POWER SYSTEM DYNAMIC STATE ESTIMATION and LOAD MODELING. Cem Bila

Transcription

1 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

2 c copyright by Cem Bila 2013 All Rights Reserved

3 Northeastern University Abstract Department of Electrical and Computer Engineering Master of Science in Electrical and Computer Engineering by Cem Bila

4 iii State estimation which constitutes the core of the Energy Management System (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, cannot capture the system dynamics very well due to the slow updating rate of SCADA systems. In mid 1980s of the 20 th 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 multimachine 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 estimation 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 transformation 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 systems. 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

5 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 calculations 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.

6 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

7 Contents Abstract ii Acknowledgments v List of Figures List of Tables viii xi 1 Introduction Motivations for the Study Related Work Contributions of the Thesis Thesis Outline Power System Model and Transient Stability Power System Dynamics and Swing Equation Numerical Integration Methods for the Swing Equation Euler Method Second order Runge-Kutta integration: Trapezoidal rule Fourth order Runge-Kutta integration Simulation Studies and Results Multimachine Stability Studies Classical Model and Assumptions Preliminary Calculations for Initialization Use of EKF and UKF Techniques in Power System DSE Mathematical Modeling Extended Kalman Filter (EKF) Algorithm Unscented Kalman Filter (UKF) Algorithm Simulation Studies and Results Description of Simulation Studies Generator 5-Bus Power System Generator 9-Bus Power System vi

8 Contents vii IEEE 5-Generator 14-Bus Power System IEEE 6-Generator 30-Bus Power System IEEE 8-Generator 37-Bus Power System IEEE 10-Generator 39-Bus Power System IEEE 7-Generator 57-Bus Power System Generator 145-Bus Power System Generator 162-Bus Power System Parameter Estimation for ZIP Load Modeling Polynomial (ZIP) Load Model Revised Power Flow Program Simulating ZIP Load ZIP Load Parameter Estimation Algorithm Concluding Remarks and Further Study Concluding Remarks Further Study A Matlab Script Example for EKF Algorithm 90 B Matlab Script Example for UKF Algorithm 132 Bibliography 157

9 List of Figures 2.1 An illustration of Euler method Plot of w 1 when t = s for the 3-generator 9-bus power system Plot of w 1 when t = 0.01s for the 3-generator 9-bus power system Plot of w 1 when t = 0.03s for the 3-generator 9-bus power system Plot of w 1 when t = 0.05s for the 3-generator 9-bus power system Multimachine power system representation (classical model) for transient stability study Classical model for simplified synchronous generator Kalman Filter is simply a two-step prediction-update process The one-line diagram of the 3-generator 5-bus power system Estimation of ω 1 by EKF for the 3-generator 5-bus power system Estimation of ω 1 by UKF for the 3-generator 5-bus power system Estimation of ω 2 by EKF for the 3-generator 5-bus power system Estimation of ω 2 by UKF for the 3-generator 5-bus power system Estimation of ω 3 by EKF for the 3-generator 5-bus power system Estimation of ω 3 by UKF for the 3-generator 5-bus power system Estimation of δ 2 1 by EKF for the 3-generator 5-bus power system Estimation of δ 2 1 by UKF for the 3-generator 5-bus power system Estimation of δ 3 1 by EKF for the 3-generator 5-bus power system Estimation of δ 3 1 by UKF for the 3-generator 5-bus power system The one-line diagram of the 3-generator 9-bus power system Estimation of ω 1 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 1 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 2 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 2 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 3 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 3 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 2 1 by EKF for the 3-generator 9-bus power system considering the transient case viii

10 List of Figures ix 3.21 Estimation of δ 2 1 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 3 1 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 3 1 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 1 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 1 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 2 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 2 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 3 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 3 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 2 1 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 2 1 by UKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 3 1 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 3 1 by UKF for the 3-generator 9-bus power system considering the transient case The one-line diagram of the 5-generator 14-bus power system Estimation of ω 5 by EKF for the 5-generator 14-bus power system Estimation of ω 5 by UKF for the 5-generator 14-bus power system Estimation of δ 5 1 by EKF for the 5-generator 14-bus power system Estimation of δ 5 1 by UKF for the 5-generator 14-bus power system The one-line diagram of the 6-generator 30-bus power system Estimation of ω 2 by EKF for the 6-generator 30-bus power system Estimation of ω 2 by UKF for the 6-generator 30-bus power system Estimation of δ 2 1 by EKF for the 6-generator 30-bus power system Estimation of δ 2 1 by UKF for the 6-generator 30-bus power system The one-line diagram of the 8-generator 37-bus power system Estimation of ω 7 by EKF for the 8-generator 37-bus power system Estimation of ω 7 by UKF for the 8-generator 37-bus power system Estimation of δ 7 1 by EKF for the 8-generator 37-bus power system Estimation of δ 7 1 by UKF for the 8-generator 37-bus power system Estimation of ω 10 by EKF for the 10-generator 39-bus power system Estimation of ω 10 by UKF for the 10-generator 39-bus power system Estimation of δ 10 1 by EKF for the 10-generator 39-bus power system Estimation of δ 10 1 by UKF for the 10-generator 39-bus power system 69

