PLACEMENT OF PHASOR MEASUREMENT UNITS: OPTIMIZATION

APPROACHES AND APPLICATIONS

By

Saleh Almasabi

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Electrical Engineering — Doctor of Philosophy

2019

ABSTRACT

PLACEMENT OF PHASOR MEASUREMENT UNITS: OPTIMIZATION

APPROACHES AND APPLICATIONS

By

Saleh Almasabi

Phasor measurement units (PMUs) provide measurements with high precision at a high

resolution (up to 50 samples per second). These measurements are synchronized and time-

stamped using the Global Positioning System. Despite these advantages, the industry has

been slow in adopting PMU technology due to the high cost of installing PMUs. Therefore,

PMU locations must be judiciously selected through optimal placement of PMUs (OPP),

which enables the minimization of installation cost.

This dissertation examines the OPP problem from several perspectives. First, the OPP

problem definition is re-examined since most OPP literature associates the PMU installation

cost with the PMU unit. Most techniques in the literature have proposed to minimize the

number of PMUs while considering the complete observability of the system. However, PMUs

require sufficient infrastructure to be in place before they can perform most of their intended

functions. Therefore, the OPP problem should be reformulated to include the supporting

infrastructure of PMUs. The proposed OPP formulation is implemented by using a bi-level

framework. This framework can accommodate different varieties of OPP, such as single-stage,

multistage, and application-based approaches. Moreover, the proposed framework achieves

the optimal solution with the maximum observability in the case of multiple optima.

Second, this dissertation examines application-based OPP approaches, where specific

technical benefits are prioritized over the cost of the OPP. Three applications are proposed.

The first application is a fault-tolerance based OPP approach, where the network fault-

tolerance is enhanced by deploying PMUs to the vulnerable elements in the network. In the

second application, a voltage stability criterion is developed and proposed, where the critical

buses are identified and prioritized for PMU allocation. The third application addresses

false data injection attacks (FDIAs), where the system topology is used by the adversary to

bypass the bad data detection of state estimators. The proposed OPP approach enhances

bad data detection against FDIAs by utilizing the PMUs as authenticators for each other.

To my parents, my brothers and my sisters.

iv

ACKNOWLEDGMENTS

First and foremost, I would like to express my sincerest and deepest appreciation to my ad-

visor, Dr. Joydeep Mitra. It has been an honor to be one of his Ph.D. students. Dr. Joydeep

Mitra has provided me with valuable guidance, support, and encouragement throughout the

past few years. I sincerely appreciate his thoughtfulness, patience and continued support to

make my Ph.D. experience productive.

I also would like to thank my committee members, Dr. Bingsen Wang, Dr. Subir Biswas,

and Dr. Ranjan Mukherjee for their inspiring and helpful classes, valuable feedback, and

being members of my committee.

I sincerely acknowledge the funding sources that helped me to complete my Ph.D. work.

I was funded by Najran University, Saudi Arabia and the Cultural Mission of Saudi Arabian.

I would also like to thank my colleagues in the ERISE lab, for their kind support and

valuable friendship, Dr. Mohammed Benidris, Dr. Salem Elsaiah, Dr. Samer Sulaeman, Dr.

Nga Nguyen, Dr. Yuting Tian, Mr. Atri Bera, and Mr. Saad Alzharahni. I would like to

express my special appreciation to Dr. Nga Nguyen, Mr. Atri Bera and Mr. Fares Theyab

Alharbi for their discussions and valuable suggestions.

Last but not least, I would like to thank my parents, my brothers and sisters for their

genuine care, support, and understanding. The thanks that I would like to express for them

is beyond words.

v

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii

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

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Motivation and Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 2 Multistage Optimal PMU Placement Considering Substation
Infrastructure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 PMU Installation Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Substation Infrastructure . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1
2.2.2 Communication Infrastructure . . . . . . . . . . . . . . . . . . . . . .
2.3 Proposed Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1.1 Priority Buses . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2.1 Multisource Dijkstra . . . . . . . . . . . . . . . . . . . . . .
2.3.2.2 O-BEBGA . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3 Multistage Installation . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.3.1 Complete observability . . . . . . . . . . . . . . . . . . . . .
2.3.3.2 Observability under single line outage
. . . . . . . . . . . .
Single PMU outage . . . . . . . . . . . . . . . . . . . . . . .
2.3.3.3
2.4 Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 No Priority Buses . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.2 Using Priority Buses . . . . . . . . . . . . . . . . . . . . . . . . . . .

x

1
2
3
4

6
6
7
7
9
10
11
12
14
15
16
19
20
20
21
22
26
27

Chapter 3 A Fault-Tolerance Based Approach to Optimal PMU Placement 28
28
30
33
35
35
38
41
43
44
46

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
3.2 Fault-Tolerance: Vulnerability Analysis . . . . . . . . . . . . . . . . . . . . .
3.3 PMU Installation Cost . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 Proposed Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.1 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.2.1 Absence of feasible solutions . . . . . . . . . . . . . . . . . .
Single-Stage Installation . . . . . . . . . . . . . . . . . . . . . . . . .
3.4.3
3.4.4 Multi-Stage Installation . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

vi

3.5.1 Considering Installation Cost
3.5.2 Vulnerability Analysis

. . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2.1
Single-Stage Installation . . . . . . . . . . . . . . . . . . . .
3.5.2.2 Multi-Stage Installation . . . . . . . . . . . . . . . . . . . .
3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Chapter 4 Multi-stage Optimal PMU Placement to Benefit System Volt-
age Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Voltage Stability Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Modeling Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Critical Buses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Case Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Voltage Stability Analysis
. . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Multi-stage OPP . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.1
5.2 Attack Model

Chapter 5 PMU Placement Against False Data Injection Attacks . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . .
5.3 Optimal PMU placement against FDIAs . . . . . . . . . . . . . . . . . . . .
5.4 Simulation and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1 RTU-based attack models
5.2.2 PMU-based attack model

Chapter 6 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . .
6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

47
50
53
54
55

56
56
57
60
60
61
63
64
65
67
69
71

73
73
74
75
76
77
79
82

83
83
85

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

86

vii

LIST OF TABLES

Table 2.1: Multistage approach, not considering effect of ZIB . . . . . . . . . .

23

Table 2.2: Multistage approach, considering effect of ZIB . . . . . . . . . . . .

24

Table 2.3:

Cost Comparison Of Proposed Approach With Other Approaches

.

25

Table 2.4: Multistage PMU Installation for the IEEE 14-bus (Using Priority
Buses), not considering Effect of ZIB . . . . . . . . . . . . . . . . .

Table 3.1:

Comparison of The Proposed Solution With Other Solutions: Effect
of ZIB is Not Considered . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.2:

Comparison of The Proposed Solution With Other Solutions: Effect
of ZIB is Considered . . . . . . . . . . . . . . . . . . . . . . . . . . .

Table 3.3:

Cost From the Proposed Approach: Effect of ZIB is Not Considered

27

47

48

49

Table 3.4:

Cost From the Proposed Approach: Effect of ZIB is Considered . . .

49

Table 3.5:

Conditional Reliability Indices –Vulnerability Indices (V.I.) . . . . .

52

Table 3.6:

Cost Comparison of the OPP; Effect of ZIB is Not Considered . . .

53

Table 3.7:

Cost Comparison The OPP; Effect of ZIB is Considered . . . . . . .

54

Table 3.8: Multi-Stage OPP; Effect of ZIB is Not Considered . . . . . . . . . .

55

Table 4.1:

LQP Results for The IEEE 14-bus system . . . . . . . . . . . . . . .

67

Table 4.2:

The Critical Bus Indices for The IEEE 14-bus system . . . . . . . .

68

Table 4.3:

LQP Results for The IEEE 30-bus system . . . . . . . . . . . . . . .

69

Table 4.4:

The Critical Bus Indices for The IEEE 30-bus system . . . . . . . .

70

Table 4.5: Multi-Stage PMU Installation for the IEEE 14-Bus System . . . . .

71

Table 4.6: Multi-Stage PMU Installation for the IEEE 30-Bus System . . . . .

71

Table 5.1:

FDIAs on the IEEE 14-bus Under Different OPP Schemes . . . . . .

81

viii

Table 5.2:

FDIAs on the IEEE 30-bus Under Different OPP Schemes . . . . . .

81

ix

LIST OF FIGURES

Figure 2.1:

Sample 9-bus system with possible communication paths

. . . . . .

9

Figure 2.2:

Sample graph to demonstrate the implementation of multisource Di-
jkstra algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

16

Figure 2.3:

Sorting function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17

Figure 2.4: Flow chart of the proposed O-BEBGA algorithm.

. . . . . . . . . .

18

Figure 2.5: Convergence of O-BEBGA for the normalized observability and cost
functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Figure 2.6: OPP solution for the IEEE 14-bus; not considering effect of ZIB.

. .

22

Figure 2.7: OPP solution for the IEEE 14-bus; considering effect of ZIB.

. . . .

24

Figure 3.1:

3-bus sample system . . . . . . . . . . . . . . . . . . . . . . . . . . .

32

Figure 3.2: Flowchart of the proposed algorithm . . . . . . . . . . . . . . . . . .

39

Figure 3.3:

Sorting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41

Figure 3.4:

IEEE RTS system . . . . . . . . . . . . . . . . . . . . . . . . . . . .

51

Figure 4.1:

Sample system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

58

Figure 4.2:

Sorting function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

65

Figure 4.3: Flow chart of the proposed algorithm . . . . . . . . . . . . . . . . .

66

x

Chapter 1

Introduction

The first phasor measurement unit (PMU) prototype was developed by Virginia Tech in the

early 1980s, and the first commercial PMU was manufactured in 1991 by Macrodyne and

Virginia Tech. [1, 2]. However, at the time PMUs were perceived to be expensive solutions.

Remote terminal units (RTUs) were used to monitor the power grid and provide sufficient

situational awareness. RTUs were used to get the conventional measurements. These mea-

surements include power injections and power flows, where the RTUs are sampled every few

seconds for state estimation. Due to the type of measurements of RTUs, non-linear and

iterative estimators were needed. Although the conventional measurements were not time-

stamped and sampled at a relatively slow rate (once every few seconds), they were sufficient

for system operations, since the system is assumed to be in quasi-steady-state.

Nevertheless, as the electrical industry is heading towards the smart grid technology, the

quasi-steady-state assumption has become less valid. The penetration of renewable energy

resources is increasing, and new smart controls are being introduced. Due to these reasons,

better measurements are needed, thereby requiring more precise and faster measurement

(units) [1]. The development of the time-synchronized PMUs has played a crucial role in

addressing these challenges.

PMUs measure the phasor components of the voltage and the current at high rates

(up to 50 Hz). These measurements are synchronized and time-stamped using the Global

Positioning System (GPS) [3]. Currently, PMUs can measure the operating frequency and

1

frequency deviation along with the voltage and current phasors.

The advent of PMUs has enabled the development of many applications. For instance,

[4–6] have used synchronized measurements to enhance wide area protection and control.

Other applications have included estimating the dynamic states (speed and angle) of ma-

chines, small signal stability, and frequency stability [7–10]. Most applications of PMUs have

depended on getting an accurate awareness of the system states. Therefore, state estimation

has become the primary application of PMUs. State estimation usually refers to estimating

the voltage magnitude and phase angle for every bus in the system. Moreover, when PMUs

provide complete observability, state estimation becomes a linear process [11, 12]. Despite

these advantages, the industry has been slow in adopting PMU technology due to the high

cost of installing PMUs. Therefore, PMU locations must be judiciously selected through

optimal placement of PMUs (OPP) which enables the minimization of installation cost.

1.1 Motivation and Challenges

Although PMUs offer huge advantages over RTUs, the industry has been slow in adopting

this technology, due to the high cost of PMUs. This slow adoption, and the high cost

associated PMU technology has driven the optimal PMU placement (OPP) problem.

In

OPP, the PMUs are judiciously selected such that maximum observability is achieved with

the minimum cost.

The OPP literature can be classified into two categories: observability-based OPP and

application-based OPP. The observability-based OPP aims to achieve complete observability

for the power grid while minimizing the total cost of installing PMUs.

On the other hand, application-based OPP focuses on other benefits besides the system

2

observability, such as transient stability, state estimation, and bad data detection [13–15].

Most of the application-based OPP approaches use the traditional definition of the OPP

problem, where the objective is to minimize the number of PMUs.

There have been several observability-based OPP approaches in the literature where

the cost of installing PMUs is associated with the PMU itself.

In these approaches, the

number of PMUs is minimized while considering the observability of the grid [16–19]. Most

OPP literature associates the PMU installation cost with the cost of the PMU unit. As

a result, most of these techniques have been proposed to minimize the number of PMUs

while considering the complete observability of the system [16–23]. However, the PMU unit

cost only 5% of the installation of the PMU and 95% of the budget is spent on upgrading

the supporting infrastructure for the PMU [24]. Therefore, the OPP problem should be

reformulated to include the installation cost of PMUs.

1.2 Contributions

The contributions of the work presented here can be summarized as follows:

• It develops a comprehensive cost model for the OPP problem, including the cost of the

PMUs as well as the infrastructure upgrade costs.

• It presents a flexible, multistage deployment plan, implemented over a period of time

depending on the budget of the utility company.

• It affords the ability to prioritize PMU placement based on specific criteria such as bus

criticality, thereby enabling application-based deployment.

• It improves the fault-tolerance of the system by enhancing the observability of critical

3

components, which are determined using vulnerability analysis. This analysis is based

on the reliability indices of the composite system.

• It integrates the vulnerability analysis into the observability function. This integration

enables cost-efficient OPP solutions while considering vulnerable buses. This integra-

tion of critical buses can be expanded to include multiple criteria for application-based

OPP approaches.

• It uses a cost model for the installation of PMUs which is derived from the industry

[24, 25]. Unlike most application-based OPP approaches which try to minimize the

number of PMUs as in [13–15, 26], this work uses the installation cost to achieve a

more realistic strategy.

• It can be used as a multi-stage process under a constrained budget, or as a single-stage

process while minimizing the total cost.

• It can achieve the optimal solution with the maximum observability in the case of

multiple optima due to the nature of the bi-level formulation.

• It enhances the security of the grid against false data injection attacks (FDIAs) through

an OPP approach.

1.3 Organization of the Thesis

This thesis is organized as follows:

Chapter 2 focuses on observability-based OPP with a realistic scenario where PMUs are

deployed over several budget periods. In this chapter, a comprehensive PMU installation cost

4

is used, where both the communication and the substation infrastructures are considered.

This chapter focuses on the observability and the financial aspects of the OPP problem.

Chapter 3 introduces an application-based OPP to enhance the fault-tolerance of the

network. In this application-based, the value of high assets is increased by deploying these

assets on vulnerable elements of the network.

Chapter 4 introduces an application-based OPP approach, where voltage stability crite-

rion is developed for prioritizing critical buses. This application-based approach assumes a

multistage scenario, where PMUs are deployed over several budget periods.

Chapter 5 examines the security of the grid against false data injection attacks (FDIA).

This chapter also introduces a secure OPP approach to enhance the resilience against such

attacks. The proposed OPP formulation does not make any assumptions about the security

of PMUs and enhances the bad data detection algorithms.

Chapter 6 provides concluding remarks and avenues for future research.

5

Chapter 2

Multistage Optimal PMU Placement

Considering Substation Infrastructure

2.1

Introduction

In this chapter, a multistage PMU placement strategy is proposed which considers two fac-

tors: substation cost (including PMU cost) and communication infrastructures. The prioriti-

zation of critical buses can be integrated with the proposed approach. This approach can be

used in an incremental way where PMUs are installed in multiple stages under a constrained

cost. Unlike most multistage approaches, this approach does not consider a predetermined

number of PMUs at each stage. Instead, the multistage installation maximizes the network

observability and prioritizes critical buses while remaining within a predetermined budget

for each stage.

