distribution network state estimation time ...distribution network state estimation, time dependency...

DISTRIBUTION NETWORK STATE ESTIMATION, TIME DEPENDENCY AND

FAULT DETECTION

Mehdi Shafiei B.Sc and M.Sc in Electrical Engineering

A Thesis Submitted in fulfilment of the requirements for the degree of

Doctor of Philosophy

Electrical Engineering and Computer Science School

Science and Engineering Faculty

Queensland University of Technology

Queensland, Australia

2019

Introduction 1

2 Chapter 1: Introduction

Keywords

Medium voltage distribution networks, Low voltage distribution networks, Photovol-

taic energy, State estimation, Distribution state estimation, Forecasting-aided state es-

timation, Conditional multivariate complex Gaussian distribution, Spatial-temporal

correlation, Pseudo measured data, Kalman filter, Augmented complex Kalman filter,

Customer loads aggregation, Multi-layer distribution state estimation, Fault detection,

Quantile regression, Instantaneous and define time thresholds.

Introduction 3


Abstract

In the traditional power networks, electricity is generated in power plants and

through transmission and passive distribution networks, it is delivered to customers.

However, due to environmental concerns, carbon emissions and energy crisis, govern-

ments are considering incentive policies to install PV rooftops, and for investors to

finance grid connected renewable farms projects.

Brisbane, the capital of Queensland is the third most populous city in Australia.

For Brisbane, PV energy is considered as the most propitious kind of renewables with

the average of 261 sunny days in a year, estimating to 2884 yearly hours of bright

sunshine.

The transition from a passive distribution network to an active one increases un-

certainty and variability in both PV generation and customer load sides. This problem

can potentially introduce various planning and operational concerns. Monitoring node

voltages, branch currents, and real data are the initial requirement for any further anal-

ysis and evaluations for any sound and stable planning and operation. However, due

to significant cost and infrastructure investment requirements for upgrade in distribu-

tion networks, installing monitoring devices is not a practical solution. Hence, consid-

ering the technical issues and the nature of requirements at distribution level, an accu-

rate and fast estimating model for distribution networks with low number of monitor-

ing devices can be a very cost effective and useful alternative.

In the last two decades, distribution state estimation (DSE) algorithms have been

applied to distribution networks estimating the states of unmonitored nodes. However,

most available approaches have mainly focused on the snapshot algorithms at a single

time instant using the pseudo measurements to approximate the customer loads. The

Introduction 5

main shortcoming of snapshot algorithms is ignoring the high time dependency factors

that exist in both customer loads and PV generations. Inclusion of this time depend-

ency of the measuring data will improve the accuracy of DSEs. Another important

factor in designing DSE algorithm is the computational time. Distribution networks

with considerable number of customer loads require DSE algorithms with low compu-

tational process time to estimate the states for different applications.

In order to consider the impact of time based data correlation in DSE formulation

two DSE algorithms with low computational time are developed in this thesis, which

account for the time dependency information of the measurement to improve the ac-

curacy of the estimated states. Inclusion of this approach for higher accuracy may in-

crease the computational time, while distribution networks require fast DSE algo-

rithms for online control and monitoring purposes. Therefore, additionally, customer

loads aggregation and layering structure are considered in the developed DSE algo-

rithms in this research thesis to decrease the estimation process time. Layering struc-

ture, divides distribution network into one main layer and several sub layers, where

customer loads in each layer are aggregated and it is seen like a large load by the main

layer. The aggregated loads in the main layer have lower variability and higher spatial

correlation that increase the accuracy of estimated states.

The anticipated high time dependency in the customer loads can also provide

DSE with enough information for the developed short-term forecasting algorithm to

detect faults in LV distribution networks. In active distribution networks, it is not cost

effective to clear fault from power substations, while distributed generating sources

are connected to the network. Furthermore, the conventional protection schemes are

not accurate enough for the future active distribution networks. For instance, as the

renewable power and load profiles considerably vary during the day and seasons, the


overcurrent relays with fixed fault current thresholds may not detect faults especially

in cases when the fault current is far lower than the nominal load consumption. Hence,

it is required to continuously update the fault current thresholds in each time step based

on the state and time dependency of the system to correctly differentiate the limit be-

tween the normal operations and the faulty conditions. This necessitates predicting the

future expectation of normal conditions of the system. As it is highly unlikely to get

zero forecast error using deterministic forecasts and due to the high accuracy required

for fault detection algorithms, it is more pragmatic to use probabilistic forecast to ob-

tain an estimation of forecast errors. Probabilistic forecast is used in this study to pre-

dict the likely ranges of the future values of the load. Then, the probable ranges are

compared with the real-time measurements to detect faults highly accurately.

Different frequency sampling rates in the measurement devices are required for

different load models in DSEs. In the presence of low frequency sampling rate, cus-

tomer loads are modelled as composite loads. In contrast, considering the forthcoming

high frequency sampling rate measurement devices in the future, it would be essential

to consider the dynamic behaviour of customer loads. Induction motors as the main

dynamic loads have a different behaviour from other type of loads. Hence, a new dy-

namic load modelling approach is required for distribution networks, which can give

a better understanding about customer loads for both state estimator and foreseeable

protection algorithms.

Introduction 7


Table of Contents

Keywords ................................................................................................................................. 2

Abstract .................................................................................................................................... 4

Table of Contents ..................................................................................................................... 8

List of Figures ........................................................................................................................ 10

List of Tables .......................................................................................................................... 13

List of Abbreviations .............................................................................................................. 16

Statement of Original Authorship .......................................................................................... 19

Acknowledgements ................................................................................................................ 21

Chapter 1: Introduction .................................................................................... 22

1.1 Background and Motivation ......................................................................................... 22

1.2 Amis and objectives of the thesis ................................................................................. 24

1.3 Significance of this research ........................................................................................ 25

1.4 Key innovations of the research ................................................................................... 26

1.5 Structure of the thesis ................................................................................................... 29

Chapter 2: Literature Review ........................................................................... 35

2.1 Introduction .................................................................................................................. 35

2.2 State Estimation and forecasting allgorithms ............................................................... 36

2.3 Distribution network protection ................................................................................... 41

2.4 Load Modelling ............................................................................................................ 46

2.5 Summary ...................................................................................................................... 48

Chapter 3: Conditional Multivariate Complex Gaussian Distribution State Estimator 53

3.1 Introduction .................................................................................................................. 53

3.2 Spatial-Temporal Correlation....................................................................................... 54

3.3 Spatial-temporal distribution state estiamtion: BASICS and formualtion ................... 61

3.4 Simulation resutls ......................................................................................................... 69

Introduction 9

3.5 Conclusion .................................................................................................................... 81

Chapter 4: Augmented Complex Kalman Filter Distribution State Estimation 84

4.1 Introduction .................................................................................................................. 84

4.2 Forecasting-aided Complex State Estimator: Basics and Formulation ......................... 85

4.3 Multi-Layer State Estimation ....................................................................................... 92

4.4 Simulation Results ........................................................................................................ 97

4.5 Protection scheme using ACKF-DSE ......................................................................... 106

4.6 Conclusion .................................................................................................................. 112

Chapter 5: Stochastic Fault Detection ........................................................... 116

5.1 Introduction ................................................................................................................ 116

5.2 Methodology ............................................................................................................... 117

5.3 Simulation Results ...................................................................................................... 124

5.4 Conclusion .................................................................................................................. 137

Chapter 6: Dynamic Load Modelling Using High Frequency Measuring Data in Distribution Network – for future work ................................................. 140

