Power system dynamic state estimation and load …...POWER SYSTEM DYNAMIC STATE ESTIMATION and LOAD MODELING A Thesis Presented by Cem Bila to The Department of Electrical and Computer

POWER SYSTEM DYNAMIC STATE

ESTIMATION and LOAD MODELING

A Thesis Presented

by

Cem Bila

to

The Department of Electrical and Computer Engineering

in partial fulfillment of the requirements

for the degree of

Master of Science

in

Electrical and Computer Engineering

Northeastern University

Boston, Massachusetts

September 2013

http://www.ece.neu.edu/

http://www.northeastern.edu/

c© copyright by Cem Bila 2013

All Rights Reserved

Northeastern University

Abstract

Department of Electrical and Computer Engineering

Master of Science in Electrical and Computer Engineering

by Cem Bila

http://www.northeastern.edu/

iii

State estimation which constitutes the core of the Energy Management Sys-

tem (EMS), plays an important role in monitoring, control and stability analysis

of electric power systems. An efficient, timely and accurate state estimation is a

prerequisite for a reliable operation of modern power grids.

Traditional state estimators, which are based on steady state system model, can-

not capture the system dynamics very well due to the slow updating rate of SCADA

systems. In mid 1980s of the 20th century, Phasor Measurement Unit (PMU)-based

Wide-Area Measurement Systems (WAMS) emerged. The introduction of this high

speed measurement systems, featured with synchronous sampling, has revolutionized

the way state estimation process is being performed. These lead to the development

of Dynamic State Estimation (DSE) techniques, which enables the dynamic view of

power systems in the control center. Various techniques are available in literature

for dynamic state estimation which can be applied to power systems.

In this thesis, the power system dynamic state estimation process, based on

Kalman Filtering techniques, is discussed. The dynamic state variables of multi-

machine power systems which are generator rotor speed and generator rotor angle

are estimated. The computational performance of Extended Kalman Filter (EKF)

and Unscented Kalman Filter (UKF) algorithms in the estimation process of the

dynamic state vector of the power systems are compared. The plots of the dynamic

state variables, rotor speed and rotor angle, are observed under various transient

conditions. It is verified that both EKF and UKF are sufficient techniques in esti-

mation of dynamic state vector elements under transient conditions. Although EKF

is one of the most widely used methods in power system dynamic state estimation

process, it is investigated that the linearization and Jacobian matrix calculation can

lead to some drawbacks. The UKF algorithm which is based on unscented transfor-

mation is introduced as a more effective method. It is demonstrated that UKF is

easier to implement and more accurate in estimation.

In addition, this thesis describes the load modeling issues in electric power sys-

tems. It is an obvious fact that the accuracy of load model is a very important factor

effecting the power system stability analysis and control. In this work, the parameter

iv

estimation for assumed ZIP load model is performed based on the Weighted Least

Square (WLS) estimation method. In order to obtain more reliable and precise cal-

culations of power system state estimation studies, a more accurate load modeling

can be developed and integrated into the dynamic state estimation process of power

systems as a future work.

Acknowledgments

First and foremost, I would like to express my deepest gratitude to my advisor,

Professor Ali Abur, for providing me the opportunity to conduct research in the

field of power systems at Northeastern University. I am grateful to his guidance,

support and patience throughout this M.S study. I am enlightened by his wealth

of knowledge and expertise. This thesis would never been accomplished without his

constructive suggestions, invaluable comments and tireless efforts on teaching. It

has been an honor and a great pleasure to be his research assistant.

I would like to extend my gratitude to Professor Hanoch Lev-Ari and Professor

Bahram Shafai for serving as my masters thesis committee members. I appreciated

that they shared their opinions, knowledge and suggestions concerning my thesis

topic.

I would like to thank Faith Crisley, Graduate Coordinator at the Department

of Electrical and Computer Engineering, for her help and invaluable advice during

the rough times of my study at Northeastern University.

I am indebted to express my gratitude to my friends who have always been there

to give me wonderful support and encouragement.

Finally, I would like to give special thanks to my parents for their endless love

and support.

v

Contents

Abstract ii

Acknowledgments v

List of Figures viii

List of Tables xi

1 Introduction 1

1.1 Motivations for the Study . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Power System Model and Transient Stability 10

2.1 Power System Dynamics and Swing Equation . . . . . . . . . . . . . 10

2.2 Numerical Integration Methods for the Swing Equation . . . . . . . . 12

2.2.1 Euler Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2.2 Second order Runge-Kutta integration: Trapezoidal rule . . . 14

2.2.3 Fourth order Runge-Kutta integration . . . . . . . . . . . . . 15

2.2.4 Simulation Studies and Results . . . . . . . . . . . . . . . . . 16

2.3 Multimachine Stability Studies . . . . . . . . . . . . . . . . . . . . . 18

2.3.1 Classical Model and Assumptions . . . . . . . . . . . . . . . . 19

2.3.2 Preliminary Calculations for Initialization . . . . . . . . . . . 20

3 Use of EKF and UKF Techniques in Power System DSE 26

3.1 Mathematical Modeling . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.2 Extended Kalman Filter (EKF) Algorithm . . . . . . . . . . . . . . . 27

3.3 Unscented Kalman Filter (UKF) Algorithm . . . . . . . . . . . . . . 30

3.4 Simulation Studies and Results . . . . . . . . . . . . . . . . . . . . . 35

3.4.1 Description of Simulation Studies . . . . . . . . . . . . . . . . 35

3.4.2 3-Generator 5-Bus Power System . . . . . . . . . . . . . . . . 38

3.4.3 3-Generator 9-Bus Power System . . . . . . . . . . . . . . . . 44

vi

Contents vii

3.4.4 IEEE 5-Generator 14-Bus Power System . . . . . . . . . . . . 56



3.4.7 IEEE 10-Generator 39-Bus Power System . . . . . . . . . . . 66


3.4.9 50-Generator 145-Bus Power System . . . . . . . . . . . . . . 73

3.4.10 17-Generator 162-Bus Power System . . . . . . . . . . . . . . 76

4 Parameter Estimation for ZIP Load Modeling 80

4.1 Polynomial (ZIP) Load Model . . . . . . . . . . . . . . . . . . . . . . 80

4.2 Revised Power Flow Program Simulating ZIP Load . . . . . . . . . . 82

4.3 ZIP Load Parameter Estimation Algorithm . . . . . . . . . . . . . . . 83

5 Concluding Remarks and Further Study 88

5.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.2 Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

A Matlab Script Example for EKF Algorithm 90

B Matlab Script Example for UKF Algorithm 132

Bibliography 157

List of Figures

2.1 An illustration of Euler method. . . . . . . . . . . . . . . . . . . . . . 13

2.2 Plot of w1 when ∆t = 0.0005s for the 3-generator 9-bus power system 16

2.3 Plot of w1 when ∆t = 0.01s for the 3-generator 9-bus power system . 17



2.6 Multimachine power system representation (classical model) for tran-sient stability study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.7 Classical model for simplified synchronous generator . . . . . . . . . . 20

3.1 Kalman Filter is simply a two-step prediction-update process . . . . . 36

3.2 The one-line diagram of the 3-generator 5-bus power system . . . . . 39

3.3 Estimation of ω1 by EKF for the 3-generator 5-bus power system . . 39

3.4 Estimation of ω1 by UKF for the 3-generator 5-bus power system . . 40





3.9 Estimation of δ2−1 by EKF for the 3-generator 5-bus power system . . 42

3.10 Estimation of δ2−1 by UKF for the 3-generator 5-bus power system . 43

3.11 Estimation of δ3−1 by EKF for the 3-generator 5-bus power system . . 43



3.14 Estimation of ω1 by EKF for the 3-generator 9-bus power systemconsidering the transient case 1 . . . . . . . . . . . . . . . . . . . . . 45

3.15 Estimation of ω1 by UKF for the 3-generator 9-bus power systemconsidering the transient case 1 . . . . . . . . . . . . . . . . . . . . . 46





3.20 Estimation of δ2−1 by EKF for the 3-generator 9-bus power systemconsidering the transient case 1 . . . . . . . . . . . . . . . . . . . . . 48

viii

List of Figures ix

3.21 Estimation of δ2−1 by UKF for the 3-generator 9-bus power systemconsidering the transient case 1 . . . . . . . . . . . . . . . . . . . . . 49
















3.37 Estimation of δ5−1 by EKF for the 5-generator 14-bus power system . 58












3.49 Estimation of ω10 by EKF for the 10-generator 39-bus power system . 67

3.50 Estimation of ω10 by UKF for the 10-generator 39-bus power system . 68

3.51 Estimation of δ10−1 by EKF for the 10-generator 39-bus power system 68

3.52 Estimation of δ10−1 by UKF for the 10-generator 39-bus power system 69

List of Figures x






3.58 Estimation of ω45 by EKF for the 50-generator 145-bus power system 74

3.59 Estimation of ω45 by UKF for the 50-generator 145-bus power system 74



3.62 Estimation of ω10 by EKF for the 17-generator 162-bus power system 77

3.63 Estimation of ω10 by UKF for the 17-generator 162-bus power system 77



4.1 An illustration of ZIP load model . . . . . . . . . . . . . . . . . . . . 82

4.2 Daily load profile for 14-bus system . . . . . . . . . . . . . . . . . . . 84

4.3 Estimation of α4 in the 14-bus power system . . . . . . . . . . . . . . 85

4.4 Estimation of β4 in the 14-bus power system . . . . . . . . . . . . . . 86

4.5 Estimation of γ4 in the 14-bus power system . . . . . . . . . . . . . . 86

List of Tables

3.1 Generator dynamic data of the 3-generator 5-bus power system . . . 38

3.2 Performance indices of EKF and UKF for the 3-generator 5-bus powersystem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44


3.4 Performance indices of EKF and UKF for the 3-generator 9-bus powersystem considering the transient case 1 . . . . . . . . . . . . . . . . . 50

3.5 Performance indices of EKF and UKF for the 3-generator 9-bus powersystem considering the transient case 2 . . . . . . . . . . . . . . . . . 56


3.7 Performance indices of EKF and UKF for the 5-generator 14-buspower system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59





3.12 Generator dynamic data of the 10-generator 39-bus power system . . 67






4.1 Power flow solution of 14-bus system with ZIP load . . . . . . . . . . 83

xi

Chapter 1

Introduction

The rapid growth and increasing complexity in recent years makes the monitor-

ing and control of power systems a very significant issue. The Energy Management

Systems (EMS) at the control centers are responsible for this task of monitoring

and control of the system. The state estimator, which is the backbone of the en-

ergy management systems, provides a optimum real time data of the system state

based on the available measurements on the assumed system model. The efficiency

and accuracy of the state estimator output is very crucial as it forms the basis for

the EMS functions such as security analysis, automatic generation control, opti-

mal power flow and load forecasting. Thus, the concept of state estimation plays a

major role in ensuring the secure and economic operation of the power systems in

large-scale interconnected power grids. Depending on the desired states (static or

dynamic), power system state estimation can be formulated as a static or dynamic

estimation problem.

1.1 Motivations for the Study

The state estimation process for a power network is determining the best esti-

mate of the present state of the system by collecting the real time measurements from

the sensors monitoring the grid. The vector consisting of bus voltage magnitudes

and bus voltage phase angles is called the static vector of an electric power system.

1

Chapter 1. Introduction 2

The real time measurement data gathered from the network and used in the esti-

mation process includes power injections, power flows on the transmission lines and

voltage magnitudes at each bus of the system. The telemetered measurement data is

received through the Supervisory Control and Data Acquisition Systems (SCADA)

and the state vector is estimated by using the predetermined state estimation al-

gorithm and power system model. If the state vector is obtained for the current

instant of time k from the set of measurement data received at the same instant of

time k, then such an estimation method is called as Static State Estimation (SSE).

In static state estimation, the snap-shot of the measurements are taken, processed

and the estimate of the state vector variables is obtained at the same point of time.

The static state estimators, having a crucial role in the reliable operation of the

transmission and distribution systems, are widely used in the power system state

estimation.

The power systems are defined as quasi-static systems. This means that they

change with time very slowly but steadily. The change of the power system is driven

by the continuous variation of the loads. As the loads change, the generators feeding

the network need to be adjusted in accordance with the load variations. This results

in the change of power injections and power flows which makes the system dynamic.

The resulting dynamic changes need to be monitored continuously and therefore

the power system state estimation needs to be performed in short interval of time.

The static state estimators can not efficiently and accurately capture this dynamic

behavior of the expanded large power networks. Consequently, another method is

developed in order to monitor the continuous dynamic changes in power systems

which is called as Dynamic State Estimation (DSE). By using the actual physical

modeling of the time varying nature of the power system, DSE algorithm predicts

the system state at the next instant of time k + 1 along with the state estimates

obtained at the previous instant of time k. The DSE method has a vital advantage

such that it allows the prediction of the system state at one time step ahead. Hence,

the forecasting ability of the DSE algorithm plays a major role in the improvement

of the overall energy management system operation and control.

The performance and the accuracy of the state estimators in power systems


heavily depends on the measurements gathered from the network. Traditionally, the

measurements including injections, flows and voltage magnitudes, are collected by

the SCADA systems via Remote Terminal Units (RTU) and processed in the state

estimation algorithms at the control center. In mid 1980s, a new device has devel-

oped which is called as Phasor Measurement Unit (PMU). The main importance of

this device lies in the fact that it can measure both the voltage phasor and current

phasor at the system buses where the device is present. By the emergency of this

synchronized measurement tool, for the first time, both the bus voltage magnitudes

and bus voltage angles can be directly measured which are the state vector ele-

ments. Moreover, PMU measurements are highly accurate compared to the SCADA

measurements. In this regard, the PMUs which work in synchronization with Global

Positioning System (GPS) satellites, are superior to the traditional SCADA systems.

The introduction of highly accurate angular measurement data by means of these

high updating rate synchronized measurement devices play an important role in the

modern day energy management systems. As a result, the PMUs can also be incor-

porated into the dynamic state estimation studies of power systems such that the

dynamic view can be captured in a more accurate and efficient way. The progress

and improvement in the way of monitoring the power networks by the emergence of

these new technologies creates a deep motivation for developing new methodologies

in the field of power system dynamic state estimation.

In the studies of power system operation and analysis, historically, generator and

transmission modeling have received the most attention. However, load modeling has

also become a considerably critical issue due to the increasing stresses on power grids.

The continually increasing complexity of the power networks and the incorporation of

various load components from many sources motivates the researchers to concentrate

on load modeling. The accurate and verified load model plays a major role in the

power system stability analysis. It is clear that more accurate load models need

also be used during the state estimation processes rather than making traditional

assumptions. The parameter estimation of reliable, complex load models by using

the measurements collected by high technology systems (PMUs) seems to become an

important concern which needs to be considered in the power system state estimation


studies.

1.2 Related Work

There has been a considerable progress in the field of power system state estima-

tion since it is firstly introduced by Fred Schweppe in the late 1960s [1–3]. The state

estimation (SE) tool has benefited from large number of theoretical developments

and practical improvements [4]. Various methodologies have been offered regarding

the mathematical formulation, numerical solution, computational procedure, real-

time implementation, measurement types, calculation of the state estimates and

identification of the modeling errors in the literature concerning both static and

dynamic power system state estimation process.

The basic concept of ’Static State Estimation (SSE)’ is defined and several nu-

merical approaches are offered as a solution in [1–3, 5, 6]. The general structure and

main functions of the static state estimator are listed in [7]. One of the most widely

used methods to solve the power system static state estimation problem appears to

be the Weighted Least Squares (WLS) approach. The WLS estimators have been

studied extensively and their numerical stability as well as computational efficiency

have been greatly improved by various techniques [8, 9]. Traditionally, the power sys-

tems are collected by low updating rate SCADA systems via Remote Terminal Units

(RTU). The widely used measurement types in the common state estimation process

can be listed as power injections, power flows, voltage and current magnitudes. More

recently, the Phasor Meausrement Units (PMUs), which provide Global Positioning

System (GPS)-synchronized measurements, among which are voltage and current

phasor magnitude and phase angles, are expected to introduce major improvements

in state estimation SE performance and capabilities [4, 10]. The impact of synchro-

nized phasor measurements on the state estimation function is well described in [11].

The problem of state estimation is investigated in very large systems using phasor

measurement units in [12]. An algorithm is developed which uses known system

topology information, together with PMU phasor angle measurements to detect sys-

tem single and double line outages [13, 14]. Although the WLS method is widely


used, well developed technique in power system state estimation, it is found to be

not robust as it fails in the presence of bad data [15]. The Least Value Estima-

tor (LAV) method is offered in [15], as an alternative to WLS estimator, which is

more robust as it can automatically reject bad-data in the absence of the leverage

measurements. The static state estimation process in power system is normally ac-

complished by without the use of time-history data or prediction [16]. The problem

of state estimation combined with the knowledge of the forecasted load is posed as

a Kalman filtering problem using a novel discrete-time model [16]. The inclusion of

dynamic models provides the basis for performing the state estimation via Kalman

filtering [16].

Estimation of the dynamic state of a power system is the first prerequisite for

control and stability prediction under transient conditions as mentioned in [17]. The

great importance of the ’Dynamic State Estimation (DSE)’ in system monitoring

and control of power systems, especially with the introduction of Phasor Measure-

ment Units (PMUs), is extensively explained in literature [18–22]. A dynamic state

estimator for power systems is firstly addressed by Debs and Larson [23]. In this

work, the state change is represented by Gaussian noise. Then, a novel method for

detecting and identifying anomalies as occurrence of bad-data, changes in network

configuration and sudden variation of states, in dynamic state estimation for electric

power systems is proposed in [24]. Simple dynamic models for the state vector be-

havior, combined with linearized measurement equations, have been proposed and

the estimations have been achieved through Kalman filtering theory [25]. Through

the application of Kalman filter techniques, the set of measurements is used to es-

timate the model state parameters at a first stage and the state vector at a second

stage [25]. New algorithms considering exponential smoothing and least-squares es-

timation techniques are used for forecasting and filtering the state vector for power

systems [25].

