Power Quality Improvement and Control Strategy for a System

POWER QUALITY IMPROVEMENT AND CONTROL STRATEGY FOR A SYSTEM

UNDER 3-PHASE UNBALANCED CONDITION

A Project

Presented to the faculty of the Department of Electrical and Electronic Engineering

California State University, Sacramento

Submitted in partial satisfaction of the requirements for the degree of

MASTER OF SCIENCE

Electrical and Electronic Engineering

David Mapinda

SPRING 2018

POWER QUALITY IMPROVEMENT AND CONTROL STRATEGY FOR A SYSTEM

UNDER 3-PHASE UNBALANCED CONDITION

A Project

David Mapinda

Approved by:

______, Committee Chair Tracy Toups, Ph.D.

______, Second Reader Preetham B. Kumar, Ph.D.

______Date

Student: David Mapinda

I certify that this student has met the requirements for format contained in the University format manual, and that this project is suitable for shelving in the Library and credit is to be awarded for the project.

______, Graduate Coordinator ______Preetham B. Kumar, Ph.D. Date

Department of Electrical and Electronic Engineering

iii

Abstract

POWER QUALITY IMPROVEMENT AND CONTROL STRATEGY FOR A SYSTEM

UNDER 3-PHASE UNBALANCED CONDITION

David Mapinda

The quality of power is the major challenging issue in the power system. In real-world

power systems, it is ordinarily impossible to maintain perfect balance in phase voltages

and currents. The variety of the loads in the networks enhances the amount of unbalance

or negative sequence components. In conjunction with asymmetrical load, further increase

in negative sequence components is introduced due to the network inherent asymmetry.

The power quality is very much affected by the magnitude and phase disturbances of three

phase voltage and current respectively. The proposed balancing compensator is built as

reactive device uses the shunt current source compensation whose instantaneous values are

determined by the instantaneous symmetrical component theory. The compensation

configuration developed in this paper is tested for its cogency on 3-phase,3-wire circuit through wide-ranging simulations for phase outages scenarios. The simulation results for the compensation theory and the ideal compensator verify the

proposed compensation method. In this paper, a general analysis of control strategy to mitigate negative-sequence and zero sequence currents in unbalanced three-phase power systems is proposed.

______, Committee Chair Tracy N. Toups, Ph.D.

______Date

TABLE OF CONTENTS

Page

List of Tables ...... viii

List of Figures ...... ix

Chapter

1. INTRODUCTION ...... 1

1.1 Electric Energy ...... 1

1.2 Balanced and Unbalanced Systems ...... 3

1.3 Literature Review ...... 4

2. SYMMETRY AND ASYMMETRY IN POWER SYSTEM ...... 6

2.1 Impacts of Asymmetry in Power Systems ...... 6

2.2 Symmetrical Components ...... 6

2.3 Three Phase System in Phasor Representation ...... 8

2.4 Three Phase System in Sequence Representation ...... 9

2.5 Unbalanced Operation ...... 11

2.6 Unbalanced Voltage ...... 12

2.7 Unbalanced Loads ...... 12

3. POWER THEORY ...... 14

3.1 Currents Physical Components Theory ...... 14

3.2 Theory of Unbalanced Power Systems ...... 20

3.3 Currents Compensation Techniques ...... 29

3.4 Numerical Solution ...... 32

4. POWER SYSTEM MODELLING ...... 34

4.1 Loads to Evaluate ...... 34

4.2 Components used in the simulation system ...... 35

4.3 Compensator design ...... 36

5. SIMULATIONS ...... 38

5.1 Case 1 Simulation Control RS=RT=TR...... 39

5.3 Case 3 simulation when TR is Open ...... 44

5.4 Case 4 simulation when ST is Open ...... 47

5.5 Simulation summary ...... ………..49

6. CONCLUSION ...... 51

Appendix ...... 53

References ...... 59

vii

LIST OF TABLES

Page

Table 3.1. Currents before and after compensation………………………………...... 32

Table 5.1. The simulation parameters of the system for both cases………….……….38

Table 5.2. Simulation results when both loads are connected………………………...40

Table 5.3. Simulation results when RS is open……………………………………….42

Table 5.4. Simulation results when TR is open……………………………………….45

Table 5.5. Simulation results when ST is open……………………………………….48

viii

LIST OF FIGURES Page

Figure 1.1. Bidirectional Active Network……………………………………………..2

Figure 1.2. Renewables 2017 courtesy of International Energy Agency (IEA).……... 3

Figure 1.3. Balanced and Unbalanced 3-Phase Systems………………………………4

Figure 2.1. Phasor diagram of three-phase system…………………………………….9

Figure 2.2. Visualization of decomposition in sequences. Positive sequence (left), negative sequence (center), zero sequence (right)...……………………………….…11

Figure 3.1. Three-phase, three-wire system...………………………………………...20

Figure 3.2. Three-phase and its equivalent circuit…….….……………………….… 23

Figure 3.3. (a)Three-phase load and (b) Its equivalent load with respect to active power P……………………………………………………………………...24

Figure 3.4. Equivalent load in ∆ configuration……………………………………….24

Figure 3.5. Rectangular box of RMS values of current physical components………..29

Figure 3.6. Circuit with shunt compensator…………………………………………. 30

Figure 3.7. Unbalanced load circuit with a balancing compensator……………….... 33

Figure 4.1. A balanced delta connected load……………………………………...….35

Figure 4.2. Three-Phase Generator…………………………………………………...35

Figure 4.3. Current positive sequence subsystem………………………………….... 36

Figure 4.4. Current negative sequence subsystem……………………………….…. .37

Figure 5.1. Case 1 circuit diagram…………………………………………………... 39

Figure 5.2. Simulation control when both loads are connected………….…………...39

Figure 5.3. Case 2 circuit diagram…………………………………………………….41

Figure 5.4. Simulation when RS is open ………………………………....…………...42

Figure 5.5. Currents comparison for loads RS, ST, and TR when RS open ……….…43

Figure 5.6. Case 3 circuit diagram……………………………………….…………....44

Figure 5.7. Simulation when TR is open…………………….………………………. 45

Figure 5.8. Currents comparison for loads RS, ST, and TR when TR open ……….…46

Figure 5.9. Case 4 circuit diagram……………………………………….…………....47

Figure 5.10. Simulation when ST is open…………………….…………………….... 48

Figure 5.11. Currents comparison for loads RS, ST, and TR when ST open.……. ….49

1. INTRODUCTION

1.1 Electric Energy

Electric energy technology is very essential in societal, economic and social development.

