real time pmu-based stability monitoring by zijie lin · monitoring algorithms for the power grids...

i

REAL TIME PMU-BASED STABILITY MONITORING

By

ZIJIE LIN

A thesis submitted in partial fulfillment of

the requirement for the degree of

MASTER OF SCIENCE IN ELECTRICAL ENGINEERING

WASHINGTON STATE UNIVERSITY

School of Electrical Engineering and Computer Science

AUGUST 2015

ii

To the Faculty of Washington State University:

The members of the Committee appointed to examine the

thesis of Zijie Lin find it satisfactory and recommend that

it be accepted.

_________________________________

Chen-Ching Liu, Ph.D., Chair

_________________________________

Anjan Bose, Ph.D.

_________________________________

Sandip, Ph.D.

iii

ACKNOWLEDGEMENT

I am extremely grateful to my supervisor Prof. Chen-Ching Liu, Director of Energy Systems

Innovation (ESI) Center of Washington State University. He provided this great opportunity

for me to participate in the project of Real Time PMU-Based Stability Monitoring for PJM

interconnection and Power Systems Engineering Research Center (PSERC S-50). His

creative guidance and invaluable suggestions helped me in completing the project. I sincerely

thank Guanqun Wang, who offered his help during the whole period of my project work

since September 2013. I also appreciate the contributions of Haosen Guo and Jie Yan since

part of my research is based on their original work. In addition, I would like to express my

appreciation for the support provided by industry members of PSERC. Finally, I am

indebted to my beloved parents for their understanding and support.

I want to thank Bonneville Power Administration (BPA) and PJM for the test data, and

the support in part by National Science Foundation. I am also very grateful to the industry

advisors of the PSERC S-50 project for their advice:

Alan Engelmann, Exelon/Commonwealth Edison

Bill Timmons, Western Area Power Administration

Clifton Black, Southern Company

David Schooley, Exelon/Commonwealth Edison

Dmitry Kosterev, Bonneville Power Administration

Eugene Litinov, ISO New England

Evangelos Farantatos, Electric Power Research Institute

George Stefopoulos, New York Power Authority

Guiseppe Stanciulescu, BC Hydro

mailto:[email protected]

iv

Jay Giri, ALSTOM Grid

Jinan Huang, Hydro-Québec Research Institute

Liang Min. Lawrence Livermore National Lab

Manu Parashar, ALSTOM Grid

Patrick Panciatici, Réseau de Transport d'Électricité

Sanjoy Sarawgi, American Electric Power

Steven Hedden, Exelon/Commonwealth Edison

Xiaochuan Luo, ISO New England

mailto:[email protected]

v

REAL TIME PMU-BASED STABILITY MONITORING

Abstract

by Zijie Lin, MS

Washington State University

August 2015

Chair: Chen-Ching Liu

The purpose of this thesis is to develop PMU-based, real-time, wide area stability

monitoring algorithms for the power grids using different methods and approaches.

Phasor Measurement Units (PMUs) are increasingly available on power grids due to

the significant investment in recent years. As a result, a priority in industry is to extract

critical information from the increasing amount of PMU data for operation, planning,

protection, and control.

This research proposes enhanced algorithms for real time stability monitoring in a

control center environment: the waveform analysis to extract the “trending” information

of system dynamics embedded in Lyapunov exponents. The algorithm will help build a

complete model for predicting system stability, which can help to avoid cascading

failures and enhance system security.

A PMU-based online waveform stability monitoring technique is proposed based on

the Maximum Lyapunov Exponent (MLE). The main idea of the MLE technique is to

calculate MLE as an index over a finite time window in order to predict unstable trending

of the operating conditions. Significant progress has been made to improve the accuracy

of MLE technique. Firstly, state calculator is applied to improve the MLE accuracy by

vi

introducing real time monitoring measurement into the computation. Secondly, the power

system model is greatly improved by adopting a dynamic structure-preserving load

model. This model takes into account the dynamics of P and Q load with respect to the

frequency/voltage variations. The purpose is to extend the MLE technique to voltage

stability analysis as well as rotor angle stability. Based on this model, the system can be

represented by a set of differential equations, which is used for MLE calculation. The

power network topology is preserved. Thirdly, parameters for the model are identified

from the results of time domain simulation based on their importance. Finally, the

methods to save memory and to improve the calculation speed are also discussed. The

proposed methods are validated by time-domain simulation of the122-bus Mini-WECC

system and the15000-bus PJM system model.

vii

TABLE OF CONTENTS

ACKNOWLEDGEMENT ................................................................................................. iii

ABSTRACT........................................................................................................................v

LIST OF TABLES .............................................................................................................. x

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

CHAPTER

1. INTRODUCTION ......................................................................................... 1

1.1.Background ......................................................................................... 1

1.2.Overview ............................................................................................. 2

1.3.Organization of the Thesis .................................................................. 4

2. LYAPUNOV EXPONENT THEORY .......................................................... 5

2.1.Maximum Lyapunov Exponent........................................................... 5

2.2.MLE Stability Criterion ...................................................................... 6

2.3.State Calculator ................................................................................... 8

2.4.Summary ........................................................................................... 10

3. DYNAMIC LOAD MODEL FOR MLE CALCULATION ....................... 11

3.1.Reduced Dynamic Model .................................................................. 11

3.1.1 Constant Current Load ........................................................ 14

3.1.2 Constant Impedance Load ................................................... 16

3.1.3 Static Load Models .............................................................. 17

3.2.Structure Preserving Model............................................................... 18

3.2.1 Frequency Dependent Model .............................................. 18

3.2.2 Load Recovery Model ......................................................... 19

viii

3.3.Partial Structure-Preserving Model .................................................... 21

3.4.Parameter Identification ..................................................................... 21

3.5.Summary ............................................................................................ 22

4. PARAMETER IDENTIFICATION ............................................................ 23

4.1.Estimation Theory .............................................................................. 23

4.2.Identifying the Parameter in Real Power Model ................................ 24

4.3.Identifying the Parameters in Reactive Power Model ....................... 26

4.3.1.Eliminating two parameters. ............................................... 26

4.3.2.Estimating the parameter . ............................................... 27

4.4.Summary ............................................................................................ 28

5. ENHANCEMENT OF THE COMPUTATIONAL SPEED........................ 29

5.1.Computation of Large Scale Power Systems ..................................... 29

5.2.Aggregation of Generators and Elimination of Buses ....................... 30

5.2.1.Eliminate Buses with No Generation or Load .................... 30

5.2.2.Combine Generators at the Same Bus ................................. 30

5.3.Use the Different Method to Calculate Differential Equations .......... 31

5.3.1.ODE 15s Method ................................................................. 31

5.3.2.Forward Euler Method ........................................................ 31

5.4.Improving the speed of MLE calculation........................................... 32

5.5.Summary ............................................................................................ 34

6. CASE STUDY AND SIMULATION RESULTS ....................................... 35

6.1.Mini-WECC System Simulation Model ........................................... 35

6.2.Application to the Reduced PJM System .......................................... 42

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6.3.Analysis and Summary ...................................................................... 46

7. CONCLUSIONS .......................................................................................... 47

REFFERENCE..................................................................................................................49

APPENDIX

I: All Test Results of Mini-WECC System ....................................................... 52

x

LIST OF TABLES

Table 6-1 System Parameter Verification in Mini-WECC System...................................36

Table 6-2 Typical Examples in Mini-WECC System.......................................................40

Table 6-3 Results of Two Load Models in PJM System...................................................43

xi

LIST OF FIGURES

Fig 2.1 Perturbation of Trajectories of a Dynamical System...............................................5

Fig 2.2 Different Trajectories and Corresponding Lyapunov Exponents............................6

Fig 2.3 Stability of a Nonlinear Dynamical System............................................................7

Fig 3.1 Parameters of a Three-bus System ....................................................................... 12

Fig 3.2 Structure of the Five-node system.........................................................................14