6.1 Introduction ................................................................................................................ 140

6.2 Dynamic load modelling............................................................................................. 141

6.3 Induction motor identification .................................................................................... 152

6.4 Conclusion .................................................................................................................. 156

Chapter 7: Conclusions ................................................................................... 159

7.1 Conclusion .................................................................................................................. 159

7.2 Future work recommendations ................................................................................... 163

Bibliography ........................................................................................................... 169

Appendices .............................................................................................................. 178

Appendix A ........................................................................................................................... 178

Appendix B List of publications ........................................................................................... 181


List of Figures

Figure 1.1. Distribution networks with various distributed generations .................... 29

Figure 2.1. Distribution networks with various distributed generations [2]. ............. 35

Figure 2.2. Typical SE Algorithm. ............................................................................. 37

Figure 2.3. The proposed fault detection method in [55]. .......................................... 44

Figure 3.1. Impact of customer numbers on spatial correlation (a) Newmarket, (b) Pecan street, and on temporal correlation (c) Newmarket, (d) Pecan street. ................................................................................................. 57

Figure 3.2. Impact of mean and snapshot values on (a) Spatial correlation, (b) Temporal correlation .................................................................................... 58

Figure 3.3. Visualization of the correlation matrix for five groups of loads and PV generations, 5×5 matrix, and black to white colors represent the highest to lowest correlation; (a) PV outputs correlation, (b) customer loads correlation, (c) net loads correlation. .................................................. 61

Figure 3.4. Spatial-temporal correlation in CMCGD. ................................................ 67

Figure 3.5. The flowchart of the spatial-temporal CMCGD state estimator. ............. 68

Figure 3.6. A six-bus distribution network. ............................................................... 70

Figure 3.7. Visualization of correlation matrix for case study 1. ............................... 71

Figure 3.8. Real-time daily load profile for case study 1. .......................................... 72

Figure 3.9. Average voltage errors for scenario 1 and 3. ........................................... 75

Figure 3.10. IEEE 123 node test feeder [103]. ........................................................... 77

Figure 3.11. Schematic of an Australian residential distribution network. ................ 78

Figure 3.12. Three phase voltage magnitudes profile at bus 8. .................................. 79

Figure 3.13. Three phase voltage magnitudes profile at bus 23. ................................ 80

Figure 4.1. Kalman gain in one day. .......................................................................... 88

Figure 4.2. (a) One step difference of the injected current, (b) Temporal correlation. ................................................................................................... 90

Figure 4.3. A typical multi-layer representation of a distribution network. ............... 93

Figure 4.4. Contribution percentage (a) individual customer, (b) group of customers. .................................................................................................... 95

Introduction 11

Figure 4.5. The flowchart of the layered state estimator. ........................................... 96

Figure 4.6. A six-bus distribution network. ............................................................... 98

Figure 4.7. Five areas voltage magnitudes in Case Study 1. .................................... 100

Figure 4.8. The estimated voltages, real and imaginary parts (a) area 1, (b) area 4. ................................................................................................................. 102

Figure 4.9. A MV/LV unbalanced distribution network. ......................................... 103

Figure 4.10. A three-layer state estimation representation. ..................................... 103

Figure 4.11. Three phase voltage magnitudes at bus 10. ......................................... 105

Figure 4.12. Three phase voltage magnitudes at bus 23. ......................................... 105

Figure 4.13. (a) Corrective error for the current measurement, (b) Corrective error in the form of histogram with fitted normal distribution. ................. 107

Figure 4.14. Examples of fault detection in (a) 3:20 am, (b) 3:20 pm and (c) 11:20 pm. ................................................................................................... 110

Figure 4.15. Corrective errors (a) Phase A, (b) Phase B and (c) Phase C................ 112

Figure 5.1. Quantiles of a typical Gaussian distribution. ......................................... 119

Figure 5.2. Measurements, and quantile 99.9% along with 17 prediction quantiles with probability level ranging from 1% to 97% in 6% increments (from the lightest to the darkest). ............................................ 122

Figure 5.3. Flowchart of the developed fault detection algorithm. .......................... 123

Figure 5.4. A six-bus distribution network. ............................................................. 125

Figure 5.5. Pavetta distribution network. ................................................................. 125

Figure 5.6. Comparison between pick up fault current and the developed IFT for (a) Case 1 (b) Case 2A, Phase A, (c) Case 2B, Phase A. ........................ 128

Figure 5.7. Examples of fault detection for Case 2A, Phase B, 30% increase in load (a) Normal condition, (b) Fault is detected by DTT, (c) Fault is detected by IFT. ......................................................................................... 132

Figure 5.8. Empirical versus Gaussian distributions fitted to the data. (a) aggregated current in Case 1, (b)-(f) the groups 1 to 5, from the lowest values to the highest values. The red and the dotted green curves represent the Gaussian distributions and the empirical distributions, respectively. ............................................................................................... 134

Figure 5.9. DTT logic .............................................................................................. 137

Figure 5.10. IFT logic .............................................................................................. 137


Figure 6.1. Frequency responses of active power-angle transfer functions. ............ 142

Figure 6.2. Frequency responses of active power-voltage transfer functions. ......... 144



Figure 6.5. Active power with respect to angle step change. ................................... 147





Figure 6.10. Case Study 1, with three induction motors. ......................................... 153

Figure 6.11. Case Study 2, with two induction motors and one fixed load. ............ 155

Introduction 13

List of Tables

Table 3.1. Pseudo measured injected current at 100% loading condition ................ 71

Table 3.2. Comparative results from Case study 1, scenario 1 – for three time steps and three loading levels ...................................................................... 73

Table 3.3. Comparison results for scenario 2 – for three time steps with a decrease and an Increase in loading ............................................................. 74

Table 3.4. Comparison results for scenario 3 with two measurement points and three Time steps ........................................................................................... 75

Table 3.5. Comparison results for case study 2, with decreasing load ..................... 77

Table 3.6. Voltage error in case study 3.................................................................... 80

Table 3.7. Voltage error in case study 3 with higher R/X ratio ................................ 81

Table 4.1. Voltage magnitude error in case study 1 ................................................ 100

Table 4.2. Voltage angle error in case study 1 ........................................................ 100

Table 4.3. Magnitude voltage error in case study 2 ................................................ 104

Table 4.4. Voltage magnitude error and computational time in case study 2 ......... 104

Table 4.5. Fault detection result in case study 1 ..................................................... 108

Table 4.6. Fault detection results for Case study 2 ................................................. 112

Table 5.1. Fault detection results for Case 1 ........................................................... 130

Table 5.2. Fault detection results for Case 2A ......................................................... 130

Table 5.3. Fault detection results for Case 2B ......................................................... 130

Table 5.4. Fault detection results for Case 1 with Gaussian assumption ................ 135

Table 5.5. Coordination of protection areas in percent (%) .................................... 136

Table 6.1. The Induction Motors Parameters .......................................................... 142

Table 6.2: Active power-angle transfer functions, zeros, poles and gains ............... 143

Table 6.3: Active power-voltage transfer functions, zeros, poles and gains ............ 144






Table 6.8: Mars Overhead Cable Parameters ........................................................... 149

Table 6.9: Active power-angle transfer functions, zeros, poles, gains and Busbars voltages ........................................................................................ 150

Table 6.10: Active power-voltage transfer functions, zeros, poles, gains and Busbars voltages ........................................................................................ 151

Table 6.11: Active power-angle transfer functions, zeros, poles, gains and Busbars voltages ........................................................................................ 152

