REAL TIME PMU-BASED STABILITY MONITORING ZIJIE LIN. A thesis submitted in partial fulfillment of the requirement for the degree of

Transcription

1 REAL TIME PMU-BASED STABILITY MONITORING By ZIJIE LIN A thesis submitted in partial fulfillment of the requirement for the degree of MASTER OF SCIENCE IN ELECTRICAL ENGINEERING WASHINGTON STATE UNIVERSITY School of Electrical Engineering and Computer Science AUGUST 215 i

2 To the Faculty of Washington State University: The members of the Committee appointed to examine the thesis of Zijie Lin find it satisfactory and recommend that it be accepted. Chen-Ching Liu, Ph.D., Chair Anjan Bose, Ph.D. Sandip, Ph.D. ii

3 ACKNOWLEDGEMENT I am extremely grateful to my supervisor Prof. Chen-Ching Liu, Director of Energy Systems Innovation (ESI) Center of Washington State University. He provided this great opportunity for me to participate in the project of Real Time PMU-Based Stability Monitoring for PJM interconnection and Power Systems Engineering Research Center (PSERC S-5). His creative guidance and invaluable suggestions helped me in completing the project. I sincerely thank Guanqun Wang, who offered his help during the whole period of my project work since September 213. I also appreciate the contributions of Haosen Guo and Jie Yan since part of my research is based on their original work. In addition, I would like to express my appreciation for the support provided by industry members of PSERC. Finally, I am indebted to my beloved parents for their understanding and support. I want to thank Bonneville Power Administration (BPA) and PJM for the test data, and the support in part by National Science Foundation. I am also very grateful to the industry advisors of the PSERC S-5 project for their advice: Alan Engelmann, Exelon/Commonwealth Edison Bill Timmons, Western Area Power Administration Clifton Black, Southern Company David Schooley, Exelon/Commonwealth Edison Dmitry Kosterev, Bonneville Power Administration Eugene Litinov, ISO New England Evangelos Farantatos, Electric Power Research Institute George Stefopoulos, New York Power Authority Guiseppe Stanciulescu, BC Hydro iii

4 Jay Giri, ALSTOM Grid Jinan Huang, Hydro-Québec Research Institute Liang Min. Lawrence Livermore National Lab Manu Parashar, ALSTOM Grid Patrick Panciatici, Réseau de Transport d'électricité Sanjoy Sarawgi, American Electric Power Steven Hedden, Exelon/Commonwealth Edison Xiaochuan Luo, ISO New England iv

5 REAL TIME PMU-BASED STABILITY MONITORING Abstract by Zijie Lin, MS Washington State University August 215 Chair: Chen-Ching Liu The purpose of this thesis is to develop PMU-based, real-time, wide area stability monitoring algorithms for the power grids using different methods and approaches. Phasor Measurement Units (PMUs) are increasingly available on power grids due to the significant investment in recent years. As a result, a priority in industry is to extract critical information from the increasing amount of PMU data for operation, planning, protection, and control. This research proposes enhanced algorithms for real time stability monitoring in a control center environment: the waveform analysis to extract the trending information of system dynamics embedded in Lyapunov exponents. The algorithm will help build a complete model for predicting system stability, which can help to avoid cascading failures and enhance system security. A PMU-based online waveform stability monitoring technique is proposed based on the Maximum Lyapunov Exponent (MLE). The main idea of the MLE technique is to calculate MLE as an index over a finite time window in order to predict unstable trending of the operating conditions. Significant progress has been made to improve the accuracy of MLE technique. Firstly, state calculator is applied to improve the MLE accuracy by v

6 introducing real time monitoring measurement into the computation. Secondly, the power system model is greatly improved by adopting a dynamic structure-preserving load model. This model takes into account the dynamics of P and Q load with respect to the frequency/voltage variations. The purpose is to extend the MLE technique to voltage stability analysis as well as rotor angle stability. Based on this model, the system can be represented by a set of differential equations, which is used for MLE calculation. The power network topology is preserved. Thirdly, parameters for the model are identified from the results of time domain simulation based on their importance. Finally, the methods to save memory and to improve the calculation speed are also discussed. The proposed methods are validated by time-domain simulation of the122-bus Mini-WECC system and the15-bus PJM system model. vi

7 TABLE OF CONTENTS ACKNOWLEDGEMENT... iii ABSTRACT...v LIST OF TABLES... x LIST OF FIGURES... xi CHAPTER 1. INTRODUCTION Background Overview Organization of the Thesis LYAPUNOV EXPONENT THEORY Maximum Lyapunov Exponent MLE Stability Criterion State Calculator Summary DYNAMIC LOAD MODEL FOR MLE CALCULATION Reduced Dynamic Model Constant Current Load Constant Impedance Load Static Load Models Structure Preserving Model Frequency Dependent Model Load Recovery Model vii

8 3.3.Partial Structure-Preserving Model Parameter Identification Summary PARAMETER IDENTIFICATION Estimation Theory Identifying the Parameter in Real Power Model Identifying the Parameters in Reactive Power Model Eliminating two parameters Estimating the parameter Summary ENHANCEMENT OF THE COMPUTATIONAL SPEED Computation of Large Scale Power Systems Aggregation of Generators and Elimination of Buses Eliminate Buses with No Generation or Load Combine Generators at the Same Bus Use the Different Method to Calculate Differential Equations ODE 15s Method Forward Euler Method Improving the speed of MLE calculation Summary CASE STUDY AND SIMULATION RESULTS Mini-WECC System Simulation Model Application to the Reduced PJM System viii

9 6.3.Analysis and Summary CONCLUSIONS REFFERENCE...49 APPENDIX I: All Test Results of Mini-WECC System ix

10 LIST OF TABLES Table 6-1 System Parameter Verification in Mini-WECC System...36 Table 6-2 Typical Examples in Mini-WECC System...4 Table 6-3 Results of Two Load Models in PJM System...43 x