11 List of Figures x 3.53 The one-line diagram of the 7-generator 57-bus power system Estimation of ω 3 by EKF for the 7-generator 57-bus power system Estimation of ω 3 by UKF for the 7-generator 57-bus power system Estimation of δ 3 1 by EKF for the 7-generator 57-bus power system Estimation of δ 3 1 by UKF for the 7-generator 57-bus power system Estimation of ω 45 by EKF for the 50-generator 145-bus power system Estimation of ω 45 by UKF for the 50-generator 145-bus power system Estimation of δ 45 1 by EKF for the 50-generator 145-bus power system Estimation of δ 45 1 by UKF for the 50-generator 145-bus power system Estimation of ω 10 by EKF for the 17-generator 162-bus power system Estimation of ω 10 by UKF for the 17-generator 162-bus power system Estimation of δ 10 1 by EKF for the 17-generator 162-bus power system Estimation of δ 10 1 by UKF for the 17-generator 162-bus power system An illustration of ZIP load model Daily load profile for 14-bus system Estimation of α 4 in the 14-bus power system Estimation of β 4 in the 14-bus power system Estimation of γ 4 in the 14-bus power system

12 List of Tables 3.1 Generator dynamic data of the 3-generator 5-bus power system Performance indices of EKF and UKF for the 3-generator 5-bus power system Generator dynamic data of the 3-generator 9-bus power system Performance indices of EKF and UKF for the 3-generator 9-bus power system considering the transient case Performance indices of EKF and UKF for the 3-generator 9-bus power system considering the transient case Generator dynamic data of the 5-generator 14-bus power system Performance indices of EKF and UKF for the 5-generator 14-bus power system Generator dynamic data of the 6-generator 30-bus power system Performance indices of EKF and UKF for the 6-generator 30-bus power system Generator dynamic data of the 8-generator 37-bus power system Performance indices of EKF and UKF for the 8-generator 37-bus power system Generator dynamic data of the 10-generator 39-bus power system Performance indices of EKF and UKF for the 10-generator 39-bus power system Generator dynamic data of the 7-generator 57-bus power system Performance indices of EKF and UKF for the 7-generator 57-bus power system Performance indices of EKF and UKF for the 50-generator 145-bus power system Performance indices of EKF and UKF for the 17-generator 162-bus power system Power flow solution of 14-bus system with ZIP load xi

13 Chapter 1 Introduction The rapid growth and increasing complexity in recent years makes the monitoring 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 energy 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, optimal 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 estimate 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

14 Chapter 1. Introduction 2 The real time measurement data gathered from the network and used in the estimation 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 algorithm 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

15 Chapter 1. Introduction 3 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 developed 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 elements. 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 incorporated 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

16 Chapter 1. Introduction 4 studies. 1.2 Related Work There has been a considerable progress in the field of power system state estimation 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, realtime 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 numerical 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 systems 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 synchronized 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 system single and double line outages [13, 14]. Although the WLS method is widely

17 Chapter 1. Introduction 5 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 Estimator (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 accomplished 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 Measurement 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 behavior, 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 estimate 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 estimation 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 estimation 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

18 Chapter 1. Introduction 6 state as mentioned in [26]. Two algorithms are proposed for dynamic state estimation 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 estimation 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 synchronous machine simultaneously in [28]. A novel framework to perform EKF based dynamic state estimation in a distributed way is proposed in [29] considering increasing complexity associated with large-scale renewable resources and novel smart-grid technologies. According to [29], DSE can be implemented in a distributed environment 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 accuracy 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, amplitude and phase tracking based on UKF is presented in [33] and it is shown that

19 Chapter 1. Introduction 7 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 increasing complexity of grids and the emergency of various load components in networks. An accurate and correct representation of loads is highly significant in power system operation, control and stability studies. The need for correct representation of electrical loads is stability studies are discussed by presenting several series of loadvoltage 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 measurementbased 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 represent 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].

20 Chapter 1. Introduction 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 multimachine 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 algorithms, 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 computation 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

21 Chapter 1. Introduction 9 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.

22 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 threephase 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], Jα m = T m T e T d = T a (2.1) where J is the moment of inertia of the rotating masses in kg.m 2, α m is rotor angular acceleration, rad/s 2, T m is the mechanical torque in N.m, T e is the electrical torque in N.m, T d is the damping torque in N.m and T a is the net accelerating torque in N.m. The rotor angular acceleration is defined as α m = dω m dt = d2 θ m dt 2 (2.2) 10

23 Chapter 2. Power System Model and Transient Stability 11 where ω m = dθm dt is rotor angular velocity in rad/s and θ m is rotor angular position 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 θ m = ω msyn t + δ m (2.3) where ω msyn is synchronous angular velocity of the rotor in rad/s and δ m is the rotor angular position with respect to a synchronously rotating reference axis in rad [49]. Then, by taking the second order derivative of θ m with respect to time, we obtain d 2 θ m dt 2 = d2 δ m dt 2 (2.4) and by substituting Eq. 2.2 and Eq. 2.4, Eq. 2.1 becomes J d2 θ m dt 2 = J d2 δ m dt 2 = T m T e T d = T a. (2.5) It is better to work with power rather than torque. Accordingly, by multiplying both sides of Eq. 2.5 by angular velocity ω m we obtain Jω m d 2 δ m dt 2 = T m ω m T e ω m T d ω m (2.6) and recalling that power is the product of torque and angular velocity, the Eq. 2.1 can be expressed as Jω m d 2 δ m dt 2 = P m P e P d (2.7) where P m is the mechanical input power in W, P e is the electrical output in W and P d is the damping power in W. At synchronous speed ω msyn, the angular momentum Jω msyn is denoted by M in joules-second per mechanical radian. By using M instead of Jω m, Eq. 2.7 can be rewritten as the following equation [47] M d2 δ m dt 2 = P m P e P d (2.8)