Table 6.12: Active power-voltage transfer functions, zeros, poles, gains and Busbars voltages ........................................................................................ 152

Table 6.13: Active power-angle/voltage transfer functions, zeros, poles and gains ........................................................................................................... 153

Table 6.14: Active power-angle/voltage transfer functions, zeros, poles and gains ........................................................................................................... 155

Introduction 15


List of Abbreviations

AAVE Average Angle Voltage Error

ACKF Augmented Complex Kalman Filter

AMVE Average Magnitude Voltage Error

AVE Average Voltage Error

BIBC Bus-Injection to the Branch-Current

CDF Cumulative Distribution Function

CMCGD Conditional Multivariate Complex Gaussian Distribution

COV Covariance

CR Correlation

CS Conditional Multivariate Complex Gaussian Distribution Spa-

tial

CST Conditional Multivariate Complex Gaussian Distribution Spa-

tial-Temporal

DER Distributed Energy Resource

DLF Direct Load Flow

DSE Distribution State Estimation

DSE-MACKF Multi-layer Distribution State Estimation based on Aug-

mented Complex Kalman Filter

DTT Definite Time Threshold

Introduction 17

FASE Forecasting-aided State Estimation

IFT Instantaneous Fault Threshold

KDE Kernel Density Estimation

KF Kalman Filter

LV Low Voltage

MAVE Maximum Angle Voltage Error

MMVE Maximum Magnitude Voltage Error

MV Medium Voltage

MVE Maximum Voltage Error

PDF Probability Distribution Function

PV Photovoltaic

RC Residential Community

RMSE Root Mean Square Error

SD Standard Deviation

SE State Estimation

VAR Variance

WLS Weighted Least Square

Introduction 19

Statement of Original Authorship

The work contained in this thesis has not been previously submitted to meet re-

quirements for an award at this or any other higher education institution. To the best

of my knowledge and belief, the thesis contains no material previously published or

written by another person except where due reference is made.

Signature: QUT Verified Signature

Date: January 2019

Introduction 21

Acknowledgements

It is my pleasure to thank those who support me during the period of my PhD.

My foremost gratitude goes toward my principal supervisor, Dr. Ghavameddin Nour-

bakhsh for his admirable supervision, encouragement and guidance. I also wish to

extend my sincere appreciation to my associate supervisors, Prof. Gerard Ledwich and

Dr. Ali Arefi for their invaluable support and advice throughout my PhD. It was my

honour to work under their supervision and to be a part of their research team.

Furthermore, I would like to convey my sincerest thanks to my QUT colleagues

and friends, especially Associate Prof. Geoff Walker, Dr. Adriana Bodnarova and Mr.

Samuel Cunningham-Nelson for their support and encouragement.

I gratefully acknowledge Queensland University of Technology (QUT) for

providing my QUTPRA scholarship, which has given me this opportunity to develop

my teaching and learning skills. Also, I would like to thank QUT Research Student

Centre Staff members, especially Ms. Janelle Fenner and Ms. Judy Liu, and staff mem-

bers of EECS School especially Ms. Joanne Kelly, Ms. Joanne Reaves and Ms. El-

lainne Steele for their support during my PhD period.

Special thanks to my lovely family, my parents, my sister and my brother in law,

for their constant encouragement and support in whole my life.

Last but not least, I would like to thank my wife Faranak for her love and con-

stant support, for all the late nights and early mornings, and for giving me hope over

last eight years. Without your everlasting encouragement, patience and pure love this

research has not been able to take this place. Thank you for always being my best

friend. I owe you everything.


Chapter 1: Introduction

1.1 BACKGROUND AND MOTIVATION

Environmental concerns and incentive-based policies supported by the network

operators have led to a rapid increase in renewable energy penetration in distribution

networks. However, a high PV penetration with intermittent generation increases the

complexity of distribution networks and calls for new control and monitoring algo-

rithms.

Although in the recent years, several methods are proposed to control and mon-

itor of medium voltage (MV) distribution networks, the challenges and the potential

solutions with the modern low voltage (LV) distribution networks are yet to be inves-

tigated. Most of the current monitoring and protection schemes rely on considerable

number of measurement and communication devices in distribution networks, which

is not feasible from the imposed cost perspective.

In order to achieve a cost-effective monitoring method, distribution state estima-

tions (DSEs) are employed to estimate the states of the unmonitored nodes. This re-

quires using pseudo data instead of measurements to make the DSE algorithms ob-

servable. Pseudo data is the historical data provided by the temporary measurement

devices or the electricity bills. The accuracy of the pseudo data affects the performance

of the DSEs. However, the problem is that the pseudo data in LV distribution networks

with unpredictable behaviour of customer loads are low in accuracy. This calls for new

methods to refine the data such that they improve the accuracy of the estimated states.

Several methods are proposed in literature for updating pseudo data based on spatial

correlation. However, temporal correlation is not well-addressed in the literature,

Introduction 23

though it provides further information for updating pseudo data to improve the accu-

racy of estimated states.

Iterative-based nature of most of the current DSE algorithms is an important

drawback of DSE algorithms. In general, the iterative-based algorithms impose a high

computational burden, which makes them inefficient for large distribution networks

with a considerable number of customer loads, while the active distribution networks

with high sampling rate measured data, rooftop PVs and storages require fast and ac-

curate state estimator for control and monitoring applications. Furthermore, in the cur-

rent literature, the real and imaginary parts of the states are estimated independently.

However, this approach can introduce inaccuracy as it ignores the interactions between

the real and imaginary parts of the states. Therefore, it is highly desirable to devise a

method that can directly include states as complex values.

In general, DSE algorithms can be divided into two main categories as snapshot

estimators and forecasting-aided state estimators (FASEs). Snapshot state estimator is

mainly used in transmission networks for balancing between different measurements

devices, while FASE algorithms are more suitable for real-time applications in distri-

bution networks. In FASE algorithms, the impact of time dependency is considered

for estimating the current states. Furthermore, through FASE, the error of estimation

can be employed to detect the abnormalities, specifically faults in distribution net-

works.

In this thesis, not only two new DSE methods are developed, but also the im-

portance of DSE algorithm in fault detection is studied. One of the main reason of

importance of DSE algorithm in fault detection is that fixed fault current thresholds in

conventional protection schemes may not be accurate at detecting high impedance

faults in LV distribution networks, where the load profile varies considerably during


the day. Hence, it is expected that using customer loads time dependency for dynami-

cally updating fault thresholds increase the accuracy of fault detection in such net-

works. FASE algorithms employ a corrective error to refine the estimated states, in

consecutive time intervals. This error signal can be used to detect faults in LV distri-

bution networks, where a large increase in corrective error can indicate a potential

abnormality in distribution networks. An alternative approach is to use probabilistic

forecasts to generate the likely ranges with probability levels for the future values of

loads. The predicted ranges can be compared with real-time measurements to detect

faults.

1.2 AMIS AND OBJECTIVES OF THE THESIS

This work develops original solutions based on customer loads time dependency

for control and monitoring in LV distribution networks with a low number of meas-

urement devices. Two DSE algorithms are established to estimate the network states

with a low number of measurement devices. Additionally, this work develops two

methods to increase the accuracy of the pseudo data. To develop a new fault detection

scheme in LV distribution networks, it is recommended to consider DSE corrective

error to determine fault thresholds. Furthermore, the quantile regression as a nonpara-

metric distribution [1] is deployed to determine the dynamic fault current thresholds

to detect faults with low fault currents. In order to achieve the main objective of this

thesis, which is improving the monitoring and protection of LV distribution networks,

the following research studies are conducted:

Developing DSE algorithms catered for distribution networks with limited

number of measurement devices. The DSE formulations are desired to esti-

mate the states in complex forms.