Fig 3.3 Structure of the System after Reduction (Current Source Load)...........................16

Fig 3.4 Structure of the System after Reduction (Admittance Load)................................17

Fig 3.5 Power Recovery Process after a Step Change of Voltage .................................... 20

Fig 4.1 Linear Model in Estimation Theory......................................................................23

Fig 5.1 Concept of MLE(Maximum Lyapunov Exponent)...............................................32

Fig 6.1 One-line Diagram of Mini-WECC System...........................................................36

Fig 6.2 Reactive Power Load at Bus 8 from PSAT Simulation.........................................38

Fig 6.3 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs=1...................................38

Fig 6.4 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs=1.9................................39

Fig 6.5 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs=2...................................39

Fig 6.6 Comparison of Different Models...........................................................................41

Fig 6.7 Comparison of Two Models for PJM System(Type 1).........................................45



1

CHAPTER ONE

INTRODUCTION

1.1 Background

Since 1960s, the scale and complexity of interconnected power grids in the United

States have increased to such an extent that large blackouts are growing in both number

and severity [1]. For example, in the 90’s, there were 66 blackouts in the U.S. that

affected over 100MW in the first five years. However, during 2006-2010, the number of

such outages reached 219, more than tripled [2]. Large scale power outages cause

significant damages to utilities and economic losses. As a result, it is critical to enhance

the technology for predicting and preventing large scale power outages.

Phasor measurement units (PMUs) provide synchronized measurements of the state

of the power system at a rate up to 120 samples per second [3]. Based on the IEEE

standard for synchronous phasors, the basic time synchronization accuracy of PMU is

±0.2μs [4]. Since communication technology is well developed, it enables online

monitoring of power systems based on PMU data. In recent years, the number of PMUs

installed on the power grids increased significantly and research on the applications of

PMUs in monitoring, protection, and control of power grid has made great progress [5].

Bonneville Power Administration (BPA) is the first utility to implement PMUs in its

wide-area monitoring system.

The main purpose of this thesis is to develop a PMU-based, real-time, wide area

stability monitoring algorithm for power grids. The proposed algorithm takes advantage

of the PMU-measured waveforms and Maximum Lyapunov Exponent stability detection

2

and it extracts the information of power system dynamics to determine if the system is

approaching instability. This algorithm can be applied to real time monitoring as a tool to

issue warning messages for system operators when a trend toward instability is detected.

1.2 Overview

Traditionally, there are two major approaches to transient stability analysis of power

systems. One is time-domain simulation methods [6-7], based on numerical integration of

the power system dynamic equations. This method provides details of the dynamic

waveforms for a large scale system if the component and network data are available.

Parallel-in-time algorithms can be performed to improve the calculation speed of the

time-domain simulation [8]. However, the high computational burden limits the online

applicability of this method. The other approach is the energy functions based direct

methods, such as relevant unstable equilibrium point (RUEP) method [9], controlling

UEP (CUEP) method [10], potential energy boundary surface (PEBS) [11], BCU method

[12], and extended equal area criteria (EEAC) method [13]. These methods have

relatively fast computational speed and provide quantitative indices for stability

assessment. A shortcoming of direct methods is that the assessment result may not

incorporate sufficient details due to simplification of the system models. There are also

knowledge-based transient stability methods. In [14], decision trees are created off-line

based on a large number of simulations. The results are used to predict transient stability

in an on-line environment. The hybrid intelligent system proposed in [15] can be used to

assess not only transient stability, but also generator tripping. In [16], on-line PMU data

3

are compared with trajectory patterns stored in the database. The Euclidean distance is

used as an index for prediction of system stability.

In this research, a new method called Maximum Lyapunov Exponent (MLE) is

proposed for monitoring of wide-area transient stability based on PMU data in an online

environment. The Maximal Lyapunov Exponent (MLE), which is a tool for prediction of

out-of-step conditions [17], is applied to the analysis of rotor angle stability [18-19].

Based on a nonlinear dynamic model of the power system, the MLE can be calculated

based on the waveforms resulting from dynamics of the system states. It is based on a

rigorous theoretical foundation and, therefore, it is able to reliably determine the stability

status using PMU streaming data within a specified time window.

However, it should be noted that certain factors could have a significant effect on

the accuracy of the prediction based on the MLE technique. Two problems need to be

solved to improve the proposed technique:

1. System model: For a large scale power system with generator and load buses

interconnected by transmission lines, the dynamic characteristics of a power system for

stability analysis is represented by a set of differential-algebraic equations (DAEs).

Previously, in order to apply the MLE technique, the power network needs to be reduced

to generator buses only and all the loads are modeled by constant impedances. This will

lead to a loss of the system topology, reducing the level of accuracy for system stability

assessment. Also, it is known that there are interactions between angle and voltage

dynamics. Therefore, the dynamic model of the system needs to be improved by retaining

the load buses and incorporating voltage dynamics of the system.

4

2. Parameter identification: This research project addresses these critical issues by

proposing new techniques. First, a structure-preserved model is applied to transient

stability analysis. This model retains the load buses as well as the network topology.

Based on this model, the MLE technique considers generator dynamics, in addition to

accounting for the load dynamics. Moreover, the structure-preserving model enables the

MLE to predict short-term voltage stability more accurately. Parameter identification for

this model has been conducted.

The methods proposed in this research are validated by extensive simulations.

1.3 Organization of the Thesis

The organization of the remaining chapters is as follows: Chapter 2 is a summary of

the theoretical basis of the MLE technique and State Calculator. Chapter 3 describes the

structure-preserving model and load recovery model for the MLE calculation. In Chapter

4, parameters are identified using linear system theory. Chapter 5 offers methods to

improve the calculation speed. In Chapter 6, the PJM system and 122-bus Mini-WECC

system are utilized to validate the proposed methods. Numerous time-domain simulations

of these systems are performed in order to validate the MLE method. The conclusions

and future work are provided in Chapter 7.

5

2. CHAPTER TWO

LYAPUNOV EXPONENT THEORY

2.1 Maximum Lyapunov Exponent

In ergodic theory [20], Lyapunov Exponent is used to characterize whether a given

system is “chaotic” and to quantify the amount of chaos. For a trajectory with

an initial value all trajectories that start out in a neighborhood of 0x will converge

toward ,or diverge as time propagates. Sensitivity of the system trajectory with

respect to a perturbation of the initial state is shown in Figure 2.1 and it is quantified by

(2.1)

Equation (2.1) describes the exponential behaviour of nearby trajectories in a short

time interval. Therefore, the time average is used to measure the rate of divergence or

convergence of trajectories resulting from infinitesimal perturbations of the initial state.

Assume that the system trajectory is defined by differential equations

(2.2)

From (2.1)

0

1ln ( ) /x t x

t (2.3)

By taking the limit as t∞, the Lyapunov exponent is given by

0x x t

0x x t

Fig2.1 Perturbation of Trajectories of a Dynamical System

6

0

1lim ln ( ) /t

x t xt

(2.4)

where represents the mean growth rate of the distance between

neighboring trajectories.

2.2 MLE Stability Criterion

A negative MLE implies that the nearby trajectories will converge to the reference

trajectory exponentially, such as the Fig.2.2(a). If the nearby trajectories diverge, such as

the Fig 2.2(b), the calculation result of MLE will be positive [28].

(a) Convergent Trajectories: Negative MLE (b) Divergent Trajectories: Positive MLE

(c) Impossible Trajectories1 (d) Impossible Trajectories2

Fig 2. 2 Different Trajectories and Corresponding Lyapunov Exponents

-0.5

0

0.5

1

1.5

2

Converg

e

Div

erg

e

Converg

e

Div

erg

e

7

The theorem of continuous-time fixed point [22] established the relationship

between the MLE and the asymptotic behavior of the dynamic system. Consider a

continuous-time dynamic system and assume all the Lyapunov exponents are non-zero.