24 Chapter 2. Power System Model and Transient Stability 12 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 = 1 2 Jω m 2 syn S B(3φ). (2.9) Replacing Jω msyn by M and solving the resulting equation for M, we obtain [47] M = 2H ω msyn S B(3φ). (2.10) Substituting Eq into Eq. 2.8 and dividing both sides by S B(3φ), the per-unit expression of the swing equation can be written as 2H d 2 δ m ω msyn dt 2 = P m P e P d S B(3φ) = P mp.u. P ep.u. P dp.u. (2.11) where P mp.u., P ep.u.. P dp.u. are per-unit mechanical, electrical and damping power respectively [47]. In Eq. 2.11, δ m is expressed in mechanical radians and ω msyn is expressed in mechanical radians per second [47]. The Eq can be rewritten as follow 2H d 2 δ ω syn dt = P 2 m P e P d in per unit. (2.12) In literature, the normalized inertia constant M is also defined as Thus, by substituting M instead of 2H ω syn equation is defined as the following equation 2H ω syn [47]. in Eq. 2.12, the convenient form of swing M d2 δ dt 2 = P m P e P d 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 dynamics as in the case of the swing equation. However, state estimation and control algorithms are almost always implemented in digital electronics [51]. Accordingly,

25 Chapter 2. Power System Model and Transient Stability 13 this often requires to transform the continuous time dynamics to discrete time dynamics [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 M dω dt = P m P e P d 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 Euler Method Consider the first-order differential equation [52] dx dt = f(x, t) (2.15) with x = x 0 at t = t 0. The Figure 2.1 illustrates the principles of applying Euler method [52]. Figure 2.1: An illustration of Euler method. At x = x 0, t = t 0 the curve representing the true solution by its tangent having a slope [52] dx dt = f(x 0, t 0 ) (2.16) x=x0

26 Chapter 2. Power System Model and Transient Stability 14 and therefore, x = dx dt t. (2.17) x=x0 The value of x at t = t 1 = t 0 + t is given by x 1 = x 0 + x = x 0 + dx dt t. (2.18) x=x0 The Euler method which is the equivalent of using the first two terms of the Taylor series expansion for x around the point (x 0, t 0 ), can be generalized as follows: x n+1 = x n + dx dt t. (2.19) x=xn 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 = t 0 + t is [52] x 1 = x 0 + x = x 0 + k 1 + k 2 2 (2.20) where k 1 = f(x 0, t 0 ) t k 2 = f(x 0 + k 1, t 0 + 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] x n+1 = x n + k 1 + k 2 2 (2.21)

27 Chapter 2. Power System Model and Transient Stability 15 where k 1 = f(x n, t n ) t k 2 = f(x n + k 1, t n + t) t Fourth order Runge-Kutta integration The general formula giving the value of x for the (n + 1) st step is [52] x n+1 = x n (k 1 + 2k 2 + 2k 3 + k 4 ) (2.22) where k 1 = f(x n, t n ) t k 2 = f(x n + k 1 2, t n + t 2 ) t k 3 = f(x n + k 2 2, t n + t 2 ) t k 4 = f(x n + k 3, t n + t) t The physical interpretation of the above solution is as follows [52]: k 1 = (slope at the beginning of time step) t k 2 = (first approximation to slope at midstep) t k 3 = (second approximation to slope at midstep) t k 4 = (slope at the end of step) t x = 1(k k 2 + 2k 3 + k 4 ) The fourth-order R-K method is equivalent to considering up to fourth derivatives in the Taylor series expansion. The three methods discussed above can be used to solve the swing equations for the multimachine power system stability.

28 Chapter 2. Power System Model and Transient Stability 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 presented. 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 ω 1 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 ω 1 can be seen in the below figures for three methods and for different time steps. In an accurate transient stability solution, the ω 1 in p.u. needs to converge to 1 after doing some oscillations starting from t = 1s. Time step=0.0005s: The plot of ω 1 when time step is s Euler 2nd order R K 4th order R K ω 1 in pu time (s) Figure 2.2: Plot of w 1 when t = s for the 3-generator 9-bus power system

29 Chapter 2. Power System Model and Transient Stability 17 Time step=0.01s: The plot of ω 1 when time step is 0.01s Euler 2nd order R K 4th order R K ω 1 in pu time (s) Figure 2.3: Plot of w 1 when t = 0.01s for the 3-generator 9-bus power system Time step=0.03s: The plot of ω 1 when time step is 0.03s nd order R K 4th order R K ω 1 in pu time (s) Figure 2.4: Plot of w 1 when t = 0.03s for the 3-generator 9-bus power system