11 LIST OF FIGURES Fig 2.1 Perturbation of Trajectories of a Dynamical System...5 Fig 2.2 Different Trajectories and Corresponding Lyapunov Exponents...6 Fig 2.3 Stability of a Nonlinear Dynamical System...7 Fig 3.1 Parameters of a Three-bus System Fig 3.2 Structure of the Five-node system...14 Fig 3.3 Structure of the System after Reduction (Current Source Load)...16 Fig 3.4 Structure of the System after Reduction (Admittance Load)...17 Fig 3.5 Power Recovery Process after a Step Change of Voltage... 2 Fig 4.1 Linear Model in Estimation Theory...23 Fig 5.1 Concept of MLE(Maximum Lyapunov Exponent)...32 Fig 6.1 One-line Diagram of Mini-WECC System...36 Fig 6.2 Reactive Power Load at Bus 8 from PSAT Simulation...38 Fig 6.3 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs= Fig 6.4 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs= Fig 6.5 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs= Fig 6.6 Comparison of Different Models...41 Fig 6.7 Comparison of Two Models for PJM System(Type 1)...45 Fig 6.8 Comparison of Two Models for PJM System(Type 2)...45 Fig 6.9 Comparison of Two Models for PJM System(Type 3)...46 xi

12 CHAPTER ONE INTRODUCTION 1.1 Background Since 196s, the scale and complexity of interconnected power grids in the United States have increased to such an extent that large blackouts are growing in both number and severity [1]. For example, in the 9 s, there were 66 blackouts in the U.S. that affected over 1MW in the first five years. However, during 26-21, the number of such outages reached 219, more than tripled [2]. Large scale power outages cause significant damages to utilities and economic losses. As a result, it is critical to enhance the technology for predicting and preventing large scale power outages. Phasor measurement units (PMUs) provide synchronized measurements of the state of the power system at a rate up to 12 samples per second [3]. Based on the IEEE standard for synchronous phasors, the basic time synchronization accuracy of PMU is ±.2μs [4]. Since communication technology is well developed, it enables online monitoring of power systems based on PMU data. In recent years, the number of PMUs installed on the power grids increased significantly and research on the applications of PMUs in monitoring, protection, and control of power grid has made great progress [5]. Bonneville Power Administration (BPA) is the first utility to implement PMUs in its wide-area monitoring system. The main purpose of this thesis is to develop a PMU-based, real-time, wide area stability monitoring algorithm for power grids. The proposed algorithm takes advantage of the PMU-measured waveforms and Maximum Lyapunov Exponent stability detection 1

13 and it extracts the information of power system dynamics to determine if the system is approaching instability. This algorithm can be applied to real time monitoring as a tool to issue warning messages for system operators when a trend toward instability is detected. 1.2 Overview Traditionally, there are two major approaches to transient stability analysis of power systems. One is time-domain simulation methods [6-7], based on numerical integration of the power system dynamic equations. This method provides details of the dynamic waveforms for a large scale system if the component and network data are available. Parallel-in-time algorithms can be performed to improve the calculation speed of the time-domain simulation [8]. However, the high computational burden limits the online applicability of this method. The other approach is the energy functions based direct methods, such as relevant unstable equilibrium point (RUEP) method [9], controlling UEP (CUEP) method [1], potential energy boundary surface (PEBS) [11], BCU method [12], and extended equal area criteria (EEAC) method [13]. These methods have relatively fast computational speed and provide quantitative indices for stability assessment. A shortcoming of direct methods is that the assessment result may not incorporate sufficient details due to simplification of the system models. There are also knowledge-based transient stability methods. In [14], decision trees are created off-line based on a large number of simulations. The results are used to predict transient stability in an on-line environment. The hybrid intelligent system proposed in [15] can be used to assess not only transient stability, but also generator tripping. In [16], on-line PMU data 2

14 are compared with trajectory patterns stored in the database. The Euclidean distance is used as an index for prediction of system stability. In this research, a new method called Maximum Lyapunov Exponent (MLE) is proposed for monitoring of wide-area transient stability based on PMU data in an online environment. The Maximal Lyapunov Exponent (MLE), which is a tool for prediction of out-of-step conditions [17], is applied to the analysis of rotor angle stability [18-19]. Based on a nonlinear dynamic model of the power system, the MLE can be calculated based on the waveforms resulting from dynamics of the system states. It is based on a rigorous theoretical foundation and, therefore, it is able to reliably determine the stability status using PMU streaming data within a specified time window. However, it should be noted that certain factors could have a significant effect on the accuracy of the prediction based on the MLE technique. Two problems need to be solved to improve the proposed technique: 1. System model: For a large scale power system with generator and load buses interconnected by transmission lines, the dynamic characteristics of a power system for stability analysis is represented by a set of differential-algebraic equations (DAEs). Previously, in order to apply the MLE technique, the power network needs to be reduced to generator buses only and all the loads are modeled by constant impedances. This will lead to a loss of the system topology, reducing the level of accuracy for system stability assessment. Also, it is known that there are interactions between angle and voltage dynamics. Therefore, the dynamic model of the system needs to be improved by retaining the load buses and incorporating voltage dynamics of the system. 3

15 2. Parameter identification: This research project addresses these critical issues by proposing new techniques. First, a structure-preserved model is applied to transient stability analysis. This model retains the load buses as well as the network topology. Based on this model, the MLE technique considers generator dynamics, in addition to accounting for the load dynamics. Moreover, the structure-preserving model enables the MLE to predict short-term voltage stability more accurately. Parameter identification for this model has been conducted. The methods proposed in this research are validated by extensive simulations. 1.3 Organization of the Thesis The organization of the remaining chapters is as follows: Chapter 2 is a summary of the theoretical basis of the MLE technique and State Calculator. Chapter 3 describes the structure-preserving model and load recovery model for the MLE calculation. In Chapter 4, parameters are identified using linear system theory. Chapter 5 offers methods to improve the calculation speed. In Chapter 6, the PJM system and 122-bus Mini-WECC system are utilized to validate the proposed methods. Numerous time-domain simulations of these systems are performed in order to validate the MLE method. The conclusions and future work are provided in Chapter 7. 4