Introduction 25

Developing a FASE algorithm for real-time studies, which is able to refine

the estimated states, gradually.

Developing a new formulation for updating pseudo data considering the im-

pact of spatial and time correlation.

Investigating the impact of load aggregation on the accuracy of DSE algo-

rithm.

Developing a new fault detector algorithm based on FASE approach as an

application of DSE methods.

Determining dynamic fault current thresholds by employing short-term prob-

abilistic forecasting.

Proposing a new dynamic load modelling for distribution networks.

1.3 SIGNIFICANCE OF THIS RESEARCH

The fast growing integration of renewable energies in distribution networks has

made these networks active. Renewables cause voltage and flow violations in distri-

bution networks. This calls for fast monitoring and protection methods to maintain the

reliability and consistent delivery of the electricity. For monitoring purposes, the high

cost of widespread monitoring installations such as measurement devices and commu-

nication platforms is not economically viable. In this regard, state estimation can play

an important role using limited measuring devices at nominated nodes while employ-

ing pseudo data for the remaining unmonitored nodes. However, usually the accuracy

of the pseudo data is very low, which decreases the accuracy of the estimated node

voltages and branch currents states in distribution network. Therefore, new methods

are needed to update and refine the pseudo data in order to obtain highly accurate re-

sults from DSE algorithms.


As an application for the developed method, FASE is employed in distribution

networks for high impedance fault detection. High impedance faults in distribution

networks will often increase the current slightly, which the conventional overcurrent

relays with fixed current thresholds may not be able to detect them. Hence, advanced

state estimation with forecasting algorithms are required to update the fault current

thresholds in each time step to increase the accuracy of the fault detection algorithms.

1.4 KEY INNOVATIONS OF THE RESEARCH

The main contribution of this research is to develop new fast DSE algorithms

with acceptable performance for distribution systems and in particular for LV net-

works with low number of measurement devices. Additionally, this work develops

new approaches for updating the pseudo data to increase the accuracy of DSEs. More-

over, two methods are proposed for fault detection for LV distributing networks. In

order to achieve the main objectives of this research, the following innovative research

developments are accomplished in this thesis and are described and listed as:

1. Customer loads show a very high time dependency, and similar load types

in different locations are highly correlated. Based on this observation, the

first innovation of this thesis is to incorporate spatial-temporal correlation

to develop a new method for updating pseudo data in MV and LV distribu-

tion networks. Conditional multivariate complex Gaussian distribution

(CMCGD) is used to characterize the spatial-temporal correlations among

consumer loads to refine the estimated states.

2. The investigations in this thesis show that the injected currents of the cus-

tomer loads can be considered as the integration of white noise. Therefore,

the novelty of the second chapter of this thesis is mainly to develop a new

DSE method based Augmented Complex Kalman Filter (ACKF) to refine

Introduction 27

the estimated states continuously. The ACKF in literature needs to update

2 states, while in this thesis the developed algorithm updates states to have a time efficient DSE algorithm.

3. The study performed in this thesis shows that the phase adjacent aggregation

not only decreases the number of estimated states, but also increases the

cross correlation between aggregated loads. One of the key contributions of

this thesis is to divide a large distribution networks into one main-layer and

several sub-layers. The customer loads in sub-layers are aggregated and act

like a large customer load seen by the main-layer. Through load aggregation,

a higher cross correlation among aggregated loads is achieved and this cor-

relation information is employed to update the pseudo data, leading to a sig-

nificant increase in the accuracy of the established framework.

4. A new probabilistic forecasting algorithm is developed to consider the time

correlation for the injected currents to predict upper limits with probability

guarantees for them. The developed framework is able to predict the upper

limits or the so-called quantiles without any restrictive assumption on the

probability distribution of the injection current. A framework is established

to link the concept of the upper limits with probability guarantees to two

fault thresholds for instantaneous and definite time tripping. One main ad-

vantage of the developed method is that it dynamically updates the fault

current thresholds for each next time step.

5. In the presence of measurement devices with higher frequency sampling in-

tervals, it is necessary to consider dynamic behaviour of induction motors

in the state estimation algorithms. The last innovative research contribution

of this thesis is to establish a new method to dynamically model a set of


induction motors in the distribution network with a first order transfer func-

tion. The gain, zero and pole of the transfer function are employed to infer

the size of the induction motors and their distance from the measurement

devices.

Introduction 29

The summery of the main contributions of each chapter is provided in Figure

1.1.

Figure 1.1. Distribution networks with various distributed generations

1.5 STRUCTURE OF THE THESIS

This thesis is presented in six chapters.


Chapter 1. Introduction:

In this chapter an overview of the thesis along with background, motivations,

significance and contributions of the work are outlined.

Chapter 2. Literature Review:

This chapter provides a comprehensive literature review towards an introduction

to snapshot and real-time DSE algorithms, load modelling and fault detection in dis-

tribution networks.

Chapter 3. Conditional Multivariate Complex Gaussian Distribution State

Estimation:

This chapter uses a model of the time correlation of loads to formulate a set of

pseudo measurements, this time correlation reduces the effective noise in the esti-

mates. In the first part of this chapter a comprehensive study on the significance of

spatial-temporal correlation is provided, which mainly focuses on:

Impact of loads aggregation on spatial-temporal correlation between two

residential communities (RCs).

Impact of two time intervals (Snapshot and mean measurement types) on

spatial-temporal correlation.

Impact of PV penetrations on spatial-temporal correlation of net customer

loads.

In the second part of chapter 3, conditional multivariate complex Gaussian dis-

tribution (CMCGD) is formulated based on spatial-temporal correlation between cus-

tomer loads to increase the accuracy of pseudo data. Finally, a new one-iteration DSE

algorithm is established to estimate the states of distribution networks. The perfor-

mance of the developed method is evaluated based on three case studies, including an

unbalance LV distribution network.

Introduction 31

Chapter 4: Augmented Complex Kalman Filter Distribution State Estima-

tion:

This chapter represents the time model of the loads as an explicit state equation

which means that state estimates have cumulative corrections. Due to the fact that the

states of distribution networks are complex values, chapter 4 is mainly focuses on aug-

mented complex Kalman filter to formulate a fast complex FASE algorithm. In chapter

4, a typical distribution network is divided into several estimation layers, with series

and parallel levels to decrease the computational time. In the main estimation layer,

the customer loads in sub layers are aggregated and act like a large load with low

variations. This allows for an increase in the accuracy of the estimated states. Low

variations and high correlation between aggregated loads comparing to the individual

loads, increase the accuracy of the developed scaling factors as the contribution of the

consumed power in each sub layer to the measured current on the LV transformer. This

helps to decrease the error of updated pseudo data. Finally, the corrective error of Kal-

man filter consecutively and continuously refines the estimated states.

Chapter 5: Stochastic Fault Detection:

The time variation of loads is limited due to system variance for residential loads,

this enables faults to be detected as an abnormal change in measured loading. The aim

of chapter 5 is to present a new fault detector algorithm for LV distribution networks

when the fault current is low, referring to high impedance fault. In this chapter it is

shown that conventional protection scheme of overcurrent relays is not accurate

enough for fault detection in LV distribution networks. Hence, quantile regression is

employed to forecast the quantiles of the current at the next time step. The forecasted

quantiles are used to continuously update the developed fault current thresholds. In


this chapter, the behaviour of the measured current in LV distribution networks is stud-

ied and it is concluded that it does follow a Gaussian distribution. The developed

framework is evaluated using data from a real distribution network with 169 houses.