30 Chapter 2. Power System Model and Transient Stability 18 Time step=0.05s: The plot of ω 1 when time step is 0.05s th order R K ω 1 in pu time (s) Figure 2.5: Plot of w 1 when t = 0.05s for the 3-generator 9-bus power system When we observe the behavior of ω 1, it is seen that the 4 th 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 = s. 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 2 nd order Runge Kutta method also gives accurate solutions but it is worse than 4 th order Runge Kutta method as expected. The 4 th order R-K still gives accurate solution even when the time step is set to 0.05s as seen in Figure 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

31 Chapter 2. Power System Model and Transient Stability 19 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) for transient stability study The phasors E i δ i stands for internal machine voltages whereas V ai represents the terminal bus voltages. The terminal currents I i are coming from the synchronous generators into the network 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 E i δ i behind a transient reactance x d connected in series which is illustrated on Figure 2.7 below: The magnitudes of the internal generator voltages E i are

32 Chapter 2. Power System Model and Transient Stability 20 Figure 2.7: Classical model for simplified synchronous generator kept constant throughout the transient simulation while the variation of angle δ i 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 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 V Li and complex power demand

33 Chapter 2. Power System Model and Transient Stability 21 S Li = P Li + jq Li, then using S Li = V Li ILi (which implies I Li = SLi /V Li ), we get y Li = I Li V Li = S Li V Li 2 = P Li jq Li V Li 2 (2.23) where y Li = g Li + jb Li is the equivalent shunt load admittance. 3. The internal voltages of the generators E i δ i 0 are calculated from the power flow data using the predisturbance terminal voltages V ai β i 0 as follows. The terminal voltage is used as reference. E i δ i = V ai + jx di I i. (2.24) Expressing I i in terms of S Gi and V ai, we have E i δ i = V ai + j x dis Gi V ai = V ai + j x di(p Gi jq Gi ) V ai = ( V ai + Q Gix di ) + j( P (2.25) Gix di V ai V ai ) Thus the angle difference between internal and terminal voltage in Figure is δ i. Since the actual terminal voltage angle is β i, we obtain the initial generator angle δ 0 i adding the predisturbance voltage angle β i to δ i, or by δ 0 i = δ i + β i. (2.26) 4. The Y bus 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 x d are connected between these nodes and the generator terminal nodes. b. In order to obtain the Y bus corresponding to the faulted system, usually the threephase to ground faults are considered. The faulted Y bus is then obtained by setting the row and column corresponding to the faulted node to zero. If there is any other

34 Chapter 2. Power System Model and Transient Stability 22 switching condition other than fault condition, the admittance matrix is determined for each switching condition. c. The postfault Y bus 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 Ŷ, is obtained as follows. The system Y bus for each network condition provides the following relationship between the voltages and currents: I = Y bus V (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 n I = (2.28) 0 We now partition the matrices Y bus and V appropriately to obtain I n Y nn. Y ns E n I = = 0 Y sn. Y ss V s (2.29) The subscript n is used to denote the internal generator nodes, and the subscript s is used for all the remaining nodes. Y nn is a diagonal matrix of inverted generator impedances; that is Y nn = 1 jx d1 0 1 jx d jx dn (2.30)

35 Chapter 2. Power System Model and Transient Stability 23 and also, the kmth element of Y ns is 1 jx Y nskm = dn if m = G n 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 I n = Y nn E n + Y ns V s 0 = Y sn E n + Y ss V s from which we eliminate V s to determine I n = (Y nn Y ns Y 1 ss Y sn )E n = Ŷ E n (2.32) The matrix Ŷ is the desired reduced admittance matrix. It has dimensions (nxn) where n is the number of the generators. From Eq 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 relationship 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] P Gi = Re(E i I i ). (2.33) The expression for the injected current at each generator bus I i in terms of the reduced admittance matrix parameters is given by Eq [53]. Using I n = Ŷ E n, the ith component of the injected current is given by I i = n Ŷ ij E i, i = 1, 2,, n. (2.34) j=1

36 Chapter 2. Power System Model and Transient Stability 24 The diagonal element Ŷii (i = 1, 2,, n) is the driving point admittance for node i and the off-diagonal element Ŷij (i = 1, 2,, n, i j) is the transfer admittance between nodes i and j [47, 49]. The diagonal and off-diagonal elements of the reduced admittance matrix Ŷ can be defined as follow [47] Ŷ ii = Ŷii θ ii = Ĝii + j ˆB ii Ŷ ij = Ŷij θ ij = Ĝij + j ˆB ij, where Ĝii = Ŷii cos(θ ii ) is the short-circuit conductance; Ĝ ij = Ŷij cos(θ ij ) is the transfer conductance; and ˆB ij = Ŷij sin(θ ij ) is the transfer susceptance. Substituting Eq into Eq. 2.33, the following expression can be written for the electric power delivered to the network by machine i [47, 53]: n P Gi = Re[ E i E j Ŷij (δ i δ j θ ij )], i = 1, 2,, n = j=1 n E i E j Ŷij cos(δ i δ j θ ij ). j=1 (2.35) The above equation can also be rewritten as P Gi = E 2 i Ŷii cos(θ ii ) + = E 2 i Ĝii + n E i E j Ŷij cos(δ i δ j θ ij ), i = 1, 2,, n j=1 j i n E i E j Ŷij cos(δ i δ j θ ij ). j=1 j i (2.36) The electric output power can also be written by using conductances and susceptances as follow P Gi = E 2 i Ĝii + = E 2 i Ĝii + n E i E j Ŷij [sin(θ ij ) sin(δ i δ j ) + cos(θ ij ) cos(δ i δ j )] j=1 j i n E i E j [ ˆB ij sin(δ i δ j ) + Ĝij cos(δ i δ j )], i = 1, 2,, n. j=1 j i (2.37)

