development of probabilistic load flow for voltage...

DEVELOPMENT OF PROBABILISTIC LOAD FLOW FOR

VOLTAGE QUALITY ANALYSIS IN THE PRESENCE OF

DISTRIBUTED GENERATION

by

Jason M. Sexauer

c© Copyright by Jason M. Sexauer, 2012

All Rights Reserved

A thesis submitted to the Faculty and the Board of Trustees of the Colorado

School of Mines in partial fulfillment of the requirements for the degree of Master of

Science (Electrical Engineering).

Golden, Colorado

Date

Signed:Jason M. Sexauer

Signed:Dr. Salman Mohagheghi

Thesis Advisor

Golden, Colorado

Date

Signed:Dr. Tyrone Vincent

Associate Professor and Interim HeadDepartment of Electrical Engineering and Computer Science

ii

ABSTRACT

There has been interest in the integration of renewable distributed generation

(DG) into the electric system. However, DG pose some significant but relatively un-

quantified risks to maintaining acceptable voltage quality, specifically voltage limits,

voltage imbalance, and flicker. The purpose of this thesis is to quantify those risks.

To do this, a probabilistic load flow (PLF) is employed which accounts for the

variations caused by DG. In order to capture how these variations affect flicker, a

PLF must be performed at a small time resolution (one second) in a quasi-steady-

state environment, which present several challenges to the stochastic models. As

such, hybrid models are developed which include a “trend randomness” component

in the form of standard statistical distributions, and a “time-correlated randomness”

component in the form of various time series approaches.

In this thesis, first an introduction to the problem is given, including motivation,

proposed solution methodology, and novelty. Next a literature survey is performed to

determine the state of the art in DG integration issues and PLF methodologies. The

nature of the statistical models used to create the load, wind speed, and irradiance

values for the PLF are then presented, followed by an overview of how the PLF is

implemented. Finally, the results of over a dozen scenarios performed on a test

system in Greeley, CO are presented and conclusions drawn.

Model results indicate that DG have strong effects on voltage regulation devices

and the interaction between the two can lead to voltage quality issues. However,

should DG be properly sized and allowed to provide VAR support, their ability to

affect the voltage profile for the better is profound. The results also show that, while

flicker levels do increase as more DG is integrated into the system, it is unlikely flicker

will reach intolerable levels even at high penetrations.

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TABLE OF CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

LIST OF SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

LIST OF ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii

DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xix

CHAPTER 1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Solution Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.3 Novelty of Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

CHAPTER 2 STATE OF THE ART: EFFECTS OF DISTRIBUTEDGENERATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1 Power Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Electricity Availability . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.2 Harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.1.3 Voltage Dips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.4 Voltage Flicker . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.1.5 Voltage Imbalance . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Protection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

iv

2.2.1 Adaptive Protection . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.2 Standing Phase Angle . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.3 Directional Overcurrent . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.4 Microgrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.5 Islanded Operations . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.6 Adaptive Protection . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Distribution System Stability . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.1 Distribution State Estimation . . . . . . . . . . . . . . . . . . . 12

2.4.2 Dynamic State Estimation . . . . . . . . . . . . . . . . . . . . . 13

2.4.3 Harmonic State Estimation . . . . . . . . . . . . . . . . . . . . 13

2.5 Voltage/Var/Watt Control . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Electric Vehicle Considerations . . . . . . . . . . . . . . . . . . . . . . 15

CHAPTER 3 STATE OF THE ART: PROBABILISTIC LOAD FLOW . . . . 16

3.1 Key Concepts of the Probabilistic Load Flow . . . . . . . . . . . . . . . 16

3.2 Paramater Considerations . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

CHAPTER 4 RESOURCE STOCHASTIC MODELS . . . . . . . . . . . . . . 22

4.1 Residential Load Model . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.1 Load Data Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1.3 Generation of Load for PLF . . . . . . . . . . . . . . . . . . . . 26

v

4.1.4 Load Unbalance . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.5 DG Deployment Random Variable . . . . . . . . . . . . . . . . . 26

4.2 Commercial and Industrial Load Model . . . . . . . . . . . . . . . . . . 27

4.3 Wind Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.3.1 Wind Data Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3.3 Generation of Wind Speed for PLF . . . . . . . . . . . . . . . . 33

4.3.4 Fidelity of Wind Model . . . . . . . . . . . . . . . . . . . . . . . 34

4.4 Solar Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.1 Solar Data Basis . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4.2 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.4.3 Generation of Irradiance for PLF . . . . . . . . . . . . . . . . . 41

4.5 Performance of Random Variables . . . . . . . . . . . . . . . . . . . . . 43

4.6 Random Variable Limitations . . . . . . . . . . . . . . . . . . . . . . . 48

CHAPTER 5 PROBABILISTIC LOAD FLOW ENGINE . . . . . . . . . . . . 49

5.1 Key Functions and Variables . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Distribution System Model . . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.1 Topology Description . . . . . . . . . . . . . . . . . . . . . . . . 51

5.2.2 Demographic Data . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Wind Turbine Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.3.1 Wind Turbine and Controller Model for Energy Capture . . . . 57

5.3.2 Development of Coefficients . . . . . . . . . . . . . . . . . . . . 58

5.4 Photovoltaic Array Model . . . . . . . . . . . . . . . . . . . . . . . . . 61

vi

5.5 Flicker Meter Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

CHAPTER 6 RESULTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6.1 Tests Performed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2 Monitoring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

6.3 Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.3.1 Voltage Considerations . . . . . . . . . . . . . . . . . . . . . . . 72

6.3.2 Flicker Considerations . . . . . . . . . . . . . . . . . . . . . . . 72

6.4 Effects of DG Penetration on Baseline . . . . . . . . . . . . . . . . . . 73



6.5 Capacity Deferment due to DG . . . . . . . . . . . . . . . . . . . . . . 75



6.6 Effects of DG Participation in Voltage Regulation . . . . . . . . . . . . 76



6.7 Effects of Season on DG . . . . . . . . . . . . . . . . . . . . . . . . . . 79



6.8 Extreme DG Integration Scenario . . . . . . . . . . . . . . . . . . . . . 81



CHAPTER 7 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

vii

7.1 Quantification of Voltage Quality Risks . . . . . . . . . . . . . . . . . . 84

7.2 Generalized Effects of DG . . . . . . . . . . . . . . . . . . . . . . . . . 88

7.3 Questions Remaining and Future Work . . . . . . . . . . . . . . . . . . 90

REFERENCES CITED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

APPENDIX A - DETAILED RESULT FIGURES . . . . . . . . . . . . . . . . . 99

A.1 Baseline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

A.2 Effects of DG Penetration on Baseline . . . . . . . . . . . . . . . . . 106

A.3 Capacity Deferment due to DG . . . . . . . . . . . . . . . . . . . . . 110

A.4 Effects of DG Participation in Voltage Regulation . . . . . . . . . . . 114

A.5 Effects of Season on DG . . . . . . . . . . . . . . . . . . . . . . . . . 118

A.6 Extreme DG Integration Scenario . . . . . . . . . . . . . . . . . . . . 122

viii

LIST OF FIGURES

Figure 4.1 Normal Distribution Parameters for Time of Day and Temperature 25

Figure 4.2 Variation of Wind Speed Weibull Coefficients Throughout theDay in Spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

Figure 4.3 Variation of Sub-Minutely Variance Weibull CoefficientsThroughout the Day in Spring . . . . . . . . . . . . . . . . . . . . . 31

Figure 4.4 Statistical Quantities for d′(t) . . . . . . . . . . . . . . . . . . . . . 32

Figure 4.5 Autocorrelations at 5000 Lags for Spring . . . . . . . . . . . . . . . 32

Figure 4.6 Autocorrelation of d′(t) with Fitted Model . . . . . . . . . . . . . . 33

Figure 4.7 Histograms for Simulated and Observed Wind Speeds . . . . . . . . 35

Figure 4.8 Error Statistical Quantities . . . . . . . . . . . . . . . . . . . . . . . 35

Figure 4.9 Cloud Event Detector on a Cloud Day . . . . . . . . . . . . . . . . 38

Figure 4.10 Statistics for Cloud Events in the Spring . . . . . . . . . . . . . . . 40

Figure 4.11 Typical Day: One Trial in Test 4 . . . . . . . . . . . . . . . . . . . 46

Figure 4.12 Wind Day: One Trial in Test 9 . . . . . . . . . . . . . . . . . . . . 46

Figure 4.13 Cloudy Day: One Trial in Test 5 . . . . . . . . . . . . . . . . . . . . 47

Figure 4.14 Calm, Clear Day: One Trial in Test 7 . . . . . . . . . . . . . . . . . 47

Figure 5.1 Block Diagram of Probabilistic Load Flow Engine . . . . . . . . . . 50

Figure 5.2 Oneline Diagram of Greeley (GRLY) Substation . . . . . . . . . . . 52

Figure 5.3 Map of Feeders in Test System . . . . . . . . . . . . . . . . . . . . . 55

Figure 5.4 Map of Load Regions in Test System . . . . . . . . . . . . . . . . . 56

Figure 5.5 Block Diagram of Wind Turbine and Controller Model . . . . . . . 57

ix

Figure 5.6 Performance Curves of Model and Skystream 3.7 . . . . . . . . . . . 60

Figure 5.7 Wind Turbine Operation in 10 Minutes of Turbulent Wind . . . . . 60

Figure 5.8 Block Diagram of the PVSystem Element of OpenDSS . . . . . . . 61

Figure 5.9 Block Diagram of Flicker Meter . . . . . . . . . . . . . . . . . . . . 64

Figure 6.1 Simplified Diagram of Test System Feeders . . . . . . . . . . . . . . 71

Figure 6.2 Voltages, Powers, and Resource Values for VAR Support Extremes . 78

Figure 6.3 Voltages, Powers, and Resource Values for Summer and Winter . . . 80

Figure 6.4 One Trial of the Extreme DG Scenario . . . . . . . . . . . . . . . . 82

Figure A.1 Baseline Voltage PDF . . . . . . . . . . . . . . . . . . . . . . . . . 102

Figure A.2 Baseline Imbalance PDF . . . . . . . . . . . . . . . . . . . . . . . 103

Figure A.3 Baseline Flicker CDF . . . . . . . . . . . . . . . . . . . . . . . . . 104

Figure A.4 Baseline Flicker PDF . . . . . . . . . . . . . . . . . . . . . . . . . 105

Figure A.5 DG Impact Scenarios’ Voltage PDF . . . . . . . . . . . . . . . . . 106

Figure A.6 DG Impact Scenarios’ Imbalance PDF . . . . . . . . . . . . . . . 107

Figure A.7 DG Impact Scenarios’ Flicker CDF . . . . . . . . . . . . . . . . . 108

Figure A.8 DG Impact Scenarios’ Flicker PDF . . . . . . . . . . . . . . . . . 109

Figure A.9 Weak Scenarios’ Voltage PDF . . . . . . . . . . . . . . . . . . . . 110

Figure A.10Weak Scenarios’ Imbalance PDF . . . . . . . . . . . . . . . . . . . 111

Figure A.11Weak Scenarios’ Flicker CDF . . . . . . . . . . . . . . . . . . . . 112

Figure A.12Weak Scenarios’ Flicker PDF . . . . . . . . . . . . . . . . . . . . . 113

Figure A.13DG VAR Support Scenarios’ Voltage PDF . . . . . . . . . . . . . 114

Figure A.14DG VAR Support Scenarios’ Imbalance PDF . . . . . . . . . . . . 115

Figure A.15DG VAR Support Scenarios’ Flicker CDF . . . . . . . . . . . . . . 116

x

Figure A.16DG VAR Support Scenarios’ Flicker PDF . . . . . . . . . . . . . . 117

Figure A.17Seasonal Scenarios’ Voltage PDF . . . . . . . . . . . . . . . . . . 118

Figure A.18Seasonal Scenarios’ Imbalance PDF . . . . . . . . . . . . . . . . . 119

Figure A.19Seasonal Scenarios’ Flicker CDF . . . . . . . . . . . . . . . . . . . 120

Figure A.20Seasonal Scenarios’ Flicker PDF . . . . . . . . . . . . . . . . . . . 121

Figure A.21Extreme Scenario’s Voltage PDF . . . . . . . . . . . . . . . . . . 122

Figure A.22Extreme Scenario’s Imbalance PDF . . . . . . . . . . . . . . . . . 123

Figure A.23Extreme Scenario’s Flicker CDF . . . . . . . . . . . . . . . . . . . 124

Figure A.24Extreme Scenario’s Flicker PDF . . . . . . . . . . . . . . . . . . . 125

xi

LIST OF TABLES

Table 4.1 Cloud Event Statistics for all Seasons . . . . . . . . . . . . . . . . . 39

Table 4.2 Long Term “Trend Randomness” and Short Term“Time-Correlated Randomness” Methods Used for Each Resource . 44

Table 6.1 Test Scenarios . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

Table 6.2 Test Batteries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Table 6.3 Conductor Sizes on Test Systems . . . . . . . . . . . . . . . . . . . 75

Table 7.1 Voltage Quality Metrics Across all Feeders for Tests Run . . . . . . 85

xii

LIST OF SYMBOLS

Secondly Load RV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L1s

Secondly Wind Speed RV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . W1s

Secondly Irradiance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I1s

Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N

Mean (Normal Distribution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µ

Variance (Normal Distribution) . . . . . . . . . . . . . . . . . . . . . . . . . . . σ2

Discrete Time at Inferred Resolution . . . . . . . . . . . . . . . . . . . . . . . . . t

Residential Load at 15 minute Resolution RV . . . . . . . . . . . . . . . . . LR,15m

Residential Load at 1 minute Resolution RV . . . . . . . . . . . . . . . . . . LR,1m

Residential Load at 1 minute Resolution Realization . . . . . . . . . . . . . . lR,1m

Residential Load at 1 second Resolution RV . . . . . . . . . . . . . . . . . . . LR,1s

Percent Load Distributed on phase x RV . . . . . . . . . . . . . . . . . . . . . . Ux

Percent Load Distributed on phase x Realization . . . . . . . . . . . . . . . . . ux

DG Deployment RV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DDG

DG Deployment Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . dDG

DG Penetration level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . qDG

DG Capacity RV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . SDG

DG Capacity Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sDG

Commercial Load at 15 minute Resolution RV . . . . . . . . . . . . . . . . . LC,15m

Commercial Load at 1 minute Resolution RV . . . . . . . . . . . . . . . . . LC,1m

xiii

Commercial Load at 1 minute Resolution Realization . . . . . . . . . . . . . . lC,1m

Commercial Load at 1 second Resolution RV . . . . . . . . . . . . . . . . . . . LC,1s

Industrial Load at 1 second Resolution RV . . . . . . . . . . . . . . . . . . . . LI,1s

Weibull Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Wbl

Wind Speed (Weibull Component) at 1 minute Resolution RV . . . . . . . . W1m,W

Wind Speed (Weibull Component) at 1 minute Resolution Realization . . . w1m,W

Scale Parameter for Wind Speed RV . . . . . . . . . . . . . . . . . . . . . . . . . a

Shape Parameter for Wind Speed RV . . . . . . . . . . . . . . . . . . . . . . . . . b

Wind Speed Random Walk at 1 minute Resolution RV . . . . . . . . . . . . W1m,N

Wind Speed Random Walk at 1 minute Resolution Realization . . . . . . . . w1m,N

Scale Parameter for Wind Speed Sub-minutely Variance RV . . . . . . . . . . . av

Shape Parameter for Wind Speed Sub-minutely Variance RV . . . . . . . . . . . bv

Wind Data Time Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . d(t)

Wind Data Time Series Less Weibull Component . . . . . . . . . . . . . . . . d′(t)

Simulated Wind Data Time Series Less Weibull Component . . . . . . . . . . d′(t)

Simulated Wind Data Time Series (at 1 minute Resolution) . . . . . . . . . . d(t)

Simulated Wind Data Time Series at 1 minute Resolution . . . . . . . . . . . w1m

Secondly Wind Speed Realization . . . . . . . . . . . . . . . . . . . . . . . . . w1s

Poisson Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poiss

Mean (Poisson Distribution) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . λ

Exponential Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . exp

Mean (Exponential Distribution) . . . . . . . . . . . . . . . . . . . . . . . . . . . µ

Number of Cloud Events Per Day RV . . . . . . . . . . . . . . . . . . . . . . . . NE

xiv

Number of Cloud Events Per Day Realization . . . . . . . . . . . . . . . . . . . nE

Inter-event Waiting Time RV . . . . . . . . . . . . . . . . . . . . . . . . . . . . TW

Inter-event Waiting Time, kth Realization . . . . . . . . . . . . . . . . . . . . tW,k

