Department of Electrical and Computer Engineering

Bus Voltage Ranking and Voltage Stability Enhancement

for Unbalanced Multiphase Networks

Parachai Juanuwattanakul

This thesis is presented for the Degree of

Doctor of Philosophy

of

Curtin University

February 2012

Declaration

To the best of my knowledge and belief this thesis contains no material previously

published by any other person except where due acknowledgment has been made.

This thesis contains no material which has been accepted for the award of any other

degree or diploma in any university.

Signature: ………………………………………….

Date: ………………………...

ABSTRACT

Voltage instabilities and subsequent system collapses are considered as growing

concerns in modern multiphase distribution networks as they are progressively

forced to operate closer to their stability limits due to many factors such as increasing

load level, lack of reactive power sources, high installation of single-phase shunt

capacitors and reverse action of voltage control devices. System operators must be

able to quickly identify trouble spots and take corrective steps to avoid critical

voltage collapses. To achieve this, suitable indices must be defined to assess system

security and take corrective control actions when predefined thresholds are reached.

In this regard, the identification and ranking of weak buses in a power system is an

important research area.

The existing conventional bus voltage ranking indices are only defined for single-

phase and balanced three-phase networks. This thesis proposes a new bus voltage

ranking index (VRI) to identify the weakest single-, two- and three-phase buses of

multiphase distribution networks. Then, applications of the proposed bus ranking

index will be tested for enhancing the voltage stability of unbalanced multiphase

distribution networks.

In the first part of this thesis, the definition of conventional voltage ranking indices

are modified and generalized to also include unbalanced and multiphase networks

using symmetrical components. For the first time, the method of symmetrical

components is applied to the three-phase voltages computed from three-phase power

flow. The new index is defined as the ratio of the (fundamental) positive-sequence

voltage at the point of voltage collapse to the positive-sequence voltage at the base-

load source. The former voltage level is determined by increasing the active power

of all loads while keeping power factor constant until the point of voltage collapse is

reached.

In the second part of this thesis, the new VRI is validated through the calculation of

grid losses and PV curves based on positive-sequence voltage. Extensive simulations

of the IEEE 13 and 34 node test feeders are performed using the DIgSILENT

PowerFactory to further validate and compare the performance of the new VRI with

three well-known conventional ranking indices.

In the third part of the thesis, the new VRI is used to identify the weakest three-phase

buses in unbalanced three-phase distribution networks. Then, the index is utilized to

place compensation devices at the weakest buses of the modified unbalanced three-

phase 13 node test feeder to improve voltage stability and increase the maximum

loading factor (MLF) under unbalanced three-phase operating conditions.

In the fourth part of the thesis, static analyses are carried out to demonstrate

applications of the proposed VRI in increasing MLF and improving voltage stability

of multiphase networks under unbalanced loading and/or network conditions. Then,

dynamic simulations are performed to further validate the accuracy of the proposed

VRI and improving voltage stability under dynamic operating conditions.

In the fifth part of the thesis, an online application of the proposed bus ranking is

introduced to identify the weakest buses in multiphase smart grids with plug-in

electric vehicle (PEV) charging stations.

Finally, the proposed voltage ranking and stability enhancement approach are

utilized to improve the performance of multiphase distribution networks by proper

placement and sizing of distributed generator (DG) units such as doubly-fed

induction generators (DFIGs) and single-phase capacitors. An iterative algorithm is

proposed for the placement and sizing of DG units and single-phase capacitors in

multiphase networks to reduce grid losses and increase MLF while keeping all bus

voltages within acceptable limits. The approach consists of utilizing the positive-

sequence voltage ratio Vcollapse/Vbase-load to identify the weakest three-phase and

single-phase buses for the installation of DG units and shunt capacitors, respectively.

DG penetration levels are increased (e.g., 40%) by evaluating their impacts on

voltage profile, grid losses, and voltage stability margin while considering the

voltage limits at all buses. The impacts of DIFG on voltage profile, active power

loss, MLF and voltage unbalance factor are highlighted.

DEDICATION

To my parents, Mamie and Papa, as well as my brother and sister for their endless

support and love.

ACKNOWLEDGMENT

I would like to express my special thanks to my supervisor, Associate Professor

Mohammad A.S. Masoum, for his invaluable advice, guidance and support all

throughout my PhD studies. I am also greatly thankful to my co-supervisor,

Professor Syed M. Islam for his assistance during the course of my study. Finally,

financial support from Sripatum University is gratefully acknowledged. Last but not

least, I wish to express my love and gratitude to my family and friends for their

endless support and love.

TABLE OF CONTENTS

Abstract ................................................................................................................... ii

Table of Contents ................................................................................................... vi

Chapter 1. Introduction .......................................................................................... 1

1.1 Statement of the problem ........................................................................... 1

1.2 Literature review ...................................................................................... 2

1.2.1 Bus ranking approaches for balanced networks .................................. 2

1.2.2 Existing bus ranking approaches for unbalanced networks ................. 7

1.2.3 Voltage stability enhancement by connecting compensation devices

considering grid losses and MLF…………………….. ............................... 7

1.3 Research objectives .................................................................................. 10

1.4 Thesis structure ....................................................................................... 11

1.5 List of publications .................................................................................. 12

Chapter 2. Proposed bus voltage ranking index (VRI) for multiphase

distribution networks ............................................................................................ 14

2.1 Introduction ............................................................................................. 14

2.2 Conventional VRI for balanced networks ................................................. 14

2.3 Derivation of proposed VRI for balanced networks .................................. 16

2.4 Proposed VRI for unbalanced multiphase distribution networks ............... 17

2.5 Derivation of proposed VRI for unbalanced multiphase networks ............. 18

2.6 Conclusions ........................................................................................... 20

Chapter 3. Validation of the proposed VRI ........................................................ 22

3.1 Introduction ............................................................................................. 22

3.2 Validation of proposed VRI using grid loss calculations………................ 22

3.3 Validation of proposed VRI using PV curves ……….. ............................. 23

3.4 Validation of proposed VRI using voltage sensitivity indices..….. ............ 23

3.4.1 Bus ranking based on sensitivity of voltage to reactive power

(V/Q)……………………. .................................................................... 24

3.4.2 Bus ranking based on sensitivity of voltage to active power

(V/P)……………………. ..................................................................... 24

3.5 Detailed simulation of IEEE multiphase 13 node test feeder to validate

proposed VRI………………………………………………… .................. 24

3.5.1 Identification of the weakest buses using proposed VRI for the IEEE

multiphase 13 node test feeder………………………………………………26

3.5.1.1 Bus ranking without/with a voltage regulator (Cases 1 and

2)…………….……………………………………..…….……….…27

3.5.1.2 Bus ranking with DG at the most suitable bus

(Case 5 )………………………...…………………………………...28

3.5.1.3 Bus ranking with DG and SVC (Case 6) .………….……….30

3.5.2 Validation of proposed VRI based on grid loss calculations for the

IEEE multiphase 13 node test feeder……………………………….……..30

3.5.2.1 Grid losses with one DG Unit for the IEEE multiphase 13

node test feeder ………………………………………….….………30

3.5.2.2 Grid losses with two DG Units for the IEEE multiphase 13

node test feeder …………………..…...………………….….……..32

3.5.3 Validation of proposed VRI based on PV curves for the IEEE

multiphase 13 node test feeder………………………………….…………32

3.5.4 Comparison of proposed VRI with other bus ranking

approaches for the IEEE multiphase 13 node test feeder .....................…...34

3.6 Detailed simulation of IEEE multiphase 34 node test feeder to validate

proposed VRI……………………………………………………… .......... 37

3.6.1 Identification of the weakest buses using proposed VRI for the IEEE

multiphase unbalanced 34 node test feeder ……………………………….39

3.6.1.1 Bus ranking without/with a voltage regulator

(Cases 8 and 9) .…………………….……………………….………40

3.6.1.2 Bus ranking with an induction generator DG unit at the

most suitable bus (Case 10)…………………………………………40

3.6.1.3 Bus ranking with a 200kW DFIG wind turbine DG unit at

the most suitable bus (Case 11)……………….……...………….….41

3.6.1.4 Bus ranking with a 2.4 MW DFIG wind turbine DG unit

(Case 12) ………………………………...…...…….……………….45

3.6.2 Validation of proposed VRI based on grid loss calculations for the

IEEE multiphase 34 node test feeder ........................................................ 46

3.6.2.1 Grid losses with one DG Unit for the IEEE multiphase 34

node test feeder .…………………….…………………………….………46

3.6.2.2 Grid losses with two DG Units for the IEEE multiphase 34

node test feeder ………………………………………………….………..47

3.6.3 Validation of proposed VRI based on PV curves for the IEEE

multiphase 34 node test feeder .................................................................. 47

3.6.4 Comparison of proposed VRI with other bus ranking

approaches for the IEEE multiphase 34 node test feeder............................48

3.7 Conclusions ........................................................................................... 51

Chapter 4. Validation and application of proposed VRI in improving voltage

stability of unbalanced three-phase networks...................................................... 52

4.1 Introduction ........................................................................................... 52

4.2 Detailed simulation of of modified IEEE unbalanced three-phase 13

node test feeder to validate proposed VRI………. .................................... 52

4.2.1 Identification of weakest three-phase buses using the proposed VRI

………………………………………………………….………………… 54

4.2.1.1 Bus ranking without/with a voltage regulator ........…………54

4.2.1.2 Bus ranking with DG at the most suitable bus .............……..55

4.2.1.3 Bus ranking with DG and SVC…………………..……….....56

4.2.2 Validation of proposed VRI based on grid loss calculations …..........58

4.2.2.1 Grid losses with one DG unit………………..........…………58

4.2.2.2 Grid losses with two DG units………………..............……..59

4.2.3 Validation of proposed VRI based on PV curves…………….......... . 59

4.3 Application of proposed VRI in improving MLF of the modified

unbalanced three-phase 13 node test feeder……………………………. ... 61

4.3.1 Enhancement of MLF by optimal sizing of one DG Unit……........... 62

4.3.2 Improving MLF by placement and sizing of compensation devices

……………………………………………………….…………………. ... 63

4.4 Conclusions ........................................................................................... 64

Chapter 5. Application of proposed VRI in improving voltage stability of

multiphase networks ............................................................................................. 65

5.1 Introduction ........................................................................................... 65

5.2 Application of proposed VRI in improving static voltage stability of the

IEEE 13 node test feeder………………………………….. ...................... 65

5.2.1 Enhancement of MLF by optimal sizing of one compensation device

in of the IEEE 13 node test feeder ………………………… ..................... 67

5.2.2 Improving MLF by placement and sizing of compensation devices in

the IEEE 13 node test feeder ........................................................................68

5.3 Application of proposed VRI in improving static voltage stability of the

IEEE 34 node test feeder………………………………. ........................... 69

5.3.1 Enhancement of MLF by optimal sizing of one compensation device

in the IEEE 34 node test feeder ……………………………………………71

5.4 Application of Proposed VRI in improving dynamic voltage stability of

the IEEE 13 node test feeder ………………………………….. ................ 73

5.5 Application of Proposed VRI in improving dynamic voltage stability of

the IEEE 34 node test feeder ………………………………….. ................ 77

5.6 Conclusions ........................................................................................... 80

Chapter 6. Online bus voltage ranking in unbalanced multiphase smart grid

with plug-in electric vehicle (PEV) charging stations .......................................... 82

6.1 Introduction ......................................................................................... 82

6.2 The modified IEEE 13 node test system with PEV charging stations ........ 83

6.3 Simulation results .................................................................................. 84

6.4 Online placement of SVC units to improve the performance of the

modified IEEE 13 node test system with PEV charging stations

………………………………….. ............................................................. 92

6.5 Conclusions ......................................................................................... 95

Chapter 7. Increasing DG penetration in multiphase distribution networks

considering grid losses, MLF and bus voltage limits ........................................... 96

7.1 Introduction ......................................................................................... 96

7.2 Impacts of DG placement on voltage profile, grid loss, and

MLF………………………….……………………………………….. ...... 97

7.2.1 Impact of DG on voltage profiles ..................................................... 97

7.2.2 Impact of DG on grid losses ............................................................. 97

7.2.3 Impact of DG on MLF ..................................................................... 97

7.2.4 Impact of DG on voltage unbalance factor ....................................... 98

7.3 Proposed algorithm for DG placement……………. .................................. 98

7.4 Simulation results .............................................................................. 100

7.4.1 Bus voltage ranking based on proposed VRI index ........................ 100

7.4.2 Placement and sizing of DG units to improve voltage profile, grid loss,

and MLF………………………………………………………………… 100

7.4.3 Placement and sizing of single-phase capacitor banks to further

improve voltage profile, grid loss, and MLF…………………………….. 104

7.4.4 Summary and analysis of simulation results ................................... 106

7.5 Conclusions ......................................................................................... 107

Chapter 8. Conclusions ....................................................................................... 109

8.1 Contributions ..................................................................................... 111

8.2 Future works ..................................................................................... 111

References ........................................................................................................... 112

Appendix A – The IEEE 13 node and 34 node test systems .............................. 118

Appendix B – Simulation parameters ............................................................... 138

Appendix C – DIgSILENT PowerFactory [32] ................................................. 140

Appendix D – Paper published in Elixir journal .............................................. 141

TABLE OF FIGURES

Figure ‎2-1 Equivalent circuit of a two bus balanced network. --------------------------14

Figure ‎2-2 PV curve based on positive-sequence voltages.------------------------------17

Figure ‎2-3 An unbalanced multiphase distribution system; network configuration

consisting of four nodes with single-, two-, and three-phase sections. ---------- 19

Figure ‎2-4 The equivalent unbalanced three-phase four-wire network for the

unbalanced multiphase distribution system of Fig. 2-3.---------------------------- 19

Figure ‎3-1 The IEEE multiphase 13 node test feeder.------------------------------------25

Figure ‎3-2 Bus ranking for Case 1 (without any voltage regulators). ----------------- 27

Figure ‎3-3 Bus ranking for Case 2 (with a voltage regulator). ------------------------- 27

Figure ‎3-4 Bus ranking for Case 5 (with one DG at bus 675). ------------------------- 29

Figure 3-5 Bus ranking for Case 6 (with one DG and one SVC at bus 675). --------30

Figure ‎3-6 Reactive and active power losses associated with DG connections at

different buses of Figure 3-1 (Case 2).----------------------------------------------- 31

Figure ‎3-7 Reactive and active power losses associated with the first DG installed at

bus 675 and the second DG connected at different buses of Figure 3-1 (Case 5).-

------------------------------------------------------------- ------------------------------- 32

Figure ‎3-8 PV curves of positive-sequence voltage at each three-phase bus for

Case 2.------------------------------------------------------------------------------------- 33

Figure ‎3-9 PV curves of positive-sequence voltage at each two-phase bus for

Case 2.------------------------------------------------------------------------------------- 33

Figure ‎3-10 PV curves of positive-sequence voltage at each single-phase bus for

Case 2.------------------------------------------------------------------------------------- 34

Figure ‎3-11 PV curves of positive-sequence voltage at each bus for Case 6.--------- 34

Figure ‎3-12 PV curves of positive-sequence voltages at buses 634 and 675 for the

modified IEEE 13 node network (Figure 3-1) with only unbalanced three-phase

networks/loads.-------------------------------------------------------------------------- 37

Figure ‎3-13 The IEEE multiphase 34 node test feeder. ----------------------------------38

Figure ‎3-14 Bus ranking for Case 8 (without any voltage regulators). ---------------- 40

Figure ‎3-15 Bus ranking for Case 9 (with a voltage regulator). ------------------------ 40

Figure ‎3-16 Bus ranking for Case 10 (with a DG type induction generator at bus

890).--------------------------------------------------------------------------------------- 40

Figure ‎3-17 Bus ranking for Case 11 (with a DFIG wind turbine DG unit at bus

890).------------------------------------------------------------------------------------- -- 41

Figure 3-18 Bus ranking for Case 12 (with DFIG wind turbines at bus 890). --------46

Figure ‎3-19 Active power loss associated with DG connections at different buses of

Figure 3-1 (Case 9).--------------------------------------------------------------------- 47

Figure ‎3-20 Active power loss associated with the first DG installed at bus 890 and

the second DG connected at different buses of Figure 3-13 (Case 10).-------------

--------------------------------------------------------------------------------------------- 47

Figure ‎3-21 PV curves of positive-sequence voltage at each three-phase bus for Case

9.------------------------------------------------------------------------------------------- 49

Figure ‎3-22 PV curves of positive-sequence voltage at each single-phase bus for

Case 9.-------------------------------------------------------------------------------------49

Figure ‎3-23 PV curves of positive-sequence voltage at each bus for Case 12.-----------

--------------------------------------------------------------------------------------------- 50

Figure 4-1 The modified unbalanced three-phase 13 node test feeder.---------------- 53

Figure ‎4-2 Bus ranking for Case 1 (without any voltage regulators). ----------------- 55

Figure ‎4-3 Bus ranking for Case 2 (with a voltage regulator). ------------------------- 55

Figure ‎4-4 Bus ranking for Case 3 (with one DG at bus 675). ------------------------- 55

Figure 4-5 Bus ranking for Case 4 (with one DG and one SVC at bus 675). ---------57

Figure 4-6 Reactive and active power losses associated with DG connections at

different buses of Figure 4-1 (Case 2).----------------------------------------------- 59

Figure ‎4-7 Reactive and active power losses associated with the first DG installed at

bus 675 and the second DG connected at different buses of Figure 4-1 (Case 3).-

------------------------------------------------------------- ------------------------------- 59

Figure ‎4-8 PV curves of positive-sequence voltage at each three-phase bus for

Case 2.------------------------------------------------------------------------------------- 60

Figure ‎4-9 PV curves of positive-sequence voltage at each bus for Case 4.-----------61

Figure ‎4-10 MLF as a function of the number of DG units placed at the weakest node

(bus 675).------------------------------------------------ -------------------------------- 63

Figure ‎4-11 Simulation results for placement and sizing of DG units in the modified

unbalanced three-phase 13 node feeder (Figure 4-1).-------------------- ---------- 63

Figure 5-1 MLF (for the IEEE 13 node test feeder) as a function of shunt capacitor

size at the weakest single-phase node (bus 611).------------------------------------ 68

Figure ‎5-2 MLF (for the IEEE 13 node test feeder) as a function of the number of

DG units placed at the weakest three-phase node (bus 675).---------------------- 68

Figure ‎5-3 Simulation results for placement and sizing of DG units in the unbalanced

IEEE 13 node test feeder (Figure 3-1).----------------------------------------------- 69

Figure ‎5-4 MLF (for the IEEE 13 node test feeder) as a function of shunt capacitor

size at the weakest single-phase node (bus 684).-------------------- --------------- 72

Figure ‎5-5 MLF (for the IEEE 13 node test feeder) as a function of the number of

DG units (DFIG wind turbines) placed at the weakest three-phase node (bus

890).--------------------------------------------------------------------------------------- 73

Figure ‎5-6 Comparison of the proposed VRI (Eq. 2-14) based on static and dynamic

approaches for Case 2 (Table 5-1).---------------------------------------------------- 74

Figure ‎5-7 Comparison of the proposed VRI (Eq. 2-14) based on static and dynamic

approaches for Case 5 (Table 5-1).---------------------------------------------------- 75

Figure ‎5-8 Voltage profiles of bus 675 under switch operation of Case 5 (Table 5-1).-

--------------------------------------------------------------------------------------------- 75

Figure ‎5-9 Active and reactive power of DG at bus 675 under switch operation of

Case 5 (Table 5-1).---------------------------------------------------------------------- 76

Figure ‎5-10 Active and reactive power of DG installed at bus 675 (of the IEEE 13

node test feeder)with/without SVC after switch closed at time 0.67s.------------- 76

Figure ‎5-11 Comparison of VRI values for dynamic operating conditions in the IEEE

13 node test feeder.------------------------------------------------------ --------------- 77

Figure ‎5-12 Comparison of the proposed VRI (Eq. 2-14) based on static and dynamic

approaches for Case 9 (Table 5-2).---------------------------------------------------- 78

Figure ‎5-13 Comparison of the proposed VRI (Eq. 2-14) based on static and dynamic

approaches for Case 10 (Table 5-2).-------------------------------------------------- 78

Figure ‎5-14 Voltage profiles of bus 890 under circuit breaker operation of Case 10

(Table 5-2).------------------------------------------------------------------------------- 79

Figure ‎5-15 Active and reactive power of DG at bus 890 under circuit breaker

operation of Case 10 (Table 5-2).----------------------------------------------------- 79

Figure ‎5-16 Active and reactive power of DG installed at bus 890 (of the IEEE 34

node test feeder) with/without SVC after circuit breaker closed at time 0.54s.----

--------------------------------------------------------------------------------------------- 80

Figure ‎5-17 Comparison of VRI values for dynamic operating conditions in the IEEE

34 node test feeder.---------------------------------------------------------------------- 80

Figure ‎6-1 The unbalanced multiphase 13 node test feeder with PEV charging

stations at bus 634 or bus 680.--------------------------------------------------------- 84

Figure ‎6-2 Daily load curves associated with Figure 6-1 for linear loads [45]. ------ 86

Figure ‎6-3 Daily load curves associated with Figure 6-1 for PEV charging stations

[36].---------------------------------------------------------------------------------------- 86

Figure ‎6-4 Simulation results for Case 1: the 24 hour voltage profile of buses 634,

675 and 680.------------------------------------------------------------------------------ 87

Figure ‎6-5 Simulation results for Case 2: the 24 hour voltage profile of buses 634,

675 and 680.------------------------------------------------------------------------------ 87

Figure ‎6-6 Simulation results for Case 3: the 24 hour voltage profile of buses 634,

675 and 680.------------------------------------------------------------------------------ 88

Figure ‎6-7 Simulation results for Case 4: the 24 hour voltage profile of buses 634,

675 and 680.----------------------------------------------------------------------------- 88

Figure ‎6-8 Simulation results for Case 5 with online placement of two SVC units:

the 24 hour voltage profile of buses 634, 675 and 680.---------------------------- 93

Figure ‎7-1 The proposed algorithm for the placement and sizing of DG units and

single-phase capacitors in multiphase networks.------------------------------------ 99

Figure ‎7-2 Simulation results for the first DG placement (stage one, iteration one);

voltage ranking index with no DFIG installation (base-case load).------------- 101

Figure ‎7-3 Simulation results for the first DG placement (stage one, iteration one);

loading factor and active power loss with different DG penetrations at bus 890.-

-------------------------------------------------------------------------------------------- 101

Figure ‎7-4 Voltage profile with 40% DG penetration at bus 890. --------------------101

Figure ‎7-5 Simulation results for the first DG placement (stage one, iteration one);

voltage profile with 30% DFIG penetration at bus 890.-------------------------- 102

Figure ‎7-6 Simulation results for the second DG placement (stage one, iteration two);

voltage ranking index with 30% DFIG units installed at bus 890.-------------- 103

Figure ‎7-7 Simulation results for the second DG placement (stage one, iteration two);

loading factor and active power loss with 30% DFIG penetration at bus 890 and

different DFIG penetration at bus 852.---------------------------------------------- 103

Figure ‎7-8 Simulation results for the second DG placement (stage one, iteration two);

voltage profile with 30% DFIG penetration at bus 890 and 30% DFIG

penetration at bus 852.---------------------------------------------------------------- 104

Figure ‎7-9 Simulation results for the third DG placement (stage one, iteration three)

showing voltage ranking index with 30% DFIG units installed at bus 890 and

30% DFIG at bus 852.----------------------------------------------------------------- 105

Figure ‎7-10 Simulation results for the single-phase capacitor placement (stage two,

iteration one); voltage ranking index with 30% DG units installed at bus 890,

30% DG at bus 852, and single-phase shunt capacitor 0.273MVAr at bus 822.---

-------------------------------------------------------------------------------------------- 105

Figure ‎7-11 Simulation results for the single-phase capacitor placement (stage two,

iteration one); voltage profile with 30% DG penetration at bus 890, 30% DG

penetration at bus 852, and single-phase 0.273MVAr shunt capacitor at bus

822.---------------------------------------------------------------------------------------106

Figure ‎7-12 Comparison of %VUF at different iterations of the proposed algorithm

(Figure 7-1).-----------------------------------------------------------------------------107

LIST OF TABLES

Table ‎3-1 Simulated case studies for the IEEE multiphase 13 node test feeder (Fig. 3-

1).------------------------------------------------------------ ----------------------------- 26

Table ‎3-2 Bus ranking for cases 1 and 2 based on the proposed VRI.----------------- 28

Table ‎3-3 Bus ranking for case 5 based on the proposed VRI.--------------------------29

Table ‎3-4 Bus ranking for case 6 based on the proposed VRI.--------------------------31

Table ‎3-5 Bus ranking results for the IEEE 13 node network with only unbalanced

three-phase networks/loads.------------------------------------------------------------ 36

Table ‎3-6 Bus ranking results for the IEEE 13 node test feeder (Fig. 3-1) with only

unbalanced three-phase networks/loads.------------------------------------------ --- 36

Table ‎3-7 Maximum loading factors with SVC.------------------------------------------ 37

Table ‎3-8 Simulated case studies for the IEEE 34 node test feeder (Fig. 3-13).------ 39

Table ‎3-9 Bus ranking for cases 8 and 9 based on the proposed VRI.----------------- 42

Table ‎3-10 Bus ranking for case 10 based on the proposed VRI. ----------------------43

Table ‎3-11 Bus ranking for case 11 based on the proposed VRI. ----------------------44

Table ‎3-12 Bus ranking for case 12 based on the proposed VRI.----------------------- 45

Table ‎3-13 Bus ranking results for the multiphase IEEE 34 node network. ---------- 50

Table ‎4-1 Simulated case studies for the modified unbalanced three-phase 13 node

test feeder (Fig. 4-1). ------------------------------------ ------------------------------ 54

Table ‎4-2 Bus ranking for cases 1 and 2 based on the proposed VRI.----------------- 56

Table ‎4-3 Bus ranking for case 3 based on the proposed VRI. -------------------------57

Table ‎4-4 Bus ranking for case 4 based on the proposed VRI. -------------------------58

Table ‎4-5 Simulation results of the modified unbalanced three-phase 13 node test

feeder (Fig. 4-1, Table 4-1): comparison of MLF without/with regulator, DG

and SVC.---------------------------------------------------------------------------------- 62

Table ‎5-1 Simulation results of the IEEE 13 node test feeder (Fig. 3-1, Table 3-1):

comparison of MLF without/with regulator, single-phase shunt capacitor, DG

and SVC. -------------------------------------------------------- ------------------------ 66

Table ‎5-2 Simulation results of the IEEE 34 node test feeder (Fig. 3-13, Table 3-1):

comparison of MLF without/with regulator, DG types IG and DFIG. ---------- 70

Table ‎5-3 Simulation results of the IEEE 34 node test feeder (Fig. 3-1): comparison

of MLF with single-phase shunt capacitor placed at different buses. ------------71

Table ‎5-4 MLF (for the IEEE 34 node test feeder) as a function of the number of DG

units (IGs) placed at the weakest three-phase node (bus 890). -------------- ----- 72

Table ‎6-1 Case 2 - Bus voltage ranking indices over 24 hours with four PEV

charging stations at bus 634.----------------------------------------------------------- 89

Table ‎6-2 Case 3 - Bus voltage ranking index for the multiphase system of Figure 6-1

with four PEV charging stations at bus 680.----------------------------------------- 90

Table ‎6-3 Case 4 - Bus voltage ranking index for the multiphase system of Figure 6-1

with four PEV charging stations at bus 680 and two PEV charging stations at

bus 634.---------------------------------------------------------------------------------- - 91

Table ‎6-4 Case 5 with online placement of two SVC units - Bus voltage ranking

index for the multiphase system of Figure 6-1.-------------------------------------- 94

Table ‎7-1 Detailed solution for DFIG and capacitor placement and sizing in the IEEE

multiphase 34 node test feeder (Figure 3-13) using the proposed algorithm of

Figure 6-1.-------------------------------------------------------------------------------108

List of abbreviations

ALR Active power loss reduction

CPF Continuation power flow

CS Charging station

DG Distributed generator

DFIG Doubly-fed induction generator

DVCI Dynamic voltage collapse index

IG Induction generator

MLF Maximum Loading Factor

PEV Plug-in electric vehicle

RLR Reactive power losses reduction

SVC Static VAR compensators

TPSI Transfer power stability index

VCPI Voltage collapse predictor indicator

VR Voltage regulator

VRI Voltage ranking index

VUF Voltage unbalance factor

List of symbols

Vectors and parameters

B, R, X, Y, Z Susceptance, resistance, reactance, admittance and impedance.