37 Chapter 2. Power System Model and Transient Stability 25 The mechanical power input P mi for each generator is constant throughout the transient stability procedure. The damping power is expressed by the damping constant,in seconds per electrical radian, times first derivative of the rotor angle which is specified as P di = D i dδ i dt = D i w i. 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]: M i dω i dt = P mi E 2 i Ĝii n E i E j [ ˆB ij sin(δ i δ j ) + Ĝij cos(δ i δ j )] P di j=1 j i dδ i dt = ω i, 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 considering the prefault, fault and postfault cases, the actual values are obtained at each computation time step for each generator in the system.

38 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 representing 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 system. 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

39 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]: x k = x k 1 + f(x k 1, u k 1, w k 1 ) t = g(x k 1, u k 1, w k 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]: z k = h(x k, v k ) (3.3) The resulting error between the measured and calculated values is given by [27] ɛ k = z k h(x k, v k ) (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 predictioncorrection process, can be summarized as [51]:

40 Chapter 3. Use of EKF and UKF Techniques in Power System DSE The discrete time system equations are presented as follows: x k = f k 1 (x k 1, u k 1, w k 1 ) y k = h k (x k, v k ) w k (0, Q k ) (3.5) v k (0, R k ) The system noise covariance matrix is represented by Q k and the measurement noise covariance matrix is represented by R k. 2. Initialize the filter: ˆx + 0 = E(x 0 ) P + 0 = E[(x ˆx + 0 )(x ˆx + 0 ) T ] (3.6) where ˆx + 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 ˆx + k 1 : Fk 1 = f k 1 x L k 1 = f k 1 w + ˆx k 1 ˆx + k 1 (3.7) 4. Perform the time update of state estimate and estimation-error covariance matrix: P k = F k 1P + k 1 F T k 1 + L k 1 Q + k 1 LT k 1 ˆx k = f k 1(ˆx + k 1, u k 1, 0) (3.8) where the subscript denotes the estimate is in an a priori estimate. Correction:

41 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Perform the following partial derivative matrices at the state update ˆx k : H k = h k x V k = h k v ˆx k ˆx k (3.9) 6. Perform the measurement update of the state estimate and estimation covariance as follows: K k = P k HT k (H k P k HT k + V k R k V T k ) 1 ˆx + k = ˆx k + K k[y k h k (ˆx k, 0)] (3.10) P + k = (I K kh k )P k where K k is the Kalman gain matrix, ˆx + k is the state estimate and P + k is the estimation 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 multimachine dynamic variables uncluding rotor speed and rotor angle. It shows sufficiently 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. Accordingly, 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 linearization 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 expansion 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.

42 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 30 (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 representing the system dynamics can create low accuracy and reduced estimation performance. 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].

43 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 31 By this theory, the statistical distribution of the state is propagated through the nonlinear 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 covariance P xx. 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 distribution of x exactly. The sigma points are then propogated to calculate the mean ȳ and covariance P yy of y. Based on the knowledge of x, the 2n + 1 sigma points can be found as x 0 = m x i = m + ( (n + λ)pxx ) i i = 1,..., n (3.12) x i+n = m ( (n + λ)pxx ) i i = 1,..., n ( ) (n i where + λ)pxx is the ith row or column of matrix square root of the (n + λ)p xx. The parameter λ can be defined as λ = α 2 (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 = AA T, 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 calculated as below: χ i = f(x i ) (3.13)

44 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 32 Then the mean and covariance of y can be calculated by using the previously calculated transformed sigma points as: ȳ = P yy = 2n i=0 2n i=0 W i my i W i c(χ i ȳ)(χ i ȳ) T (3.14) where the weights W i m and W i c are defined as W 0 m = λ n + λ Wc 0 = λ n + λ + (1 α2 + β) Wm i = Wc i 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 calculation, 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: x k = f k 1 (x k 1, u k 1, w k 1 ) y k = h k (x k, v k ) w k (0, Q k ) (3.16) v k (0, R k ) The system noise covariance matrix is represented by Q k and the measurement noise covariance matrix is represented by R k. 2. Initialize the filter: ˆx + 0 = E(x 0 ) P + 0 = E[(x ˆx + 0 )(x ˆx + 0 ) T ] (3.17)

45 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 33 where ˆx + 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 ˆx i k 1 are specified according to the following formula: ˆx (i) k 1 = ˆx+ k 1 + x(i), i = 1,..., 2n ( ) T x (i) = (n + λ)p + k 1, i = 1,..., n ( x (n+i) = (n + λ)p + k 1 i ) T i, i = 1,..., n (3.18) (b) Use the known nonlinear system equation f(.) to transform the sigma points into ˆx (i) k vectors as shown in Eq with appropriate changes since our nonlinear transformation is f(.) rather than h(.): ˆx (i) k 1 f( ) ˆx (i) k, ˆx(i) k = f(ˆx (i) k 1, u k, t k ) (3.19) (c) Combine the ˆx (i) k given by the following formula: vectors to obtain the a priori state estimate at time k which is ˆx k = 1 2n 2n i=1 ˆx (i) k (3.20) (d) Estimate the a priori error covariance by adding Q k 1 to the end of the equation in order to take the process noise into account: P k = 1 2n 2n i=1 (ˆx (i) k ˆx k )(ˆx(i) k ˆx k )T + Q k 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.