Waiting Time to First Event . . . . . . . . . . . . . . . . . . . . . . . . . . . . tW,0

Mean Waiting Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µW

Event Duration RV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . TD

Event Duration, kth Realization . . . . . . . . . . . . . . . . . . . . . . . . . . tD,k

Mean Event Duration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . µD

Number of ARMA Models . . . . . . . . . . . . . . . . . . . . . . . . . . . nARMA

Irradiance at 1 minute resolution RV . . . . . . . . . . . . . . . . . . . . . . . I1m

Inertia of Wind Turbine Rotor, Gearbox, and Generator . . . . . . . . . . . . . J

Rotational Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ω

Aerodynamic Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . τaero

Generator (or Control) Torque . . . . . . . . . . . . . . . . . . . . . . . . . . . τc

Air Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ρ

Blade Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R

Coefficient of Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cp

Tip Speed Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . λ

Design Tip Speed Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . λ∗

Wind Speed Observed by Wind Turbine . . . . . . . . . . . . . . . . . . . . . . . u

Blade Pitch Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . β

Short Term Flicker Value . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pst

xv

LIST OF ABBREVIATIONS

Autoregressive Moving-Average . . . . . . . . . . . . . . . . . . . . . . . . ARMA

Component Object Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . COM

Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . CDF

Deterministic Load Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DLF

Distributed Energy Resource . . . . . . . . . . . . . . . . . . . . . . . . . . . DER

Distributed Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . DG

Eaton Substation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EATO

Electric Power Research Institute . . . . . . . . . . . . . . . . . . . . . . . . EPRI

Electric Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . EV

Flexible AC Transmission System . . . . . . . . . . . . . . . . . . . . . . . FACTS

Greeley Substation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GRLY

Intelligent Electronic Device . . . . . . . . . . . . . . . . . . . . . . . . . . . . IED

Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MC

National Renewable Energy Lab . . . . . . . . . . . . . . . . . . . . . . . . . NREL

Over-Ambient Flicker . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . OAF

Phasor Measurement Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . PMU

Photovoltaic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PV

Plug-in Hybrid Electric Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . PHEV

Power Quality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PQ

Probabilistic Load Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PLF

xvi

Probability Density Function . . . . . . . . . . . . . . . . . . . . . . . . . . . PDF

Probability Mass Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . PMF

Random Variable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . RV

University of Northern Colorado . . . . . . . . . . . . . . . . . . . . . . . . UNC

Vehicle to Grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . V2G

Volt-Amperes Reactive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . VAR

Volt/VAR/Watt Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . VVWC

xvii

ACKNOWLEDGMENTS

Above all, I would like to thank my advisor Dr. Salman Mohagheghi for his

incalculable assistance in the creation of this work. I would also like to extend thanks

to my thesis committee which includes Dr. Ravel Ammerman and Dr. P.K. Sen.

It is with their constant support and advice that I was able to complete this work.

Finally, I would like to thank the faculty of the Colorado School of Mines for their

instruction; specifically, I extend special thanks to Dr. Mandy Hering for her consult

on the statistical models for wind, Dr. Katie Johnson for her advise on the wind

turbine model, and Dr. Tyrone Vincent for his instruction on spectral factorization.

With these people’s assistance, I was able to accomplish the breadth and depth of

this work.

I am also grateful for my experience as an intern at Xcel Energy, which has

allowed me to make this work as realistic as possible. I would like to expressly thank

my manager, Kelly Bloch, for authorizing the use of the SmartGridCity load data

and Molly Dimond for proofing this document.

I would like to thank my parents, Jane and Walter Sexauer, whose constant sup-

port and encouragement throughout my life has culminated in the production of this

work. My father also graciously provided for the large computing capacity required to

run the simulations. Thank you for always pushing me to pursue my dreams; without

them I would not be where I am today.

I am also in debt to my friends for their poise and strength in which I could take

refuge in moments of uncertainty. Among others, I would like to specifically name

Ashley Bell, Katie Frenis, Jerad Hughes, Chris Lorenzini, Kerry McBee, Brady Nunn,

Nic Shugart, and Daniel Suhr. Thank you for continuously encouraging me to be a

better me.

xviii

The forceps of our minds are clumsy forceps, and

crush the truth a little in taking

hold of it.

— H.G. Wells

xix

CHAPTER 1

INTRODUCTION

The work presented in this thesis is the culmination of over a year of research

into the impacts of distributed generation on the distribution system, probabilistic

load flow methodologies, and load, wind, and solar statistical modeling techniques.

The goal of this research is to understand how the random variations in resources

on the distribution system (that is, load, wind, and solar) affect voltage quality with

special emphasis on flicker. The results indicate that DG have significant effects on

voltage regulation devices and the interaction between the two can lead to voltage

quality issues. However, when DG is properly sized and allowed to provide VAR

support, their ability to affect the voltage profile for the better is profound, especially

in reducing the impacts DG may cause in reduced capacity reinforcement. Finally,

while flicker levels do increase as more DG is integrated into the system, it is unlikely

flicker will reach intolerable levels even at high penetrations.

1.1 Motivation

There has been an interest within the utility sector and the public at large to

integrate a potentially large number of distributed generation (DG) sources into the

electric grid. Because of their small and dispersed nature, this generation is likely

to be interconnected at the distribution level. This poses several challenges to the

existing distribution system, which was never designed for such power flows. One

of the problems associated with these power flows, due the intermittent nature of

renewable DG, is reduced voltage quality. The purpose of this thesis is to investigate

and quantify the risk to voltage quality associated with DG.

1

1.2 Solution Methodology

To quantify this risk, probabilistic load flow (PLF) techniques are employed. A

probabilistic load flow captures the variations in generation and load due to the

random processes which drive them. This is in contrast to a standard load flow,

or deterministic load flow (DLF), which assumes values for generation and load.

Currently, the most common way to assess the impacts of DG is to run a DLF at

the extremes (ie, high load, low generation and low load and high generation – both

of which are improbable events). However, a PLF adds to this analysis how often a

specific voltage condition is likely to exist on the system and allows system planners

to decide if the risk is acceptable.

1.3 Novelty of Research

One of the most common uses for PLF is the analysis of the voltage limits seen

at a given bus and how often they are exceeded. Voltage limits are only scratching

the surface of voltage quality issues. Two other aspects of voltage quality, flicker and

voltage imbalance, will also be addressed in this thesis; flicker specifically has never

– to the author’s knowledge – been explored using PLF techniques. The use of PLF

is extremely relevant as flicker analysis includes a time and magnitude component

which is virtually impossible to capture in a deterministic analysis. Likewise, because

DG tend to interconnect without the system being specifically planned around them,

voltage imbalance are likely to creep in; however, a deterministic analysis is unable

to capture to what extent voltage imbalance is likely to manifest itself.

The analysis of these voltage quality issues, specifically flicker, require a level of

detail also not employed in any PLF to the author’s knowledge. This is because

flicker is a relatively short term phenomena; flicker issues caused by DG are likely

to be observed in the one second to a few minute time frame. This means a PLF

needs to be run at a one second interval, which brings the problem into the quasi-

2

steady-state realm. A time frame of this granularity has made this work challenging

because, while some common assumptions of load flow still hold true, many others

need to be challenged because of the dynamic nature imposed by this time frame.

These challenges include incorporating time-correlation and time of day trends in

the random variables, modeling regulator and capacitor controls, and including the

dynamics of wind turbines.

3

CHAPTER 2

STATE OF THE ART: EFFECTS OF DISTRIBUTED GENERATION

Distributed generation (DG) alter the fundamental nature in which the distri-

bution system operates. With the notable exception of secondary network systems,

most distribution systems are designed to be operated radially, that is with a single

source (the distribution substation) feeding loads via a well understood feed-path.

This design allows several simplifications to the electric system (compared to a mesh

feed transmission system) such as simplified protection and reduced monitoring needs.

With the advent of large levels of DG, several of these aspects require revaluation.

2.1 Power Quality

Distributed generation is, from a power quality perspective, the introduction of a

whole new type of “load” to the distribution system [1]. Like all loads, several power

quality (PQ) phenomena are improved and hindered by the wide-scale implementation

of DG on the power system [1].

2.1.1 Electricity Availability

Historically, the principle use of DG is as a means to increase on-site energy

reliability. In industries where power availability is highly desirable, DG can be used

to augment grid power during fault and outage events, greatly adding to the reliability

seen by the customer [2]. Also, one of the greatest advantages to a microgrid system

is the use of DG to increase reliability.

2.1.2 Harmonics

Harmonics are the presence of non-60 Hz components in voltage and current sine

wave on an electric system. Harmonics decrease power quality because they cause

4

the malfunction of protection devices, overheating of transformers, motors, and other

magnetic equipment.

Gauging the impact on the harmonic profile of a feeder due to DG is difficult

because performing a sophisticated harmonic load-flow study is arduous due to the

large number of small and insignificant DG. Therefore, simplified methods have been

adopted to perform the analysis. These methods must account for how the impact of

harmonics generated might be affeced by parallel resonance, and more rarely series

resonance, in the utility impedance. They should also include the dependence of

the harmonic emission during partial-generation conditions for DG equipment, as at

low power of the inverter, the harmonics can be several times higher than at rated

conditions [3].

2.1.3 Voltage Dips

The term voltage dip is commonly used to described two related forms of voltage

phenomena. The first, voltage sag, is an event where the voltage drops to between

10% and 90% of its nominal value for between 0.5 and 60 cycles. The second, a

momentary voltage interruption, is an event where the voltage drops to less than

10% of its nominal value for between 0.5 cycles and 3 seconds [2]. Both of these

events are commonly caused by faults in a neighbouring part of the system, motor

starting, or rapid load changes. As such, the severity of voltage dips is closely tied to

the short-circuit availability of the system at a given bus.

The effects of DG on voltage dips is somewhat conflicting in the literature. The

authors in [1] note that adding generation causes the fault level to be increased at

the bus, thus improving the power quality. However, [3] observes that, in general,

DG units do not appreciably change the short-circuit availability. It is likely that

changes in the fault level at a bus are dependent on the rating of the DG attached

to it. Currently, most DG are required to drop off-line during a voltage dip and as

such they provide little to no support for voltage dip events; however, in the future,

5

automatic tripping of the DG for voltage dips will be unacceptable as the loss of

generation during such an event only serves to amplify its magnitude [4].

2.1.4 Voltage Flicker

Flicker is voltage variations that are perceived by the human eye, causing irri-

tation. These voltage fluctuations can also inhibit motor starting, and cause motor

speed and torque to vary, causing an undesirable effect in the load which can lead to

manufacturing defects in products. Flicker can also affect actions in control systems,

reduces the lifetime of electronics, and cause lighting variations in both incandescent

and fluorescent lamps. Flicker comes from two sources. The first, repetitive events,

include routine motor starting (such as in air conditioners and refrigerators), photo-

copiers, and capacitor switching. The second, fast variations, come from devices with

rapidly changing loads, such as arc furnaces and welders, interharmonics from VFDs,

rolling and saw mills, and traction loads [5].

Voltage fluctuations caused by DG can be either direct or indirect in nature.

Indirect fluctuations include increased transformer load-tap-changer operations and

the malfunction of FACTs and other voltage regulating devices [3, 5]. Direct flicker is

caused by the stochastic nature of the prime mover for renewable DG, such as varying

irradiation patterns. Unfortunately, the modulation of solar irradiance from cloud

movement has not been widely studied [3]. Speaking practically, the fluctuations in

wind or irradiance are likely to occur at close to the same time across a distribution

system because of how physically close the buses of the system are to one another

[3]. Distributed generation will mostly impact short-term (Pst, 1 day) flicker analysis

and is unlikely to be seen in long-term (Plt, 1 week) observation windows [3, 5]. The

net impact of DG on flicker in the literature is conflicting: [5] observes that “with

the proliferation of these DGs expected within the next 3 years to reach 30% of all

new generation, it is believed that the voltage flicker severity level will increase as a

consequence of the combined effect of these DGs on voltage fluctuations throughout

6

the power system” while [3] notes, “although one cannot exclude flicker problems due

to the operation of PV-DG units, from a review of previous works, it is concluded that

flicker is an issue of rather marginal importance when dealing with such installations.”

It is the hope of this work to further add to this discussion and help settle how much

of a flicker problem DG pose.

2.1.5 Voltage Imbalance

Voltage imbalance is a measure of how far off the voltage phasors are from their

nominal values (ie, a magnitude of 1 pu with a 120◦ phase shift between phases).

Two ways to measure voltage imbalance are shown in (2.1) and (2.2), where V+ is the

positive sequence voltage and V− is the negative sequence voltage. Equation (2.1) is

the method used in this thesis.

%VImbal =Vavg − Vmaxdev

Vavg(2.1)

%VImbal =V+V−

(2.2)

Voltage imbalance results in the overheating and derating of induction motors, pro-

tection device misoperation, the generation of non-characteristic harmonics in power

electronics, voltage regulation equipment to be ineffective, and load imbalance. The

probability of a feeder with no DG interconnected on it to have a voltage imbalance

of more than 3% (the most common limit for imbalance) is 2-5% [6]. It is worth

noting that most of a system’s voltage imbalance can be from non-DG factors, such

as a lack of line transposition and load imbalance. In fact, 60-70% of total imbalance

is due to untransposed lines in the distribution system [3].

Three-phase DG is unlikely to create significant voltage imbalance; as such, single-

phase DG is the cause of concern. The interconnection of DG is often random and,

as such, cannot be modeled using deterministic analysis. A study found that the

7

probability of unacceptable voltage imbalance at the beginning of a feeder with high

DG penetration was 0%, but at the end of the feeder was 30% (in contrast to the

2-5% for a non-DG feeder). Ways to improve voltage imbalance include increasing

the feeder cross-section, installing capacitors, or allowing DG to participate in voltage

regulation. If DG may inject VARs into the system, the probability of a feeder having

unacceptable voltage unbalance drops from 30% to 1.8% with an increased capacity

of 16% in the DG need for the VAR injection [6]. It is the hope of this work to further

add to this discussion and help settle how much of a voltage imbalance problem DG

pose.

2.2 Protection

Protection is an important aspect of distribution systems in which various pro-

tective devices – such as fuses, sectionalizers, reclosers, and breakers – coordinate to

prevent equipment damage and reduce safety risks by interrupting faults.

2.2.1 Adaptive Protection

Adaptive protection is a methodology where solid-state relays are dynamically

reconfigured to have settings which match the current system configuration. This is

an important concept in a distribution system with DG due to variable nature of the

generation, and thus the equivalent fault impedance and fault source. Use of adaptive

relaying has helped find a better balance between security and dependability on the

transmission system and is likely to do the same on the distribution system [7].

2.2.2 Standing Phase Angle

Another concern in protection is in regards to the standing phase angle that de-

velops across a recloser during the reclose operation. This is an especially difficult

problem with large DG penetrations that do not trip off-line gracefully as well as

with loads with large quantities of spinning mass. Some relays and Phasor Measure-

8

ment Units (PMUs) measure the phase angle and close the recloser at a time when

the systems are most closely synchronized. This may also allow for quicker reclose

operations, reducing the inconvenience of momentary power interruptions [7, 8].

2.2.3 Directional Overcurrent

High penetration of distributed energy resources (DERs) in the distribution sys-

tem may create scenarios where the system configuration changes to a closed loop

or at best a doubly fed system. Here, the overcurrent protection devices need to be

modified into directional ones that detect not only the current magnitudes in excess of

the normal operating condition but also the current direction. This is often achieved

by simultaneously looking at the voltage and current waveforms [9]. In an attempt

to avoid installing voltage measurement devices, some methodologies have been pro-

posed that consider the change in the current phase for detecting the fault direction

[10]. This is specifically important at the distribution level. However, the shift in the

phase angle is in the range of microseconds and accurate phase angle measurement is

necessary.

2.2.4 Microgrids

Microgrids are subsets of the standard distribution system which can use their

generation to independently supply their loads in the event of a system fault in

the larger system. Microgrids can therefore both parallel the distribution system

(operating in “grid connected mode”) and function independently of the distribution

system (operating in “islanded mode”).

2.2.5 Islanded Operations

Because of the “fragility” of the islanded system compared to the normal gird,

special control is needed. Information provided by PMUs can be used to improve

the control and management systems so that island and reconnection operations are

9

more straightforward and reliable. The PMU information can also be used by utility

operators to make more informed decisions regarding microgrid operation on the

system during critical system events [11]. However, in islanded mode, poor power

quality and voltage rise may occur that can damage equipment [5].

2.2.6 Adaptive Protection

The protection advantages brought about by adaptive protection as outlined in

Section 2.2.1 become even more important in the protection system of a microgrid.