P, Q, S, V Real power, reactive power, complex power and voltage.

Phase angle.

𝛼 Phase angle of the load impedance.

𝛽 Phase angle of the Thevenin impedance.

Subscripts

Base-load Base-load condition.

Collapse The point of voltage collapse.

i, j,k,m,N Bus number.

No-load No-load condition.

Thev Thevenin.

Superscripts

+ Positive-sequence.

* Complex conjugate operator.

init Initial operating state

limit Voltage stability limit

1

Chapter 1. Introduction

1.1 STATEMENT OF THE PROBLEM

Modern distribution networks are being operated closer to their voltage stability

limits due to many factors such as increasing load levels [1], lack of reactive power

sources [2], high installation of single-phase shunt capacitors [3] and reverse action

of voltage control devices [4]. Under these stressed operating conditions, voltage

instability and voltage collapse may occur if suitable monitoring and control

measures are not engaged.

The analyses of voltage stability are divided in two categories depending on the type

of disturbance; static approaches based on the power flow calculation and dynamic

approaches based on time-domain simulation. Static analyses are simple and fast

solutions widely used to identify the weakest buses and determine voltage stability

margins for small disturbances. However, dynamic analyses for large disturbances

should also be performed and compared to verify the results of static approaches [5]-

[7].

Unbalanced operation of distribution networks significantly decreases the voltage

stability margins. Analyses of unbalanced networks indicate that there is at least one

phase with clockwise rotation (e.g., as the load levels increase on the PV curves, the

voltage magnitudes decrease) and much lower voltage level than the other two

phases [9]. This considerably complicates the voltage stability analysis of unbalanced

and multiphase networks. Therefore, it is very complicated to rank the buses and

identify the weakest bus under different voltage level and phase-voltage stability

margins unless these conditions are somehow merged to simplify the procedure.

This thesis proposes a new voltage ranking index (VRI) based on the (fundamental)

positive- sequence voltage ratio of Vcollapse/Vbase-load to identify the weakest single-,

two- and three-phase buses in unbalanced and multiphase distribution networks. The

proposed VRI is compared with three conventional indices and validated by grid loss

2

calculations and PV curves. Further validations of the new index are presented

through extensive simulations of the IEEE unbalanced multiphase 13 node and 34

node test feeders without/with voltage regulators, single-phase shunt capacitors,

distribution generations (DGs) and static VAR compensators (SVCs). Finally,

simulations are performed to demonstrate the application of the new VRI in

increasing maximum loading factor (MLF) and improving voltage stability under

static and dynamic operating conditions.

1.2 LITERATURE REVIEW

In this section, a comprehensive literature review is carried out to (1) examine

different existing approaches to rank the buses for balanced networks, (2) study the

sensitivity method proposed to rank the buses for unbalanced networks, (3)

investigate voltage stability enhancement by connecting compensation devices

considering grid losses and MLF.

1.2.1 Bus ranking approaches for balanced networks

In balanced networks, there are several techniques based on static approaches to

identify the weakest bus. The current bus ranking approaches include:

Modal analysis [5-6], [9-10], [14]- In this method the eigenvalues and

eigenvectors of the reduced Jacobian matrix are first calculated. The

magnitudes of the eigenvalues are then used to provide a relative proximity of

the system to voltage instability. Positive eigenvalues represent voltage

stability of system and the smaller the magnitude, the closer the relevant

modal voltage is to being unstable. Finally, the weakest bus of the system is

determined by computing the eigenvector for different buses in the system.

The node with the highest eigenvector is identified as the weakest bus of the

system. This method is very useful as it provides a relative proximity of the

system to voltage instability, as well as the key contributing factors to

instability such as the weakest buses and branches. However, the approach is

based on the assumption that the active power is kept constant and can only

be applied to balanced systems. When the system is under stressed, both

dV/dQ and dV/dP are important parameters to determine the stability of the

3

system. Therefore, this method can be invalid when the system is under stress

conditions.

Sensitivity analysis [6], [14]- This method is based on the relative sensitivity

of voltage magnitude with respect to the reactive power. Load flow

calculations are utilized to compute the relationship between voltage changes

and reactive power changes at different buses (dV/dQ). The magnitude of the

sensitivity index becomes large when the system is close to MLF. Therefore,

the weakest bus is classified as the one with the maximum value of the

voltage sensitivity index. However, according to reference [14] the

magnitudes of the sensitivities for different system conditions may not

accurately provide a direct measure of the relative degree of stability.

The V/V0 index [10-11]- This conventional and well-known index is based on

the ratio of the voltage magnitude at certain load obtained from load flow

study to the voltage magnitude at an identical state but with all the loads set

to zero. This index allows immediate detection of the weakest bus and

corrective action can be taken to prevent the voltage instability. In this

proposal, both the magnitude and phase of bus voltage will be applied to

modify this index and extend its application to multiphase networks using

symmetrical components.

Bus voltage change index [12]- A bus voltage change index is defined for

each load bus as:

𝑉𝐶𝑖 =𝑉𝑖

𝑖𝑛𝑖𝑡 −𝑉𝑖𝑙𝑖𝑚𝑖𝑡

𝑉𝑖𝑙𝑖𝑚𝑖𝑡 (1-1)

where 𝑉𝑖𝑖𝑛𝑖𝑡 and 𝑉𝑖

𝑙𝑖𝑚𝑖𝑡 are the voltage magnitudes at bus i at the initial

operating state and at the voltage stability limit, respectively. The order of the

bus ranking can be sorted based on this index. The largest bus voltage change

index will correspond to the weakest bus.

L index [10]- This index calculates the distance between the present state of

the system and the stability limit. L-index describes the stability of the

complete system as it varies from 0 to 1 corresponding to no load and voltage

4

collapse conditions, respectively. Buses with highest index values are

identified as the weakest buses in the system.

Reactive power margin [48]- Reactive power margin of load buses is

determined as a margin between the voltage axis and the lowest MVAr point

of the Q-V curve. This index indicates how further the loading on a particular

bus can be increased before its loading limit is exhausted and voltage collapse

takes place. Reference [11] proves that reactive power margin is a suitable

index to identify the weakest bus irrespective of the load pattern in a

distribution system.

PV curve [6], [10-11], [14]- PV curve is generated by calculating a series of

power flow calculations. With a load increasing, its voltage magnitude will

become lower until reaching a point of voltage collapse. This curve can be

utilized to determine voltage stability margins. The margin between the

voltage collapse point and the current operating point can be used as an index

for bus ranking. The weakest bus is identified as the bus with the lowest

voltage stability margin.

QV curve [6], [10-11]- This curve is also generated by calculating a series of

power flow. With the help of QV curve, it is possible for the operators to

know the maximum reactive power that can be compensated before reaching

minimum voltage limit. The MVAr distance from the operating point to the

bottom of the QV curve is called the reactive power margin. QV curve can be

used as an index for voltage stability limit. The bus, which has the lowest

margin of reactive power, is the weakest bus in the system.

Integrated bus voltage change index with reactive power margin [13]- This

method combines the bus voltage change index with the reactive power

margin index. The bus with largest value of the two indices is identified as

the weak buses.

For dynamic analyses of balanced networks, the following indices can be used to

predict and indicate voltage collapse:

5

Voltage collapse prediction index (VCPI) [14]- The voltage magnitude and

voltage angle information of the participating buses in the system and the

network admittance matrix are required to calculate this index and predict

voltage collapse. The technique relies on basic power flow equations to

compute voltage phasors and the network admittance matrix and calculate the

VCPI index at every participating bus. The voltage collapse prediction index

at bus k is obtained as:

𝑉𝐶𝑃𝐼𝑘 =

1 −

𝑌𝑘𝑚

𝑌𝑘𝑗𝑁𝑗=1

𝑗≠𝑘

𝑁

𝑚 =1𝑚 ≠𝑘

𝑉𝑚

𝑉𝑘

(1-2)

where N is the number of participating buses; j, k, and m are positive integer

numbers; 𝑌𝑘𝑚 is the admittance between buses k and m;𝑌𝑘𝑗 is the admittance

between buses k and j; 𝑉𝑚 is the voltage phasor at bus m; and 𝑉𝑘 is the

voltage phasor at bus k.

The VCPI index varies from 0 to 1. The values of this index determine the

proximity to voltage collapse at a bus. The buses with VCPI index values of 0

are stable while a bus with an index value of 1 is experiencing a voltage

collapse.

Power transfer stability index (PTSI) [15]- The PTSI is calculated at every

bus by using information of the load apparent power, Thevenin voltage,

Thevenin impedances and phase angles of the load. The PTSI requires

voltage phasor information of the participating buses in a system and network

admittance matrix:

𝑃𝑇𝑆𝐼 = 2𝑆𝐿𝑍𝑇𝑕𝑒𝑣 (1+cos (𝛽−𝛼))

𝐸𝑇𝑕𝑒𝑣2 (1-3)

where 𝑆𝐿 is the load apparent power, 𝑍𝑇𝑕𝑒𝑣 is the Thevenin impedance,

𝐸𝑇𝑕𝑒𝑣 is the Thevenin voltage, 𝛼 is the phase angle of the load impedance,

and 𝛽 is the phase angle of the Thevenin impedance.

6

The value of PTSI will vary between 0 and 1. When PTSI value reaches 1, it

indicates that a voltage collapse has occurred. Compared to VCPI, PTSI is

more sensitive in detecting dynamic voltage collapse because its value will be

very near to unity during a collapse condition. Therefore, the PTSI index can

be considered to be a more accurate indicator compared to the VCPI for

voltage collapse prediction.

Dynamic voltage collapse index (DVCI) [16]- The DVCI is implemented by

measuring power and voltage at the sending end:

𝐷𝑉𝐶𝐼 =𝑉𝑖

2 − 𝑅𝑖𝑗 𝑃𝑗 + 𝑋𝑖𝑗 𝑄𝑗 + 𝑅𝑖𝑗2 +𝑋𝑖𝑗

2 (𝑃𝑗2+𝑄𝑗

2 )

2(𝑋𝑖𝑗 𝑃𝑗 − 𝑅𝑖𝑗 𝑄𝑗 )2 (1-4)

where 𝑉𝑖 is the magnitude of voltage at sending bus i, 𝑅𝑖𝑗 is the resistance

between buses i and j, 𝑋𝑖𝑗 is the reactance between buses i and j, 𝑃𝑗 is the

active power flowing at receiving bus j, and 𝑄𝑗 is the reactive power flowing

at receiving bus j.

When the power-flow equations are solvable, this index is greater than or

equal to 1.0. The index equals 1.0 when the system is reaching the maximum

loading level. On loading the feeder more than its capacity, the index assumes

a value less than 1.0.

The bus ranking problem becomes very complicated under unbalanced and

multiphase operating conditions and has not been addressed in the literature. None of

the above-mentioned indices for dynamic analysis and prediction of voltage collapse

identifies the weakest bus of the system. In term of static voltage stability analysis,

all above-mentioned bus ranking indices are only capable of identifying the weakest

buses of balanced systems and do not apply to unbalanced and multiphase networks.

Therefore, there is much need and interest to define a reliable bus ranking index for

unbalanced and multiphase networks that may be used for static and dynamic

analyses. In this proposal, definition of the V/V0 index will be modified by including

both the voltage magnitude and the voltage phase information and its application will

be extended to unbalanced and multiphase networks using symmetrical components.

7

1.2.2 Existing bus ranking approaches for unbalanced networks

Many three-phase continuation power flow (CPF) methods have been used to

analyze voltage stability margins of unbalanced distribution networks [8, 17-19]. The

usual approach is to run three-phase power flow and generate PV curves by

increasing the active power at selected loads. Reference [18] shows that PV curves of

phases „b‟ and „c‟ at bus 675 in the IEEE 13 node test feeder (Fig. 3-1) have counter-

clockwise rotations while phase „a‟ has a clockwise rotation. In addition, the PV

curves for the unbalanced networks and/or unbalanced loads have shown different

voltage stability margins on each phase. Reference [19] applies the voltage

sensitivities index V/P associated with the maximum loading factor (MLF) to

perform stability analysis on unbalanced three-phase networks. However, according

to reference [14] and [20], the magnitudes of the voltage sensitivity indices (V/P)

do not provide a correct measure of voltage stability for unbalanced three-phase

networks. In addition, references [10-11] state that bus voltage ranking allows

immediate detection of weakest bus. However, in the literature, bus voltage ranking

index is only defined for balanced three-phase networks. Therefore, it is essential to

define a new VRI that also includes unbalanced three-phase and multiphase

networks.

1.2.3 Voltage stability enhancement by connecting compensation devices

considering grid losses and MLF

An important application of bus ranking in distribution networks is for voltage

stability enhancement. The purpose of bus ranking is to determine which node is the

weakest bus for connecting DG and/or reactive power compensation devices. DGs

can be allocated at the first bus reaching the voltage limit to improve voltage profile

and reduce grid losses [21]. In addition, the best location for reactive power

compensation to improve voltage stability margins is the weakest bus in the network

[9]. Hence, it might also be sufficient and reasonable to enhance voltage stability

margins in unbalanced multiphase distribution networks by connecting DG and/or

reactive power compensation devices at the weakest single-, two- and three-phase

buses [3]. The present integration of Distributed Generation (DG) units in power

systems has many advantages, but also challenges the performance of the old

8

networks. One of these challenges is to investigate the location and the penetration

level of DG units which can easily be absorbed in the system without major

structural changes while keeping all bus voltage levels within permissible limits. Due

to the high penetration of DG, voltage instability problems have become important

issues in power systems. Most studies confirm that 10-15% penetration of DG can be

absorbed in the electricity network [22]. It is well-known that high penetration levels

of DG (e.g., 40%) may have significant impacts on voltage profile, grid loss, and

voltage stability margin [29-30]. These impacts may appear either positively or

negatively, depending on the type of distribution networks, nature of distributed

generation sources and load characteristics. It seems reasonable to expect that the

connection of DG to the utility grid might improve the voltage profile and will

enhance the voltage stability of a distribution system while reducing active and

reactive losses [23]. Even though DG has a variety of benefits, it also imposes some

problems and limitations. These problems become highly significant as the

penetration level of DG increases and its impact will become worse. This will

eventually require voltage stability analysis to ensure a proper and reliable operation

of the power system with large amounts of DG [6]. When the power system becomes

stressed (e.g., as a result of increasing load), voltage instability can easily occur. This

type of voltage instability mostly occurs at the weakest bus [24]. Therefore, both the

location and the penetration level of DG become a challenging task for system

planning and operation. Several methods to place DG units have been reported in the

literature including:

Voltage sensitivity analysis [19]- The voltage sensitivities can be considered

as an indicator of voltage instability. In principle, the larger the voltage

sensitivity is, the lower the maximum loading factor will be. Therefore, the

best locations to install compensation devices for voltage stability

enhancement are the buses with the highest voltage sensitivity values.

Continuation of power flow for determination of the most sensitive bus to

voltage collapse [19], [25-26]- This method is based on the analysis of

power-flow continuation and determination of most sensitive buses based on

𝜕𝑉/𝜕𝑃 at the point of voltage collapse. After that, the compensation devices

9

with certain capacity will be placed at the most sensitive buses with the

highest on 𝜕𝑉/𝜕𝑃 values.

Voltage stability index (VSI) [27], [49]- The locations of DG units will be

identified by means of the voltage stability index of buses. The computation

of voltage stability index of all the buses in the system is defined as:

𝑉𝑆𝐼𝑗 = 𝑉𝑖4 − 4 𝑃𝑗𝑅𝑖𝑗 + 𝑄𝑗𝑋𝑖𝑗 )𝑉𝑖

2 − 4 𝑃𝑗𝑋𝑖𝑗 − 𝑄𝑗𝑅𝑖𝑗 2 (1-5)

where 𝑉𝑖 is the magnitude of voltage at bus i, 𝑅𝑖𝑗 is the resistance between

buses i and j, 𝑋𝑖𝑗 is the reactance between buses i and j, 𝑃𝑗 is the active power

flowing at bus j, and 𝑄𝑗 is the reactive power flowing at bus j.

Clearly, the most appropriate locations for DG placement and voltage

stability enhancement are the buses with the minimum VSI values.

Optimization approaches [27-28]- In [27], optimization approaches based on

genetic algorithms (GAs) are used to determine the size of compensation

devices placed at the weakest bus as identified by VSI index to minimise the

network power loss and maximise the voltage regulation in a given network.

According to [28], optimum placement of DG units with reactive power

capability can enhance voltage stability and maximize voltage stability

margin in the entire power network.

Voltage profile and loss calculations [29-30]- In [29], a system unbalanced

voltage variance index which is more reasonable and more accurate than the

system average voltage index is proposed for considering voltage profiles

and grid losses in order to find the optimal location of DG. The buses with

the lowest voltage profile and active power loss were selected for placing

compensation devices. According to [30], the voltage profile and loss studies

confirmed that DG unit could provide a significant improvement to the

voltage profile of the system. These studies also revealed the significance of

properly locating and sizing DG units. They also demonstrated that increasing

DG power output does not necessarily correlate to an improved voltage

profile.

10

References [28, 31] show that the proper sizing and location of DG can significantly

influence the voltage profile and should be well planned to maintain the node

voltages within the permissible limits. Detailed analyses of unbalanced networks

based on continuation three-phase power flow show that the three PV curves on each

phase of the unbalanced networks are different [18-19]. Therefore, to determine the

voltage stability margins, the method of symmetrical components has been employed

in this thesis to merge the three PV curves to one PV curve based on positive-

sequence voltage. Furthermore, in order to extend and generalize the conventional

definition of bus voltage ranking index for multiphase networks, symmetrical

components are also applied to the three-phase voltages computed from three-phase

power flow [3].

1.3 RESEARCH OBJECTIVES

The main objective of this research is to develop a new ranking index for unbalanced

multiphase distribution networks and to propose a new algorithm for the placement

and sizing of DG units and single-phase capacitors in multiphase networks in order

to reduce grid losses and increase MLF while keeping all bus voltages within

acceptable limits. In particular, the new index must identify the weakest single-

phase, two-phase and three-phase buses suitable for reactive power compensation

and voltage stability enhancement. Therefore, the main objectives of this thesis can

be summarized as follows:

1- Define a new bus ranking index for balanced and unbalanced multiphase

networks. This new index will help researchers to identify the weakest bus

for voltage stability enhancement.

2- Validate the proposed bus ranking index and confirm its accuracy using grid

losses calculations and PV curves based on positive-sequence voltage.

3- Validate the accuracy of the proposed bus ranking index in dynamic system

conditions.

4- Application of the proposed bus ranking index to improve and increase MLF

by placing single-phase shunt capacitor, DG and FACTS devices.

11

5- Propose an iterative algorithm to improve the performance of multiphase

distribution networks by proper placement and sizing of DG units and single-

phase capacitors.

1.4 THESIS STRUCTURE

This thesis consists of seven chapters. Chapter 1 discusses bus ranking and voltage

stability enhancement approaches. Chapter 2 presents the conventional bus ranking

approach for balanced networks and proposes new bus ranking approaches to

identify the weakest buses of balanced and unbalanced networks. The method of

symmetrical component has been applied to three-phase voltages resulting from

three-phase power flow. These indices will be compared and applied to different

situations in later chapters through extensive simulations.

Chapter 3 compares the performance and accuracy of the conventional and the new

ranking indices with the well-known voltage sensitivity ratios V/P and V/Q

defined for balanced and unbalanced three-phase distribution networks. Grid loss

calculations, PV curves based on positive-sequence voltage and voltage sensitivity

methods are compared through simulation studies.

In Chapter 4, the proposed VRI is applied to the modified unbalanced three-phase 13

node test feeder to improve voltage stability and increase MLF under unbalanced

three-phase conditions.

In Chapter 5, an application of proposed VRI for improving voltage stability margins

is utilized to improve MLF in multiphase distribution networks. It is revealed that the

proposed VRI can fulfill both the static and dynamic voltage stability criteria.

In Chapter 6, a bus ranking approach for online applications is proposed to identify

the weakest buses during the 24 hour period in order to study and compensate the

detrimental impacts of PEV charging stations on voltage profiles and voltage

stability of smart grid. Simulations results show that the proposed online bus ranking

approach can be used to control and improve the detrimental impacts of large PEV

charging stations.

12

Chapter 7 presents an iterative algorithm for the placement and sizing of DG units

and single-phase capacitors in multiphase networks to reduced grid losses, increase

MLF while keeping all bus voltage within acceptable limits. Simulation results

including locations and the maximum penetration levels of DG units (DFIGs) as well

as the locations and sizes of single-phase capacitors are presented for the IEEE

multiphase 34-node test feeder.

Finally, conclusions and suggestions for future research are presented in Chapter 7.

1.5 LIST OF PUBLICATIONS

The main content of the thesis is based on the following published/submitted articles:

Journal articles:

J1. P. Juanuwattanakul and M.A.S Masoum “Bus Voltage Ranking for

Unbalanced Three-phase Distribution Networks and Voltage Stability

Enhancement,” Elixir on electrical engineering, pp. 5976-5981, Dec 2011.

J2. P. Juanuwattanakul and M.A.S Masoum “Bus Voltage Ranking

Index for Multiphase Distribution Networks,” Submitted to IET Science,

Measurement & Technology (Manuscript ID SMT-2011-0063).

J3. P. Juanuwattanakul and M.A.S Masoum “Increasing DG

Penetration in Multiphase Distribution Networks Considering Grid Losses,

Maximum Loading Factor and Bus Voltage Limits,” Submitted to IET

Generation, Transmission & Distribution (Manuscript ID GTD-2011-0841).

Conference papers:

C1. P. Juanuawattanakul and M.A.S Masoum “Voltage Stability

Enhancement for Unbalanced Multiphase Distribution Networks,” in the

proceedings of IEEE PES General Meeting 2011, Detroit, USA, July 2011.

C2. P. Juanuawattanakul and M.A.S Masoum, “Analysis and

Comparison of Bus Ranking Indices for Balanced and Unbalanced Three-Phase

Distribution Networks,” in the proceedings of AUPEC 2011, Brisbane,

Australia, September 2011.

13

C3. P. Juanuawattanakul, M.A.S Masoum, and Pual. S. Moses “Voltage

Analysis for Placement of DG in Unbalanced Distribution Networks,” in the

proceedings of EPQU 2011, Lisbon, Portugal, October 2011.

C4. P. Juanuawattanakul and M.A.S Masoum, “Identification of the

weakest buses in Unbalanced Multiphase Smart Grids with Plug-In Electrical

Vehicle Charging Stations,” in the proceeding of ISGT 2011, Perth, Australia,

November 2011.

C5. P. Juanuawattanakul, M.A.S Masoum, C. Niyomsak, and M.

Mohseni “Voltage Analysis for Placement of DG in Multiphase Distribution

Networks,” Accepted for presentation and publication, IEEE PES General

Meeting 2012, San Diego, USA, July 22-26, 2012.

14

Chapter 2. Proposed bus voltage ranking index (VRI)

for multiphase distribution networks

2.1 INTRODUCTION

In balanced networks, all conventional methods mentioned in chapter 1 can be used

to identify the weakest bus. However, in unbalanced three-phase networks, the only

existing method available is based on voltage sensitivities index V/P associated

with MLF to perform bus ranking on unbalanced three-phase networks. However,

this method does not provide a correct measure of voltage stability under unbalanced

three-phase networks [14], [20]. Furthermore, the conventional voltage bus ranking

index as shown in equation (2-1) is only defined for balanced three-phase networks.

Therefore, the rationale is toward defining a new VRI that can be applied to

unbalanced three-phase and multiphase networks. The development of such an index

constitutes the main focus of this chapter.