Since the industrial revolution started, by keeping the power flowing to homes and

businesses is a critical necessity for everyday life and economic vitality [1]. The main

sources of electric power generation are renewable sources, fossil fuels and nuclear power

plants.

The main challenge of an electrical power system is to supply electric energy to the end-

users with a reliable and efficient way. In traditional power systems, large power

generation plants supply most of the power in the unidirectional power flow and passive

electrical distribution network. This makes the energy efficiency improvements a

complicated task. Nowadays, many utility companies are looking to find new solutions

and adds more improvements to this huge market by utilizing small units for power

generation in distributed side.

As a result, passive unidirectional networks become bidirectional active networks by

using renewable sources distribution generator (DG) units designed for the supply of

commercial power into the electricity grid as illustrated in Figure 1.1.

Figure 1.1. Bidirectional Active Network [15].

According to International Energy Agency (IEA), boosted by a strong solar PV market, renewables accounted for almost two-thirds of net new power capacity around the world in 2016, with almost 165 gigawatts (GW) coming online. This was another record year, largely because of booming solar PV deployment in China and around the world, driven by sharp cost reductions and policy support. Last year, new solar PV capacity around the world grew by 50%, reaching over 74 GW, with China accounting for almost half of this expansion. For the first time, solar PV additions rose faster than any other fuel, surpassing the net growth in coal [16].

Figure 1.2. Renewables 2017 courtesy of International Energy Agency (IEA) [16].

Consequently, renewable energy resources (RESs) become more common to penetrate and support the main grid to improve the reliability and efficiency of all systems by using

DG units instead of centralized large generation units. However, the high penetration level of sporadic renewable energy sources (RES) in distribution systems may pose a threat to the network in terms of stability, voltage regulation and power quality issues.

1.2 Balanced and Unbalanced Systems

A purely balance system can be defined as, when voltage magnitude, current magnitude of all three phases are equal and 120° phase apart. In general, three phase loads are

considered balanced if the voltages, currents and power factors in all three phases are

identical. Unbalanced voltages and currents in a network are of the concerns under the

power quality issues. The unbalanced occurs mainly when the great number of single- phase loads which are unequally distributed over the phases. The unbalanced voltages can cause extra losses in components of the network, such as generators, motors and transformers, while unbalanced currents cause extra losses in components like transmission lines and transformer.

Active power filters and power factor corrector can be applied to compensate the unbalance at the load side. However, their contribution to transmission is not large because their usage is focused on a single load. Flexible Alternating Current Transmission

System (FACTS) devices can be employed to compensate the unbalanced currents and voltages in transmission systems.

Figure 1.3. Balanced and Unbalanced 3-Phase Systems

1.3 Literature Review

Many research studies have been conducted to investigate and analyze MGs issues with

different standpoints; configurations, operation, management and control improvements.

The microgrid (MG) is proficient of operating either in a grid connected mode (connected to a power grid) or in an island mode (uses distributed energy resources to supply power to the loads). In most of these research studies, the analysis has been carried out with the assumption of symmetrical operation or balanced three phase loads in MGs. However, in practice, single phase generation, like rooftop photovoltaic inverters, and single phase residential loads are commonly used in LV MGs. The output power of single phase DGs in LV MGs and residential loads are various and highly dependent on the surrounding conditions and the end-users’ behaviors. Therefore, the unbalanced operations are more common to take place, especially in low voltage MGs with majority of single phase inverters and loads which will increase the negative currents to flow in MGs and causes serious problems and abnormal operation of sensitive equipment in the MGs.

Also, an extensive use of nonlinear loads is common in distribution systems because the user required the compacted and low energy expendable devices. Largely, those devices are based on power electronics and microelectronics. These two technologies are substantially improving the quality of modern life. But, at the time of the nonlinearity and unbalance condition, same sensitive technologies are incompatible with each other and create a challenge to maintain quality of service in the form of poor power factor, increased heating losses, transient and steady state disturbance nearby point of common coupling (PCC).

Ultimately end users suffer from poor power quality problems and pay additional electricity penalty and in an industry, it may be loose customer productivity.

Conventionally, passive LC filters have been used to solve this type of problem.

2. SYMMETRY AND ASYMMETRY IN POWER SYSTEM

The unbalanced conditions are very common for Low Voltage (L V) in microgrid setting

due to single phase loads and single phase inverters usage. The unbalanced conditions

create unbalanced currents circulating between MGs and the main grid, which adds

negative affects into grid and the MGs.

2.1 Impacts of Asymmetry in Power Systems

Asymmetry in current and voltage add negative effects throughout the process of power

delivery. The negative sequence voltage increases the heat and losses of induction motors.

The negative sequence currents reduce the system efficiency.

Asymmetry also affects the power system components with the following [9]:

• Protection relays and measuring instruments become a complicated issue.

• Motors and generators: losses, vibration, temperature.

• Transmission lines: the capacity of the lines is reduced.

• Transformer: the zero sequence currents circulating in the delta side, causing heat

and losses.

2.2 Symmetrical Components

In 1918, Charles Legeyt Fortescue described how a set of three unbalanced phasors could

be expressed as the sum of three symmetrical sets of balanced phasor [10]. The method

of symmetrical components is used to simplify fault analysis by converting a three-phase unbalanced system into two sets of balanced phasors system and a set of phasors with no phase difference, or symmetrical components. These sets of phasors are called the

positive, negative, and zero sequence components. These components allow for the

simple analysis of power systems under faulted or other unbalanced conditions.

According to the International Electro Technical Commission (IEC) gives a definition of

the degrees of the unbalance: the unbalance factor is described in terms of the negative-

sequence unbalanced factor and zero-sequence unbalance factor expressed in the

following equations.

