Online Analysis and Control of Electric Power Distribution Systems
A Thesis
Submitted to the Faculty
of
Drexel University
by
Nicholas S. Coleman
in partial fulfillment of the
requirements for the degree
of
Doctor of Philosophy
March 2018 © Copyright 2018
Nicholas S. Coleman. All Rights Reserved. ii
Dedication
This thesis is dedicated to the
2017–2018 Philadelphia Eagles
USA Today Sports / Matthew Emmons
Super Bowl LII Champions
“Without failure, who would you be?”
– Nick Foles, Super Bowl LII MVP iii
Acknowledgements
I would like to thank and acknowledge my advisor, Dr. Karen Miu. Thank you for the guidance, direction, and support that you have provided me throughout my time in the lab. Your thoroughness is unmatched.
I give thanks to Dr. Hande Benson, Dr. Thomas Halpin, Dr. Harry Kwatny, and Dr. Steven
Weber for serving on my committee. Thank you for spending time reviewing and commenting on my work, and for sharing your expertise and insights both now and throughout my undergraduate and graduate studies as Drexel.
Thank you to the students and staff of the Center for Electric Power Engineering who have helped me learn, work, and experiment in the lab.
Finally, I thank my mother and stepfather, who are the hardest working people I know, and my wife Jessica, whose constant encouragement has kept me on track over the years. iv
Table of Contents
List of Tables ...... viii
List of Figures ...... ix
Abstract ...... x
1. Introduction ...... 1
1.1 Motivation ...... 2
1.2 Objectives...... 3
1.3 Research Contributions ...... 3
1.4 Organization ...... 4
2. Distribution System Modeling ...... 6
2.1 Distribution System Structure...... 6
2.2 System State and Parameters (Decision Variables) ...... 8
2.3 Constraints ...... 8
2.3.1 Electrical Constraints ...... 8
2.3.2 Operating Constraints...... 9
2.3.3 Control Constraints...... 10
2.4 Branch Component Models ...... 11
2.5 Power Injection Models ...... 12
2.5.1 Load Models ...... 12
2.5.2 Distributed Generation Model...... 13
2.5.3 Shunt Battery Model...... 14
2.5.4 Shunt Capacitor Model...... 14
2.6 Injection Variation Modeling ...... 15
2.6.1 Spatial Injection Variation Model...... 16
2.6.2 Spatiotemporal Injection Variation Model ...... 17
3. Distribution Load Capability ...... 20 v
3.1 Literature Review ...... 20
3.2 General Load Capability Problem Formulation ...... 21
3.2.1 Solution Methodology ...... 22
3.2.2 General DLC Solution Algorithm...... 23
3.3 Power Factor-Based Load Capability Estimator...... 24
3.3.1 Constraint Transformation...... 24
3.3.2 Analytical Relationship ...... 25
3.3.3 Predictor-Corrector Method...... 26
3.4 Existence, Finiteness, and Non-negativity ...... 28
3.5 Power Factor-Based Estimator Solution Algorithm...... 31
3.6 Simulation Results...... 34
3.6.1 Simulation Parameters...... 35
3.6.2 Variation Case Identification...... 37
3.6.3 Numerical Results ...... 37
3.6.4 Convergence Behavior and Series Acceleration...... 38
3.7 Convergence of the Power Factor Based Estimator ...... 39
3.8 Concluding Remarks...... 40
4. Time Window Selection for Quasi-Static Time Series Analysis ...... 41
4.1 Literature Review ...... 41
4.2 Design of QSTS Studies for Distribution System Analysis ...... 43
4.3 Implicit Temporal Load Capability...... 44
4.3.1 ITLC Inputs...... 45
4.3.2 Critical Times and Sub-Windows...... 46
4.3.3 Unscheduled Control Actions ...... 47
4.3.4 Scheduled Control Actions...... 48
4.3.5 ITLC Outputs...... 48
4.3.6 Detailed ITLC Example...... 48 vi
4.3.7 Problem Formulation...... 50
4.3.8 Main Implicit Temporal Load Capability Solution Algorithm ...... 51
4.3.9 Control Sub-Algorithms ...... 54
4.3.10 Simulation Results ...... 56
4.3.11 Additional Applications ...... 65
4.4 Risk-Aware Time Window Selection ...... 65
4.4.1 Forecast Model with Uncertainty...... 67
4.4.2 Risk Model ...... 68
4.4.3 Problem Formulation...... 71
4.4.4 Simulation Results ...... 72
4.4.5 Control Selection...... 75
4.5 Concluding Remarks...... 76
5. Asynchronous Distribution State Estimation ...... 77
5.1 Literature Review ...... 77
5.2 Contemporary Operating Environment...... 79
5.3 General Problem Formulation...... 79
5.4 Specific Problem Formulation...... 80
5.5 Solution Methodology ...... 84
5.6 Pseudomeasurements and Virtual Measurements ...... 85
5.7 Simulation Results...... 86
5.7.1 Load Classification Procedure...... 87
5.7.2 Measurement Data...... 90
5.7.3 Simulation Cases...... 91
5.7.4 Numerical Results ...... 92
5.8 Concluding Remarks...... 93
6. Conclusion ...... 95
6.1 Summary of Contributions...... 96 vii
6.2 Extensions and Future Work ...... 96
6.2.1 Distribution Load Capability ...... 96
6.2.2 Time Window Selection Methods...... 97
6.2.3 Asynchronous Distribution State Estimation...... 97
Bibliography ...... 99
Appendix A: Convergence Analysis for the PF-Based DLC Estimator ...... 104
A.1 Estimator and Update Equations...... 104
A.2 Illustrative Example ...... 105
A.3 Convergence Conditions...... 107
A.3.1 Notation Definitions ...... 108
A.3.2 Explicit Statement of the Fixed Point Iteration...... 109
A.3.3 Statement of the Exact Convergence Condition...... 110
A.3.4 Convergence with Typical Distribution System Assumptions ...... 113
A.3.5 Convergence Region Example...... 116
Appendix B: Implicit Temporal Load Capability: Tap Operation Analysis . . . . 119
B.1 Test Circuit and Injection Profile...... 119
B.1.1 Injection Profile Development...... 120
B.1.2 Control Modeling...... 121
B.2 Simulation Results...... 122
Vita ...... 124 viii
List of Tables
2.1 PPL distribution test circuit summary...... 7
2.2 Distribution branch component models...... 12
3.1 Variation cases for the power factor-based estimator...... 30
3.2 DLC simulation parameters...... 36
3.3 Aggregated injections and injection variation parameters for the DLC simulations. . . . 38
3.4 DLC series acceleration results...... 39
4.1 ITLC simulation results, organized by sub-window...... 64
4.2 Risk-aware time window selection simulation results: time window optimization and post-processed capacitor control settings...... 74
5.1 Elements of the ADSE measurement sensitivity matrix...... 83
5.2 Measurment data types for the ADSE simulations...... 90
5.3 ADSE simulation cases...... 91
5.4 ADSE simulation results with equal measurement ages of ten minutes...... 92
5.5 ADSE simulation results with random measurement ages...... 92
A.1 Parameters for the convergence behavior example...... 105
A.2 Numerical results of the convergence example...... 106
B.1 Voltage regulator settings...... 122 ix
List of Figures
2.1 One-line diagram of the three-phase portion of a 2556-node PPL distribution circuit... 7
3.1 Overview of Algorithm 3.2...... 32
3.2 One-line diagram of the modified 2556-node PPL distribution circuit showing the photo- voltatic DG units and the energized capacitors...... 36
4.1 ITLC Example...... 49
4.2 Overview of Algorithm 4.1...... 52
4.3 Net three-phase substation real and reactive demand forecasts...... 57
4.4 Critical variable forecasts construted by computing the power flow solution at each point in the time series forecast...... 58
4.5 Critical variable forecasts with preventative control actions at critical times identified by ITLC...... 58
4.6 The risk-limiting dispatch timeline, adapted from [4]...... 70
4.7 Risk function at different urgency levels...... 71
4.8 Forcasted total three-phase substation demand with 95% confidence interval bands. . . 74
4.9 Risk associated with the injection forecast...... 74
5.1 One-line diagram of the radial, three-phase, 394 bus distribution test circuit [72]. . . . . 86
5.2 Load classification procedure...... 89
A.1 One-line diagram of the two node system...... 105
A.2 Convergence behavior example...... 107
A.3 Convergence region based on the satisfaction of inequality (A.3.5) for the example pa- rameter set given in Section A.2...... 118
A.4 Emperically determined convergence rate for the example parameter set given in Sec- tion A.2 over a range of initial conditions...... 118
B.1 IEEE 34-node distribution test feeder...... 119
B.2 Day-ahead injection profile for the test feeder...... 121
B.3 Voltage magnitude ranges at monitored buses...... 123
B.4 Tap position forecast...... 123 x
Abstract
Online Analysis and Control of Electric Power Distribution Systems Nicholas S. Coleman Karen N. Miu, Ph.D.
Historically, electric power distribution systems were considered passive subsystems served by the larger transmission grid. Recently, smart grid initiatives have driven the evolution of distribution systems into active systems with market-aware customers and distributed power generation. Along with more diverse and complex injections, contemporary distribution systems are equipped with additional sensing equipment, two-way communications networks, and advanced metering infras- tructure (AMI). These are essential technologies that enable several core functions of a smart grid, including real-time monitoring and online control.
This thesis presents several tools for the online analysis and control of modern electric power distribution systems. “Online” refers to a control framework that can react to changing system con- ditions in order to maintain static security and meet different operating objectives. Specifically, the objective of this research is to integrate temporal information (i.e., forecasts) into distribution sys- tem analysis tools while maintaining fundamental engineering requirements by re-examining classical problems through a contemporary lens.
Connected by an underlying injection forecast model, three research topics are explored: 1) dis- tribution load capability, 2) analytical time window selection for quasi-static time series (QSTS) analysis, and 3) distribution state estimation with explicit consideration of non-synchronized mea- surements. The work proposed here is a necessary step towards distribution system optimization in an online setting with uncertain and/or bidirectional power flows.
1
Chapter 1: Introduction
The construction of the trans-continental electric power grid during the twentieth century drove a period of unprecedented technological and economic growth. Reliable, on-demand electric power has evolved from a luxury into an absolute necessity, and today, the power grid is America’s most critical infrastructure.
As we progress through the twenty-first century, changing energy needs are driving major grid modernization efforts. The need to modernize the grid is widely recognized; tens of billions of dollars are spent each year on smart grid projects. The Electric Power Research Institute (EPRI) estimates that fully realizing the “smart grid” vision set forth in the Energy Independence and Security Act of 2007 could have an economic value in excess two trillion dollars [1,2].
One of the major areas of smart grid investment is the deployment of sensing equipment, two-way communications networks, and advanced metering infrastructure (AMI) within power distribution networks [3]. These are essential technologies that enable several core functions of a smart grid, including real-time monitoring online control.
This thesis presents several timely tools for the online analysis and control of modern electric power distribution systems. In this thesis, traditional analysis tools have been revisited in the context of the emerging grid environment that includes smart grid sensing and communications infrastructures. In particular, the following topics have been investigated:
Distribution load capability with power factor-based constraints.
Analytical time window selection for quasi-static time series analysis.
Distribution state estimation with non-synchronized measurement data.
This chapter discusses the motivation, objectives, contributions, and organization of this thesis. 2
1.1 Motivation
Historically, distribution system control schemes have been developed using seasonal loading averages and a limited number of measurements. Individual device actuation has relied on time-of- day settings and/or localized sensing (e.g, automatic capacitor switching based on a locally sensed voltage magnitude).
In emerging system environments, distributed energy resources and interactive energy markets contribute to less predictable distribution network behavior [4]. In order to economically and reliably supply power under these conditions, grid operators must be able to actively identify and respond to changing system conditions. Technologies that enable the necessary shift from seasonally planned control schemes to real-time monitoring and online control include:
Advanced metering infrastructure (AMI), which records customer energy consumption at
hourly (or more frequent) intervals, and transmits this data to centralized processing units
at least once a day. AMI systems allow operators to understand near-real-time system condi-
tions with outage alarms or on-demand data pings [5].
Distribution automation (DA) devices, which enable remote control of feeder or capacitor
switches, for example, send real-time voltage, current, and/or power flow data to system
operators. Automatic reporting of alarms and switch operations also allows operators to
update system models to reflect actual field conditions [3].
Utility-scale storage systems and demand response programs, which provide system operators
with new options for managing the flow of power and energy in response to changing system
conditions.
Communications networks, which enable centralized operators to obtain real-time data and to
dispatch control actions remotely.
Several distribution operators have already begun to adopt and deploy these smart grid tech- nologies. For example, PPL Electric Utilities has deployed thousands of DA devices and an AMI network covering 1.4 million customers to its circuits in Pennsylvania. Coupled with a reliable 3 cellular communications network, these investments have enabled PPL to develop advanced feeder reconfiguration and capacitor control functions [6,7]. With these tools, operators may perform optimization for loss reduction or reliability improvement more frequently than previously possible.
While spatial optimization tools are valuable, there is far more potential for leveraging the data collected and transmitted by AMI and DA devices. Specifically, the next step is integrating temporal information into distribution operations. This is a necessary step towards the optimization of distribution systems with uncertain, bidirectional power flows, and is also the motivation for this thesis.
1.2 Objectives
This thesis aims to develop practical tools for the analysis and control of distribution systems in emerging power system environments. The specific objectives of this thesis are:
To incorporate temporal information (i.e., power injection forecasts) into distribution system
analysis tools.
To maintain fundamental engineering requirements by re-examining classical problems through
a contemporary lens.
1.3 Research Contributions
The research contributions of this thesis cover three different topics:
Distribution load capability (DLC)
– A power factor-based estimator of load capability
– Convergence analysis of the power factor-based estimator
– The application of series acceleration to distribution load capability
Analytical time window selection methods for quasi-static time series (QSTS) analysis 4
– Implicit temporal load capability (ITLC): a method that uses forecasted injection char-
acteristics to identify critical times and identify a feasible control schedule
– Risk-aware time window selection: a method that concentrates computational power on
times when operating risk is forecasted to be relatively greater
– A demonstration of significant computational efficiency improvements compared to tra-
ditional QSTS analyses
Asynchronous distribution state estimation (ADSE)
– A mathematical model with explicit consideration of non-synchronized measurements
– A distribution measurement sensitivity matrix defined with respect to time-varying power
injections
1.4 Organization
Chapter2 provides background information related to power distribution system modeling. The general structure of power distribution systems is discussed. System constraints and relevant math- ematical models are presented, including injection variation models that tie together Chapters3-5.
Chapter3 reviews load capability, and presents the power factor-based load capability estimator.
A solution algorithm for the power factor-based estimator is presented. Motivated by the observation of linear convergence behavior, a series acceleration technique is studied for both power factor- and current-based load capability estimator. Simulation results show that series acceleration can yield significant computational efficiency improvements. A formal convergence analysis for the PF-based estimator with constant power injection models is presented in AppendixA and summarized in
Chapter3.
Chapter4 presents analytical time window selection methods for quasi-static time series (QSTS) analysis. First, a design philosophy for QSTS analyses of distribution systems is discussed. Two different time window selection methods that adhere to this philosophy are then presented: implicit temporal load capability (ITLC), and a risk-aware method. Both methods perform analytical time 5 window selection and produce feasible control paths through a time-varying nodal injection forecast.
Solution algorithms and simulation results are presented, demonstrating significant improvements in both computational efficiency and solution quality when compared to traditional QSTS methods that use predetermined, uniform time window divisions.
Chapter5 presents a mathematical model for asynchronous distribution state estimation (ADSE).
Measurements in contemporary distribution environments are discussed. A general ADSE model is formulated, and then a specific model is defined and described. By design, the ADSE measurement
Jacobian is the same as the measurement Jacobian that appears in the classical formulation for syn- chronous static state estimation, so traditional solution methodologies may be applied. Simulation results demonstrate that the ADSE produces more accurate state estimates in a variety of cases.
Chapter6 summarizes the contributions of this thesis. Possible extensions and future research directions are also discussed. 6
Chapter 2: Distribution System Modeling
This chapter provides background information related to the modeling and analysis of electric power distribution systems. The notation and models presented here will be utilized throughout the re- mainder of this thesis. First, the general structure of a multi-phase distribution system is presented and typical system constraints are stated. Next, steady-state component models are reviewed. Fi- nally, the injection variation models are presented; these are central to the technical contributions of this thesis, and link together the problem formulations presented in Chapters refch-dlc,4, and5.
2.1 Distribution System Structure
This thesis considers n-node distribution systems that have a single three-phase electrical inter- connection to a transmission system though a substation. The distribution substation serves as the slack bus with balanced three-phase voltages.
Distribution systems typically have unbalanced, multi-phase1 networks, which are often radial in structure. Network branches include lines, transformers, switches, and voltage regulators. Loads, distributed generation units, energy storage systems, and capacitor banks may be connected at any of the n − 3 nodes downstream of the substation.
The distribution network illustrated in Figure 2.1 is indicative of the typical scale and structure of power distribution systems. This network, which is located in Enola, Pennsylvania and operated by PPL Electric Utilities, will be referred to several times throughout this thesis. Network properties are summarized in Table 2.1.
1Multi-phase refers to one-, two-, or three-phase 7 5 A B C C A B C A B C 4 C 6 C A B C B A 2 C B A B A B C B A A B 3 A B C A B A A B C C A B C A A A C A B A C A A B C A B C C A B Comments 12.47 kV nominal, 10 MVA rating - - - Gang-operated banks3.9 supply MVAR (nominal a at 12.47 total kV) of up to B A 1 B C B C A A B C 1 6 C 426 C 2055 2556 PPL distribution test circuit summary. A B C C Number C B A B C C B Table 2.1: C r A B C o t i c a e e e p n n n a A i i i L L L C
C Φ Φ Φ Φ 3 2 1 3 A C 2 C ) A B A n C C o i t a Multi-phase buses Nodes Multi-phase loads Three-phase capacitor banks Component Three-phase substation t h s B c h t b i c t u n i w C o S i w S ( t
S a d
1 t
e n s d s s e b a o One-line diagram of the three-phase portion of a 2556-node PPLdistribution circuit. The circuit is located in Enola, PA and u p l u o B S C O L C C A A Figure 2.1: operated by PPL Electric426 Utilities. multi-phase Phase loads labels and at 6 terminal three-phase, buses gang-operated indicate capacitors. the presence of a single- or two-phase branches. The circuit includes 8
2.2 System State and Parameters (Decision Variables)
T 2n The system state x = [θV , |V |] ∈ R is composed of the voltage phase angles and magnitudes at each node. Along with the network model, the system state can be used to compute other relevant system quantities (e.g. currents, or power flows).
The system parameters include u, a vector of control settings, and λ, the injection parameters.
Generally, u is integer-valued and λ is real-valued; their dimensions are problem dependent and will be defined accordingly.
2.3 Constraints
Electrical, operating, and control constraints may be imposed. This section defines these con- straints and their compact representations.
2.3.1 Electrical Constraints
Electrical constraints, imposed by the conservation of complex power, are commonly represented by:
F (x, u, λ) = 0 (2.1)
Several different forms of the electrical constraints exist (e.g., [8–10]). Here, they are stated using the following system of nonlinear algebraic multi-phase power flow equations [11]:
n X 0 = −Pi + |Vi||Vk| (Gik cos θik + Bik sin θik) (2.2) k=1 n X 0 = −Qi + |Vi||Vk| (Gik sin θik − Bik cos θik) (2.3) k=1 for nodes i = 1, ··· , n, where
Pi = Pi(x, u, λ) is the net real power injection at node i,
Qi = Qi(x, u, λ) is the net reactive power injection at node i,
Gik is the real part of Yik,
Bik is the imaginary part of Yik, 9
Yik is the entry of the nodal admittance matrix corresponding to nodes i and k,
|Vi|∠θi is the complex voltage phasor at node i, and
θik = θi − θk is the voltage phase angle difference between nodes i and k.
The relationship between the net nodal power injections and x, u, and λ is summarized in Section 2.5.
The solution to the nonlinear system defined by (2.2)-(2.3) is the power flow solution required to check the operating constraints. Solving the power flow equations is often the most computationally demanding task in the analysis of power systems; reducing the number of power flow solutions required to solve a problem is a recurring theme in Chapters3 and4 of this thesis.
2.3.2 Operating Constraints
The operating (or engineering) constraints are defined by safe operating ranges and equipment ratings. Typical operating constraints include voltage magnitude limits at each of the n nodes, and branch current and/or kVA ratings on each of the nb branches. In contemporary distribution systems, where power factor control is of particular interest, power factor (PF) limits may be imposed on certain branch flows or injections.
In this thesis, the operating constraints are compactly represented by:
G(x, u, λ) ≤ 0 (2.4)
which may include the following real-valued inequality constraints:
min max (2.5) Vi ≤ |Vi| ≤ Vi , ∀i = 1, ..., n
max (2.6) |Ib| ≤ Ib , ∀b = 1, ..., nb
max (2.7) |Sb| ≤ Sb , ∀b = 1, ..., nb
L U (2.8) θVI,b ≤ θVI,b ≤ θVI,b, ∀b ∈ BPF where 10
Ib is the complex current flow on branch b,
Sb is the complex power flow on branch b,
θVI,b ∈ [−π/2, π/2] is the power factor angle of the flow on branch b,
min max Vi and Vi are minimum and maximum voltage magnitude constraints at node i,
max Ib is the current rating of branch b,
max Sb is the apparent power limit (or thermal rating) of branch b,
L U θVI,b and θVI,b are lower and upper PF angle constraints on the power flow on branch b, and
BPF is the set of branches with power factor constraints (typically, |BPF | << n).
Remarks on Power Factor Constraints
“Lower” and “upper” is used (instead of “minimum” and “maximum”) when referring to power
factor and power factor angle constraints in order to avoid confusion related to leading and
lagging terminology. For example, if an acceptable power factor range is unity to 0.9 lagging,
then the “upper” constraint is 0.9 lagging, corresponding to a PF angle of 25.84◦, and the
“lower” constraint is unity, corresponding to a PF angle of 0◦.
Imposing power factor constraints on branch flows is a general approach that requires consider-
ation of downstream real and reactive losses. Power factor constraints may also be formulated
as constraints on individual power injections without the need to consider nonlinear loss terms.
Distribution power factor constraints are further discussed in Section 3.1.
