535+2.631j 1.2350+5.9791j 5.1661-12.9338j 684 0.0851+0.4107j 0.1001+0.8661j 0.1439+1.5067j 0.2910+2.7231j 1.4130+6.9237j 5.9187-14.7165j 671 0.2821+0.6125j 0.2925+1.2613j 0.3303+2.0848j 0.4654+3.4602j 1.5277+7.7121j 5.9565-12.9205j 675 0.0985+0.4820j 0.1155+1.0158j 0.1660+1.7650j 0.3357+3.4182j 1.6292+8.0493j 6.8447-16.9031j 680 0.1239+0.5248j 0.1394+1.0970j 0.1876+1.8780j 0.3521+3.3112j 1.6123+8.1210j 6.7400-16.2505j

Table B-72 Node driving impedance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.0209+0.2217j 0.0190+0.3213j 0.0203+0.4026j 0.0225+0.4846j 0.0254+0.5745j 0.0296+0.6788j 645 0.0553-0.0046j 0.0299+0.3114j 0.0259+0.5001j 0.0263+0.6570j 0.0288+0.8102j 0.0338+0.9773j 632 0.2104-2.2012j 0.0726-1.0801j 0.0436-0.5618j 0.0362-0.2108j 0.0376+0.0940j 0.0468+0.4186j 634 0.2339-1.8361j 0.09797-0.6972j 0.06900-0.1493j 0.0613+0.2338j 0.0623+0.5697j 0.0708+0.9223j 611 0.3015-3.5367j 0.0961-1.99571j 0.0495-1.3422j 0.0331-0.9564j 0.0268-0.6799j 0.0257-0.4479j 684 0.3465-3.9551j 0.1105-2.1861j 0.0562-1.4386j 0.0365-1.0036j 0.0279-0.7022j 0.0244-0.4657j 671 0.5427-2.3788j 0.3069-0.4835j 0.2507+0.4278j 0.2290+1.3591j 0.2187+1.5132j 0.2136+1.9257j 675 0.4079-4.4473j 0.1368-2.3562j 0.0774-1.4219j 0.0600-0.8149j 0.0591-0.3096j 0.0713+0.2116j 680 0.4289-4.0233j 0.1615-1.9358j 0.1024-0.9807j 0.0847-0.3481j 0.0831+0.1824j 0.0943+0.7244j

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Table B-73 Node driving impedance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.0362+0.8075j 0.0492+0.9834j 0.0915+1.2926j 1.3622+2.7684j 0.1194+0.6561j 0.0719+1.2209j 645 0.0435+1.1817j 0.0670+1.4789j 0.1627+2.0941j 3.5560+5.8989j 0.2506-0.3245j 0.0737+0.6971j 632 0.0697+0.8406j 0.1329+1.5371j 0.4149+3.2297j 11.1716+15.2478j 0.7812-5.2552j 0.1610-2.4371j 634 0.0924+1.3652j 0.1521+2.0654j 0.4185+3.7024j 10.5512+15.0516j 0.7579-4.1992j 0.1757-1.5043j 611 0.0298-0.2178j 0.0452+0.0703j 0.1193+0.6203j 3.0277+3.9923j 0.2342-1.5182j 0.0628-0.6624j 684 0.0246-0.2562j 0.0314-0.0354j 0.0686+0.3028j 1.5291+2.0295j 0.1163-0.6668j 0.0320-0.2044j 671 0.2123+2.3094j 0.2171+2.6997j 0.2492+3.1932j 1.5623+4.93821j 0.2925+2.688j 0.2152+3.2867j 675 0.1056+0.8783j 0.2035+1.9763j 0.6483+4.6638j 17.8377+23.9292j 1.2667-9.0050j 0.2613-4.5415j 680 0.1265+1.4011j 0.2191+2.4822j 0.6391+5.0582j 16.8314+23.2281j 1.2121-7.7184j 0.2686-3.4721j

Table B-74 Node driving impedance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 19 20 650 0.0861+1.6558j 0.1225+2.1864j 645 0.0574+1.1582j 0.0662+1.5545j 632 0.0696-1.5400j 0.0413-1.0649j 634 0.0902-0.6186j 0.0639j-0.1310j 611 0.0395-0.3174j 0.0362-0.0662j 684 0.0194-0.0079j 0.0157+0.1281j 671 0.2031+3.6433j 0.1991+3.9434j 675 0.1106-3.1646j 0.0626-2.4760j 680 0.1276-2.1322j 0.0727-1.4409j

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Table B-75 Node driving reactance in ohms (harmonic frequencies 1 – 6) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 1 2 3 4 5 6 650 0.06 0.1075 0.1639 0.2366 0.3891 -0.1418 645 0.13 0.2587 0.4123 0.6453 1.3038 -1.7925 632 0.28 0.5727 0.9837 1.7558 4.4891 -10.1438 634 0.34 0.6648 1.1103 1.9096 4.6276 -9.6481 611 0.36 0.7339 1.2823 2.3496 6.2445 -14.9396 684 0.43 0.8786 1.5216 2.7566 7.2264 -17.0127 671 0.82 1.3679 2.1602 3.5357 8.0288 -15.1123 675 0.50 1.0299 1.7819 3.2202 8.3972 -19.5556 680 0.56 1.1182 1.8991 3.3515 8.4617 -18.8473

Table B-76 Node driving reactance in ohms (harmonic frequencies 7 – 12) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 7 8 9 10 11 12 650 0.2222 0.3222 0.4035 0.4856 0.5756 0.6801 645 -0.0069 0.3117 0.5007 0.6575 0.8112 0.9786 632 -2.2158 -1.0818 -0.5616 -0.2103 0.0949 0.4201 634 -1.8481 -0.6969 -0.1476 0.2358 0.572 0.9251 611 -3.5586 -1.9987 -1.3428 -0.9565 -0.6797 -0.4475 684 -3.9786 -2.1894 -1.4393 -1.0037 -0.7021 -0.4656 671 -2.3748 -0.4625 0.4481 1.0545 1.5301 1.9412 675 -4.4756 -2.3600 -1.4222 -0.8142 -0.3082 0.2141 680 -4.0486 -1.937 -0.9791 -0.3458 0.1852 0.7281

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Table B-77 Node driving reactance in ohms (harmonic frequencies 13 – 18) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 13 14 15 16 17 18 650 0.8092 0.9862 1.3013 3.7340 0.6468 1.226 645 1.1836 1.4829 2.1118 8.4592 -0.3573 0.6955 632 0.8438 1.5459 3.2784 23.3578 -5.3703 -2.4465 634 1.3695 2.0748 3.7495 22.6934 -4.3061 -1.5122 611 -0.2170 0.0725 0.6329 6.1806 -1.5502 -0.6648 684 -0.2556 -0.0341 0.3097 3.1322 -0.6824 -0.2057 671 2.3239 2.7139 3.2112 5.9406 2.685 3.2956 675 0.8833 1.9900 4.7405 36.8839 -9.1929 -4.5576 680 1.407 2.4963 5.1318 35.434 -7.8935 -3.4861

Table B-78 Node driving reactance in ohms (harmonic frequencies 19 – 20) on Expanded IEEE 13 Node Test Feeder (Overhead construction) with Cap 1, 3, 4, and 6 energized, each rated at 800 kVAr Harmonic Frequency Node 19 20 650 1.6594 2.2027 645 1.1591 1.5569 632 -1.5421 -1.0655 634 -0.6198 -0.1308 611 -0.3175 -0.0655 684 -0.0081 0.1281 671 3.6526 3.9523 675 -3.1686 -2.4774 680 -2.135 -1.4413

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B.3 Analysis of X/R Ratio

Results of the previous section indicate that reactance is the dominant attribute in determining the occurrence of parallel harmonic resonance on a distribution feeder, which validates the information stated in [81].

Although customer harmonic impedance and transformer impedance affect the occurrence of harmonic resonance, the infrastructure of the distribution system also has a significant impact on the X/R ratio of driving impedances as evaluated on the mainline. An X/R ratio above two indicates that the resistive component of the driving impedance is no more than half the value of the reactance. The work performed in [99] indicates that fluctuations in customer harmonic impedance have little impact on driving reactance at high levels of harmonic current injection. Therefore, when high levels of harmonic currents are being injected into the system by the majority of customers, the driving impedance is more a function of the feeder construction than it is related to customer’s harmonic impedance.

Electric distribution feeders deliver energy to customers through a system of laterals that are comprised of overhead conductors and/or underground cables sized between #4 and 1000 kcmil. The cables and conductors are installed in fewer parallel paths than the secondary distribution conductors/cables that serve customers; therefore, their impedance has more of a direct influence on driving impedances as calculated from the mainline of the feeder.

To evaluate how system construction can affect the X/R ratio at locations on the feeder backbone, the X/R ratio of overhead conductors and underground cables rated 5 kV and 15 kV were analyzed. Cable and conductor impedance information was provided by Southwire Company. For the overhead conductor, vertical construction with adequate spacing per voltage level was assumed.

Results indicate that for overhead conductors and 5 kV and 15 kV underground cables, the reactance at the

3rd, 5th, and 7th harmonic frequencies is the dominant attribute for mainline and sub-distribution systems. The X/R ratios for all cables and conductors analyzed are listed in Tables B-1 through B-5. These results suggest that most distribution feeders comprised of mostly residential customers should have an X/R ratio greater than 1 for the 3rd,

5th, and 7th harmonic frequencies.

Based upon the augmented frequency scan analysis and the X/R ratio of overhead conductors and underground cables rated 5 kV and 15 kV, it appears that the resistive component has little impact on identifying the occurrence of harmonic resonance. Although the resistive component will dampen the magnitude of the driving impedance, the frequency in which it occurs will remain the same. The goal of the HRI algorithm is to identify the energized state of capacitor banks that will produce harmonic resonance conditions. Therefore, any increase in

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driving reactance that is greater than non-harmonic resonance conditions, regardless of the accuracy of the value, will be identified by the HRI algorithm.

Table B-79 Typical X/R Ratio of 5 kV Underground AL Cables Harmonic Freq. Conductor Size 3 5 7 8 0.11 0.19 0.26 6 0.17 0.29 0.41 4 0.25 0.42 0.59 2 0.37 0.61 0.85 1 0.44 0.74 1.03 1/0 0.54 0.90 1.26 2/0 0.66 1.11 1.55 3/0 0.82 1.36 1.91 4/0 1.00 1.67 2.33 250 1.18 1.97 2.75 300 1.37 2.29 3.21 350 1.56 2.60 3.64 400 1.73 2.88 4.03 500 2.16 3.60 5.03 600 2.51 4.18 5.85 750 3.04 5.07 7.09 1000 4.05 6.76 9.46

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Table B-80 Typical X/R Ratio of 5 kV Underground CU Cables Harmonic Freq. Conductor Size 3 5 7 8 0.19 0.31 0.44 6 0.29 0.48 0.67 4 0.42 0.69 0.97 2 0.61 1.01 1.42 1 0.73 1.21 1.70 1/0 0.90 1.50 2.09 2/0 1.09 1.81 2.54 3/0 1.35 2.24 3.14 4/0 1.65 2.76 3.86 250 1.95 3.25 4.55 300 2.27 3.78 5.29 350 2.57 4.29 6.00 400 2.85 4.75 6.65 500 3.56 5.93 8.30 600 4.13 6.89 9.64 750 5.00 8.33 11.67 1000 6.67 11.11 15.56

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Table B-81 Typical X/R Ratio of 15 kV Underground AL Cables Harmonic Freq. Conductor Size 3 5 7 2 0.94 1.56 2.19 1 1.21 2.02 2.83 1 1.15 1.92 2.69 1/0 1.49 2.48 3.47 1/0 1.41 2.35 3.28 2/0 1.81 3.02 4.23 2/0 1.74 2.90 4.07 3/0 2.14 3.56 4.98 4/0 2.63 4.38 6.13 250 3.00 5.00 7.00 300 3.52 5.86 8.21 350 4.02 6.69 9.37 400 4.53 7.55 10.58 500 5.53 9.21 12.90 600 6.39 10.65 14.91 750 7.80 13.01 18.21 1000 10.00 16.67 23.33

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Table B-82 Typical X/R Ratio of 15 kV Underground CU Cables Harmonic Freq. Conductor Size 3 5 7 2 1.56 2.60 3.64 1 1.90 3.17 4.43 1/0 2.34 3.90 5.46 2/0 2.85 4.75 6.66 3/0 3.52 5.86 8.20 4/0 4.35 7.24 10.14 250 4.95 8.26 11.56 300 5.80 9.67 13.53 350 6.62 11.04 15.45 400 7.48 12.46 17.45 500 9.11 15.19 21.26 600 10.53 17.56 24.58 750 12.83 21.39 29.94 1000 16.44 27.41 38.37

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Table B-83 Typical X/R Ratio of Vertical Overhead Conductors Harmonic Freq. Conductor Size 3 5 7 8 CU 0.71 1.19 1.66 6 CU 1.10 1.83 2.56 4 CU 1.68 2.80 3.93 2 CU 2.50 4.17 5.84 1 CU 3.10 5.16 7.22 1/0 3.82 6.37 8.92 2/0 4.73 7.88 11.03 4/0 7.12 11.87 16.62 4 ASCR 1.05 1.75 2.45 2 ASCR 1.61 2.69 3.76 1/0 ASCR 2.43 4.05 5.67 2/0 ASCR 2.95 4.92 6.89 4/0 ASCR 4.15 6.92 9.69 336 ASCR 6.83 11.39 15.94 795 AL 14.64 24.39 34.15

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APPENDIX C AMPL CODE FOR MODELS 1, 2, AND 3

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# The following code maximizes the driving reactance of the Expanded IEEE 13 Node Test Feeder # with 6 capacitor banks. The integer variables represent the states of each capacitor bank switch # The 3rd, 5th and 7th harmonic frequencies are analyzed # Parameters z,y,x,w replace constants in interactive terms for the 5th and 7th harmonic frequency

#Define integers

var a:=0 integer; # defines integer variable for capacitor bank 1 var f:=0 integer; # defines integer variable for capacitor bank 2 var b:=0 integer; # defines integer variable for capacitor bank 3 var g:=0 integer; # defines integer variable for capacitor bank 4 var h:=0 integer; # defines integer variable for capacitor bank 5 var l:=0 integer; # defines integer variable for capacitor bank 6

#Define parameters that replace constants that are repeated

param z = 0.0172; # replaces squared variable constant @ 5th frequency param y = 0.0003; # replaces cubed variable constant @ 5th frequency param x = 0.0241; # replaces squared variable constant @ 7th frequency param w = 0.00058; # replaces cubed variable constant @ 7th frequency

#maximize objective function that identifies the states of capacitor banks that produce the highest driving reactance

maximize profit:(((.93172-1.511*a*z-.23641*b*z+.36462*a*b*y-.69302 *f*z+.98597*a*f*y+.17477*b*f*y-2.3373*g*z+3.778*a*g*y+.5502*b*g*y+1.737*f*g*y- 3.66*h*z+5.9407*a*h*y+.9016*b*h*y+2.7273*f*h*y+5.261*g*h*y- 4.747*l*z+7.695*a*l*y+1.182*b*l*y+3.53102*f*l*y+9.1222*g*l*y+8.47212*h*l*y)^2)+((1.62516-3.94415*a*z- 4.5576*b*z+7.0293*a*b*y-1.28259*f*z+2.57269*a*f*y+3.3695*b*f*y-4.5851*a*b*f*z*y- 5.03433*g*z+10.409*a*g*y+8.384*b*g*y-12.9301*a*b*g*z*y+3.87107*f*g*y-6.789*a*f*g*z*y- 6.19795*b*f*g*z*y-13.1131*a*b*h*z*y-3.558*h*z+8.1687*a*h*y+8.50223*b*h*y+2.7813*f*h*y- 5.328*a*f*h*z*y-6.2856*b*f*h*z*y+7.01*g*h*y-14.49*a*g*h*z*y-11.674*b*g*h*z*y-5.3905*f*g*h*z*y- .42645*l*z+1.03125*a*l*y+1.1842*b*l*y-1.826*a*b*l*z*y+.33635*f*l*y-.67266*a*f*l*z*y- .87547*b*f*l*z*y+1.2891*g*l*y-2.665*a*g*l*z*y-2.1467*b*g*l*z*y-.99124*f*g*l*z*y+.87153*h*l*y- 2*a*h*l*z*y-2.082*b*h*l*z*y-.68139*f*h*l*z*y-1.717*g*h*l*z*y)^2)) /((.25774-a*.67595*z- .88271*b*z+1.36141*a*b*y-.20626*f*z+.44091*a*f*y+.65258*b*f*y-.888*a*b*f*z*y- 1.23322*g*z+2.5498*a*g*y+2.053*b*g*y-3.1674*a*b*g*z*y+0.94826*f*g*y-1.66319*a*f*g*z*y- 1.40862*h*z+3.2344*a*h*y+3.2344*b*h*y-5.19218*a*b*h*z*y+1.1013*f*h*y-2.10976*a*f*h*z*y- 2.4888*b*f*h*z*y+2.7757*g*h*y-5.739*a*g*h*z*y-4.622*b*g*h*z*y-2.13438*f*g*h*z*y- 1.5922*l*z+3.85043*a*l*y+4.4215*b*l*y-6.81936*a*b*l*z*y+1.2558*f*l*y-2.51156*a*f*l*z*y- 3.26881*b*f*l*z*y+4.8132*g*l*y-9.95178*a*g*l*z*y-8.01531*b*g*l*z*y-3.70103*f*g*l*z*y+3.254*h*l*y- 7.472*a*h*l*z*y-7.77697*b*h*l*z*y-2.544*f*h*l*z*y-6.41234*g*h*l*z*y)^2)

(((100*(39.911-39.1077*a*.0103-6.2721*b*.0103+5.832*a*b*.0103- 17.9096*f*.0103+15.37*a*f*.00011+2.7964*b*f*.00011- 63.5706*g*.0103+62.0338*a*g*.00011+9.209*b*g*.00011+28.5079*f*g*.00011- 106.025*h*.0103+104.025*a*h*.00011+16.11*b*h*.00011+47.715*f*h*.00011+91.888*g*h*.00011- 144.935*l*.0103+141.843*a*l*.00011+22.29*b*l*.00011+65.0186*f*l*.00011+168.339*g*l*.00011+157.755*h*l *.00011)^2)+(100*(75.051-113.9*a*.0103-139*b*.0103+129.29*a*b*.00011- 35.96*f*.0103+44.762*a*f*.00011+61.988*b*f*.00011-

190

146.3*g*.0103+184.95*a*g*.00011+154.46*b*g*.00011+67.979*f*g*.00011- 102.1*h*.0103+145.16*a*h*.00011+158.47*b*h*.00011+48.362*f*h*.00011+123.94*g*h*.00011- 12.17*l*.0103+18.383*a*l*.00011+22.293*b*l*.00011+5.826*f*l*.00011+23.093*g*l*.00011+15.399*h*l*.00011 )^2))/(100*(15.487-a*26.41*.0103-37.89*b*.0103+35.231*a*b*.00011- 7.585*f*.0103+10.383*a*f*.00011+16.891*b*f*.00011- 52.68*g*.0103+66.615*a*g*.00011+55.632*b*g*.00011+24.484*f*g*.00011- 62.72*h*.0103+89.176*a*h*.00011+97.358*b*h*.00011+29.711*f*h*.00011+76.142*g*h*.00011- 73.52*l*.0103+111.04*a*l*.00011+134.66*b*l*.00011+35.195*f*l*.00011+139.49*g*l*.00011+93.015*h*l*.0001 1)^2))

(((((0.1273-.41978*a*x-.46*b*x+.99276*a*b*w-.1398*f*x+.38*a*f*w+.4758*b*f*w-.9056*a*b*g*x*w- .532*g*x+1.52*a*g*w+1.18*b*g*w-2.547*a*f*g*x*w-1.22*b*f*g*x*w+.571*f*g*w- 0.38*h*x+1.19*a*h*w+1.18*b*h*w-2.559*a*b*h*x*w+.414*f*h*w-1.0896*a*f*h*x*w-1.22684*b*f*h*x*w- 2.279*b*g*h*x*w-1.1032*f*g*h*x*w+1.027*g*h*w-.0462*l*x+.152*a*l*w+.165*b*l*w-.35745*a*b*l*x*w- .1386*a*f*l*x*w-.1713*b*f*l*x*w-.5398*a*g*l*x*w-.4191*b*g*l*x*w-.2028*f*g*l*x*w-.40596*a*h*l*x*w- .4033*b*h*l*x*w-.1407*f*h*l*x*w-.349*g*h*l*x*w+.0507*f*l*w+.189*g*l*w+.129*h*l*w))) /((.01745-a*.6065*x-.07327*b*x+.15807*a*b*w-.0193*f*x+.05533*a*f*w+.07575*b*f*w- .14419*a*b*f*x*w-.1038*g*x+.2964*a*g*w+.2302*b*g*w-.4965*a*b*g*x*w+.11141*f*g*w- .27047*a*f*g*x*w-.23799*b*f*g*x*w-.1146*h*x+.36*a*h*w+.35777*b*h*w-.7717*a*b*h*x*w+.1248*f*h*w- .3285*a*f*h*x*w-.3698*b*f*h*x*w+.3099*g*h*w-.8851*a*g*h*x*w-.6872*b*g*h*x*w-.33261*f*g*h*x*w- .12479*l*x+.41013*a*l*w+.447*b*l*w-.96435*a*b*l*x*w+.13696*f*l*w-.37414*a*f*l*x*w-.4621*b*f*l*x*w- 1.456*a*g*l*x*w-1.1307*b*g*l*x*w-.54726*f*g*l*x*w+.50988*g*l*w+.34869*h*l*w-1.0952*a*h*l*x*w- 1.08808*b*h*l*x*w-.3799*f*h*l*x*w-.9424*g*h*l*x*w )))^2) ; ;

#binary and compensation constraints.