In OPP, buses having one or more of the following criteria: high voltage buses, high

impact on transient stability, or sensitive loads, are considered critical buses [13, 14]. These

buses are sometimes given higher priority to enhance system awareness from a stability

perspective. Other researchers have proposed prioritizing buses based on different criteria

such as reliability and state estimation [27–29]. Bus prioritization can be integrated with

The content of this chapter has been reproduced with permission from Saleh Almasabi and Joydeep
Mitra, “Multistage Optimal PMU Placement Considering Substation Infrastructure,” in IEEE Transactions
on Industry Applications, vol. 54, no. 6, pp. 6519-6528, Nov.-Dec. 2018. doi: 10.1109/TIA.2018.2862401

6

the proposed approach while considering both observability and the actual cost of the PMU

installation. The major contributions of this work may be summarized as follows.

• It develops a comprehensive cost model for the OPP problem, including the cost of the

PMUs as well as the infrastructure upgrade costs.

• It presents a flexible, multistage deployment plan, implemented over a period of time

depending on the budget of the utility company.

• It affords the ability to prioritize PMU placement based on specific criteria such as bus

criticality, thereby enabling application-based deployment.

2.2 PMU Installation Cost

This section discusses the cost of PMU installation. It presents the cost model for upgrading

a substation and the cost of the communication infrastructure for PMU installation.

2.2.1 Substation Infrastructure

Most of the cost is associated with the installation process of the PMU and not the PMUs

themselves.

In fact, the PMUs cost about 5% of the total installation cost [24]. Most

of the cost is spent on upgrading the substation and communication infrastructures. A

report recently published by the U.S. Department of Energy (DOE) showed that the PMU

installation cost ranges from $40,000 to $180,000 per PMU [24].

The cost varies depending on the infrastructure support for the PMUs. Typically, PMUs

need sufficient communication infrastructure to send the measurement data to the PDC. The

substation infrastructure also needs to be sufficient to utilize the functionalities of PMUs.

7

Formulating the installation cost for the PMUs is a complicated process. Although PMU in-

stallation requires the same infrastructure upgrades, such as communication, cyber-security,

and other equipment upgrades, the approach to installing PMUs can differ depending on the

utility and existing infrastructure support for the PMUs. For instance, a utility can install

new stand-alone PMUs, or upgrade existing digital relays to enable PMU functionality [24].

Moreover, installing PMUs also depends on the availability of CTs and PTs [30]. Once the

infrastructure of the substation is in place to support the PMUs, the installation cost can

go down to 35% of the initial cost [24].

The proposed cost model of PMU installation in (2.1) considers the difference between

prepared buses, where minimal upgrades are needed, and unprepared buses, by introducing

gi index. The index takes the value of 4.5 for unprepared buses and 1 for prepared buses.

The model also includes the cost of adding additional measurement channels by including

the cost of PTs and CTs.

N(cid:88)

i=1

Cost =

where

N number of buses in the system;

gi prepared bus index;

gi(aPi + biPi) + K(P )

(2.1)

a cost of installing PMU and basic upgrades at the substation;

bi cost for installing additional PT or CT at substation i;

K(P ) cost function for the communication infrastructure for

PMUs;

P

= [P1, P2, . . . , Pi, . . . , PN ]T ;

8

where Pi takes the value of zero or one and indicates if a PMU is to be installed at substation

i.

Figure 2.1: Sample 9-bus system with possible communication paths

2.2.2 Communication Infrastructure

The measurement data obtained by the PMUs are sent to the PDC, where the PDC sorts

the data and processes it for other applications. Mohammadi et al. [31] have proposed to

reduce the distance between the PMUs and the PDC to lower the total cost. In [31], it is also

have proposed to place the PDC on a non-PMU bus to minimize the total communication

distance. The work in [31], however, has not considered the cost of upgrading the substation

for PMU installation. In this chapter, the PDC is assumed to be installed at one of the

substations where a PMU is to be placed. Then, the path connecting all PMUs is minimized

to lower the communication infrastructure cost (2.2).

There can be several communication paths connecting all PMUs at different substations.

Consider the 9-bus system in Fig. 2.1; the PMUs are placed at buses 4 and 7 to make the

system observable. However, there are two communication paths to connect both PMUs with

9

15 milesBUS 1BUS 2BUS 314 milesBUS 4BUS 5BUS 620 milesBUS 9BUS 8BUS 712 miles20 miles10 miles14 miles26 miles16 miles13 milesG 1G 2G 3Communication path#1Communication path#2PMU locationPPPDCPthe PDC located at bus 7. As seen in Fig. 2.1, the first communication path is about 32 miles,

and the second one is about 24 miles. Therefore, in order to minimize the communication

infrastructure cost, the path connecting all PMUs with PDC needs to be minimized. The

communication infrastructure is assumed to be passive optical network (PON) with optical

ground wire (OPGW). The cost model in (2.2) is derived from [32, 33].

n(cid:88)

j=1

min K(P ) =

leni,j · cci,j + Ne · Pj + Nb · Pj + PDC

(2.2)

where n is the number of PMUs and leni,j is the length of the transmission line between

buses i and j. The communication cost cc is either $2,414 or $0 per mile [34]. Ne represents

the passive cost of the communication infrastructure such as the housing chassis, optical

switch, wave filters. Nb, on the other hand, represents the cost per additional channel. Ne

and Nb are assumed to be $5,530 and $125 respectively [32]. PDC is assumed to have a total

cost of $7,500 [35].

2.3 Proposed Approach

This section presents the approach for the multistage OPP. As discussed in the previous

sections, the PMU installation cost plays a critical role in determining the installation pro-

cess. In the multistage approach, maximizing the benefits of PMU installation takes higher

priority over the cost function. This approach assumes that the utility sets a budget for the

installation of PMUs and the first objective is to maximize the observability and priority

buses while minimizing the installation cost and not exceeding the predetermined budget for

the current stage.

10

2.3.1 Problem Statement

As mentioned earlier, the optimal placement for PMUs highly depends on the installation cost

and available budget. In the proposed multistage approach, the observability is maximized,

subject to the observability constraint described below, and the cost function is minimized.

The observability in (2.3) needs to have enough redundancies for the desired observability

conditions. For instance, under normal operating conditions, the observability constraint in

(2.4) must be satisfied.

O = H × P

O ≥ I(cid:48)

(2.3)

(2.4)

where P is a vector of length equal to the number of buses N , as described in section 2.2.1;
I(cid:48) is a vector of length N with all its elements equal to 1; and H is an N × N connectivity

matrix. The entries for P and H are defined in (2.5) and (2.6) respectively.





hij =

Pi =

1, if a PMU is installed at bus i

0, if no PMU is at bus i

(2.5)

1, if i = j

1, if there is a branch connecting bus i and bus j

(2.6)

0, otherwise

In the proposed approach, the observability function in (2.7) is treated as a higher level

objective function, and is subjected to minimizing the cost function. The cost function in

(2.9) is treated as the lower level objective function, subjected to the higher level objective

11

(observability function). This setup allows maximizing the observability while minimizing

the cost without violating the budget constraint, thereby reaching the optimal solution for the

given budget. It should be noted that during the multistage process complete observability in

(2.4) cannot be achieved; therefore the observability constraint is changed to the multistage

condition in (2.8).

N(cid:88)

i=1

O(cid:48)

i

max

subject to

O ≤ ˙I “multistage observability condition”

N(cid:88)

i=1

min Cost =

gi(aiPi + biPi) + K(P )

subject to

C ≤ Cbudget

(2.7)

(2.8)

(2.9)

2.3.1.1 Priority Buses

For application-based OPP schemes, some buses are prioritized for PMU installation regard-

less of their contribution to the overall observability. These schemes range from stability

criteria to reliability, and many others [13–15, 27]. In the proposed OPP scheme, priority

buses can be chosen using any criterion.

The priority buses for the network are embedded in the observability function as bias

using a priority vector R (2.10). The R vector has the length of the number of buses N . If

all buses are treated equally then all elements of R are set to zero. The higher priority buses

are determined based on the utility criteria, then arranged in descending order in a vector

12

L. Then the priority bias is assigned using the following algorithm.

Procedure 1 Priority Vector R
Initialize priority buses (L), R = zeros1×N

N = number of buses

M = maximum number of branches in (H)

kk = length of L

for j = 1 : kk

i = l(j)

if i (cid:54)= 0 then
ri = M (kk − j + 1)

else

ri = 0

endif

endfor

Maximizing the modified observability vector O(cid:48) gives bias to the higher priority buses.

However, this vector cannot be used to test the observability of the network since the R vector

skews the observability. As a result, the skewed observability in (2.7) is used for optimizing

the OPP, and the original observability in (2.3) is used as a constraint for observability



testing.

where

h(cid:48)
ij =

1 + ri, if i = j

1, if there is a branch connecting bus i and bus j

(2.10)

0, otherwise

13

O(cid:48) = H(cid:48) × P ;

ri bus i priority index;

N number of buses.

The cost of the overall PMU installation can be reduced by considering the effect of

zero-injection buses (ZIB). Considering the effect of ZIBs improves the overall observability

of the system, thereby reducing the number of PMUs needed to achieve the observability

constraint. The effect of ZIB can be summarized into two points. If all buses connected to a

ZIB are observable, the ZIB is considered observable by applying KCL. Also, an unobservable

bus, when connected to an observable ZIB, is considered observable only if all of the other

buses connected to the ZIB are observable.

2.3.2 Algorithm

The optimal placement problem in subsection 2.3.1 is a discontinuous bi-level problem. It

also involves optimizing the communication infrastructure cost K(P ) within the installation

cost in (2.9). A multisource Dijkstra algorithm is used to obtain the shortest path connecting

all PMUs and PDC. Since the proposed model involves optimizing three objective functions

(2.2), (2.7) and (2.9), evolutionary algorithms are the appropriate tools for solving such a

problem. The proposed algorithm uses an opposition-based elitist binary genetic algorithm

(O-BEBGA) to solve the bi-level OPP in subsection 2.3.1. The opposition element is added

to the algorithm to enhance the overall performance since opposition-based methods have

proven their superiority in terms of convergence speed and results [36–38].

14

2.3.2.1 Multisource Dijkstra

The design of the communication infrastructure involves finding the most cost effective path

(2.2) between the PMUs and PDC. There are several algorithms that can be used to find this

path such as Floyd-Warshal, Bellman-Ford, Johnson and Dijkstra algorithms. The Dijkstra

algorithm is among the most efficient algorithms for single source undirected weighted graphs

[39]. However, the communication network design problem is not a single source/destination

problem; rather, it is a multisource single destination problem, or a single source/destination

with a must pass nodes. The traditional Dijkstra algorithm can still be used to solve this

problem. This entails using the Dijkstra algorithm n times to establish one communication

line between two source nodes out of n source nodes. The multisource Dijkstra algorithm,

on the other hand, can be used to pair up source nodes in one run.

The multisource Dijkstra is used to find the shortest paths Px1 connecting every source
node si to the nearest source node sj, where (si, sj ∈ S). This step generates a set G =
{φ1, φ2, . . . , φn1} with n1 subsets, where n1 = f loor(Ns/2) and Ns is the number of source

nodes. Each subset φ has at least two connected source nodes. The next step is to connect

the subsets in G to each other. First, the weights for the paths in Px1 are set to zero.

The multisource Dijkstra is then used to obtain new paths Px2 that connect the subsets in

G. It should be noted that the new paths Px2, may have redundant routes, however these

redundant routes have zero weight. The process of finding new paths and updating their

weights is repeated until all the subsets in G are connected with one path. This path is the

union of all paths Pxi obtained from the multisource Dijkstra algorithm.

The sample graph in Figure 2.2 demonstrates the implementation of multisource Dijkstra

algorithm for the communication network design. The source nodes in S are a, d, f and j.

15

Figure 2.2: Sample graph to demonstrate the implementation of multisource Dijkstra algo-
rithm.

The first loop of the multisource Dijkstra generates the G set with subsets φ1 and φ2. The

φ1 subset contains the source nodes a and j; the φ2 subset contains source nodes d and f .
The Px1 paths for the subsets in G are {a–k–j, d–f}. The next step is to connect the subsets
φ1 and φ2, which produce the path Px2={a–b–c–d}. The union of the paths P x1 and P x2

produces the shortest path connecting all the source nodes in S.

2.3.2.2 O-BEBGA

As discussed previously, multistage approaches can lead to higher cost for the overall instal-

lation of PMUs, since the solutions for each stage are often sub-optimal for the complete

observability. This is because maximizing the observability at each stage increases the in-

stallation cost [40]. To overcome this issue, the O-BEBGA solves the OPP for the desired

observability condition first. Then, the optimal solution for the complete observability (Xs)

is used as optimal solution for the multistage installation. To enhance the performance fur-

ther, the search space multistage installation is reduced to include only optimal location of

16

jhakgbeih96314910121336527Step # 1X1PStep # 2X2PcdfFigure 2.3: Sorting function.

PMUs in Xs.

The proposed algorithm solves the optimal placement problem in a parallel manner by

initializing random candidates where each candidate xi has the length of the number of

buses N , thereby evaluating the candidate buses simultaneously instead of using systematic

increments. By maximizing the observability function (higher objective), while minimizing

the cost of PMU installation, the predetermined budget Ck

budget for each stage (k) is optimally

utilized.

The higher and lower objectives share the same decision variables, meaning there are no

decision variables exclusive to one objective or the other. The proposed approach exploits this

advantage to evaluate both objectives simultaneously without using different search spaces

for each objective. The proposed approach uses a sorting function to handle the simultaneous

evaluation in the same search space. This function sorts all candidates according to their

feasibility and fitness of the higher and the lower objectives, as seen in Fig. 2.3. As a result,

the algorithm is guided towards the optimal solution where the cost is minimized and the

observability is maximized.

The overall flowchart for the proposed algorithm is shown in Fig. 2.4. The crossover

and mutation probabilities are Pc = 0.7 and Pm = 0.3 respectively. The Orn donates the

17

CandidatesCostRankBus 1Bus 2Bus 3Bus 4…1011…6200018.311 010…5400014.621 100…5500014.63        0001…30002.9n-20011…24002.2n-10101…14001.4nFeasible Not Feasible  Figure 2.4: Flow chart of the proposed O-BEBGA algorithm.

18

YesYesNoNoYesNoYesNoBinary mutation to get (Xm)  Xp = Parent Selection (Tournament Selection) Generate dynamic opposition population Has termination criteria been reached? Compare Xp, Xc and XmThen select the best for next generationEvaluate candidates for both objectives (3,4)Obtain optimal solution (Xs) for the complete observability Set SF=1Initialize random candidates XoRank solutions using the sort functionRank solutions using the sort functionOptimize the communication path using multi-source Dijkstra algorithmInitializenumber of stages k, budget for each stage           , system data (H) , priority buses (L)    kbudgetCSet the priority vector (R),SF=0;     Check for pre-installed    Calculate the pre-cost            Set the new budget         k-1CFinal solution for stage kk=k+1SF=1Complete observability?Final solutionNoYesIs SF=1?NoYesReduce search space Binary crossover to get (Xc)  kk-1new,budgetbudgetC=C+Ck-1PMUsrnmIs M< PrnoIs O< Pr ncIs C< Popposition random variable; the crossover and mutation random variables are denoted by Crn

and Mrn respectively. Double point crossover is used to generate the offspring population Xc,

and single point mutation is used to generate the mutated population Xm. The algorithm

uses dynamic opposition with probability of Po = 0.4.

There are many variations of opposition techniques in the literature. The two most

common are the global opposition and the dynamic opposition. In the proposed algorithm, a

modified dynamic opposition is used to generate the opposition population when applicable.

Instead of generating the total opposite of the chosen individual Xi, only a third of the

variables in Xi are selected for the opposition process (2.11).