16 2. CHAPTER TWO LYAPUNOV EXPONENT THEORY 2.1 Maximum Lyapunov Exponent In ergodic theory [2], Lyapunov Exponent is used to characterize whether a given system is chaotic and to quantify the amount of chaos. For a trajectory with an initial value all trajectories that start out in a neighborhood of x will converge toward,or diverge as time propagates. Sensitivity of the system trajectory with respect to a perturbation of the initial state is shown in Figure 2.1 and it is quantified by (2.1) x x xt xt Fig2.1 Perturbation of Trajectories of a Dynamical System Equation (2.1) describes the exponential behaviour of nearby trajectories in a short time interval. Therefore, the time average is used to measure the rate of divergence or convergence of trajectories resulting from infinitesimal perturbations of the initial state. Assume that the system trajectory is defined by differential equations (2.2) From (2.1) 1 ln x( t) / x (2.3) t By taking the limit as t, the Lyapunov exponent is given by 5

17 Converge Diverge Converge Diverge 1 lim ln x( t) / x t t (2.4) where represents the mean growth rate of the distance between neighboring trajectories. 2.2 MLE Stability Criterion A negative MLE implies that the nearby trajectories will converge to the reference trajectory exponentially, such as the Fig.2.2(a). If the nearby trajectories diverge, such as the Fig 2.2(b), the calculation result of MLE will be positive [28] (a) Convergent Trajectories: Negative MLE (b) Divergent Trajectories: Positive MLE (c) Impossible Trajectories1 (d) Impossible Trajectories2 Fig 2. 2 Different Trajectories and Corresponding Lyapunov Exponents 6

18 The theorem of continuous-time fixed point [22] established the relationship between the MLE and the asymptotic behavior of the dynamic system. Consider a continuous-time dynamic system and assume all the Lyapunov exponents are non-zero. Then the steady state behavior of the system consists of a fixed point. In particular, if all Lyapunov Exponents are negative, i.e., MLE is negative, this fixed point is an attracting fixed point. In summary, it is impossible that original trajectory is stable but MLE is positive, which means updated trajectory converge to the original one if the original trajectory is stable. The following is the proof of the theorem[17]. Consider the trajectory x f x as 2 2 a nearby trajectory of (2.2), with an initial value x () 2, as shown in Fig Give the x1 a disturbance dx and let x2() x1() dx, which means that dx x2() x1() According to the definition of Lyapunov exponent, if the MLE of exists an such that, for any x x, x1 t is negative, there lim x t x t (2.5) t x1(t) x2(t)=x1(t+ t) x1() x2()=x1( t) Fig 2.3 Stability of a Nonlinear Dynamical System Since, then x t x t t x x t 2 1 xt is continuous, there must exist a T 2 1. Since, such that x t x. Let 7

19 it is obtained that x x x t x (2.6) lim x t x t lim x t t x t lim f x t T (2.7) t t t Since T, it follows that lim f x t (2.8) t It means when t tends to infinity, is smaller than. x1 t will approach an stable state k if MLE of this case From the proof above, it is also clear that possibility of the situation in Fig2.2(c) and (d) is. It means that during the MLE simulation, it is impossible that a original trajectory oscillates less and less but the update trajectory diverges from it. Similarly, if a original trajectory diverges, it cannot be true that update trajectory still converge to it. 2.3 State Calculator State Calculator is driven from the available PMU measurements. In this project, the MLE calculation is based on the State Calculator. As introduced in Chapter 1, PMU allows online measurement of multiple points at the same time. It can measure frequency, current, voltage, and power as well as phasor angle, at both 5 and 6 Hz waveforms for 288 samples per second. A phase-lock oscillator along with a Global Positioning System (GPS) provides the high-speed synchronized sampling. A PMU can be a dedicated device or it can be incorporated into a protective relay, installed at substations or power plants. Phasor Data Concentrators (PDC) are used to collect the information. A Supervisory Control And Data Acquisition (SCADA) system connects the central control facility with various substations. In summary, in a power system monitoring network, PMUs are 8

20 installed at various buses throughout the entire power system to acquire the measurements. It is known from the linear and observability theory [26] and [27], that uniform placement of the observation devices will give the largest observability. Thus, in this thesis, the PMUs are uniformed distributed among the 15, buses and then simulated using the TSAT. The PMU reports the data every several cycles. After calculating a time point in Matlab, the result will be replaced with PMU measurement data if this bus has a PMU. The PJM system has 15, buses and among them, 1,9 buses have PMUs. These 1,9 load buses calculation results (frequency or voltage) will be replaced by the real time PMU data. When calculations are completed, the MLE (Maximum Lyapunov Exponent) method is employed to analyze these waveforms to see whether the system is stable or not. If the waveforms converge, they are considered to be stable. If they diverge, they are judged to be unstable. Algorithm 1.Place the PMUs at heavily loaded buses or generator buses. 2. Calculate the waveform of every bus for the test system under a disturbance, including load bus's phase angle, generator bus's rotor angle and rotor speed. 3. The data from PMU have a higher level of accuracy. PMU reports data every 2 cycles (.33s) while the time step in the calculation is.33s, so check the PMU data every 1 calculation loops. Then replace the calculation result with the PMU measurements. 9

21 4. Use the updated waveform to calculate the value of next time point. Continue doing that until the simulation time is over. 5. Calculate the MLE. If the MLE is smaller than, it means the system is stable after this fault. If the MLE is larger than, it means that system cannot remain stable under the disturbance. 2.4 Summary The MLE can be applied to analyze transient stability of a nonlinear power system model. It extracts the trending information from the system trajectory and does not require the knowledge of the equilibrium points of the system. When the dynamical system model is known, the MLE can achieve an accurate prediction of the transient stability behaviour. However, several factors, such as the model of the system and the window size, will affect the accuracy of MLE calculation. These issues will be addressed in the following chapters. 1

