UNIVERSITY OF CALGARY. Adaptive Simplified Neuro-Fuzzy Controller as Supplementary Stabilizer for SVC. Anas M. AlBakkar A THESIS

Transcription

1 UNIVERSITY OF CALGARY Adaptive Simplified Neuro-Fuzzy Controller as Supplementary Stabilizer for SVC by Anas M. AlBakkar A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN ELECTRICAL AND COMPUTER ENGINEERING CALGARY, ALBERTA NOVEMBER, 2014 ANAS ALBAKKAR 2014

2 ABSTRACT A simplified version of an Adaptive Neuro-Fuzzy Controller (ANFC) applied to a FACTS device, namely a Static VAr Compensator (SVC), is presented in this dissertation. The proposed Adaptive Simplified Neuro-Fuzzy Controller (ASNFC), used as a supplementary controller to damp power system oscillations, consists of a reduced number of input Membership Functions (MFs) and Consequent Parameters (CPs). Unlike the common techniques of using the generator speed or the power angle deviations as inputs to the controller, the input to the ASNFC is the power deviation at the bus where the SVC is located. A Neuro Identifier is used to track the behaviour of the system in real-time and update the controller on-line. The effectiveness of the proposed controller is tested on a single machine infinite bus system, and a multi-machine system. Results of simulation studies demonstrate that the performance with the proposed ASNFC is practically the same as with ANFC, but with a smaller number of parameters to optimize that reduces computation time for real-time application. In addition, the proposed ASNFC is further tested on a physical model power system where the controller is applied to the generation unit. The results obtained indicate a successful implementation of the ASNFC in damping power system oscillations over the Conventional Power System Stabilizer (CPSS). Furthermore, similar dynamic performance is provided by the ASNFC, as compared to the detailed ANFC. ii

3 ACKNOWLEDGMENTS I would like to express my sincere gratitude to my supervisor Dr. O. P. Malik for his constant support, guidance and encouragement throughout the program. My enthusiasm in this subject is inspired by his profound knowledge in this area and his refined research style. Due to his belief in my capabilities I have learned so much, not only in the area of my specialty but also with regards to teaching methods, conducting research and professional interactions. I would also like to thank all the professors and supporting staff from the Department of Electrical and Computer Engineering at the University of Calgary for their help during my studies, particularly Mr. Garwin Hancock and Rob Thomson. Many thanks are also due to my thesis committee members for their involvement and the time they spared to review my thesis. I would like to acknowledge the financial support provided the Ministry of Higher Education in Saudi Arabia represented by the Saudi Arabian Cultural Bureau in Ottawa during my enrolling in the PhD program. Special appreciation goes to Dr. Ali Alattiyah who made this scholarship feasible. Many thanks go to my advisor at the Saudi Cultural Bureau, Mr. Ahmed Ismail. My sincere and profound gratitude is due to my parents whose kind care and interest in my success I could never replace. Their prayers and moral support are invaluable and will always contribute to my progress. iii

4 My special thanks go to my parents in-law Mr. Callum and Mrs. Lynda MacGregor for their constant encouragement and support. Last but not least, I am deeply indebted to my beloved wife, Sarah-Faye MacGregor, who patiently devoted herself and time helping me throughout the program. She has been a source of encouragement and optimism in my life. Without her constant and unlimited spiritual support, feedback and suggestions, I would not have been able to complete my dissertation. iv

5 DEDICATION To my beloved wife Sarah, my sons: Danyal & Elijah, and family v

6 TABLE OF CONTENTS ABSTRACT... ii ACKNOWLEDGMENTS... iii DEDICATION...v TABLE OF CONTENTS... vi LIST OF TABLES... ix LIST OF FIGURES AND ILLUSTRATIONS...x LIST OF ABBREVIATIONS AND ACRONYMS... xiv LIST OF SYMBOLS... xviii CHAPTER INTRODUCTION Power System Stability Power System Damping Techniques Damping in the Transmission Path Damping at the generator location Power System Stabilizers (PSSs) Conventional Power System Stabilizer (CPSS) Adaptive Power System Stabilizer (APSS) Self-Tuning Control Based Adaptive PSS Artificial Intelligence Based Adaptive PSS Statement of the Problem Dissertation Objectives and Contributions Outlines of the Dissertation...14 CHAPTER SVC BASED- POWER SYSTEM STABILIZER Introduction Static VAr Compensator (SVC) Steady State Power Transfer Capacity Enhancement of Transient Stability Mathematical Model of an SVC Auxiliary Control Conclusions...24 CHAPTER ADAPTIVE NEURO-IDENTIFIER Introduction System Identification and Neural Networks Artificial Neural Networks Neural Network Topologies...32 vi

