Computationally efficient PMU-based L1 estimators for ...cj82ss08h/... · Computationally...

Computationally Efficient PMU-based L1 Estimators for Large Power

Systems

A Dissertation Presented

by

Chenxi Xu

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements

for the degree of

Doctor of Philosophy

in

Electrical and Computer Engineering

Northeastern University

Boston, Massachusetts

May 2018

To my parents.

ii

Contents

List of Figures vi

List of Tables vii

List of Acronyms ix

Acknowledgments x

Abstract of the Dissertation xi

1 Introduction 11.1 Operating States and Security Analysis of Power Systems . . . . . . . . . . . . . . 11.2 State Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 Weighted Least Squares (WLS) SE . . . . . . . . . . . . . . . . . . . . . 31.2.2 Least Absolute Value (LAV) SE . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Phasor Measurement Unit (PMU) . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Dissertation Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature Review 92.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Centralized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Distributed Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 LAV Estimator with Equality Constraints 133.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Direct Addition of Zero Injection Measurements . . . . . . . . . . . . . . . . . . . 143.3 Explicit Addition of Zero Injection Equality Constraints by Kron Reduction . . . . 163.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.4.1 Impact on Bad Data Rejection . . . . . . . . . . . . . . . . . . . . . . . . 173.4.2 Impact on Computational Performance . . . . . . . . . . . . . . . . . . . 18

3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

iii

4 Linear LAV Estimator using Dantzig-Wolfe Decomposition 214.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.2 Dantzig-Wolfe Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Decomposition Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.3.1 Master Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3.2 Sub-problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4.1 Robustness under Single Bad Data . . . . . . . . . . . . . . . . . . . . . . 324.4.2 Robustness under Multiple Bad Data . . . . . . . . . . . . . . . . . . . . 364.4.3 Computational Time Performance . . . . . . . . . . . . . . . . . . . . . . 36

4.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Two-stage Multi-area LAV Estimation 385.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.2 Two-stage State Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 40

5.2.1 First Stage LAV State Estimation . . . . . . . . . . . . . . . . . . . . . . 445.2.2 Second Stage LAV State Estimation . . . . . . . . . . . . . . . . . . . . . 44

5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.3.1 Robustness under Bad Data . . . . . . . . . . . . . . . . . . . . . . . . . 485.3.2 Statistical Analysis of Robustness under Single Bad Data . . . . . . . . . . 525.3.3 Statistical Analysis of Robustness under Multiple Bad Data . . . . . . . . 525.3.4 Computational Performance . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

6 Multi-area LAV SE with Zone Generation 556.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2 Multi-zone State Estimation Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 57

6.2.1 Existing Zone LAV State Estimation . . . . . . . . . . . . . . . . . . . . . 606.2.2 Newly Generated Zone LAV State Estimation . . . . . . . . . . . . . . . . 60

6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3.1 IEEE 30-bus test system . . . . . . . . . . . . . . . . . . . . . . . . . . . 626.3.2 2917-bus Utility System . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7 Multi-area LAV SE with Multiple Copies 697.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 697.2 Distributed Multi-copy LAV SE with Automatic Zone Generation . . . . . . . . . 717.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7.3.1 Robustness under Gross Error . . . . . . . . . . . . . . . . . . . . . . . . 807.3.2 Computational Performance . . . . . . . . . . . . . . . . . . . . . . . . . 84

7.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

iv

8 Implementation on High-performance Computer 868.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 868.2 Discovery Cluster . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 888.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

8.3.1 Robustness under gross error . . . . . . . . . . . . . . . . . . . . . . . . . 908.3.2 Computational performance . . . . . . . . . . . . . . . . . . . . . . . . . 91

8.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

9 Conclusions and Future Work 949.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 949.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Bibliography 97

A List of Publications 102

v

List of Figures

4.1 IEEE 30-bus system partitioned into 3 areas . . . . . . . . . . . . . . . . . . . . . 274.2 Flowchart of the modified Dantzig-Wolfe decomposition . . . . . . . . . . . . . . 314.3 Bad Data Location Illustration for IEEE 30 bus system . . . . . . . . . . . . . . . 33

5.1 Flowchart for Two-stage State Estimation Algorithm . . . . . . . . . . . . . . . . 425.2 IEEE 30-bus system divided into 3 areas . . . . . . . . . . . . . . . . . . . . . . . 435.3 Three-area System Diagram and Measurement Configuration . . . . . . . . . . . . 465.4 Measurements Used in the First Stage Estimation . . . . . . . . . . . . . . . . . . 475.5 Measurements Used in the Second Stage Estimation . . . . . . . . . . . . . . . . . 475.6 Areas and Bad Data Locations for IEEE 30 bus system . . . . . . . . . . . . . . . 49

6.1 IEEE 14 bus system with 3 zones . . . . . . . . . . . . . . . . . . . . . . . . . . . 586.2 Flowchart for Two-stage ZG State Estimation Algorithm . . . . . . . . . . . . . . 596.3 New Zone Partition for IEEE 30 bus system . . . . . . . . . . . . . . . . . . . . . 63

7.1 7-bus system divided into two zones with multiple copies . . . . . . . . . . . . . . 727.2 Automatic Copy Generation Algorithm Flowchart . . . . . . . . . . . . . . . . . . 757.3 IEEE 14 bus system with 2 zones . . . . . . . . . . . . . . . . . . . . . . . . . . . 767.4 Copy 1 current measurement allocation . . . . . . . . . . . . . . . . . . . . . . . 787.5 Copy 2 current measurement allocation . . . . . . . . . . . . . . . . . . . . . . . 787.6 Copy 3 current measurement allocation . . . . . . . . . . . . . . . . . . . . . . . 797.7 Zone Partition for IEEE 30 bus system . . . . . . . . . . . . . . . . . . . . . . . . 817.8 30-bus System Centralized vs MC Distributed without Guassian Error . . . . . . . 817.9 30-bus System Centralized vs MC Distributed with Guassian Error . . . . . . . . . 827.10 Centralized and MC Distributed Results for NPCC test system . . . . . . . . . . . 837.11 Centralized and MC Distributed Results for 2917-bus system (Single Bad Data) . . 837.12 Centralized and MC Distributed Results for 2917-bus system (Multiple Bad Data) . 84

8.1 Discovery Cluster schematic [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . 898.2 Cluster MSEs for 2916-bus system . . . . . . . . . . . . . . . . . . . . . . . . . . 908.3 Cluser MSEs for 16216-bus system . . . . . . . . . . . . . . . . . . . . . . . . . 918.4 Efficiency for simulation cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

vi

List of Tables

3.1 ZI Simulation Results of NPCC Test System . . . . . . . . . . . . . . . . . . . . . 183.2 ZI CPU Times for 181-Bus Test System . . . . . . . . . . . . . . . . . . . . . . . 193.3 ZI CPU Times For 2917-Bus Test System . . . . . . . . . . . . . . . . . . . . . . 20

4.1 30-Bus DW Results for BD Measurement Type I . . . . . . . . . . . . . . . . . . 334.2 30-Bus DW Results for BD Measurement Type II . . . . . . . . . . . . . . . . . . 344.3 30-Bus DW Results for BD Measurement Type III . . . . . . . . . . . . . . . . . . 344.4 2917-Bus DW Results for BD Measurement Type I . . . . . . . . . . . . . . . . . 354.5 2917-Bus DW Results for BD Measurement Type II . . . . . . . . . . . . . . . . . 354.6 2917-Bus DW Results for BD Measurement Type III . . . . . . . . . . . . . . . . 354.7 2917-Bus DW Results for Multiple Bad Data . . . . . . . . . . . . . . . . . . . . 364.8 2917-Bus DW Computational Time Performance . . . . . . . . . . . . . . . . . . 37

5.1 30-Bus TS Results for BD Measurement Type I . . . . . . . . . . . . . . . . . . . 485.2 30-Bus TS Results for BD Measurement Type II . . . . . . . . . . . . . . . . . . . 505.3 30-Bus TS Results for BD Measurement Type III . . . . . . . . . . . . . . . . . . 505.4 2917-Bus TS Results for BD Measurement Type I . . . . . . . . . . . . . . . . . . 515.5 2917-Bus TS Results for BD Measurement Type II . . . . . . . . . . . . . . . . . 515.6 2917-Bus TS Results for BD Measurement Type III . . . . . . . . . . . . . . . . . 515.7 2917-Bus TS Results for Multiple Bad Data . . . . . . . . . . . . . . . . . . . . . 535.8 TS Computational Time Performance . . . . . . . . . . . . . . . . . . . . . . . . 53

6.1 30-Bus ZG Results for BD Measurement Type I . . . . . . . . . . . . . . . . . . . 636.2 30-Bus ZG Results for BD Measurement Type II . . . . . . . . . . . . . . . . . . 646.3 30-Bus ZG Results for BD Measurement Type III . . . . . . . . . . . . . . . . . . 646.4 2917-Bus ZG Results for BD Measurement Type I . . . . . . . . . . . . . . . . . 656.5 2917-Bus ZG Results for BD Measurement Type II . . . . . . . . . . . . . . . . . 656.6 2917-Bus ZG Results for BD Measurement Type III . . . . . . . . . . . . . . . . . 666.7 Failed ZG SE % with Multiple Bad Data . . . . . . . . . . . . . . . . . . . . . . . 666.8 ZG Computational Time Performance . . . . . . . . . . . . . . . . . . . . . . . . 67

7.1 Partitions of 14-Bus System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777.2 Boundary/internal buses of the 14-bus system . . . . . . . . . . . . . . . . . . . . 777.3 MC Computation Times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

vii

8.1 Information on the two test systems . . . . . . . . . . . . . . . . . . . . . . . . . 898.2 CPU Times for Cluster Simulation Cases . . . . . . . . . . . . . . . . . . . . . . . 918.3 Efficiency for simulation cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

viii

List of Acronyms

AGC Automatic Generation Control.

BD Bad Data.

BFS Basic Feasible Solution.

DW Dantzig-Wolfe.

EMS Energy Management System.

ISO Independent System Operator.

LAV Least Absolute Value.

LP Linear Programming.

OPF Optimal Power Flow.

PMU Phasor Measurement Unit.

PP Parallel Processing

RTO Regional Transmission Organization.

SCADA Supervisory Control and Data Acquisition.

SE State Estimator.

VLSI Very Large Scale Interconnected.

WLS Weighted Least Squares.

ZI Zero Injection.

ix

Acknowledgments

I would like to express my sincere appreciation to my research advisor, Dr. Ali Abur, for his

constant guidance and encouragement, without which this work would not have been possible. Being

one of the most knowledgeable and respected professors in the area of power system state estimation,

Dr. Abur has taught me not only the theoretical and practical knowledge on state estimation, but

also the way of doing research and becoming a better researcher. It is a great honor to be a research

assistant to Dr. Abur.

I would also like to thank Dr. Mahshid Amirabadi and Dr. Bradley Lehman for serving as

my dissertation committee members. They gave me constant inspiration and encouragement during

my graduate studies.

Finally I would like to express my heartfelt love and gratitude to my parents. This work

cannot be done without their continuous support and encouragement.

This work made use of Engineering Research Center shared facilities supported by the

Engineering Research Center Program of the National Science Foundation and the Department of

Energy under NSF Award Number EEC-1041877 and the CURENT Industry Partnership Program.

x

Abstract of the Dissertation

Computationally Efficient PMU-based L1 Estimators for Large Power

Systems

by

Chenxi Xu

Doctor of Philosophy in Electrical and Computer Engineering

Northeastern University, May 2018

Dr. Ali Abur, Advisor

Phasor Measurement Units (PMUs) are increasingly deployed in power systems because

of their nice characteristics like fast data acquisition rate and GPS clock synchronization. With the

explicit usage of PMU measurements, Least Absolute Value (LAV) State Estimator (SE), together

with its built-in Bad Data (BD) rejection capability, can be formulated as a Linear Programming

(LP) problem and solved efficiently by high-performance LP solvers. This dissertation reviews the

foundational research on power system state estimation and proposes several novel LAV SEs with

high robustness and computational performance for Very Large Scale Interconnected (VLSI) power

grids when the system is measured by only PMUs.

The first part of this dissertation presents two centralized LAV SEs incorporating Zero

Injection (ZI) measurements into the LAV state estimation formulation using direct enforcement and

Kron reduction, respectively.

Based on the current circumstance that VLSI power grids are usually divided into several

independent and non-overlapping zones, the second part of this dissertation presents several multi-

area distributed LAV SEs.

The first algorithm combines a well-known LP decomposition method: Dantzig-Wolfe

(DW) decomposition with the LAV SE considering the motivation that LAV can be formulated as an

xi

LP problem and multi-area state estimation measurement matrix has the exact structure required by

DW.

The second algorithm uses a two-stage set-up to assure adequate robustness around zone

boundaries. All zones run their own SEs and the estimated boundary bus states, together with

measurements between zones, are both used as measurements for the second stage SE run by a

central coordinator.

The third algorithm generates one or several additional zones covering all boundary buses

and their direct neighbors. This new zone and all existing zones run their SEs simultaneously in

parallel. Results are collected and reconciled to provide a full set of state estimates.

The fourth algorithm creates one or several ”copies” of the system. Each copy contains

one way of system zone partitioning. All buses appear at least once as an internal bus in these copies.

All zones in all copies run independent SEs. An algorithm is developed for the automatic copy

generation .

The above multi-copy algorithm is implemented and further tested on a high-performance

multi-core computer using parallel processing.

Above algorithms are implemented on different test systems with sizes ranging from 30-bus

to 16216-bus and the corresponding simulation results are presented in this dissertation.

xii

Chapter 1

Introduction

1.1 Operating States and Security Analysis of Power Systems

Power systems are composed of generation, transmission, and distribution systems. The

transmission system includes various substations which are interconnected by transmission lines,

transformers and other devices.

Three states of power system operation are defined: normal state, emergency state and

restorative state. In the normal state, all loads can be supplied by existing generators without violating

any operational constraints. The normal state is defined as secure when the system remains in the

normal state with the potential occurrence of any contingencies, otherwise, the system is identified as

insecure. In the emergency state, all loads can still be supplied. However, one or several operational

limits are violated. Immediate corrective action is needed to bring the system back to the normal

state, or the system will go into the restorative state. In the restorative state, not all loads can be

supplied and this will cause partial or total blackouts.

Power systems are operated by system operators in area control centers. North American

power grids are normally monitored by multiple regional transmission organizations (RTOs) or

independent system operators (ISOs) based on geography, administrative areas or control zones.

The main goal for system operators is to make sure the system maintains secure normal operating

condition. In order to achieve this goal, continuous system monitoring, system state determination

and corresponding preventive actions application are needed. These steps are known as the security

analysis of the system.

The first step of the security analysis requires frequent monitoring of the system. To

achieve this, measurements are placed around the system in substations and their data are collected

1

CHAPTER 1. INTRODUCTION

and transmitted to the control center. Measurement data received by the control center include

branch power flows, bus voltage and branch current magnitudes, generator outputs, etc. These raw

measurements will be processed by the SE and the measurement noise and gross errors will be

eliminated. The SE then will provide an optimal estimation of system states, which are the voltage

magnitude and angle for all system buses. The estimated system states will be passed to the energy

management system (EMS) for further applications like contingency analysis, automatic generation

control (AGC), optimal power flow (OPF), etc.

The monitoring of power system is commonly done by supervisory control and data

acquisition (SCADA) measurements. However, information provided by SCADA measurements

may not always be correct due to the appearance of measurement noise and potential gross errors.

Moreover, the available set of measurement may not be able to provide the complete AC system

states. Also, it is not realistic to have all buses directly measured in the system.

1.2 State Estimation

In order to address the above concerns, Fred Schweppe proposed the idea of state estimation

in the 1970s following the major blackout in the Northeast [2, 3, 4]. Nowadays, SE has become an

important part of the EMS. The SE normally includes the following functions [5]:

• Topology processor: Gathers information from circuit breakers and switches in the system to

create the accurate one-line diagram of the system.

• Observability analysis: Determines if a complete set of SE solution can be obtained by the

available set of measurements.

• State estimation solution: Determines the optimal state estimate for the entire system. Also,

provides the estimates for all branch flows, loads, transformer taps, generator outputs, etc.

• Bad data processing: Detects the existence of gross errors in the measurement set. Identifies

and eliminates bad measurements provided there is enough redundancy in the measurement

configuration.

• Parameter and structural error processing: Estimates various network parameters and detect

structural errors in the network configuration if there is enough measurement redundancy.

2


1.2.1 Weighted Least Squares (WLS) SE

The Weighted Least Squares (WLS) SE is one of the most commonly implemented SEs.

The iterative solution steps are shown below [5]:

1. Set the iteration index k = 0.

2. Initialize the state vector xk.

3. Calculate the gain matrix, G(xk).

4. Calculate the right-hand side tk = H(xk)TR−1(z− h(xk)).

5. Decompose G(xk) and solve ∆xk.

6. Is the convergence criterion satisfied?

If no, update xk+1=xk+∆xk, k = k + 1, and go back to step 3.

Else, stop.

It is well documented that the WLS SE is not robust and will be biased by even a single

bad measurement. In order to make the results of keeping WLS robust, an additional post-estimation

bad data processing is needed [6, 7]. However, implementing such a bad data removal process

will become increasingly time-consuming with growing system size. There are several alternative

methods which have better robustness compared to WLS [8, 9, 10, 11, 12]. Among these, this work

will mainly focus on the Least Absolute Value (LAV) SE, which will be briefly described next.

1.2.2 Least Absolute Value (LAV) SE

The LAV SE minimizes the L1 norm of measurement residuals instead of L2 [11]. LAV

can be conveniently formulated as a linear programming (LP) problem and solved by an efficient LP

solver. While LAV estimator has this very useful built-in bad data rejection capability, it has not been

commonly implemented since it remains vulnerable to the ”leverage” measurements when using

SCADA measurements.

Another and probably more significant reason why LAV SE has not been widely adopted by

the commercial power system software developers so far is its high computational burden compared

to that of the WLS SE. It is noted that when using SCADA measurements, there will be two nested

loops in the LAV SE solution algorithm, one of the inner LP iterations to solve the linearized problem,

and one for the outer state estimation iterations similar to the WLS SE iterations.

3


1.3 Phasor Measurement Unit (PMU)

Phasor measurement units (PMUs) were first introduced in 1988 as devices which could

measure voltage and current signals in a synchronized manner. PMU measurements have two main

advantages: firstly, all PMUs in the system are GPS clock synchronized, which leads to linear

measurement equations; secondly, refresh rate of PMU measurements is much faster (over 30

times/sec) compared to SCADA measurements (several seconds).

Even though currently the number of PMUs installed in most power systems is not sufficient

to make the system fully observable, it is not unrealistic to assume that in the near future, power

systems will be fully observable by only PMU measurements. Thus in this work, all investigations

will assume full observability by PMU measurements only in the studied systems.

The explicit usage of PMU measurements will perfectly resolve those two main disadvan-

tages of LAV estimator stated in Section 1.2.2. It is shown in [13, 14] that leverage points can be

readily eliminated by a simple scaling of the measurement equations when only PMU measurements

are used. Furthermore, the linearity of measurement equations eliminates the outer state estimation

iterations which help to speed up the LAV SE solution.