22 3. CHAPTER THREE DYNAMIC LOAD MODEL FOR MLE CALCULATION The dynamic model of a power system can be formulated as a set of differentialalgebraic equations (DAEs) [23]. Differential equations represent the dynamic characteristics of power system components, such as generators, governors, and exciters. Algebraic equations model the steady state power flow on the network. In previous work, the MLE technique is applied to the assessment of rotor angle stability. The system needs to be reduced to retain only the generator buses and, consequently, DAE is reduced to a set of ordinary differential equations (ODEs). The disadvantage of this method is that load bus information as well as the system topology is implicit and it causes errors in the stability assessment. Moreover, it is known that there are interactions between angle and voltage dynamics. In this thesis, a structure-preserving model is proposed for the MLE computation taking into account both generator and load dynamics. 3.1 Reduced Dynamic Model Consider a power system with a total of n buses with generators at m buses. Hence there are n m load buses. The classical 2 nd -order swing equations are used to represent the synchronous generators. Such generators are modeled as a constant voltage source in series with the transient reactance x d '. Therefore, the power system can be augmented by m fictitious buses representing the generator internal buses. The total number of buses in the augmented network is n, which is n + m. The swing equations of the system are given by 11

23 i i 1, i1,..., m. (3.1) 1 PM P i e D i ii 2H i The algebraic equations of the system are the power flow relationship, i.e., n i i j ij i j ij j1 i i j ij i j ij j1 P V V Y cos i 1,..., n n Q V V Y sin i 1,..., n (3.2) Since the MLE can only be applied to ODEs, the algebraic equations at the load bus need to be reduced. A 3-bus system is used to illustrate the Ward equivalent. A 3-bus system with two generator buses and one load bus is shown in Fig 3.1. Parameters of a three-bus system are listed in Fig 3.1. V j.2 I1 I 2 j.5 j.5 I g 2 I l 2 P G 2.5, V2 1.3+j.1 j.5 j.3 j.25 j.5 j.5 j.5 I 3 3 S=1+j.4 I l3 The admittance matrix of the system is Fig3.1 Parameters of a Three-Bus System 12

24 y11 y12 y13 j8.23 j5 j3.33 Y y y y j5 j8.9 j y31 y32 y 33 j3.33 j4 j7.23 (3.3) The power flow is computed by the Newton-Raphson algorithm 1 1. j. V j.522 (3.4) j.1616 Based on the node analysis, the current injections at each bus are: I j.1397 I I2 YV.1819 j.1676 I j.61 (3.5) Based on the current direction noted in Figure 3.1, it is seen that I I g I (3.6) 2 2 l2 S 2.3 j.1 I l l j.1155 (3.7) V.9986 j I I I (3.8) g2 2 l2 j (3.9) I l 3 I 3 When the internal admittance of the generators is considered, two more buses are added to the system. Therefore, the structure of the five-bus system is shown in Fig

25 Fig. 3.2 Structure of the Five-Bus System The internal voltage of the two generator buses are V V I * j j.1598 (3.1) V V I g * j (3.11) j The internal admittance of generator is y d 1 j5 (3.12) j.2 While performing network reduction of a power system, load buses can be treated as either current sources or admittances Constant Current Load The admittance matrix of the five-bus system is 14

26 Y 5* 5 j8.23 y j5 j3.33 y d d j5 j8.9 y j4 y d d j3.33 j4 j7.23 y y d d y d a c y d b d (3.13) Based on the node analysis, current injections at each bus are j.1155 I 2l I I Y *V.9594 j.61 I i 5*1 5*5 5*1 3 (3.14) Ie.7989 j.1397 I j.2831 I g 2 Take I 5* 1 1 as an example: I * V (3.15) 1 y11 V1 y12 * V2 y13 * 1 y * V y * V y * V y * V y * V I y * V y * V (3.16) I5 * d d 4 1 d 1 d 4 Perform the network reduction and one obtains Y * 2 c. j j1.932 *b d a (3.17) I c *I j eq a i.751 j.3964 (3.18).1975 j.214 V 4 I reduce I e Ieq Y2*2 *.2289 j.1133 V 5 (3.19) 2 j I eq is the equivalent current sources of the two loads in the previous system. Fig. 3.3 shows the structure of the system after network reduction. 3 15

27 I1 I g I eq1 I eq2 Fig. 3.3 Structure of the System after Reduction (Current Source Load) Constant Impedance Load The Y matrix of the 5-node system is j8.23 y j5 j8.9 y Y5*5 j3.33 j4 j7.23 yl3 y d d j5 y l 2 d y d j3.33 j4 y y d d y d a c y d b d (3.2) The equivalent admittance of the two loads are Il 2 yl 2.3 j.1 V 2 Il3 yl j.4524 V 3 (3.21) (3.22) Based on the node analysis, current injections at the buses are I 5*1 Y 5*5 *V 5*1 I I.7989 j.1397 I j.2831 I g 2 e i (3.23) Take I 5* 1 2 as an example I * V (3.24) 2 y21 V1 y22 * V2 y33 * 3 16

28 17 * * * * * * * * * * * * 2 I *1 g l l g d d l l g d d l d d l I I I I V y V y V y I I V y V y V y I V y V y V y V y V y V y (3.25) Perform network reduction to obtain *b a c d Y 2 * 2 j j j (3.26) * I a c I e eq (3.27) In this case, loads are converted into admittances, so no equivalent current sources need to be added to the internal buses. However, the new Y matrix is different with the previous case, because the load admittance has changed the structure of the system. The structure of the system after reduction is shown in Fig *2 i reduce * Y I I I V V j j eq (3.28) Fig. 3.4 Structure of the System after Reduction (Admittance Load) Static Load Models Besides the constant current load and constant impedance load, there are several load models, such as the ZIP model (3.29) (3.3) 4 5 I 1 I g 2

29 These models consider the influence of voltage or frequency at load buses on the load amount. However, these models are static; they cannot represent dynamic response of the load. A dynamic load model is required to represent the dynamic characteristics of loads under frequency and voltage variations. 3.2 Structure Preserving Model Frequency Dependent Model The structure preserving model with frequency-dependent load is first proposed in [24]. It assumes that the real power load contains two parts: a static part and a frequency dependent part, i.e., P P D (3.31) D D Where is the angle at the load bus and D is the damping factor describes the effect of frequency on real power. With this model, the original network topology is explicitly represented. The complete dynamic model of the system containing both generators and loads can be described as: n M D b sin P P, i 1,..., n (3.32) i i i i ij i j Mi Di j1 ji For generator buses, the parameters satisfy M D P (3.33) i, i, Di For load buses, the conditions on the parameters are M D P (3.34) i, i, Mi 18