2.2 CONVENTIONAL VRI FOR BALANCED NETWORKS

This section starts with the definition and derivation of the conventional voltage

ranking index (VRI=V/Vo) using the two bus balanced network of Figure 2-1 and

continues to extend its definition to multiphase networks using symmetrical

components [3].

Rij+jXij

i j

ViÐi VjÐj

Pi+jQi

Figure 2-1 Equivalent circuit of a two bus balanced network.

The conventional VRI is only defined for single-phase and balanced three-phase

networks [9-10]:

15

𝑉𝑅𝐼𝑗𝑐𝑜𝑛𝑣𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 =

𝑉

𝑉0=

𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑

𝑉𝑗 ,𝑛𝑜 −𝑙𝑜𝑎𝑑 (2-1)

where j is the bus number, 𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑 and 𝑉𝑗 ,𝑛𝑜−𝑙𝑜𝑎𝑑 are the bus voltages for the

base-load and no-load operating conditions, respectively.

Balanced three-phase load flow can be used to compute 𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑 by setting the

complex power at bus j to zero:

𝑆𝑗 = 𝑓 𝛿, 𝑉 = 𝑃𝑗 − 𝑗𝑄𝑗 = 𝑉𝑗 ∠𝛿𝑗 ∗

𝑉𝑖∠𝛿𝑖−𝑉𝑗 ∠𝛿𝑗

𝑅𝑖𝑗 +𝑗 𝑋𝑖𝑗 = 0 (2-2)

where 𝑉𝑖∠𝛿𝑖 and 𝑉𝑗 ∠𝛿𝑗 are the voltages at buses i and j, respectively. Separating real

and imaginary parts of (2-2):

𝑅𝑒𝑎𝑙 𝑆𝑗 = 0

𝐼𝑚𝑎𝑔 𝑆𝑗 = 0 ⇒

𝑊𝑟𝑒𝑎𝑙 𝛿𝑖𝑗 , 𝑉𝑗 = 𝑃𝑗𝑅𝑖𝑗 + 𝑄𝑗 𝑋𝑖𝑗 = 𝑉𝑖𝑉𝑗 𝑐𝑜𝑠𝛿𝑖𝑗 − 𝑉𝑗 2

𝑊𝑖𝑚𝑎𝑔 𝛿𝑖𝑗 , 𝑉𝑗 = 𝑃𝑗𝑋𝑖𝑗 − 𝑄𝑗 𝑅𝑖𝑗 = 𝑉𝑖𝑉𝑗 𝑠𝑖𝑛𝛿𝑖𝑗 (2-3)

where 𝛿𝑖𝑗 = 𝛿𝑖 − 𝛿𝑗 . The voltage 𝑉𝑗 is computed by squaring and adding the real and

imaginary parts of (2-3):

𝑉𝑗4 + 2 𝑃𝑗𝑅𝑖𝑗 + 𝑄𝑗 𝑋𝑖𝑗 − 0.5𝑉𝑖

2 𝑉𝑗2 + 𝑃𝑗

2 + 𝑄𝑗2 𝑅𝑖𝑗

2 + 𝑋𝑖𝑗2 = 0 (2-4)

There are four solutions to (2-4),

𝑉𝑗 = ± 1

2 −𝑏 ± 𝑏2 − 4𝑐 (2-5)

where 𝑏 = − 𝑉𝑖2 − 2𝑃𝑗𝑅𝑖𝑗 − 2𝑄𝑗 𝑋𝑖𝑗 and 𝑐 = 𝑃𝑗

2 + 𝑄𝑗2 𝑅𝑖𝑗

2 + 𝑋𝑖𝑗2 . However,

−𝑏 is always positive because the term (−2𝑃𝑗𝑅𝑖𝑗 − 2𝑄𝑗 𝑋𝑖𝑗 ) is small as compared

to (𝑉𝑖2) and also (4𝑐) is small as compared to (𝑏2); therefore, the unique positive

and stable solution of (2-5) is

𝑉𝑗 = 𝑉𝑗 ,𝑏𝑎𝑠𝑒𝑑 −𝑙𝑜𝑎𝑑 = + 1

2 −𝑏 + 𝑏2 − 4𝑐 (2-6)

Substituting (2-6) in (2-1) results in

𝑉𝑅𝐼𝑗𝑐𝑜𝑛𝑣𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 =

𝑉

𝑉0=

0.5𝑉𝑖2−𝑃𝑗 𝑅𝑖𝑗 −𝑄𝑗 𝑋𝑖𝑗 +𝐴

𝑉𝑖 (2-7)

16

where A = 0.25 𝑉𝑖2 − 2𝑃𝑗𝑅𝑖𝑗 − 2𝑄𝑗 𝑋𝑖𝑗

2− 𝑃𝑗

2 + 𝑄𝑗2 𝑅𝑖𝑗

2 + 𝑋𝑖𝑗2

The conventional VRI (2-7) is based on the ratio of the voltage magnitude at the

base-load (obtained from power flow calculation) to the voltage magnitude at an

identical state but with all loads set to zero. This index works well under balanced

operating conditions. For unbalanced conditions, the conventional VRI will be

modified and extended to unbalanced networks using symmetrical components.

2.3 DERIVATION OF PROPOSED VRI FOR BALANCED NETWORKS

The proposed VRI for balanced networks is defined as:

𝑉𝑅𝐼𝑗𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑 =

𝑉𝑗 ,𝑐𝑜𝑙𝑙𝑎𝑝𝑠𝑒

𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑

(2-8)

To compute the proposed VRI for balanced three-phase networks; the bus voltage at

the point of voltage collapse (𝑉𝑗 ,𝑐𝑜𝑙𝑙𝑎𝑝𝑠𝑒 ) is computed based on the Newton-Raphson

load flow by forcing (2-3) to zero. The Jacobian corresponding to (2-3) is defined as

follows:

𝐽 = −𝑉𝑖𝑉𝑗 𝑠𝑖𝑛𝛿𝑖𝑗 𝑉𝑖𝑉𝑗 𝑐𝑜𝑠𝛿𝑖𝑗 − 2𝑉𝑗

𝑉𝑖𝑉𝑗 𝑐𝑜𝑠𝛿𝑖𝑗 𝑉𝑖𝑠𝑖𝑛𝛿𝑖𝑗 (2-9)

At the collapse point, the Jacobian matrix is singular, therefore:

𝑑𝑒𝑡 𝐽 = 0 ⇒ 𝑉𝑗 𝑐𝑜𝑠 𝛿𝑖𝑗

𝑉𝑖= 0.5 ⇒ 𝑉𝑗 ,𝑐𝑜𝑙𝑙𝑎𝑝𝑠𝑒 =

0.5𝑉𝑖

𝑐𝑜𝑠 𝛿𝑖𝑗 (2-10)

Substituting (2-6) and (2-10) in (2-8) results in

𝑉𝑅𝐼𝑗𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑 =

𝑉𝑗 ,𝑐𝑜𝑙𝑙𝑎𝑝𝑠𝑒

𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑

=0.5𝑉𝑖

𝑐𝑜𝑠 𝛿𝑖𝑗 0.5𝑉𝑖2−𝑃𝑗𝑅𝑖𝑗 −𝑄𝑗 𝑋𝑖𝑗 +𝐴

(2-11)

Note that

𝑉𝑅𝐼𝑗𝑏𝑎𝑙𝑎𝑛𝑐𝑒𝑑 =

𝑐𝑜𝑠𝛿𝑖𝑗

0.5 𝑉𝑅𝐼𝑗

𝑐𝑜𝑛𝑣𝑒𝑛𝑡𝑖𝑜𝑛𝑎𝑙 (2-12)

17

Therefore, compared to the conventional index (2-1), the proposed index (2-12) is

sensitive to both voltage magnitude (e.g., 𝑉/𝑉0) and voltage phase angle (𝛿𝑖𝑗 ). The

phase angle is computed as:

𝛿𝑖𝑗 = 𝑡𝑎𝑛−1 𝑃𝑗𝑋𝑖𝑗 −𝑄𝑗 𝑅𝑖𝑗

𝑃𝑗 𝑅𝑖𝑗 +𝑄𝑗 𝑋𝑖𝑗 + 𝑉𝑗 2 (2-13)

2.4 PROPOSED VRI FOR UNBALANCED MULTIPHASE

DISTRIBUTION NETWORKS

To extend and generalize the definition of VRI for unbalanced three-phase and

multiphase networks, symmetrical components are applied to the three-phase

voltages resulting from three-phase power flow calculations. The new index is

defined as the ratio of the positive-sequence voltage at the point of voltage collapse

to the positive-sequence voltage at the base-load.

𝑉𝑅𝐼𝑗unbalanced & 𝑚𝑢𝑙𝑡𝑖𝑝 𝑕𝑎𝑠𝑒

=𝑉𝑗 ,𝑐𝑜𝑙𝑙𝑎𝑝𝑠𝑒

+

𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑+ (2-14)

where 𝑉𝑗 ,𝑐𝑜𝑙𝑙𝑎𝑝𝑠𝑒+ and 𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑

+ are the positive-sequence bus voltages at the point

of voltage collapse and the base case load, respectively. The positive-sequence

voltage at the point of voltage collapse is determined by increasing the active power

of all loads while keeping the power factor constant until the point of voltage

collapse is reached as demonstrated in Figure 2-2.

Posit

ive S

eque

nce

Volta

ge (V

+ )

Active Power (P)

Collapse

Base-load

Figure 2-2 PV curve based on positive-sequence voltages.

18

This new index can be used to reveal the weakest buses of single-phase and

(un)balanced three-phase networks, as well as the weakest single-, two- and three-

phase buses of multiphase networks with unbalanced loads and/or line

configurations.

Symmetrical components can also be applied to the conventional VRI (2-1) to extend

its application to online identification of the weakest buses in unbalanced multiphase

distribution networks, as it is fast to detect the weakest bus. Therefore, the

conventional VRI for online application is defined as:

𝑉𝑅𝐼𝑗online =

𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑+

𝑉𝑗 ,𝑛𝑜 −𝑙𝑜𝑎𝑑+ (2-15)

However, VRI for online application (2-15) in balanced networks is less sensitive

than the proposed VRI (2-14) with the factor of 2

𝑐𝑜𝑠𝛿𝑖𝑗 (see (2-12)).

2.5 DERIVATION OF PROPOSED VRI FOR UNBALANCED

MULTIPHASE NETWORKS

For unbalanced three-phase networks, equation (2-14) can be easily applied to

unbalanced three-phase networks using symmetrical components. Derivation of the

symmetrical components based on (2-14) to multiphase networks is not straight

forward and can be performed as follows:

The multiphase network (Figure 2-3) can be represented by an equivalent

unbalanced four line three-phase network (Figure 2-4) using dummy nodes and

lines. This is done to hypothetically complete the missing phases and missing

lines of the unbalanced multiphase network [17]. The line parameters in Figure 2-

4 can be obtained by Carson's Equation:

19

i j

a

k l

aaijZ

b

c

n

a

b

c

n

c

n

c

n

bbijZ

ccijZ bc

ijZ

caijZ

nnijZ

banijZ

bnijZ

cnijZ

abijZ bb

jkZ

ccjkZ

nnjkZ nn

klZ

ccklZ

bcjkZ

cnjkZ cn

klZ

bnjkZ

Figure 2-3 An unbalanced multiphase distribution system; network configuration

consisting of four nodes with single-, two-, and three-phase sections.

𝑍𝑖𝑗

𝑎𝑎 𝑍𝑖𝑗𝑎𝑏 𝑍𝑖𝑗

𝑎𝑐 𝑍𝑖𝑗𝑎𝑛

𝑍𝑖𝑗𝑏𝑎 𝑍𝑖𝑗

𝑏𝑏 𝑍𝑖𝑗𝑏𝑐 𝑍𝑖𝑗

𝑏𝑛

𝑍𝑖𝑗𝑐𝑎 𝑍𝑖𝑗

𝑐𝑏 𝑍𝑖𝑗𝑐𝑐 𝑍𝑖𝑗

𝑐𝑛

𝑍𝑖𝑗𝑛𝑎 𝑍𝑖𝑗

𝑛𝑏 𝑍𝑖𝑗𝑛𝑐 𝑍𝑖𝑗

𝑛𝑛

(2-16)

i j

aaaijZ

b

c

n

a

b

c

n

bbijZ

ccijZ bc

ijZ

caijZ

nnijZ

anijZ

bnijZ

cnijZ

abijZ

Figure 2-4 The equivalent unbalanced three-phase four-wire network for the unbalanced

multiphase distribution system of Fig. 2-3.

Applying Kron’s reduction, the effects of the neutral or ground wire are still

included in this model:

𝑍𝑖𝑗𝑎𝑏𝑐 =

𝑍𝑖𝑗𝑎𝑎−𝑛 𝑍𝑖𝑗

𝑎𝑏−𝑛 𝑍𝑖𝑗𝑎𝑐−𝑛

𝑍𝑖𝑗𝑏𝑎−𝑛 𝑍𝑖𝑗

𝑏𝑏−𝑛 𝑍𝑖𝑗𝑏𝑐−𝑛

𝑍𝑖𝑗𝑐𝑎−𝑛 𝑍𝑖𝑗

𝑐𝑏−𝑛 𝑍𝑖𝑗𝑐𝑐−𝑛

(2-17)

20

For any missing phases, the corresponding rows and columns in (2-17) will

contain zero entries.

For the equivalent unbalanced three-phase network (Figure 2-4), the relationship

between bus voltages and branch currents can be expressed as:

𝑉𝑖𝑎

𝑉𝑖𝑏

𝑉𝑖𝑐

=

𝑉𝑗𝑎

𝑉𝑗𝑏

𝑉𝑗𝑐

+

𝑍𝑖𝑗𝑎𝑎−𝑛 𝑍𝑖𝑗

𝑎𝑏−𝑛 𝑍𝑖𝑗𝑎𝑐−𝑛

𝑍𝑖𝑗𝑏𝑎−𝑛 𝑍𝑖𝑗

𝑏𝑏−𝑛 𝑍𝑖𝑗𝑏𝑐−𝑛

𝑍𝑖𝑗𝑐𝑎−𝑛 𝑍𝑖𝑗

𝑐𝑏−𝑛 𝑍𝑖𝑗𝑐𝑐−𝑛

𝐼𝑖𝑗𝑎

𝐼𝑖𝑗𝑏

𝐼𝑖𝑗𝑐

(2-18)

Finally, symmetrical components are applied to the equivalent unbalanced three-

phase voltages resulting from three-phase power flow.

𝑉𝑗

𝑎0

𝑉𝑗𝑎1

𝑉𝑗𝑎2

=1

3 1 1 11 𝑎2 𝑎1 𝑎 𝑎2

𝑉𝑗𝑎

𝑉𝑗𝑏

𝑉𝑗𝑐

(2-19)

Therefore, to extend the application of the proposed VRI index (2-8) to

multiphase networks, we take the following steps: i) draw the equivalent

unbalanced four line three-phase network (Figure 2-4) using dummy nodes and

lines, ii) use symmetrical components to find the equivalent positive-sequence

components (2-19) of the multiphase systems, (iii) compute the new VRI index

for multiphase networks:

𝑉𝑅𝐼𝑗𝑚𝑢𝑙𝑡𝑖𝑝 𝑕𝑎𝑠𝑒

=𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑉𝑗 ,𝑐𝑜𝑙𝑙𝑎𝑝𝑠𝑒

+

𝐸𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑉𝑗 ,𝑏𝑎𝑠𝑒 −𝑙𝑜𝑎𝑑+ =

0.5𝑉𝑖

𝑐𝑜𝑠 𝛿𝑖𝑗 0.5𝑉𝑖2−𝑃𝑗𝑅𝑖𝑗 −𝑄𝑗 𝑋𝑖𝑗 +𝐴

(2-20)

2.6 CONCLUSIONS

A new idea is presented in this chapter that applies the method of symmetrical

components to the conventional bus ranking problem to extend its definition and

applications to both unbalanced three-phase and (un)balanced multiphase networks.

Main conclusions regarding the proposed bus ranking indices are as follows:

The new VRI can be utilized to identify the weakest single-phase, two-phase, and

three-phase bus under multiphase operating conditions.

Compared to the conventional bus ranking definition, the proposed bus ranking

index is sensitive to both voltage magnitudes and voltage phase angles.

21

In subsequent chapters, the new VRI will be compared to other bus ranking

methods and then applied to different power system configurations. The ability of

the new VRI to assist in voltage stability improvement by identifying appropriate

DG locations and sizes will also be investigated.

22

Chapter 3. Validation of the proposed VRI

3.1 INTRODUCTION

In Chapter 2, a new bus voltage ranking index (2-14) is defined as the ratio of the

positive-sequence voltage at the point of voltage collapse to the positive-sequence

voltage at the base-load. This new index can be used to reveal the weakest buses of

single-phase and (un)balanced three-phase networks, as well as the weakest single-,

two- and three-phase buses of multiphase networks with unbalanced loads and/or line

configurations.

In this chapter, the performance and accuracy of the conventional and the new bus

ranking indices are validated using grid loss calculations, PV curves and the well-

known voltage sensitivity approaches V/P and V/Q for balanced and unbalanced

three-phase distribution networks. Validations of the proposed VRI based on the

above mentioned four approaches are performed through detailed simulations of the

IEEE 13 node and IEEE 34 node test feeders under balanced and unbalanced

operating conditions using the DIgSILENT PowerFactory software package [32]. In

addition, applications of the generalized index of V/V0 to improve voltage stability of

unbalanced networks are also demonstrated.

3.2 VALIDATION OF PROPOSED VRI USING GRID LOSSES

CALCULATIONS

In order to validate the proposed VRI, grid losses associated with the placement of a

DG unit at different buses (e.g., all possible locations of DG) are computed and

compared with the losses without a DG unit. The active power losses reduction

(ALR) and the reactive power losses reduction (RLR) associated with the application

of one DG unit are calculated from:

23

%100,

loss

DGVRlossloss

PPPALR (3-1)

%100,

loss

DGVRlossloss

QQQ

RLR (3-2)

where lossP and lossQ are the total active and reactive power losses without voltage

regulator or DG unit, respectively. DGVRlossP , and DGVR

lossQ , are the total active and

reactive power losses with voltage regulator or DG unit, respectively.

The weakest bus of the system after the placement of a DG unit will be the bus with

the lowest ALR and RLR values. Validation of the proposed ranking approach can

be performed by comparing the weakest buses identified by grid losses (according to

ALR and RLR values) and by (2-14).

3.3 VALIDATION OF PROPOSED VRI USING PV CURVES

To further validate the proposed VRI, a continuation three-phase power flow is

utilized to plot the PV curves for unbalanced three-phase distribution networks. The

method of symmetrical components will then be applied to merge the three

individual PV curves into one based on positive-sequence voltage. Finally, the

maximum loading factor (MLF) will be determined using the PV curves based on

positive-sequence voltage. MLF is defined as the ratio of the maximum system load

(at the voltage collapse point) to the base load:

loadbase

collapse

PP

MLF (3-3)

The bus that has the lowest voltage stability margin or lowest MLF factor will be

considered as the weakest bus of the system. Therefore, further validation of the

proposed ranking approach can be performed by comparing the weakest buses

identified by PV curves and by index (2-14).

24

3.4 VALIDATION OF PROPOSED VRI USING VOLTAGE

SENSITIVITY INDICES

The next approach to validate the proposed VRI is through voltage sensitivity

analysis. This will be done using the sensitivities of voltage magnitude to reactive

and active powers at each bus.

3.4.1 Bus ranking based on sensitivity of voltage to reactive power (V/Q)

The sensitivity of voltage to reactive power injection (V/Q) at each bus is first

calculated. Then, the weakest bus is identified as the one with the maximum value of

the voltage sensitivity index [6]. Validation of the proposed ranking approach can be

performed by comparing the weakest buses according to the sensitivity index V/Q

and the proposed index of (2-14).

3.4.2 Bus ranking based on sensitivity of voltage to active power (V/P)

Reference [19] utilized the voltage sensitivities (V/P) along with the PV curves as

an indicator of voltage instability. In general, the higher the voltage sensitivity is, the

lower MLF will be. Therefore, the bus with the highest voltage sensitivity index

value (V/P) at the maximum loading factor point can be considered as the weakest

bus. Therefore, validation of the proposed ranking approach can also be performed

by comparing the weakest buses identified by V/P and (2-14).

3.5 DETAILED SIMULATION OF IEEE MULTIPHASE 13 NODE

TEST FEEDER TO VALIDATE PROPOSED VRI

In this section, detailed simulations of the IEEE multiphase 13 node test feeder

(Figure 3-1) is performed to: (1) find the weakest buses of the feeder (without/with

shunt capacitors, voltage regulators, DGs and SVCs) based on the proposed VRI, (2)

validate the identified weakest buses through grid losses calculations, PV curves and

voltage sensitivity indices (𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃). The IEEE multiphase 13 node test

feeder is simulated using DIgSILENT PowerFactory software [32]. The system data,

simulation parameters and general information on DIgSILENT PowerFactory

software are presented in Appendixes A1-A2 [33], B and C, respectively. This

25

multiphase unbalanced feeder consists of three-phase (buses 650, RG60, 632, 634,

634, 671, 692 and 675), two-phase (buses 645, 646 and 684) and single-phase (buses

611 and 652) sections with overhead lines, two underground lines (through buses

684, 652 and 692, 675), unbalanced spot loads (Y-PQ, D-PQ, Y-I, D-I, Y-Z, D-Z),

distributed loads (Y-PQ) between buses 632 and 671, a single-phase shunt capacitor

(at buses 611), a three-phase shunt capacitor (at buses 675), and an in-line

transformer (between buses 633 and 634). There is also a three-phase voltage

regulator connected between buses 650 and RG60.

646 645 632 633 634

650

692 675611 684

652

671

680

RG604.16 kV

0.48 kV4.16 kV

Switch

Two-phase

Single-phaseThree-phase

115 kV

Figure 3-1 The IEEE multiphase 13 node test feeder.

Simulations are performed on the IEEE multiphase 13 node test feeder (Figure 3-1)

for the following seven cases (Table 3-1):

Case 1: without any voltage regulators and fixed transformer tap ratio set to 1.0.

Case 2: with a voltage regulator and variable transformer tap ratio.

Case 3: similar to Case 2 with the addition of a single-phase shunt capacitor

(0.1MVar) connected at the weakest node (bus 611, identified by the proposed VRI

index).

Case 4: similar to Case 2 with the addition of a single-phase shunt capacitor

(0.1MVar) at bus 652.

Case 5: similar to Case 2 with a DG (Appendix B, Table B1) injecting 358 kW

active power (e.g., 10% of the total load) installed at the weakest three-phase node

(bus 675, identified by the proposed VRI index).

26

Case 6: similar to Case 2 with one DG (358 kW) and one SVC (0.36 MVAr, acting

as an unbalanced voltage controller) installed at the weakest three-phase node (bus

675, identified by the proposed VRI index).

Case 7: similar to Case 6 with the DG and SVC installed at bus 680.

TABLE 3-1 SIMULATED CASE STUDIES FOR THE IEEE MULTIPHASE 13 NODE TEST FEEDER (FIG. 3-1).

Case

number

System operating condition of the IEEE multiphase 13

node test feeder

Simulation results

1 No voltage regulators, transformer tap ratio set to

1.0

Fig. 3-2, Table 3-2

(column 1)

2 A voltage regulator, variable transformer tap ratio Figs. 3-3, 3-8, 3-9

and 3-10, Table 3-2

(column 2)

3 Case 2 with a single-phase shunt capacitor at the

weakest single-phase bus (bus 611)

Table 4-1

4 Case 2 with a single-phase shunt capacitor at bus

652

Table 4-1

5 Case 2 with a DG at the weakest three-phase bus

(bus 675)

Fig. 3-4, Table 3-3

6 Case 2 with a DG and a SVC at the weakest three-

phase bus (bus 675)

Figs. 3-5 and 3-11,

Table 3-4

7 Case 6 with the DG and SVC installed at bus 680 Table 4-1

3.5.1 Identification of the weakest buses using the proposed VRI for the

IEEE multiphase 13 node test feeder

The proposed VRI (2-14) will be utilized to locate the weakest single-phase and

three-phase buses for the placement of single-phase shunt capacitors and three-phase

DGs with SVC to enhance voltage stability. At each compensation level, the

27

proposed index (2-14) is calculated and the bus ranking is updated since the system

configuration is changed.

3.5.1.1 Bus ranking without/with a voltage regulator (Cases 1 and 2)

Figures 3-2 and 3-3 (and Table 3-2, columns 2-3) show the bus rankings for Cases 1

and 2 based on (2-14) without and with a voltage regulator, respectively. According

to these figures, the voltage regulator has no effect on the order of bus ranking.

0

0.2

0.4

0.6

0.8

1

1.2

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

VR

I

Weakest bus (single-phase)Weakest bus (three-phase)

Weakest bus (two-phase)

Figure 3-2 Bus ranking for Case 1 (without any voltage regulators).

Note that the four nodes with the lowest positive-sequence voltage ratios (2-14) are

buses 611, 684, 652 and 675. Therefore, the best single-phase node for the capacitor

connection is bus 611, while the most appropriate location for the installation of

three-phase DG and SVC compensators is bus 675 since nodes 611, 684 and 652 are

single-phase, two-phase and single-phase buses, respectively.

0

0.2

0.4

0.6

0.8

1

1.2

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

VR

I

Weakest bus (single-phase)Weakest bus (three-phase)

Weakest bus (two-phase)

Figure 3-3 Bus ranking for Case 2 (with a voltage regulator).

28

3.5.1.2 Bus ranking with DG at the most suitable bus (Case 5)

It is well-known that DG devices (e.g., induction generators) should be connected at

the most suitable three-phase buses (e.g., weakest buses with the lowest VRI values)

to improve the voltage stability. Simulation results of Figure 3-4 and Table 3-3

indicate that the application of one DG (an induction generator) at bus 675 does not

change the order of VRI values and therefore has no impact on the order of bus

ranking.

TABLE 3-2 BUS RANKING FOR CASES 1 AND 2 BASED ON THE PROPOSED VRI.