46 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 34 (a) Choose sigma points ˆx (i) k for the mean and covariance of x k are ˆx k and P k : with appropriate changes since the current best guess ˆx (i) k = ˆx k + x(i), i = 1,..., 2n ( ) T x (i) = (n + λ)p k, i = 1,..., n ( x (n+i) = (n + λ)p k i ) T i, i = 1,..., n (3.22) (b) Use the known nonlinear measurement equation h(.) to transform the sigma points into ŷ (i) k vectors as follow: ŷ (i) k h( ) ˆx (i) k, ŷ(i) k = h(ˆx (i) k, t k) (3.23) (c) Combine the ŷ (i) k vectors to obtain the predicted measurement at time k: ŷ k = 1 2n 2n i=1 ŷ (i) k (3.24) (d) Estimate the covariance of the predicted measurement by adding R k to the end of the equation in order to take the measurement noise into account: P y = 1 2n 2n i=1 (ŷ (i) k ŷ k)(ŷ (i) k ŷ k) T + R k (3.25) (e) Estimate the cross covariance between ˆx k and ŷ k: P xy = 1 2n 2n i=1 (ˆx (i) k ˆx k )(ŷ(i) k ŷ k) T (3.26)

47 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 35 (f) Finally, the measurement update of the state estimate can be performed by using the normal Kalman filter equations: K k = P xy P 1 y ˆx + k = ˆx k + K k(y k ŷ k ) (3.27) P + k = P k (K kp y K T k ) where K k is the Kalman gain matrix, ˆx + k is the state estimate and P + k is the estimation error covariance matrix. 3.4 Simulation Studies and Results 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.

48 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 36 The dynamic state vector x and the measurement vector z can be shown as below: ω 1 ω 2. ω x = n δ 1 δ 2. δ n P G1. P Gn Q G1. Q z = Gn V 1. V s θ 1. θ s (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 measurements and the state estimates at time instant k along with the mathematical

49 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 37 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 measurement noise. This means that the diagonal elements of the Q (system noise covariance) and R (measurement noise covariance) matrix are set to σ 2 = 10 4 during the simulations. The diagonal elements of the initial error covariance matrix P 0 is also set to σ 2 = The initial values of the state estimate vector ˆx + 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 integration 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 S base = 100 MV A 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]: ξ k = 1 2n 2n i=1 x i k ˆx i k (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 ˆx 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

50 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 38 each computation step: ξ = 1 k max ξ k (3.30) k max k=1 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: ξ ω = 1 k max ξ δ = 1 k max k max k=1 k max k=1 [ 1 n [ 1 n ] n ωk i ˆω k i i=1 ] (3.31) n δk i ˆδ k i i=1 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 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 G 1 G 2 G 3 x d p.u H s D p.u

51 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 39 Figure 3.2: The one-line diagram of the 3-generator 5-bus power system Generator 1 : The rotor speed ω 1 is estimated by using EKF as shown on Figure 3.3 and UKF as shown on Figure Estimation of ω 1 by EKF ω 1 ω est ω 1 in pu time (s) Figure 3.3: Estimation of ω 1 by EKF for the 3-generator 5-bus power system

52 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of ω 1 by UKF ω 1 ω est ω 1 in pu time (s) Figure 3.4: Estimation of ω 1 by UKF for the 3-generator 5-bus power system Generator 2 : The rotor speed ω 2 is estimated by using EKF as shown on Figure 3.5 and UKF as shown on Figure Estimation of ω 2 by EKF ω 2 ω est ω 2 in pu time (s) Figure 3.5: Estimation of ω 2 by EKF for the 3-generator 5-bus power system

53 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of ω 2 by UKF ω 2 ω est ω 2 in pu time (s) Figure 3.6: Estimation of ω 2 by UKF for the 3-generator 5-bus power system Generator 3 : The rotor speed ω 3 is estimated by using EKF as shown on Figure 3.7 and UKF as shown on Figure Estimation of ω 3 by EKF ω 3 ω est ω 3 in pu time (s) Figure 3.7: Estimation of ω 3 by EKF for the 3-generator 5-bus power system

54 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of ω 3 by UKF ω 3 ω est ω 3 in pu time (s) Figure 3.8: Estimation of ω 3 by UKF for the 3-generator 5-bus power system Generator 2 : The relative rotor angle δ 2 1 is estimated by using EKF as shown on Figure 3.9 and UKF as shown on Figure Estimation of δ 2 1 by EKF 0 δ 2 1 est δ δ 2 1 in rad time (s) Figure 3.9: Estimation of δ 2 1 by EKF for the 3-generator 5-bus power system

55 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of δ 2 1 by UKF 0.05 δ 2 1 est δ δ 2 1 in rad time (s) Figure 3.10: Estimation of δ 2 1 by UKF for the 3-generator 5-bus power system Generator 3 : The relative rotor angle δ 3 1 is estimated by using EKF as shown on Figure 3.11 and UKF as shown on Figure Estimation of δ 3 1 by EKF 0.3 δ 3 1 est δ δ 3 1 in rad time (s) Figure 3.11: Estimation of δ 3 1 by EKF for the 3-generator 5-bus power system The comparison of the performance indices of EKF and UKF is illustrated in Table 3.2.