The results suggest that the developed model can be very promising for LV residential

distribution networks.

Chapter 6: Dynamic Load Modelling Using Measured Data in Distribution

Networks:

In the presence of measured data with a high frequency sampling intervals, it is

necessary to consider dynamic load models as the part of domain load prediction in

the state estimation algorithms. The aim of chapter 6 is to develop a new dynamic load

modelling and identification framework as initial step for future research work on its

own. Induction motors as the main dynamic loads in the distribution networks can be

modelled as a first order transfer function that gain, pole and zero can be considered

to infer the size of induction motors. The developed method can be employed in the

state estimation algorithm where the estimated states represent the dynamic behaviour

of the LV distribution network for control and protection studies. This chapter repre-

sents the first step toward dynamic load modelling for monitoring, control and protec-

tion purposes for future LV distribution networks.

Chapter 7: Conclusions and Future Works:

Conclusion drawn from this thesis along with the future work directions are pro-

vided in this chapter.

1.5.1 Thesis road map

The aim of this thesis is to develop new solutions for monitoring distribution

systems and particularly LV networks with a low number of measurement devices.

Introduction 33

The analysis in this thesis show that customer loads have high time dependency, and

the temporal correlation information can be used to design highly accurate monitoring,

control and protection algorithms for the LV distribution networks. For LV distribu-

tion networks with limited number of measurement devices, pseudo data are employed

for the unmonitored nodes to give information about customer loads. However, pseudo

data often comes with a high error rate, which decreases the accuracy of the state esti-

mator algorithms. Therefore, a new method based on CMCGD is developed to incor-

porate time and spatial correlation information to update the pseudo data. To increase

the spatial-temporal correlation, load aggregation is considered, where increasing the

number of aggregated customer loads leads to an increase in the accuracy of estimated

states. In this method, spatial correlation of the measured data at time t as well as its

temporal correlation with the previous time steps increases accuracy of the pseudo

data. In the developed method, in each time step a combination of the measured data

at t and a window of previous measured data are employed to update the pseudo data.

In the first developed DSE method, the corrective error of each window of meas-

ured data is independent from other corrective errors. Therefore, an alternative DSE

method that considers the possibility of continuously refining the estimated states is

established in this thesis. Investigations show that the differences between injected

current in two successive time steps act like a white noise. Therefore, injected currents

are considered as the states in a new established DSE based ACKF algorithm, which

refines the estimated states gradually.

In order to study the practicality of the developed method in the protection ap-

plications, the Kalman filter corrective errors are considered to determine the fault

current thresholds. It is shown that fault conditions in the distribution networks will


cause a considerable jump in the corrective error. For a network without historical data

the error of ACKF can be used to determine the fault thresholds.

In the presence of historical data from few measurement points, a new probabil-

istic forecasting algorithm is established to capture the high time dependency in the

injected currents and develop a model to predict upper limits with probability guaran-

tees for the injected currents. The instantaneous and definite time thresholds are deter-

mined based on the predicted upper limits. In the established method, the upper limits

are predicted continuously and the fault thresholds are updated dynamically for each

next time step. One significant advantage of the established method is that it is distri-

bution-free. Hence, there is no restrictive assumption about the probability distribution

of the injected currents. This is important because the empirical investigations pro-

vided in this thesis show that the inject current does not follow a Gaussian distribution.

All the developed methods in this research are designed in such a way that they

work efficiently in applications with low sampling frequencies such as one-minute

sampling interval. With the low frequency sampling data, the particular characteristic

of induction motor loads is lost and appear like any other composite loads. By increas-

ing the sampling frequency, the dynamic behaviour of these loads can be observed and

can impact the load modelling in distribution systems. Therefore, in chapter 6, a new

framework to model the dynamic load behaviour for customer loads is established.

This framework is a significant first step for the design of state estimator algorithms

for the future distribution networks equipped with new measurement devices capable

of high frequency sampling rate.

Literature Review 35

Chapter 2: Literature Review

2.1 INTRODUCTION

The phrases Renewable Energy Resources (RERs) and distributed generations

(DGs) are commonly used for local supplied generation such as photovoltaic panels,

wind turbines, fuel cells and gas engines which are connected to the distribution net-

works as shown in Figure 2.1, [2].

Figure 2.1. Distribution networks with various distributed generations [2].

Increasing penetration of RERs and DGs in distribution networks introduces new

challenges for monitoring of these networks. The problems relating to the protection

of these networks are:

1- In distribution networks it is not cost effective to have measurement points in

all nodes. Consequently, the new monitoring methods should be based on a

few measurement points for control and monitoring.

2- Active distribution networks require efficient monitoring method for the fast

decision making required in designing the control and protection algorithms.

3- Conventional protection schemes with fixed threshold currents may not be

accurate enough for fault detection in distribution networks.

36 Chapter 2: Literature Review

In this chapter a comprehensive literature review is provided to address:

1- State Estimation in distribution networks: In section 2.2, a comprehensive

review of state estimators in distribution networks are provided. Recent stud-

ies in both snapshot and forecasting-aided state estimators are provided in

this section. Furthermore, in the last part of this section, a study about short

term forecasting algorithm is provided, to report the effectiveness of the de-

veloped method for distribution networks.

2- To have state estimators in distribution networks with low number of real

measurements, the load models play an important role. Hence, in section 2.3,

dynamic and static load modellings are introduced based on the results of

several articles.

3- It is tried to give an overview about distribution networks protection algo-

rithms in section 2.4. To explain the fundamental concepts and developments

in the areas of protection schemes, several methods based on conventional

protection algorithms, communication based fault detectors, state estimators

and forecasting algorithms are studied and reviewed in this section.

2.2 STATE ESTIMATION AND FORECASTING ALLGORITHMS

Power system state estimation has been used extensively for transmission sys-

tems operation and control since its first introduction in 1970 [3-5]. As shown in Figure

2.2, the application of SE can be divided into three steps, namely; inputs, State Esti-

mation (SE) process and output [6], as follows:

Inputs:

Network Parameters

System Measurements


Pseudo measurements

DSE Process

Processing the networks based on the first part

Analysis the Observability of the networks

DSE algorithm

Bad data identification

Output

Voltage magnitude of busbars

Voltage angle of busbars

Node injection current

Branch current

Figure 2.2. Typical SE Algorithm.

SE in distribution networks has become important in last decade, especially after

proposing networks with DERs. Distribution State Estimation (DSE) in comparison

with SE in transmission network has new challenges [6]. Firstly, the number of meas-

urements in distribution networks is limited for economic reasons. The type of meas-

urements is another difference between these two networks. In distribution networks,

the operators employ pseudo measurements data mainly for node voltages and current

Input SE Process Output


injections, while in transmission networks the operators have access to accurate meas-

urement data of various quantities. The third and the most important difference is the

diversity in distribution loads, which causes two problems [7-10]. Firstly, due to the

dynamic changes in distribution networks, the DSE algorithm requires low computa-

tional burden. Secondly, diversity in customer loads causes pronounced phase imbal-

ances, which should be considered in DSE algorithm. The early development of DSE

goes back to 1990 when the weighted least square (WLS) algorithm was designed and

applied to a distribution system [11]. Later in 1996, a DSE was designed and employed

as a real-time monitoring in distribution management system (DMS) for applications

such as volt/var control considering the impacts of DERs, feeder reconfiguration, bat-

tery storage management and protection [12, 13]. Multiphase DSE approaches suitable

for LV network in the presence of DERs are established in [14].