subject to one: a>= 0; #lower binary bounds constraint for integer a subject to two: b>= 0; #lower binary bounds constraint for integer b subject to three: f>= 0; #lower binary bounds constraint for integer f subject to four: g>= 0; #lower binary bounds constraint for integer g subject to five: h>= 0; #lower binary bounds constraint for integer h subject to six: l>= 0; #lower binary bounds constraint for integer l subject to seven: 1>= a; #upper binary bounds constraint for integer a subject to eight: 1>= b; #upper binary bounds constraint for integer b subject to nine: 1>= f; #upper binary bounds constraint for integer f subject to ten: 1>= g; #upper binary bounds constraint for integer g subject to eleven: 1>= h; #upper binary bounds constraint for integer h subject to twelve: 1>= l; #upper binary bounds constraint for integer l subject to thirteen: 5=a+l+b+f+g+h; #compensation restraint

191

# The following code maximizes the driving reactance of Feeder 1358 with 5 capacitor banks # The integer variables represent the states of each capacitor bank switch # The 3rd, 5th and 7th harmonic frequencies are analyzed # Parameters S, T, U, M, N, P, Q, replace constants in interactive terms for the 3rd, 5th and 7th harmonic frequency

#Define integers

var F integer; # defines integer variable for capacitor bank 1 var A integer; # defines integer variable for capacitor bank 2 var B integer; # defines integer variable for capacitor bank 3 var G integer; # defines integer variable for capacitor bank 4 var H integer; # defines integer variable for capacitor bank 5

#Define parameters that replace constants that are repeated

param S = 0.1206; # replaces single variable constant @ 7th frequency param T = 0.0145; # replaces squared variable constant @ 7th frequency param U = 0.00175; # replaces cubed variable constant @ 7th frequency param M = 0.0517; # replaces squared variable constant @ 3rd frequency param N = 0.00266; # replaces cubed variable constant @ 3rd frequency param P = 0.0861; # replaces squared variable constant @ 5th frequency param Q = 0.007415; # replaces cubed variable constant @ 5th frequency

#maximize objective function that identifies the states of capacitor banks that produce the highest driving reactance

maximize profit:((((2.0768-.4113*A*M-.0644*B*M+.0111*A*B*N- 1.607*F*M+.3183*A*F*N+0.0499*B*F*N -2.433*G*M+.4819*A*G*N+.0755*B*G*N+1.5332*F*G*N- 2.9881*H*M+.5918*A*H*N+.0927*B*H*N +2.0264*F*H*N +1.1698*G*H*N)^2)+(( 5.5627-3.1226*A*M- 2.652*B*M+.4582*A*B*N-4.5088*F*M+2.2238*A*F*N+1.7738*B*F*N - 1.3208*G*M+.7363*A*G*N+.6238*B*G*N+1.0135*F*G*N -1.3371*H*M+.7502*A*H*N+.6373*B*H*N +1.0837*F*H*N +.3175*G*H*N)^2))/(( 4.169-2.3998*A*M-2.0630*B*M+.3565*A*B*N- 4.1293*F*M+2.0366*A*F*N+1.6245*B*F*N-5.1342*G*M+2.8621*A*G*N+2.424*B*G*N+3.9396*F*G*N -6.2011*H*M+3.4943*A*H*N+2.9748*B*H*N+5.2071*F*H*N +2.468*G*H*N)^2))+((((346614- 323185*A*P-274292*B*P+78819*A*B*Q-466526*F*P+382482*A*F*Q+304767*B*F*Q- 136680*G*P+126623*A*G*Q+107157*B*G*Q+174177*F*G*Q- 138507*H*P+129145*A*H*Q+109607*B*H*Q+186424*F*H*Q +54617*G*H*Q)^2)+((129550-42672*A*P- 6680*B*P+1919*A*B*Q-166435*F*P+54821*A*F*Q+8582*B*F*Q- 251396*G*P+82806*A*G*Q+12963*B*G*Q+26318*F*G*Q- 309275*H*P+101871*A*H*Q+15947*B*H*Q+348410*F*H*Q+200992*G*H*Q)^2))/((156538-149872*A*P- 128679*B*P+36977*A*B*Q-257511*F*P+211122*A*F*Q+168224*B*F*Q- 319539*G*P+296026*A*G*Q+250518*B*G*Q+407201*F*G*Q- 386609*H*P+362040*A*H*Q+307860*B*H*Q+539065*F*H*Q+255473*G*H*Q)^2))+((((11962-15728*A*S- 13306*B*S+5366*A*B*T-22538*F*S+26036*A*F*T+20682*B*F*T-8340*A*B*F*U- 6605*G*S+8629*A*G*T+7280*B*G*T-2935*A*B*G*U+11781*F*G*T-13610*A*F*G*U-10810*B*F*G*U- 6687*H*S+8793*A*H*T+7439*B*H*T-2999*A*B*H*U+12600*F*H*T-14556*A*F*H*U- 11563*B*F*H*U+3693*G*H*T-4824*A*G*H*U-4070*B*G*H*U-6586*F*G*H*U)^2)+ ((4473-2068*A*S- 322.8*B*S+130.16*A*B*T-8048*F*S+3721*A*F*T+580.7*B*F*T-234*A*B*F*U- 12203*G*S+5642*A*G*T+880.6*B*G*T-355*A*B*G*U+17868*F*G*T-8261*A*F*G*U-1289*B*F*G*U- 14991*H*S+6931*A*H*T+1081*B*H*T-436*A*B*H*U+23626*F*H*T-10924*A*F*H*U- 1705*B*F*H*U+13651*G*H*T-6312*A*G*H*U-985*B*G*H*U-19988*F*G*H*U)^2))/((3844-5192*A*S- 4444*B*S+1792*A*B*T-8866*F*S+10243*A*F*T+8137*B*F*T-3281*A*B*F*U- 11038*G*S+14419*A*G*T+12164*B*G*T-4905*A*B*G*U+19686*F*G*T-22742*A*F*G*U-

192

18066*B*F*G*U-13334*H*S+17609*A*H*T+14927*B*H*T-6019*A*B*H*U+26030*F*H*T- 30071*A*F*H*U-23888*B*F*H*U+12347*G*H*T-16130*A*G*H*U-13607*B*G*H*U-22021*F*G*H*U)^2)) ;

#Binary and compensation constraints

subject to one: A>= 0; #Lower binary bound constraint for integer A subject to two: B>= 0; #Lower binary bound constraint for integer B subject to three: F>= 0; #Lower binary bound constraint for integer F subject to four: G>= 0; #Lower binary bound constraint for integer G subject to five: H>= 0; #Lower binary bound constraint for integer H subject to seven: 1>= A; #Upper binary bound constraint for integer A subject to eight: 1>= B; #Upper binary bound constraint for integer B subject to nine: 1>= F; #Upper binary bound constraint for integer F subject to ten: 1>= G; #Upper binary bound constraint for integer G subject to eleven: 1>= H; #Upper binary bound constraint for integer H subject to twelve: 3=A+B+F+G+H; #Compensation constraint

193

# The following code maximizes the driving reactance of Feeder 1358 tied Feeder 1444 with 8 capacitor banks # The integer variables represent the states of each capacitor bank switch # The 3rd, 5th and 7th harmonic frequencies are analyzed # Parameters S, T, U, V, N, O, P, Q, and R replace constants in interactive terms for the 3rd, 5th and 7th harm. freq.

#Define Integers

var F integer; # cap 1 # defines integer variable for capacitor bank 1 var A integer; # cap 2 # defines integer variable for capacitor bank 2 var B integer; # cap 3 # defines integer variable for capacitor bank 3 var G integer; # cap 4 # defines integer variable for capacitor bank 4 var H integer; # cap 5 # defines integer variable for capacitor bank 5 var J integer; # cap 6 # defines integer variable for capacitor bank 6 var L integer; # cap 7 # defines integer variable for capacitor bank 7 var M integer; # cap 8 # defines integer variable for capacitor bank 8

#Define parameters that replace constants that are repeated

param S = 0.217; # replaces single variable constant @ 7th frequency param T = 0.047; # replaces squared variable constant @ 7th frequency param U = 0.0102; # replaces cubed variable constant @ 7th frequency param V = 0.093; #3 # replaces single variable constant @ 3rd frequency param N = 0.0086; # replaces squared variable constant @ 3rd frequency param O = 0.000804; # replaces cubed variable constant @ 3rd frequency param P = 0.15509; #5 # replaces single variable constant @ 5th frequency param Q = 0.02402; # replaces squared variable constant @ 5th frequency param R = 0.003723; # replaces cubed variable constant @ 5th frequency

#maximize objective function that identifies the states of capacitor banks that produce the highest driving reactance maximize profit:(((((1.5878-1.3477*A*V-1.2274*B*V+.6394*A*B*N- 1.0486*F*V+.6995*A*F*N+.3616*B*F*N-.1883*A*B*F*O-1.1693*G*V+.9464*A*G*N+.7954*B*G*N- .4143*A*B*G*O+.6045*F*G*N-.4033*A*F*G*O-.2085*B*F*G*O-.8113*H*V+.6741*A*H*N+.5932*B*H*N- .3089*A*B*H*O+.4831*F*H*N-.3223*A*F*H*O-.1666*B*F*H*O+.3270*G*H*N-.2646*A*G*H*O- .2224*B*G*H*O-.1690*F*G*H*O)*(6.0309-L*V)*(2.15-M*V))+(-6.4046+5.4363*A*V+4.9525*B*V- 2.57939*A*B*N+4.2297*F*V-2.8216*A*F*N-1.4587*B*F*N+.75978*A*B*F*O+4.7169*G*V-3.8175*A*G*N- 3.2086*B*G*N+1.6710*A*B*G*O-2.4386*F*G*N+1.6269*A*F*G*O+.8409*B*F*G*O+3.2727*H*V- 2.7193*A*H*N-2.3928*B*H*N+1.2462*A*B*H*O-1.9487*F*H*N+1.3000*A*F*H*O+.6720*B*F*H*O- 1.3189*G*H*N+1.0674*A*G*H*O+.8917*B*G*H*O+.6818*F*G*H*O))^2)/((11.6582-10.0786*A*V- 9.4463*B*V+4.9199*A*B*N-8.3644*F*V+5.5801*A*F*N+2.8850*B*F*N-1.5026*A*B*F*O- 12.0017*G*V+9.7135*A*G*N+8.1639*B*G*N-4.2520*A*B*G*O+6.2047*F*G*N-4.1393*A*F*G*O- 2.1400*B*F*G*O-11.1917*H*V+9.2997*A*H*N+8.1828*B*H*N-4.2618*A*B*H*O+6.6645*F*H*N- 4.4460*A*F*H*O-2.2986*B*F*H*O+4.5109*G*H*N-3.6509*A*G*H*O-3.0685*B*G*H*O-2.3321*F*G*H*O- 14.2563*J*V+12.1009*A*J*N+11.0239*B*J*N-5.7416*A*B*J*O+9.4151*F*J*N-6.2810*A*F*J*O- 3.2473*B*F*J*O+10.4995*G*J*N-8.4977*A*G*J*O-7.1421*B*G*J*O-5.4281*F*G*J*O+7.2847*H*J*N- 6.0532*A*H*J*O-5.3262*B*H*J*O-4.3379*F*H*J*O-2.9361*G*H*J*O- 16.1221*L*V+13.7286*A*L*N+12.5704*B*L*N-16.5470*A*B*L*O+10.8072*F*L*N-7.2097*A*F*L*O- 3.7275*B*F*L*O+12.6944*G*L*N-10.2742*A*G*L*O-8.6352*B*G*L*O-6.5628*F*G*L*O+9.4960*H*L*N- 7.8906*A*H*L*O-6.9430*B*H*L*O-5.6547*F*H*L*O-3.8274*G*H*L*O+3.4258*J*L*N-2.9079*A*J*L*O- 2.6491*B*J*L*O-2.2625*F*J*L*O-2.5230*G*J*L*O-0.474*H*J*L*O- 19.3728*M*V+16.5668*A*M*N+15.2701*B*M*N-7.9531*A*B*M*O+13.2410*F*M*N-8.8333*A*F*M*O- 4.5669*B*F*M*O+16.5619*G*M*N-13.4045*A*G*M*O-11.2661*B*G*M*O-

194

8.5623*F*G*M*O+13.4152*H*M*N-11.1473*A*H*M*O-9.8085*B*H*M*O-7.9885*F*H*M*O- 5.4071*G*H*M*O+9.5758*J*M*N-8.1281*A*J*M*O-7.4047*B*J*M*O-6.3241*F*J*M*O-7.0524*G*J*M*O- 4.8931*H*J*M*O+7.4722*L*M*N-6.3628*A*L*M*O-5.8260*B*L*M*O-5.0088*F*L*M*O-5.8835*G*L*M*O- 4.4011*H*L*M*O-.04718*J*L*M*O)^2))+ (((((4.718-5.58*A*P-4.88*B*P+3.99*A*B*Q-3.869*F*P+3.917*A*F*Q+2.123*B*F*Q- 1.7376*A*B*F*R-5.20*G*P+6.01*A*G*Q+4.97*B*G*Q-4.07*A*B*G*R+3.56*F*G*Q-3.606*A*F*G*R- 1.95*B*F*G*R-3.50*H*P+4.10*A*H*Q+3.508*B*H*Q-2.87*A*B*H*R+2.67*F*H*Q-2.70*A*F*H*R - 1.467*B*F*H*R+2.26*G*H*Q-2.62*A*G*H*R-2.16*B*G*H*R-1.55*F*G*H*R)*(L*M*Q-1.356*L*P- 3.627*M*P+4.92)+(-6.88+8.153*A*P+7.1304*B*P-5.837*A*B*Q+5.647*F*P-5.717*A*F*Q- 3.097*B*F*Q+2.5358*A*B*F*R+7.6026*G*P-8.785*A*G*Q-7.258*B*G*Q+5.9412*A*B*G*R- 5.1997*F*G*Q+5.2552*A*F*G*R+2.8528*B*F*G*R+5.1078*H*P-5.9886*A*H*Q- 5.1194*B*H*Q+4.1897*A*B*H*R-3.9027*F*H*Q+3.9501*A*F*H*R+2.1408*B*F*H*R- 3.309*G*H*Q+3.8236*A*G*H*R+3.1592*B*G*H*R+2.264*F*G*H*R))^2))/((11.053-13.15*A*P- 11.644*B*P+9.5315*A*B*Q-9.40*F*P+9.5153*A*F*Q+5.157*B*F*Q-4.2213*A*B*F*R- 14.925*G*P+17.246*A*G*Q+14.249*B*G*Q-11.663*A*B*G*R+10.209*F*G*Q-10.333*A*F*G*R- 5.60*B*F*G*R-12.62*H*P+14.801*A*H*Q+12.653*B*H*Q-10.357*A*B*H*R+9.64*F*H*Q-9.763*A*F*H*R- 5.2912*B*F*H*R+8.17*G*H*Q-9.45*A*G*H*R-7.80*B*G*H*R-5.59*F*G*H*R- 16.34*J*P+19.34*A*J*Q+16.91*B*J*Q-13.84*A*B*J*R+13.39*F*J*Q-13.56*A*F*J*R- 7.349*B*F*J*R+18.03*G*J*Q-20.83*A*G*J*R-17.216*B*G*J*R-12.336*F*G*J*R+12.118*H*J*Q- 14.20*A*H*J*R-12.14*B*H*J*R-9.25*F*H*J*R-7.85*G*H*J*R-19.016*L*P+22.54*A*L*Q+19.76*B*L*Q- 16.182*A*B*L*R+15.72*F*L*Q-15.92*A*F*L*R-8.62*B*F*L*R+22.06*G*L*Q-25.492*A*G*L*R- 21.062*B*G*L*R-15.091*F*G*L*R+15.842*H*L*Q-18.571*A*H*L*R-15.875*B*H*L*R-12.10*F*H*L*R- 10.24*G*H*L*R+6.40*J*L*Q-7.578*A*J*L*R-6.62*B*J*L*R-5.24*F*J*L*R-7.07*G*J*L*R-4.75*H*J*L*R- 23.22*M*P+27.57*A*M*Q+24.26*B*M*Q-19.859*A*B*M*R+19.403*F*M*Q-19.639*A*F*M*R- 10.644*B*F*M*R+28.498*G*M*Q-32.929*A*G*M*R-27.20*B*G*M*R-19.493*F*G*M*R+21.871*H*M*Q- 25.638*A*H*M*R-21.917*B*H*M*R-16.707*F*H*M*R-14.16*G*H*M*R+17.116*J*M*Q-20.266*A*J*M*R- 17.73*B*J*M*R-14.04*F*J*M*R-18.89*G*J*M*R-12.69*H*J*M*R+14.017*L*M*Q-16.617*A*L*M*R- 14.572*B*L*M*R-11.594*F*L*M*R-16.26*G*L*M*R-11.67*H*L*M*R-4.718*J*L*M*R)^2))+ ((((.5034*L*M*T-.736*A*L*M*U-.6292*B*L*M*U-.465*F*L*M*U-.724*G*L*M*U-.4672*H*L*M*U) +(- .509*L*S+.7444*A*L*T+0.6363*B*L*T-.69352*A*B*L*U+.4703*F*L*T-.61493*A*F*L*U- .34792*B*F*L*U+.73225*G*L*T-1.0559*A*G*L*U-.86475*B*G*L*U-.5853*F*G*L*U+.47253*H*L*T- .68745*A*H*L*U-.57832*B*H*L*U-.41386*F*H*L*U-.41983*G*H*L*U)+(- 1.30532*M*S+1.90845*A*M*T+1.63152*B*M*T-1.77802*A*B*M*U+1.20575*F*M*T-1.5765*A*F*M*U- .89199*B*F*M*U+1.87733*G*M*T-2.707*A*G*M*U-2.21702*B*G*M*U- 1.50057*F*G*M*U+1.21145*H*M*T-1.76246*A*H*M*U-1.4826*B*H*M*U-1.06106*F*H*M*U- 1.07635*G*H*M*U)+(1.32027-1.93031*A*S-1.6502*B*S+1.79839*A*B*T- 1.21956*F*S+1.5946*A*F*T+.90221*B*F*T-.98351*A*B*F*U-1.89883*G*S+2.7381*A*G*T+2.2424*B*G*T- 2.44514*A*B*G*U+1.51776*F*G*T-1.98911*A*F*G*U-1.11233*B*F*G*U- 1.22533*H*S+1.78265*A*H*T+1.499*B*H*T-1.63441*A*B*H*U+1.07321*F*H*T-1.4069*A*F*H*U- .79468*B*F*H*U+1.08868*G*H*T-1.568*A*G*H*U-1.28643*B*G*H*U-.87021*F*G*H*U)+(- .37537+.549*A*S+.4692*B*S-.51128*A*B*T+.34644*F*S-.45409*A*F*T- .25647*B*F*T+.27949*A*B*F*U+.5397*G*S-.77853*A*G*T-.63787*B*G*T+.69512*A*B*G*U- .43151*F*G*T+.5655*A*F*G*U+.3194*B*F*G*U+.3484*H*S-.50687*A*H*T- .4264*B*H*T+.46467*A*B*H*U-.30512*F*H*T+.3998*A*F*H*U+.22585*B*F*H*U - .30948*G*H*T+.44634*A*G*H*U+.36569*B*G*H*U+.24739*F*G*H*U))^2)/((.566-.829*A*S- .72*B*S+.777*A*B*T-.5325*F*S+.698*A*F*T+.39415*B*F*T-.4295*A*B*F*U- .9313*G*S+1.343*A*G*T+1.10*B*G*T-1.19*A*B*G*U+.7444*F*G*T-.9755*A*F*G*U-.55099*B*F*G*U- .7234*H*S+1.0525*A*H*T+.8854*B*H*T-.965*A*B*H*U+.63357*F*H*T-.8303*A*F*H*U- .4689*B*F*H*U+.643*G*H*T-.9269*A*G*H*U-.7595*B*G*H*U-.5138*F*G*H*U- .9449*J*S+1.3802*A*J*T+1.1812*B*J*T-1.2872*A*B*J*U+.8723*F*J*T-1.1431*A*F*J*U- .6456*B*F*J*U+1.356*G*J*T-1.9599*A*G*J*U-1.606*B*G*J*U-1.0863*F*G*J*U+.8771*H*J*T- 1.27*A*H*J*U-1.07*B*H*J*U-.7681*F*H*J*U-.77923*G*H*J*U-1.1257*L*S+1.6473*A*L*T+1.4101*B*L*T - 1.5366*A*B*L*U+1.0445*F*L*T-1.3699*A*F*L*U-.77315*B*F*L*U+1.6822*G*L*T-2.426*A*G*L*U- 1.98*B*G*L*U-1.3445*F*G*L*U+1.15*H*L*T-1.6753*A*H*L*U-1.40*B*H*L*U-1.0085*F*H*L*U- 1.02*G*H*L*U+.509*J*L*T-.74469*A*J*L*U-.6364*B*J*L*U-.469*F*J*L*U-.7322*G*J*L*U-

195

.47259*H*J*L*U-1.38*M*S+2.02*A*M*T+1.7328*B*M*T-1.8883*A*B*M*U+1.2878*F*M*T- 1.68*A*F*M*U-.95327*B*F*M*U+2.147*G*M*T-3.0923*A*G*M*U-2.539*B*G*M*U- 1.7161*F*G*M*U+1.5547*H*M*T-2.2617*A*H*M*U-1.9027*B*H*M*U-1.3616*F*H*M*U- 1.3813*G*H*M*U+1.3054*J*M*T-1.9093*A*J*M*U-1.6318*B*J*M*U-1.205*F*J*M*U-1.8774*G*J*M*U- 1.2117*H*J*M*U+1.113*L*M*T-1.6287*A*L*M*U-1.3942*B*L*M*U-1.0328*F*L*M*U-1.6632*G*L*M*U- 1.1386*H*L*M*U-.50343*J*L*M*U)^2)) ;