2.3.3 Control Constraints
Additional constraints may be imposed on controllable devices; static control constraints are com- pactly represented by:
H(x, u, λ) ≤ 0 (2.9)
Examples of constraints that may be embedded within (2.9) include switching set points for automatic capacitor banks, transformer taps, or smart inverters. However, some control constraints 11 may require “memory” of previous events; for example, switching dead bands or battery state-of- charge constraints. Such constraints may be represented by:
H(x, U, Λ) ≤ 0 (2.10) where U is a sequence of control settings spanning some operating horizon and Λ is a corresponding sequence of injection parameters. For example, consider a distribution system that includes nbatt battery banks and ncap capacitor banks (i.e., a total of nbatt + ncap controllable devices) that are gang-operated according to a K interval control schedule. Then, U may take the following form:
U = (u1, u2, . . . , uK ) where the control setting for interval k is:
uk = uB1,k, . . . , uBnbatt,k, uC1,k, . . . , uCncap,k
In practice, it may be convenient to concatenate the vector elements of U into a matrix of dimen- sion [K × (nbatt + ncap)]. Continuing the example, a dead band constraint that restricts capacitor bank Ci from switching on consecutive control intervals can be checked using the second order dif- ference of a vector containing the interval-by-interval control settings of capacitor bank Ci. This constraint may be stated as follows:
uCi,k − 2uCi,k−1 + uCi,k−2 ≤ 1 , k = 3,...,K −uCi,k + 2uCi,k−1 − uCi,k−2 ≤ 1
2.4 Branch Component Models
Branch types include multi-phase lines, switches, transformers, and voltage regulators. Table 2.2 lists the branch component models used in this thesis. 12
Table 2.2: Distribution branch component models.
Branch Component Comments Reference Distribution lines Short and medium (π) models. [10] Sectionalizing switches Zero impedance model. [10] Transformers Various connection models. [10] Modeled as wye-connected transformers with secondary Voltage regulators [12] taps only.
2.5 Power Injection Models
Injections refers to loads, distributed generation (DG) units, shunt battery banks, and shunt capacitor banks. At node i, the net real and reactive power injections are:
Pi(x, u, λ) = PLi(x, λ) + PGi(λ) + PBi(u) (2.11)
Qi(x, u, λ) = QLi(x, λ) + QGi(λ) + QBi(u) + QCi(x, u) (2.12)
where
PLi and QLi are the real and reactive load demand at node i,
PGi and QGi are the real and reactive DG injections (e.g., inverted PV) at node i,
PBi and QBi are the real and reactive shunt battery injections through an inverter at node i, and
QCi is the reactive capacitor injection at at node i.
The remainder of this chapter contains detailed mathematical models for the individual injection terms. Relationships between the injections and x, u will be explicitly defined along with each individual injection model. Relationships between the injections and λ are described alongside the injection variation models in Section 2.6.
2.5.1 Load Models
Load models define a relationship between node voltages and the complex power drawn by the load.
Let n`d denote the number of loads. For load Li, connected at node i, a nominal operating voltage
VLi,nom and a nominal complex power SLi,nom = PLi,nom + jQLi,nom are specified. A general load 13 equation may be stated with the following polynomial or ZIP model:
" 2# Vi Vi PLi = PLi,nom(λ) a1 + a2 + a3 (2.13) VLi,nom VLi,nom " 2# Vi Vi QLi = QLi,nom(λ) b1 + b2 + b3 (2.14) VLi,nom VLi,nom where
Vi is the voltage at node i,
VLi,nom is the nominal voltage of load Li,
a1, b1 ≥ 0 are the constant power coefficients,
a2, b2 ≥ 0 are the constant current coefficients,
a3, b3 ≥ 0 are the constant impedance coefficients,
a1 + a2 + a3 = 1 and,
b1 + b2 + b3 = 1.
Setting the appropriate coefficients to unity yields a purely constant power, constant current, or constant impedance load model. Grounded and ungrounded multi-phase load models, constructed from the individual phase-to-ground or line-to-line connected load components, are defined in [10].
Polynomial load models are an attractive choice for this thesis because they have a constant power factor specified by the complex argument of SLi,nom. This is of particular importance in
Chapters3 and4, where accurate power factor information is required to solve load capability with respect to power factor constraints.
2.5.2 Distributed Generation Model
Distributed generation (DG) units, such as photovoltaic arrays, are modeled as negative, grounded, constant power loads that may vary with λ. Let ndg denote the number of distributed generation 14 injections. The nodal injection model is:
PGi = −PGi,nom(λ) (2.15)
QGi = −QGi,nom(λ) (2.16)
where PGi,nom and QGi,nom are the nominal real and reactive powers of DG Gi.
2.5.3 Shunt Battery Model
Shunt battery banks are modeled as bidirectional, grounded, constant complex power devices. Bat- tery systems may inject or absorb with non-unity power factor through the use of smart inverters [13].
Let nbatt denote the number of shunt battery banks. The nodal injection model is:
PBi = uBiPBi,nom (2.17)
QBi = uBiQBi,nom (2.18)
where
uBi ∈ {−1, 0, 1} is the control setting of battery bank Ci, and
PBi,nom and QBi,nom are the nominal real and reactive powers of battery bank Bi.
The possible control settings for battery bank Bi are uBi = −1 for discharging, uBi = 0 for off, and uBi = 1 for charging.
2.5.4 Shunt Capacitor Model
Standard ratings for shunt capacitor banks at distribution-level voltages vary in discrete steps, typically between 50 and 800 kVAR per phase [14]. Capacitor banks are modeled as grounded, constant susceptance (impedance), devices that inject purely reactive power into the distribution system when energized. Let ncap denote the number of shunt capacitor banks. The nodal injection model is: 2 Vi QCi = −uCiQCi,nom (2.19) VCi,nom 15 where
uCi ∈ {0, 1} is the control setting of capacitor bank Ci,
QCi,nom is the capacitive reactive power rating of capacitor bank Ci, and
VCi,nom is the nominal voltage of capacitor bank Ci.
The possible control settings for capacitor bank Ci are uCi = 0 for off or de-energized and uCi = 1 for on or energized.
Device Indexing
The power injection models defined in this section have leveraged a node indexing scheme (e.g., capacitor Ci is connected to node i). However, since many controllable devices are gang-operated across phases at a bus, it is more convenient to use a device indexing scheme in which devices of a common type are numbered consecutively. For example, capacitor bank C2 in the PPL circuit
(Figure 2.1) collectively refers to capacitive injections at nodes 1457a, 1457b, and 1457c. Throughout the remainder of this thesis, use of the device indexing scheme will be indicated by the use of boldface type. Circuit diagrams, provided throughout this thesis, will indicate the locations of these devices.
2.6 Injection Variation Modeling
This thesis studies how systems will or should be controlled in response to varying injection conditions captured by the injection parameter λ. Thus, an explicit model of how λ influences
(2.13)-(2.16) is important. Two models are described:
The spatial model captures variations in non-discrete injections, including loads and DG. λ is
unitless. This model applies directly to the load capability studies found in Chapter3.
The spatiotemporal model is an extension of the spatial model that includes non-monotonic
injections linked to time. λ has units of time. This model is leveraged for the quasi-static
time series analyses in Chapter4 and the asynchronous state estimation model presented in
Chapter5.
Note that this section does not deal with changing control settings, u. 16
2.6.1 Spatial Injection Variation Model
The spatial injection variation model for those injections which are parameterized by λ (loads and
DG) is stated as follows:
SXi,nom(λ) = SXi,ref + λSˆXi (2.20)
for Xi ∈ {Li, Gi} and nodes i = 1, ··· , n, where
λ ∈ R is the unitless injection variation factor,
SXi,ref = PXi,ref + jQXi,ref is the initial complex injection from device Xi, and
SˆXi = PˆXi + jQˆXi is the complex injection variation direction for device Xi.
Equations (2.13)-(2.20) provide the relationships necessary to express the net nodal real and reactive injections in terms of x, u, and λ by substitution into (2.11) and (2.12).
The various injections and the injection variation direction information are now collected into complex vectors of dimension ninj = n`d + ndg + nbatt + ncap.
T Snom(λ) = SL,nom(λ), SG,nom(λ), SB,nom, SC,nom (2.21) T Sref = SL,init, SG,init, SB,ref, SC,ref (2.22) T ˆ n ncap S = SˆL, SˆG, 0 batt , 0 (2.23)
where
Sref is a vector of reference power values,
subscripts L, G, B, and C refer to sub-vectors of load, distributed generation, battery, and capacitor
injections, respectively,
0d is a zero-vector of dimension d, and
superscript T denotes transpose.
Sref will typically contain initial conditions for loads and DG injections, and nameplate ratings for the discretely switched battery and capacitor bank injections. 17
With (2.21)-(2.23), a single vector equation may be defined to capture all variation in the nominal injections as a function of λ:
Snom(λ) = Sref + λSˆ (2.24)
Constant Sˆ may be used to model injection variation in a specific direction; for example, this is done in load capability studies when studying long-term average load growth [15–19]. More generally, the inclusion of DG injections may be used to study increasing DG penetration over time.
2.6.2 Spatiotemporal Injection Variation Model
When daily or seasonal variations are of interest, (2.24) may not suffice for the following reasons:
An assumption of constant or averaged load growth is generally impractical on these timescales.
It is desirable to include time in the model directly.
In such cases, it is appropriate to replace (2.24) with a different continuous function describing nominal injection variation. In this thesis, it is assumed that a time series forecast of load and DG injection levels is available. A continuous forecast is derived by interpolating between elements of the time series. Linear interpolation, which has been used in QSTS studies including [20–22], is used in the model described below.
ninj Let SF(tk) ∈ C , for k = 1, ..., K + 1 be a time series of forecasted nominal injection levels:
T SF(tk) = SL,F(tk), SG,F(tk), SB,ref, SC,ref (2.25)
where SL,F(tk) ∈ n`d and SG,F(tk) ∈ ndg are the sub-vectors corresponding to load and DG forecasts.
The time series forms K time windows, numbered k = 1, ..., K. Time window k consists of the interval [tk, tk+1), and the duration of time window k is:
τk = tk+1 − tk (2.26)
ˆ ninj Each time window is associated with its own variation vector Sk ∈ C , with units of complex 18
power per unit time. Sˆk is estimated as a constant vector using linear interpolation (higher-order approximations are possible):
SF(tk+1) − SF(tk) Sˆk = Pˆ k + jQˆ k = (2.27) tk+1 − tk
Note that the elements of Sˆk corresponding to controllable devices are equal to zero, as the nominal injections from these devices do not vary continuously (or at all).
The forecasted nominal injection vector over the horizon t ∈ [t1, tK+1] is given as follows:
ˆ SF(tk) + (t − tk)Sk, t ∈ [tk, tk+1), k = 1,...,K Snom(t) = (2.28) SF(tk), t = tK+1
ninj Snom(t) ∈ C is a continuous, piecewise-differentiable function of t, where, by construction:
dS (t) nom = Sˆ , t ∈ (t , t ), k = 1,...,K (2.29) dt k k k+1
In the spatiotemporal model, λ has units of time, and appears when identifying specific injection sets of interest that may occur throughout the forecast. These injection conditions are mapped to specific times using (2.28). Actual nodal injections (as opposed to nominal) are obtained by making appropriate substitutions of the real and reactive parts of the elements of Snom(t) into (2.13)-(2.19).
In Chapter4, a forecast model based on (2.26)-(2.28) will be used to formulate quasi-static time series analysis problems. Two different analytical methods are used to divide an initial injection forecast into appropriate analysis and control windows. Both methods seek feasible control paths in order to maintain static security over the forecast horizon. In Chapter5, the same model is applied (over shorter time scales, generally) in order to formulate the asynchronous distribution state estimation model. Modeling details specific to these problem formulations are reserved for
Chapters4 and5. 19
Remarks on Injection Forecasting
Although not the focus of this work, injection forecasting is a widely researched area, and is of great importance in contemporary power system operating environments. As injections become more stochastic and diverse, accurate forecast data will become increasingly valuable in maintaining the reliability of existing systems [23], and will aid in planning economically efficient new systems [4].
Load and distributed generation forecasts may be divided into several classes. Classical time series methods include autoregressive moving average (ARMA), autoregressive integrated moving average (ARIMA), and other techniques. Regression-based methods, which can account for external influences such as weather, seasonality, and customer class, are also common [24]. Machine learning and evolutionary programming approaches have been implemented as well; examples include neural networks [25] and genetic algorithms [26].
The spatiotemporal injection variation model is intended for use with forecast horizons of minutes to days. Such forecasts may be referred to as very short-term (on the order of one minute to one hour) or short-term (on the order of one hour to several days) [23,27,28]. In this thesis, it is assumed that appropriate nodal injection forecasts are available in advance, and are specified either through the time series SF, or through a sequence of one or more injection variation directions in Sˆ. 20
Chapter 3: Distribution Load Capability
This chapter presents results related to distribution load capability analysis. First, transmission and distribution load capability literature is reviewed, and then the general load capability problem is presented. Next, a power factor-based estimator of distribution load capability is formulated and presented alongside solution algorithms and simulation results. A series acceleration technique is then applied to improve the convergence speed of the power factor-based and current-based esti- mators. Finally, a condition for linear convergence of the power factor-based estimator is briefly discussed, which further strengthens the case for the use of series acceleration. A full derivation and of the convergence condition and an illustrative example are presented in AppendixA.
3.1 Literature Review
In general, load capability2 (LC) studies are used to estimate how far injections can vary in a certain direction while a set of static security constraints remains satisfied. Early load capability work focused on power transmission systems, and can be classified as the branch of steady-state stability analysis formulated using non-linear optimization, as opposed to direct methods, [29, 30], continuation methods [31–33], or approximations with linear optimization [34], for example.
The original presentation of what is now refereed to as load capability was used to estimate the maximum loading limits of transmission systems with respect to the steady-state voltage stability limit (i.e., the electrical constraints, (2.1)) [15]. Later transmission LC work incorporated inequality constraints as well [16,35, 36].
Transmission load capability is usually computed with respect to generator reactive power or voltage stability limits [15, 35, 36]. Distribution system loading, on the other hand, is more often limited by equipment ratings than by stability limits. Distribution load capability (DLC) formula- tions, therefore, typically focus on operating constraints including branch current ratings [17, 18],
2The term load capability is used for historical reasons; the work presented in this thesis could be more generally referred to as injection capability. 21 nodal voltage magnitude limits [18], or thermal limits [19]. This chapter will present a DLC prob- lem formulation with nodal power factor constraints. Power factor (PF) constraints are of particular interest in evolving distribution system environments, where:
(Near) unity PF injections through grid-tie inverters can substantially reduce real power de-
mand at the substation with a minimal impact on reactive power demand, reducing the ratio
of billable real power to generated apparent power.
Distribution system operators may correct substation power factors as an ancillary service to
upstream transmission systems.
Customers may be subject to low PF penalties (e.g. [37]).
Options for controlling real and reactive power flows are becoming more prevalent, including:
– Switched capacitor banks (reactive power delivery).
– Grid-scale batteries (real power delivery or absorption).
– “Smart” grid-tie inverters, which can control their injection PF over a specified range [13].
3.2 General Load Capability Problem Formulation
The general load capability problem relies on the spatial injection variation model defined in
Section 2.6.1. The reference vector, Sref, is populated with a set of initial injection conditions and nominal shunt device ratings. Let the fixed control setting be u =u ¯. The solution to the problem is that λ ∈ R which satisfies the following nonlinear optimization problem:
max λ x,λ s.t. F (x, u,¯ λ) = 0 (3.1)
G(x, u,¯ λ) ≤ 0
Recall that the constraints stated in (3.1) are functions of x, u(=u ¯), and λ because the nodal power injections are functions of x, u, and λ as defined in Sections 2.5 and 2.6. The equality constraints will 22 ensure that the system state x is statically secure (stable). The inequality constraints will ensure that equipment ratings are not exceeded and that other relevant quantities fall within their safe or economical operating ranges. The solution to (3.1) represents the maximum injection variation factor λ for which these constraints remain satisfied. Note that λ < 0 indicates an infeasible initial condition, requiring variation in the direction of −Sˆ to restore static security.
3.2.1 Solution Methodology
Various techniques have been used to solve the optimization problem (3.1). The chosen solution methodology may depend on factors such as system size and properties, the number and/or type of constraints, or the qualitative objectives of the study (e.g., speed vs. accuracy). Examples include interior point methods [16], analytically-based predictor-corrector methods [17, 18], and repeated power flow (essentially a line search on λ)[19].
This thesis adopts a modular approach in which individual estimators are defined for each type3 of applicable operating constraint. Different methods may be used to estimate DLC with respect to each operating constraint type, so long as satisfaction of the electrical constraints is also guaranteed.
For example, consider a DLC problem in which the operating constraints include voltage, current, thermal, and power factor constraints. Then G may be partitioned as follows:
T G = [ GV | GI | GS | GPF ] (3.2)
where sub-vectors GV ,GI ,GS, and GPF contain the voltage, current, thermal, and power factor constraints respectively. Then, the following sub-program is defined for the voltage constraints:
λV = max λ x,λ s.t. F (x, u,¯ λ) ≤ 0 (3.3)
GV (x, u,¯ λ) ≤ 0 along with analogous sub-problems for the current, thermal, and power factor constraints.
3Current ratings, minimum voltage magnitude limits, maximum voltage magnitude limits, etc. are different types of operating constraints. 23
The solution to the overall problem is:
λ = min {λV , λI , λS, λPF } (3.4)
This method has the following benefits:
The optimization is effectively broken into several sub-problems with simpler constraint sets.
Any methodology may be used to solve each sub-problem, so long as it guarantees satisfaction
of the electrical constraints.
Oscillatory behavior caused by the active constraint type alternating between iterations is
avoided.
3.2.2 General DLC Solution Algorithm
Algorithm 3.1 is the general distribution load capability solution algorithm. The inputs are the vectors Sref and Sˆ. The algorithm calls independent, iterative sub-algorithms that compute indi- vidual load capability estimates with respect to the different types of operating constraints. Each sub-algorithm is required to update the state, and in turn, the net nodal power injections, by solving the nonlinear power flow equations. The output of Algorithm 3.1 is λ, the solution to (3.1).
Algorithm 3.1: General Distribution Load Capability Solution Algorithm
1: Initialize Sref and Sˆ. 2: Run DLC sub-algorithms to obtain λV , λI , λS, and λPF . 3: Choose the result that satisfies all constraints (3.4):
λ = min{λV , λI , λS, λPF }
4: Update the injections using (2.24). 5: Return the load capability solution λ and the corresponding state x.
This chapter presents a methodology for computing λPF in particular. Additional load capability estimators will be referenced from the literature. 24
3.3 Power Factor-Based Load Capability Estimator
This section presents an analytically-based predictor-corrector method for estimating distribution load capability with respect to power factor constraints. Specifically, Snom(λPF ), the nominal injec- tion set that yields the solution to the optimization problem, is what is “predicted” and “corrected” on each iteration.
In this section, the discussion is restricted to Pb > 0 for branches b ∈ BPF . That is, net reverse real power flow4 is assumed not to occur on the branches with power factor constraints, because there is no practical situation in which such a constraint could be enforced on a branch with bidirectional power flow. Power factor constraints could be enforced on branches with net negative real power
flow so long as it is unidirectional; in such a case, the sign convention on the branch may be reversed in order to apply the method below.
3.3.1 Constraint Transformation
First, power factor angles are transformed into a more convenient measure, referred to here as the power ratio:
Qb βb , tan θVI,b = (3.5) Pb
where Sb = Pb + jQb is the complex branch flow on branch b. The tangent function is a monotonic bijection from the domain θVI,b ∈ (−π/2, π/2) to the range βb ∈ (−∞, ∞); thus, β constraints may
5 unambiguously replace θVI constraints on branches b = 1, . . . , nb with:
L U βb ≤ βb ≤ βb (3.6)
L U where βb and βb are lower and upper power ratio limits on the power flow on branch b. Power ratios are convenient because they may expressed in terms of λ without the use of trigonometric functions, as is shown in the next subsection.
4“Reverse” is defined with respect to the “normal” power flow direction associated with a branch; in a radial system, positive power flow typically radiates outward from the substation. 5While PF constraints may apply to either branch flows or to individual injections, this thesis utilizes the more general branch flow constraints. 25
3.3.2 Analytical Relationship
The relationship between the independent decision variable λ and the constrained quantity βb may be expressed as follows:
Qb Qb,init + λQˆb + νQ βb = = (3.7) Pb Pb,init + λPˆb + νP where
Pb = Pb(x, u,¯ λ) is the real power flow on branch b,
Qb = Qb(x, u,¯ λ) is the reactive power flow on branch b,
Pb,init = Pb(x, u,¯ 0) is the initial real power flow on branch b,
Qb,init = Qb(x, u,¯ 0) is the initial reactive power flow on branch b,
ˆ P ˆ Pb = br Pj is the variation direction of Pb due to real injection variation, j∈Db ˆ P ˆ Qb = br Qj is the variation direction of Pb due to reactive injection variation, j∈Db
br Db is the set of nodes downstream of branch b,
νP = νP (x, u,¯ λ) is the nonlinear real power error term, and
νQ = νQ(x, u,¯ λ) is the nonlinear reactive power error term.
Note the following:
Pˆb and Qˆb are constants; the λPˆb and λQˆb terms account for variation in Pb and Qb due to
nominal downstream injection variation.
ˆ ˆ Pb + jQb 6= 0 (otherwise, the problem is trivial).
νP and νQ account for losses and nonlinear load models; the presence of these terms requires
an iterative approach.
L U U L Let λb and λb be in the injection variation factors that yield βb = βb and βb = βb respectively.
U Without loss of generality, assume that the constraint of interest is βb ≤ βb . Then:
Q + λU Qˆ + ν βU = b,init b b Q (3.8) b U ˆ Pb,init + λb Pb + νP 26
U U Assuming that λb exists, the DLC objective is to find λb . It is important to note that the exis-
U 6 tence of λb is not guaranteed ; this can be detected in advance, and will be discussed in Section 3.4.
U For now, assume that λb exists and is finite.
Remark on Notation
L U The symbols λb and λb have been introduced to represent injection variation factors that yield satisfaction of lower and upper power factor constraints on the flow Sb with equality; these quantities may or may not be equal to λPF . λPF is analogous to λV , which appears in (3.3); that is, λPF is the load capability solution that considers all of the power factors constraints within GPF .
3.3.3 Predictor-Corrector Method
The predictor equation is derived from a linear approximation of (3.8). The approximation is made by ignoring nonlinear error terms, which are assumed to be are small relative to the initial power
flows.