The following LAV state estimation algorithm is used in this research work for its auto-

matic bad data rejection capability and fast computing performance when explicitly using PMU

measurements. LAV SE can be formulated as the following optimization problem where the vector

xr represents the system state in rectangular coordinates, including the real and imaginary parts of the

bus voltage phasors. The measurement vector z contains the real and imaginary parts of voltage and

current phasor measurements received from PMUs, and is assumed to have additive Gaussian noise.

The measurement Jacobian is denoted by H which depends only on the network model parameters.

Note that all vectors and matrices are denoted in boldface in the rest of this dissertation.

min ‖r‖1s.t. z−H xr = r

(1.1)

where:

m is the number of measurements,

n is the number of system states,

rT = [ rr1 rr2 ... rrm ri1 ri2 ... rim ],

rri and rii are real and imaginary parts of the ith measurement residual, respectively,

4


H is a (2m× 2n) constant measurement Jacobian matrix,

xTr = [ e1 e2 . . . en f1 f2 . . . fn ] is a (2n× 1) estimated system state vector,

ei and fi are the real and imaginary parts of the ith state variable respectively,

zT = [ zr1 zr2 . . . zrm zi1 zi2 . . . zim ] is a (2m× 1) vector of measurements, zriand zii are the real and imaginary parts of the ith measurement, which may be a voltage or current

phasor.

When only PMU measurements are used, the problem (1.1) can be formulated as an

equivalent LP problem [14]:

min cT x

s.t A x = z

x ≥ 0

(1.2)

where:cT =

[02n 12m

]xT =

[Xa Xb U V

]A =

[H −H I −I

]I is a (2m× 1) identity matrix,

02n is a (1× 2n) zero vector,

12m is a (1× 2m) one vector,

Xa, Xb are (1× 2n) and U, V are (1× 2m) non-negative vectors defined as:

xr = XTa −XT

b

r = UT −VT(1.3)

The LP problem in (1.2) can be efficiently solved by any available LP solver.

1.4 Motivation

It is described above that PMU measurements can be acquired at a much higher scan rate

compared to traditional SCADA measurements. Thus in order to fully exploit the benefits of having

these measurements at such high scan rates, the estimation process needs to be completed within a

time period commensurate with the PMU scan rates.

This dissertation is intended to present fast and robust SEs in order to fully exploit the fast

scan rate of PMU measurements and meanwhile achieve high robustness against bad data. After

5


investigating the existing robust state estimation approaches, some issues were found when they were

attempted to be implemented on VLSI power grids.

The first issue is that even with very efficient LP solvers and high-performance computers,

the computational time will still inevitably grow with the system size and will gradually become the

limiting factor.

The second issue is that ISOs may be reluctant to share their system parameters and

measurement configurations with a central processor which solves the wide-area SE problem.

The third issue is that when a two-stage distributed computational approach is used, it

will inadvertently hit a hard limit as the number of zones and consequently the number of boundary

measurements increases with increasing system size. This limitation cannot be readily removed even

when one considers an unlimited number of available processors due to the implicitly sequential

processing of the two stages.

The fourth issue is that blindly dividing the system into zones and running distributed SEs

will result in the reduction of redundancy around zone boundaries because measurements between

zones have to be ignored. Thus a carefully-designed distributed algorithm is needed to utilize all

available measurements.

Therefore the main goal of this dissertation is to address these important issues to make the

CPU times of SEs independent of to system sizes and at the same time maintain robustness against

gross errors in the measurements. In order to achieve this, several centralized and distributed SEs are

proposed.

1.5 Dissertation Contributions

The main contributions of this dissertation are summarized below:

1. Two centralized robust SEs are introduced. Two different ways of incorporating equality

constraints (primarily but not limited to ZI measurements) into power system static LAV

state estimation problem formulation are presented and their performances are comparatively

evaluated with respect to computational efficiency and robustness against bad data. Primary

contribution of the algorithm is to identify the conditions (network size and configuration of

zero injections) under which these methods become more or less favorable compared to each

other. These results would enable choice of the method that would perform most efficiently for

the specific system under investigation.

6


2. A robust multi-area distributed linear SE combining the robustness properties of LAV estimator

with the well documented Dantzig-Wolfe (DW) decomposition principle in order to facilitate

utilization of LAV estimators for monitoring very large scale multi-area systems is proposed.

Three main contributions of the proposed implementation are: (1) robustness against bad

measurements irrespective of their location, (2) potential computational efficiency via the use of

parallel processors in a multi-area setting and (3) capability to solve wide area state estimation

problem while each area maintains confidentiality of its network data and measurements.

3. A two-stage multi-area distributed linear SE is developed. This algorithm extends a previously

developed two-stage multi-area state estimation approach to the case of large scale robust

phasor-only linear LAV SE. In the presented approach, it will be assumed that the entire power

grid is measured only by PMU measurements and all control areas (or zones) will have their

own LAV state estimators installed and capable of estimating their area state variables, i.e.

each area is observable by its internal measurement set. It will be assumed that a system-wide

coordinator will exist and will have access to PMU measurements of the areas as well those

estimated system states computed by individual area SEs. The coordinator will execute a

second stage SE using these two types of data as measurements. The proposed implementation

has three main contributions: (1) robustness against gross errors regardless of their location, (2)

high computational performance with the distributed processing among areas and (3) ability to

solve SE problem for very large, multi-area systems without having individual areas to share

their internal system and measurement information with others.

4. A novel multi-area LAV SE with zone generation is introduced. The proposed solution involves

creation of zones in addition to the existing ones by identifying the boundary buses of existing

zones and making sure that they appear as internal buses in these newly created zones. The

intention for those newly created zones is to make full use of those measurements on tie-lines

between zones (areas) which are inadvertently disregarded by individual area state estimators.

It is shown that when measured exclusively by PMUs, power grids can be efficiently monitored

by the proposed estimator which will (1) remain robust against gross errors appearing on

different locations around the system and (2) will be computationally efficient due to its

naturally distributed computational framework.

5. A multi-area LAV SE with multiple zone copies are developed. The proposed approach makes

multiple copies of the system model where each copy is configured using a different zone

7


configuration. The idea is to design the system copies and their zone configurations in such a

way that each bus appears as an internal bus (not a zone boundary bus) in at least one system

copy. Each zone in each system copy will be assigned a processor which will execute its

own state estimation. As will be illustrated below, the results of all processors can thus be

reconciled in a simple manner to obtain a robust solution as fast as the scan rate of the PMU

measurements provided that there is sufficient number available of processors. An automatic

algorithm to generate such partitioning copies are also proposed. The main contributions of

the proposed implementation are (1) robustness against bad measurements irrespective of their

location, (2) computational speed of overall solution via the use of parallel processors in a

multi-area setting.

6. The multi-copy LAV SE algorithm is implemented on a high-performance computer cluster

namely the ”Discovery Cluster” managed by the Northeastern University and is tested using a

2917-bus and a 16216-bus large utility system. Test results validate the expected CPU time

savings provided by the proposed algorithm when using multiple computing cores.

8

Chapter 2

Literature Review

2.1 Overview

After the deregulation of power industry back in the 1970s, the complexity of power

system operation increased dramatically. Meanwhile, renewable energy sources started to play a

more important role in the 21st century. Such sources are being connected to the grid at lower

voltages. All these require close monitoring of the system state in multi-area large-scale power grids.

This is usually carried out by SEs using measurement data gathered from different locations around

the power system. Due to the finite accuracy of meters and the telecommunication medium, raw

measurement data usually contain random errors. Also, when the meter has biases, drifts or wrong

connections, gross errors will also appear within measurements. Gross errors may be also caused

by failures in telecommunication systems or noise due to unexpected interference. Thus, one of the

biggest challenges SE is facing is to provide accurate estimates of system state even in the presence

of bad measurements.

Monitoring has been commonly done by SCADA measurements. While SCADA measure-

ments are still widely used, PMUs are also rapidly populating the substations. PMU has significantly

faster data acquisition rate compared to traditional SCADA measurements. With the growing dimen-

sion and complexity of modern power grids, the computational time for SEs to run also increases

exponentially. This brings up another challenge for SEs to be fast enough to be able to keep up with

the pace of the data refreshing rate of PMU measurements when implemented on VLSI power grids.

So to sum up, the question becomes: how could we develop an SE with not only high

robustness level against bad data, but also short enough computational time to fully utilize the high

data refreshing rate of PMU measurements on large-scale, multi-area power grids?

9

CHAPTER 2. LITERATURE REVIEW

Various researchers have contributed voluminous literature on this topic and their ap-

proaches can be categorized into either centralized or distributed solution. Both categories will be

discussed in detail in the following sections.

2.2 Centralized Solutions

The WLS SE is so far the most commonly used SE. However, WLS method is known

to be non-robust, i.e. it will fail to provide an unbiased estimate even in the presence of a single

bad measurement. In order to resolve this issue, a separate post-estimation bad data detection,

identification and removal process is needed. The maximum normalized residual rNmax test is one of

the most commonly used methods to detect and identify a single bad measurement [7]. Multiple bad

data, on the other hand, are more difficult to handle. One alternative method is Hypothesis Testing

Identification (HTI) method. First presented in [15, 16], HTI differs from the largest normalized

residual method as bad data are identified based on the computed estimates of measurement errors,

which resolves the weakness of the largest normalized residual method when residuals are strongly

correlated. Hence, robust alternative formulations for SE are widely investigated.

One of the robust SEs presented are the M-estimators [17]. An M-estimator minimizes

an objective function expressed by a function of measurement residuals, subject to measurement

equations as constraints. The first M-estimator proposed was [18]. Several other variations in the

objective function were made yielding corresponding sets of M-estimators [6, 8, 12, 19, 20]. These

SEs can be solved by two alternative methods: Newton’s method and the iteratively re-weighted

least squares method [5]. There has always been a trade-off between robustness and computational

efficiency in most of these methods. Furthermore, the existence of leverage measurements in

typical power system measurement configurations presented additional challenges to robustness in

implementing these alternative methods [12]. An observation (measurement) is called a leverage

point if it lies far away from the bulk of the rest of the measurements in the sample space [21].

Specifically in power systems, these measurements act like ”critical” measurements which will be

forced to be satisfied and will bias the estimation result if any of them carry gross errors.

The LAV estimation problem, introduced in Section 1.2.2, can be shown to be equivalent to

an LP problem, which can be easily solved by one of the available LP solvers [22]. This property is

exploited to implement and test LAV SE on power systems in subsequent studies [10, 23]. While LAV

SE’s robustness against bad measurements was seen as a significant advantage, it remained vulnerable

10


to leverage measurements. Furthermore, its computational performance was not competitive with

WLS in the absence of bad data when SCADA measurements were used.

With the increasing number of PMUs being installed in power systems, the LAV SE was

revisited by researchers as a viable and robust alternative. Those above-mentioned shortcomings

of the LAV SE are conveniently removed by the exclusive usage of PMU measurements whose

measurement equations become strictly linear. Furthermore, leverage measurements can be avoided

by applying a simple scaling transformation to the phasor measurement equations [14]. The linearity

of the measurement equations also eliminates the need to use iterative methods, yielding a faster

computational performance for the LAV SE. Solution times become comparable to the WLS estimator

in the absence of bad data, and they even become shorter than those of the WLS when a large number

of bad data are present in the measurement set, as shown in [14, 24].

Equality constraints, normally zero power injections for SCADA-based measurements

or zero current injections for PMU-based measurements, constitute not only free but also perfect

measurements which ought to be strictly satisfied by their nature. Incorporating these ZIs into SEs

will definitely increase the robustness level since these perfect measurements are strictly enforced.

The idea of incorporating equality constraints into SE has been considered by a large number of

researchers in the past. Starting with the paper [25], it was studied by numerous groups worldwide in

[26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37]. These publications provide invaluable algorithmic

innovations and detailed implementation methods which have been successfully applied to various

production grade EMS software globally. Majority of these methods explored ways of implementing

equality constraints into WLS SE. It has also been noted that equality constraints could be readily

introduced in LAV SEs without significantly modifying the estimation formulation. In fact, it could

possibly accelerate the solution due to the way Simplex iterations were carried out starting from an

appropriate basis which could be determined directly from the results of observability analysis [38].

2.3 Distributed Solutions

While the above factors in Section 2.2 make LAV SE competitive with WLS and other

robust SEs, it is still not practical to implement any centralized SE for VLSI power grids because

a centralized SE will inevitably slow down with increasing system size and complexity. On the

other hand, it may be possible to distribute the computational burden by making the SE process

distributed. Some excellent papers presented different approaches with either parallel or hierarchical

computational architectures using SCADA measurements [39, 40, 41, 42, 43, 44]. Some of these

11


approaches attempt to make each zone run their state estimators independently with minimal data

exchange between them. However, blindly dividing the system into zones will lead to significant

loss of measurement redundancy around zone boundaries. This shortcoming is identified by several

researchers and various algorithms [45, 46, 47, 48, 49] are proposed to address it by using a hierar-

chical SE based on SCADA measurements or a hybrid SE using SCADA and PMU measurements.

For certain algorithms using PMU measurements, at least one PMU is installed in every control area

for synchronizing voltage angle references among areas.

It is well-known and documented that LP problems with a bordered block diagonal structure

with a single linking set of rows can be efficiently solved by an algorithm initially proposed by

Dantzig and Wolfe [50] and referred as the Dantzig-Wolfe (DW) decomposition algorithm. It is

already shown in [44] that the SCADA measurements, when linearized around an operating point,

yield the above mentioned special structure suitable for DW decomposition.

So conclusions can be drawn that as a centralized solution, the LAV SE is highly robust

under the assumption that future power grids will be made fully observable by only PMUs. Fur-

thermore, including ZIs into LAV SE may further increase its robustness level. Meanwhile, the

explicit usage of PMU measurements will make above distributed algorithms more relevant, since

the estimation problem can be decomposed into several completely independent sub-problems and

solved in parallel. Aforementioned advantages of the LAV SE with purely PMUs can be combined

with these parallel processing structures.

This dissertation presents two ways of incorporating ZI measurements into the centralized

LAV SE, together with several novel distributed LAV SEs. All algorithms are proposed under the

assumption of the explicit usage of PMU measurements. Details of these algorithms will be shown

in the following chapters.

12

Chapter 3

LAV Estimator with Equality

Constraints

3.1 Introduction

Power systems have historically been monitored via measurements provided by the Su-

pervisory Control and Data Acquisition (SCADA) systems in substations. These measurements are

communicated to control center computers where they are processed by a static state estimator. Most

commonly used method for estimating the system state has so far been the weighted least squares

(WLS) estimation. However, WLS method is known to be non-robust, i.e. it will fail to provide an

unbiased estimate even in the presence of a single bad measurement. Hence, every WLS estimator

has to be equipped with a post-estimation bad data processing function in order to properly detect,

identify and remove gross errors from the measurement set. Alternative robust SEs, at the same

time, are not widely implemented with SCADA measurements due to their heavy computational

burden.With the recent trend of PMU measurements, one of these robust SEs, the LAV SE, shows

some merits. It is proven in Section 2.2 to be robust and computational efficient with the explicit

usage of PMU measurements.

The objective of this work is to investigate the use of equality constraints in this setting and

determine how best to incorporate such constraints in the robust linear estimation formulation. While

such equality constraints can be used to enforce various types of variables and measurements, this

work will mainly consider zero injections (ZIs) which are the so-called free and perfect measurements

of any state estimator, be it SCADA-based (zero power injections) or PMU-based (zero current

13

CHAPTER 3. LAV ESTIMATOR WITH EQUALITY CONSTRAINTS

injections).

As described in Section 2.2, using ZI in SEs has been studied by various researchers.

However, these studies did not explicitly consider PMU measurements; hence their performance

evaluations were mostly based on SCADA-based iterative estimation procedures. Unlike the case

of SCADA-based state estimation where ZIs are defined as the power balance equations with a net

zero value, in this work, ZI measurements refer to the Kirchhoffs current law at those ZI buses. As

such, these are not really measured quantities and they constitute physical constraints imposed by

the circuit theory. Hence, they are not only perfect (error free) measurements but they also come

free without any installation costs. Furthermore, in the case of LAV estimation, they can be readily

introduced into the state estimation formulation without the need to modify the problem formulation.

This is not the case when using WLS estimators where the problem formulation needs to be somehow

revised for instance by using the method of Lagrange multipliers.

This chapter implements two alternative methods in order to incorporate ZI measurements

as equality constraints in LAV estimation in Section 3.2 and Section 3.3. Simulation results using

140-bus, 181-bus and 2917-bus test systems are shown in Section 3.4. The conclusions are shown in

Section 3.5.

3.2 Direct Addition of Zero Injection Measurements

An attractive feature of the LAV estimation formulation is that ZI measurements can be

directly incorporated into the optimization formulation of (1.2). This is accomplished by simply

removing the non-negative slack variables corresponding to the ZI measurement equations. It can

be assumed without loss of generality, that all ZI measurements are ordered first in building the

measurement equations and thus, the constraints associated with the problem of (1.1) can be written

as:

z = H x + r 0Tnz

zr

=

Hz

Hr

[x] +

0Tnz

rr

(3.1)

where:

0nz is a (1× 2nz) vector of zeros,

nz is the number of ZI measurements,

Hz is the measurement Jacobian matrix associated with ZI measurements,

14


Hr is the measurement Jacobian matrix associated with measurements excluding all ZI

measurements,

zr and rr are measured values and residuals corresponding to the regular measurements

respectively.

The updated version of the LP problem given in (1.2) will then be written as follows:

Min. c1Ty1

s.t. M1y1 = b1

y1 ≥ 0

(3.2)

where:c1

T = [ 0n 0n 1nr 1nr ]

y1T = [ XT

a XTb UT

r VTr ]

M1 =

Hz

Hr

−Hz

−Hr

Ωnz

Inr

Ωnz

−Inr

bT1 =

[0Tnz zTr

]UT =

[0Tnz UT

r

]VT =

[0Tnz VT

r

]nr is the number of regular measurements,

nz is the number of ZI measurements,

0n is a zero vector of dimension (1× n),1nr is a vector of 1s of dimension (1× 2nr),

Ωnz is a zero matrix of dimension (2nz × 2nr),

Inr is an identity matrix of dimension (2nr × 2nr),

0nz is a zero matrix of dimension (2nz × 1),

Ur and Vr are non-negative (2nr × 1) vectors where:

rTr = UrT −Vr

T (3.3)

Note that ZI measurements are modeled as complex variables (with zero real and imaginary

parts).

This is actually a much smaller problem in terms of the number of unknown variables

since all the slack variables associated with the ZIs are left out of the formulation forcing them to be

exactly zero.

15


3.3 Explicit Addition of Zero Injection Equality Constraints by Kron

Reduction

An alternative method to enforce ZIs (or equality constraints) is via the application of

Kron reduction to eliminate ZI measurements from the set of all measurement equations. This

is accomplished by ordering the columns of H in (3.1) in such a way that the (nz × nz) square

sub-matrix at the top left corner is non-singular. This is done by partial LU decomposition with

column pivoting which yields the following re-ordered version of the original H matrix:

Hord =

Hzz Hzr

Hrz Hrr

(3.4)

Hzz Hzr

Hrz Hrr

xz

xr

=

0Tnz

zr−rr

(3.5)

where:

Hzz is a (2nz × 2nz) non-singular square matrix,

xz and xr are the sub-vectors that constitute the system state and ordered consistent with

the column ordering of Hord.