#Binary, Compensation, and Solution Constraints

subject to one: A>= 0; #Lower bounds constraint for A subject to two: B>= 0; #Lower bounds constraint for B subject to three: F>= 0; #Lower bounds constraint for F subject to four: G>= 0; #Lower bounds constraint for G subject to five: H>= 0; #Lower bounds constraint for H subject to six: J>= 0; #Lower bounds constraint for J subject to seven: L>= 0; #Lower bounds constraint for L subject to eight: M>= 0; #Lower bounds constraint for M subject to nine: 1>= A; #Upper bounds constraint for A subject to ten: 1>= B; #Upper bounds constraint for B subject to eleven: 1>= F; #Upper bounds constraint for F subject to twelve: 1>= G; #Upper bounds constraint for G subject to thirteen: 1>= H; #Upper bounds constraint for H subject to fourteen: 1>= J; #Upper bounds constraint for J subject to fifteen: 1>= L; #Upper bounds constraint for L subject to sixteen: 1>= M; #Upper bounds constraint for M subject to seventeen: 3=A+B+F+G+H+J+L+M; #Compensation constraint subject to eighteen: 2>=H+L+M; #Solution Constraint subject to nineteen: 2>=J+L+M; #Solution Constraint subject to twenty: 2>=J+L+H; #Solution Constraint subject to twentyone: 2>=J+L+G; #Solution Constraint subject to twentytwo: 2>=G+L+M; #Solution Constraint subject to twentythree: 2>=B+L+M; #Solution Constraint subject to twentyfour: 2>=F+L+M; #Solution Constraint subject to twentyfive: 2>=A+L+M; #Solution Constraint subject to twentysix: 2>=H+J+M; #Solution Constraint subject to twentyseven: 2>=G+J+M; #Solution Constraint subject to twentyeight: 2>=B+J+L; #Solution Constraint subject to twentynine: 2>=A+J+L; #Solution Constraint subject to thirty: 2>=F+J+L; #Solution Constraint subject to thirtyone: 2>=B+J+M; #Solution Constraint subject to thirtytwo: 2>=F+J+M; #Solution Constraint subject to thirtythree: 2>=A+J+M; #Solution Constraint subject to thirtyfour: 2>=G+L+H; #Solution Constraint subject to thirtyfive: 2>=G+M+H; #Solution Constraint subject to thirtysix: 2>=G+J+H; #Solution Constraint

196

# The following code maximizes the driving reactance of Feeder 1358 tied Feeder 1444 with 8 capacitor banks # The integer variables represent the states of each capacitor bank switch # The 3rd, 5th and 7th harmonic frequencies are analyzed # Parameters S, T, U, V, N, O, P, Q, & R replace interactive term constants for 3rd, 5th & 7th harm. freq. on A-Phase # Parameters C, D, E, K, W, X, Y, I, & Z replace interactive term constants for 3rd, 5th & 7th harm. freq. on B-Phase

#Define Integers

var F:=0 integer; # defines integer variable for capacitor bank 1 var A:=0 integer; # defines integer variable for capacitor bank 2 var B:=0 integer; # defines integer variable for capacitor bank 3 var G:=0 integer; # defines integer variable for capacitor bank 4 var H:=0 integer; # defines integer variable for capacitor bank 5 var J:=0 integer; # defines integer variable for capacitor bank 6 var L:=0 integer; # defines integer variable for capacitor bank 7 var M:=0 integer; # defines integer variable for capacitor bank 8

#Define parameters that replace constants that are repeated

param S = 0.145; # replaces single variable constant @ 7th frequency A-Phase param T = 0.021; # replaces squared variable constant @ 7th frequency A-Phase param U = 0.00303; # replaces cubed variable constant @ 7th frequency A-Phase param V = 0.062; # replaces single variable constant @ 3rd frequency A-Phase param N = 0.004; # replaces squared variable constant @ 3rd frequency A-Phase param O = 0.00024; # replaces cubed variable constant @ 3rd frequency A-Phase param P = 0.103; # replaces single variable constant @ 5th frequency A-Phase param Q = 0.0107; # replaces squared variable constant @ 5th frequency A-Phase param R = 0.00110; # replaces cubed variable constant @ 5th frequency A-Phase

param C = 0.180834; # replaces single variable constant @ 7th frequency B-Phase param D = 0.032701; # replaces squared variable constant @ 7th frequency B-Phase param E = 0.005913; # replaces cubed variable constant @ 7th frequency B-Phase param K = 0.0775; #3 # replaces single variable constant @ 3rd frequency B-Phase param W = 0.006006; # replaces squared variable constant @ 3rd frequency B-Phase param X = 0.000465; # replaces cubed variable constant @ 3rd frequency B-Phase param Y = 0.129167; # replaces single variable constant @ 5th frequency B-Phase param Z = 0.016684; # replaces squared variable constant @ 5th frequency B-Phase param I = 0.002155; # replaces cubed variable constant @ 5th frequency B-Phase

197

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198

.43151*F*G*T+.5655*A*F*G*U+.3194*B*F*G*U+.3484*H*S-.50687*A*H*T- .4264*B*H*T+.46467*A*B*H*U-.30512*F*H*T+.3998*A*F*H*U+.22585*B*F*H*U - .30948*G*H*T+.44634*A*G*H*U+.36569*B*G*H*U+.24739*F*G*H*U))^2)/((.566-.829*A*S- .72*B*S+.777*A*B*T-.5325*F*S+.698*A*F*T+.39415*B*F*T-.4295*A*B*F*U- .9313*G*S+1.343*A*G*T+1.10*B*G*T-1.19*A*B*G*U+.7444*F*G*T-.9755*A*F*G*U-.55099*B*F*G*U- .7234*H*S+1.0525*A*H*T+.8854*B*H*T-.965*A*B*H*U+.63357*F*H*T-.8303*A*F*H*U- .4689*B*F*H*U+.643*G*H*T-.9269*A*G*H*U-.7595*B*G*H*U-.5138*F*G*H*U- .9449*J*S+1.3802*A*J*T+1.1812*B*J*T-1.2872*A*B*J*U+.8723*F*J*T-1.1431*A*F*J*U- .6456*B*F*J*U+1.356*G*J*T-1.9599*A*G*J*U-1.606*B*G*J*U-1.0863*F*G*J*U+.8771*H*J*T- 1.27*A*H*J*U-1.07*B*H*J*U-.7681*F*H*J*U-.77923*G*H*J*U-1.1257*L*S+1.6473*A*L*T+1.4101*B*L*T - 1.5366*A*B*L*U+1.0445*F*L*T-1.3699*A*F*L*U-.77315*B*F*L*U+1.6822*G*L*T-2.426*A*G*L*U- 1.98*B*G*L*U-1.3445*F*G*L*U+1.15*H*L*T-1.6753*A*H*L*U-1.40*B*H*L*U-1.0085*F*H*L*U- 1.02*G*H*L*U+.509*J*L*T-.74469*A*J*L*U-.6364*B*J*L*U-.469*F*J*L*U-.7322*G*J*L*U- .47259*H*J*L*U-1.38*M*S+2.02*A*M*T+1.7328*B*M*T-1.8883*A*B*M*U+1.2878*F*M*T- 1.68*A*F*M*U-.95327*B*F*M*U+2.147*G*M*T-3.0923*A*G*M*U-2.539*B*G*M*U- 1.7161*F*G*M*U+1.5547*H*M*T-2.2617*A*H*M*U-1.9027*B*H*M*U-1.3616*F*H*M*U- 1.3813*G*H*M*U+1.3054*J*M*T-1.9093*A*J*M*U-1.6318*B*J*M*U-1.205*F*J*M*U-1.8774*G*J*M*U- 1.2117*H*J*M*U+1.113*L*M*T-1.6287*A*L*M*U-1.3942*B*L*M*U-1.0328*F*L*M*U-1.6632*G*L*M*U- 1.1386*H*L*M*U-.50343*J*L*M*U)^2))

(((((1.5878-1.3477*A*K-1.2274*B*K+.6394*A*B*W-1.0486*F*K+.6995*A*F*W+.3616*B*F*W- .1883*A*B*F*X-1.1693*G*K+.9464*A*G*W+.7954*B*G*W-.4143*A*B*G*X+.6045*F*G*W- .4033*A*F*G*X-.2085*B*F*G*X-.8113*H*K+.6741*A*H*W+.5932*B*H*W- .3089*A*B*H*X+.4831*F*H*W-.3223*A*F*H*X-.1666*B*F*H*X+.3270*G*H*W-.2646*A*G*H*X- .2224*B*G*H*X-.1690*F*G*H*X)*(6.0309-L*K)*(2.15-M*K))+(-6.4046+5.4363*A*K+4.9525*B*K- 2.57939*A*B*W+4.2297*F*K-2.8216*A*F*W-1.4587*B*F*W+.75978*A*B*F*X+4.7169*G*K- 3.8175*A*G*W-3.2086*B*G*W+1.6710*A*B*G*X- 2.4386*F*G*W+1.6269*A*F*G*X+.8409*B*F*G*X+3.2727*H*K-2.7193*A*H*W- 2.3928*B*H*W+1.2462*A*B*H*X-1.9487*F*H*W+1.3000*A*F*H*X+.6720*B*F*H*X- 1.3189*G*H*W+1.0674*A*G*H*X+.8917*B*G*H*X+.6818*F*G*H*X))^2)/((11.6582-10.0786*A*K- 9.4463*B*K+4.9199*A*B*W-8.3644*F*K+5.5801*A*F*W+2.8850*B*F*W-1.5026*A*B*F*X- 12.0017*G*K+9.7135*A*G*W+8.1639*B*G*W-4.2520*A*B*G*X+6.2047*F*G*W-4.1393*A*F*G*X- 2.1400*B*F*G*X-11.1917*H*K+9.2997*A*H*W+8.1828*B*H*W-4.2618*A*B*H*X+6.6645*F*H*W- 4.4460*A*F*H*X-2.2986*B*F*H*X+4.5109*G*H*W-3.6509*A*G*H*X-3.0685*B*G*H*X-2.3321*F*G*H*X- 14.2563*J*K+12.1009*A*J*W+11.0239*B*J*W-5.7416*A*B*J*X+9.4151*F*J*W-6.2810*A*F*J*X- 3.2473*B*F*J*X+10.4995*G*J*W-8.4977*A*G*J*X-7.1421*B*G*J*X-5.4281*F*G*J*X+7.2847*H*J*W- 6.0532*A*H*J*X-5.3262*B*H*J*X-4.3379*F*H*J*X-2.9361*G*H*J*O- 16.1221*L*K+13.7286*A*L*W+12.5704*B*L*W-16.5470*A*B*L*X+10.8072*F*L*W-7.2097*A*F*L*X- 3.7275*B*F*L*X+12.6944*G*L*W-10.2742*A*G*L*X-8.6352*B*G*L*X-6.5628*F*G*L*X+9.4960*H*L*W- 7.8906*A*H*L*X-6.9430*B*H*L*X-5.6547*F*H*L*X-3.8274*G*H*L*X+3.4258*J*L*W-2.9079*A*J*L*X- 2.6491*B*J*L*X-2.2625*F*J*L*X-2.5230*G*J*L*X-0.474*H*J*L*X- 19.3728*M*K+16.5668*A*M*W+15.2701*B*M*W-7.9531*A*B*M*X+13.2410*F*M*W-8.8333*A*F*M*X- 4.5669*B*F*M*X+16.5619*G*M*W-13.4045*A*G*M*X-11.2661*B*G*M*X- 8.5623*F*G*M*X+13.4152*H*M*W-11.1473*A*H*M*X-9.8085*B*H*M*X-7.9885*F*H*M*X- 5.4071*G*H*M*X+9.5758*J*M*W-8.1281*A*J*M*X-7.4047*B*J*M*X-6.3241*F*J*M*X-7.0524*G*J*M*X- 4.8931*H*J*M*X+7.4722*L*M*W-6.3628*A*L*M*X-5.8260*B*L*M*X-5.0088*F*L*M*X- 5.8835*G*L*M*X-4.4011*H*L*M*X-.04718*J*L*M*X)^2))+ (((((4.718-5.58*A*Y-4.88*B*Y+3.99*A*B*Z-3.869*F*Y+3.917*A*F*Z+2.123*B*F*Z-1.7376*A*B*F*I- 5.20*G*Y+6.01*A*G*Z+4.97*B*G*Z-4.07*A*B*G*I+3.56*F*G*Z-3.606*A*F*G*R-1.95*B*F*G*I- 3.50*H*Y+4.10*A*H*Z+3.508*B*H*Z-2.87*A*B*H*R+2.67*F*H*Q-2.70*A*F*H*I -

199

1.467*B*F*H*I+2.26*G*H*Z-2.62*A*G*H*I-2.16*B*G*H*I-1.55*F*G*H*I)*(L*M*Z-1.356*L*Y- 3.627*M*Y+4.92)+(-6.88+8.153*A*Y+7.1304*B*Y-5.837*A*B*Z+5.647*F*Y-5.717*A*F*Z- 3.097*B*F*Z+2.5358*A*B*F*I+7.6026*G*Y-8.785*A*G*Z-7.258*B*G*Z+5.9412*A*B*G*I- 5.1997*F*G*Z+5.2552*A*F*G*I+2.8528*B*F*G*I+5.1078*H*Y-5.9886*A*H*Z- 5.1194*B*H*Z+4.1897*A*B*H*I-3.9027*F*H*Z+3.9501*A*F*H*I+2.1408*B*F*H*I- 3.309*G*H*Q+3.8236*A*G*H*I+3.1592*B*G*H*I+2.264*F*G*H*I))^2))/((11.053-13.15*A*Y- 11.644*B*Y+9.5315*A*B*Z-9.40*F*Y+9.5153*A*F*Z+5.157*B*F*Z-4.2213*A*B*F*I- 14.925*G*Y+17.246*A*G*Z+14.249*B*G*Z-11.663*A*B*G*I+10.209*F*G*Z-10.333*A*F*G*I- 5.60*B*F*G*I-12.62*H*Y+14.801*A*H*Z+12.653*B*H*Z-10.357*A*B*H*I+9.64*F*H*Z-9.763*A*F*H*I- 5.2912*B*F*H*I+8.17*G*H*Z-9.45*A*G*H*I-7.80*B*G*H*I-5.59*F*G*H*I- 16.34*J*Y+19.34*A*J*Z+16.91*B*J*Z-13.84*A*B*J*I+13.39*F*J*Q-13.56*A*F*J*I- 7.349*B*F*J*I+18.03*G*J*Z-20.83*A*G*J*I-17.216*B*G*J*I-12.336*F*G*J*Y+12.118*H*J*Z- 14.20*A*H*J*I-12.14*B*H*J*I-9.25*F*H*J*I-7.85*G*H*J*I-19.016*L*Y+22.54*A*L*Z+19.76*B*L*Z- 16.182*A*B*L*I+15.72*F*L*Z-15.92*A*F*L*I-8.62*B*F*L*I+22.06*G*L*Z-25.492*A*G*L*I- 21.062*B*G*L*I-15.091*F*G*L*I+15.842*H*L*Z-18.571*A*H*L*I-15.875*B*H*L*I-12.10*F*H*L*I- 10.24*G*H*L*Y+6.40*J*L*Z-7.578*A*J*L*I-6.62*B*J*L*I-5.24*F*J*L*I-7.07*G*J*L*I-4.75*H*J*L*I- 23.22*M*Y+27.57*A*M*Z+24.26*B*M*Z-19.859*A*B*M*I+19.403*F*M*Z-19.639*A*F*M*I- 10.644*B*F*M*I+28.498*G*M*Z-32.929*A*G*M*I-27.20*B*G*M*I-19.493*F*G*M*I+21.871*H*M*Z- 25.638*A*H*M*I-21.917*B*H*M*I-16.707*F*H*M*I-14.16*G*H*M*I+17.116*J*M*Z-20.266*A*J*M*I- 17.73*B*J*M*I-14.04*F*J*M*I-18.89*G*J*M*I-12.69*H*J*M*I+14.017*L*M*Z-16.617*A*L*M*I- 14.572*B*L*M*I-11.594*F*L*M*I-16.26*G*L*M*I-11.67*H*L*M*I-4.718*J*L*M*I)^2))+ ((((.5034*L*M*D- .736*A*L*M*E-.6292*B*L*M*E-.465*F*L*M*E-.724*G*L*M*E-.4672*H*L*M*E) +(- .509*L*C+.7444*A*L*D+0.6363*B*L*D-.69352*A*B*L*E+.4703*F*L*D-.61493*A*F*L*E- .34792*B*F*L*E+.73225*G*L*D-1.0559*A*G*L*E-.86475*B*G*L*E-.5853*F*G*L*E+.47253*H*L*D- .68745*A*H*L*E-.57832*B*H*L*E-.41386*F*H*L*E-.41983*G*H*L*E)+(- 1.30532*M*C+1.90845*A*M*D+1.63152*B*M*D-1.77802*A*B*M*E+1.20575*F*M*D-1.5765*A*F*M*E- .89199*B*F*M*E+1.87733*G*M*D-2.707*A*G*M*E-2.21702*B*G*M*E- 1.50057*F*G*M*E+1.21145*H*M*D-1.76246*A*H*M*E-1.4826*B*H*M*E-1.06106*F*H*M*E- 1.07635*G*H*M*E)+(1.32027-1.93031*A*C-1.6502*B*C+1.79839*A*B*D- 1.21956*F*C+1.5946*A*F*D+.90221*B*F*D-.98351*A*B*F*E- 1.89883*G*C+2.7381*A*G*D+2.2424*B*G*D-2.44514*A*B*G*E+1.51776*F*G*D-1.98911*A*F*G*E- 1.11233*B*F*G*E-1.22533*H*C+1.78265*A*H*D+1.499*B*H*D-1.63441*A*B*H*E+1.07321*F*H*D- 1.4069*A*F*H*E-.79468*B*F*H*E+1.08868*G*H*D-1.568*A*G*H*E-1.28643*B*G*H*E- .87021*F*G*H*E)+(-.37537+.549*A*C+.4692*B*C-.51128*A*B*D+.34644*F*C-.45409*A*F*D- .25647*B*F*D+.27949*A*B*F*E+.5397*G*C-.77853*A*G*D-.63787*B*G*D+.69512*A*B*G*E- .43151*F*G*D+.5655*A*F*G*E+.3194*B*F*G*E+.3484*H*C-.50687*A*H*D- .4264*B*H*D+.46467*A*B*H*E-.30512*F*H*D+.3998*A*F*H*E+.22585*B*F*H*E - .30948*G*H*D+.44634*A*G*H*E+.36569*B*G*H*E+.24739*F*G*H*E))^2)/((.566-.829*A*C- .72*B*C+.777*A*B*D-.5325*F*C+.698*A*F*D+.39415*B*F*D-.4295*A*B*F*E- .9313*G*C+1.343*A*G*D+1.10*B*G*D-1.19*A*B*G*E+.7444*F*G*D-.9755*A*F*G*E-.55099*B*F*G*E- .7234*H*C+1.0525*A*H*D+.8854*B*H*D-.965*A*B*H*E+.63357*F*H*D-.8303*A*F*H*E- .4689*B*F*H*E+.643*G*H*D-.9269*A*G*H*E-.7595*B*G*H*E-.5138*F*G*H*E- .9449*J*C+1.3802*A*J*D+1.1812*B*J*D-1.2872*A*B*J*E+.8723*F*J*D-1.1431*A*F*J*E- .6456*B*F*J*E+1.356*G*J*D-1.9599*A*G*J*E-1.606*B*G*J*E-1.0863*F*G*J*E+.8771*H*J*D- 1.27*A*H*J*E-1.07*B*H*J*E-.7681*F*H*J*E-.77923*G*H*J*E-1.1257*L*C+1.6473*A*L*D+1.4101*B*L*D - 1.5366*A*B*L*E+1.0445*F*L*D-1.3699*A*F*L*E-.77315*B*F*L*E+1.6822*G*L*D-2.426*A*G*L*E- 1.98*B*G*L*E-1.3445*F*G*L*E+1.15*H*L*D-1.6753*A*H*L*E-1.40*B*H*L*E-1.0085*F*H*L*E- 1.02*G*H*L*E+.509*J*L*D-.74469*A*J*L*E-.6364*B*J*L*E-.469*F*J*L*E-.7322*G*J*L*E- .47259*H*J*L*E-1.38*M*C+2.02*A*M*D+1.7328*B*M*D-1.8883*A*B*M*E+1.2878*F*M*D- 1.68*A*F*M*E-.95327*B*F*M*E+2.147*G*M*D-3.0923*A*G*M*E-2.539*B*G*M*E- 1.7161*F*G*M*E+1.5547*H*M*D-2.2617*A*H*M*E-1.9027*B*H*M*E-1.3616*F*H*M*E- 1.3813*G*H*M*E+1.3054*J*M*D-1.9093*A*J*M*E-1.6318*B*J*M*E-1.205*F*J*M*E-1.8774*G*J*M*E- 1.2117*H*J*M*E+1.113*L*M*D-1.6287*A*L*M*E-1.3942*B*L*M*E-1.0328*F*L*M*E-1.6632*G*L*M*E- 1.1386*H*L*M*E-.50343*J*L*M*E)^2)) ;

200

#Binary, Compensation, and Solution Constraints

subject to one: A>= 0; subject to two: B>= 0; subject to three: F>= 0; subject to four: G>= 0; subject to five: H>= 0; subject to six: J>= 0; subject to seven: L>= 0; subject to eight: M>= 0; subject to nine: 1>= A; subject to ten: 1>= B; subject to eleven: 1>= F; subject to twelve: 1>= G; subject to thirteen: 1>= H; subject to fourteen: 1>= J; subject to fifteen: 1>= L; subject to sixteen: 1>= M; subject to seventeen: 4=A+B+F+G+H+J+L+M; subject to eighteen: 3>=J+L+M+H; subject to nineteen: 3>=J+L+M+G; subject to twenty: 3>=J+L+M+B; subject to twentyone: 3>=J+L+M+F; subject to twentytwo: 3>=J+L+M+A; subject to twentythree: 3>=F+B+H+A; subject to twentyfour: 3>=F+B+M+A; subject to twentyfive: 3>=F+B+L+A; subject to twentysix: 3>=F+L+M+A; subject to twentyseven: 3>=F+J+B+A; subject to twentyeight: 3>=F+J+M+A; subject to twentynine: 3>=F+J+L+A; subject to thirty: 3>=F+B+G+A; subject to thirtyone: 3>=F+J+G+A; subject to thirtytwo: 3>=F+A+G+L; subject to thirtythree: 3>=F+A+G+M; subject to thirtyfour: 3>=F+A+G+H; subject to thirtyfive: 3>=F+A+M+H; subject to thirtysix: 3>=F+A+L+H; subject to thirtyseven: 3>=F+A+J+H; subject to thirtyeight: 3>=G+H+L+M; subject to thirtynine: 3>=B+H+L+M; subject to forty: 3>=F+H+L+M; subject to fortyone: 3>=A+H+L+M; subject to fortytwo: 3>=B+G+L+M; subject to fortythree: 3>=F+G+L+M; subject to fortyfour: 3>=A+G+L+M; subject to fortyfive: 3>=F+B+L+M; subject to fortysix: 3>=A+B+L+M; subject to fortyseven: 3>=A+B+J+M; subject to fortyeight: 3>=F+B+J+M; subject to fortynine: 3>=A+B+J+L; subject to fifty: 3>=F+B+J+L; subject to fiftyone: 3>=G+B+J+M; subject to fiftytwo: 3>=F+G+J+M;