Then the steady state behavior of the system consists of a fixed point. In particular, if all

Lyapunov Exponents are negative, i.e., MLE is negative, this fixed point is an attracting

fixed point. In summary, it is impossible that original trajectory is stable but MLE is

positive, which means updated trajectory converge to the original one if the original

trajectory is stable.

The following is the proof of the theorem[17]. Consider the trajectory 2 2x f x as

a nearby trajectory of (2.2), with an initial value 2 (0)x , as shown in Fig. 2.3. Give the x1

a disturbance dx and let 2 1(0) (0)x x dx , which means that 2 1(0) (0)dx x x

According to the definition of Lyapunov exponent, if the MLE of 1x t is negative, there

exists an 0 such that, for any 2 10 0x x ,

2 1lim 0t

x t x t

(2.5)

Fig 2.3 Stability of a Nonlinear Dynamical System

Since x t is continuous, there must exist a 0T , such that 0x t x . Let

2 10x x t , then 2 1x t x t t . Since

x2(0)=x1(∆t)

x2(t)=x1(t+∆t)

x1(0)

x1(t)

8

2 1 1 10 0 0x x x t x (2.6)

it is obtained that

2 1 1 1lim lim lim 0t t t

x t x t x t t x t f x t T

(2.7)

Since 0T , it follows that

lim 0t

f x t

(2.8)

It means when t tends to infinity, 1x t will approach an stable state k if MLE of this case

is smaller than 0.

From the proof above, it is also clear that possibility of the situation in Fig2.2(c) and

(d) is 0. It means that during the MLE simulation, it is impossible that a original

trajectory oscillates less and less but the update trajectory diverges from it. Similarly, if a

original trajectory diverges, it cannot be true that update trajectory still converge to it.

2.3 State Calculator

State Calculator is driven from the available PMU measurements. In this project, the

MLE calculation is based on the State Calculator. As introduced in Chapter 1, PMU

allows online measurement of multiple points at the same time. It can measure frequency,

current, voltage, and power as well as phasor angle, at both 50 and 60 Hz waveforms for

2880 samples per second. A phase-lock oscillator along with a Global Positioning System

(GPS) provides the high-speed synchronized sampling. A PMU can be a dedicated device

or it can be incorporated into a protective relay, installed at substations or power plants.

Phasor Data Concentrators (PDC) are used to collect the information. A Supervisory

Control And Data Acquisition (SCADA) system connects the central control facility with

various substations. In summary, in a power system monitoring network, PMUs are

9

installed at various buses throughout the entire power system to acquire the

measurements.

It is known from the linear and observability theory [26] and [27], that uniform

placement of the observation devices will give the largest observability. Thus, in this

thesis, the PMUs are uniformed distributed among the 15,000 buses and then simulated

using the TSAT.

The PMU reports the data every several cycles. After calculating a time point in Matlab,

the result will be replaced with PMU measurement data if this bus has a PMU. The PJM

system has 15,000 buses and among them, 1,090 buses have PMUs. These 1,090 load

buses calculation results (frequency or voltage) will be replaced by the real time PMU

data. When calculations are completed, the MLE (Maximum Lyapunov Exponent)

method is employed to analyze these waveforms to see whether the system is stable or

not. If the waveforms converge, they are considered to be stable. If they diverge, they are

judged to be unstable.

Algorithm

1.Place the PMUs at heavily loaded buses or generator buses.

2. Calculate the waveform of every bus for the test system under a disturbance,

including load bus's phase angle, generator bus's rotor angle and rotor speed.

3. The data from PMU have a higher level of accuracy. PMU reports data every 2

cycles (0.033s) while the time step in the calculation is 0.00033s, so check the PMU data

every 100 calculation loops. Then replace the calculation result with the PMU

measurements.

10

4. Use the updated waveform to calculate the value of next time point. Continue

doing that until the simulation time is over.

5. Calculate the MLE. If the MLE is smaller than 0, it means the system is stable

after this fault. If the MLE is larger than 0, it means that system cannot remain stable

under the disturbance.

2.4 Summary

The MLE can be applied to analyze transient stability of a nonlinear power system

model. It extracts the “trending” information from the system trajectory and does not

require the knowledge of the equilibrium points of the system. When the dynamical

system model is known, the MLE can achieve an accurate prediction of the transient

stability behaviour. However, several factors, such as the model of the system and the

window size, will affect the accuracy of MLE calculation. These issues will be addressed

in the following chapters.

11

3. CHAPTER THREE

DYNAMIC LOAD MODEL FOR MLE CALCULATION

The dynamic model of a power system can be formulated as a set of differential-

algebraic equations (DAEs) [23]. Differential equations represent the dynamic

characteristics of power system components, such as generators, governors, and exciters.

Algebraic equations model the steady state power flow on the network. In previous work,

the MLE technique is applied to the assessment of rotor angle stability. The system needs

to be reduced to retain only the generator buses and, consequently, DAE is reduced to a

set of ordinary differential equations (ODEs). The disadvantage of this method is that

load bus information as well as the system topology is implicit and it causes errors in the

stability assessment. Moreover, it is known that there are interactions between angle and

voltage dynamics. In this thesis, a structure-preserving model is proposed for the MLE

computation taking into account both generator and load dynamics.

3.1 Reduced Dynamic Model

Consider a power system with a total of 0n buses with generators at m buses. Hence

there are 0n m load buses. The classical 2nd

-order swing equations are used to represent

the synchronous generators. Such generators are modeled as a constant voltage source in

series with the transient reactance 'dx . Therefore, the power system can be augmented by

m fictitious buses representing the generator internal buses. The total number of buses in

the augmented network is n , which is 0n + m .

The swing equations of the system are given by

12

1

, 1,...,1

2 i i

i i

M e i i

i

i mP P D

H

. (3.1)

The algebraic equations of the system are the power flow relationship, i.e.,

1

1

cos 1,...,

sin 1,...,

n

i i j ij i j ij

j

n

i i j ij i j ij

j

P V V Y i n

Q V V Y i n

(3.2)

Since the MLE can only be applied to ODEs, the algebraic equations at the load bus

need to be reduced. A 3-bus system is used to illustrate the Ward equivalent.

A 3-bus system with two generator buses and one load bus is shown in Fig 3.1.

Parameters of a three-bus system are listed in Fig 3.1.

Fig3.1 Parameters of a Three-Bus System

The admittance matrix of the system is

j0.25

j0.2

j0.3

j0.05 j0.05

j0.05

j0.05 j0.05

j0.05

0.3+j0.1

S=1+j0.4

1 2

3

1,5.0 22 VPG 011 V

1I 2I

3I

2lI

2gI

3lI

13

11 12 13

21 22 23

31 32 33

8.23 5 3.33

Y 5 8.9 4

3.33 4 7.23

y y y j j j

y y y j j j

y y y j j j

(3.3)

The power flow is computed by the Newton-Raphson algorithm

1616.09301.0

0522.09986.0

0000.00000.1

172.0944.0

052.01

01

V

j

j

j

(3.4)

Based on the node analysis, the current injections at each bus are:

1

2

3

0.7989 0.1397

I YV 0.1819 0.1676

0.9594 0.6001

I j

I j

I j

(3.5)

Based on the current direction noted in Figure 3.1, it is seen that

222 lg III (3.6)

1155.02944.00522.09986.0

1.03.0

2

22 j

j

j

V

SI l

l

(3.7)

2831.04763.0222 jIII lg (3.8)

33 II l (3.9)

When the internal admittance of the generators is considered, two more buses are added

to the system. Therefore, the structure of the five-bus system is shown in Fig. 3.2.

14

Fig. 3.2 Structure of the Five-Bus System

The internal voltage of the two generator buses are

4 1 1 * 0.2 1.0279 0.1598V V I j j (3.10)

0431.00553.12.0*225 jjIVV g (3.11)

The internal admittance of generator is

52.0

1j

jyd (3.12)

While performing network reduction of a power system, load buses can be treated as

either current sources or admittances.