X

opp
i

= Xmin + Xmax − Xi

(2.11)

The algorithm is terminated if the conditions in (2.12) are met. The terms α and β

are constants; where γ(1) indicates how much of the current population is feasible and γ(2)

indicates if the population is converging to an optimum. The variance of the cost (V ar[C])

is used to determine γ(2), where k is the index for the current population.



γ =

γ(1) =

γ(2) =

(cid:12)(cid:12)(cid:12)O − max(O)
(cid:12)(cid:12)(cid:12) ≤ α
(cid:12)(cid:12)(cid:12)V ar[Ck−1] − V ar[Ck]

(cid:12)(cid:12)(cid:12)

(2.12)

2.3.3 Multistage Installation

In realistic scenarios, PMUs are installed in a multistage manner. The majority of existing

multistage methods assume the minimum number of PMUs per stage. This assumption is

unrealistic since PMU installation is restricted by the financial burden, substation infrastruc-

19

ture, and technical benefits at each stage. The proposed approach uses a financial capital

Cbudget, as the limit for each stage instead of using the number of PMUs as the limit.

At each stage k, the previously installed PMUs (P M U k−1) are initiated, and the pre-

installation cost Ck−1 is calculated. The budget for each stage Ck

budget is set. Then the
budget is modified to include the pre-installed PMUs cost Ck−1. The pre-installed
PMU locations are maintained during the initialization of the random population Xo and

budget Ck

during the mutation step. Since the pre-installed PMU locations are maintained for the

initial and parent populations, the crossover population Xc maintains the pre-installed PMU

locations by default.

Ck
new,budget = Ck

budget + Ck−1

(2.13)

2.3.3.1 Complete observability

Initially, the multistage approach cannot achieve complete observability; however as stages

are added, or more money is added to the budget, the complete observability constraint in

(2.14) is satisfied.

H × P = O ≥ ˙I

(2.14)

2.3.3.2 Observability under single line outage

It is required to have at least two measurement redundancies for every bus in the network

to achieve observability under a single line outage. Therefore, the observability constraint is

changed to (2.15).

H × P = Os ≥ ˝I

20

(2.15)



Os,i =

(cid:80)(aij × pi)

2,

if a PMU is installed at bus i

(2.16)

where Os is the observability vector for all buses in the system and ˝I is a vector of length

equal to the number of buses, with all entries equal to 2.

Figure 2.5: Convergence of O-BEBGA for the normalized observability and cost functions.

2.3.3.3 Single PMU outage

In a single PMU outage, every bus needs to have two independent measurements, either by

two different PMUs or if ZIB effect is considered through KCL and a PMU. Therefore, the

observability constraint in (2.8) is changed to the following:

H × P = Op ≥ ˝I

(cid:88)
(aij × pi)

Op,i =

21

(2.17)

(2.18)

051015200.750.80.850.90.951  observability   cost2.4 Simulation and Results

In this section, the proposed approach is tested on the IEEE reliability test system (RTS),

IEEE 14-bus, 30-bus and 118-bus test systems.

In subsection 2.4.1 all buses are treated

equally and no prioritization is given to any bus. The higher priority buses and other

observability conditions are tested in subsection 2.4.2. The buses are divided into two cat-

egories: prepared buses and unprepared buses. The prepared buses are assumed to have

sufficient infrastructure, require basic security, network upgrades, and cost 75% less than

the unprepared buses [24].

The base cost per PMU is assumed to be $40,000, and the cost per additional PT or

CT is assumed to be $2,380 [41]. The cost of PDC is assumed to be $7,500. The length of

the transmission lines are obtained from [42]. The communication links are assumed to be

running along the transmission lines where the cost of the communication links is assumed

to be $2,414 per mile [34] or $0 if the communication link already exists.

Figure 2.6: OPP solution for the IEEE 14-bus; not considering effect of ZIB.

22

Bus 14Bus 13Bus 12Bus 1Bus 6Bus 9Bus 8Bus 3Bus 2Bus 5Bus 4Bus 11Bus 10Bus 7   + PDCSTAGE #1 PMU locationsSTAGE #2 PMU locationsP2P2P1P1P2P2P1P1P1P1P2P2P1Table 2.1: Multistage approach, not considering effect of ZIB

System

PMU locations

Stage budget

Ck

budget

Cost

Remaining

Unprepared

buses

First Stage (k = 1); Effect of ZIB Is Not Considered

IEEE 14-bus

IEEE RTS

IEEE 30-bus

IEEE 118-bus

IEEE 14-bus

IEEE RTS

IEEE 30-bus

IEEE 118-bus

IEEE RTS

IEEE 30-bus

IEEE 118-bus

4, 5, 11

15, 16, 21

15, 16, 26

2, 5, 15, 19, 21, 30, 34, 45, 49, 66, 68,

77, 84, 89, 92, 105

$300,000

$252,189

$47,811

7, 9

$350,000

$213,522

$136,478

10, 11, 17, 24

$450,000

$407,745

$42,255

9, 12, 25, 27, 28

$4,000,000

$3,910,268

$89,732

—

Second Stage (k = 2); Effect of ZIB Is Not Considered

4, 5, 8, 11, 13 a

5, 6, 8, 15, 16, 21

6, 7, 11, 15, 16, 22, 26

2, 5, 9, 15, 19, 21, 27, 30, 34, 40,
45, 49, 52, 56, 59, 66, 68, 71, 77,

80, 84, 89, 92, 105, 110

$200,000

$350,000

$350,000

$121,656

$286,227

$78,344

$63,773

7, 9

10, 11, 17, 24

$235,776

$114,224

9, 12, 25, 27, 28

$2,000,000

$1,987,498

$41,932

—

Third Stage (k = 3); Effect of ZIB Is Not Considered

5, 6, 8, 9, 15, 16, 21, 23 a

3, 6, 7, 11, 13, 15, 16, 20, 22, 26, 29 a

$350,000

$350,000

$266,245

$83,755

10, 11, 17, 24

$317,741

$132,259

9, 12, 25, 27, 28

2, 5, 9, 12, 15, 19, 21, 27, 30, 31, 32, 34, 36,
40, 45, 49, 52, 56, 59, 63, 66, 68, 70, 71, 77,
80, 84, 86, 89, 92, 94, 100, 105, 110, 118 a

$2,000,000

$1,022,043

$977,957

—

aComplete observability is achieved.

The proposed algorithm is used with a population size of 3 × N , where N is the number

of buses. The performance of the O-BEBGA is shown in Fig. 2.5. Although the algorithm

maximizes the observability, the minimization of the PMU installation cost drives the ob-

servability to a cost effective solution.

It should be noted that maximizing observability

often increases the cost. However, there exist cases where the same or better observability

can be achieved at a better cost, as is the case for generations 5 and 12 in Fig. 2.5.

23

Figure 2.7: OPP solution for the IEEE 14-bus; considering effect of ZIB.

Table 2.2: Multistage approach, considering effect of ZIB

System

PMU locations

Stage budget

Ck

budget

Cost

Remaining

Unprepared

buses

First Stage (k = 1); Effect of ZIB Is Considered

IEEE 14-bus

IEEE RTS

IEEE 30-bus

4, 5, 11

5, 20

$300,000

$252,189

$47,811

7, 9

$400,000

$352,366

$47,634

10, 11, 17, 24

3, 4, 10, 15, 20

$500,000

$426,236

$73,764

9, 12, 25, 27, 28

IEEE 118-bus

45, 49, 53, 72, 80, 84, 86, 94

$2,500,000

$2,357,860

$142,140

Second Stage (k = 2); Effect of ZIB Is Considered

—

7, 9

4, 5, 11, 13a

2, 5, 14, 20, 21

$150,000

$73,448

$76,552

$350,000

$222,514

$127,486

10, 11, 17, 24

3, 4, 10, 13, 15, 20, 29

$350,000

$238,861

$111,139

9, 12, 25, 27, 28

2, 8, 12, 19, 21, 27, 34, 37, 45, 49,

53, 56, 68, 72, 75, 77, 80, 84, 86, 92, 94

$2,000,000

$1,924,513

$75,487

—

Third Stage (k = 3); Effect of ZIB Is Considered

2, 5, 8, 14, 20, 21a

$350,000

$142,932

$207,068

10, 11, 17, 24

3, 4, 7, 10, 13, 15, 16, 20, 29a

$350,000

$230,720

$119,280

9, 12, 25, 27, 28

2, 8, 11, 12, 19, 21, 27, 31, 32, 34, 37,
40, 45, 49, 53, 56, 62, 68, 72, 75, 77,
80, 84, 86, 89, 92, 94, 100, 105, 110a

$2,000,000

$1,924,513

$75,487

—

IEEE 14-bus

IEEE RTS

IEEE 30-bus

IEEE 118-bus

IEEE RTS

IEEE 30-bus

IEEE 118-bus

aComplete observability is achieved.

24

Bus 14Bus 13Bus 12Bus 1Bus 6Bus 9Bus 8Bus 3Bus 2Bus 5Bus 4Bus 11Bus 10Bus 7   + PDCSTAGE #1 PMU locationsSTAGE #2 PMU locationsP1P2P2P2P2P1P1P1P1P1P1System

Ref. [30]a

2, 8, 10, 13
$554, 650

2, 6, 7, 9
$870, 400

Ref. [16]

2, 6, 7, 9
$870, 400

IEEE 14-bus

IEEE RTS

IEEE 30-bus

3, 4, 7, 10, 13, 16, 20, 21

2, 3, 8, 10, 16, 21 ,23

2, 3, 8, 10, 16, 21, 23

$1, 342, 300

$1, 222, 100

$1, 222, 100

3, 6, 7, 11, 13, 15, 17

1, 2, 6, 9, 10, 12,

2, 3, 6, 9, 10, 12

20, 21, 24, 26, 30

$1, 236, 400

15, 19, 25, 27
$1, 825, 100

15, 19, 25, 27
$1, 802, 100

IEEE 118-bus

—

—

1, 5, 9, 12, 13, 17, 21, 23,
26, 28, 34, 37, 41, 45, 49,
53, 56, 62, 63, 68, 71, 75,

77, 80, 85, 86, 90, 94,

101, 105, 110, 114

$8, 224, 600

—

—

—

—

—

—

—

—

Ref. [19]b

2, 6, 8, 9
$679, 320

—

—

Proposed

4, 5, 8, 11, 13

$373, 845

5, 6, 8, 9, 15, 16, 21, 23

$765, 994

3, 6, 7, 11, 13, 15,
16, 20, 22, 26,29

$961, 262

2, 5, 11, 12, 15, 17, 21, 24,
25, 28, 34, 37, 40, 45, 49,

52, 56, 62, 63, 68, 73,
75, 77, 80, 85, 86, 90,
94, 101, 105, 110, 114

2, 5, 9, 12, 15, 19, 21, 27,
30, 31, 32, 34, 36, 40, 45,
49, 52, 56, 59, 63, 66, 68,
70, 71, 77, 80, 84, 86, 89,
92, 94, 100, 105, 110, 118

$7, 988, 600

$6, 890, 379

—

—

—

—

4, 5, 11
$325, 637

2, 5, 8, 14, 20, 21

$895, 817

3, 4, 7, 10, 13, 15

16, 20, 29
$717, 812

2, 8, 11, 12, 19, 21, 27, 31,
32, 34, 37,40, 45, 49, 53, 56,
62, 68, 72, 75, 77, 80, 84, 86,

89, 92, 94, 100, 105, 110

$6, 063, 617

Table 2.3: Cost Comparison Of Proposed Approach With Other Approaches

Ref. [43]

Ref. [44]

Ref. [31]

Under Normal Operating Conditions; Effect of ZIB Is Not Considered

Under Normal Operating Conditions; Effect of ZIB Is Considered

IEEE 14-bus

IEEE RTS

IEEE 30-bus

2, 8, 10, 13
$554, 650

2, 6, 9

$631, 110

2, 6, 9

$631, 110

2 , 8, 10, 15, 22, 23

2, 8, 10, 15, 20, 21

1, 2, 8, 16, 21, 23

$1, 076, 000

3, 7, 8, 10, 13,
15, 17, 19, 29

$976, 260

$1, 032, 300

1, 2, 10, 12,
15, 19, 27
$1, 239, 600

$941, 410

2, 3, 10, 12,
18, 24, 30
$1, 159, 400

2, 6, 9
$631, 110

—

2, 4, 10, 12,
15 ,18, 27
$1, 176, 600

—

—

3, 7, 10, 12,
15, 20, 27
$1, 099, 700

IEEE 118-bus

—

—

1, 6, 8, 12, 15 , 17, 21, 25, 29,

34, 40, 45, 49, 53, 56, 62,
72, 75, 77, 80, 85, 86, 90,

2, 8, 11, 12, 15, 19, 21, 27, 31,
32, 34, 40, 45, 49, 52, 56, 62,

65, 72, 77, 80, 85, 86, 90,

94, 101, 105, 110, 114

94, 101, 105, 110

2, 8, 11, 12, 17, 21, 24,
27, 29, 43, 47, 49, 52, 56,
62, 71, 75, 77, 80, 85, 86,
90, 94, 102, 105, 110, 114

$7, 840, 900

$7, 523, 100

$8, 173, 500

aHas two optimal solutions, one for minimum number of PMUs and the other for a cost model.
bHas multiple optimal solutions, only the solution with the minimum cost is presented.

25

2.4.1 No Priority Buses

The model in section 2.2 is used and no priority is given to any bus. The multistage OPP is

performed as a two-stage process for the IEEE 14-bus and a three-stage process for the IEEE

RTS, the IEEE 30-bus and the IEEE 118-bus. Each stage is treated independently budget-

wise, meaning the remainder of the budget from each stage is not added to the next stage

budget. Complete observability is achieved for all systems within three stages. It should be

noted that the number of stages in which complete observability is achieved depends on the

budget specified by the utility.

The OPP is performed in two different cases. In the first case, ZIBs are treated as normal

buses. The result of the multistage OPP is shown in Table 2.1. The OPP solution for the

IEEE 14-bus is shown in Fig. 2.6 and Fig. 2.7. The ZIB effect is considered in the second

case as shown in Table 2.2. A brute force approach was used to test the optimality of the

results for the IEEE 14-bus test system for normal conditions and single line outage. The

results of the brute force showed that the proposed approach found the global optima for

the 14-bus IEEE test system.

The proposed approach is compared with some of the recent approaches in OPP liter-

ature, as seen in Table 2.3. These approaches include classical and evolutionary methods,

mainly particle swarm optimization (PSO), Cellular Learning Automata (CLA) and binary

imperialistic competition algorithm (BICA). The results show that the proposed approach

achieves better overall cost for the OPP. This enhancement is partially due to the compre-

hensive installation cost model of the proposed approach, which considers the substation

and communication infrastructure. Most of the other methods have not considered such a

comprehensive model.

26

2.4.2 Using Priority Buses

The multistage PMU installation is performed on the IEEE 14-bus test system with prede-

termined priority buses. The higher priority buses for the IEEE 14-bus are chosen to be the

high voltage buses, L = [1, 2]. The installation is performed as a four-stage process.

The first stage has a budget limit of $400,000, and the remaining stages have budget limits

of $500,000 each. The first and second stages are used to achieve complete observability

under normal conditions (2.14). The third stage is used to achieve observability for single

line outage contingencies (2.16). The single PMU outage in (2.18) is chosen as the desired

observability for the final stage. The results in Table 3.8 show the optimal PMU installation

at each stage. The results show a comparison between treating all buses equally. The results

show that prioritizing buses can drive up the PMU installation cost as seen in Table 3.8.