201

subject to fiftythree: 3>=A+G+J+M; subject to fiftyfour: 3>=F+G+H+M; subject to fiftyfive: 3>=B+G+H+M; subject to fiftysix: 3>=B+G+H+L; subject to fiftyseven: 3>=M+G+H+J; subject to fiftyeight: 3>=A+G+H+M; subject to fiftynine: 3>=L+G+H+J; subject to sixty: 3>=B+G+H+J; subject to sixtyone: 3>=A+G+H+J; subject to sixtytwo: 3>=F+G+H+J; subject to sixtythree: 3>=B+H+J+M; subject to sixtyfour: 3>=A+H+J+M; subject to sixtyfive: 3>=F+H+J+M; subject to sixtysix: 3>=F+H+G+L; subject to sixtyseven: 3>=A+H+G+L; subject to sixtyeight: 3>=F+B+H+L; subject to sixtynine: 3>=A+B+H+L; subject to seventy: 3>=B+H+J+L; subject to seventyone: 3>=A+H+J+L; subject to seventytwo: 3>=F+H+J+B; subject to seventythree: 3>=F+H+J+L; subject to seventyfour: 3>=F+H+B+G; subject to seventyfive: 3>=F+H+B+M; subject to seventysix: 3>=A+H+B+J;

202

APPENDIX D TEST RESULTS FOR ACCURACY AND CONVERGENCE ANALYSIS

203

Table D-1 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 800 kVAr, Compensation = 1 Cap

Table D-2 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 800 kVAr, Compensation = 2 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 0 1 0 1 2 1 250050 6.5217 3.0050 14.6040 5.6663 485.8704 122.3257 5 3 0 0 0 1 1 0 2 2 4708 5.9917 3.0475 11.9946 6.1127 -59.9155 -30.1045 4 3 0 0 0 0 1 1 2 3 1841 7.0246 2.9763 20.3862 5.9100 -36.5141 -1.9687 4 2 0 0 1 0 0 1 2 4 1238 6.1945 3.0325 11.8954 5.4887 29.4366 12.0326 4 3 1 0 0 0 0 1 2 5 694 5.9917 2.7299 10.7937 4.1037 21.8757 5.8230 2 2 0 0 1 0 1 0 2 6 631 5.6853 3.0706 9.5495 5.7616 17.2608 12.7497 4 3 0 0 1 1 0 0 2 7 432 5.3654 3.1286 8.2497 6.2193 14.3548 9.0579 4 2 1 0 0 0 1 0 2 8 332 5.4960 2.7604 8.5010 4.2499 13.1022 6.0308 3 2 1 0 0 1 0 0 2 9 214 5.1874 2.8065 7.4098 4.4838 8.2542 6.1093 1 0 0 1 0 0 0 1 2 10 190 6.0512 2.8187 11.0967 4.4850 -1.7013 -0.4953 1 1 1 0 1 0 0 0 2 11 163 4.9809 2.8538 6.6668 4.6740 7.0109 3.7991 0 0 0 1 0 0 1 0 2 12 140 5.5515 2.8514 8.8593 4.6622 -1.2574 -0.6399 1 1 0 1 0 1 0 0 2 13 119 5.2395 2.9008 7.6348 4.9489 -1.0281 -0.8451 1 0 0 1 1 0 0 0 2 14 108 5.0286 2.9515 6.8409 5.1845 -1.1091 -1.2534 0 0 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

204

Table D-3 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 800 kVAr, Compensation = 3 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 0 1 1 1 3 1 5169 7.8746 3.3696 67.9177 19.4537 -8.4211 0.6085 4 3 1 0 0 1 0 1 3 2 39305 6.5265 3.0117 14.6538 5.7325 170.9214 99.4000 5 3 0 1 0 0 1 1 3 3 1271 7.1222 3.0910 30.0625 17.5195 -1.0867 -0.0670 3 2 0 0 1 0 1 1 3 4 1271 7.3471 3.3518 27.1271 9.6440 -19.1446 -1.5266 4 3 1 0 0 1 1 0 3 5 4183 5.9965 3.0542 12.0584 6.1737 -57.2894 -24.5515 4 3 1 0 0 0 1 1 3 6 1819 7.0298 2.9829 20.4663 5.9857 -36.2056 -1.9647 4 2 1 0 1 0 0 1 3 7 1261 6.1988 3.0389 11.9271 5.5278 29.9350 12.2859 4 3 0 0 1 1 0 1 3 8 1112 6.8150 3.3889 18.8529 8.9952 -22.3473 -10.5871 5 4 1 0 1 0 1 0 3 9 646 5.6897 3.0771 9.5935 5.8040 17.5044 12.9848 4 3 0 0 1 1 1 0 3 10 484 6.2683 3.4435 15.0892 9.2092 -8.0088 -7.5188 4 3 1 0 1 1 0 0 3 11 445 5.3696 3.1354 8.2902 6.2676 17.3683 0.4870 3 1 0 1 0 1 0 1 3 12 340 6.6105 3.1221 15.6050 6.5769 -0.9112 -0.2643 3 1 0 1 1 0 0 1 3 13 246 6.2739 3.1509 12.5223 6.2755 -0.9738 -0.4566 3 1 0 1 0 1 1 0 3 14 265 6.0760 3.1676 12.9993 7.1483 -0.6240 -0.3182 2 2 0 1 1 0 1 0 3 15 191 5.7603 3.1921 10.1425 6.6366 -0.7194 -0.5821 2 1 1 1 0 0 0 1 3 16 191 6.0552 2.8249 11.1223 4.5292 -1.4971 -0.4404 1 1 0 1 1 1 0 0 3 17 169 5.4363 3.2551 8.7958 7.2570 -0.4886 -0.7542 2 2 1 1 0 0 1 0 3 18 141 5.5555 2.8576 8.8937 4.7050 -1.1218 -0.5681 1 1 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

205

Table D-4 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 800 kVAr, Compensation = 4 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 1 1 1 1 4 1 20249 8.3664 3.8669 -130.085 -57.8891 -4.6826 0.6066 3 3 0 1 0 1 1 1 4 2 17316 8.0228 3.5217 125.922 38.1316 -0.5858 0.0551 3 3 1 0 0 1 1 1 4 3 5499 7.8830 3.3791 69.5280 20.3360 -8.4591 0.6377 4 3 1 0 1 0 1 1 4 4 1262 7.3545 3.3603 27.4074 9.8304 -18.4343 -1.4576 4 3 1 0 1 1 0 1 4 5 999 6.8218 3.3975 17.9702 9.7496 -20.8498 -9.2690 5 4 1 0 1 1 1 0 4 6 477 6.2756 3.4521 14.8470 9.0751 -9.0693 -6.6644 5 2 0 1 1 0 1 1 4 7 1359 7.4746 3.5001 33.9008 12.2569 -0.6664 -0.0619 3 3 1 1 0 0 1 1 4 8 607 7.1289 3.0995 22.2385 7.0974 -0.9865 -0.0565 3 2 1 1 1 0 0 1 4 9 249 6.2793 3.1588 12.5957 6.3613 -0.8722 -0.4069 3 1 1 1 1 0 1 0 4 10 194 5.7662 3.2001 10.2490 6.7234 -0.6845 -0.5189 2 2 0 1 1 1 0 1 4 11 662 6.9313 3.5407 21.5310 11.5644 -0.4708 -0.2454 3 2 0 1 1 1 1 0 4 12 622 6.2756 3.4521 16.8016 10.8838 -10.6934 -7.6644 5 4 1 1 0 1 0 1 4 13 346 6.6167 3.1305 15.7263 6.7179 -0.8316 -0.2323 3 2 1 1 0 1 1 0 4 14 270 6.0825 3.1759 13.1641 7.2803 -0.6102 -0.2828 2 2 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

206

Table D-5 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 600 kVAr, Compensation = 1 Caps Variable s Driving Reactance (Ohms) Points of Concern Objective 3rd 5th 7th Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 0 0 0 1 1 1 354 5.6535 2.6816 9.1353 3.9119 14.0713 4.1688 2 2 0 0 0 0 1 0 1 2 231 5.3080 2.7043 7.7925 4.0149 9.4895 5.3832 2 1 0 0 0 1 0 0 1 3 180 5.0899 2.7386 7.0354 4.1796 7.4933 5.0889 0 0 0 0 1 0 0 0 1 4 154 4.9424 2.7749 6.5509 3.6344 5.7415 5.8133 0 0 1 0 0 0 0 0 1 5 136 4.8476 2.5804 6.3092 3.6259 5.9030 4.4630 0 0 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

207

Table D-6 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 600 kVAr, Compensation = 3 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 1 0 1 1 3 1 23379 6.4811 3.0898 14.1744 6.0036 145.3317 43.8553 5 3 0 0 1 1 1 0 3 2 3336 5.7938 3.1563 10.4197 6.4770 43.5917 35.2064 4 3 1 0 0 0 1 1 3 3 2713 6.3028 2.8478 12.7827 4.6953 48.8829 10.7548 4 3 0 0 1 1 0 1 3 4 2053 6.1626 3.1219 11.5939 6.1188 37.9441 20.2442 4 3 0 0 0 1 1 1 3 5 2039 6.7644 3.0796 17.1651 6.5261 -40.0373 -7.1359 5 2 1 0 0 1 0 1 3 6 769 5.9948 2.8747 10.8217 4.7623 22.4692 8.6252 3 3 1 0 0 1 1 0 3 7 571 5.6333 2.9037 9.3474 4.9683 18.6347 9.0466 3 3 1 0 1 0 0 1 3 8 440 5.7894 2.9020 9.6885 4.8021 14.8191 7.7028 3 3 1 0 1 0 1 0 3 9 293 5.4374 2.9289 8.0299 4.9620 10.4825 7.4667 3 2 0 1 0 0 1 1 3 10 251 6.3555 2.9196 13.1674 5.0695 -1.9299 -0.4554 2 1 1 0 1 1 0 0 3 11 231 5.2137 2.9696 7.5012 5.2235 8.6009 7.0175 3 0 0 1 0 1 0 1 3 12 198 6.0444 2.9480 11.1302 5.1478 -1.7480 -0.6895 1 1 0 1 1 0 0 1 3 13 171 5.8353 2.9766 9.9269 5.1855 -1.8792 -0.9744 1 1 0 1 0 1 1 0 3 14 164 5.6809 2.9784 9.6483 5.3859 -1.3493 -0.7696 2 1 1 1 0 0 0 1 3 15 150 5.6971 2.7524 9.3069 4.1922 -2.5613 -0.9772 1 1 0 1 1 0 1 0 3 16 142 5.4813 3.0049 8.5246 5.3736 -1.5092 -1.1252 2 1 0 1 1 1 0 0 3 17 130 5.2558 3.0479 7.7120 5.6842 -1.2440 -1.3313 2 0 1 1 0 0 1 0 3 18 122 5.3495 2.7765 7.9515 4.3104 -2.0909 -1.1404 1 0 1 1 0 1 0 0 3 19 109 5.1297 2.8126 7.1819 4.5008 -1.8594 -1.3702 0 0 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

208

Table D-7 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 600 kVAr, Compensation = 4 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 1 0 1 1 1 0 4 1 3869 5.7976 3.1616 10.4731 6.5225 48.1775 35.5234 4 3 1 0 1 0 1 1 4 2 29966 6.4849 3.0950 14.2218 6.0493 170.0042 35.2088 5 3 1 0 1 1 0 1 4 3 2222 6.1662 3.1271 11.9788 6.1646 37.4144 22.1526 4 3 1 0 0 1 1 1 4 4 1938 6.7686 3.0851 17.7222 6.6012 -39.3982 -7.1182 5 2 0 0 1 1 1 1 4 5 937 7.0041 3.3737 22.1514 9.6134 -17.3373 -4.4248 4 3 0 1 0 1 1 1 4 6 468 6.8381 3.1707 18.8531 7.4161 -1.2062 -0.2368 3 2 0 1 1 0 1 1 4 7 324 6.5482 3.1808 14.9099 6.7016 -1.3347 -0.4276 3 2 0 1 1 1 0 1 4 8 254 6.2259 3.2149 12.5378 6.8431 -1.1235 -0.6376 2 2 0 1 1 1 1 0 4 9 220 5.8551 3.2513 11.0446 7.2827 -0.8803 -0.7081 2 2 1 1 0 0 1 1 4 10 252 6.3591 2.9247 13.2057 5.1197 -1.7748 -0.4171 2 1 1 1 0 1 0 1 4 11 193 6.0478 2.9531 11.1643 5.1950 -1.6060 -0.6302 1 1 1 1 1 0 0 1 4 12 172 5.8384 2.9816 9.9517 5.2210 -1.7121 -0.8902 1 1 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

209

Table D-8 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 600 kVAr, Compensation = 5 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 1 0 1 1 1 1 5 1 911 7.0096 3.3803 22.3415 9.7603 -15.9511 -1.0805 4 3 0 1 1 1 1 2 5 2 849 7.0947 3.4844 25.0834 11.9884 -2.9526 -0.2223 3 2 1 1 0 1 1 3 5 3 475 6.8432 3.1772 18.9898 7.5501 -1.1409 -0.2126 3 2 1 1 1 0 1 4 5 4 327 6.5529 3.1869 14.9985 6.7847 -1.2504 -0.3913 3 2 1 1 1 1 0 5 5 5 255 5.8599 3.2575 12.5001 7.3612 -0.8661 -0.6427 2 2 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

Table D-9 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 1 Cap Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 1 0 0 0 0 1 1 1794 4.8555 2.5967 6.3276 3.6801 32.2716 25.8486 2 2 0 0 0 0 0 0 1 2 171 5.0866 2.6082 7.0232 3.6903 7.4396 4.5642 0 0 0 0 0 0 1 0 1 3 155 4.8860 2.6157 6.3976 3.6720 7.3105 4.4636 0 0 0 0 0 1 0 0 1 4 146 4.8611 2.5827 6.3305 3.6692 7.3601 4.4611 0 0 0 0 1 0 0 0 1 5 140 4.8763 2.6394 6.3769 3.6245 7.3112 4.4477 0 0 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

210

Table D-10 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 2 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 1 0 0 0 1 2 1 3316309 5.0971 2.6283 7.0510 3.7584 1540.045 971.843 2 2 0 1 0 0 1 0 2 2 150533 5.2532 2.6704 7.5525 3.8865 -355.173 -155.807 3 2 0 1 0 1 0 0 2 3 43748 4.9313 2.6472 6.5312 3.8405 160.720 133.520 2 2 0 1 1 0 0 0 2 4 8555 4.8864 2.6600 6.4020 3.8981 74.1900 54.4236 2 2 1 1 0 0 0 0 2 5 2340 4.8560 2.5978 6.3290 3.6840 37.0352 29.7419 2 2 0 0 0 0 1 1 2 6 202 5.2416 2.6496 7.5180 3.8130 8.4857 4.8157 2 0 0 0 0 1 0 1 2 7 187 5.1697 2.6603 7.2738 3.8589 8.0028 4.9328 0 0 0 0 1 0 0 1 2 8 178 5.1202 2.6722 7.1128 3.9096 7.4076 5.0524 0 0 1 0 0 0 0 1 2 9 172 5.0871 2.6093 7.0247 3.6940 7.4418 4.5720 0 0 0 0 0 1 1 0 2 10 169 5.0679 2.6683 6.8418 3.8951 7.3665 4.5033 0 0 0 0 1 0 1 0 2 11 161 5.0191 2.6800 6.4049 3.9438 7.3208 4.5020 0 0 1 0 0 0 1 0 2 12 155 4.9866 2.6168 6.3699 3.7242 7.3048 4.5064 0 0 0 0 1 1 0 0 2 13 153 4.8954 2.5918 6.3159 3.7091 7.3179 4.4992 0 0 1 0 0 1 0 0 2 14 147 4.8516 2.6279 6.3162 3.6731 7.3622 4.4697 0 0 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

211

Table D-11 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 3 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 1 1 0 0 1 0 3 1 1550729 4.9970 2.6371 6.7263 3.7941 -999.300 -742.985 2 2 1 1 0 1 0 0 3 2 606540 4.9319 2.6484 6.5328 3.8449 599.118 497.501 2 2 1 1 0 0 0 0 3 3 146875 5.0977 2.6295 7.0527 3.7626 -322.877 -206.220 2 2 1 1 1 0 0 0 3 4 18349 4.8870 2.6612 6.4035 3.9026 98.8601 92.1292 2 2 0 1 1 1 0 0 3 5 8445 4.9646 2.7133 6.6200 4.0803 -65.2791 -63.9350 2 2 0 1 1 0 1 0 3 6 6914 5.0301 2.7013 6.8156 4.0226 -63.0323 -53.3477 2 2 0 1 1 0 0 1 3 7 6105 5.1313 2.6933 7.1444 3.9870 -62.8801 -45.3057 2 2 0 1 0 1 0 1 3 8 2849 5.1810 2.6813 7.3071 3.9344 -43.8477 -28.6811 2 2 0 1 0 1 1 0 3 9 2774 5.0791 2.6894 6.9744 3.9719 -41.1805 -31.2788 2 2 0 1 0 0 1 1 3 10 2294 5.2532 2.6704 7.5525 3.8865 -40.4138 -23.5066 3 2 0 0 0 1 1 1 3 11 226 5.3325 2.7049 7.8300 4.0070 9.2900 5.2914 2 1 0 0 1 0 1 1 3 12 211 5.2784 2.7158 7.6292 4.0489 8.6650 5.3681 2 0 1 0 0 0 1 1 3 13 202 5.2423 2.6508 7.5199 3.8171 8.6706 4.8244 2 0 0 0 1 1 0 1 3 14 196 5.2059 2.7270 7.3814 4.1013 7.7810 5.5166 0 0 1 0 0 1 0 1 3 15 187 5.1703 2.6615 7.2756 3.8632 8.0702 4.9420 0 0 1 0 1 0 0 1 3 16 178 5.1208 2.6734 7.1146 3.9139 7.2900 5.0621 0 0 0 0 1 1 1 0 3 17 177 5.1036 2.7355 7.0470 4.1424 7.0181 5.6449 0 0 1 0 0 1 1 0 3 18 169 5.0685 2.6695 6.9436 3.8993 7.0900 5.0429 0 0 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

212

Table D-12 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 4 Caps Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 1 1 1 1 0 0 4 1 4632 4.9652 2.7145 6.6218 4.0852 -48.0399 -47.2214 2 2 1 1 1 0 1 0 4 2 4070 5.0307 2.7026 6.8175 4.0274 -48.0208 -40.8046 2 2 1 1 1 0 0 1 4 3 3816 5.1319 2.6946 7.1463 3.9917 -49.4081 -35.7027 2 2 1 1 0 1 0 1 4 4 2056 5.1817 2.6825 7.3091 3.9393 -37.0000 -24.2370 2 2 1 1 0 1 1 0 4 5 1966 5.0798 2.6906 6.9763 3.9768 -34.3563 -26.1902 2 2 1 1 0 0 1 1 4 6 1736 5.2539 2.6716 7.5545 3.8913 -34.8800 -20.3015 3 2 0 1 1 1 0 1 4 7 924 5.2178 2.7492 7.4194 4.1875 -22.9023 -17.1101 2 2 0 1 1 0 1 1 4 8 873 5.2905 2.7377 7.6684 4.1327 -23.1331 -15.1193 3 2 0 1 1 1 1 0 4 9 840 5.1154 2.7578 7.0842 4.2303 -20.3883 -17.9994 2 2 0 1 0 1 1 1 4 10 642 5.3450 2.7267 7.8719 4.0896 -19.8020 -11.7222 3 2 0 0 1 1 1 1 4 11 239 5.3722 2.7741 7.9653 4.2705 9.4215 5.9779 3 1 1 0 0 1 1 1 4 12 227 5.3332 2.7061 7.8323 4.0119 9.1479 5.3022 2 0 1 0 1 0 1 1 4 13 212 5.2791 2.7170 7.6313 4.0537 8.7150 5.3793 2 0 1 0 1 1 0 1 4 14 196 5.2065 2.7283 7.3835 4.1061 7.8505 5.5274 0 0 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

213

Table D-13 HRI Algorithm results for Expanded IEEE 13 Node Test Feeder, Cap = 200 kVAr, Compensation = 5 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 1 1 1 0 1 1 5 1 735 5.2913 2.7390 7.6708 4.1382 -20.9501 -13.7106 3 2 1 1 0 1 1 1 5 2 559 5.3457 2.7279 7.8744 4.0953 -18.1811 -10.7902 3 2 1 1 1 1 0 1 5 3 767 5.2185 2.7505 7.4217 4.1931 -20.5060 -15.4022 2 2 0 1 1 1 1 1 5 4 386 5.3729 2.7754 7.9679 4.2761 15.1504 5.9912 3 1 1 1 1 1 1 0 5 5 696 5.1161 2.7591 7.0864 4.2359 -18.3078 -16.1682 2 2 Non-harmonic Resonance Values 156.23 4.8562 2.5863 6.3108 3.6104 7.3061 4.4406