3.1.1 Constant Current Load

The admittance matrix of the five-bus system is

15

dc

ba

000

000

0023.7433.3

049.85

033.3523.8

Y 5*5

dd

dd

dd

dd

yy

yy

jjj

yjyjj

yjjyj

(3.13)

Based on the node analysis, current injections at each bus are

e

g

l

I

I

I

I

j

j

j

j

I

I

0

2831.04763.0

1397.07989.0

6001.09594.0

1155.02944.0-

0

V*YI i

2

1

3

2

1*55*51*5

(3.14)

Take 1I 1*5 as an example:

3132121111 *** VyVyVyI (3.15)

0*******1I 411431321211111*5 VyVyIVyVyVyVyVy dddd (3.16)

Perform the network reduction and one obtains

7466.10000.09032.10000.0

9032.10000.07492.10000.0b*

acdY 2*2

j

jj (3.17)

3964.07051.0

3501.06014.0-I*

acI i

j

jeq

(3.18)

5

4

2*2reduce *Y1133.02289.0

2104.01975.0III

V

V

j

jeqe

(3.19)

eqI is the equivalent current sources of the two loads in the previous system. Fig. 3.3

shows the structure of the system after network reduction.

16

Fig. 3.3 Structure of the System after Reduction (Current Source Load)

3.1.2 Constant Impedance Load

The Y matrix of the 5-node system is

dc

ba

000

000

0023.7433.3

049.85

033.3523.8

Y 3

2

5*5

dd

dd

l

ddl

dd

yy

yy

yjjj

yjyyjj

yjjyj

(3.20)

The equivalent admittance of the two loads are

1000.03000.02

22 j

V

Iy l

l

(3.21)

4524.01102.13

33 j

V

Iy l

l

(3.22)

Based on the node analysis, current injections at the buses are

i

e

2

1

1*55*51*5I

I0

0

0

2831.04763.0

1397.07989.0

0

0

0

V*YI

gI

I

j

j

(3.23)

Take 2I 1*5 as an example

3332221212 *** VyVyVyI (3.24)

4 5

1I 2gI

1eqI 2eqI

17

0

******

******2I

2222

52222252222

53232222221211*5

gllg

ddllgddl

ddl

IIII

VyVyVyIIVyVyVyI

VyVyVyVyVyVy

(3.25)

Perform network reduction to obtain

9812.13466.06991.12886.0

6991.12886.09314.12519.0b*

acdY 2*2

j

jj (3.26)

0

0I*

acI e

eq (3.27)

In this case, loads are converted into admittances, so no equivalent current sources

need to be added to the internal buses. However, the new Y matrix is different with the

previous case, because the load admittance has changed the structure of the system. The

structure of the system after reduction is shown in Fig.3.4.

5

4

2*2ireduce *Y2831.04763.0

1397.07989.0III

V

V

j

jeq

(3.28)

Fig. 3.4 Structure of the System after Reduction (Admittance Load)

3.1.3 Static Load Models

Besides the constant current load and constant impedance load, there are several

load models, such as the ZIP model

(3.29)

(3.30)

4 5

1I2gI

18

These models consider the influence of voltage or frequency at load buses on the

load amount. However, these models are static; they cannot represent dynamic response

of the load. A dynamic load model is required to represent the dynamic characteristics of

loads under frequency and voltage variations.

3.2 Structure Preserving Model

3.2.1 Frequency Dependent Model

The structure preserving model with frequency-dependent load is first proposed in

[24]. It assumes that the real power load contains two parts: a static part and a frequency

dependent part, i.e.,

0

D DP P D (3.31)

Where is the angle at the load bus and D is the damping factor describes the

effect of frequency on real power. With this model, the original network topology is

explicitly represented. The complete dynamic model of the system containing both

generators and loads can be described as:

0 0

1

sin , 1,...,n

i i i i ij i j Mi Dijj i

M D b P P i n

(3.32)

For generator buses, the parameters satisfy

00, 0, 0i i DiM D P (3.33)

For load buses, the conditions on the parameters are

00, 0, 0i i MiM D P (3.34)

19

3.2.2 Load Recovery Model

The frequency dependent model does not consider the voltage behavior at the load

buses closely related to the reactive power load. Moreover, it does not consider the load

recovery process.

Measurements in the laboratory and on power system buses indicate that a typical

MW load response to a step change in voltage is of the general form in Fig. 3.5. The

responses for real and reactive power are similar qualitatively. Intuitively, this behavior

can be interpreted as follows. A step change in voltage produces a step change in MW

load. In a longer time-scale, the lower voltage tap changers and other control devices act

to restore voltages and, as a result, the load also recovers. With a certain recovery time on

a time-scale of a few seconds, this behavior captures the characteristics of induction

machines. In a time-scale of minutes, the role of tap-changers and other control devices is

included. Over hours, the load recovery and possible overshoot may emanate from

heating load. The importance of load models which capture the more general response in

Fig. 3.5 is evident. Indeed, the dynamic changes in load may substantially affect the

voltage dynamics.

20

Fig 3.5 Power Recovery Process after a Step Change of Voltage

Based on the above discussion, the load recovery model is proposed in [25]. This work

applies the load recovery model to reactive power load, i.e.,

(3.35)

Where is a time constant that describes the recovery response of the load. This model

can also be expressed as

q q q s tT x x Q V Q V (3.36)

in which

q d tx Q Q V (3.37)

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

t(s)

Voltage(u

nit)

step change of voltage

0 1 2 3 4 5 6 7 8 9 100

0.5

1

1.5

t(s)

P/Q

(unit)

power variant

21

sQ and tQ represent the steady-state response and transient response respectively. By

combining (3.6) and (3.8), the structure preserving model considering reactive load

recovery is formulated for the MLE analysis.

3.3 Partial Structure-Preserving Model

An effective model named the "partial structure-preserving" model is further

developed. This model only preserves the heavily loaded buses while deleting the lightly

loaded buses. The proposed partial preserved model has a similar level of accuracy as

that of the complete structure preserving model with a much higher calculation speed.

3.4 Parameter Identification

In the structure-preserving model, a few parameters are needed for each load bus. In

order to simplify the parameter identification process, the models of sQ and

tQ in a

single-index form [25] are used, i.e.,

0 0

0 0

/

/

qs

qt

n

s

n

t

Q V Q V V

Q V Q V V

(3.38)

Hence, 4 parameters need to be identified, i.e., damping factor D for which is

relative to the frequency variation, time constant for voltage recovery time, and qsn and

qtn for static and transient voltage exponents. In the frequency dependent model, only D

is needed while in the load recovery model, all four parameters are necessary.

A practical way to determine the parameters by simulation is adopted. First,

waveforms are computed for each load bus. The frequency and voltage curves are viewed

22

as inputs, and the real and reactive power load curve are outputs. Then for a fault scenario,

record the inputs and outputs at N discrete time instants, i.e., 1, … , N.

For the damping factor D, since this is a linear coefficient between real power and

frequency from (3.5), one can take the N points, i.e., 1 1,P , 2 2,P , …, ,N NP as

N samples. Then, the least square method is used to determine the slope D.

The parameter and qsn can be estimated based on the time response of the curve.

By linearizing (3.7), the relationship between dQ and V can be derived as

0 0/

1

qt q qs

d

q

n T s nQ Q V V

T s

. (3.39)

Therefore, by using the same input value V , the outputs dQ produced are

compared to the outputs. Typically, a least square quadratic criterion can be used to

obtain the most suitable values of qtn .

3.5 Summary

A structure-preserving model is developed which includes the frequency dependent

“P” model and load recovery “Q” model. The structure-preserving model retains the load

buses for MLE calculation. Previously a reduced model was used where load buses are

reduced for the computation of Lyapunov Exponents. By using the structure-preserving

model, useful information from both generator and load buses can be obtained. The

effectiveness of the three models is validated by time-domain simulation.