Bus number Case 1 Case 2

RG60 0.98593 1.04609

632 0.88131 0.90252

633 0.87531 0.89467

634 0.80824 0.80699

645 0.88264 0.91523

646 0.88035 0.91147

671 0.80418 0.79744

680 0.80418 0.79744

684 0.70594** 0.64939**

611 0.62933* 0.55389*

652 0.77406 0.73490

692 0.80418 0.79744

675 0.79423*** 0.78377***

*) The weakest single-phase bus.

**) The weakest two-phase bus.

***) The weakest three-phase bus.

29

0

0.2

0.4

0.6

0.8

1

1.2

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

VR

I

Weakest bus (single-phase)Weakest bus (three-phase)

Weakest bus (two-phase)

Figure 3-4 Bus ranking for Case 5 (with one DG at bus 675).

TABLE 3-3 BUS RANKING FOR CASE 5 BASED ON THE PROPOSED VRI.

Bus number Case 5

RG60 1.04587

632 0.90546

633 0.89770

634 0.81110

645 0.91791

646 0.91415

671 0.80186

680 0.80186

684 0.66207**

611 0.57571*

652 0.73902

692 0.80186

675 0.78862***

*) The weakest single-phase bus.

**) The weakest two-phase bus.

***) The weakest three-phase bus.

30

3.5.1.3 Bus ranking with DG and SVC (Case 6)

One DG and one SVC are connected at bus 675 (e.g., the three-phase node with the

lowest VRI) and the proposed index (2-14) is recalculated. As a result, the order of

the weakest nodes are changed to buses 611, 684, 634, 652, 671 (or 692), and 680 as

shown in Figure 3-5 and Table 3-4. This means the next suitable bus for connecting

additional DG and SVC units is bus 634. However, this bus has a different voltage

level (0.46 kV) compared to other buses (4.16 kV) and therefore, bus 671 (or 692)

will be selected to properly compare grid losses.

0

0.2

0.4

0.6

0.8

1

1.2

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

VR

I

Weakest bus (single-phase)Weakest bus (three-phase) Weakest bus (two-phase)

Figure 3-5 Bus ranking for Case 6 (with one DG and one SVC at bus 675).

3.5.2 Validation of proposed VRI based on grid loss calculations for the

IEEE multiphase 13 node test feeder

Grid losses associated with the placement of DG units at each node (e.g., all possible

locations of DG) are computed and compared with the losses generated with the DG

unit connected at the weakest bus as identified by the proposed VRI.

3.5.2.1 Grid losses with one DG unit for the IEEE multiphase 13 node test feeder

A three-phase induction generator is placed at different buses of the IEEE multiphase

13 node feeder (Figure 3-1) and system active and reactive losses are plotted in

Figure 3-6. This figure confirms that bus 675 (resulting in the lowest grid losses) is

the most suitable bus for DG placement, as was previously identified by the proposed

VRI (2-14).

31

TABLE 3-4 BUS RANKING FOR CASE 6 BASED ON THE PROPOSED VRI.

Bus number Case 6

RG60 1.05177

632 0.96139

633 0.94595

634 0.77360***

645 0.94095

646 0.93346

671 0.98381

680 0.98381

684 0.89518**

611 0.77550*

652 1.00702

692 0.98381

675 1.00000

*) The weakest single-phase bus.

**) The weakest two-phase bus.

***) The weakest three-phase bus.

Bus Number

Re

acti

ve P

ow

er

Lo

ss (

MV

Ar)

Ac

tive

Po

wer

Lo

ss (

MW

)

0.285

0.295

0.305

0.315

0.325

0.335

0.345

0.355

RG60 632 633 671 680 692 675

0.090

0.095

0.100

0.105

0.110

0.115

Active power loss

Reactive power loss0.120

0.125

Figure 3-6 Reactive and active power losses associated with DG connections at different buses of Figure 3-1 (Case 2).

32

3.5.2.2 Grid losses with two DG units for the IEEE multiphase 13 node test feeder

According to (2-14), with the addition of one DG (at bus 675, Figure 3-4), the most

suitable location for the connection of a second DG unit is still at bus 675. This is in

agreement with the grid loss plots of Figure 3-7 generated by connecting the first DG

at bus 675 and placing a second DG at different buses of the IEEE 13 node feeder.

These results further confirm the accuracy of the proposed bus ranking index.

3.5.3 Validation of proposed VRI based on PV curves for the IEEE

multiphase 13 node test feeder

The PV curves based on positive-sequence voltages are plotted and compared with

the PV curve generated when DG and SVC units are connected at the weakest bus.

Figures 3-8, 3-9, and 3-10 show the PV curves of positive-sequence voltages at each

three-, two- and single-phase bus for Case 2, respectively. According to these

figures, buses 675, 684 and 611 are the weakest three-, two- and single-bus as

previously recognized by (2-14).

After connecting a combination of DG and SVC units at bus 675, PV curves for Case

6 are regenerated and plotted in Figure 3-11. As expected and previously recognized

by the proposed VRI, the lowest stability margins occur at bus 634.

Re

acti

ve P

ow

er

Lo

ss (

MV

Ar)

Acti

ve

Po

wer

Lo

ss (

MW

)

0.250

0.260

0.270

0.280

0.290

0.300

0.310

Bus NumberRG60 632 633 671 680 692 675

0.084

0.088

0.092

0.096

0.100

Active power loss

Reactive power loss0.104

0.108

Figure 3-7 Reactive and active power losses associated with the first DG installed at bus 675 and the second DG connected at different buses of Figure 3-1 (Case 3).

33

114669466746654663466

1.2

1.1

1.0

0.9

0.8

0.7

Total Load of Selected Loads (kW)

Bus 675; the weakest three-phase bus

632671

633 634675

RG60680 692

Po

sit

ive-S

eq

uen

ce V

olt

ag

e (

p.u

.)

Figure 3-8 PV curves of positive-sequence voltage at each three-phase bus for Case 2.

1146694667466

0.70

0.60

684645 646Total Load of S elected L oads (kW)

5466

Bus 684; the weakest two- phase bus

0.50

0.403466

Po

sit

ive

-Se

qu

en

ce

Vo

lta

ge

(p

.u.)

Figure 3-9 PV curves of positive-sequence voltage at each two-phase bus for Case 2.

34

11466946674665466

0.36

0.32

0.28

0.24

3466

0.20

0.16

611652Total Load of S elected L oads (kW)

Bus 611; the weakest single- phase bus

Po

sit

ive

-Se

qu

en

ce

Vo

lta

ge

(p

.u.)

Figure 3-10 PV curves of positive-sequence voltage at each single-phase bus for Case 2.

19466154661146674663466

1.2

1.1

1.0

0.9

0.8

0.7

Total Load of Selected Loads (kW)

Bus 634; the weakest three-phase bus

632671

633 634675

RG60680 692

Po

sit

ive-S

eq

uen

ce V

olt

ag

e (

p.u

.)

Figure 3-11 PV curves of positive-sequence voltage at each bus for Case 6.

3.5.4 Comparison of proposed VRI with other bus ranking approaches for

the IEEE multiphase 13 node test feeder

Table 3-5 compares the performance of the proposed VRI (2-14) with three well-

known bus ranking indices; 𝑉/𝑉0, 𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃 for the IEEE 13 node network

35

of Figure 3-1 under balanced three-phase, unbalanced three-phase and unbalanced

multiphase operating conditions.

Under balanced three-phase conditions (Table 3-5, column 2), all methods

identify the same weakest bus (e.g., node 634). However, the conventional

ranking approaches are not applicable to unbalanced three-phase and multiphase

systems and fail to identify the correct weakest buses.

Under unbalanced three-phase conditions (Table 3-5, column 3), the weakest bus

is node 675 as identified by the proposed VRI and confirmed by the calculated

PV curves (Figure 3-12) and grid losses (Table 3-6, columns 2-3). However,

based on the two voltage sensitivity methods (𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃), the weakest

bus is node 634 which is not correct. This is further confirmed by placing SVC

units at buses 634 and 675 and computing the corresponding maximum loading

factors as demonstrated in Table 3-7. Therefore, the magnitudes of the voltage

sensitivity methods do not provide a correct measure of voltage stability under

unbalanced three-phase networks.

For multiphase operation (Table 3-5, column 4), the weakest three-, two- and

single- phase buses are nodes 675, 684 and 611, respectively; as identified by the

proposed VRI (Figure 3-3) and confirmed by grid losses (Figure 3-6) and PV

curves PV curves (Figures 3-8 to 3-10). Therefore, the conventional indices

(𝑉/𝑉0, 𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃) cannot properly identify the weakest buses of

multiphase networks.

36

TABLE 3-5 BUS RANKING RESULTS FOR THE IEEE 13 NODE NETWORK WITH ONLY UNBALANCED THREE-PHASE NETWORKS/LOADS.

Bus ranking approach

The weakest buses of the unbalanced IEEE 13 node test feeder

(Figure 3-1)

Balanced

three-phase *

Unbalanced three-

phase **

Unbalanced

multiphase***

Grid losses 634 675 (Table 3-6) 675(3p), see Figure 3-7

PV curves 634 675 (Figure 3-12) 611 (1p, Figure 3-10),

684 (2p, Figure 3-9),

675 (3p, Figure 3-8)

𝑉/𝑉0 (2-1)

[10-11]

634 N/A N/A

Index 𝜕𝑉/𝜕𝑄 [6] 634 634**** (Table 3-6) N/A

Index 𝜕𝑉/𝜕𝑃 [19] 634 634**** (Table 3-6) N/A

Proposed VRI (2-14) 634 675 (Table 3-6) 611(1p), 684 (2p),

675(3p), see Fig 3-3

*) Modified Figure 3-1 with only balanced networks/loads.

**) Modified Figure 3-1 with only unbalanced three-phase networks/loads.

***) 1p, 2p and 3p correspond to single-, two- and three-phase buses.

****) Calculated by positive sequence.

TABLE 3-6 BUS RANKING RESULTS FOR THE IEEE 13 NODE TEST FEEDER (FIG. 3-1) WITH ONLY UNBALANCED THREE-PHASE NETWORKS/LOADS.

Bus

number

Grid losses 𝜕𝑉/𝜕𝑃

[p.u./MW]

𝜕𝑉/𝜕𝑄

[p.u./MVAr]

Proposed VRI

(2-14) P

[MW]

Q

[MVAr]

634 0.13555 0.47702 -0.26223 0.36904 0.61776

675 0.13448 0.47281 -0.23591 0.30041 0.60153

37

TABLE 3-7 MAXIMUM LOADING FACTORS WITH SVC.

MLF

Base-case 2.687

SVC at bus 634 3.282

SVC at bus 675 5.095

971684667216596647163466

Total Load of S elected L oads (kW)

675634

1.00

0.90

0.80

0.70

0.60

0.50

Bus 675

Bus 634

Po

sit

ive-S

eq

uen

ce V

olt

ag

e (

p.u

.)

Figure 3-12 PV curves of positive-sequence voltages at buses 634 and 675 for the

modified IEEE 13 node network (Figure 3-1) with only unbalanced three-phase

networks/loads.

3.6 DETAILED SIMULATION OF IEEE MULTIPHASE 34 NODE

TEST FEEDER TO VALIDATE PROPOSED VRI

In this section, detailed simulations of the IEEE multiphase unbalanced 34 node test

feeder (Figure 3-13) is performed to: (1) find the weakest buses of the feeder

(without/with voltage regulators, induction generator and DFIG wind turbine) based

on the proposed VRI (2-14), (2) validate the identified weakest buses through grid

losses calculations, PV curves and voltage sensitivity indices (𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃).

38

The system data for the IEEE multiphase 34 node test feeder is presented in

Appendixes A3 and A4 [33]. This unbalanced multiphase feeder consists of three-

phase and single-phase sections with unbalanced spot loads (Y-PQ, D-PQ, Y-I, D-I,

Y-Z, and D-Z), distributed loads (Y-PQ, Y-I, Y-Z, D-I, D-Z, and D-PQ), three-phase

shunt capacitors (at buses 844 and 848), and an in-line transformer (between buses

832 and 888).

There are also two automatic voltage regulators. Bus 800 is treated as a slack bus

with a voltage set point of 1.05 p.u. At a base-case load condition, the voltage at bus

890 is lower than the permissible voltage limit because the line between buses 888

and 890 is relatively long. However, other bus voltages are in the acceptable range of

0.95p.u. to 1.05p.u.

800

806 808 812 814

810

802 850818

824 826816

820

822

828 830 854 856

852

832888 890

838

862

840836860834

842

844

846

848

864

858

Single-phase (phase-a)

Single-phase (phase-b)

Single-phase (phase-b)

Figure 3-13 The IEEE multiphase 34 node test feeder.

Simulations are performed on the multiphase unbalanced IEEE 34 node test feeder

(Figure 3-13) for the following cases (Table 3-8):

Case 8: without a voltage regulator (fixed transformer tap ratio set to 1.0).

Case 9: with a voltage regulator (variable transformer tap ratio).

Case 10: Case 9 with a DG (three-phase induction generator) injecting 200 kW

active power (e.g., 10% of the total load) installed at the weakest three-phase node

(bus 890).

Case 11: Case 9 with one DG (200 kW DFIG wind turbine) installed at the weakest

three-phase node (bus 890).

39

Case 12: Case 9 with DGs (2.4 MW DFIG wind turbines) installed at the weakest

three-phase node (bus 890).

TABLE 3-8 SIMULATED CASE STUDIES FOR THE IEEE 34 NODE TEST FEEDER (FIG. 3-13).

Case

number

System operating condition of the IEEE 34 node

test feeder Simulation results

8 No voltage regulators, transformer tap ratio

set to 1.0

Fig. 3-14, Table 3-9

(column 1)

9 A voltage regulator, variable transformer tap

ratio

Figs. 3-15, 3-21 and 3-

22, Table 3-9

(column 2)

10 Case 9 with a DG (200 kW IG) at the

weakest three-phase node (bus 890)

Fig. 3-16, Table 3-10

11 Case 9 with a DG (200 kW DFIG wind

turbine) at the weakest three-phase node

(bus 890)

Fig. 3-17, Table 3-11

12 Case 9 with DGs (2.4 MW DFIG wind

turbines) at the weakest three-phase node

(bus 890)

Figs. 3-18 and 3-23,

Table 3-12

3.6.1 Identification of the weakest buses using the proposed VRI for the

IEEE multiphase unbalanced 34 node test feeder

In the following sections, the proposed VRI (2-14) will be utilized to locate the

weakest three-phase buses for the placement of three-phase induction generator and

DFIG wind turbine to enhance voltage stability. At each compensation level, the

proposed index (2-14) is calculated and the bus ranking is updated since the system

configuration is changed. To show the validity of the proposed bus ranking and the

effectiveness of the compensation devices (induction generator and DFIG wind

turbine), grid losses, PV curves (based on positive-sequence voltages) and voltage

stability margins are calculated and compared for the aforementioned cases.

40

3.6.1.1 Bus ranking without/with a voltage regulator (Cases 8 and 9)

Figures 3-14 (corresponding to columns 2 of Table 3-9) and 3-15 (corresponding to

column 3 of Table 3-9) show the bus rankings for Cases 8 and 9 based on (2-14)

without and with a voltage regulator, respectively. According to these figures, the

voltage regulator has no effect on the order of bus ranking.

0

0.2

0.4

0.6

0.8

1

1.2

80

0

80

2

80

6

80

8

81

0

81

2

81

4

85

0

81

6

81

8

82

0

82

2

82

4

82

6

82

8

83

0

85

4

85

2

83

2

85

8

83

4

84

2

84

4

84

6

84

8

86

0

83

6

84

0

86

2

83

8

86

4

88

8

89

0

85

6

Weakest bus (single-phase)Weakest bus (three-phase)

Bus Number

VR

I

Single-phase Three-phase

Figure 3-14 Bus ranking for Case 8 (without any voltage regulators).

Note that the four nodes with the lowest positive-sequence voltage ratios (2-14) are

buses 890, 864, 822 and 888. Therefore, the most appropriate location for the

installation of a three-phase DG is bus 890 since nodes 864 and 822 are single-phase

buses and nodes 890 and 888 are three-phase buses.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

80

0

80

2

80

6

80

8

81

0

81

2

81

4

85

0

81

6

81

8

82

0

82

2

82

4

82

6

82

8

83

0

85

4

85

2

83

2

85

8

83

4

84

2

84

4

84

6

84

8

86

0

83

6

84

0

86

2

83

8

86

4

88

8

89

0

85

6

Weakest bus (single-phase)Weakest bus (three-phase)

Bus Number

VR

I

Single-phase Three-phase

Figure 3-15 Bus ranking for Case 9 (with a voltage regulator).

3.6.1.2 Bus ranking with an induction generator DG unit at the most suitable bus (Case 10)

As mentioned before, installation of DG devices (e.g., induction generators) at the

most suitable three-phase buses (e.g., weakest buses with the lowest VRI values) can

improve the voltage stability. Simulation results of Figure 3-16 and Table 3-10

41

indicate that the application of one DG (an induction generator) at bus 890 does not

change the order of VRI values and therefore has no impact on the order of bus

ranking.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

80

0

80

2

80

6

80

8

81

0

81

2

81

4

85

0

81

6

81

8

82

0

82

2

82

4

82

6

82

8

83

0

85

4

85

2

83

2

85

8

83

4

84

2

84

4

84

6

84

8

86

0

83

6

84

0

86

2

83

8

86

4

88

8

89

0

85

6

Weakest bus (single-phase)Weakest bus (three-phase)

Bus Number

VR

I

Single-phase Three-phase

Figure 3-16 Bus ranking for Case 10 (with a DG type induction generator at bus 890).

3.6.1.3 Bus ranking with a 200kW DFIG wind turbine DG unit at the most

suitable bus (Case 11)

One DFIG wind turbine DG unit is connected at bus 890 (e.g., the three-phase node

with the lowest VRI) and the proposed index (2-14) is recalculated. As a result, the

order of the weakest nodes are changed to buses 890, 864, 822, 888, and 620 as

shown in Figure 3-17 and Table 3-11. Simulation results indicate that the application

of one DG (200 kW DFIG wind turbine) at bus 890 does not change the order of VRI

values and therefore has no impact on the order of bus ranking.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

80

0

80

2

80

6

80

8

81

0

81

2

81

4

85

0

81

6

81

8

82

0

82

2

82

4

82

6

82

8

83

0

85

4

85

2

83

2

85

8

83

4

84

2

84

4

84

6

84

8

86

0

83

6

84

0

86

2

83

8

86

4

88

8

89

0

85

6

Weakest bus (single-phase)Weakest bus (three-phase)

Bus Number

VR

I

Single-phase Three-phase

Figure 3-17 Bus ranking for Case 11 (with a DFIG wind turbine DG unit at bus 890).

42

TABLE 3-9 BUS RANKING FOR CASES 8 AND 9 BASED ON THE PROPOSED VRI.

Bus number Case 8 Case 9

800 1.00000 1.00000

802 0.99799 0.99573

806 0.99666 0.99290

808 0.97176 0.93948

810 (single-phase) 0.97495 0.94645

812 0.94172 0.87464

814 0.91684 0.82070

850 1.00154 0.83477

816 0.90944 0.83414

818 (single-phase) 0.88203 0.79616

820 (single-phase) 0.84634 0.71782

822 (single-phase) 0.84140 0.70916

824 0.89911 0.81571

826 (single-phase) 0.91036 0.84105

828 0.89828 0.81422

830 0.87777 0.77750

854 0.87725 0.77657

852 0.83959 0.70887

832 0.83322 0.74432

858 0.82963 0.73901

834 0.82548 0.73287

842 0.82538 0.73271

844 0.82488 0.73197

846 0.82442 0.73130

848 0.82437 0.73124

860 0.82491 0.73202

836 0.82455 0.73150

840 0.82452 0.73145

862 0.82454 0.73149

838 (single-phase) 0.83164 0.75167

864 (single-phase) 0.80473* 0.68868*

888 0.80342 0.69969

890 0.69640*** 0.54248 ***

856 (single-phase) 0.88758 0.80031

*) The weakest single-phase bus.

***) The weakest three-phase bus.

43

TABLE 3-10 BUS RANKING FOR CASE 10 BASED ON THE PROPOSED VRI.

Bus number Case 10

800 1.00000

802 0.99582

806 0.99305

808 0.94083

810 (single-phase) 0.94775

812 0.87803

814 0.82600

850 0.85138

816 0.85077

818 (single-phase) 0.79710

820 (single-phase) 0.73635

822 (single-phase) 0.72791

824 0.83300

826 (single-phase) 0.85770

828 0.83156

830 0.79632

854 0.79543

852 0.73097

832 0.77219

858 0.76691

834 0.76079

842 0.76063

844 0.75988

846 0.75920

848 0.75918

860 0.75996

836 0.75946

840 0.75941

862 0.75944

838 (single-phase) 0.77859

864 (single-phase) 0.71722*

888 0.73057

890 0.59503***

856 (single-phase) 0.81841

*) The weakest single-phase bus.

***) The weakest three-phase bus.

44

TABLE 3-11 BUS RANKING FOR CASE 11 BASED ON THE PROPOSED VRI.

Bus number Case 11

800 1.00000

802 0.99498

806 0.99498

808 0.99165

810 (single-phase) 0.92914

812 0.93821

814 0.85398

850 0.79201

816 0.82070

818 (single-phase) 0.74986

820 (single-phase) 0.66849

822 (single-phase) 0.65709

824 0.79889

826 (single-phase) 0.83131

828 0.79720

830 0.75550

854 0.75444

852 0.67831

832 0.71712

858 0.71062

834 0.70310

842 0.70291

844 0.70201

846 0.70117

848 0.70109

860 0.70204

836 0.70150

840 0.70134

862 0.70138

838 (single-phase) 0.72512

864 (single-phase) 0.64766*

888 0.67604

890 0.53690***

856 (single-phase) 0.78413

*) The weakest single-phase bus.

***) The weakest three-phase bus.

45

3.6.1.4 Bus ranking with a 2.4 MW DFIG wind turbine DG unit (Case 12)

A 2.4 MW DFIG wind turbine (Case 12) is connected at bus 890 (e.g., the three-

phase node with the lowest VRI) and the proposed index (2-14) is recalculated. As a

result, the order of the weakest three-phase nodes are changed to buses 852, 890 and

814 as shown in Figure 3-18 and Table 3-12. This means the next suitable bus for

connecting a compensation device is bus 852.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

800

802

806

808

810

812

814

850

816

818

820

822

824

826

828

830

854

852

832

858

834

842

844

846

848

860

836

840

862

838

864

888

890

856

Weakest bus (three-phase)

Bus Number

VR

I

Weakest bus (single-phase)Single-phase Three-phase

Figure 3-18 Bus ranking for Case 12 (with DFIG wind turbines at bus 890).

3.6.2 Validation of proposed VRI based on grid loss calculations for the

IEEE multiphase 34 node test feeder

Grid losses associated with the placement of DG units at each node (e.g., all possible

locations of DG) are computed and compared with the losses generated with the DG

unit connected at the weakest bus as identified by the proposed VRI.

3.6.2.1 Grid losses with one DG unit for the IEEE multiphase 34 node test feeder

A three-phase DFIG wind turbine is placed at different buses of the unbalanced IEEE

34 node feeder (Figure 3-13) and the system active power losses are plotted in Figure

3-19. This figure confirms that bus 890 (resulting in the lowest grid losses) is the

most suitable bus for DG placement, as was previously identified by the proposed

VRI (2-14).

46

TABLE 3-12 BUS RANKING FOR CASE 12 BASED ON THE PROPOSED VRI.

Bus number Case 12

800 1.00000

802 0.99298

806 0.98835

808 0.90360

810 (single-phase) 0.92369

812 0.80679

814 0.73076

850 0.85460

816 0.85357

818 (single-phase) 0.71349

820 (single-phase) 0.54996

822 (single-phase) 0.52667*

824 0.82593

826 (single-phase) 0.89795

828 0.82374

830 0.77118

854 0.76989

852 0.68017***

832 0.76769

858 0.75772

834 0.74618

842 0.74588

844 0.74448

846 0.74326

848 0.74314

860 0.74459

836 0.74364

840 0.74355

862 0.74362

838 (single-phase) 0.79897

864 (single-phase) 0.63886

888 0.73966

890 0.68601

856 (single-phase) 0.83478

*) The weakest single-phase bus.

***) The weakest three-phase bus.

47

0

0.05

0.10

0.15

0.20

0.25

0.30

800

802

806

808

812

814

850

816

824

828

830

854

852

832

858

834

842

844

846

848

860

836

840

862

888

890

Acti

ve

Po

wer

Lo

ss (

MW

)

Bus Number

Figure 3-19 Active power loss associated with DG connections at different buses of Figure 3-13 (Case 9).

3.6.2.2 Grid losses with two DG units for the IEEE multiphase 34 node test

feeder

According to (2-14), with the addition of one DG (at bus 890, Figure 3-16), the most

suitable location for the connection of a second DG unit is still at bus 890. This is in

agreement with the grid loss plots of Figure 3-20 generated by connecting the first

DG at bus 890 and placing a second DG at different buses of the IEEE 34 node

feeder. These results further confirm the accuracy of the proposed bus ranking index.

0

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20

800

802

806

808

812

814

850

816

824

828

830

854

852

832

858

834

842

844

846

848

860

836

840

862

888

890

Acti

ve

Po

wer

Lo

ss (

MW

)

Bus Number

Figure 3-20 Active power loss associated with the first DG installed at bus 890 and the second DG connected at different buses of Figure 3-13 (Case 10).

3.6.3 Validation of proposed VRI based on PV curves for the IEEE

multiphase 34 node test feeder

Figure 3-21 shows the PV curves of positive-sequence voltages at each three-phase

bus for Case 9. According to this figure, bus 890 has the lowest stability margin.

48

Therefore, this is the weakest three-phase bus as previously recognized by (2-14).

The PV curves based on positive-sequence voltages of single-phase bus are plotted

as shown in Figure 3-22. As expected and previously recognized by the proposed

VRI, the weakest single-phase bus is bus 864.

After connecting a 2.4 MW DFIG wind turbine at bus 890, PV curves for Case 12

are regenerated and plotted in Figure 3-23. As expected and previously recognized

by the proposed VRI, the lowest stability margins occur at bus 852.