Table 2.4: Multistage PMU Installation for the IEEE 14-bus (Using Priority Buses), not
considering Effect of ZIB

All buses are treated

Priority buses

Unprepared

equally L = [φ]

are used L = [1, 2]

buses

Stage One
O ≤ ˙I
Stage Two
O ≥ ˙I

Stage Three
Os ≥ ˝I

Stage Four
Op ≥ ˝I

4, 6, 8

$219,799

4, 5, 6, 8,
11, 13, 14
$300,966

1, 2

$185,777

1, 2, 8,
10, 13

$491,476

2, 4, 5, 6, 8, 10,

1, 2, 4, 6, 8, 10,

11, 13, 14
$133,860

11, 13, 14
$255,764

2, 4, 5, 6, 8, 10,

1, 2, 4, 6, 8, 10,

11, 13, 14a

$0b

11, 13, 14a

$0b

7, 9

7, 9

7, 9

7, 9

aNo additional installation of PMUs.
bObservability already achieved at the previous stage.

27

Chapter 3

A Fault-Tolerance Based Approach to

Optimal PMU Placement

3.1

Introduction

The concepts of reliability and fault-tolerance are closely related, yet distinct. While re-

liability is the ability of a system to perform its required function within a specific pe-

riod, fault-tolerance is defined as the ability of a system to perform despite a failure (fault)

event [45, 46]. Nevertheless, both concepts share some common metrics such as failure rate,

repair rate, and availability. The proposed strategy of deploying PMUs in the proximity of

higher probability contingencies increases the likelihood of more effective remedial actions,

both preventive and corrective, thereby enhancing fault-tolerance. The work presented in

this chapter is motivated by the idea that the value of a high cost asset such as PMU can

be increased by deploying it so as to improve network fault-tolerance.

A new criterion for PMU placement is proposed that incorporates fault-tolerance into the

OPP framework while minimizing the installation cost of PMUs. Reliability indices along

with conditional probabilities are used to determine the critical components and assess the

network vulnerability. The results from the vulnerability analysis are integrated into the

The content of this chapter has been reproduced with permission from S. Almasabi and J. Mitra,“A
Fault-Tolerance Based Approach to Optimal PMU Placement,” in IEEE Transactions on Smart Grid. doi:
10.1109/TSG.2019.2896211

28

OPP problem. Therefore, critical buses are given higher priority while considering both

the network connections and the installation cost of PMUs. This enhances the situational

awareness of network vulnerabilities and enables better operational control during faults.

The main contributions of the proposed approach are:

• It improves the fault-tolerance of the system by enhancing the observability of critical

components, which are determined using vulnerability analysis. This analysis is based

on the reliability indices of the composite system.

• It integrates the vulnerability analysis into the observability function. This integra-

tion enables for cost-efficient OPP solutions while considering vulnerable buses. This

integration of critical buses can be expanded to include multiple criteria for application-

based OPP approaches.

• It uses a cost model for the installation of PMUs which is derived from the industry

[24, 25]. Unlike most application-based OPP approaches which try to minimize the

number of PMUs as in [13–15, 26], this work uses the installation cost to achieve a

more realistic strategy.

• It can be used as a multi-stage process under a constrained budget, or as a single-stage

process while minimizing the total cost.

• It can achieve the optimal solution with the maximum observability in the case of

multiple optima due to the nature of the bi-level formulation.

This chapter is organized as follows. Section 3.2 discusses the fault-tolerance and the

vulnerability assessment.

In Section 3.3, the cost of installing PMUs is presented. The

29

proposed approach is presented in Section 3.4. Sections 3.5 and 3.6 present the case studies

and the conclusion respectively.

3.2 Fault-Tolerance: Vulnerability Analysis

The vulnerability analysis applied here has two primary steps. The first step is to identify the

critical components of the system. The second step is to rank and validate monitoring these

components. There are several approaches to assess the criticality of the system components.

In [47, 48] Markov cut-set method along with DC-optimal power flow (DC-OPF) is used to

assess the reliability of composite systems and to determine the critical components of the

system. In [49], the critical components of the system are identified based on the impact

of component outages on the interruption cost under predetermined loading conditions and

forced outages.

The network vulnerability is determined by using AC-optimal power flow (AC-OPF) and

determining the reliability indices of the conditional probability for the composite system

S, given that component ck has failed (3.2). The system failure is considered to be a load

curtailment event, which can be considered as a conditional loss of load probability (LOLP),

conditioned by a contingency, as shown in (3.2).

This approach assesses the direct impact of each component on the system. Each com-

ponent ck in the system has a failure rate λk and a repair rate µk. The probability of failure

Qk is shown in (3.1). To properly determine the criticality of a component ck on the system,

the failure rate of that component is inflated so that the probability of failure is almost one
Qk ≈ 1. Then the reliability indices of the system are obtained by using AC-OPF along

with Monte Carlo simulations. The results from this analysis represent the criticality of

30

component ck to the system.

Qk =

λk

Γk = Pk,f

λk + µk

(cid:0)Sf|ck,f

(cid:1)

(3.1)

(3.2)

where

Γk criticality of component ck;

Pk,f the probability of a system failure event given that

component ck has failed;

Sf

the event of the composite system failure;

ck,f the component ck failure event.

However, since the purpose of this analysis is to prioritize the location of the PMUs for

installation, ranking components by their criticality is not enough. The failure rate of the

individual components needs to be considered. For instance, consider the 3-bus system in Fig.

3.1 which has one generator, two transmission lines and two loads. It can be observed that

the generator and transmission line L1 are the critical components for this system. However,

if these components are highly reliable and transmission line L2 is unreliable, monitoring the

transmission line T2 takes priority over transmission line T1. Therefore, it is not enough

to consider the criticality of a component, but the reliability of each component needs to

be considered too. The vulnerability indices in (3.3) consider both the criticality of each

component and its failure rate. The pseudo algorithm for the vulnerability analysis is shown

in Procedure 2.

where

Vk = N (Γk , Qk)

(3.3)

31

Vk the vulnerability index for component k;

Qk the failure probability of component k;
N the cumulative distribution function (CDF)

for the normal distribution.

Figure 3.1: 3-bus sample system

Procedure 2 Network Vulnerability
Initialization load system and reliability data

Nc : the number of components

for k = 1 : Nc

Inflate the probability of failure Qk for component ck

Perform Monte Carlo simulation

Obtain Γk
Calculate Vk = N (Γk , Qk), and reset Qk

endfor

The results from this analysis are used to give the vulnerable buses higher priority by forming

a bias matrix containing the vulnerability indices. The elements of the bias matrix are

determined as in (3.4). When applying the vulnerability analysis to the sample system in

Fig.1, the components of the system will get a vulnerability index Vk. These indices are

32

T1T2L1L2BUS 1BUS 2BUS 3G1used to build the bias matrix, a 3 × 3 matrix in this case, where the elements of the bias

matrix are determined using (3.4). These elements are incorporated into the observability

in section 3.4.2.

vi,j = Vk, if component ck is connecting bus i to bus j

Hi,j

(3.4)

j=N(cid:88)

j=N(cid:88)

vi,i =

1 + vi,j −

j=1

j=1

where N is the number of buses and H is the N × N connectivity matrix defined in (3.5).



Hij =

1, if a PMU is installed at bus i

1, if there is a branch connecting bus i and bus j

(3.5)

0, else

3.3 PMU Installation Cost

Most of the cost is associated with the installation process and the substation infrastructure

rather than the PMUs themselves. A report published by the U.S. Department of Energy

(DOE) has showed that the PMU installation cost ranges from $40, 000 to $180, 000 per

PMU [24]. The cost varies depending on the infrastructure support for the PMUs. Therefore,

it stands to reason that installation cost should be considered instead of minimizing the

number of PMUs.

Although PMU installation requires the same infrastructure upgrades, such as communi-

cation, cyber-security and other equipment upgrades, the approach to install them can differ

depending on the utility and the existing infrastructure support for the PMUs. For instance,

a utility can install new stand alone PMUs, or upgrade existing digital relays to enable PMU

33

functionality [24]. Moreover, installing PMUs also depends on the availability of CTs and

PTs [30]. Once the substation’s infrastructure is in place to support PMUs, the installation

cost can go down to 35% of the initial cost [24]. This chapter uses a model that includes the

basic cost of each PMU unit in addition to the infrastructure cost, such as communication,

security, PT and CT which may differ from one substation to another. The installation cost

for PMUs is modeled as follows:

C =

N(cid:88)

i=1

where

ai unit cost for PMU at substation i

(ai + bi + k(i))Pi + PDC

(3.6)

bi cost for installing additional channels at substation i

k(i) cost function for the communication network infrastructure

at substation i;

PDC cost of phase data concentrator.

The Pi terms form a vector P = [P1, P2, . . . , Pi, . . . , PN ]T , where Pi assumes the value of

zero or one indicating if a PMU is to be installed at substation i. The cost coefficient

for the communication network k(i) at substation i, can take values between zero—if the

communication infrastructure is sufficient for the installation—or $8,000 per branch if the

communication infrastructure are to be updated, as described in (3.7).

n(cid:88)

k(i) =

leni,j cci,j

(3.7)

j=1

where n is the number of buses, and leni,j is the transmission line between buses i and j.

The communication cost cc is either $8,000 or $0.

34

The PMU installation cost model in (3.6) considers additional PMU channels and their

associated costs, by considering the additional equipment needed for these channels such

as PTs and CTs. It is assumed that the PMU will utilize all available channels. However,

the cost of these channels is considered by the variable bi, which differs from one substation

to another depending on the number of channels and the type of available equipment. For

instance, instead of using PTs for voltage measurements, capacitively coupled voltage trans-

formers can be used. This can be accommodated by changing the value of the veritable bi

to correspond to the specific equipment available at substation i.

3.4 Proposed Approach

This section presents the approach for the optimal placement of PMUs. As discussed in

the previous section the cost plays a critical role in determining the optimal installation

process.

In the proposed approach, the cost is given a higher priority over the network

and vulnerability. The approach assumes that the utility sets a budget for the installation

of PMUs and the higher objective is to minimize the cost and to make sure the optimal

placement does not exceed the budget.

In the lower objective, the network observability

is maximized while given a bias for the vulnerable buses or branches. This bias is used to

prioritize the vulnerable buses for the OPP.

3.4.1 Problem Statement

As discussed earlier, the OPP highly depends on the installation cost and the available

budget. Therefore, the cost function in (3.8) is treated as higher level objective function.

The observability function in (3.9) is treated as a lower level objective, subjected to the higher

level objective (cost function). This bi-level setup allows maximizing the observability while

35

minimizing the cost without violating the budget constraint, thereby reaching the optimal

solution for the given budget. Moreover, this bi-level formulation enables for the optimal

solution with the maximum observability to be selected in the case of multiple optimal

solutions. The vulnerability of the network is incorporated into the observability function as

bias for vulnerable buses or branches as in (3.12). These vulnerability indices are obtained

using Procedure 2 and (3.4).

min C =

subjtect to

N(cid:88)

i=1

(ai + bi + k(i))Pi + PDC

(3.8)

(3.9)

(3.10)

(3.11)

(3.12)

where

C ≤ Cbudget
(cid:88)N

i=1

max

O(cid:48)

i

O(cid:48) = H(cid:48) × B

1, if a PMU is installed at bus i

0, else



Bi =



H(cid:48)
ij =

1 + vi,i, if a PMU is installed at bus i

1 + vi,j, if there is a branch connecting

bus i to bus j

0, else

vi,j the vulnerability element (i, j) in the bias matrix of (3.4);

36

vi,i the vulnerability element (i, i) in the bias matrix of (3.4);

N number of buses.

Maximizing the modified observability vector O(cid:48) gives priority to the vulnerable elements

in the network. However, this vector cannot be used to test the observability of the network

since the observability is skewed by the vulnerability indices. Therefore, the constraints in

(3.13) and (3.14) are added to the second objective function in (3.9). The constraint in

(3.13) is used to give high priority to vulnerable buses and the constraint in (3.14) is used

to check for the desired observability.

O(cid:48)
O ≥ γ

O ≥ 1 “complete observability constraint”

O = H × B

where

γ priority index;

1 a vector of length N , with all elements equal to 1;

2 a vector of length N , with all elements equal to 2.

(3.13)

(3.14)

Zero Injection buses (ZIB) are buses with no load or generation units, and can be taken

advantage of to reduce the number of PMUs required for achieving observability. Since

the sum of all currents flowing to the ZIB is zero, Kirchhoff’s current law can be used to

determine the current flow through the adjacent branches. The ZIB effect can be summarized

as follows: A ZIB is considered observable if all adjacent buses of the ZIB are observable.

An unobservable bus that is adjacent to a ZIB is considered observable if the ZIB and all of

37

the other buses adjacent to the ZIB are observable.

3.4.2 Algorithm

The optimal placement problem in subsection 3.4.1 is a discontinuous bi-level problem.

Therefore, evolutionary algorithms are the appropriate tools for solving such a problem.

Several evolutionary algorithms can be used for solving the OPP problem; the proposed

algorithm uses opposition-based elitist genetic algorithm (O-BEGA) to solve the bi-level

OPP in subsection 3.4.1. The opposition element is added to the algorithm to enhance the

overall performance since opposition-based methods have proven their superiority in terms

of convergence-speed and results [36, 37]. The overall flowchart for proposed algorithm is

shown in Fig. 3.2.

The proposed algorithm solves the OPP problem in a parallel manner by initializing ran-

dom candidates where each candidate X has the length of the number of buses N . However,

this approach necessitates using of higher objective (cost objective); otherwise, the solution

would converge to installing the PMUs on all buses. The complexity rises from minimizing

the cost (higher objective) while maximizing the observability function.

The higher and the lower objectives share the same decision variables, meaning there

are no decision variables exclusive to one objective or the other. The proposed approach

exploits this advantage to evaluate both objectives simultaneously without using different

search spaces for each objective.

The proposed approach uses a sorting function to handle the simultaneous evaluation in

the same search space. The sorting function sorts all candidates according to the feasibility,

and fitness of both the higher and lower objectives, as shown in Fig. 3.3. As a result,

the algorithm is guided towards the optimal solution where the cost is minimized and the

38

Figure 3.2: Flowchart of the proposed algorithm

observability is maximized.

The algorithm starts by randomly generating an initial population XO, and then evaluat-

ing and ranking this population. The dynamic opposition is used when the random variable

OR is lower than opposition probability (PO = 0.4); where the dynamic opposition is ap-

plied on a third of the randomly selected variables. The parent population (XP ) is selected

using tournament selection, where two random candidates are matched against each other,

and the winner is selected as a parent candidate for generating the crossover population

(XC ). The probability of generating the crossover population PC is 0.8, and the XC is

generated using binary double point crossover. The mutation population XM is generated

with a probability PM of 0.1 using one-bit binary mutation, where one variable is randomly

chosen and changed. The crossover and mutation steps are performed after comparing the

39

YesYesNoNoYesNoYesNoIs MR < PMBinary mutation to get the mutation population (XM)  Binary crossover to get the crossover population (XC) Is the CR < PCUse tournament selection to get the parent population (XP) Generate dynamic opposition population Is OR < POHas termination criteria in (15) been reached? Compare XP, XC and XMThen select the best for next generationEvaluate candidates for both objectivesFinal SolutionInitialization random population XORank solutions using the sorting functionRank solutions using the sort functionYesprobabilities of PC and PM against the random variables CR and MR. After all the popu-

lations (Xp, XM , Xc) have been generated, they are evaluated and ranked using the sorting

function, then the termination criteria in (3.15) are checked. If the termination criteria have

not been reached, the best among the current population is sent for the next iteration. The

algorithm is summarized in Fig. 3.2

The algorithm is terminated if the conditions in (3.15) are met. In Υ(1), the mean of the

population observability is measured against the member with the maximum observability in

the current population. As the algorithm converges, the left-hand side of Υ(1) will converge

to zero. Since the member with the maximum observability is more likely to be feasible due

to the nature of the OPP problem, measuring the population mean observability against the

maximum observability serves as an effective measure of feasibility.