23

4. CHAPTER FOUR

PARAMETER IDENTIFICATION

4.1 Estimation Theory

In a linear model, the outputs have a linear relationship with the unknowns:

Z(k)=H(k)θ+V(k) (4.1) where vector θ(n*1) contains the unknown parameters that need to be estimated. Z(k)

is a N*1 vector called the measurement (output) vector. H(k), which is N*n, is the

observation matrix; and V(k), which is N*1, is the measurement noise vector. Usually,

V(k) is random. The argument k of Z(k), H(k), and V(k) denotes that the last

measurement available is kth. All other measurements occur before the kth.

Fig 4.1 Linear Model in Estimation Theory

The mismatch is given by the least square method based on the linear model,

θ θ θ θ (4.2)

(4.3)

Thus

(4.4)

where W=W(k), is the weight matrix, used to indicate which measurements are more

important than others.

In this project, the data is measured within several seconds. In fact, the gap between

24

the two consecutive time points is only 0.03s (dt=0.03s). It is assumed that datum at

every time point has equal importance. Thus W= . Equation (4.4) can be written as:

(4.5)

Matrix H'H must be nonsingular for the inverse matrix in the above equation to exist.

After getting the estimated value of , the residual, R, is used to determine whether the

estimation result is reasonable:

(4.6)

The following is the mathematical description of tested load model:

; (4.7)

(4.8)

(4.9)

(4.10)

(4.11)

where is the rotor angle of the load bus. P is the real power at the load bus and is the

power at the load bus before the disturbance.

In the structure-preserving model (dynamic load model), four parameters need to be

estimated for each load bus: damping factor D that connects frequency with active power

and voltage related parameters , , Tq.

4.2 Identifying the Parameter in Real Power Model

To guarantee the accuracy of system simulation, these parameters should be

calculated from the real world measurements. However, the actual PMU measurements

q q q s tT x x Q V Q V

q d tx Q Q V

25

are not available. As a result, data from a commercial software tool called TSAT is used.

TSAT has complete power system models and a high level of simulation accuracy. It is

assumed that the data from time domain simulation using TSAT serve as a good

approximation of the actual power system measurements from PMU. This approximation

is also applied in section 4.3.

In this section, the parameter D used in the active power part is determined first.

A practical way to determine the parameters can be summarized by the following

steps: First, suppose that the measurements from PMUs at load buses are available.

Measurements of the frequency curves are used as inputs while the measurements of

active power are outputs. Or use voltage as inputs, reactive power as outputs. Consider a

fault scenario, record the input and outputs at N discrete time instants, i.e., 1, … , N.

For damping factor D, since the coefficient between real power and frequency is

linear, N points, , …, are taken as

sample points. Then, the least square method is used to obtain the slope D.

Algorithm to get Parameter D:

Step 1: Run the simulation in TSAT for 20 seconds. The sampling interval is 0.033s.

Step 2: Export the data of real power and frequency of all load buses into an excel

table. Assume that this system has m load buses in total. Hence, for every fault, there are

m columns and 600 rows.

Step 3: From a linear model, D=(P-P0)/ and so P-P0= ; this is a

linear equation. The (P-P0) is the measurement.

Step 4: Use the least square method. An unknown X= ; Thus:

D= (4.12)

26

where H= . F is the column of frequency f from all time points.

4.3 Identifying the Parameters in Reactive Power Model

For parameter identification of the reactive power model, 3 parameters need to be

estimated. However, the measurements can only provide information to estimate one

parameter. Thus two parameters need to be eliminated.

4.3.1 Eliminating two parameters.

Recall that (4.8)~(4.11) are given by:

Thus (4.13)

(4.14)

In the real world monitoring, the power system is a discrete-time system. Therefore,

(4.14) can be written as (4.15) in kth time point:

(4.15)

(4.16)

Recall

and

27

(4.17)

(4.18)

is 0.0001~0.00033s, but the range of is 2~5s. Therefore, one can ignore the

items that are multiplied by .

(4.19)

Compare the left side with the right side, an approximation can be made:

and

Finally, it is obtained that:

(4.20)

where is the current reactive power measured in this bus, is the original

reactive power, V is the voltage at the present time, is the initial voltage.

At this step, the most important parameter is retained while the two less important

parameters are eliminated.

4.3.2 Estimating the parameter .

For factor , it is a coefficient of a non-linear equation, The logarithmic method is

used to change the relationship to be a linear one. That is,

; (4.21)

where Q and V are the current reactive power and voltage of the load while and

are the initial data before the disturbance. Q0 and V0 are from the power flow results

and V and Q are from the simulation results with TSAT.

28

Algorithm to Determine Parameter

Step 1 and Step 2 are the same as the algorithm above.

Step 3: It is known that

; this is a linear equation. One can view

the voltage as an input and Q as an output. The measurement matrix H is

and the

Z(measurement) is

in the estimation theory. is the unknown X.

Step 4: Use the least square method to calculate the unknown variable

X= .

Thus in this case:

=

; (4.22)

where H=

; i=1,2...n.

4.4 Summary

Parameter identification is performed to improve the accuracy of the structure

preserving model and to make the prediction of the system behavior more accurate. There

are two parts of parameter identification. First, identify the coefficient relating the active

power and frequency. Secondly, the coefficient between reactive power and voltage is

identified. For the active power part, the linear estimation method is applied. For the

reactive part, a non-linear estimation procedure is used. An algorithm is developed to

eliminate two less important parameters. Key parameters are determined successfully in

this way. Through the simulation results in Chapter 7, it is shown that key parameters

lead to a highly accurate system model.

29

5. CHAPTER FIVE

ENHANCEMENT OF THE COMPUTATIONAL SPEED

5.1 Computation of Large Scale Power Systems

In this thesis, the computational results of two systems are used: one is the Mini-

WECC system with 122 buses and the other one is the reduced PJM system model with

15000 buses. In the Mini-WECC system, the speed of calculation and required computer

memory are both manageable since it only has 122 buses with 34 generators and 25 load

buses. There are only 150 variables in the structure-preserving model of this system. The

matrix size is smaller than 200*200. The computational speed is very high in Matlab. For

a 10 seconds simulation, the calculation time is 10-20 secondsin Matlab. Reliable

prediction is achieved in 5-8 seconds.

For a large scale system, the PJM system, significant hurdles need to be overcome.

The PJM system has 15,000 buses with 1,090 generator buses, more than 3,000

generators and about 10,000 load buses. Among these buses, 2,544 load buses are heavily

loaded. The total number of variables involved in the differential equations is over 40

thousands, thus the matrix size exceeds 40 thousand by 40 thousand, which means more

than 1.6 billion units memory is needed. It is a very high memory requirement for the

computer. The calculation sometimes cannot be completed due to the lack of sufficient

memory. Besides, for some cases, it needs a very long time to report the correct results.

The problem must be solved before the next step of the simulation can continue.

30

5.2 Aggregation of Generators and Elimination of Buses

5.2.1 Eliminate Buses with No Generation or Load

In some cases, a bus has neither generation nor load. For the PJM system model,

non-generator buses all have load. However, some buses have very light load. Thus, a

reduction method is applied to combine the lightly loaded buses with other load buses.

After careful testing, load buses with load over 0.5 per unit are retained while buses

loaded at less than 0.5 per unit are combined with other load buses. By this procedure, the

total number of load buses is reduced to 2,544, instead of over 10 thousand before the

reduction. The reduction solves the memory problem.

Another option to reduce the dimension of the system model is to retain only those

buses with PMUs installed. However, the accuracy of prediction is lower compared with

the reduction of lightly loaded buses. This is because some buses with PMU installation

is very lightly loaded. These buses are not good indicators for the load behavior. These

lightly loaded buses have a low impact on the system relative to heavy loaded buses.