One of the most widely used method in power system dynamic state estima-

tion is Extended Kalman Filter (EKF) which takes into account both the incoming

measurements and the predicted state to obtain an optimal estimate of the system


state as mentioned in [26]. Two algorithms are proposed for dynamic state estima-

tion which incorporate the measurement function nonlinearities in the EKF scheme

in [26]. The feasibility of applying Extended Kalman Filter techniques to include

dynamic state variables (generator rotor speed and rotor angle) in the state estima-

tion process is well investigated in [27]. The proposed EKF based dynamic state

estimation procedure is tested on a multi-machine system with both large and small

disturbances [27]. The extended Kalman filter with unknown inputs, referred to as

EKF-UI, is proposed for estimating the states and the unknown inputs of the syn-

chronous machine simultaneously in [28]. A novel framework to perform EKF based

dynamic state estimation in a distributed way is proposed in [29] considering increas-

ing complexity associated with large-scale renewable resources and novel smart-grid

technologies. According to [29], DSE can be implemented in a distributed environ-

ment by decomposing the systems into subsystems to increase the computational

speed of DSE process in large scale power systems.

Although the EKF is one of the most widely used estimation algorithm for

nonlinear systems, more than 35 years of experience in estimation community has

shown that it is difficult to implement, difficult to tune and only reliable for systems

that are almost linear on the time scale of the updates [30]. In order to overcome

the difficulties and drawbacks of the EKF algorithm which mainly arise from its use

of linearization, the Unscented Transformation (UT) is developed as a method to

propogate mean and covariance information through nonlinear transformation [30].

The motivation, use and advantages of proposed Unscented Kalman Filter (UKF),

based on unscented transformation, is illustrated in literature [30–36]. The accu-

racy and easier implementation of UKF in estimating the dynamics of generators

is investigated in [31, 34] with appropriate simulation results. The performance of

the UKF technique is derived, demonstrated and compared with the performance of

classical EKF technique by using three different test power systems under typical

network and measurement conditions in [32]. It is proved by using the performance

indices that the UKF has higher filtering capacities during slow dynamic changes

than EKF estimator [32]. A new parameter estimation method for frequency, am-

plitude and phase tracking based on UKF is presented in [33] and it is shown that


UKF method has high estimation accuracy both under normal and noisy conditions

[33]. A derivative free approach to Kalman filtering is introduced and applied to

state estimation-based control of a class of nonlinear dynamical systems in [35]. A

new method for the simultaneous estimation of power components and frequency is

presented based on UKF method [36].

The concept of load modeling becomes an important issue with the increas-

ing complexity of grids and the emergency of various load components in networks.

An accurate and correct representation of loads is highly significant in power sys-

tem operation, control and stability studies. The need for correct representation of

electrical loads is stability studies are discussed by presenting several series of load-

voltage tests performed on the Southern California Edison system in [37]. In [38], it

is emphasized that the parameters of a second order state space load model can be

identified form actual system load measurements by using a Weighted Least Squares

(WLS) parameter identiifcation process. The development of dynamic load models

for the Taipower system is described in [39]. The development of measurement-

based composite load model which has the structure of a motor in combination with

a static ZIP type is discussed in [40]. The identification of the parameters of this

model is presented based on the multicurve identification technique [40]. The work

[41] indicates that the multi-state physical load models can be used to properly rep-

resent the behavior of end-use loads for distribution system analysis. A multistate

ZIP model of PEVs is discussed and formulated in [42]. The performance of two

algorithms is investigated for complex load model parameter estimation by using

phasor measurements [43]. The load modeling at distribution level with complete

load model structure is discussed in [44]. The composite load modeling is reviewed

and an augmented state estimator based on Local Iterative Extended Kalman Filter

(LIEKF) is proposed to estimate dominant parameters of the load in [45].


1.3 Contributions of the Thesis

The main objective of this thesis is to demonstrate how the application of

Kalman Filtering techniques are valid in estimating the dynamic variables of multi-

machine power systems. As illustrated in earlier studies, the Extended Kalman Filter

(EKF) and Unscented Kalman Filter (UKF) are both sufficient tools to be included

in power system dynamic state estimation studies. In comparison of these two algo-

rithms, it is realized that UKF is more accurate, robust and easy to implement in

the state estimation of dynamic states under various transition cases.

In performing the dynamic state estimation of the multi-machine systems, the

load models are assumed to be constant impedance as in the case of most of the

related work in the literature. A composite load model namely ZIP load model is

also discussed in this study. It is verified that the parameters of the polynomial

which represents the ZIP load model can be estimated by means of the Wighted

Least Square (WLS) algorithm.

1.4 Thesis Outline

This thesis comprises five chapters and it is organized as follows. In the current

chapter, motivations for conducting this research are presented including general

background information, the relevant literature is reviewed and the contributions of

the thesis is briefly discussed.

In the succeeding chapter (Chapter 2), the derivation of the swing equation

is briefly summarized. The classical multi-machine model of power systems based

on several assumptions is explained and the steps for initialization of the system is

listed. The numerical integration methods used in the transient stability computa-

tion procedure of the power systems are explained and compared.

Chapter 3 covers the Extended Kalman Filter (EKF) and Unscented Kalman

Filter (UKF) techniques applied in the estimation of dynamic variables. It compares


the application of EKF and UKF algorithms in power system DSE based on the

results of various simulation studies.

Chapter 4 provides a brief review of the polynomial ZIP load modeling in power

systems. It expresses the mathematical model of composite ZIP loads and discusses

the ZIP load model parameter estimation based on Weighted Least Square (WLS)

technique.

Finally, in Chapter 5, the main contributions of this study to power system

dynamic state estimation and load modeling are discussed. Also, some possible

ideas are mentioned about what can be done for further study.

Chapter 2

Power System Model and

Transient Stability

2.1 Power System Dynamics and Swing Equation

The dynamic equation governing the motion of the machine rotor of a three-

phase synchronous generator is called the swing equation [46, 47]. In rotational

systems, the net accelerating torque acting on a rotating body is the product of

the moment of inertia of the rotor times its angular acceleration which is based

on Newton’s second law [47, 48]. The equation for the rotor motion is given by

[47, 49, 50],

Jαm = Tm − Te − Td = Ta (2.1)

where J is the moment of inertia of the rotating masses in kg.m2, αm is rotor angular

acceleration, rad/s2, Tm is the mechanical torque in N.m, Te is the electrical torque

in N.m, Td is the damping torque in N.m and Ta is the net accelerating torque in

N.m. The rotor angular acceleration is defined as

αm =dωmdt

=d2θmdt2

(2.2)

10

Chapter 2. Power System Model and Transient Stability 11

where ωm = dθmdt

is rotor angular velocity in rad/s and θm is rotor angular posi-

tion with respect to a stationary axis in rad [49]. It is convenient to measure the

rotor angular position with respect to a synchronously rotating reference axis and

accordingly it is defined as

θm = ωmsynt+ δm (2.3)

where ωmsyn is synchronous angular velocity of the rotor in rad/s and δm is the rotor

angular position with respect to a synchronously rotating reference axis in rad [49].

Then, by taking the second order derivative of θm with respect to time, we obtain

d2θmdt2

=d2δmdt2

(2.4)

and by substituting Eq. 2.2 and Eq. 2.4, Eq. 2.1 becomes

Jd2θmdt2

= Jd2δmdt2

= Tm − Te − Td = Ta. (2.5)

It is better to work with power rather than torque. Accordingly, by multiplying

both sides of Eq. 2.5 by angular velocity ωm we obtain

Jωmd2δmdt2

= Tmωm − Teωm − Tdωm (2.6)

and recalling that power is the product of torque and angular velocity, the Eq. 2.1

can be expressed as

Jωmd2δmdt2

= Pm − Pe − Pd (2.7)

where Pm is the mechanical input power in W , Pe is the electrical output in W and

Pd is the damping power in W . At synchronous speed ωmsyn, the angular momentum

Jωmsyn is denoted by M in joules-second per mechanical radian. By using M instead

of Jωm, Eq. 2.7 can be rewritten as the following equation [47]

Md2δmdt2

= Pm − Pe − Pd (2.8)


The normalized inertia constant H, in joules/VA or per unit-seconds, is defined

as [49]

H =stored kinetic energy at synchronous speed

three-phase rating of the generator=

12Jωm

2syn

SB(3φ)

. (2.9)

Replacing Jωmsyn by M and solving the resulting equation for M , we obtain [47]

M =2H

ωmsynSB(3φ). (2.10)

Substituting Eq. 2.10 into Eq. 2.8 and dividing both sides by SB(3φ), the per-unit

expression of the swing equation can be written as

2H

ωmsyn

d2δmdt2

=Pm − Pe − Pd

SB(3φ)

= Pmp.u. − Pep.u. − Pdp.u. (2.11)

where Pmp.u., Pep.u.. Pdp.u. are per-unit mechanical, electrical and damping power

respectively [47]. In Eq. 2.11, δm is expressed in mechanical radians and ωmsyn is

expressed in mechanical radians per second [47]. The Eq. 2.11 can be rewritten as

follow2H

ωsyn

d2δ

dt2= Pm − Pe − Pd in per unit. (2.12)

In literature, the normalized inertia constant M is also defined as 2Hωsyn

[47].

Thus, by substituting M instead of 2Hωsyn

in Eq. 2.12, the convenient form of swing

equation is defined as the following equation

Md2δ

dt2= Pm − Pe − Pd in per unit. (2.13)

2.2 Numerical Integration Methods for the Swing

Equation

In general, systems in the real world are described with continuous-time dy-

namics as in the case of the swing equation. However, state estimation and control

algorithms are almost always implemented in digital electronics [51]. Accordingly,


this often requires to transform the continuous time dynamics to discrete time dy-

namics [51]. In order to perform the numerical integration, it is more convenient to

convert the swing equation into a set of coupled first order differential equations as

follow

Mdω

dt= Pm − Pe − Pd

dδ

dt= ω.

(2.14)

In this section, three numerical integration methods are discussed and compared in

order to obtain a solution for the set of differential equations of the above form.

2.2.1 Euler Method

Consider the first-order differential equation [52]

dx

dt= f(x, t) (2.15)

with x = x0 at t = t0. The Figure 2.1 illustrates the principles of applying Euler

method [52].

Figure 2.1: An illustration of Euler method.

At x = x0, t = t0 the curve representing the true solution by its tangent having

a slope [52]dx

dt

∣∣∣∣x=x0

= f(x0, t0) (2.16)


and therefore,

∆x =dx

dt

∣∣∣∣x=x0

∆t. (2.17)

The value of x at t = t1 = t0 + ∆t is given by

x1 = x0 + ∆x = x0 +dx

dt

∣∣∣∣x=x0

∆t. (2.18)

The Euler method which is the equivalent of using the first two terms of the

Taylor series expansion for x around the point (x0, t0), can be generalized as follows:

xn+1 = xn +dx

dt

∣∣∣∣x=xn

∆t. (2.19)

2.2.2 Second order Runge-Kutta integration: Trapezoidal

rule

Referring to the differential equation Eq. 2.15, the second order R-K method

for the value of x at t = t0 + ∆t is [52]

x1 = x0 + ∆x = x0 +k1 + k2

2(2.20)

where

k1 = f(x0, t0)∆t

k2 = f(x0 + k1, t0 + ∆t)∆t.

This method is equivalent to considering first and second derivative terms in

the Taylor series and the general formula giving the value of x for the (n+ 1)st step

is [52]

xn+1 = xn +k1 + k2

2(2.21)


where

k1 = f(xn, tn)∆t

k2 = f(xn + k1, tn + ∆t)∆t.

2.2.3 Fourth order Runge-Kutta integration

The general formula giving the value of x for the (n+ 1)st step is [52]

xn+1 = xn +1

6(k1 + 2k2 + 2k3 + k4) (2.22)

where

k1 = f(xn, tn)∆t

k2 = f(xn + k12, tn + ∆t

2)∆t

k3 = f(xn + k22, tn + ∆t

2)∆t

k4 = f(xn + k3, tn + ∆t)∆t

The physical interpretation of the above solution is as follows [52]:

k1 = (slope at the beginning of time step)∆t

k2 = (first approximation to slope at midstep)∆t

k3 = (second approximation to slope at midstep)∆t

k4 = (slope at the end of step)∆t

∆x = 16(k1 + 2k2 + 2k3 + k4)

The fourth-order R-K method is equivalent to considering up to fourth deriva-

tives in the Taylor series expansion.

The three methods discussed above can be used to solve the swing equations

for the multimachine power system stability.


2.2.4 Simulation Studies and Results

In this section, the three numerical integration methods used in the transient

stability computation procedure are compared and some simulation results are pre-

sented. The WSCC 3-Generator 9-Bus test power system [50] is considered in order

to compare the performance of the three methods in transient analysis.

The behavior of the rotor speed of the first generator ω1 can be used to observe

the performances of the methods under different time steps. As a transient case, a

bus fault at bus 4 is considered between t = 1s and t = 1.15s. The plot of ω1 can

be seen in the below figures for three methods and for different time steps. In an

accurate transient stability solution, the ω1 in p.u. needs to converge to 1 after doing

some oscillations starting from t = 1s.

Time step=0.0005s:

0 5 10 15 20 25 30 35 40 45 500.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

The plot of ω1 when time step is 0.0005s

time (s)

ω 1 in p

u

Euler2nd order R−K4th order R−K

Figure 2.2: Plot of w1 when ∆t = 0.0005s for the 3-generator 9-bus powersystem


Time step=0.01s:

5 10 15 20 25 30 35 40 45 50

0.98

0.99

1

1.01

1.02

1.03

1.04

1.05

1.06


time (s)

ω 1 in p

u

Euler2nd order R−K4th order R−K

Figure 2.3: Plot of w1 when ∆t = 0.01s for the 3-generator 9-bus power system

Time step=0.03s:

5 10 15 20 25 30 35 40 45

1

1.002

1.004

1.006

1.008

1.01

1.012


time (s)

ω 1 in p

u

2nd order R−K4th order R−K



Time step=0.05s:

5 10 15 20 25 30 35 40 45 50

1

1.002

1.004

1.006

1.008

1.01

1.012


time (s)

ω 1 in p

u

4th order R−K


When we observe the behavior of ω1, it is seen that the 4th order Runge Kutta

method gives better transient stability solution compared to others. The Euler

method which only considers the first two terms of the Taylor series expansion, gives

sufficient accuracy if and only if the time step is set to be very small as ∆t = 0.0005s.

If the time step is set to be very small, the computational effort required will be very

high [52]. Therefore Euler method is not very appropriate method to be used in

transient stability studies of power systems as it requires very small time steps and

accordingly high computer storage. The 2nd order Runge Kutta method also gives

accurate solutions but it is worse than 4th order Runge Kutta method as expected.

The 4th order R-K still gives accurate solution even when the time step is set to

0.05s as seen in Figure 2.5.

2.3 Multimachine Stability Studies

A multimachine power system can schematically be represented as in Figure 2.6

below with n number of synchronous generators and m-constant impedance loads

[47, 53]. The transmission network, together with the transformers, connects the


various nodes. The loads, which are modeled as constant impedances, connect the

load buses to the reference node.

Figure 2.6: Multimachine power system representation (classical model) fortransient stability study

The phasors |Ei|∠δi stands for internal machine voltages whereas Vai represents

the terminal bus voltages. The terminal currents Ii are coming from the synchronous

generators into the network.

2.3.1 Classical Model and Assumptions

It is an obvious fact that, the electic power systems are continually growing in

size with the addition of many interconnected transmission networks and emerging

new technologies. This high dimensionality and the increase in complexity of the

power systems make the transient stability study a complicated issue. It is more

convenient to simplify the complex power system and involve some assumptions while

doing the mathematical modeling. The classical model is considered throughout the

transient analysis and the following set of assumptions are considered [49, 53]:

1. The three-phase synchronous machines are represented by a constant internal

voltage Ei∠δi behind a transient reactance xd connected in series which is illustrated

on Figure 2.7 below: The magnitudes of the internal generator voltages |Ei| are


Figure 2.7: Classical model for simplified synchronous generator

kept constant throughout the transient simulation while the variation of angle δi is

observed.

2. The motion of each synchronous machine rotor (relative to a synchronously

rotating reference frame) is at a fixed angle relative to the angle of the voltage

behind the transient reactance.

3. The loads are represented by constant impedances.

4. The mechanical power input to each of the synchronous generators is constant.

5. The damping power is represented as proportional to the rotor speed ω and it is

considered negligible in some cases.

2.3.2 Preliminary Calculations for Initialization

To prepare the multimachine system for transient stability study, the following

set of preliminary calculations are made [46, 53]:

1. The system data are converted to a common system base; a system base of 100

MVA is conventionally chosen. The mechanical, electrical and damping power values

are represented in per unit.

2. The load data from the prefault power flow are converted to equivalent impedances

or admittances. The data for this step are obtained from the result of the power flow

study. If a certain load bus has a voltage solution VLi and complex power demand


SLi = PLi + jQLi, then using SLi = VLiI∗Li (which implies ILi = S∗Li/V

∗Li), we get

yLi =ILiVLi

=S∗Li|VLi|2

=PLi − jQLi

|VLi|2(2.23)

where yLi = gLi + jbLi is the equivalent shunt load admittance.