X Unbalance (N) % = n x100 X p

X Unbalance (Z) % = 0 x100 X p

Where XXXpn,,0 are positive, negative and zero sequence components.

The Symmetrical Component Theory (SCT) is widely used in power system fault analysis. According to SCT, an asymmetrical three-phase signal (either current or voltage) can be represented as a sum of positive, negative and zero sequence components.

Symmetrical components allow unbalanced phase quantities such as currents and voltage to be replaced by three separate balanced symmetrical components. In three phase system the sequence is defined as the order in which they pass through a positive maximum. The transformation is given by

2 X p 1 aaX a 1 2  Xnb= 1 a aX 3    XX0 11 1c

Where X could be voltage (V ) or current (I ) and is an operator and has the value of

24ππ jj ae=33, a2 = e

    X a XXXap_ an_ a_0     =++ XXb bp_  X bn_  X b_0  X     c XXXcp_  cn_  c_0 

 2 X ap_ 1 aaX a 1 2  Positive sequence: Xbp_ =  a1 aXb 3  aa2 1 X X cp_ c

 2 X an_ 1 aaX a 1 2  Negative sequence: Xbn_ =  a1 aXb 3  aa2 1 X X cn_ c

 X a _0 111X a 1  Zero sequence: XXbb_0 = 111 3  111X X b _0 c

2.3 Three Phase System in Phasor Representation

In a three-phase system, the three phases are denoted a , b , and c . The frequency is the same in all three phases. During ideal conditions, the phase components are distributed

by 120° and their amplitudes are equal. If phase a is taken as reference, phase b lags

120° behind phase a . The phasors rotate counterclockwise. In an ideal situation, the

three-phase system has equal amplitudes in all three phases and exactly 120° phase distribution. The system is then called symmetric or balanced.

Vtaa( )= V cos(ω t)

^ 2π Vtbb( )= V cosω t − 3

^ 4π Vtcc( )= V cosω t − 3

Figure 2.1. Phasor diagram of three-phase system.

2.4 Three Phase System in Sequence Representation

When phase b lags 120° behind phase a , the system is said to have a positive sequence.

And if phase b is lagging 2400 behind phase a :the system is said to have a negative

sequence. The voltage and current positive sequence is demonstrated in the equations

bellow. Positive and negative sequence can be visualized as rotating counterclockwise

and clockwise, respectively.

Vtaa( )= V cos(ω t)

^ 2π Vtbb( )= V cosω t − 3

^ 4π Vtcc( )= V cosω t − 3

Itaa( )= I cos(ωφ t − )

^ 2π Itbb( )= I cosωφ t −− 3

^ 4π Itcc( )= I cosωφ t −− 3

An important property of a three-phase system with only positive sequence, negative

sequence, or a sum of both, is that the instantaneous sum of the phase components is zero.

vtvtvtabc()++= () () 0

The mean value:

vtvtvt()++ () () v = abc 0 3

is called the zero-sequence component. The zero- sequence component v0 represents an

Asymmetry component which is the same in all three phases. In a 3-phase, three wire system with no neutral conductor there is no zero-sequence current. However, in a four-

wire system the possibility of zero sequence currents exists. In an asymmetric, or

unbalanced, three phase systems can be decomposed into a positive sequence component,

a negative sequence component and a zero sequence component:

Figure 2.2 Visualization of decomposition in sequences. Positive sequence (left),

negative sequence (center), zero sequence (right).

2.5 Unbalanced Operation

Utilities networks struggle to maintain three phase balanced voltages and currents in all

terminals. Generator terminal voltage, impedance of the transmission and distribution

lines and loads are the main contributors of unbalance. The voltage unbalance in a 3-

phase system is mainly caused by network devices such as rotating machines,

transmission and distribution lines, and transformers. While asymmetrical loads

contribute significantly in unbalanced current in the system. Voltage and current

unbalance cooperates with each other in case of asymmetrical loads and faultlessly

balanced supply. For example, asymmetrical load draws unequal current from each phase

which will cause different voltage drop in each phase of the supply system impedance

resulting in both voltage and current unbalance. On the other hand, current unbalance

response also to voltage unbalance when the load is perfectly balanced and the supply

12 system is unbalanced. Therefore, there is a relationship between voltage and current unbalance in the 3-phase system that needs to be considered.

2.6 Unbalanced Voltage

In the systems that utilize large central generators usually do not experience voltage unbalance since their control system and ability to maintain balanced three phase voltages at their terminals (positive sequence voltage). However, their currents at the connection point with transmission might comprise of both positive sequence and negative sequence components due to the unbalance from the untransposed transmission. For economic reasons, transposition in not fully implemented in practical network. Thus, continuous voltage and current unbalance is caused by transmission lines and distribution feeders due to the unequal mutual and self-capacitance and inductance. There are many small distributed generators (DG’s) in the networks that are usually connected at the medium voltage (MV) and low voltage (LV) level. For instance, small renewable sources such as solar cells and wind turbines present a signiﬁcant voltage unbalance at their connection points. Most of MV and LV connection points have comparatively high impedance which enhance the degree of voltage unbalance [7].

2.7 Unbalanced Loads

Generally, are the main cause in large part of unbalance in 3-phase system especially unbalanced currents. The unbalanced load is caused by large single phase, number of small loads connected to only one phase, three-phase induction motor with unbalanced windings, unequal impedance in the power transmission or distribution system, and traction loads. The unbalance occurs due to unequal phase current drawn by the above-

mentioned loads at their point of common coupling (PCC), which propagates to other

terminals in the network. Industrial load, mainly consist of numerous induction motors,

is usually linked to the high voltage (HV) network. Induction motor loads tends to reduce the degree of the pre-existing voltage unbalance at their point of common coupling (PCC) by acting as a compensator. However, the attenuation to the voltage unbalance is very sensitive to the machine loading [8]. The connection of numerous low voltage (LV) loads with a single phase might not guarantee the balance distribution between the supply system phases. Even with equal single-phase load distribution, most LV loads vary continuously with time which will cause unequal current drawn from the supply system.

In the future works we will discuss more on unbalanced voltage supply, however, this paper focused more on unbalanced currents due to unbalanced loads.