5.2.2 Combine Generators at the Same Bus

This step is focused on improvement of the computational speed. Since generators at

the same bus will have the same frequency and same voltage, it is practical to combine

generators located at the same bus to an equivalent generator. The combination of

generators will simplify the simulation significantly. Simplifications are performed by:

(5.1)

(5.2)

(5.3)

31

(5.4)

Equation (5.1)~(5.4) must under the same common base value. After the

simplification, every generator bus only has one large generator. Therefore there are

1,090 generator buses with 1,090 generators.

The reduced PJM system model has 3,634 buses in total after the procedure of

model reduction, with 1,090 generator buses and 2,544 load buses. The speed of

calculation decreases from a large number to about 20 minutes.

5.3 Use the Different Method to Calculate Differential Equations

5.3.1 ODE 15s Method

For 122 mini-WECC system, a calculation function called ODE45 is used to solve

the differential equations. This routine uses a variable step Runge-Kutta Method to solve

differential equations. The syntax for ODE45 for first-order differential equations and

second-order differential equations are basically the same. However, for the large scale

PJM system model, ODE45 is not a practical tool. ODE45 has a high level of accuracy

with a low speed of calculation. For a large scale system such as PJM, every calculation

step will generate truncation errors that cannot be ignored. As a result, the ODE45

function needs a very long time to reach its accuracy, which means it is not a practical

tool for the PJM system.

The ODE15 function, which is similar to ODE45 but with a larger mismatch, is used

to replace ODE45. But unfortunately, the result did not improve much.

5.3.2 Forward Euler Method

The second approach is to apply the Forward Euler method:

32

(5.5)

The Forward Euler method is easier and has a lower accuracy than the previous

methods. However, its calculation speed is very good. Due to the fewer number of

iterations, the cumulative error of truncation is smaller than the two previous methods.

Thus, the calculation time decreases to 2 minutes.

5.4 Improving the speed of MLE calculation

As introduced in Chapter 2, Maximum Lyapunov Exponent (MLE) is a stability

characteristic of the power system. There are different ways to calculate the MLE with

different levels of computational speed and accuracy.

Fig 5.1 Concept of MLE(Maximum Lyapunov Exponent)

1. Jacobian method. [17]

In this method, calculation of the Jacobian matrix is needed. However, there are

many approximations and truncations when the Jacobian is computed. The small errors in

every step will accumulate to form a large bias term that could not be ignored.

Furthermore, the error will result in wrong predictions of the future system behavior.

Also, calculation of the Jacobian is very time consuming. Calculation of cos or sin will

takes 10 times the effort more than multiplication, and 40 times more than addition. In a

large system such as the PJM system, the approach is not practical. The author uses the

1x 2x

1x

2sx

1sx2x

tt

Small disturbed trajectory

Original trajectory x f x

33

Taylor expansion first to develop an equation that includes the Jacobian matrix for

calculation of the MLE:

1 12

1 1

ln lnt

I J x t xx

x x

(5.6)

=

+

; (5.7)

The problem is that a real system is not simple compared with the 3-bus system in [17].

The level of accuracy in other part of the program is about The truncation

error of the Taylor expansion is higher than , so it is not accurate to use the

Jacobian method to calculate the MLE.

2. Direct Calculation[28]

This method is called a directed method because it does not require a system model.

The availability of voltage measurements is sufficient. The direct method is not

applicable to the PJM system model because it is aimed at voltage stability while the

focus of this project is on rotor stability. To guarantee a high level of accuracy for this

method, all data must be obtained from PMU measurements. However, for the PJM

system, not all preserved buses have PMU. For the bus without PMUs, its voltage will

not be measured. Consequently, its stability cannot be calculated by the direct method.

Even For those buses with PMUs, the measurements are not available for all time points.

In other words, the direct method can only obtain Lyapunov Exponent from buses which

have PMU and choose the maximum one among them as MLE. Thus, this MLE cannot

represent the state of the buses without PMU. Furthermore, the MLE cannot represent the

state of the entire system. Therefore, the MLE obtained by direct calculation is called

"local MLE", which cannot be used as the stability indicator for the entire system.

34

3.Simplied Jacobian method

This is used for the PJM system model. The waveforms of variables are obtained

after calculation of the power system condition following a fault. Several seconds of

these waveforms are selected and a small disturbance is introduced. The updated

waveforms will be different from the original waveforms. where

is the differential matrix of variables instead of the Jacobian matrix and is the

original waveform. The detailed algorithm is given as follow:

(5.8)

(5.9)

(5.10)

(5.11)

(5.12)

=

+

(5.13)

5.5 Summary

The speed of calculation speed is improved step by step. First, the exact reasons

behind the low calculation speed are determined for the large scale PJM system model.

Then generators at the same bus are combined to simplify the topology. Buses without

generator or load are also eliminated in this step for the same purpose. Next, a method for

calculation of differential equations is identified. The Euler method is chosen for this case

and it works well. At this time, the speed of calculation has improved by about 90%.

Finally, a new MLE calculation algorithm is developed. It reduces time and improves the

accuracy significantly by reducing truncation errors.

35

6. CHAPTER SIX

CASE STUDY AND SIMULATION RESULTS

6.1 Mini-WECC System Simulation Model

In this research, the Mini-WECC system is used to validate the proposed models and

algorithms. The Mini-WECC system is a reduced-order model of the WECC system

designed for the study of power oscillation issues. The system has 122 buses with 34

generators, 171 lines, and 25 loads. Fig.6.1 shows the one-line diagram of this system,

but it only includes high-voltage buses. Loads and generators are connected via step-up

transformers.

The Matlab toolbox PSAT is used to perform time-domain simulation of this

system. The detailed 6-order model is used for generators. That is,

(6.1)

(6.2)

(6.3)

(6.4)

(6.5)

(6.6)

The loads are modeled as a generalized exponential voltage frequency dependent

model [29], i.e.,

(6.7)

(6.8)

36

Fig 6.1 One-line Diagram of Mini-WECC System

37

Load Bus

Number

8 11 16 21 24 26 29 36 43

D 1.482 0.874 1.118 0.972 1 0.144 0.862 0.489 5.31

qT 3 5 3 3 3 3 3 3 3

qsn 1.9 1.8 2 2 2 2 2 2 2

qtn 2 2 2 2 2 2 2 2 2

Load Bus

Number

49 50 55 56 64 69 70 73 78

D 1 4.313 1.127 2.626 4.237 1 1.984 2.809 0.762

qT 3 3 3 3 3 3.5 3.5 4 4

qsn 2 2 2 2 2 2 2 2 2

qtn 2 2 2 2 2 1.9 1.9 2 2

Load Bus

Number

89 95 109 112 113 120 121

D 1 0.869 0.619 3.522 1 3.839 1

qT 4 4 4 4 4 4 4

qsn 2 2 2 2 2 2 2

qtn 2 2 2 2 2 1.9 2

Table 6-1 System Parameter Verification

Take bus 8 as an example. The waveform of its reactive power from TSAT is shown

in Figure 6.2. The reactive load response of the structure preserving model in Matlab

under the same voltage input with different parameters is shown from Fig.6.3-Fig.6.5.

is calculated using a non-linear parameter identification method. It is observed

that with the parameter (qT ,

qtn , qsn ) =(3, 2, 1.9), the reactive load response is the closest

to the simulation result. Note that in this Mini-WECC system, are not

calculated from measurements. They are estimated from the comparison of Matlab

programming and TSAT simulation. Thus, accuracy cannot be guaranteed. Note: in the

following, the SP model means structure preserving model.