3. The internal voltages of the generators |Ei|∠δi0 are calculated from the power flow

data using the predisturbance terminal voltages |Vai|∠βi0 as follows. The terminal

voltage is used as reference.

|Ei|∠δi′ = |Vai|+ jxdiIi. (2.24)

Expressing Ii in terms of SGi and Vai, we have

|Ei|∠δi′ = |Vai|+ jxdiS

∗Gi

|Vai|= |Vai|+ j

xdi(PGi − jQGi)

|Vai|

= (|Vai|+QGixdi|Vai|

) + j(PGixdi|Vai|

)

(2.25)

Thus the angle difference between internal and terminal voltage in Figure is δ′i. Since

the actual terminal voltage angle is βi, we obtain the initial generator angle δ0i by

adding the predisturbance voltage angle βi to δ′i, or

δ0i = δ′i + βi. (2.26)

4. The Ybus matrices for the prefault, faulted and postfault network conditions are

calculated. In obtaining these matrices, the following steps are involved:

a. The equivalent load admittances calculated in step 2 are connected between the

load busses and the reference node. Additional nodes are provided for the internal

generator nodes and the appropriate values of admittances corresponding to xd are

connected between these nodes and the generator terminal nodes.

b. In order to obtain the Ybus corresponding to the faulted system, usually the three-

phase to ground faults are considered. The faulted Ybus is then obtained by setting

the row and column corresponding to the faulted node to zero. If there is any other


switching condition other than fault condition, the admittance matrix is determined

for each switching condition.

c. The postfault Ybus is obtained by removing the line that would have been switched

following the protective relay operation.

5. In the final step, the reduced bus admittance matrix, which is denoted by Y ,

is obtained as follows. The system Ybus for each network condition provides the

following relationship between the voltages and currents:

I = YbusV (2.27)

where the current vector I is given by the injected currents at each bus. In the

classical model considered, injected currents exist only at the n-internal generator

buses. All other currents are zero. As a result, the injected current vector has the

form

I =

In

· · ·

0

(2.28)

We now partition the matrices Ybus and V appropriately to obtain

I =

In

· · ·

0

=

Ynn

... Yns

· · · · · · · · ·

Ysn... Yss

En

· · ·

Vs

(2.29)

The subscript n is used to denote the internal generator nodes, and the subscript s

is used for all the remaining nodes.

Ynn is a diagonal matrix of inverted generator impedances; that is

Ynn =

1

jxd10

1jxd1

. . .

0 1jxdn

(2.30)


and also, the kmth element of Yns is

Ynskm =

−1jxdn

if m = Gn and k = n,

0 otherwise.

(2.31)

Note that the voltage at the internal generator nodes are given by the internal emf ’s.

Expanding Eq. 2.29, we get

In = YnnEn + YnsVs 0 = YsnEn + YssVs

from which we eliminate Vs to determine

In = (Ynn − YnsY −1ss Ysn)En = Y En (2.32)

The matrix Y is the desired reduced admittance matrix. It has dimensions (nxn)

where n is the number of the generators. From Eq. 2.32 we also observe that the

reduced admittance matrix provides us a complete description of all the injected

currents in terms of the internal generator bus voltages. We will now use this re-

lationship to derive an expression for the (active) electrical power output of each

generator and hence obtain the differential equations governing the dynamics of the

system.

The power injected into the network at node i, which is the electrical power

output of machine i, is given by [49, 53]

PGi = Re(EiI∗i ). (2.33)

The expression for the injected current at each generator bus Ii in terms of the

reduced admittance matrix parameters is given by Eq. 2.32 [53].

Using In = Y En, the ith component of the injected current is given by

Ii =n∑j=1

YijEi, i = 1, 2, · · · , n. (2.34)


The diagonal element Yii (i = 1, 2, · · · , n) is the driving point admittance for

node i and the off-diagonal element Yij (i = 1, 2, · · · , n, i 6= j) is the transfer admit-

tance between nodes i and j [47, 49]. The diagonal and off-diagonal elements of the

reduced admittance matrix Y can be defined as follow [47]

Yii = |Yii|∠θii = Gii + jBii

Yij = |Yij|∠θij = Gij + jBij,

where Gii = |Yii| cos(θii) is the short-circuit conductance; Gij = |Yij| cos(θij) is the

transfer conductance; and Bij = |Yij| sin(θij) is the transfer susceptance.

Substituting Eq. 2.34 into Eq. 2.33, the following expression can be written for

the electric power delivered to the network by machine i [47, 53]:

PGi = Re[n∑j=1

|Ei||Ej||Yij|∠(δi − δj − θij)], i = 1, 2, · · · , n

=n∑j=1

|Ei||Ej||Yij| cos(δi − δj − θij).(2.35)

The above equation can also be rewritten as

PGi = |E2i ||Yii| cos(θii) +

n∑j=1j 6=i

|Ei||Ej||Yij| cos(δi − δj − θij), i = 1, 2, · · · , n

= E2i Gii +

n∑j=1j 6=i

|Ei||Ej||Yij| cos(δi − δj − θij).(2.36)

The electric output power can also be written by using conductances and suscep-

tances as follow

PGi = E2i Gii +

n∑j=1j 6=i

|Ei||Ej||Yij|[sin(θij) sin(δi − δj) + cos(θij) cos(δi − δj)]

= E2i Gii +

n∑j=1j 6=i

|Ei||Ej|[Bij sin(δi − δj) + Gij cos(δi − δj)], i = 1, 2, · · · , n.

(2.37)


The mechanical power input Pmi for each generator is constant throughout the

transient stability procedure. The damping power is expressed by the damping con-

stant,in seconds per electrical radian, times first derivative of the rotor angle which

is specified as Pdi = Didδidt

= Diwi. The first order differential equations representing

the second order swing equation for a synchronous generator in a multimachine power

system can be expressed can be expressed as follow by using the above description

of the electric output power [53]:

Midωidt

= Pmi − E2i Gii −

n∑j=1j 6=i

|Ei||Ej|[Bij sin(δi − δj) + Gij cos(δi − δj)]− Pdi

dδidt

= ωi, i = 1, 2, · · · , n.(2.38)

The difference form of the above first order differential equations are determined

by using the three numerical integration methods described in the Section 2.2. The

actual solution of the above equations for both the rotor speed ω and the rotor angle

δ are found by discrete time simulation of the transient operation. By consider-

ing the prefault, fault and postfault cases, the actual values are obtained at each

computation time step for each generator in the system.

Chapter 3

Use of EKF and UKF Techniques

in Power System DSE

3.1 Mathematical Modeling

The dynamic state estimation algorithms calculate the dynamic states of the

system which are the state variables in the non-linear algebraic equations represent-

ing the power system. The first step in the dynamic state estimation process is the

identification of the mathematical modeling for the time behavior of the power sys-

tem. By using the mathematical model of the system and the collected measurement

data, DSE predicts the dynamic state vector one step ahead.

A dynamic system can generally be modeled as set of non-linear differential

equations [27, 28]:dx

dt= f(x, u, w) (3.1)

where f(.) is the system function, x vector represents the state variables, the u

vector represents the algebraic variables and w stands for process (system) noise.

26

Chapter 3. Use of EKF and UKF Techniques in Power System DSE 27

The difference form of Eq. 3.1 can be written as [27]:

xk = xk−1 + f(xk−1, uk−1, wk−1)∆t

= g(xk−1, uk−1, wk−1)(3.2)

where k − 1 is the present instant of time index, k is the next instant of time index

and ∆t is the time step. The measurements at time step k can be represented as a

vector of non-linear functions h(.) in terms of the state variables x and measurement

noise v as below [27]:

zk = h(xk, vk) (3.3)

The resulting error between the measured and calculated values is given by [27]

εk = zk − h(xk, vk) (3.4)

Since not all of the dynamic variables of power systems can directly be measured,

they need to be computed and estimated. The execution of Kalman Filter techniques

in the power system dynamic state estimation can solve this problem. Kalman Filter

has the ability of incorporating the noise characteristics into the computations. The

Extended Kalman Filter (EKF) and Unscented Kalman Filter (UKF) algorithms

can be applied to estimate the dynamics of a multimachine power system which

are the state variables in the non-linear differential equations. The EKF and UKF

algorithms are illustrated in the following sections.

3.2 Extended Kalman Filter (EKF) Algorithm

The Extended Kalman Filter (EKF) algorithm which is a two step prediction-

correction process, can be summarized as [51]:


1. The discrete time system equations are presented as follows:

xk = fk−1(xk−1, uk−1, wk−1)

yk = hk(xk, vk)

wk ∼ (0, Qk)

vk ∼ (0, Rk)

(3.5)

The system noise covariance matrix is represented by Qk and the measurement

noise covariance matrix is represented by Rk.

2. Initialize the filter:

x+0 = E(x0)

P+0 = E[(x− x+

0 )(x− x+0 )T ]

(3.6)

where x+0 represents the initial state and P+

0 represents the initial state covariance

matrix. The subscript + indicates the estimate is in an a posteriori estimate.

For k = 1, 2, ..., perform the following:

Prediction:

3. Compute the following partial derivative matrices at the current state estimate

x+k−1:

Fk−1 =∂fk−1

∂x

∣∣∣∣x+k−1

Lk−1 =∂fk−1

∂w

∣∣∣∣x+k−1

(3.7)

4. Perform the time update of state estimate and estimation-error covariance matrix:

P−k = Fk−1P+k−1F

Tk−1 + Lk−1Q

+k−1L

Tk−1

x−k = fk−1(x+k−1, uk−1, 0)

(3.8)

where the subscript − denotes the estimate is in an a priori estimate.

Correction:


5. Perform the following partial derivative matrices at the state update x−k :

Hk =∂hk∂x

∣∣∣∣x−k

Vk =∂hk∂v

∣∣∣∣x−k

(3.9)

6. Perform the measurement update of the state estimate and estimation covariance

as follows:

Kk = P−k HTk (HkP

−k H

Tk + VkRkV

Tk )−1

x+k = x−k +Kk[yk − hk(x−k , 0)]

P+k = (I −KkHk)P

−k

(3.10)

where Kk is the Kalman gain matrix, x+k is the state estimate and P+

k is the estima-

tion error covariance matrix.

The Extended Kalman Filter (EKF), as illustrated above, is one of the most

widely used estimation algorithm for estimating the non-measurable state variables

of non-linear systems. It can also be successfully applied to the estimation of multi-

machine dynamic variables uncluding rotor speed and rotor angle. It shows suffi-

ciently valid performance in both small disturbance conditions and large disturbance

conditions. However, it is obvious that the system state equations f(.) and some of

the measurement equations h(.) are non-linear functions of state variables. Accord-

ingly, the linearization and Jacobian matrix calculation are indispensible. The EKF

method linearizes all of the non-linear trnasformations and it calculates Jacobian

matrices during the estimation process. It is a well known fact that, although the

EKF method is computationally efficient in dynamic state estimation, the lineariza-

tion and Jacobian matrix calculation can lead to some serious drawbacks [30, 34]:

(1) First of all, the linearization is reliable if the higher order terms in Taylor ex-

pansion can be ignored. If this condition can not hold, the linearized approximation

can be extremely poor. As a result, if the time step is not set sufficiently small, the

filter performance will be highly unstable.


(2) The linearization process can only be applied if the Jacobian matrices exist.

However, in some cases, Jacobian matrices do not exist and it is impossible to

perform linearization during the filtering process.

(3) The other disadvantage about EKF is that the Jacobian matrix calculation which

includes many partial derivations, is an error-prone process. The Jacobian matrix

calculation can be really CPU intensive. It can require pages of dense algebra which

will later on need to be converted into a code while running the simulations. If

there is some incorrect parts in the algebra or in the code, this can lead to serious

problems.

In order to overcome the above drawbacks of EKF method existing due to the

linearization and Jacobian matrix calculation, another method can be offered which

does not include linearization and Jacobian matrix computation. The Unscented

Kalman Filter (UKF) method, as illustrated in the following section, can be used

instead of EKF in the dynamic studies of muti-machine power systems. UKF which

is based on unscented transformation is a more efficient, straightforward and easy

to implement in the dynamic state estimation process.

3.3 Unscented Kalman Filter (UKF) Algorithm

As mentioned, the linear approximations applied to non-linear equations repre-

senting the system dynamics can create low accuracy and reduced estimation perfor-

mance. There occurs unstability in the filtering due to the reason that the high order

terms in the Taylor series expansion are neglected. The Unscented Transform (UT),

which offers an opportunity to overcome these limitations of linearization based EKF

algorithm, can be summerized as:

Unscented Transform (UT):

The unscented transformation is an effective method which claims that it is

easier to approximate a Guassian distribution than a non-linear function [32, 33].


By this theory, the statistical distribution of the state is propagated through the non-

linear equations which provides better approximation of state vector and covariance

matrix [32]. The Unscented Transformation theory can be summarized as follow[32–

34]:

Suppose that x is an n-dimensional random variable with mean m and covari-

ance Pxx. Then suppose another random variable y which is related to x through a

non-linear function

y = f(x). (3.11)

The main idea of unscented transformation is to obtain a set of deterministically

chosen sigma points which capture the mean and covariance of the original distri-

bution of x exactly. The sigma points are then propogated to calculate the mean y

and covariance Pyy of y.

Based on the knowledge of x, the 2n+ 1 sigma points can be found as

x0 = m

xi = m+

(√(n+ λ)Pxx

)ii = 1, . . . , n

xi+n = m−(√

(n+ λ)Pxx

)ii = 1, . . . , n

(3.12)

where

(√(n+ λ)Pxx

)iis the ith row or column of matrix square root of the (n +

λ)Pxx. The parameter λ can be defined as λ = α2(n + κ) − n. It is suggested

to use 10−4 ≤ α ≤ 1 and κ = 3 − n or κ = 0. The square root matrix can be

approximated by P = AAT , where A is lower triangular matrix obtained from the

Cholesky factorization of P [32].

In the next state, the previously obtained sigma points can be transformed

through non-linear function and as a result the transformed sigma points are calcu-

lated as below:

χi = f(xi) (3.13)


Then the mean and covariance of y can be calculated by using the previously

calculated transformed sigma points as:

y =2n∑i=0

W imy

i

Pyy =2n∑i=0

W ic(χ

i − y)(χi − y)T

(3.14)

where the weights W im and W i

c are defined as

W 0m =

λ

n+ λ

W 0c =

λ

n+ λ+ (1− α2 + β)

W im = W i

c =1

2(n+ λ)

(3.15)

where the parameter β takes a value 2 which is typical for a Guassian distribution.

The UKF algorithm consists of three main parts which are sigma points calcu-

lation, state prediction and state correction respectively.

The Unscented Kalman Filter (UKF), based on unscented transformation (UT)

theory, can be summarized as follows [51]:

1. The discrete time system equations are presented as follows:

xk = fk−1(xk−1, uk−1, wk−1)

yk = hk(xk, vk)

wk ∼ (0, Qk)

vk ∼ (0, Rk)

(3.16)

The system noise covariance matrix is represented by Qk and the measurement

noise covariance matrix is represented by Rk.

2. Initialize the filter:

x+0 = E(x0)

P+0 = E[(x− x+

0 )(x− x+0 )T ]

(3.17)


where x+0 represents the initial state and P+

0 represents the initial state covariance

matrix. The subscript + indicates the estimate is in an a posteriori estimate.

3. The following time update equations are used to propagate the state estimate

and covariance from one measurement time to the next.

(a) Firstly, to propagate from time step k−1 to k, the sigma points xik−1 are specified

according to the following formula:

x(i)k−1 = x+

k−1 + x(i), i = 1, . . . , 2n

x(i) =

(√(n+ λ)P+

k−1

)Ti

, i = 1, . . . , n

x(n+i) = −(√

(n+ λ)P+k−1

)Ti

, i = 1, . . . , n

(3.18)

(b) Use the known nonlinear system equation f(.) to transform the sigma points

into x(i)k vectors as shown in Eq. 3.18 with appropriate changes since our nonlinear

transformation is f(.) rather than h(.):

x(i)k−1

f(·)−−→ x(i)k , x

(i)k = f(x

(i)k−1, uk, tk) (3.19)

(c) Combine the x(i)k vectors to obtain the a priori state estimate at time k which is

given by the following formula:

x−k =1

2n

2n∑i=1

x(i)k (3.20)

(d) Estimate the a priori error covariance by adding Qk−1 to the end of the equation

in order to take the process noise into account:

P−k =1

2n

2n∑i=1

(x(i)k − x

−k )(x

(i)k − x

−k )T +Qk−1 (3.21)

4. The time update equations are completed at this point and the measurement

update equations need to be implemented in the final part of the UKF algorithm.


(a) Choose sigma points x(i)k with appropriate changes since the current best guess

for the mean and covariance of xk are x−k and P−k :

x(i)k = x−k + x(i), i = 1, . . . , 2n

x(i) =

(√(n+ λ)P−k

)Ti

, i = 1, . . . , n

x(n+i) = −(√

(n+ λ)P−k

)Ti

, i = 1, . . . , n

(3.22)

(b) Use the known nonlinear measurement equation h(.) to transform the sigma

points into y(i)k vectors as follow:

y(i)k

h(·)−−→ x(i)k , y

(i)k = h(x

(i)k , tk) (3.23)

(c) Combine the y(i)k vectors to obtain the predicted measurement at time k:

yk =1

2n

2n∑i=1

y(i)k (3.24)

(d) Estimate the covariance of the predicted measurement by adding Rk to the end

of the equation in order to take the measurement noise into account:

P−y =1

2n

2n∑i=1

(y(i)k − yk)(y

(i)k − yk)

T +Rk (3.25)

(e) Estimate the cross covariance between x−k and yk:

P−xy =1

2n

2n∑i=1

(x(i)k − x

−k )(y

(i)k − yk)

T (3.26)


(f) Finally, the measurement update of the state estimate can be performed by using

the normal Kalman filter equations:

Kk = PxyP−1y

x+k = x−k +Kk(yk − yk)

P+k = P−k − (KkPyK

Tk )

(3.27)

where Kk is the Kalman gain matrix, x+k is the state estimate and P+

k is the estima-

tion error covariance matrix.