Q + λU Qˆ βU ≈ b,init b b (3.9) b U ˆ Pb,init + λb Pb
U Solving for λb : Q − βU P λU = b,init b b,init (3.10) b U ˆ ˆ βb Pb − Qb
ζ, a dummy variable for λ, and iteration indicies denoted by k, are now introduced to produce the predictor equation: Qk − βU P k ζk = b b b (3.11) U ˆ ˆ βb Pb − Qb
0 0 U ˆ ˆ where Pb = Pb,init, Qb = Qb,init and βb Pb − Qb 6= 0. Note that:
The numerator of (3.11) changes on each iteration, while the denominator is fixed.
U k (3.11) is one-to-one over the domain βb ∈ (−∞, ∞), so ζ is unique on each iteration.
6 U U For example, λb does not exist when, for the given the initial condition and injection variation direction, βb = βb requires Pb < 0. 27
Next, the nominal injections are updated using the spatial injection variation model:
k+1 k ˆ Snom = Sref + λ S (3.12)
where : k X λk = ζs (3.13) s=1
The corrector step reconciles the error caused by ignoring νP and νQ in the predictor step, and
k+1 is performed by solving the power flow equations with Snom. This provides a corrected injection set
k+1 k+1 k+1 k+1 S and an updated state x . Then, the real and reactive flows, Pb and Qb , are calculated and returned to the predictor equation for the next iteration.
K Given a convergence tolerance λ > 0, convergence occurs on iteration K where ζ < λ, and:
K U K X k λb = λ = ζ (3.14) k=1
Note that ζk will be scaled by the magnitude of the denominator of (3.11). For consistent
U ˆ ˆ implementation, the tolerance may be set to λ = δλ βb Pb − Qb for a fixed δλ > 0. This approach is useful when solving DLC multiple times with different parameter sets, for example.
Assuming 1) that the initial condition is feasible, 2) that there are multiple upper and/or lower
L U power factor constraints (e.g., across phases at a single bus), and 3) that multiple λb and/or λb exist, λPF is obtained with:
L U λPF = min min λb , min λb (3.15) L,U b∈BPF b∈BPF
L U If λb or λb does not exist for any b ∈ BPF , then the corresponding entry in (3.15) is set to ∅. The next subsection discusses the existence and other properties of these solutions. 28
3.4 Existence, Finiteness, and Non-negativity
L U When they exist, λb and λb always provide mathematically valid measures of distance to con- straint boundaries, but they do not always provide practical information related to forward injection variation along Sˆ. In particular, finite, non-negative solutions are most useful when DLC is being used to study variation from a feasible initial condition along an injection variation vector Sˆ that has been derived from forecast data. Is it, therefore, useful to identify when finite, non-negative so- lutions to the power factor-based DLC problem exist. Practical reasons for this are listed here using notation applicable to upper PF constraints (analogous statements hold for lower PF constraints):
U Existence: by definition, λb does not exist if there is no injection variation factor λ that
U yields βb = βb with Pb > 0. This implies that forward variation will not yield constraint
satisfaction equality.
U Finiteness: if λb = ∞ , then the predictor-corrector method will not converge. Furthermore,
U U λb = ∞ indicates that the βb ≤ βb constraint itself does not limit injection capability.
U Non-negativity: 0 ≤ λb < ∞ implies that the initial condition is feasible, and that forward
variation will eventually yield constraint satisfaction with equality.
L U Additionally, initial infeasibility can be identified using λb and λb together.
Initial infeasibility: initial infeasibility requires immediate corrective control; the following
condition implies that the initial condition is infeasible:
L U L U {0} ∩ min min λb , min λb , max max λb , max λb = ∅ b∈BPF b∈BPF b∈BPF b∈BPF
where the square brackets denote the construction of an interval of the form [a, b] ∈ R. The problem’s parameters may be used to detect whether any of the above conditions will occur before running 29 the predictor-corrector method. Two definitions are required:
Qb,init βb,init , (3.16) Pb,init ˆ ˆ Qb βb , (3.17) Pˆb
L U ˆ The ordering of βb , βb , βb,init, and βb is used to determine initial feasibility and estimated ranges of non-negative λ values for which PF constraints are continuously satisfied (i.e., feasible regions).
Fourteen parameter orderings (later referred to as “variation cases” or simply “cases”) are listed in
Table 3.1 on the following page and are indexed by r. Table 3.1 was populated with the following assumptions:
L U Lower and upper PF constraints are imposed ∀b ∈ BPF (if not, set βb = −∞ or βb = ∞).
L U The feasible region has a non-empty interior (βb < βb ).
ˆ The power ratio is expected to vary (βb,init 6= βb). 30
Table 3.1: Variation cases for the power factor-based estimator.
† Case, r Parameter Ordering Feasible Region , Er Initial Condition L U ˆ L U 1 βb,init ≤ βb < βb ≤ βb max λb , min λb Infeasible ˆ L U U L 2 βb ≤ βb < βb ≤ βb,init max λb , min λb Infeasible L ˆ U L 3 βb,init ≤ βb ≤ βb ≤ βb max λb , ∞ Infeasible L ˆ U U 4 βb ≤ βb ≤ βb ≤ βb,init max λb , ∞ Infeasible ˆ L U L 5 βb ≤ βb ≤ βb,init ≤ βb 0, min λb Feasible L U ˆ U 6 βb ≤ βb,init ≤ βb ≤ βb 0, min λb Feasible L ˆ U 7 βb ≤ βb,init ≤ βb ≤ βb [0, ∞) Feasible L ˆ U 8 βb ≤ βb ≤ βb,init ≤ βb [0, ∞) Feasible ˆ L U 9 βb < βb,init = βb < βb {0} Feasible L U ˆ 10 βb < βb = βb,init < βb {0} Feasible ˆ L U 11 βb < βb,init < βb < βb ∅ Infeasible L U ˆ 12 βb < βb < βb,init < βb ∅ Infeasible ˆ L U 13 βb,init < βb ≤ βb < βb ∅ Infeasible L U ˆ 14 βb < βb ≤ βb < βb,init ∅ Infeasible † Minimizations and maximizations are over the set b ∈ BPF ∩Br, where Br is the set of branches described by the parameter ordering associated with case r.
Remarks on Feasible Regions
The feasible regions listed in Table 3.1 are estimates based on the linearization of (3.5) to (3.9).
It is possible that the variation case r appears to change between iterations, or that a power flow solution vanishes as a feasible region supremum is approached. Nevertheless, these approximate intervals will be leveraged in order to reduce average computational burden. If case switching causes oscillatory behavior or vanishing solutions, then additional steps may be required to ensure that a valid solution is obtained. For example, a step size limit may be placed on ζk in order to limit the magnitude of the nonlinear error that is accrued on iteration k.
On average, checking feasible regions will improve computational efficiency by helping to select which constraints should be considered on each iteration. For example, if the true system-wide feasible region is [0, ∞), then nothing has to be computed, as PF constraints do not limit injection 31 variation. The estimated system-wide feasible region can is formed as follows:
\ Esys = {Er|BPF ∩ Br 6= ∅} (3.18) r where
Esys is the system-wide feasible region,
Er is the feasible region associated with case r, and
BPF ∩ Br is the set of power factor-constrained branches described by case r.
Table 3.1 includes eight cases (r = 1, 2, 3, 4, 11, 12, 13, 14) with infeasible initial power ratios.
These cases, especially the latter four, imply the need for a corrective control action. In cases r = 1, 2, 3, 4, λ = min Er will restore feasibility to branches b ∈ BPF ∩ Br. There are also two degenerate cases (r = 9, 10) in which the power factor constraint is initially satisfied with equality but no variation in the forecasted direction is permissible. Practically, the degenerate cases require preventative control as well.
3.5 Power Factor-Based Estimator Solution Algorithm
Algorithm 3.2 on page 33 is the power factor-based distribution load capability solution algorithm.
Algorithm 3.2 is called by Algorithm 3.1 when solving the general DLC problem. A flow chart describing Algorithm 3.2 is presented in the Figure 3.1 on the following page. The following notation is introduced for use with the algorithm:
? superscript is a dummy variable that may take on values L or U (referring lower and upper
qualifiers) and
ζmax is a step size limit used to prevent solutions from vanishing. 32
Initialize logicals and parameters, obtain the power flow so- 1 lution x , and compute the branch flows Sb for b ∈ BPF .
Estimate the system-wide Does Esys feasible region, Esys contain zero?
No Yes
Exit the algorithm and Yes Does perform corrective or E = {0}? preventative control. sys
No
Solve DLC with- Yes Is Esys un- out PF constraints bounded?
No
PF constraints No Implement the predictor- satisfied? corrector method.
Yes
Terminate and return the solution if converged.
Figure 3.1: Overview of Algorithm 3.2 33
Algorithm 3.2: PF Based Load Capability Estimator Solution Algorithm
1: Set CONV = F ALSE, define convergence tolerance λ, and define iteration limit kmax. 0 2: Initialize the nominal injection vector Snom = Sref. 1 3: Solve the multi-phase power flow equations to obtain the state x and compute Sb , ∀b ∈ BPF . List possible bounds on the system-wide feasible region:
4: ∀b ∈ BPF , use Table 3.1 to determine the variation case, r. 5: Find Esys using (3.18). Check for shortcuts:
6: if 0 ∈/ Esys (infeasible) or Esys = {0} (degenerate) 7: Exit the algorithm and perform corrective or preventative control. 8: else if sup{Esys} = ∞ 9: Solve DLC while temporarily ignoring PF constraints and denote the solution as λtemp. 10: if GPF (x, u,¯ λtemp) ≤ 0 11: PF constraints do not limit injection capability. Set λPF and exit the algorithm. 12: end if 13: end if Implement the predictor-corrector method, if applicable:
14: for each Element of GPF 15: Set k = 1. ? 16: Set βb to the constraint boundary corresponding to the active element of GPF . 17: while not CONV and k < k n max o k k ? k ? ˆ ˆ max 18: Set ζ = max (Qb − βb Pb )/(βb Pb − Qb), ζ k+1 19: Update the nominal injection set Snom using (3.12). 20: Solve the multi-phase power flow equations to update the state x. 21: if F x, u,¯ λk = 0 (i.e., the power flow equations converge) k+1 k+1 22: Compute the power flows Pb and Qb 23: else 24: PF constraints do not limit load capability. Set ζk = 0. 25: end if k 26: if |ζ | < λ 27: CONV = TRUE 28: end if 29: Set k = k + 1. 30: end while Convergence check:
31: if CONV = F ALSE ? 32: The algorithm did not converge. Use an alternative method to find λb (e.g. bisection). 33: else ? Pk s 34: Record λb = s=1 ζ 35: end if 36: end for Return solution:
37: Return the PF-based load capability solution λPF using (3.15) with ? = L, U. 38: if GPF (x, u,¯ λPF ) > 0 39: The solution is invalid. Check for oscillatory behavior (case switching), or try an alterna- tive method to find λPF . 40: end if 34
3.6 Simulation Results
Load capability studies were simulated on the 2556-node PPL distribution circuit shown in
Figure 2.1, modified to include hypothetical photovoltaic installations. The modified test circuit is shown in Figure 3.2 on page 36, and has the following features:
Three-phase substation rated for 10 MVA.
Unbalanced loads (individual two- and three-phase loads are balanced across the phases) with
the following minimum and peak nominal total demand levels:
Minimum demand: 6.368 + j0.475 MVA
Peak demand: 9.235 + j2.604 MVA
Capacitor banks C1, C5, and C6 are energized, supplying a total of 1.8 MVAR (nominal).
This information is captured inu ¯.
Hypothetical photovoltaic DG units, collectively rated for 1.9 MW with unity power factor,
were added to the circuit as follows:
Three-phase: one unit with a three-phase rating of 270 kW, at bus 825.
Phase a: six residential installations rated for 90 kW each.
Phase b: six residential installations rated for 90 kW each.
Phase c: three commercial installations rated for 180 kW each.
The operating constraints include voltage magnitude limits at every node, current ratings on every branch, and power factor limits at the substation. In the simulations, the operating constraints that become active are the current rating of 426 A on branch 3, phase b (branch 3b), and the upper
PF constraint of 0.98 lagging at the substation, phase c (branch 1c). With the PF constraint written as a power ratio constraint, the constraints of interest are mathematically stated as follows:
|I3b| ≤ 426 A
β1c ≤ 0.2031 (unitless) 35
The load capability estimators used here include the PF-based estimator described in this thesis, and the current-based estimator presented in [17, 18].
3.6.1 Simulation Parameters
Two different cases are presented. In both cases, the initial condition is the minimum demand level with zero PV injection (i.e. Sref is the same in both cases). The corresponding initial values of the critical variables are as follows:
|I3b,init| = 322.93 A
β1c,init = 0.0782
The load variation vector, SˆL, is also the same in both cases, but the PV injection variation vector, SˆG, is different in each case. This yields different net injection variation directions. The cases could model a typical loading day with different levels of cloud cover, for example. Table 3.2 on the following page provides the following aggregated simulation parameters for Cases 1 and 2:
Pb,init + jQb,init, the initial power flow on branch b,
ˆ ˆ P ˆ PLb + jQLb = br SLj, the variation in Sb due to downstream load variation, j∈Db ˆ P ˆ PGb = br SGj, the variation in Pb due to downstream distributed generation variation, and j∈Db
Sˆb = (PˆLb + PˆGb) + jQˆLb, the net variation in Sb due to downstream injection variation.
Note that the reactive portion of the net complex power variation pattern is the same for both cases because the PV injections have unity power factor and only affect the net real power injections. 36 5 A B C C A B G C A B C 1896 1581 2191 . . . 6 0 0 0 C j j j b ˆ S C B Case 2 A 5293 + 5383 + 3089 + C . . . 0 0 0 B G C B B A B C C 5 2 C B 2 8 1896 1581 2191 B
. . . s 0 0 0 j j j u Net MVA Variation, B C C C B Case 1 G B A B C 5993 + 6083 + 4689 + A . . . 0 0 0 C A B C C A A G G C A B C C C A B C Gb 3500 3500 5600 ˆ . . . P 0 0 0 Case 2 2 − − − m a A B C e r t s C A n w B o D
G D
2800 2800 4000 . . . B C 2 1 PV MW Variation, 0 0 0 B Case 1 2 C − − − C DLC simulation parameters. A B A B C d e t A A c m A 2 e a n e A B r C B n t A s o c B n - Table 3.2: w s 1896 1581 2191 Lb o u . . . ˆ 0 0 0 A B B D Q C A
j j j j G G + D D B G G G Lb ˆ G P A 8793 + 8883 + 8689 + . . . r A B C 0 0 0 o t i Load MVA Variation c a e e e p C n n n a i i i L L L C
A Φ Φ Φ Φ G init 3 2 1 3 1175 1979 1597 A C b, . . . 2 0 0 0 C ) j j j A jQ B n A C + o G C i t a t init h s B G c b, 3 0119 + 3143 + 0422 + h t
. . . b i c P One-line diagram of the modified 2556-node PPL distribution circuit showing the photovoltatic DG units and the energized t h 2 2 2 u n i w Initial MVA Flow A c o S i w S ( n t
S a a d
1 t r
e n s d s s B e b a o u p l u o B b c a S C O L -2 -2 -2 b c b a 1 1 1 Branch Figure 3.2: capacitors. The phase(s) of DG units match those of the buses to which they are connected or downstream of. C C A A 37
3.6.2 Variation Case Identification
Details for identifying the variation case are presented for Case 1. At the branches exiting the
U L substation, the upper power ratio is βb = 0.2031. No lower constraint was specified, so βb = −∞ is assigned. For all three phases, this corresponds to case r = 6 from Table 3.1 on page 30:
L U ˆ βb ≤ βb,init ≤ βb ≤ βb
U The system-wide feasible region is Esys = E6 = [0, minb λb ]. Since the initial power ratios are
L feasible and tending towards the upper power ratio limit, computing λb is not necessary. With
−8 λ = 10 , Algorithm 3.2 returns the following per-phase solutions:
U U U λ1a = 1.9122 λ1b = 2.1505 λ1c = 1.5514
Therefore, λ ∈ Esys = [0, 1.5514] admit feasible injection sets with respect to power factor con- straints. Solving the power flow equations confirms that F (x, u,¯ 1.5514) = 0. It is determined that:
U λPF = min λb = 1.5514 (3.19) b
3.6.3 Numerical Results
Load capability results are given in Table 3.3. Simulations were performed using the constant impedance, constant current, and constant power load models presented in Section 2.5.1. Using a
−8 −8 power flow solver tolerance x = 10 and load capability tolerance λ = 10 , LC results for each load model were identical to within 10−4 per unit (which corresponds to approximately 333 VA per phase at the substation), but converged at different speeds. 38
Table 3.3: Aggregated injections and injection variation parameters for the DLC simulations.
Injection Variation Factors Critical Variables Simulation λPF λI λ β1c |I3b| Case 1 1.5514 1.5042 1.5042 0.2000 462.00 A Case 2 1.4347 1.6609 1.4347 0.2031 443.08 A
The following remarks are made regarding the results in Table 3.3:
In Case 1, the current rating on branch 3b limits load capability to a variation factor of
λI = 1.5042. The branch 1c power factor constraint limits load capability to λPF = 1.5514.
Overall load capability is therefore limited by current ratings to λ = 1.5042.
In Case 2, increased PV supply released additional branch capacity and λI increased by more
than 10% to 1.6609. However, the reduction in net real power demand increased the substation
power ratios, and λPF dropped by about 7.5% to 1.4347, thus reducing overall load capability
to λ = 1.4347.
The results illustrate the fact that while distributed PV installations (and distributed gen-
eration in general) release branch capacity, they may result in PF constraint violations that
reduce overall load capability. This circuit exemplifies the need for power factor-based load
capability studies in distribution systems with growing DG penetration.
3.6.4 Convergence Behavior and Series Acceleration
Linear convergence behavior was observed for both the current- and PF-based estimators in studies using each of the three load models [38]. It is possible to leverage this behavior by applying a convergence acceleration technique; specifically, Aitken’s delta-squared process was applied to reduce the number of iterations required to obtain the load capability solutions [39].
Aitken’s delta-squared process transforms a linearly converging series into an “accelerated” series, which converges to the same solution with fewer iterations. Note that separate series exist for the
k K PF- and current-based estimators. For the PF-based estimator, the original series is {λ }k=1 as defined in Section 3.3.3. The terms of an accelerated series {ξk} are generated from three terms of 39 the original series as follows:
k+2 k+1 2 k+2 k+2 (λ − λ ) ξ = λ − k+2 k+1 (3.20) λ − 2λ + λk
Note that ξ1 and ξ2 are empty.
Aikten’s method was applied to the simulation results on iterations k > 2, and in all cases, the transformed series converged to the correct load capability solution. Aitken’s method reduced the iteration count for the current-based estimator by an average of 85% across all load models.
While the PF-based estimator converged rapidly on its own, Aitken’s method reduced the number of iterations required with constant power load models by an average of 41%. Results are shown in
Table 3.4.
Table 3.4: DLC series acceleration results.
Standard Algorithms Accelerated Algorithms Simulation Load Model λPF λI λPF λI Z 15 142 15 29 Case 1 I 12 143 12 26 P 25 146 14 13 Z 13 156 13 30 Case 2 I 11 157 11 27 P 13 157 8 13
3.7 Convergence of the Power Factor Based Estimator
Motivated by the observation that the PF-based estimator appears to converge linearly and rapidly, a formal convergence analysis for the power factor based estimator of load capability has been conducted. The convergence analysis is presented in AppendixA. Using constant power injection models, the following results are presented:
An exact condition for linear convergence of Algorithm 3.2, based on the contraction mapping 40
theorem [40].
An approximate convergence condition that leverages typical assumptions regarding distribu-
tion systems.
A discussion of convergence regions and convergence rates, given reasonable distribution system
approximations.
A detailed example of the convergence of the power factor-based load capability estimator for
a two node (worst-case) network, including tracing of the convergence regions (e.g. the set of
initial conditions for which convergence is achieved with the given constraints and parameters).
The results of the example problem reveal a convergence region that covers the entirety of a feasible region defined by a substation apparent power constraint of 1.0 per unit, with the exception of the origin, where power ratio is undefined.
3.8 Concluding Remarks
This chapter presented a model for estimating distribution load capability with respect to power factor constraints. The original power factor angle constraints were transformed into computationally- friendly power ratio constraints, and feasible regions were defined for various combinations of con- straint and forecast parameters.
The PF-limited LC estimator provides a tool for evaluating distribution system performance, and can be used to predict undesirable power factor conditions. The estimator can be included in existing load capability formulations without requiring additional inputs.
Aitken’s delta-squared process was leveraged to improve the computational efficiency of DLC algorithms. This is of particular interest when there is a large constraint set, or when DLC solu- tions are required rapidly (e.g., for online analysis) [41]. In Chapter4, series acceleration provides valuable computational savings when used with the implicit temporal load capability method, which effectively solves many DLC problems in succession as part of an online quasi-static time series analysis. 41
Chapter 4: Time Window Selection for Quasi-Static Time Series Analysis
The previous chapter focused on the analysis of distribution systems subject to spatially varying injections. Now, temporal information is integrated into the discussion in order to address online system operating issues. Specifically, this chapter presents analytical time window selection methods for quasi-static time series (QSTS) studies of power distribution systems. The methods presented in this chapter accelerate QSTS studies, enabling use as a practical online analysis and control tool.
First, background information and a literature review are presented to motivate the use of analytically-driven time window selection. Then, the structure and design of an online QSTS analy- sis are discussed. Next, the implicit temporal load capability (ITLC) time window selection method is introduced; the presentation includes the following:
Descriptions of the ITLC inputs and outputs.
A detailed example problem.
The mathematical problem formulation.
The ITLC solution algorithm and control decision sub-algorithms.
Select simulation results.
A discussion of additional applications.
Finally, an alternative perspective is explored. A risk-aware time window selection model, in- spired by the concept of risk-limiting dispatch, is presented. A risk function is defined in terms of a dispatch timeline and constraint violation probability. Optimal control settings are selected as a post processing step. A problem formulation and illustrative simulation results are presented.
4.1 Literature Review
In smart distribution systems, widespread sensing and communication infrastructures enable utility companies to rapidly, and in some cases, autonomously respond to changing system conditions. 42
Real-time fault location, isolation, and service restoration (FLISR) is a major application of these technologies [42]. Smart grid monitoring and control systems also allow grid operators to leverage classical distribution control functions, such as network reconfiguration or capacitor control, to meet real-time control objectives [7].
While real-time control is an essential feature of a smart distribution system, it is not practical to rely entirely on real-time decision making; some level of near-future planning (e.g., day-ahead) is necessary in order to effectively and economically manage network behavior that is growing increas- ingly complex and stochastic. This is defined as the “online” control problem, and generally requires knowledge of real-time system conditions (e.g., measurement data and a system model based on the active control setting), and a forecast spanning some control horizon.