In Υ(2), the mean of the installation cost is measured against the member with the

maximum installation cost in the current population. Notice that the member with the

maximum installation cost is more likely to represent a feasible yet sub-optimal solution.

Nevertheless, as the population converges, this member will start drifting towards the mean

of the population—since all the population members drift toward the optimal solution upon

convergence. Therefore, the left-hand side of Υ(2) tends to zero as the algorithm progresses.

The terms α and β are constants. These constants are inversely proportional to the size

of the system. Therefore, the α and β values that work for a big system will work for a

smaller system, but the converse may not hold. The α and β are also proportional to the

size of the population; the smaller the size of the population the smaller these constants

40

must be.



Υ =

Υ(1) = |O(cid:48) − max(O(cid:48))| ≤ α

Υ(2) = |C − max(C)| ≤ β

(3.15)

Figure 3.3: Sorting function

3.4.2.1 Absence of feasible solutions

Normally the OPP problem has multiple feasible solutions. Therefore, once the algorithm is

terminated the top-ranked members will have the same objective function values. More likely

these top-ranked members will duplicate each other unless multiple optima exist with same

the objective function values. Therefore, the scenario where the member ranked number 1

is infeasible should not occur.

A scenario where the top-ranked members are infeasible can happen if there are no fea-

sible solutions for the problem. The nonexistence of feasible solutions is attributed to the

constraints of the problem. For the OPP problem, there are two types of constraints observ-

ability constraints and cost constraint. The observability constraints are well established,

and we know there are feasible solutions for these constraints. Therefore, infeasibility can

only happen due to the cost constraint defined in (3.8). In the following, we consider the

41

Variables     RankBus 1Bus 2Bus 3…Bus N011…16200017.410 10…05400014.42010…15500013.23        100… 30003.9N-2010… 24002.6N-1000…114001.4NFeasibleNot feasibleCost  possibility of infeasibility due to the cost constraint.

In the following example, we refer to the number of PMUs as the cost, and the objective

is to minimize this cost. We use the number of PMUs for simplicity as this holds for the

installation cost.

Suppose we solve for the IEEE 14-bus complete observability under normal conditions,

and we chose to minimize the number of PMUs. And suppose that cost constraint in (3.8)
is chosen to be C ≤ 3. We know for a fact that no such solution exists since the optimal

solution has a total number of PMUs equal to 4. In such a scenario, there are no feasible

solutions, and therefore the upper part (feasible solutions) of the Sorting function in Fig. 3.3.

would not exist. As this happens, the infeasible solutions can be divided into two groups.

Group A in which the solutions are infeasible due to observability constraints. Group B

in which the solutions are infeasible due to cost constraints. These two groups should be

handled differently by the sorting function as follows:

• All members of group B must have a higher ranking than any member of group A.

• The members of group B are ranked according to the least cost constraint violation.

• The members of group A are ranked in descending order according to the observability.

These rules apply only to the infeasible solutions! By the using rules above, the solution

ranked number 1 will be infeasible but satisfies the observability constraint, and this solution

will have 4 PMUs (the optimal solution for the IEEE 14-bus). Notice that this solution still

violates the cost constraint! Moreover, the algorithm will converge, and termination criteria

in (3.15) will still work.

42

3.4.3 Single-Stage Installation

In a single-stage installation process, the OPP is solved for achieving the desired observability

within one budget period. In other words, it is assumed that there are enough resources to

deploy the PMUs at the required locations in a relatively short period. It should be noted

that most of the OPP approaches are single-stage processes. The desired observability such

as normal conditions, single line outage or a single PMU outage.

In any of the desired observability conditions the priority index γ of (3.13) is used. This

index gamma assumes values in the interval [1, 2]. When γ is set to 1, all buses are treated

equally from the vulnerability perspective. When γ is greater than 1, higher priority is

given to the vulnerable buses. For the complete observability under normal conditions, the

placement of PMUs must satisfy the observability constraint in (3.14).

As for observability under a single line outage, it is necessary to have at least two mea-

surement redundancies for every bus in the network. Therefore, the observability constraint

in (3.14) is changed to (3.16).

H × B = Os ≥ 2

(3.16)

where Os is the observability vector for all buses in the system under single line outage

contingency, as shown in (3.17).



Os
i =

if a PMU is installed at bus i

2,

(cid:80)(aij × bi), otherwise

(3.17)

In a single PMU outage, every bus needs to have at least two independent measurements.

Either by two different PMUs, or if ZIB effect is considered, through KCL and a PMU.

43

Therefore, the observability constraint in (3.14) is changed to (3.18).

H × B = Op ≥ 2

(3.18)

where Op is the observability vector for all buses in the system under single PMU outage

contingency, as shown in (3.19).

(cid:88)

Op
i =

(aij × bi)

(3.19)

3.4.4 Multi-Stage Installation

In multi-stage installation OPP approaches, a more realistic scenario is assumed, where

existing resources may not be enough to achieve the desired observability in one budget

period. Therefore, the desired observability is achieved over several stages or years due to

limited resources or budget.

The majority of existing multi-stage methods assume the minimum number of PMUs per

stage. This assumption is unrealistic since PMU installation is restricted by the financial

burden, infrastructure of the substations, and technical benefits at each stage.

The proposed approach uses a financial capital Cbudget, as the limit for each stage instead

of using the number of PMUs as the limit. The buses are prioritized using the vulnerability

indices as in (3.12). The same bi-level model in section 3.4.1 is used, except the modified
observability function O(cid:48) is treated as the higher objective, and the cost objective is treated

as the lower objective. In the final stage, the objectives are switched back to the original

formulation with the observability as the lower objective and the installation cost as the

44

higher one. The observability constraint in (3.14) is changed to the following.

O ≤ 1

At each stage k, the pre-installed PMUs (PMUk−1) are initiated, and the pre-installation
cost Ck−1 is calculated using (3.6). The budget for each stage Ck

budget is set. Then, the
budget is modified to include the pre-installed PMUs cost Ck−1. The pre-installed
PMU locations are maintained for the initial random population XO and for the mutation

budget Ck

population XM . Since the pre-installed PMU locations are maintained for the initial and

parent populations, the crossover population XC maintains the pre-installed PMU locations

by default. The multi-stage approach is summarized in Procedure 3.

Ck
new,budget = Ck

budget + Ck−1

(3.20)

45

Procedure 3 Multi-Stage Installation
Initialization number of stages (n), Cbudget for each stage

for k = 1 : number of stages

set P M U K−1, Ck
calculate Ck−1, Ck

budget

new budget using (3.20)

if k = number of stages

Switch the higher and lower objectives

endif

Optimize the PMUs installation using the O-BEGA

in section 2.3.2

k = k + 1

endfor

3.5 Simulation and Results

This section illustrates the application of the proposed OPP approach. In the first part of

this section, the OPP approach is presented and compared with other approaches in the

literature. Then, the OPP solutions for the cost criteria are presented. The second part

of this section presents the vulnerability analysis and the OPP solution when incorporating

this analysis. The algorithm is implemented in MATLAB, and the computation times for

all three cases are under 2 seconds on a single processor.

46

Single Line Outage; Effect of ZIB Is Not Considered

IEEE 14-bus

IEEE 30-bus

IEEE RTS

IEEE 14-bus

IEEE 30-bus

IEEE RTS

IEEE 14-bus

IEEE 30-bus

IEEE RTS

4 PMUs

7 PMUs

7 PMUs

—-

—-

7 PMUs

7 PMUs

4 PMUs

4 PMUs

10 PMUs

10 PMUs

10 PMUs

7 PMUsa

(cid:80)O = 19 (cid:80)O = 16 (cid:80)O = 19
(cid:80)O = 50 (cid:80)O = 50 (cid:80)O = 47
(cid:80)O = 34 (cid:80)O = 34 (cid:80)O = 29
(cid:80)O = 23a (cid:80)O = 25
(cid:80)O = 62a (cid:80)O = 59, 54
(cid:80)O = 45a (cid:80)O = 45, 48
(cid:80)O = 36, 37
(cid:80)O = 80, 81
(cid:80)O = 65

15 PMUsa

11 PMUsa

16 PMUs

12 PMUs

21 PMUs

—-

—-

14 PMUs

—-

—-

—-

—-

—-

9 PMUs

7 PMUs

7 PMUs

4 PMUs

10 PMUs

(cid:80)O = 19
(cid:80)O = 52
(cid:80)O = 34
(cid:80)O = 25
(cid:80)O = 60a, 62
(cid:80)O = 57a, 49
(cid:80)O = 39
(cid:80)O = 85
(cid:80)O = 65

21 PMUs

14 PMUs

9 PMUs

16 PMUs, 15 PMUsa

12 PMUs, 11 PMUsa

Table 3.1: Comparison of The Proposed Solution With Other Solutions: Effect of ZIB is Not
Considered

System

Ref. [43]

Ref. [16]

Ref. [30]b

Proposed

Under Normal Operating Conditions; Effect of ZIB Is Not Considered

Single PMU Outage; Effect of ZIB Is Not Considered

a Ignores radial buses in contingency cases.
b Has multiple optimal solutions, maximum observability is shown.

3.5.1 Considering Installation Cost

In this section, the analysis is done without including the vulnerability analysis. The model in

section 2.2 is used, and no bias is given to any bus. The buses are divided into two categories:

prepared buses and unprepared buses. The prepared buses are assumed to have sufficient

infrastructure and require basic security and network upgrades, which are covered under the

basic PMU installation. The unprepared buses, on the other hand, require more upgrades

such as additional PTs and CTs for the added measurements. Where as the prepared buses

are assumed to cost 35% of the initial cost. The base cost per PMU is assumed to be $40,000,

and the cost per additional PT or CT is assumed to be $4,000. New communication lines

and software upgrades are assumed to cost $8,000, and PDC are assumed to cost $8,000 [25].

Three test systems are used for the analysis: the IEEE 14-bus, IEEE 30-bus and IEEE

47

Table 3.2: Comparison of The Proposed Solution With Other Solutions: Effect of ZIB is
Considered

System

Ref. [43]

Ref. [20]

Ref. [16]

Ref. [50]

Ref. [30] b

Proposed

Under Normal Operating Conditions; Effect of ZIB Is Considered

6 PMUs

6 PMUs

7 PMUs

7 PMUsa

—-
—-

—-

6 PMUs

6 PMUs

7 PMUs

7 PMUs

3 PMUs

3 PMUs

—-

—-

7 PMUs

7 PMUs

3 PMUs

3 PMUs

7 PMUs

7 PMUs

3 PMUs

3 PMUs

(cid:80)O = 16 (cid:80)O = 16 (cid:80)O = 16 (cid:80)O = 16 (cid:80)O = 16
(cid:80)O = 16
(cid:80)O = 47 (cid:80)O = 44, 47 (cid:80)O = 53 (cid:80)O = 55 (cid:80)O = 55, 56 (cid:80)O = 56
(cid:80)O = 38
(cid:80)O = 33, 36 (cid:80)O = 40
(cid:80)O = 34
(cid:80)O = 25, 25 (cid:80)O = 27
(cid:80)O = 27 (cid:80)O = 35a
(cid:80)O = 57 (cid:80)O = 55a (cid:80)O = 59 (cid:80)O = 58, 61 (cid:80)O = 66
(cid:80)O = 50, 52 (cid:80)O = 46
(cid:80)O = 35 (cid:80)O = 33, 35 (cid:80)O = 35
(cid:80)O = 61 (cid:80)O = 65, 69 (cid:80)O = 72
(cid:80)O = 61

(cid:80)O = 33 (cid:80)O = 35
(cid:80)O = 61 (cid:80)O = 63
(cid:80)O = 62

(cid:80)O = 59

(cid:80)O = 52a

9 PMUsa

15 PMUs

15 PMUs

13 PMUs

14 PMUs

7 PMUs

7 PMUs

14 PMUs

15 PMUs

10 PMUs

8 PMUs

12 PMUs

11 PMUs

7 PMUs

7 PMUs

10 PMUsa

—-

—-

—-

7 PMUs

7 PMUs

—-

—-

7 PMUs

14 PMUs

13 PMUs

—-

—-

—-

13 PMUs,
10 PMUsa

Single PMU Outage; Effect of ZIB Is Considered

Single Line Outage; Effect of ZIB Is Considered

—-
—-

13 PMUs

IEEE 14-bus

IEEE 30-bus

IEEE RTS

IEEE 14-bus

IEEE 30-bus

IEEE RTS

IEEE 14-bus

IEEE 30-bus

IEEE RTS

a Ignores radial buses in contingency cases.
b Has multiple optimal solutions, maximum observability is shown.

RTS systems. First, the bi-level optimization approach is tested using minimum number of

PMUs as the cost function. The results are compared with other approaches as seen in Table

3.1 and Table 3.2. The proposed algorithm always achieve better or comparable performance

with the existing OPP schemes.

There can be several optimal solutions that satisfy the observability conditions while

requiring same number of PMUs. However, these optimal solutions may have different

observability levels (redundancies)(cid:80)O. The higher the number of redundancies, the better

the solution is, as long as the cost of the solution is the same. For instance, the IEEE 14-bus

single PMU outage in Table I shows two optimal solutions. Both solutions require the same

number of PMUs (7 PMUs); however the proposed approach achieves a better observability

((cid:80)O = 39) with PMU locations {2, 4, 5, 6, 7, 8, 9, 10, 13}. The optimal solution of [30]
achieves an observability of ((cid:80)O = 37) with PMU locations {1, 2, 4, 6, 7, 8, 9, 10, 13}.