3.4 Simulation Studies and Results

3.4.1 Description of Simulation Studies

In this section, the dynamic variables which are the generator rotor speed ω and

the generator rotor angle δ are estimated by using both Extended Kalman Filter

(EKF) and Unscented Kalman Filter (UKF) techniques. The estimation process

is simulated along with the transient stability computation procedure on various

multi-machine test power systems. During the simulations, in order to test the

estimation performance of both EKF and UKF techniques, different scenarios were

considered as bus fault, sudden load change, line switch. These different transient

cases are applied to the multi-machine test systems in a definite time interval and the

change of the behaviors of the dynamic state variables are observed. The EKF and

UKF algorithms are applied in order to estimate the actual behavior of the dynamic

variables under these transient conditions. In order to compare the performances of

EKF and UKF, the simulation and filtering specifications are definitely kept same

for both algorithm. The bus real power injections, reactive power injections, bus

voltage magnitudes and bus voltage angles are used as measurement vector during

the estimation process.


The dynamic state vector x and the measurement vector z can be shown as

below:

x =

ω1

ω2

...

ωn

δ1

δ2

...

δn

z =

PG1

...

PGn

QG1

...

QGn

|V1|...

|Vs|

θ1

...

θs

(3.28)

where n represents the number of generators and s represents the number of buses

in the power power system.

The general principle of the Kalman filtering process which is applied during

the dynamic estimation process of the test power systems can be summarized on the

Figure 3.1 below.

Figure 3.1: Kalman Filter is simply a two-step prediction-update process

As summarized on Figure 3.1, in the DSE process, by using the coming mea-

surements and the state estimates at time instant k along with the mathematical


model of the test power system, the state vector for the next instant of time k + 1

is predicted.

During the simulations, a random Gaussian noise is assumed with mean equal

to zero and standard deviation equal to σ = 10−2 for both system noise and mea-

surement noise. This means that the diagonal elements of the Q (system noise

covariance) and R (measurement noise covariance) matrix are set to σ2 = 10−4 dur-

ing the simulations. The diagonal elements of the initial error covariance matrix

P0 is also set to σ2 = 10−4. The initial values of the state estimate vector x+0 are

arbitrarily chosen but the same values are used for both EKF and UKF case. In this

study, the simulations are carried out by using all of the numerical integration inte-

gration methods which are Euler method, Second Order Runge Kutta and Fourth

Order Runge Kutta method. However, the results for Fourth Order Runge Kutta

method are presented for all of the test systems during the comparison of EKF and

UKF as it performs more accurate transient stability solution. The base is assumed

as Sbase = 100 MVA and the system frequency is assumed as 60 Hz for all of the

simulations and for all of the test systems.

The comparison of the performances of EKF and UKF is made based on the

following performance indices:

Estimation Error (ξ):

The estimation error for the time instant k is calculated by using the following

formula [32]:

ξk =1

2n

2n∑i=1

|xik − xik| (3.29)

where 2n is the number of states (n rotor speed and n rotor angle), x is the actual

state vector calculated at the end of the transient stability analysis and x represents

the estimated state vector as a result of the Kalman filtering process.

The overall estimation error is defined by using the following formula which

calculates the mean of the estimation error vector including the estimation error for


each computation step:

ξ =1

kmax

kmax∑k=1

ξk (3.30)

The estimation error is calculated for the generator rotor speed ω variables and

the generator rotor angle δ variables by using the same formula separately as shown

below:

ξω =1

kmax

kmax∑k=1

[1

n

n∑i=1

|ωik − ωik|

]

ξδ =1

kmax

kmax∑k=1

[1

n

n∑i=1

|δik − δik|

] (3.31)

where n is the number of ω and δ in the state vector.

The EKF and UKF performances are presented in the following section including

the plots of the dynamic states and the performance indices for different test power

systems.

3.4.2 3-Generator 5-Bus Power System

The one-line diagram of the 3-generator 5-bus test power system is shown on

Figure 3.2 [53]. The generator dynamic data is given in Table 3.1 [53]. The time

step is set as ∆t = 0.02s. The generator rotor speed ω and generator relative rotor

angle δ are estimated considering the transient case described below:

Transient Case : Fault at Bus 4 from t = 0s to t = 0.1s. The Line 3− 4 is removed

at t = 0.1s.

Table 3.1: Generator dynamic data of the 3-generator 5-bus power system

Para. Unit G1 G2 G3

xd p.u. 0.08 0.18 0.12H s 10 3.01 6.4D p.u. 0 0 0


Figure 3.2: The one-line diagram of the 3-generator 5-bus power system

Generator 1 : The rotor speed ω1 is estimated by using EKF as shown on Figure 3.3

and UKF as shown on Figure 3.4.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

Estimation of ω1 by EKF

time (s)

ω 1 in p

u

ω

1

ωest1

Figure 3.3: Estimation of ω1 by EKF for the 3-generator 5-bus power system


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

Estimation of ω1 by UKF

time (s)

ω 1 in p

u

ω

1

ωest1

Figure 3.4: Estimation of ω1 by UKF for the 3-generator 5-bus power system



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045


time (s)

ω 2 in p

u

ω

2

ωest2



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045


time (s)

ω 2 in p

u

ω

2

ωest2




0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

time (s)


ω 3 in p

u

ω

3

ωest3



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 51

1.005

1.01

1.015

1.02

1.025

1.03

1.035

1.04

1.045

time (s)


ω 3 in p

u

ω

3

ωest3


Generator 2 : The relative rotor angle δ2−1 is estimated by using EKF as shown on

Figure 3.9 and UKF as shown on Figure 3.10.

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

Estimation of δ2−1

by EKF

time (s)

δ 2−1 in

rad

δ

2−1

δ2−1est

Figure 3.9: Estimation of δ2−1 by EKF for the 3-generator 5-bus power system


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1


by UKF

time (s)

δ 2−1 in

rad

δ

2−1

δ2−1est

Figure 3.10: Estimation of δ2−1 by UKF for the 3-generator 5-bus power system



0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4


by EKF

time (s)

δ 3−1 in

rad

δ

3−1

δ3−1est


The comparison of the performance indices of EKF and UKF is illustrated in

Table 3.2.


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4


by UKF

time (s)

δ 3−1 in

rad

δ

31

δ3−1est


Table 3.2: Performance indices of EKF and UKF for the 3-generator 5-bus powersystem

PerformanceIndex

EKF UKF

ξ 0.0109 0.0879ξω 4.8579× 10−4 7.6836× 10−4

ξδ 0.0213 0.1750


The one-line diagram of the 3-generator 9-bus test power system is shown on

Figure 3.13 [50]. The generator dynamic data is given in Table 3.3 [50]. The time


angle δ for three generators are estimated considering the two transient cases.

Transient Case 1 : Fault at Bus 4 from t = 1s to t = 1.15s [27].


Para. Unit G1 G2 G3

xd p.u. 0.0608 0.1198 0.1813H s 13.64 6.4 3.01D p.u. 9.6 2.5 1





0 1 2 3 4 5 6 7 8 9 101

1.005

1.01

1.015


time (s)

ω 1 in p

u

ω

1

ωest1

Figure 3.14: Estimation of ω1 by EKF for the 3-generator 9-bus power systemconsidering the transient case 1


0 1 2 3 4 5 6 7 8 9 101

1.005

1.01

1.015


time (s)

ω 1 in p

u

ω

1

ωest1

Figure 3.15: Estimation of ω1 by UKF for the 3-generator 9-bus power systemconsidering the transient case 1



0 1 2 3 4 5 6 7 8 9 100.995

1

1.005

1.01

1.015

1.02


time (s)

ω 2 in p

u

ω

2

ωest2



0 1 2 3 4 5 6 7 8 9 100.995

1

1.005

1.01

1.015

1.02


time (s)

ω 2 in p

u

ω

2

ωest2




0 1 2 3 4 5 6 7 8 9 100.998

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016

1.018

time (s)


ω 3 in p

u

ω

3

ωest3



0 1 2 3 4 5 6 7 8 9 100.995

1

1.005

1.01

1.015

1.02

time (s)


ω 3 in p

u

ω

3

ωest3




0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1

1.5


by EKF

time (s)

δ 2−1 in

rad

δ

2−1

δ2−1est

Figure 3.20: Estimation of δ2−1 by EKF for the 3-generator 9-bus power systemconsidering the transient case 1


0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5

1


by UKF

time (s)

δ 2−1 in

rad

δ

2−1

δ2−1est

Figure 3.21: Estimation of δ2−1 by UKF for the 3-generator 9-bus power systemconsidering the transient case 1



0 1 2 3 4 5 6 7 8 9 10−0.4

−0.2

0

0.2

0.4

0.6

0.8

1


by EKF

time (s)

δ 3−1 in

rad

δ

3−1

δ3−1est



Table 3.4 for the transient case 1.


0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7


by UKF

time (s)

δ 3−1 in

rad

δ

3−1

δ3−1est


Table 3.4: Performance indices of EKF and UKF for the 3-generator 9-bus powersystem considering the transient case 1

PerformanceIndex

EKF UKF

ξ 0.0242 0.0224ξω 6.1186× 10−4 0.0011ξδ 0.479 0.0437

Transient Case 2 : Load change of 100 W at Bus 4 from t = 1s to t = 1.15s [27].




0 1 2 3 4 5 6 7 8 9 100.996

0.997

0.998

0.999

1

1.001

1.002

1.003

1.004

1.005


time (s)

ω 1 in p

u

ω

1

ωest1


0 1 2 3 4 5 6 7 8 9 100.996

0.997

0.998

0.999

1

1.001

1.002

1.003

1.004

1.005


time (s)

ω 1 in p

u

ω

1

ωest1





0 1 2 3 4 5 6 7 8 9 100.995

1

1.005

1.01

1.015


time (s)

ω 2 in p

u

ω

2

ωest2


0 1 2 3 4 5 6 7 8 9 100.995

1

1.005

1.01

1.015


time (s)

ω 2 in p

u

ω

2

ωest2





0 1 2 3 4 5 6 7 8 9 100.996

0.998

1

1.002

1.004

1.006

1.008

time (s)


ω 3 in p

u

ω

3

ωest3


0 1 2 3 4 5 6 7 8 9 100.996

0.998

1

1.002

1.004

1.006

1.008

time (s)


ω 3 in p

u

ω

3

ωest3





0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5


by EKF

time (s)

δ 2−1 in

rad

δ

2−1

δ2−1est


0 1 2 3 4 5 6 7 8 9 10−1.5

−1

−0.5

0

0.5


by UKF

time (s)

δ 2−1 in

rad

δ

2−1

δ2−1est





0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4


by EKF

time (s)

δ 3−1 in

rad

δ

3−1

δ3−1est


0 1 2 3 4 5 6 7 8 9 10−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4


by UKF

time (s)

δ 3−1 in

rad

δ

3−1

δ3−1est



Table 3.5 for the transient case 2.


Table 3.5: Performance indices of EKF and UKF for the 3-generator 9-bus powersystem considering the transient case 2

PerformanceIndex

EKF UKF

ξ 0.0115 0.0078ξω 5.5429× 10−4 7.9425× 10−4

ξδ 0.0225 0.0149

3.4.4 IEEE 5-Generator 14-Bus Power System

The one-line diagram of the IEEE 5-generator 14-bus test power system is shown

on Figure 3.34. The generator dynamic data is given in Table 3.6. The time step is

set as ∆t = 0.02s. The generator rotor speed ω and generator relative rotor angle δ

for three generators are estimated considering the transient case described below:

Transient Case : Line 4− 5 is removed at t = 2s.




Para. Unit G1 G2 G3 G4 G5

xd p.u. 0.2995 0.185 0.185 0.232 0.232H s 5.148 6.54 6.54 5.06 5.06D p.u. 2 2 2 2 2



2 4 6 8 10 12 14 16 18

0.997

0.998

0.999

1

1.001

1.002

1.003

1.004

1.005

1.006


time (s)

ω 5 in p

u

ω

5

ωest5



0 2 4 6 8 10 12 14 16 18 200.9965

0.997

0.9975

0.998

0.9985

0.999

0.9995

1

1.0005


time (s)

ω 5 in p

u

ω

5

ωest5




0 2 4 6 8 10 12 14 16 18 20−1.5

−1.4

−1.3

−1.2

−1.1

−1

−0.9

−0.8

−0.7


by EKF

time (s)

δ 5−1 in

rad

δ

5−1

δ5−1est



Table 3.7.


0 2 4 6 8 10 12 14 16 18 20−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0


by UKF

time (s)

δ 5−1 in

rad

δ

5−1

δ5−1est


Table 3.7: Performance indices of EKF and UKF for the 5-generator 14-buspower system

PerformanceIndex

EKF UKF

ξ 0.0119 0.0075ξω 3.0873× 10−4 4.7698× 10−5

ξδ 0.0235 0.0150




set as ∆t = 0.02s. The generator rotor speed ω and generator relative rotor angle δ

for three generators are estimated considering the transient case described below:



Para. Unit G1 G2 G3 G4 G5 G6

xd p.u. 0.2995 0.185 0.185 0.232 0.232 0.232H s 5.148 6.54 6.54 5.06 5.06 5.06D p.u. 2 2 2 2 2 2






2 4 6 8 10 12 14 16 18 20

0.9995

1

1.0005

1.001

1.0015

1.002

1.0025

1.003


time (s)

ω 2 in p

u

ω

2

ωest2


2 4 6 8 10 12 14 16 18 20

0.9995

1

1.0005

1.001

1.0015

1.002

1.0025

1.003

1.0035

1.004


time (s)

ω 2 in p

u

ω

2

ωest2





0 2 4 6 8 10 12 14 16 18 20−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2


by EKF

time (s)

δ 2−1 in

rad

δ

2−1

δ2−1est


0 2 4 6 8 10 12 14 16 18 20−1.2

−1

−0.8

−0.6

−0.4

−0.2

0

0.2


by UKF

time (s)

δ 2−1 in

rad

δ

2−1

δ2−1est



Table 3.9.



PerformanceIndex

EKF UKF

ξ 0.0054 0.0044ξω 2.3019× 10−4 4.1161× 10−4

ξδ 0.0130 0.0085



on Figure 3.44. The generator dynamic data is given in Table 3.10. The time step

is set as ∆t = 0.04s. The generator rotor speed ω and generator relative rotor angle

δ for three generators are estimated considering the transient case described below:

Transient Case : Line 32− 35 is removed between t = 2s and t = 2.5s.




Generator xd (p.u.) H (s) D (p.u.)

G1 0.08 7.5 2.6G2 0.19 3 2.1G3 0.16 6.2 1.45G4 0.1408 4 2.23G5 0.32 4 0.93G6 0.152 3.32 2.12G7 0.195 2.72 1.7G8 0.08 6.2 1.88



2 4 6 8 10 12 14 16 18 20

1

1.0005

1.001

1.0015

1.002


time (s)

ω 7 in p

u

ω

7

ωest7



2 4 6 8 10 12 14 16 18

1

1.0002

1.0004

1.0006

1.0008

1.001

1.0012

1.0014

1.0016

1.0018


time (s)

ω 7 in p

u

ω

7

ωest7




2 4 6 8 10 12 14 16 18 20

−0.228

−0.226

−0.224

−0.222

−0.22

−0.218

−0.216


by EKF

time (s)

δ 7−1 in

rad

δ

7−1

δ7−1est



Table 3.11.


2 4 6 8 10 12 14 16 18 20

−0.23

−0.225

−0.22

−0.215

−0.21

−0.205

−0.2


by UKF

time (s)

δ 7−1 in

rad

δ

7−1

δ7−1est



PerformanceIndex

EKF UKF

ξ 0.0015 3.8377× 10−4

ξω 1.7485× 10−5 4.5823× 10−5

ξδ 0.0029 7.2172× 10−4


The generator dynamic data is given in Table 3.12. The damping coefficient D

is set to zero for all generators. The time step is set as ∆t = 0.02s. The generator

rotor speed ω and generator relative rotor angle δ for three generators are estimated

considering the transient case described below:

Transient Case : Line 4− 14 and 16− 17 is removed at t = 2s.




G1 0.006 500 0G2 0.0697 30.3 0G3 0.0531 35.8 0G4 0.0436 28.6 0G5 0.132 26 0G6 0.05 34.8 0G7 0.049 26.4 0G8 0.057 24.3 0G9 0.057 34.5 0G10 0.031 42 0

Generator 10 : The rotor speed ω10 is estimated by using EKF as shown on Fig-

ure 3.49 and UKF as shown on Figure 3.50.

2 4 6 8 10 12 14 16 18 20

1

1.001

1.002

1.003

1.004

1.005

1.006

1.007

Estimation of ω10

by EKF

time (s)

ω 10 in

pu

ω

10

ωest10



2 4 6 8 10 12 14 16 18 20

1

1.001

1.002

1.003

1.004

1.005

1.006

1.007

Estimation of ω10

by UKF

time (s)

ω 10 in

pu

ω

10

ωest10


Generator 10 : The relative rotor angle δ10−1 is estimated by using EKF as shown

on Figure 3.51 and UKF as shown on Figure 3.52.

0 2 4 6 8 10 12 14 16 18 20−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


by EKF

time (s)

δ 10−1

in r

ad

δ

10−1

δ10−1est

Figure 3.51: Estimation of δ10−1 by EKF for the 10-generator 39-bus powersystem


Table 3.13.