Eliminating xz from 3.5 yields the reduced Jacobian:

Hred = Hrr −Hrz H−1zz Hzr (3.6)

LAV state estimation problem can thus be rewritten in the following form with the reduced

number of variables:

Min. cT2 |rr|s.t. zr −Hred xr = rr

(3.7)

where xz can be recovered from the solution xr as follows:

xz = −H−1zz Hzr xr (3.8)

Modified LAV problem of (3.7) can be solved using any LP solver. Being exactly equivalent

to the original problem, the solutions of both problems are expected to be identical, yet given the

reduced number of unknowns in problem (3.7), there may be computational advantages which are

explored next.

16


3.4 Simulation Results

In this section, the performance of the two alternative implementations of the robust linear

LAV state estimator with equality constraints is tested using two test systems. Simulations are carried

out on a PC with Intel R© CoreTM i7-5500U CPU, 16GB RAM and Windows 8.1 64-bit operating

system. LP solutions are obtained in MATLAB R© R2015a and using GUROBI optimizer solver

version 6.0.5.

Two cases are simulated. The first case is intended to illustrate the impact of introducing

ZIs as equality constraints in identifying and automatically removing bad data compared to the case

where these injections are treated as regular measurements. This case is implemented using the

140-bus NPCC test system. The second case demonstrates CPU time performance improvement

of introducing ZI equality constrains on two test systems: (1) 181-bus WECC test system and (2)

2917-bus utility test system.

3.4.1 Impact on Bad Data Rejection

Using the NPCC test system with 140 buses, 171 branch PMUs are placed using the

placement method of [51]. It is noted that this system has 51 ZI buses. Furthermore, 8 current

injection measurements located at buses 3, 7, 11, 21, 22, 36, 83, and 98 are added for extra

redundancy.

LAV state estimation solution for this system is obtained by first treating ZIs as regular

injection measurements (method 0) and then by explicitly forcing ZIs as equality constraints (method

1&2). As described in Section 3.2 and Section 3.3, equality constraints can be incorporated into the

equivalent LP problem formulation in two different ways. However, both methods yield identical

solutions, so the results for these two separate implementations of equality constrained LAV are not

shown as duplicates, but as a single column under method 1&2 in Table 3.1. Regular (non-zero)

current injection measurements are modified one at a time, each time setting their values both the

magnitude and phase angle equal to zero.

As evident from the results shown in Table I, when single bad datum is introduced in these

regular injection measurements, method 0 fails to automatically reject bad measurements in almost

half of the cases. Meanwhile method 1&2 stays robust against bad data and successfully identifies

and rejects all bad data. These results validate the advantages of formulating ZI measurements as

strict equality constraints (method 1&2) when executing linear phasor-based LAV state estimation.

17


Inj. Original Measurement Residual (p.u.) Reject BD?

at Bus Measurements (p.u.) Method 0 Method 1&2 Mtd. No. Y/N

3 -0.0898+0.8985i 0-0.8985i 0.0898-0.8985i 0:N / 1:Y

7 -3.6794+0.2255i 3.6794-0.2255i 3.6794-0.2255i 0:Y / 1:Y

11 -6.7892+0.2253i 6.7892-0.2253i 6.7892-0.2253i 0:Y / 1:Y

21 6.4348+0.7582i 0+0i -6.4348-0.7582i 0:N / 1:Y

22 6.2387-0.6103i 0+0i -6.2387+0.6103i 0:N / 1:Y

36 5.4947+1.1057i 0+0i -5.4947-1.1057i 0:N / 1:Y

83 -3.7578-1.1899i 0+0i 3.7578+1.1899i 0:N / 1:Y

98 5.7766+2.8258i -5.5025+0i -5.7766-2.8258i 0:N / 1:Y

Table 3.1: ZI Simulation Results of NPCC Test System

3.4.2 Impact on Computational Performance

GUROBI solver which is used in both of these two test cases fully exploits sparsity of the

network equations.

3.4.2.1 181-bus WECC Test System

181-bus WECC test system is used first to compare the computation time performance

between direct addition of ZI equality constraints (Section 3.2) and use of Kron reduction to eliminate

extra variables (Section 3.3) in LAV state estimation.

WECC test system has 313 branches and 81 ZI buses. It is measured by 258 PMUs. In

order to gauge the dependency of computation time on the number of ZI constraints, the number

of ZI buses included as equality constraints in the LAV formulation is varied from 0 to 80 with

increments of 10. For all test cases, a single error in the voltage magnitude of the voltage at bus 49 is

introduced into the measurement set.

Referring to the two alternative implementations of Section 3.2 and Section 3.3 as method

1 and 2 respectively, Table 3.2 presents CPU times required for both methods. In all test cases both

estimators successfully identified and rejected bad data and have the yielded identical state estimates.

Given CPU times include the overall solution time for the LAV estimator using the GUROBI solver.

18


No. of CPU Time (seconds)

ZIs Method 1 Method 2

0 0.0176 0.0181

10 0.0180 0.0180

20 0.0179 0.0193

30 0.0186 0.0152

40 0.0188 0.0150

50 0.0188 0.0146

60 0.0198 0.0147

70 0.0209 0.0147

80 0.0211 0.0142

Table 3.2: ZI CPU Times for 181-Bus Test System

3.4.2.2 2917-bus Utility Test System

These two methods are also tested on a larger 2917-bus utility test system with 3826

branches and 910 ZI buses. The system is measured by 2586 PMUs. The degree (number of

neighbors) of ZI buses varied among them. As expected, those having higher degrees resulted in large

numbers of fill-ins during Kron reduction causing loss of sparsity in the reduced Jacobian matrix. To

minimize the impact on sparsity, ZI buses to be eliminated are chosen based on their degrees. The

degree threshold d is varied from 2 to 10, i.e. only those ZI buses with less than degree d are treated

as equality constraints. Remaining ZIs are processed as regular measurements with high weights.

For all test cases, a single error in the voltage magnitude of the voltage at bus 583 is introduced.

Table 3.3 presents CPU times for both methods. In all test cases both estimators successfully

identified and rejected bad data and have the yielded identical state estimates. Given CPU times

include the overall solution time for the LAV estimator using the GUROBI solver.

Table 3.3 implies that increasing the number of ZI constraints, LAV state estimation

with Kron reduced formulation yields faster solutions in small-scale systems. However, Table 3.3

demonstrates that when these two methods are applied to large systems, increased number of

neighbors will lead to loss of sparsity for method 2, which offsets the advantage of the reduced

dimension of Jacobian matrix. This issue can be addressed by limiting the reduction only to those

ZI buses with less than a certain degree, such as 2, 3 or 4. As can be seen in Table 3.3, under these

19


conditions, method 2 will perform faster than method 1. Hence, ZI buses and their degrees can be

used as a guide to decide on the best method to employ for a given system.

ZI Bus CPU Time (seconds) No. of

Neighbors (Less than) Method 1 Method 2 ZI Used

2 0.2377 0.1979 81

3 0.2320 0.2011 313

4 0.2291 0.2270 383

5 0.2277 0.2482 472

6 0.2135 0.2592 550

7 0.2039 0.2921 645

8 0.2018 0.4507 727

9 0.1982 0.9952 793

10 0.1931 6.1484 847

Table 3.3: ZI CPU Times For 2917-Bus Test System

Thus, the computational performance of method 1 is better for large size system with ZI

buses having large number of neighbors. On the other hand, method 2 may be preferable for small

systems or with any size system where ZI buses have very few neighbors.

3.5 Conclusions

This chapter investigates the phasor-based linear LAV state estimation problem when

there are a substantial number of ZI buses in the system. ZIs constitute no-cost and also error

free measurements and hence are quite beneficial in state estimation. Two different methods of

introducing ZI measurements into the LAV SE are proposed. Both methods’ performance against

the appearance of BD and CPU time performances are evaluate using two different test systems.

It is shown that the incorporation of ZI and LAV SE is rather straight forward. Simulation results

show that by including ZI measurements, the robustness level of the SE will increase. What’s more,

both of these two methods have their own computational time advantages depending on the specific

measurement and network configuration.

20

Chapter 4

Linear LAV Estimator using

Dantzig-Wolfe Decomposition

4.1 Introduction

Two centralized SEs using ZIs are introduced in the previous chapter. Simulation results

show that both these two algorithms have high robustness against gross errors. Meanwhile, their

computational performance varies with different system dimensions and measurement configurations.

However, as described in Section 2.3, with the increasing dimension and complexity of modern

power systems, a centralized SE’s computational time will grow dramatically. So in this chapter, we

will combine the robust linear LAV SE shown in Section 1.3 with a well-known LP decomposition

method and extend the SE to a distributed structure.

Following the deregulation of power industry, power system operators are confronted with

the difficulties of coordinating and monitoring power exchanges over long distances between remote

areas. This required wide-area large scale monitoring capability at regional control centers and thus

motivated development of several innovative multi-area solutions to the state estimation problem.

These approaches are reviewed in detail in Section 2.3. Initial studies focused on methods that used

measurements provided by the Supervisory Control and Data Acquisition (SCADA) system.

The goals of these approaches were two folds: (1) to address the high dimensionality of

the estimation problem by formulating the problem in either parallel and/or hierarchical framework;

(2) to address the reluctance of individual area system operators to share real-time network and

measurement data by developing multi-area solutions with minimal exchange of data among area

21

CHAPTER 4. LINEAR LAV ESTIMATOR USING DANTZIG-WOLFE DECOMPOSITION

operators.

While it is important for state estimators to provide estimates of system states that are as

close to the actual states as possible, given the inadvertent uncertainties in the measurements and

network parameters, the main goal of the estimator is to detect and remove bad data in order to

provide an unbiased estimate. This is commonly referred to as statistical robustness of the estimator

against bad data. There have been several studies towards developing robust alternatives to the

statistically non-robust Weighted Least Squares (WLS) estimator. One such estimator is the so called

Least Absolute Value (LAV) estimator which minimizes the L1 norm of the measurement residuals.

LAV estimator has not been adopted initially due to its vulnerability to leverage points when the

estimation is strictly based on SCADA measurements. However, this issue is resolved in by a simple

scaling transformation when using only PMU measurements and a linear estimator [14]. As a result

of this development LAV estimator is shown to be not only computationally competitive with the

WLS counterpart, but more importantly highly robust against bad data unlike WLS.

In this chapter, the afore mentioned robustness properties of LAV estimator will be com-

bined with the well documented Dantzig-Wolfe (DW) decomposition principle [50] in order to

facilitate utilization of LAV estimators for monitoring very large scale multi-area systems. DW

method was used with SCADA based LAV estimators in the past [5]. In this chapter, it will be shown

that having PMU measurements not only makes the DW approach easier to utilize but also more

reliable and robust against bad data, irrespective of measurement locations in individual areas or at

their boundaries.

Implementation of DW decomposition method requires several changes which are specific

to phasor measurement equations and power system topology. Furthermore, it is still assumed that

individual areas remain reluctant to share their real-time measurements and/or network data. Hence,

in the proposed approach, all areas initially find their own state estimation results with no sharing

of network and measurement information. Then each area calculates and sends its own relative

cost and a generated column back to the master computer. The master computer receives all data

from individual area computers and determines whether the optimal solution is found in which case

the procedure terminates. Else, master computer continues the solution procedure by sending the

required signals back to slave computers of each area. Details of this exchange will be clarified in the

subsequent sections. Note that use of only PMU measurements eliminates the need for a reference

(slack) bus unlike the case of SCADA measurements.

The DW algorithm is illustrated in Section 4.2. The formulation, partitioning and proposed

implementation of the algorithm that are illustrated using the small IEEE 30-bus test system is shown

22


in in Section 4.3. Simulation results for a large 2917-bus multi-area utility system are shown in

Section 4.4, followed by the conclusions in Section 4.5.

4.2 Dantzig-Wolfe Decomposition

Dantzig-Wolfe (DW) decomposition is an iterative process for solving large LP prob-

lems when those problems have a special bordered block-diagonal structure. Consider the linear

programming problem shown in standard form [50] below:

min cTx

s.t. A x = b

x ≥ 0

(4.1)

and assume that matrix A has the following form:

A =

L1 L2 . . . LN

A1

A2

. . .

AN

(4.2)

where N is indicating the number of partitions, for instance the number of areas in the

case of power grids. By using the same partitioning on vectors x, cT and b , the LP problem of (4.2)

can be rewritten as:

minN∑i=1

ciTxi

s.t.N∑i=1

Lixi = b0

Aixi = bi

xi ≥ 0, i = 1, 2, . . . , N

(4.3)

where:

b0 is the vector corresponding to linking constraints in b,

bi is the vector corresponding to constraints in area i in b.

23


Problem (4.3) can be viewed as N independent sub-problems minimizing the sum of their

own total costs except the first block constraint, which will be referred as the linking constraint.

Every sub-problem will have the following form:

min ciTxi

s.t. Aixi = bi

xi ≥ 0

(4.4)

The constraint set for the ith sub-problem is Si = xi : Aixi = bi,xi ≥ 0. For a stable

operating power system, every variable xi will be assumed to be bounded by a large artificial upper

bound. In mathematics, an extreme point of a convex set S in a real vector space is a point in S

which does not lie in any open line segment joining two points of S. Furthermore, any point xi ∈ Sican be shown to be expressible as a linear combination of all the extreme points of Si, i.e. the set

xi1, xi2, , xiKi in the following form [50]:

xi =Ki∑j=1

αijxij,

whereKi∑j=1

αij = 1

and αij ≥ 0 j = 1, 2, . . . ,Ki

(4.5)

where:

Ki is the number of extreme points in area i,

αijs are the weighting coefficients of the extreme points.

Next, let us define pij = cTi xij as the equivalent cost of the extreme point xij and

qij = Lixij as the equivalent activity vector of xij in the set of linking constraints. The original LP

problem can now be converted to an equivalent master problem as given below [50]:

minN∑i=1

Ki∑j=1

pijαij

s.t.N∑i=1

Ki∑j=1

qijαij = b0

Ki∑j=1

αij = 1

αij ≥ 0, j = 1, 2, . . . ,Ki

i = 1, 2, ..., N

(4.6)

Note that the unknown variables in the master problem are transformed from the xi’s of

the original problem to αij’s which are defined as:

24


αij = (α11, . . . , α1K1 , α21, . . . , α2K2 , . . . , αN1, . . . , αNKN)

and the equivalent master problem of (4.6) can be written in compact form as:

min pTα

s.t. Qα = g

α ≥ 0

(4.7)

where:

gT = [bT0 , 1, 1, . . . , 1], the number of 1’s is equal to the number of areas or partitions of

the original problem,

p is a vector of pij’s each associated with an αij ,

Q is a matrix whose columns are given by:

qij

ei

where each of the above columns are associated with an αij , and ei represents a (1×N)

singleton vector with all zero elements except the ith entry which is equal to 1.

The relative cost vector associated with each αij can be calculated by:

rij = pij − λT qij

ei

(4.8)

where λT = pBTB−1; B represents the basis matrix composed of the appropriate columns

of Q ; pB is the vector composed of the entries of vector p corresponding to the basis columns.

Thus, the ith sub-problem which minimizes an adjusted relative cost function ri can be

written as follows [50]:

min ri = (ciT − λ0TLi)xi

s.t. Aixi = bi

xi ≥ 0

(4.9)

where: mL is the number of linking constraints (the number of rows in Li ).

λ0 is a vector containing the first mL entries in λ.

Let us define r∗ as the vector of relative cost coefficients corresponding to all possible

basis column vectors. If all entries of r∗ are nonnegative, then the procedure will be terminated and

the current solution will be declared as optimal. Else, the column corresponding to the most negative

25


entry in r∗ will be chosen as the one to enter the basis in the next iteration. If the chosen columns

index is i then the entering column will have the following form: Lixi

ei

(4.10)

Under certain circumstances, one of the sub-problems may become unbounded. This may

be caused by the value of the adjusted cost function and/or the structure of the problem due to the

measurement configuration. In such cases, the optimal solution of the sub-problem is substituted

by the unbounded ray of that unbounded sub-problem. ”Unbounded ray”, or sometimes referred to

as ”extreme ray”, is a nonzero element x of a polyhedral cone C ⊆ Rn if there are n− 1 linearly

independent constraints binding at x. For an unbounded LP problem. Unbounded ray can be easily

obtained by any available LP solver. Therefore, the entering column will appear as [52]: Lixi

0

(4.11)

Following the revised simplex method, the exiting column of the basis B will be determined

by pivoting. It is replaced by the column chosen to enter by (4.10) or (4.11). This iterative process of

updating the basis and obtaining a new basic feasible solution will continue until an optimal solution

is reached.

4.3 Decomposition Implementation

This section will use the IEEE 30-bus test system as an example in order to demonstrate

the details of formulating the solution of LAV state estimation problem using DW decomposition.

As will be shown in this section, owing to the special characteristics of the LAV estimator, several

modifications will have to be made to the standard DW decomposition procedure. IEEE 30-bus

test system will be assumed to have three non-overlapping areas as shown in Figure 4.1. Tie-lines

whose terminal buses are defined as boundary buses interconnect these areas. Tie-line current flow

measurements and boundary bus current injection measurements are denoted as linking measurements

(or constraints) in (4.1) because these constraints are functions of state variables belonging to different

areas. Rest of the measurements are labeled as internal measurements (or constraints) belonging to

only one (their own) area.

26


Figure 4.1: IEEE 30-bus system partitioned into 3 areas

By properly re-ordering the measurements as well as state variables, the LAV state estima-

tion problem of (1.2) can be rewritten as follows:

min cTx

s.t. A x = z

x ≥ 0

(4.12)

where:

cT = [ wL wL 0 0 w1 w1 0 0 w2 w2 0 0 w3 w3 ]

xT = [ UL VL X1a X1b U1 V1 X2a X2b U2 V2 X3a X3b U3 V3 ]

A =

L

1

2

3

I −I HL1 −HL1 0 0 HL2 −HL2 0 0 HL3 −HL3 0 0

H1 −H1 I −I

H2 −H2 I −I

H3 −H3 I −I

zT = [zL

T z1T z2

T z3T]

j is area index, j = 1, 2, 3,

27


mL is the number of linking measurements,

mj is the number of internal measurements in area j,

nj is the number of system states in area j,

wL is (1×mL) vector of 1’s,

wj are (1×mj) vectors of 1’s,

UL and VL are slack variables in 1.3 corresponding to linking measurements,

Uj and Vj are area j’s slack variables in 1.3,

Xja and Xjb are area j’s system states in 1.3,

HLj is a constant measurement Jacobian matrix of linking measurements corresponding to

states in area j,

Hj is a constant measurement Jacobian matrix of internal measurements in area j,

zL and zj are re-ordered vectors of linking measurements and internal measurements

respectively.