7 The Back-propagation Learning Algorithm Proposed Adaptive Neuro-Identifier (ANI) Conclusions...40 CHAPTER ADAPTIVE NEURO-FUZZY CONTROL SYSTEM Introduction Fuzzy Logic Systems Fuzzy Set Theory Linguistic Variable Fuzzy IF-THEN Rules Structure of a Fuzzy Logic System Fuzzification Knowledge Base Fuzzy Inference Defuzzification Neuro-Fuzzy Systems Adaptive Neuro-Fuzzy Inference System (ANFIS) Structure of the Neuro-Fuzzy Controller (NFC) On-line Adaptation Technique Adaptive Simplified Neuro-Fuzzy Controller (ASNFC) Simplification of the Rule-Base Structure Control System Design of the proposed ASNFC Conclusions...63 CHAPTER APPLICATION OF AN ADAPTIVE SIMPLIFIED NEURO-FUZZY CONTROLLER IN POWER SYSTEM Introduction SMIB System Configuration Simulation Studies Normal Load Condition Light Load Condition Leading Power Factor Operation Three-Phase to Ground Short Circuit Change in Operating Conditions from the Normal Load Condition to the Full Load Condition Change in Operating Conditions from the Light Load Condition to the Normal Load Condition Stability Margin Test Multi-Machine System Configuration Simulation Studies Test A Test B...83 vii

8 Test C Test D Test E Test F Test G Conclusions...94 CHAPTER EXPERIMENTAL STUDIES OF AN ADAPTIVE SIMPLIFIED NEURO-FUZZY CONTROLLER IN A REAL-TIME SYSTEM Introduction Physical Model of a Power System Real-Time Hardware and Software Proposed Control Structure and Training Experimental Test Results and Discussion Experimental Results of the Adaptive Neuro-Identifier Input Torque Reference Step Change Under Light Load Condition Three-Phase to Ground Short Circuit Test Experimental Results of the Proposed ASNFC Input Torque Reference Step Change in a 300 km Double Transmission Line System Input Torque Reference Step Change in a 300 km Transmission Line System Under a Leading Power Factor Operation Three-Phase to Ground Short Circuit Test at the Middle of a 300 km Transmission Line Three-Phase to Ground Short Circuit Test at the Middle of a 200 km Transmission Line Input Torque Reference Step Change in a 200 km Double Transmission Line System Single-Phase to Ground Short Circuit Test at the Middle of a 200 km Transmission Line Conclusions CHAPTER CONCLUSIONS AND FUTURE WORK Conclusions Suggested Future Work REFERENCES APPENDIX A APPENDIX B APPENDIX C viii

9 LIST OF TABLES Table 4.1 Sugeno-type rule-base table with 49 rules Table 4.2 Reduced fuzzy rule-base table Table 5.1 Mechanical power values for different types controllers when the system lost its stability Table 5.2 Multi-machine system case studies Table 6.1 Time integral performance criteria with a 20% step increase in the input mechanical torque (P=0.80 p.u. and 0.75 p.f. lag) Table 6.2 Time integral performance criteria when a 20% step increase in the input mechanical torque (P=0.50 p.u. and 0.65 p.f. lead) Table 6.3 Time integral performance criteria when a three-phase to ground short circuit is applied (P=0.80 p.u. and 0.75 p.f. lag) Table 6.4 Time integral performance criteria when a three-phase to ground short circuit is applied (P=0.97 p.u. and 0.93 p.f. lag) Table 6.5 Time integral performance criteria when a 15% step Increase in the input mechanical torque (P=0.80 p.u. and 0.75 p.f. lag) Table 6.6 Time integral performance criteria when a single-phase to ground short circuit is applied (P=0.97 p.u. and 0.93 p.f. lag) ix