0 2 4 6 8 10 12 14 16 18 20−0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9


by UKF

time (s)

δ 10−1

in r

ad

δ

10−1

δ10−1est

Figure 3.52: Estimation of δ10−1 by UKF for the 10-generator 39-bus powersystem


PerformanceIndex

EKF UKF

ξ 0.0205 0.0168ξω 4.3964× 10−4 8.1861× 10−4

ξδ 0.0406 0.0329




set as ∆t = 0.02s. The damping coefficient D is set to zero for all generators. The

generator rotor speed ω and generator relative rotor angle δ for three generators are

estimated considering the transient case described below:






G1 0.152 18.3 0G2 0.150 10.2 0G3 0.234 6.5 0G4 0.251 4.5 0G5 0.262 4.5 0G6 0.252 5.4 0G7 0.252 5.4 0




1 2 3 4 5 6 7 8 9 10

1

1.002

1.004

1.006

1.008

1.01

1.012

1.014

1.016


time (s)

ω 3 in p

u

ω

3

ωest3


1 2 3 4 5 6 7 8 9 10

1

1.005

1.01

1.015


time (s)

ω 3 in p

u

ω

3

ωest3





0 1 2 3 4 5 6 7 8 9 10−0.6

−0.58

−0.56

−0.54

−0.52

−0.5

−0.48

−0.46

−0.44

−0.42

−0.4


by EKF

time (s)

δ 3−1 in

rad

δ

3−1

δ3−1est


0 1 2 3 4 5 6 7 8 9 10−0.6

−0.58

−0.56

−0.54

−0.52

−0.5

−0.48

−0.46

−0.44

−0.42

−0.4


by UKF

time (s)

δ 3−1 in

rad

δ

3−1

δ3−1est



Table 3.15.



PerformanceIndex

EKF UKF

ξ 0.0412 0.0242ξω 0.0015 0.0023ξδ 0.0808 0.0462


The generator dynamic data of the 50-generator 145-bus test power system is

taken from [54]. The damping coefficient D is set to zero for all generators. The time


angle δ for three generators are estimated considering the transient case described

below:

Transient Case : Line 59− 107 and line 136− 140 are removed at t = 1s.




2 4 6 8 10 12 14 16 18 20

1

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006

Estimation of ω45

by EKF

time (s)

ω 45 in

pu

ω

45

ωest45

Figure 3.58: Estimation of ω45 by EKF for the 50-generator 145-bus powersystem

2 4 6 8 10 12 14 16 18

1

1.0001

1.0002

1.0003

1.0004

1.0005

1.0006

Estimation of ω45

by UKF

time (s)

ω 45 in

pu

ω

45

ωest45

Figure 3.59: Estimation of ω45 by UKF for the 50-generator 145-bus powersystem




2 4 6 8 10 12 14 16 18 20

−0.458

−0.456

−0.454

−0.452

−0.45

−0.448

−0.446

−0.444

−0.442

−0.44


by EKF

time (s)

δ 45−1

in r

ad

δ

45−1

δ45−1est


2 4 6 8 10 12 14 16 18

−0.455

−0.45

−0.445

−0.44

−0.435

−0.43

−0.425

−0.42


by UKF

time (s)

δ 45−1

in r

ad

δ

45−1

δ45−1est



Table 3.16.



PerformanceIndex

EKF UKF

ξ 0.0043 2.5621× 10−4

ξω 4.4831× 10−5 1.3326× 10−5

ξδ 0.0085 4.9910× 10−4


The generator dynamic data of the 17-generator 162-bus test power system is

taken from [54]. The damping coefficient D is set to zero for all generators. The time


angle δ for three generators are estimated considering the transient case described

below:

Transient Case : Line 95− 99 and line 159− 160 are removed at t = 2s.




2 4 6 8 10 12 14 16 18 200.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

1.0012

1.0014

1.0016

Estimation of ω10

by EKF

time (s)

ω 10 in

pu

ω

10

ωest10

Figure 3.62: Estimation of ω10 by EKF for the 17-generator 162-bus powersystem

2 4 6 8 10 12 14 16 18

0.9996

0.9998

1

1.0002

1.0004

1.0006

1.0008

1.001

1.0012

1.0014

1.0016

Estimation of ω10

by UKF

time (s)

ω 10 in

pu

ω

10

ωest10

Figure 3.63: Estimation of ω10 by UKF for the 17-generator 162-bus powersystem




2 4 6 8 10 12 14 16 18 20

0.293

0.2935

0.294

0.2945

0.295

0.2955

0.296


by EKF

time (s)

δ 10−1

in r

ad

δ

10−1

δ10−1est


2 4 6 8 10 12 14 16 18

0.294

0.296

0.298

0.3

0.302

0.304

0.306

0.308

0.31


by UKF

time (s)

δ 10−1

in r

ad

δ

10−1

δ10−1est




Table 3.17.


PerformanceIndex

EKF UKF

ξ 0.0022 9.0924× 10−4

ξω 4.4752× 10−5 5.6862× 10−5

ξδ 0.0043 0.0018

Chapter 4

Parameter Estimation for ZIP

Load Modeling

It is an obvious fact that the load modeling plays an important role in power

systems analysis which is mentioned in the literature [37–45]. One of the most

commonly accepted power system load model is the polynomial load model which is

also called as ZIP load model. Power system analysis software tools often use this

traditional and commonly accepted ZIP load model.

4.1 Polynomial (ZIP) Load Model

A polynomial (ZIP) load model can be defined as a quadratic curve load for

real and reactive power that is based on the load voltage. In this model, both

the real and reactive power consumed by the load bus is expressed as a voltage

dependent polynomial which is one of the most common method in end-use load

modeling. In the ZIP model, all loads in the power system network can be assumed

as a combination of constant power, constant current and constant impedance load.

The mathematical representation of the polynomial ZIP load model can be

expressed as below [41, 42, 55]:

80

Chapter 4. Parameter Estimation for ZIP Load Modeling 81

Pi =

[|V 2a ||V 2n |.|Sn|.Z%.cos(Zθ) +

|Va||Vn|

.|Sn|.I%.cos(Iθ) + |Sn|.P%.cos(Pθ)

]Qi =

[|V 2a ||V 2n |.|Sn|.Z%.sin(Zθ) +

|Va||Vn|

.|Sn|.I%.sin(Iθ) + |Sn|.P%.sin(Pθ)

]100 = Z% + I% + P%

(4.1)

where

Pi : Real power consumption of ith load;

Qi : Reactive power consumption of ith load;

Va : Actual terminal voltage;

Vn : Nominal terminal voltage;

Sn : Apparent power consumption at nominal voltage;

Z% : Percent of load that is constant impedance;

I% : Percent of load that is constant current;

P% : Percent of load that is constant power;

Zθ : Phase angle of constant impedance component;

Iθ : Phase angle of constant current component;

Pθ : Phase angle of constant power component;

In a time-variant load representation, the coefficients of the ZIP model, Vn, Sn,

Z%, I%, P%, Zθ, Iθ and Pθ remain constant, but the power consumption, Pi and Qi,

of the ith load varies with the actual terminal voltage, Va [41]. In the polynomial

representation of the ZIP load, the constant coefficient is equivalent to P%, the linear

coefficient is equivalent to I% and the quadratic coefficient is equivalent to Z% [41].

The ZIP load model only varies the power consumption as a function of actual

terminal voltage, Va [41].

In this study the mathematical representation of the ZIP load model, as illus-

trated in Figure 4.1, is simplified as below:

PLi= αPLi0

+ β|Vi|+ γ|Vi|2, i = 1, . . . ,m

QLi= αQLi0

+ β|Vi|+ γ|Vi|2, i = 1, . . . ,m(4.2)


Figure 4.1: An illustration of ZIP load model

where

PLi: Real power consumption of ith load;

QLi: Reactive power consumption of ith load;

|Vi| : Voltage magnitude of the ith load bus

α : Coefficient of constant power component

β : Coefficient of constant current component

γ : Coefficient of constant impedance component

m : Number of load buses

4.2 Revised Power Flow Program Simulating ZIP

Load

In this section, a program which gives a power flow solution for power systems

whose loads are represented as polynomial ZIP model, is revised. A 14-Bus power

system is used as a test system which has composite ZIP load model at each load

bus. The following table illustrates a power flow solution of a 14-Bus system. The

ZIP load parameters (α, β and γ) are specified for each load bus in the test system

and the input file containing the ZIP parameters is given to the power flow program

as an input. The voltage magnitude |Vi|, phase angle θi, real power PL and reactive

power QL are obtained as an output of the power flow program. The program gives

the results based on the given input file containing the specified ZIP parameters and


initial powers (PLi0 and QLi0) for each bus. By using the bus voltage magnitudes

|Vi|, obtained as an output of the program, and the specified ZIP load parameters

of each load bus, the real power PLi and reactive power QLi of each load bus are

calculated. The calculated values of real and reactive power are compared to the

values obtained in the solution of the power flow program. The Table 4.1 illustrates

that the power flow solution obtained as an output of the program is correct.

Table 4.1: Power flow solution of 14-bus system with ZIP load

BusNo

|Vi|(V)

θi in deg PLiin

puQLi

inpu

αPLi0+ β|Vi| +

γ|Vi|2 in puαQLi0

+ β|Vi| +γ|Vi|2 in pu

1 1.0600 0.00 0 0 0 02 1.0237 -4.13 0.2149 0.1900 0.2149 0.19003 0.9971 -9.25 0.4105 0.2020 0.4105 0.20214 0.9524 -8.81 0.4183 0.2822 0.4183 0.28225 0.9633 -7.58 0.0558 0.0399 0.0557 0.03996 0.9640 -14.98 0.1258 0.1155 0.1257 0.11557 0.9408 -14.65 0.0913 0.0913 0.0913 0.09138 0.9412 -17.30 0.2325 0.2325 0.2325 0.23259 0.9373 -15.52 0.1444 0.1087 0.1444 0.108710 0.9260 -15.86 0.1079 0.1012 0.1079 0.101211 0.9299 -15.75 0.0970 0.0935 0.0970 0.093512 0.9277 -15.82 0.1021 0.0927 0.1021 0.092713 0.9313 -15.79 0.1180 0.1020 0.1180 0.102014 0.9089 -16.37 0.1177 0.0971 0.1177 0.0971

4.3 ZIP Load Parameter Estimation Algorithm

The ZIP load parameters of 14-Bus power system are estimated by using a

technique based on Weighted Least Square (WLS) algorithm. In order apply the

estimation algorithm to the load bus ZIP parameters of 14-Bus system, the following

steps are followed:

The daily load profile L(MW) of the 14-Bus test system is assumed as shown

on the Figure 4.2. The values of the load profile are chosen for every 15 minutes in a

day. Therefore, also considering the initial point, there occurs 60x2415

+ 1 = 97 points

in the load profile for each 15 minute in 24 hours.


0 200 400 600 800 1000 1200 1400200

300

400

500

600

700

800

900

1000Daily Real Power Variation

time (min)

L (M

W)

Figure 4.2: Daily load profile for 14-bus system

The base load for the system is assumed as 700 MW and each point of load

profile data is divided by this base load 700 MW. Then, the 97 coefficients are

obtained and 97 input data file for 14-Bus test system are produced by multiplying

the power injections and load powers by these coefficients. The produced 97 input

data files, containing also the ZIP parameters for each load bus, are given to the

power flow program as an input. Then 97 power flow solutions are obtained for each

input data file as an output of the program. By using the voltage magnitudes |Vi|,

real load power solutions PLi from the solutions and initial load powers |PLi0| of the

load buses from the input files, the ZIP load paramaters are estimated and compared

to their original values entered in the input files. The mathematical representation

of the estimation algorithm can be summarized as below

PLi1

PLi2...

PLik...

PLiN

︸︷︷︸

z

=

PLi01 |Vi|1 |Vi|21

PLi02 |Vi|2 |Vi|22

......

...

PLi0k |Vi|k |Vi|2k...

......

PLi0N |Vi|N |Vi|2N

︸︷︷︸

H

αi

βi

γi

︸︷︷︸x

(4.3)


where i represents the number of load bus in the system and N represents the

maximum number of points in the load profile which is 97 in this case.

The estimation of the ZIP parameters can be obtained by using the following

formula:

x = (HTH)−1HT z

x = (HTWH)−1HTWz(4.4)

The weight matrix used in the estimation of the ZIP parameters is defined as

W =

w1 0 · · · 0

0 w2 · · · 0...

.... . .

...

0 0 · · · wWL

(4.5)

where WL is the window length. The ZIP parameters are estimated by sliding the

window of length WL through the polynomial equations.

The ZIP parameter estimation of bus 4 is presented as an example as below:

0 200 400 600 800 1000 1200 1400−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6Actual and Estimated plot of Alpha for Load Bus 4

time (min)

Alp

ha

AlphaAlpha

est

Figure 4.3: Estimation of α4 in the 14-bus power system

The technique summarized above can be used to estimated the ZIP load pa-

rameters which is based on Weighted Least Squares(WLS) algorithm. By sliding


200 400 600 800 1000 1200

0

2

4

6

8

10

Actual and Estimated plot of Beta for Load Bus 4

time (min)

Bet

a

BetaBeta

est

Figure 4.4: Estimation of β4 in the 14-bus power system

200 400 600 800 1000 1200

−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Actual and Estimated plot of Gama for Load Bus 4

time (min)

Gam

a

GamaGama

est

Figure 4.5: Estimation of γ4 in the 14-bus power system


the window having a specified length, the parameters are estimated by using the

Eq 4.4. It is seen on the figures that the three ZIP parameters of the load bus 4 are

approximately estimated but there occurs a considerable error during the transition

of the parameters.

Chapter 5

Concluding Remarks and Further

Study

5.1 Concluding Remarks

The Dynamic State Estimation (DSE) which calculates the state by consider-

ing the time varying behavior of the system, plays a major role in real monitoring

and control of large scale power grids. The introduction of the Phasor Measure-

ment Units (PMUs) into power systems makes a great progress in the dynamic state

estimation process. This thesis presents the Extended Kalman Filter (EKF) and

Unscented Kalman Filter (UKF) techniques in the aplication of dynamic state esti-

mation for multi-machine power systems. As verified by the simulation results, the

UKF algorithm is very promising method compared to EKF as it is more efficient,

robust, easy to implement and has lower computational demand. The estimation

error values of EKF and UKF algorithms illustrates that UKF gives more accurate

performance under different transient conditions.

In addition, the importance of using accurate load models in power system

stability analysis and control is also addressed. The composite polynomial ZIP load

model is explained which is one of the most common model used in power system

analysis tools. An algorithm based on Weighted Least Squares (WLS) parameter

88

Chapter 5. Concluding Remarks and Further Study 89

identification process is presented in order to estimate the parameters of ZIP load

model. By using the voltage magnitudes and real or reactive power values of the load

buses as measurements, it is possible to estimate the three parameters representing

the ZIP model of the load buses.

5.2 Further Study

In the process of Dynamic State Estimation (DSE) applied to various multima-

chine power systems, several assumptions are considered. For instance, a classical

model is used for generators and constant impedance model is assumed for all the

load buses in the system. As a further study, the dynamic parameters, other than

rotor speed and rotor angle, of the generators can also be considered in the dynamic

state estimation and the dynamic state vector can be extended. The loads, repre-

sented by more accurate and reliable models, can be incorporated into the power

system dynamic state estimation.