48

Table 3.3: Cost From the Proposed Approach: Effect of ZIB is Not Considered

Minimum number of PMUs

Proposed

PMU locations

Cost From PMU installation

PMU locations

Under Normal Operating Conditions; Effect of ZIB Is Not Considered (O ≥ 1)

Cost of PMU installation

IEEE 14-bus

IEEE RTS

2, 6, 7, 9

2, 3, 8, 10, 16, 21, 23

IEEE 30-bus

2, 4, 6, 9, 10, 12, 15, 19, 25, 27

IEEE 14-bus

1, 3, 6, 8, 9, 10, 13

IEEE RTS

IEEE 30-bus

1, 2, 3, 4, 7, 10, 11, 12

15, 16, 20, 21, 22

2, 3, 7, 8, 10, 11, 12, 13, 15

16, 19, 22, 24, 26, 27, 29

IEEE 14-bus

2, 4, 5, 6, 7, 8, 9, 11, 13

IEEE RTS

1, 2, 3, 7, 8, 9, 10, 11,
15, 16, 17, 20, 21, 23

IEEE 30-bus

2, 3, 4, 6, 7, 9, 10, 11, 12, 13, 15, 16,

18, 20, 22, 24, 25, 26, 27, 28, 29

$224,800

$239,600

2, 8, 10, 13

2, 3, 5, 7, 14, 16, 21, 23

$510,600

3, 5, 8, 11, 13, 15, 17, 19, 22, 26, 30

Single Line Outage; Effect of ZIB Is Not Considered (Os ≥ 2)

$216,400

$437,800

$488,200

1, 3, 6, 7, 10, 12, 14

3, 4, 5, 6, 7, 9, 13, 14,

15, 18, 20, 22

1, 3, 5, 6, 8, 10, 11, 13, 14, 15,

16, 19, 21, 24, 26, 29, 30

Single PMU Outage; Effect of ZIB Is Not Considered (Op ≥ 2)
1, 2, 3, 7, 8, 10, 11, 12, 13, 14

$329,800

$527,400

$789,200

1, 2, 3, 4, 5, 6, 7, 8, 13, 14

15, 16, 19, 21, 22, 23

1, 3, 5, 6, 7, 8, 9, 10, 11, 12, 13, 15,

16, 18, 20, 22, 24, 25, 26, 29, 30

Unprepared buses

7, 9

10, 11, 17, 24

9, 12, 25, 27, 28

7, 9

10, 11, 17, 24

$81,200

$168,000

$215,600

$184,800

$247,800

$345,800

9, 12, 25, 27, 28

$243,600

$333,200

$640,000

7, 9

10, 11, 17, 24

9, 12, 25, 27, 28

Table 3.4: Cost From the Proposed Approach: Effect of ZIB is Considered

Minimum number of PMUs

Proposed

PMU locations

Cost of PMU installation

PMU locations

Cost of PMU installation

Under Normal Operating Conditions; Effect of ZIB Is Considered (O ≥ 1)

IEEE 14-bus

2, 6, 9

IEEE RTS

2, 8, 10, 16, 21, 23

IEEE 30-bus

2, 4, 10, 12, 19, 24, 27

IEEE 14-bus

1, 3, 6, 8, 9, 11, 13

IEEE RTS

1, 2, 7, 9, 10, 16, 20, 21

IEEE 30-bus

1, 2, 4, 7, 10, 11, 12, 13,

15, 16, 19, 23, 26, 30

$140,800

$234,000

2, 8, 10, 13

2, 5, 7, 16, 22, 23

$314,600

3, 5, 10, 13, 15, 17, 19, 29

Single Line Outage; Effect of ZIB Is Considered (Os ≥ 2)

$216,400

$260,600

$330,600

1, 3, 6, 8, 10, 12, 14

2, 5, 6, 7, 9, 16, 20, 21

1, 3, 5, 10, 11, 13, 14,
16, 18, 19, 23, 26, 30

Single PMU Outage; Effect of ZIB Is Considered (Op ≥ 2)

IEEE 14-bus

2, 4, 5, 6, 9, 11, 13

IEEE RTS

IEEE 30-bus

1, 2, 3, 7, 8, 9, 10,

18, 19, 20, 21

1, 2, 4, 6, 7, 10, 12, 13
15, 16, 18, 19, 24, 27

$227,600

$322,200

$460,200

1, 2, 3, 8, 10, 11, 12, 13, 14

1, 2, 4, 5, 6, 7, 8, 13,

15, 16, 19, 21

1, 3, 5, 7, 10, 11, 12, 13,
15, 16, 18, 19, 23, 26, 29

$81,200

$123,200

$163,800

$138,600

$170,800

$256,200

$179,200

$250,600

$386,600

Unprepared buses

7, 9

10, 11, 17, 24

9, 12, 25, 27, 28

7, 9

10, 11, 17, 24

9, 12, 25, 27, 28

7, 9

10, 11, 17, 24

9, 12, 25, 27, 28

The proposed approach always achieve higher redundancy level because of the bi-level

formulation, where the observability is treated as a lower objective and is being maximized,

while the being subjected to the higher objective (minimize the cost). Other OPP approaches

treat the observability as a constraint and try to minimize the cost (number of PMUs) as

follows:

min C = number of PMUs

s.t. O ≥ 1

(3.21)

Using the formulation in (3.21) should lead to an optimal solution. However, in the

presence of multiple optimal solutions, this formulation may not lead to the solution with

49

the higher number of redundancies as is evident from the results in Table 3.1 and Table 3.2.

After verifying the performance of the bi-level optimization, the cost model in section

(2.2) is used for the OPP. The PMU placement is performed for two different cases. In the

first case, the PMUs are placed under normal conditions, under a single line outage and

single PMU outage. Also, ZIBs are treated as normal buses. The result of the placement is

shown in Table 3.3. The ZIB effect is considered in the second case under normal operation,

single line outage and single PMU outage as shown in Table 3.4.

The results show that the using minimum number of PMUs does not necessitate a lower

installation cost, especially when each substation has its own cost factors. For instance,

consider the IEEE RTS solution for normal conditions in Table III. When the minimum

number of PMUs is used as the objective, the installation cost is about $239,600 with a total

of a 7 installed PMUs. On the other hand, when the installation cost is used as the objective

more PMUs are installed (8 PMUs) at a lower cost ($81,200). The same phenomenon can

be observed for the IEEE 14-bus case for normal conditions in Table IV.

The impact of different approaches (minimum number of PMUs approach and the pro-

posed approach) on the level of redundancy of the solution can be observed from the cost.

Consider the IEEE 14-bus under normal conditions in Table III; both solutions have the

same number of PMUs (4 PMUs). However, the cost obtained by the proposed solution

of {2, 8, 10, 13} with an observability of (cid:80)O = 14 is much lesser compared to the other
solution of {2, 6, 7, 9} which uses more channels and has a higher observability ((cid:80)O = 19).

3.5.2 Vulnerability Analysis

In this section, the vulnerability indices obtained from the vulnerability analysis are used

to give bias to the vulnerable buses. Monte Carlo simulation along with AC-OPF is used

50

to obtain the conditional reliability for the IEEE RTS system [51]. Since the system failure

is considered to be a load curtailment event, the loss of load probability (LOLP) is used as

the indicator of the criticality of components. The results in Table 3.5 show the conditional

LOLP of the transmission lines. The results also show the vulnerability indices of these

components.

Figure 3.4: IEEE RTS system

The results from the conditional probability show that transmission lines T10 and T11

are highly critical to the system operation. Transmission line number T10 connects buses
{6} and {10}, and transmission line number T11 connects buses {7} and {8} as seen in

51

Bus 17T2Bus 18Bus 21Bus 22Bus 19Bus 16Bus 14Bus 13Bus 12Bus 11Bus 24Bus 15Bus 3Bus 9Bus 10Bus 6Bus 8Bus 7Bus 2Bus 1T3T1T4T5T11T13T12T10T9T6T8T14T7T17T15T16T18T20T21T22T36T37T27T19T24T26T23T25T38T31T33T32T30T28T29T34T35Bus 5Bus 4Bus 23Bus 20 Fig. 3.4. To account for the reliability of the transmission lines: the probability of failure is

multiplied by the conditional reliability indices. Then the CDF of the normal distribution is

used to obtain the network vulnerability indices (V.I.) as in (3.3). The results are shown in

Table 3.5.

Table 3.5: Conditional Reliability Indices –Vulnerability Indices (V.I.)

Transmission Conditional
Line
Number

LOLP,

Γk

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

0.1034

0.1061

0.1072

0.1058

0.1074

0.1069

0.1187

0.1046

0.1088

1.0000

1.0000

0.1082

0.1105

0.1053

0.1075

0.1074

0.1126

0.1057

0.1054

V.I.

0.40822

0.40822

0.4006

0.41097

0.42925

0.40984

0.65915

0.40461

0.40344

0.99987

0.82297

0.42223

0.42403

0.62489

0.63061

0.63035

0.64372

0.42027

0.41798

Transmission Conditional
Line
Number

LOLP,

Γk

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

0.1105

0.1141

0.1040

0.1125

0.1067

0.1096

0.1086

0.1156

0.1093

0.1068

0.1050

0.1134

0.1047

0.1038

0.1048

0.1057

0.1111

0.1052

0.1065

V.I.

0.42403

0.45381

0.43729

0.42119

0.40651

0.42547

0.42466

0.43029

0.41242

0.40864

0.40338

0.45758

0.40929

0.40868

0.41548

0.41614

0.41149

0.40759

0.43133

The vulnerability indices show that transmission lines T10 and T11 are still highly critical

to the system operation. Therefore, the buses connected to these transmission lines are

critical. The vulnerability indices are integrated into the observability function as described

in subsection 3.4.1. By integrating the vulnerability indices into the observability, the OPP

is solved to generate a more resilient solution while minimizing the total installation cost.

52

3.5.2.1 Single-Stage Installation

For the single-stage installation, the PMU placement is considered under all observability

conditions. The results for the PMU placement are shown in Table 3.6 and Table 3.7. The

results show that higher priority buses are monitored directly by the PMUs. However, the

OPP cost is higher than normal PMU installation. This is due to the constraint on the

priority index γ, which forces the OPP solution to a more secure and resilient solution,

where the fault-tolerance of the system is enhanced. For instance, for normal conditions the

critical components are directly monitored when the priority index is used by placing PMUs
on buses {6, 9}. A better option, from fault-tolerance perspective, is to use buses {8, 10},

however this option has a much higher cost.

Table 3.6: Cost Comparison of the OPP; Effect of ZIB is Not Considered

γ = 1

γ = 1.5

Unprepared

buses

Normal Operating Conditions; Effect of ZIB Is Not Considered

2, 3, 5, 7, 14, 16, 21, 23

3, 5, 6, 7, 9, 16, 21, 23

IEEE RTS

10, 11, 17, 24

$168,000

$170,800

IEEE RTS

IEEE RTS

Single Line Outage; Effect of ZIB Is Not Considered

3, 4, 5, 6, 7, 9, 13, 14,

2, 3, 5, 7, 9, 10, 13, 14,

15, 18, 20, 22

15, 18, 20, 22

10, 11, 17, 24

$247,800

$337,600

Single PMU Outage; Effect of ZIB Is Not Considered

1, 2, 3, 5, 6, 7, 9, 13, 14, 15,

1, 2, 3, 5, 6, 7, 8, 9, 12, 14,

16, 18, 19, 21, 22, 23

15, 16, 19, 21, 22, 23

10, 11, 17, 24

$329,000

$338,800

53

Table 3.7: Cost Comparison The OPP; Effect of ZIB is Considered

γ = 1

γ = 1.5

Unprepared

buses

Normal Operating Conditions; Effect of ZIB Is Not Considered

1, 2, 8, 14, 20, 21

5, 6, 7, 9, 20, 21, 22, 23

IEEE RTS

10, 11, 17, 24

$123,200

$162,400

Single Line Outage; Effect of ZIB Is Considered

2, 5, 6, 7, 9,

3, 4, 5, 6, 7, 9

IEEE RTS

16, 20, 21

$170,800

16, 20, 21

11, 12, 17, 24

$190,400

Single PMU Outage; Effect of ZIB Is Considered

1, 2, 5, 6, 7, 9,

1, 2, 5, 6, 7, 8, 9,

IEEE RTS

15, 16, 19, 21, 23

14, 16, 20, 21, 22, 23, 24

11, 12, 17, 24

$228,200

$339,400

3.5.2.2 Multi-Stage Installation

The multi-stage PMU installation is performed as a three-stage process; each stage has its

own budget. All the stages have a budget limit of $70,000, where each stage is treated

independently budget-wise. The results in Table 3.8 show the optimal PMU installation at

each stage. The third stage presents the complete observability solution for both cases. The

results show a comparison between treating all buses equally and using vulnerability indices.

The results show that prioritizing vulnerable buses, produce a better observability and a

more resilient OPP solution.

54

Table 3.8: Multi-Stage OPP; Effect of ZIB is Not Considered

All buses are treated

Vulnerability indices Unprepared

equally

16, 21, 23

are used

9, 21, 23

O = 13,O(cid:48) = 24.35

O = 13,O(cid:48) = 27.39

buses

10, 11,

$64,400

$65,800

17, 24

1, 3, 8, 16, 21, 23
O = 25, O(cid:48) = 41.29

2, 3, 8, 9, 21, 23
O = 19,O(cid:48) = 44.33

10, 11,

$63,000

$63,000

17, 24

1, 2, 3, 8, 14, 16, 21, 23
O = 32, O(cid:48) = 50.39

2, 3, 5, 8, 9, 16, 21, 23
O = 34,O(cid:48) = 55.97

10, 11,

$40,600

$42,000

17, 24

Stage one

C1

budget = $7e4

Stage two

C2

budget = $7e4

Stage Three

C3

budget = $7e4

3.6 Conclusion

In this chapter, a new approach for application-based OPP is presented. Traditionally, for

application-based OPP approaches, the objective is to minimize the number of PMUs. This

chapter, however, considers the installation cost of PMUs, the observability, and the vulner-

ability of the system. The fault-tolerance aspect is enhanced through the prioritization of

the critical buses for the installation of PMUs, where the criticality of the buses is quantified

and integrated into the observability function using vulnerability analysis.

The proposed approach can be used as a single-stage or a multi-stage installation. The

bi-level formulation enables the optimal solution with the maximum observability to be se-

lected in the case of multiple optima. Moreover, the integration of the vulnerability indices

into the observability function allows for a solution that can attain a trade-off between the

technical benefits and the cost-effectiveness of the OPP. This process is controlled by a prior-

ity index, which can be determined by the utility at the planning stage. This incorporation

of priority indices into the observability function can be used for other application-based

OPP approaches or for multiple criteria application-based OPP approaches.

55

Chapter 4

Multi-stage Optimal PMU Placement

to Benefit System Voltage Stability

4.1

Introduction

The OPP literature can be classified into two categories: observability-based OPP and

application-based OPP. The observability-based OPP aims to achieve complete observability

for the power grid while minimizing the total cost of installing PMUs. The application-based

OPP, on the other hand, seeks to achieve other technical benefits out of the PMUs such as

bad data detection, while minimizing the cost of PMU installation.

Transient stability has been considered for PMU deployment by [13], where generators

are ranked by determining the individual machine trajectory and the proximity of this tra-

jectory to the transient energy function stability margin. In this chapter, a multi-stage OPP

approach is proposed, where voltage stability is prioritized. The proposed criterion can be

used in addition to other criteria of PMU deployment. Voltage stability analysis is necessary

to mitigate voltage collapse, which occurs when the voltage of a bus falls below acceptable

margins, thus causing power system instability. This may happen due to several reasons in-

cluding disturbance, sudden increase of load, or the inability of the system to supply sufficient

The content of this chapter has been reproduced with permission from S. Almasabi, N. Nguyen, F. T.
Alharbi and J. Mitra, “Multi-stage Optimal PMU Placement to Benefit System Voltage Stability,” 2018 North
American Power Symposium (NAPS), Fargo, ND, USA, 2018, pp. 1-6. doi: 10.1109/NAPS.2018.8600666

56

reactive power to the load. This instability often lies with critical buses and transmission

lines with heavy loads. Critical buses are identified using the proposed voltage stability

criterion, and a cost model is used for each stage. There are several voltage stability indices

that can be used to create the voltage stability criterion such as fast voltage stability index,

line stability index and line stability factor [52–54]. The line stability factor is chosen for its

superior performance compared to the other indices in dynamic operation [55]. The proposed

voltage stability creation can be integrated with other OPP criteria for application-based

OPP.

The rest of the chapter is organized as follows: section 4.2, describes the voltage stability

criterion, section 4.3, describes the modeling approach for the OPP. The analysis for the

case studies and conclusion are presented in sections 4.4 and 4.5, respectively.

4.2 Voltage Stability Criterion

In this section, the voltage stability criterion is developed using the line stability factor

(LQP) in [54]. As mentioned earlier, the line stability factor is chosen for its accuracy in

predicting the voltage collapse compared to other indices [52]. The LQP is derived for the

real and reactive power of the system. Consider the sample system in Fig. 4.1, the apparent

power at bus j is a function of the real and reactive power (S = P + jQ). The apparent

power can also be expressed as a function of the current and voltage, Sj = I∗

j Vj. The current

I can be expressed as follows:

Vi − Vj

Z

Vi − Vj
R + jX

=

I =

(4.1)

57

Figure 4.1: Sample system

By decomposing the current in (4.1) into its real and imaginary parts as

Vi cos δi − Vj cos δj + j(Vi sin δi − Vj sin δj)

R + jX

I =

,

(4.2)

the real power Pj and reactive power Qj can be expressed as follows:

Pj =(cid:2)(Vi cos δi − Vj cos δj)

− (Visinδi − Vj sin δj)

Qj =(cid:2)(Vi cos δi − Vj cos δj)

+ (Vi sin δi − Vj sin δj)

R

R2 + X2

R

R2 + X2
X

(cid:3)Vj

R2 + X2

X

R2 + X2

(cid:3)Vj.