Using partitioning with respect to the three areas, (4.12) can be rewritten as done in (4.3)

as follows:

min cLTxL + c1

Tx1 + c2Tx2 + c3

Tx3

s.t.∑3

j=1 Ljxj + L0xL = zL

A1x1 = z1

A2x2 = z2

A3x3 = z3

xj ≥ 0, j = 1, 2, 3

xL ≥ 0

(4.13)

where:

cTL = [ wL wL] , 2mL array of variables,

cTj = [ 0 wj wj] , (2(mj + nj)) array of variables,

Lj = [ HLj −HLj 0 0 ], (mL × 2(mj + nL)) matrix,

L0 = [ I −I ], (mL × 2mL) matrix,

Aj = [ Hj −Hj I −I ], (mj × 2(mj + nj)) matrix,

xLT = [ UL VL ], (2mL) array of variables,

xjT = [ Xja Xjb Uj Vj ], (2(mj + nj)) array of variables.

According to (4.3)-(4.9), (4.13) can be solved by solving the following master problem

and the three independent sub-problems:

28


4.3.1 Master Problem

min pTα

s.t. Qα = g

α ≥ 0

(4.14)

where:

pT = [ c1xb1 c2xb2 c3xb3 1mL]

1mL is a (1×mL) vector,

gT = [ zL 1 1 1 ]

Note that is the basic feasible solution (BFS) for area j, which can be calculated by simply

solving the following LAV state estimation problem for area j only:

min cjTxbj

s.t. Aj xbj = zj

xbj ≥ 0, j = 1, 2, 3

(4.15)

The column of Q associated with area j will then be given by: Ljxbj

ej

(4.16)

For very large power systems, it is very time consuming and also not necessary to find all

extreme points in (4.5). Instead, it is sufficient to generate only one basic feasible solution (BFS) for

each zone and use this BFS to create the column of Q given in (4.16). Thus, for the above 3-area

example, three columns will be created, one per area. Note that Q is the initial basis of the master

problem and therefore should be a square and nonsingular matrix. It will therefore include (mL + 3)

rows and thus an additional set of mL columns need to be added to the already generated three

columns of Q.

Here, the linking variables xL appearing in (4.13) will be moved into the master problem

augmenting the variables. Note that LAV formulation implicitly guarantees at least one of the u, v

pairs as defined in (1.3), to be zero. The non-zero variable u (or v) will yield the positive (or negative)

residual. Hence, a specially designed matrix SI, whose columns correspond to a mixture of u and v

variables, will be incorporated into the basis Q. SI will be a (mL×mL) diagonal matrix with entries

equal to ±1, the sign will match the sign of the corresponding entry r(i) in:

29


r = zL −[

L1 xb1 L2 xb2 L3 xb3

]T(4.17)

The structure of Q for the three area example will be:

Q =

L1xb1 L2xb2 L3xb3 SI

1

0

0

0

1

0

0

0

1

0

0

0

(4.18)

4.3.2 Sub-problems

min (cjT − λ0TLj)xj

s.t. Aj xj = zj

xj ≥ 0, j = 1, 2, 3

(4.19)

where:

λT0 is is the vector made up of the first mL entries of λ ,

λ is the vector of simplex multiplier, λT = pTQ−1

When trying to solve each sub-problem using any available LP solver, some of the sub-

problems may occasionally become unbounded. This problem will be caused by the adjusted cost

function and is intrinsic to the LAV state estimation formulation. Such cases are readily handled by

using unbounded rays as described above in (4.11).

Once optimal solutions for all areas are obtained, the following relative cost is calculated

for all sub-problems:

r∗j = (cjT − λ0TLi) x∗j − λmL+1

j = 1, 2, 3(4.20)

If all r∗j ≥ 0, the current basis will be optimal. The optimal solution in terms of the original

variables as well as the optimal objective function value can be recovered [50]. Otherwise, the most

nonnegative one among all three areas will be chosen and the corresponding column will enter the

basis Q. A flowchart for the entire iterative solution process is given in Figure 4.2.

30


Figure 4.2: Flowchart of the modified Dantzig-Wolfe decomposition

31



The proposed approach is tested with three different types of bad data in this system. Two

test systems are used: IEEE 30-bus test system and a large size (2917-bus) utility system. The

PC used for simulations has Intel R© CoreTM i7-4910MQ CPU, 32GB RAM and Windows 7 64-bit

operating system. All simulations are carried out in MATLAB R© R2015a environment and the

GUROBI optimizer solver version 6.5.0 is used as the efficient LP solver.

Three solution algorithms are implemented and comparatively evaluated:

• Approach 1: Modified DW decomposition based LAV state estimation solution developed in

this chapter.

• Approach 2: Independent LAV state estimation implemented for every area, discarding the

measurements linking one area to its neighbors.

• Approach 3: LAV state estimation implemented for the integrated system and solved as a

single large system problem.

Gaussian errors with standard deviation of 0.0001 are introduced to all phasor measure-

ments. Simulations focus on two aspects of state estimation: robustness under bad data, i.e. capability

to reject bad measurements and computational efficiency, i.e. CPU time required for the state estima-

tion solution.

There are three different types of measurements in the system: internal measurements with

no incidence to boundary buses (Type I), internal measurements that are incident to boundary buses

(Type II) and linking measurements (Type III). Bad data test considers bad data appearing on these

three different types on measurements. The mean squared error (MSE) as defined below will be used

as a metric to quantify the mismatch between estimated and true states:

MSE =1

Nx

N∑i=1

(xi − xi)2 (4.21)

where Nx is the dimension of system state vector xi .

4.4.1 Robustness under Single Bad Data

4.4.1.1 IEEE 30-bus test system

The IEEE 30-bus test system is divided into three zones and its partitions are shown in

Figure 4.3. Measurement system includes 32 branch PMU measurements and 5 current phasor

32


injection measurements. Every branch PMU includes a voltage phasor measurement on the from-end

of the branch and a branch current phasor measurement. These measurements are optimally placed

by [51].

Figure 4.3: Bad Data Location Illustration for IEEE 30 bus system

A gross error of zero is introduced in the real part of the current phasor measurement

measuring current from bus 14 to bus 12, marked as Bad Data 1 in Figure 4.3. Computed MSE

values given in Table 4.1 indicate the successful rejection of this bad data for all three approaches.

This simulation validates the uniformity of performance between the proposed as well as existing

LAV solutions for Type I measurements.

Approach MSE Reject BD?

1 4.5814e-05 Y

2 4.8593e-05 Y

3 4.6636e-05 Y

Table 4.1: 30-Bus DW Results for BD Measurement Type I

Bad data zero is introduced in the real part of the current phasor measurement along branch

23-15 which happens to be an internal measurement but incident to a boundary bus. MSE results

33


shown in Table 4.2 illustrate the failure of approach 2 (isolated area LAV state estimation solution).

The reason for this is the loss of redundancy when running isolated LAV SE. Disregarding of linking

measurements when executing individual area estimation causes some internal measurements (in

this case branch current 23-15 is one such measurement) become critical. This illustrates dangers of

blindly splitting the system and running distributed LAV SE. On the other hand, since they do not

remove any measurements, both approaches 1 and 3 remain robust against bad data as expected.


1 5.0520e-05 Y

2 0.0019 N

3 3.3664e-05 Y

Table 4.2: 30-Bus DW Results for BD Measurement Type II

Bad data appearing on linking measurements is tested. Specifically, the real part of current

phasor along branch 16-17 is intentionally corrupted by bad data zero. Note that in this case approach

2 is not applicable since it discards linking measurements and does not use them at all. Thus, only

approaches 1 and 3 will be compared in this case. The MSE values for both approaches are shown in

Table 4.3. Note that both approaches successfully identify and reject this bad data, validating the

robustness of the proposed algorithm on par with the integrated solution.


1 4.0048e-05 Y

3 3.7882e-05 Y

Table 4.3: 30-Bus DW Results for BD Measurement Type III

4.4.1.2 2917-bus Utility system

Next, a large utility system with 2917 buses and 11 zones is considered. An optimal PMU

placement algorithm [51] is used to place 4393 PMUs in this system making sure that redundancy

level is commensurate with the requirements of robustness.

Real part of branch current phasor measurement on branch 849-846 is set to zero. MSE

values computed for this case are given in Table 4.4. As evident from these values, bad measurement

is automatically rejected in all three approaches similar to the case of Table 4.1.

34



1 4.9780e-05 Y

2 5.0641e-05 Y

3 4.9715e-05 Y

Table 4.4: 2917-Bus DW Results for BD Measurement Type I

Real part of voltage phasor measurement at bus 2883 is intentionally set equal to the wrong

value of zero. Table 4.5 shows computed MSE values which are quite similar to the case of Table 4.2,

where isolated LAV for individual areas fails to reject bad measurement due to lack redundancy and

both approaches 1 and 3 stay robust against bad data.


1 5.1604e-05 Y

2 0.0129 N

3 5.0755e-05 Y

Table 4.5: 2917-Bus DW Results for BD Measurement Type II

One of the linking measurements (real part of current phasor along branch 448-2883) is

intentionally corrupted with gross error zero in this case. Computed MSE values shown in Table 4.6

are consistent with results of Table 4.3, indicating that approaches 1 and 3 can both successfully

identify and reject bad data.


1 4.9465e-05 Y

3 5.0342e-05 Y

Table 4.6: 2917-Bus DW Results for BD Measurement Type III

Results of simulations in both small and large scale power systems validate strongly robust

behavior of the proposed DW decomposition based LAV state estimator as well as the integrated

system LAV state estimator against all types of bad data.

35


4.4.2 Robustness under Multiple Bad Data

In this section, multiple zero gross bad data Type I are randomly introduced into the

2917-bus utility system. All three approaches are tested and their MSEs under different bad data

scenarios are shown in Table 4.7.

Bad Data Amount Approach 1 MSE Approach 2 MSE Approach 3 MSE

11 4.9435e-05 5.0508e-05 4.9598e-05

22 5.0136e-05 5.0789e-05 4.9925e-05

33 5.7655e-05 5.8155e-05 5.6969e-05

44 5.0046e-05 5.1094e-05 5.0538e-05

55 9.2480e-05 0.0092 9.2486e-05

Table 4.7: 2917-Bus DW Results for Multiple Bad Data

Results shown that both Approach 1 and Approach 3 stays robust under multiple bad data

scenarios. Due to the reduced redundancy level around boundaries, Approach 2 fails when 55 bad

data are introduced into the system.

4.4.3 Computational Time Performance

Finally, computational aspects of the proposed algorithm are tested comparatively with the

integrated solution for the entire system. Given the similar robustness capabilities of both approaches

1 and 3, only those two are tested for their computational time performance in this section.

As mentioned earlier, one advantage of approach 1 is that all zones can run their sub-

problems in parallel. Thus, this can be exploited by employing multi-core parallel computing. This

is assumed to be done for approach 1 and the expected computational times are calculated:

ttotal = (tsub + tmaster)× iter (4.22)

where:

tsub is the maximum computational time for one sub-problem,

tmaster is the computational time for master problem,

iter is the number of iterations.

Computational times for the state estimation solution implemented by approaches 1 and 3

are recorded for the small and large scale test systems and shown in Table 4.8. It is evident from

36


these results that approach 3 (integrated solution of LAV state estimation problem) is significantly

faster than approach 1 (proposed DW decomposition based solution) for the 30-bus small size system.

However, when system size grows, approach 1 becomes computationally more efficient (with the

assumption of multi-core parallel implementation). In fact, for the 2917-bus large scale system, the

solution time for the proposed DW based LAV solution is about 60% of the corresponding time taken

up by approach 3.

It should also be noted that, if the individual zone dimensions remain bounded, the solution

time required by the proposed algorithm (approach 1) for very large scale multi-area system state

estimation problem will remain quite insensitive to growing system size and number of zones. On the

contrary, the computational burden for approach 3 will rapidly increase with increasing system size.

Bus No. Approach 1 Time (s) Approach 3 Time (s)

30 0.0910 0.0082

2917 1.3403 2.6575

Table 4.8: 2917-Bus DW Computational Time Performance

4.5 Conclusions

This chapter introduced a distributed linear LAV SE using the well-known Dantzig-Wolfe

decomposition. The proposed algorithm takes advantage of the readily LP format of (1.1) and

decompose the SE problem into a master problem and several independent slave problems by DW.

This approach not only makes the solution time insensitive to system size and number of areas

(provided that individual area sizes remain bounded) but it also provides an attractive computational

architecture where individual areas do not have to exchange their network or measurement informa-

tion among themselves. The chapter contains simulation results illustrating the robustness of the

proposed algorithm to gross errors and its computational advantages in particular when used for very

large-scale systems.

37

Chapter 5

Two-stage Multi-area LAV Estimation

5.1 Introduction

A distributed linear LAV SE using Dantzig-Wolfe decomposition is proposed in Chapter 4.

It combines the LP-formulated linear LAV with the well-known DW decomposition. Robustness and

computational advantages are validated by simulation results. However, because of the characteristic

of DW algorithm, the SE needs to go through several iterations between the master and slave

problems until the problem converges. In order to further speed up the SE process and fully utilize

the advantage of linearity in LAV, a two-stage distributed linear LAV algorithm is presented in this

chapter.

Power transmission systems have been growing in size and complexity due to the necessity

to monitor lower and lower voltage levels in detail in order to track power flows manipulated by

renewable sources. These sources are increasingly being connected in large numbers at lower voltage

sections of the grid. Furthermore, due to the possibilities brought up by wide-area control and

optimization over long distances, monitoring very large scale power grids covering multiple control

areas is becoming a necessity. Monitoring has been commonly carried out by state estimators that

used measurements provided by the supervisory control and data acquisition (SCADA) systems.

While SCADA measurements are still widely used, synchronized phasor measurements are also

rapidly populating the substations with the installation of large numbers of phasor measurement units

(PMU).

Large multi-area power grid monitoring is challenging due to the increased size and

complexity of the system model and measurement volume. Each area may have a system operator and

its own regional control center and network applications running by its own operator. Individual area

38

CHAPTER 5. TWO-STAGE MULTI-AREA LAV ESTIMATION

operators may be reluctant to share real-time data and measurements or may not have the bandwidth

to do it due to the large volume of data involved. Thus a distributed computational structure may

better address the challenges involved. This was recognized earlier by various researchers and several

novel multi-area state estimation (SE) algorithms have been developed by using hierarchical or

parallel computing methods and SCADA measurements, as described earlier in Section 2.3. Some of

these approaches assume at least one PMU measurement per area and avoid coordination of state

estimation results among individual area reference bus angles. Also, the data interchange between

control areas is minimized or even eliminated with the intention of data protection.

These methods typically made use of the well-known weighted least squares (WLS) SE

algorithm, which requires a separate post-estimation bad data processing function in order to detect

and remove gross errors. An alternative to WLS estimator is the least absolute value (LAV) estimator

which minimizes the L1 norm (rather than L2) of measurement residuals. LAV estimators can be

efficiently implemented using existing high-performance linear programming (LP) solvers and they

can automatically process gross errors and remove them as part of the estimation process. Despite

these desirable properties, LAV estimators have not been implemented in the past for two reasons:

when using SCADA measurements the SE problem becomes nonlinear requiring multiple LAV

solutions in an iterative manner making it computationally slow, and more importantly the existence

of leverage points in SCADA measurements causes the LAV estimation to fail to automatically

eliminate gross errors thus defeating the main purpose of choosing it over WLS.

It has recently been shown that both of these shortcomings would readily be eliminated if

the power grid were made observable by only PMU measurements [13, 14]. The SE problem will

become linear eliminating the need for repeated LAV solutions and the vulnerability of ”leverage

points” can be also resolved by a simple use of scaling [14].

While the above factors make LAV SE competitive with WLS counterpart, it is still not

practical to implement any state estimator for very large scale power grids due to two reasons de-

scribed in Section 1.4. In order to solve those problems, this chapter implement a previous developed

SCADA-based two-stage multi-area SE [43] to the environment with purely PMU measurements.

Under the assumption described in Section 1.3, the whole power system will be made fully

observable only by PMUs. It is assumed that all control zones are equipped with their own SEs and

all of their state variables can be estimated locally. This is not an unreasonable assumption given

the fact that most control areas have well measured internal systems but less than complete set of

real-time measurements received from their neighbors. In the proposed approach, all those local

estimated states will be gathered by a central coordinator supervising the operation of the whole

39


system. The second stage SE will be executed by this coordinator combining linking measurement

data with estimated internal bus states from the first stage.

This chapter shows that in an all-PMU environment, this two-stage estimator will not only

remain robust against gross errors but also be computationally efficient due to the approximately

even distribution of computational burden among the areas. A further simplification compared to

[43] is the removal of area reference angles from the unknown vector in the second stage since all

PMU measurements are GPS synchronized and thus have the same global reference.

The chapter is organized such that the proposed two-stage SE algorithm is introduced in

detail and a practical example of its implementation using the IEEE 30-bus system is illustrated in

Section 5.2. Simulation results for both IEEE 30-bus and a 2917-bus utility system are provided in

Section 5.3, followed by the conclusions in Section 5.4.

5.2 Two-stage State Estimation Algorithm

It is assumed that a large-scale power system consists of n independent control areas and

for a single area j, buses in that area belong to either one of the following two categories:

• Internal bus: all of its neighbors belong to area j;

• Boundary bus: at least one of its neighbors belongs to a different area other than area j.

Assuming explicit use of PMU measurements, all measurements will be GPS clock syn-

chronized. This makes it possible for every area to run its own state estimation by using its own

measurement data and not use any of its bus voltage phase angles as a reference. The result of

state estimation for area j contains the estimated states for all buses within that area, including

both internal and boundary buses. It is noted that under the unrealistic error-free conditions, the

system-wide solution can simply be obtained by combining the state estimation results from all areas.

This however will never be possible in practice due to the existence of errors in all measurements.

Multi-area solution will address the issue of measurement errors and ensure a robust state estimation

solution irrespective of the location of any gross errors.

Individual area measurement configurations can be designed to meet the required local

redundancy levels for bad data rejection at any location within the areas. This can be accomplished us-

ing the previous developed PMU placement method described in [51]. Nonetheless, when individual

areas locally solve their state estimation problems, some of those measurements incident to inter-area

40


tie-lines will have to be discarded since they will be unusable by either neighbor. Such measurements

will be referred as linking measurements while the rest of measurements will be labeled as internal

measurements associated with their own area. Ignoring those linking measurements may cause some

internal measurements to become critical, which implies that the state estimation algorithm will fail

to detect and remove gross errors in these measurements during the first stage estimation leading to

biased solutions.

The two-stage multi-area state estimation algorithm presented in this chapter aims to

address the above described problem. In the first stage, every area runs its own state estimation and

sends its estimated states to the central coordinator for its use in the second stage. The coordinator

also collects measurements that are incident to area tie-lines from all areas, and executes the second

stage state estimation. This allows detection and correction of any potential erroneous estimated

states for area boundary buses. The final state estimation solution for all buses in the system will be a

combination of internal bus voltages estimated in first stage and all boundary bus voltages estimated

in the second stage. A flowchart of the overall algorithm is shown in Figure 5.1.