214

Table D-14 HRI Algorithm results for Feeder 1358, Cap = 800 kVAr, Compensation = 1 Cap Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 Req'd No. Value NODE 4 NODE 11 NODE 4 NODE 11 NODE 4 NODE 11 110% 120% 0 0 0 0 1 1 1 36.7 0.5000 0.3295 0.8351 2.7156 1.1916 4.9724 2 1 0 0 0 1 0 1 2 30.7 0.5003 0.3252 0.8363 2.5952 1.1941 4.4326 2 1 1 0 0 0 0 1 3 20.6 0.5043 0.2808 0.8575 2.2807 1.2615 3.3996 0 0 0 0 1 0 0 1 4 19.3 0.5105 0.2965 0.8867 2.2205 1.3393 3.1306 0 0 Non-harmonic Resonance Values 17.01 0.5343 0.2496 0.8303 2.2098 1.1664 3.1184

215

Table D-15 HRI Algorithm results for Feeder 1358, Cap = 800 kVAr, Compensation = 2 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 Req'd No. Value NODE 4 NODE 11 NODE 4 NODE 11 NODE 4 NODE 11 110% 120% 0 0 0 1 1 2 1 106 0.5018 0.3343 0.8477 3.3119 1.2815 9.6494 2 2 1 0 0 0 1 2 2 43 0.5058 0.2861 0.8677 2.8162 1.3175 5.5536 2 1 0 0 1 0 1 2 3 43 0.5119 0.3030 0.8953 2.7251 1.3766 5.5021 2 1 0 1 0 0 1 2 4 37 0.5097 0.2981 0.8847 2.7234 1.3473 5.0133 2 1 1 0 0 1 0 2 5 36 0.5061 0.2824 0.8694 2.6938 1.3226 4.9643 2 1 0 0 1 1 0 2 6 31 0.5165 0.2548 0.9211 2.2893 1.4708 4.8366 1 1 Non-harmonic Resonance Values 17.01 0.5343 0.2496 0.8303 2.2098 1.1664 3.1184

216

Table D-16 HRI Algorithm results for Feeder 1358, Cap = 800 kVAr, Compensation = 3 Caps Variable s Driving Reactance (Ohms) Points of Concern Objective 3rd 5th 7th Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 Req'd No. Value NODE 4 NODE 11 NODE 4 NODE 11 NODE 4 NODE 11 110% 120% 1 0 0 1 1 3 1 169 0.5100 0.2900 0.8900 3.4700 1.5400 12.3200 2 2 0 0 1 1 1 3 2 110 0.5138 0.3060 0.9098 3.3270 1.4980 9.6993 2 2 0 1 0 1 1 3 3 109 0.5116 0.3007 0.8990 3.3243 1.4634 9.6663 2 2 1 0 1 0 1 3 4 45 0.5181 0.2580 0.9329 2.8293 1.5473 5.6564 2 1 1 1 0 0 1 3 5 45 0.5158 0.2528 0.9215 2.8270 1.5105 5.6399 2 1 1 1 0 1 0 3 6 38 0.5161 0.2495 0.9234 2.7044 1.5173 5.0437 2 1 0 1 1 0 1 3 7 39 0.5222 0.2702 0.9532 2.7342 1.5930 5.0779 2 1 1 0 1 1 0 3 8 38 0.5181 0.2580 0.9329 2.8293 1.5047 5.0306 2 1 0 1 1 1 0 3 9 33 0.5225 0.2668 0.9548 2.6134 1.5975 4.5288 2 1 Non-harmonic Resonance Values 17.01 0.5343 0.2496 0.8303 2.2098 1.1664 3.1184

217

Table D-17 HRI Algorithm results for Feeder 1358, Cap = 800 kVAr, Compensation = 4 Caps Variable s Driving Reactance (Ohms) Points of Concern 3rd 5th 7th Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 Req'd No. Value NODE 4 NODE 11 NODE 4 NODE 11 NODE 4 NODE 11 110% 120% 1 0 1 1 1 4 1 187 0.5215 0.2603 0.9510 3.5035 1.8619 12.9630 3 2 1 1 0 1 1 4 2 184 0.5204 0.2503 0.9409 3.4935 1.8017 12.8929 3 2 0 1 1 1 1 4 3 116 0.5216 0.2703 0.9710 3.3437 1.7617 9.9600 3 2 1 1 1 0 1 4 4 48 0.5304 0.2202 1.0010 2.8428 1.8317 5.7858 3 1 Non-harmonic Resonance Values 17.01 0.5343 0.2496 0.8303 2.2098 1.1664 3.1184

218

Table D-18 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 2 Caps Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 0 0 0 1 1 0 2 1 44 0.7294 1.2402 0.8780 2.0330 1.0410 5.9845 2 2 0 0 0 0 0 0 1 1 2 2 68 0.7393 1.2443 0.8815 1.9754 1.0656 7.7894 2 1 0 0 0 0 0 1 0 1 2 3 37.9 0.7351 1.2402 0.8829 1.9740 1.0374 5.4493 2 1 0 0 0 0 1 1 0 0 2 4 16 0.7327 1.2457 0.9215 1.6538 1.0097 3.1215 1 1 0 0 0 1 0 0 1 0 2 5 14.4 0.7387 1.2236 0.9172 1.6232 1.0025 2.7646 1 1 0 0 0 1 0 1 0 0 2 6 14.7 0.7385 1.2323 0.9127 1.5328 1.0032 2.8666 1 1 0 0 0 0 1 0 1 0 2 7 15.6 0.7480 1.2232 0.9095 1.5201 1.0623 2.9985 1 1 0 0 0 0 1 0 0 1 2 8 14.4 0.7459 1.2224 0.8989 1.5320 1.0327 2.8466 1 1 0 0 0 1 0 0 0 1 2 9 13.3 0.7545 1.2023 0.8424 1.4995 1.0021 2.7033 1 1 1 1 0 0 0 1 0 0 2 10 6.5 0.7258 1.2236 0.8322 1.4277 0.9406 0.9599 0 0 1 0 0 0 0 0 0 1 2 11 11 0.7549 1.2557 0.9033 1.5945 1.0235 2.1005 1 0 1 0 0 0 0 0 1 0 2 12 12 0.7895 1.2658 0.9124 1.4857 1.0230 2.3510 1 1 1 0 1 0 0 1 0 0 2 13 6 0.7322 1.2033 0.9073 1.5234 0.9686 0.1247 0 0 1 0 0 0 0 1 0 0 2 14 12.6 0.7546 1.2349 0.8565 1.5263 1.0147 2.5614 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

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Table D-19 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 3 Caps, Optimizations 1 - 25 Variable s Driving Reactance (Ohms) Points of Concern Objective 3rd 5th 7th Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 1 1 0 0 0 0 0 1 3 1 204 0.7252 1.2026 0.8734 3.3331 1.1695 13.7457 2 2 1 1 0 0 0 0 1 0 3 2 153 0.7175 1.2229 0.8696 2.3942 1.0854 -11.9847 2 2 1 1 0 0 0 1 0 0 3 3 93 0.7333 1.4206 0.8751 2.7625 1.0803 8.9746 2 2 1 1 0 0 1 0 0 0 3 4 71.6 0.7122 1.2540 0.9353 1.9660 1.2890 7.8829 2 1 0 1 0 0 0 0 1 1 3 5 134 0.7356 1.4931 0.8877 3.3212 1.2254 10.9041 2 2 0 1 0 0 0 1 0 1 3 6 74 0.7220 1.2441 0.9058 1.9629 1.2771 8.1151 2 1 0 1 0 0 0 1 1 0 3 7 72 0.7813 1.2540 0.9353 1.9660 1.2890 7.8829 2 1 0 1 0 0 1 0 0 1 3 8 69 0.7558 1.2320 0.8766 1.9599 1.0596 7.7497 2 1 0 1 0 0 1 1 0 0 3 9 69.6 0.7216 1.4972 0.8735 2.0698 1.0621 7.7715 2 2 0 1 0 1 0 0 0 1 3 10 57 0.7214 1.4084 0.8878 2.0292 1.1341 6.9410 2 2 0 1 0 1 0 0 1 0 3 11 46.9 0.7215 1.2441 0.9073 2.0178 1.1483 6.2001 2 2 0 1 0 1 0 1 0 0 3 12 45.3 0.7846 1.2320 0.8780 2.0147 1.1373 5.9965 2 2 0 1 0 1 1 0 0 0 3 13 46.3 0.7214 1.2540 0.9369 2.0211 1.0255 6.0401 2 2 0 1 1 0 0 0 0 1 3 14 39 0.7231 1.2441 0.9059 1.9648 1.0142 5.4385 2 1 0 1 1 0 0 0 1 0 3 15 39 0.7402 1.2540 0.9354 1.9680 1.0247 5.4707 2 1 0 1 1 0 0 1 0 0 3 16 38.4 0.7620 1.2320 0.8767 1.9619 1.0344 5.4123 2 1 0 1 1 0 1 0 0 0 3 17 21 0.7216 1.6582 0.8958 1.7227 1.0152 3.2785 2 1 1 0 0 0 0 0 1 1 3 18 19 0.7203 1.4590 0.8944 1.7985 1.0142 3.0255 2 1 1 0 0 0 0 1 0 1 3 19 22 0.7317 1.4582 0.8965 1.8643 1.0153 3.3842 2 1 1 0 0 0 0 1 1 0 3 20 16.4 0.7481 1.5445 0.8913 1.4479 1.0125 2.9890 2 1 1 0 0 0 1 0 0 1 3 21 16.2 0.7162 1.4018 0.8800 1.6042 1.0111 2.9821 1 1 1 0 0 0 1 1 0 0 3 22 15.9 0.7225 1.5118 0.8659 1.5569 1.0120 2.8798 1 1 1 0 0 1 0 0 0 1 3 23 16.3 0.7414 1.4960 0.8608 1.5087 1.0953 3.0236 1 1 1 0 0 1 0 0 1 0 3 24 14.2 0.7266 1.4896 0.8785 1.5525 1.0543 2.6236 1 1 1 0 0 1 0 1 0 0 3 25 15.5 0.7213 1.2286 0.8608 1.5245 1.0285 2.9861 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

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Table D-20 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 3 Caps, Optimizations 26 - 30 Variable s Driving Reactance (Ohms) Points of Concern Objective 3rd 5th 7th Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 1 0 0 1 1 0 0 0 3 26 18.4 0.7224 1.2390 0.8799 1.6369 1.1112 3.2403 1 1 1 0 1 0 0 0 0 1 3 27 14.2 0.7229 1.2124 0.8797 1.5164 1.0033 2.7848 1 1 1 0 1 0 0 0 1 0 3 28 15.3 0.7218 1.2235 0.8602 1.5001 1.0037 3.0229 1 1 1 0 1 0 0 1 0 0 3 29 15.8 0.7211 1.2249 0.8503 1.5023 1.0012 3.0622 1 1 1 0 1 0 1 0 0 0 3 30 15.1 0.7219 1.2235 0.8504 1.5024 0.9855 2.9895 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

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Table D-21 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 4 Caps, Optimizations 1 – 25 Variable s Driving Reactance (Ohms) Points of Concern Objective 3rd 5th 7th Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 0 1 1 1 1 0 4 1 502 0.8513 2.9254 0.8985 4.9660 1.0238 21.7129 3 3 0 0 0 1 1 0 1 1 4 2 272334 0.8619 3.1066 2.3306 214.0040 3.5195 465.3331 5 5 0 0 1 0 1 0 1 1 4 3 316 0.8280 1.5487 0.9915 4.1929 1.0235 17.1289 3 2 0 0 0 1 1 1 0 1 4 4 310 0.8211 1.4512 0.9815 3.3307 1.0217 17.3033 2 2 0 1 0 0 1 0 1 1 4 5 278 0.8100 1.3849 0.9816 3.2619 1.0265 16.3384 2 2 1 0 0 0 1 0 1 1 4 6 265 0.8014 1.3948 0.9842 3.1487 1.0235 15.8385 2 2 0 0 1 1 0 0 1 1 4 7 229 0.8202 1.3489 0.9716 1.6205 0.9902 15.1275 1 1 0 1 0 1 0 0 1 1 4 8 225 0.8120 1.3977 0.8461 1.5692 0.9995 14.6510 1 1 0 1 0 0 0 1 1 1 4 9 138 0.7985 1.5075 0.9058 3.1234 1.0227 10.7092 2 2 1 0 0 0 0 1 1 1 4 10 139 0.8374 1.4931 0.8877 3.3212 1.0225 11.0414 2 2 0 0 1 0 0 1 1 1 4 11 129 0.8267 1.4598 0.9353 3.3956 1.0289 10.5983 2 2 0 1 1 0 0 0 1 1 4 12 95 0.7222 1.2009 0.8450 1.5930 0.9828 9.4193 1 1 0 0 0 1 0 1 1 1 4 13 88.8 0.8204 1.4206 0.8751 2.7625 1.0803 8.7511 2 2 1 0 1 0 0 0 1 1 4 14 88 0.7100 1.2088 0.8489 1.5020 0.9892 9.1025 1 1 0 0 1 0 1 1 0 1 4 15 90 0.7299 1.2199 0.8538 1.5109 0.9433 9.2025 1 1 0 0 1 0 1 1 1 0 4 16 107.99 0.8204 1.4452 0.9755 3.0625 1.0352 9.5121 2 2 1 0 0 0 1 1 1 0 4 17 104 0.8111 1.4244 0.9865 3.0012 1.0236 9.3912 2 2 0 1 0 0 1 1 0 1 4 18 85 0.8009 1.2320 0.8614 1.9599 1.0232 8.7275 2 1 0 1 0 0 1 1 1 0 4 19 100 0.7720 1.2764 0.9440 3.1809 1.0117 9.2069 2 2 0 0 0 0 1 1 1 1 4 20 83 0.7742 1.2203 0.9312 3.2035 1.0125 8.2207 2 2 1 1 0 0 0 0 1 1 4 21 84 0.7629 1.2292 0.8664 2.4439 0.9899 8.6428 2 2 1 0 0 0 1 1 0 1 4 22 83 0.7900 1.4296 0.8949 3.3341 1.0217 8.1437 2 2 0 0 1 1 0 1 0 1 4 23 78 0.7238 1.2290 0.8531 1.4906 0.9897 8.4892 1 1 0 0 1 1 0 1 1 0 4 24 85 0.7671 1.2133 0.8569 2.5461 0.9901 8.6598 2 2 0 1 0 1 0 1 0 1 4 25 75 0.7598 1.2198 0.8460 1.8939 0.9918 8.1909 2 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

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Table D-22 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 4 Caps, Optimizations 26 - 50 Variable s Driving Reactance (Ohms) Points of Concern Objective 3rd 5th 7th Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 1 0 1 0 1 1 0 4 26 80 0.7901 1.4299 0.8800 2.8734 1.0556 8.2511 2 2 1 0 0 1 0 1 0 1 4 27 70 0.7227 1.2299 0.8660 1.5322 0.9478 7.9929 1 1 1 0 0 1 0 1 1 0 4 28 82 0.7578 1.2015 0.8561 2.4351 0.9890 8.5521 2 2 0 1 1 0 0 1 0 1 4 29 46 0.7345 1.2276 0.8539 1.4990 1.0002 6.4239 1 1 0 1 1 0 0 1 1 0 4 30 50 0.7520 1.4206 0.8751 2.9763 1.1908 5.9977 2 2 1 0 1 0 0 1 0 1 4 31 43.39 0.7917 1.3957 0.8602 2.8482 1.0845 5.5507 2 2 0 1 0 1 0 0 1 1 4 32 50 0.7209 1.2395 0.8602 1.5121 1.0279 6.5971 1 1 0 1 0 0 1 1 0 1 4 33 85 0.8009 1.2320 0.8614 1.9599 1.0232 8.7275 2 1 0 1 0 1 0 1 0 1 4 34 75 0.7598 1.2198 0.8460 1.8939 0.9918 8.1909 2 1 0 0 1 1 1 0 1 0 4 35 24 0.7438 1.4084 0.8352 2.0292 0.9875 4.0133 2 2 0 0 1 1 1 1 0 0 4 36 24.44 0.7284 1.2287 0.8502 1.4809 1.0032 4.2920 1 1 1 0 0 1 1 0 1 0 4 37 24 0.7320 1.2287 0.8502 1.9937 0.9905 3.9902 2 2 1 0 0 1 1 1 0 0 4 38 24.22 0.7327 1.2399 0.8510 1.9836 0.9915 4.0233 2 2 1 0 0 1 1 0 0 1 4 39 21.45 0.7238 1.2284 0.8535 2.1301 1.0023 3.5899 2 2 0 1 0 1 1 0 0 1 4 40 21.4 0.7653 1.2571 0.8874 1.9924 0.9786 3.6781 2 2 0 1 0 1 1 0 1 0 4 41 23.3 0.7291 1.2349 0.8735 2.0018 0.9948 3.8827 2 2 0 1 0 1 1 1 0 0 4 42 23.6 0.7748 1.2238 0.8749 1.9737 0.9985 3.9408 2 1 0 0 1 1 1 0 0 1 4 43 22.3 0.7352 1.2399 0.8655 1.9836 0.9899 3.7928 2 2 1 1 0 0 1 0 0 1 4 44 15.45 0.7227 1.2288 0.8534 1.7738 0.9790 2.8782 2 1 1 0 1 0 1 0 1 0 4 45 16.9 0.7227 1.2286 0.8530 1.7737 0.6848 3.1011 2 1 0 1 1 0 1 0 1 0 4 46 17 0.7231 1.2287 0.8534 1.7737 0.9783 3.1439 2 1 1 0 1 0 1 1 0 0 4 47 17.24 0.7229 1.2283 0.8535 1.7727 0.9905 3.1884 2 1 0 1 1 0 1 1 0 0 4 48 17.1 0.7232 1.2288 0.8532 1.7873 0.9891 3.1308 2 1 1 1 0 0 1 0 1 0 4 49 16.65 0.7227 1.2283 0.8534 1.7726 0.9985 3.0178 2 1 1 1 0 0 1 1 0 0 4 50 16.9 0.7232 1.2282 0.8535 1.7927 0.9891 3.0922 2 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

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Table D-23 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1800 kVAr, Compensation = 4 Caps, Optimizations 51 - 67 Variable s Driving Reactance (Ohms) Points of Concern Objective 3rd 5th 7th Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 1 1 1 0 1 0 0 4 51 16.25 0.7227 1.2292 0.8533 1.7826 0.9756 2.9978 2 1 1 1 0 1 0 1 0 0 4 52 15.87 0.7229 1.2291 0.8534 1.7738 0.9836 2.9184 2 1 1 0 1 1 0 1 0 0 4 53 16.2 0.7231 1.2287 0.8528 1.8733 0.9748 2.9422 2 1 1 0 1 0 1 0 0 1 4 54 15.7 0.7227 1.2293 0.8530 1.7722 0.9514 2.9248 2 1 0 1 1 1 0 0 1 0 4 55 15.91 0.7221 1.2347 0.8544 1.6949 0.9488 2.8966 1 1 0 1 1 1 0 1 0 0 4 56 16.25 0.7227 1.2292 0.8533 1.7826 0.9756 2.9978 2 1 1 0 1 1 0 0 0 1 4 57 14.73 0.7223 1.2202 0.8562 1.7103 0.9615 2.8492 1 1 1 0 1 1 0 1 0 0 4 58 16.2 0.7231 1.2287 0.8528 1.8733 0.9748 2.9422 2 1 1 1 0 1 0 0 0 1 4 59 14.42 0.7200 1.2248 0.8620 1.7301 0.9503 3.0220 1 1 1 1 0 1 0 1 0 0 4 60 15.87 0.7229 1.2291 0.8534 1.6665 0.9836 2.9184 1 1 1 1 1 0 0 0 1 0 4 61 12.99 0.7258 1.2237 0.8510 1.6481 0.9550 2.4099 1 1 1 1 1 0 0 0 0 1 4 62 12.14 0.7230 1.2203 0.8521 1.5032 0.9445 2.3901 1 1 1 1 1 0 0 1 0 0 4 63 13.21 0.7300 1.2203 0.8567 1.4919 0.9615 2.3903 1 1 0 1 1 1 1 0 0 0 4 64 9.48 0.7222 1.2295 0.8513 1.5023 0.9606 1.7840 0 0 1 1 0 1 1 0 0 0 4 65 9.43 0.7213 1.2203 0.8577 1.4913 0.9615 1.7151 0 0 1 1 1 0 1 0 0 0 4 66 7.929 0.7203 1.2249 0.8651 1.4799 0.9516 1.7154 0 0 1 1 1 1 0 0 0 0 4 67 7.59 0.7251 1.2203 0.8554 1.4790 0.9410 1.6032 0 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

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Table D-24 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 2 Caps, 6 locations of concern Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 0 0 0 1 1 0 2 1 16.35 0.7210 1.2326 0.8579 1.4778 0.9420 3.2674 1 1 0 0 0 0 0 1 0 1 2 2 14.82 0.7213 1.2305 0.8578 1.6877 0.9421 3.0109 1 1 0 0 0 0 0 0 1 1 2 3 15.52 0.7243 1.2242 0.8563 1.4887 0.9454 3.1232 1 1 0 0 0 0 1 1 0 0 2 4 10.93 0.7217 1.2321 0.8548 1.4987 0.9425 2.2497 1 1 0 0 0 1 0 1 0 0 2 5 10.44 0.7178 1.2274 0.8573 1.6991 0.9429 2.1287 1 1 0 0 0 1 0 0 1 0 2 6 10.144 0.7176 1.2342 0.8549 1.5004 0.9449 2.1353 1 1 0 0 0 0 1 0 1 0 2 7 10.61 0.7215 1.2282 0.8589 1.5436 1.0465 2.1297 1 1 0 0 0 1 0 0 0 1 2 8 9.47 0.7178 1.2345 0.8562 1.5110 0.9793 2.1229 1 0 0 0 1 0 0 1 0 0 2 9 9.69 0.7177 1.2258 0.8584 1.5015 0.9496 2.0249 1 0 1 0 0 0 0 1 0 0 2 10 9.69 0.7202 1.2229 0.8573 1.5300 0.9397 2.0250 1 0 0 0 0 0 1 0 0 1 2 11 9.89 0.7234 1.2222 0.8523 1.5083 0.9423 2.0299 1 0 1 0 0 0 0 0 0 1 2 12 8.84 0.7273 1.2457 0.8578 1.4907 0.9450 2.0101 1 0 1 0 0 0 0 0 1 0 2 13 9.43 0.7217 1.2269 0.8531 1.4985 0.9949 2.0192 1 0 0 0 1 0 0 0 1 0 2 14 9.44 0.7243 1.2240 0.8594 1.4900 0.9588 2.0192 1 0 0 1 0 0 0 1 0 0 2 15 9.63 0.7175 1.2315 0.8560 1.4994 0.9445 2.0125 1 0 0 1 0 0 0 0 1 0 2 16 9.38 0.7229 1.2317 0.8575 1.4859 0.9411 2.0120 1 0 0 0 1 0 0 0 0 1 2 17 8.83 0.7245 1.2256 0.8585 1.4862 0.9458 2.0689 1 0 0 1 0 0 0 0 0 1 2 18 8.79 0.7226 1.2251 0.8523 1.4816 0.9421 2.0734 1 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