(4.3)

(4.4)

By making δ = δi − δj, and assuming the network is predominantly inductive (R << X),

the P and Q are expressed as follows:

(cid:21)
(cid:20)Vi sin δ
Qj =(cid:2)(−Vi cos δ + Vj)

Pj =

X

Vj

1
X

58

(cid:3)Vj.

(4.5)

(4.6)

SibusibusjSjR+jXINow by using the trigonometric property sin2δ + cos2δ = 1 the relationships in (4.5) and

(4.6) can be expressed as

(cid:20)XPj

(cid:21)2

ViVj

+

(cid:20)XQj − V 2

j

(cid:21)2

ViVj

= 1.

j − (2XQj − V 2
V 4

i )V 2

j + (X2Q2

j ) + P 2

j X2 = 0

(4.7)

(4.8)

Therefore, V 2 is expressed as the quadratic relationship in (4.8). By assuming V 2 has a real
solution, the discriminant ((2XQj − V 2
P 2
i X
V 2
i

to zero. As a result, 4(cid:0) X

i X2) needs to be greater or equal

(cid:1)(cid:0)Qj +

V 2
i

j + P 2

(cid:1) ≤ 1, which is the LQP in (4.9).
i ) − 4(X2Q2
(cid:18) X

(cid:19)(cid:18)

(cid:19)

V 2
i

P 2
i X
V 2
i

LQPij = 4

Qj +

(4.9)

where i and j are the subscripts for the buses. The higher the LQP, the closer the line is to

voltage collapse since the discriminant gets closer to being imaginary. Since these LQP line

indices, indicates the criticality of the lines in the system, these indices can be utilized to

rank critical buses for the OPP. By summing up adjacent LQP indices of each bus, a critical

bus index (Cr) is expressed as follows:

Cri =

1(cid:80)j=N

j=1 Ai,j

N(cid:88)

j=1

LQPij

(4.10)

where N is the number of uses in the system, Cri represent bus i critical index. A is an N×N

connectivity matrix of the system in (4.11). These indices in (4.10) are used to determine

the critical buses for the OPP. These critical buses are incorporated into the observability

59

function, where these buses are given higher priority over other buses in the system.



Aij =

1, if i = j

1, if there is a branch connecting bus i and bus j

(4.11)

0, otherwise

4.3 Modeling Approach

This section describes the observability criterion, incorporation of critical buses, and the

overall model. The optimization approach is also presented.

4.3.1 Observability

The observability of the system can be measured using the connectivity matrix A and the

vector of PMU locations as in (4.12). The system is considered observable under normal

conditions if the constraint in (4.13) is met, which ensures that all the buses of the system

are measured by the PMU, either directly or indirectly.

O = H × P

O ≥ I

(4.12)

(4.13)

where P is a vector of length equal to the number of buses (N ), I is a vector of ones of
length N , and A is the N × N connectivity matrix defined in (4.11). The entries for the

PMU locations vector P is defined in (4.14).

60



Pi =

1, if a PMU is installed at bus i

0, if no PMU is at bus i

(4.14)

In the multi-stage approach, the complete observability constraint cannot be met at the

initial stages. Therefore, this constraint is changed to the multi-stage condition in (4.15)

O ≤ I

(4.15)

The PMUs can measure the buses of the network indirectly by considering the zero-injection

buses (ZIB) effect. When the ZIB effect is considered, the overall cost for observability is

reduced. This ZIB effect takes advantage of the PMU current measurements, to apply KCL

thereby reducing the number of PMUs required for complete observability. This effect is

summarized as follows:

• If all buses adjacent to a ZIB are observable, the ZIB is considered observable.

• If all buses adjacent to a ZIB are observable—except for one—and the ZIB is observable;

then the unobservable bus is also considered observable.

4.3.2 Critical Buses

The voltage stability criterion in section 4.2 is used to rank the buses and determine the

critical buses of the system.

In the proposed approach, the critical buses criterion is

embedded into the observability function (4.12), by integrating the priority vector R =

[Cr1Cr2 . . . CrN ]T . The critical buses criterion is embedded into the observability to prior-

itize the critical buses while enhancing the observability of the system. The higher priority

61

buses are determined based on the voltage stability criterion, then arranged in descending

order in a vector L. Then the priority bias is assigned using the following algorithm.

Procedure 4 Priority Bus Vector L
Initialize priority buses (L)

N = number of buses

M = maximum number of branches in (A)

kk = length of L

R = [Cr1 . . . CrN ]T

Sort R in descending vector

Normalize and filter R

for j = 1 : kk

i = R(j)

if i (cid:54)= 0 then
Li = M ∗ (kk − j + 1)

else

Li = 0

endif

endfor

The priority vector L is incorporated into the connectivity matrix A, to set a bias for the
critical buses, as in (4.16). However, by skewing the connectivity matrix A(cid:48), the observability
tests in (4.13) and (4.15) are also skewed. Therefore, the skewed observability O(cid:48) is used
of the optimization of PMU locations, and the original observability O is used to check the

62

observability condition.



A(cid:48)
ij =

1 + Li, if i = j

1,

0,

if there is a branch connecting bus i and bus j

(4.16)

otherwise

where

O(cid:48)

Li

N

= A(cid:48) × P

= bus (i) priority index

= number of buses

4.3.3 Problem Statement

The proposed multi-stage OPP uses a PMU installation cost model to optimize the PMU

locations. The cost model is based on the report published by the U.S. Department of

Energy [24], which indicates that a PMU cost about $40, 000 on average. The cost model

also considers the number of channels used by the PMUs, by considering the cost of additional

measurements, $4, 000 per channel (4.19).

In the multi-stage OPP, the observability is maximized at each stage. However, maxi-

mizing the observability is limited by the budget of each stage. Therefore, the problem is

set up as a bi-level optimization problem as follows:

n(cid:88)

max O(cid:48) =

O(cid:48)

i

(4.17)

Subject to

i=1

63

O ≤ I “multi-stage observability condition”

N(cid:88)

i=1

min PCost =

(a × Pi + bi × Pi)

(4.18)

(4.19)

Subject to

C ≤ Cbudget

where

Pi = the PMU at bus i, either 0 and 1

a

= the cost of PMU installation

bi = the cost of additional channels at bus i

4.3.4 Optimization

Several evolutionary optimization approaches can be applied to the bi-level optimization

problem in 4.3.3. The opposition-based elitist binary genetic algorithm (O-BEBGA) is cho-

sen for its performance since it uses opposition elements to enhance the convergence speed

and the solution quality [36–38, 56].

Traditionally, bi-level problem objectives are solved separately, and the variables are

exchanged between the objective functions to sort out the bi-level objectives. In this problem,

however, both objective functions share the same search space; therefore, both functions can

be evaluated at the same time, and the need for variable exchange is eliminated. This

simultaneous evaluation can be done by using a sorting function which sorts the variables in

terms of feasibility, higher objective maximization, and lower objective minimization. The

64

Figure 4.2: Sorting function

sorting function is shown in Fig. 4.2.

The overall algorithm for the proposed approach is shown in Fig. 7. The algorithm

solves for the complete observability, then uses the budget of each stage to determine the

optimal solution for the stage, where the observability is maximized taking advantage of

the available budget for the current stage. The algorithm assumes that each k stage is

independent; therefore at each stage, the budget Ck

budget is utilized to the maximum as long

as the observability objective justifies it. The SF in the flowchart is used to determine the

current mode of the algorithm. When SF is set to zero, the algorithm solves for complete

observability; and when SF is set to one, the algorithm solve for the optimal solution within

the current stage k. The crossover probability (Pc) is set to 0.7 and the mutation probability

(Pm) is set to 0.1. The Orn, Crn, and Mrn are random variables.

4.4 Case Studies

This section illustrates the application of the proposed OPP on the IEEE 14-bus and IEEE

30-bus test systems. First, the voltage creation results are presented; then the critical buses

are incorporated into the multi-stage OPP framework.

65

Variables     RankBus 1Bus 2Bus 3…Bus N011…16200017.410 10…05400014.42010…15500013.23        100… 30003.9N-2010… 24002.6N-1000…114001.4NFeasibleNot feasible Figure 4.3: Flow chart of the proposed algorithm

66

YesYesNoNoYesNoYesNoBinary mutation to get (Xm)  Xp = Parent Selection (Tournament Selection) Generate dynamic opposition population Has termination criteria been reached? Compare Xp, Xc and XmThen select the best for next generationEvaluate candidates for both objectives (3,4)Obtain optimal solution (Xs) for the complete observability Set SF=1Initialize random candidates XoRank solutions using the sort functionRank solutions using the sort functionOptimize the communication path using multi-source Dijkstra algorithmInitializenumber of stages k, budget for each stage           , system data (H) , priority buses (L)    kbudgetCSet the priority vector (R),SF=0;     Check for pre-installed    Calculate the pre-cost            Set the new budget         k-1CFinal solution for stage kk=k+1SF=1Complete observability?Final solutionNoYesIs SF=1?NoYesReduce search space Binary crossover to get (Xc)  kk-1new,budgetbudgetC=C+Ck-1PMUsrnmIs M< PrnoIs O< Pr ncIs C< P4.4.1 Voltage Stability Analysis

The voltage stability criterion in section 4.2 is used to determine the critical buses of the

system. First, the LQP is evaluated for the system, then the results are used to determine

the critical bus index vector in the network as in (4.10). The peak loads for the systems are

increased uniformly, until the system is on the verge of voltage collapse. By stressing out

the system, the vulnerable branches can be identified using the LQP indices.

The results of the LQP for the IEEE 14-bus are shown in Table 4.1. The vulnerable

branches for the IEEE 14-bus are the branches connecting buses 1-5 and 2-3, as shown in

Table 4.1. The critical bus indices Cr for the IEEE 14-bus are shown in

Table 4.1: LQP Results for The IEEE 14-bus system

Line

Line

From Bus To Bus

LQP

From Bus To Bus

LQP

1

1

2

2

2

4

5

4

4

5

2

5

3

4

5

3

4

7

9

6

0.649446

0.978306

0.884747

0.26333

0.16056

0.440793

0.102306

0.240604

0.225796

0.723623

6

6

6

7

7

9

9

11

12

13

11

12

13

8

9

10

14

10

13

14

0.14243

0.072266

0.113537

0.357901

0.128808

0.003097

0.030256

0.112265

0.025232

0.155987

Table II. It can be seen that buses 1 and 3 are buses with high criticality form a voltage

stability perspective. The same analysis is applied for the IEEE 30-bus. The results for the

LQP and the critical bus indices are shown in Tables 4.6 and 4.4 respectively. The results

show that buses 1 and 2 are the critical buses for this system.

67

Table 4.2: The Critical Bus Indices for The IEEE 14-bus system

Bus

Cri

Bus

Cri

1

2

3

4

5

6

7

0.542584

0.391616

0.441847

0.212138

0.392959

0.210371

0.181828

8

9

10

11

12

13

14

0.17895

0.077591

0.038454

0.084899

0.032499

0.073689

0.062081

68

Table 4.3: LQP Results for The IEEE 30-bus system

Line

Line

From Bus To Bus

LQP

From bus To bus

LQP

1

1

2

3

2

2

4

7

6

6

6

6

11

9

4

13

12

12

12

14

16

2

3

4

4

5

6

6

5

7

8

9

10

9

10

12

12

14

15

16

15

17

0.500684

0.488809

0.136807

0.084614

0.747487

0.259297

0.139397

0.203464

0.078182

0.151653

0.097725

0.225193

0.258252

0.133004

0.273633

0.215200

0.057810

0.086120

0.069457

0.015862

0.038807

15

18

20

10

10

10

10

22

15

22

23

25

25

27

28

27

27

29

28

6

18

19

19

20

17

21

22

21

23

24

24

24

26

25

27

29

30

30

8

28

0.040394

0.012413

0.014889

0.061605

0.024886

0.066151

0.06171

0.002366

0.065136

0.056142

0.049225

0.015551

0.086484

0.039594

0.306153

0.075241

0.112366

0.03269

0.130139

0.003284

4.4.2 Multi-stage OPP

In the multi-stage OPP, the PMUs are installed in the system over a span of number of

years. Each stage has its own independent budget, and the OPP utilizes the current budget

69

Table 4.4: The Critical Bus Indices for The IEEE 30-bus system

Bus

Cri

Bus

Cri

Bus

Cri

1
2
3
4
5
6
7
8
9
10
11

0.329831
0.328855
0.191141
0.12689
0.316984
0.119341
0.093882
0.093931
0.122245
0.081793
0.129126

11
12
13
14
15
16
17
18
19
20
26

0.129126
0.117036
0.1076
0.024557
0.041503
0.036088
0.021231
0.017602
0.009101
0.025498
0.043242

21
22
23
24
25
26
27
28
29
30
30

0.022839
0.030055
0.03812
0.03023
0.035407
0.043242
0.106671
0.109894
0.035977
0.048352
0.048352

within the framework presented in subsection 4.3.3. The critical indices obtained from the

voltage stability analysis are used to prioritize the PMUs installation.

The OPP solution for the IEEE 14-bus is shown in Table 4.5. The complete observability

is achieved in three stages. The budget for each stage is set to $100,000; each stage has its

own independent budget. The critical buses for the IEEE 14-bus are presented in the priority

vector L, and the PMUs are installed on these buses first, as long as its feasible.

The OPP solution for IEEE 30-bus is shown in Table 4.6. The budget for each stage is

set to $200,000 and the critical buses from the voltage stability analysis are L = [1, 2]. The

PMUs are installed on the higher priority buses first, since these buses provide the maximum
skewed observability O(cid:48).

70

Table 4.5: Multi-Stage PMU Installation for the IEEE 14-Bus System

ZIB effect is NOT considered

ZIB effect is considered

Stage One

P 1

Cost

Stage Two

P 2

Cost

L = [1, 3]

1, 3

$96,000

1, 3, 8, 10

$92,000

Stage Three

1, 3, 8, 10, 13

L = [1, 3]

1, 3

$96,000

1, 3, 9

$56,000

1, 3, 6, 9
$56,000

P 3

Cost

$52,000

’Complete Observability’

’Complete Observability’

Table 4.6: Multi-Stage PMU Installation for the IEEE 30-Bus System

ZIB effect is NOT considered

ZIB effect is considered

Stage One

P 1

Cost

L = [1, 2]

1, 2, 10

$168,000

L = [1, 2]

1, 2

$104,000

Stage Two

1, 2, 6, 10, 11, 12

1, 2, 10, 24

P 2

Cost

$172,000

$116,000

Stage Three

1, 2, 6, 10, 11, 12,

1, 2, 10, 12,

P 3

Cost

18, 23, 26, 29

$188,000

19, 24, 30

$156,000

’Complete Observability’

’Complete Observability’

4.5 Conclusion

This chapter presented a multi-stage OPP approach, where the voltage stability criterion is

used to determine the critical buses in the system. The proposed voltage stability criterion is

based on the line stability index. By using this criterion, the vulnerable branches and critical

buses can be identified. These buses should be given higher measurement redundancy, since

they represent the vulnerable parts of the system; thereby ensuring voltage collapse and

system contingency are identified promptly to start remedial actions. The proposed creation

71

can be integrated with other criteria for application-based OPP.

72

Chapter 5

PMU Placement Against False Data

Injection Attacks

5.1

Introduction

The false data injection attack (FDIA) is carefully designed to bypass the bad data detection

(BDD) test, which is residually based, of the state estimators [57–60]. The BDD is used

initially to detect anomalies in the measured data due to noise or malfunctioning devices. The

most popular defense against FDIAs is to secure a minimum subset of network measurements

to render FDIAs infeasible [58, 61, 62]. Switching the network topology as a new defense

mechanism has been proposed by [63]. The authors of [64] proposed two detection algorithms

for FDIAs, one based on the distribution of the measurements and the other is based on the

probability over time. The authors of [65] used online data from the PMUs and load forecast

to detect FDIAs but still assumed the PMU data to be uncompromisable. The references

mentioned above used DC estimators and remote terminal units (RTU) measurements to

evaluate their models and detection approaches.