The IEEE 30-bus test system will be used as an example in this section for further illustra-

tion of the proposed algorithm. The system will be divided into three non-overlapping areas as shown

in Figure 5.2. By assigning all internal measurements to their own areas, the LAV state estimation

formulation in 1.2 can be rewritten as follows:

min cTx

s.t. A x = z

x ≥ 0

(5.1)

where:

cT = [ wL wL 0 0 w1 w1 0 0 w2 w2 0 0 w3 w3 ]

xT = [ UL VL X1a X1b U1 V1 X2a X2b U2 V2 X3a X3b U3 V3 ]

A =

L

1

2

3

I −I HL1 −HL1 0 0 HL2 −HL2 0 0 HL3 −HL3 0 0

H1 −H1 I −I

H2 −H2 I −I

H3 −H3 I −I

zT = [zL

T z1T z2

T z3T]

j is area index, j = 1, 2, 3,

41


Figure 5.1: Flowchart for Two-stage State Estimation Algorithm

42


Figure 5.2: IEEE 30-bus system divided into 3 areas







UL and VL are slack variables in (1.3) corresponding to linking measurements,

Uj and Vj are area j’s slack variables in (1.3),

Xja and Xjb are area j’s system states in (1.3),


states in area j,

Hj is a constant measurement Jacobian matrix of internal measurements in area j,


respectively.

In the first stage, all areas will solve the LAV state estimation for their own systems

independent of each other.

43


5.2.1 First Stage LAV State Estimation

The first stage of the algorithm is the distributed LAV state estimation for all individual

areas. At this stage, all linking measurements are discarded since they are unusable. Using the

notation in (5.1), formulation of the first stage LAV SE for area j will take the following form:

min cjTxj

s.t. Aj xj = zj

xj ≥ 0, j = 1, 2, 3

(5.2)

where:



cjT = [02nj

12mj],



Since individual state estimation solutions are completely independent of each other in the

first stage, the associated computations can be carried out in parallel by individual area computers.

After the completion of first stage estimation solutions by all areas, these solutions will be sent to the

central coordinator to be used in the second stage. It should be noted that having PMU measurements

in all areas eliminates the need to use a separate reference bus in each area. All solutions will be

readily synchronized with respect to the same global reference, i.e. the GPS clock.

5.2.2 Second Stage LAV State Estimation

After receiving all estimated states from the first stage as well as all linking raw measure-

ments, the central coordinator will solve the second stage LAV state estimation problem.

The entire set of estimated system states in the first stage can be rearranged as: xbdy est 1

xint est 1

(5.3)

where:

Xbdy est 1 and Xint est 1 are the boundary and internal bus voltages respectively, esti-

mated in the first stage.

44


Note that, since none of the linking measurements are incident to internal buses, only those

system states associated with the boundary buses (Xbdy) are estimated in the second stage. The

second stage LAV state estimation will then be formulated as the following linear programming

problem:

min csTxs

s.t. As xs = zs

xs ≥ 0

(5.4)

where:

mb is the number of boundary buses,


csT = [04mb

12(mL+mb)],

Hs =

Ls −Ls

Is −Is

, a (2(mL +mb)× 4mb) matrix,

Ls is the modified Jacobian matrix of linking constraints in (5.1) by removing all zero

columns corresponding to internal system states, making its dimension (2mL × 2mb),

Is is a (2mb × 2mb) identity matrix,

As = [ Hs IL −IL], a (2(mL +mb)× (4mL + 8mb)) matrix,

IL is a (2(mb +mL)× 2(mb +mL)) identity matrix,

xsT = [ Xbdy a Xbdy b Ubdy Vbdy] .

The updated estimates for the boundary buses are obtained after solving (5.4) using an LP

solver. Then, by combining the voltage estimates for internal buses from the first stage and the newly

estimated voltages of boundary buses Xbdy est 2 in the second stage, a complete set of system states

can be obtained as: xbdy est 2

xint est 1

(5.5)

In both first and second stage estimation solutions, possible creation of leverage points

are avoided through the use of simple scaling as described in [14]. Thus, both stage estimators

are made free of leverage measurements and consequently they maintain their robustness against

gross errors. The second stage estimation results will include the updated estimated system states

for all boundary buses. The first stage estimated voltages for individual area internal buses will

not be changed by the second stage estimator since internal buses are not incident to the linking

45


measurements. Therefore, the boundary bus voltages estimated in the second stage and area internal

bus voltages estimated in the first stage are combined to achieve the final estimation result for the

proposed two-stage multi-area state estimator.

In order to illustrate the algorithm better, a sample one-line diagram for a three-area system

together with its measurement configuration is shown in Figure 5.3. In the first stage state estimation

that is carried out by individual areas, only the internal measurements as shown in Figure 5.4 will be

taken into consideration by the area state estimators. The measurements that will be processed in the

second stage state estimation by the central coordinator are shown in Figure 5.5. A quick glance at

these figures will clearly illustrate that some measurements are discarded by the first stage of the

algorithm.

Figure 5.3: Three-area System Diagram and Measurement Configuration


The presented approach is tested on two test systems: IEEE 30-bus test system and an

actual 2917-bus utility system. The testing platform is a PC with Intel R© CoreTM i7-4910MQ CPU,

32GB RAM and Windows 7 64-bit operating system. The software environment is MATLAB R©

R2015a and the LP solutions are obtained by GUROBI optimizer version 6.5.0.

46


Figure 5.4: Measurements Used in the First Stage Estimation

Figure 5.5: Measurements Used in the Second Stage Estimation

47


The proposed approach is implemented along with two others and their performances are

comparatively evaluated:

• Approach 1: Two-stage LAV SE proposed in this chapter (Figure 5.4 and Figure 5.5).

• Approach 2: First stage LAV SE run in every area ignoring linking measurements (Figure 5.4).

• Approach 3: LAV state estimation for the entire power system, including all available mea-

surements (Figure 5.3).

Gaussian errors with standard deviation σ = 10−4 are added to all phasor measurement

data. Measurements used in test systems are divided into three categories: internal measurements not

incident to any boundary buses (Type I), internal measurements that are incident to boundary buses

(Type II) and linking measurements (Type III). Bad data on all these three types of measurements

are simulated. Mean squared error (MSE) defined below is used to quantify the mismatch between

estimated and true system states (4.21).

5.3.1 Robustness under Bad Data

5.3.1.1 IEEE 30-bus test system

The 3-area partition of IEEE 30-bus system is shown in Figure 5.6. 37 PMU measurements

are placed in this system including 32 branch PMUs and 5 current injection PMUs using the optimal

PMU placement algorithm [51]. One voltage and one current phasor measurement are placed at the

sending-end of every branch PMU.

A zero gross error is introduced to the real part of the current phasor measurement on branch

14 - 12, labeled as Bad Data 1 in Figure 5.3. MSE values in Table 5.1 show that all three approaches

successfully reject this bad data and remain robust. This confirms that all three implementation

approaches perform equally as expected for scenarios involving Type I bad data.


1 3.7390e-05 Y

2 4.5100e-05 Y

3 4.1717e-05 Y

Table 5.1: 30-Bus TS Results for BD Measurement Type I

48


Figure 5.6: Areas and Bad Data Locations for IEEE 30 bus system

49


Current phasor measurement on branch 23-15 is intentionally set equal to zero to simulate

a gross error. This measurement is an internal measurement which is also incident to a boundary bus.

Table 5.2 shows the MSE results for all approaches where approach 2 clearly appears to fail. This is

expected since the redundancy level is reduced significantly in the vicinity of this measurement when

disregarding the linking measurements in approach 2. This may cause some internal measurements

to become critical measurements and become vulnerable to bad data. Hence, this case illustrates

the potential risk of simply partitioning the system and executing state estimation solutions for each

partition. Meanwhile, both the proposed approach 1 and the full system estimation (approach 3) give

correct results and remain robust.


1 4.7054e-05 Y

2 0.0018 N

3 3.8191e-05 Y

Table 5.2: 30-Bus TS Results for BD Measurement Type II

The real part of current phasor measurement on branch 16-17 is replaced by zero simulating

a gross error. In this scenario approach 2 is not considered since the gross error is in a linking

measurement which is discarded by approach 2.


1 5.4103e-05 Y

3 3.8645e-05 Y

Table 5.3: 30-Bus TS Results for BD Measurement Type III

As shown in Table 5.3 MSEs for approach 1 and 3 demonstrate both approaches remain

robust against gross errors appearing in linking measurements.

5.3.1.2 2917-bus Utility system

A large scale utility transmission grid with 11 areas, 2917 buses and 3826 branches is used

to test the proposed approach. This system is populated with 4393 PMUs by using a previously

developed optimal PMU placement algorithm which can design the measurement configuration to

guarantee a desired level of local measurement redundancy [51].

50


Bad data is introduced to the current phasor measurement on branch 849-846. MSEs for

this case are shown in Table 5.4. It is evident from the results that bad data is successful rejected by

all three approaches.


1 5.0476e-05 Y

2 5.3794e-05 Y

3 5.0170e-05 Y

Table 5.4: 2917-Bus TS Results for BD Measurement Type I

In this case gross error is added to the voltage phasor measurement at bus 4359. Table 5.5

validates the expectation of failure for approach 2 which discards linking measurements and thus

fails to reject gross error due to low local redundancy. On the other hand, both approach 1 and 3

successfully reject the gross error and provide unbiased SE solutions.


1 4.9140e-05 Y

2 0.0493 N

3 4.8955e-05 Y

Table 5.5: 2917-Bus TS Results for BD Measurement Type II

This case considers bad data appearing in a linking measurement, namely the current

phasor on branch 448-2883. MSE results in Table 5.6 confirm robustness of both approaches 1 and 3,

similar to the case of Table 5.3.


1 4.9572e-05 Y

3 4.8995e-05 Y

Table 5.6: 2917-Bus TS Results for BD Measurement Type III

Above simulations demonstrate similar robust performance for the proposed two-stage

LAV SE algorithm and the integrated LAV SE solution both for small and large scale power systems

considering errors in all possible types of measurements.

51


5.3.2 Statistical Analysis of Robustness under Single Bad Data

Single gross error is injected into the measurement set by adding a fix bias equal to 250σ

(σ = 10−4) representing the standard deviation of measurement errors) to a randomly chosen

measurement within the measurement set. This is repeated 100 times and approach 1 is used to

estimate the state of the system using the generated measurement sets. The MSE values are computed

for each SE solution in order to statistically quantify the robustness of approach 1. The results show

that for all 100 cases, 100% of the MSE values remained below σ = 10−4, validating the high level

of robustness of the proposed approach 1 against single gross error in the measurements.

It is noteworthy to mention that the above Monte Carlo simulations are repeated for

approach 1 this time without using the previously mentioned scaling on the measurement Jacobian

for both estimation stages. The results show a drop in robustness where only 85% of the MSE values

remain below threshold. This once again illustrates the effectiveness of scaling on eliminating the so

called leverage measurements and facilitating implementation of a highly robust state estimator.

5.3.3 Statistical Analysis of Robustness under Multiple Bad Data

Possibility of having simultaneous errors in more than one measurement is also considered.

Multiple gross errors of each with magnitude 250σ are introduced to multiple measurements selected

at random in the 2917-bus utility system. Such gross errors are added to a fixed number (NB) of

randomly selected internal measurements of each zone as well as in linking measurements. The

number NB is varied from 1 to 7 to simulate cases of increasing numbers of multiple errors. For

each of these cases MSEs are computed and those resulting in MSE values larger than the threshold

σ = 10−4 are declared failed estimates since these imply that the SE fails to automatically reject one

or more of the introduced multiple bad data. The results are shown in Table 5.7.

As expected, with increasing number of multiple bad data, the two stage LAV estimator

hits a breakdown point. However, this breakdown point is a function of local redundancy as well as

the location of the considered multiple errors. Optimization of measurement design for best results

and limits on the number of multiple errors are important considerations but are not covered in this

work

5.3.4 Computational Performance

Having confirmed the robustness of the proposed approach under gross errors in the

previous sections, this section considers its computational performance in comparison to approach 3,

52


Bad Data Amount Number of Failed SE Cases

12 0

24 0

36 0

48 2

60 7

72 8

84 15

Table 5.7: 2917-Bus TS Results for Multiple Bad Data

i.e. centralized LAV SE for the entire system.

It is noted that the first stage of the proposed algorithm is highly suitable for parallel

processing (PP) given the independence of individual area state estimation solutions. Hence, consid-

ering the use of potential multi-core parallel computing, the expected computational times can be

calculated by:

ttotal = tfirst/Na+ tsecond (5.6)

where:

tfirst is the processing time for first stage,

tsecond is the processing time for second stage,

Na is the number of areas.

Please note that the data communication time is not taken into consideration in this

simulation.

Computational times of both 30-bus and 2917-bus systems for approach 1 (with or without

simulated parallel processing) and approach 3 are obtained and shown in Table 5.8.

No. of Buses Approach 1 w/o PP (s) Approach 1 w/ PP (s) Approach 3(s)

30 0.0117 0.0055 0.0041

2917 0.2348 0.0335 2.5468

Table 5.8: TS Computational Time Performance

53


In the case of small 30-bus system approach 3 is found to be faster that approach 1, but if

parallel processing is employed in implementing approach 1, then their computational performances

become comparable. However, even the slowest approach (approach 1 without parallel processing) is

fast enough for accommodating standard PMU data sampling rate, which is normally 30Hz.

Meanwhile, approach 3 is implemented using a large 2917-bus utility power system and

the computation time is obtained for the entire system solution as 2.5468 seconds. On the other hand,

use of approach 1 for the same system yields very significant computational gains compared to the

integrated solution by approach 3. It is observed that approach 1 could bring more than 90% savings

in CPU time even without parallel processing. When implemented using parallel processors (one

per area) CPU savings increase to more than 98% compared to that of approach 3. Note that all of

the reported CPU times are obtained using an off the shelf laptop. Using more powerful multi-core

workstations may show further improvements in computational performance. Using approach 1 and

assuming that first stage estimation is carried out by individual area processors in parallel, the largest

size area will be the determining factor for the CPU time of the overall solution. In contrast, the

solution time for approach 3 will increase with increasing system size and complexity.

As evident from the above results, the proposed two-stage LAV algorithm not only is

highly robust against gross errors in measurements, but also is computationally efficient to keep up

with the scan rates of PMU measurements even for very large scale power systems. Furthermore, its

multi-area distributed design makes this performance nearly independent of system size, bounded

mainly by the size of the largest area.

5.4 Conclusions

A robust and linear state estimator for very large scale power grids is presented in this

chapter. The estimator is designed to execute in two stages, where the first stage involves simultaneous

state estimation solutions carried out by multiple area processors. The second stage is a central

estimator which solves a much smaller problem, yet handles gross errors which might have been

missed by individual area estimators in the first stage. The attractive feature of the two-stage design

is that gross error rejection is built-into the estimation logic via the use of LAV estimators for both

stages. Also, the distributed scheme of this algorithm makes it possible for all zones in the first stage

to run their own SE simultaneously in parallel, which yields better computational performance. The

developed prototype program is successfully tested on small and large scale power systems using

Monte Carlo simulations involving single and multiple bad data.

54

Chapter 6

Multi-area LAV SE with Zone

Generation

6.1 Introduction

A two-stage linear LAV SE is proposed in the last chapter. It demonstrates good robustness

against bad data and high computational performance. There is also parallel computing potential in

the first stage. However, as described in Section 2.3, the two-stage sequential set-up still prevent the

algorithm to run in full parallel among zones. When system dimension grows, the complexity of the

second stage will become a computational burden. In order to provide one possible solution for this,

a multi-area linear LAV SE with zone generation is presented in this chapter.

State estimators typically use measurements provided by the supervisory control and data

acquisition (SCADA) system. As the system size, model complexity and measurement volume

increase so will the computational burden on the estimators. Large-scale power systems are usually

divided into several control zones. Taking advantage of this natural partitioning, several novel

multi-area state estimation (SE) algorithms have been proposed in the past shown in Section 2.3.

Those algorithms are developed using either a hierarchical or a parallel structure using SCADA

measurements. The Weighted Least Square (WLS) estimator is commonly used in most of these

algorithms. Since WLS estimator itself is not robust, a separate bad data processing is needed

following the SE in order to remove any existing errors [6]. Least Absolute Value (LAV) estimator

which minimizes the L1 norm instead of the L2 norm of measurement residuals, is a more robust

alternative to WLS [11]. It can be formulated as a Linear Programming (LP) problem and solved

55

CHAPTER 6. MULTI-AREA LAV SE WITH ZONE GENERATION

efficiently by one of several available LP solvers. LAV estimator is robust against gross errors and can

automatically remove them during the SE solution. Despite this attractive feature, LAV estimators

were not widely implemented in the past due to two main reasons: (1) Use of SCADA measurements

make the SE problem nonlinear, thus requiring multiple solutions of large LP problems in an iterative

fashion which in turn makes the overall solution computationally slow [38]; (2) So called leverage

points commonly exist among SCADA measurements which cannot be rejected by LAV SE if they

carry gross errors, thus offsetting its robustness advantage over WLS [12].

Recent proliferation of Phasor Measurement Units (PMU) in power grids brings certain

advantages over SCADA measurements when used for state estimation. Since PMUs can directly

measure complex voltage and current phasors synchronized with respect to the same global reference,

measurement equations and consequently the SE problem formulation become linear. This eliminates

the need to solve multiple LP problems. Furthermore, when using PMU measurements, the structure

of the measurement Jacobian readily lends itself to simple scaling which can be shown to eliminate

any existing leverage points in the measurement set [14].

Above observations favor the use of LAV estimator when measurements are exclusively

provided by PMUs. However, growing system size will still be a limiting factor when implementing

it for very large scale power grids. This chapter tries to address this problem by proposing a different

multi-area state estimation scheme. In this scheme, it is assumed that a system-wide coordinator

will exist and it has access to all system data. The main idea is the following: Given enough

local redundancy and using the afore mentioned scaling approach to eliminate leveraging effects of

measurements, LAV estimators can provide robust solutions for all bus voltages. The exception to

this are those buses at the system boundaries where one has to inadvertently reduce local redundancy

by disregarding the measurements incident to neighboring zone buses. Thus, if a partitioning can be

used where every bus appears as an internal (not boundary) bus for one of the partitions, then overall

solution can be guaranteed to be robust to any gross errors in any part of the system. Therefore,

one of several additional zones will be created on top of the current zone configuration. Boundary

buses of existing zones will be covered as internal buses in the newly generated zones. These newly

generated zones will make sure that measurements between original zones (on tie-lines) can be fully

utilized.

This chapter will start with a introduction of the proposed multi-zone SE algorithm with a

tutorial example in Section 6.2. Then simulation results for both IEEE 30-bus system and a 2917-bus

utility system are shown in Section 6.3, followed by the conclusions in Section 6.4.

56


6.2 Multi-zone State Estimation Algorithm

A large-scale power system is assumed to have N independent control zones. For any

single zone j, buses in that zone can be categorized into either one of the following three types:

1. Boundary bus: at least one of its neighbors belongs to a different zone other than zone j;

2. Sub-boundary bus: all its neighbors belong to zone j and at least one of its neighbors is a

boundary bus;

3. Internal bus: all its neighbors belong to zone j and none of them is a boundary bus.

When the power grid is assumed to be measured exclusively by GPS synchronized PMU

measurements, there will be no need to choose a reference bus. This implies that all zones can run their

own state estimators and obtain their internal estimated system states. Measurement configurations

of individual zones are assumed to be optimally designed to ensure the required redundancy level for

bad data rejection. However, blindly dividing the system and running distributed state estimations

will reduce the redundancy level around boundary buses since those measurements on tie-lines

between zones, as known as linking measurements, will be discarded by individual estimators.