10 LIST OF FIGURES AND ILLUSTRATIONS Figure 1.1 General scheme of self-tuning adaptive control... 7 Figure 2.1 Schematic diagram of an SVC Figure 2.2 A Transmission line with SVC connected at the middle of a SMIB system Figure 2.3 Effect of an SVC on P-δ diagram Figure 2.4 P-δ Diagram curves illustrating transient stability margins in a SMIB system: (a) without SVC (b) with SVC Figure 2.5 Block diagram of SVC with a supplementary controller Figure 3.1 Direct adaptive control scheme Figure 3.2 Indirect adaptive control scheme Figure 3.3 Simple model of an artificial neuron Figure 3.4 Single-layer feed-forward network Figure 3.5 Multi-layer perceptron network Figure 3.6 Recurrent Neural Network Figure 3.7 Three-layer neural network Figure 3.8 Relationship between the error function and weights of the network Figure 3.9 Adaptive neuro-identifier architecture Figure 3.10 Overall plant structure with the ANI Figure 4.1 Structure of a fuzzy system Figure 4.2 Defuzzification methods for a fuzzy function Figure 4.3 ANFIS structure with two inputs and one output Figure 4.4 Architecture of an adaptive neuro-fuzzy controller Figure 4.6 The distance, d, between the switching line and actual state x

11 Figure 4.7 Illustration of d1, d2 and d3 between the switching line and actual states in the phase-plane Figure 4.8 Perpendicular distance between point P (e, e) and the switching line in the phase-plane Figure 4.9 Input membership functions for the SFLC Figure 4.10 Overall control system structure of ASNFC Figure 5.1 ASNFC for an SVC device in an SMIB system Figure 5.2 Generator speed deviation under the normal load condition in response to a 0.10 p.u. step increase in the input mechanical torque and return to the normal condition Figure 5.3 Voltage at the middle bus under normal load condition in response to a 0.10 p.u. step increase in input mechanical torque and return to normal condition Figure 5.4 Example of membership functions before and after adaptation. before adaptation, after adaptation Figure 5.5 Example of consequent parameters before and after adaptation. before adaptation, after adaptation Figure 5.6 System output and ANI responses to a p.u. step increase in the input mechanical torque and return to the normal condition Figure 5.7 Generator speed deviation under the light load condition in response to a 0.10 p.u. step increase in the input mechanical torque and return to the normal condition Figure 5.8 Generator speed deviation under the leading power factor operating condition in response to a 10% increase in the input mechanical torque and return to the original condition Figure 5.9 Generator speed deviation in response to a three-phase fault at the middle of a transmission line connecting the generator to the middle bus with a successful re-closure Figure 5.10 Generator speed deviation in response to a 10% step increase in the input mechanical torque under the normal load condition and return to the fully loaded generator condition xi

12 Figure 5.11 Generator speed deviation in response to a 10 % step increase in the input mechanical torque under the light load condition and return to the normal load condition Figure 5.12 Generator speed deviation in response to a continuous mechanical torque increase at the rate of 0.05 p.u. per second Figure 5.13 Schematic model of a multi-machine power system with an SVC device installed at the middle of the tie-line connecting bus 6 and bus Figure 5.14 Inter-area mode oscillations between G1 and G2 under the first operating condition in response to Test A Figure 5.15 Local mode oscillations between G2 and G3 under the first operating condition, in response to Test A Figure 5.16 Inter-area Mode oscillations between G1 and G2 under the first operating condition, in response to Test B Figure 5.17 Local mode oscillations between G2 and G3 under the first operating condition, in response to Test B Figure 5.18 Inter-area mode oscillations between G1 and G2 under the first operating condition in response to Test C Figure 5.19 Local mode oscillations between G2 and G3 under the first operating condition, in response to Test C Figure 5.20 Inter-area mode oscillations between G1 and G2 under the second operating condition in response to Test D Figure 5.21 Local mode oscillations between G1 and G2 under the second operating condition in response to Test D Figure 5.22 Inter-area mode oscillations between G1 and G2 under the second operating condition in response to Test E Figure 5.23 Local mode oscillations between G2 and G3 under the second operating condition in response to Test E Figure 5.24 Inter-area mode oscillations between G1 and G2 under the first operating condition in response to Test F Figure 5.25 Local mode oscillations between G2 and G3 under the first operating condition in response to Test F Figure 5.26 Power deviation at bus 9 in Test F xii