The scheme must work – despite selectivity and sensitivity issues – in both the grid

connected configuration, when fault current is high due the fault availability of the

transmission system, and in the islanded configuration when the dominant feed di-

rection may be altered and the fault current availability is reduced [12]. In a simple

microgrid, it may be possible to create a protection coordination scheme that operates

in both islanded and grid-connected configurations. However, as demonstrated in [12],

most static protection schemes will either favor unnecessary tripping of DG sources

or unnecessary tripping of the line protective devices. As such, adaptive protection

is a virtual necessity in microgrids.

2.3 Distribution System Stability

Stability is the ability for “an electric power system, for a given initial operating

condition, to regain a state of operating equilibrium after being subjected to a phys-

ical disturbance, with most system variables bounded so that practically the entire

system remains intact” [13]. There are three major stabilities for system operators

to consider: power angle (or rotor angle) stability, which includes small disturbance

angle stability and transient stability; voltage stability, which includes large and small

disturbance perturbations; and frequency stability.

Traditionally stability issues were considerations purely at the transmission level;

however, in a weak distribution system (for example, when capital investments have

10

been deferred due to high DER penetration), stability may become an important

issue. PMUs are a virtual necessity for analyzing distribution stability.

Stability analysis of the distribution system is the ability to monitor stability

margins in multi-sourced circuits, such as those with DG. In the presence of DG,

transient (ie, first swing stability), long term dynamic stability, and voltage collapse

are the most in need of study [1], with frequency stability being a factor only at very

high penetration levels [14].

Stability for power-electronic controlled DG is strongly dependent on the control

strategy employed. There are three control strategies in use: sinewave converters,

pure resistive converters, and programmable dampening resistance converters. The

sinewave converter, which tries to output a sine wave regardless of the waveform

already on the system, does not contribute to power system stability. A pure resistive

converter in essence looks like a negative resistance to the power system which has

the effect of reducing the dampening at harmonic frequencies, causing instabilities.

However, the programmable dampening resistance converter addresses these issues

and as such maintains the damping potential of the generator over a wide range of

power levels [4].

With the presently conservative status of IEEE 1547 and utility interconnection

guidelines with regards to providing ancillary services, many stability issues are not

a problem; however, there are several movements within the industry to relax these

standards. This practice of passive operation also limits the capacity of distributed

generation that can be connected to an existing system because utilities plan for a

worst-case scenario that has a low probability of occurring [1]. With greater observ-

ability into the system through PMUs, this may be mitigatable by granting system

operators the ability to have more dynamic control, thus increasing the instanta-

neous capacity of the system instead of being forced to assume worst case values for

all variables and circumstances.

11

2.4 State Estimation

State estimation is a statistical method in which noisy or inaccurate measure-

ments from across a system are processed to find the most likely true condition of the

system, including the ability to predict the conditions of buses without measurements

on them [15]. State Estimation has been used extensively on the transmission sys-

tem to provide more accurate load flow information, perform parameter estimation,

implement bad data and fault measurement detection, and perform topology error

processing. Many of these applications could be extended to the distribution system

especially in the presence of distributed generation. Several other benefits could be

realized, as outlined below.

2.4.1 Distribution State Estimation

While distribution state estimation is not widely implemented at this time, it

brings several benefits to the system. First and foremost are those that are already

used at the transmission level – most notably bad data detection and parameter es-

timation – which can be used to verify devices such as capacitors and regulators are

functioning correctly, as well as to estimate the conductor sizes along the feeder. In

the presence of distributed generation, a distribution state estimator can estimate the

loads demands and DG’s generation, even with limited measurements of the system

[16]. This information can then be extended to optimally dispatch DER or optimize

other operational characteristics of the system (capacitor switching, regulator set-

tings, etc...) [11, 17]. Finally, a distribution state estimator can work to improve the

accuracy of an adaptive protection scheme [17].

Several cautions must be observed with regard to distribution state estimators.

The estimators must also take into account certain non-linearities common in the

distribution system, namely VAR compensators, voltage regulators, and load-tap

changers [16].

12

2.4.2 Dynamic State Estimation

A dynamic state estimator predicts the state of the system one time-step ahead by

using a mathematical model of the dynamic behaviors of the system. It achieves this

by using historic data, the previous time step, and power angle information which

PMUs provide [18–21]. The emphasis on time-step ahead data delivers more realistic

estimates of the state compared to a traditional estimator [18] which is useful to

security analysis, control, and risk-prevention applications allowing them to make

more informed decisions [19]. While these are applications that are not crucial to

distribution system operations, monitoring their affects at the distribution level will

be beneficial for transmission operations as higher granularity load models provide

more realistic dynamic and voltage collapse studies [22].

Presently, dynamic state estimators are not widely in use; many choose to use

standard state estimation techniques which are more widely available but provide

poorer quality data. One of the major obstacles in implementing a dynamic state

estimator is the high quality dynamic system model which is required; this model is

difficult to create and verify [23].

2.4.3 Harmonic State Estimation

Harmonic state estimation allows one to pinpoint the source of harmonics on a

distribution feeder as well as capture the harmonic profile throughout the system. The

harmonic estimator can also predict the injections of non-constant DG and harmonic

loads, however it must assume that DG do not have a harmonic contribution [24].

The complex bus voltages for each harmonic are the states of the system and the

measurement model of the system is defined uniquely from other state estimation

techniques because the concept of reactive power is not well defined for harmonic

power flow [25].

13

2.5 Voltage/Var/Watt Control

The problem of controlling the voltage regulating transformers and shunt capaci-

tors in a distribution system is often referred to as the Volt/Var control (VVC) prob-

lem. The main objective here is to devise a control policy for regulating the shunt

capacitors and transformer tap positions in order to minimize the peak hour demand,

reduce losses, or release the congestion of the system, while all the voltages are kept

within the permissible range. Peripheral objectives may also be accounted for, such

as reducing the reactive power flow through the main distribution transformer and

minimizing the total number of switching/tap operations on the capacitors and/or

transformers. With DER becoming more common in the distribution grid, the prob-

lem is now extended in the form of Voltage/Var/Watt Control (VVWC) which also

takes the active power provided by the DERs into account.

In the traditional distribution grid, the active and reactive power balance equa-

tions have always been described based on the average values of active and reactive

power of components. Moreover, bus voltage magnitudes have been considered to be

accurate representatives of the distribution buses. However, the modern grid equipped

with DER and active loads may require a more dynamic approach for VVWC. A dy-

namic modeling of the problem requires use of instantaneous active and reactive power

quantities, defined based on the instantaneous values of the currents and voltages.

Traditionally, the large size of the distribution network and sometimes the lack

of an accurate network model make dynamic modeling a difficult task. Nevertheless,

the approach can be successfully adopted for small scale systems, specifically isolated

power systems such as microgrids and all-electric shipboard systems. In this case,

phase angles cannot be ignored anymore, and PMU installations or PMU functional-

ities embedded in the IEDs will become a necessity.

14

2.6 Electric Vehicle Considerations

The high dependence of traditional vehicles on the finite sources of fossil fuels,

the environmental concerns on vehicular pollution, and the need for higher energy

efficiency, fuel economy, and fuel flexibility have all paved the way for the introduction

of plug-in hybrid electric vehicles (PHEV) and electric vehicles (EV). Such a fleet of

vehicles can be powered by the underutilized electric power grid during the off-peak

hours with little need to increase the capacity of the existing grid infrastructure [26]

although the multi-cycle ratings of transformers may need to be re-evaluated due to

the increased capacity factor. Traditionally, these vehicles have been considered as

nonlinear loads for the grid, whose impacts on stability and quality of supply have

been studied in detail [26–28]. However, as new technological advances in power

electronics and machine design – as well as the government mandates and subsidies

for energy independence and resilience of the transportation system, further accelerate

the penetration rate of PHEV/EVs into the transportation fleet – the possibility of

utilizing these vehicles not just as loads but also as sources of energy arises. The

energy stored in the batteries of the PHEV/EVs can be extracted by discharging the

battery for a relatively short duration of time and injecting its energy back into the

grid. This service, often referred to as vehicle-to-grid (V2G), can in principle provide

peak load shaving, smoothing generation from non-dispatchable renewable energy

resources and act as a reserve against unexpected outages [29]. The V2G application

can be implemented for groups of vehicles parked at a charging station, or a large

number of individual vehicles connected to their corresponding charging poles.

However, some researchers argue using EV in day-ahead generation scheduling or

peak shaving is not nearly as beneficial as using the technology to provide ancillary

services, such as providing spinning reserves and voltage and frequency regulation

[30–32]. Traditionally, due to relatively slow V2G communication and activation

system, EVs are not expected to participate in primary reserve service.

15

CHAPTER 3

STATE OF THE ART: PROBABILISTIC LOAD FLOW

A Probabilistic Load Flow (PLF) expands upon the traditional Deterministic

Load Flow (DLF) by incorporating uncertainty directly into the solution process. In

theory, nearly any value expressed in a DLF could be probabilistically analysed in a

PLF, although load, generation, and topology variations are the most common. The

objective of a PLF is to determine the sensitivity of the electric system to the modeled

uncertainties so more intelligent risk analysis and mitigation can be performed.

3.1 Key Concepts of the Probabilistic Load Flow

A standard Deterministic Load Flow (DLF) ignores the uncertainties that chang-

ing network configuration, load and DG variation, equipment outage rates, and other

such uncertainties have on the power system [33]. Consider instead a Probabilistic

Load Flow (PLF), which does not assume the generation and load is deterministic,

but instead follows a known probability distribution [34]. The objective then of the

Probabilistic Load Flow (PLF) is to create a statistically valid model of the system

so that the system states (ie, voltages) can be stochastically described.

Currently, most public utility commissions hold distribution providers to a set

range of voltage values. These values are hard set, with the possible exception of

flicker variations. However, several researchers [1, 35] note that stochastic voltage

limits may be more appropriate; in fact, this method has been proposed in the new

European standard for voltage characteristics on public distribution systems, EN-

50160.

One of the more major (and less explored) areas of concern for a PLF is in the

existence of correlations between the various random variables (ie, wind output, PV

output, and load). Nearly every PLF paper reviewed neglected to account for the

16

intrinsic interdependence between the random variables, or would mention them but

not attempt to mitigate the effect, simply stating that they are a source of error.

This is acceptable, to a degree, because the random variables are neither completely

independent or completely linearly correlated with respect to each other [33].

Regardless, some common loose dependencies do exist in resource random vari-

ables. In [34], the author observes a correlation factor between -0.3 and 0.3 between

EV load/generation and wind generation, which he attributes to the diurnal aspect

of wind. Distributed Generation such as combined heat and power facilities and wind

turbines are also correlated with the load due to weather conditions [33]. This author

would also propose that solar generation is dependent on wind speeds as high wind

speeds can cause significant cloud volatility.

It is difficult to model a multi-variable, dependent stochastic process if only the

marginal, and not joint, distributions are known [33] as is most often the case. The

most common way around this is to design the problem in such a way that the random

variables are more independent of each other. For example, the interdependence

between load demand and renewable DG can be modeled through two levels: the

time of day or the season of the year (depending on the desired results from the PLF)

and the weather of that specific day/season [33]. Others have decided to try to directly

model the interdependence of the random variables, such as [36], who proposed short-

term load demand be modeled with partial correlation: the load demands’ mean

values rise and fall in step with a small independent, normally distributed random

variable.

3.2 Paramater Considerations

Each parameter (ie, load, wind, and PV) can be represented by a random variable

with a given probability distribution that is created based on historical record, statis-

tical analysis, or engineering judgment [35]. It is well documented that wind speeds

are Weibull in nature and as such the power from a wind turbine can be modeled as a

17

Weibull random variable. All papers reviewed also assume that the load is uniformly

normally distributed, at least on larger time frames. The nature of solar resources,

however, still remain relatively unknown.

Several papers in the literature address the interconnection of wind resources in

the PLF problem [34, 37, 38]. It is worth noting that wind variability can be observed

both spatially and temporally [37], however all PLFs researched considered only the

temporal variation of wind. Averaging periods of one hour are commonly seen in

wind speed measurements as they capture enough wind information to be useful, are

feasible to collect and store, and allow the variations due to turbulence (over short

time periods) and weather fronts (over long time periods) to be factored out [38].

Some researchers choose to model wind speeds using the normal distribution [38]

so that the simpler Stochastic Load Flow method can be employed [33]. However,

wind speeds are more closely Weibully distributed, which is the most common way

to represent wind data [34, 37, 38]. The authors in [37] have also developed a way to

model the power output of a wind turbine given a wind speed Weibull distribution

by modeling the power as a Weibull with shape and scale parameters similar to the

wind speed distribution, but adjusted for the specifications of a given wind turbine.

Another fairly accurate if not as common way to model wind data is using an auto-

regressive or auto-regressive-moving-average technique [38]. Finally, a Markov model

can also be employed with the limitation that it can only be used for long time frames

[38].

The statistical distribution of solar resources is not as clearly addressed in the

literature. The authors in [39] note that cloudiness creates the major difference

between an ideal irradiance shape (as would been seen from outer space) and those

observed by PV arrays. They go on to use the clearness index, a measure of the

observed irradiance over the extraterrestrial irradiance, as the basis for the creation

of a random variable for the solar resource. However, the distribution of the clearness

18

index is not trivially found: [40] models it as a modified Gamma distribution while

[41] models it as a Poisson distribution. This author feels that the clearness index

approach is inherently inadequate for flicker analysis, as shorter time frames must be

considered and clearness indexes are realistically only defined over an hourly or daily

time frames [39]. As such, some novel ideas are needed in this area.

In the spirit of a fully forward looking PLF, some work has been done concerning

the probability distributions of Electric Vehicles (both as loads and sources). The

authors in [34] have chosen to model an EV charging park in two stages. First, the

number of charging and discharging EV can be modeled through queuing theory as

a Poisson process. Once it is known how many EV are charging and discharging,

the service time – ie, load duration – follows an exponential distribution due to the

exponential charging nature of chemical battery technology. Because the charging

value over time is also well understood, modeling EV in a short-interval PLF, as

proposed in this work, would be relatively simple to implement and could be the

subject of future work.

Finally, the statistical model of the load parameter must be created. It is uni-

versally understood that the load can be modeled as a normally distributed random

variable [34, 37, 39]. This author infers the reason: in essence, every individual load

in a load center (ie, home, office space, distribution transformer, feeder, substation,

or balancing authority) can be modeled as a Bernoulli trial. When several Bernoulli

trials are conducted concurrently, they can be modeled as a binomial distributed

random variable. Several binomial random variables, summed together, can be mod-

eled as a normal random variable. So in essence, by the time that at least 30 or

so Bernoulli trials are conducted, the random variable becomes normal; nearly any

load center bigger than a house will have at least 30 individual loads in it. However,

the observation that there exists an intermediate binomial distribution becomes quite

salient when synthesizing the load data for this thesis (see Section 4.1.2).

19

3.3 Methods

There are two major methods to conduct a PLF: analytical and numerical. The

analytical method takes the individual probability density functions (PDFs) for each

random variable and convolves them together according to the load flow equations.

Because the load flow equations are non-linear, they must be linearized in order to

make the convolution solvable. The convolution will then result in a PDF for each

variable, including the system states. The analytical method, while quite mathemat-

ically elegant, is difficult to perform on a realistic system [33].

The numerical method performs a large number of DLFs with inputs determined

from realizations (that is, samplings) of the random variables in the PLF [33]. The

most common numerical method is the Monte Carlo, which preforms the PLF by run-

ning DLF with inputs sampled using a specific sampling technique (simple random

sampling, stratified random sampling, etc...) so that the desired outcome may be

more easily found (ie, found in fewer iterations). Another approach implemented in

[34] is the Learning Automata approach, which is more robust to incomplete stochas-

tic information, has a greater tendency to find global optima, and the “degree of

satisfaction” is more easily expressed over a Monte Carlo approach.

It is worth noting at least one researcher [37] has considered a hybrid approach

which uses convolution to simplify the power flow problem so that Monte Carlo sim-

ulations arrive more quickly at the correct outcome.

3.4 Applications

The PLF is capable of considering the distribution system operation in light of

load variations, DG injection variations, network reconfiguration, and other stochastic

processes present in the system. This provides a more complete representation of the

voltage profile on a distribution circuit by considering all possible uncertainties in the

system over the planning and operation time frames [35].

20

One of the greatest factors currently limiting the capacity of DG on a distribution

system is that all control problems are solved deterministically at the planning stage

[1]. As such, one of the best uses of the PLF is to determine the control settings of

voltage regulation devices (ie, capacitor banks and voltage regulators) that are robust

to different operational states [33, 34]. To find the ideal settings, a sensitivity analysis

is preformed on the constrained variables with respect to the control variables [34].