Those measurements that are not incident to any buses outside a given zone will be defined as

internal measurements for that zone. In the process of categorizing measurements, some internal

measurements thus may become critical due to the discarding of some boundary measurements, in

other words, they will become vulnerable to bad data. The state estimator will be biased by any

existing gross errors and fail to yield accurate estimates for the system states. To solve this problem,

a multi-zone state estimation algorithm is presented in this chapter. In the proposed approach, one

or several new zones are automatically generated including all boundary and sub-boundary buses.

Linking measurements, as well as those internal measurements related to boundary buses become

internal measurements and are included in the SE formulation and all boundary buses in the original

zone configuration become internal buses in this new zone.

As shown in Figure 6.1, the IEEE 14-bus system is used as an example to illustrate the

proposed approach. Note that even though this newly-generated zone overlaps with the existing

zone 1 and zone 2, the SE for those three zones are still totally independent. Only internal bus state

estimates of all zones will be saved and become part of the final estimated state of the entire system.

Estimated states for boundary buses in all zones will be discarded due to their potential unreliability.

If the size of the newly generated zone is too large compared to existing zone, the new zone can be

57


further split into several smaller zones. An additional zone will be generated to cover those boundary

buses created by the new zone partition within the new generated zone. For example, if a newly

generated zone is divided into n small zones, the total new zone number will be n+1. The flowchart

of the proposed approach is shown in Figure 6.2.

Figure 6.1: IEEE 14 bus system with 3 zones

The LAV state estimation formulation (1.2) can be rewritten by assigning all measurements

to their own zones:

min cTx

s.t. A x = z

x ≥ 0

(6.1)

58


Figure 6.2: Flowchart for Two-stage ZG State Estimation Algorithm

where:

cT = [ wL wL 0 0 w1 w1 0 0 w2 w2 ]

xT = [ UL VL X1a X1b U1 V1 X2a X2b U2 V2 ]

A =

L

1

2

I −I HL1 −HL1 0 0 HL2 −HL2 0 0

H1 −H1 I −I

H2 −H2 I −I

zT = [zL

T z1T z2

T]


mj is the number of internal measurements in zone j,


nj is the number of system states in zone j,



UL and VL are slack variables in (1.3) corresponding to linking measurements,

Uj and Vj are zone j’s slack variables in (1.3),

Xja and Xjb are zone j’s system states in (1.3),

59



states in zone j,

Hj is a constant measurement Jacobian matrix of internal measurements in zone j,


respectively.

6.2.1 Existing Zone LAV State Estimation

As described above, states of existing zones are independently estimated. Linking mea-

surements in the original zone partition are discarded. State estimation for zones 1 and 2 can be

formulated with the notation in (6.1):

min cjTxj

s.t. Aj xj = zj

xj ≥ 0, j = 1, 2

(6.2)

where:

mj is the number of internal measurements in zone j,

nj is the number of system states in zone j,

cjT = [02nj

12mj],



6.2.2 Newly Generated Zone LAV State Estimation

When a new zone 3 is generated, as shown in Fig. 2, all measurements need to be reassigned

to the new zone configuration. All remaining system buses beside zone 3 is assigned to zone 4. The

updated formulation (6.1) is shown below:

min cTx

s.t. A x = z

x ≥ 0

(6.3)

60


where:

cT = [ wL 1 wL 1 0 0 w3 w3 0 0 w4 w4 ]

xT = [ UL 1 VL 1 X3a X3b U3 V3 X4a X4b U4 V4 ]

A =

L

1

2

I −I HL3 −HL3 0 0 HL4 −HL4 0 0

H3 −H3 I −I

H4 −H4 I −I

zT = [zL

T z3T z4

T]

All notations are the same with (6.1) except now j = 3, 4. Please note that (6.1) and (6.3)

are exactly the same problems. Problem (6.3) is just a rearranged version of (6.1).

Once (6.2) is set up, the state estimation formulation for the newly generated zone 3 will

be:

min c3Tx3

s.t. A3 x3 = z3

x3 ≥ 0

(6.4)

where:

m3 is the number of internal measurements in zone 3,

n3 is the number of system states in zone 3,

c3T = [02n3 12m3 ],

A3 = [ H3 −H3 I −I ], (m3 × 2(m3 + n3)) matrix,

x3T = [ X3a X3b U3 V3 ], (2(m3 + n3)) array of variables.

All variables related to zone 4 are not used in the state estimation. If more than one new

zone is generated, separate SE problems (6.4) need to be generated for every new zone. When all

existing zone LAV state estimation results (6.2) and newly generated zone LAV state estimation

results (6.4) are ready, those SEs can run independently to get their sets of estimated system states

for their own zone. The estimated results can simply be a combined set. As stated before, only the

estimated states for internal buses of every zone are considered in the final set of system states. All

those estimated states for boundary buses are discarded due to their potential criticality. If a certain

bus is estimated multiple times in different zones, the average value of all estimates will be taken for

the state of that bus.

61



The proposed approach is tested on two test systems: IEEE 30-bus test system and a

2917-bus utility system. The testing platform used is a PC with Intel R© CoreTM i7-4910MQ CPU,

32GB RAM and Windows 7 64-bit operating system. The software environment is MATLAB R©

R2015a and the LP solver is GUROBI optimizer version 6.5.0.

Three different algorithms are applied to those two test systems and their corresponding

robustness and computational performances are evaluated:

• Approach 1: LAV SE with zone generation technique as proposed in this chapter.

• Approach 2: LAV SE run in every zone ignoring linking measurements.

• Approach 3: LAV state estimation for the entire power system, including all available mea-

surements.

In all three algorithms, Gaussian errors with a standard deviation of σ = 10−4 are added to

all measurements. Measurements in multi-zone systems can be categorized into three different types:

internal measurements not incident to any boundary bus (Type I), internal measurements incident

to a boundary bus (Type II) and linking measurements between zones (Type III). Bad data on these

three different types of measurements are simulated in both test systems. Mean squares error (MSE)

defined below is used to quantify the mismatch between estimated and true system states, in other

words, show the level of robustness of the simulated algorithm in (4.21).

6.3.1 IEEE 30-bus test system

The IEEE 30-bus system is originally partitioned into two zones, 1 and 2 as shown in

Figure 6.3. 37 PMUs, including 32 branch PMUs and 5 current injection PMUs, are placed in this

system using the optimal PMU placement algorithm [51]. Every branch PMU includes one voltage

phasor measurement at the sending end and one current phasor measurement. In this case, the size of

the newly-generated Zone 3 is 22, which is also shown in Figure 6.3.

6.3.1.1 Internal measurement (Type I)

A gross error (zero current) is introduced to real part of the current phasor measurement on

branch 1-3. MSE values in Table 6.1 shows that all three approaches remain robust. This shows that

when Type I bad data occurs, all three approaches remain equally robust.

62


Figure 6.3: New Zone Partition for IEEE 30 bus system


1 4.7770e-05 Y

2 5.6161e-05 Y

3 4.8412e-05 Y

Table 6.1: 30-Bus ZG Results for BD Measurement Type I

63


6.3.1.2 Boundary Incident Measurements (Type II)

A gross error of zero is introduced to the real part of the current phasor measurement on

branch 14-12. MSE values in Table 6.2 show that approach 2 fails to reject this bad data. This case

demonstrates the potential risk of running individual zone state estimators by simply ignoring linking

measurements. Both approaches 1 and 3 remain robust under this scenario.


1 5.3906e-05 Y

2 0.0024 N

3 5.7492e-05 Y

Table 6.2: 30-Bus ZG Results for BD Measurement Type II

6.3.1.3 Linking Measurements (Type III)

A gross error of zero is introduced to the real part of the current phasor measurement on

branch 14-15. Approach 2 is not considered in this case because it ignores linking measurements. As

shown in Table 6.3, both approaches 1 and 3 remain robust despite bad data in linking measurements.


1 4.1545e-05 Y

3 4.2555e-05 Y

Table 6.3: 30-Bus ZG Results for BD Measurement Type III

6.3.2 2917-bus Utility System

Three algorithms are applied to a large-scale utility system with 2917 buses, divided into

11 zones. 4393 PMUs are placed in this system using an optimal PMU placement algorithm [51]. In

this system, there are 161 boundary buses and 562 sub-boundary buses, which yields the size of the

newly-generated Zone 12 as 723.

64


6.3.2.1 Internal measurement (Type I)

A gross error of zero is introduced to the current phasor measurement on branch 846-849.

As evident from the given MSEs in Table 6.4 all three approaches remain robust in this case.


1 6.9691e-05 Y

2 6.9289e-05 Y

3 6.9670e-05 Y

Table 6.4: 2917-Bus ZG Results for BD Measurement Type I

6.3.2.2 Boundary Incident Measurements (Type II)

A gross error of zero is introduced to the voltage phasor measurement at bus 2883. Table 6.5

shows the results for all three approaches. As expected, approach 2 fails and both approaches 1 and 3

remain robust.


1 7.3682e-05 Y

2 0.0129 N

3 7.8955e-05 Y

Table 6.5: 2917-Bus ZG Results for BD Measurement Type II

6.3.2.3 Linking Measurements (Type III)

A gross error of zero is introduced to the current phasor on branch 448-2883, which is a

linking measurement. MSE results in Table 6.6 show both approaches 1 and 3 successfully reject bad

data.

All above simulations validate that the proposed LAV SE algorithm with zone generation

technique perform identically as the centralized LAV SE on both systems when bad data on all types

of measurements are intentionally introduced.

65



1 6.9078e-05 Y

3 6.9025e-05 Y

Table 6.6: 2917-Bus ZG Results for BD Measurement Type III

6.3.2.4 Statistical Analysis of Robustness under Single Bad Data

Previous cases considered isolated scenarios for validation. In this section, a more system-

atic testing is carried out using Monte Carlo simulations. A single gross error is introduced into the

measurement set by adding a fixed bias equal to 250σ (σ = 10−4 representing the standard deviation

of measurement errors) to a randomly chosen measurement within the measurement set. Monte Carlo

simulations are carried out by estimating the system states 100 times using approach 1. The results

show that among 100 test cases, 100% of the MSE values remain below σ = 10−4. This illustrates

that the proposed approach 1 has high level of robustness against single bad data.

6.3.2.5 Statistical Analysis of Robustness under Multiple Bad Data

The performance of the proposed approach is also tested under multiple bad data. Multiple

gross errors of magnitude 250σ are randomly spread among the measurement set in the 2917-bus

utility system. A fixed number (NB) of randomly selected bad data in each zone is selected. MSEs

for different NB scenarios are calculated and cases with MSEs larger than the threshold σ = 10−4

are defined as failed cases. The simulation results are shown in Table 6.7, where NB varies from 1 to

6.

NB Approach 1 Approach 2

1 0 0

2 0 1%

3 3% 6%

4 5% 10%

5 20% 28%

6 39% 46%

Table 6.7: Failed ZG SE % with Multiple Bad Data

66


With the increasing number of bad data, both approach 1 and 2 reach their own limits.

However, results show that with the same number of bad data, approach 1 has better performance

than approach 2. Moreover, the failure limit is related to both local redundancy level and multiple

bad data locations.

6.3.2.6 Computational Performance

The computational performance of the proposed approach 1 is also compared with the

centralized LAV SE in this section. As described above, approach 1 consists of several independent

sub-problems and is extremely suitable for parallel processing. Considering the potential usage of

high performance multi-core computer, the expected parallel computing time can be predicted as:

ttotal = tmax zone (6.5)

where:

tmax zone is the maximum processing time for all independent zones.

Computational time for the 2917-bus system is shown in Table 6.8. This system has 11

original zones and 1 newly generated zone. The maximum processing time is for the newly generated

zone, which has the largest dimension among all zones. Note that, as described in previous sections,

the computational time can be further reduced by introducing additional layers of newly generated

zones. As shown in Table 6.8, under the assumption of parallel processing, the proposed approach 1

saves almost 98% of the estimated processing time compared to approach 3, which solves the LAV

SE for the entire system.

No. of Buses Est. Approach 1 (s) Approach 3(s)

2917 0.0564 2.5468

Table 6.8: ZG Computational Time Performance

6.4 Conclusions

This chapter presents a robust and linear state estimator designed to run parallel computing

for all independent zones. One or more new zones are generated to ensure the redundancy level

around those boundary buses of the original zone partition. The LAV SE is used for all zones, which

67


has the desirable built-in bad data rejection capability. The proposed algorithm is tested successfully

on a small IEEE 30 bus system and a large 2917 bus system. Several specific bad data rejection cases,

as well as two sets of Monte Carlo simulations with both single and multiple bad data, are carried

out using this algorithm.

68

Chapter 7

Multi-area LAV SE with Multiple Copies

7.1 Introduction

In the previous chapter, we presented a distributed linear LAV SE which generates one

or several additional zones overlapping with existing zones. The new zones together with original

zones can run their SEs independently and simultaneously. Even though this algorithm achieves full

parallel computing, one problem of this algorithm is that the size of the newly generated zone can be

large compared with original zones and may become the bottleneck of the whole state estimation

process. In this chapter, we will extend the previous distributed linear LAV SE to make the CPU

time of the SE totally independent of the system size.

As the power systems grow in size and complexity, monitoring their operating states

will present computational challenges. The increasing appearance of renewable sources urges the

necessity of monitoring lower voltage levels. Furthermore, the monitoring of very large scale power

grids containing multiple control zones becomes crucial due to the possibilities raised by ideas of

wide-area control and optimization. Monitoring of power grids has been traditionally done by the

supervisory control and data acquisition (SCADA) systems. While SCADA measurements are still

widely used, the number of installed synchronized phasor measurements units (PMUs) in substations

is also rapidly increasing. It is reasonable to make the assumption that in the near future, power

grids will be measured by only PMUs. This chapter is based on this assumption that redundant

configurations of PMUs exist in the power grid.

Despite the existence of well-developed computationally efficient state estimators, computa-

tional loads of many of these algorithms increase almost proportional to the system and measurement

sizes. Scalable algorithms that remain computationally efficient despite large increases in system

69

CHAPTER 7. MULTI-AREA LAV SE WITH MULTIPLE COPIES

size and model complexity are needed in order to facilitate implementation of much talked about

wide-area control and protection applications. Taking all the above mentioned concerns into con-

sideration, a distributed computational structure for power system state estimation (SE) may be a

possible solution. This was well recognized, worked on and documented by various researchers, as

described in Section 2.3. These approaches consider hierarchical or parallel computing methods and

use existing SCADA measurements. Some of these approaches assume one PMU per zone to avoid

synchronization of reference angles among zones. These methods commonly use the well-known

weighted least squares (WLS) algorithm which requires execution of a separate post-estimation bad

data processing function [7].

High refresh rates for PMU measurements (every 33msec) compared to SCADA (every

3-5 sec) must be matched by the state estimator. It is also important to have a robust estimator

which remains insensitive to bad data. Previous studies have shown that the least absolute value

(LAV) estimator could be a viable and more robust alternative to the WLS method unless there were

leverage measurements [38]. More recently, it is shown that when using only PMU measurements it

is possible to eliminate leverage measurements by simple scaling [14]. Thus, LAV estimator can be

used as a robust alternative estimator as long as its computational performance is acceptable.

The computational time for LAV estimator is expected to grow rapidly with system size and

will gradually reach the measurement refresh time even with advanced high-performance computers.

Hence, even for highly computationally efficient and robust estimators, there will be a need to make

their implementation in such a way that their computational performance will not be affected by

growing system size. This chapter reports on one possible solution for addressing this scalability

issue. The solution is based on the following observations:

(1) A large system can be split into many small size subsystems which will naturally be all

interconnected. In power grids, due to the natural sparsity of connections, identified subsystems are

expected to have relatively small number of connections among them compared to the total number

of buses in the overall system.

(2) SE of observable islands can be carried out simultaneously and independent of each

other, yet they will be synchronized if each observable island contains at least one error-free phasor

measurement. In the case of all PMU-based state estimators, synchronization will already be built-in.

The only short-coming will be that some of the bad data may be missed at or around the boundary

buses due to the forced reduction of local redundancy resulting from splitting of the system into

subsystems. Several measurements that are incident to boundary buses will have to be discarded by

individual observable islands because they become unusable as a result of the zone splitting. This

70


may cause some measurements around boundary buses become critical or experience very low local

redundancy and become vulnerable to potential bad data.

(3) Unlike the measurements incident to boundary buses, internal buses are not affected by

the system partitioning. The redundancy levels are maintained around those internal buses before

and after the system split. Hence, if a robust linear estimator such as the one described in [14] is

used, internal state estimates are expected to be free of biases even when there are gross errors in the

measurements.

Based on the above observations, this chapter will propose an unconventional SE approach

for solving a very large scale power system. The approach will involve simultaneous solution of

several copies of the same large scale system, using a strategically designed partitioning of the system

in each copy. The system partitioning will ensure that each bus in the large system will appear as an

internal system bus in at least one of these copies. The success of the approach strongly depends

on designing the appropriate zone configurations in multiple copies of the system model. So, an

algorithm for automatically creating such zone partitioning in multiple system copies for any given

system is also developed.

This chapter starts with describing the distributed SE algorithm together with the algorithm

developed for automatically generating zone partitioning in multiple copies in Section 7.2. Simulation

results for three test systems (the IEEE 30-bus system, a 140-bus NPCC test system and a 2917-bus

utility test system) are given in Section 7.3, followed by conclusions in Section 7.4.

7.2 Distributed Multi-copy LAV SE with Automatic Zone Generation

Large-scale power systems are normally divided into several independent and non-overlapping

control zones. The partitioning is determined by ISO’s control areas, geography, or administrative

divisions. In this work, it is assumed that the state estimation can be carried out using only PMU

measurements, which are considered to be placed around the system providing observability and

sufficient redundancy for the detection and identification of all measurement errors using a previously

developed PMU placement method [51]. Note that, while the method developed in [51] is used in

this work, implementation of the proposed state estimation algorithm does not require the use of this

specific PMU placement method. Any PMU placement method can be applied as long as it ensures

that the system will be observable with no critical measurements or critical pairs.

The exclusive usage of PMU measurements enables individual zones to execute their state

estimators simultaneously independent of each other since all measurements are synchronized by the

71


GPS. This is true with the caveat that those linking measurements between zones that are incident

to neighboring zones will have to be discarded. This will cause the redundancy level around zone

boundaries to significantly decrease and some measurements to become critical, which means they

will be vulnerable to bad data.

In order to remove this shortcoming, an unconventional state estimation algorithm is

recently developed. This algorithm creates several copies of the system model where each copy is

assigned a different and strategically designed zone partitioning.

Figure 7.1: 7-bus system divided into two zones with multiple copies

The algorithm can best be described on a small 7-bus system shown in Figure 7.1 where

the three created copies of the system are shown. The base case zone partitioning of the power system

is kept in the first copy which is also referred as the original copy, shown as copy 1 in Figure 7.1.