3. POWER THEORY

Power theory is a collection of information about the properties of propagation of energy in electrical circuits. It is a combination of research works and experience of several generations of scientists and electrical engineers. The purpose of power theory is to optimize the operating point of electrical systems by minimizing losses, and eventually reducing operating costs. These costs are caused by:

• Increased losses in resistive elements

• Increased losses in engines

• Capacitor failures

• Low efficiency of power sources

• The increase of current in the neutral wire

• Resonance phenomena caused by higher harmonics

• Productions shutdowns caused by improper operation of protection systems.

3.1 Currents Physical Components Theory

The development of Currents’ Physical Components (CPC) power theory started in 1984.

The CPC power theory starts with Budeanu and Fryze’s attempts, however, the CPC theory takes the importance of harmonics and frequency properties of a circuit for power properties of electrical loads and power definition into consideration. ut( )= 2 U sinω t , it( )= 2 I sin(ωϕ t − )

The active power or real power which is defined as mean value of instantaneous power,

1 T = = ϕ ϕϕ= − ϕ P∫ u()() t i t dt UI cos where ui T 0

The reactive power which is caused by phase shift between voltage and current,

1 T T Q=∫ u( t ) i ( t −= ) dt UI sinϕ T 0 4

Apparent power which is the product of voltage and current RMS values of the system,

11TT S= ∫∫ u22() t dt i () t dt= u i TT00

The apparent, active, and reactive power in a single-phase system

S222= PQ +

Electric energy conversion into other forms of energy needed for the energy consumer is

determined by the active power P , while equipment for energy delivery has to be rated

with respect to voltage and current rms values, apparent power S .

The power factor is defined as the ratio of the active and apparent power,

P λ = S

and within sinusoidal system simplified to λϕ= cos . Power factor is a term used to

describe

the effectiveness of utilization of energy delivery equipment.

In 1927, Budeanu published his remarkable work a reference book that led the power

theory over a long period of time, based on the concepts of reactive power Q and distortion power D in terms of the rectangular currents decomposition of the main current

16 and harmonic power decomposition. Both Budeanu’s concepts ( Q and D ) are still supported in the actual IEEE Standard Dictionary of Electrical and Electronical Terms

(IEEE, 1997). Budeanu theory defines the total reactive power QB as the sum of the contributions of each harmonic component:

Q= ∑∑ UInnsinϕ n= Q n = Q B nN∈∈nN

Clearly,

∞∞22 22   2 P+= QB ∑∑ UIkkcosϕϕ k UIkksin k≤ S kk=11= 

Therefore, Budeanu suggested that there is another power in the circuit. Because it only occurs in the presence of the voltage and current distortion, Budeanu introduced the name of the distortion power for this power. So, Budeanu had to define the power difference, named the distortion power:

2 22 D= S −+() PQB

One of the main faults of the Budeanu theory is that it cannot support the distortion power as defined by its name; distortion power D appears to be not related to the waveform distortion of the load voltage and/or current (Czarnecki, 2005a). The lack of distortion is not always associated with zero distortion power. That is the reason for the absence of a method of compensation of the distortion power as defined by Budeanu.

Neither the reactive power QB is associated directly to the energy oscillation between

the source and the load, so the power factor cannot be improved using the Budeanu

definition of Q .

The Fryze’s definition of the reactive power based on the load current decomposition into the active and reactive currents in a time-domain:

it()= ia () t + i rF () t

Active current (current component) is proportional to the supply voltage,

= P iae() t Gut (), Ge = 2 u

The active and reactive currents are mutually orthogonal and their RMS value satisfy the

equation,

22 2 ii=a + i rF

2 Multiplying with the square RMS voltage u then,

222 S= PQ + F

Fryze’s definition of reactive power

QFr= ui

Provides no solution to reduce the reactive power by mean of the load compensation.

Fryze’s definition of the reactive power QF includes all the power that is not related to

the direct energy conversion. The compensation of the reactive power QF is not supported by the Fryze’s definition, because a separation of the main causes of energy oscillation

and current distortion is not made. The compensation of the reactive current component

irF by mean of active compensators may produce more distortion in the source current,

affecting directly the quality of energy supplied the other consumers.

With non-sinusoidal voltage and current such that the both have the form with added

notation of Un describing a complex RMS value or magnitude and phase angle:

jnω1 t ut( )= U0 + 2 Re ∑ Uen nN∈

jnω1 t i( t )= YU00 + 2 Re ∑YUnn e nN∈

An LTI load can be described by its admittance in the form:

Ynn= G + jB n

Now the current can be categorized into fictitious currents. The active current

component ita () which is related with the permanent energy flow P (active power) with

the equivalent conductance of the load Ge with respect to the active power.

ω P = = jn1 t = ia() t Gu e () t 2Re∑ GUen e , where Ge 2 nN∈ u

After subtracting the active current from the remaining component of the load current:

jnω1 t it()−=− ita () ( Y00 GUe ) + 2Re∑ ( Yn − GUe en ) nN∈

The active current ita () is the minimum current of a load at voltage ut() that has

contributed to the active power. There are two components of the current that are

19 remaining which do not contribute to permanent energy transmission. The first one is reactive current which is the current component with 90° phase shift within the load current in the form of:

jnω1 t ir( t )= 2 Re ∑ jBnn U e nN∈

itr () occurs in load currents if and only if one of the load susceptance Bn for at least one current harmonic is not equal to zero, or at least one shifted current harmonic with respect to the supply voltage harmonic. A reactive power is given by,

2 Qn= U nn Isinϕ n = − BU n n

The second remaining current is called scattered current which is caused by the change

of load Gn conductance with respect to equivalent conductanceGe per harmonic order n

, when GGne> orGGne< .

jnω1 t its( )=−+ ( G00 GUe ) 2 Re∑ ( Gn − GUe en ) nN∈

Then load current,

itititit()=++asr () () ()

Therefore, these currents are called current’s physical components.