Another application is to estimate the annual energy lost due to over-voltage

protection on DG and other applications related to the estimate of bus over-voltage

(such as customer complaints, over energy utilization, etc...) [39]. Probabilistic load

flows are also commonly found in the analysis of voltage sag, faults, and reliability

[6].

The author of [3] observes that since DG caused voltage quality disturbances are

described stochastically with time, probabilistic techniques are needed to characterize

the resulting disturbance level. This observation is the foundation of this work and

the reason a PLF is an integral part of quantifying the risk DG pose to voltage quality

on a power system.

21

CHAPTER 4

RESOURCE STOCHASTIC MODELS

The added value given by a probabilistic load flow is founded in the stochastic

models created for the random variables (RV) employed. At its heart, three random

variables are used in the PLF presented in this work.

L1s – Secondly Load – This is the load at each bus for each second of the simu-

lated day. It is composed by adding the load for the residential, commercial,

and industrial (or spot) loads which are formulated through data analysis and

information presented in the literature. Thus, L1s = LR,1s + LC,1s + LI,1s.

W1s – Secondly Wind Speed – This is the wind for each second of the day and is

composed through a combination autocorrelation and Weibull methods with a

random walk.

I1s – Secondly Irradiance – This is the irradiance on the PV panels and is composed

through a deterministic baseline irradiance component added to a stochastically

determined cloud event component. The cloud event component is determined

through a series of Poisson and exponential distributions married to an ARMA

model for intra-event irradiance.

One of the greatest challenges in the production of this work and a significant

aspect of its novelty and contribution to the body of scientific knowledge is in the

production of small resolution (one second) statistical models for wind speed and

solar irradiance suitable for use in power system analysis. Nearly all DG integration

methods using a PLF are looking at a “planning” time frame, which considers averages

over anywhere between an hour and a year. At this temporal resolution, the wind

speed or irradiance at one moment can reasonably be assumed to be independent

22

of the value at another. However, at the small resolutions needed for this work,

statistical methods like autocorrelation analysis and time series methodologies must

be employed because the value of a resource at one second is highly dependent on

the values over the previous few seconds. Yet, because the ultimate goal of this work

is still to support planning (from the stand point of how to build a system tolerant

of DG impacts), seasonality and trend aspects of the data must still be taken into

account. This calls for a hybrid methodology which employs time series analysis

at small resolutions while using statistical analysis to capture the overall trend of a

resource throughout the time of day. A summary of this aspect of the RVs is presented

in Table 4.2 at the end of this chapter (page 44).

It is clear then that while only three random variables are required for the PLF,

a whole host of random variables are developed to create them. The details of these

stochastic models are presented in the following sections.

4.1 Residential Load Model

The residential load model is based on real data collected from Smart Meters

deployed in Boulder, CO. These loads are randomly combined in groups and the

mean and standard deviation of the loading groups found for every 15 minute period

in a day. These 15 minute models are then interpolated down to 1 minute models, at

which point random load deviations are added. With the noise corrupted 1 minute

data, another interpolation is performed to bring the data to 1 second, the resolution

of the PLF simulator.

4.1.1 Load Data Basis

As mentioned previously, the creation of the load is based on the number of

residential and commercial properties (as recorded by the government of Weld county)

located within a specific load zone, as discussed in Section 5.2.1 and show in Figure 5.4.

Residential data will be shaped based on the load curve found from SmartGridCity

23

data, explained below1.

It is commonly assumed and has been verified by this author, with some caveats,

that load follows a normal distribution. This distribution, denoted by N has two

parameters, the mean µ and the variance σ2. The Probability Density Function

(PDF) for this distribution is given in [42] as shown in (4.1).

fN(x;µ, σ2) =1

σ√

2πe−

(x−µ)2

2σ2 (4.1)

4.1.2 Data Analysis

In order to be as realistic as possible, the load shapes will be derived from data

taken from Smart Meters in SmartGridCity, a next-generation distribution system in

Boulder, CO. Data is recorded on a 15 minute interval from every residential meter in

the system. Because several of the simulations require higher resolution data, some

statistical trickery is performed on the data to emulate the random properties of

loads.

For each season, 50 homes of data is collected. The homes are then placed in 1,000

groups with 5 to 20 homes per group. This is done to help model the diversity of loads

up to the transformer level. The demand is then divided by the number of homes in

the group so that the probability model can be developed for an “equivalent” home

(that is, a home with the affects of diversity captured). This is necessary because

when less than roughly 30 devices are in a load center (as tends to be the case for a

home), the load is not quite normal and is more closely binomial; however, creating

these equivalent homes through aggregation allows them to be modeled as normal.

These are then aggregated to each 15 minute window within a day and, working

under the assumption the load is normally distributed and that the load is fully

coincident within a 15 minute period, the mean and variance for each 15 minute

period is found. These parameters for the normal distribution form the basis from

1Unfortunately, smart meters are not enabled to collect demand information on commercialproperties, so a separate source must be used as explained in section 4.2.

24

which load is randomly synthesized.

Research shows that the load is much more dependent on time of day than the

weather in all but the most extreme weather circumstances. Take for example the

residential load data for Spring 2011. A normal regression was taken with respect

to time and with respect to temperature. The results are shown Figure 4.1. The

results for time are as expected: the load has a textbook residential load shape (large

peak in the evening with a smaller peak in the morning). The results for temperature

are more interesting: it shows that for all but the coldest weather (less than 0◦ C)

the average load seems to be constant. In fact, the value the mean hovers around

is 0.614 kW, which is the mean load in the Spring for the data set. This indicates

that the load is temperature independent except at low (and, in the summer, high)

temperatures. Thus, even though there is some temperature dependence in the load,

it is not a strong factor and, as such, is not included in the PLF. However, as a point

of future work, temperature dependence (including a temperature model) would be

a worthwhile addition to this work.

(a) Mean and Variance of Load as a Function ofTime of Day

(b) Mean and Variance of Load as a Functionof Temperature

Figure 4.1: Normal Distribution Parameters for Time of Day and Temperature

25

Thus, a statistical model for load at 15 minute intervals is created as shown in

(4.2) where µ(t) and σ2(t) are as described in Figure 4.1(a).

LR,15m(t) ∼ N(µ(t), σ2(t)) (4.2)

4.1.3 Generation of Load for PLF

To generate the load of each house on the system, the time of day dependent

normal distribution (ie, equation (4.2)) is sampled for each home, creating lR,15m the

realization of LR,15m. A simple linear interpolation is then performed to bring the 15

minute data to a 1 minute resolution. A random walk with a variance of 0.05% of the

load is added as shown in (4.3) to create the 1 minute data. No other randomness is

added to create the one second data because the sub-minutely temporal correlation

is high.

LR,1m(t) = Lr,15m(t) +N(0, 0.005 · lr,15m(t)) (4.3)

From 1 minute load intervals, a simple linear interpolation is performed to bring

the data down to the desired resolution of 1 second creating LR,1s.

4.1.4 Load Unbalance

Load unbalance is modeled as a normally distributed PDF with a mean of 33.3%

of the load per phase and variance of 3%. Imbalance is then trivially calculated as

shown in (4.4).

Ua ∼ N(0.333, 0.03)

Ub ∼ N(0.333, 0.03) (4.4)

uc = 1− ua − ub

4.1.5 DG Deployment Random Variable

While not an aspect of the load, at this stage the future deployment of the DG

in the system is generated because it is calculated as a ratio of the peak load. The

26

penetrations are based on the peak load of each phase of each bus multiplied by the

scenario’s penetration level assumption, qDG, multiplied by a normally distributed

random variable DDG ∼ N(1, 0.2) which is realized for each bus and phase combina-

tion to find the installed capacity of DG on the system, SDG as shown in (4.5). This

process ensures that the mean of SDG is the desired penetration. The assumed PV

and wind DG penetration ratio is 75% PV, 25% wind.

sDG = max(lR,15m)(qDG)(dDG) (4.5)

4.2 Commercial and Industrial Load Model

Because commercial and industrial loads are not monitored in SmartGridCity,

a model from the literature was needed to create these loads. The authors in [43]

present typical values for the peak demand of a commercial load and [44] gives typical

daily load shapes (both mean and standard deviation) for four generalized classifi-

cations of commercial loads. The information presented in these papers was used to

generate four typified commercial load shapes at a one hour resolution. A simple lin-

ear interpolation is then performed to bring the data down to a 15 minute resolution,

creating LC,15m.

From the 15 minute resolution data, the exact same process is performed as de-

scribed for the residential load in section 4.1.2 producing LC,1s.

For the industrial loads, each was assumed to be balanced three-phase with a

specific load shape class depending on the type of spot load it is. Due to the higher

diversity within an industrial load, the variance of these shapes was scaled significantly

down to 7% of the value given in [44]. From here, the same process used to create

LC,1s is used to create LI,1s.

4.3 Wind Model

The wind model is based on data collected by the National Renewable Energy Lab

(NREL) in Boulder, CO. A series of time dependent Weibull curves are found for the

27

data. The data is then subtracted from a sampled Weibull distribution to find the

“noise” in that data for modelling via an autocorrelation analysis. The intra-minute

variance is also recorded. To simulate the wind, the minutely Weibull probability is

combined with the minutely autocorrelation trend. To increase the resolution down

to one second, randomly distributed white noise with Weibully distributed variance

based on the recorded intra-minute variance is added to the linear interpolation be-

tween minutes.

4.3.1 Wind Data Basis

The wind data for this project is collected from NREL’s National Wind Technology

Center M2 tower [45]. The measurements used in analysis include the average wind

speed at a height of 10m and the standard deviation of the wind speed at a height

of 10m, both of which are collected at one minute resolution. For each season, data

from 2007 to 2011 was used.

Many sources indicate that wind can be modeled with a Weibull distribution

[34, 37, 38] denoted by Wbl(a, b) with scale parameter a and shape parameter b and

PDF as given in [42] shown in (4.6).

fWbl(x; a, b) = ba−bxb−1e−(xa)b

x ≥ 0 (4.6)

However, inherent in modelling wind data statistically is the assumption that

the wind observed at one moment is independent of the wind observed at another.

For longer period studies, such as wind resource planning studies, this assumption

is acceptable. However, for the purposes of this work one needs to account for the

fact that the wind speed at the current moment is strongly correlated to wind speeds

of moments ago. The most appropriate method to do this is to create a time-series

using the autocorrelation method which models a wide sense stationary, normally

distributed process. To do this,

28

1. Remove any deterministic component from the data set (such as seasonality

and trend).

2. The autocorrelations of the data needs to be found over a large number of lags.

3. If the data can be modeled using a time-series, the autocorrelations will closely

follow a function R(τ). The parameters for this function need to be estimated

through curve fitting.

4. The estimated autocorrelation function R(τ) can be transformed into a power

spectral density G(z) via spectral factorization.

5. When the estimated power spectral density G(z) is perturbed with white noise

a time series which resembles the original data set is created.

By combining the “trend randomness” aspect of the Weibull statistical approach

with the “time-correlated randomness” aspect of the autocorrelation approach, a good

model for wind can be created.

Because the NREL data contains sub-minutely wind variation information, a nor-

mal distribution (as given in (4.1)) in the form of a random walk W1s,N ∼ N(0, σ2)

is used to generate the sub-minutely variation.

4.3.2 Data Analysis

In order to stochastically generate the Wind for the PLF, three pieces of informa-

tion need to extracted from the data set:

1. The Weibull wind speed distribution for each minute of the day.

2. The autocorrelation function (and associated power spectral density) that re-

lates the wind at one minute with the previous and future wind values.

3. The variance for the random walk used to model sub-minutely wind.

29

To create the time of day dependent Weibull distributions, all the days in the

dataset are sampled for the given minute of the day. All of these wind speeds are

fitted to a Weibull distribution using a Maximum Likelihood estimator, which is used

to estimate the coefficients a(t) and b(t) for the Weibull distribution. How these

coefficients vary in time for Spring can be seen in Figure 4.2.

Figure 4.2: Variation of Wind Speed Weibull Coefficients Throughout the Day inSpring

In a similar way the intra-minute variances are arranged and sampled. Experi-

mentation has shown that the variances – which are used in a normal distribution

for the random walk – are Weibully distributed and as such the Weibull coefficients

av(t) and bv(t) are estimated for the sub-minutely variances. How these coefficients

vary in time for Spring can be seen in Figure 4.3.

While the derivation of the Weibull wind speed distribution and random walk

variance (items 1 and 3 of the list on page 29) have been relatively simple, the

derivation of the autocorrelation function (item 2) is intricate and is the focus of the

remainder of this section.

30

Figure 4.3: Variation of Sub-Minutely Variance Weibull Coefficients Throughout theDay in Spring

Recall the purpose of an autocorrelation method is to model the “time-correlated

randomness” of a dataset once a deterministic “trend” has been removed. Suppose

though that instead of using a deterministic trend a probabilistic trend is used. What

this means is that if we take the wind data d(t) and subtract out a Weibull component

W1m,W ∼ Wbl(a, b) (and thus its realization w1m,W (t)) we will have “errors” which

should be wide-sense stationary and normally distributed. These “errors” can then be

modeled using an autocorrelation method. The distribution of d′(t) = d(t)−w1m,W (t)

and the normal probability plot are given in Figure 4.4. Although the data looks quite

normal in Figure 4.4(a), Figure 4.4(b) shows it is only quasi-normal.

It is worth noting that this process of creating d′(t) has done little to affect the

autocorrelation of d(t), as shown in Figure 4.5 for Spring. Most importantly the

strong correlation seen within a day time frame (which is about 1500 lags) is still

apparent. This matches intuition about weather patterns – ie, some days are windy

and some are not (and thus the wind speed one hour is closely related to the wind

31

(a) Distribution of d′(t) (b) Normal Probability Plot of d′(t)

Figure 4.4: Statistical Quantities for d′(t)

speed last hour) but the wind speed of yesterday tells one little about the wind speed

today.

(a) Autocorrelation of d(t) (b) Autocorrelation of d′(t)

Figure 4.5: Autocorrelations at 5000 Lags for Spring

A function, in the form of (4.7), is now fitted to the autocorrelations.

R(τ) = ξ2e−α|τ | (4.7)

Two parameters ξ and α must be estimated using the non-linear curve fitting

toolbox of MATLAB. The results are shown in Figure 4.6.

32

Figure 4.6: Autocorrelation of d′(t) with Fitted Model

With these parameters estimated, the power spectral density is found as shown

in (4.8) [46].

G(z) =ξ√

1− e−2αz−1

1− e−αz−1(4.8)

4.3.3 Generation of Wind Speed for PLF

To generate wind using the Weibull wind speed distribution, autocorrelation, and

sub-minutely variances gathered above, the following is performed to create W1s.

1. A time-series is generated using ξ and α to create a simulated d′(t), d′(t).

2. The time of day dependent Weibull wind speed distribution is added, creating

d(t). This creates a realized version of W1m.

3. d(t) is interpolated down to a one second time frame and the intra-second

variance random walk added in. This creates a realization of W1s.

To create the time-series d′(t), a state-space representation of (4.8) is created, as

shown in (4.9) [46]. It is worth noting because of the large correlations at thousands

of lags, a full day of wind needs to be simulated before accurate results are created.

33

x(t) = Φx(t) + w(t) (4.9)

d′(t) = Hx

where...

Φ = e−α

H = σ√

1− e−2α

w(t) = While noise distributed N(0, 1)

A random variable W1m,W (t) ∼ Wbl(a(t), b(t)) is created where a(t) and b(t) are

as found in the previous section. A realization w1s,W (t) is then found and used to

create d(t) – a realization of W1m – as shown in (4.10).

d(t) = w1m = d′(t) + w1m,W (t) (4.10)

Observe that d(t) is at a one minute resolution. To bring the simulation down to

the one second time frame, the one minute data is interpolated. A random variable

W1s,N(t) ∼ N(0, v(t)) is created where v(t) is a realization of V (t) ∼ Wbl(av(t), bv(t)).

W1s,N(t) is then sampled to create w1s,N which is used in (4.11) to create w1s(t) which

is the realization of W1s(t). A 10 second smoother is applied to w1s,N to mimic the

strong autocorrelation secondly wind has.

w1s(t) = d(t) + w1s,N(t) (4.11)

It is through this methodology that W1s(t) is created.

4.3.4 Fidelity of Wind Model

In order to evaluate how well this convoluted method of wind generation mimics

true wind speeds, an experiment was set up where six years of Spring wind was

compared against six years of simulated Spring wind data. The histograms for the

simulated and observed wind are given in Figure 4.7.