30 3.2.2 Load Recovery Model The frequency dependent model does not consider the voltage behavior at the load buses closely related to the reactive power load. Moreover, it does not consider the load recovery process. Measurements in the laboratory and on power system buses indicate that a typical MW load response to a step change in voltage is of the general form in Fig The responses for real and reactive power are similar qualitatively. Intuitively, this behavior can be interpreted as follows. A step change in voltage produces a step change in MW load. In a longer time-scale, the lower voltage tap changers and other control devices act to restore voltages and, as a result, the load also recovers. With a certain recovery time on a time-scale of a few seconds, this behavior captures the characteristics of induction machines. In a time-scale of minutes, the role of tap-changers and other control devices is included. Over hours, the load recovery and possible overshoot may emanate from heating load. The importance of load models which capture the more general response in Fig. 3.5 is evident. Indeed, the dynamic changes in load may substantially affect the voltage dynamics. 19

31 P/Q(unit) Voltage(unit) 1.5 step change of voltage t(s) power variant t(s) Fig 3.5 Power Recovery Process after a Step Change of Voltage Based on the above discussion, the load recovery model is proposed in [25]. This work applies the load recovery model to reactive power load, i.e., (3.35) Where is a time constant that describes the recovery response of the load. This model can also be expressed as T x x Q V Q V (3.36) q q q s t in which q d t x Q Q V (3.37) 2

32 Q and s Q represent the steady-state response and transient response respectively. By t combining (3.6) and (3.8), the structure preserving model considering reactive load recovery is formulated for the MLE analysis. 3.3 Partial Structure-Preserving Model An effective model named the "partial structure-preserving" model is further developed. This model only preserves the heavily loaded buses while deleting the lightly loaded buses. The proposed partial preserved model has a similar level of accuracy as that of the complete structure preserving model with a much higher calculation speed. 3.4 Parameter Identification In the structure-preserving model, a few parameters are needed for each load bus. In order to simplify the parameter identification process, the models of single-index form [25] are used, i.e., Q s and Q in a t / Q V Q V V s / Q V Q V V t nqs nqt (3.38) Hence, 4 parameters need to be identified, i.e., damping factor D for which is relative to the frequency variation, time constant for voltage recovery time, and n qs and n qt for static and transient voltage exponents. In the frequency dependent model, only D is needed while in the load recovery model, all four parameters are necessary. A practical way to determine the parameters by simulation is adopted. First, waveforms are computed for each load bus. The frequency and voltage curves are viewed 21

33 as inputs, and the real and reactive power load curve are outputs. Then for a fault scenario, record the inputs and outputs at N discrete time instants, i.e., 1,, N. For the damping factor D, since this is a linear coefficient between real power and frequency from (3.5), one can take the N points, i.e., P, P,,,, 1, N samples. Then, the least square method is used to determine the slope D. P as N N The parameter and n qs can be estimated based on the time response of the curve. By linearizing (3.7), the relationship between Q and d V can be derived as d n qttq s nqs / Ts q 1 Q Q V V. (3.39) Therefore, by using the same input value V, the outputs Q produced are d compared to the outputs. Typically, a least square quadratic criterion can be used to obtain the most suitable values of n qt. 3.5 Summary A structure-preserving model is developed which includes the frequency dependent P model and load recovery Q model. The structure-preserving model retains the load buses for MLE calculation. Previously a reduced model was used where load buses are reduced for the computation of Lyapunov Exponents. By using the structure-preserving model, useful information from both generator and load buses can be obtained. The effectiveness of the three models is validated by time-domain simulation. 22

34 4. CHAPTER FOUR PARAMETER IDENTIFICATION 4.1 Estimation Theory In a linear model, the outputs have a linear relationship with the unknowns: Z(k)=H(k)θ+V(k) (4.1) where vector θ(n*1) contains the unknown parameters that need to be estimated. Z(k) is a N*1 vector called the measurement (output) vector. H(k), which is N*n, is the observation matrix; and V(k), which is N*1, is the measurement noise vector. Usually, V(k) is random. The argument k of Z(k), H(k), and V(k) denotes that the last measurement available is kth. All other measurements occur before the kth. Fig 4.1 Linear Model in Estimation Theory The mismatch is given by the least square method based on the linear model, θ θ θ θ (4.2) (4.3) Thus (4.4) where W=W(k), is the weight matrix, used to indicate which measurements are more important than others. In this project, the data is measured within several seconds. In fact, the gap between 23

35 the two consecutive time points is only.3s (dt=.3s). It is assumed that datum at every time point has equal importance. Thus W=. Equation (4.4) can be written as: (4.5) Matrix H'H must be nonsingular for the inverse matrix in the above equation to exist. After getting the estimated value of, the residual, R, is used to determine whether the estimation result is reasonable: The following is the mathematical description of tested load model: (4.6) ; (4.7) T x x Q V Q V q q q s t (4.8) (4.9) (4.1) x Q Q V q d t (4.11) where is the rotor angle of the load bus. P is the real power at the load bus and is the power at the load bus before the disturbance. In the structure-preserving model (dynamic load model), four parameters need to be estimated for each load bus: damping factor D that connects frequency with active power and voltage related parameters,, Tq. 4.2 Identifying the Parameter in Real Power Model To guarantee the accuracy of system simulation, these parameters should be calculated from the real world measurements. However, the actual PMU measurements 24

36 are not available. As a result, data from a commercial software tool called TSAT is used. TSAT has complete power system models and a high level of simulation accuracy. It is assumed that the data from time domain simulation using TSAT serve as a good approximation of the actual power system measurements from PMU. This approximation is also applied in section 4.3. In this section, the parameter D used in the active power part is determined first. A practical way to determine the parameters can be summarized by the following steps: First, suppose that the measurements from PMUs at load buses are available. Measurements of the frequency curves are used as inputs while the measurements of active power are outputs. Or use voltage as inputs, reactive power as outputs. Consider a fault scenario, record the input and outputs at N discrete time instants, i.e., 1,, N. For damping factor D, since the coefficient between real power and frequency is linear, N points,,, are taken as sample points. Then, the least square method is used to obtain the slope D. Algorithm to get Parameter D: Step 1: Run the simulation in TSAT for 2 seconds. The sampling interval is.33s. Step 2: Export the data of real power and frequency of all load buses into an excel table. Assume that this system has m load buses in total. Hence, for every fault, there are m columns and 6 rows. Step 3: From a linear model, D=(P-P)/ and so P-P= ; this is a linear equation. The (P-P) is the measurement. Step 4: Use the least square method. An unknown X= ; Thus: D= (4.12) 25