With proper structuting, quasi-static time series (QSTS) analyses provide a suitable framework for approaching the online control problem. With QSTS, a time series of steady-state operating points is used to describe an inherently dynamic system. Long-term dynamics are captured, while short-term dynamics (e.g., transient events caused by capacitor switching) are assumed to be sta- ble and represented as algebraic states [43]. It is assumed that no large disturbances occur, thus preserving long-term dynamical relationships.
In distribution systems, QSTS has typically been used as a long-term planning tool. Various
DG integration studies have used QSTS to investigate how photovoltaics [20], wind generation [44], and energy storage systems [45] will impact control operations on distribution feeders. These works used a one year planning horizon and fixed time steps of one second [20], one minute [46], or one hour [44, 45].
The QSTS planning studies cited above are very computationally intensive. For example, the analysis performed in [20] (one year horizon, one second resolution) required 31,536,000 chonological power flow solutions; even with tremendous processing power and a relatively simple network, it can take hours or days to obtain this many solutions to the 2n nonlinear power flow equations [47].
Clearly, this will not suffice for the aims of this thesis. For any system size, a useful online analysis tool must produce results in seconds or minutes. Then, even if the forecast is inaccurate or if the 43 system model changes unexpectedly, a new control schedule can be rapidly generated.
Several researchers have acknowledged the computational issues associated with QSTS power system studies and have proposed methods for reducing time and/or memory requirements [48]; examples include statistical techniques, which assign matching power flow solutions to times with similar forecasted injection conditions [22, 49], and variable time-step methods, which reduce the overall size of the time series [47]. These methods, however, do not necessarily translate into effective tools for online analysis. All of them lack one fundamental element that is required for an online control scheme: feedback. That is, in the event that a constraint violation is identified, these methods provide no means of capturing the changes to the system model that would likely be initiated by grid operators in order to maintain static security.
This thesis presents QSTS solutions that meet the requirements of an effective online analysis and control scheme for power distribution systems; the methods presented here are consistent with the following philosophy of online QSTS analysis and control:
Time window selection should be analytically-driven.
There is an underlying requirement of static security, which operators will attempt to satisfy
whenever possible.
Quasi-static time series analyses must be carefully designed in order to properly represent an
inherently dynamic power system.
These points are further discussed in the following section.
4.2 Design of QSTS Studies for Distribution System Analysis
There are a number of considerations to take into account when designing a QSTS study for distribution system analysis. Specifically, it is important to select analysis times that will yield mathematically meaningful and practically useful results.
The most fundamental considerations stem from that fact that QSTS analysis is based on the quasi-steady state (QSS) approximation of a dynamical system [50], in which it is assumed that 44
“fast” dynamics (e.g., switching transients) quickly settle to stable equilibrium points, and thus can be represented as algebraic states. In order to satisfy this assumption, a minimum time window between consecutive power flow solutions may be imposed. An appropriate minimum time window duration may depend on the types and sizes of switched devices, or the system inertia, for example.
The minimum time window duration could also account for controller delay or actuation deadbands.
On the other hand, maximum time window durations may be imposed. In order to sufficiently capture the slower dynamics and potentially important system events, consecutive power flow solu- tions should not be too far apart in time. External factors, such as the structure of the local energy market, or regulatory requirements, may also influence the the maximum time window duration.
For an online analysis, the trade off between accuracy and efficiency is also very important.
Therefore, the number of analysis times included in the time series and/or the duration of the operating horizon itself may be limited or determined in advance. The distribution of the analysis times across the operating horizon can also be considered. While many QSTS studies simply use uniform time windows, it may be more cost-effective to spend more computational power on intervals where operating risk or dispatch urgency is relatively high, for example.
The time window selection methods presented in this thesis have been designed with this design philosophy in mind. In general, minimum time window constraints are imposed in order to maintain the validity of the underlying QSS approximation, and non-uniform time window selection strategies are presented in order to concentrate analysis times to where they are needed most.
4.3 Implicit Temporal Load Capability
This section presents implicit temporal load capability (ITLC), the first of two analytical time window selection methods presented in this chapter. ITLC is an extension of the distribution load capability concepts discussed in Chapter3. It uses forecasted nodal injection characteristics to identify critical times when control actions are necessary in order to maintain feasibility. The initial time window structure is refined by sub-dividing the time windows at critical times, and in turn, implicit temporal load capability produces the following: 45
A sequence of nonuniform analysis and control intervals, and
A statically secure control path that spans the injection horizon, if possible.
Note that the objective of ITLC is not to produce an optimal online control sequence; instead,
ITLC seeks to identify when control actions are necessary in order to maintain feasibility over a forecast horizon. ITLC serves as a first step towards optimal online control.
4.3.1 ITLC Inputs
ITLC requires a time-varying forecast of loads and distributed generation injections, and a schedule of planned control actions. In terms of the spatiotemporal injection variation model, this may be fully specified with an initial injection vector Sinit, and the following sequences:
Sˆ = Sˆ1, Sˆ2,..., SˆK (4.1)
T = τ1, τ2, . . . , τK (4.2)
U = u1, u2, . . . , uK (4.3)
where
K is the number of time windows in the forecast,
ˆ ˆ ninj S is a sequence of complex-valued injection variation vectors, Sk ∈ C ,
T is a sequence of real-valued time window durations, τk ∈ R, and
ncap+n U is a sequence of integer-valued control setting vectors, uk ∈ {−1, 0, 1} batt .
Sˆ may be specified directly or computed from a time series forecast SF(tk). In the latter case,
Sinit = SF(t1). The number and duration of time windows, embedded within T , may rely on forecasted power injection characteristics [51], scheduled control actions, or a regulatory framework, for example. At a minimum, ITLC requires a new time window whenever the active control setting changes. These characteristic-driven intervals provide a good initial structure for QSTS studies. 46
4.3.2 Critical Times and Sub-Windows
ITLC sequentially traverses the forecast specified by Sinit and (4.1)-(4.3). Within each time window, distribution load capability metrics are used to identify critical conditions (i.e., feasible injection sets for which one or more constraints are satisfied with equality) that occur within an injection forecast.
Critical conditions are mapped to specific critical times using (2.28). A subset of the initial time windows are sub-divided at the critical times to create sub-windows.7
Time window k will contain Lk ≥ 1 sub-windows, denoted (k, 1), (k, 2),..., (k, Lk). Lk is initially unknown, and is determined as part of the solution. Define tk,l as the start time of sub-window
(k, l) (consequently, tk = tk,1). Let λk,l be the duration of sub-window (k, l). λk,l is the maximum non-negative scalar such that:
1. Snom(tk,l + λk,l) yields a statically secure state x, and
2. tk,l + λk,l ≤ tk+1.
The first condition stated above is satisfied by finding constraint-limited upper bounds of λk,l using the distribution load capability (DLC) metrics. This is how λ enters the spatiotemporal injection variation model. The second condition preserves the initial time window structure, and is satisfied by imposing the following non-negative upper bound on λk,l:
l−1 e X λk,l = τk − λk,s = tk+1 − tk (4.4) s=1
If no non-negative scalar satisfies the conditions listed above, then λk,l = 0. Therefore, the duration of sub-window (k, l) is:
V I S PF e λk,l = max min λk,l, λk,l, λk,l, λk,l , . . . , λk,l , 0 (4.5) where
7With ITLC, “time window selection” refers the process analytically “selecting” constraint-driven analysis times that sub-divide arbitrary or characteristic-driven time windows into sub-windows. 47
V I S PF λk,l, λk,l, λk,l, and λk,l are DLC estimates computed with respect to voltage, current, apparent
power, and power factor constraints, respectively, in sub-window (k, l),
the ellipsis indicates that more DLC estimates may be included, and
e λk,l is the bound imposed by the time window structure.
Remarks
Snom(tk,l + λk,l) is the nominal injection vector that reflects the forecasted injections at time
tk,l + λk,l. The actual injections, which are also a function of the active control setting and
the system state, are given by the nonlinear injection functions (2.13)-(2.19).
e e It follows from (4.4) that λk,1 = τk automatically. If λk,1 = λk,1, then time window k contains
no critical times and sub-window (k, 1) spans the full time window; otherwise, the time window
is sub-divided.
A negative DLC estimate indicates that Snom(tk,l) yields an infeasible state. Using (4.5), the
sub-window duration is λk,l = 0, and corrective control is required before further variation.
4.3.3 Unscheduled Control Actions
When a critical time is identified, an unscheduled control action is attempted in order to maintain feasibility. The specific control selection algorithm is modular and may be chosen by the user. After the injection space is updated to account for the selected control action, a new sub-window begins, and the ITLC continues to traverse the injection forecast.
If a critical time is identified and no available control action yields a feasible result, then ITLC will continue to traverse the forecast and identify sustained constraint violations. An operator may use this information to determine whether a violation ride-through is viable, or if an emergency condition exists within the forecast.
∗ The realized control setting in sub-window (k, l) is uk,l. Since λk,l can be zero, one or more unscheduled control actions may occur at the beginning of a time window in order to relieve an initial constraint violation. 48
4.3.4 Scheduled Control Actions
At the beginning of time window k, ITLC compares the scheduled control settings uk−1 and uk in order to determine whether or not any actions were planned in advance. If these scheduled control vectors are the same, then no action is taken. Otherwise, a scheduled control algorithm is entered, which will determine if either of the following conditions are true:
1. The scheduled action has already been executed by the unscheduled control algorithm, or
2. The scheduled action is not feasible with respect to the constraints.
If either of the above conditions are true, then the scheduled actuation is simply ignored. Other implementations are possible; for example, in [52], scheduled control actions are delayed by one or more time windows when they are determined to be infeasible at their originally scheduled time.
4.3.5 ITLC Outputs
Implicit temporal load capability returns a sequence of sub-window durations Λ and an associated sequence of realized control settings U ∗. The solution provides a time-stamped forecast of critical times and control actions, which accounts for the operator preferences or schedule defined in U.
4.3.6 Detailed ITLC Example
A detailed example is provided to illustrate the progression of an implicit temporal load capability study. While ITLC is generally an n-dimensional problem, it is represented here using only the net three-phase substation power injections. The example is illustrated in Figure 4.1, and described below.
Sinit is marked on the vertical axis, and the table at the top of the figure shows the input sequences (4.1)-(4.3). The initial forecast is divided into K = 5 uniform three-hour time windows
(separated by solid vertical lines). In time window k, the Sˆk (represented by aggregated variation at the substation) is equal to the slope of the net load and DR forecast. uk represents scheduled control settings for time window k; here, the control vector has the following form:
uk = [uB uC ] 49
ITLC Example INPUTS
7
5 init Net kVA
3 Uncontrollable Injection Forecast Scheduled Control Critical Condition 1 Resulting Forecast with Control
OUTPUTS 0 3 6 7.2 9 12 13.8 15 Time (hr)
Figure 4.1: ITLC Example. Top: input sequences. Middle: initial and adjusted injection profiles. Bottom: output sequences. The main text of this subsection presents a detailed description of this figure.
where uB is a battery control setting and uC is the capacitor control setting. The scheduled con- trol setting u1 = [1 0] indicates that during time window k = 1, the battery is charging and the capacitor is not energized. At t2 = 3, the battery is disconnected, and no more control actions are scheduled; thus, uk = [0 0] for k = 2,..., 5.
In time window 3, a critical condition is identified. This could represent a branch current rating being met, for example. The critical condition is mapped to a critical time at t = 7.2 hr. Time window 3 is divided into sub-windows (3, 1) and (3, 2) (separated by a vertical dash-dot line). In order to maintain feasibility, the capacitor is energized at t3,2 = 7.2 hr. No further critical conditions are identified with time window 3, and the appropriate entries are recorded in the output sequences shown at the bottom of the figure: the sub-windows have durations of λ3,1 = 1.2 hr and λ3,2 = 1.8
∗ hr, respectively; the control setting uk,l = [0 1] is realized in sub-window (3, 2), and remains the same during sub-windows (4, 1), and (5, 1). 50
Another critical condition is identified in time window 5. This could represent a power factor constraint being met due to the (originally unplanned) capacitor energization, for example. The time window is sub-divided at t = t5,2 = 13.8 hr, and the capacitor is turned off to prevent a constraint
∗ violation: u5,2 = [0 0]. The analysis horizon ends at tK+1 = t6 = 15 hr.
4.3.7 Problem Formulation
Given Sinit and sequences Sˆ, T , and U, the goal of implicit temporal load capability is to find:
Λ = (λ1,1, . . . , λ1,L1 , ...... , λK,1, . . . , λK,LK ) (4.6)
U ∗ = (u∗ , . . . , u∗ , ...... , u∗ , . . . , u∗ ) (4.7) 1,1 1,L1 K,1 K,LK
such that ∀(k, l):
∗ F (x, uk,l, λk,l) = 0 (4.8)
∗ G(x, uk,l, λk,l) ≤ 0 (4.9)
H(x, U, Λ) ≤ 0 (4.10)
where (4.8) represents the power flow equations, (4.9) represents the operating constraints, and
(4.10) represents the control constraints. Each of these constraint sets were described in Chapter2.
Remarks on the ITLC Problem Formulation
Finding Λ requires solving a sequence of nonlinear DLC optimization problems; series acceler-
ation, therefore, is valuable for ITLC.
Control constraints may depend on operational objectives or on the number and type of con-
trollable devices in the system. Possible control constraints include:
– a system-wide switching time deadband (to preserve the QSS approximation).
– device-specific switching deadbands (e.g., in order to prevent chattering).
– switchgear actuation frequency limits (e.g., to limit maintenance or replacement costs). 51
– spatial actuation triggers for individual devices (in order to capture automatic switching
of devices such as transformer taps).
4.3.8 Main Implicit Temporal Load Capability Solution Algorithm
This section presents Algorithm 4.1, the constraint-driven solution algorithm for implicit temporal load capability. Algorithm 4.1 is modular: sub-algorithms for scheduled and unscheduled control actions may be selected by the user. The sub-algorithms employed here use greedy heuristics to return feasible options when possible. Integrating optimal or robust control is of interest and is reserved for future work. Specific control sub-algorithms are presented in Section 4.3.9.
In addition to the necessary calls to control sub-algorithms, the main ITLC algorithm also calls individual DLC estimators such as the power factor-based estimator (Algorithm 3.2). A system- wide deadband of duration D (units of time) is imposed in order to preserve the quasi-steady state approximation; this is tracked using a deadband timer d ∈ [0,D).
The remainder of this section includes:
A flowchart describing Algorithm 4.1 appears in Figure 4.2 on the following page.
Algorithm 4.1 itself appears on on page 53.
Specific remarks on steps of the ITLC solution algorithm appear on page 54. 52
Initialize the injections and the deadband timer, and compute the initial state x. Set (k, l) = (1, 1).
e Set λk,1 = 0 and λk,1 = τk.
Yes Run Alg. 4.2 k = k + 1 Scheduled control? ∗ and set uk,l =u ˜.
No No
Yes e Yes k = K? λk,l = λk,l? Update Snom and x.
No
Run Algorithm 3.1.
LC V I S PF λk,l = min λk,l, λk,l, λk,l, λk,l , ... Update Snom and x. LC λk,l = max λk,l , d
Run Alg. 4.3, set Update Snom and x. ∗ uk,l =u ˜ and d = D.
No
e No λk,l = λk,l? l = l + 1 λk,l ≥ d?
Yes
e Yes λk,l = max{0, d − λk,l}
Return Λ and U ∗.
Figure 4.2: Overview of Algorithm 4.1 53
Algorithm 4.1: Implicit Temporal Load Capability 1: Initialize the deadband timer d = 0. ∗ 2: Initialize the nominal injections Snom(t1,1) = SF(t1) and the control setting u1,1 = u1. 3: Update the nonlinear power injection functions using (2.13)-(2.19). 4: Solve the multi-phase power flow equations to compute the initial state x.
Begin time window k:
5: for k = 1 to K 6: Set l = 1. e 7: Initialize λk,1 = 0 and λk,1 = τk. Check for a scheduled control action:
8: if k > 1 and uk 6= uk−1 and d = 0 then ∗ 9: Run Algorithm 4.2 and set uk,1 =u ˜. 10: Update the nonlinear power injection functions using (2.13)-(2.19). 11: Solve the multi-phase power flow equations to compute x. 12: end if
Begin sub-window (k, l):
e 13: while λk,l < λk,l V I S PF 14: Run Algorithm 3.1 to compute λk,l, λk,l, λk,l, λk,l ,... LC V I S PF 15: Set λk,l = min{λk,l, λk,l, λk,l, λk,l ,...}. LC e 16: Set λk,l = max{min{λk,l , λk,l}, d}. ˆ 17: Update the nominal injections Snom(tk,l + λk,l) = SF(tk) + λk,lS. 18: Update the nonlinear power injection functions using (2.13)-(2.19). 19: Solve the multi-phase power flow equations to compute x.
Check whether the time window ended, or if a new sub-window is required:
e 20: if λk,l = λk,l (i.e., the time window is over) then
21: Set d = max{0, d − λk,l}. 22: else 23: Set l = l + 1 (initialize a new sub-window). e 24: Set λk,l = 0 and λk,l = tk+1 − tk,l. Check the deadband timer to see if a control action is permitted:
25: if λk,l ≥ d then 26: Reset d = D (the deadband timer has expired). ∗ 27: Run Algorithm 4.3: uk,l =u ˜. 28: Update the nonlinear power injection functions using (2.13)-(2.19). 29: Solve the multi-phase power flow equations to compute x. 30: else
31: Set d = max{0, d − λk,l}. 32: end if 33: end if 34: end while 35: end for
Return the ITLC solution:
36: Return Λ and U ∗. 54
Remarks on Algorithm 4.1
Lines3-4, 10-11, 18-19, and 28-29: following any variation along Sˆk and/or discrete
change(s) in the control setting, the injections and the state vector are updated.
Line9: if control is scheduled and the deadband timer is not active (d = 0), then Algorithm 4.2
is called to perform the scheduled action(s).u ˜ is the output of Algorithm 4.2. Details are
provided in Section 4.3.9.
Line 14: distribution load capability estimates are computed at the beginning of each sub-
window. DLC estimation necessarily includes constraint checking steps, ensuring that constraint-
limited sub-window durations are feasible.
LC Lines 15- 16: the minimum constraint-limited LC estimate is recorded in λk,l , and the
realized sub-window duration is recorded in λk,l. These lines are equivalent to (4.5), with d
replacing zero to enforce the deadband constraint.
Line 27: when a constraint boundary is encountered before the end of a time window, Al-
gorithm 4.3 is used to make unscheduled control actions.u ˜ is the output of Algorithm 4.3.
Details are provided in Section 4.3.9.
4.3.9 Control Sub-Algorithms
ITLC was designed to allow users to select control strategies that suit their specific system or oper- ating objectives. This section presents one possible set of choices for the control sub-algorithms; the algorithms described here were used to generate the simulation results that appear in Section 4.3.10.
Algorithm 4.2 is the scheduled control sub-algorithm. Scheduled actions are embedded within
U. It is assumed that u1 is the active control setting at the beginning of the analysis so that no scheduled switching occurs at t1. This main ITLC algorithm will call Algorithm 4.2 to conduct scheduled switching at the beginning of time window k > 1 only if the deadband timer is not active
(i.e., if d = 0) at tk. Within Algorithm 4.2, the feasibility of scheduled control actions is tested; if they are not feasible, then the scheduled control actions are rejected. 55
Algorithm 4.2: Scheduled Control 1: Chooseu ˜ to reflect the scheduled control action. 2: if u˜ requires switching then 3: Update the nonlinear power injection functions using (2.13)-(2.19). 4: Solve the multi-phase power flow equations to compute x. 5: if u˜ is feasible then 6: Keepu ˜ and reset the deadband timer: d = D. 7: else ∗ 8: Restore the previous control settingu ˜ = uk−1,Lk−1 . 9: end if 10: end if
Algorithm 4.3 is the unscheduled control sub-algorithm. When a critical condition is encoun- tered, an unscheduled control action is selected with local optimization. If a constraint violation is unavoidable due to the switching deadband or a lack of available control options, a violation ride-through is attempted. Algorithm 4.3 employs the following:
Constraint-dependent control objectives, which are listed in the steps of Algorithm 4.3.
A local search space including settings that are reachable with a single switching operation.
A greedy selection strategy, which ranks options based on the estimated ability to clear the
violation, and selects the highest ranked option with a feasible result.
Algorithm 4.3: Unscheduled Control LC V 1: if λk,l = λk,l then max min 2: Chooseu ˜ = arg minu{|Vi|−0.5(Vi +Vi )}, where i is the node with a voltage violation LC I 3: else if λk,l = λk,l then max 4: Chooseu ˜ = arg maxu{Ib − |Ib|}, where b is the branch with a current violation LC S 5: else if λk,l = λk,l then max 6: Chooseu ˜ = arg maxu{Sb − |Sb|}, where b is the branch with a thermal violation LC PF 7: else if λk,l = λk,l then L U 8: Chooseu ˜ = arg minu{θVI,i − 0.5(θVI,i + θVI,i)}, where i is the node with a power factor violation 9: end if 10: if u˜ is feasible then 11: Reset the deadband timer: d = D 12: else ∗ 13: Keep the previous control settingu ˜ = uk,l−1 and attempt to ride-through the violation 14: end if 56
4.3.10 Simulation Results
This section presents two simulations: a “traditional” QSTS simulation and an ITLC simulation. In the QSTS simulation, the steady-state operating condition is obtained for each point in a time series injection forecast; in the ITLC simulation, critical times are identified in order to refine the initial time window structure, and control actions are selected in order to avoid constraint violations.
The test system is the 2556-node, multi-phase, radial PPL distribution circuit illustrated in
Figure 2.1. The circuit contains 426 multi-phase loads and 6 three-phase, gang-operated capacitors.
All six capacitor banks are initially de-energized. Utility-captured nodal injection data was available at peak and light daily average loading levels. The three-phase complex substation demand levels corresponding to peak and light loading are 9.191+j3.121 MVA and 3.068+j1.018 MVA, respectively.
The considered operating constraints include current ratings on every branch and an upper power factor constraint of 0.98 lagging on each of the per-phase substation injections, corresponding to an upper power factor angle of 0.2003 radians. Two particular constraints of interest8 are stated as follows:
|I3b| ≤ 462 A
θVI,1c ≤ 0.2003 rad.
With no corrective control, the branch current constraint is violated under peak demand, while the power factor constraint is violated under both peak and light loading conditions.