56 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of δ 3 1 by UKF 0.3 δ 31 est δ δ 3 1 in rad time (s) Figure 3.12: Estimation of δ 3 1 by UKF for the 3-generator 5-bus power system Table 3.2: Performance indices of EKF and UKF for the 3-generator 5-bus power system Performance EKF UKF Index ξ ξ ω ξ δ Generator 9-Bus Power System 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 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 two transient cases. Transient Case 1 : Fault at Bus 4 from t = 1s to t = 1.15s [27]. Table 3.3: Generator dynamic data of the 3-generator 9-bus power system Para. Unit G 1 G 2 G 3 x d p.u H s D p.u

57 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 45 Figure 3.13: The one-line diagram of the 3-generator 9-bus power system Generator 1 : The rotor speed ω 1 is estimated by using EKF as shown on Figure 3.14 and UKF as shown on Figure Estimation of ω 1 by EKF ω 1 ω est ω 1 in pu time (s) Figure 3.14: Estimation of ω 1 by EKF for the 3-generator 9-bus power system considering the transient case 1

58 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of ω 1 by UKF ω 1 ω est ω 1 in pu time (s) Figure 3.15: Estimation of ω 1 by UKF for the 3-generator 9-bus power system considering the transient case 1 Generator 2 : The rotor speed ω 2 is estimated by using EKF as shown on Figure 3.16 and UKF as shown on Figure Estimation of ω 2 by EKF ω 2 ω est ω 2 in pu time (s) Figure 3.16: Estimation of ω 2 by EKF for the 3-generator 9-bus power system considering the transient case 1

59 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of ω 2 by UKF ω 2 ω est ω 2 in pu time (s) Figure 3.17: Estimation of ω 2 by UKF for the 3-generator 9-bus power system considering the transient case 1 Generator 3 : The rotor speed ω 3 is estimated by using EKF as shown on Figure 3.18 and UKF as shown on Figure Estimation of ω 3 by EKF ω 3 ω est ω 3 in pu time (s) Figure 3.18: Estimation of ω 3 by EKF for the 3-generator 9-bus power system considering the transient case 1

60 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of ω 3 by UKF ω 3 ω est ω 3 in pu time (s) Figure 3.19: Estimation of ω 3 by UKF for the 3-generator 9-bus power system considering the transient case 1 Generator 2 : The relative rotor angle δ 2 1 is estimated by using EKF as shown on Figure 3.20 and UKF as shown on Figure Estimation of δ 2 1 by EKF δ 2 1 est δ δ 2 1 in rad time (s) Figure 3.20: Estimation of δ 2 1 by EKF for the 3-generator 9-bus power system considering the transient case 1

61 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 49 1 Estimation of δ 2 1 by UKF δ 2 1 est δ δ 2 1 in rad time (s) Figure 3.21: Estimation of δ 2 1 by UKF for the 3-generator 9-bus power system considering the transient case 1 Generator 3 : The relative rotor angle δ 3 1 is estimated by using EKF as shown on Figure 3.22 and UKF as shown on Figure Estimation of δ 3 1 by EKF 0.8 δ 3 1 est δ δ 3 1 in rad time (s) Figure 3.22: Estimation of δ 3 1 by EKF for the 3-generator 9-bus power system considering the transient case 1 The comparison of the performance indices of EKF and UKF is illustrated in Table 3.4 for the transient case 1.

62 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of δ 3 1 by UKF 0.6 δ 3 1 est δ δ 3 1 in rad time (s) Figure 3.23: Estimation of δ 3 1 by UKF for the 3-generator 9-bus power system considering the transient case 1 Table 3.4: Performance indices of EKF and UKF for the 3-generator 9-bus power system considering the transient case 1 Performance EKF UKF Index ξ ξ ω ξ δ Transient Case 2 : Load change of 100 W at Bus 4 from t = 1s to t = 1.15s [27].

63 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 51 Generator 1 : The rotor speed ω 1 is estimated by using EKF as shown on Figure 3.24 and UKF as shown on Figure Estimation of ω 1 by EKF ω 1 ω est ω 1 in pu time (s) Figure 3.24: Estimation of ω 1 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 1 by UKF ω 1 ω est ω 1 in pu time (s) Figure 3.25: Estimation of ω 1 by UKF for the 3-generator 9-bus power system considering the transient case 2

64 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 52 Generator 2 : The rotor speed ω 2 is estimated by using EKF as shown on Figure 3.26 and UKF as shown on Figure Estimation of ω 2 by EKF ω 2 ω est ω 2 in pu time (s) Figure 3.26: Estimation of ω 2 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 2 by UKF ω 2 ω est ω 2 in pu time (s) Figure 3.27: Estimation of ω 2 by UKF for the 3-generator 9-bus power system considering the transient case 2

65 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 53 Generator 3 : The rotor speed ω 3 is estimated by using EKF as shown on Figure 3.28 and UKF as shown on Figure Estimation of ω 3 by EKF ω 3 ω est ω 3 in pu time (s) Figure 3.28: Estimation of ω 3 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of ω 3 by UKF ω 3 ω est ω 3 in pu time (s) Figure 3.29: Estimation of ω 3 by UKF for the 3-generator 9-bus power system considering the transient case 2