Appendix A

Matlab Script Example for EKF

Algorithm

clear all;

close all;

clc;

%%

%%

% IEEE 14 Bus System

% Power System Dynamic State Estimation by EKF

% Measurements : Pinj, Qinj, |V|, theta

% Line is removed after t=1s

%%

%%

%System Frequency

90

Chapter A. Appendix A 91

f0=60;

w_syn=2*pi*f0;

%System Base (MVA)

Sbase=100;

%Number of Generators and Buses

n=5;

s=14;

%Simulation Time Values in sec

t_step=0.02;

t_SW=2;

t_end=20;

t=(0:t_end/t_step)*t_step;

%Machine Dynamic Data

H=[5.148;6.54;6.54;5.06;5.06];

Xd=[0.2995;0.185;0.185;0.232;0.232];

D=[2;2;2;2;2];

M=2*H/w_syn;

%Loaded Bus

bus_load=[2;3;4;5;6;9;10;11;12;13;14];

%Line removed

a=4;

b=5;

%%


%%

%Power Flow Solution PET

Vbus_abs0=zeros(s,1);

theta0=zeros(s,1);

E_abs=zeros(n,1);

delta0=zeros(n,1);

PG0=zeros(n,1);

QG0=zeros(n,1);

PL=zeros(length(bus_load),1);

QL=zeros(length(bus_load),1);

fname_BUSOTP=’BUSOTP.dat’;

fid_BUSOTP=fopen(fname_BUSOTP);

cell_BUSOTP = textscan(fid_BUSOTP, ’%s’, ’delimiter’, ’\n’);

size_BUSOTP=size(cell_BUSOTP{1});

rownum_BUSOTP = size_BUSOTP(1);

fclose(fid_BUSOTP);


readline=fgets(fid_BUSOTP);


gennum=1;

for ii=1:s

%Bus Voltages

busnum=str2num(readline(1:6));

Vbus_abs0(busnum)=str2num(readline(7:14));

theta0(busnum)=(pi/180)*str2num(readline(15:23));


%Generator Real and Reactive Power

if str2num(readline(24:35)) ~= 0 | str2num(readline(36:47)) ~= 0

PG0(gennum)=str2num(readline(24:35))/Sbase;

QG0(gennum)=str2num(readline(36:47))/Sbase;

gennum=gennum+1;

end

%Real and Reactive Load

for jj=1:length(bus_load)

if str2num(readline(1:6)) == bus_load(jj)

PL(jj)=(str2num(readline(24:35))/Sbase)

-str2num(readline(48:59));

QL(jj)=(str2num(readline(36:47))/Sbase)


end

end


end

fclose(fid_BUSOTP);

%%

%%

%Admittance Matrices before the line is removed

Yss=zeros(s,s);

Ynn=zeros(n,n);

Yns=zeros(n,s);

Ysn=zeros(s,n);


fname_ymatrix=’ymatrix.dat’;

fid_ymatrix=fopen(fname_ymatrix);

cell_ymatrix = textscan(fid_ymatrix, ’%s’, ’delimiter’, ’\n’);

size_ymatrix=size(cell_ymatrix{1});

rownum_ymatrix = size_ymatrix(1);

fclose(fid_ymatrix);


for rownum = 1 : rownum_ymatrix

readline=fgets(fid_ymatrix);

bus_i=str2num(readline(1:6));

bus_j=str2num(readline(7:15));

%Yss

G=str2num(readline(17:31));

B=str2num(readline(33:49));

Yss(bus_i, bus_j)=G+i*B;

end


%Load Admittance

SL=PL+j*QL;

VL=zeros(length(bus_load),1);

for ii=1:length(bus_load)

VL(ii)=Vbus_abs0(bus_load(ii));

end

yL=conj(SL)./(VL.^2);


%Add load admittances to Yss

for ii=1:length(yL)

Yss(bus_load(ii),bus_load(ii))=Yss(bus_load(ii),bus_load(ii))+yL(ii);

end

%Add inverted generator impedences to Yss

for ii=1:n

Yss(ii,ii)=Yss(ii,ii)+1/(j*Xd(ii));

end

%Ynn

Ynn=diag(1./(j*Xd));

%Yns

Yns=[-Ynn zeros(n,s-n)];

Yns(4,6)=Yns(4,4);Yns(4,4)=0;

Yns(5,8)=Yns(5,5);Yns(5,5)=0;

%Ysn

Ysn=transpose(Yns);

%Bus Reconstruction Matrix

RV=-inv(Yss)*Ysn;

%System Admittance Matrix

Y_bus=[Ynn Yns; Ysn Yss];

%Reduced Admittance Matrix

Y=Ynn-Yns*inv(Yss)*Ysn;

%Admittance Matrices after the line is removed

yab=-Yss(a,b);

Yss_SW=Yss;

Yss_SW(a,a)=Yss_SW(a,a)-yab;

Yss_SW(b,b)=Yss_SW(b,b)-yab;

Yss_SW(a,b)=Yss_SW(a,b)+yab;

Yss_SW(b,a)=Yss_SW(b,a)+yab;


Ynn_SW=Ynn;

Yns_SW=Yns;

Ysn_SW=Ysn;


RV_SW=-inv(Yss_SW)*Ysn_SW;


Y_bus_SW=[Ynn_SW Yns_SW; Ysn_SW Yss_SW];


Y_SW=Ynn_SW-Yns_SW*inv(Yss_SW)*Ysn_SW;

%%

%%

%Initialization:

%Initial values of state variables

Va_abs=Vbus_abs0(1:n);

beta=theta0(1:n);

w0=[0;0;0;0;0];

delta0=angle(Va_abs+QG0.*Xd./Va_abs + j*PG0.*Xd./Va_abs)+beta;

%Initial values of generator voltages and currents

E_abs=abs(Va_abs+QG0.*Xd./Va_abs + j*PG0.*Xd./Va_abs);

E0=E_abs.*exp(j*delta0);

I0=Y*E0;

%Initial power injections

PG0=real(E0.*conj(I0)); PM=PG0;

QG0=imag(E0.*conj(I0));

%%


%%

%Actual state:

kSW=t_SW/t_step+1;

kend=t_end/t_step+1;

%State

x=zeros(2*n,kend);

w=zeros(n,kend);

w(:,1)=w0;

delta=zeros(n,kend);

delta(:,1)=delta0;

%Measurements

PG1=zeros(n,kend);

QG1=zeros(n,kend);

Vmag=zeros(s,kend);

Vangle=zeros(s,kend);

z=zeros(2*n+2*s,kend);

for k=2:kend

if k<kSW

%Generator Voltage and Current

E1=E_abs.*exp(j*delta(:,k-1));

I1=Y*E1;

%Measurements : PG, QG

PG1(:,k-1)=real(E1.*conj(I1));

QG1(:,k-1)=imag(E1.*conj(I1));


Vmag(:,k-1)=abs(RV*E1);

Vangle(:,k-1)=angle(RV*E1);

z(:,k-1)=[PG1(:,k-1);QG1(:,k-1);Vmag(:,k-1);Vangle(:,k-1)];

%States

%k1

k1_w = t_step * (M.^-1.*(PM-PG1(:,k-1)-(D.*((w(:,k-1))/w_syn))));

k1_delta = t_step * (w(:,k-1));

%k2

E2=E_abs.*exp(j*(delta(:,k-1)+k1_delta/2));

I2=Y*E2;

PG2=real(E2.*conj(I2));

k2_w = t_step * (M.^-1.*(PM-PG2-(D.*((w(:,k-1)+k1_w/2)/w_syn))));

k2_delta = t_step * (w(:,k-1)+k1_w/2);

%k3


I3=Y*E3;




%k4

E4=E_abs.*exp(j*(delta(:,k-1)+k3_delta));

I4=Y*E4;


k4_w = t_step * (M.^-1.*(PM-PG4-(D.*((w(:,k-1)+k3_w)/w_syn))));

k4_delta = t_step * (w(:,k-1)+k3_w);

w(:,k) = w(:,k-1) + (1/6)*(k1_w + 2*k2_w + 2*k3_w + k4_w);


delta(:,k) = delta(:,k-1) + (1/6)*(k1_delta + 2*k2_delta

+ 2*k3_delta + k4_delta);

else



I1=Y_SW*E1;




Vmag(:,k-1)=abs(RV_SW*E1);

Vangle(:,k-1)=angle(RV_SW*E1);


%States

%k1



%k2


I2=Y_SW*E2;




%k3



I3=Y_SW*E3;




%k4


I4=Y_SW*E4;




w(:,k) = w(:,k-1) + (1/6)*(k1_w + 2*k2_w + 2*k3_w + k4_w);



end

end

%Actual State x

%w in pu

%delta in rad

x=[1+w/w_syn; delta];

x=x(:,1:kend-1);

%%

%%

%Estimated state:


%Number of states and measurements

ns=2*n;

nm=2*n+2*s;

%Covariance Matrices

sig=1e-2;

P_plus=sig^2*eye(ns);%Error covariance matrix

Q=sig^2*eye(ns);%System noise covariance matrix

R=sig^2*eye(nm);%Measurement noise covariance matrix

%State

xhat_plus=zeros(ns,kend);

xhat_plus(:,1)=[1.5;3.5;3.2;2.1;2.3;4.5;2.3;2.2;1.5;3.2];

what_plus=xhat_plus(1:n,:);

deltahat_plus=xhat_plus(n+1:ns,:);

%Measurements

y=zeros(nm,kend);

h=zeros(nm,kend);

%Extended Kalman Filter (EKF) Algorithm

for k=2:kend

if k<kSW

%Compute the partial derivative matrices F and L

%%%%%%%%%%%%%%%%%%%%%%%%%% k1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%k1w:

%PM

d_k1w_PM=0;


%PG1

E1=E_abs.*exp(i*deltahat_plus(:,k-1));

I1=Y*E1;

dE1=[zeros(n),diag(i*E_abs.*exp(i*deltahat_plus(:,k-1)))];

dI1=Y*dE1;

dPG1=real(conj(diag(I1))*dE1+diag(E1)*conj(dI1));

d_k1w_PG1=(-1*t_step*diag(M.^-1))*dPG1;

%w(k-1):

d_k1w_w=[(-1*t_step/w_syn)*diag(D).*diag(M.^-1),zeros(n)];

%d_k1w:

d_k1w=d_k1w_PM+d_k1w_PG1+d_k1w_w;

%k1delta:

%w(k-1):

d_k1delta_w=[t_step*eye(n),zeros(n)];

%d_k1delta

d_k1delta=d_k1delta_w;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%% k2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%k2w:

%PM

d_k2w_PM=0;

%PG1


I1=Y*E1;


dI1=Y*dE1;



%PG2


E2=E_abs.*exp(i*(deltahat_plus(:,k-1)+(0.5)*t_step*what_plus(:,k-1)));

I2=Y*E2;

dE2_1=[zeros(n),diag(i*E_abs.*exp(i*deltahat_plus(:,k-1)))];

dE2_2=[diag(i*(0.5)*t_step*E_abs.*exp(0.5*i*t_step

*what_plus(:,k-1))),zeros(n)];

dE2=dE2_1+dE2_2;

dI2=Y*dE2;



%w(k-1)

d_k2w_w=[(0.5)*(t_step^2/w_syn^2)*(diag(D)^2.*diag(M.^-1)^2)

-(t_step/w_syn)*(diag(D).*diag(M.^-1)) zeros(n)];

%d_k2w

d_k2w=d_k2w_PM+d_k2w_PG1+d_k2w_PG2+d_k2w_w;

%dk2delta:

%PM:

d_k2delta_PM=0;

%PG1


I1=Y*E1;


dI1=Y*dE1;


d_k2delta_PG1=(-1*(0.5)*t_step^2*diag(M.^-1))*dPG1;

%w(k-1)

d_k2delta_w=[diag(t_step)-((0.5)*t_step^2/w_syn)*diag(D)

.*diag(M.^-1),zeros(n)];

%d_k2delta

d_k2delta=d_k2delta_PM+d_k2delta_PG1+d_k2delta_w;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%% k3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%k3w:

%PM

d_k3w_PM=0;

%PG1


I1=Y*E1;


dI1=Y*dE1;


d_k3w_PG1=(((-1*t_step^3)/(4*w_syn^2))*diag(D.^2).*diag(M.^-3))*dPG1;

%PG2


I2=Y*E2;




dE2=dE2_1+dE2_2;

dI2=Y*dE2;


d_k3w_PG2=((t_step^2/(2*w_syn))*diag(D)*diag(M.^-2))*dPG2;

%PG3


I1=Y*E1;

PG1_k2delta=real(E1.*conj(I1));

k1w=t_step * (M.^-1.*(PM-PG1_k2delta-(D.*((what_plus(:,k-1))/w_syn))));

k2delta=t_step*what_plus(:,k-1)+0.5*t_step*k1w;

E3=E_abs.*exp(i*(deltahat_plus(:,k-1)+0.5*k2delta));

I3=Y*E3;

dE3_1=[zeros(n), diag(i*E_abs.*exp(i*deltahat_plus(:,k-1)))];

dE3_2=[zeros(n), dPG1(:,n+1:2*n)*-1*0.5*t_step^2*diag(M.^-1)


*diag(i*0.5*E_abs.*exp(-1*i*0.5*t_step^2*diag(M.^-1)*PG1_k2delta))];

dE3_3=[(t_step*eye(n)-0.5*t_step^2*w_syn^-1*diag(D)*diag(M.^-1))

*diag(i*0.5*E_abs.*exp(i*0.5*what_plus(:,k-1))), zeros(n)];

dE3=dE3_1+dE3_2+dE3_3;

dI3=Y*dE3;


d_k3w_PG3=-1*t_step*diag(M.^-1)*dPG3;

%w(k-1)

d_k3w_w=[((t_step^2/(2*w_syn))*diag(D.^2).*diag(M.^-2))

-((t_step^3/(4*w_syn^3))*diag(D.^3).*diag(M.^-3))

-((t_step/w_syn)*diag(D).*diag(M.^-1)), zeros(n)];

%d_k3w

d_k3w=d_k3w_PM+d_k3w_PG1+d_k3w_PG2+d_k3w_PG3+d_k3w_w;

%k3delta:

%PM

d_k3delta_PM=0;

%PG1


I1=Y*E1;


dI1=Y*dE1;


d_k3delta_PG1=((0.25*t_step^3/w_syn)*diag(D).*diag(M.^-2))*dPG1;

%PG2


I2=Y*E2;




dE2=dE2_1+dE2_2;

dI2=Y*dE2;



d_k3delta_PG2=(-0.5*t_step^2*diag(M.^-1))*dPG2;

%w(k-1)

d_k3delta_w=[((0.25*t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-2))

+t_step*eye(n)-0.5*(t_step^2/w_syn)*diag(D).*diag(M.^-1), zeros(n)];

%d_k3delta

d_k3delta=d_k3delta_PM+d_k3delta_PG1+d_k3delta_PG2+d_k3delta_w;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%% k4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%k4w:

%PM

d_k4w_PM=0;

%PG1


I1=Y*E1;


dI1=Y*dE1;


d_k4w_PG1=(t_step^4/(4*w_syn^3))*diag(D.^3)*diag(M.^-4)*dPG1;

%PG2


I2=Y*E2;




dE2=dE2_1+dE2_2;

dI2=Y*dE2;


D_k4w_PG2=(-0.5*t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-3)*dPG2;

%PG3



I1=Y*E1;





I3=Y*E3;





*diag(i*0.5*E_abs.*exp(i*0.5*(t_step*eye(n)-0.5*t_step^2

*w_syn^-1*diag(D)*diag(M.^-1))*what_plus(:,k-1))), zeros(n)];

dE3=dE3_1+dE3_2+dE3_3;

dI3=Y*dE3;


d_k4w_PG3=(t_step^2/w_syn)*diag(D)*diag(M.^-2)*dPG3;

%PG4


I1=Y*E1;



I2=Y*E2;


k3delta=(0.5*t_step^2*diag(M.^-1)-0.25*(t_step^3/w_syn)*diag(D)

*diag(M.^-2))*PM-0.5*t_step^2*diag(M.^-1)*PG2_k3delta+0.25

*(t_step^3/w_syn)*diag(D)*diag(M.^-2)*PG1_k3delta

+(0.25*(t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-2)+t_step*eye(n)

-0.5*(t_step^2/w_syn)*diag(D).*diag(M.^-1))*what_plus(:,k-1);

E4=E_abs.*exp(i*(deltahat_plus(:,k-1)+k3delta));

I4=Y*E4;



dE4_2=[zeros(n), dPG1(:,n+1:2*n)*0.25*(t_step^3/w_syn)*diag(D)

*diag(M.^-2)*diag(i*0.5*E_abs.*exp(i*0.25*(t_step^3/w_syn)

*diag(D)*diag(M.^-2)*PG1_k3delta))];


*diag(i*E_abs.*exp(-1*i*0.5*t_step^2*diag(M.^-1)*PG2_k3delta))];

dE4_4=[((0.25*t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-2)+t_step*eye(n)

-0.5*(t_step^2/w_syn)*diag(D)*diag(M.^-1))*diag(i*E_abs

.*exp(i*((0.25*t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-2)+t_step*eye(n)

-0.5*(t_step^2/w_syn)*diag(D)*diag(M.^-1))*what_plus(:,k-1))), zeros(n)];

dE4=dE4_1+dE4_2+dE4_3+dE4_4;

dI4=Y*dE4;



%w(k-1)

d_k4w_w=[(t_step^4/(4*w_syn^4))*diag(D.^4)*diag(M.^-4)

+(t_step^2/(w_syn^2))*diag(D.^2)*diag(M.^-2)-(t_step^3/(2*w_syn^3))

*diag(D.^3)*diag(M.^-3)-(t_step/w_syn)*diag(D)*diag(M.^-1), zeros(n)];

%d_k4w

d_k4w=d_k4w_PM+d_k4w_PG1+D_k4w_PG2+d_k4w_PG3+d_k4w_PG4+d_k4w_w;

%k4delta:

%PM

d_k4delta_PM=0;

%PG1


I1=Y*E1;


dI1=Y*dE1;


d_k4delta_PG1=-1*(t_step^4/(4*w_syn^2))*diag(D.^2).*diag(M.^-3)*dPG1;

%PG2



I2=Y*E2;




dE2=dE2_1+dE2_2;

dI2=Y*dE2;


d_k4delta_PG2=(t_step^3/(2*w_syn))*diag(D)*diag(M.^-2)*dPG2;

%PG3


I1=Y*E1;





I3=Y*E3;







dE3=dE3_1+dE3_2+dE3_3;

dI3=Y*dE3;


d_k4delta_PG3=-1*t_step^2*diag(M.^-1)*dPG3;

%w(k-1)

d_k4delta_w=[(t_step^3/(2*w_syn^2))*diag(D.^2)*diag(M.^-2)

-(t_step^4/(4*w_syn^3))*diag(D.^3)*diag(M.^-3)-((t_step^2/w_syn)

*diag(D)*diag(M.^-1)), zeros(n)];

%d_k4delta

d_k4delta=d_k4delta_PM+d_k4delta_PG1+d_k4delta_PG2


+d_k4delta_PG3+d_k4delta_w;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%dw

d_w=[eye(n),zeros(n)]+(1/6)*(d_k1w+2*d_k2w+2*d_k3w+d_k4w);

%d_delta

d_delta=[zeros(n),eye(n)]+(1/6)*(d_k1delta+2*d_k2delta

+2*d_k3delta+d_k4delta);

%F

F=[d_w;d_delta];

%L

L=eye(ns);

%Prediction:

%Time update of state estimate and estimation error covariance

%Compute P_minus and x_minus


I1=Y*E1;

PG_plus1=real(E1.*conj(I1));

k1_w=t_step*M.^-1.*(PM-PG_plus1-(D.*((xhat_plus(1:n,k-1))/w_syn)));

k1_delta=t_step*xhat_plus(1:n,k-1);

E2=E_abs.*exp(i*(deltahat_plus(:,k-1)+0.5*k1_delta));

I2=Y*E2;



k2_w=t_step*M.^-1.*(PM-PG_plus2-(D.*((xhat_plus(1:n,k-1)

+0.5*k1_w)/w_syn)));

k2_delta=t_step*(xhat_plus(1:n,k-1)+0.5*k1_w);


I3=Y*E3;