Keep in mind the purpose of this exercise is not to reproduce the six years of data

identically but instead to generally mimic wind in a way which is significant to both

34

Figure 4.7: Histograms for Simulated and Observed Wind Speeds

short and long term trends. How well the model mimics the observed data is judged

in the errors. Ideally, the errors will be normally distributed white noise. The error

distribution and normal probability plot are given in Figure 4.8.

(a) Error Distribution (b) Error Normal Probability Plot

Figure 4.8: Error Statistical Quantities

Although the errors are not normally distributed, they appear to be quasi-normal,

as is the case for the “errors” d′(t) = d(t)−w1m,W (t) used to create the model. Also,

while the simulated wind is somewhat Weibull, it is not the same Weibull distribution

35

as the source wind; this is acceptable for this work as interest is in mimic wind

resources and not necessarily the specific wind at NREL. As such, the model seems

to be operating fairly.

As a point of future work, improvement of the wind model could be investigated.

This author feels that the wind model does extremely well at micro-level wind simu-

lations and fairly well at macro-level studies. It is likely the methodology employed

here could be adapted to include mixed distributions (instead of pure Weibull dis-

tributions) in the creation of d′(t) and d(t); doing this would likely make the errors

more normal and potentially increase the fidelity of the autocorrelation component

of the wind speed generation engine.

4.4 Solar Model

The solar model is based on data collected by NREL in Golden, CO. An algorithm

is used to detect when a cloud event has occurred and collect meaningful statistics

on each event including how many events occur per day, the duration of the event,

the inter-event waiting time, and the irradiance shape within the event. Statistical

distributions are fitted to the data. Using the fact that, in essence, solar irradiance is

deterministic without cloud events, the irradiance profile of a day is created by taking

the ideal irradiance and subjecting it to cloud events with known probabilities.

4.4.1 Solar Data Basis

The solar data for this project is collected from the Solar Radiation Research

Laboratory of NREL located in Golden, CO [47]. The measurements used in analysis

include the “Global LI-200” irradiance measurement (referred to as the “measured”

or “observed” irradiance) and the “Global Extraterrestrial” irradiance measurement

(referred to as the “ideal” or “extera” irradiance). For each season, data from 2006

to 2011 was used.

36

The irradiance seen by a PV is a strong function of time and could in fact be mod-

eled deterministically if not for cloud cover, which must be modeled as a stochastic

process. Cloud events are modeled as Poisson processes. That is to say the number

of cloud events that are expected to be observed in a given day can be modeled using

a Poisson random variable denoted by Poiss(λ) with mean λ and Probability Mass

Function (PMF) as given in [42] shown in (4.12).

fPoiss(x;λ) =λx

x!e−λ x = {0, 1, 2, ...} (4.12)

The inter-event waiting time (that is, the time between events) is modeled as an

exponential distribution2 where µ is the mean, with PDF as given in (4.13).

fexp(x;µ) =e−

xµ

µx ≥ 0 (4.13)

Usually, for a Poisson process, if the number of events occurring over an observed

time (in this case, one day) can be modeled as NE ∼ Poiss(λ) then the waiting time is

TW ∼ exp(λ) [42]; that is to say there is a relationship between the expected number

of events E[NE] = λ and the expected waiting time between the events E[TW ] = λ.

Note that Poisson processes are instantaneous events. However, cloud events are

non-instantaneous. As such, while the inter-event waiting time is still exponentially

distributed, it will not have the characteristics of an ideal Poisson process. As such,

the inter-event waiting time will need to be analysed separately with mean µW .

Finally, the duration of an event TD is exponentially distributed with mean µD

and properties as previously discussed in equation (4.13).

4.4.2 Data Analysis

To create a model of the irradiance shape from the raw data, several statistics

about cloud events must be collected. The first trick is to detect a cloud event has

2This formulation is slightly different than the typical way in which the exponential distributionis defined: it is common that the argument be the reciprocal of the mean. In order to facilitate aneasier discussion in this work, the argument is re-defined as the mean itself and the PDF in (4.13)denoted by exp(µ) reformed to reflect this change

37

occurred. Cloud events are characterized by a large derivative in the solar irradiance.

However, there are moderately large derivatives in the solar data regardless because

of the sun’s movement through the sky (ie, at dawn and dusk). To avoid the false

detection of these events, a 100 minute robust smoother is applied to the irradiance

data with the intent that only the largest cloud events will not be smoothed out.

A discrete derivative is applied to both the real data and the smoothed data and

the difference between these two derivatives found at every time point. A 15 minute

smoother is applied to the difference data to only capture moderately large cloud

events. Any time the value of the smoothed difference of derivatives is above a

threshold (experimentation has found 10 to be a good number), the cloud event is

registered with its start and stop time. This process is illustrated in figure Figure 4.9

which shows a sample day’s irradiance pattern (in green) and when a cloud event is

detected (in blue).

Figure 4.9: Cloud Event Detector on a Cloud Day

For each cloud event detected, event statistics are gathered. The number of events

per day is the number of starts observed. For all but the first event, the inter-event

waiting time is the start time of this event less the stop time of the previous event.

38

The event duration is the stop time minus the start time.

Once the entire data set has been analysed, the appropriate distribution is fitted

to each statistic. The results are shown in Table 4.1. Sample fits for Spring are also

given in Figure 4.10.

Table 4.1: Cloud Event Statistics for all Seasons

StatisticSpring Summer Winter

Name Symbol

Number of Events Per Day λ 7.4178 7.6844 6.5537

Waiting Time Between Events (min) µW 46.5186 29.1841 25.1106

Event Duration (min) µD 54.0616 47.1998 33.4991

The intra-event irradiance shape must also be estimated. The nature of the vari-

ations within a cloud event are subject to the density of the clouds, the rate of cloud

movement, and other factors well beyond the predictive capability of this model. As

such, it is assumed that intra-event irradiance is independent of every other random

variable.

The largest factor of irradiance variation that is modelable is the previous values

of irradiance; because of the strong correlation in irradiance values with previous

irradiances, an auto-regressive moving average (ARMA) model is most appropriate

for simulating this randomness. Therefore, a second order ARMA model at five lags

(which gives the form of (4.14)) is fitted to each event’s irradiance shape less the

deterministic component. After the models have been checked for stability, they are

catalogued into a library for use in the generation of intra-event irradiances.

39

(a) Number of Cloud Events Per Day (b) Time Between Cloud Events

(c) Duration of Cloud Events

Figure 4.10: Statistics for Cloud Events in the Spring

40

A(q)y(t) = C(q)e(t) (4.14)

where...

A(q) = 1 + a1q−1 + a2q

−2

C(q) = 1 + c1q−1 + c2q

−2 + c3q−3 + c4q

−4 + c5q−5

and...

q−n = y(t− n) (ie, value of y at lag n)

e(t) ∼ N(0, 1) (ie, white noise)

y(t) Output time series

a1, a2 Order Coefficients

c1, ..., c5 Lag Coefficients

With the three descriptors for the random variables NE, TD, and TW – which de-

scribe when cloud events occur – and the library of ARMA models for intra-irradiance

shapes – which describe what the irradiance looks like inside a cloud event – a stochas-

tic model can be derived for the solar characteristics affecting a PV array.

4.4.3 Generation of Irradiance for PLF

To generate the irradiance for a given day, first a baseline extraterrestrial irradi-

ance curve is selected from the library of those available for the season in question.

Using the data from Table 4.1, a random variable describing the number of events in

the day is created as shown in (4.15). Similarly, random variables are created for the

event duration (equation (4.16)) and event waiting time (equation (4.17)).

NE ∼ Poiss(λ) (4.15)

TD ∼ exp(µD) (4.16)

TW ∼ exp(µW ) (4.17)

NE is then sampled to create nE. For each event {1, 2, ..., nE}, TD and TW are

sampled to create {tD,1, tD,2, ..., tD,nE} and {tW,1, tW,2, ..., tW,nE}. A check is then per-

formed to see if less than 10% or more than 97% of the day is covered in cloud events

as shown in (4.18); if this is the case, TD and TW are re-sampled until the criteria is

41

met.

0.1 ≤nE∑k=1

tD,k +

nE∑k=1

tW,k ≤ 0.97 (4.18)

With a valid event set created, the time to begin simulating events tW,0 – which

is in essence the wait time between sunrise and the first event – is randomly selected

from the remaining time not mapped to an event less the length of all events. The

time of sunrise is also randomly selected uniformly between the appropriate minutes

of the day for the selected season.

Now that the starting and stopping times for each cloud event have been defined,

the intra-event irradiance shape must be created. A uniform random variable is

used to select which irradiance ARMA model to select. The library of shapes is

relatively vast (for example, the Spring library has nARMA = 74 intra-event ARMA

models) and thus selecting an ARMA model uniformly provides sufficient randomness.

Once selected, an ARMA model is perturbed with white noise to randomly generate

the intra-event irradiance shape. Some simple checks are also performed to ensure

the intra-event irradiance remains above zero and below the ideal irradiance. This

procedure creates I1m as given in (4.19).

I1m(t) = Iideal(t) +

nE∑k=1

Cuk(q)

Auk(q)ek(t) (4.19)

where...

ek(t) =

N(0, 1) for tW,k−1 ≤ t ≤ tD,k

0 otherwise

tW,0 Randomly Chosen Start Time

uk For event k, the sampled value from a uniform

random distribution between 1 and nARMA

An(q), Cn(q) The nth ARMA model where

n = {1, 2, ..., nARMA}

42

Because cloud movements tend to be “slow,” it would be impractical to assume

any large amount of irradiance variation on the sub-minutely level. As such, I1s is

created from I1m through a simple linear interpolation; no random walk is added.

4.5 Performance of Random Variables

As mentioned in the introduction to this chapter, one of the biggest goals of this

author in the production of this work was to capture the time dependent aspects of

resource variations. This has been accomplished through a hybrid approach for each

resource which includes a statistical component for long term (minutely to hourly)

variations dependent on time of day and a time series component which accounts

for the dependence of short term variations (secondly to minutely). The specific

methodologies employed have been explained in detail in the previous sections and

are summarized in Table 4.2.

Panning out to the big picture, the RVs described in this chapter are used in a

PLF (Chapter 5) in a series of tests (Chapter 6) to analyse the effects of DG on

voltage quality (Chapter 7). While a detailed discussion of these other aspects of the

thesis has not yet occurred, in order to understand how well the RVs are performing,

it is necessary to discuss how they fit into the larger work. To do this, a series of

plots known as the “One Trial” plots are shown in Figure 4.11 to Figure 4.14. These

plots show the results of the PLF for a single iteration of the Monte Carlo (ie, one

full day, 86400 seconds) and are meant to exemplify typical and extreme realizations

of the various RVs over time.

The first pane in a plot shows over a dozen voltages from all across the system and

is meant to provide a general feeling for what the voltage regulation and flicker on the

system looks like. The second pane show the powers (both active and reactive) for

all feeders in the system, show the realizations of L1s, and how DG impact the power

flow on the system. The third panel shows how the wind speed W1s (in green) and

solar irradiance I1s (in blue) have been realized for a specific trial. The significance

43

Tab

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44

of the “test numbers” for each figure are enumerated in Table 6.1.

As can be seen in the Figure 4.11 to Figure 4.14, the two-tiered approach of

Table 4.2 has provided several benefits in modeling the dependence between load,

wind, and irradiance.

• Wind tends to blow stronger at night, as commonly observed in Colorado

(see Figure 4.11).

• It is common to see the wind pattern change dramatically at sunrise and sunset,

as would be expected due to the large change in thermals seen at dawn and dusk

(see Figure 4.11).

• Unusually windy days (gale force winds) occur, but extremely infrequently

(see Figure 4.12).

• Unusually cloudy days occur, but relatively infrequently (see Figure 4.13).

• Unusually calm and/or clear days occur more frequently (see Figure 4.14).

• Wind gusts and large cloud movements occur at an intuitively reasonable rate

(see Figure 4.11 to Figure 4.14).

• The lack of a strong overlap between solar and peak load is clear and the affect

of increased ramp rates to meet peak demand apparent. On the distribution

system this manifests itself as a significant number of regulator and capacitor

operations occurring in the later afternoon and early evening (see Figure 4.11

to Figure 4.14).

• Short term rapid variations (wind gusts and dense clouds) cause noticeable

voltage deviations on the system (see Figure 4.11 to Figure 4.14).

45

Figure 4.11: Typical Day: One Trial in Test 4

Figure 4.12: Wind Day: One Trial in Test 9

46

Figure 4.13: Cloudy Day: One Trial in Test 5

Figure 4.14: Calm, Clear Day: One Trial in Test 7

47

4.6 Random Variable Limitations

While effort has been taken to capture some degree of interdependency between

irradiance and wind speed at a time of day level, more macroscopic dependencies

are not modeled. For example, windy days may also be cloudy days because of

macroscopic weather fronts. Unfortunately, this would require a significantly more

comprehensive meteorological model. A point of future work would be to integrate

such a model.

Another limit to the random variables presented is in capturing weather variations

due to geo-spacial considerations. The distribution system proposed is spread over

nearly 200 square miles. Topographical features and weather front movements mean

that the wind speed and irradiance observed by the DG across the footprint are non-

uniform. A methodology which involves modeling the front movements along with

the resource values would be a good addition to this work.

Additional tuning is also needed in the irradiance simulator to account for very

cloudy or very clear days as 1) the event detector has a hard time detecting these

kinds of days and 2) the constraint parameters of equation (4.18) bias the simulator

against these kinds of days.

48

CHAPTER 5

PROBABILISTIC LOAD FLOW ENGINE

To run simulations of a meshed distribution system as necessary when consider-

ing DG, a power flow is necessary. In order to concentrate the research efforts on

more novel aspects, an existing power flow package has been selected. This pack-

age, OpenDSS, is a well proven power circuit simulator maintained by Electric Power

Research Institute (EPRI). It has several advantages over other simulation packages

considered, including:

1. The ability to provide out-of-the-box imbalanced, three-phase power flow.

2. A strong heritage in “smart-grid” applications, including distributed generation.

3. A Component Object Model (COM) interface, which provides the ability for

a third-party software (ie, MATLAB) to interface with the solution engine di-

rectly.

4. An aptitude towards “quasi-static solution modes,” that is, sequential time

simulations.

5. A realistic capacitor and regulator control simulator

In total, the simulation engine is composed of a MATLAB program which creates

the network topology, weather scenarios, load shapes, and parameter variations for

the various simulations; the MATLAB program interacts with OpenDSS’s solution

engine through the COM interface and single-precision binary page files to perform

the simulations.

49

Figure 5.1: Block Diagram of Probabilistic Load Flow Engine

5.1 Key Functions and Variables

At a high level, the operation of the code is illustrated in Figure 5.1. The net-

work topology can also be programmed, in a general form, into MATLAB as the

“Advanced Network Description” (variable an) which is made through the “create-

Topology” fucntion. Once models have been created for wind, PV, and load the

“generateResources” function is called.

All of these generalized variables are then loaded into the OpenDSS environment

via the an2dss and load2dss functions. This allows one, should it be necessary in

the future, to completely replace OpenDSS with a different simulation engine; all

that would need to be created are interface functions for the new solution engine.

Command and small variables are imported and exported through the COM interface.

Bulk data transactions are imported into OpenDSS through single precision binary

files and exported through the COM interface directly.

When the parameters for the simulation have been encoded into the simulation

engine, function “rundss” works with OpenDSS to solve the circuit and retrieve the

necessary results. They are then analysed in “analyzeResults” and output as needed.

50

This process is iterated several hundred times to create a simply sampled Monte

Carlo.

5.2 Distribution System Model

In order to study the effects of distributed generation on the distribution system,

a test system was composed. The goals of the test-bed are:

• To present a realistic test system that both captures the current nature of distri-

bution systems as well as presents opportunities to study future modifications

that may be needed

• To capture all forms of distribution systems, including

– long rural feeders with few customers where voltage regulation is the lim-

iting factor to load capacity and DG penetration

– shorter urban feeder with many customers where amapacity is the limiting

factor to load capacity and DG penetration

– mixed feeders where both ampacity and voltage regulation come into play

– an asymmetrically fed secondary spot network zone

– a sub-distribution system

• To create enough specific information to provide realistic load and DG data,

but keep the provided data general enough that it could be interpreted as a

generalized distribution system.

5.2.1 Topology Description

In order to provide a platform for a realistic distribution system, a specific ge-

ographic location was selected. This allows real distance measurements and demo-

graphic data to be collected. The location selected was Central Weld County, Col-

orado to include the Northern extent of the city of Greeley and the town of Eaton.