66 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 54 Generator 2 : The relative rotor angle δ 2 1 is estimated by using EKF as shown on Figure 3.30 and UKF as shown on Figure Estimation of δ 2 1 by EKF δ 2 1 est δ δ 2 1 in rad time (s) Figure 3.30: Estimation of δ 2 1 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 2 1 by UKF δ 2 1 est δ δ 2 1 in rad time (s) Figure 3.31: Estimation of δ 2 1 by UKF for the 3-generator 9-bus power system considering the transient case 2

67 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 55 Generator 3 : The relative rotor angle δ 3 1 is estimated by using EKF as shown on Figure 3.32 and UKF as shown on Figure Estimation of δ 3 1 by EKF 0.3 δ 3 1 est δ δ 3 1 in rad time (s) Figure 3.32: Estimation of δ 3 1 by EKF for the 3-generator 9-bus power system considering the transient case Estimation of δ 3 1 by UKF 0.3 δ 3 1 est δ δ 3 1 in rad time (s) Figure 3.33: Estimation of δ 3 1 by UKF for the 3-generator 9-bus power system considering the transient case 2 The comparison of the performance indices of EKF and UKF is illustrated in Table 3.5 for the transient case 2.

68 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 56 Table 3.5: Performance indices of EKF and UKF for the 3-generator 9-bus power system considering the transient case 2 Performance EKF UKF Index ξ ξ ω ξ δ 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 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. Figure 3.34: The one-line diagram of the 5-generator 14-bus power system

69 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 57 Table 3.6: Generator dynamic data of the 5-generator 14-bus power system Para. Unit G 1 G 2 G 3 G 4 G 5 x d p.u H s D p.u Generator 5 : The rotor speed ω 5 is estimated by using EKF as shown on Figure 3.35 and UKF as shown on Figure Estimation of ω 5 by EKF ω 5 ω est ω 5 in pu time (s) Figure 3.35: Estimation of ω 5 by EKF for the 5-generator 14-bus power system

70 Chapter 3. Use of EKF and UKF Techniques in Power System DSE Estimation of ω 5 by UKF ω 5 ω est ω 5 in pu time (s) Figure 3.36: Estimation of ω 5 by UKF for the 5-generator 14-bus power system Generator 5 : The relative rotor angle δ 5 1 is estimated by using EKF as shown on Figure 3.37 and UKF as shown on Figure Estimation of δ 5 1 by EKF 0.8 δ 5 1 est δ δ 5 1 in rad time (s) Figure 3.37: Estimation of δ 5 1 by EKF for the 5-generator 14-bus power system The comparison of the performance indices of EKF and UKF is illustrated in Table 3.7.

71 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 59 0 Estimation of δ 5 1 by UKF 0.1 δ 5 1 est δ δ 5 1 in rad time (s) Figure 3.38: Estimation of δ 5 1 by UKF for the 5-generator 14-bus power system Table 3.7: Performance indices of EKF and UKF for the 5-generator 14-bus power system Performance EKF UKF Index ξ ξ ω ξ δ IEEE 6-Generator 30-Bus Power System The one-line diagram of the IEEE 6-generator 30-bus test power system is shown on Figure The generator dynamic data is given in Table 3.8. 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 3 4 is removed at t = 1s. Table 3.8: Generator dynamic data of the 6-generator 30-bus power system Para. Unit G 1 G 2 G 3 G 4 G 5 G 6 x d p.u H s D p.u

72 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 60 Figure 3.39: The one-line diagram of the 6-generator 30-bus power system Generator 2 : The rotor speed ω 2 is estimated by using EKF as shown on Figure 3.40 and UKF as shown on Figure 3.41.

73 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 61 Estimation of ω 2 by EKF ω 2 ω est ω 2 in pu time (s) Figure 3.40: Estimation of ω 2 by EKF for the 6-generator 30-bus power system Estimation of ω 2 by UKF ω 2 ω est ω 2 in pu time (s) Figure 3.41: Estimation of ω 2 by UKF for the 6-generator 30-bus power system

74 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 62 Generator 2 : The relative rotor angle δ 2 1 is estimated by using EKF as shown on Figure 3.42 and UKF as shown on Figure Estimation of δ 2 1 by EKF 0 δ 2 1 est δ δ 2 1 in rad time (s) Figure 3.42: Estimation of δ 2 1 by EKF for the 6-generator 30-bus power system 0.2 Estimation of δ 2 1 by UKF 0 δ 2 1 est δ δ 2 1 in rad time (s) Figure 3.43: Estimation of δ 2 1 by UKF for the 6-generator 30-bus power system The comparison of the performance indices of EKF and UKF is illustrated in Table 3.9.

75 Chapter 3. Use of EKF and UKF Techniques in Power System DSE 63 Table 3.9: Performance indices of EKF and UKF for the 6-generator 30-bus power system Performance EKF UKF Index ξ ξ ω ξ δ IEEE 8-Generator 37-Bus Power System The one-line diagram of the IEEE 8-generator 37-bus test power system is shown on Figure The generator dynamic data is given in Table 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 is removed between t = 2s and t = 2.5s. Figure 3.44: The one-line diagram of the 8-generator 37-bus power system