The AMI data provides no temporal information, so a 24-hour injection profile with uniform one hour time windows (τk = 1 hr, k = 1,..., 24) was constructed by fitting typical demand curves around the known load levels. Minimum loading occurs at t1 = 0 hr (midnight) and peak loading occurs at t17 = 16 hr (4 PM). The resulting injection variation vectors Sˆk for k = 1,..., 24, are complex vectors of dimension ninj = 426. The real and reactive injection forecasts are represented by the net three-phase substation demand in Figure 4.3.
8These are the same constraints encountered in the simulations presented in Section 3.6. 57
Net Substation Demand: Time Series Forecast
10 MW MVAR 8
6
4
2 Net Power (MW / MVAR) / (MW Power Net 0 0 4 8 12 16 20 24 Time (hr)
Figure 4.3: Net three-phase substation real and reactive demand forecasts. Vertical lines de- note time window divisions. The data markers are forecasted demand levels; linear interpolation is used between data markers.
Traditional Quasi-Static Time Series Simulation
Traditional QSTS results are obtained by computing the power flow solution at the beginning of each hourly time window and also at t24 + τ24 (at the end of the final time window). Figure 4.4 on the next page shows the resulting time series of the critical variables, |I3b| and θVI,1c. With hourly time windows, the QSTS simulation captured useful information that can not be extracted from the
AMI data alone; specifically:
The substation, phase c (branch 1c) PF violation is in place at every point in the time series
but experiences relatively small variations throughout the day.
The branch 3, phase b (branch 3b) current rating is violated only under peak loading (4 PM).
On the adjacent forecast samples, (3 PM and 5 PM), |I3b| is within 3% of the 462 A rating.
Shorter, uniform windows (e.g., one second) would help better time-localize critical conditions, but at greatly increased computational cost. Instead, ITLC uses the forecasted nodal injection characteristics to identify critical times in a significantly more computationally efficient manner.
These critical times sub-divide only those hourly time windows in which violations are anticipated, rather than requiring a large number of shorter time windows that span the entire forecast.
58 PF Angle (rad.) Angle PF 0.4 0.3 0.2 0.1 0 -0.1 24 24 20 20 16 16 460 A 16 hr 2.31 hr 12 12 462 A Time (hr) Time (hr) 2 hr 0.2003 rad. 15.93 hr . Bottom: power factor (angle) of the 8 8 b . Solid vertical lines denote time window c 458.1 A Branch 1c Power Factor Lagging Leading 0.1902 rad. Branch 3b Current Magnitude 4 4 Critical variable forecasts with preventative control ac- 0 0 500 400 300 200 100
0.921 0.955 0.995 1.000 0.995 PF Current (A) Current : 0.980 U Figure 4.5: tions at criticalthrough times branch identified by 3,flow ITLC. phase on Top: branch current 1,divisions. magnitude phase Dashed vertical linesvisions. denote critical times/sub-window di-
PF Rating: 462 PF Angle (rad.) Angle PF 0.4 0.3 0.2 0.1 0 -0.1 24 24 20 20 17 hr 16 16 460.7 A 16 hr 12 12 Time (hr) Time (hr) 480.8 A 15 hr 8 8 Branch 1c Power Factor 451.7 A Branch 3b Current Magnitude 4 4 . Bottom: power factor (angle) of the flow on branch b Critical variable forecasts construted by computing the Lagging Leading . Solid vertical lines denote time window divisions. 0 0 c 500 400 300 200 100
0.921 0.955 0.995 1.000 0.995 PF Current (A) Current : 0.980 U power flow solutiondotted at lines each are pointbetween visual in points aide the in only time thebranch and 3, series time do phase series. forecast. not Top: indicate The the current magnitude true through path Figure 4.4: 1, phase PF Rating: 462 59
Implicit Temporal Load Capability Simulation
For the ITLC simulation, all six gang-operated capacitor banks are available and a system-wide switching deadband of D = 0.1 hr is imposed. For illustrative purposes, it is assumed that there is no delay between a critical time and the resulting control operation. Initially, all capacitors are off. Capacitor C2 (300 kVAR/phase) is scheduled to turn on at t14 = 13 hr (1 PM) and off at t20 = 19 hr (7 PM) in order to support the system under peak loading conditions. Therefore, initial control schedule U consists of:
[0 0 0 0 0 0], k = 1, ..., 13, 20,..., 24 uk = [0 1 0 0 0 0], k = 14,..., 19
Note that capacitor locations are indicated on Figure 2.1.
The following paragraphs will outline the steps of Algorithm 4.1 with respect to the simulated scenario. Detailed descriptions are provided for the initialization steps and the first three time windows; an abbreviated description is presented for the remainder of the forecast. The implicit temporal load capability simulation results are illustrated in Figure 4.5 on the preceding page.
Initialization: The main ITLC algorithm begins by initializing the deadband timer at d = 0 hr and initializing the nodal injection vector Snom(t1,1 = 0) to reflect the net midnight injections.
Time window 1 (initial violation): The first for-loop iteration of Algorithm 4.1 covers time window k = 1. The scheduled control action steps are not visited in the first time window, but the QSTS results indicate an initial constraint violation, which means that an unscheduled control action will occur at t1. In sub-window (1, 1), the current-based [18] and power factor-based load capability estimates are as follows:
I PF λ1,1 = 15.002 λ1,1 = −54.522 60 with units of hours, where the limiting current rating is on branch 3, phase b and the limiting PF constraint is on substation phase c. The upper bound on sub-window duration imposed by the time window structure is:
e λ1,1 = τk = 1
Lines 15-16 of the main ITLC algorithm are used to find appropriate sub-window duration λ1,1:
LC I PF PF λ1,1 = min{λ1,1, λ1,1 } = λ1,1
LC e λ1,1 = max{min{λ1,1 λ1,1}, d} = d = 0 hr
e e Since d ≤ λ1,1 < λ1,1, lines 23-29 are executed. Sub-window (1, 2) is created and λ1,2 = 1.
∗ Algorithm 4.3 is called, where the deadband timer is set to d = D = 0.1 hr, and u1,2 = [1 0 0 0 0 0] is selected as a new feasible control setting. This requires energizing capacitor C1, which nominally supplies 200 kVAR/phase.
The non-linear injection functions are updated to reflect the control action, and then the state is updated by solving the multi-phase power flow equations. Energizing C1 relieves the initial PF violation; see the jump in PF angle at t = 0 in Figure 4.5 on page 58.
The next while-loop iteration, which covers sub-window (1, 2), begins with the updated state and
∗ the initially feasible control setting u1,2 = [1 0 0 0 0 0]. With these conditions, the load capability estimates are:
I PF λ1,2 = 15.984 λ1,2 = 3.403 and the sub-window duration is computed as follows:
LC I PF PF λ1,2 = min{λ1,2, λ1,2 } = λ1,2
LC e e λ1,2 = max{min{λ1,2 λ1,2}, d} = λ1,2 = 1 hr
e λ1,2 = λ1,2 indicates that the time window will end, and the deadband timer expires (d = 0). The 61
nominal injections vary along Sˆ1 for a duration of λ1,1 = 1 hr:
Snom(1) = Snom(0) + 1 × Sˆ1
and the updated state is computed by solving the multi-phase power flow equations. Following the corrective control at t = 0, no further critical times were identified in k = 1. The while-loop is exited, and the algorithm passes to the next for-loop iteration, which will cover time window k = 2.
Time window 2 (no violation): No control actions are scheduled at t2 (i.e., u2 = u1), and no critical times are identified within time window k = 2. Thus, no new sub-windows are created, no control actions take place, and d does not reset. At the end of k = 2, the following is recorded:
e λ2,1 = λ2,1 = 1
∗ u2,1 = [1 0 0 0 0 0]
Snom(2) = Snom(1) + 1 × Sˆ2
e Time window 3 (PF violation): No control actions are scheduled at t3. λ3,1 = τ3 = 1 hr. The relevant load capability estimates are:
I PF λ3,1 = 5.2615 λ3,1 = 0.3098 where the limiting current rating is on branch 3b and the limiting PF constraint is on branch 1c.
Therefore: LC I PF PF λ3,1 = min{λ3,1, λ3,1 } = λ3,1
LC e e λ3,1 = max{min{λ3,1 λ3,1}, d} = λ3,1 = 0.3098 hr
e ˆ Since λ3,1 < λ3,1, a critical time has been identified within k = 3. Injections vary along S3 until the critical time:
Snom(2.3098) = Snom(2) + 0.3098 × Sˆ3
e Following the logic of Algorithm 4.1 for d ≤ λ3,1 < λ3,1, sub-window (3, 2) is initialized with 62
e λ3,2 = 0.6902 hr. Algorithm 4.3 is called to perform an unscheduled control action, where the
∗ deadband timer is reset to d = D = 0.1 hr. The new control setting u3,2 = [1 1 0 0 0 0] requires energizing capacitor C2 (300 kVAR/phase). The nonlinear injection functions and the state are updated to reflect this change. Energizing capacitor bank C2 alleviates the PF violation; see the jump in PF angle during time window k = 3 in Figure 4.5. Note that C2 has been energized earlier than originally scheduled in U.
The while-loop repeats, and no further critical times are encountered in k = 3. The deadband timer expires (d = 0), and the following is recorded:
e λ3,2 = λ3,2 = 0.6902
∗ u3,2 = [1 1 0 0 0 0]
Snom(3) = Snom(2.3098) + 0.6902 × Sˆ3
Time windows 4 to 24 (current violation in time window 16): In k = 4,..., 13, no control actions are scheduled and no critical times are identified. These windows proceed in a fashion similar to time window k = 2.
In k = 14, capacitor C2 is scheduled to turn on. Sub-Algorithm 4.2 is entered, but since C2 is already on, there is no change. Moving forward, no additional critical times occur until k = 16. The following outputs are recorded:
e λk,1 = λk,1 = 1 , k = 4,..., 15 ∗ uk,1 = [1 1 0 0 0 0]
A critical time is identified within k = 16 at t = 15.9322 hr when the current rating at branch 3b limits load capability. To prevent a constraint violation, a new sub-window is initialized and Al- gorithm 4.3 returns a decision to energize capacitor bank C3 (200 kVAR/phase). The outputs for 63 time window k = 16 are:
I ∗ λ16,1 = λ16,1 = 0.9322 u16,1 = [1 1 0 0 0 0]
e ∗ λ16,2 = λ16,2 = 0.0678 u16,2 = [1 1 1 0 0 0]
When k = 16 ends, the unexpired deadband timer is at d = 0.0322 hr. This carries over to time window k = 17, in which the deadband expires. No scheduled actions or critical times are encountered in time windows k = 18, 19.
At the beginning of time window 20, capacitor bank C2 is scheduled to turn off. Sub-Algorithm
4.2 is entered, where it is determined that this change is infeasible because it would yield a PF constraint violation on branch 1c. Therefore, this scheduled action is ignored, and the control
∗ setting uk,1 = [1 1 1 0 0 0] is realized in sub-windows (k, 1), k = 17,..., 24.
ITLC Simulation Results Summary: Table 4.1 on the following page summarizes the ITLC simulation results. Sub-windows are listed along with their respective time intervals, durations, limiting operating constraints (where applicable), and realized control settings.
ITLC identified critical times at 12:00 AM, 2:18 AM and 3:56 PM. Time windows k = 1, 3, 16 were each divided into two sub-windows. Algorithm 4.3 made feasible capacitor switching decisions to avoid constraint violations.
Comparison with traditional QSTS analysis: ITLC extracted temporal information that the
QSTS simulation could not provide. For example, ITLC results show that the capacitor actuation at
2:18 AM frees enough branch capacity to delay the overcurrent violation until 3:56 PM, four minutes before forecasted peak demand. This type of information could allow operators to make more informed decisions when choosing whether to switch a capacitor or to ride-through the temporary overcurrent violation. 64 - 1 ··· ··· ··· ··· ··· ··· (24,1) [23,24) [1 1 1 0 0 0] [11 P, 12 A) - - 1 1 [3,4) (4,1) (23,1) [22, 23) [3, 4 A) [10, 11 P) [1 1 0 0 0 0] [1 1 1 0 0 0] before variation. - C1 ··· ··· ··· ··· ··· ··· (3,2) 0.6902 [2.3098,3) [1 1 0 0 0 0] [2:18, 3 A) c 1 VI, - 1 θ (3,1) (17,1) 0.3098 [16,17) [4, 5 P) cos [2,2.3098) [1 0 0 0 0 0] [1 1 1 0 0 0] [2, 2:18 A) - - 1 [1,2) (2,1) (16,2) 0.0678 [1, 2 A) [1 0 0 0 0 0] [1 1 1 0 0 0] [3:56, 4 P) [15.9322 , 16) | b - 3 1 I | [0,1) (1,2) (16,1) 0.9322 ITLC simulation results, organized by sub-window. [1 0 0 0 0 0] [1 1 0 0 0 0] [3, 3:56 P) [12 P, 1 A) [15,15.9322) c 1 VI, Table 4.1: - - - 0 1 θ (1,1) (15,1) [14,15) [2, 3 P) cos [0 0 0 0 0 0] [1 1 0 0 0 0] ∗ k,l ∗ k,l u u ) ) ) (hr) ) (hr) (hr) (hr) k, l k, l k,l k,l k,l k,l λ λ λ λ + + k,l k,l , t , t k,l k,l PF violation caused by an initially infeasible control setting was corrected by energizing capacitor bank t t Time of day (HH:MM AM/PM) Duration Sub-window ( [ Time of day (HH:MM AM/PM) Duration Critical variable Realized control setting Sub-window ( [ Critical variable Realized control setting 65
The simulation required 3,569 power flow solutions, less than 5% of the 86,400 solutions required for a day-ahead QSTS analysis with one second time steps. This computational demand level is comparable to traditional QSTS with uniform 24 second time windows. In any case, ITLC provides the following benefits that are fundamentally excluded from the traditional QSTS method:
Analytical critical time identification — traditional QSTS can only indicate that a critical time
has occurred within the preceding time window.
A feasible control path — QSTS can identify constraint violations, but offers no means of
actively avoiding them.
4.3.11 Additional Applications
Section 4.3.10 presented results for a distribution system that experiences branch current overloading with typical daily peak loading and is subject to substation power factor constraints. These results represent one possible application of the implicit temporal load capability critical time identification method. Examples of additional applications of ITLC include the following:
Detailed analysis of ride-through capability (i.e., the ability to withstand a sustained constraint
violation for some period of time).
Time-localization of transformer tap operations — an example is provided in AppendixB.
Generation of alternative feasible control paths that provide operators with increased control
flexibility; for example, [53] identifies a capacitor-only control path through a day-ahead injec-
tion forecast, and then takes direct load control options into consideration in order to develop
alternative control paths.
4.4 Risk-Aware Time Window Selection
With implicit temporal load capability, time window selection is primarily based on feasibility requirements. While feasibility is a requirement, it is not the sole factor that may drive time window selection. This section concisely presents an alternative “risk-aware” time window selection method. 66
Risk-based constraints and objectives have recently appeared in power systems literature as a means to better handle increasing levels of uncertainty [4,54–56]. Effective online risk management is made possible by smart grid technologies that allow system operators to obtain and react to accurate real-time system information. [4] introduced the concept of risk-limiting dispatch, which is a planning and control framework that accounts for the presence of uncertain injections by accepting a certain level of operational risk. This is a fundamental shift from the traditional practice of maximizing reliability in the worst-case scenario, no matter how unlikely that scenario may be. “Worst-case dispatch” can become prohibitively expensive in a smart grid that serves many stochastic and/or bidirectional injections.
Risk-limiting dispatch has provided inspiration for the risk-aware time window selection model; in particular, the risk quantification model presented in [4] is similar to the one that is utilized here. The stochastic nature of smart grid injections is accounted for by considering time-varying forecast inaccuracy. Time windows are selected by divided the operating horizon into non-uniform intervals in order to “equalize” the risk level in each time window. Analysis times (the time window endpoints) are selected with via constrained nonlinear optimization. Feasible or optimal control settings may then be selected for each time window.
The risk-aware model is similar to implicit temporal load capability in that they are both analyt- ical in nature, and both account for the design considerations discussed in Section 4.2; however, the underlying features driving time window selection are completely different. The risk-aware method is presented as a possible alternative to ITLC, and may be valuable for systems with relatively high levels of injection uncertainty.
The objective of this section is to illustrate the alternative risk-aware time window selection method. In order to do so concisely, a simplified injection model and a small constraint set are employed. The following subsections present a forecast model with uncertainty, a risk quantification model, a risk-aware time window selection problem formulation, a post-processing optimal control step, and simulation results. 67
4.4.1 Forecast Model with Uncertainty
Rather than separately considering real and reactive injections, the injection forecast is assumed to be specified in terms of apparent power. Furthermore, the controllable injections are not included in the forecast, as the control settings are later determined as a post-processing step. Forecast uncertainty is accounted for by including a time-varying error term. The expected load and DG power injections at time t ∈ [tk, tk+1) are contained in the following vector:
S (t ) Sˆ ν (t) ¯ L,F k L,k L Snom(t) = + (t − tk) + (4.11) S (t ) Sˆ ν (t) G,F k G,k G where
¯ n`d+ndg Snom(t) ∈ R is a vector containing the expected apparent power injections from loads and
distributed generation at time t,
SˆL,k and SˆG,k are the sub-vectors of Sˆk corresponding to loads and DG, and
νL(t) and νG(t) are the load and DG error vectors, which may depend on forecast data accuracy,
weather conditions, or real-time electrical data, for example.
For notational compactness, absolute value signs are dropped from apparent power quantities for the remainder of this chapter; that is, S and S quantities will refer to scalar and vector apparent power values, respectively.
The nominal apparent power injection at node i is assumed to be distributed according to:
¯ 2 Si,nom(t) ∼ N Si,nom(t), σi (t) (4.12)
where
i = 1, . . . , n, and
2 2 y ∼ N y,¯ σy is a normally distributed random variable with meany ¯ and variance σy.
The variance of (4.12) is specified along with the forecast, and may vary over time according to historical forecast performance, for example. 68
Remarks on the Choice of Probability Distribution
In this section, Gaussian forecast error is assumed, which by the law of large numbers, is appropriate when aggregating groups of similar injections. For example, empirical studies in [57] have validated the Gaussian error assumption for aggregations of twenty or more homes.
Different error distributions may be more appropriate for point forecasts generated in a different manner [58]. Examples include the following:
Forecasts that individually model every injection from AMI data; it is generally accepted that
forecast errors for individual customers are not Gaussian [57].
Forecasts that aggregate dissimilar loads over a large geographical area; [59] proposes a non-
parametric model for forecasting total demand in Victoria, Australia, while [60] proposes a
hyperbolic error distribution for total load in the California Independent System Operator
(CAISO) and New York Independent System Operator (NYISO) areas.
The risk model presented below only requires that the probability of constraint violation be com- puted from the forecast data. Different distributions may be implemented so long as this probability can be computed.
4.4.2 Risk Model
Like in [4], risk is quantified in terms of:
the probability of violating select operating constraints at a given time, and
a dispatch timeline, which is defined on the next page and illustrated in Figure 4.6.
Let the operating constraints in G be partitioned into G1 and G2, where G1 is the subset of operating constraints that are considered when computing risk. G1 includes constraints whose satisfaction can be evaluated, at least approximately, using only the circuit and forecast parameters.
For example, substation reactive power demand or branch power flows in a radial network can estimated from the nominal injection forecast with only algebraic relationships; evaluating voltage magnitude constraints, on the other hand, requires solving the nonlinear power flow equations. 69
In this section, G1 includes a total three-phase substation apparent power constraint (i.e., a thermal overload constraint), which allows for a straightforward calculation of the overload proba- bility at time t. For the purposes of computing this probability, it is assumed that all capacitors and batteries are de-energized, so a fixed control setting ofu ¯ = 0nbatt+ncap is used. The overload probability at time t is:
Ψ(x, u,¯ t) = Pr [G1(x, u,¯ t) > 0]
max = Pr [S3Φ,nom(t) > S3Φ ] (4.13)
where X S3Φ,nom(t) = Si,nom(t) is the estimated three-phase substation demand at time t, and i=1,...,n max S3Φ is the three-phase substation apparent power rating.
If multiple constraints are included in G1, then a weighted probability of constraint violation may be used instead. Note the operating constraints are defined in (2.4) as a function of x, u, and
λ; here, G1 and Ψ are written as functions of x, u(=u ¯), and t. This is because injection variation parameters λ are not computed with the risk-aware model; instead, nominal injection levels are drawn directly from the (interpolated) continuous-time injection forecast.
The dispatch timeline divides operating decisions into different levels of urgency based on the time available to react to identified system conditions. More urgent conditions are associated with greater risk. For example, an “emergency” response is associated with less reaction time, and therefore greater risk than a “scheduled” action. The dispatch timeline is depicted in Figure 4.6.
The corresponding time horizons used here are as follows:
Emergency: s1 = 0.25 hr
Recourse: s2 = 2 hr
Scheduling: s3 = 12 hr 70
emergency recourse scheduling time, t
real time s1 s2 s3
Figure 4.6: The risk-limiting dispatch timeline, adapted from [4].
Risk Function Definition
The risk function maps the violation probability and the dispatch urgency at time t (as defined in terms of the dispatch timeline) to a scalar measure of risk. The specific choice of risk function is modular; here, a logistic function is used. The logistic function is an attractive choice because it is monotonic and has adjustable parameters to allow the user to penalize certain conditions with different degrees of severity. The risk function is defined as follows:
−1 ρ(x, u,¯ t) = 1 + exp − αΨ(x, u,¯ t) − κ(t) (4.14)
where
α > 0 is the slope or steepness parameter, and
κ(t) is the shift parameter associated different sections of the dispatch timeline.
In Figure 4.7, the risk function is illustrated with the following slope and shift parameters:
α = 10 (4.15)
0.05, 0 ≤ t ≤ s1 (Emergency) κ(t) = 0.15, s1 < t ≤ s2 (Recourse) (4.16) 0.40, s2 < t ≤ s3 (Scheduling) 71
Risk Function at Different Urgency Levels
1
0.75
)
t
;
7 u
; 0.5
x
( ;
0.25 5(z1) = 0.05 (Emergency) 5(z2) = 0.15 (Recourse)
5(z3) = 0.40 (Scheduling) 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 *(x; u7; t)
Figure 4.7: The risk function at times t = z1, z2, z3, which correspond to the emergency, recourse, and scheduling urgency levels as defined in (4.16). Note that the ρ-axis intercepts represent the minimum risk associated with the different urgency levels.
4.4.3 Problem Formulation
A constrained least-squares optimization problem, with a nonlinear objective and linear constraints is used to select the analysis times. The objective is to divide the area under ρ(x, u,¯ t), into M regions that are as close as possible to being equal in area, subject to the constraints. This results in a denser allocation of analysis times where the risk is greater.