3.6.4 Comparison of proposed VRI with other bus ranking approaches for

the IEEE multiphase 34 node test feeder

Table 3-13 compares the performance of the proposed VRI (2-14) with three well-

known bus ranking indices; 𝑉/𝑉0, 𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃 for the IEEE 34 node network

of Figure 3-13 under unbalanced multiphase operating conditions. For multiphase

operation (Table 3-13), the weakest three- and single- phase buses are nodes 890 and

864, respectively; as identified by the proposed VRI (Figure 3-15) and confirmed by

grid losses (Figure 3-19) and PV curves (Figures 3-21 and 3-22). Therefore, the

conventional indices (𝑉/𝑉0 , 𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃) cannot properly identify the

weakest buses of multiphase networks.

49

4769376927691769

1.125

1.000

0.875

0.750

0.625

0.500

0.375

Total Load of Selected Loads (kW)

Bus 890; the weakest three-phase bus

848840

852846858

888 890

808812 814

850

816 824832830828

854

800

834842 844

860

836

802

862

806

Po

sit

ive

-

Se

qu

en

ce

Vo

lta

ge

(p.u

.

Figure 3-21 PV curves of positive-sequence voltage at each three-phase bus for Case 9.

Total Load of Selected Loads (kW)

)

4769376927691769

0.39

0.36

0.33

0.30

0.27

0.24

0.21

810 818 820 822826 856 864 838

Bus 864; the weakest single-phase bus

Po

sit

ive

-

Se

qu

en

ce

Vo

lta

ge

(p.u

.

Figure 3-22 PV curves of positive-sequence voltage at each single-phase bus for Case 9.

50

801967695519426930191769

1.40

1.20

1.00

0.80

0.60

Total Load of Selected Loads (kW)

Po

sit

ive-

S

eq

uen

ce

Vo

lta

ge

(p.u

.)

Bus 852; the weakest three-phase bus

848840

852846858

888 890

808812 814

850

816 824832830828

854

800

834842 844

860

836

802

862

806

Figure 3-23 PV curves of positive-sequence voltage at each bus for Case 12.

TABLE 3-13 BUS RANKING RESULTS FOR THE MULTIPHASE IEEE 34 NODE NETWORK.

Bus ranking approach The weakest buses of the multiphase IEEE 34 node test

feeder* (Figure 3-13)

Grid losses 890(3p), see Figure 3-19

PV curves 890 (3p, Figure 3-21), 864 (1p, Figure 3-22)

𝑉/𝑉0 (2-1) [10-11] N/A

Index 𝜕𝑉/𝜕𝑄 [6] N/A

Index 𝜕𝑉/𝜕𝑃 [19] N/A

Proposed VRI (2-14) 864(1p), 890(3p), see Figure 3-15

*) 1p and 3p correspond to the weakest single- and three-phase buses.

51

3.7 CONCLUSIONS

This chapter utilized the proposed new voltage ranking index (2-14) to identify the

weakest single-, two- and three-phase buses of unbalanced and multiphase

distribution networks. The validity of the new index is demonstrated for the IEEE

unbalanced multiphase 13 node and IEEE 34 node test feeders based on grid losses,

PV curves and voltage sensitivity analysis. Main conclusions are as follows:

The proposed ranking index can accurately identify the weakest single-, two- and

three-phase buses under different operating conditions without and with voltage

regulators, capacitors and DGs (without/with SVCs).

The performance and the accuracy of the proposed VRI for system operations

without/with single-phase capacitors, voltage regulators, DGs and SVCs have

been clearly validated through detailed simulations and analysis.

52

Chapter 4. Validation and application of proposed VRI

in improving voltage stability of unbalanced three-phase

networks

4.1 INTRODUCTION

In this chapter, a new bus positive-sequence voltage index of Vcollapse/Vbase-load is

introduced to identify the weakest three-phase buses in unbalanced three-phase

distribution networks. First, the proposed ranking index is validated based on grid

losses and PV curves without and with compensation devices. Then, the index is

utilized to place three-phase DG units without and with SVC devices at the weakest

buses of the modified IEEE unbalanced three-phase 13 node test feeder. Finally,

simulation results and extensive case studies without/with a voltage regulator, DGs

and SVCs are presented to show the application of the proposed approach in

improving voltage stability and increasing MLF under unbalanced three-phase

conditions.

4.2 DETAILED SIMULATION OF MODIFIED IEEE UNBALANCED

THREE-PHASE 13 NODE TEST FEEDER TO VALIDATE PROPOSED VRI

In this section, detailed simulations of the modified unbalanced three-phase 13 node

test feeder shown in Figure 4-1 is performed to: (1) find the weakest buses of the

feeder (without/with voltage regulators, DGs and SVCs) based on the proposed VRI,

(2) validate the identified weakest buses through grid losses calculations and PV

curves, (3) improve MLF of the networks using compensation devices.

The unbalanced three-phase 13 node test feeder (Figure 4-1) has been modified to

have unbalanced three-phase conditions. The feeder specifications and data are

presented in Appendixes A-5 and A-6.

53

Simulations are performed on the modified unbalanced three-phase 13 node test

feeder (Figure 4-1) for the following case studies:

Case 1: without a voltage regulator (fixed transformer tap ratio set to 1.0).

Case 2: with a voltage regulator (variable transformer tap ratio).

Case 3: Case 2 with a DG (three-phase induction generator) injecting 358 kW active

power (e.g., 10% of the total load) installed at the weakest three-phase node (bus

675).

Case 4: Case 2 with one DG (358 kW) and one SVC (0.36 MVar, acting as an

unbalanced voltage controller) installed at the weakest three-phase node (bus 675).

Case 5: similar to Case 4 with the DG and SVC installed at bus 680.

646 645 632 633 634

650

692 675611 684

652

671

680

RG604.16 kV

0.48 kV4.16 kV

Switch

Three-phase

Three-phaseThree-phase

115 kV

Figure 4-1 The modified unbalanced three-phase 13 node test feeder.

The proposed VRI (2-14) will be utilized to locate the weakest three-phase buses for

the placement of three-phase DGs with SVC to enhance voltage stability. At each

compensation level, the proposed index (2-14) is calculated and the bus ranking is

updated since the system configuration is changed. To show the validity of the

proposed bus ranking and the effectiveness of the compensation devices (DG and

SVC), grid losses, PV curves (based on positive-sequence voltages) and voltage

stability margins are calculated and compared for the aforementioned cases.

54

TABLE 4-1 SIMULATED CASE STUDIES FOR THE MODIFIED UNBALANCED THREE-PHASE 13 NODE TEST FEEDER (FIG. 4-1).

Case

number

System operating condition of the modified

unbalanced three-phase 13 node test feeder

Simulation results

1 No voltage regulators, transformer tap ratio set to 1.0 Fig. 4-2, Table 4-2

(column 1)

2 A voltage regulator, variable transformer tap ratio Figs. 4-3, 4-6 and 4-8,

Table 4-2

(column 2)

3 Case 2 with a DG at the weakest three-phase bus

(bus 675)

Figs. 4-4 and 4-7,

Table 4-3

4 Case 2 with a DG and a SVC at the weakest three-

phase bus (bus 675)

Figs. 4-5 and 4-9,

Table 4-4

5 Case 4 with the DG and SVC installed at bus 680 Table 4-5

4.2.1 Identification of weakest three-phase buses using the proposed VRI

The proposed VRI (2-14) will be utilized to locate the weakest three-phase buses for

the placement of DGs with/without SVC to enhance voltage stability. The proposed

VRI is applied to a practical scenario of improving the voltage stability of the

modified unbalanced three-phase 13 node test feeder (Figure 4-1).

4.2.1.1 Bus ranking without/with a voltage regulator

Figures 4-2 and 4-3 (and Table 4-2, columns 2-3) show the bus rankings for Cases 1

and 2 based on (2-14) without and with a voltage regulator, respectively. According

to these figures, the voltage regulator has no effect on the order of bus ranking.

Note that the four nodes with the lowest VRI are buses 675, 652, 611 and 684.

Therefore, the most appropriate location for the installation of three-phase DG and

SVC compensators is bus 675.

55

0.75

0.80

0.85

0.90

0.95

1.00

1.05

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

VR

I

Weakest Bus

Figure 4-2 Bus ranking for Case 1 (without any voltage regulators).

0.75

0.80

0.85

0.90

0.95

1.00

1.05

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

VR

I Weakest Bus

Figure 4-3 Bus ranking for Case 2 (with a voltage regulator).

4.2.1.2 Bus ranking with DG at the most suitable bus

DG devices (e.g., induction generators) are to be connected at the weakest three-

phase buses (e.g., weakest buses with the lowest VRI values) to improve the voltage

stability. Simulation results of Figure 4-4 and Table 4-3 indicate that the application

of one DG (an induction generator) at bus 675 does not change the order of VRI

values and therefore has no impact on the order of bus ranking.

0.75

0.80

0.85

0.90

0.95

1.00

1.05

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

VR

I Weakest Bus

Figure 4-4 Bus ranking for Case 3 (with one DG at bus 675).

56

TABLE 4-2 BUS RANKING FOR CASES 1 AND 2 BASED ON THE PROPOSED VRI.

Bus number Case 1 Case 2

RG60 0.95790 1.00004

632 0.85406 0.91581

633 0.84810 0.91099

634 0.78130 0.85749

645 0.85087 0.91252

646 0.85000 0.91148

671 0.77672 0.85468

680 0.77672 0.85468

684 0.77610 0.85403

611 0.77562 0.85359

652 0.7756 0.85329

692 0.77672 0.85468

675 0.76663 0.84700

4.2.1.3 Bus ranking with DG and SVC

One DG and one SVC are connected at bus 675 (e.g., the node with the lowest VRI

value) and the proposed index (2-14) is recalculated (Figure 4-5 and Table 4-4). As a

result, the orders of the weakest nodes are changed to buses 634, 633, 646, 645, 632,

and 652. This means the next suitable bus for connecting additional DG and SVC

units is bus 634.

57

TABLE 4-3 BUS RANKING FOR CASE 3 BASED ON THE PROPOSED VRI.

Bus number Case 3

RG60 1.00005

632 0.91926

633 0.91460

634 0.86290

645 0.91604

646 0.91503

671 0.86000

680 0.86000

684 0.85937

611 0.85895

652 0.8586

692 0.86000

675 0.85272

0.75

0.80

0.85

0.90

0.95

1.00

1.05

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

VR

I

Weakest Bus

Figure 4-5 Bus ranking for Case 4 (with one DG and one SVC at bus 675).

58

TABLE 4-4 BUS RANKING FOR CASE 4 BASED ON THE PROPOSED VRI.

Bus number Case 4

RG60 1.00002

632 0.93408

633 0.91952

634 0.75705

645 0.9250

646 0.92233

671 0.98042

680 0.98042

684 0.97848

611 0.97755

652 0.97496

692 0.98042

675 1.00000

4.2.2 Validation of proposed VRI based on grid loss calculations

Grid losses associated with the placement of DG units at each node (e.g., all possible

locations of DG) are computed and compared with the losses generated with the DG

unit connected at the weakest bus as identified by the proposed VRI (2-14).

4.2.2.1 Grid losses with one DG unit

A three-phase induction generator is placed at different buses of the modified

unbalanced three-phase 13 node test feeder (Figure 4-1) and system active and

reactive losses are plotted in Figure 4-6. This figure confirms that bus 675 (resulting

in the lowest grid losses) is the most suitable bus for DG placement, as was

previously identified by the proposed VRI (2-14).

59

0.100

0.105

0.110

0.115

0.120

0.125

0.130

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

Ac

tiv

e P

ow

er

Lo

ss

(M

W)

0.345

0.365

0.385

0.405

0.425

0.445

0.465R

ea

ctiv

e P

ow

er L

os

s (M

VA

r)

Active power loss

Reactive power loss

Figure 4-6 Reactive and active power losses associated with DG connections at different buses of Figure 4-1 (Case 2).

4.2.2.2 Grid losses with two DG units

According to (2-14), with the addition of one DG (at bus 675, Figure 4-4), the most

suitable location for the connection of a second DG unit is still at bus 675. This is in

agreement with the grid loss plots of Figure 4-7 generated by connecting the first DG

at bus 675 and placing a second DG at different buses of the modified unbalanced

three-phase 13 node feeder. These results further confirm the accuracy of the

proposed bus ranking index.

0.088

0.092

0.096

0.100

0.104

0.108

0.112

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

Ac

tiv

e P

ow

er

Lo

ss

(M

W)

0.300

0.315

0.330

0.345

0.360

0.375

0.390

Re

ac

tive

Po

we

r Lo

ss

(MV

Ar)

Active power loss

Reactive power loss

Figure 4-7 Reactive and active power losses associated with the first DG installed at bus 675 and the second DG connected at different buses of Figure 4-1 (Case 3).

4.2.3 Validation of proposed VRI based on PV curves

The PV curves based on positive-sequence voltages are plotted and compared with

the PV curve generated when DG and SVC units are connected at the weakest bus.

Figure 4-8 shows the PV curves of positive-sequence voltages at each bus for Case 2.

60

According to this figure, bus 675 has the lowest stability margin. Therefore, this is

the weakest bus as previously recognized by (2-14).

After connecting a combination of DG and SVC units at bus 675, PV curves for

Case 6 are regenerated and plotted in Figure 4-9. As expected and previously

recognized by the proposed VRI, the lowest stability margins occur at bus 634. This

will further reveal the validity of the proposed voltage ranking index for unbalanced

three-phase networks.

846674666466546644663466

1.04

1.00

0.96

0.92

0.88

0.84

0.80

Po

sit

ive-S

eq

uen

ce V

olt

ag

e [

p.u

.]

Total Load of Selected Loads (kW)

Bus 675

646 680 611 634 675 652

Figure 4-8 PV curves of positive-sequence voltage at each three-phase bus for Case 2.

61

184661546612466946664663466

1.10

1.00

0.90

0.80

0.70

Total Load of Selected Loads (kW)646 680 611 634 675 652

Bus 634

Po

sit

ive-S

eq

uen

ce V

olt

ag

e [

p.u

.]

Figure 4-9 PV curves of positive-sequence voltage at each bus for Case 4.

4.3 APPLICATION OF PROPOSED VRI IN IMPROVING MLF OF

THE MODIFIED UNBALANCED THREE-PHASE 13 NODE TEST FEEDER

Application of the proposed bus ranking index for the placement of DGs

without/with SVCs at the weakest three-phase buses will improve the maximum

loading factors as demonstrated in Table 4-5. According to the tabulated results:

A comparison of the maximum loading factors for Cases 1 and 2 indicates that the

voltage stability margin is higher with a voltage regulator. Therefore, voltage

regulators can help to improve the voltage stability margins of unbalanced

distribution systems.

After connecting DG at bus 675 (Case 3), the voltage stability margin has slightly

decreased from 2.375 to 2.343.

There is a significant improvement in MLF when a combination of DG and SVC

units is placed at the weakest three-phase bus. For example, after connecting DG

and SVC (358 kW and 0.36 MVar) at buses 680 (Case 5) and 675 (Case 4) the

maximum loading factor is improved (from 2.375 with no compensation) to 4.390

and 4.967, respectively.

62

TABLE 4-5 SIMULATION RESULTS OF THE MODIFIED UNBALANCED THREE-PHASE 13 NODE TEST FEEDER (FIG. 4-1, TABLE 4-1): COMPARISON OF MLF WITHOUT/WITH REGULATOR, DG

AND SVC.

Case

number Description Order of bus ranking

(2-14) MLF*

1 No regulation 675, 652, 611, 684, 680 2.199

2 With regulation 675, 652, 611, 684, 680 2.375

3 DG at bus 675 675, 652, 611, 684, 680 2.343

4 Combination of DG

and SVC at bus 675

634, 633, 646, 645, 632 4.967

5 Combination of DG

and SVC at bus 680

634, 633, 646, 645, 675 4.390

*) Computed by increasing the active power of all loads until the power flow solution

becomes unstable.

4.3.1 Enhancement of MLF by optimal sizing of one DG Unit

The maximum loading factors of Table 4-5 are computed for DG compensation

values of 358 kW. These factors can be improved by proper sizing of the

compensation devices as shown in Figure 4-10.

Figure 4-10 shows the impact of increasing the number of DG units on MLF. Each

DG unit injects 358 kW of active power. According to this figure, MLF can be

improved from 4.967 (Case 4) to 5.223 if the level of DG compensation at the

weakest three-phase node (bus 675) is increased from 358 kW to 5.012 MW.

63

4.80

4.85

4.90

4.95

5.00

5.05

5.10

5.15

5.20

5.25

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

The number of DG units

Lo

ad

ing

Facto

r

Figure 4-10 MLF as a function of the number of DG units placed at the weakest node

(bus 675).

4.3.2 Improving MLF by placement and sizing of compensation devices

A relatively simple procedure is used to properly place and size the compensation

devices to further improve MLF of the unbalanced distribution system. The approach

is to place one compensation unit (e.g., a 358 kW DG with SVC) at the weakest bus

and compute the corresponding MLF. The procedure is then repeated by relocating

the weakest bus (based on Eq. 2-14 with all previous units in service) and placing

more compensation devices.

With the above-mentioned approach for placement of DG (with a 0.36 MVAr SVC

used for voltage regulation) are shown in Figure 4-11. The selected size of the unit

DG is 358 kW. Based on Figure 4-11, MLF can be further improved to 6.119 with a

total DG of 716 kW (consisting of 358 kW and 358 kW units at buses 675 and 634,

respectively).

0

1

2

3

4

5

6

7

No DG DG at bus

675

DG at bus

675, 634

DG at bus

675, 634, 646

Lo

ad

ing

Fa

cto

r

Figure 4-11 Simulation results for placement and sizing of DG units in the modified

unbalanced three-phase 13 node feeder (Figure 4-1).

64

4.4 CONCLUSIONS

This chapter employed the new VRI of (2-14) to identify the weakest three-phase

buses of unbalanced three-phase distribution networks. The validity of the new index

is demonstrated for the modified unbalanced three-phase 13 node feeder based on

grid losses and PV curves. The proposed index is used to improve MLF by placing

DGs (without and with SVCs) at the weakest three-phase buses. Main conclusions

regarding the stability of unbalanced distribution networks are as follows:

Voltage regulators have positive impacts on voltage stability margins under

unbalanced three-phase conditions.

After connecting induction generator at bus 675 (Case 3), the voltage stability

margin has slightly decreased from 2.375 to 2.343. However, a combination of

DG and SVC devices at the weakest three-phase bus will considerably increase

MLF and significantly improve the voltage stability.

The order of bus ranking cannot be changed without reactive power compensation

devices. The order of bus ranking is changed when SVC with voltage controller is

installed at the weakest bus (bus 675) and at bus 680.

Both the new VRI and PV curves based on positive-sequence voltage can be

properly utilized to identify the weakest bus under unbalanced conditions.

Proper sizing of one compensation device (DG with SVC) at the weakest bus will

improve MLF.

65

Chapter 5. Application of proposed VRI in improving

voltage stability of multiphase networks

5.1 INTRODUCTION

Voltage stability enhancement has always been a main concern of the distribution

system operators who are unremittingly trying to optimize energy resources and

maximize the overall profits. As discusses in Chapters 3 and 4, the voltage stability

of a system can be improved by increasing the maximum loading factor (MLF)

defined in equation (3-3) as the ratio of the maximum system load (at the voltage

collapse point) to the base load. The proposed VRI of (2-14) can be used to properly

place shunt capacitors and DG units (without/with SVCs) at the weakest single-phase

and three-phase buses to increase MLF. In this chapter, the proposed VRI is applied

to a practical scenario of improving the voltage stability of the IEEE 13 node test

feeder (Figure 3-1) and the IEEE 34 node test feeder (Figure 3-13).

5.2 APPLICATION OF PROPOSED VRI IN IMPROVING STATIC

VOLTAGE STABILITY OF THE IEEE 13 NODE TEST FEEDER

Application of the proposed VRI for the placement of shunt capacitors and DGs

without/with SVCs at the weakest single-phase and three-phase buses will improve

MLF as demonstrated in Table 5-1. According to the tabulated results:

A comparison of the maximum loading factors for Cases 1 and 2 (Table 5-1)

indicates that the voltage stability margin is higher with a voltage regulator.

Therefore, voltage regulators can help to improve the voltage stability margins of

unbalanced distribution systems.

Application of the single-phase shunt capacitor at the weakest single-phase bus

will improve MLF as demonstrated in rows 3-4 (Table 5-1) where MLF is

increased from 3.155 (without a shunt capacitor) to 3.189 (with a shunt capacitor

66

placed at the weakest node; bus 611). However, the stability margin will decrease

if the weakest bus is not selected. For example, MLF is decreased to 3.127 if the

shunt capacitor is placed at bus 652 (Case 4, Table 5-1).

TABLE 5-1 SIMULATION RESULTS OF THE IEEE 13 NODE TEST FEEDER (FIG. 3-1, TABLE 3-1): COMPARISON OF MLF WITHOUT/WITH REGULATOR, SINGLE-PHASE SHUNT CAPACITOR,

DG AND SVC.

Case

Number

(Table 3-1)

Description Order of bus ranking (2-14)* MLF**

1 No regulation 611(1p), 684(2p), 652(1p),

675(3p), Figure 3-2

2.359

2 With regulation 611(1p), 684(2p), 652(1p),

675(3p), Figure 3-3

3.155

3 Shunt capacitor at

bus 611

611(1p), 684(2p), 652(1p),

675(3p)

3.189

4 Shunt capacitor at

bus 652

611(1p), 684(2p), 652(1p),

675(3p)

3.127

5 DG at bus 675 611(1p), 684(2p), 652(1p),

675(3p), Figure 3-4

3.158

6 Combination of

DG and SVC at bus

675

611(1p), 684(2p), 634(3P),

652(1p), 671(3p) or 692(3p),

Figure 3-5

5.415

7 Combination of

DG and SVC at bus

680

611(1p), 634(3p), 684(2P),

646(2p), 675(3p)

5.126

*) 1p, 2p and 3p correspond to single-phase, two-phase and three-phase buses,

respectively.

**) Computed according to (3-3) by increasing the active power of all loads while keeping

the power factor constant until the power flow solution diverges.

67

The reason is that the voltage at bus 652 is increased whereas the voltage at bus

611 is decreased. In addition, the phase voltage unbalance rate (defined as the

ratio of the max voltage deviation from the average phase voltage to average

phase voltage [34]) at bus 684 is 0.022526 which is higher than the base-case

value of 0.006294 due to the unbalanced multiphase loading at buses 611 (phase

c), 652 (phase a) and 684 (phase ac). On the other hand, the phase voltage

unbalance rate at bus 684 (with a shunt capacitor placed at bus 611) is decreased

to 0.000448 which is lower than the base-case value.

After connecting DG at bus 675 (Case 5, Table 5-1), the voltage stability margin

has slightly increased from 3.155 to 3.158.

There is a significant improvement in MLF when a combination of DG and SVC

units is placed at the weakest three-phase bus. For example, after connecting DG

and SVC (358 kW and 0.36 MVAr) at buses 680 (Case 7, Table 5-1) and 675

(Case 6, Table 5-1), MLF is improved (from 3.155 with no compensation) to

5.126 and 5.415, respectively.

5.2.1 Enhancement of MLF by optimal sizing of one compensation device in

the IEEE 13 node test feeder

The maximum loading factors of Table 5-1 are computed for fixed shunt capacitor

and DG compensation values of 0.1 MVAr and 358 kW, respectively. These factors

can be improved by optimal sizing of the compensation devices as shown in Figures

5-1 and 5-2. The approach is to place one compensation device at the weakest bus

and compute the corresponding MLF. The procedure is then repeated by increasing

the size of compensation devices until reaching the maximum possible loading

factor. Based on Figure 5-1, MLF can be improved from 3.189 (Case 3, Table 5-1) to

3.446 if the size of the shunt capacitor placed at the weakest single-phase node (bus

611) is increased from 0.1 MVAr to 2.8 MVAr.

Figure 5-2 shows the impact of increasing the number of DG units on MLF. Each

DG unit injects 358 kW of active power. According to this figure, MLF can be

improved from 5.415 (Case 6, Table 5-1) to 5.826 if the level of DG compensation at

the weakest three-phase node (bus 675) is increased from 358 kW to 3.938 MW.

68

Furthermore, MLF of Case 5 (Table 5-1) can also be improved from 3.158 to 3.159 if

the size of DG is increased from 358 kW to 716 kW.

0

0.5

1

1.5

2

2.5

3

3.5

4

0.1 0.3 0.5 0.7 0.9 1.1 1.3 1.5 1.7 1.9 2.1 2.3 2.5 2.7 2.9 3.1

Capacitor size (MVAr)

Lo

ad

ing

Facto

rs

Figure 5-1 MLF (for the IEEE 13 node test feeder) as a function of shunt

capacitor size at the weakest single-phase node (bus 611).

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

1

Lo

ad

ing

Fa

cto

r

2 3 4 5 6 7 8 9 10 11 12 13

The Number of DG Units

Figure 5-2 MLF (for the IEEE 13 node test feeder) as a function of the number of

DG units placed at the weakest three-phase node (bus 675).

5.2.2 Improving MLF by placement and sizing of compensation devices in

the IEEE 13 node test feeder

A relatively simple procedure is used to properly place and size the compensation

devices to further improve MLF of the unbalanced distribution system. The approach

is to place one compensation unit (e.g., a 0.1 MVAr capacitor or a 358 kW DG with

SVC) at the weakest bus and compute the corresponding MLF. The procedure is then

repeated by relocating the weakest bus (based on Eq. 2-14 with all previous units in

service) and placing more compensation devices.

With the above-mentioned approach for placement and sizing of single-phase shunt

capacitors (with a unit size of 0.1 MVAr), the weakest nodes will be buses 611 and

652. MLF of the IEEE 13 node test feeder will be increased from 3.155 (Case 2 of

69

Table 5-1, one capacitor unit at bus 611) to 3.623 with a total shunt capacitor size of

3.6 MVAr (consisting of 2.7 MVAr and 0.9 MVAr at buses 611 and 652,

respectively). The accuracy can be improved by selecting a smaller unit capacitor

size.