Buses with at least one of neighbor in a different zone are defined as boundary buses, e.g. bus 2

in copy 1. Buses with all their neighbors in the same zone are defined as internal buses, e.g. bus 2

in copy 2. The main idea of this algorithm is to make sure all boundary buses in the original copy

appear at least once as internal buses in other copies. The term ”target bus” is defined describing

those boundary buses in the original copy, which are buses 2, 3, 4 and 5 in copy 1. Note that buses 2

and 4 appear as internal buses in copy 2 and buses 3 and 5 appear as internal buses in copy 3. So

creating these three copies with the chosen partitions ensures that each bus appears as an internal bus

72


in at least one of the three copies.

When system dimension grows, it becomes a challenge to manually find partitions that

will place each of the target buses as an internal bus in at least one copy. Therefore, an iterative copy

generation algorithm is developed to automatically create zone partitions making sure all target buses

appear at least once as internal buses.

Before describing the steps of the developed algorithm, several definitions will be given:

• Let k be the system copy index, where 1 ≤ k ≤ K, K being the total number of copies.

• Let Nb be the number of buses in the original copy of the system.

• Let qk be the number of zones in copy k.

• Let the set of boundary buses in copy k be ck =bk1, b

k2, . . . , b

kLk

, where Lk is the number

of boundary buses in copy k, and bkl be the lth boundary bus in copy k.

• Let the target set be T = b11, b12, . . . , b1L1.

• Let W (bkl ) be the set of neighboring buses of bus bkl .

• Let mk be the zone index for copy k.

The detailed steps of the algorithm can now be described as follows:

1. Set k = 1. Determine Nb, q1 and T.

2. Set k = k + 1, mk = 1, l = 1.

3. Consider bus b1l .

If all buses in W (b1l ) are not assigned to any zone yet, then:

Assign all buses in W (b1l ) and b1l to zone mk and remove b1l from T. Set L1 =L1 − 1.

Else, l = l + 1.

Endif

4. If l > L1, then

73


Find all unassigned buses in copy k and assign them evenly among the remainingzones mk to q1 in copy k based on the bus number sequence.If T is empty, then go to step 5.Else, go to step 2.Endif

Else

If No. of buses in zone mk≥Nb/q1, then

mk = mk + 1.If mk = q1, then

Find all unassigned buses in copy k and assign them to mk, go tostep 2.

Else, go to step 3Endif

Else, go to step 3Endif

Endif

5. Eliminate isolated buses. Isolated bus is a bus with no connection to any other buses in its

zone. Identify isolated buses in all copies and assign them to one of its neighboring zones.

6. Remove the redundant copy. The definition of a redundant copy is a copy whose internal

buses already appeared at least once as internal buses in either the original copy or other newly

generated copies.

7. Terminate the algorithm.

The flowchart of the automatic copy generation algo-rithm is shown in Figure 7.2.

After generating new system copies, LAV SE will be executed on every zone in every copy

independently. If a bus appears to be a boundary bus in a copy, the estimated state for this bus in this

copy will be discarded. A union of all estimated states corresponding to internal buses in all copies,

forms the full set of estimated states for the whole system. In this set, there may be several estimates

for the same bus, since states of some internal buses may be estimated multiple times in different

copies. Hence, the final result will be obtained by taking the averages of all multiple estimates of

buses which appear as internal buses in respective copies. The final estimated states∧xi will be given

by:

74


Figure 7.2: Automatic Copy Generation Algorithm Flowchart

75


∧xi = Average

∧xki

For bus i in copy k /∈ ck(7.1)

The number of processors needed NP for full parallel processing will be:

NP =K∑k=1

qk (7.2)

The IEEE 14-bus system with 2 zones as shown in Figure 7.3 will be used as an example

to demonstrate the implementation of the proposed parallel LAV SE solution. 18 branch PMUs are

placed in this system to ensure full observability and sufficient redundancy for bad data processing.

Each branch PMU contains a voltage phasor measurement at the sending end and a current phasor

measurement measuring the current from the sending end to the receiving end.

Figure 7.3: IEEE 14 bus system with 2 zones

After applying the automatic copy generation algorithm, 2 new copies together with the

original copy are created, where each copy has 2 zones. The bus partitioning for all created copies of

the 14-bus system are given in Table 7.1. Furthermore, the boundary and internal buses in all copies

are shown in Table 7.2. Boundary buses are marked by ”B” and internal buses are shaded.

76


HHHHHH

HHCopy

Zone1 2

1 1 2 3 4 5 6 7 8 9 10 11 12 13 14

2 4 5 6 7 8 9 10 11 12 13 14 1 2 3

3 6 10 11 12 13 14 1 2 3 4 5 7 8 9

Table 7.1: Partitions of 14-Bus System

HHHHH

HHHCopy

Bus1 2 3 4 5 6 7 8 9 10 11 12 13 14

1 B B B B B B B

2 B B B B B

3 B B B B B

Table 7.2: Boundary/internal buses of the 14-bus system

As described above, every zone in every copy runs its own SE independently using its

local measurements. All voltage phasor measurements will be assigned to the zone of their buses and

current phasor measurements between zones will be discarded. The current measurement allocations

for three copies of the 14-bus system are shown in Figure 7.4, Figure 7.5, and Figure 7.6.

The LAV state estimation problem in (1.2) will be solved by all zones in all copies as

3× 2 = 6 independent subproblems:

min cTkmkxkmk

s.t. Akmkxkmk

= zkmk

xkmk≥ 0

k = 1, 2, 3

mk = 1, 2

(7.3)

where:cTkmk

=[

0 0 1 1]

xTkmk

=[

Xkmka Xkmkb UkmkVkmk

]Akmk

=[

Hkmk−Hkmk

I −I]

Ukmkand Vkmk

are copy k zone mk’s slack variables in (1.3),

Xkmka and Xkmkb are copy k zone mk’s system states in (1.3),

77


Figure 7.4: Copy 1 current measurement allocation


78



Hkmkis a constant measurement jacobian of internal measurements in copy k zone mk,

Zkmkare vectors of internal measurements in copy k zone mk, including voltage phasor

measurements measuring buses in copy k zone mk and corresponding current phasor measurements

shown in Figure 7.4, Figure 7.5, and Figure 7.6.

Solving (7.3) for all zones in all copies will yield the estimated states xkmkfor all copies.

The full set of estimated states can then be calculated as described above.

In this algorithm, since the copy generation is a one-time task prior to the state estimation

process, the total computation time can be assumed to be independent of this preliminary off-line

process and only related to the solution complexity of the sub-problems. Thus, assuming unlimited

number of processors, the total computation time for the state estimation solution will be determined

by the size of the largest zone in all copies and will be completely independent of the overall system

size.

79



The proposed approach is tested on three test systems having very different sizes: a small

IEEE 30-bus test system, a medium 140-bus NPCC test system and a large 2917-bus utility test

system. The simulations are done using a PC with Intel R© CoreTM i7-4910MQ CPU, 32GB RAM,

and Windows 7 64-bit operating system. The algorithm is implemented using MATLAB R© version

R2015a and GUROBI optimizer version 6.5.0 as the LP solver.

Additive Gaussian errors with a standard deviation of σ = 10−4 are added to all measure-

ments. Mean Squared Error (MSE) (as defined below) is used to evaluate the accuracy of estimated

states with respect to the true states (4.21).

7.3.1 Robustness under Gross Error

7.3.1.1 IEEE 30-Bus Test System

As shown in Figure 7.7, the IEEE 30-bus system with 41 branches is originally divided

into two zones. 42 branch PMUs are optimally placed around the system by a previously developed

optimal PMU placement algorithm [51]. Each PMU includes one voltage phasor measurement at the

sending end of the branch and one current phasor measurement measuring current flow.

By applying the automatic partitioning copy generation algorithm described in Section 7.2,

two additional copies are generated.

A single additive gross error is introduced to a random measurement within the measure-

ment set. The magnitude of the error is set equal to 250σ (σ = 10−4 representing the standard

deviation of measurement errors). Monte Carlo simulations are carried out using 100 repeated runs

to test the robustness level of the proposed approach. The pass/fail thresholds of MSEs are chosen as

10−4, i.e. the estimator will be declared robust if the MSE stays below the 10−4 threshold. In order

to make a comparison of the centralized solution versus the proposed distributed solution, MSEs for

both solutions with and without Gaussian errors are shown in Figure 7.8 and Figure 7.9.

The simulation results not only show that the proposed approach stays robust in all 100

simulation runs, but also demonstrate that the distributed solution has the similar level of robustness

compared with the centralized solution.

80


Figure 7.7: Zone Partition for IEEE 30 bus system

Figure 7.8: 30-bus System Centralized vs MC Distributed without Guassian Error

81


Figure 7.9: 30-bus System Centralized vs MC Distributed with Guassian Error

7.3.1.2 140-Bus NPCC Test System

Next, a medium-size 140-bus NPCC test system is considered. This test system includes

233 branches. A set of 186 PMUs is optimally placed for the testing of the proposed method [51].

The system is originally divided into three zones. 65 buses are identified as boundary buses

in the original zonal partitioning. After applying the copy generation algorithm of Section 7.2, two

new copies are generated. So, altogether three copies will be used for the state estimation of this test

system.

Bad data testing with single bad data of 250σ is carried out with this test system. The

results for both the centralized solution and the proposed distributed solution are shown in Figure 7.10.

These plots confirm that the proposed approach remains fully robust rejecting 100% of randomly

introduced gross errors in all 100 runs.

7.3.1.3 2917-Bus Utility Test System

The approach is also tested on a large scale 2917-bus utility test system. The system has

3826 branches and is originally divided into 11 zones. 4393 branch PMUs are optimally placed in

the system [51]. Three new copies are automatically generated and altogether four copies are used.

Similar Monte Carlo simulations with single bad data of 250σ are carried out for the system.

The simulation results are shown in Figure 7.11. The results validate the high level of robustness

against single bad measurement, i.e. 100% of the simulation MSEs stay below the threshold of 10−4.

82


Figure 7.10: Centralized and MC Distributed Results for NPCC test system

Figure 7.11: Centralized and MC Distributed Results for 2917-bus system (Single Bad Data)

83


Next, a more difficult but less likely case is considered where two gross errors of magnitude

250σ are randomly introduced in the measurement set. The results of these simulations are shown in

Figure 7.12 where it is evident that for the simulated bad data scenarios involving two gross errors,

the proposed approach still maintains robustness yielding MSEs well below the acceptable threshold

of 10−4.

Figure 7.12: Centralized and MC Distributed Results for 2917-bus system (Multiple Bad Data)

By comparing the results for all three test systems with different sizes, the conclusion

can be drawn that the proposed copy generation technique works well under different sizes and the

multi-copy LAV state estimator remains robust under randomly created single and double bad data

scenarios.

7.3.2 Computational Performance

Another potential advantage of the proposed approach is that the computation time will

not be proportional to the size of the system because the centralized SE is partitioned into several

independent zonal SE problems. Assuming availability of a high-performance multi-core computer,

those independent SEs can be executed in parallel. Assuming similar sizes for each zone in every

copy and neglecting communication delays, parallel computing time can be roughly predicted as:

tpar =ttotalNP

(7.4)

where:

84


tpar is the total CPU time when using parallel processors,

ttotal is the total CPU time when using a single processor,

NP is the total number of processors.

No. of Buses No. of Copies NP ttotal (sec) tpar (sec) Centralized (sec)

30 3 6 0.0174 0.0029 0.0046

140 3 9 0.0534 0.0045 0.0106

2917 4 44 0.8193 0.0186 2.5468

Table 7.3: MC Computation Times

The computational time results for all three test systems are shown in Table 7.3. The

results show that when the system dimension grows, the computational time for centralized solution

increases dramatically. However, the predicted parallel computation time remains low because it is

related to the largest size of the single zone. If the size of every zone can be kept roughly equal, the

overall computation time will stay the same even when the system size grows. Also, the number of

copies for every system and the number of independent parallel processors needed are also listed

in Table 7.3. For the large 2917-bus system, this algorithm only requires 44 independent cores to

achieve full parallel computation.

7.4 Conclusions

This chapter presents an unconventional computational framework. First, multiple copies

of the system model are generated where each copy is strategically partitioned into different zones to

ensure that every system bus appears as an internal bus in at least one system copy. Each zone in

each copy is then assigned to a processor and each processor can thus execute its own state estimator

independent of the others. By this, the overall computation time will remain bounded independent of

the system size. The proposed computational framework is implemented and tested on small and

large test systems. Test results validate the viability of this approach for large-scale robust state

estimation.

85

Chapter 8

Implementation on High-performance

Computer

8.1 Introduction

The previous chapter proposed a promising distributed linear LAV SE with multiple

partition copies. Every copy consists a way of system partitioning and all zones in all copies can

run their SEs at the same time. This makes the total CPU time irrelevant to system size. In this

chapter, that algorithm will be actually implemented on a multi-core high-performance computer

using parallel computing technique to further validate the algorithm’s performance.

Power system operator needs to maintain the power system in the normal and secure state,

and thus continuously monitors the system state, determines any potential preventive actions if the

system state appears to be insecure [5]. The monitoring of the power system is accomplished by the

state estimators (SEs) using the Supervisory Control and Data Acquisition (SCADA) measurements.

Detection and elimination of gross errors within the measurement set is one of the important functions

of the SE. It is well documented that the commonly used weighted least squares (WLS) estimator

is not robust and will be biased by even a single bad measurement, so a separate function for

post-estimation bad data processing is needed [6, 7] to detect, identify and remove any existing bad

data. However, implementing such a bad data processing function will add extra computational

burden to the SE solution and due to its sequential nature, bad data removal will become increasingly

time-consuming with growing system size. Meanwhile, the least absolute value (LAV) estimator has

been proposed as an alternative to WLS, which minimizes the L1 norm of measurement residuals

86

CHAPTER 8. IMPLEMENTATION ON HIGH-PERFORMANCE COMPUTER

instead of L2 [11]. LAV can be conveniently formulated as a linear programming (LP) problem and

solved by an efficient LP solver. While LAV estimator has this very useful built-in bad data rejection

capability, it remains vulnerable to the so called ”leverage” measurements when using SCADA

measurements. When a measurement lies far away from the bulk of the rest of the measurements

in the sample space, it is referred as a ”leverage” point [21]. Specifically in power systems, these

measurements act like ”critical” measurements which will be forced to be satisfied and will bias the

estimation result if any of them carry gross errors. One other and probably more significant reason

why LAV SE has not been widely adopted by the commercial power system software developers so

far is because of its higher computational burden compared to that of the WLS SE. It is noted that

when using SCADA measurements, there will be two nested loops in the LAV SE solution algorithm,

one for the inner LP iterations to solve the linearized problem, and one for the outer state estimation

iterations similar to the WLS SE iterations.

Although SCADA measurements continue to dominate the measurement systems, syn-

chronized phasor measurements are also increasingly becoming available from phasor measurement

units (PMU) installed at substations. Since all PMUs are GPS clock synchronized, they can directly

measure voltage and current phasors, which makes the measurement equations linear in state esti-

mation problem formulation. Even though currently the number of PMUs installed in most power

systems is not sufficient to make the system fully observable, it is not unrealistic to assume that in

the near future, power systems will be fully observable by only PMU measurements. This chapter’s

work is based on that assumption, that is a redundant set of PMU measurements is available to be

used by the state estimator. It is shown in [13, 14] that leverage points can be readily eliminated by

a simple scaling of the measurement equations. Furthermore, linearity of measurement equations

eliminates the outer state estimation iterations which help to speed up the LAV SE solution. PMU

measurements can be acquired at much higher scan rates (up to 30 times/sec) compared to traditional

SCADA measurements (every 2-3 seconds). Thus, in order to fully exploit the benefits of having

these measurements at such high scan rates, the estimation process needs to be completed within a

time period commensurate with the PMU scan rates.

Even with high-performance computers and efficient LP solvers, a centralized SE will

still become too slow when system dimension keeps growing. In order to solve this issue, various

researchers tried to distributed the computational burden to local zones. Several previous works

has been described in Section 2.3. These works use either a parallel or hierarchical structure trying

to minimize the data exchange between independent zones. However, with the increasing number

of zones, the CPU requirements for all these algorithms will still increase even with an unlimited

87


number of processors.

In this chapter, a different approach will be considered which was recently developed in

order to avoid the above mentioned limitations. This approach creates multiple copies of the network

model where zone partitions are strategically chosen to be different in each copy of the system. A

single bus may appear in several different zones in different copies. Copies and zones are deliberately

designed to ensure every bus in the system to appear as an internal bus (with no connection to buses in

another zone) in at least one of the copies. Each zone in each copy will be assigned to an independent

processor which will execute that zone’s state estimator in parallel with all others. The results from

all processors will be collected and a final solution will be obtained via a simple reconciliation. An

algorithm that strategically generates the required number of copies and designs associated zones

is also developed. This algorithm is presented in detail in Chapter 7. In this chapter, the proposed

approach is implemented on a high-performance computer cluster namely the ”Discovery Cluster”

managed by the Northeastern University.

The chapter will start with a brief introduction of the high-performance computer Discovery

Cluster in Section 8.2, followed by the results of testing this implementation on the high-performance

computer using a 2917-bus and a 16216-bus large utility system in Section 8.3. The conclusions are

presented in Section 8.4.

8.2 Discovery Cluster

The main goal of this chapter is to test and validate the performance of the developed

state estimation algorithm in Chapter 7 using a multi-core high-performance computer. Hence, the

above described algorithm is implemented on the ’Discovery Cluster’, a high-performance computer

managed by Northeastern University’s information technology services for research in computing.

Discovery Cluster has two login nodes and two administrative nodes. These nodes have

dual Intel R© XeonTM E5-2670 [email protected] GHz and 256 GB RAM. It consists of several independent

compute nodes with dual Intel R© XeonTM E5-2650 [email protected] GHz or higher and 128 GB RAM or

higher. Discovery Cluster also has a 50 TB Hadoop cluster available and several large memory nodes

for large memory simulation cases on the 10 Gb’s backplane. Each compute node has 700 GB local

disk storage space for user data. The facility connects back to the Northeastern main campus via

multiple dedicated 10 Gbps optical fiber connections [1]. The method to connect to Discovery cluster

is via SSH (secure shell). Figure 8.1 demonstrates the overall schematic of this high-performance

computer.

88


Figure 8.1: Discovery Cluster schematic [1]


All simulations are carried out in MATLAB R© version 2016a. The proposed algorithm

is used to find the state estimation solution for two utility systems, having 2,917 and 16,216 buses

respectively. Using an optimal PMU placement algorithm [51], sets of sufficiently redundant PMUs

are placed in both systems. The placed PMUs are all assumed to be branch PMUs, namely those

that are incident to a single branch (line or transformer) and measure one voltage and one current

phasor at the sending end of the branch. More details about both systems with regards to the number

of zones and the size of the largest zone in their original copies, and the number of branch PMUs

placed for state estimation are given in Table 8.1. The table also gives the number of extra copies

that had to be created in addition to the original copy in order to implement the developed algorithm.