+0.5*k2_w)/w_syn)));


E4=E_abs.*exp(i*(deltahat_plus(:,k-1)+k2_delta));

I4=Y*E4;


k4_w=t_step*M.^-1.*(PM-PG_plus4-(D.*((xhat_plus(1:n,k-1)+k3_w)/w_syn)));

k4_delta=t_step*(xhat_plus(1:n,k-1)+k3_w);

what_minus=xhat_plus(1:n,k-1)+(1/6)*(k1_w+2*k2_w+2*k3_w+k4_w);

deltahat_minus=xhat_plus(n+1:2*n,k-1)+(1/6)*(k1_delta+2*k2_delta

+2*k3_delta+k4_delta);

P_minus=F*P_plus*transpose(F)+L*Q*L’;

xhat_minus=[what_minus;deltahat_minus];

%Correction:

%Compute the partial derivative matrices H and V

dE=[zeros(n),diag(i*E_abs.*exp(i*deltahat_minus))];


dI=Y*dE;

E=E_abs.*exp(i*deltahat_minus);

I=Y*E;

h1=real(E.*conj(I));

h2=imag(E.*conj(I));

h3=abs(RV*E);

h4=angle(RV*E);

h(:,k)=[h1;h2;h3;h4];

HH1=real(conj(diag(I))*dE+diag(E)*conj(dI));

HH2=imag(conj(diag(I))*dE+diag(E)*conj(dI));

HH3=zeros(s,2*n);

HH4=[zeros(s,n) ones(s,n)];

HH=[HH1;HH2;HH3;HH4];

V=eye(nm);

%Measurement update of state estimate and estimation error covariance

%Compute K, x_plus and P_plus

K=P_minus*transpose(HH)*inv(HH*P_minus*transpose(HH)+V*R*transpose(V));

v=sig^2*randn(nm,1);

y1=PG1(:,k);

y2=QG1(:,k);

y3=Vmag(:,k);

y4=Vangle(:,k);

y(:,k)=[y1;y2;y3;y4]+v;

xhat_plus(:,k)=xhat_minus+K*(y(:,k)-h(:,k));


deltahat_plus=xhat_plus(n+1:2*n,:);

P_plus=(eye(ns)-K*HH)*P_minus;

else


%Compute the partial derivative matrices F and L

%%%%%%%%%%%%%%%%%%%%%%%%%% k1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%k1w:

%PM

d_k1w_PM=0;

%PG1


I1=Y_SW*E1;


dI1=Y_SW*dE1;



%w(k-1):

d_k1w_w=[(-1*t_step/w_syn)*diag(D).*diag(M.^-1),zeros(n)];

%d_k1w:

d_k1w=d_k1w_PM+d_k1w_PG1+d_k1w_w;

%k1delta:

%w(k-1):

d_k1delta_w=[t_step*eye(n),zeros(n)];

%d_k1delta

d_k1delta=d_k1delta_w;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%% k2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%k2w:

%PM

d_k2w_PM=0;

%PG1



I1=Y_SW*E1;


dI1=Y_SW*dE1;



%PG2


I2=Y_SW*E2;




dE2=dE2_1+dE2_2;

dI2=Y_SW*dE2;



%w(k-1)

d_k2w_w=[(0.5)*(t_step^2/w_syn^2)*(diag(D)^2.*diag(M.^-1)^2)

-(t_step/w_syn)*(diag(D).*diag(M.^-1)) zeros(n)];

%d_k2w

d_k2w=d_k2w_PM+d_k2w_PG1+d_k2w_PG2+d_k2w_w;

%dk2delta:

%PM:

d_k2delta_PM=0;

%PG1


I1=Y_SW*E1;


dI1=Y_SW*dE1;


d_k2delta_PG1=(-1*(0.5)*t_step^2*diag(M.^-1))*dPG1;

%w(k-1)


d_k2delta_w=[diag(t_step)-((0.5)*t_step^2/w_syn)

*diag(D).*diag(M.^-1),zeros(n)];

%d_k2delta

d_k2delta=d_k2delta_PM+d_k2delta_PG1+d_k2delta_w;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%% k3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%k3w:

%PM

d_k3w_PM=0;

%PG1


I1=Y_SW*E1;


dI1=Y_SW*dE1;


d_k3w_PG1=(((-1*t_step^3)/(4*w_syn^2))*diag(D.^2).*diag(M.^-3))*dPG1;

%PG2


I2=Y_SW*E2;




dE2=dE2_1+dE2_2;

dI2=Y_SW*dE2;


d_k3w_PG2=((t_step^2/(2*w_syn))*diag(D)*diag(M.^-2))*dPG2;

%PG3


I1=Y_SW*E1;






I3=Y_SW*E3;





*diag(i*0.5*E_abs.*exp(i*0.5*what_plus(:,k-1))), zeros(n)];

dE3=dE3_1+dE3_2+dE3_3;

dI3=Y_SW*dE3;



%w(k-1)

d_k3w_w=[((t_step^2/(2*w_syn))*diag(D.^2).*diag(M.^-2))

-((t_step^3/(4*w_syn^3))*diag(D.^3).*diag(M.^-3))

-((t_step/w_syn)*diag(D).*diag(M.^-1)), zeros(n)];

%d_k3w

d_k3w=d_k3w_PM+d_k3w_PG1+d_k3w_PG2+d_k3w_PG3+d_k3w_w;

%k3delta:

%PM

d_k3delta_PM=0;

%PG1


I1=Y_SW*E1;


dI1=Y_SW*dE1;


d_k3delta_PG1=((0.25*t_step^3/w_syn)*diag(D).*diag(M.^-2))*dPG1;

%PG2



I2=Y_SW*E2;




dE2=dE2_1+dE2_2;

dI2=Y_SW*dE2;


d_k3delta_PG2=(-0.5*t_step^2*diag(M.^-1))*dPG2;

%w(k-1)

d_k3delta_w=[((0.25*t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-2))

+t_step*eye(n)-0.5*(t_step^2/w_syn)*diag(D).*diag(M.^-1), zeros(n)];

%d_k3delta

d_k3delta=d_k3delta_PM+d_k3delta_PG1+d_k3delta_PG2+d_k3delta_w;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%%%%%%%%%%%%%%%%%%%%%%%%%% k4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%k4w:

%PM

d_k4w_PM=0;

%PG1


I1=Y_SW*E1;


dI1=Y_SW*dE1;


d_k4w_PG1=(t_step^4/(4*w_syn^3))*diag(D.^3)*diag(M.^-4)*dPG1;

%PG2


I2=Y_SW*E2;





dE2=dE2_1+dE2_2;

dI2=Y_SW*dE2;


D_k4w_PG2=(-0.5*t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-3)*dPG2;

%PG3


I1=Y_SW*E1;





I3=Y_SW*E3;







dE3=dE3_1+dE3_2+dE3_3;

dI3=Y_SW*dE3;


d_k4w_PG3=(t_step^2/w_syn)*diag(D)*diag(M.^-2)*dPG3;

%PG4


I1=Y_SW*E1;



I2=Y_SW*E2;


k3delta=(0.5*t_step^2*diag(M.^-1)-0.25*(t_step^3/w_syn)*diag(D)

*diag(M.^-2))*PM-0.5*t_step^2*diag(M.^-1)*PG2_k3delta+0.25


*(t_step^3/w_syn)*diag(D)*diag(M.^-2)*PG1_k3delta

+(0.25*(t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-2)+t_step*eye(n)

-0.5*(t_step^2/w_syn)*diag(D).*diag(M.^-1))*what_plus(:,k-1);

E4=E_abs.*exp(i*(deltahat_plus(:,k-1)+k3delta));

I4=Y_SW*E4;


dE4_2=[zeros(n), dPG1(:,n+1:2*n)*0.25*(t_step^3/w_syn)*diag(D)

*diag(M.^-2)*diag(i*0.5*E_abs.*exp(i*0.25*(t_step^3/w_syn)

*diag(D)*diag(M.^-2)*PG1_k3delta))];


*diag(i*E_abs.*exp(-1*i*0.5*t_step^2*diag(M.^-1)*PG2_k3delta))];

dE4_4=[((0.25*t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-2)+t_step*eye(n)

-0.5*(t_step^2/w_syn)*diag(D)*diag(M.^-1))*diag(i*E_abs

.*exp(i*((0.25*t_step^3/w_syn^2)*diag(D.^2)*diag(M.^-2)+t_step*eye(n)

-0.5*(t_step^2/w_syn)*diag(D)*diag(M.^-1))*what_plus(:,k-1))), zeros(n)];

dE4=dE4_1+dE4_2+dE4_3+dE4_4;

dI4=Y_SW*dE4;



%w(k-1)

d_k4w_w=[(t_step^4/(4*w_syn^4))*diag(D.^4)*diag(M.^-4)

+(t_step^2/(w_syn^2))*diag(D.^2)*diag(M.^-2)-(t_step^3/(2*w_syn^3))

*diag(D.^3)*diag(M.^-3)-(t_step/w_syn)*diag(D)*diag(M.^-1), zeros(n)];

%d_k4w

d_k4w=d_k4w_PM+d_k4w_PG1+D_k4w_PG2+d_k4w_PG3+d_k4w_PG4+d_k4w_w;

%k4delta:

%PM

d_k4delta_PM=0;

%PG1


I1=Y_SW*E1;



dI1=Y_SW*dE1;


d_k4delta_PG1=-1*(t_step^4/(4*w_syn^2))*diag(D.^2).*diag(M.^-3)*dPG1;

%PG2


I2=Y_SW*E2;




dE2=dE2_1+dE2_2;

dI2=Y_SW*dE2;


d_k4delta_PG2=(t_step^3/(2*w_syn))*diag(D)*diag(M.^-2)*dPG2;

%PG3


I1=Y_SW*E1;





I3=Y_SW*E3;





*diag(i*0.5*E_abs.*exp(i*0.5*(t_step*eye(n)-0.5*t_step^2*w_syn^-1

*diag(D)*diag(M.^-1))*what_plus(:,k-1))), zeros(n)];

dE3=dE3_1+dE3_2+dE3_3;

dI3=Y_SW*dE3;


d_k4delta_PG3=-1*t_step^2*diag(M.^-1)*dPG3;


%w(k-1)

d_k4delta_w=[(t_step^3/(2*w_syn^2))*diag(D.^2)*diag(M.^-2)

-(t_step^4/(4*w_syn^3))*diag(D.^3)*diag(M.^-3)-((t_step^2/w_syn)

*diag(D)*diag(M.^-1)), zeros(n)];

%d_k4delta

d_k4delta=d_k4delta_PM+d_k4delta_PG1+d_k4delta_PG2

+d_k4delta_PG3+d_k4delta_w;

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%dw

d_w=[eye(n),zeros(n)]+(1/6)*(d_k1w+2*d_k2w+2*d_k3w+d_k4w);

%d_delta

d_delta=[zeros(n),eye(n)]+(1/6)*(d_k1delta+2*d_k2delta

+2*d_k3delta+d_k4delta);

%F

F=[d_w;d_delta];

%L

L=eye(ns);

%Prediction:

%Time update of state estimate and estimation error covariance

%Compute P_minus and x_minus


I1=Y_SW*E1;



k1_w=t_step*M.^-1.*(PM-PG_plus1-(D.*((xhat_plus(1:n,k-1))/w_syn)));

k1_delta=t_step*xhat_plus(1:n,k-1);


I2=Y_SW*E2;



+0.5*k1_w)/w_syn)));



I3=Y_SW*E3;



+0.5*k2_w)/w_syn)));


E4=E_abs.*exp(i*(deltahat_plus(:,k-1)+k2_delta));

I4=Y_SW*E4;



+k3_w)/w_syn)));

k4_delta=t_step*(xhat_plus(1:n,k-1)+k3_w);

what_minus=xhat_plus(1:n,k-1)+(1/6)*(k1_w+2*k2_w+2*k3_w+k4_w);

deltahat_minus=xhat_plus(n+1:2*n,k-1)+(1/6)*(k1_delta+2*k2_delta


P_minus=F*P_plus*transpose(F)+L*Q*L’;


xhat_minus=[what_minus;deltahat_minus];

%Correction:

%Compute the partial derivative matrices H and V

dE=[zeros(n),diag(i*E_abs.*exp(i*deltahat_minus))];

dI=Y_SW*dE;

E=E_abs.*exp(i*deltahat_minus);

I=Y_SW*E;

h1=real(E.*conj(I));

h2=imag(E.*conj(I));

h3=abs(RV_SW*E);

h4=angle(RV_SW*E);

h(:,k)=[h1;h2;h3;h4];

HH1=real(conj(diag(I))*dE+diag(E)*conj(dI));

HH2=imag(conj(diag(I))*dE+diag(E)*conj(dI));

HH3=zeros(s,2*n);

HH4=[zeros(s,n) ones(s,n)];

HH=[HH1;HH2;HH3;HH4];

V=eye(nm);

%Measurement update of state estimate and estimation error covariance

%Compute K, x_plus and P_plus

K=P_minus*transpose(HH)*inv(HH*P_minus*transpose(HH)+V*R*transpose(V));


y1=PG1(:,k);

y2=QG1(:,k);

y3=Vmag(:,k);

y4=Vangle(:,k);

y(:,k)=[y1;y2;y3;y4]+v;


xhat_plus(:,k)=xhat_minus+K*(y(:,k)-h(:,k));


deltahat_plus=xhat_plus(n+1:2*n,:);

P_plus=(eye(ns)-K*HH)*P_minus;

end

end

%Estimated State

%w in pu

%delta in rad

x_est=[1+xhat_plus(1:n,:)/w_syn; xhat_plus(n+1:ns,:)];

x_est=x_est(:,1:kend-1);

%%

%%

%Overall Estimation Error

err_est=(1/size(x,2))*sum((1/ns)*sum(abs(x-x_est)));

%Estimation Error for w

err_est_w=(1/size(x,2))*sum((1/n)*sum(abs(x(1:n,:)-x_est(1:n,:))));

%Estimation Error for delta

err_est_delta=(1/size(x,2))*sum((1/n)*sum(abs(x(n+1:2*n,:)-x_est(n+1:2*n,:))));

%%


%%

t=t(1:kend-1);

%Figures:

figure(1)%w1 in pu

plot(t,x(1,:),’linewidth’,2);

hold on;

plot(t,x_est(1,:),’linestyle’,’--’,’color’,’r’,’linewidth’,2);

grid on;

xlabel(’time (s)’,’fontsize’,15);

ylabel(’\omega_1 in pu’,’fontsize’,15);

legend(’\omega_1’,’\omegaê^s^t_1’);

figure(2)%w2 in pu

plot(t,x(2,:),’linewidt’,2);

hold on;

plot(t,x_est(2,:),’linestyle’,’--’,’color’,’r’,’linewidt’,2);

grid on;




figure(3)%w3 in pu


hold on;


grid on;




figure(4)%w4 in pu


hold on;



grid on;




figure(5)%w5 in pu


hold on;


grid on;

title(’Estimation of \omega_5 by EKF’,’fontsize’,15);




figure(6)%delta1 in rad

plot(t,x(n+1,:),’linewidth’,2);

hold on;

plot(t,x_est(n+1,:),’linestyle’,’--’,’color’,’r’,’linewidth’,2);

grid on;


ylabel(’\delta_1 in rad’,’fontsize’,15);

legend(’\delta_1’,’\delta_1ê^s^t’);



hold on;


grid on;







hold on;


grid on;






hold on;


grid on;






hold on;


grid on;




figure(11)%delta2-1 in rad

plot(t,x(n+2,:)-x(n+1,:),’linewidth’,2);

hold on;

plot(t,x_est(n+2,:)-x_est(n+1,:),’linestyle’,’--’,’color’,’r’,’linewidth’,2);

grid on;


ylabel(’\delta_2_1 in rad’,’fontsize’,15);


legend(’\delta_2_1’,’\delta_2_1ê^s^t’);



hold on;


grid on;






hold on;


grid on;






hold on;


grid on;

title(’Estimation of \delta_5_-_1 by EKF’,’fontsize’,15);


ylabel(’\delta_5_-_1 in rad’,’fontsize’,15);

legend(’\delta_5_-_1’,’\delta_5_-_1ê^s^t’);

figure(15)%delta1 in deg

plot(t,x(n+1,:)*(180/pi),’linewidth’,2);

hold on;

plot(t,x_est(n+1,:)*(180/pi),’linestyle’,’--’,’color’,’r’,’linewidth’,2);

grid on;



ylabel(’\delta_1 in deg’,’fontsize’,15);




hold on;


grid on;






hold on;


grid on;






hold on;


grid on;






hold on;


grid on;





figure(20)%delta2-1 in deg

plot(t,x(n+2,:)*(180/pi)-x(n+1,:)*(180/pi),’linewidth’,2);

hold on;

plot(t,x_est(n+2,:)*(180/pi)-x_est(n+1,:)*(180/pi),’linestyle’,’--’,

’color’,’r’,’linewidth’,2);

grid on;


ylabel(’\delta_2_1 in deg’,’fontsize’,15);




hold on;



grid on;






hold on;



grid on;







hold on;



grid on;




%%

Appendix B

Matlab Script Example for UKF

Algorithm

clear all;

close all;

clc;

%%

%%

% IEEE 14 Bus System

% Power System Dynamic State Estimation by UKF

% Measurements : Pinj, Qinj, |V|, theta

% Line is removed after t=1s

%%

%%

%System Frequency

132

Chapter B. Appendix B 133

f0=60;

w_syn=2*pi*f0;

%System Base (MVA)

Sbase=100;

%Number of Generators and Buses

n=5;

s=14;

%Simulation Time Values in sec

t_step=0.02;

t_SW=2;

t_end=20;

t=(0:t_end/t_step)*t_step;