38

Fig. 6.2 Reactive Power Load at Bus 8 from PSAT Simulation

Fig. 6.3 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs=1

0 1 2 3 4 5 6 7-980

-960

-940

-920

-900

-880

-860

-840

-820

-800

-780

time(s)

reac

tive

pow

er o

f ps

at

0 1 2 3 4 5 6 7-980

-960

-940

-920

-900

-880

-860

-840

-820

-800

-780

time(s)

reactive p

ow

er

of

recovery

model

39

Fig. 6.4 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs=1.9

Fig. 6.5 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs=2

0 1 2 3 4 5 6 7-980

-960

-940

-920

-900

-880

-860

-840

-820

-800

-780

time(s)

reactive p

ow

er

of

recovery

model

0 1 2 3 4 5 6 7-1000

-950

-900

-850

-800

-750

time(s)

reactive p

ow

er

of

recovery

model

40

In this research, 120 fault scenarios have been considered. The results of time-

domain simulation, reduced dynamic model, structure preserving model with frequency

dependent load (“P model"), and structuring preserving model with both frequency-

dependent load and voltage dependent load (“PQ model") are compared for each

scenario, together with the MLE. Detailed results are shown in Appendix A. Some typical

cases are listed in Table 6.2. In this table, the green color means that the MLE result of

this model is consistent with the time-domain simulation result, while the orange color

means the opposite.

Fault Type Fault

Location

Clear

Time Reduced

Model MLE “P”

Model MLE “PQ”

Model MLE Simulation

Results

Generator trip 1 0.1s stable -0.2408 stable -0.2708 stable -0.2816 stable

Generator trip 5 1s stable -0.1189 unstable 0.2101 unstable 2.2087 unstable

Generator trip 10 1s stable -0.1232 unstable -0.1201 unstable 0.1247 unstable

Generator trip 17 1s stable -0.1225 stable -0.1435 unstable 4.7173 unstable

Line Outage 19-20 0.2s stable -0.0599 stable -0.1425 unstable 7.7455 unstable

Line Outage 66-67 0.5s stable -0.0390 unstable 1.8246 unstable 0.2894 unstable

Line Outage 89-90 0.5s stable -0.0464 unstable 0.693 unstable 0.2825 unstable

Line Outage 105-104 0.5s stable -0.0491 stable -0.1444 stable -0.1451 unstable

Table 6-2 Typical Examples

The results of these cases can be categorized into 4 types.

Type 1: All three models, i.e., load-reduced model, “P” model and “PQ” model are

consistent with the results of time domain simulation. Specifically, the assessments of the

MLEs are consistent with the simulation results.

Type 2: The structure preserving “P” model and “PQ” model are accurate, while the

load-reduced model fails to produce an accurate prediction. Fig.6.6 shows one of the

typical cases, in which a three phase fault occurs near bus 17 and it is cleared after 0.1s

by open breaker at line17-18. From the waveforms of relative rotor angle and angular

speed of generators, it can be observed that the results of structure preserving models are

41

unstable, which is consistent with time domain simulation results, while the assessment

by the load-reduced model is not correct.

Type 3: Only the structure preserving “PQ” model produces an accurate prediction.

Specifically, only the assessment of the MLE by the "PQ" model is consistent with the

simulation results.

Type 4: All three models fail to achieve an accurate prediction. Specifically, the

assessments of the MLEs are not consistent with the simulation results.

Within the total 120 cases, the load-reduced model leads to 72 cases that are

consistent with time-domain simulation results, which represents an accuracy rate of

60%. The structure preserving “P model" results in 85 consistent cases, which is a

70.83% accuracy rate. The “PQ model" has 110 consistent cases, which is a 91.33%

accuracy rate. Besides, in 10 out of the 120 cases all three models fail. From the result,

the structure preserving “PQ model" has the highest accuracy level compared to the other

two models. As a result, the MLE technique should adopt the structure preserving “PQ

model" to achieve an accurate assessment of system stability.

Fig. 6.6 shows the situation of type 2.

Relative rotor angle Relative rotor angular speed

(a) Load-Reduced Model

0 1 2 3 4 5 6-1

-0.5

0

0.5

1

1.5

time(sec)

Rela

tive R

oto

r A

ngle

(Radia

n)

0 1 2 3 4 5 6-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

time(sec)

Rela

tive R

oto

r S

peed o

f G

enera

tors

(Radia

n/s

)

time(s) time(s)

42

(b) Structure-Preserving “P Model"

(c) Structure-Preserving “PQ Model"

(d) Time-Domain Simulation Result

Fig. 6.6 Comparison of Different Models

6.2 Application to the Reduced PJM System

As explained in Chapter 5, PJM is a transmission operator in the Eastern

Interconnection that provides electric power for District of Columbia and 13 states, such

0 1 2 3 4 5 6-20

0

20

40

60

80

100

120

140

time(sec)

Rela

tive R

oto

r A

ngle

of

Genera

tors

(Radia

n)

0 1 2 3 4 5 6-5

0

5

10

15

20

25

30

time(sec)

Rela

tive R

oto

r S

peed o

f G

enera

tors

(Radia

n/s

)

0 0.5 1 1.5 2 2.5-10

-5

0

5

10

15

20

25

30

35

time(sec)

Rela

tive R

oto

r A

ngle

of

Genera

tors

(Radia

n/s

)

0 0.5 1 1.5 2 2.5-10

-5

0

5

10

15

20

25

time(sec)

Rela

tive R

oto

r S

peed o

f G

enera

tors

(Radia

n/s

)

0 1 2 3 4 5 6-10

0

10

20

30

40

50

60

70

0 1 2 3 4 5 6-5

0

5

10

15

20

25

time(s) time(s)

time(s) time(s)

time(s) time(s)

43

as Delaware, Illinois, Indiana. PJM manages the high-voltage electric power grid and

serves more than 61 million people. The system has 15,000 buses with 1,390 generator

buses and over 10,000load buses. It has both DC and AC load, single and multiple

generators buses. It has very complex structure and requires1.6 billion memory units.

The structure preserving load model is applied to the PJM system model. Here are

the simulation results for the PJM system:

Fault Location TSAT Partial SP Reduced Model

Line outage Fault-on time = 0.2s

Bus 1 Bus2 Simulation Results MLE Prediction MLE Prediction

1 5485 stable -0.284 2.5258

1 13125 stable -0.2827 2.5264

2 2481 unstable 2.901 2.5251

2 9065 unstable 3.1188 2.5244

2 13003 unstable 2.8516 2.5252

2 13473 unstable 3.2702 2.5255

3 899 unstable -0.2925 2.5254

3 5483 unstable -0.3458 2.5256

3 8489 unstable -0.3064 2.5262

4 6384 stable 7.3888 2.5251

4 6896 unstable 165 2.5257

4 10108 unstable 18.225 2.5257

5 5964 unstable 31.6393 2.5257

5 15139 stable 20.8166 2.5256

6 5754 stable 8.6514 0.000384

7 2315 stable 7.4794 2.5256

7 8877 stable 6.1016 2.5255

8 6847 stable -0.2767 2.5257

8 11024 stable -0.2776 2.5257

9 436 stable -0.2827 2.5258

9 3568 stable -0.2828 2.5258

9 6182 stable -0.2828 2.5257

9 13660 stable -0.2826 2.5257

10 2788 stable -0.2826 2.5257

10 10490 stable -0.2825 2.5257

Table 6-3 Results of Two Load Models in PJM System

44

The green color means the MLE is consistent with the time domain simulation result

while the black color means they are inconsistent. It is clear that partial structure-

preserving has a much higher level of accuracy than that of the reduced load model. Only

40% results of the reduced load model are consistent with the simulation results from

TSAT. In contrast, 72% of the results of the structure-preserving model are consistent

with those obtained from TSAT.

The results of these cases can be categorized into 3 types.

Type1: Both load-reduced model and partial structure-preserving model are

consistent with the results of time domain simulation. Specifically, the assessments of the

MLEs are consistent with the simulation results. Fig 6.7 shows one typical case. A three

phase fault occurs on line 2-9025 and it is cleared after 0.1s by open breaker at the line.