13 Figure 5.27 ASNFC signal applied to an SVC device in Test F Figure 5.28 Inter-area mode oscillations between G1 and G2 under the second operating condition in response to Test G Figure 5.29 Local mode oscillations between G2 and G3 under the second operating condition in response to Test G Figure 6.1 Schematic diagram of lab setup Figure 6.2 system output and ANI responses to a 0.2 p.u. increase in the input mechanical torque (P=0.50 p.u. and 0.85 p.f. lag) Figure 6.3 System output and ANI response to a three-phase to ground short circuit test (P=0.80 p.u. and 0.75 p.f. lag) Figure 6.4 Generator speed deviation in response to a 20% step increase in the torque reference (P=0.80 p.u. and 0.75 p.f. lag) Figure 6.5 Generator speed deviation in response to a 20% step increase in the torque reference (P=0.50 p.u. and 0.65 p.f. lead) Figure 6.6 Generator speed deviation in response to a three-phase to ground short circuit test at the middle of a 300 km transmission line with a successful reclosure (P=0.80 p.u. and 0.75 p.f. lag) Figure 6.7 Generator speed deviation in response to a three-phase to ground short circuit test at the middle of a 200 km transmission line with an unsuccessful re-closure (P=0.97 p.u. and 0.93 p.f. lag) Figure 6.8 Generator speed deviation in response to a 15% step increase in the torque reference (P=0.80 p.u. and 0.75 p.f. lag) Figure 6.9 Generator speed deviation in response to a single-phase to ground short circuit test at the middle of a 200 km transmission line (P=0.97 p.u. and 0.93 p.f. lag) Figure A.1 AVR and Excitation Model Type ST1A, IEEE Standard P Figure A.2 CPSS model Type PSS1A, IEEE Standard P421.5/D xiii

14 LIST OF ABBREVIATIONS AND ACRONYMS AC AI ANFC ANFIS ANI ANN APSS ARMA ASNFC B CMAC COA COG CP CPSS DC FACTS FALCON FL FLC Alternating Current. Artificial Intelligence Adaptive Neuro-Fuzzy Controller Adaptive Neuro-Fuzzy Inference System Adaptive Neuro-Identifier Artificial Neural Network Adaptive Power System Stabilizer Auto Regressive Moving Average Adaptive Simplified Neuro-Fuzzy Controller Big Cerebella Model Articulation Controller Center of Area Center of Gravity Consequent Parameter Conventional Power System Stabilizer Direct Current Flexible AC Transmission Systems Fuzzy Adaptive Learning Control Network Fuzzy Logic Fuzzy Logic Controller xiv

15 FS FUN GCPSS HVDC IAE IEEE ISE LN LOM LP M MF MLP MN MOM MP MRAC MV NN NFC PA P.f. Fuzzy System Fuzzy Net Generator Conventional Power System Stabilizer High Voltage Direct Current Integral of the Absolute Error The Institute of Electrical and Electronic Engineers Inc. Integral of the Square Error Large Negative Largest of Maximum Large Positive Medium Membership Function Multi-Layer Perceptron Medium Negative Mean of Maximum Medium Positive Model Reference Adaptive Control Minimum Variance Neural Network Neuro-Fuzzy Controller Pole Assignment Power Factor xv

16 POD PS PSS p.u. RBF RLS RNN S SCPSS SCRC S SFLC SMIB SN SOM SP SSSC STATCOM SVC TCR TCSC TCSR Power Oscillation Damping Pole Shifting Power System Stabilizer Per-Unit representation of electrical variables Radial Basis Function Recursive Least Squares Recurrent Neural Network Small SVC Conventional Power System Stabilizer Thyristor Switched Series Capacitor Seconds Simplified Fuzzy Logic Controller Single Machine Infinite Bus Small Negative Smallest of Maximum Small Positive Static Synchronous Series Compensator Static Synchronous Compensator Static Var Compensator Thyristor-Controlled Reactor Thyristor Controlled Series Capacitor Thyristor Controlled Series Reactor xvi