The BDD of AC-estimators, on the other hand, is harder to bypass without detection [58].

The authors of [66] investigated the FDIAs on AC-estimators in which the authors concluded

that the adversaries need to collect online data in addition to topology knowledge for a

successful attack. In [67], the authors used graph theory to design an attack algorithm and

73

determine vulnerable measurements. These proposed approaches considered RTUs as the

measurement devices for their models. Zhao et al. [68] used PMUs as a secure platform to

develop a detection method for FDIA on nonlinear estimators.

The vulnerability of PMU to FDIAs has been introduced in [69, 70]. The ability to mask

line outages via FDIA was investigated in [69]. The ability to spoof the global positioning

system (GPS) signal of the PMUs and the consequential impact has been examined in [70]. In

[71], a protection scheme against FDIAs was proposed, in which a small number of PMUs are

deployed as a secure platform to render FDIAs infeasible, under a DC-estimator paradigm.

Most studies have used DC-estimator, which is a simplified linear estimator [57, 58, 60–

64]. In the studies which consider nonlinear estimators, the PMU measurements and their

effect were not considered. The studies that consider FDIAs under PMU paradigm did not

consider the state estimation part of the process; they either assumed a DC attack scenario

or proposed deploying PMUs as secure measurement units to guard against FDIAs [68–71].

As mentioned earlier, most FDIA literature considers PMUs to be a secure and uncom-

promisable platform. In literature that considers computerizing PMUs, no defense mecha-

nism was presented. This chapter describes a formulation for the optimal placement of PMUs

(OPP) to guard against FDIAs. This formulation, assumes that PMUs can be compromised,

and the deployed PMUs serves as monitors and authenticators for each other.

5.2 Attack Model

This section briefly reviews the attack model for different state estimators and describes the

attack model for PMU based state estimation.

74

5.2.1 RTU-based attack models

The FDIA rely on manipulating the measurements by an attack vector (a), where the attack

vector is constructed by taking advantage of the system topology. Most of the FDIA litera-

ture use a DC-estimator model, however, the same strategy can be used for AC-estimators.

Consider the following model

z = hx + v;

(5.1)

where z and v are vectors representing the measurement and noise respectively with a size

of m by one. In DC-estimators, z consists of the real power flows and injections. In AC-

estimators, the z consists of both real and reactive power measurements. The x vector

consists of the bus angles for DC-estimators, and bus voltages and angles for AC-estimators.

h is a constant Jacobian matrix with a size of m by n; where m is the number of measurements

and n is the number of states.

The attack vector a is constructed to avoid detection by either the largest normalized

residual (LNR) or the chi-square test as follows:

zcomp. = ztru + a;

(cid:107)z − hx(cid:107) ≤ τ1;

J(ˆx) ≤ τ2;

(5.2)

(5.3)

(5.4)

where τ1 is a tolerance constant and τ2 is the confidence level of the chi-square test. As seen

in (5.5), by using such vector the regular BDD test can no longer detect the FDIA [61–63].

75

However, the adversaries need to know the topology of the grid, to use this vector.

(cid:107)zcomp. − hxcomp.(cid:107) = (cid:107)ztru + a − h(x + c)(cid:107)
= (cid:107)ztru + ch − hx − hc)(cid:107) = (cid:107)ztru − hx)(cid:107) ≤ τ1

(5.5)

5.2.2 PMU-based attack model

The AC- and DC-estimators discussed in 5.2.1 rely on RTU measurements; therefore, the

adversaries need to compromise several RTUs to launch a successful attack. However, the

adversaries need only to compromise one PMU for a successful FDIA.

Unlike the RTU, a single PMU can have as many channels as the number of adjacent

buses. As a result, the PMU can measure the current flow to all adjacent buses in phasor

form. This feature makes state estimation a straightforward process, but creates a vulnera-

bility where compromising a single unit is enough for a successful attack.

To launch a successful attack, the measurements vector in (5.2), which consists of the

voltage and current flows in rectangular form, can be manipulated using the same strategy

for DC-estimators, and the attack vector should use the grid topology to mask the data.

Moreover, the adversaries do not need to know the complete topology of the grid (h); they

need to know the local topology, which can be estimated by monitoring the PMU measure-
ment data. Consider hs to be a subset of h, (hs ∈ h), corresponding to the local topology.

The attack vector is constructed as follows:

zcomp. = ztru + a;

where

a = c × [0 . . . hs,1 hs,2 . . . hs,i 0 . . . 0]T ;

(5.6)

zcomp. = [ztru1ztru2 . . . zcomp1zcomp2 . . . zcompiztrui+1...]T .

76

By using this attack vector the BDD tests in (5.5) and (5.4) can be bypassed without

detection.

5.3 Optimal PMU placement against FDIAs

Most approaches propose securing a subset of the measurements to defend against FDIA [58].

Others deploy PMUs as the secure measurement subset, within a DC-estimator or a nonlinear

estimator framework [68, 71, 72]. However, this chapter the proposes an OPP formulation

to guard against FDIAs, while assuming PMUs to be compromisable. In this scenario, the

adversaries can manipulate all channel of the compromised PMU. Thereby, manipulate serval

states instead of just one. Moreover, these states will not only pass the BDD test but behave

as confirming errors, which makes harder to detect even when the adversaries do not have

accurate knowledge of the local topology hs.

Before discussing the proposed OPP approach, the complete observability for the system

using PMUs is presented. The constraint in (5.7) must be met for achieving full observability,

where 1 is a vector of length N with all its elements equal to 1, and N is the number of

buses.

where

O = A P ≥ 1

1, if a PMU is installed at bus (i)

0, otherwise



Pi =

(5.7)

(5.8)

77



0, otherwise

1, if a PMU is installed at bus i

1, if there is a branch connecting

Aij =

bus i to bus j

(5.9)

The proposed OPP formulation guards against a compromised PMU, by utilizing the

other deployed PMUs. There needs to be enough independent redundancies, which would

prevent the FDIA from bypassing the BDD tests in (5.4) and (5.5). However, these redun-

dancies need to be kept at minimum to reduce the overall installation cost.

Suppose we have the deployed PMUs in a set d = {P1, P2, . . . , Pn}, where the attacked

PMU is a member of this set d. The other deployed PMUs of this set serve as secure

measurement units that can guard against the compromised measurements of the attacked

PMU if and only if (5.10) holds [61].

Rank(Hp) = Rank(Hp,2N−2) + 2 xk ∈ X;

(5.10)

where Hp is the transition matrix with m rows corresponding to the p set of PMU mea-

surements and 2N columns; and N is the number of the states in the system. The matrix

Hp,2N−2 is the same as the Hp matrix except for the number of columns, where the columns
corresponding to state xk, (xk ∈ X), are removed. It should be noted that the estimated

states in X consist of the bus voltage magnitudes and angles in rectangular form. Therefore,

each bus has two columns in the Hp matrix corresponding to the real imaginary parts of the

bus voltage.

In this framework, the rest of the deployed PMUs (not-attacked) serve as members of

78

the p set (protected measurements), and the attacked PMU can be any member of the d set.

The deployed PMUs in the d set are chosen in such a way that any member of the set can

be removed—due to an attack— and the rest of the members will serve as the p set such

that (5.10) holds for all buses that have PMUs.

To facilitate this scheme, the OPP in (5.11) is proposed. By using this criterion, the

BDD of the system will be able to detect any FDIA on any single PMU. Since each PMU

offers np independent measurements and the corresponding Hp will have a rank equals to np.

The criterion in (5.11) guarantee that there will be at least one independent measurement

supporting or contradicting any all-channel attack on a PMU.

where

O ≥ ˘l

2, if a PMU is installed at bus i

1, otherwise



˘li =

(5.11)

(5.12)

5.4 Simulation and Results

In this section, the proposed approach is tested on the IEEE 14-bus and IEEE 30-bus test

systems. To test the validity of the proposed approach, the FDIA is used on all the PMUs

in the system, one at a time. The FDIA is used as an all-channel attack, where all channels

of the attacked PMU are manipulated as in (5.6). The system states X are estimated using

wighted least square (WLS) method. The significance level for the chi-square test is set to

0.05.

In this scenario, the adversaries take control of a single PMU and can manipulate all

79

the channels of the attacked PMU. Two schemes are used for deploying PMUs, the normal

observability in (5.7), and the proposed scheme in (5.11). The FDIA is tested on all the

deployed PMUs, one at a time.

Table 5.1 shows that deploying PMUs under normal conditions makes the PMUs at buses
{2} and {6} vulnerable to FDIAs; therefore, the FDIAs can bypass the BDD test. However,
for the same scheme PMUs at buses {7} and {9} are not vulnerable to FDIA, since the

observability for these buses satisfies the criterion in (5.11).

As for the proposed approach, the FDIAs do not succeed on any of the deployed PMUs,

since these attacks attempt to change at least one of the system states which are protected

according to (5.10), (5.11). For the IEEE 30-bus, Table 5.2 shows that PMUs at buses
{11}, {12} and {19} are vulnerable to FDIAs when deploying PMUs for normal conditions

observability. The proposed approach, on the other hand, protects against FDIAs of a single

PMU attack. Moreover, the new OPP scheme does not necessitate increasing the number of

PMUs to guard against FDIAs as is the case with the IEEE 30-bus test system in Table 5.2.

80

Table 5.1: FDIAs on the IEEE 14-bus Under Different OPP Schemes

PMU Locations Attacked PMU J( ˆX)

X (95%) BDD Test

2, 6, 7, 9

4, 5, 6, 7, 9a

2

6

7

9

4

5

6

7

9

003.15 003.94

Passed

003.11 003.94

Passed

169.91

003.94

Failed

168.76

003.94

Failed

1270.3

12.338

Failed

1246.2

12.338

Failed

64.022

12.338

Failed

240.501

12.338

Failed

191.039

12.338

Failed

aProposed OPP approach.

Table 5.2: FDIAs on the IEEE 30-bus Under Different OPP Schemes

PMU Locations Attacked PMU

J( ˆX)

X (95%) BDD Test PMU Locations Attacked PMU J( ˆX)

X (95%) BDD Test

1, 2, 6, 10, 11, 12,

19, 24, 25, 27

1

2

6

10

11

12

19

24

25

27

1,200.1

21.664

Failed

1,375.9

21.664

Failed

226.955

21.664

Failed

050.060

21.664

Failed

010.229 21.664

Passed

2, 4, 6, 9, 10, 12,

011.800 21.664

Passed

15, 18, 25, 27a

010.010 21.664

Passed

043.514

21.664

Failed

117.734

21.664

Failed

097.362

21.664

Failed

2

4

6

9

10

12

15

18

25

27

261.592

29.788

Failed

2713.91

29.788

Failed

2833.82

29.788

Failed

563.695

29.788

Failed

446.976

29.788

Failed

308.635

29.788

Failed

311.102

29.788

Failed

87.9443

29.788

Failed

093.003

29.788

Failed

098.649

29.788

Failed

aProposed OPP approach.

As shown in Table 5.1 and Table 5.2, the proposed OPP approach guards against FDIA

81

for a single PMU all channels attack, since this attack is guaranteed to violate the conditions

in (5.10) for at least one of the protected states. However, the proposed OPP approach might

be susceptible to a single channel attack, where the adversaries take full control of a PMU

but manipulate only a single channel. In this single channel attack, some of the adjacent

buses of the attacked PMU might be susceptible to this attack. Yet, this single channel
attack can be guarded against if the criterion in (5.11) is changed to O ≥ ˘2, where ˘2 is a

vector of length N with all its elements equal to 2.

5.5 Conclusion

An attack on a PMU can have a high impact, as the adversary can manipulate the states

of all connected buses. This chapter investigates the FDIAs on PMUs form an OPP per-

spective. The proposed OPP approach utilizes the PMUs as authenticators for each other.

This approach does not require any modification to existing BDD tests or state estimation

methods and can be used with new BDD algorithms.

82

Chapter 6

Conclusions and Future Work

This chapter provides a general conclusion about the methods that have been used, the

general outcomes of this work and suggestions. Possible future developments that can be

built on or added to the presented work are also provided.

6.1 Conclusions

This dissertation has presented a comprehensive approach to the OPP problem. In chapter

2, a realistic approach for the observability-based OPP has been presented. A comprehensive

installation cost of PMUs has been used, where the substation and communication infras-

tructures are considered. Chapter 2 has also presented a multistage approach, where PMUs

are deployed over several budget periods to achieve complete observability by the final pe-

riod. The presented multistage approach achieves the optimal PMU allocation for the whole

process, instead of targeting the optima for the current budget period. The results in this

chapter show that OPP problems should be solved to minimize the installation cost, and

not to minimize the number of PMUs. Solving for the minimum number of PMUs does not

reflect the actual cost since it does not consider the infrastructure of the substations nor does

it consider the additional cost of current channels. In cases where multistage OPP approach

is chosen due to budget limitations, decision-makers should not target maximum observ-

ability for the current budget but aim for observability solution that is part of the targeted

83

observability at the final stage. The expectation to the previous statement is when decision

makers are pursuing application-based OPP and not observability-based OPP approach.

In chapter 3 a fault-tolerance based OPP approach is developed in which the vulnera-

bility of the network is assessed for PMU deployment. This strategy of deploying PMUs in

the proximity of higher probability contingencies increases the likelihood of more effective

remedial actions, both preventive and corrective. This framework achieves the optima with

the maximum observability in the case of multiple optima. The results show that embed-

ding the vulnerability into the connectivity matrix enable for cost-effective results. This

approach of quantifying the targeted application and integrating this quantified application

into the observability can be extended to handle multiple applications simultaneously in a

cost-effective manner.

The rest of the dissertation focuses on the application-based OPP approaches while con-

sidering the installation cost of PMUs. Chapter 4 uses the multistage approach to solve

the OPP problem while enhancing the voltage stability for the power grid. A criterion for

voltage stability has been developed, where critical buses are identified and prioritized for

PMU deployment. The results form the voltage stability criterion identified critical buses to

be the PV buses (buses with generators). Installing PMUs at these buses can also be used

to estimate and monitor the dynamics of the generators mainly the torque, rotor angle, and

speed.

Chapter 5 addresses the emerging cyber-security risks of FDIAs from the perspective of

PMU deployment. The proposed approach does not make assumptions about the security

of the grid and utilizes the PMUs as authenticators for each other. This approach does not

require any modification to existing BDD tests or state estimation methods and can be used

with new BDD algorithms.

84

6.2 Future Work

The framework shows the potential for handling multiple criteria for application-based

PMUs. As the PMU deployment requires substantial investment by the utilities, more

benefits should be expected from the PMUs. Currently, most application-based approaches

pursue one technical benefit besides the observability such as voltage stability. There is a

need to investigate multi-application-based approaches where several applications of PMUs

can be pursued cost-effectively.

One of the advantages of PMUs is the high resolution of measurement data. While

this high rate of data sampling enables for better situational awareness and development of

advanced controls, it creates a problem of handling large data efficiently without consuming

vast resources. This big data issue emphasizes the need for faster computational tools to

process this data efficiently.

Data analysis is another promising field for power systems using PMU data. The large

volume of data being provided by PMUs can present new insights and opportunities for power

grid operations. Data analysis can be used to forecast the lifetime of the grid components

and their current conditions. New insights into the grid contingencies and the dynamics of

the generators can also be used to enhance existing controls.

The cyber-security aspect of the grid is a significant concern. The FDIAs which has

been presented in this work reflects one of these concerns. There is a need to develop new

schemes to defend against such attacks. Moreover, there are other cyber-security attacks

such as man-in-the-middle and denial of service. As the current power grid incorporates

more and more smart grid technologies, these concerns will become increasingly critical and

must be addressed.

85

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