Simulation results for the placement and sizing of DG (with a 0.36 MVAr SVC used

for voltage regulation) are shown in Figure 5-3. The selected size of the unit DG is

358 kW. Based on Figure 5-3, MLF can be further improved to 6.758 with a total DG

of 716 kW (consisting of 358 kW and 358 kW units at buses 675 and 634,

respectively).

0

1

2

3

4

5

6

7

8

No DG DG at bus 675 DGs at buses

675, 634

DGs at buses

675, 634, 633

Lo

ad

ing

Fa

cto

r

Figure 5-3 Simulation results for placement and sizing of DG units in the IEEE 13 node

test feeder (Figure 3-1).

5.3 APPLICATION OF PROPOSED VRI IN IMPROVING STATIC

VOLTAGE STABILITY OF THE IEEE 34 NODE TEST FEEDER

Application of the proposed VRI for the placement of DGs at the weakest three-

phase buses will improve MLF as demonstrated in Table 5-2. According to the

tabulated results:

A comparison of the maximum loading factors for Cases 8 and 9 (Table 5-2)

indicates that the voltage stability margin is higher with voltage regulators.

Therefore, voltage regulators can help to improve the voltage stability margins of

unbalanced distribution systems.

70

After connecting DG at bus 890 (Case 10, Table 5-2), the voltage stability margin

has slightly decreased from 2.518 to 2.411. However, after connecting DG at bus

890 (Case 11, Table 5-2), MLF is improved to 2.862.

There is a significant improvement in MLF when a DG unit (2.4 MW DFIG wind

turbines) is installed at the weakest three-phase node (bus 890). For example,

after connecting DG at buses 890 (Case 12, Table 5-2), MLF is improved (from

2.518 with no compensation) to 3.890.

TABLE 5-2 SIMULATION RESULTS OF THE IEEE 34 NODE TEST FEEDER (FIG. 3-13, TABLE 3-1): COMPARISON OF MLF WITHOUT/WITH REGULATOR, DG TYPES IG AND DFIG.

Case

Number

(Table 3-8)

System operating condition of the

IEEE 34 node test feeder MLF*

8 No voltage regulators, transformer tap ratio set to 1.0 1.895

9 Two voltage regulator, variable transformer tap ratio 2.518

10 Case 9 with a DG (200 kW IG, Appendix B, Table B2)

at the weakest three-phase node (bus 890)

2.411

11 Case 9 with a DG (200 kW DFIG wind turbine,

Appendix B, Table B3) at the weakest three-phase

node (bus 890)

2.862

12 Case 9 with DGs (2.4 MW DFIG wind turbines) at the

weakest three-phase node (bus 890)

3.890

*) Computed according to (3-3) by increasing the active power of all loads while keeping the

power factor constant until the power flow solution diverges.

Application of the proposed VRI for the placement of single-phase shunt capacitors

(0.1 MVAr) at the weakest single-phase buses will further improve MLF as

demonstrated in Table 5-3. According to the tabulated results:

Application of the single-phase shunt capacitor at the weakest single-phase bus

will improve MLF as demonstrated in Table 5-3 where MLF is increased from

2.518 (without a shunt capacitor) to 2.574 (with a shunt capacitor placed at the

71

weakest node; bus 864). However, the stability margin will decrease if the

weakest bus is not selected. For example, MLF is decreased to 2.502 if the shunt

capacitor is placed at bus 826 (Table 5-3).

TABLE 5-3 SIMULATION RESULTS OF THE IEEE 34 NODE TEST FEEDER

(FIG. 3-13): COMPARISON OF MLF WITH SINGLE-PHASE SHUNT CAPACITOR PLACED AT

DIFFERENT BUSES.

Bus number MLF*

810 2.522

818 2.554

820 2.546

822 2.526

826 2.502

838 2.502

864 2.574

856 2.530

*) Computed according to (3-3) by increasing the active power of all loads while

keeping the power factor constant until the power flow solution diverges.

5.3.1 Enhancement of MLF by optimal sizing of one compensation device in

the IEEE 34 node test feeder

The maximum loading factors of Table 5-3 are computed for fixed shunt capacitor

0.1 MVAr. These factors can be improved by optimal sizing of the compensation

devices as shown in Figures 5-4. The approach is to place one compensation device

at the weakest bus and compute the corresponding MLF. The procedure is then

repeated by increasing the size of compensation devices until reaching the maximum

possible loading factor. Based on Figure 5-4, MLF can be improved from 2.574

(Table 5-3) to 2.682 if the size of the shunt capacitor placed at the weakest single-

phase node (bus 864) is increased from 0.1 MVAr to 0.275 MVAr.

72

Table 5-4 shows the impact of increasing the number of DG units (IGs) on MLF.

Each DG unit injects 200 kW of active power. According to this table, MLF is

getting worse with the increasing in the number of DG units.

However, MLF of Case 11 (Table 5-2) continues to increase from the base-case if the

size of DG is increased as shown in Figure 5-5.

2.30

2.35

2.40

2.45

2.50

2.55

2.60

2.65

2.70

2.75

0.05 0.075 0.1 0.125 0.15 0.175 0.2 0.225 0.25 0.275 0.3

Capacitor size (MVAr)

Lo

ad

ing

Fac

tor

Figure 5-4 MLF (for the IEEE 34 node test feeder) as a function of shunt capacitor size

at the weakest single-phase node (bus 684).

TABLE 5-4 MLF (FOR THE IEEE 34 NODE TEST FEEDER) AS A FUNCTION OF THE NUMBER OF DG UNITS (IGS) PLACED AT THE WEAKEST THREE-PHASE NODE (BUS 890).

Number of DG unit MLF*

1 2.411

2 2.410

3 2.301

4 2.200

5 2.071

*) Computed according to (3-3) by increasing the active power of all loads while

keeping the power factor constant until the power flow solution becomes unstable.

73

0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

1 2 3 4 5 6 7 8 9 10

The number of DG units

Lo

ad

ing

Fac

tor

Figure 5-5 MLF (for the IEEE 34 node test feeder) as a function of the number of

DG units (DFIG wind turbines) placed at the weakest three-phase node (bus 890).

5.4 APPLICATION OF PROPOSED VRI IN IMPROVING DYNAMIC

VOLTAGE STABILITY OF THE IEEE 13 NODE TEST FEEDER

Dynamic simulations are performed to further validate the accuracy of the proposed

VRI. The IEEE 13 node test feeder (Figure 3-1) will be subjected to small and large

disturbances.

We will first consider a small disturbance caused by the operation of the voltage

regulator connected between buses 650 and RG60:

With a voltage regulator (Case 2, Table 5-1), the proposed VRI (2-14) is

calculated based on static (using power flow calculations) and dynamic (using

time-domain simulation) approaches and compared in Figure 5-6. Note that the

order of bus rankings based on the static and dynamic simulations are almost the

same. This confirms that the proposed VRI can be utilized in both static and

dynamic analyses.

With DG placed at bus 675 (Case 5, Table 5-1), the proposed VRI (2-14) is

applied to each bus at t=3sec. Although there is a clear distinction in the

calculated VRI values at some buses (Figure 5-7), the order of dynamic and static

bus voltage rankings are still the same. This confirms that the application of one

DG (type IG) has no significant impact and will not change the order of bus

ranking.

74

We now consider the impact of large disturbances caused by the operation of the

switch (Figure 3-1) connected between buses 671 and 692. Figure 5-8 shows the

voltage profiles of bus 675 at the static voltage collapse point (MLF=3.158) for Case

5 (Table 5-1) for two critical operating conditions corresponding to the switch being

opened at t=0.50sec and closed at t=0.66sec and t=0.67sec, respectively.

It can be seen that after the switch is closed at t=0.67sec, the voltage profile

cannot recover to its initial value (Figure 5-8). Furthermore, DG remains stable

after the switch is closed at t=0.66sec and becomes unstable if it is closed at

t=0.67sec (Figure 5-9).

However, with the placement of SVC at bus 675, DG becomes stable as shown in

Figure 5-10.

Figure 5-11 compares the calculated values of VRI for the above mentioned

critical operating conditions. Based on this figure, it is clear that VRI of the

critical case with the switched being closed at t=0.66sec can be used as an

indicator to identify the stability of the system.

0

0.2

0.4

0.6

0.8

1

1.2

Static Dynamic at time 200sec

VR

I

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

Figure 5-6 Comparison of the proposed VRI (Eq. 2-14) based on static and dynamic

approaches for Case 2 (Table 5-1).

75

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

0

0.2

0.4

0.6

0.8

1

1.2

VR

I

Static Dynamic at time 3sec

Figure 5-7 Comparison of the proposed VRI (Eq. 2-14) based on static and dynamic

approaches for Case 5 (Table 5-1).

2.97112.37691.78271.18850.5942 [sec]

1.0

0.8

0.6

0.4

SW closed at time 0.66sec

SW closed at time 0.67sec

Po

sit

ive

-

Se

qu

en

ce

Vo

lta

ge

(p.u

.)

675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.

0

0.2

0.0

Figure 5-8 Voltage profiles of bus 675 under switch operation of Case 5 (Table 5-1).

76

2.97112.37691.78271.18850.5942 [sec]

0.80

0.40

0.00

-0.40

SW closed at time 0.66sec

SW closed at time 0.67sec

DG at bus 675: Total Active Power in MW

DG at bus 675: Total Reactive Power in MVArDG at bus 675: Total Reactive Power in MVArDG at bus 675: Total Active Power in MW

SW closed at time 0.66sec

SW closed at time 0.67sec

0

-0.80

-1.20

Acti

ve P

ow

er

[MW

] / R

eacti

ve

Po

wer

[MV

Ar]

Figure 5-9 Active and reactive power of DG at bus 675 under switch operation of Case

5 (Table 5-1).

2.97112.37691.78271.18850.5942 [sec]

0.80

0.40

0.00

-0.40

With SVC at bus 675

Without SVC

DG/SVC at bus 675: Total Active Power in MW

DG at bus 675: Total Reactive Power in MVAr

DG at bus 675: Total Active Power in MWDG/SVC at bus 675: Total Reactive Power in MVAr

Without SVC

With SVC at bus 675

0

-0.80

-1.20

Acti

ve P

ow

er

[MW

] / R

eacti

ve P

ow

er

[MV

Ar]

Figure 5-10 Active and reactive power of DG installed at bus 675 (of the IEEE 13 node

test feeder) with/without SVC after switch closed at time 0.67s.

77

0

0.2

0.4

0.6

0.8

1

1.2

RG60 632 633 634 645 646 671 680 684 611 652 692 675

Bus Number

(Stable) SW closed 0.66s (Unstable) SW closed 0.67s (Stable) SVC+ SW closed 0.67s

VR

I

Figure 5-11 Comparison of VRI values for dynamic operating conditions in the IEEE 13

node test feeder.

5.5 APPLICATION OF PROPOSED VRI IN IMPROVING DYNAMIC

VOLTAGE STABILITY OF THE IEEE 34 NODE TEST FEEDER

Dynamic simulations are performed to further validate the accuracy of the proposed

VRI. The IEEE 34 node test feeder (Figure 3-13) will be subjected to small and large

disturbances.

We will first consider a small disturbance caused by the operation of the two voltage

regulators connected between buses 814-RG10 and 852-RG11:

With two voltage regulators (Case 9, Table 5-2), the proposed VRI (2-14) is

calculated based on static (using power flow calculations) and dynamic (using

time-domain simulation) approaches and compared in Figure 5-12. Note that the

order of bus rankings based on the static and dynamic simulations are almost the

same. This confirms that the proposed VRI can be utilized in both static and

dynamic analysis.

With DG placed at bus 890 (Case 10, Table 5-2), the proposed VRI (2-14) is

applied to each bus at t=3sec. Although there is a clear distinction in the

calculated VRI values at some buses (Figure 5-12), the order of dynamic and

static bus voltage rankings are still the same. This confirms that the application of

one DG (type IG) has no significant impact and will not change the order of bus

ranking.

78

0

0.2

0.4

0.6

0.8

1.0

1.2

80

0

80

2

80

6

80

8

81

0

81

2

81

4

85

0

81

6

81

8

82

0

82

2

82

4

82

6

82

8

83

0

85

4

85

2

83

2

85

8

83

4

84

2

84

4

84

6

84

8

86

0

83

6

84

0

86

2

83

8

86

4

88

8

89

0

85

6

Bus Number

VR

IStatic Dynamic at time 400 sec

Figure 5-12 Comparison of the proposed VRI (Eq. 2-14) based on static and dynamic

approaches for Case 9 (Table 5-2).

0

0.2

0.4

0.6

0.8

1.0

1.2

800

802

806

808

810

812

814

850

816

818

820

822

824

826

828

830

854

852

832

858

834

842

844

846

848

860

836

840

862

838

864

888

890

856

Bus Number

VR

I

Static Dynamic at time 3 sec

Figure 5-13 Comparison of the proposed VRI (Eq. 2-14) based on static and dynamic

approaches for Case 10 (Table 5-2).

We now consider the impact of large disturbances caused by the operation of the

circuit breaker between buses 888 and 890 (Figure 3-13). Figure 5-14 shows the

voltage profiles of bus 890 at the static voltage collapse point (MLF=2.411) for Case

10 (Table 5-2) for two critical operating conditions corresponding to the circuit

breaker being opened at t=0.50sec and closed at t=0.53sec and t=0.54sec,

respectively.

It can be seen that after the switch is closed at t=0.54sec, the voltage profile

cannot recover to its initial value (Figure 5-14). Furthermore, DG remains stable

after the circuit breaker is closed at t=0.53sec and becomes unstable if it is closed

at t=0.54sec (Figure 5-15).

However, with the placement of SVC at bus 890, DG becomes stable as shown in

Figure 5-16.

79

Figure 5-17 compares the calculated values of VRI for the above mentioned

critical operating conditions. Based on this figure, it is clear that VRI of the

critical case with the circuit breaker being closed at t=0.53sec can be used as an

indicator to identify the stability of the system.

3.002.001.000.00 [sec]

0.60

0.50

0.40

0.30

0.20

0.10

CB closed at time 0.53sec

CB closed at time 0.54sec

Po

sit

ive

-

Se

qu

en

ce

Vo

lta

ge

(p.u

.)

675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.

Figure 5-14 Voltage profiles of bus 890 under circuit breaker operation of Case 10

(Table 5-2).

3.002.001.000.00 [sec]

0.250

0.125

0.000

-0.125

-0.250

-0.375

CB closed at time 0.53sec

CB closed at time 0.54sec

DG at bus 890: Total Active Power in MW

DG at bus 890: Total Reactive Power in MVArDG at bus 890: Total Reactive Power in MVArDG at bus 890: Total Active Power in MW

CB closed at time 0.53sec

CB closed at time 0.54sec

Ac

tiv

e P

ow

er

[MW

] /

Re

ac

tiv

e P

ow

er

[MV

Ar]

Figure 5-15 Active and reactive power of DG at bus 890 under circuit breaker operation

of Case 10 (Table 5-2).

80

3.002.001.000.00 [sec]

0.250

0.125

0.000

-0.125

-0.250

-0.375

With SVC at bus 890

Without SVC

DG at bus 890: Total Active Power in MW

DG at bus 890: Total Reactive Power in MvarDG at bus 890: Total Reactive Power in MvarDG at bus 890: Total Active Power in MW

With SVC at bus 890Without SVC

Ac

tiv

e P

ow

er

[MW

] / R

ea

cti

ve

Po

we

r [M

VA

r]

Figure 5-16 Active and reactive power of DG installed at bus 890 (of the IEEE 34

node test feeder) with/without SVC after circuit breaker closed at time 0.54s.

0

0.2

0.4

0.6

0.8

1.0

1.2

800

802

806

808

810

812

814

850

816

818

820

822

824

826

828

830

854

852

832

858

834

842

844

846

848

860

836

840

862

838

864

888

890

856

Bus Number

VR

I

(Stable) CB closed 0.53s (Unstable) CB closed 0.54s (Stable) SVC+CB closed 0.54s

Figure 5-17 Comparison of VRI values for dynamic operating conditions in the IEEE 34

node test feeder.

5.6 CONCLUSIONS

This chapter employed the new VRI of (2-14) to identify the weakest single-, two-

and three-phase buses of unbalanced and multiphase distribution networks for

voltage stability enhancement. Main conclusions are as follows:

Application of the proposed bus ranking index for the placement of shunt

capacitors and DGs without/with SVCs at the weakest single-phase and three-

phase buses will improve the maximum loading factors.

81

Installation of a single-phase shunt capacitor at the bus which is not the weakest

single-phase bus can reduced the overall loading factor.

The new index can be applied to both static and dynamic approaches. The

proposed index based on static approach can be used to determine which bus is

the weakest bus for the voltage stability enhancement whereas the proposed index

based on dynamic approach can be used as an indicator to identify the stability of

the system.

Time domain simulation is recommended in critical cases.

DG units with controllable reactive power such as a synchronous generator may

perform better than an induction generator in terms of voltage stability

enhancement.

The application of symmetrical components as employed in this thesis may

require values that are difficult to obtain without full three-phase measurements at

all levels of the system.

82

Chapter 6. Online bus voltage ranking in unbalanced

multiphase smart grids with plug-in electric vehicle

(PEV) charging stations

6.1 INTRODUCTION

Plug-in electric vehicles (PEVs) are expected to become popular in the near future as

alternatives to conventional fuel-based automobiles in order to reduce the emission to

the environment [35-40]. However with the random charging behaviors and

unpredictable penetration levels of PEVs in the residential feeders, voltage drop

issues and voltage stability problems are anticipated in the future smart grid

configurations [35-37, 41]. A possible solution will be to shift a portion of PEV

loading to the distribution networks by intelligently siting and sizing PEV charging

stations or PEV smart parks.

To promote and support the increasing penetration of PEVs entering into smart grid,

many counties are planning to increase the number of charging stations and/or smart

parks [39-40]. However, there are also important issues associated with increasing

the number of charging stations and smart parks in term of line overloading, bus

voltage regulations and stability problems. PEV charging stations can affect system

voltage profile, load flow and stability of the smart grid. Therefore, electric utilities

are very interested in investigating the possible impacts and drawbacks of PEV

charging demand on their distribution networks [38, 42].

In smart parks, PEV charging operation can be performed in charging mode and

discharging mode. To increase the effectiveness of smart parks, PEVs should be

charged from the grid during off-peak load hours (charging modes) and discharged to

the grid during the peak load hours (discharging mode). The electric utilities may

require load shedding if there is a high demand charging during the peak load hours

[38, 40]. In [41, 43], smart parks are placed at the lowest voltage lines under normal

83

operating conditions as reactive power and voltage supports to enhance voltage

stability in discharging modes. However, in charging modes, the system is less

stable. In addition, the bus which has the lowest voltage may not be the most suitable

location for connecting smart parks as reactive and voltage supports.

Identification of weakest buses through the bus ranking indices will play an

important role for the analysis and voltage stability enhancement of smart grids. The

purpose of bus ranking in smart grid is to determine which nodes are the weakest

buses during 24 hours for connecting compensation devices [44]. Furthermore, it can

provide insights for properly placing and sizing future PEV charging stations and

smart parks. It has been shown that the best location for reactive power

compensation to improve voltage stability margin is the weakest bus [3, 10].

In this chapter, symmetrical components are applied to the conventional bus voltage

ranking index V/Vo to extend its application to online (for example every one hour)

identification of the weakest buses of unbalanced multiphase smart grids during the

24 hours considering the impacts of charging stations. Simulations are performed and

compared using DIgSILENT PowerFactory software to identify the weakest three-

phase buses of the modified unbalanced multiphase 13 node test feeder without/with

PEV charging stations.

6.2 THE MODIFIED IEEE 13 NODE TEST SYSTEM WITH PEV

CHARGING STATIONS

For the analysis of this chapter, the IEEE unbalanced multiphase 13 node test feeder

of Figure 6-1 [33] is considered with four PEV charging stations connected at bus

634 or bus 680. The network has been simulated using DIgSILENT PowerFactory

software [32]. The system is identical to Figure 3-1 with exception of the PEV

charging stations. System data is available in the Appendixes A1 and A2.

For the dynamic analysis of this chapter, the daily P and Q load curves of Figure 6-2

are assumed and utilized for the linear loads [45]. For the PEV charging stations (at

buses 634 and 680), the daily load curve of Figure 6-3 with two peaks at 7am and

6pm is employed [36].

84

646 645 632 633 634

650

692 675611 684

652

671

680

RG604.16 kV

0. 48 kV4. 16 kV

Switch

Two- phase

Single-phaseThree- phase

115 kV

CS1

CS4CS3CS2

CS8

CS5

CS6

CS7

Case 2

Case 3

Figure 6-1 The modified unbalanced multiphase 13 node test feeder with PEV charging

stations at bus 634 or bus 680.

6.3 SIMULATION RESULTS

Simulations are performed for the modified IEEE unbalanced multiphase 13 node

test feeder of Figure 6-1 without and with PEV charging stations to investigate their

impacts on voltage profiles and bus voltage ranking indices. Simulation results are

presented for four case studies.

Case 1: No PEV charging stations.

The VRI index for an online application (2-15) is calculated and ranked to locate the

weakest three-phase buses of Figure 6-1 without any PEV charging stations. Figure

6-4 shows the impact of the dynamic daily load curves of Figures 6-2 and 6-3 on the

voltage profiles of selected nodes (buses 634, 675 and 680). According to this figure,

bus 634 has the lowest voltage profile. However the three-phase buses over 24 hours

which have the lowest bus voltage ranking indices are buses 675, 634, and 680.

Therefore, the three-phase weakest bus for Case 1 is bus 675.

85

Case 2: Four PEV charging stations at bus 634.

In the multiphase unbalanced system of Figure 6-1 four 0.2MW PEV charging

stations with the total peak charge level of 0.8MW are included at bus 634. The peak

charging (0.8MW) is about 25% of total load (3.46MW). Figure 6-5 shows the

impact of placing the PEV charging stations at bus 634 on the voltage profiles of

buses 634, 675 and 680. With the pattern charging of PEVs (Figure 6-3) at buses

634, the voltage levels at bus 634 is lower than other buses as shown in Figure 6-5.

Table 6-1 shows bus voltage ranking indices with PEV charging stations at bus 634

over 24 hours. According to this Table, the weakest three-phase bus has changed

from bus 675 (Case 1) to bus 634 as a result of PEV charging activities at bus 634.

Case 3: Four PEV charging stations at bus 680.

Case 2 is repeated, except the four 0.2MW PEV charging stations with the daily load

curves of Figure 6-3 located at bus 680. Figure 6-6 shows the impact of the charging

stations on voltage profiles with PEV charging stations at bus 680. Compared to

Case 2, the voltage profile is improved during 11am to 17pm. Table 6-2 shows the

bus voltage ranking indices with PEV charging stations at bus 680 over 24 hours.

According to this Table, the three lowest bus voltage ranking indices are associated

with buses 675, 680, and 634.

Case 4: Four PEV charging stations at bus 680 and two PEV charging stations at bus

634.

Case 3 is repeated with the addition of two 0.2MW PEV charging stations with the

daily load curves of Figure 6-3 located at bus 634. Figure 6-7 shows the impact of

the charging stations on voltage profiles with four PEV charging stations at bus 680

and two PEV charging stations at bus 634. Table 6-3 shows the bus voltage ranking

indices over 24 hours with PEV charging stations at bus 680. According to this

Table, the locations of the weakest bus have changed between buses 675 and 680.

For example, the weakest three-phase bus has changed to bus 680 at 7-9 a.m. and 6-9

p.m.

86

24.019.214.49.64.80.0

100

80

60

40

20

0

Time [Hour]

Per

cent

age

of P

eak

Load

[%]

P daily load curve

Q daily load curve

Figure 6-2 Daily load curves associated with Figure 6-1 for linear loads [45].

24.019.214.49.64.80.0

100

80

60

40

20

0

Time [Hour]

Per

cent

age

of P

eak

Load

[%]

P load curve

Figure 6-3 Daily load curves associated with Figure 6-1 for PEV charging stations [36].

87

24.019.214.49.64.80.0

1.10

1.00

0.90

0.80

0.70

634: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.680: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.

Time [Hour]

Pos

itive

-Seq

uenc

e V

olta

ge [p

.u.]

Figure 6-4 Simulation results for Case 1: the 24 hour voltage profile of buses 634, 675

and 680.

24.019.214.49.64.80.0

1.10

1.00

0.90

0.80

0.70

Pos

itive

-Seq

uenc

e V

olta

ge [p

.u.]

634: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.680: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.

Time [Hour]

Figure 6-5 Simulation results for Case 2: the 24 hour voltage profile of buses 634, 675

and 680.

88

24.019.214.49.64.80.0

1.10

1.00

0.90

0.80

0.70

Pos

itive

-Seq

uenc

e V

olta

ge [p

.u.]

634: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.680: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.

Time [Hour]

Figure 6-6 Simulation results for Case 3: the 24 hour voltage profile of buses 634, 675

and 680.

24.019.214.49.64.80.0

1.10

1.00

0.90

0.80

0.70

Pos

itive

-Seq

uenc

e V

olta

ge [p

.u.]

Time [Hour]634: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.680: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.

Figure 6-7 Simulation results for Case 4: the 24 hour voltage profile of buses 634, 675

and 680.

89

TABLE 6-1 CASE 2 - BUS VOLTAGE RANKING INDICES OVER 24 HOURS WITH FOUR PEV CHARGING STATIONS AT BUS 634.

Time

VRI

(Eq. 2-15)

at Bus 634

VRI

(Eq. 2-15)

at Bus 675

VRI

(Eq. 2-15)

at Bus 680

Weakest bus

0.00 0.952573 0.976249 0.979515 634

1.00 0.903801 0.929840 0.933893 634

2.00 0.887627 0.913419 0.917316 634

3.00 0.877407 0.902948 0.906692 634

4.00 0.864649 0.889935 0.893688 634

5.00 0.859166 0.884163 0.887809 634

6.00 0.852665 0.878688 0.882301 634

7.00 0.830194 0.862575 0.866267 634

8.00 0.799218 0.841839 0.845514 634

9.00 0.766849 0.802819 0.807022 634

10.00 0.750888 0.775109 0.779623 634

11.00 0.735064 0.756605 0.761184 634

12.00 0.721301 0.741535 0.746147 634

13.00 0.716600 0.736621 0.741095 634

14.00 0.710610 0.730538 0.735009 634

15.00 0.708724 0.728681 0.733081 634

16.00 0.709924 0.730943 0.735222 634

17.00 0.705164 0.731868 0.736030 634

18.00 0.714965 0.748383 0.752134 634

19.00 0.718777 0.756636 0.76027 634

20.00 0.732496 0.766829 0.770428 634

21.00 0.755066 0.781949 0.785535 634

22.00 0.769885 0.794123 0.797730 634

23.00 0.781631 0.804035 0.807678 634

90

TABLE 6-2 CASE 3 - BUS VOLTAGE RANKING INDEX FOR THE MULTIPHASE SYSTEM OF FIGURE 6-1 WITH FOUR PEV CHARGING STATIONS AT BUS 680.