The original time series SF(tk) was defined at times denoted tk, for k = 1,...,K + 1. The
∗ analysis times are denoted tm ∈ [t1, tK+1], for m = 1,...,M + 1. The endpoints of the forecast
∗ ∗ horizon are required to be analysis times; that is, t1 = t1 and tM+1 = tK+1. The decision variables,
∗ ∗ therefore, are t2, . . . , tM . The problem is stated as follows:
minimize: ΣM (A − µ )2 ∗ ∗ i=1 m A t2 ,··· ,tM
∗ ∗ subject to: tm+1 − tm ≥ τmin (4.17)
∗ ∗ tm+1 − tm ≤ τmax
where 72
∗ Z tm+1 Am = ρ(x, u,¯ s)ds is the area of region m, ∗ tm
µA is the mean area of regions m = 1, ··· ,M,
τmin is the minimum time window duration, and
τmax is the maximum time window duration.
Remarks
In ITLC, the minimum set of analysis times includes the K +1 time window endpoints defined
by the initial forecast time window structure. In the risk-aware formulation, analysis times are
not selected directly from the time window structure of the forecast, and the final M analysis
and control time windows do not necessarily align with the K initial time windows (except for
at the forecast horizon endpoints, t = t1 and t = tK+1).
Because the number of power flow solutions does not grow with K, a more detailed initial
injection forecast (i.e., large K) may be used with little additional computational overhead.
M is selected in advance, and is necessarily an integer between dtK+1/τmaxe and btK+1/τminc
(inclusive)9. The specific value of M may be determined based on the available processing
power, for example.
4.4.4 Simulation Results
Simulations were performed on the n = 2556 node PPL distribution circuit appearing in Figure 2.1 with a 12-hour injection profile where t1 = 0 hr and tK+1 = 12 hr. The aggregate injection forecast and the results are shown in Figure 4.8, Figure 4.9, and Table 4.2, all of which appear on page 74.
The substation (bus 1) three-phase apparent power forecast with 95% confidence interval bands is shown in Figure 4.8. Substation overload is forecasted to occur under peak loading conditions.
The 95% confidence interval has a width of 3σS(t), where:
9d·e and b·c denote the ceiling and floor functions, respectively. 73
1 σ (t) = v(t)S¯ (t) (4.18) S 3 3Φ,nom t tK+1 0.4 + 0.1, t ∈ 0, t 2 v(t) = K+1 (4.19) t 0.3, t ∈ K+1 , t 2 K+1
Note that v(t) linearly increases from v(0) = 0.1 to v(tK+1/2) = 0.3, then remains constant through v(tK+1) = 0.3. This models increasing relative uncertainty for times that are further into the future, until a maximum relative uncertainty level is reached halfway through the injection horizon.
The slope and shift parameters used in the risk function are given in (4.15) and (4.16). There are M = 10 time windows, and the minimum and maximum time window durations are τmin = 0.25 hr and τmax = 1.8 hr, respectively. The time-varying risk function and the resulting time windows divisions are shown in Figure 4.9.
Quantitative results are summarized in Table 4.2; the top half of the table shows the results of the optimization problem (4.17). Subject to the constraints, analysis times are concentrated where risk is greatest. All time window durations exceed τmin. Time windows 3, 4, 8, 9, and 10, where risk is relatively low, have a duration of τmax.
The bottom half of Table 4.2 shows the post-processed capacitor control results, which are described in Section 4.4.5. 74 0 0 0 0 0 0 10 1.80 12 10.20 0.032 = 10 non- 9 0 0 0 0 0 0 10 M 8.40 1.80 0.032 8 t 8 0 0 0 1 1 1 ) s 6.60 1.80 r 0.033 a h c ( e r t o 6 , F e k m 7 1 1 1 0 1 1 i s i 5.79 0.81 T 0.197 R 4 6 1 1 1 1 1 1 5.51 0.28 Risk associated with the injection forecast. 0.197 2 5 0 0 1 0 1 1 5.22 0.29 0 0.197 1 0
0.8 0.6 0.4 0.2
( x ; u ; t ) ; 7 Figure 4.9: Dashed vertical linesuniform divide time the windows. horizon into 4 0 0 0 0 1 1 3.42 1.80 0.171 3 0 0 0 0 1 0 1.62 1.80 0.095 2 0 0 0 0 0 0 0.68 0.94 0.173 12 1 0 0 0 0 0 0 10 0.00 0.68 0.173 t s a c e r o 8 F m ) A r m V h ( A M (hr) t n 6 , o ∗ m e i A t t Area — 600 kVAR — 900 kVAR — 600 kVAR — 300 kVAR — 900 kVAR — 600 kVAR m V a i t Duration (hr) s T M b Time Window C1 C2 C3 C4 C5 C6 6 u : 4 . S 9 I . ) l t a = C ( t x ) o a 3 % ; ) T m 1 3 Risk-aware time window selection simulation results: time window optimization and post-processed capacitor control settings. 5 7 2 Forcasted total three-phase substation demand S 9 S 0
8 6 4 2 0
optimization 12 10 Table 4.2: Time window
S u b s t a t i o n D e m a n d ( M V A ) 0 = off, 1 = on Figure 4.8: with 95% confidence interval bands. Capacitor settings: 75
4.4.5 Control Selection
Following time window selection, a control setting um is selected for each time window m = 1,...,M as a post-processing step. Here, the control objective for time window m is as follows:
∗ 6 ∗ ∗ min J(x, um, tm) = 10 × JS(x, um, tm) + JV (x, um, tm) (4.20) um where
∗ max JS = max Pr S3Φ(x, um, tj ) > S3Φ (4.21) j=m,m+1 ¯ ∗ max JV = max 0, max |Vi(tj ))| − Vi (4.22) i=1,...,n j=m,m+1 where
∗ ∗ S3Φ(x, um, tj ) is the three-phase substation demand at time tj associated with the probabilistic
forecast and control setting um, and
¯ ∗ ¯ ∗ Vi(tj ) is the voltage at node i associated the with expected injections Snom(tj ).
The nodal voltages required to evaluate the control objective become available after solving the
∗ ∗ power flow solutions at the selected analysis times t1, . . . , tM+1.
JS serves to minimize the probability of substation overload at either endpoint of time window m and is heavily weighted in (4.20). JV penalizes overvoltage violations at either endpoint of time window m, and serves as a “secondary” competing objective. Without JV , all of the capacitors would remain on at all times in order to minimize substation apparent power demand, resulting in undesirable voltage conditions and low power factors throughout the distribution circuit.
Because there were only 26 possible control settings, optimal settings were found via exhaustive search. Three-phase nominal kVAR ratings and the resulting control schedule are shown in the bottom half of Table 4.2. Capacitors were gradually switched on between time windows 3 and 6 to counteract the growing probability of overload. In time window 6, all six capacitors (totaling 3.9
MVAR) were switched on. Capacitor bank C4 (0.3 MVAR) was turned off in time window 7 due to 76 localized voltage rise at the end of the time window. As the overload probability (and risk) dropped off sharply, all capacitors turned off over the next two time windows.
4.5 Concluding Remarks
This chapter presented two time window selection methods for online quasi-static time series analysis of power distribution systems. The methods presented in this chapter follow the design phi- losophy that QSTS analyses should be analytically-driven, and that they must be carefully designed in order to satisfy the assumptions of a quasi-steady state (QSS) approximation of a dynamical system.
First, the implicit temporal load capability (ITLC) model was presented. Distribution load capability metrics were used to identify critical injection conditions for which operating constraints are expected to be satisfied with equality. Using a time-varying nodal injection forecast, critical conditions are mapped to critical times. An arbitrary or characteristic-driven initial time window structure is refined by sub-dividing the time windows at critical times. The result is a sequence of time consecutive sub-windows, and, if possible, an associated sequence of feasible control settings.
Next, an alternative approach was taken with the risk-aware time window selection model. The risk-aware model considers time-varying forecast error, and analysis times are more densely packed where operational risk is relatively greater. Feasible or optimal control setting are then selected for each time-window as a post-processing step.
The online QSTS analysis and control framework provides distribution system operators with a practical tool for rapidly generating and evaluating feasible control paths through injection variation forecasts spanning finite time horizons (e.g., day-ahead). Specific control selection subroutines are modular, and my vary based on the available control options or operating objectives. Computation- ally efficient design allows for additional processing, such as the generation of alternative control paths [53], or complete re-evaluation in the event that improved forecast data becomes available.
Analytical time window selection provides a first step towards computationally-tractable optimal online control for electric power distribution systems. 77
Chapter 5: Asynchronous Distribution State Estimation
The spatiotemporal injection variation model will now be applied to another fundamental tool in power systems analysis: state estimation. This chapter presents a distribution state estimator with explicit consideration of measurement asynchronicity10. A time-varying injection forecast is used to estimate changes in measured quantities that cannot be captured by classical state estimation problem formulations.
First, classical state estimation (SE) literature and several more recent works with applications in distribution systems are reviewed. Next, the background behind the asynchronous distribution state estimator (ADSE) is discussed. Then, the mathematical model for the ADSE is formulated and simulation results are presented. Finally, specific applications and possible directions for deeper analysis are presented.
State estimation (SE) is an integral part of grid monitoring and control systems at both the transmission and distribution levels. While this thesis generally focuses on distribution systems, the problem formulation presented in this chapter is more general, and can also be applied to transmission systems.
5.1 Literature Review
The classical static state estimation formulation for power systems was introduced in [63]. Static state estimation is a widely investigated and highly nuanced topic; as such, this review is by no means exhaustive. More comprehensive surveys of SE literature are provided in [64, 65].
Many different factors may be considered when designing a state estimator. The most funda- mental choice is that of the state variables themselves. The classical, and most common choice of state vector includes the nodal voltage magnitudes and phase angles. Augmented state vectors have also included transformer tap settings or branch power flows, for example [64]. Several distribution-
10In this thesis, “asynchronus” refers to sets of measurements with timestamps that differ by seconds or minutes; this work does not address asychronicity associated with communications delays, or asynchronous parallel data processing (e.g, [61, 62]). See Section 5.2 for more information. 78 focused works have instead elected to use the real and imaginary branch current flows as the state variables, citing improved computational efficiency when handling multiple phases [66, 67].
In general, the number of states is typically far greater than the number of available measurement devices. Therefore, observability is an important consideration. Pseudomeasurements derived from historical loading data are typically included in order to achieve full observability. Observable zones may be identified by examining network topology. This can be used to significantly reduce the number of physical measurement devices required in order to estimate the state. For example, observability can be achieved in a radial system with just one independent measurement per lateral per phase [10].
The types of actual measurements that are usually available include nodal voltage magnitudes, branch current flow magnitudes, real and reactive branch power flows, and real and reactive nodal power injections. Measurement devices may be single-phase or multi-phase. Direct measurement of voltage phase angles is also possible where phasor measurement units (PMUs) are installed.
Well-placed PMUs have been shown to improve observabilitiy indicies and overall state estimation solution quality in tranmission systems [68, 69], but PMUs are presently uncommon in distribution systems [70].
Researchers have also explored different solution methodologies, including the traditional uncon- strained weighted least squares (WLS) approach [62, 63, 66], constrained WLS [71, 72], or heuristic methods such as particle swarm optimization [73].
Regardless of the formulation, all power system state estimators are built with the fundamental assumption that the measurements are synchronized in time. However, this assumption is false. Even with the advent of GPS clock-synchronized PMUs, the vast majority of real-time measurements in distribution systems are not PMU based.
Of course, the lack of truly synchronized measurements does not invalidate existing state esti- mators; reliable solutions are regularly obtained even with a complete lack of synchronization and a large number of pseudomeasurements. Nevertheless, there is room for improvement, especially in smart grid operating environments. The objective of this chapter is to better leverage the informa- 79 tion available to operators of smart distribution systems in order to improve the quality of static state estimates. The state estimator presented here combines time-stamped measurement data from smart grid devices (as discussed in Chapter2) with power injection forecasts in order to approximate a synchronized measurement vector. This vector is then used to perform state estimation using the traditional WLS method. By design, the measurment Jacobian is not affected (i.e., the original and adjusted measurement vectors have the same sensitivity with respect to the states), allowing the use of the classical normal equations and solution methodologies.
5.2 Contemporary Operating Environment
Consider, for example, the PPL circuits discussed in Chapter1. PPL distribution systems include smart meters (AMI) that typically record energy consumption data on fifteen minute intervals, and then transmit recorded information to a data concentrator on an hourly basis. Meanwhile, DA devices typically record and transmit data on intervals of three to fifteen minutes. Even if some devices are synchronized initially, synchronization will eventually be lost when alarm reporting or manual polling occurs. Therefore, centralized operators at PPL are constantly receiving data, but their “current” measurement vector will contain entries of varying ages, with some data as old as fifteen minutes. As AMI data becomes available, it can be used to update forecasts and/or pseudomeasurements.
5.3 General Problem Formulation
The asynchronous distribution state estimation (ADSE) model is a modification the classical model, which is given as follows:
z = h(x) + νc (5.1) where 80
z ∈ Rm is the measurement vector,
x ∈ R2n is the system state,
h : x → z is the (generally nonlinear) measurement function, and
m νc ∈ R is the error vector, which captures meter inaccuracy and modeling error, for example.
In the ADSE model, the measurement vector is adjusted to account for injection variation be- tween the measurement time stamps and the state estimation time. This is accomplished by in-
m troducing an additional error term, νf ∈ R , which is used to relate changes in the injections to changes in the measured quantities:
z = h(x) + νc + νf (5.2)
y ∈ Rm is an adjusted measurement vector, and is defined as follows:
y = z − ν¯f (5.3)
y = h(x) + νc + νf0 (5.4)
where
ν¯f = E[νf ] is the expected value of νf , and
νf0 = νf − ν¯f is a zero-mean error term associated with forecasted injection variation.
(5.4) is the general form of the ADSE model. As is also true with the classical formulation, a key feature that differentiates between implementations of the model is the characterization of the error terms. In the next section, the error terms defined more specifically in order to develop a specialized
ADSE model that specifically applies to radial power distribution systems.
5.4 Specific Problem Formulation
In (5.1), νc captures all of the error, but not all of the possible error sources are actually charac- terized when solving the state estimation problem. The following assumptions are typically applied: 81
m νc ∼ N (0 , Σc), where Σc is a diagonal measurement covariance matrix.
2 The diagonal elements of Σc are σj,c for j = 1, . . . , m, where σj,c is proportional to a manufacturer-
specified accuracy level of measurement j.
The physical models are perfect (i.e., load and network modeling error is not characterized).
The measurements are synchronized and transmitted without delay.
With ADSE, the assumption of measurement synchronization is eliminated, and νf is extracted from νc (the remaining assumptions listed above, and hence the effective treatment of νc as a metering error term, remain the same). The new error terms are assumed to have the following form:
νf ∼ N (¯νf , Σf ), where:
– ν¯f =ν ¯f (t, t0) is the measurement forecast, or the expected change in the measured quan-
m tities due to injection variation between the measurement times stored in t0 ∈ R and
the estimation time t ∈ R.
– Σf = Σf (t, t0) is a diagonal covariance matrix.
2 – The diagonal elements of Σf (t, t0) are σj,f for j = 1, . . . , m, where σj,f = σj,f (t, t0)
characterizes the accuracy of adjusted measurement yj.
m νf0 ∼ N (0 , Σf ) is the measurement forecast error.
The measurement forecastν ¯f (t, t0) is explicitly defined as follows:
↓ ν¯f (t, t0) = −λ ◦ KSˆ (5.5)
where
m λ = t−t0 ∈ R is a vector of measurement “ages”, which will act as injection variation parameters,
K ∈ Rm×2ninj is the measurement sensitivity matrix, h iT Sˆ↓ = Pˆ , Qˆ ∈ R2ninj is a constant vector of real and reactive injection variation directions, and
◦ is the Hadamard product (element-wise multiplication). 82
Substituting (5.5) into (5.4) yields an ADSE model in terms of the varying power injections:
↓ y = z + λ ◦ KSˆ = h(x) + νc + νf0(λ) (5.6)
K contains measurement sensitivities computed with respect to the nominal power injections
(i.e., a first order approximation of ∂z/∂Snom). The same concept was used in the formulation of a load estimation problem in [72]. K takes on the following general form:
∂ |V | /∂P ∂ |V | /∂Q M1 nom M1 nom ∂ |I | /∂P ∂ |V | /∂Q M2 nom M2 nom ∂P /∂P ∂P /∂Q M3 nom M3 nom K = (5.7) ∂Q /∂P ∂Q /∂Q M3 nom M3 nom ∂P /∂P ∂P /∂Q M4 nom M4 nom ∂QM4/∂Pnom ∂QM4/∂Qnom where
|VM1| is the set of metered nodal voltage magnitudes,
|IM2| is the set of metered branch current magnitudes,
PM3 is the set of metered real branch power flows,
QM3 is the set of metered reactive branch power flows,
PM4 is the set of metered real power injections, and
QM4 is the set of metered reactive power injections.
Voltage phase angle measurements are not included in (5.7) because PMUs are uncommon in distribution systems, but they could be included in the general case. Individual entries of K that apply in radial power distribution systems are listed in Table 5.1; the sources or derivations of these entries appear beneath the table. 83
Table 5.1: Elements of the ADSE measurement sensitivity matrix.
† † Measurement Type Real Injection Pj Imaginary Injection Qj
Nodal voltage magnitude, |Vj| 0 0
Branch current magnitude, |Ib| Pb/ |Sb| Qb/ |Sb|
Branch real power flow, Pb 1 0
Branch reactive power flow, Qb 0 1
Nodal real power injection, Pj 1 0
Nodal reactive power injection, Qj 0 1 br † For branch measurements, j ∈ Db . All unlisted relationships are zeros.
The first row of Table 5.1 indicates that voltage magnitude measurements are insensitive to changing injections. These relationships were derived in [10] and later leveraged in [72].
The elements of the second row, which relate branch current magnitude measurements to down-
11 stream injections , are derived here. The sensitivities of interest are ∂ |Ib| /∂Pj and ∂ |Ib| /∂Qj.
The branch current magnitude can be expressed as:
∗ X X P` + jQ` I = I = (5.8) b ` V br br ` `∈Db `∈Db
br Assuming that V` ≈ 1.0 p.u. ∀` ∈ Db , the branch current magnitude can be expressed as:
r P 2 P 2 |Ib| = br P` + br Q` (5.9) `∈Db `∈Db
br The derivatives of (5.9) with respect to individual real or reactive injections at node j ∈ Db are given as follows:
P br P` ∂ |Ib| `∈Db = r (5.10) ∂Pj P 2 P 2 br P` + br Q` `∈Db `∈Db P br Q` ∂ |Ib| `∈Db = r (5.11) ∂Qj P 2 P 2 br P` + br Q` `∈Db `∈Db
11 br Recall that Db is the set of nodes downstream of branch b. 84
The numerator of (5.10) is the sum of all the real power injections downstream of branch b, and the numerator of (5.11) is the sum of all the reactive power injections downstream of branch b. Assuming that the losses are much smaller than the sum of the injections, (5.10)-(5.11) are approximately equal to the following expressions which are subsequently used in K:
∂ |I | P b ≈ b (5.12) ∂Pj |Sb| ∂ |I | Q b ≈ b (5.13) ∂Qj |Sb|
The third and fourth rows relate branch power flows to downstream injections, and are also derived in [10]. Nonlinear error terms (such as those discussed in Chapter3) are ignored in the first order approximation.
The fifth and sixth rows relate measured injections to themselves, and are straightforward.
5.5 Solution Methodology
By keeping νf independent of x, the ADSE measurement Jacobian, which relates the adjusted measurements to the states, is the same as the measurement Jacobian that appears in the classical model. That is:
∂y ∂(z − ν¯ ) ∂z = f = (5.14) ∂x ∂x ∂x
This property allows the classical normal equations to remain in use. In other words, the adjustment to the measurement vector has no effect on the solution procedure. The form of the iterative normal equations associated with (5.6) is:
HT WH ∆x = HT W z − h(x) + λ ◦ KSˆ↓ (5.15) where
H ∈ Rm×2n is the measurement Jacobian, 85
−1 m×m W = (Σc + Σf ) ∈ R is the weight matrix, and
superscript T denotes the matrix transpose.
The normal equations (5.15) can be solved using Netwon’s method, for example. A typical termination criterion is to require ||∆x|| to converge to within a small tolerance x > 0.
5.6 Pseudomeasurements and Virtual Measurements
When the number of actual measurements is not sufficient to achieve observability, or when more redundancy is desired, the adjusted measurement vector may be augmented with pseudomea- surements and/or virtual measurements. A convenient set of pseudomeasurements could include forecasted nodal injections that are already being used to adjust the measurement vector; in this manner, an injection variation forecast can be directly embedded into the pseudomeasurements.
Virtual measurements are pieces of information that can be inferred from physical circuit laws, such as zero current injections at non-existent branches extended beyond feeder terminals. Virtual measurements are not affected by varying injections.
When pseudomeasurements and/or virtual measurements are includes, the measurement weight matrix is a diagonal matrix composed as follows:
−1 W Σ + Σ a c f W = W = Σ (5.16) p p Wv Σv where
Wa is the diagonal weight matrix associated with the actual measurements,
Wp is the diagonal weight matrix associated with the pseudomeasurements, and
Wv is the diagonal weight matrix associated with the virtual measurements.
Typically, virtual measurements have the greatest weights, and pseudomeasurements have the lowest weights. 86
5.7 Simulation Results
Simulations were performed using a data from a 394 bus, radial, three-phase distribution test circuit located in Elmira, New York that is operated by New York State Electric & Gas Corporation
(NYSEG). A one-line diagram of the test circuit is shown in Figure 5.1.
Figure 5.1: One-line diagram of the radial, three-phase, 394 bus distribution test circuit [72]. The circuit is located in Elmira, New York and operated by NYSEG. Note that 396 buses appear in the diagram; two of these buses are disconnected in the simulations. 87
With three phases at every bus, 43 closed three-phase switches (modeled as zero impedance branches), and 59 ungrounded buses, the NYSEG circuit has 991 distinct electrical nodes, yielding a state vector length of 1982.
Average loading data was available for each of the 199 three-phase loads. Of the 199 loads,
21 are single customers with balanced real and reactive nominal power demand, while the rest are unbalanced aggregated loads. There are no DG or storage units within the circuit, and all capacitors are de-energized.