Type 2: The partial structure preserving model is accurate, while the load-reduced

model fails to produce an accurate prediction. Specifically, the assessment of the MLE

from structure preserving model is consistent with the simulation results but the

assessment of the MLE from load reduced model is not consistent with the simulation

results. Fig.6.8 shows one of the typical cases, in which a three phase fault occurs on line

1-5485 and it is cleared after 0.2s by open breaker at line1-5485.

Type 3: Both two models give an inaccurate prediction. Specifically, the

assessments of the MLEs are not consistent with the simulation results. Fig.6.9 shows one

of the typical cases, in which a three phase fault occurs on line 7-2315 and it is cleared

after 0.1s by open breaker at line7-2315.

Type 4: The partial structure preserving model is inaccurate, while the load-reduced

model produce an accurate prediction. Specifically, the assessment of the MLE from

45

structure preserving model is not consistent with the simulation results but the assessment

of the MLE from load reduced model is consistent with the simulation results.

Type1:

Fig 6.7 Comparison of Two Models for PJM System(Type 1)

Reduce Load Model, MLE=2.524. Prediction: unstable. Right!

Partial Structure-Preserving P Model, MLE= 3.1188, Prediction: unstable. Right!

Type2:


46

Reduce Load Model, MLE2.5258. Prediction: unstable. Wrong!

Partial Structure-Preserving P Model, MLE= =-0.284, Prediction: stable. Right!

Type3:


Reduce Load Model, MLE2.526. Prediction: unstable. Wrong!

Partial Structure-Preserving P Model, MLE= =7.4794, Prediction: unstable. Right!

6.3 Analysis and Summary

The structure preserving model shows a much higher level of accuracy realtive to

reduced model in both Mini-WECC system and the large scale PJM system. The

partial structure preserving model that only preserves heavily loaded buses also shows

high accuracy. It means that there is no need to preserve all buses of the entire power

system.

47

7. CHAPTER SEVEN

CONCLUSIONS

A PMU-based online waveform stability monitoring technique is proposed based on

the Maximum Lyapunov Exponent (MLE). The main idea of the MLE technique is to

calculate MLE as an index over a finite time window in order to predict unstable trending

of the operating conditions. The MLE technique is based on solid analytical foundation

and it is computationally efficient. Significant progress has been made to improve the

accuracy of MLE technique in the following aspects:

1. The online prediction of the power system is much improved by adopting a

structure-preserving model that takes the dynamical correlation between real/reactive

power load and the load bus frequency/voltage variations into account. An effective

model called a partial structure-preserving model is further developed. This model only

preserves the heavily loaded buses while removing the lightly loaded buses. The partial

structure preserving model has the same accuracy as that of the complete structure

preserving model while the speed of calculation is much higher.

2. A highly efficient MLE prediction method is developed. The MLE technique is

extended to voltage stability analysis as well as rotor angle stability. Based on this model,

the system can be represented by a set of differential equations, which is suitable for

MLE calculation. There are several techniques for the calculation of MLE. The Tylor

first order expansion is used for the MLE calculation in this project.

3. Traditionally, the ODE or DAE method is applied in calculation of differential

equations. These routines use a variable step Runge-Kutta Method to solve differential

equations. However, this method is not practical for the large scale PJM system. ODE has

48

a very high level of accuracy but slow speed of calculation. For a large scale system such

as the PJM system, every calculation step will generate truncation errors that cannot be

ignored. As a result, ODE is too slow to be applied to the PJM system. Therefore, the

Forward Euler method is applied in this research. The Forward Euler method is easier and

has a lower level of accuracy relative to the ODE or DAE method. However, its

calculation speed is much better. Due to the fewer number of iterations, the cumulation of

the truncation errors is smaller.

4. The proposed methods are validated by time domain dynamic simulations. The

dynamic model of a 122-bus Mini-WECC system has been established in Matlab-based

simulation toolbox PSAT. The large scale PJM system model has been established in

TSAT. PSAT and TSAT are used to perform time-domain simulation as a reference for

validation of the MLE method. From the simulation results, it is clearly demonstrated that

the structure-preserving model with load recovery process is the most accurate one

among the three models. Thus, the structure-preserving model is reliable and it can be

used for online rotor angle stability analysis as well as the short-term voltage stability

based on the MLE technique. It is also clear that, with the new method, the accuracy of

prediction is improved significantly.

5. Parameter identification is developed in this research to improve accuracy in the

structure preserving model. Both linear and non-linear estimation methods are applied in

this research. Parameters are determined according to their importance. Key parameters

are identified successfully. Small residuals show that the proposed estimation method is

accurate. Hundreds of the simulation results from Chapter 6 also validate that.

49

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52

APPENDIX

53

APPENDIX I. All Tested Results of Mini-WECC System

Fault Type Fault

location

Clear

Time

Reduced Model and

MLE

“P” Model and MLE “PQ” model and MLE Simulatio

n Results

Generator

fault 1 0.1 stable -0.2408 stable -0.2708 stable -0.2816 stable

0.2 stable -0.2118 stable -0.2611 unstable 0.6446 unstable

1 unstable 0.0167 unstable 0.3720 unstable 0.1670 unstable

Generator

Trip 2 0.1 stable -0.1630 stable -0.2708 unstable 0.7365 unstable


1 stable -0.1590 stable -0.2669 unstable 3.6259 unstable

Generator

Trip 3 0.1 stable -0.0617 stable -

0.4078

stable -0.1432 stable


1 stable -0.1142 unstable 0.0466 unstable 0.1345 unstable

Generator

Trip 4 0.1 stable -0.0622 stable -0.2748 stable 1.7748 stable

0.2 stable -0.0664 stable -0.2748 stable -0.2754 stable


Generator

Trip 5 0.1 stable -0.0608 stable -0.3718 stable -

0.2573

stable



Generator

Trip 6 0.1 stable -0.0610 stable -0.1441 stable -0.3745 stable


1 stable -0.1237 stable -0.3679 stable -0.2636 stable

Generator



1 unstable 0.3064 stable -0.2501 unstable 0 unstable

Generator




Generator




Generator



1 stable -0.1232 unstable -0.1201 unstable 0.1247 unstable

Generator




Generator 13 0.1 stable -0.0605 stable -0.2716 stable -0.2786 stable

54

Trip



Generator




Generator

Trip 17 0.1 stable -0.0606 stable -0.2721 stable 5.6360 unstable

0.2 stable -0.0555 stable -0.2751 stable 7.4599 unstable


Generator




Generator




Generator




Generator




Generator

Trip 27 0.1 stable -0.0602 stable -0.2721 stable -7.7126 unstable



Generator




Line

Outage 37-38 0.1 stable -0.1203 stable -7.7157 stable -0.2715 stable



Line

Outage 3-4 0.1 stable -0.0519 stable -0.1397 stable -0.3885 unstable

0.2 unstable 0.0578 unstable 0.2739 unstable 3.0669 unstable


Line




Line

Outage 27-28 0.1 stable -0.1149 stable -

11.0276

stable -0.2625 stable

55


0.5 unstable -0.1183 stable -7.7235 stable -0.2588 stable

Line

Outage 12-13 0.1 unstable 0.0407 unstable 0.3794 unstable 0.1120 unstable



Line

Outage 105-

104

0.1 stable -0.0487 stable -0.3670 stable -0.2809 unstable



Line



0.5 stable -0.2367 unstable 7.6671 unstable 0.0021 unstable

Line




Line




Line

Outage 17-18 0.1 stable -0.0414 unstable 0.3892 unstable 3.0291 unstable



Line




Line




Line

Outage 19-20 0.1 stable -0.0290 stable -0.3866 unstable 9.8170 unstable



Line




Line




Line

Outage 48-49 0.1 stable -0.0718 stable -0.3881 unstable 10.050 unstable


56


Line




Line




Line



0.5 stable -0.0464 unstable +0.693 unstable 0.2825 unstable

Line




real time pmu-based stability monitoring by zijie lin · monitoring algorithms for the power grids...

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