225

Table D-25 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 3 Caps, 3 locations of concern, Optimizations 1 – 25 Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 0 0 0 0 0 1 1 1 3 1 39.4 0.7230 1.2338 0.8994 1.5875 1.0873 5.8483 1 1 0 0 0 0 1 0 1 1 3 2 18.62 0.7204 1.2329 0.8672 1.4999 0.9489 3.6999 1 1 0 0 0 1 0 0 1 1 3 3 17.42 0.7191 1.2283 0.8562 1.5029 0.9828 3.5507 1 1 0 0 0 0 1 1 1 0 3 4 19.7 0.7243 1.2293 0.8684 1.5203 0.9794 3.8763 1 1 0 0 0 1 0 1 1 0 3 5 18.43 0.7245 1.2338 0.8685 1.5095 0.9589 3.7275 1 1 0 0 0 0 1 1 0 1 3 6 17.65 0.7214 1.2276 0.8785 1.4906 1.0040 3.4297 1 1 0 0 1 0 0 1 1 0 3 7 16.55 0.7239 1.2248 0.8678 1.5094 0.9993 3.6042 1 1 1 0 0 0 0 1 1 0 3 8 16.52 0.7187 1.2312 0.8890 1.5290 0.9678 3.3979 1 1 0 1 0 0 0 1 1 0 3 9 16.39 0.7211 1.2291 0.8658 1.5291 1.0094 3.3498 1 1 0 0 1 0 0 0 1 1 3 10 15.72 0.7221 1.2300 0.8520 1.4931 0.9636 3.2983 1 1 1 0 0 0 0 0 1 1 3 11 15.67 0.7238 1.2239 0.8560 1.4820 0.9574 3.2939 1 1 0 1 0 0 0 0 1 1 3 12 15.5 0.7243 1.2299 0.8541 1.4980 0.9748 3.2419 1 1 0 0 1 0 0 1 0 1 3 13 15 0.7236 1.2277 0.8541 1.4984 1.0093 3.4200 1 1 0 0 0 1 0 1 0 1 3 14 16.56 0.7211 1.2333 0.8558 1.4984 0.9938 3.4297 1 1 1 0 0 0 0 1 0 1 3 15 14.97 0.7219 1.2270 0.8527 1.4984 0.9502 3.1976 1 1 0 1 0 0 0 1 0 1 3 16 14.85 0.7232 1.2286 0.8567 1.4880 0.9988 3.2013 1 1 0 0 0 1 1 1 0 0 3 17 12.26 0.7220 1.2298 0.8537 1.5109 1.0103 2.7849 1 1 0 0 0 1 1 0 1 0 3 18 11.87 0.7212 1.2342 0.8572 1.5209 0.9809 2.6858 1 1 0 0 0 1 1 0 0 1 3 19 11 0.7246 1.2244 0.8559 1.4890 0.9984 2.4983 1 1 0 0 1 0 1 0 1 0 3 20 10.72 0.7238 1.2237 0.8571 1.4797 0.9642 2.4092 1 1 1 0 0 0 1 1 0 0 3 21 11.04 0.7238 1.2247 0.8569 1.4863 0.9788 2.4996 1 1 0 0 1 0 1 1 0 0 3 22 11.04 0.7239 1.2247 0.8570 1.4863 0.9788 2.4996 1 1 0 1 0 0 1 1 0 0 3 23 10.95 0.7213 1.2247 0.8535 1.4847 0.9700 2.3909 1 1 1 0 0 0 1 0 1 0 3 24 10.72 0.7211 1.2254 0.8562 1.4895 0.9805 2.2846 1 1 0 1 0 0 1 0 1 0 3 25 10.63 0.7215 1.2268 0.8542 1.4898 0.9947 2.2749 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

226

Table D-26 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 2 Caps, 3 locations of concern, Optimizations 26 - 52 Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE NODE NODE NODE NODE NODE Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value 692A 684B 692A 684B 692A 684B 110% 120% 1 0 0 1 0 1 0 0 3 26 10.56 0.7219 1.2284 0.8591 1.4887 0.9956 2.2945 1 1 0 0 1 1 0 1 0 0 3 27 10.55 0.7215 1.2245 0.8563 1.4898 0.9870 2.2186 1 1 0 1 0 1 0 1 0 0 3 28 10.46 0.7215 1.2263 0.8543 1.4952 0.9899 2.1986 1 1 1 0 0 1 0 0 1 0 3 29 10.25 0.7218 1.2251 0.8577 1.4964 0.9900 2.1982 1 1 0 0 1 1 0 0 1 0 3 30 10.25 0.7216 1.2289 0.8582 1.4982 0.9816 2.0990 1 0 0 1 0 1 0 0 1 0 3 31 10.16 0.7221 1.2254 0.8492 1.4990 0.9899 2.0746 1 0 0 0 1 0 1 0 0 1 3 32 9.99 0.7218 1.2267 0.8466 1.4898 0.9895 2.0413 1 0 1 0 0 0 1 0 0 1 3 33 9.98 0.7225 1.2238 0.8501 1.4986 0.9899 2.0054 1 0 0 0 1 1 0 0 0 1 3 34 9.58 0.7212 1.2352 0.8512 1.4815 0.9520 2.0155 1 0 1 0 0 1 0 0 0 1 3 35 9.57 0.7215 1.2215 0.8575 1.4887 0.9542 2.0165 1 0 1 0 1 0 0 0 1 0 3 36 9.52 0.7223 1.2321 0.8574 1.4896 0.9437 2.2032 1 1 0 1 0 1 0 0 0 1 3 37 9.49 0.7212 1.2262 0.8592 1.4990 0.9603 2.1065 1 0 0 1 1 0 0 0 1 0 3 38 9.46 0.7219 1.2205 0.8549 1.4989 0.9580 2.0095 1 0 1 1 0 0 0 0 1 0 3 39 9.45 0.7186 1.2281 0.8564 1.4991 0.9789 2.0904 1 0 1 0 1 0 0 0 0 1 3 40 8.93 0.7230 1.2320 0.8519 1.4888 0.9455 2.0128 1 0 0 1 1 0 0 0 0 1 3 41 8.87 0.7213 1.2218 0.8542 1.4846 0.9482 2.0042 1 0 1 1 0 0 0 0 0 1 3 42 8.86 0.7190 1.2264 0.8572 1.4814 0.9453 2.0598 1 0 1 0 0 1 1 0 0 0 3 43 7.82 0.7178 1.2277 0.8514 1.4891 0.9454 1.6755 0 0 0 0 1 1 1 0 0 0 3 44 7.81 0.7215 1.2214 0.8518 1.4813 0.9425 1.6827 0 0 0 1 0 1 1 0 0 0 3 45 7.74 0.7222 1.2215 0.8520 1.4820 0.9405 1.6784 0 0 1 0 1 0 1 0 0 0 3 46 7.73 0.7198 1.2344 0.8550 1.4839 0.9495 1.6851 0 0 0 1 1 0 1 0 0 0 3 47 7.18 0.7142 1.2242 0.8543 1.4871 0.9428 1.6681 0 0 1 1 0 0 1 0 0 0 3 48 7.16 0.7219 1.2261 0.8521 1.4897 0.9457 1.6726 0 0 1 0 1 1 0 0 0 0 3 49 6.99 0.7186 1.2252 0.8512 1.4810 0.9425 1.6740 0 0 0 1 1 1 0 0 0 0 3 50 6.94 0.7187 1.2245 0.8541 1.4807 0.9422 1.6740 0 0 1 1 0 1 0 0 0 0 3 51 6.94 0.7199 1.2299 0.8536 1.4779 0.9445 1.6705 0 0 1 1 1 0 0 0 0 0 3 52 6.55 0.7239 1.2314 0.8512 1.4874 0.9414 1.6799 0 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

227

Table D-27 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 4 Caps, 6 locations of concern, Optimizations 1 – 25 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 110% 120% 0 0 0 0 1 1 1 1 4 1 57.38 0.7240 1.2324 1.0094 1.8983 1.1240 6.9094 2 1 0 0 0 1 0 1 1 1 4 2 49.96 0.7177 1.2282 0.9838 1.7046 1.1246 6.2076 1 1 0 0 1 0 0 1 1 1 4 3 40.63 0.7198 1.2284 0.9378 1.6040 1.1256 5.6753 1 1 1 0 0 0 0 1 1 1 4 4 40.23 0.7222 1.2296 0.9474 1.5989 1.1146 5.5939 1 1 0 1 0 0 0 1 1 1 4 5 39.73 0.7190 1.2297 0.9769 1.6094 1.1588 5.7890 1 1 0 0 0 1 1 0 1 1 4 6 22.94 0.7233 1.2305 0.8531 1.4885 0.9577 4.2099 1 1 0 0 0 1 1 1 1 0 4 7 23.73 0.7195 1.2234 0.9094 1.5093 1.0928 4.3099 1 1 0 0 0 1 1 1 0 1 4 8 20.94 0.7201 1.2280 0.8553 1.5004 0.9768 4.0939 1 1 0 0 1 0 1 0 1 1 4 9 19.02 0.7229 1.2323 0.8530 1.4900 0.9685 3.7689 1 1 1 0 0 0 1 0 1 1 4 10 18.94 0.7241 1.2263 0.8585 1.5009 0.9560 3.7049 1 1 0 0 1 1 0 1 1 0 4 11 18.84 0.7196 1.2313 0.9472 1.5030 1.0594 3.3927 1 1 0 1 0 0 1 0 1 1 4 12 18.78 0.7190 1.2264 0.8520 1.7993 1.0119 3.5988 2 1 0 0 1 0 1 1 1 0 4 13 20.09 0.7215 1.2218 0.8591 1.7895 1.0425 3.6053 2 1 1 0 0 0 1 1 1 0 4 14 20.5 0.7242 1.2325 0.8876 1.5039 1.0493 3.8720 1 1 0 1 0 0 1 1 1 0 4 15 19.84 0.7241 1.2251 0.8894 1.5193 1.0490 3.8017 1 1 1 0 0 1 0 1 1 0 4 16 18.76 0.7188 1.2272 0.9382 1.4921 1.0340 3.5937 1 1 0 1 0 1 0 1 1 0 4 17 18.55 0.7182 1.2307 0.8524 1.4860 1.0298 3.7295 1 1 0 0 1 0 1 1 0 1 4 18 18.04 0.7184 1.2345 0.8750 1.5390 1.1931 3.4973 1 1 1 0 0 0 1 1 0 1 4 19 17.94 0.7204 1.2331 0.8582 1.5038 1.0240 3.6370 1 1 0 1 0 0 1 1 0 1 4 20 17.8 0.7240 1.2253 0.8544 1.4921 1.0128 3.5935 1 1 0 0 1 1 0 0 1 1 4 21 17.87 0.7205 1.2276 0.8666 1.5040 1.0245 3.4296 1 1 1 0 0 1 0 0 1 1 4 22 17.77 0.7230 1.2262 0.8884 1.5383 1.2040 3.4298 1 1 0 1 0 1 0 0 1 1 4 23 17.61 0.7213 1.2339 0.8776 1.5040 1.0239 3.5026 1 1 0 0 1 1 0 1 0 1 4 24 16.98 0.7235 1.2317 0.9373 1.4993 1.0498 3.2398 1 1 1 0 0 1 0 1 0 1 4 25 16.94 0.7216 1.2332 0.9297 1.5190 1.0295 3.2929 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

228

Table D-28 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 4 Caps, 6 locations of concern, Optimizations 26 – 50 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 110% 120% 0 1 0 1 0 1 0 1 4 26 16.77 0.7186 1.2269 0.8573 1.4923 0.9938 3.4693 1 1 1 0 1 0 0 0 1 1 4 27 15.95 0.7209 1.2239 0.8738 1.4926 1.0294 3.2938 1 1 0 1 1 0 0 0 1 1 4 28 15.86 0.7182 1.2240 0.8674 1.4983 1.0329 3.3282 1 1 1 0 1 0 0 1 1 0 4 29 16.78 0.7231 1.2327 0.8629 1.4924 1.0303 3.4020 1 1 0 1 1 0 0 1 1 0 4 30 16.66 0.7239 1.2244 0.8794 1.5104 1.0210 3.3093 1 1 1 1 0 0 0 0 1 1 4 31 15.76 0.7195 1.2308 0.8581 1.4820 1.2040 3.2389 1 1 1 1 0 0 0 1 1 0 4 32 16.59 0.7239 1.2244 0.8794 1.5104 1.0210 3.3093 1 1 1 0 1 0 0 1 0 1 4 33 15.22 0.7215 1.2248 0.8426 1.4985 0.9983 3.2905 1 1 0 1 1 0 0 1 0 1 4 34 15.15 0.7255 1.2239 0.8453 1.4909 0.1005 3.2986 1 1 1 1 0 0 0 1 0 1 4 35 15.07 0.7218 1.2248 0.8578 1.4980 0.9981 3.2975 1 1 1 0 0 1 1 1 0 0 4 36 12.48 0.7214 1.2295 0.8674 1.4909 1.0937 2.8486 1 1 0 0 1 1 1 1 0 0 4 37 12.48 0.7216 1.2295 0.8676 1.4909 1.0938 2.8486 1 1 0 1 0 1 1 1 0 0 4 38 12.33 0.7218 1.2251 0.8655 1.4993 1.0869 2.8342 1 1 0 0 1 1 1 0 1 0 4 39 12.08 0.7189 1.2274 0.8558 1.5029 1.0383 2.7619 1 1 1 0 0 1 1 0 1 0 4 40 12.08 0.7203 1.2299 0.8575 1.5059 1.0403 2.7675 1 1 0 1 0 1 1 0 1 0 4 41 11.94 0.7233 1.2237 0.8551 1.4991 1.0030 2.7684 1 1 0 0 1 1 1 0 0 1 4 42 11.2 0.7214 1.2252 0.8642 1.4899 1.0200 2.7675 1 1 1 0 0 1 1 0 0 1 4 43 11.19 0.7213 1.2295 0.8677 1.4927 1.0003 2.4893 1 1 0 1 1 0 1 1 0 0 4 44 11.11 0.7252 1.2225 0.8597 1.5121 1.0122 2.4926 1 1 1 1 0 0 1 1 0 0 4 45 11.09 0.7225 1.2215 0.8565 1.5039 1.0251 2.4863 1 1 0 1 0 1 1 0 0 1 4 46 11.07 0.7184 1.2238 0.8520 1.5039 1.0238 2.4839 1 1 1 0 1 0 1 0 1 0 4 47 10.86 0.7184 1.2349 0.8652 1.4808 1.0190 2.2985 1 1 0 1 1 0 1 0 1 0 4 48 10.79 0.7246 1.2272 0.8657 1.4913 1.0104 2.4903 1 1 1 1 0 0 1 0 1 0 4 49 10.76 0.7200 1.2285 0.8726 1.4990 1.0025 2.4863 1 1 1 0 1 1 0 1 0 0 4 50 10.72 0.7216 1.2268 0.8948 1.5037 1.0320 2.4104 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

229

Table D-29 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 4 Caps, 6 locations of concern, Optimizations 51 – 69 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 110% 120% 0 1 1 1 0 1 0 0 4 51 10.64 0.7195 1.2291 0.8894 1.4904 0.9917 2.2948 1 1 1 1 0 1 0 1 0 0 4 52 10.62 0.7202 1.2249 0.8586 1.4990 0.9862 2.2830 1 1 1 0 1 1 0 0 1 0 4 53 10.41 0.7228 1.2241 0.8564 1.4927 0.9831 2.4903 1 1 1 1 1 0 0 0 1 0 4 54 10.34 0.7230 1.2258 0.8525 1.4925 0.9937 2.3926 1 1 1 1 0 1 0 0 1 0 4 55 10.31 0.7200 1.2250 0.8463 1.4987 0.9982 2.3849 1 1 1 0 1 0 1 0 0 1 4 56 10.12 0.7200 1.2255 0.8678 1.4986 0.9898 2.3748 1 1 0 1 1 0 1 0 0 1 4 57 10.06 0.7225 1.2325 0.8542 1.4996 0.9703 2.3088 1 1 1 1 0 0 1 0 0 1 4 58 10.03 0.7215 1.2349 0.8510 1.5074 0.9538 2.2931 1 1 1 1 1 0 0 1 0 0 4 59 9.83 0.7232 1.2459 0.8520 1.4804 0.9680 2.2508 1 1 1 0 1 1 0 0 0 1 4 60 9.73 0.7223 1.2254 0.8549 1.4917 0.9493 2.1040 1 0 0 1 1 1 0 0 0 1 4 61 9.66 0.7223 1.2321 0.8539 1.4897 0.9448 2.0394 1 0 1 1 0 1 0 0 0 1 4 62 9.63 0.7240 1.2339 0.8540 1.4816 0.9837 2.0193 1 0 1 1 1 0 0 0 1 0 4 63 9.57 0.7229 1.2308 0.8512 1.5000 0.9455 2.0139 1 0 1 1 1 0 0 0 0 1 4 64 8.97 0.7212 1.2297 0.8503 1.4980 0.9436 2.0158 1 0 1 0 1 1 1 0 0 0 4 65 7.93 0.7239 1.2306 0.8502 1.4848 0.9568 2.0008 1 0 0 1 1 1 1 0 0 0 4 66 7.85 0.7238 1.2459 0.8547 1.4870 0.9633 1.7503 0 0 1 1 0 1 1 0 0 0 4 67 7.84 0.7191 1.2234 0.8546 1.4793 0.9462 1.6806 0 0 1 1 0 1 0 1 0 0 4 68 7.25 0.7238 1.2314 0.8572 1.4801 0.9501 1.8456 0 0 1 1 1 1 0 0 0 0 4 69 7.02 0.7211 1.2201 0.8575 1.4803 0.9480 1.6380 0 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

230

Table D-30 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 5 Caps, 6 locations of concern, Optimizations 1 - 25 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 110% 120% 0 0 0 1 1 1 1 1 5 1 91.62 0.7227 1.2284 0.8653 2.0396 0.9948 7.9284 2 2 0 0 1 0 1 1 1 1 5 2 62.3 0.7176 1.2331 0.8845 1.7540 0.9809 6.5828 2 1 1 0 0 0 1 1 1 1 5 3 60.77 0.7205 1.2347 0.8852 1.7403 0.9871 6.4920 1 1 0 1 0 0 1 1 1 1 5 4 60.67 0.7187 1.2307 0.8845 1.7303 0.9986 6.4973 1 1 0 1 0 1 0 1 1 1 5 5 53.98 0.7197 1.2232 0.9032 1.6393 0.9850 6.1903 1 1 0 0 1 1 0 1 1 1 5 6 55.25 0.7199 1.2328 0.8929 1.6590 0.9899 6.2903 1 1 1 0 0 1 0 1 1 1 5 7 53.53 0.7197 1.2333 0.8956 1.6481 0.9782 6.2019 1 1 0 1 1 0 0 1 1 1 5 8 42.75 0.7242 1.2283 0.8867 1.5847 0.9786 5.6385 1 1 1 0 1 0 0 1 1 1 5 9 42.45 0.7184 1.2347 0.8813 1.5672 0.9592 5.6584 1 1 1 1 0 0 0 1 1 1 5 10 41.79 0.7234 1.2326 0.8795 1.5500 1.0024 5.5735 1 1 0 0 1 1 1 0 1 1 5 11 23.71 0.7205 1.2265 0.8546 1.7095 0.9863 4.1504 1 1 0 0 1 1 1 1 1 0 5 12 24.79 0.7230 1.2324 0.8520 1.5050 0.9686 4.3032 1 1 1 0 0 1 1 0 1 1 5 13 23.34 0.7175 1.2268 0.8589 1.7934 0.9988 4.3049 2 1 1 0 0 1 1 1 1 0 5 14 24.6 0.7180 1.2315 0.8634 1.5094 0.9600 4.2986 1 1 0 1 0 1 1 0 1 1 5 15 23.31 0.7246 1.2332 0.8563 1.7737 0.9755 4.2050 2 1 0 1 0 1 1 1 1 0 5 16 24.36 0.7217 1.2277 0.8587 1.5110 0.9897 4.1204 1 1 0 0 1 1 1 1 0 1 5 17 22.16 0.7183 1.2269 0.8534 1.5939 0.9923 4.0409 1 1 1 0 0 1 1 1 0 1 5 18 21.87 0.7227 1.2278 0.8508 1.5034 0.9913 3.9922 1 1 0 1 0 1 1 1 0 1 5 19 21.86 0.7189 1.2326 0.8533 1.4825 0.9679 3.9475 1 1 1 1 0 0 1 0 1 1 5 20 19.8 0.7217 1.2281 0.8527 1.4982 0.9759 3.8039 1 1 0 1 1 0 1 0 1 1 5 21 19.76 0.7215 1.2238 0.8582 1.4790 0.9894 3.6928 1 1 0 1 1 0 1 1 1 0 5 22 20.56 0.7194 1.2238 0.8557 1.4868 0.9653 3.8587 1 1 1 0 1 0 1 0 1 1 5 23 19.68 0.7245 1.2327 0.8587 1.4900 0.9683 3.6893 1 1 1 0 1 0 1 1 1 0 5 24 20.64 0.7215 1.2270 0.8584 1.4917 0.9452 3.7488 1 1 1 1 0 0 1 1 1 0 5 25 20.43 0.7185 1.2262 0.8509 1.4836 0.9635 3.8594 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