%Machine Data

H=[5.148;6.54;6.54;5.06;5.06];

Xd=[0.2995;0.185;0.185;0.232;0.232];

D=[2;2;2;2;2]*1;

M=2*H/w_syn;

%Loaded Bus

bus_load=[2;3;4;5;6;9;10;11;12;13;14];

%Line removed

a=4;

b=5;

%%


%%

%Power Flow Solution PET

Vbus_abs0=zeros(s,1);

theta0=zeros(s,1);

E_abs=zeros(n,1);

delta0=zeros(n,1);

PG0=zeros(n,1);

QG0=zeros(n,1);

PL=zeros(length(bus_load),1);

QL=zeros(length(bus_load),1);

fname_BUSOTP=’BUSOTP.dat’;


cell_BUSOTP = textscan(fid_BUSOTP, ’%s’, ’delimiter’, ’\n’);

size_BUSOTP=size(cell_BUSOTP{1});

rownum_BUSOTP = size_BUSOTP(1);

fclose(fid_BUSOTP);




gennum=1;

for ii=1:s

%Bus Voltages

busnum=str2num(readline(1:6));

Vbus_abs0(busnum)=str2num(readline(7:14));

theta0(busnum)=(pi/180)*str2num(readline(15:23));


%Generator Real and Reactive Power

if str2num(readline(24:35)) ~= 0 | str2num(readline(36:47)) ~= 0

PG0(gennum)=str2num(readline(24:35))/Sbase;

QG0(gennum)=str2num(readline(36:47))/Sbase;

gennum=gennum+1;

end

%Real and Reactive Load

for jj=1:length(bus_load)

if str2num(readline(1:6)) == bus_load(jj)

PL(jj)=(str2num(readline(24:35))/Sbase)


QL(jj)=(str2num(readline(36:47))/Sbase)


end

end


end

fclose(fid_BUSOTP);

%%

%%

%Admittance Matrices before the line is removed

Yss=zeros(s,s);

Ynn=zeros(n,n);

Yns=zeros(n,s);

Ysn=zeros(s,n);


fname_ymatrix=’ymatrix.dat’;


cell_ymatrix = textscan(fid_ymatrix, ’%s’, ’delimiter’, ’\n’);

size_ymatrix=size(cell_ymatrix{1});

rownum_ymatrix = size_ymatrix(1);



for rownum = 1 : rownum_ymatrix

readline=fgets(fid_ymatrix);

bus_i=str2num(readline(1:6));

bus_j=str2num(readline(7:15));

%Yss

G=str2num(readline(17:31));

B=str2num(readline(33:49));

Yss(bus_i, bus_j)=G+i*B;

end


%Load Admittance

SL=PL+j*QL;

VL=zeros(length(bus_load),1);

for ii=1:length(bus_load)

VL(ii)=Vbus_abs0(bus_load(ii));

end

yL=conj(SL)./(VL.^2);


%Add load admittances to Yss

for ii=1:length(yL)

Yss(bus_load(ii),bus_load(ii))=Yss(bus_load(ii),bus_load(ii))+yL(ii);

end

%Add inverted generator impedences to Yss

for ii=1:n

Yss(ii,ii)=Yss(ii,ii)+1/(j*Xd(ii));

end

%Ynn

Ynn=diag(1./(j*Xd));

%Yns

Yns=[-Ynn zeros(n,s-n)];

Yns(4,6)=Yns(4,4);Yns(4,4)=0;

Yns(5,8)=Yns(5,5);Yns(5,5)=0;

%Ysn

Ysn=transpose(Yns);


RV=-inv(Yss)*Ysn;


Y_bus=[Ynn Yns; Ysn Yss];


Y=Ynn-Yns*inv(Yss)*Ysn;

%Admittance Matrices after the line is removed

yab=-Yss(a,b);

Yss_SW=Yss;

Yss_SW(a,a)=Yss_SW(a,a)-yab;

Yss_SW(b,b)=Yss_SW(b,b)-yab;

Yss_SW(a,b)=Yss_SW(a,b)+yab;

Yss_SW(b,a)=Yss_SW(b,a)+yab;


Ynn_SW=Ynn;

Yns_SW=Yns;

Ysn_SW=Ysn;


RV_SW=-inv(Yss_SW)*Ysn_SW;


Y_bus_SW=[Ynn_SW Yns_SW; Ysn_SW Yss_SW];


Y_SW=Ynn_SW-Yns_SW*inv(Yss_SW)*Ysn_SW;

%%

%%

%Initialization:

%Initial values of state variables

Va_abs=Vbus_abs0(1:n);

beta=theta0(1:n);

w0=[0;0;0;0;0];

delta0=angle(Va_abs+QG0.*Xd./Va_abs + j*PG0.*Xd./Va_abs)+beta;

%Initial values of generator voltages and currents

E_abs=abs(Va_abs+QG0.*Xd./Va_abs + j*PG0.*Xd./Va_abs);

E0=E_abs.*exp(j*delta0);

I0=Y*E0;

%Initial power injections

PG0=real(E0.*conj(I0)); PM=PG0;

QG0=imag(E0.*conj(I0));

%%


%%

%Actual state:

kSW=t_SW/t_step+1;

kend=t_end/t_step+1;

%State

x=zeros(2*n,kend);

w=zeros(n,kend);

w(:,1)=w0;

delta=zeros(n,kend);

delta(:,1)=delta0;

%Measurements

PG1=zeros(n,kend);

QG1=zeros(n,kend);

Vmag=zeros(s,kend);

Vangle=zeros(s,kend);

z=zeros(2*n+2*s,kend);

for k=2:kend

if k<kSW



I1=Y*E1;





Vmag(:,k-1)=abs(RV*E1);

Vangle(:,k-1)=angle(RV*E1);


%States

%k1



%k2


I2=Y*E2;




%k3


I3=Y*E3;




%k4


I4=Y*E4;





w(:,k) = w(:,k-1) + (1/6)*(k1_w + 2*k2_w + 2*k3_w + k4_w);



else



I1=Y_SW*E1;




Vmag(:,k-1)=abs(RV_SW*E1);

Vangle(:,k-1)=angle(RV_SW*E1);


%States

%k1



%k2


I2=Y_SW*E2;




%k3



I3=Y_SW*E3;




%k4


I4=Y_SW*E4;




w(:,k) = w(:,k-1) + (1/6)*(k1_w + 2*k2_w + 2*k3_w + k4_w);



end

end

%Actual State x

%w in pu

%delta in rad

x=[1+w/w_syn; delta];

x=x(:,1:kend-1);

%%

%%


%Estimated state:

%Number of states and measurements

ns=2*n;

nm=2*n+2*s;

%Covariance Matrices

sig=1e-2;

P=sig^2*eye(ns);%Error covariance matrix

Q=sig^2*eye(ns);%System noise covariance matrix

R=sig^2*eye(nm);%Measurement noise covariance matrix

%Estimated State

x_hat=zeros(2*n,kend);

xhat_plus(:,1)=[1.5;3.5;3.2;2.1;2.3;4.5;2.3;2.2;1.5;3.2];

%Sigma points

x_sigma=zeros(ns,2*ns);

%Unscented Kalman Filter (UKF) Algorithm

for k=2:kend

if k<kSW

%Sigma Points

A=chol(ns*P);

x_tilda=[A, -A];

x_sigma=repmat(x_hat(:,k-1),1,2*ns)+x_tilda;

w_sigma=x_sigma(1:n,:);

delta_sigma=x_sigma(n+1:ns,:);


%Update Sigma Points

E1=repmat(E_abs,1,2*ns).*exp(j*delta_sigma);

I1=Y*E1;

Pi1=real(E1.*conj(I1));

PM_rep=repmat(PM,1,2*ns);

M_rep=repmat(M,1,2*ns);

D_rep=repmat(D,1,2*ns);

%k1

k1_w = t_step * (M_rep.^-1.*(PM_rep-Pi1-(D_rep.*(w_sigma./w_syn))));

k1_delta = t_step * w_sigma;

%k2

E2=repmat(E_abs,1,2*ns).*exp(j*(delta_sigma+k1_delta/2));

I2=Y*E2;


k2_w = t_step * (M_rep.^-1.*(PM_rep-Pi2-(D_rep.*((w_sigma

+k1_w/2)./w_syn))));

k2_delta = t_step * (w_sigma+k1_w/2);

%k3


I3=Y*E3;


k3_w = t_step * (M_rep.^-1.*(PM_rep-Pi3-(D_rep.*((w_sigma

+k2_w/2)./w_syn))));


%k4

E4=repmat(E_abs,1,2*ns).*exp(j*(delta_sigma+k3_delta));

I4=Y*E4;



k4_w = t_step * (M_rep.^-1.*(PM_rep-Pi4-(D_rep

.*((w_sigma+k3_w)./w_syn))));

k4_delta = t_step * (w_sigma+k3_w);

w_sigma_up = w_sigma + (1/6)*(k1_w+2*k2_w+2*k3_w+k4_w);

delta_sigma_up = delta_sigma + (1/6)*(k1_delta+2*k2_delta


w_sigma=w_sigma_up;

delta_sigma=delta_sigma_up;

x_sigma=[w_sigma;delta_sigma];

%Prioiri State Estimate

x_minus=(1/(2*ns))*(sum(x_sigma’))’;

%Priori Covariance Matrix

x_minus_rep=repmat(x_minus,1,2*ns);

P_minus=(1/(2*ns))*(x_sigma-x_minus_rep)*(x_sigma-x_minus_rep)’+Q;

%New Sigma Points

A=chol(ns*P_minus);

x_tilda=[A’,-A’];




%y Sigma Points

E=repmat(E_abs,1,2*ns).*exp(j*delta_sigma);

I=Y*E;

y1=real(E.*conj(I));

y2=imag(E.*conj(I));

y3=abs(RV*E);


y4=angle(RV*E);

y_sigma=[y1;y2;y3;y4];

%y Predict

y_predict=(1/(2*ns))*(sum(y_sigma’))’;

%Covariance of predicted measurements Py

y_predict_rep=repmat(y_predict,1,2*ns);

P_y=(2*ns)^-1*(y_sigma-y_predict_rep)*(y_sigma-y_predict_rep)’+R;

%Cross covariance Pxy

P_xy=(2*ns)^-1*(x_sigma-x_minus_rep)*(y_sigma-y_predict_rep)’;

%Measurement update of state estimate

K=P_xy*P_y^-1;

y_PG=PG1(:,k);

y_QG=QG1(:,k);

y_Vmag=Vmag(:,k);

y_Vangle=Vangle(:,k);


y=[y_PG;y_QG;y_Vmag;y_Vangle]+v;

x_hat(:,k)=x_minus+K*(y-y_predict);

P=P_minus-K*P_y*K’;

else

%Sigma Points

A=chol(ns*P);

x_tilda=[A, -A];





%Update Sigma Points

E1=repmat(E_abs,1,2*ns).*exp(j*delta_sigma);

I1=Y_SW*E1;


PM_rep=repmat(PM,1,2*ns);

M_rep=repmat(M,1,2*ns);

D_rep=repmat(D,1,2*ns);

%k1

k1_w = t_step * (M_rep.^-1.*(PM_rep-Pi1-(D_rep.*(w_sigma./w_syn))));

k1_delta = t_step * w_sigma;

%k2


I2=Y_SW*E2;


k2_w = t_step * (M_rep.^-1.*(PM_rep-Pi2

-(D_rep.*((w_sigma+k1_w/2)./w_syn))));


%k3


I3=Y_SW*E3;


k3_w = t_step * (M_rep.^-1.*(PM_rep-Pi3

-(D_rep.*((w_sigma+k2_w/2)./w_syn))));


%k4


E4=repmat(E_abs,1,2*ns).*exp(j*(delta_sigma+k3_delta));

I4=Y_SW*E4;


k4_w = t_step * (M_rep.^-1.*(PM_rep-Pi4-

(D_rep.*((w_sigma+k3_w)./w_syn))));

k4_delta = t_step * (w_sigma+k3_w);

w_sigma_up = w_sigma + (1/6)*(k1_w+2*k2_w+2*k3_w+k4_w);

delta_sigma_up = delta_sigma + (1/6)*(k1_delta+2*k2_delta


w_sigma=w_sigma_up;

delta_sigma=delta_sigma_up;

x_sigma=[w_sigma;delta_sigma];

%Priori State Estimate

x_minus=(1/(2*ns))*(sum(x_sigma’))’;

%Priori Covariance Matrix

x_minus_rep=repmat(x_minus,1,2*ns);

P_minus=(1/(2*ns))*(x_sigma-x_minus_rep)*(x_sigma-x_minus_rep)’+Q;

%New Sigma Points

A=chol(ns*P_minus);

x_tilda=[A’,-A’];




%y Sigma Points

E=repmat(E_abs,1,2*ns).*exp(j*delta_sigma);

I=Y_SW*E;


y1=real(E.*conj(I));

y2=imag(E.*conj(I));

y3=abs(RV_SW*E);

y4=angle(RV_SW*E);

y_sigma=[y1;y2;y3;y4];

%y Predict

y_predict=(1/(2*ns))*(sum(y_sigma’))’;

%Covariance of predicted measurements Py

y_predict_rep=repmat(y_predict,1,2*ns);

P_y=(2*ns)^-1*(y_sigma-y_predict_rep)*(y_sigma-y_predict_rep)’+R;

%Cross covariance Pxy

P_xy=(2*ns)^-1*(x_sigma-x_minus_rep)*(y_sigma-y_predict_rep)’;

%Measurement update of state estimate

K=P_xy*P_y^-1;

y_PG=PG1(:,k);

y_QG=QG1(:,k);

y_Vmag=Vmag(:,k);

y_Vangle=Vangle(:,k);


y=[y_PG;y_QG;y_Vmag;y_Vangle]+v;

x_hat(:,k)=x_minus+K*(y-y_predict);

P=P_minus-K*P_y*K’;

end

end


%Estimated State

%w in pu

%delta in rad

x_est=[1+x_hat(1:n,:)/w_syn; x_hat(n+1:ns,:)];

x_est=x_est(:,1:kend-1);

%%

%%

%Overall Estimation Error

err_est=(1/size(x,2))*sum((1/ns)*sum(abs(x-x_est)));

%Estimation Error for w

err_est_w=(1/size(x,2))*sum((1/n)*sum(abs(x(1:n,:)-x_est(1:n,:))));

%Estimation Error for delta

err_est_delta=(1/size(x,2))*sum((1/n)*sum(abs(x(n+1:2*n,:)-x_est(n+1:2*n,:))));

%%

%%

t=t(1:kend-1);

%Figures:

figure(1)%w1 in pu


hold on;


grid on;





figure(2)%w2 in pu

plot(t,x(2,:),’linewidt’,2);

hold on;

plot(t,x_est(2,:),’linestyle’,’--’,’color’,’r’,’linewidt’,2);

grid on;




figure(3)%w3 in pu


hold on;


grid on;




figure(4)%w4 in pu


hold on;


grid on;




figure(5)%w5 in pu


hold on;


grid on;

title(’Estimation of \omega_5 by UKF’,’fontsize’,15);







hold on;


grid on;






hold on;


grid on;






hold on;


grid on;






hold on;


grid on;







hold on;


grid on;






hold on;


grid on;






hold on;


grid on;






hold on;



grid on;






hold on;


grid on;

title(’Estimation of \delta_5_-_1 by UKF’,’fontsize’,15);


ylabel(’\delta_5_-_1 in rad’,’fontsize’,15);

legend(’\delta_5_-_1’,’\delta_5_-_1ê^s^t’);



hold on;


grid on;






hold on;


grid on;







hold on;


grid on;






hold on;


grid on;






hold on;


grid on;






hold on;



grid on;







hold on;



grid on;






hold on;



grid on;






hold on;



grid on;




%%

Bibliography

[1] F. C. Schweppe and J. Wildes, “Power system static-state estimation, part

i: Exact model,” IEEE Transactions on, Power Apparatus and Systems, vol.

PAS-89, no. 1, pp. 120–125, 1970.

[2] F. C. Schweppe and D. Rom, “Power system static-state estimation, part ii:

Approximate model,” IEEE Transactions on, Power Apparatus and Systems,

vol. PAS-89, no. 1, pp. 125–130, 1970.

[3] F. C. Schweppe, “Power system static-state estimation, part iii: Implementa-

tion,” IEEE Transactions on, Power Apparatus and Systems, vol. PAS-89, no. 1,

pp. 130–135, 1970.

[4] A. Gomez-Exposito, A. Abur, A. de la Villa Jaen, and C. Gomez-Quiles, “A

multilevel state estimation paradigm for smart grids,” Proceedings of the IEEE,

vol. 99, no. 6, pp. 952–976, 2011.

[5] R. Larson, W. Tinney, and J. Peschon, “State estimation in power systems

part i: Theory and feasibility,” IEEE Transactions on, Power Apparatus and

Systems, vol. PAS-89, no. 3, pp. 345–352, 1970.

[6] R. Larson, W. Tinney, L. Hajdu, and D. Piercy, “State estimation in power sys-

tems part ii: Implementation and applications,” IEEE Transactions on, Power

Apparatus and Systems, vol. PAS-89, no. 3, pp. 353–363, 1970.

[7] F. Schweppe and E. Handschin, “Static state estimation in electric power sys-

tems,” Proceedings of the IEEE, vol. 62, no. 7, pp. 972–982, 1974.

157

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[8] A. Abur and M. K. Celik, “A fast algorithm for the weighted least absolute value

state estimation [for power systems],” IEEE Transactions on, Power Systems,

vol. 6, no. 1, pp. 1–8, 1991.

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Power system dynamic state estimation and load …...POWER SYSTEM DYNAMIC STATE ESTIMATION and LOAD MODELING A Thesis Presented by Cem Bila to The Department of Electrical and Computer