37 where H=. F is the column of frequency f from all time points. 4.3 Identifying the Parameters in Reactive Power Model For parameter identification of the reactive power model, 3 parameters need to be estimated. However, the measurements can only provide information to estimate one parameter. Thus two parameters need to be eliminated Eliminating two parameters. Recall that (4.8)~(4.11) are given by: Thus (4.13) (4.14) In the real world monitoring, the power system is a discrete-time system. Therefore, (4.14) can be written as (4.15) in kth time point: (4.15) Recall and (4.16) 26

38 (4.17) (4.18) is.1~.33s, but the range of is 2~5s. Therefore, one can ignore the items that are multiplied by. (4.19) Compare the left side with the right side, an approximation can be made: and Finally, it is obtained that: (4.2) where is the current reactive power measured in this bus, is the original reactive power, V is the voltage at the present time, is the initial voltage. At this step, the most important parameter is retained while the two less important parameters are eliminated Estimating the parameter. For factor, it is a coefficient of a non-linear equation, The logarithmic method is used to change the relationship to be a linear one. That is, ; (4.21) where Q and V are the current reactive power and voltage of the load while and are the initial data before the disturbance. Q and V are from the power flow results and V and Q are from the simulation results with TSAT. 27

39 Algorithm to Determine Parameter Step 1 and Step 2 are the same as the algorithm above. Step 3: It is known that ; this is a linear equation. One can view the voltage as an input and Q as an output. The measurement matrix H is and the Z(measurement) is in the estimation theory. is the unknown X. Step 4: Use the least square method to calculate the unknown variable X=. Thus in this case: = ; (4.22) where H= ; i=1,2...n. 4.4 Summary Parameter identification is performed to improve the accuracy of the structure preserving model and to make the prediction of the system behavior more accurate. There are two parts of parameter identification. First, identify the coefficient relating the active power and frequency. Secondly, the coefficient between reactive power and voltage is identified. For the active power part, the linear estimation method is applied. For the reactive part, a non-linear estimation procedure is used. An algorithm is developed to eliminate two less important parameters. Key parameters are determined successfully in this way. Through the simulation results in Chapter 7, it is shown that key parameters lead to a highly accurate system model. 28

40 5. CHAPTER FIVE ENHANCEMENT OF THE COMPUTATIONAL SPEED 5.1 Computation of Large Scale Power Systems In this thesis, the computational results of two systems are used: one is the Mini- WECC system with 122 buses and the other one is the reduced PJM system model with 15 buses. In the Mini-WECC system, the speed of calculation and required computer memory are both manageable since it only has 122 buses with 34 generators and 25 load buses. There are only 15 variables in the structure-preserving model of this system. The matrix size is smaller than 2*2. The computational speed is very high in Matlab. For a 1 seconds simulation, the calculation time is 1-2 secondsin Matlab. Reliable prediction is achieved in 5-8 seconds. For a large scale system, the PJM system, significant hurdles need to be overcome. The PJM system has 15, buses with 1,9 generator buses, more than 3, generators and about 1, load buses. Among these buses, 2,544 load buses are heavily loaded. The total number of variables involved in the differential equations is over 4 thousands, thus the matrix size exceeds 4 thousand by 4 thousand, which means more than 1.6 billion units memory is needed. It is a very high memory requirement for the computer. The calculation sometimes cannot be completed due to the lack of sufficient memory. Besides, for some cases, it needs a very long time to report the correct results. The problem must be solved before the next step of the simulation can continue. 29

41 5.2 Aggregation of Generators and Elimination of Buses Eliminate Buses with No Generation or Load In some cases, a bus has neither generation nor load. For the PJM system model, non-generator buses all have load. However, some buses have very light load. Thus, a reduction method is applied to combine the lightly loaded buses with other load buses. After careful testing, load buses with load over.5 per unit are retained while buses loaded at less than.5 per unit are combined with other load buses. By this procedure, the total number of load buses is reduced to 2,544, instead of over 1 thousand before the reduction. The reduction solves the memory problem. Another option to reduce the dimension of the system model is to retain only those buses with PMUs installed. However, the accuracy of prediction is lower compared with the reduction of lightly loaded buses. This is because some buses with PMU installation is very lightly loaded. These buses are not good indicators for the load behavior. These lightly loaded buses have a low impact on the system relative to heavy loaded buses Combine Generators at the Same Bus This step is focused on improvement of the computational speed. Since generators at the same bus will have the same frequency and same voltage, it is practical to combine generators located at the same bus to an equivalent generator. The combination of generators will simplify the simulation significantly. Simplifications are performed by: (5.1) (5.2) (5.3) 3

42 (5.4) Equation (5.1)~(5.4) must under the same common base value. After the simplification, every generator bus only has one large generator. Therefore there are 1,9 generator buses with 1,9 generators. The reduced PJM system model has 3,634 buses in total after the procedure of model reduction, with 1,9 generator buses and 2,544 load buses. The speed of calculation decreases from a large number to about 2 minutes. 5.3 Use the Different Method to Calculate Differential Equations ODE 15s Method For 122 mini-wecc system, a calculation function called ODE45 is used to solve the differential equations. This routine uses a variable step Runge-Kutta Method to solve differential equations. The syntax for ODE45 for first-order differential equations and second-order differential equations are basically the same. However, for the large scale PJM system model, ODE45 is not a practical tool. ODE45 has a high level of accuracy with a low speed of calculation. For a large scale system such as PJM, every calculation step will generate truncation errors that cannot be ignored. As a result, the ODE45 function needs a very long time to reach its accuracy, which means it is not a practical tool for the PJM system. The ODE15 function, which is similar to ODE45 but with a larger mismatch, is used to replace ODE45. But unfortunately, the result did not improve much Forward Euler Method The second approach is to apply the Forward Euler method: 31