51

This was chosen because of the author’s personal experience with the distribution

system in the area, which aids in making the model as realistic as possible. The area

also contains a good mixture of rural, urban, residential, commercial, and industrial

load types. The author wishes to emphasis the system presented in this thesis is not

in any way a true representation of the distribution system in use in Weld County; it

is simply a distribution system designed for the area which conforms, to the best of

the author’s ability, to known utility guidelines and planning philosophies regarding

substation locating, feeder routes, contingency mitigation, capacitor and regulator

placement, and load balancing. However, the system was specially designed to be

“marginal” so that the effects of DG can be more readily studied.

The distribution system centres around Greeley (abbreviated GRLY) substation

with two 30 MVA 69kV-12.47kV transformers placed in a split-bus arrangement. The

transformers are fed from an infinite 69kV bus. From GRLY substation, six feeders

supply the load of the area numbered GRLY 1001 to GRLY 1006; odd numbered

feeders are on bank 1, even on bank 2. Each bus is also provided bus capacitors, 4

stages at 3 MVAR per stage, to provide power factor correction to bring the substation

into interconnection compliance (98% lag on the high side of the transformer). A

oneline of the substation can be seen in Figure 5.2.

Figure 5.2: Oneline Diagram of Greeley (GRLY) Substation

The distribution system is broken into 99 buses with between 5 and 15 buses

per feeder. There are a total of 4 voltage regulators and 12 capacitors. The use of

52

relatively few buses is acceptable because 1) utilities tend to only concern themselves

with the power quality on the feeder main level, unless a specific incident is reported,

and 2) the aggregation of smaller distributed generation into a larger one does not

change the global disturbance level of voltage altering emissions [3]. The system also

includes a spot network area ate the University of Northern Colorado (UNC) campus

and a sub-distribution zone which serves the town of Eaton through the 4kV Eaton

substation (abbreviated EATO). Each feeder tries to capture a different aspect of the

distribution system.

GRLY 1001 – 1466 Residential Customers, 1181 Commercial Customers

This feeder is a generic rural feeder. It serves the Lucern Plant (a large industrial

load) about mid way through its length. It is also the primary feed for the EATO

substation.


This feeder is a long rural feeder. It serves mostly farmhouses (classified as

residential customers). It is about 28 miles long, includes two voltage regulators

and three capacitors. This is the alternate feed for EATO substation.


This feeder is a mixed duty feeder, serving initially urban load via overhead

before entering unincorporated Weld county. It is also quite heavily loaded.


This feeder is an urban underground feeder. It serves mostly residential cus-

tomers and the UNC spot network.


This feeder is an urban underground feeder. It serves the commercial district

and the UNC spot network.

53


This feeder is a lightly-loaded rural feeder. It serves the Weld County Regional

Airport and also serves as the back-door to the system, providing some capacity

for the first contingency.

4kV system – 1136 Residential Customers, 73 Commercial Customers

Composed of two feeders, EATO 471 and EATO 472 from Eaton substation,

this system serves the small town of Eaton, CO. It is present to give the test-bed

a sub-distribution aspect.

A map of these feeders can be found in Figure 5.3.

5.2.2 Demographic Data

In order to create realistic load data, demographic information for the region was

collected from the county records [48]. First, for each bus, a load zone was created

with the centroid of the load region being the bus when possible. These load zones

can be seen in Figure 5.4.

For each load zone, the county records are used to find the number of residential

and commercial properties located in the region. While more detailed information

is available, such as the number of bedrooms and square footage for the properties

as well as detailed building utilization information, mining this data is considerably

more arduous than simply establishing if the property is residential or commercial.

Also, because tens to hundreds of loads are represented by a single bus, and thus

diversity comes into play, such detail is unnecessary.

The loads at theses buses are created by sampling L1s,R or L1s,C , which represent

a single residential or commercial load, for each load at the bus. The loads are then

summed, thus assuming full coincidence, and the net load L1s found for each phase

of each bus. This aggregation is what is placed as a load in OpenDSS. Industrial

customers are modeled as spot loads with known load shapes at a predetermined bus.

54

Figure 5.3: Map of Feeders in Test System

55

Figure 5.4: Map of Load Regions in Test System

56

5.3 Wind Turbine Model

While the description of the wind has been covered in detail in Section 4.3, taking

the simulated wind speed and creating a power is a different matter. In a physical

system, this energy conversion is done by a wind turbine which has dynamics of its

own with time constants on the order of seconds. As such, the wind turbine itself

must be modeled dynamically for the purposes of this work.

The wind turbine model is based on a simplified controller and idealized aerody-

namic torque calculation. The model also assumes the only dynamic introduced by

the turbine is inertial. Despite these simplifications, this model is sufficient to capture

the salient effects of wind variation on power production. As a point of future work,

this model could be expanded to include a better model of the turbine (perhaps

through NREL’s FAST code [49]), generator (perhaps through SimPowerSystem’s

Doubly Fed Induction Generator Model [50]), and controller.

5.3.1 Wind Turbine and Controller Model for Energy Capture

The wind turbine and controller is modeled as simple first-order system as pre-

sented in [51]. This model can be seen in Figure 5.5.

Figure 5.5: Block Diagram of Wind Turbine and Controller Model [51]

This model assumes that the largest contribution to the dynamics of the wind

turbine are the rotor, gearbox, and generator inertia J . Other dynamics are ignored.

As such, the inertia absorbs the difference between aerodynamic torque imposed on

57

the turbine τaero and the torque the controller anticipates (and thus, torque imposed

on the generator) τc. This relationship is shown in (5.1).

Jω = τaero − τc (5.1)

The controller is attempting to keep the tip speed ratio λ as close to the design

tip speed ratio λ∗ as possible. Assuming an ideal controller, the control torque can

be given by (5.2).

τc =1

2ρπR5Cp,max

λ∗3ω2 (5.2)

The aerodynamic torque is calculated from the wind speed u as given by (5.3).

τaero =1

2ρπR3Cp

λu2 (5.3)

Note that the coefficient of performance Cp in (5.3) is a function of the actual tip

speed ratio λ and the blade pitch angle β. The function for Cp as presented in [52] is

shown in (5.4).

Cp(λ, β) = 0.5176

(116

λi− 0.4β − 5

)e−21/λi + 0.0068λ (5.4)

where

λi =

(1

λ+ 0.08β− 0.035

β3 + 1

)−1Finally, the power output of the turbine is given by (5.5).

P = τc ω (5.5)

Using equations (5.1) to (5.5) a wind turbine can be modeled from its basic phys-

ical and design parameters and input wind speed.

5.3.2 Development of Coefficients

In order to create realistic parameters for the wind turbine model, a DG scale wind

turbine was selected and the coefficients reverse engineered from the specifications

sheet. The turbine selected is the Southwest Windpower Skystream 3.7, a 2.4kW

58

turbine marketed for residential applications. The specifications for this turbine are

provided in [53].

Because this is a small turbine, it does not have pitch control. As such, it is

assumed that β = 0◦. The air density ρ for the Greeley area is about 1.0 kg/m2. The

blade length R for this turbine is 1.86 m. Because the specification sheet produces

a rated tip speed and rated wind, the design tip speed ratio λ∗ can be easily found

to be 5.07. Assuming the turbine operates at Cp,max at rated wind speed and rated

power output, Cp,max can be reverse engineered by finding the rated power over the

maximum power extractable from the wind. Doing this yields a Cp,max of 0.201.

The only remaining physical parameter is the rotor, gearbox, and generator inertia

J . This value cannot be derived from information in [53]. To determine this value,

it is assumed that the turbine will reach rated power 10 seconds after a wind speed

step input to rated wind (13 m/s) is applied. This yields a J of 0.25 kg·m2. As a

point of future work, finding a better value for this number is high on the list as it is

one of the most important aspects to determining how short-term fluctuations from

wind affect the distribution system.

Finally, region 3 operation of the turbine – between rated wind speed and cut out

wind speed – is controlled through aerodynamic stalling. This effect is not modeled

by equation (5.3) and as such is captured external to it through a simple linear trend.

To test the performance of the model against the characteristics given in [53], a unit

ramp wind was applied to the model to try to replicate the performance curve as

shown in Figure 5.6.

High speed cut-out (25 m/s) and low speed cut-in (3.5 m/s) are included in the

model a-priori by setting wind speeds outside of these limits to 0 m/s. This causes

some problems in the dynamics near the cut-out speed as the controller does not

anticipate this. The effect of this is readily apparent in Figure 5.6(a). However, this

effect is of little concern as the modeled wind rarely exceeds 25 m/s.

59

(a) Model Performance Curve (b) Performance Curve provided in [53]

Figure 5.6: Performance Curves of Model and Skystream 3.7

The performance of the model showing both region 2 and region 3 operation can

be seen in Figure 5.7 in which the model is perturbed with turbulent wind. The

smoothing effects of the turbine are apparent.

Figure 5.7: Wind Turbine Operation in 10 Minutes of Turbulent Wind

60

5.4 Photovoltaic Array Model

While the description of the irradiance has been covered in detail in Section 4.4,

taking the simulated irradiance and creating a power is a different matter. In a

physical system, this energy conversion is done by a photovoltaic (PV) array which

has dynamics of its own with time constants on the order of milliseconds. Because

of these small time constants, the dynamics of the PV array can be ignored for the

purposes of this work. However, other aspects, such as array/inverter efficiency and

temperature dependence should be taken into account.

Built into OpenDSS is a comprehensive photovoltaic model. This model takes

several constants, a few inputs, and automatically injects into its connected bus the

appropriate power. Figure 5.8, from the OpenDSS documentation [54], shows con-

ceptually how the PV element model works.

Figure 5.8: Block Diagram of the PVSystem Element of OpenDSS [54]

The constants to be set are Pmpp, the rated power output at one sun, the inverter

efficiency shape, the temperature de-rating curve, interconnection kV, interconnection

type, and what VAR control mode should be used (constant VAR, constant power-

61

factor, and in a future version bus voltage regulation).

The variables to be set are T , the temperature of the panel, and I the irradiance

seen by the panel. Both of these values can be input as “loadshape” curves, allowing

the duty-cycle analysis that will be used for voltage and power fluctuation studies to

be performed with ease. The documentation indicates this model is most accurate for

time-steps over one second (as it is assuming the inverter finds the max power point

instantaneously), which is sufficient for the needs of this thesis.

There is however one limitation of this model which needed to be worked around.

Regarding reactive power, the model presently only supports two modes: 1) constant

power factor control and 2) constant VAR control. However, one of one key aspects

of DG integration that needs to be further explored (and is tackled to a degree in this

work) is allowing DG to regulate the voltage of its bus. The creators of OpenDSS are

aware of this shortcoming and are planning to implement a constant voltage control

option in the future; in the interim, all PVSystem objects had to be replaced with

standard generator objects in any scenarios which required DG to perform voltage

regulation.

5.5 Flicker Meter Model

To perform the flicker calculations, a model of a flicker meter was implemented.

The code used to simulate the flicker meter is taken from an open source MATLAB

code titled “FlickerSim” [55]. This code implements a flicker meter as described in the

IEC standards for flicker measurement [56] and produces an estimate of short-term

flicker Pst.

In order to use the flicker meter code, the one second resolution voltage phasor

magnitude samples need to be up-scaled to 2000 Hz sine waves. In this up-scaling it is

assumed that no voltage flicker is occuring below the one second threshold, although

in a real system some flicker due to DG may occur here due to power conditioning

equipment.

62

The measurement is conditioned in a five step process as enumerated below and

shown in Figure 5.9.

1. Input Voltage Adaptor – Any DC component is removed from the signal and

the input measurement normalized with respect to the peak-amplitude value.

2. Quadratic Demodulator – The sampled values are squared.

3. Bandpass and Weighting Filter – High-pass and low-pass Butterworth filters

are applied to the input. A weighting filter is then applied. This step models

the reaction of the human eye and brain to lighting variations.

4. Squaring and Smoothing – Another low-pass Butterworth filter is applied to the

signal after it has been scaled.

5. Statistical Evaluation – A cumulative distribution function (CDF) is formed by

binning the filter signal into 10,000 bins. Percentiles (ie, the percent of time

over a certain threshold) are evaluated from the CDF for equation (5.6) [57].

Pst =√

0.0314P0.1 + 0.0525P1s + 0.0657P3s (5.6)

+ 0.28P10s + 0.08P50s

where...

P1s =P0.7 + P1 + P1.3

3

P3s =P2.2 + P3 + P4

3

P10s =P6 + P8 + P10 + P13 + P17

5

P50s =P30 + P50 + P80

3Pn The nth percentile of the CDF

63

Figure 5.9: Block Diagram of Flicker Meter [56]

It is worth emphasizing this process of extensive filtering and sampling is quite

computationally intensive. In order for the work presented here to become realistically

applied to planning DG integration strategies on distribution systems, a method to

estimate Pst is needed which is not so arduous. The IEC standards provide a method

for flicker estimation [56], however they would need to be adapted for this application.

64

CHAPTER 6

RESULTS

Using the statistical models and load flow engine created in the previous sections, a

series of test scenarios were run and results extracted. The three key results considered

from the simulation include the following random variables’ PDFs/CDFs:

• Voltage PDF – This is the probability of a given bus’s voltage taking on a

value between 0.9 and 1.1 pu at a resolution of ±0.001 pu. The purpose of this

metric is to gauge how much more often over-voltage occurs due to DG.

• Voltage PDF – This is the probability of a given bus’s voltage imbalance

taking on a value between 0% and 5% at a resolution of ±0.04%. The purpose

of this metric is to gauge how DG affects voltage imbalance.

• Flicker PDF – This is the probability of a given bus’s short term flicker value,

Pst, taking on a value at a resolution of ±0.001%. The purpose of this metric

is to gauge how DG affects flicker.

• Flicker Duration Curve (Flicker CDF3) – This is the percent of time (or

probability) that the bus’s Pst is above a given value. It is analogous to a “load

duration curve” commonly used by distribution planning engineers to determine

at-risk loading.4 This curve is formed by taking the Kaplan-Meier estimate of

the CDF for the Pst value from all trials, forming the Pst survival function. The

flicker duration curve is then found by taking the survival function’s complement

3As explained, this is not technically the CDF of the flicker but the CDF’s complement ontransposed axes. The term “CDF” is used purely for convenience.

4It is common practice in distribution planning to evaluate 1) how many hours per year and 2)the level of loading that exists which causes the violation of certain N-0 or N-1 criterion using loadduration curves so that risks can be compared at a system-wide level [58].

65

and transposing the axes so that it resembles a load duration curve in nature

and application.

Note that all PDFs/CDFs described above are technically “empirical distribu-

tion functions,” not “true distribution functions” (the true PDF/CDF can only be

obtained through convolution). However, they are referred to as PDFs/CDFs for

convenience.

6.1 Tests Performed

In order to get a fairly diverse and accurate grasp on the alternate futures that

distributed generation could take on in the electrical system, several parameters of

the system were permuted over in order to gauge the impact DG has on voltage limits,

imbalance, and flicker. These are:

• DG Penetration Level – Penetration levels were varied at 10%, 25%, and

50% increments with a base penetration of 0%. For the purposes of this thesis

penetration level is defined as the “capacity penetration level,” which is the

installed capacity of DG on the system over the peak load demanded on the

system for a given simulation day. It should be noted that due to the relatively

low capacity factors of the solar and wind resources, this is significantly less than

the “energy penetration level” sometimes also use to measure DG penetration.

• Season – Seasons analysed include Winter (December 1st to February 28th),

Spring (March 1st to April 30th), and Summer (June 1st to September 30th).

The base season was chosen to be Spring because, from a DG integration stand

point, it is the most interesting: loads are at their lowest and PV is at its highest

with a fair amount of wind. This should pose the most significant threat to the

system from an over-voltage standpoint.

66

• System Strength – The system strength can be “strong” or “weak.” The

strong system has been sized appropriately for peak load without any DG.

The weak system has had most of the conductor sizes reduced to simulate the

effects of capital deferment due to additional capacity gained through large DG

penetrations on a feeder.

• DG VAR Support – VAR support for the DG can be either enabled or dis-

abled. The baseline assumption, disabled, assumes that all DG are following

IEEE 1547-2008 which prescribes that DG provide no VAR injection and thus

operate at unity power factor [59]. However, there has been some interest in

allowing DG to participate in VAR support [1] and thus regulate the voltage

at their bus just like a conventional generator5. An important issue to address

when considering DG reactive power participation is how much must the in-

verter be oversized. The authors in [6] found a substantial decrease in voltage

imbalance with a modest 16% over-sizing of the DG. To investigate this issue

two reactive power capacities are considered: 20% and 100% of nameplate active

power capacity.