A high penetration of DERs in a distribution network on one hand, and unpre-

dictable customer loads behaviour, on the other hand, require fast and accurate DSE

methods for online generation/load demand considerations and planning [15-18].

However, a high number of customer loads and a huge amount of measured data from

smart meters make centralized DSE algorithms complicated and computationally de-

manding [19]. An enhanced form of DSE with a significant reduction in measurement

points, while retaining accurate estimations, can be an attractive alternative. Generally,

decreasing the number of measurement points leads to an under-determined system,

meaning that the measurements cannot provide sufficient information required for an

accurate state estimation algorithm [20]. This problem is resolved using pseudo meas-

urements for unmeasured buses, satisfying the distribution network observability con-

ditions [21]. Pseudo measurements can be obtained from advanced metering infra-


structure (AMI), historical data or customers’ billing data [22], or they can be calcu-

lated based on the nominal customers’ load power, consumed power and daily load

profiles [23]. Although the use of pseudo measurement has become an essential part

of the DSE algorithms, the associated error with this type of measurement data is still

substantial, leading to DSE with low accuracy and reliability. Spatial correlation in-

formation between real measured and unmeasured points is deployed to improve DSE

accuracy in [24]. In [25], spatial correlation between loads are used to further increase

the accuracy of the estimated injected currents in unmeasured buses. Similarly, in [26],

spatial dependencies is modelled to determine pseudo load profile for unmeasured cus-

tomers where the essential load patterns are extracted from the smart meters data using

clustering techniques.

Although, the impact of incorporating spatial correlation information on the ac-

curacy of DSE is addressed in the literature, there is no developed DSE formulation

capable of including temporal dependency or spatial-temporal dependencies. This is

while customer loads in general show high correlations in successive time steps. The

temporal correlation can offer a measure for the similarity of load(s) variations in time,

greatly enhancing estimation quality. Recently, in the forecasting literature, the great

potential of involving spatial-temporal dependencies in improving PV power predic-

tion performance has been verified [27]. Also, spatial-temporal correlations is consid-

ered for load growth forecasting and load demand in electrical vehicles (EVs) charging

patterns [28]. A Vector Auto-Regressive (VAR) model is considered in [29] to inte-

grate time and space correlations present in measured data into a DSE algorithm. In

this article WLS as an iterative algorithm is employed for state estimation in presence

of several phasor measurement units. The iterative based algorithms with the large

amount of required data in a distribution network makes these estimation processes


computationally time-consuming, while an active distribution network with DERs re-

quires fast and accurate state estimators, updating data in less than one second [30].

To deal with a large amount of measured data, a new algorithm based on compressed

measurements is developed in [31]. However, the iterative DSE algorithm developed

in [31] computationally is highly demanding.

Traditionally, SE algorithms are formulated with a static nature in which states

are estimated based on single instantaneous measurements, and previous measured

values are incorporated in the estimation. However, due to high cross-correlations,

valuable information can be extracted from previous measured data to be used in fore-

casting-aided SE (FASE) methods. In general, FASE algorithms are used to detect

unexpected variation in the system states for control and protection analysis, network

configuration errors and bad data detection [32]. For instance, in the security and con-

trol analysis, a recursive FASE method based on measurements is reported in [33], and

also a comprehensive study of this method is detailed in [32, 34]. In [35], it is shown

that Kalman filter as the most common time series forecasting-aided state estimator

[36] has a better performance compared with WLS for distribution networks. In [37],

two decoupled FASE algorithms are established for estimating voltage magnitude and

voltage angle independently. These algorithms are not efficient due to their high com-

putational cost and ignoring the dependencies between magnitude and angle estima-

tion noises [38]. A complex formulation for the state estimator can potentially improve

the accuracy of estimation [39].

Distribution networks contain an extremely large number of customer nodes, and

it is not computationally efficient to process state estimation in a single layer. Instead,

the network can be divided into several subareas, where DSE is carried out in sequence


or parallel [40]. In a multi-layers state estimation approach, several factors can be con-

sidered to determine the boundaries of the subareas such as overlapping buses, coor-

dination and synchronization [41]. A multi-layers DSE is presented in [16, 17] consid-

ering parallel and series zones [14, 18]. In parallel zones, DSE is employed for several

zones simultaneously allowing for a lower computation time. Similarly, in series

zones, network schematic matrix reduction leads to a higher computational efficiency.

Although this multi-layers state estimation method reduces the computational time, it

requires a significant number of real measurements, making them economically unat-

tractive and computationally inefficient for real distribution networks.

2.3 DISTRIBUTION NETWORK PROTECTION

In the past decade, distribution networks have witnessed rapid changes in local

generation and network monitoring, control and operation. Distribution networks in

the presence of local DERs have become more complex to monitor and operate. This

necessitates development of more advanced protection frameworks tailored for the

specific limitations and structure of distribution networks.

Protection and smart switch devices play an important role in active distribution

networks. Protection devices identify faults and send commands to smart switches to

isolate a fault, while other switches may facilitate alternative route of power supply to

reduce outages. The role of protective devices and smart switches become more im-

portant as penetration of renewable energy resources increase in distribution networks

[42].

Protection is a critical aspect of future distribution network operation. In partic-

ular that; DERs with power electronics interface, bidirectional fault currents, and high

vulnerability factor of network devices require new protection schemes in comparison


with the traditional distribution networks. Furthermore, for operating in both grid-con-

nected and islanding modes new reliable protection schemes should be designed [43].

In [44] the problems associated with the protection in active distribution networks are

presented, mostly related to the operation of protection relays as well as the behaviour

of distribution networks in fault situations. In the noted article, it has been shown that

traditional protection schemes are not applicable in distribution networks with DERs.

Therefore the authors in [45] have established digital relays with a communication

scheme for distribution networks protection. For the high fault current protection, [46]

recommends to isolate DGs from the network even in the situation with the faults in

the remote area. In [47, 48] a novel idea is presented to detect and discriminate fault

based on the measured three-phase current and voltage waveforms when faults occur

in the power transmission network. In this approach, Negative-sequence components

of the three-phase current and voltage quantities are employed in order to have fast

online fault detection.

In some other approaches, multilevel wavelet transform, principal component

analysis, support vector machines (SVM), and adaptive structure neural networks are

employed at the same time for fault detection. In [49] wavelet is employed for fault

detection based on the comparison between the nominal values and extracted positive

and negative sequence components of the voltage. It is of vital importance to limit the

ground current in distribution networks due to power electronics interface, hence in

[50] a grounding electrode system is applied to limit grounding current. In [51] Dijks-

tra’s method is developed to find the relay hierarchy and thereby update new settings

due to the operation condition of distribution networks. Dijkstra’s method is a search

algorithm which finds the shortest path between nodes. Based on the condition of

switches (close or open), this algorithm models distribution networks as a graph which


relays represent nodes and connections consider as edges. Consequently, the relays

settings are coordinated based on the distances between the nodes.