Time

VRI

(Eq. 2-15)

at Bus 634

VRI

(Eq. 2-15)

at Bus 675

VRI

(Eq. 2-15)

at Bus 680

Weakest bus

0.00 0.979404 0.972515 0.974186 675

1.00 0.933280 0.925467 0.927765 675

2.00 0.916540 0.909093 0.911267 675

3.00 0.905937 0.898683 0.900723 675

4.00 0.892817 0.885717 0.88779 675

5.00 0.887124 0.880024 0.882001 675

6.00 0.881562 0.874371 0.876254 675

7.00 0.865036 0.856942 0.858496 675

8.00 0.843357 0.833764 0.834641 675

9.00 0.804699 0.795358 0.797155 675

10.00 0.777989 0.769874 0.772719 675

11.00 0.760107 0.752256 0.755335 675

12.00 0.745374 0.737803 0.741009 675

13.00 0.740763 0.733249 0.736329 675

14.00 0.734825 0.727404 0.730494 675

15.00 0.733023 0.725723 0.728745 675

16.00 0.735255 0.727916 0.730753 675

17.00 0.735924 0.727764 0.730107 675

18.00 0.751547 0.742872 0.744385 675

19.00 0.758997 0.749838 0.750951 675

20.00 0.768964 0.760327 0.761631 675

21.00 0.784309 0.776794 0.778571 675

22.00 0.796762 0.789638 0.791612 675

23.00 0.807028 0.800091 0.802215 675

91

TABLE 6-3 CASE 4 - BUS VOLTAGE RANKING INDEX FOR THE MULTIPHASE SYSTEM OF FIGURE 6-1 WITH FOUR PEV CHARGING STATIONS AT BUS 680 AND TWO PEV CHARGING

STATIONS AT BUS 634.

Time

VRI

(Eq. 2-15)

at Bus 634

VRI

(Eq. 2-15)

at Bus 675

VRI

(Eq. 2-15)

at Bus 680

Weakest bus

0.00 0.980186 0.969367 0.969430 675

1.00 0.931868 0.919640 0.920164 675

2.00 0.915040 0.903259 0.903690 675

3.00 0.904979 0.893448 0.893767 675

4.00 0.892726 0.881386 0.881763 675

5.00 0.888124 0.876815 0.877107 675

6.00 0.882623 0.871067 0.871200 675

7.00 0.861427 0.847999 0.847394 680

8.00 0.830174 0.813725 0.811798 680

9.00 0.792331 0.777062 0.776437 680

10.00 0.774322 0.761940 0.763098 675

11.00 0.762916 0.751094 0.752668 675

12.00 0.753490 0.742078 0.743883 675

13.00 0.752773 0.741401 0.743099 675

14.00 0.749586 0.738288 0.740013 675

15.00 0.749811 0.738618 0.740282 675

16.00 0.752786 0.741394 0.742800 675

17.00 0.749838 0.736729 0.737238 675

18.00 0.759273 0.744732 0.743968 680

19.00 0.760036 0.744424 0.742983 680

20.00 0.768626 0.754182 0.753168 680

21.00 0.787815 0.775682 0.775632 680

22.00 0.803453 0.792100 0.792427 675

23.00 0.816681 0.805766 0.806365 675

92

6.4 ONLINE PLACEMENT OF SVC UNITS TO IMPROVE THE

PERFORMANCE OF THE MODIFIED IEEE 13 NODE TEST SYSTEM WITH

PEV CHARGING STATIONS

This section introduces a new online approach to improve the performances of the

emerging smart grids with renewable energy resources and smart appliances. For

these systems, prediction and forecasting of the daily load curves may not be feasible

as the location, time and duration of the smart loads (such as PEVs and smart

appliances) are randomly changing during the 24 hour period. Therefore, the

conventional approaches of locating and sizing of compensation devices based on the

forecasted daily load curves are not accurate.

The proposed approach is to place compensation devices at the weakest buses,

perform online VRI ranking, and then switch these devices in (and out of) service

according to the lowest VRI values. The approach will be demonstrated for the

modified unbalanced multiphase 13 node test feeder of Figure 6-1 through the

following case study.

Case 5: Online placement of SVC units for Case 4.

Online VRI ranking of the modified unbalanced multiphase 13 node test feeder with

four PEV charging stations at bus 680 and two PEV charging stations or bus 634

(Case 4) indicates that the weakest bus changed between nodes 675 and 680 over the

24 hour period (Table 6-3). Therefore, compensation devices which are installed at

buses 675 and 680 will be switched on and off according to the time in Table 6-3.

Figure 6-8 shows the impact of online placement of two SVC units on voltage

profiles with four PEV charging stations at bus 680 and two PEV charging stations at

bus 634.

Compared to Case 4 (Figure 6-7), the voltage profiles at all buses are improved,

especially at buses 675 and 680. Table 6-4 shows the bus voltage ranking indices

after the online placement of SVC units installed at bus 675 and 680. According to

this Table, the weakest node (after online SVC placement) is changed from buses

675 and 680 to bus 634.

93

24.019.214.49.64.80.0

1.10

1.00

0.90

0.80

0.70

Pos

itive

-Seq

uenc

e V

olta

ge [p

.u.]

Time [Hour]634: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.675: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.680: Line-Ground Positive-Sequence Voltage, Magnitude in p.u.

Figure 6-8 Simulation results for Case 5 with online placement of two SVC units: the

24 hour voltage profile of buses 634, 675 and 680.

94

TABLE 6-4 CASE 5 BUS VOLTAGE RANKING INDEX FOR THE MULTIPHASE SYSTEM OF FIGURE 6-1 WITH FOUR PEV CHARGING STATIONS AT BUS 680 AND TWO PEV CHARGING

STATIONS AT BUS 634 AFTER ONLINE PLACEMENT OF TWO SVC UNITS

Time

VRI

(Eq. 2-15)

at Bus 634

VRI

(Eq. 2-15)

at Bus 675

VRI

(Eq. 2-15)

at Bus 680

Weakest bus

0.00 0.998229 1.007018 1.003504 634

1.00 0.901377 0.918712 0.919228 634 2.00 0.885086 0.902328 0.902751 634 3.00 0.875946 0.893086 0.893397 634 4.00 0.864844 0.881851 0.88222 634 5.00 0.861389 0.878238 0.878524 634 6.00 0.854892 0.872471 0.872597 634 7.00 0.823465 0.845201 0.844589 634 8.00 0.774945 0.80296 0.801047 634 9.00 0.744331 0.767646 0.767017 634 10.00 0.744796 0.760438 0.761584 634 11.00 0.741155 0.755132 0.756705 634 12.00 0.737111 0.750354 0.752172 634 13.00 0.739296 0.752539 0.754253 634 14.00 0.737999 0.751276 0.753022 634 15.00 0.739474 0.752874 0.754561 634 16.00 0.741581 0.755811 0.757236 634 17.00 0.729233 0.747394 0.747900 634 18.00 0.726917 0.749654 0.748874 634 19.00 0.718291 0.743732 0.742282 634 20.00 0.729811 0.752813 0.751791 634 21.00 0.760041 0.778158 0.778101 634 22.00 0.781170 0.797583 0.797903 634 23.00 0.798652 0.813863 0.814459 634

95

6.5 CONCLUSIONS

This chapter has extended the application of the conventional bus voltage ranking

index of V/Vo defined for balanced three-phase systems to online identification of

the weakest buses of the unbalanced multiphase smart grid over 24 hours considering

the impacts of PEV charging stations. Furthermore, the impacts of PEV charging

stations on voltage profiles and bus ranking indices have been investigated. The

approach is demonstrated on an unbalanced multiphase 13 node test feeder using

DIgSILENT PowerFactory software considering two locations for the PEV charging

stations. Main conclusions are:

PEV charging stations with relatively large power ratings can have detrimental

impacts of smart grid loading and voltage profiles over the 24 hour period.

As the smart grid loads (such as PEV chargers, smart appliances, charging

stations and smart parks) and renewable resources (such as rooftop PVs and

wind generators) have inherent discontinuous characteristics, the locations of

the weakest voltage buses will change over the 24 hour period. Therefore,

online dynamic bus ranking approaches are required in smart gird systems.

The proposed dynamic bus ranking approach of this chapter can be utilized to

identify the weakest buses over the 24 hour period.

To control and improve the detrimental impacts of large PEV charging

stations, they should be located at the strongest buses.

96

Chapter 7. Increasing DG penetration in multiphase

distribution networks considering grid losses, maximum

loading factor and bus voltage limits

7.1 INTRODUCTION

This chapter proposes a new algorithm to improve the performance of multiphase

distribution networks by properly locating DG units and single-phase capacitors in

the three-phase and single-phase sections and increasing their ratings. The approach

consists of utilizing the positive-sequence voltage ratio Vcollapse/Vbase-load (2-14) to

identify the weakest three-phase and single-phase buses for the installation of DG

units and shunt capacitors, respectively. DG penetration levels are increased by

evaluating their impacts on voltage profile, grid losses, and MLF while considering

the voltage limits at all buses. Detailed simulations are performed for the placement

and sizing of a doubly-fed induction generator (DFIG) and single-phase capacitors in

the IEEE multiphase 34 node test feeder using DIgSILENT PowerFactory software.

The impacts of DFIG on voltage profile, active power loss and voltage stability

margin are highlighted.

An iterative algorithm is proposed for the placement and sizing of DG units and

single-phase capacitors in multiphase networks to reduced grid losses and increase

MLF while keeping all bus voltage within acceptable limits. Simulation results

including locations and the maximum penetration levels of DG units as well as the

locations and sizes of single-phase capacitors are presented for the IEEE multiphase

34 node test feeder as shown in Figure 3-13.

97

7.2 IMPACTS OF DG PLACEMENT ON VOLTAGE PROFILE, GRID

LOSS, AND MLF

7.2.1 Impact of DG on voltage profiles

In balanced three-phase networks, voltage profiles are usually plotted using the rms

bus voltage values. For unbalanced networks, system unbalanced voltage variance

index [46] has been proposed for considering voltage profiles instead of using system

rms voltage [26, 30]. However, for multiphase networks, voltage magnitudes in some

phases are missing. Therefore, in this chapter, the voltage profiles of all phases will

be plotted in the range of 0.95-1.05p.u. (see Figures 7-5, 7-8 and 7-11).

7.2.2 Impact of DG on grid losses

Grid losses associated with the placement and the penetration level of a DG unit

(e.g., at the weakest bus) are computed and compared with the losses without any

compensation device. The active power loss reduction (ALR) (for example due to the

installation of DG units or compensation devices) is calculated by (3-1).

The DG penetration level is defined as

𝑃𝑒𝑛𝑒𝑡𝑟𝑎𝑡𝑖𝑜𝑛 𝐿𝑒𝑣𝑒𝑙 =𝑃𝐷𝐺

𝑃𝑙𝑜𝑎𝑑× 100% (7-1)

where 𝑃𝐷𝐺 and 𝑃𝑙𝑜𝑎𝑑 are the total active power of the DG units and system loads,

respectively.

7.2.3 Impact of DG on MLF

Using a continuation three-phase power flow, PV curves for multiphase distribution

networks will be plotted. The method of symmetrical components will then be

applied to merge the three individual PV curves into a single PV curve based on

positive-sequence voltage. Finally, the maximum loading factor (MLF) will be

determined using the single PV curve based on positive-sequence voltage [3]. MLF

is defined as the ratio of the maximum system load (at the voltage collapse point) to

the base load.

98

7.2.4 Impact of DG on voltage unbalance factor

The voltage unbalance factor (VUF) is defined as the ratio of negative-sequence

voltage component to positive-sequence voltage component [47]:

% 𝑉𝑈𝐹 =negative −sequence voltage component

positive −sequence voltage component× 100% (7-2)

7.3 PROPOSED ALGORITHM FOR DG PLACEMENT

The proposed iterative algorithm of Figure 7-1 is designed to increase the penetration

level of DG units in multiphase networks in order to reduce total active power loss

and enhance voltage stability margins considering voltage limits at all buses. In

addition, single-phase shunt capacitors are also utilized to further improve the

performance of the systems.

Stage one of the algorithm consists of an iterative procedure to properly place and

increase the penetration of DG units in multiphase system. DG units are located one

at a time and their corresponding sizes are increased until a voltage violation is

detected in the system. To find the best location and rate of the first DG, a small

DFIG is temporary placed at the weakest three-phase bus as identified by the

calculated VRI (2-14). The size of DFIG is then increase (to reduce total system loss

and increase MLF) until one of the bus voltages is increased above the permissible

level. The first iteration terminates by permanently connecting the first DG at BusDG

with PLDG. This procedure is repeated to place more DG units as long as no voltage

violations are noticed and there are improvements in the total system loss and MLF.

Stage two of the proposed algorithm is similar to stage one with the exception of

selecting the weakest single-phase buses (identified by VRI) and connecting single-

phase capacitor banks to the single-phase sections of the multiphase network.

99

run three-phase load flow, calculate ALR (Eq. 3-1) and MLF

yes

no

0.95£Vbus£1.05 ?

run three-phase power flow, calculate VRI (Eq. 2-14), and identify the weakest three-phase bus (exclude any buses with DG),

set BusDG=the weakest three-phase bus

S

tage one: placement and sizing of D

G units

initialize parameters, location of DG (BusDG=0) and penetration level of DG (PLDG=0)

start

increase PLDG

by 1% of PLoad

temporary placement of one DG unit at BusDG with PLDG

run three-phase load flow, calculate ALR (Eq. 3-1) and MLF

0.95£Vbus£1.05 ?

run three-phase power flow, calculate VRI (Eq. 2-14), and identify the weakest single-phase bus (exclude any buses with single-phase shunt

capacitor), Set BusCap=the weakest single-phase bus

initialize parameters, location of single-phase shunt capacitor (BusCap=0) and sizing of single-phase shunt capacitor (QCap=0)

temporary connection of a single-phase shunt capacitor at BusCap with QCap

connect DG at BusDG with PLDG

yes

noPLDG¹0 ?

connect a single-phase shunt capacitor at BusCap with QCap

yesnoQCap¹0 ?

Stage tw

o: placement and sizing of single-phase shunt capacitorsstop

increase QCap

by 1% of Qload

yes

no

Figure 7-1 The proposed algorithm for the placement and sizing of DG units and

single-phase capacitors in multiphase networks.

100

7.4 SIMULATION RESULTS

For the analysis of this chapter, the IEEE multiphase 34 node test feeder of Figure 3-

13 [33] is considered. The network has been simulated using DIgSILENT

PowerFactory software [32]. The system data are available in [33].

7.4.1 Bus voltage ranking based on proposed VRI index

Figure 7-2 shows the bus voltage ranking for the base-case load with two automatic

voltage regulators which regulate the voltages in the range of 0.95-1.05p.u. The

weakest three-phase and single-phase buses are 890 and 864, respectively.

7.4.2 Placement and sizing of DG units to improve voltage profile, grid loss, and MLF

Stage one of the proposed iterative algorithm (Figure 7-1) consists of the installation

of DFIG wind turbines.

Iteration One- A DFIG wind turbine with power factor control is installed at the

weakest three-phase bus (bus 890) through a 4.16kV/0.69kV transformer. The

size of DFIG output is gradually increased to determine its impacts on loading

factor, active power loss reduction, and voltage profile. Simulations results are

presented in Figure 7-3 indicating that active power loss is lowest (ALR =

62.31%) at a DG penetration level of 40% while the loading factor escalates as

the DG penetration increases. However, with 40% DG penetration at bus 890, as

identified by grid loss calculation, there will be a voltage violation (at bus 890, all

phases) for a DG penetration of 40% as shown in Figure 7-4. According to the

algorithm of Figure 7-1, with 30% DG penetration at bus 890, all the bus voltage

profiles are in the range of 0.95-1.05p.u. (Figure 7-5). Notice that the voltage

profile of phase c at bus 890 is 1.0499p.u., which is very close to the upper

voltage limit of 1.05p.u. Any further increase in the DG penetration level at this

bus beyond 30% will cause an overvoltage condition at bus 890. Therefore, the

maximum penetration of the first DFIG that can be installed at bus 890 is 30%

(600kW, 666.66kVA). Furthermore, the total active power loss is reduced from

0.2641MW to 0.1053MW and MLF is increased from 2.518 to 3.150. These

101

results indicate that voltage limits should be considered as a constraint for the

methods of DG placement.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

80

0

80

2

80

6

80

8

81

0

81

2

81

4

85

0

81

6

81

8

82

0

82

2

82

4

82

6

82

8

83

0

85

4

85

2

83

2

85

8

83

4

84

2

84

4

84

6

84

8

86

0

83

6

84

0

86

2

83

8

86

4

88

8

89

0

85

6

Weakest bus (single-phase)Weakest bus (three-phase)

Bus Number

VR

I

Single-phase Three-phase

Figure 7-2 Simulation results for the first DG placement (stage one, iteration one);

voltage ranking index with no DFIG installation (base-case load).

0.00

0.10

0.20

0.30

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

% DG Penetration

Lo

ad

ing

Facto

r

Acti

ve

Po

wer

Lo

ss (

MW

)

Loading FactorActive Power Loss

Figure 7-3 Simulation results for the first DG placement (stage one, iteration one);

loading factor and active power loss with different DG penetrations at bus 890.

1.085

1.058

1.031

1.004

0.977

0.950

808

810

812

814

850

816

818

824

820

822

826

858

832

852

830

828

854

856

800

864

888

890

834

842

844

846

848

860

836

840

802

862

838

806

Bus Number

Vo

lta

ge

Pro

file

[p

.u.]

Line-Ground Voltage, phase-aLine-Ground Voltage, phase-bLine-Ground Voltage, phase-c

Figure 7-4 Voltage profile with 40% DG penetration at bus 890.

102

808

810

812

814

850

816

818

824

820

822

826

858

832

852

830

828

854

856

800

864

888

890

834

842

844

846

848

860

836

840

802

862

838

806

1.050

1.030

1.010

0.990

0.970

0.950

Bus Number

Vo

lta

ge

Pro

file

[p

.u.]

Line-Ground Voltage, phase-aLine-Ground Voltage, phase-bLine-Ground Voltage, phase-c

Figure 7-5 Simulation results for the first DG placement (stage one, iteration one);

voltage profile with 30% DFIG penetration at bus 890.

Iteration Two- With 30% DFIG connected at bus 890, a similar procedure is

implemented in the second iteration to properly locate and size the second DFIG

and increase the penetration of DG units. According to Figure 7-6, the four

weakest buses are now 890, 852, 888, and 814. That is the weakest three-phase

bus is still bus 890. However, according to the results of the first iteration, the

DG penetration level is restricted at this bus due to a voltage violation at bus 890.

As a result, the most appropriate position for the second DFIG is bus 852. The

algorithm continues by increasing the size of DG while considering MLF, active

power loss (Figure 7-7) and voltage profiles (Figure 7-8). Iteration two is

terminated at a maximum DG penetration of 30% at bus 852. This will result in a

further active power loss reduction of 76.92% and MLF will be increased to

3.519.

103

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6

Weakest bus (three-phase)

Bus Number

VR

IWeakest bus (single-phase) Single-phase Three-phase

Figure 7-6 Simulation results for the second DG placement (stage one, iteration two);

voltage ranking index with 30% DFIG units installed at bus 890.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100

0.00

0.10

0.20

0.30

Lo

ad

ing

Fa

cto

r

% DG Penetration

Ac

tiv

e P

ow

er

Lo

ss

(M

W)

Loading FactorActive Power Loss

Figure 7-7 Simulation results for the second DG placement (stage one, iteration two);

loading factor and active power loss with 30% DFIG penetration at bus 890 and different

DFIG penetration at bus 852.

Iteration Three- With the two DFIGs in service at buses 890 and 852, the four

weakest three-phase buses are buses 890, 814, 888, and 848 (Figure 7-9). As

there is already a DG unit in service at bus 890, the best location for the third

DFIG connection is bus 814. However, with only 1% penetration of DG at bus

814, there will be a voltage violation at bus 808 (e.g., phase c voltage is increased

to 1.050142p.u.). The first stage of the algorithm (Figure 7-1) will be terminated

as any further DFIG connection will result in a voltage violation. Therefore,

according to the results of iterations 1-3, the maximum DG penetration can be

safely increased to 60% (30% DFIG units installed at bus 890 and 30% DFIG at bus

852) without any voltage violations.

104

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6

1.050

1.030

1.010

0.990

0.970

0.950

Bus Number

Vo

ltag

e P

rofi

le [

p.u

.]

Line-Ground Voltage, phase-aLine-Ground Voltage, phase-bLine-Ground Voltage, phase-c

Figure 7-8 Simulation results for the second DG placement (stage one, iteration two);

voltage profile with 30% DFIG penetration at bus 890 and 30% DFIG penetration at bus

852.

7.4.3 Placement and sizing of single-phase capacitor banks to further improve voltage profile, grid loss, and MLF

Stage two of the proposed algorithm (Figure 7-1) aims at further improvements in

voltage unbalanced factor (VUF), total power loss, MLF and voltage profiles through

the installation of capacitor banks in the single-phase sections of the multiphase

network.

Iteration One- The first capacitor bank is connected at the weakest single-phase

bus and its size is increased until a voltage violation is spotted. According to

Figure 7-9, the weakest single-phase location is bus 822 and the capacitor size

can be safely increased to 273kVAr while all bus voltage profiles are kept in the

range of 0.95-1.05p.u. (Figure 7-10). Note that any further increase of this

capacitor size beyond 273kVAr will cause an overvoltage condition at bus 802

(phase c). The inclusion of the two DFIGs (at busses 890 and 852) and a single-

phase capacitor (at bus 822) has increased the total active power loss from

0.0610MW to 0.0778MW while MLF is further increased to 3.575.

105

Iteration Two- The iterative procedure is repeated to install more single-phase

shunt capacitors. According to Figure 7-10, the four weakest single-phase

locations are buses 822, 820, 864 and 818. The next location for capacitor

placement is bus 820. However, installation of a 3kVAr (1% of Qload) single-

phase shunt capacitor at this bus 820 will cause an overvoltage condition at bus

802 (phase a) as shown in Figure 7-11. Therefore, the second stage of the

algorithm terminates with only one capacitor bank connected to bus 822.

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6

Weakest bus (three-phase)

Bus Number

VR

I

Weakest bus (single-phase)Single-phase Three-phase

Figure 7-9 Simulation results for the third DG placement (stage one, iteration three)

showing voltage ranking index with 30% DFIG units installed at bus 890 and 30% DFIG at

bus 852.

0

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Bus Number

VR

I

Weakest bus (single-phase)Single-phase Three-phase

Figure 7-10 Simulation results for the single-phase capacitor placement (stage two,

iteration one); voltage ranking index with 30% DG units installed at bus 890, 30% DG at bus

852, and single-phase shunt capacitor 0.273MVAr at bus 822.

106

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0.970

0.950

Bus Number

Vo

ltag

e P

rofi

le [

p.u

.]

Line-Ground Voltage, phase-aLine-Ground Voltage, phase-bLine-Ground Voltage, phase-c

Figure 7-11 Simulation results for the single-phase capacitor placement (stage two,

iteration one); voltage profile with 30% DG penetration at bus 890, 30% DG penetration at

bus 852, and single-phase 0.273MVAr shunt capacitor at bus 822.

7.4.4 Summary and analysis of simulation results

Simulation results for increasing the penetration of DFIG and single-phase capacitors

in the IEEE multiphase 34 node test feeder of Figure 3-13 based on the proposed

algorithm (Figure 7-1) are summarized and compared in Table 7-1. The impacts of

DG and capacitor installations on the performance (total active power loss, MLF and

VUF) of the multiphase network are highlighted in rows 3-6 and 9-11 of Table 7-1,

respectively. With the proposed algorithm, a total DG penetration level of 60% (30%

DFIG units installed at bus 890 and 30% DFIG at bus 852) is achieved and a 0.273MVar

shunt capacitor is placed at bus 822 without any voltage violations which will

reduced the total active power loss to 0.0778MW and increased MLF to 3.575. In

addition, the percentage of VUF at the weakest three-phase bus has been

considerably improved from 2.99 to 0.36 as shown in Figure 7-12.

107

0.00

0.50

1.00

1.50

2.00

2.50

3.00

800 802 806 808 812 814 850 816 824 828 830 854 852 832 858 834 842 844 846 848 860 836 840 862 888 890

%VUF Base-case%VUF with 30% DG penetration at bus 890%VUF with 30% DG penetration at buses 890 and 852%VUF with 30% DG penetration at buses 890 and 852and single-phase shunt capacitor at bus 822

Vo

ltag

e U

nb

ala

nce

Facto

r [%

]

Bus Number

Figure 7-12 Comparison of %VUF at different iterations of the proposed algorithm

(Figure 7-1).

7.5 CONCLUSIONS

This chapter has extended the definition of the conventional bus voltage ranking

index (VRI) of V/Vo defined for balanced three-phase systems to identify the

weakest buses of the multiphase systems. The new VRI is utilized through a

proposed iterative procedure to increase the penetration levels of DG and single-

phase capacitors in order to improve the performance of the multiphase networks.

The proposed algorithm is relatively simple and can effectively reduce total active

power loss, increase MLF and decrease VUF while keeping all bus voltages within

the allowable lower and higher limits. Main conclusions are:

The proposed bus ranking approach based on the positive-sequence voltage

ratio Vcollapse/Vno-load can effectively identify the weakest three-phase and

single-phase buses for DG and shunt capacitors placements, respectively.