17 TCPST TSC UPFC ZO Thyristor Controlled Phase Shifting Transformer Thyristor-Switched Capacitor Unified Power Flow Controller Zero xvii

18 LIST OF SYMBOLS Chapter 1 ˆ ( t ) Parameter vector estimate u(t) Control system output y(t) System output Chapter 2 A1, A2 Accelerating energy of the uncompensated system Ap1, Ap2 BSVC Bmax Bmin I1 I2 Isvc Ksvc P Pm Tsvc Vm Vr Vref Vs X1 X2 Accelerating energy of the compensated system SVC variable susceptance SVC variable susceptance upper limit SVC variable susceptance lower limit Current of the first segment of the transmission line Current of the second segment of the transmission line Input current to the SVC device SVC gain Active power Mechanical input SVC transfer function time constant Midpoint of the transmission line voltage Receiving end bus voltage Reference voltage Sending end bus voltage First segment of the transmission line Second segment of the transmission line δ 1, δ2 Generator rotor angles of the uncompensated system xviii

19 δ p1, δp2 δ Crit δp Crit m Generator rotor angles of the compensated system Critical limit of the rotor angle in the uncompensated system Critical limit of the rotor angle in the compensated system Angle of the terminal voltage Angle of the middle bus voltage Chapter 3 b d pk E e i f( ) J i net j O j, O k P ΔPsvc ΔP svc u(t) wn W xn yp y p y pk yr η Bias of the neuron Desired output Error function Error signal between the estimated ANI output and the desired output Neuron activation function Performance index of the ANI Local field Output of neuron j or k Training data set Power deviation at the SVC bus Estimated power deviation at the SVC bus Control system output Weights of the neural network The gradient with respect to the weight matrix Input signals to the neuron System output Estimated system output Actual system output Output of the model reference of the system Learning rate Partial derivative xix

20 Chapter 4 A, B Fuzzy sets a,b,c ANFIS parameter set d1,2,3 e(t) e(t) e(t-1) e c e c (k + 1) f 1, f 2 f i J c Perpendicular distance from the switch line in the phase-plane Error signal Change-of-error signal previous error value Error signal between the ASNFC estimated and the desired outputs Error signal between the estimated and the desired outputs one step ahead Input Function blocks of the ASNFC Output of the consequent layer Performance index of the Adaptive controller K1, K2 NFC Input scaling factors K3 L1,2,3,4,5 μa(x) μb(x) Output scaling factor Number of layer in the ANFC or ASNFC Membership function of a fuzzy set A defined in the universe x Membership function of a fuzzy set B defined in the universe x O 1,i Output of the node i in layer 1 O * l O i P P (e, e) Output of nodes belong to S Output of the i th node of the l th layer Number of nodes in the (l+1) layer A given point in the phase-plane ΔP svc (k + 1) Estimated power deviation at time step k+1 ΔP d (k + 1) Desired power value at time step k+1 pi, qi, ri Consequent Parameters S Set of nodes Su uus w i Sign of the control signal Unsigned control action Firing strength w i Normalized firing strength xx

21 x y yd y(t) y(t-1) θ θ η Fuzzy universe of discourse Output of the ANFIS network Desired value Output of the plant Previous value of the plant output An arbitrary parameter in the NFC The gradient descent with respect to θ Partial derivative Learning rate Chapter 5 Bsvc G SVC susceptance Generator K1, K2 Input scaling factors K3 P ΔPSVC ΔP svc u(k) Output scaling factor Active power Power deviation at the SVC bus Derivative the power deviation at the SVC bus Output of the controller at time step k Chapter 6 e n W(k) W Δω Δω Δω Error signal Weight matrix at time step k The gradient descent with respect to the weight matrix Generator speed deviation Derivative of generator speed deviation Estimation of generator speed deviation Appendices ed Generator d-axis voltage xxi