5.7.1 Load Classification Procedure
Each load in the NYSEG circuit was assigned to one of four classes. Balanced three-phase loads were classified as industrial loads (subset IL). The remaining loads were classified as commercial
(CL), residential (RL), or auxiliary (AL) (e.g., street lighting) based on the mean per-customer real power demand on the feeder to which they were connected.
The injection variation vector was designed to model behavior between 7:00 AM and 8:00 AM on a weekday. The scenario features a decrease in residential loading, increases in commercial and industrial loading, and no change in auxiliary loading. The variation of load Li can be described by the spatiotemporal model with one time window (T = 1). This simplifies to (2.20) with units of time assigned to λ and units of complex power per unit time assigned to SˆLi:
SLi,nom(λ) = SLi,ref + λSˆLi (5.17)
where SLi,nom(λ) is taken from the available average loading data and is considered the “true” injection set at time t. The reference power SLi,ref will represent the injections at a time prior to t, that is associated with a specific value of λ, as explained in the next subsection. The magnitudes of 88 the injection variation direction for loads in each class are:
−0.315 kVA/hr, Li ∈ RL 0.222 kVA/hr, Li ∈ CL ˆ SLi = (5.18) 0.115 kVA/hr, Li ∈ IL 0 kVA/hr, Li ∈ AL
Load power factors remain constant as the load magnitude varies; that is, ∀Li, QˆLi/PˆLi = QLi,ref/PLi,ref.
The classification algorithm is presented as a flowchart in Figure 5.2 on the following page. The device indexing convention is used for consecutively numbered three-phase loads, L1 through Ln`d.
Additional notation used in Figure 5.2 is defined as follows:
n`at is the number of laterals,
Lm is the set of loads located on lateral m,
PL,m is the nominal three-phase peak real power load on lateral m, and
P¯L,m is the mean per-customer nominal three-phase peak real power on lateral m. 89 \IL} \IL} m ? m L at L + 1 ` m = CL ∪ { RL ∪ { m > n m = = CL RL Yes Yes \IL} ,m m \IL| L No Yes L m P L No No 50 kW? | 150 kW? = 1 ≥ ≥ = m AL ∪ { ,m ,m ,m L L = ¯ L ¯ P P ¯ P AL . Terminate Load classification procedure. j L Yes = 1. Yes IL ∪ j Figure 5.2: = ; define IL at =1 ` m n } ? m d ` L + 1 { j balanced? = , j j as empty sets; set j > n p j L L ∅ P : partition the loads by Is AL 6 = , j a,b,c X = L RL p , ∩ + Initialize CL m their laterals into , ,m L L No : No P IL m = ,m for L P 90
5.7.2 Measurement Data
Voltage magnitudes and real and reactive power flows are sensed at branching buses (i.e., the buses that have multiple immediate downstream buses). Voltage magnitudes are also sensed at the sub- station. A total of 87 physical devices provide 390 entries into the measurement vector. Addi- tional pseudomeasurements and virtual measurements were added to ensure observability of the three-phase, 394 bus (991 electrical node) distribution system. The measurement data sources are summarized in Table 5.2.
Table 5.2: Measurment data types for the ADSE simulations.
Measurement type Meter count Data points Total data per meter points
Metered voltage magnitudes: |Vi| 44 3 132
Metered power flows: Pb, Qb 43 6 258
Pseudomeasurements: forecasted Pi, Qi 199 6 1194
Virtual measurements: |Ib| = 0 151 3 453 Total: 2037
Asynchronous measurement data was synthesized using Algorithm 5.1. The NYSEG loading data was assumed to represent a snapshot at time t, and the value associated with load Li was assigned to SLi,nom(λ) and collected into the vector Snom(λ). Elements of z were generated by
m moving backwards in time to find “true” injection sets at each time in t0 = t − λ ∈ R .
Algorithm 5.1: Asynchronous Measurement Data Synthesis Algorithm
1: Assume the snapshot (time t) data to Snom(λ). 2: for Measurement j = 1 to m 3: Assign a finite age for measurement j to λj. 4: Use (5.17) to find Sref at tj = t − λj. 5: Update the nonlinear power injection functions using (2.13)-(2.19). 6: Solve the multi-phase power flow equations to compute the state xj. 7: Add Gaussian noise to the value of measurement j associated with time tj. 8: Store the noisy measurement in zj. 9: end for 91
Remarks on Algorithm 5.1
Line3: measurement ages may be assigned randomly over some finite range, or according to
a specific pattern. Note that some measurements may actually be synchronized because they
come from the same device, such as multiple voltage magnitudes sensed at the same bus, or
real and reactive power flows on the same branch.
j Lines7-8: x is the “true” system state at time tj, and zj ∈ R is a single piece of (noisy)
information extracted from that state. Following from the previous remark, there may be one
or more j for which states xj are identical.
5.7.3 Simulation Cases
Table 5.3 outlines the parameters of ten different simulation cases. Note that Table 5.3 refers only to measurements j that belong to the set of physical measurements. The pseudomeasurements and virtual measurements do not vary by case. U[a, b] refers to a uniform distribution over the interval
[a, b]. Forecast error is imposed by perturbing the actual injection variation at each node by a random amount within ±5% of its forecasted value.
Table 5.3: ADSE simulation cases.
Case Ages Measurement Values Weights† Forecast Accuracy
1 yj = zj waj = 1 N/A ↓ 2 yj = zj + λ ◦ KSˆ waj = 1 Perfect ↓ 3 λj = 0.167 hr yj = zj + λ ◦ KSˆ waj = 1 − λj Perfect ↓ 4 yj = zj + λ ◦ KSˆ waj = 1 ± 5% random error ↓ 5 yj = zj + λ ◦ KSˆ waj = 1 − λj ± 5% random error
6 y = z waj = 1 N/A ↓ 7 yj = zj + λ ◦ KSˆ waj = 1 Perfect ↓ 8 λj ∼ U[0, 0.167 hr] yj = zj + λ ◦ KSˆ waj = 1 − λj Perfect ↓ 9 yj = zj + λ ◦ KSˆ waj = 1 ± 5% random error ↓ 10 yj = zj + λ ◦ KSˆ waj = 1 − λj ± 5% random error
† waj is the diagonal element of Wa associated with measurement j. 92
In all cases, the weights for the pseudomeasurements and virtual measurements were set to wp = 0.2 and wv = 5, respectively. Since λj ∈ [0, 0.167] hr, waj ∈ [0.833, 1], where the oldest measurements have the least weight.
5.7.4 Numerical Results
The numerical results of the simulations are given in Tables 5.4 and 5.5. Measurement residuals are listed to show that precision is similar across all cases. The performance metric is the accuracy of
12 the state estimate xest when compared to the “true” state xtrue using a two-norm . xtrue is identical in all ten cases, and was computed by solving the power flow equations with injection set Snom(λ).
Overall, the results indicate that the ADSE model results in more accurate state estimation in a variety of scenarios.
Table 5.4: ADSE simulation results with equal measurement ages of ten minutes.
Case Residual Error: ||xtrue − xest||2 Improvement (vs. Case 1) 1 9.404 × 10−4 0.4089 — 2 9.834 × 10−4 0.4074 0.385% 3 8.508 × 10−4 0.3980 2.666% 4 9.828 × 10−4 0.4074 0.365% 5 8.519 × 10−4 0.3981 2.661%
Table 5.5: ADSE simulation results with random measurement ages.
Case Residual Error: ||xtrue − xest||2 Improvement (vs. Case 6) 6 8.138 × 10−4 0.2950 — 7 8.183 × 10−4 0.2805 4.944% 8 7.661 × 10−4 0.2751 6.750% 9 8.181 × 10−4 0.2805 4.934% 10 7.655 × 10−4 0.2751 6.758%
12 Note that ||xtrue − xest||2 is not normalized to account for the length of the state vector; the two-norms appearing here are sums of 1982 squared measurement error values. 93
Cases 1 and 6 represent implementations of the classical model in which aged, asynchronous measurements are assumed to be synchronized at time t. In Case 1, all of the measurement ages were
fixed at 0.167 hr (10 min). In Case 6, the measurement ages were randomly assigned between zero and ten minutes. The error metric for Case 6 was 0.2950, which represents a 27.86% improvement over Case 1. This demonstrates that measurement age has a significant impact on the accuracy of distribution system state estimates obtained using the classical model.
Tables 5.4 and 5.5 provide direct comparisons between the two classical model cases and the asynchronous distribution state estimator cases. Table 5.4 lists data for cases with uniform mea- surement ages of 10 minutes. Cases 2 and 4 show modest improvements when only adjusting the measurements by adding the λ ◦ KSˆ↓ term to the measurement vector to account for forecasted measurement changes that correspond to forecasted injection variation. Cases 3 and 5 show fur- ther improvements when both adjusting the measurements and reducing the weights of the adjusted measurements (uniformly). The weight adjustment effectively acknowledges a degradation in the quality of the information over time.
Table 5.5 lists data for cases with randomly distributed measurement ages. Qualitatively, the results are similar to those shown in Table 5.4. Some improvement is found by adjusting the measurements, and further improvement occurs when applying the simple weight adjustment.
Both Tables 5.4 and 5.5 show that adding two-sided random error to the injection forecasts
(±5%) had little effect. Adding symmetrical zero-mean error to many data points will, on average, have little or no impact on the overall two-norm error metric. Therefore, this outcome was not surprising.
5.8 Concluding Remarks
In this chapter, the asynchronous distribution state estimator was presented. The ADSE model is a natural and straightforward extension of the classical model. A measurement sensitivity matrix has been defined in order to estimate changes in measured quantities driven by evolving power in- jections. Simulation results show that the ADSE model yields more accurate state estimates, even 94 with relatively old (10 min) measurement data. The formulation has been designed in a manner that allows for the use of classical solution techniques with no adjustment to the classical measure- ment Jacobian. The availability of high-accuracy power injection forecasts enables the practical implementation of the ADSE model in smart power distribution systems. 95
Chapter 6: Conclusion
This thesis has presented tools for the online analysis and control of electric power distribution systems. Online analysis and control methods are needed to handle the increasingly diverse and uncertain injections that are inherent to smart grid operating environments. Fundamental engi- neering requirements have been preserved through careful reformulation of classical problems in a contemporary distribution systems framework. The tools presented in this thesis are modular and computationally efficient, which allows for use in a wide range of systems and operational conditions.
First, the power factor-based estimator of distribution load capability was presented. Power factor (PF) constraints are increasingly relevant in systems that include energy storage systems or distributed generation. The PF-based estimator can be used to obtain system-wide PF information or identify injection conditions for which the set of power factor constraints remains satisfied.
Next, implicit temporal load capability (ITLC), an extension of distribution load capability, was presented as a constraint-driven time window selection method for online quasi-static time series
(QSTS) analyses of distribution systems. With ITLC, arbitrary or characteristic-driven initial time window divisions are refined in order to produce a sequence of analysis and control windows with an associated sequence of feasible control settings. An alternative risk-aware time window selection method was also presented. This method considers time-varying forecast accuracy in order to more densely pack analysis times when operational risk is relatively greater.
Finally, the asynchronous distribution state estimator (ADSE) was presented. ADSE makes static state estimates with explicit consideration of the fact that the available measurement data is not synchronized in time. The vector of asynchonized measurements was adjusted to account for forecasted changes in nodal power injections between the measurement timestamps and the state estimation time. Measured quantities were related to the forecasted power injections through the measurement sensitivity matrix. Simulation results demonstrate improved accuracy compared to the classical state estimation, which relies on the false assumption that distribution system measurements are synchronized in time. 96
6.1 Summary of Contributions
Specific contributions of this thesis include the following:
The power-factor based estimator of distribution load capability.
Application of series acceleration techniques to distribution load capability.
A convergence analysis of the power factor-based estimator.
The implicit temporal load capability time window selection method for QSTS.
A risk-aware time window selection method for QSTS.
The asynchronous distribution state estimator for static state estimation with explicit consid-
eration of non-synchronized measurements.
A measurement sensitivity matrix defined with respect to time-varying nodal power injections.
Simulation results that demonstrate each of the above contributions.
6.2 Extensions and Future Work
This thesis aimed to integrate temporal information into distribution system analysis and control tools; the research presented here presents a necessary step on a path towards optimal online control of smart electric power distribution systems. This section discusses related research and potential technical extensions that may be explored in the future.
6.2.1 Distribution Load Capability
Distribution load capability is a powerful analysis tool, which, in this thesis, has been applied to develop an online operating tool. Since ITLC requires many DLC solutions, each of which can be quite computationally intensive on its own, accelerating DLC studies is of interest. This work presented a convergence study for the PF-based estimator with constant power injection models; future work could include more comprehensive studies that consider other load models or mixed load models. 97
In initial studies presented in this thesis, linear convergence behavior was observed for the current- based DLC estimator, and series acceleration with Aikten’s delta-squared method proved particularly useful. Convergence analysis of the current-based DLC estimator (and others) is also of interest.
6.2.2 Time Window Selection Methods
Potential improvements to the time window selection models include incorporating dynamic con- straints (i.e., constraints that vary over time) and more detailed forecast models, or using analytical methods to determine when the analysis should be recomputed. More detailed forecasts may include very-short term load or DG forecasts, for example. Knowing when to recompute the analysis in order to ensure that the control schedule is based on accurate forecast data is valuable for online control.
The ITLC formulation includes control constraints with memory and modular control sub- algorithms that may be used to select control settings that are optimal within each time window; future work could include optimizing the control schedule over the full injection profile. Since no forecast is perfect, risk-aware ITLC is another interesting extension. Using probabilistic forecasts, critical times could be placed where acceptable risk thresholds are crossed, rather than at critical times associated with a perfect injection forecast.
The risk-aware formulation presented in this thesis could benefit from more detailed modeling of real and reactive injection uncertainty as discussed in Section 4.4.1.
A useful extension to the risk-aware model would be a pre-processing step that selects the optimal number of time windows. This may be done, for example, with one-dimensional k-means clustering along the time series [74]. This could be an important addition when balancing computational expense with solution quality, as at some point, the addition of more time windows will provide little or no additional benefit.
6.2.3 Asynchronous Distribution State Estimation
There are numerous opportunities to apply and further study the asynchronous distribution state estimator. The model presented in this thesis could be improved by incorporating weight optimiza- tion, or by further tuning the measurement sensitivity matrix to account for losses and/or modeling 98 approximations. Characterizing the measurement arrival process could also provide useful insights.
ADSE could be used in smart grid environments to improve overall metering schemes. Potential applications include:
Polling time optimization: intelligently choosing which meters to poll and when could im-
prove average state estimate accuracy over time, subject to data processing and transmission
constraints.
Cyclical performance detection: except for when alarms occur, AMI and DA meters are pro-
grammed to send data at regular intervals; this could impose cyclical patterns onto accuracy
or observability metrics.
Bad data detection
Asynchronous state estimation could also be applied to transmission systems. This would require modifying the ADSE measurement sensitivity matrix to account for a meshed topology and/or the presence of phasor measurement units. 99
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Appendix A: Convergence Analysis for the PF-Based DLC Estimator
Power system nonlinearity requires an iterative solution for the power factor-based estimator of distribution load capability (DLC), which was presented in Chapter3 and in [38]. This appendix presents a convergence analysis for Algorithm 3.2, the power factor-based DLC solution algorithm, with a radial distribution network and constant power injection models. This appendix includes:
A restatement of relevant equations from Chapter3.
A detailed example of the behavior of Algorithm 3.2 on a two node (worst-case) network.
The derivation of an exact linear convergence condition, via the contraction mapping theorem.
An approximate linear convergence condition with typical distribution system assumptions.
A demonstration that with typical distribution system assumptions, the feasible region of
initial power injections defined by a substation apparent power rating of 1.0 p.u. lies within
the region of convergence for most parameter sets, except at the origin (where PF is undefined).
A.1 Estimator and Update Equations
Relevant equations from Chapter3 are repeated in this section. Without loss of generality, assume that an upper PF constraint on branch b is the constraint of interest (i.e., that the objective
U to find λ that yields θVI,b = θVI,b); analogous notation applies when a lower PF constraint is active.
The iterative estimator equation is as follows:
Qk − βU P k ζk = b b b (3.11) U ˆ ˆ βb Pb − Qb
The equation used to update the injections on each iteration is:
k+1 k ˆ Snom = Sref + λ S (3.12) 105 where λk is computed as follows: k X λk = ζs (3.13) s=1
K The estimator algorithm terminates on iteration K when |ζ | < λ, where λ > 0 is a small convergence tolerance. The estimated solution to the power factor-based DLC problem is λ = λK .
A.2 Illustrative Example
An example is provided to illustrate the behavior of Algorithm 3.2 for the two node system shown in Fig. A.1. This represents a network in which all of the injections are lumped together at a node j, and the power factor of the branch flow Sb (flowing from node i to node j) is constrained.
Simulation parameters are listed in Table A.1. Results are presented in Figure A.2 and Table A.2.
i j
Sb P b jQ b R jX ij ij Sj,nom P j ,nom jQ j ,nom
Figure A.1: One-line diagram of the two node system.
Table A.1: Parameters for the convergence behavior example.
Parameter Symbol Value 0 0 Initial injection Snom = Sj,nom 0.56 + j0.12 p.u. Injection variation direction Sˆj 1.00 + j0.80 p.u.
Branch Impedance Rij + jXij 0.01 + 0.04 p.u.
Node i Voltage Vi 1.0 p.u. U Upper Power Factor Constraint cos θVI,b 0.97 Lagging U U Upper Power Ratio Constraint βb = tan θVI,b 0.2506 † −3 λ convergence tolerance λ 10 −8 Power flow convergence tolerance x 10 †A relatively large tolerance has been selected so that the results in Figure A.2 are easy to read. 106
In the n-node case, the convergence of Snom to a fixed point can not be easily illustrated, but in this example, convergence can be shown on the complex plane. Figure A.2 on the following page is interpreted as follows:
0 1. Snom = Sref, is indicated by the “0” marker. For any iteration, net nominal demand will lie
0 ˆ on the hyperplane (here, a line) defined by Snom + λS.
k k 2. Given Snom, the nonlinear power flow equations are solved to obtain Sb , the true complex
k power flow on branch b. Graphically, this flow is connected to Snom by a dashed line.
k k U 3. ζ , calculated with (3.11), is equal to the distance between Sb and the Qb = βb Pb line as
measured in the injection variation direction. The direction of injection variation is indicated
by the vector field in the background.
k+1 0 ˆ 4. Snom is computed with (3.12) and plotted on Snom + λS.
K 5. Steps2-4 repeat until iteration K, where |ζ | < λ.
0 ˆ 6. The final injection set is indicated by a star on the Snom + λS line, and the corresponding
K U power flow of interest Sb is indicated by a star on the Qb = βb Pb line.
Table A.2 indicates that after three iterations, the load capability solution converged to a value of λ = 0.0129. Using (3.12), this corresponds to a final injection set of Snom = 0.5729 + j0.1304 p.u.
Solving the power flow equations and subsequently computing the branch flows yields a power flow
U of Sb = 0.5765 + j0.1445 p.u., for which βb = 0.2506 as required by the constraint.
Table A.2: Numerical results of the convergence example.
k k Pk s Iteration, k ζ λ = s=1 ζ 0 0.0142 0.0142 1 −0.0013 0.0128 2 0.0001 0.0129 107
Convergence Behavior Example
0.145 U - Pb Qb = b 0.140
0.135
^ 2 1 0.130 6S + 3 0 om
0.125 S n Reactive Power (p.u.) Power Reactive
0.120 0 0.560 0.565 0.570 0.575 0.580 Real Power (p.u.)
Legend k Sb : converged power .ow solution on iteration k
k k Snom : .ow estimated from net demand on iteration k Stars indicate -nal iteration
Figure A.2: Convergence behavior example. The vector field indicates the direction of injec- 0 ˆ tion variation. The Snom + λS line contains all possible demand levels. For each iteration, a k k dashed line connects the net demand level Snom to the resulting power flow Sb (which includes U losses). The power flow associated with the converged injection set lies on the Qb = βb Pb line.
A.3 Convergence Conditions
A single iteration of the while-loop within Algorithm 3.2 may be described with the equation below, which follows from (3.12)-(3.13):
k+1 k k ˆ Snom = Snom + ζ S (A.1)
k ∞ This section will show that (A.1) performs a contraction mapping, which implies that {Snom}k=1 converges to a fixed point [40]. A function h is a contraction mapping on metric space (T, d) if there is a scalar 0 ≤ p < 1 such that ∀a, b ∈ T :
d(h(a), h(b)) ≤ pd(a, b) (A.2) 108
Since (A.1) is iterative, let b = h(a):
d(h(a), h2(a)) ≤ pd(a, h(a)) (A.3)
Here, T ⊆ R2ninj , d is the infinity norm, and:
Pk+1 Pk Pk Pˆ nom nom nom k = h = + ζ (A.4) k+1 k k ˆ Qnom Qnom Qnom Q
k k k where Pnom and Qnom are, respectively, the real and reactive components of Snom. Eq. (A.4) is a real-valued restatement of (A.1). This section proceeds by:
1. Defining additional notation.
2. Manipulating h into an explicit fixed point iteration equation.
3. Deriving the exact condition for linear convergence based on (A.3).
4. Showing that, given typical distribution system assumptions and constraints, the convergence
condition is satisfied for a wide range of circumstances.
5. Continuing the example of Section A.2 to trace the applicable convergence region T .
A.3.1 Notation Definitions
For notational compactness, the scripts will be dropped from power ratio constraint symbols, which will be now represented simply by β. The following definitions are also introduced:
−1 γ = (βPˆb − Qˆb) (A.5)
Mij = Xij − βRij (A.6)
2 2 P k + Qk |Ik|2 = j,nom j,nom (A.7) j k 2 |Vj | where 109
Zij = Rij + jXij is the impedance of the path between nodes i and j, and
Vj is the voltage at node j.