Analysis of simulation results indicates that the penetration level of DG is

limited by considering the bus voltage limits rather than grid losses and/or

MLF. Therefore, at high penetration levels of DG units, it is necessary to take

voltage limits into account.

Placements of shunt capacitors at the weakest single-phase buses will not only

increase MLF, but also further improve the unbalanced voltage factor.

108

TABLE 7-1 DETAILED SOLUTION FOR DFIG AND CAPACITOR PLACEMENT AND SIZING IN THE IEEE MULTIPHASE 34 NODE TEST FEEDER (FIGURE 3-13) USING THE PROPOSED

ALGORITHM OF FIGURE 7-1.

Stage one: Placement and sizing of DFIGs

Itera

tion

Weakest

three-

phase bus

Penetration

of DFIG [%]

Total

loss

[MW]

MLF VUF at

bus 890

[%]

Simulation

results

0 - - 0.2641 2.518 2.985199 Fig. 7-12

1 890 30 0.1053 3.150 0.492043 Figs. 7-2, 7-3,

7-12

2 852 30 0.0610 3.519 0.361299 Figs. 7-6, 7-7,

7-12

3 814 - - - - Fig. 7-9

Stage two: Placement and sizing of single-phase shunt capacitors

Itera

tion

Weakest

single-

phase bus

Capacitor

size [kVAr]

Total

loss

[MW]

MLF VUF at

bus 890

[%]

Simulation

results

0 - - 0.0610 3.519 0.361299 Fig. 7-12

1 822 273 0.0778 3.575 0.356531 Figs. 7-10 and

7-12

2 820 - - - - Fig. 7-11

Final Solution: 30% DFIG penetration at bus 890, 30% DFIG penetration at bus 852

and 273kVAr capacitor at bus 822.

109

Chapter 8. Conclusions

This thesis proposes a new bus voltage ranking index (VRI) and applies it to improve

the voltage stability of unbalanced three-phase and multiphase networks. After a

literature review conducted in Chapter 1, Chapter 2 proposed a new bus ranking

approach based on positive-sequence voltage of Vcollapse/Vbase-load for unbalanced and

multiphase networks (2-14) to identify the weakest single-phase, two-phase, and

three-phase buses. Another bus ranking approach is also introduced for online

applications such as the emerging smart grids.

Chapter 3 compares the performance and accuracy of the conventional and the

proposed VRI for multiphase networks. The new index is validated using the well-

known voltage sensitivity approaches 𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃 for balanced and

unbalanced three-phase distribution networks. Further validations are performed

through grid loss calculations and generation of PV curves based on positive-

sequence voltage and voltage sensitivity methods. Detailed simulation results for the

modified IEEE multiphase 13 node and 34 node test feeders show the validity and

accuracy of the new bus ranking approach. The main outcomes of this chapter are as

follows: (1) the proposed ranking index can accurately identify the weakest single-,

two- and three-phase buses under different operating conditions without and with

voltage regulators, capacitor banks and DG units (without/with SVCs); (2) the

conventional bus voltage ranking index and the two voltage sensitivity methods

(𝜕𝑉/𝜕𝑄 and 𝜕𝑉/𝜕𝑃) are able to accurately identify the weakest buses of balanced

networks. However, in unbalanced networks, the conventional VRI and the two

voltage sensitivity methods failed to detect the weakest bus.

Chapter 4 presents the application of the proposed VRI in improving the voltage

stability and increasing the MLF of unbalanced three-phase networks. Detailed

simulation results including five case studies are presented for the modified IEEE

unbalanced three-phase 13 node test feeder. Main conclusions are: (1) the proposed

ranking index can be properly utilized to identify the weakest bus under unbalanced

110

three-phase operating conditions; (2) the proposed VRI can be used as an index to

place compensation devices at the weakest buses of unbalanced three-phase networks

to enhance the voltage stability and improve MLF; (3) placement of compensation

devices at the weakest bus may change the location of the weakest bus.

In Chapter 5, the proposed VRI in utilized to improve the voltage stability margins

and MLF of multiphase distribution networks. Extensive simulation results are

carried on for the IEEE multiphase 13 and 34 node test feeders. It is revealed that the

proposed VRI can fulfill both the static and dynamic voltage stability criteria. Static

voltage stability improvements are achieved by using the proposed VRI to identify

the weakest single-, two- and three-phase buses of multiphase distribution networks,

while the proposed index based on dynamic approach at the critical time can be used

as an indicator to identify the stability of the system.

In Chapter 6, an online bus ranking approach is proposed to identify the weakest

buses over the 24 hour period considering active and reactive daily loads curves. The

approach is used to study and compensate the detrimental impacts of PEV charging

stations on voltage profiles and voltage stability of the distribution network.

Simulations results indicate that the location of the weakest buses can be changed

over 24 hours with PEV charging stations. Then, the switching strategy of

compensation devices connected at the weakest buses according to the lowest hourly

VRI values can perform better than the conventional installations of the

compensation devices in terms of voltage profiles.

In Chapter 7, the proposed VRI is utilized to improve the performance of multiphase

distribution networks by properly increasing the penetration levels and ratings of DG

units such as DFIGs and single-phase capacitors. Simulation results show that high

penetration levels of DG units can cause overvoltage problems. Consequently, at

high penetration levels of DGs, it is necessary to also take voltage limits into

account. Therefore, an iterative algorithm for the placement and sizing of DG units

and single-phase capacitors is introduced to reduced grid losses and increase MLF in

multiphase networks while keeping all bus voltages within acceptable limits. It is

shown that the new algorithm for the placement and sizing of three-phase DG units

and single-phase capacitors can effectively reduce total active power loss, increase

111

MLF and decrease VUF of multiphase distribution networks while keeping all bus

voltages levels within the permissible limits.

8.1 CONTRIBUTIONS

The main results of this thesis have been released in five conference papers and three

journal articles (two under review) as listed in Section 1.5. The primary contributions

are as follow.

1. The proposed bus voltage ranking index can be used to identify the weakest

single-, two- and three-phase buses in multiphase networks for voltage stability

enhancement.

2. The proposed index can be applied to both static and dynamic approaches. Static

voltage stability can be improved by using the proposed VRI to identify the

weakest single-, two- and three-phase buses of multiphase distribution networks,

while the proposed index based on dynamic approach at the critical time can be

used as an indicator to identify the stability of the system.

3. The proposed index is modified for online bus voltage ranking and voltage

stability improvement in distribution systems with dynamic loads to identify the

weakest buses over the 24 hour period considering active and reactive daily loads

curves.

4. A new iterative algorithm is proposed and tested for properly placing and

increasing the penetration levels of three-phase DG units and single-phase

capacitor banks in multiphase networks to reduced grid losses, increase MLF and

decrease VUF while keeping all bus voltages within acceptable limits.

8.2 FUTURE WORKS

The following areas are suggested for future research in continuation of this work.

1. Online bus voltage ranking and control of compensation devices for voltage

stability enhancement in the emerging smart grid configurations with renewable

energy resources and dynamic loads such as PEVs and smart appliances.

112

2. Online application of the proposed iterative algorithm for the placement and

sizing of DG units and single-phase capacitors in smart grids for dynamically

increasing the penetration levels of compensation devices.

113

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Every reasonable effort has been made to acknowledge the owners of copyright

material. I would be pleased to hear from any copyright owner who has been omitted

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Station," in Proc. Power and Energy Engineering Conference (APPEEC), 2010

Asia-Pacific, pp. 1-4.

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index in a power system with Plug-in Electric Vehicles," in Proc. Bulk Power

System Dynamics and Control (iREP) - VIII (iREP), 2010 iREP Symposium, pp.

1-7.

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stability of a power system with plug-in electric vehicles," Generation,

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119

Appendix A - The IEEE 13 Node and 34 Node Test

Systems

A1-IEEE 13 Node Test Feeder Data

TABLE A1-1 OVERHEAD LINE CONFIGURATION DATA.

Configuration Phasing Phase Neutral Spacing

ACSR ACSR ID

601 B A C N 556,500 26/7 4/0 6/1 500

602 C A B N 4/0 6/1 4/0 6/1 500

603 C B N 1/0 1/0 505

604 A C N 1/0 1/0 505

605 C N 1/0 1/0 510

TABLE A1-2 UNDERGROUND LINE CONFIGURATION DATA.

Configuration Phasing Cable Neutral Space ID

606 A B C N 250,000 AA, CN None 515

607 A N 1/0 AA, TS 1/0 Cu 520

TABLE A1-3 TRANSFORMER DATA.

kVA kV-high kV-low R - % X - %

Substation 5,000 115 - D 4.16 Gr. Y 1 8

XFM -1 500 4.16 – Gr.W 0.48 – Gr.W 1.1 2

120

TABLE A1-4 LINE SEGMENT DATA.

Node A Node B Length(ft.) Configuration

632 645 500 603

632 633 500 602

633 634 0 XFM-1

645 646 300 603

650 632 2000 601

684 652 800 607

632 671 2000 601

671 684 300 604

671 680 1000 601

671 692 0 Switch

684 611 300 605

692 675 500 606

TABLE A1-5 CAPACITOR DATA.

Node Ph-A Ph-B Ph-C

kVAr kVAr kVAr

675 200 200 200

611 100

Total 200 200 300

121

TABLE A1-6 REGULATOR DATA.

Regulator ID 1

Line Segment 650 - 632

Location 50

Phases A - B -C

Connection 3-Ph,LG

Monitoring Phase A-B-C

Bandwidth 2.0 volts

PT Ratio 20

Primary CT Rating 700

Compensator Settings Ph-A Ph-B Ph-C

R - Setting 3 3 3

X - Setting 9 9 9

Voltage Level 122 122 122

TABLE A1-7 DISTRIBUTED LOAD DATA.

Node A Node B Load Ph-1 Ph-1 Ph-2 Ph-2 Ph-3 Ph-3

Model kW kVAr kW kVAr kW kVAr

632 671 Y-PQ 17 10 66 38 117 68

122

TABLE A1-8 SPOT LOAD DATA.

Node Load Ph-1 Ph-1 Ph-2 Ph-2 Ph-3 Ph-3

Model kW kVAr kW kVAr kW kVAr

634 Y-PQ 160 110 120 90 120 90

645 Y-PQ 0 0 170 125 0 0

646 D-Z 0 0 230 132 0 0

652 Y-Z 128 86 0 0 0 0

671 D-PQ 385 220 385 220 385 220

675 Y-PQ 485 190 68 60 290 212

692 D-I 0 0 0 0 170 151

611 Y-I 0 0 0 0 170 80

TOTAL 1158 606 973 627 1135 753

123

A2-IEEE 13 Node Test Feeder Impedance

For overhead line configuration, impedance matrix given below can be input directly

to DiGSILENT software. Except for underground line configuration data,

DiGSILENT requires in R0+jX0 and B0 format. Therefore, the phase impedance

matrix has to be converted to the sequence impedance matrix by the modified

Carson‟s equation (A-1) [50].

𝒁𝟎𝟏𝟐 = 𝑨 −𝟏 𝒁𝒂𝒃𝒄 𝑨 = 𝒁𝟎𝟎 𝒁𝟎𝟏 𝒁𝟎𝟐

𝒁𝟏𝟎 𝒁𝟏𝟏 𝒁𝟏𝟐

𝒁𝟐𝟎 𝒁𝟐𝟏 𝒁𝟐𝟐

(A-1)

where

𝑨 = 𝟏 𝟏 𝟏𝟏 𝒂𝟐 𝒂𝟏 𝒂 𝒂𝟐

𝑨 −𝟏 =𝟏

𝟑 𝟏 𝟏 𝟏𝟏 𝒂 𝒂𝟐

𝟏 𝒂𝟐 𝒂

For example, the phase impedance matrix of underground line configuration 606

connecting between buses 692 and 675 can be converted to the sequence impedance

matrix as shown in (A-2).

𝒁𝟎𝟏𝟐 = 𝟏. 𝟒𝟏𝟎𝟕 + 𝟎. 𝟒𝟔𝟔𝟒𝒊 −𝟎. 𝟎𝟎𝟐𝟖 − 𝟎. 𝟎𝟎𝟖𝟏𝒊 −𝟎. 𝟎𝟎𝟓𝟔 + 𝟎. 𝟎𝟎𝟔𝟓𝒊

−𝟎. 𝟎𝟎𝟓𝟔 + 𝟎. 𝟎𝟎𝟔𝟓𝒊 𝟎. 𝟒𝟖𝟕𝟒 + 𝟎. 𝟒𝟏𝟓𝟏𝒊 −𝟎. 𝟎𝟐𝟔𝟒 + 𝟎. 𝟎𝟒𝟓𝟐𝒊−𝟎. 𝟎𝟎𝟐𝟖 − 𝟎. 𝟎𝟎𝟖𝟏𝒊 𝟎. 𝟎𝟓𝟐𝟑 + 𝟎. 𝟎𝟎𝟎𝟑𝒊 𝟎. 𝟒𝟖𝟕𝟒 + 𝟎. 𝟒𝟏𝟓𝟏𝒊

(A-2)

Configuration 601:

Z (R +jX) in ohms per mile

0.3465 1.0179 0.1560 0.5017 0.1580 0.4236

0.3375 1.0478 0.1535 0.3849

0.3414 1.0348

B in micro Siemens per mile

6.2998 -1.9958 -1.2595

5.9597 -0.7417

5.6386

124

Configuration 602:

Z (R +jX) in ohms per mile

0.7526 1.1814 0.1580 0.4236 0.1560 0.5017

0.7475 1.1983 0.1535 0.3849

0.7436 1.2112

B in micro Siemens per mile

5.6990 -1.0817 -1.6905

5.1795 -0.6588

5.4246

Configuration 603:

Z (R +jX) in ohms per mile

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.3294 1.3471 0.2066 0.4591

1.3238 1.3569

B in micro Siemens per mile

0.0000 0.0000 0.0000

4.7097 -0.8999

4.6658

Configuration 604:

Z (R +jX) in ohms per mile

1.3238 1.3569 0.0000 0.0000 0.2066 0.4591

0.0000 0.0000 0.0000 0.0000

1.3294 1.3471

B in micro Siemens per mile

4.6658 0.0000 -0.8999

0.0000 0.0000

4.7097

125

Configuration 605:

Z (R +jX) in ohms per mile

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

1.3292 1.3475

B in micro Siemens per mile

0.0000 0.0000 0.0000

0.0000 0.0000

4.5193

Configuration 606:

Z (R +jX) in ohms per mile

0.7982 0.4463 0.3192 0.0328 0.2849 -0.0143

0.7891 0.4041 0.3192 0.0328

0.7982 0.4463

B in micro Siemens per mile

96.8897 0.0000 0.0000

96.8897 0.0000

96.8897

Configuration 607:

Z (R +jX) in ohms per mile

1.3425 0.5124 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000

B in micro Siemens per mile

88.9912 0.0000 0.0000

0.0000 0.0000

0.0000

126

A3-IEEE 34 Node Test Feeder Data

TABLE A3-1 LINE SEGMENT DATA.

Node A Node B Length(ft.) Configuration 800 802 2580 300 802 806 1730 300 806 808 32230 300 808 810 5804 303 808 812 37500 300 812 814 29730 300 814 850 10 301 816 818 1710 302 816 824 10210 301 818 820 48150 302 820 822 13740 302 824 826 3030 303 824 828 840 301 828 830 20440 301 830 854 520 301 832 858 4900 301 832 888 0 XFM-1 834 860 2020 301 834 842 280 301 836 840 860 301 836 862 280 301 842 844 1350 301 844 846 3640 301 846 848 530 301 850 816 310 301 852 832 10 301 854 856 23330 303 854 852 36830 301 858 864 1620 303 858 834 5830 301 860 836 2680 301 862 838 4860 304 888 890 10560 300

127

TABLE A3-2 OVERHEAD LINE CONFIGURATION.

Configuration Phasing Phase Neutral Spacing ID

ACSR ACSR

300 B A C N 1/0 1/0 500

301 B A C N #2 6/1 #2 6/1 500

302 A N #4 6/1 #4 6/1 510

303 B N #4 6/1 #4 6/1 510

304 B N #2 6/1 #2 6/1 510

TABLE A3-3 TRANSFORMER DATA.

kVA kV-high kV-low R [%] X [%]

Substation 2500 69 - D 24.9 -Gr. W 1 8

XFM -1 500 24.9 - Gr.W 4.16 - Gr. W 1.9 4.08

TABLE A3-4 SPOT LOADS.

Model kW kVAr kW kVAr kW kVAr

860 Y-PQ 20 16 20 16 20 16

840 Y-I 9 7 9 7 9 7

844 Y-Z 135 105 135 105 135 105

848 D-PQ 20 16 20 16 20 16

890 D-I 150 75 150 75 150 75

830 D-Z 10 5 10 5 25 10

Total 344 224 344 224 359 229

128

TABLE A3-5 SHUNT CAPACITORS.

Node Ph-A Ph-B Ph-C

kVAr kVAr kVAr

844 100 100 100

848 150 150 150

Total 250 250 250

129

TABLE A3-6 DISTRIBUTED LOADS.

Node Node Load Ph-1 Ph-1 Ph-2 Ph-2 Ph-3 Ph-3

A B Model kW kVAr kW kVAr kW kVAr

802 806 Y-PQ 0 0 30 15 25 14

808 810 Y-I 0 0 16 8 0 0

818 820 Y-Z 34 17 0 0 0 0

820 822 Y-PQ 135 70 0 0 0 0

816 824 D-I 0 0 5 2 0 0

824 826 Y-I 0 0 40 20 0 0

824 828 Y-PQ 0 0 0 0 4 2

828 830 Y-PQ 7 3 0 0 0 0

854 856 Y-PQ 0 0 4 2 0 0

832 858 D-Z 7 3 2 1 6 3

858 864 Y-PQ 2 1 0 0 0 0

858 834 D-PQ 4 2 15 8 13 7

834 860 D-Z 16 8 20 10 110 55

860 836 D-PQ 30 15 10 6 42 22

836 840 D-I 18 9 22 11 0 0

862 838 Y-PQ 0 0 28 14 0 0

842 844 Y-PQ 9 5 0 0 0 0

844 846 Y-PQ 0 0 25 12 20 11

846 848 Y-PQ 0 0 23 11 0 0

Total 262 133 240 120 220 114

130

TABLE A3-7 REGULATOR DATA.

Regulator ID 1

Line Segment 814 - 850

Location 814

Phases A - B -C

Connection 3-Ph,LG

Monitoring Phase A-B-C

Bandwidth 2.0 volts

PT Ratio 120

Primary CT Rating 100

Compensator Settings Ph-A Ph-B Ph-C

R - Setting 2.7 2.7 2.7

X - Setting 1.6 1.6 1.6

Voltage Level 122 122 122

Regulator ID 2

Line Segment 852 - 832

Location 852

Phases A - B -C

Connection 3-Ph,LG

Monitoring Phase A-B-C

Bandwidth 2.0 volts

PT Ratio 120

Primary CT Rating 100

Compensator Settings Ph-A Ph-B Ph-C

R - Setting 2.5 2.5 2.5

X - Setting 1.5 1.5 1.5

Voltage Level 124 124 124

131

A4-IEEE 34 Node Test Feeder Impedance

Configuration 300

Z (R +jX) in ohms per mile

1.3368 1.3343 0.2101 0.5779 0.2130 0.5015

1.3238 1.3569 0.2066 0.4591

1.3294 1.3471

B in micro Siemens per mile

5.3350 -1.5313 -0.9943

5.0979 -0.6212

4.8880

Configuration 301

Z (R +jX) in ohms per mile

1.9300 1.4115 0.2327 0.6442 0.2359 0.5691

1.9157 1.4281 0.2288 0.5238

1.9219 1.4209

B in micro Siemens per mile

5.1207 -1.4364 -0.9402

4.9055 -0.5951

4.7154

Configuration 302

Z (R +jX) in ohms per mile

2.7995 1.4855 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000

0.0000 0.0000

B in micro Siemens per mile

4.2251 0.0000 0.0000

0.0000 0.0000

0.0000

132

Configuration 303

Z (R +jX) in ohms per mile

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

2.7995 1.4855 0.0000 0.0000

0.0000 0.0000

B in micro Siemens per mile

0.0000 0.0000 0.0000

4.2251 0.0000

0.0000

Configuration 304

Z (R +jX) in ohms per mile

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

1.9217 1.4212 0.0000 0.0000

0.0000 0.0000

B in micro Siemens per mile

0.0000 0.0000 0.0000

4.3637 0.0000

0.0000

133

A5- The Modified Unbalanced Three-Phase 13 Node Test Feeder Data

TABLE A5-1 OVERHEAD LINE CONFIGURATION DATA.

Configuration Phasing Phase Neutral Spacing

ACSR ACSR ID

601 B A C N 556,500 26/7 4/0 6/1 500

602 C A B N 4/0 6/1 4/0 6/1 500

TABLE A5-2 UNDERGROUND LINE CONFIGURATION DATA.

Configuration Phasing Cable Neutral Space ID

606 A B C N 250,000 AA, CN None 515

TABLE A5-3 TRANSFORMER DATA.

kVA kV-high kV-low R [%] X [%]

Substation 5,000 115 - D 4.16 Gr. Y 1 8

XFM -1 500 4.16 – Gr.W 0.48 – Gr.W 1.1 2

134

TABLE A5-4 LINE SEGMENT DATA.

Node A Node B Length(ft.) Configuration

632 645 500 602

632 633 500 602

633 634 0 XFM-1

645 646 300 602

650 632 2000 601

684 652 800 606

632 671 2000 601

671 684 300 601

671 680 1000 601

671 692 0 Switch

684 611 300 601

692 675 500 606

TABLE A5-5 CAPACITOR DATA.

Node Ph-A Ph-B Ph-C

kVAr kVAr kVAr

675 200 200 200

611 100

Total 200 200 300

135

TABLE A5-6 REGULATOR DATA.

Regulator ID 1

Line Segment 650 - 632

Location 50

Phases A - B -C

Connection 3-Ph,LG

Monitoring Phase A-B-C

Bandwidth 2.0 volts

PT Ratio 20

Primary CT Rating 700

Compensator Settings Ph-A Ph-B Ph-C

R - Setting 3 3 3

X - Setting 9 9 9

Voltage Level 122 122 122

TABLE A5-7 DISTRIBUTED LOAD DATA.

Node A Node B Load Ph-1 Ph-1 Ph-2 Ph-2 Ph-3 Ph-3

Model kW kVAr kW kVAr kW kVAr

632 671 Y-PQ 17 10 66 38 117 68

136

TABLE A5-8 SPOT LOAD DATA.

Node Load Ph-1 Ph-1 Ph-2 Ph-2 Ph-3 Ph-3

Model kW kVAr kW kVAr kW kVAr

634 Y-PQ 160 110 120 90 120 90

645 Y-PQ 0 0 170 125 0 0

646 D-Z 0 0 230 132 0 0

652 Y-Z 128 86 0 0 0 0

671 D-PQ 385 220 385 220 385 220

675 Y-PQ 485 190 68 60 290 212

692 D-I 0 0 0 0 170 151

611 Y-I 0 0 0 0 170 80

TOTAL 1158 606 973 627 1135 753

137

A6- The Modified Unbalanced Three-Phase 13 Node Test Feeder Impedance

Configuration 601:

Z (R +jX) in ohms per mile

0.3465 1.0179 0.1560 0.5017 0.1580 0.4236

0.3375 1.0478 0.1535 0.3849

0.3414 1.0348

B in micro Siemens per mile

6.2998 -1.9958 -1.2595

5.9597 -0.7417

5.6386

Configuration 602:

Z (R +jX) in ohms per mile

0.7526 1.1814 0.1580 0.4236 0.1560 0.5017

0.7475 1.1983 0.1535 0.3849

0.7436 1.2112

B in micro Siemens per mile

5.6990 -1.0817 -1.6905

5.1795 -0.6588

5.4246

Configuration 606:

Z (R +jX) in ohms per mile

0.7982 0.4463 0.3192 0.0328 0.2849 -0.0143

0.7891 0.4041 0.3192 0.0328

0.7982 0.4463

B in micro Siemens per mile

96.8897 0.0000 0.0000

96.8897 0.0000

96.8897

138

Appendix B – Simulation parameters

TABLE B1 SIMULATION PARAMETERS FOR THE INDUCTION GENERATOR (400 KW).

Generator Parameters

Nominal voltage 4.16 kV

Power factor 0.9 lagging

Nominal apparent power 475 kVA

Rated mechanical power 400 kW

Nominal frequency 50 Hz

No of pole pairs 1

Connection D

TABLE B2 SIMULATION PARAMETERS FOR THE INDUCTION GENERATOR (200 KW).

Generator Parameters

Nominal voltage 0.69 kV

Power factor 0.92 lagging

Nominal apparent power 226.92 kVA

Rated mechanical power 200 kW

Nominal frequency 50 Hz

No of pole pairs 1

Connection Y

139

TABLE B3 SIMULATION PARAMETERS FOR THE DFIG WIND TURBINE.

Generator Parameters

Nominal voltage 0.69 kV

Power factor 0.92 lagging

Nominal apparent power 226.92 kVA

Rated mechanical power 200 kW

Nominal frequency 50 Hz

No of pole pairs 1

Connection Y

Efficiency at nominal operation 95.8 %

Nominal speed 2980 rpm

Zero-sequence resistance 0.01 p.u.

Zero-sequence reactance 0.1 p.u.

140

Appendix C –DIgSILENT PowerFactory [32]

DIgSILENT is a computer aided engineering tool for the analysis of industrial,

utility, and commercial electrical power systems. It has been designed as an

advanced integrated and interactive software package dedicated to electrical power

system and control analysis in order to achieve the main objectives of planning and

operation optimization. DIgSILENT was the world's first power system analysis

software with an integrated graphical one-line interface. That interactive one-line

diagram included drawing functions, editing capabilities and all relevant static and

dynamic calculation features. The accuracy and validity of the results obtained with

this package has been confirmed in a large number of implementations, by

organizations involved in planning and operation of power systems.

141

Appendix D – Elixir Journal: Bus Voltage Ranking for

Unbalanced Three-phase Distribution Networks and

Voltage Stability Enhancement

142

143

144

145

146

147