P i= Gu = , aa u

2 is=∑ () Gn − GU en, nN∈ 0

2 22 Qn ir=∑∑ BUnn =  nN∈∈nNUn

Due to the mutual orthogonality of all components [2], rms value of this current components satisfy the relationship,

2 222 ii=++asr i i

2 Multiplying with by square of the voltage RMS value u ,

22 22 22 22 iuiuiuiu=++ars,

2222 S=++ PDQs where,

Dss= ui and Q= uir

Scattered and reactive powers respectively.

3.2 Theory of Unbalanced Power Systems

The power theory based on the current’s physical components theory proposed by L.S

Czarnecki has been developed in the frequency domain. This theory is a proposal for the physical interpretation of power phenomena occurring in electric circuits under unbalanced conditions and the presence of non-sinusoidal waveforms.

Figure 3.1. Three-phase, three-wire system

The active and reactive powers in three-phase, three-wire systems, shown in Figure.3.1, with both sinusoidal supply voltage and line currents are defined:

1 T = ++ = ϕ Active power: P∫ ( uRR i u SS i u TT i) dt∑ Uf I fcos f T 0 f= RST,,

Reactive power: Q= ∑ UIffsinϕ f f= RST,,

According to IEEE Standard Dictionary of Electrical and Electronics Terms, there are two different definitions of the apparent power,

Arithmetic apparent power: SA=++ UI RR UI SS UI TT

22 Geometric apparent power: SG = PQ +

Also, there is a third definition that is not referred as standard,

2 2 2 222 Buchholz apparent power: SB= U R + U S + U TRST III ++

These three definition works on obtaining the same value of apparent power S ,Only when the line currents are symmetrical.

Three-phase, sinusoidal line-to-ground voltages uuRS, and uT and line currents iiRS,

and iT , denoted generally by xxRS, and xT , could be arranged into three-phase vectors:

xXRR     jtωωjt = = 11= x xSS2 Re X  e2 Re Xe    xXTT  

When these quantities are sinusoidal, but with the period T , and three-phase vector

harmonics are denoted by xn , then

jnω1 t x=∑∑ xnn = 2 Re Xe nN∈∈nN

Assuming that the voltages and currents in the systems do not contain a DC component.

However, it could be included.

The scalar product of three-phase periodic quantities xt() and yt()of the same period

T is defined as,

T 11T ( x, y) = xT () t y () t dt = ( x y ++ x y x y) dt ∫∫0 RR ss TT TT0

Where xT is a transposed matrix xt().Calculated from scalar product to frequency- domain as

T * (xy , )= Re ∑ XYnn nN∈

The asterisk * denotes a conjugate of Yn .

The active power in three-phase systems,

T * P=( ui , ) = Re ∑ Unn I nN∈

The RMS value of three-phase vector is given by

1 T = = TT= * x( x, x) ∫ x ().() t x t dt∑ Xnn . X T 0 nN∈

Re-arranging,

* X Rn T * * 222 x=∑∑ XXn. n = [ XRn ,, X Sn X Tn] . X Sn = ∑( XRn ++ X Sn X Tn ) ∈∈ ∈ nN nN * nN XTn

222 =xxxRST ++

Therefore, the RMS value of three-phase quantity, is equal to the root of the sum of squares of RMS value of individual phase quantities.

Figure 3.2. Three-phase and its equivalent circuit

With assumption of line resistances RRRS, and RT ,do not change with harmonic

frequency, then the active power is given by,

222 P3φ =++ iRRRS iR STT iR

Since, RRRRRST= = = then,

222 2 P= R i ++ i i = Ri 3φ RST

Thus, for any three-phase symmetrical device with resistance R and asymmetrical

currentsiiRS, , and iT is equal to active power P in a single phase equivalent circuit. The

current RMS value equal to

222 iiii=RST ++

Three-phase, three-wire circuits with linear, time-invariant loads(LTI) supplied with a sinusoidal symmetrical voltage of positive sequence. For those loads, there is an equivalent resistive and balanced load as shown in the circuits below

Figure 3.3. (a)Three-phase load and (b) Its equivalent load with respect to active power P The active power of the load on circuit the circuit above in Figure 3.3 is

22 2 PuGuGuG=Re ++ se Te,

P Ge = 222’ uuuRST++

P Ge = 2 u

Where Ge is called the equivalent conductance of a three-phase load.

Figure 3.4. Equivalent load in ∆ configuration

The active power for the above load in Figure 3.4 is

=222 ++ PRe{ YuRS RS Yu ST ST Yu TR TR }

Assuming the supply voltage is sinusoidal and symmetrical, then

uuuRS= ST = TR =3u uR =

Hence,

2 P=Re{ YYYRS ++ ST TR } u

2 PG= e u

The equivalent admittance of three-phase load Ye consists of a real and imaginary part.

Its real part is equal to the load equivalent conductance Ge , and imaginary part Be , is referred to the equivalent susceptance of three-phase loads.

YYe=++ RS Y ST Y TR

Yee= G + jB e

The line current of the equivalent resistive load:

iGRa eU R    ωω = = jt11= jt ia i Sa 2 Re  Ge U S  e 2 Re{ Gee U }    iGTa eU T 

iae= Gu

Active current ia is the smallest current required for energy permanent conversion in the

load with power P .The remaining part of supply current ii− a is useless and does not contribute to energy conversion. However, it is responsible for an increase in supply current RMS value.

IR− GU eR ω −= − jt1 i ia 2 Re  IS GU eS e  IT− GU eT

The complex RMS(CRMS) value of the line current:

IR=−( U R UY S) RS −−( U T UY R) TR

IR=( Y RS ++ Y ST Y TR) U R −( YU ST R + YU TR T + YU RS S )

With positive sequence supply voltage systems

* j2π /3 UT=α UU RS, = αα U R ,1 = e

The CRMS value IR becomes,

* IR=( Y RS ++ Y ST YU TR) R −( Y ST +αα Y TR + YU RS) R

IR= Y eR U + AU R

Where,

* AY=−++( STαα Y TR Y RS )

A is unbalanced admittance.

Likewise, the CRMS value of the line S and T currents are given by,

IS= Y eS U + AU T

IT= Y eT U + AU S

Combining the above formula, then the useless current ii− a can be written as:

YeR U+− AU R G eR U ω −= + − jt1 i ia 2 Re  YeS U AU T G eR U e  YeT U+− AU S G eR U

# jtω1 i−= iae2 Re( jB U + AU) e

Where,

U R U R =  # =  UUS , UUT   UT U S

The useless current in supply lines above contains two components. One occurs when the

equivalent susceptance Be of the load has a non-zero value.