A.3.2 Explicit Statement of the Fixed Point Iteration
Substituting (3.11) into (A.4) yields the following:
k+1 k ˆ Pnom Pnom Qk − βP k P = + b b (A.8) ˆ ˆ k+1 k βPb − Qb ˆ Qnom Qnom Q
The numerator of the fraction in (A.8) changes on each iteration, while the denominator remains
k k constant. Pb and Qb are nonlinear functions of the injections downstream of branch b, which can be expressed as follows:
k ˜k X k 2 Pb = Pb + Rij|Ij | (A.9) br j∈Db
k ˜k X k 2 Qb = Qb + Xij|Ij | (A.10) br j∈Db where
˜k X k Pb = Pj,nom (A.11) br j∈Db
˜k X k Qb = Qj,nom (A.12) br j∈Db
The first terms of (A.9)-(A.10) represent power attributed to downstream loads/injections, and the latter terms represent losses. Substituting (A.5), (A.9), and (A.10) into (A.8) yields:
Pk+1 Pk Pˆ nom nom ˜k X k 2 ˜k X k 2 = + γ Qb + Xij|Ij | − β(Pb + Rij|Ij | ) (A.13) k+1 k j∈Dbr j∈Dbr ˆ Qnom Qnom b b Q 110
Comparing (A.4) and (A.13) yields the following equation which will be leveraged later on:
k ˜k X k 2 ˜k X k 2 ζ = γ Qb + Xij|Ij | − β(Pb + Rij|Ij | ) (A.14) br br j∈Db j∈Db
Using (A.6) to simplify (A.13)
Pk+1 Pk Pˆ nom nom ˜k ˜k X k 2 = + γ Qb − βPb + Mij|Ij | (A.15) k+1 k j∈Dbr ˆ Qnom Qnom b Q
Finally, an explicit statement of the fixed point iteration is obtained by expanding the squared current magnitude with (A.7).
2 2 Pk+1 Pk P k + Qk Pˆ nom nom ˜k ˜k X j,nom j,nom = + γ Qb − βPb + Mij (A.16) |V k|2 k+1 k j∈Dbr j ˆ Qnom Qnom b Q
Note the distinctions between the different power quantities that appear in (A.16):
k k Pnom and Qnom are length ninj real-valued vectors of real and reactive injections.
k k br Pj,nom and Qj,nom are the individual real and reactive injections at nodes j ∈ Db .
˜k ˜k Pb and Qb are the sums of the real and reactive injections downstream of branch b.
A.3.3 Statement of the Exact Convergence Condition
Eq. (A.16) is a contraction mapping on (T, || · ||∞) if for some 0 ≤ ρ < 1:
Pk+2 Pk+1 Pk+1 Pk nom nom nom nom − ≤ ρ − (A.17) k+2 k+1 k+1 k Qnom Qnom Qnom Qnom ∞ ∞ 111
The constant ρ is now omitted and strict inequality is used. Using (A.15):
Pˆ ˜k+1 ˜k+1 X k+1 2 γ Q − βP + Mij|I | b b j br ˆ j∈Db Q ∞ Pˆ ˜k ˜k X k 2 < γ Qb − βPb + Mij|Ij | (A.18) br ˆ j∈Db Q ∞
Dividing both sides by |γ| and moving scalar terms outside of the vector norm operation yields the following:
Pˆ ˜k+1 ˜k+1 X k+1 2 Q − βP + Mij|I | · b b j br ˆ j∈Db Q ∞
Pˆ ˜k ˜k X k 2 < Q − βP + Mij|I | · (A.19) b b j br ˆ j∈Db Q ∞
Since the vector norms on either side of the inequality are identical and equal to positive scalars, both sides may be divided by the infinity norm of [Pˆ , Qˆ ]T , leaving a comparison of scalars:
˜k+1 ˜k+1 X k+1 2 ˜k ˜k X k 2 Qb − βPb + Mij|Ij | < Qb − βPb + Mij|Ij | (A.20) br br j∈Db j∈Db
Without loss of generality, assume the “worst-case scenario”: that all of the demand is lumped at a single node j that is the maximum electrical distance away from node i. This is equivalent to the two node example presented earlier. This case maximizes the error between iterations because
br it results in the greatest magnitude of power losses. Since Db = {j}, the summation sign may be dropped:
˜k+1 ˜k+1 k+1 2 ˜k ˜k k 2 Qb − βPb + Mij|Ij | < Qb − βPb + Mij|Ij | (A.21)
br ˜k k ˜k k Furthermore, Db = {j} also yields Pb = Pj,nom and Qb = Qj,nom because j is the only injection 112 node downstream of node i. Therefore:
k+1 k+1 k+1 2 k k k 2 Qj,nom − βPj,nom + Mij|Ij | < Qj,nom − βPj,nom + Mij|Ij | (A.22)
The focus now shifts to the left hand side, denoted L.H.S., which will be transformed in order to obtain an expression entirely in terms of variables on iteration k:
k+1 k+1 k+1 2 L.H.S. = Qj,nom − βPj,nom + Mij|Ij | (A.23)
br The following identities, derived from (A.15) with Db = {j}, will be used to expand the real and reactive power terms:
k+1 k k k k 2 ˆ Pj,nom = Pj,nom + γ Qj,nom − βPj,nom + Mij|Ij | Pj (A.24)
k+1 k k k k 2 ˆ Qj,nom = Qj,nom + γ Qj,nom − βPj,nom + Mij|Ij | Qj (A.25)
The expansion of (A.23) is as follows:
Qk + γ(Qk − βP k + M |Ik|2)Qˆ j,nom j,nom j,nom ij j j L.H.S. = (A.26) −β P k + γ(Qk − βP k + M |Ik|2)Pˆ + M |Ik+1|2 j,nom j,nom j,nom ij j j ij j
Some rearrangement yields:
L.H.S. = Qk − βP k + γ(Qˆ − βPˆ )(Qk − βP k + M |Ik|2) + M |Ik+1|2 (A.27) j,nom j,nom j j j,nom j,nom ij j ij j
From (A.5), γ(Qˆj − βPˆj) = −1. Then:
L.H.S. = Qk − βP k − (Qk − βP k + M |Ik|2) + M |Ik+1|2 (A.28) j,nom j,nom j,nom j,nom ij j ij j
k+1 2 k 2 = Mij |Ij | − |Ij | (A.29) 113
Expanding the current terms using (A.7):
k+1 2 k+1 2 k 2 k 2 Pj,nom + Qj,nom Pj,nom + Qj,nom L.H.S. = Mij − (A.30) |V k+1|2 |V k|2 j j k+1 2 k+1 2 k 2 k 2 Pj,nom + Qj,nom Pj,nom + Qj,nom = Mij − (A.31) |V k + ∆V k|2 |V k|2 j j j
k where ∆Vj is the change in voltage at node j between iterations k and k + 1. Once again expanding the k + 1 real and reactive power terms with (A.24) and (A.25) yields the following:
h i2 k k k k 2 ˆ Pj,nom + γ Qj,nom − βPj,nom + Mij|Ij | Pj k k 2 |Vj + ∆Vj | h i2 k k k k 2 ˆ L.H.S. = M Qj,nom + γ Qj,nom − βQj,nom + Mij|Ij | Qj (A.32) ij + k k 2 |Vj + ∆Vj | 2 2 P k + Qk − j,nom j,nom k 2 |Vj |
Note that the current terms may be expanded again using (A.7), but this step is omitted for now.
The following inequality represents the exact condition for convergence:
2 2 P k + Qk L.H.S. < Qk − βP k + M j,nom j,nom (A.33) j,nom j,nom ij k 2 |Vj |
∆Vj is a nonlinear function of the injections, but distribution system properties can be used to approximate ∆Vj for a given load perturbation. Additionally, |∆Vj| is known to vary monotonically with a constant injection variation direction [75].
A.3.4 Convergence with Typical Distribution System Assumptions
This subsection will show that the convergence condition is typically satisfied with the following reasonable assumptions regarding the power distribution system:
1. Using the rated substation voltage and apparent power as per unit base quantities, branch 114
impedance values are small (<< 1.0 p.u.).
k 2 2. Vj ≈ 1.0 p.u. and |∆Vj| << 1.0 p.u.
3. Distribution line X/R ratios are typically between 2.5 and 4.0 [8].
4. The range of practical power ratio constraints is approximately β ∈ [−0.62, 0.62], which corre-
sponds PF constraints between 0.85 leading and 0.85 lagging.13.
k k k k 2 From (A.14) for the two node case, ζ = γ(Qj,nom − βPj,nom + Mij|Ij | ). Using Assumption2 above, (A.32) can be restated as follows:
2 2 h k k ˆ i h k k ˆ i k 2 k 2 L.H.S. = Mij P + ζ Pj + Q + ζ Qj − P − Q (A.34) j,nom j,nom j,nom j,nom
h i 2 h i k k ˆ k ˆ k ˆ2 ˆ2 = Mij 2ζ Pj,nomPj + Qj,nomQj + ζ Pj + Qj (A.35)
Substituting (A.35) into the convergence condition (A.33), and applying Assumption2 again:
h i 2 h i 2 2 k k ˆ k ˆ k ˆ2 ˆ2 k k k k Mij 2ζ Pj,nomPj + Qj,nomQj + ζ Pj + Qj < Qj,nom − βPj,nom + Mij(Pj,nom + Qj,nom ) (A.36)
Replacing ζk:
h i 2γ Qk − βP k + M |Ik|2 P k Pˆ + Qk Qˆ j,nom j,nom ij j j,nom j j,nom j Mij 2 h i + γ2 Qk − βP k + M |Ik|2 Pˆ2 + Qˆ2 j,nom j,nom ij j j j
2 2 k k k k < Qj,nom − βPj,nom + Mij Pj,nom + Qj,nom (A.37)
13When imposed, power factor constraints on consumers typically fall between 0.9 lagging and unity [76, 77]; con- straints on DG injections typically fall between 0.85 lagging and 0.85 leading [78–80]. 115
And finally expanding with squared current magnitude terms with Assumption2:
2 2 h i 2γ Qk − βP k + M P k + Qk P k Pˆ + Qk Qˆ j,nom j,nom ij j,nom j,nom j,nom j j,nom j Mij 2 22 h i + γ2 Qk − βP k + M P k + Qk Pˆ2 + Qˆ2 j,nom j,nom ij j,nom j,nom j j
2 2 k k k k < Qj,nom − βPj,nom + Mij Pj,nom + Qj,nom (A.38)
Numerical values of Pˆj, Qˆj, γ, Mij, and β may now be substituted in to (A.38) in order to
0 0 find a region of convergence (i.e., the set of Pj,nom and Qj,nom for which the inequality is satisfied).
However, one more step is taken in order to restate the condition as a multivariate polynomial inequality:
4 M 3 (Pˆ2 + Qˆ2) × P k ij j j j,nom
3 − 2M 2 Qˆ (Pˆ + βQˆ ) × P k ij j j j j,nom
2 + M β(−βPˆ2 + 2Pˆ Qˆ + βQˆ2) × P k ij j j j j j,nom
2 2 + 2M 2 Pˆ (Pˆ + βQˆ ) × P k Qk M × P k ij j j j j,nom j,nom ij j,nom
2 k k k + 2MijPˆjQˆj(β + 1) × P Q − β × P 2 j,nom j,nom j,nom γ < (A.39) 2 2 − 2M 3 (Pˆ2 + Qˆ2) × P k Qk + 1 × Qk ij j j j,nom j,nom j,nom
2 2 − 2M 2 Qˆ (Pˆ + βQˆ ) × P k Qk − M × Qk ij j j j j,nom j,nom ij j,nom
2 + M (Pˆ2 + 2βPˆ Qˆ − Qˆ2) × Qk ij j j j j j,nom
3 + 2M 2 Pˆ (Pˆ + βQˆ ) × Qk ij j j j j,nom
4 + M 3 (Pˆ2 + Qˆ2) × Qk ij j j j,nom
Note the following:
Both sides of (A.39) are real scalars.
Based on the definition (A.6) and Assumptions3 and4 on the preceding page, Mij ∈ [1.88Rij, 4.62Rij].
Then, using Assumption1, 0 < Mij << 1. 116
k k Since the substation apparent power rating is 1.0 p.u., Pj,nom and Qj,nom will normally be
smaller than 1.0 p.u. (even with all the loads lumped at a single node); the terms on the
left-hand side all contain a product or two or more of these terms.
Based on the points listed above, the inequality (A.39) will be satisfied for typical distribution
k k system properties and most initial conditions. Even if Qj,nom − βPj,nom > 0 is small, the squared nominal power terms on the right-hand side will usually be significantly larger than the terms on the
ˆ ˆ left-hand side. Divergence may occur if βPj − Qj is small (i.e., if γ is very large), as the variation vector is nearly parallel to the constraint boundary, and iterative ζk values may be very large. In this case, however, it is likely that a different constraint (e.g., electrical constraints, or a branch current ratings) will limit load capability more strictly than PF constraints. This type of behavior
k can be captured by checking for Snom sets that stray far from the feasible region defined by the substation apparent power rating. Capturing cases in which load capability is more strictly limited by electrical constraints than by power factor constraints is discussed in Chapter3.
A.3.5 Convergence Region Example
Recall that the parameters from the example in Section A.2 are β = 0.2506, Zij = 0.01 + j0.04, and
Sˆj = 1.0 + j0.8. The corresponding constants are γ = −1.8202 and Mij = 0.0375. For this example, the convergence condition in the form of (A.39) is as follows:
4 3 0.0003 × P k − 0.0089 × P k j,nom j,nom 2 0.0375 × P k 2 2 j,nom + 0.0470 × P k + 0.0112 × P k Qk j,nom j,nom j,nom − 0.2506 × P k 2 2 j,nom − 0.2112 × P k Qk + 0.0006 × P k Qk < j,nom j,nom j,nom j,nom + 1 × Qk 2 2 j,nom − 0.0089 × P k Qk + 0.0945 × Qk j,nom j,nom j,nom 2 − 0.0375 × Qk 3 4 j,nom + 0.0112 × Qk + 0.0003 × Qk j,nom j,nom
The region of initial power injections that satisfies the inequality (A.3.5) is illustrated in Fig- ure A.3 on page 118. The colored regions satisfy the convergence condition, and the colors indicate the Lipschitz constant ρ (the ratio of the left-hand side to the right-hand side), which measures the 117 strength with which the condition is satisfied. The black circle centered at the origin marks the feasible region defined by an apparent power constraint of 1.0 p.u. Figure A.3 indicates that for the example parameter set, the convergence condition is satisfied for all points within this feasible region except for the origin (where power factor is undefined). Note that an initial condition at the origin represents a de-energized system and is not of practical importance in this context.
Figure A.4, also on the following page, illustrates empirically determined convergence rates for the example parameter set over a range of different initial conditions. Initial conditions with the colored region resulted in a converged load capability solution, and the colors indicate the rate of convergence. Linear convergence at a rate of 0.50 or faster occurred for all points within the feasible region denoted by the black circle.
Note that the extreme points (>> 1.0 p.u. apparent power loading), Figures A.3 and A.4 are colored differently. This reflects the fact that Assumption2, and therefore the validity of the convergence condition A.3.5, breaks down at these extreme points.
Note that Figures A.3 and A.4 do not show basins of attraction to one specific fixed point; the
0 0 ˆ fixed point depends on Snom. All convergent points on the hyperplane Snom + λS do converge to a
0 ˆ common fixed point. Therefore, if the initial condition is divergent, but the Snom + λS hyperplane passes through the region of convergence, a different (convergent) point on this hyperplane could be used as the initial condition in order to find the fixed point within the injection space that corresponds
0 ˆ to the power factor-based DLC solution. Note that in the present example, the Snom +λS hyperplane is a line through the complex plane. 118 Emperically determined convergence rate for the ex- Figure A.4: ample parameter set givenditions. in Section A.2 over a range of initial con- Convergence region based on the satisfaction of in- Figure A.3: equality ( A.3.5 ) for the example parameter set given in Section A.2 . 119
Appendix B: Implicit Temporal Load Capability: Tap Operation Analysis
In this appendix, implicit temporal load capability (ITLC) is used to identify when local voltage set-points will trigger automatic voltage regulator tap operations. The purpose of the analysis is to demonstrate the following:
The ability to account for voltage magnitude constraints.
The ability to model automatic control actions triggered by local sensing.
A significant improvement over traditional QSTS when solving the same problem.
B.1 Test Circuit and Injection Profile
Simulations were performed using the IEEE 34-bus test feeder [81]. The test feeder represents an unbalanced three-phase distribution circuit. The single line diagram illustrated in Figure B.1 shows the positions of capacitors, voltage regulators, and a transformer. The circuit also includes spot loads and distributed loads, which are not pictured in Figure B.1.
Figure B.1: IEEE 34-node distribution test feeder. 120
The test feeder experiences significant voltage drop at the nodes distant from the substation.
The following features are used to combat this problem:
The substation voltage is fixed at 1.05 p.u.
(2) three-phase capacitor banks provide voltage support:
– C1: 300 kVAR bank at bus 844.
– C2: 450 kVAR bank at bus 848.
(2) three-phase voltage regulators are installed along the main feeder.
– VR1: branch connecting buses 814 and 850.
– VR2: branch connecting buses 852 and 832.
B.1.1 Injection Profile Development
Nominal branch data, load models, and capacitor data is specified in [81]. To appropriately adapt the circuit to a temporal analysis, the following assumptions were made:
Load sizes: kW and kVAR values in [81] are assumed to represent maximum load levels.
Load classification: the algorthim in [20] was used to classify each load as residential,
commercial, industrial, or street lighting.
Static load profiles: for each class, hourly load profiles were obtained from Southern Cali-
fornia Edison historial loadings data between midnight on September 17, 2010 and midnight
on September 18, 2010 [82]. Load profiles were scaled according to the maximum load values
from [81]. The file descriptions for each class as they appear in [82] are as follows:
– Residential: domestic, single/multiple
– Commercial: general service, non-demand metered, small commercial
– Industrial: general service, demand metered, medium commercial/industrial
– Street Lighting: street and area lighting 121
The 24-hour injection profile, represented by net three-phase apparent power demand, is illus- trated in Figure B.2. The data markers are elements of the time series forecast, while the lines connecting the markers illustrate the linear interpolation between points of the time series.
Net Substation Demand: Time Series Forecast 3
2.5
2
1.5
1
0.5 Three Phase Apparent Power (MVA) Power Apparent Phase Three
0 0 4 8 12 16 20 24 Time (hr)
Figure B.2: Day-ahead injection profile for the test feeder.
B.1.2 Control Modeling
The test circuit contains two three-phase capacitor banks and two voltage regulators (VRs). For the purposes of this study, both capacitor banks are fixed on. The voltage regulators operated on a per-phase basis, and are assumed to operated with no delay or dead band. The voltage magnitudes sensed at the buses downstream of each VR are used to trigger the tap operations. Integer-valued tap steps fall between −16 and +16 (inclusive); one tap step increments the effective tap ratio by
0.00625, with a tap setting of 0 corresponding to a 1 : 1 tap ratio. The VR settings are outlined in
Table B.1. 122
Table B.1: Voltage regulator settings.
Voltage Regulator Reference Bus Minimum Voltage (p.u.) Maximum Voltage (p.u.) VR1 850 1.000 1.033 VR2 832 1.017 1.050
B.2 Simulation Results
The 24-hour simulation required 3066 power flow solutions and identified 181 tap changing oper- ations. Solid vertical lines in Figure B.3 represent initial hourly time window divisions while dashed vertical lines represent sub-window divisions (i.e., tap changing actions). Note that the resulting sub-window divisions are non-uniform. Several time windows did not require any sub-division at all, so very little computational effort was expended during these time windows.
Figure B.3 also shows the voltage profiles of the monitored buses over the forecast horizon.
Voltage regulators VR1 and VR2 operate based on sensed nodal voltage magnitudes at buses 850 and 832, respectively. Each phase operates independently. The bands in Figure B.3 contain the range of voltage magnitudes across the phases at each of these buses. Note the that voltage ranges stay within their corresponding limits.
The tap position forecast is shown in Figure B.4. This figure shows how all six per-phase tap positions are expected to change in accordance with their pre-programmed settings.
This ITLC simulation required 96.5% fewer power flow solutions than a than traditional QSTS with one second time windows, and is computationally comparable to a QSTS analysis with uniform
28-second time windows. Time windows of 28 seconds, however, introduce significant error because the tap changing times can only be localized to within certain windows rather than assigned to specific operating times. 123
Monitored Nodal Voltage Ranges 1.06
1.05
1.04
1.03
1.02
1.01 VR1 Limits Voltage Magnitude (p.u.) Magnitude Voltage VR2 Limits 1 VR1 Range VR2 Range 0.99 0 4 8 12 16 20 24 Time (hr)
Figure B.3: Voltage magnitude ranges at monitored buses.
Voltage Regulator 1: Tap Position Forecast 8 Ph. a 4 Ph. b 0 Ph. c
-4 Tap Position Tap
-8 0 4 8 12 16 20 24 Time (hr) Votlage Regulator 2: Tap Position Forecast 8 Ph. a 4 Ph. b 0 Ph. c
-4 Tap Position Tap
-8 0 4 8 12 16 20 24 Time (hr)
Figure B.4: Tap position forecast. 124
Vita
Name Nicholas S. Coleman Place of Birth Philadelphia, PA, USA Citizenship United States of America
Education Drexel University
Doctor of Philosophy, Electrical Engineering 2018 Graduate Minor in Computational Engineering 2018 Master of Science, Electrical Engineering 2013 Graduate Certificate in Engineering Management 2013 Bachelor of Science, Electrical Engineering 2013
Employment
Research Fellow, Drexel University Jan. 2014 - Mar. 2018 Teaching Assistant, Drexel University Jan. 2014 - Mar. 2018 Adjunct Instructor, Salem Community College (NJ) Aug. 2013 - Dec. 2013 Research Assistant, Technical University of Berlin July 2012 - Sep. 2012 Project Management Co-op, Children’s Hospital of Phila. Sep. 2010 - Mar. 2011 Electrical Engineering Co-op, EwingCole Sep. 2009 - June 2010
Selected Publications
N. S. Coleman, K. N. Miu, “Identification of Critical Times for Distribution System Time Series Analysis,” IEEE Transactions on Power Systems, vol. 33, no. 2, pp. 1746–1754, Mar. 2018.
N. S. Coleman and K. N. Miu, “Distribution Load Capability with Nodal Power Factor Constraints,” IEEE Transactions on Power Systems, vol. 32, no. 4, pp. 3120–3126, July 2017.
N. S. Coleman, et. al., Hardware Setup of a Solar Microgrid Laboratory,” in Proc. 2017 IEEE PES General Meeting, Chicago, IL, USA, 16-20 July 2017.
N. S. Coleman, C. Schegan, K. N. Miu, “A study of power distribution system fault classification with machine learning techniques,” in Proc. 2015 North American Power Symposium (NAPS), Charlotte, NC, USA, 4-6 Oct. 2015.
N. S. Coleman, K. N. Miu, “A Study of Time Window Selection for Electric Power Distribution System Analysis,” in Proc. 2015 IEEE International Symposium on Circuits and Systems (ISCAS), pp. 1891–1894, Lisbon, Portugal, 24-27 May 2015.
N. S. Coleman, “Load Capability for Smart Distribution Systems,” M.S. thesis, Dept. of Electrical and Computer Engineering, Drexel University, Philadelphia, PA, USA, 2013.