231

Table D-31 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 5 Caps, 6 locations of concern, Optimizations 26 - 50 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 110% 120% 1 0 1 1 0 1 1 0 5 26 19.42 0.7222 1.2256 0.8521 1.4799 0.9685 3.6940 1 1 0 1 1 1 0 1 1 0 5 27 19.4 0.7176 1.2288 0.8527 1.4907 0.9895 3.8407 1 1 1 1 0 1 0 1 1 0 5 28 19.02 0.7232 1.2244 0.8640 1.6554 0.9983 3.6705 1 1 0 1 1 1 0 0 1 1 5 29 18.74 0.7239 1.2286 0.8540 1.6039 0.9799 3.6098 1 1 0 1 1 0 1 1 0 1 5 30 18.65 0.7181 1.2330 0.9038 1.6309 0.9760 3.6029 1 1 1 0 1 1 0 0 1 1 5 31 18.59 0.7224 1.2300 0.8583 1.5039 0.9685 3.6741 1 1 1 0 1 0 1 1 0 1 5 32 18.59 0.7198 1.2252 0.8520 1.4803 0.9457 3.6788 1 1 1 1 0 0 1 1 0 1 5 33 18.423 0.7240 1.2325 0.8589 1.4928 0.9936 3.7048 1 1 1 1 0 1 0 0 1 1 5 34 18.42 0.7225 1.2333 0.8586 1.5249 0.9911 3.7039 1 1 0 1 1 1 0 1 0 1 5 35 17.78 0.7190 1.2320 0.8558 1.4799 0.9657 3.6509 1 1 1 0 1 1 0 1 0 1 5 36 17.65 0.7181 1.2294 0.8580 1.4844 0.9653 3.5096 1 1 1 1 0 1 0 1 0 1 5 37 17.51 0.7180 1.2289 0.9035 1.7499 0.9645 3.4029 2 1 1 1 1 0 0 1 1 0 5 38 17.49 0.7183 1.2287 0.8579 1.5104 0.9767 3.6029 1 1 1 1 1 0 0 0 1 1 5 39 16.33 0.7220 1.2289 0.8583 1.5209 0.9604 3.4927 1 1 1 1 1 0 0 1 0 1 5 40 15.59 0.7244 1.2298 0.8837 1.4893 0.9899 3.3045 1 1 1 0 1 1 1 1 0 0 5 41 12.83 0.7235 1.2320 0.8583 1.4904 0.9749 2.8559 1 1 0 1 1 1 1 1 0 0 5 42 12.74 0.7218 1.2262 0.8516 1.4915 0.9785 2.6895 1 1 1 1 0 1 1 1 0 0 5 43 12.68 0.7231 1.2258 0.8513 1.4809 0.9586 2.8304 1 1 1 0 1 1 1 0 1 0 5 44 12.42 0.7238 1.2338 0.8571 1.4900 0.9698 2.6904 1 1 0 1 1 1 1 0 1 0 5 45 12.35 0.7205 1.2234 0.8534 1.4870 0.9705 2.7940 1 1 1 1 0 1 1 0 1 0 5 46 12.29 0.7229 1.2268 0.8544 1.4904 0.9648 2.7985 1 1 1 0 1 1 1 0 0 1 5 47 11.52 0.7180 1.2247 0.8579 1.4847 0.9759 2.5837 1 1 0 1 1 1 1 0 0 1 5 48 11.47 0.7205 1.2304 0.8552 1.4861 0.9690 2.6300 1 1 1 1 0 1 1 0 0 1 5 49 11.39 0.7231 1.2239 0.8527 1.4863 0.9769 2.5937 1 1 1 1 1 0 1 1 0 0 5 50 11.33 0.7182 1.2248 0.8521 1.4870 0.9496 2.6674 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

232

Table D-32 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr, Compensation = 5 Caps, 6 locations of concern, Optimizations 51 - 56 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 110% 120% 1 1 1 0 1 0 1 0 5 51 11.02 0.7231 1.2297 0.8519 1.4835 0.9731 2.4864 1 1 1 1 1 1 0 1 0 0 5 52 10.91 0.7242 1.2231 0.8528 1.4806 0.9583 2.2094 1 1 1 1 1 1 0 0 1 0 5 53 10.63 0.7201 1.2310 0.8512 1.4875 0.9687 2.3902 1 1 1 1 1 0 1 0 0 1 5 54 10.27 0.7219 1.2285 0.8564 1.4903 0.9674 2.3983 1 1 1 1 1 1 0 0 0 1 5 55 9.89 0.7234 1.2340 0.8561 1.4894 0.9657 2.1940 1 1 1 1 1 1 1 0 0 0 5 56 7.99 0.7245 1.2293 0.8536 1.4872 0.9503 1.8460 0 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

233

Table D-33 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 2 Caps, 12 locations of concern, Phase A Reactances Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) 105% 110% 0 0 0 0 0 1 1 0 2 1 40.89 0.7210 1.2326 0.8579 1.4778 0.9420 3.2674 1 1 0 0 0 0 0 0 1 1 2 2 40.94 0.7243 1.2242 0.8563 1.4887 0.9454 3.1232 1 1 0 0 0 0 0 1 0 1 2 3 36.32 0.7213 1.2305 0.8578 1.6877 0.9421 3.0109 1 1 0 0 0 0 1 0 1 0 2 4 23.26 0.7215 1.2282 0.8589 1.5436 1.0465 2.1297 1 1 0 0 0 0 1 1 0 0 2 5 23.95 0.7217 1.2321 0.8548 1.4987 0.9425 2.2497 1 1 0 0 0 1 0 0 1 0 2 6 22.02 0.7176 1.2342 0.8549 1.5004 0.9449 2.3529 1 1 0 0 0 1 0 1 0 0 2 7 22.66 0.7178 1.2274 0.8573 1.6991 0.9429 2.1287 1 1 0 0 0 0 1 0 0 1 2 8 21.55 0.7234 1.2222 0.8523 1.5083 0.9423 2.0299 1 0 0 0 0 1 0 0 0 1 2 9 20.46 0.7178 1.2345 0.8562 1.5110 0.9793 2.1229 1 0 0 0 1 0 0 0 1 0 2 10 20.14 0.7243 1.2240 0.8594 1.4900 0.9588 2.0192 1 0 0 0 1 0 0 1 0 0 2 11 20.68 0.7177 1.2258 0.8584 1.5015 0.9496 2.2493 1 1 1 0 0 0 0 0 1 0 2 12 20.12 0.7217 1.2269 0.8531 1.4985 0.9949 2.0192 1 0 1 0 0 0 0 1 0 0 2 13 20.74 0.7202 1.2229 0.8573 1.5300 0.9397 2.0250 1 0 0 1 0 0 0 0 1 0 2 14 19.98 0.7229 1.2317 0.8575 1.4859 0.9411 2.0120 1 0 0 1 0 0 0 1 0 0 2 15 20.52 0.7175 1.2315 0.8560 1.4994 0.9445 2.0125 1 0 0 0 1 0 0 0 0 1 2 16 18.81 0.7245 1.2256 0.8585 1.4862 0.9458 2.0689 1 0 1 0 0 0 0 0 0 1 2 17 18.79 0.7273 1.2457 0.8578 1.5007 0.9450 2.0101 1 0 0 1 0 0 0 0 0 1 2 18 18.66 0.7226 1.2251 0.8523 1.4816 0.9421 2.0734 1 0 0 0 0 1 1 0 0 0 2 19 16.00 0.7224 1.2208 0.8525 1.4833 0.9399 1.5846 0 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

234

Table D-34 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 2 Caps, 12 locations of concern, Phase B Reactances Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) 105% 110% 0 0 0 0 0 1 1 0 2 1 40.89 0.7183 1.2398 0.8554 1.4828 0.9583 4.4622 1 1 0 0 0 0 0 0 1 1 2 2 40.49 0.7271 1.2207 0.8569 1.4761 0.9650 4.3351 1 1 0 0 0 0 0 1 0 1 2 3 36.32 0.7255 1.2254 0.8592 1.6932 0.9538 3.9954 1 1 0 0 0 0 1 0 1 0 2 4 23.26 0.7209 1.2231 0.8563 1.5530 0.9466 2.7104 1 1 0 0 0 0 1 1 0 0 2 5 23.95 0.7211 1.2227 0.8517 1.4960 0.9640 2.7948 1 1 0 0 0 1 0 0 1 0 2 6 22.02 0.7172 1.2383 0.8536 1.4981 0.9861 2.6895 1 1 0 0 0 1 0 1 0 0 2 7 22.66 0.7135 1.2241 0.8502 1.6953 0.9569 2.7017 1 1 0 0 0 0 1 0 0 1 2 8 21.55 0.7286 1.2263 0.8456 1.5116 0.9561 2.4152 1 1 0 0 0 1 0 0 0 1 2 9 20.46 0.7186 1.2311 0.8633 1.5015 0.9611 2.5398 1 1 0 0 1 0 0 0 1 0 2 10 20.14 0.7242 1.2233 0.8570 1.4903 0.9525 2.2356 1 1 0 0 1 0 0 1 0 0 2 11 20.68 0.7124 1.2211 0.8579 1.4983 0.9678 2.5164 1 1 1 0 0 0 0 0 1 0 2 12 20.12 0.7231 1.2292 0.8522 1.4974 0.9568 2.4158 1 1 1 0 0 0 0 1 0 0 2 13 20.74 0.7243 1.2178 0.8590 1.5245 0.9648 2.4166 1 1 0 1 0 0 0 0 1 0 2 14 19.98 0.7184 1.2290 0.8629 1.4889 0.9568 2.4590 1 1 0 1 0 0 0 1 0 0 2 15 20.52 0.7175 1.2236 0.8617 1.4966 0.9652 2.5149 1 1 0 0 1 0 0 0 0 1 2 16 18.81 0.7248 1.2303 0.8572 1.4800 0.9661 2.3956 1 1 1 0 0 0 0 0 0 1 2 17 18.79 0.7299 1.2502 0.8592 1.4970 0.9682 2.2416 1 1 0 1 0 0 0 0 0 1 2 18 18.66 0.7216 1.2220 0.8535 1.4830 0.9678 2.2365 1 1 0 0 0 1 1 0 0 0 2 19 16.00 0.7247 1.2215 0.8640 1.4933 0.9371 1.6285 0 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

235

Table D-35 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 3 Caps, 12 locations of concern, Phase A Reactances, Optimizations 1 - 25 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) 105% 110% 0 0 0 0 0 1 1 1 3 1 384.00 0.7230 1.2338 0.8994 1.5875 1.0873 5.8483 1 1 0 0 0 0 1 1 1 0 3 2 54.25 0.7243 1.2293 0.8684 1.5203 0.9794 3.8763 1 1 0 0 0 1 0 1 1 0 3 3 48.85 0.7245 1.2338 0.8685 1.5095 0.9589 3.7275 1 1 0 0 0 0 1 0 1 1 3 4 55.07 0.7204 1.2329 0.8672 1.4999 0.9489 3.6999 1 1 1 0 0 0 0 1 1 0 3 5 41.63 0.7187 1.2312 0.8890 1.5290 0.9678 3.3979 1 1 0 0 1 0 0 1 1 0 3 6 41.61 0.7239 1.2248 0.8678 1.5094 0.9993 3.4200 1 1 0 1 0 0 0 1 1 0 3 7 41.01 0.7211 1.2291 0.8658 1.5291 1.0094 3.3498 1 1 0 0 0 1 0 0 1 1 3 8 49.34 0.7191 1.2283 0.8562 1.5029 0.9828 3.5507 1 1 0 0 0 0 1 1 0 1 3 9 46.48 0.7214 1.2276 0.8785 1.4906 1.0040 3.6042 1 1 0 0 1 0 0 0 1 1 3 10 41.82 0.7221 1.2300 0.8520 1.4931 0.9636 3.2983 1 1 0 0 0 1 0 1 0 1 3 11 42.75 0.7211 1.2333 0.8558 1.4984 0.9938 3.4297 1 1 1 0 0 0 0 0 1 1 3 12 41.54 0.7238 1.2239 0.8560 1.4820 0.9574 3.2939 1 1 0 1 0 0 0 0 1 1 3 13 41.05 0.7243 1.2299 0.8541 1.4980 0.9748 3.2419 1 1 1 0 0 0 0 1 0 1 3 14 36.95 0.7219 1.2270 0.8527 1.4984 0.9502 3.1976 1 1 0 0 1 0 0 1 0 1 3 15 36.95 0.7236 1.2277 0.8541 1.4984 1.0093 3.2019 1 1 0 0 0 1 1 0 1 0 3 16 29.99 0.7212 1.2342 0.8572 1.5209 0.9809 2.6858 1 1 0 0 0 1 1 1 0 0 3 17 28.07 0.7220 1.2298 0.8537 1.5109 1.0103 2.7849 1 1 1 0 0 1 0 1 0 0 3 18 25.99 0.7221 1.2245 0.8533 1.5008 0.9653 2.5016 1 1 0 0 0 1 1 0 0 1 3 19 24.92 0.7246 1.2244 0.8559 1.4890 0.9984 2.4983 1 1 0 0 1 0 1 0 1 0 3 20 23.57 0.7238 1.2237 0.8571 1.4797 0.9642 2.4092 1 1 0 0 1 0 1 1 0 0 3 21 24.27 0.7225 1.2209 0.8561 1.5026 0.9704 2.4165 1 1 1 0 0 0 1 0 1 0 3 22 23.52 0.7226 1.2283 0.8536 1.5016 0.9717 2.4085 1 1 1 0 0 0 1 1 0 0 3 23 24.32 0.7238 1.2247 0.8569 1.4863 0.9788 2.4996 1 1 0 1 0 0 1 0 1 0 3 24 23.32 0.7215 1.2203 0.8555 1.5021 0.9685 2.3514 1 1 0 1 0 0 1 1 0 0 3 25 24.02 0.7226 1.2203 0.8540 1.4990 0.9733 2.2158 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

236

Table D-36 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 3 Caps, 12 locations of concern, Phase A Reactances, Optimizations 26 – 50 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) 105% 110% 0 0 1 1 0 0 1 0 3 26 22.35 0.7290 1.2216 0.8516 1.4896 0.9716 2.1127 1 0 0 0 1 1 0 1 0 0 3 27 23.00 0.7224 1.2204 0.8502 1.4985 0.9682 2.1380 1 1 1 0 0 1 0 0 1 0 3 28 22.33 0.7222 1.2263 0.8579 1.4983 0.9607 2.1305 1 1 0 1 0 1 0 0 1 0 3 29 22.08 0.7220 1.2249 0.8546 1.4862 0.9785 2.0281 1 0 0 1 0 1 0 1 0 0 3 30 22.74 0.7228 1.2205 0.8532 1.4985 0.9560 2.1203 1 0 0 0 1 0 1 0 0 1 3 31 21.84 0.7229 1.2256 0.8516 1.4726 0.9409 2.0298 1 0 1 0 0 0 1 0 0 1 3 32 21.77 0.7225 1.2281 0.8546 1.4792 0.9685 2.1243 1 1 0 1 0 0 1 0 0 1 3 33 21.06 0.7215 1.2249 0.8516 1.4862 0.9762 2.1535 1 1 1 0 1 0 0 1 0 0 3 34 21.01 0.7256 1.2349 0.8505 1.4851 0.9776 2.1502 1 1 1 1 0 0 0 1 0 0 3 35 20.89 0.7213 1.2345 0.8556 1.4821 0.9841 2.1001 1 0 0 0 1 1 0 0 0 1 3 36 20.78 0.7216 1.2346 0.8597 1.4863 0.9701 2.0252 1 0 1 0 0 1 0 0 0 1 3 37 20.75 0.7281 1.2265 0.8503 1.4800 0.9615 2.0222 1 0 0 1 0 1 0 0 0 1 3 38 20.52 0.7210 1.2261 0.8502 1.4806 0.9671 2.0312 1 0 1 0 1 0 0 0 1 0 3 39 20.39 0.7223 1.2339 0.8574 1.4896 0.9437 2.3093 1 1 0 1 1 0 0 0 1 0 3 40 20.22 0.7230 1.2350 0.8580 1.4816 0.9488 2.0155 1 0 0 1 1 0 0 1 0 0 3 41 20.77 0.7229 1.2295 0.8549 1.4875 0.9482 2.0032 1 0 1 1 0 0 0 0 1 0 3 42 20.17 0.7186 1.2281 0.8564 1.4991 0.9789 2.0904 1 0 1 0 1 0 0 0 0 1 3 43 19.02 0.7230 1.2320 0.8519 1.4888 0.9455 2.0128 1 0 0 1 1 0 0 0 0 1 3 44 18.89 0.7222 1.2341 0.8502 1.4917 0.9401 2.0016 1 0 1 1 0 0 0 0 0 1 3 45 18.82 0.7190 1.2264 0.8572 1.4814 0.9453 2.0598 1 0 0 0 1 1 1 0 0 0 3 46 16.23 0.7186 1.2254 0.8548 1.4882 0.9348 1.6049 0 0 1 0 0 1 1 0 0 0 3 47 16.22 0.7178 1.2277 0.8514 1.4891 0.9454 1.6755 0 0 0 1 0 1 1 0 0 0 3 48 16.05 0.7185 1.2250 0.8512 1.4823 0.9402 1.6003 0 0 1 0 1 0 1 0 0 0 3 49 14.69 0.7198 1.2344 0.8550 1.4839 0.9495 1.6851 0 0 1 1 0 0 1 0 0 0 3 50 14.56 0.7219 1.2261 0.8521 1.4897 0.9457 1.6726 0 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

237

Table D-37 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 3 Caps, 12 locations of concern, Phase B Reactances, Optimizations 1 – 25 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) 105% 110% 0 0 0 0 0 1 1 1 3 1 384 0.7266 1.2345 0.8865 2.6786 0.9846 15.8234 2 2 0 0 0 0 1 1 1 0 3 2 54.25 0.7226 1.2306 0.8670 1.8490 0.9705 5.4287 2 1 0 0 0 1 0 1 1 0 3 3 48.85 0.7213 1.2388 0.8652 1.8324 0.9782 5.0232 2 1 0 0 0 0 1 0 1 1 3 4 55.07 0.7231 1.2421 0.8623 1.8228 0.9635 5.5809 2 1 1 0 0 0 0 1 1 0 3 5 41.63 0.7184 1.2290 0.8885 1.5389 0.9983 4.5070 1 1 0 0 1 0 0 1 1 0 3 6 41.61 0.7227 1.2176 0.8715 1.5151 0.9769 4.5526 1 1 0 1 0 0 0 1 1 0 3 7 41.01 0.7217 1.2246 0.8632 1.5298 0.9765 4.4590 1 1 0 0 0 1 0 0 1 1 3 8 49.34 0.7198 1.2293 0.8568 1.4933 0.9748 5.2406 1 1 0 0 0 0 1 1 0 1 3 9 46.48 0.7218 1.2224 0.8767 1.4915 0.9753 4.9533 1 1 0 0 1 0 0 0 1 1 3 10 41.82 0.7212 1.2328 0.8555 1.4958 0.9786 4.5962 1 1 0 0 0 1 0 1 0 1 3 11 42.75 0.7221 1.2293 0.8610 1.4902 0.9913 4.5689 1 1 1 0 0 0 0 0 1 1 3 12 41.54 0.7221 1.2163 0.8551 1.4902 0.9899 4.5810 1 1 0 1 0 0 0 0 1 1 3 13 41.05 0.7283 1.2254 0.8534 1.4933 0.9899 4.4968 1 1 1 0 0 0 0 1 0 1 3 14 36.95 0.7196 1.2202 0.8529 1.5037 0.9511 4.1421 1 1 0 0 1 0 0 1 0 1 3 15 36.95 0.7208 1.2307 0.8539 1.4855 0.9882 4.2037 1 1 0 0 0 1 1 0 1 0 3 16 29.99 0.7187 1.2417 0.8504 1.5145 0.9768 3.6528 1 1 0 0 0 1 1 1 0 0 3 17 28.07 0.7269 1.2278 0.8571 1.5079 0.9768 3.3200 1 1 1 0 0 1 0 1 0 0 3 18 25.99 0.7213 1.2281 0.8511 1.4992 0.9867 3.0026 1 1 0 0 0 1 1 0 0 1 3 19 24.92 0.7242 1.2305 0.8517 1.4966 0.9863 3.0216 1 1 0 0 1 0 1 0 1 0 3 20 23.57 0.7197 1.2227 0.8525 1.4870 0.9855 2.8742 1 1 0 0 1 0 1 1 0 0 3 21 24.27 0.7154 1.2244 0.8517 1.4895 0.9790 2.8763 1 1 1 0 0 0 1 0 1 0 3 22 23.52 0.7121 1.2206 0.8590 1.4958 0.9895 2.6782 1 1 1 0 0 0 1 1 0 0 3 23 24.32 0.7261 1.2248 0.8572 1.4792 0.9905 2.9303 1 1 0 1 0 0 1 0 1 0 3 24 23.32 0.7218 1.2230 0.8517 1.4885 0.9862 2.7482 1 1 0 1 0 0 1 1 0 0 3 25 24.02 0.7200 1.2291 0.8514 1.4800 0.9985 2.7962 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