43 (5.5) The Forward Euler method is easier and has a lower accuracy than the previous methods. However, its calculation speed is very good. Due to the fewer number of iterations, the cumulative error of truncation is smaller than the two previous methods. Thus, the calculation time decreases to 2 minutes. 5.4 Improving the speed of MLE calculation As introduced in Chapter 2, Maximum Lyapunov Exponent (MLE) is a stability characteristic of the power system. There are different ways to calculate the MLE with different levels of computational speed and accuracy. x 2s Small disturbed trajectory x f x x 1s t x x 2 1 t x1 x2 Original trajectory Fig 5.1 Concept of MLE(Maximum Lyapunov Exponent) 1. Jacobian method. [17] In this method, calculation of the Jacobian matrix is needed. However, there are many approximations and truncations when the Jacobian is computed. The small errors in every step will accumulate to form a large bias term that could not be ignored. Furthermore, the error will result in wrong predictions of the future system behavior. Also, calculation of the Jacobian is very time consuming. Calculation of cos or sin will takes 1 times the effort more than multiplication, and 4 times more than addition. In a large system such as the PJM system, the approach is not practical. The author uses the 32

44 Taylor expansion first to develop an equation that includes the Jacobian matrix for calculation of the MLE: x 2 ln ln t I J x t x 1 1 x x 1 1 (5.6) = + ; (5.7) The problem is that a real system is not simple compared with the 3-bus system in [17]. The level of accuracy in other part of the program is about The truncation error of the Taylor expansion is higher than, so it is not accurate to use the Jacobian method to calculate the MLE. 2. Direct Calculation[28] This method is called a directed method because it does not require a system model. The availability of voltage measurements is sufficient. The direct method is not applicable to the PJM system model because it is aimed at voltage stability while the focus of this project is on rotor stability. To guarantee a high level of accuracy for this method, all data must be obtained from PMU measurements. However, for the PJM system, not all preserved buses have PMU. For the bus without PMUs, its voltage will not be measured. Consequently, its stability cannot be calculated by the direct method. Even For those buses with PMUs, the measurements are not available for all time points. In other words, the direct method can only obtain Lyapunov Exponent from buses which have PMU and choose the maximum one among them as MLE. Thus, this MLE cannot represent the state of the buses without PMU. Furthermore, the MLE cannot represent the state of the entire system. Therefore, the MLE obtained by direct calculation is called "local MLE", which cannot be used as the stability indicator for the entire system. 33

45 3.Simplied Jacobian method This is used for the PJM system model. The waveforms of variables are obtained after calculation of the power system condition following a fault. Several seconds of these waveforms are selected and a small disturbance is introduced. The updated waveforms will be different from the original waveforms. is the differential matrix of variables instead of the Jacobian matrix and where is the original waveform. The detailed algorithm is given as follow: (5.8) (5.9) (5.1) (5.11) (5.12) = + (5.13) 5.5 Summary The speed of calculation speed is improved step by step. First, the exact reasons behind the low calculation speed are determined for the large scale PJM system model. Then generators at the same bus are combined to simplify the topology. Buses without generator or load are also eliminated in this step for the same purpose. Next, a method for calculation of differential equations is identified. The Euler method is chosen for this case and it works well. At this time, the speed of calculation has improved by about 9%. Finally, a new MLE calculation algorithm is developed. It reduces time and improves the accuracy significantly by reducing truncation errors. 34

46 6. CHAPTER SIX CASE STUDY AND SIMULATION RESULTS 6.1 Mini-WECC System Simulation Model In this research, the Mini-WECC system is used to validate the proposed models and algorithms. The Mini-WECC system is a reduced-order model of the WECC system designed for the study of power oscillation issues. The system has 122 buses with 34 generators, 171 lines, and 25 loads. Fig.6.1 shows the one-line diagram of this system, but it only includes high-voltage buses. Loads and generators are connected via step-up transformers. The Matlab toolbox PSAT is used to perform time-domain simulation of this system. The detailed 6-order model is used for generators. That is, (6.1) (6.2) (6.3) (6.4) (6.5) (6.6) The loads are modeled as a generalized exponential voltage frequency dependent model [29], i.e., (6.7) (6.8) 35

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

48 Load Bus Number D T q n qs n qt Load Bus Number D T q n qs n qt Load Bus Number D T q n qs n qt Table 6-1 System Parameter Verification Take bus 8 as an example. The waveform of its reactive power from TSAT is shown in Figure 6.2. The reactive load response of the structure preserving model in Matlab under the same voltage input with different parameters is shown from Fig.6.3-Fig.6.5. is calculated using a non-linear parameter identification method. It is observed that with the parameter ( T q, n, qt n ) =(3, 2, 1.9), the reactive load response is the closest qs to the simulation result. Note that in this Mini-WECC system, are not calculated from measurements. They are estimated from the comparison of Matlab programming and TSAT simulation. Thus, accuracy cannot be guaranteed. Note: in the following, the SP model means structure preserving model. 37

49 reactive power of recovery model reactive power of psat time(s) Fig. 6.2 Reactive Power Load at Bus 8 from PSAT Simulation time(s) Fig. 6.3 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs=1 38

50 reactive power of recovery model reactive power of recovery model time(s) Fig. 6.4 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs= time(s) Fig. 6.5 SP Model from Matlab at Bus 8 with Tq=3, nqt=2, nqs=2 39