Bus Branch Zone Copy PMU Biggest Zone

2917 3826 11 3+1 4393 388

16216 21125 69 4+1 22658 1500

Table 8.1: Information on the two test systems

89


Simulations are done using a compute node of the Discovery Cluster with dual Intel R©

XeonTM E5-2670 [email protected] GHz and 256 GB RAM. That node has 16 physical cores so up to 16

parallel sessions are simulated. Two different LP solvers are used: the LP solver of MATLAB R©, and

GUROBI 7.0.0.

Additive Gaussian errors with standard deviation of σ = 10−4 are added to all measure-

ments which are produced using a power flow program.

8.3.1 Robustness under gross error

For both test systems, a single gross error of magnitude 250σ is introduced in a random

measurement each time the state estimator is executed. Execution is repeated 100 times for both test

systems. Mean squared error (MSE) as defined below is used as the metric to evaluate the robustness

of the algorithm in (4.21).

Robustness criterion for evaluating the algorithm’s performance is chosen as the above

defined MSE which is compared with the standard deviation of the measurement errors σ = 10−4

introduced in generating the noisy measurement set. Corresponding results are plotted in Figure 8.2

and Figure 8.3. As evident from the MSE values remaining below the chosen σ threshold in these

figures, the algorithm shows satisfactory level of robustness for both systems under single bad data.

Figure 8.2: Cluster MSEs for 2916-bus system

90


Figure 8.3: Cluser MSEs for 16216-bus system

8.3.2 Computational performance

Next, the computational performance of the proposed approach when using the high-

performance computer cluster is evaluated for both systems. The CPU times for the proposed

algorithm with 1, 4, 8, 12 and 16 computing cores are tested. The CPU time for the centralized LAV

solution for both systems are also obtained for comparison. Two different LP solvers are used in all

simulations: MATLAB R© solver and GUROBI solver.

CPU times corresponding to the state estimation solution of 2917-bus and 16216-bus test

systems when using the central algorithm for the entire system versus using the proposed algorithm

with 1, 4, 8, 12, 16 cores are given in Table 8.2 for both MATLAB R© and GUROBI solvers.

Simulation Centralized Distributed tNp (s)

Case tcent. (s) t1 t4 t8 t12 t16

2917(M) 46.2113 27.9725 17.0372 11.0531 9.1106 7.9199

2917(G) 3.2849 1.4106 1.3651 1.2826 1.2573 1.2286

16216(M) 321.6568 185.43 52.8358 33.8721 26.8171 23.2358

16216(G) 12.1713 8.2513 4.3946 2.6566 2.4161 2.2948

Table 8.2: CPU Times for Cluster Simulation Cases

The results show that even without using multiple cores, the proposed algorithm may lead

91


to 39.47%, 57.06%, 42.35% and 32.21% time savings compared with the centralized solution. When

multiple computing cores are used, the CPU times for the proposed algorithm are further reduced.

When the maximum number (16) of cores are used, the proposed algorithm yields CPU time savings

of 82.86%, 62.59%, 92.78% and 81.15% for the above four test cases.

Next, consider the so called parallel processing efficiency (PPE) ENp which is defined as:

ENp =t1

tNp ×Np(8.1)

where tNp is the cpu time obtained using Np cores.

PPEs for all simulation cases are shown in Table 8.3 and plotted in Figure 8.4.

Simulation Efficiency Index ENp (s)

Cases E1 E4 E8 E12 E16

2917(M) 100% 41.05% 31.64% 25.59% 22.07%

2917(G) 100% 25.83% 13.75% 9.35% 7.18%

16216(M) 100% 87.74% 68.43% 57.62% 49.88%

16216(G) 100% 46.94% 38.82% 28.46% 22.47%

Table 8.3: Efficiency for simulation cases

Figure 8.4: Efficiency for simulation cases

The following observations are made:

92


• PPE decreases when the number of cores increases for both solvers.

• Even though MATLAB R© solver is slower than GUROBI solver, it has better PPE.

• The implementation of the algorithm on larger scale systems has better PPE.

8.4 Conclusions

This chapter presents the implementation results of a proposed state estimation algorithm

whose computational performance remains insensitive to growing system size. The algorithm

automatically generates different partition copies of the system, creates independent state estimation

sub-problems, and utilizes the parallel processing capability of the multi-core computer to solve

the overall system. This algorithm is implemented and tested on very large scale systems using a

high-performance multi-core computer. Test results validate the expected CPU time savings provided

by the proposed algorithm when using multiple computing cores.

93

Chapter 9

Conclusions and Future Work

9.1 Concluding Remarks

Research work presented in this dissertation focus on the application of Phasor Measure-

ment Units (PMU) in Very large-Scale Interconnected (VLSI) power systems. Several linear robust

SEs are presented, including both centralized and distributed algorithms. A summary of the major

contributions of this dissertation is given below.

• In Chapter 3, two different linear centralized state estimation approaches of incorporating

equality constraints, mainly but not limited to zero injection (ZI) measurements, in power sys-

tem state estimation are presented. This chapter of the dissertation shows that both approaches

have high robustness against bad data and these two approaches have their own computational

advantage with different systems and measurement set-ups.

• A robust distributed linear SE is proposed in Chapter 4. This approach addresses the fact

that the multi-area LAV SE with purely PMU measurements has the special block-diagonal

structure which can be easily fitted into the well-known LP decomposition method: Dantzig-

Wolfe (DW) decomposition. The proposed algorithm has the same robustness and better

computational performance compared with the centralized solution.

• A novel two-stage robust linear SE is presented in Chapter 5. This approach is based on a

previously developed two-stage solution using a combination of SCADA and PMU measure-

ments and extends it to the measurement configuration of purely PMUs. This approach has the

same robustness as the centralized SE. It also runs much faster compared with the centralized

94

CHAPTER 9. CONCLUSIONS AND FUTURE WORK

LAV SE. Its distributed configuration in the first stage makes it possible to facilitate the state

estimation process by using parallel computing.

• Chapter 6 introduces a distributed linear LAV SE. This SE considers that fact that large-scale

power grids are normally divided into multiple zones and generates one or several additional

zones to ensure the redundancy level around zone boundaries in the original zone configuration.

This algorithm ensures high robustness. The computational performance is also superior to the

centralized approach. This scheme has the potential to be implemented in parallel.

• In Chapter 7, a distributed linear LAV SE is presented with multiple zone partition copies.

Every copy includes a way of zone partitioning of the system. This algorithm maintains the

redundancy around boundary buses and makes the whole algorithm robust against gross errors.

The CPU time is much better compared to the centralized solution and it will be unrelated to

the system size with enough computing cores used.

• The multi-copy distributed linear LAV SE is implemented to very large-scale power systems

using a multi-core high-performance computer in Chapter 8. The feasibility of the algorithm is

validated on VLSI power systems using multi-core computer parallel computing.

9.2 Future Work

This dissertation can be extended in several new research directions as outlined below:

• Implementation of distributed SEs using actual PMU data

One potential extension of the current work is to implement these proposed SEs to large-scale

power grids using high-performance computers with real PMU data. Above simulations

are based on emulated PMU data with artificially introduced noises and gross errors. The

performance of these algorithms can be further tested using actual PMU measurement noise

and bad data. The implementation of those approaches using real PMU data will further

validate the feasibility of practical application of these algorithms.

• Computational time optimization

Even though currently proposed algorithms are much faster compared with the centralized

solution, their computational performance may be further improved when exploiting matrix

sparsity, algorithm modeling, etc. The final goal is to make these algorithms applicable to

95

CHAPTER 9. CONCLUSIONS AND FUTURE WORK

large-scale power grids and run in real-time. This means the SE needs to be fast enough to be

commensurate with the pace of the high data acquisition rate of PMU measurements.

• Optimal copy generation algorithm

An automatic copy generation algorithm is proposed in Section 7.2. This method can be further

improved by developing a modified version that can generate such copies with a minimal

number of newly generated copies to make sure all buses in the system appear at least once

as internal buses in these new copies together with the original copy. This will be another

potential extension of the presented work.

96

Bibliography

[1] Northeastern university information technology servicesresearch computing. [Online].

Available: https://www.northeastern.edu/rc/

[2] F. C. Schweppe and J. Wildes, “Power system static-state estimation, part i: Exact model,”

IEEE Transactions on Power Apparatus and Systems, no. 1, pp. 120–125, 1970.

[3] F. C. Schweppe and D. B. Rom, “Power system static-state estimation, part ii: Approximate

model,” IEEE Transactions on Power Apparatus and Systems, no. 1, pp. 125–130, 1970.

[4] F. C. Schweppe, “Power system static-state estimation, part iii: Implementation,” IEEE Trans-

actions on Power Apparatus and systems, no. 1, pp. 130–135, 1970.

[5] A. Abur and A. G. Exposito, Power system state estimation: theory and implementation. CRC

press, 2004.

[6] L. Mili, M. G. Cheniae, and P. J. Rousseeuw, “Robust state estimation of electric power systems,”

IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 41,

no. 5, pp. 349–358, 1994.

[7] A. Monticelli and A. Garcia, “Reliable bad data processing for real-time state estimation,” IEEE

Transactions on Power Apparatus and Systems, no. 5, pp. 1126–1139, 1983.

[8] R. Baldick, K. Clements, Z. Pinjo-Dzigal, and P. Davis, “Implementing nonquadratic objective

functions for state estimation and bad data rejection,” IEEE Transactions on Power Systems,

vol. 12, no. 1, pp. 376–382, 1997.

[9] M. K. Celik and A. Abur, “A robust wlav state estimator using transformations,” IEEE Transac-

tions on Power Systems, vol. 7, no. 1, pp. 106–113, 1992.

97

https://www.northeastern.edu/rc/

BIBLIOGRAPHY

[10] M. Irving, R. Owen, and M. Sterling, “Power-system state estimation using linear programming,”

in Proceedings of the Institution of Electrical Engineers, vol. 125, no. 9. IET, 1978, pp. 879–

885.

[11] W. Kotiuga, “Development of a least absolute value power system tracking state estimator,”

IEEE transactions on Power Apparatus and Systems, no. 5, pp. 1160–1166, 1985.

[12] L. Mili, M. Cheniae, N. Vichare, and P. J. Rousseeuw, “Robust state estimation based on

projection statistics [of power systems],” IEEE Transactions on Power Systems, vol. 11, no. 2,

pp. 1118–1127, 1996.

[13] M. Gol and A. Abur, “Pmu based robust state estimation using scaling,” in North American

Power Symposium (NAPS), 2012. IEEE, 2012, pp. 1–5.

[14] M. Gol and A. Abur, “Lav based robust state estimation for systems measured by pmus,” IEEE

Transactions on Smart Grid, vol. 5, no. 4, pp. 1808–1814, 2014.

[15] T. Van Cutsem, M. Ribbens-Pavella, and L. Mili, “Hypothesis testing identification: A new

method for bad data analysis in power system state estimation,” IEEE Transactions on Power

Apparatus and Systems, no. 11, pp. 3239–3252, 1984.

[16] L. Mili and T. Van Cutsem, “Implementation of the hypothesis testing identification in power

system state estimation,” IEEE Transactions on Power Systems, vol. 3, no. 3, pp. 887–893,

1988.

[17] P. J. Huber et al., “Robust estimation of a location parameter,” The Annals of Mathematical

Statistics, vol. 35, no. 1, pp. 73–101, 1964.

[18] H. M. Merrill and F. C. Schweppe, “Bad data suppression in power system static state estima-

tion,” IEEE Transactions on Power Apparatus and Systems, no. 6, pp. 2718–2725, 1971.

[19] E. Handschin, F. C. Schweppe, J. Kohlas, and A. Fiechter, “Bad data analysis for power system

state estimation,” IEEE Transactions on Power Apparatus and Systems, vol. 94, no. 2, pp.

329–337, 1975.

[20] L. Mili, V. Phaniraj, and P. J. Rousseeuw, “Least median of squares estimation in power systems,”

IEEE Transactions on Power Systems, vol. 6, no. 2, pp. 511–523, 1991.

98

BIBLIOGRAPHY

[21] P. J. Rousseeuw and A. M. Leroy, Robust regression and outlier detection. John wiley & sons,

2005, vol. 589.

[22] H. M. Wagner, “Linear programming techniques for regression analysis,” Journal of the

American Statistical Association, vol. 54, no. 285, pp. 206–212, 1959.

[23] W. W. Kotiuga and M. Vidyasagar, “Bad data rejection properties of weughted least absolute

value techniques applied to static state estimation,” IEEE Transactions on Power Apparatus

and Systems, no. 4, pp. 844–853, 1982.

[24] M. Gol and A. Abur, “A fast decoupled state estimator for systems measured by pmus,” IEEE

Transactions on Power Systems, vol. 30, no. 5, pp. 2766–2771, 2015.

[25] F. Aschmoneit, N. Peterson, and E. Adrian, “State estimation with equality constraints,” in

Tenth PICA Conference Proceedings, 1977, pp. 427–430.

[26] A. Gjelsvik, S. Aam, and L. Holten, “Hachtel’s augmented matrix method-a rapid method

improving numerical stability in power system static state estimation,” IEEE Transactions on

Power Apparatus and Systems, no. 11, pp. 2987–2993, 1985.

[27] W. Liu, F. Wu, L. Holten, A. Gjelsvik, and S. Aam, “Computational issues in the hachtel’s aug-

mented matrix method for power system state estimation,” Proc. of Power System Computation,

Lisbon, Portugal, 1987.

[28] F. F. Wu, W.-H. Liu, and S.-M. Lun, “Observability analysis and bad data processing for state

estimation with equality constraints,” IEEE Transactions on Power Systems, vol. 3, no. 2, pp.

541–548, 1988.

[29] F. L. Alvarado and W. F. Tinney, “State estimation using augmented blocked matrices,” IEEE


[30] R. R. Nucera and M. L. Gilles, “A blocked sparse matrix formulation for the solution of

equality-constrained state estimation,” IEEE Transactions on Power Systems, vol. 6, no. 1, pp.

214–224, 1991.

[31] K. A. Clements, G. W. Woodzell, and R. C. Burchett, “A new method for solving equality-

constrained power system static-state estimation,” IEEE Transactions on Power Systems, vol. 5,

no. 4, pp. 1260–1266, 1990.

99

BIBLIOGRAPHY

[32] A. Gjelsvik, “The significance of the lagrange multipliers in wls state estimation with equality

constraints,” in Proceedings of the 11th Power Systems Computation Conference, 1993, pp.

619–625.

[33] A. Abur and M. K. Celik, “Least absolute value state estimation with equality and inequality

constraints,” IEEE Transactions on Power Systems, vol. 8, no. 2, pp. 680–686, 1993.

[34] W.-M. Lin and J.-H. Teng, “State estimation for distribution systems with zero-injection

constraints,” IEEE Transactions on Power Systems, vol. 11, no. 1, pp. 518–524, 1996.

[35] E. Kliokys and N. Singh, “Minimum correction method for enforcing limits and equality

constraints in state estimation based on orthogonal transformations,” IEEE Transactions on

Power Systems, vol. 15, no. 4, pp. 1281–1286, 2000.

[36] G. N. Korres, “A robust algorithm for power system state estimation with equality constraints,”


[37] Y. Guo, W. Wu, B. Zhang, and H. Sun, “An efficient state estimation algorithm considering

zero injection constraints,” IEEE Transactions on Power Systems, vol. 28, no. 3, pp. 2651–2659,

2013.

[38] A. Abur and M. K. Celik, “A fast algorithm for the weighted least absolute value state estimation

(for power systems),” IEEE Transactions on Power Systems, vol. 6, no. 1, pp. 1–8, 1991.

[39] T. Van Cutsem and M. Ribbens-Pavella, “Critical survey of hierarchical methods for state

estimation of electric power systems,” IEEE Transactions on Power Apparatus and Systems,

no. 10, pp. 3415–3424, 1983.

[40] V. Kekatos and G. B. Giannakis, “Distributed robust power system state estimation,” IEEE


[41] G. N. Korres, “A distributed multiarea state estimation,” IEEE Transactions on Power Systems,

vol. 26, no. 1, pp. 73–84, 2011.

[42] A. J. Conejo, S. de la Torre, and M. Canas, “An optimization approach to multiarea state

estimation,” IEEE Transactions on Power Systems, vol. 22, no. 1, pp. 213–221, 2007.

[43] L. Zhao and A. Abur, “Multi area state estimation using synchronized phasor measurements,”


100

BIBLIOGRAPHY

[44] A. Abur and M. Celik, “Multi-area linear programming state estimator using dantzig-wolfe

decomposition,” in Proceedings Tenth Power Systems Computations Conference, 1990, pp.

1038–1044.

[45] W. Jiang, V. Vittal, and G. T. Heydt, “A distributed state estimator utilizing synchronized phasor

measurements,” IEEE Transactions on Power Systems, vol. 22, no. 2, pp. 563–571, 2007.

[46] R. Ebrahimian and R. Baldick, “State estimation distributed processing [for power systems],”


[47] T. Van Cutsem, J.-L. Horward, and M. Ribbens-Pavella, “A two-level static state estimator

for electric power systems,” IEEE transactions on power apparatus and systems, no. 8, pp.

3722–3732, 1981.

[48] L. Xie, D.-H. Choi, S. Kar, and H. V. Poor, “Fully distributed state estimation for wide-area

monitoring systems,” IEEE Transactions on Smart Grid, vol. 3, no. 3, pp. 1154–1169, 2012.

[49] E. Caro, A. J. Conejo, and R. Minguez, “Decentralized state estimation and bad measurement

identification: An efficient lagrangian relaxation approach,” IEEE Transactions on Power

Systems, vol. 26, no. 4, pp. 2500–2508, 2011.

[50] D. G. Luenberger, Y. Ye et al., Linear and nonlinear programming. Springer, 1984, vol. 2.

[51] M. Gol and A. Abur, “Optimal pmu placement for state estimation robustness,” in Innovative

Smart Grid Technologies Conference Europe (ISGT-Europe), 2014 IEEE PES. IEEE, 2014,

pp. 1–6.

[52] R. H. Kwon, Introduction to linear optimization and extensions with MATLAB R©. CRC Press,

2013.

101

Appendix A

List of Publications

• Journal Papers

[1] C. Xu and A. Abur, ”A fast and robust linear state estimator for very large scale intercon-

nected power grids” IEEE Transactions on Smart Grid (2017).

[2] C. Xu and A. Abur, ”A massively parallel framework for very large scale linear state

estimation” IEEE Transactions on Power Systems, under review, (2017).

• Conference Papers

[3] C. Xu and A. Abur, ”Robust linear state estimation with equality constraints.” In Power

and Energy Society General Meeting (PESGM), 2016, pp. 1-5. IEEE, 2016.

[4] C. Xu and A. Abur, ”Robust linear state estimation for large multi-area power grids.” In

Innovative Smart Grid Technologies Conference (ISGT), 2016 IEEE Power & Energy

Society, pp. 1-5. IEEE, 2016.

[5] C. Xu and A. Abur, ”Robust linear state estimation using multi-level power system

models with different partitions.” In PowerTech, 2017 IEEE Manchester, pp. 1-5. IEEE,

2017.

[6] C. Xu and A. Abur, ”Robust state estimation via network partitioning” In 49th North

American Power Symposium (NAPS), pp. 1-5. IEEE, 2017.

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