=−++222 =− 2 From the equation: QIm{ YRS u RS Y ST u ST Y TR u TR } Bue

jtω1 ire= 2 Re{ jB Ue }

Therefore, ir is referred to reactive current due to its association with reactive power Q .

Second component is:

# jtω1 iu = 2 Re{ AU e }

iu is called unbalanced current because it causes the supply current asymmetry due to a negative sequence of voltage vector U # , only occurs in the supply current when the

coefficient A is not equal to zero ( A ≠ 0 ). In balanced systems, A =0 when YYYRS= ST = TR

The decomposition of the supply current in three phase power systems:

ii=++aru i i

categorized in three equivalent parameters of the load which are Ge , Be and A .

These three currents also known as physical components of the supply current linked individually with three distinctive physical phenomena in the circuit as follows:

ia -Permanent energy conversion in the load due to its active power

ir -Current phase shift with respect to the supply voltage due to the load reactive power

iu -Supply current asymmetry due to the load imbalance.

RMS values of physical components of the supply current:

iae= Gu

ire= Bu

iu = Au

The active and reactive currents are mutually orthogonal due to their 900 mutually shift.

Also, the scalar products of the reactive and unbalanced currents are

* i, i= Re G UT AU# =Re G A* 1 ++αα*2U =0 ( au) { e ( ) } { e }( ) R

* i, i= Re jB UT AU# =Re jB A* 1 ++αα*2U =0 ( ru) { e ( ) } { e }( ) R

Hence, the physical components are mutually orthogonal and their RMS values satisfy the relationship

2 222 ii=++aru i i

Multiplying by the square of the supply voltage RMS value u , then we get the power equation

S2222=++ PQD

Where,

2 Q= u. ire = − Bu

2 D= u. iu = Au

Q and D are reactive and unbalanced powers respectively.

Figure 3.5. Rectangular box of RMS values of current physical components

The power equation provides quantifiable evidence on the effect of power factor, and this method makes it conceivable to express power factor in terms of not only the load, but also load equivalent parameters.

PP i G Power factor: λ = = = a = e S 222++ 2 2 2 222+ + PQD iiiaru++ GBAee

3.3 Currents Compensation Techniques

The unbalanced current in the power system contributes to the supply current RMS value and apparent power increase which resulted in the power factor decreasing in the same way as reactive current as shown in the previous power equation. In the contrary, the reduction of these currents would contribute to power factor improvement in the system.

The shunt balancing compensator can be used to reduce both currents. The load is

compensated entirely when a vector of the line current ic of a balancing compensator is

equal to the negative value of the sum of the load reactive and unbalanced current.

ic=−− ii ru

Balancing compensator can be designed in two forms. As reactive devices comprising of

capacitors and inductors, or as fast switching devices composed of a three-phase inverter, a measurement and a control system. In a reactive compensator, the line currents are specified by the structure and LC parameters of the compensator while in the switching compensator are shaped by fast switching of inverters switches. However, in this paper will discuss a balancing compensator as reactive device.

Figure 3.6. Circuit with shunt compensator

A reactive compensator in Figure.52.6, one-ports connected in delta V configuration and

of branch susceptances TRS ,TST and TTR .If these one-ports are ideal and lossless devices, then the compensator modifies only the reactive and unbalanced currents as follows:

ω ' jt1 ir =2 Re{ j Be +( T ST ++ T TR T RS ) Ue }

ω '= − ++αα*#jt1 iu 2 Re{ A jT( ST T TR T RS ) Ue }

The reactive current is compensated completely only if,

BTTTe+( ST ++ TR RS ) =0

The unbalanced current is compensated completely only if,

* A− jT( ST ++αα T TR T RS ) =0

The left side of the above equation is a complex number, then the equation is satisfied only if is satisfied both real and imaginary parts. Then,

* Re{A− jT( ST ++αα T TR T RS )} =0

* Im{A− jT( ST ++αα T TR T RS )} =0

The compensator properties are specified by three susceptances TRS ,TST and TTR which satisfy the above equations, if they are equal to

( 3 ReA−− Im ABe ) T = RS 3

(2 Im AB− ) T = e ST 3

(−3 ReA −− Im ABe ) T = TR 3

When the susceptance TXY calculated from the above equations is positive, then a capacitor should be selected as the compensator for that branch. And when is negative, then an inductor should be selected. Their capacitance and inductance are equal to

TXY CXY = ω1

1 LXY = − ω1TXY

3.4 Numerical Solution

Load Values Compensator Current no filter Current w/filter

RS 50 Ω 0.4492 nF 5.451 A 3.191 A

ST 0 30.64 µF -5.451 A 3.197 A

TR Open 229.65 mH 0 A 3.192 A

Table 3.1. Currents before and after compensation.

The compensator susceptanceTTRS, ST and TTR was calculated using MATLAB code listed in the appendix A chapter.

Figure 3.7. Unbalanced load circuit with a balancing compensator.

4. POWER SYSTEM MODELLING

The project concerns the phase outage of the three-phase 480V AC voltage, including a

neutral connection with three loads 7ohm each. Both three loads are connected in delta

∆.The simulations shows the unbalanced currents in each scenerio.After the compensator

was applied the three currents in phase ABC were able to be balanced. The method and

concepts used are based on power theory compensation of unbalanced currents.

The analysis of the compensation requirements of a general unbalanced load is done in

terms of symmetrical component method which facilitate proper mathematical basis.The

compensating currents are transformed to a symmetrical components to find out the

required balancing susceptance of the compensator.The unbalanced three phase load is

represented by a delta-connected network in which load admittance YYRS, ST , and YTR are

real value since the load is purely resistive.