In [52] an overcurrent relay is employed to detect the fault and limit the output

of DERs and finally reclose the breakers after fault clearance. Furthermore, the over-

load relays are considered in the developed protection scheme to limit the output power

of DERs, when several loads are out of service during fault conditions. In [45] numeric

relays are employed to implement differential protection scheme in a distribution net-

works. Furthermore, the authors proposed a new method to simulate high impedance

fault to test their proposed protection algorithm. In this method, the magnitude of fault

resistance is randomly changes between 50 and 1000 Ω. The fault duration also varies

between 10µs and 5ms. With this method a true model for high impedance fault is

simulated. In [53] time-frequency transform algorithm is employed to retrieve the

spectral energy of the fault current at both ends of the feeder. Differential algorithm is

considered to find the threshold difference and thereby send a command to the breaker

soon as fault is detected. In this paper the results shows that the established method

can be useful in both radial and mesh network and can identify high impedance faults

as well. Islanding detection is one of the most important issues in active distribution

networks. Several methods have been reported in research literature regarding island-

ing detection. For example; in [54] the authors concluded that positive feedback (PF),

voltage unbalance and harmonic distortion are the most valuable islanding detection

methods. However, normally there are several DERs exit in a distribution network and

therefore they can contribute to frequency and voltage errors, consequently PF method

could destabilize the network. In [55] the authors proposed to divide the network into

small segments, containing transformers, line segments and circuit breaker at the


boundary between each segment, as shown in Figure 2.3. Although this method can be

useful, the cost of allocating relays for each segment may not be acceptable.

Figure 2.3. The proposed fault detection method in [55].

Several papers proposed communication platforms in active distribution net-

works for monitoring and control purposes [56], [57]. Power Line Communication

(PLC) is considered as an attractive communication system for distribution network

operators due to its low investment cost [58]. In [59], a new fault detection and location

algorithm is proposed using PLC devices installed at the beginning point of the Me-

dium Voltage (MV) distribution line. In this method, the deviations in detection met-

rics related to the network impedances and frequencies are used to detect the high

impedance faults. Although the developed method alleviates the shortcoming of the

conventional overcurrent protection schemes for the case of detecting high impedance

faults, it is not cost effective approach in LV distribution networks due to its large

range of required measurement devices and communication platforms.

In the distribution networks with insufficient communication platforms, fore-

casting-aided state estimators can be employed for real-time applications [60]. Kalman


filter as the most common real-time state estimator is considered to formulate a new

fault detector in [61]. In this article, the Kalman filter in the time domain estimates

the amplitude, phase and frequency of fundamental components of the traveling waves

to detect the singularity points. However, the method requires investment in installing

highly accurate Phasor Measurement Units (PMUs) in distribution networks, while the

high cost of PMUs limits the scale of their widespread installation. The measurement

data is employed in Bayesian and Gaussian prior distribution frameworks to identify

outages in the power networks. Bayesian inference is also employed in [62] to form

posterior distribution condition for voltage dip state estimation caused by the presence

of DERs.

In the distribution networks using conventional measurement devices with low

sampling rates, forecasting algorithms in both regression and classification processes

play an important role to reduce the operation costs and to improve the reliability.

SVM as a statistical data classification method is employed in [63] and [64], while

voltage disturbances and fault current signals are considered in SVM as independent

variables. Different fault scenarios are classified by SVM to detect and identify faults

in a power system. The problem with fault classification algorithms is that they require

protection relay, and disturbance or event recorder data in different fault conditions,

which are not available in LV distribution networks. In LV distribution networks,

model-based fault detectors can be used to continuously monitor the difference be-

tween the measured data and the model-predicted outputs [65]. In [66], the output of

deterministic load forecasts and the hypothesized fault conditions are compared with

the measured line flows to detect outages in a distribution network. However, highly

intermittent generation and consumption profiles in a distribution network can easily

cause non-negligible prediction errors in deterministic forecasts. While the probability


of getting zero forecast errors in deterministic forecasts is zero, it is more reasonable

to use probabilistic forecasts to provide an estimation of forecast errors. Probabilistic

forecasts can output likely ranges with probability guarantees for the future values of

the load. The probable ranges can be compared with real-time measurements to detect

faults. Bayesian quickest change detection and statistical process control methods are

recommended in literatures [67-70] as probabilistic short-term forecasting algorithms

to detect unpredicted changes in the behaviour of the system. As part of these two

methods, there are parameters that are determined based on Gaussian assumption and

it can be difficult to determine those parameters where the underlying distribution is

non-parametric. The pattern of the changes in injected currents in distribution networks

in the presence of renewable resources does not fit to any probability distributions.

Hence, a distribution free algorithm is required for short term forecasting to detect

faults in distribution networks.

2.4 LOAD MODELLING

The utilities are responsible for careful design, adequate planning and safe oper-

ation of their current distribution networks with renewable generation resources. Part

of this responsibility is to have advance techniques and technologies in place to mini-

mise customer outages while managing the network with minimum permissible quality

issues. However, considering the extensive distribution networks, it is not cost effec-

tive for utilities to upgrade their networks with the current traditional standard protec-

tion devices. All these changes and the consequent challenges and constraints need to

be addressed properly. One of the critical problems arises in such networks is the pro-

tection, where renewable resources with different types of static and dynamic loads

are present.


Load modelling can be introduced as active and reactive power changes to volt-

age and frequency changes [71]. Although identifying the accurate load modelling is

important for analysing the power system network, several factors reduce the accuracy

of load modelling such as: diversity of load modelling, uncertainty of load behaviour

for each costumer in each day, lack of information in distribution networks [72].

Static and dynamic load modelling can be employed for power system analysis.

It is worth to note that the static type of load modelling can be employed to approxi-

mate the components of dynamic load modelling [73]. For dynamic load modelling a

set of nonlinear equations are considered as described in [74].

Dynamic and static load modelling play an important role in power system anal-

ysis, because the characteristics of loads have a major effect in power systems espe-

cially the stability of inter-area modes [75]. In [76] dynamic load modelling is em-

ployed to study the effect of active and reactive loads on the oscillation of multi ma-

chine power system. Placement of the dynamic loads in the network is the topic of

[77], which influences the damping of inter-area oscillation. In this paper, the authors

claim that dynamic load model can affect the parameters of Power System Stabilizer

(PSS).

Also, Load modelling is important in stability analysis. Characteristic of dy-

namic load modelling is employed in power system analysis in [78]. In this paper,

evolutionary algorithm is used to identify the parameters of load modelling. The load

modelling is applied in [79] for analysing voltage stability. In this paper, both active

and reactive powers have been modelled as constant current and constant impedance

respectively. Furthermore, induction motors are considered as dynamic loads which

obviously affect the voltage stability. Induction motors cause changes in dynamic be-

haviour of the network. In dynamic study of induction motor, inertia ( ) and torque


damping factor ( ) should be considered. In [80], the authors estimate these parame-

ters based on the measured data from the Phase Measurement Unit (PMU). To estimate

these parameters the transfer function between the changes in active power and fre-

quency is considered.

The next step after load modelling in power system study is load aggregating. In

[81] constant impedance and constant power loads are considered as static loads, while

induction motors are considered as a dynamic loads of an aggregated load model. The

authors of [82] claimed that the induction motors in a network can be modelled by one

or two aggregated induction motors. In [83] it is claimed that based on the measure-

ments on the busbars, the induction motors can be divided into three groups: small,

medium, large induction motors. Least square algorithm is employed to identify the

percentage of small, medium and large induction motors and constant impedance and

constant power loads based on the measurements on busbars.