Constrained by time available to run simulations, 14 simulations were run as

demonstrated in Table 6.1. A darkened cell indicates an off baseline parameter.

5Due to the present limitations of OpenDSS, the PV arrays had to be modeled as generatorobjects instead of PVarray objects so that the aspect of bus voltage regulation control could beenabled.

67

Table 6.1: Test Scenarios

TestBattery

Test # DG Pene-tration

Season SystemStrength

DG VARSupport

1 1 None Spring Strong No

2 2 10% Spring Strong No



3 5 10% Spring Weak No



4 8 25% Spring Strong 20%




6 12 50% Winter Strong No

6 13 50% Summer Strong No

7 14 50% Summer Weak 100%

68

Table 6.2: Test Batteries

Test Test DG Penetration Season System DG VAR Title

Battery # Strength Support

1 1 None Spring Strong No Baseline

2 1, 2-4 10% to 50% Spring Strong No Effects of DG on Baseline

3 1, 5-7 10% to 50% Spring Weak No Capacity Deferment due to DG

4 1, 8-9 25% to 50% Spring Strong Yes DG with Voltage Regulation(20% VAR Capacity)

5 1, 10-11 25% to 50% Spring Strong Yes DG with Voltage Regulation(100% VAR Capacity)

6 1, 4, 12-13 50% Summer & Winter Strong No Effects of Season on DG

7 1, 7, 11, 14 50% Summer Weak Yes Extreme DG Integration

69

Each test is composed of 600 simulation trials which – based on experimentation –

seems to be an appropriate value for convergence of all the PDFs. On the 8 core, 3.2

GHz, 32 GB RAM server the simulations were run on, a trial with flicker calculations

takes about 40 minutes while without flicker calculations the trial takes about 5

minutes. Because the flicker PDFs converge faster than the other PDFs and the

flicker calculation is so computationally intensive, flicker is only found for the first

180 of the 600 trials.

The tests in Table 6.1 can be grouped into six batteries of tests as shown in

Table 6.2. Each battery is looking into a specific aspect of DG integration while

holding the other aspects constant. These test batteries form the basis for the result

analysis presented in the following sections.

6.2 Monitoring

All scenarios include the measurement of 15 bus voltages. These voltages are the

end of feeder voltages for all eight feeders plus the buses before voltage regulators for

the four regulators on the system. A voltage monitor is also included at the UNC

bus for the two feeders (GRLY 1004 and GRLY 1005), which is the approximate

midpoint of both feeders. Finally, because GRLY 1006 serves North and South of the

substation, an extra monitor is added so that both endpoints can be monitored.

For flicker measurements, the number of monitored buses is reduced to 11 to speed

computation time. The removed monitors are both 4kV sub-distribution feeders and

both GRLY 1005 monitors because it so closely mirrors GRLY 1006.

Recall the following salient aspects of the test-bed: GRLY 1001 and GRLY 1002

are the overhead rural feeders. GRLY 1003 is an overhead mixed urban and rural

feeder. GRLY 1004 and GRLY 1005 are underground urban feeders. Finally, GRLY

1006 is a lightly loaded feeder that serves a large industrial load. For reference, a

simplified diagram of the test feeders is shown in Figure 6.1.

70

Figure 6.1: Simplified Diagram of Test System Feeders

71

Due the quantity of large figures needed to support the results, most of the figures

for this section are included in Appendix A which starts on page 99.

6.3 Baseline

For the baseline test battery, the distribution system was simulated during Spring

with standard loading but no DG interconnected.

6.3.1 Voltage Considerations

In the baseline scenario, the voltages for all feeders are between the ANSI C84.1

range of acceptable voltages for medium voltage primary distribution feeders, 0.95 to

1.05 pu. This can be clearly seen in Figure A.1. Observe that the voltage PDF is

quite distinct between the different feeders but is actually quite similar for buses on

the same feeder. It is also common for the voltage PDF to have two “humps,” which

are direct reflections of the bimodal nature of the load (see Figure 4.1(a)). This is

especially apparent in feeders without voltage regulators or at the meter before the

first voltage regulator. It is clear then that the voltage regulator has a significant

impact on the shape of the voltage PDF.

The imbalance PDF for the baseline scenario is given in Figure A.2. Note that all

feeders have below 1.5% voltage imbalance at all times and most feeders tend to have

below 0.5% most of the time. The biggest reason that some of the PDFs are more

densely concentrated than others (specifically, “GRLY 1002: Before Reg 1,” and the

GRLY 1004, 1005, and 1006 plots) is due to large spot loads on these feeder, which

are assumed to be perfectly balanced.

6.3.2 Flicker Considerations

The only sources of flicker on the baseline are the regulators and capacitors switch-

ing due to load variations. The baseline flicker CDF and PDF can be seen in Fig-

ure A.3 and Figure A.4. The largest flicker observed on any feeder is Pst = 0.45 on

72

“GRLY 1002: End of Feeder.” Note that the feeders with regulators tend to have

higher flickers than those without. Also note that the flicker probabilities in Fig-

ure A.4 tend to clump around ≤ 0.02 (which is the “ambinet” flicker and is, for all

intents and purposes, no flicker), 0.03, and 0.07; these are the flicker levels created

by capacitor and regulator switching. Thus an increase in probability density around

these values signifies increased device switching. Flicker limits are never over tolera-

tion limits (Pst = 1.0) and the flicker is greater than ambient only between 1% and

12% of the time. The end of GRLY 1002, which is the bus most prone to large flicker

values sees a flicker over ambient about 9% of the time.

6.4 Effects of DG Penetration on Baseline

To look into the effects of DG penetration, the test system was simulated during

Spring with standard loading and DG interconnected at 10%, 25%, and 50% capacity

as a ratio of peak demand.


For this scenario, all of the bus voltages remained within ANSI C84.1 limits, as

seen in Figure A.5. For urban feeders, an increase in DG corresponds to a shift in the

probability density (for example, the mode (peak) of “GRLY 1004: End of Feeder”

PDF shifts from ≈1.01 pu to ≈1.035 pu as the amount of DG on the feeder goes

from 0% to 50%). Contrast this with the rural feeders which – instead of manifest-

ing the increased DG penetration with mode shifts – manifest it through increased

probabilities around the existing mode (for example, the mode of “GRLY 1001: End

of Feeder” is at about 1.03 pu for all scenarios, but the probability at that voltage

increases from 0.05 to 0.11). As “GRLY 1001: Before Reg” shows, this effect is most

likely do to the voltage regulators.

Note that the PDFs’ extreme values (minimum and maximum voltages) do not

significantly change for most buses as DG penetration is increased. This is an ex-

73

tremely important result for system planning as it indicates that DG penetration is

unlikely to cause voltages to go significantly out of bounds. Instead what tends to

occur is the shape of the PDF changes between set bounds.

There are some exceptions to this trend (the end of GRLY 1001 and the 4kV feed-

ers): in these cases the minimum observed voltage shifts upward while the maximum

stays the same. The shape of the PDF also change significantly as DG penetration is

increased. The reason this might occur is unknown at this time, however it is worth

noting that all three of these buses are within a mile of each other.

The voltage imbalance as seen in Figure A.6 does not seem to change significantly

as DG penetration increases.


The flicker duration curve and flicker PDF can be seen in Figure A.7 and Fig-

ure A.8. Like voltage, a pretty clear distinction can be drawn between the urban and

rural feeders and how they manifest flicker as DG penetration increases. For rural

feeders, on the buses after a voltage regulator, the flicker associated with regulator

tap changes shifts to higher and higher probabilities (on the CDF, this looks like a

shift in the “transition point;” on the PDF, this looks like an increase in probabilities

around the Pst value’s associated with regulators). For urban feeders, higher flicker

values see a higher probability of occurring as DG penetration increases. While this

effect is also visible on the rural feeders, it is more pronounced on urban feeders.

It seems that feeder length and loading per unit length may also be an aspect in

flicker intensity in the presence of distributed generation. The short, lightly loaded

GRLY 1001 and 1006 have less flicker problems than the long GRLY 1002 and heavily

loaded GRLY 1003. This may also be because more densely loaded feeders also have

more DG on them due to how DG is allocated across the system.

74

6.5 Capacity Deferment due to DG

To look into the effects of capacity deferment that might occur due to large DG

penetrations, the test system was simulated during Spring with standard loading

and DG penetrations, but with all conductors downgraded by one to three sizes.

Conductor sizes are given in Table 6.3. This emulates the effects of a utility choosing

to defer capital investment on their system because of the capacity DG provides.

Table 6.3: Conductor Sizes on Test Systems

Level Overhead Underground

1 795 AAC, 397 Neutral 750 CU XLPE

2 556 AAC, 2/0 Neutral 500 CU XLPE

3 266 AAC, 2/0 Neutral 250 CU XLPE

4 2/0 AAC, 2/0 Neutral 2/0 AL XLPE

5 #2 ACSR, #4 Neutral #2 AL XLPE

6† #4 ACSR, #4 Neutral —

† Only used in weakened scenarios


For this scenario, several of the voltages were below the ANSI C84.1 limit of 0.95

pu, as seen in Figure A.9. Several of the voltage PDFs change shape compared to

the base case, although voltages which tend to have a bimodal shape continued to

have such a shape, usually with a widening between the two “humps” (as in “GRLY

1006: South End of Feeder”). As DG penetration increases, the effects of the weak

system are much the same as on the strong system – as observed in section 6.4.1 and

Figure A.5 – but with more dramatic results. It is clear that the benefits and risks

granted by DG in voltage are amplified in a weak system.

Regarding voltage imbalance as seen in Figure A.10, weakening the system tended

to increase the voltage imbalance at all buses. However, DG penetration levels con-

75

tinued to show little effect on voltage imbalance.



ure A.12. By weakening the system, flicker levels have increased. However, unlike

in the case of DG impacts on the strong system – where flicker tended to increase

the probability of voltage regulators creating flicker – in the weak system most flicker

appears to be the direct result of resource variation. Even in cases where regulators

operated more often, increasing the probability density at the regulator prone Pst

locations, the “humps” tended to be elongated instead of rising due to both regulator

operation and resource variation. This effect can most easily be seen in “GRLY 1002:

End of Feeder.”

Again, most of the same observations that were made about DG impacts in section

6.4.2 regarding DG penetration impacts on flicker can be made for these scenarios, but

with amplified results. It seems then that capacity deferment will weaken the ability

for a utility to provide high power quality in the presence of distributed generation.

6.6 Effects of DG Participation in Voltage Regulation

To look into the effects of VAR participation by DG, the test system was simulated

during Spring with standard loading and DG interconnected at 25% and 50% capacity

as a ratio of peak demand. However, where as previously the DG were in power factor

control mode regulating to unity, the DG were set to bus voltage regulation control

mode with bus regulation at 1.04 pu. Two VAR capacity levels are considered, 20%

and 100%. This is to say the generators or inverters for the DG are oversized by an

additional 20% or 100% so that they can provide VARs.

76


For this scenario, bus voltages remained within ANSI C84.1 limits most of the

time, as seen in Figure A.13, however there are some cases where the bus voltage

exceeded the 1.05 pu upper threshold. A key observation to make about the voltage

PDFs in the VAR case over the regular penetration case is that the extreme values

taken on by the PDF actually shift as penetration or VAR capacity are increased. In

Figure A.5, the PDFs changed shape constrained within the extrema of the baseline

scenario; contrast this with Figure A.13, where the shape and extrema change.

It is thus important to recognize that, should DG be allowed to participate in

VAR regulation, their effect on the feeder voltage is significantly more pronounced

than when at unity power factor. This poses a double-edged sword to the industry:

while VAR regulating DG are more effective at maintaining an ideal voltage than

traditional regulation devices, they need to be careful controlled. What happens if

this control is not exercised is most easily seen in “GRLY 1003: Before Reg 1.” In

this case, at high DG penetration and VAR participation, it is quite common for the

voltage to exceed the 1.05 pu threshold. This seems to be because DG in one part of

the feeder are supplying large quantities of VARs to try to regulate their bus’s voltage

while the DG at the other end of the feeder do not have sufficient capacity to sink all

of the VARs created to keep the bus voltage below 1.05 pu.

However, the benefit of keeping the voltage profile level across the feeder is readily

visible in the contrast between the voltage profile in Figure 6.2(a) – which varies widely

as the day progresses due to resource (especially load) variations – and the voltage

profiles in Figure 6.2(b) – where the voltage profile is quite level except during the

peak hour or two of the day. It is clear then that voltage regulation is improved with

DG VAR Support.

Regarding voltage imbalance as seen in Figure A.14, VAR supporting DG have

little impact on imbalance, but it is more impact than the previous tests consid-

77

(a) One Trial at 50% Penetration

(b) One Trial at 50% Penetration, 100% VAR Capacity

Figure 6.2: Voltages, Powers, and Resource Values for VAR Support Extremes

78

ered. It appears that increasing the VAR capability of the DG fleet increases voltage

imbalance, which is a counter-intuitive result.



ure A.16. For most of the feeders, the flicker level at the buses increased as more DG

was available to provide more VAR support. However, on GRLY 1002 the highest

flicker levels were observed for the “50% penetration, 20% VAR” scenario. The cause

for this is unknown. Increased flicker levels were mostly due to increased tap opera-

tions, as is apparent by taller “humps” in Figure A.16. However, some scenarios did

create new flicker sources, such as the “25% DG, 100% VAR” test in GRLY 1001 and

1003. This is likely due to capacitor switching which were not previously switching

on the feeder, although this cannot be said conclusively unless deeper analysis of the

results is performed.

6.7 Effects of Season on DG

To look into the effects of seasonality on DG and the distribution system, the test

system was simulated during Spring,6 Winter, and Summer with standard loading

and DG interconnected at 50% capacity as a ratio of daily peak demand. The intent

of these scenarios is to capture how seasonal variations in the resources affects the

distribution system. Some of the salient differences posed by seasons – such as the

difference in load shapes, the fact that Summers tend to be less windy, and the days

are shorter in the Winter – are exemplified in Figure 6.3.


For this scenario, bus voltages remained within ANSI C84.1 limits most of the

time, as seen in Figure A.17, however there are some cases where the bus voltage

6The 50% Spring penetration data has already been created in Test 4 and is repeated in theseresults for convenience

79

(a) One Trial During the Winter

(b) One Trial During the Summer

Figure 6.3: Voltages, Powers, and Resource Values for Summer and Winter

80

exceeded the 1.05 pu upper threshold, especially during the summer. In general,

Summer and Winter voltages tended to have wider PDFs than the Spring case, with

Summer being the widest. The results reflect anticipated differences between seasons

and the penetration of DG has done little to change the general trends.

The voltage imbalance, as seen in Figure A.18, sees little variation across the sea-

sons. However, in general, load is more more imbalanced during the summer and more

balanced during the winter, although the variation is practically indistinguishable.



ure A.20. Regardless of season, flicker levels (both flicker duration and peak Pst) are

increased over the baseline due to the presence of DG. Summers tend to experience an

increased probability of high Pst values while winters tend to experience an increased

probability of lower Pst values. This may simply be a reflection of the fact that the

system is in general “weaker” during the summer and thus more susceptible to higher

flicker values. The flicker continues to manifest itself as either larger, concentrated Pst

values from increased tap changer operations or smaller Pst values caused by resource

variations.

6.8 Extreme DG Integration Scenario

To look into the effects of an extreme DG integration, the test system was simu-

lated during Summer with standard loading on a weak system with DG interconnected

at 50% capacity with 100% VAR support. Tests 7 and 11 are also included for conti-

nence in comparison. The idea of this test is to see how much different a feeder which

relies on DG as an integral asset to maintaining acceptable voltage is compared to a

feeder which provides all of its own support, as is the case with the baseline scenario.

The susceptibility of this configuration is exemplified in Figure 6.4.

81

Figure 6.4: One Trial of the Extreme DG Scenario


For this scenario, bus voltages violated ANSI C84.1 limits fairly often, as seen

in Figure A.21. Compared to the 50% penetration, weak system case the addition

of VAR support seems to be helping the feeder considerably, shifting nearly every

bus up at least 0.005pu. The shape of the voltage PDF rarely resembles any of the

other PDFs and even the bi-modal nature that was so prevalent in previous PDFs

is diminished; this is likely caused by a combination of weak system and summer

loading. Most interesting is the fact that both high voltage and low voltages are

readily found on different buses in the system and that general range of voltages a

specific bus takes on is quite large.