4.1 Loads to Evaluate

The ultimate goal is to create a balanced three-phase voltage source that, independent of

the load condition, always provides the correct voltage. Now, this is more of a target to

take aim at, than an in practice achievable goal. The presented simulations of the abc

phase outages will be limited by a number of 3 different load conditions. A total number

of 4 different load cases are to be considered for simulations .Both loads are resistive

with 7ohm values. Unbalanced loads are assumed to be very common for the multi-phase power systems and will consequently be carefully studied. The unbalance may be caused

by unevenly distributed single-phase loads, or by a combination of single-phase loads and

three-phase loads.

4.2 Components used in the simulation system

The loads used for simulation are delta ∆ connected:

Figure 4.1. A balanced delta connected load

Three-Phase- Generator:

Figure 4.2. Three-Phase Generator

The voltages and currents are equal in magnitude for phases abc,, respectively. These voltages and currents are 1200 apart phase shift with 60Hz operating frequency.

4.3 Compensator design

The balancing compensator design is based on passive elements such as capacitors and inductors, consists of fixed shunt susceptance connected to an asymmetrical load. The

main purpose of the compensators is to reduce unbalance currents ir and iu to zero. The values of susceptance were determined by the formula using the MATLAB program listed in the appendix chapter. If the susceptance obtained is positive, then a capacitor was used in the compensator branch. And when the susceptance is negative, then an inductor was used for the branch. The compensators have Δ configuration with the

assumption that it is constructed of lossless reactive elements of susceptances TRS ,TST

and TTR .

Current-Positive sequence subsystem:

Figure 4.3. Current positive sequence subsystem

Current negative-sequence subsystem:

Figure 4.4. Current negative sequence subsystem

5. SIMULATIONS

This chapter shows the simulations of symmetrical and asymmetrical system that is caused due to unbalanced load conditions. The MATLAB software was used to run the simulations in four different cases. Case 1 is the control system where the system is symmetrical with balanced loads. Thus, load currents which converted to RMS value is balanced in both phases. Case 2 the system is asymmetrical with open RS which caused by unbalanced load condition.

The simulation is run without applying the filter to obtain the RMS currents, negative- sequence, and positive-sequence currents of the system. Then the filter with passive elements is applied to mitigate negative sequence current that would make the system symmetrical. Case 3 is the simulation of the system with open TR. Likewise, the system is run without a filter then the filter is applied to balance the currents. Case 4 is the simulation of the system when ST is open. Then the filter is applied to bring the reactive and unbalanced currents to zero. Below are the parameters used for both cases.

LOAD CASE1 CASE2 CASE3 CASE4 RS 7Ω OPEN 7Ω 7Ω ST 7Ω 7Ω 7Ω OPEN TR 7Ω 7Ω OPEN 7Ω Table 5.1. The simulation parameters of the system for both cases

5.1 Case 1 Simulation Control RS=RT=TR

Figure 5.1. Case 1 circuit diagram.

Figure 5.2. Simulation control when both loads are connected.

Results:

Load Values Current

RS 7 Ω 118.8 A

ST 7 Ω 118.8 A

TR 7 Ω 118.8 A

Table 5.2. Simulation results when both loads are connected.

As shown in the simulation results table the RMS currents are equal in both phases. The load is balanced and the system is symmetrical and therefore only positive sequence currents exists. The positive sequence components equal the corresponding phase currents in this case.

5.2 Case 2 simulation when RS is open

Figure 5.3. Case 2 circuit diagram.

Figure 5.4. Simulation when RS is open

Results:

Load Values Compensator Current no filter Current w/filter

RS open 2192.708 mH 68.57 A 79.5 A

ST 7 Ω 32.152 mH 68.57 A 78.9 A

TR 7 Ω 218.83 µF 118.8 A 79.2 A

Table 5.3. Simulation results when RS is open.

Figure 5.5. Currents comparison for loads RS, ST, and TR when RS open

During the unbalance condition, both the magnitude of RMS currents for phase RS and

ST were the same. However, the current for phase TR was slightly higher than the other phases RS and ST respectively. With the compensation, the reactive compensator compensates entirely the reactive and unbalanced currents, reducing the three-phase RMS value of the phase TR current from i = 118.18 A to i = 79.2 A.

5.3 Case 3 simulation when TR is open

Figure 5.6. Case 3 circuit diagram

Figure 5.7. Simulation when TR is open

Results:

Load Values Compensator Current no filter Current w/filter

RS 7 Ω 0.032152 H 68.57 A 79.2 A

ST 7 Ω 218.83 µF 118.8 A 79.18 A

TR open 3.2088 nF 68.57 A 79.2 A

Table 5.4. Simulation results when TR is open.

Figure 5.8. Currents comparison for loads RS, ST, and TR when TR open

The simulation results show when the system is unbalanced, both the magnitude of RMS currents for phase RS and TR were the same. However, the current for phase ST was slightly higher than the other phases RS and TR respectively. With the compensation, the reactive compensator compensates entirely the reactive and unbalanced currents, reducing the three-phase RMS value of the phase ST current from i = 118.8 A to i =

79.18 A.

5.4 Case 4 simulation when ST is open

Figure 5.9. Case 4 circuit diagram

Figure 5.10. Simulation when ST is Open

Results:

Load Values Compensator Current no filter Current w/filter

RS 7 Ω 218.83 µF 118.8 A 79.65 A

ST open 0 µF 68.57 A 79.56 A

TR 7 Ω 32.152 mH 68.57 A 79.19 A

Table 5.5. Simulation results when ST is open.

Figure 5.11. Currents comparison for loads RS, ST, and TR when ST open

Before applied the compensator when the system was unbalanced, the RS phase current

RMS value was higher than ST and TR phases respectively. After the compensator was applied the unbalanced currents were reduced to zero. The system was completely compensated and RS RMS phase current was reduced from i = 118.8 A to i = 79.6A.

5.5 Simulation summary

The zero and negative sequence currents are undesirable in the system. Delta connection in the simulated systems eliminated the zero-sequence current but the negative sequence current is directly fed back to the source. Then after applying the compensator the simulation results show the currents were balanced for both scenarios and reduce the effect of negative sequence current due to unbalanced load. In each case when the system was asymmetrical due to the phase outage at least one phase maintained the RMS phase

current value like when the system was symmetrical which was i =118.8 A with an

error of less than 1% for both cases.