51 In this research, 12 fault scenarios have been considered. The results of timedomain simulation, reduced dynamic model, structure preserving model with frequency dependent load ( P model"), and structuring preserving model with both frequencydependent load and voltage dependent load ( PQ model") are compared for each scenario, together with the MLE. Detailed results are shown in Appendix A. Some typical cases are listed in Table 6.2. In this table, the green color means that the MLE result of this model is consistent with the time-domain simulation result, while the orange color means the opposite. Fault Type Fault Clear Reduced MLE P MLE PQ MLE Simulation Location Time Model Model Model Results Generator trip 1.1s stable stable stable stable Generator trip 5 1s stable unstable.211 unstable unstable Generator trip 1 1s stable unstable unstable.1247 unstable Generator trip 17 1s stable stable unstable unstable Line Outage s stable stable unstable unstable Line Outage s stable -.39 unstable unstable.2894 unstable Line Outage s stable unstable.693 unstable.2825 unstable Line Outage s stable stable stable unstable Table 6-2 Typical Examples The results of these cases can be categorized into 4 types. Type 1: All three models, i.e., load-reduced model, P model and PQ model are consistent with the results of time domain simulation. Specifically, the assessments of the MLEs are consistent with the simulation results. Type 2: The structure preserving P model and PQ model are accurate, while the load-reduced model fails to produce an accurate prediction. Fig.6.6 shows one of the typical cases, in which a three phase fault occurs near bus 17 and it is cleared after.1s by open breaker at line From the waveforms of relative rotor angle and angular speed of generators, it can be observed that the results of structure preserving models are 4

52 Relative Rotor Angle(Radian) Relative Rotor Speed of Generators(Radian/s) unstable, which is consistent with time domain simulation results, while the assessment by the load-reduced model is not correct. Type 3: Only the structure preserving PQ model produces an accurate prediction. Specifically, only the assessment of the MLE by the "PQ" model is consistent with the simulation results. Type 4: All three models fail to achieve an accurate prediction. Specifically, the assessments of the MLEs are not consistent with the simulation results. Within the total 12 cases, the load-reduced model leads to 72 cases that are consistent with time-domain simulation results, which represents an accuracy rate of 6%. The structure preserving P model" results in 85 consistent cases, which is a 7.83% accuracy rate. The PQ model" has 11 consistent cases, which is a 91.33% accuracy rate. Besides, in 1 out of the 12 cases all three models fail. From the result, the structure preserving PQ model" has the highest accuracy level compared to the other two models. As a result, the MLE technique should adopt the structure preserving PQ model" to achieve an accurate assessment of system stability. Fig. 6.6 shows the situation of type 2. Relative rotor angle Relative rotor angular speed time(sec) time(s) (a) Load-Reduced Model time(sec) time(s) 41

53 Relative Rotor Angle of Generators(Radian/s) Relative Rotor Speed of Generators(Radian/s) Relative Rotor Angle of Generators(Radian) Relative Rotor Speed of Generators(Radian/s) time(sec) time(s) time(sec) (b) Structure-Preserving P Model" time(s) time(sec) time(s) time(sec) (c) Structure-Preserving PQ Model" time(s) time(s) (d) Time-Domain Simulation Result time(s) Fig. 6.6 Comparison of Different Models 6.2 Application to the Reduced PJM System As explained in Chapter 5, PJM is a transmission operator in the Eastern Interconnection that provides electric power for District of Columbia and 13 states, such 42

54 as Delaware, Illinois, Indiana. PJM manages the high-voltage electric power grid and serves more than 61 million people. The system has 15, buses with 1,39 generator buses and over 1,load buses. It has both DC and AC load, single and multiple generators buses. It has very complex structure and requires1.6 billion memory units. The structure preserving load model is applied to the PJM system model. Here are the simulation results for the PJM system: Fault Location TSAT Partial SP Reduced Model Line outage Fault-on time =.2s Bus 1 Bus2 Simulation Results MLE Prediction MLE Prediction stable stable unstable unstable unstable unstable unstable unstable unstable stable unstable unstable unstable stable stable stable stable stable stable stable stable stable stable stable stable Table 6-3 Results of Two Load Models in PJM System 43

55 The green color means the MLE is consistent with the time domain simulation result while the black color means they are inconsistent. It is clear that partial structurepreserving has a much higher level of accuracy than that of the reduced load model. Only 4% results of the reduced load model are consistent with the simulation results from TSAT. In contrast, 72% of the results of the structure-preserving model are consistent with those obtained from TSAT. The results of these cases can be categorized into 3 types. Type1: Both load-reduced model and partial structure-preserving model are consistent with the results of time domain simulation. Specifically, the assessments of the MLEs are consistent with the simulation results. Fig 6.7 shows one typical case. A three phase fault occurs on line and it is cleared after.1s by open breaker at the line. Type 2: The partial structure preserving model is accurate, while the load-reduced model fails to produce an accurate prediction. Specifically, the assessment of the MLE from structure preserving model is consistent with the simulation results but the assessment of the MLE from load reduced model is not consistent with the simulation results. Fig.6.8 shows one of the typical cases, in which a three phase fault occurs on line and it is cleared after.2s by open breaker at line Type 3: Both two models give an inaccurate prediction. Specifically, the assessments of the MLEs are not consistent with the simulation results. Fig.6.9 shows one of the typical cases, in which a three phase fault occurs on line and it is cleared after.1s by open breaker at line Type 4: The partial structure preserving model is inaccurate, while the load-reduced model produce an accurate prediction. Specifically, the assessment of the MLE from 44

56 structure preserving model is not consistent with the simulation results but the assessment of the MLE from load reduced model is consistent with the simulation results. Type1: Fig 6.7 Comparison of Two Models for PJM System(Type 1) Reduce Load Model, MLE= Prediction: unstable. Right! Partial Structure-Preserving P Model, MLE= , Prediction: unstable. Right! Type2: Fig 6.8 Comparison of Two Models for PJM System(Type 2) 45

57 Reduce Load Model, MLE Prediction: unstable. Wrong! Partial Structure-Preserving P Model, MLE= =-.284, Prediction: stable. Right! Type3: Fig 6.9 Comparison of Two Models for PJM System(Type 3) Reduce Load Model, MLE Prediction: unstable. Wrong! Partial Structure-Preserving P Model, MLE= =7.4794, Prediction: unstable. Right! 6.3 Analysis and Summary The structure preserving model shows a much higher level of accuracy realtive to reduced model in both Mini-WECC system and the large scale PJM system. The partial structure preserving model that only preserves heavily loaded buses also shows high accuracy. It means that there is no need to preserve all buses of the entire power system. 46