238

Table D-38 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 3 Caps, 12 locations of concern, Phase B Reactances, Optimizations 26 - 50 Variable s Driving Reactance (Ohms) Points of 3rd 5th 7th Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) 105% 110% 0 0 1 1 0 0 1 0 3 26 22.35 0.7222 1.2241 0.8500 1.4683 0.9983 2.7812 1 1 0 0 1 1 0 1 0 0 3 27 23.00 0.7226 1.2230 0.8542 1.4989 0.9785 2.7800 1 1 1 0 0 1 0 0 1 0 3 28 22.33 0.7221 1.2205 0.8503 1.4986 0.9845 2.7790 1 1 0 1 0 1 0 0 1 0 3 29 22.08 0.7200 1.2198 0.8596 1.4893 0.9788 2.7865 1 1 0 1 0 1 0 1 0 0 3 30 22.74 0.7230 1.2199 0.8595 1.4794 0.9816 2.7721 1 1 0 0 1 0 1 0 0 1 3 31 21.84 0.7226 1.2291 0.8518 1.4799 0.9256 2.7185 1 1 1 0 0 0 1 0 0 1 3 32 21.77 0.7222 1.2262 0.8555 1.4799 0.9783 2.5932 1 1 0 1 0 0 1 0 0 1 3 33 21.06 0.7295 1.2284 0.8516 1.4760 0.9642 2.4119 1 1 1 0 1 0 0 1 0 0 3 34 21.01 0.7298 1.2246 0.8512 1.4762 0.9600 2.3947 1 1 1 1 0 0 0 1 0 0 3 35 20.89 0.7262 1.2316 0.8526 1.4748 0.9642 2.5867 1 1 0 0 1 1 0 0 0 1 3 36 20.78 0.7205 1.2299 0.8562 1.4792 0.9641 2.4900 1 1 1 0 0 1 0 0 0 1 3 37 20.75 0.7206 1.2265 0.8550 1.4850 0.9400 2.4813 1 1 0 1 0 1 0 0 0 1 3 38 20.52 0.7158 1.2205 0.8541 1.4877 0.9415 2.3995 1 1 1 0 1 0 0 0 1 0 3 39 20.39 0.7197 1.2282 0.8586 1.4899 0.9686 2.5487 1 1 0 1 1 0 0 0 1 0 3 40 20.22 0.7195 1.2205 0.8578 1.4817 0.9562 2.4182 1 1 0 1 1 0 0 1 0 0 3 41 20.77 0.7215 1.2216 0.8564 1.4896 0.9485 2.3847 1 1 1 1 0 0 0 0 1 0 3 42 20.17 0.7234 1.2356 0.8533 1.4905 0.9768 2.4898 1 1 1 0 1 0 0 0 0 1 3 43 19.02 0.7232 1.2286 0.8450 1.4866 0.9632 2.3827 1 1 0 1 1 0 0 0 0 1 3 44 18.89 0.7216 1.2159 0.8416 1.4662 0.9789 2.2595 1 1 1 1 0 0 0 0 0 1 3 45 18.82 0.7165 1.2193 0.8533 1.4799 0.9875 2.3746 1 1 0 0 1 1 1 0 0 0 3 46 16.23 0.7156 1.2295 0.8502 1.4801 0.9816 -0.3185 0 0 1 0 0 1 1 0 0 0 3 47 16.22 0.7156 1.2360 0.8469 1.4966 0.9769 -0.2185 0 0 0 1 0 1 1 0 0 0 3 48 16.05 0.7216 1.2250 0.8491 1.4963 0.9842 -0.2400 0 0 1 0 1 0 1 0 0 0 3 49 14.69 0.7183 1.2379 0.8499 1.4829 0.9654 -0.8547 0 0 1 1 0 0 1 0 0 0 3 50 14.56 0.7240 1.2336 0.8540 1.4865 0.9808 -0.3582 0 0 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

239

Table D-39 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase A Reactances, Optimizations 1 - 25 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) 105% 110% 0 0 0 0 1 1 1 1 4 1 14500.00 0.7240 1.2324 0.8800 1.8983 1.1240 6.9094 2 1 0 0 0 1 0 1 1 1 4 2 2000.00 0.7177 1.2282 0.8869 1.7046 1.1246 6.2076 1 1 0 0 1 0 0 1 1 1 4 3 460.00 0.7198 1.2284 0.8678 1.6040 1.1256 5.6753 1 1 1 0 0 0 0 1 1 1 4 4 428.00 0.7222 1.2296 0.8610 1.5989 1.1146 5.5939 1 1 0 1 0 0 0 1 1 1 4 5 412.00 0.7190 1.2297 0.8693 1.6094 1.1588 5.7890 1 1 0 0 0 1 1 0 1 1 4 6 91.56 0.7233 1.2305 0.8531 1.4885 0.9577 4.2099 1 1 0 0 0 1 1 1 1 0 4 7 85.90 0.7195 1.2234 0.9094 1.5093 1.0928 4.3099 1 1 0 0 0 1 1 1 0 1 4 8 64.51 0.7201 1.2280 0.8553 1.5004 0.9768 4.0939 1 1 1 0 0 1 0 1 1 0 4 9 59.40 0.7212 1.2295 0.8549 1.5002 0.9810 3.8016 1 1 0 0 1 0 1 0 1 1 4 10 58.12 0.7229 1.2323 0.8530 1.4900 0.9685 3.7689 1 1 1 0 0 0 1 0 1 1 4 11 57.09 0.7241 1.2263 0.8585 1.5009 0.9560 3.7049 1 1 0 1 0 0 1 0 1 1 4 12 56.64 0.7190 1.2264 0.8520 1.7993 1.0119 3.5988 2 1 0 0 1 1 0 0 1 1 4 13 52.73 0.7205 1.2276 0.8666 1.5040 1.0245 3.4296 1 1 1 0 0 1 0 0 1 1 4 14 51.69 0.7230 1.2262 0.8884 1.5383 1.2040 3.4298 1 1 0 0 1 1 0 1 1 0 4 15 50.95 0.7260 1.2261 0.8784 1.5234 1.0032 3.3510 1 1 0 1 0 1 0 1 1 0 4 16 49.85 0.7219 1.2331 0.8469 1.5151 1.0235 3.3402 1 1 1 0 0 1 0 1 0 1 4 17 53.54 0.7216 1.2332 0.9297 1.5190 1.0295 3.2929 1 1 0 1 0 1 0 0 1 1 4 18 51.39 0.7213 1.2339 0.8776 1.5040 1.0239 3.5026 1 1 0 0 1 0 1 1 0 1 4 19 48.95 0.7184 1.2345 0.8750 1.5390 1.1931 3.4973 1 1 0 0 1 0 1 1 1 0 4 20 56.12 0.7130 1.2300 0.8516 1.5283 1.0006 3.6915 1 1 1 0 0 0 1 1 0 1 4 21 48.54 0.7204 1.2331 0.8582 1.5038 0.9933 3.6370 1 1 1 0 0 0 1 1 1 0 4 22 55.94 0.7242 1.2325 0.8876 1.5039 1.0493 3.8720 1 1 0 1 0 0 1 1 0 1 4 23 48.00 0.7240 1.2253 0.8544 1.4921 1.0128 3.5935 1 1 0 1 0 0 1 1 1 0 4 24 54.90 0.7241 1.2251 0.8894 1.5193 1.0490 3.8017 1 1 0 0 1 1 0 1 0 1 4 25 45.02 0.7235 1.2317 0.9373 1.4993 1.0498 3.2398 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

240

Table D-40 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase A Reactances, Optimizations 26 – 50 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) 105% 110% 0 1 0 1 0 1 0 1 4 26 44.37 0.7186 1.2269 0.8573 1.4923 0.9938 3.4693 1 1 1 0 1 0 0 0 1 1 4 27 43.06 0.7209 1.2239 0.8738 1.4926 1.0294 3.2938 1 1 0 1 1 0 0 0 1 1 4 28 42.98 0.7182 1.2240 0.8674 1.4983 1.0329 3.3282 1 1 1 1 0 0 0 0 1 1 4 29 42.27 0.7195 1.2308 0.8581 1.4820 1.2040 3.2389 1 1 1 0 1 0 0 1 0 1 4 30 38.11 0.7230 1.2316 0.8629 1.4903 1.0404 3.1298 1 1 1 0 1 0 0 1 1 0 4 31 42.74 0.7231 1.2327 0.8629 1.4924 1.0303 3.4020 1 1 0 1 1 0 0 1 0 1 4 32 37.90 0.7246 1.2248 0.8712 1.4849 1.0228 3.2517 1 1 0 1 1 0 0 1 1 0 4 33 42.21 0.7223 1.2239 0.8600 1.5196 0.9862 2.8401 1 1 1 1 0 0 0 1 0 1 4 34 37.62 0.7235 1.2327 0.8517 1.5030 0.9840 3.1298 1 1 1 1 0 0 0 1 1 0 4 35 42.03 0.7200 1.2245 0.8557 1.5039 1.0296 3.3864 1 1 1 0 0 1 1 1 0 0 4 36 33.71 0.7214 1.2295 0.8674 1.4909 1.0937 2.8486 1 1 0 0 1 1 1 0 1 0 4 37 31.62 0.7241 1.2299 0.8750 1.4847 1.0200 2.7044 1 1 1 0 0 1 1 0 1 0 4 38 31.34 0.7189 1.2274 0.8558 1.5029 1.0383 2.7619 1 1 0 1 0 1 1 0 1 0 4 39 31.21 0.7233 1.2237 0.8551 1.4991 1.0030 2.7684 1 1 0 0 1 1 1 1 0 0 4 40 28.91 0.7215 1.2289 0.8592 1.5003 0.9908 2.6399 1 1 0 1 0 1 1 1 0 0 4 41 28.41 0.7238 1.2312 0.8543 1.4838 0.9827 2.7437 1 1 1 0 1 1 0 1 0 0 4 42 27.48 0.7223 1.2324 0.8513 1.4791 0.9806 2.5008 1 1 1 1 0 1 0 1 0 0 4 43 27.26 0.7238 1.2314 0.8572 1.4801 0.9501 1.8456 0 0 0 0 1 1 1 0 0 1 4 44 25.74 0.7232 1.2269 0.8517 1.4986 0.9613 2.1523 1 1 1 0 0 1 1 0 0 1 4 45 25.57 0.7221 1.2280 0.8500 1.4793 0.9762 2.2216 1 1 0 1 0 1 1 0 0 1 4 46 25.30 0.7233 1.2240 0.8519 1.4795 0.9752 2.2082 1 1 1 0 1 0 1 0 1 0 4 47 24.00 0.7184 1.2349 0.8652 1.4808 1.0190 2.2985 1 1 1 0 1 0 1 1 0 0 4 48 24.80 0.7212 1.2201 0.8575 1.4706 0.9921 2.1285 1 1 0 1 1 0 1 0 1 0 4 49 23.90 0.7246 1.2272 0.8657 1.4913 1.1040 2.4903 1 1 0 1 1 0 1 1 0 0 4 50 24.58 0.7259 1.2375 0.8563 1.5022 1.0040 2.3099 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

241

Table D-41 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase A Reactances, Optimizations 51 – 64 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) (A Ph) 105% 110% 1 1 0 0 1 0 1 0 4 51 23.72 0.7236 1.2339 0.8519 1.5183 0.9989 2.4393 1 1 1 1 0 0 1 1 0 0 4 52 24.52 0.7182 1.2403 0.8560 1.4937 0.9990 2.4370 1 1 1 0 1 1 0 0 1 0 4 53 22.90 0.7190 1.2352 0.8545 1.4891 1.0024 2.4109 1 1 0 1 1 1 0 0 1 0 4 54 22.74 0.7123 1.2358 0.8500 1.4800 1.0121 2.1591 1 1 0 1 1 1 0 1 0 0 4 55 23.39 0.7221 1.2335 0.8519 1.4660 1.0221 2.2109 1 1 1 1 0 1 0 0 1 0 4 56 22.63 0.7183 1.2310 0.8536 1.4849 1.0193 2.4904 1 1 1 0 1 0 1 0 0 1 4 57 22.24 0.7240 1.2330 0.8543 1.5130 0.9787 3.3983 1 1 0 1 1 0 1 0 0 1 4 58 22.18. 0.7210 1.2259 0.8500 1.5033 0.9842 2.0315 1 0 1 1 0 0 1 0 0 1 4 59 21.97 0.7230 1.2292 0.8567 1.4790 0.9329 2.3919 1 1 1 0 1 1 0 0 0 1 4 60 21.28 0.7236 1.2249 0.8529 1.4890 0.9318 2.0200 1 0 0 1 1 1 0 0 0 1 4 61 21.16 0.7221 1.2236 0.8515 1.4799 0.9472 2.0618 1 0 1 1 0 1 0 0 0 1 4 62 21.01 0.7240 1.2339 0.8540 1.4816 0.9837 2.1929 1 1 1 1 1 0 0 0 1 0 4 63 20.56 0.7229 1.2308 0.8512 1.5000 0.9455 2.1393 1 1 1 1 1 0 0 1 0 0 4 64 21.21 0.7206 1.2333 0.8520 1.4906 0.9456 2.3095 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

242

Table D-42 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase B Reactances, Optimizations 1 – 25 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) 105% 110% 0 0 0 0 1 1 1 1 4 1 14500.00 0.7198 1.2334 0.9816 3.6321 1.9839 120.3654 3 3 0 0 0 1 0 1 1 1 4 2 2000.00 0.7109 1.2238 0.9682 2.4961 1.8402 43.7168 3 3 0 0 1 0 0 1 1 1 4 3 460.00 0.7198 1.2302 0.9588 1.6161 1.4030 20.3928 2 1 1 0 0 0 0 1 1 1 4 4 428.00 0.7203 1.2252 0.9488 1.5938 1.0295 19.5925 1 1 0 1 0 0 0 1 1 1 4 5 412.00 0.7222 1.2275 0.9735 1.6128 1.1023 19.0950 1 1 0 0 0 1 1 0 1 1 4 6 91.56 0.7235 1.2265 0.8582 1.4838 1.0060 7.9221 1 1 0 0 0 1 1 1 1 0 4 7 85.90 0.7223 1.2178 0.9041 1.5121 0.9961 7.2085 1 1 0 0 0 1 1 1 0 1 4 8 64.51 0.7214 1.2351 0.8526 1.4960 0.9763 6.0285 1 1 1 0 0 1 0 1 1 0 4 9 59.40 0.7205 1.2312 0.8505 1.4899 0.9782 5.7195 1 1 0 0 1 0 1 0 1 1 4 10 58.12 0.7256 1.2241 0.8563 1.4954 1.0285 5.8432 1 1 1 0 0 0 1 0 1 1 4 11 57.09 0.7205 1.2331 0.8545 1.5029 1.0122 5.7938 1 1 0 1 0 0 1 0 1 1 4 12 56.64 0.7218 1.2337 0.8514 1.7971 0.9937 5.6939 2 1 0 0 1 1 0 0 1 1 4 13 52.73 0.7150 1.2295 0.8719 1.5011 0.9957 5.3903 1 1 1 0 0 1 0 0 1 1 4 14 51.69 0.7238 1.2179 0.8846 1.5407 1.0059 5.2939 1 1 0 0 1 1 0 1 1 0 4 15 50.95 0.7216 1.2251 0.8575 1.5480 1.0285 5.2088 1 1 0 1 0 1 0 1 1 0 4 16 49.85 0.7257 1.2241 0.8697 1.5465 1.1523 5.1825 1 1 1 0 0 1 0 1 0 1 4 17 53.54 0.7163 1.2390 0.9282 1.5207 1.0076 5.0497 1 1 0 1 0 1 0 0 1 1 4 18 51.39 0.7225 1.2375 0.8754 1.5038 1.0516 5.3019 1 1 0 0 1 0 1 1 0 1 4 19 48.95 0.7125 1.2447 0.8820 1.5386 0.9971 4.9585 1 1 0 0 1 0 1 1 1 0 4 20 56.12 0.7216 1.2346 0.8566 1.5244 0.9788 5.4160 1 1 1 0 0 0 1 1 0 1 4 21 48.54 0.7162 1.2418 0.8614 1.5078 0.9798 5.0130 1 1 1 0 0 0 1 1 1 0 4 22 55.94 0.7255 1.2293 0.8844 1.5070 0.9918 5.5380 1 1 0 1 0 0 1 1 0 1 4 23 48.00 0.7291 1.2273 0.8565 1.4951 0.9879 5.0101 1 1 0 1 0 0 1 1 1 0 4 24 54.90 0.7214 1.2314 0.8861 1.5136 0.9920 5.4991 1 1 0 0 1 1 0 1 0 1 4 25 45.02 0.7235 1.2272 0.9364 1.5096 1.0193 4.5298 1 1

243

Table D-43 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase B Reactances, Optimizations 26 - 50 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) 105% 110% 0 1 0 1 0 1 0 1 4 26 44.37 0.7167 1.2350 0.8621 1.4926 1.0037 4.3209 1 1 1 0 1 0 0 0 1 1 4 27 43.06 0.7193 1.2168 0.8814 1.5030 0.9921 4.5490 1 1 0 1 1 0 0 0 1 1 4 28 42.98 0.7166 1.2182 0.8620 1.4998 0.9934 4.6940 1 1 1 1 0 0 0 0 1 1 4 29 42.27 0.7200 1.2261 0.8595 1.4872 0.9928 4.5948 1 1 1 0 1 0 0 1 0 1 4 30 38.11 0.7214 1.2280 0.8662 1.4992 0.9956 4.1391 1 1 1 0 1 0 0 1 1 0 4 31 42.74 0.7205 1.2355 0.8647 1.4951 1.0022 4.2050 1 1 0 1 1 0 0 1 0 1 4 32 37.90 0.7216 1.2382 0.8649 1.4862 0.9916 4.1555 1 1 0 1 1 0 0 1 1 0 4 33 42.21 0.7242 1.2262 0.8542 1.7852 0.9915 4.5002 2 1 1 1 0 0 0 1 0 1 4 34 37.62 0.7230 1.2262 0.8477 1.6480 1.0030 4.2341 1 1 1 1 0 0 0 1 1 0 4 35 42.03 0.7150 1.2271 0.8542 1.5021 1.0069 4.3912 1 1 1 0 0 1 1 1 0 0 4 36 33.71 0.7282 1.2258 0.8671 1.4946 0.9976 3.8465 1 1 0 0 1 1 1 0 1 0 4 37 31.62 0.7247 1.2216 0.8767 1.4803 0.9859 3.7029 1 1 1 0 0 1 1 0 1 0 4 38 31.34 0.7187 1.2236 0.8605 1.5008 1.0502 3.4281 1 1 0 1 0 1 1 0 1 0 4 39 31.21 0.7235 1.2154 0.8561 1.5042 1.0024 3.3120 1 1 0 0 1 1 1 1 0 0 4 40 28.91 0.7222 1.2250 0.8500 1.4861 1.0020 3.1207 1 1 0 1 0 1 1 1 0 0 4 41 28.41 0.7249 1.2320 0.8548 1.4769 1.0028 3.3287 1 1 1 0 1 1 0 1 0 0 4 42 27.48 0.7205 1.2255 0.8649 1.4700 1.0120 3.4100 1 1 1 1 0 1 0 1 0 0 4 43 27.26 0.7209 1.2258 0.8570 1.4850 1.0160 3.9205 1 1 0 0 1 1 1 0 0 1 4 44 25.74 0.7214 1.2255 0.8570 1.4827 1.0040 3.1847 1 1 1 0 0 1 1 0 0 1 4 45 25.57 0.7206 1.2301 0.8507 1.4983 1.0025 3.0022 1 1 0 1 0 1 1 0 0 1 4 46 25.30 0.7214 1.2248 0.8500 1.5008 1.0022 3.0148 1 1 1 0 1 0 1 0 1 0 4 47 24.00 0.7189 1.2322 0.8631 1.4865 1.0040 2.8022 1 1 1 0 1 0 1 1 0 0 4 48 24.80 0.7221 1.2203 0.8522 1.4892 1.0033 3.0615 1 1 0 1 1 0 1 0 1 0 4 49 23.90 0.7202 1.2204 0.8660 1.4816 0.9940 2.8039 1 1 0 1 1 0 1 1 0 0 4 50 24.58 0.7210 1.2205 0.8513 1.4752 1.6205 -2.2154 2 2 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

244

Table D-44 HRI Algorithm results for Feeder 1358 tied to Feeder 1444, Cap = 1200 kVAr (Phase A) and 1500 kVAr (Phase B), Compensation = 4 Caps, 12 locations of concern, Phase B Reactances, Optimizations 51 – 64 Variable s Driving Reactance (Ohms) 3rd 5th 7th Points of Concern Objective Experiencing Caps Optimization Function NODE 4 NODE 15 NODE 4 NODE 15 NODE 4 NODE 15 Resonance w1 w2 w3 w4 w5 w6 w7 w8 Req'd No. Value (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) (B Ph) 105% 110% 1 1 0 0 1 0 1 0 4 51 23.72 0.7232 1.2338 0.8531 1.5320 1.5400 -2.3104 2 1 1 1 0 0 1 1 0 0 4 52 24.52 0.7182 1.2403 0.8560 1.4937 1.5564 -2.4983 2 1 1 0 1 1 0 0 1 0 4 53 22.90 0.7221 1.2480 0.8590 1.4900 0.9803 2.5616 1 1 0 1 1 1 0 0 1 0 4 54 22.74 0.7200 1.2450 0.8592 1.4880 0.9913 2.0519 1 0 0 1 1 1 0 1 0 0 4 55 23.39 0.7191 1.2481 0.8517 1.4983 0.9981 2.2802 1 1 1 1 0 1 0 0 1 0 4 56 22.63 0.7174 1.2252 0.8601 1.4822 1.0106 2.7940 1 1 1 0 1 0 1 0 0 1 4 57 22.24 0.7214 1.2328 0.8508 1.5006 1.0495 3.0737 1 1 0 1 1 0 1 0 0 1 4 58 22.18. 0.7165 1.2265 0.8503 1.4998 0.9920 2.5998 1 1 1 1 0 0 1 0 0 1 4 59 21.97 0.7260 1.2365 0.8569 1.4763 1.7187 -2.3802 2 2 1 0 1 1 0 0 0 1 4 60 21.28 0.7220 1.2460 0.8512 1.4985 0.9916 2.6033 1 1 0 1 1 1 0 0 0 1 4 61 21.16 0.7226 1.2337 0.8503 1.4985 1.7488 -2.0027 2 1 1 1 0 1 0 0 0 1 4 62 21.01 0.7230 1.2352 0.8558 1.4836 1.1744 2.4409 1 1 1 1 1 0 0 0 1 0 4 63 20.56 0.7215 1.2238 0.8542 1.4926 0.9938 2.5240 1 1 1 1 1 0 0 1 0 0 4 64 21.21 0.7155 1.2337 0.8501 1.5022 1.0959 2.6188 1 1 Non-harmonic Resonance Values 8.593 0.7175 1.2229 0.8506 1.4778 0.9409 1.6703

245