Regarding voltage imbalance as seen in Figure A.22, the extreme scenario usually

takes a shape close to the 50% penetration, weak system scenario, indicating a large

voltage imbalance. Interestingly, it is common for the voltage imbalance to be worse

82

in the extreme scenario, despite the fact it should improve due to the increased VAR

support on the feeder.



ure A.24. The flicker in the extreme scenario is improved over the weak scenario

quite noticeably causing a reduced flicker duration. Close analysis of the flicker PDF

indicates this is due to a reduction in indirect flicker; this effect is most easily seen in

the GRLY 1002 feeder’s PDFs. Direct flicker is nearly identical in the extreme and

weak cases.

83

CHAPTER 7

CONCLUSION

Through the integration of stochastic models of the three major resources on the

distribution system – load, wind power, and solar power – with probabilistic load

flow techniques, a methodology has been developed which can analyse and quantify

the voltage quality risk which distributed generation pose to the electric distribution

system. The results indicate that DG have strong effects on voltage regulation devices

and the interaction between the two can lead to voltage quality issues. However, when

DG is allowed to provide VAR support, their ability to affect the voltage profile for

the better is profound, especially in reducing the impacts DG may cause in reduced

capacity due to deferred capital investment. Finally, while flicker levels do increase

as more DG is integrated into the system, it is unlikely flicker will reach intolerable

levels even at high penetrations.

7.1 Quantification of Voltage Quality Risks

Given the assumptions made regarding the test-bed, resource quantities, and ran-

domness models several conclusions can been drawn regarding the voltage quality

risk of DG. These risks are enumerated in Table 7.1 which shows the minimum (if

applicable), maximum, and average bus voltage, percent voltage imbalance, flicker

value, and percent of the time an over-ambient flicker (OAF) is observed. Note that

while an over-ambient flicker value is defined as any flicker greater than or equal to

0.02, a large over-ambient flicker does not necessitate that there is a flicker problem

on the system. For there to be a detectable, harmful flicker problem, Pst would need

to be greater than 1.0, which never occurs in any scenario. As such, DG do not pose

a significant flicker problem.

84

Tab

le7.

1:V

olta

geQ

ual

ity

Met

rics

Acr

oss

all

Fee

der

sfo

rT

ests

Run

Tes

t#

Tes

tT

itle

Vol

tage

Lim

it(p

u)

%Im

bal

ance

Pst

Val

ue

%O

ver

-Am

bie

nt

Fli

cker

1

Min

Avg

Max

Avg

Max

Avg

Max

Min

Avg

Max

1B

ase

lin

e0.9

570

1.01

891.

079

0.28

963.

280.

0086

0.46

590.

3535

4.25

9513

.813

3

210%

DG

0.95

61.

0199

1.07

900.

2796

2.6

0.00

880.

4635

0.34

574.

8380

19.3

762

325%

DG

0.9

550

1.02

051.

0790

0.27

473.

00.

0089

0.46

200.

6798

7.12

6021

.031

1

450%

DG

0.9

570

1.02

191.

0730

0.27

242.

80.

0094

0.47

240.

6876

9.69

0525

.455

7

510%

DG

Wea

k0.9

040

0.99

551.

0780

0.48

715.

0†0.

0102

0.33

600.

9867

11.7

585

46.5

329

625%

DG

Wea

k0.

9‡0.

9966

1.08

0.47

325.

0†0.

0108

0.36

111.

1964

14.7

641

51.6

610

750%

DG

Wea

k0.9

080

1.00

051.

0890

0.46

5.0†

0.01

260.

6053

3.28

6319

.820

560

.085

4

825%

DG

,20

%V

AR

0.9

590

1.02

161.

0690

0.26

123.

120.

0087

0.46

250.

5322

4.49

2617

.022

1

950%

DG

,20

%V

AR

0.9

610

1.02

381.

0830

0.24

803.

040.

0089

0.46

021.

3363

6.17

3523

.078

3

10

25%

DG

,10

0%

VA

R0.9

640

1.02

751.

0660

0.24

433.

040.

0087

0.51

191.

7169

5.43

0217

.771

8

11

50%

DG

,10

0%

VA

R0.9

830

1.03

321.

0760

0.29

602.

840.

0089

0.43

061.

0333

6.18

7319

.838

5

12

50%

DG

,W

inte

r0.

931.

0185

1.08

700.

3197

4.2

0.01

030.

4472

0.00

7813

.093

737

.621

9

13

50%

DG

,S

um

mer

0.9

330

1.01

781.

0910

0.34

284.

280.

0096

0.35

730.

4545

11.2

422

29.1

770

14

Extr

eme

DG

0.9

0‡1.

0149

1.10†

0.58

495.

0†0.

0112

0.56

632.

1054

17.6

994

51.2

219

1O

ver-

amb

ient

flic

ker

(OA

F)

isa

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85

In general the following observations are made regarding voltage quality metrics as

DG penetration levels increase. First, from a voltage limit perspective, as penetration

levels increase, the average voltage on the system goes up, the maximum voltage (with

some exceptions) goes up slightly, but the minimum voltage is stable. From a voltage

imbalance perspective, as penetration levels increase, the average imbalance decreases

but the maximum imbalance is unaffected. For flicker, as penetration levels increase,

the average Pst value, average OAF, and maximum OAF increase and the changes

in OAF are especially dramatic. These results match the literature and intuition on

DG impacts.

Contrasting the weak system with the strong system, another set of intuitive

results are found. In general, while keeping in step with the effects of DG penetration,

the weaker system tends to have lower voltage metrics, higher imbalance metrics, and

higher flicker metrics. Again, the result of weakening the system on OAF is dramatic,

causing the weaker system to have more than double the OAF of the strong system

for equivalent DG penetrations.

Some of the results for utilization of VAR injection in DG are surprising. One

would expect as more VAR capacity is available to the system the minimum voltage

limits to rise, the maximum voltage limits to fall, and the average to approach the

voltage set-point of 1.04pu. This is the case when comparing metrics within the VAR

support tests, however when comparing the VAR tests with the DG penetration tests

it is observed that the VAR support tests’ maximum voltage limit increase over the

unregulated DG penetration tests’. While more study is needed in order be conclusive,

this author speculates this is do to an inconsistent amount of VAR injection across the

feeder; some DG are supplying VARs to the system while others are trying to sink the

VARs created by the other DG. Another surprise is in the effects of VAR regulating

DG on voltage imbalance. While the average imbalance on the feeder went down as

more VAR is available, the maximum voltage imbalance went up. The reason for

86

this is unknown. Finally, for flicker, the average flicker remained relatively constant

regardless of DG VAR support, however the maximum Pst value decreases as more

VAR support is available, as expected. Even so, the OAF metrics increased as VAR

support is added to the feeder, but the OAFs for tests without VAR support are

worse than the OAF with VAR support. This result suggests there may be a “saddle

point” minimum in the OAF contour as DG penetration and VAR support rating are

varied.

Regarding seasonality, some interesting results are observed for the metrics be-

tween Spring, Winter, and Summer. Spring is the season with the worst minimum

voltage, average voltage, and maximum Pst. Winter has the worst average Pst and

OAF. Summer has the worst voltage imbalance and maximum voltage. These re-

sults confirm some of the literature while also raising new questions. It is believed

that Spring is the most challenging time from a DG integration perspective because

loads are at their lowest and DG is at its highest. And while Spring does pose some

challenges, the fact the largest maximum voltage is observed during the Summer is

surprising and more research is needed to determine why this is the case. Also in-

teresting is that Winter has most of the worst flicker problems (and Spring the best

OAFs), although they are not significantly worse than the other seasons. At the least,

this study brings forward the recommendation that all seasons, not just Spring, be

considered when performing DG integration studies.

Finally, the extreme DG scenario has shed some light into the bigger picture

of DG integration. This scenario has by far the widest voltage limits of any test

performed. However, the average voltage is a full 0.015 pu higher than test 7, the

50% penetration weak DG test which demonstrates the value of VAR injection in

a weak grid. The imbalance metrics are also the highest of any test performed.

Regarding flicker, the only test with worse Pst and OAF values was test 7, but the

fact that improvement was made over test 7 again extols the virtues of DG voltage

87

regulation. However, when one considers the cost of providing the extreme scenario’s

level of VAR regulation compared to the meager benefit over test 7, it seems a more

intelligent VAR regulation scheme is necessary. Regardless, one must conclude that

DG, when equipped with VAR regulation, are capable of mitigating some of the

voltage quality risks they inflict.

7.2 Generalized Effects of DG

From the results and conclusions presented above, some generalized conclusions

about how DG affects voltage quality are drawn.

• VAR Impacts – There is much interest, and much concern, in the industry

regarding allowing DG to participate in the voltage regulation of a feeder. This

work exemplifies why allowing DG to participate in VAR regulation is in need of

careful, unprejudiced consideration. On one hand, the effect of VAR regulating

DG to help shape the voltage profile of a feeder is profound and the possible

benefits vast. However, it is clear that without careful, centralized management

of the VAR injections it is easy to get unexpected, less-desirable results in

voltage quality. How to manage this challenge will be a defining aspect of the

industry in the 21st century.

• Capital Deferment due to DG – The analysis performed indicates that the

degradation of voltage quality due to capacity deferment spurred by the excess

capacity provided by DG can be mitigated partially by the DG itself, if the DG

is allowed to provide VAR support.

• Direct and Indirect Flicker – The flicker PDFs presented in Appendix A

show how the increased flicker DG create on a system tend to clump in one

of two places: near low flicker values and near existing “humps” in the flicker

PDF. These correspond to direct and indirect flicker.

88

– This work demonstrates that DG’s direct flicker is unlikely to have a major

impact as it concentrates near lower Pst values, although the Pst value’s

magnitude and probability do increase noticeable as DG penetration in-

creases.

– Of greater concern should be the impact of indirect flicker. Not only do

increases in indirect flicker occur at higher Pst values, it is also the ma-

jor driver in a large OAF, and signifies an increased operation of voltage

regulator tap changers and capacitor switches. This opens the door to

a centralized, intelligent control scheme for voltage regulating devices on

feeders with high penetrations to increase device life and reduce the most

noticeable aspect of DG induced flicker.

• Rural vs Urban Impacts – The results indicate that urban feeders are likely

to see DG flicker impacts in the form of direct flicker. Urban feeders’ average

voltage will also rise as more DG is integrated on the system. Contrast this with

rural feeders, which are likely to see DG flicker impacts manifest themselves as

indirect flicker. Rural feeders’ average voltage magnitude will stay the same

but the probability of this voltage occurring will increase, meaning voltages

tend towards the mode instead of the extreme. The key difference between

urban and rural feeders which creates this contrast is the presence of voltage

regulators on rural feeders.

• Flicker Impacts – Even with the increase in indirect flicker, the effects of DG

never cause the flicker on a feeder to exceed the limit of 1.0. The largest Pst

observed was 0.6053, which occurred during the most susceptible test (Test 7:

50% DG, Weak System). It is worth noting this is the flicker on the feeder main

and that the flicker on the laterals and customer services is likely to be higher.

89

• Voltage Limit Impacts – As observed by other researchers utilizing PLFs

for DG impact studies, the impacts of non-regulating DG on voltage are sig-

nificantly less than a DLF indicates when making worst-case assumptions. In

fact, as discussed previously and as the figures in Appendix A demonstrate, the

extreme voltage values change little as more DG is integrated into the system.

This work confirms that a PLF is a more realistic approach to assessing the

voltage limit risk DG pose.

• Voltage Imbalance Impacts – While in contradiction to other researchers’

findings (notably [6]), DG seems to pose little risk to the voltage imbalance on

a system. An exception is if DG provide VAR support, in which case the DG

risk increasing the voltage imbalance if not properly controlled.

7.3 Questions Remaining and Future Work

While the work presented in this document has attempted to make significant

strides into determining the voltage quality risk DG pose to the distribution system,

much room for improvement and expansion remains. At a least, the following avenues

of future work exist:

• Further verify and improve the resource models so that they more accurately

reflect the weather and load

– The use of mixed distributions in the wind model is likely to increase its

fidelity at replicating a specific wind resource

– The introduction of conditional probabilities in the solar model could be

used to more easily model extreme events – like a large number of cloud

events in a day or quickly successive cloud events – which are causing some

of the poorness of fit in the solar RVs

90

– The use of joint probabilities to model any relationships between the re-

sources

• Verify the simple turbine model’s inertial component is accurate, or replace with

a more comprehensive wind turbine model

• Extend the test-bed to include laterals, where flicker values are more apparent

• Further research is needed into why the voltage imbalance effects found are in

opposition to other’s findings

• Modeling of electric vehicles and/or community attached storage devices and

analysis of how they could be used to mitigate the voltage quality effects found

in this work

• Extending the VAR research to include more tests, especially test on the weak

system

• Extending the VAR research to include centralized (or otherwise more intelli-

gent) DG VAR control

91

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APPENDIX A - DETAILED RESULT FIGURES

Detailed figures illustrating the results of the test batteries performed are given

in this appendix. These are provided as a direct supplement to the discussion in

Chapter 6.

Each figure in this appendix contains an empirical distribution function for the

15 buses (11 buses in the case of flicker) monitored in the simulations. The first pane

is the key to all the other panes in the figure and contains the overall title of the

distribution function displayed, a legend for the various test in the test battery, and

the x-axis and y-axis axis labels.

For each test battery run a separate subsection is created and a series of four

figures are presented. These four figures are

1. Voltage PDF – This figure shows how probable a per-unit voltage level is on a

given bus. There are x-axis graduations at 0.95 and 1.05 pu to signify the ANSI

C84.1 limits. The x-axis and y-axis are consistent across all panes in the plot.

Ideally, all probabilities are clustered close to each other at a value between the

ANSI C84.1 limits.

2. Voltage Imbalance PDF – This figure shows how probable a voltage imbal-

ance is on a given bus. Graduations are given out to a 1.5% voltage imbalance.

The x-axis and y-axis are consistent across all panes in the plot. Ideally, all

probabilities are clustered close to 0% imbalance.

3. Flicker CDF – This figure shows for what percent of time (ie, how often) the

flicker level on a given bus is at or above a given Pst value and can be thought

of as a “flicker duration curve.” This graph is read by choosing a specific flicker

level on the y-axis, seeing where it meets the curve, and reading down to the

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x-axis to find out how often a flicker that bad or worse is on the bus. The x-axis

is consistent across all panes in the plot, however the y-axis varies from pane to

pane. Ideally, the graph has a small peak flicker value (the top left most point

on the graph) followed by a quick, sharp decay to Pst = 0.

4. Flicker PDF – This figure shows how probable a specific flicker value is on a

given bus. The purpose of this plot is to see how increased flickers are mani-

festing themselves on the system: direct flicker will cause a widening in the left

side of the figure; indirect flicker will cause a raising of the “humps” at which

voltage regulator switching occurs. The x-axis and y-axis are consistent across

all panes in the plot. Ideally, all probabilities are clustered close to the left side

of the graph.

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A.1 Baseline

Figure A.1: Baseline Voltage PDF

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Figure A.2: Baseline Imbalance PDF

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Figure A.3: Baseline Flicker CDF

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Figure A.4: Baseline Flicker PDF

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A.2 Effects of DG Penetration on Baseline

Figure A.5: DG Impact Scenarios’ Voltage PDF

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Figure A.6: DG Impact Scenarios’ Imbalance PDF

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Figure A.7: DG Impact Scenarios’ Flicker CDF

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Figure A.8: DG Impact Scenarios’ Flicker PDF

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A.3 Capacity Deferment due to DG

Figure A.9: Weak Scenarios’ Voltage PDF

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Figure A.10: Weak Scenarios’ Imbalance PDF

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Figure A.11: Weak Scenarios’ Flicker CDF

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Figure A.12: Weak Scenarios’ Flicker PDF

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A.4 Effects of DG Participation in Voltage Regulation

Figure A.13: DG VAR Support Scenarios’ Voltage PDF

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Figure A.14: DG VAR Support Scenarios’ Imbalance PDF

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Figure A.15: DG VAR Support Scenarios’ Flicker CDF

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Figure A.16: DG VAR Support Scenarios’ Flicker PDF

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A.5 Effects of Season on DG

Figure A.17: Seasonal Scenarios’ Voltage PDF

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Figure A.18: Seasonal Scenarios’ Imbalance PDF

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Figure A.19: Seasonal Scenarios’ Flicker CDF

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Figure A.20: Seasonal Scenarios’ Flicker PDF

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A.6 Extreme DG Integration Scenario

Figure A.21: Extreme Scenario’s Voltage PDF

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Figure A.22: Extreme Scenario’s Imbalance PDF

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Figure A.23: Extreme Scenario’s Flicker CDF

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Figure A.24: Extreme Scenario’s Flicker PDF

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