The compensators passive components values changed in some instances, or just keeps

alternating within the branches of the compensator. The negative sequence currents were

virtually zero after the compensator was applied to the systems.

Simulations presented the principle of load balancing using different load conditions and

recommends the suitable susceptance of the compensator to be connected in parallel with

delta connected load. This simulation was performed only with the resistive load, but it

can easily be used to compensate RL loads and reactive power (Q). However, in actual system the design takes into consideration the economic aspects to meet the required standard.

6. CONCLUSION

It is impractical to sustain perfect balanced voltages and currents in three phase systems.

Hereafter, the representation of the network as a sequence component (Positive Sequence,

Negative Sequence, and Zero Sequence) is crucial in the analysis. When deﬁning and measuring the degree of unbalance (Negative Sequence), the shunt reactive compensator method is the most precise one because it relies on sequence components rather than phase and line quantities which are not inﬂuenced by an angle variation while the former varies with the phase angle.

The supply system contribution to voltage and current unbalance is much less than the load contribution. When designing a power system network, unbalance assessment methods are essential to avoid risk of Negative Sequence components in the network.

However, they necessitate very sophisticated analysis due to the complex interaction between several sources of unbalance in the network. Both unbalance currents and voltage impact the operation of rotating machines and reduce their efﬁciency. Therefore, they should be protected against excessive Negative Sequence components from the network or the load.

For both cases the simulation results show the negative sequence currents effect were mitigated by load balancing using compensators comprised with an inductor or capacitor elements obtained by calculation depending upon the degree of unbalance in the system.

Moreover, the phase currents were also balanced in both cases. Hence, the simulations results validate the compensation strategy proposed.

Appendix

%Calculation of Zero, Negative and Positive Sequence Current of a balanced Load clear; a=-0.5+0.866i; va=V/sqrt(3); vb=va*a*a; vc=va*a; Za=50; Ia=(va)/(Za); Ib=(vb)/(Za); Ic=(vc)/(Za); I_z=(Ia+Ib+Ic)/3; abs(Io) I_p=(Ia+(a*Ib)+((a^2)*Ic))/3; abs(I_p) I_n=(Ia+((a^2)*Ib)+(a*Ic))/3; abs=(I_n)

CASE1: %% Compensator Case1 clear; % Load impedance %ZRS=inf; ZST=7; ZTR=7; a=-0.5+0.866i; % Load admittance %YRS=1/ZRS; YST=1/ZST; YTR=1/ZTR; Ye=YST+a*YTR;% equivalent admittance Ge=real(Ye);% equivalent conductance Be=imag(Ye);% equivalent susceptance %a=-0.5+0.866i;%(e^j(4*pi/3) % Unbalanced admittance A=-Ye; % The Susceptances Calculation TRS=(sqrt(3)*real(A)-imag(A)-Be)/3 TST=(2*imag(A)-Be)/3 TTR=(-sqrt(3)*real(A)-imag(A)-Be)/3 % Selecting a capacitor or an Inductor if TRS<0; fprintf('Select an Inductor= % f \n',TRS); else fprintf('Select a Capacitor=% f \n',TRS); if TST<0; fprintf('Select an Inductor=% f \n',TST); else fprintf('Select a Capacitor=% f \n',TST);

if TTR<0; fprintf('Select an Inductor=% f \n',TTR); else fprintf('Select a Capacitor=% f \n',TTR); end end end

CASE2: %% Compensator Case2 clear; % Load impedance ZRS=7; ZST=7; %ZTR=inf; w1=2*pi*60; alpha=-0.5+0.866i; alpha_2=-0.5-0.866i; % Load admittance YRS=1/ZRS; YST=1/ZST; %YTR=1/ZTR; Ye=YRS+YST; %(Ye=YRS+YST+YTR) equivalent admittance Ge=real(Ye);% equivalent conductance Be=imag(Ye);% equivalent susceptance % Unbalanced admittance A=-(YST+(alpha_2)*YRS);%-(YST+(alpha)*YTR+(alpha_2)*YRS; % The Susceptances Calculation TRS=(sqrt(3)*real(A)-imag(A)-Be)/3 TST=(2*imag(A)-Be)/3 TTR=(-sqrt(3)*real(A)-imag(A)-Be)/3 % Selecting a capacitor or an Inductor if TRS<0; fprintf('Select an Inductor= %.18f \n',TRS)

else fprintf('Select a Capacitor=%.18f \n',TRS)

if TST<0; fprintf('Select an Inductor=%.18f \n',TST)

else fprintf('Select a Capacitor=%.18f \n',TST)

if TTR<0; fprintf('Select an Inductor=%.18f \n',TTR)

55 else fprintf('Select a Capacitor=%.18f \n',TTR)

end end end

CASE 3: %% Compensator Case3 clear; % Load impedance ZRS=7; %ZST=inf; ZTR=7; w1=2*pi*60; alpha=-0.5+0.866i; alpha_2=-0.5-0.866i; % Load admittance YRS=1/ZRS; %YST=1/ZST; YTR=1/ZTR; Ye=YRS+YTR; %(Ye=YRS+YST+YTR) equivalent admittance Ge=real(Ye);% equivalent conductance Be=imag(Ye);% equivalent susceptance % Unbalanced admittance A=-((alpha)*YTR+(alpha_2)*YRS);%-(YST+(alpha)*YTR+(alpha_2)*YRS; % The Susceptances Calculation TRS=(sqrt(3)*real(A)-imag(A)-Be)/3 TST=(2*imag(A)-Be)/3 TTR=(-sqrt(3)*real(A)-imag(A)-Be)/3 % Selecting a capacitor or an Inductor if TRS<0; fprintf('Select an Inductor= %.18f \n',TRS)

else fprintf('Select a Capacitor=%.18f \n',TRS)

if TST<0; fprintf('Select an Inductor=%.18f \n',TST)

else fprintf('Select a Capacitor=%.18f \n',TST)

if TTR<0; fprintf('Select an Inductor=%.18f \n',TTR) else fprintf('Select a Capacitor=%.18f \n',TTR)

end end end

Numerical solution system:

Current subsystem:

Positive, negative, Zero currents subsystem:

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