2.5 SUMMARY

As reviewed in this chapter, enhancing state estimation and forecasting ap-

proaches are proposed in several articles for monitoring distribution networks. In ac-

tive distribution networks with DERs, the focus of most articles has been placed on

fast control and monitoring algorithms. In order to design accurate control and moni-

toring algorithms, a considerable number of measurement devices are required. In-

stalling measurement devices with communication platforms is not cost-effective in

distribution networks. Hence, DSEs can be very effective to estimate the states of un-

measured nodes in distribution networks. The state estimation methods currently avail-

able in literature have the following drawbacks:

1- The snapshot algorithms such as WLS as the common approach to estimate

the states of the distribution networks consider the states in each time step


independent of the previous time steps [7-10, 84-86]. This is while in the

distribution networks with low number of measurement points, incorporating

the impact of time dependency can improve the accuracy of the estimated

states. Consequently, the time dependency can be deployed in the FASE al-

gorithms such as Kalman filter [39] or recursive least square to refine the

estimated states.

2- For complex states, in various articles [14-18, 87] the estimation is designed

via two formulations, to estimate the magnitude and the phase separately.

These iterative methods are practical for high and medium voltage networks,

in presence of long transmission lines with considerable angle difference be-

tween node voltages. In contrast, the low voltage distribution networks do

not have long overhead lines or underground cables that can cause significant

angle difference between node voltages. Therefore, a linear complex state

estimation formulation can be considered to estimate states in a single itera-

tion and in a complex format. Non-iterative complex state estimation algo-

rithm decreases the computational time for real time monitoring or control

applications.

3- Complex regressive least square methods exist in the literature [38, 88, 89]

for state estimation. However, these methods do not consider the relative

certainty between the measured data and estimated states. In the developed

complex state estimation formulation in this thesis, the gains are considered

to represent the relative gain between current measured data and the esti-

mated states, and can be tunned based on the value of the corrective error. If

there is a high confidence in the measurement relative to the covariance of


the propagated state, the gain places more weight on the most recent meas-

ured data, and force the estimated states to follow the measured data more

responsively. Otherwise, the estimated states follow the pattern of estimation

process more closely and do not change this pattern.

Due to the high uncertainties in both customer loads and DERs in distribution

networks, conventional protection algorithms with fixed fault thresholds may not be

sensitive enough for fault detection, especially with low fault currents. Therefore, new

distribution network fault detectors are proposed in the literature to detect faults with

low short circuit currents. Communication based protection algorithms are proposed

in several articles [45, 51, 58, 59, 90]. However, the main issues related to these meth-

ods are the cost and the security of these methods. To have a cost benefit and accurate

protection schemes, DSEs and forecasting algorithms are considered as a basis for de-

veloping new protection schemes. In these methods, measured data, pseudo data and

the disturbance of protection relays in different fault scenarios are considered to clas-

sify different fault scenarios, for fault detection and identification.

Conditional Multivariate Complex Gaussian Distribution State Estimator 53

Chapter 3: Conditional Multivariate Com-plex Gaussian Distribution State Estimator

3.1 INTRODUCTION

The increasing complexity of distribution networks calls for advancement in

DSE to monitor the operating conditions more accurately. A sufficient number of

measurements is imperative for a reliable and accurate state estimation. The limitation

on the measurement devices is generally tackled using the so-called pseudo measured

data. However, the errors in pseudo data by current techniques are quite high leading

to a poor DSE. As customer loads in distribution networks show high cross-correlation

in various locations and over successive time steps, it is plausible that employing the

spatial and time dependencies can improve the pseudo data accuracy and estimation.

Although, the role of spatial dependency in DSE has been addressed in the literature,

one can hardly find an efficient DSE framework capable of incorporating temporal

dependencies present in customer loads. Consequently, to obtain a more efficient and

accurate state estimation, a new non-iterative DSE framework is developed to involve

spatial-temporal dependencies together. The spatial-temporal dependencies are mod-

elled by conditional multivariate complex Gaussian distributions and are studied for

both static and real-time state estimations, where information at preceding time steps

are employed to increase the accuracy of DSE. The efficiency of the established ap-

proach is verified based on quality and accuracy indices, standard deviation and com-

putational time.

This chapter is organised as follow: Firstly, the spatial-temporal correlation is

introduced. This is followed by introducing the formulation of state estimation based

54 Chapter 3: Conditional Multivariate Complex Gaussian Distribution State Estimator

spatial-temporal correlation; then three case studies are considered to evaluate and val-

idate the performance of the developed DSE algorithm.

3.2 SPATIAL-TEMPORAL CORRELATION

Correlation is a statistical relationship between two random variables. Spatial

correlation is computed based on the data from different locations, while temporal cor-

relation represents the degree of similarities between data in different time steps [91].

The temporal and spatial correlation coefficients are usually defined as [92]:

, ∗, var x

(3.1)

where , and denote covariance, standard deviation and variance, respec-

tively. For spatial correlation, and represent two sets of data from two different

geographic locations, while temporal correlation describes the dependency at a given

location and between time intervals of and . In this section, the aim is to

study the spatial-temporal correlation strength between net loads in a distribution sys-

tem based on two important factors:

1. The number of loads in customer communities (RCs),

2. Time interval for which load data is averagely available,

3. Presence of DERs at load buses.

3.2.1 Spatial-temporal correlation-Impact of the number of loads in RCs

To study the first factor, two RCs is considered, while the number of customers

are gradually increased in each community. The spatial and temporal correlation coef-

ficients are computed and shown in Figure 3.1. For the analysis, one-minute active


power data of two real datasets from Newmarket suburb in Brisbane, Australia in

summer season, and Pecan street, Texas, USA in winter season is considered [93]. The

spatial correlations are calculated for the aggregated load in each community over the

course of 7 days. The temporal correlation coefficients represent the correlation

strength between successive time steps of the aggregated loads. According to Figure

3.1, as the number of customers increases, the spatial correlation between the aggre-

gated loads in two RCs increases. This happens due to smoothing effects of aggrega-

tion. As one observes in Figure 3.1 (a) and Figure 3.1 (b), the spatial correlation coef-

ficient between two individual houses is less than 20% and it increases to more than

80% when the number of houses in each RC rises to 25. The temporal correlation in

Figure 3.1 (c) and Figure 3.1 (d) shows a similar trend. Note that; in Figure 3.1 (c), the

irregular jump at the case with two houses is accidental and does not imply to be a

general trend.

(a)


(b)

(c)


(d)

Figure 3.1. Impact of customer numbers on spatial correlation (a) Newmarket, (b) Pecan street, and on temporal correlation (c) Newmarket, (d) Pecan street.

3.2.2 Spatial-temporal correlation-Impact of time interval

The second statement is the time interval of available data. Some applications

require the correlation calculation between loads’ average values over longer time in-

tervals. Whereas other algorithms are using the correlation of snapshot values. In order

to show the impact of time interval, three sample load buses, each with twenty house-

holds are considered. Figure 3.2 (a) shows the spatial correlation between load bus 1

and other two buses, for two time intervals of thirty minutes (Snapshot) and half an

hour (Mean of thirty one minute samples). The load data for half an hour time interval

obtained by averaging 30 one-minute data. As seen, the spatial correlation has changed

slightly, for example from 0.77 to 0.80 for one minute and half an hour time intervals,

between bus 1 and 2.


(a)

(b)

Figure 3.2. Impact of mean and snapshot values on (a) Spatial correlation, (b) Temporal correlation

RC1-RC2 RC1-RC3

0.75

0.85

0.95Mean Instantaneous

Corr(t,t-30) Corr(t,t-60)

0.75

0.8

distribution network state estimation time ...distribution network state estimation, time dependency...

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