22 ef g H id if iq Generator field voltage Governor output Generator inertia Generator d-axis current Generator field current Generator q-axis current KA, KC, KF AVR gains Kd KLF, ILF Q ra RC, XC re rf rkd rkq Generator damping ratio coefficient AVR gains Reactive power Generator armature resistance Voltage transducer compensation constants Transmission line resistance Generator field winding resistance Generator d-axis damper winding resistance Generator q-axis damper winding resistance T1, T2, T3, T4 AVR time constants Tdo Tdo" Te Tg Tm Tqo" Tsvc VAMAX VAMIN VIMAX VIMIN VOEL Transient d-axis time constant Sub-transient d-axis time constant Generator electric torque output Governor time constant Generator mechanical torque input Sub transient q-axis time constant SVC transfer function time constant AVR command signal upper limit AVR command signal lower limit AVR input signal upper limit AVR input signal lower limit AVR over-excitation limit xxii

23 VPSS VRef VRMAX VRMIN VSTMAX VSTMIN Xd Xd Xd" Xe Xf Xkd Xkq Xq Xq" λd λf λkd λkq λq PSS output AVR voltage reference setting AVR regulator upper limit AVR regulator lower limit PSS output upper limit PSS output lower limit Generator d-axis reactance Transient d-axis reactance Sub-transient d-axis reactance Transmission line reactance Generator field reactance Generator d-axis damper winding reactance Generator q-axis damper winding reactance Generator q-axis reactance Sub-transient q-axis reactance Generator d-axis flux linkage Generator field flux linkage Generator d-axis damper winding flux linkage Generator q-axis damper winding flux linkage Generator q-axis flux linkage xxiii

24 1 CHAPTER 1 INTRODUCTION 1.1 Power System Stability Electric power systems evolved from single generating plant to highly interconnected networks that generate, transmit and distribute electricity from one place to another. Modern power systems are complex dynamic systems containing numerous interconnected elements [1]. Many components in power systems are highly nonlinear and some are combinations of electrical and mechanical parts, each having unique dynamic behavior. Interactions between electrical and mechanical parts result in sophisticated system dynamics, transient behavior, and consequently, different kinds of unstable characteristics. In addition, the fact that power systems are spread over vast geographical areas subjects them to many types of disturbances [2], [3]. It is important for a power system to maintain its stability after the occurrence of a disturbance; otherwise, system oscillations will arise. This might lead to a loss of synchronism, cascaded outages, and a shutdown of a major portion of the power system. A reliable electric power system must be able to withstand wide variety of disturbances over a wide range of operating conditions. This topic has been of continuous major concern in power system operations. Improving the stability of power systems and overcoming their instability problems have been investigated for many years [4], [5], [6], [7].

25 2 A common problem in power systems is the possibility of power system oscillations, resulting from various types of disturbances. These disturbances vary from sudden load change, loss of a line, loss of load or a short circuit of a generating unit or transmission line. Oscillations in power systems can be categorized into local and inter-area oscillations. A local oscillation, or plant mode, refers to a single generator oscillating against a large system with frequency in the range of 0.7 to 2 Hz. Alternatively, inter-area oscillation refers to a group of generators oscillating against one another and is usually in the range of 0.2 to 0.8 Hz. A large system has lower oscillation frequency when compared to a smaller system because of the higher effective reactance of tie lines between large systems [8], [9]. 1.2 Power System Damping Techniques Over the years, researchers have addressed many methodologies and control algorithms to damp the oscillations in the power system and enhance its dynamic performance [4], [8], [9], [10], [11]. Oscillation damping control, in general, can be divided into two groups [12]: Damping in the transmission path Damping at the generator location Damping in the Transmission Path High Voltage Direct Current (HVDC) and Flexible AC Transmission Systems (FACTS) have been utilized to damp power system oscillations and enhance power system stability. Besides supplementary modulation of generator excitation systems, damping of

26 3 power system oscillations can be effectively achieved through control of FACTS and HVDC systems [12], [13], [14], [15]. HVDC systems have the ability to improve power system stability and reduce low frequency oscillations between areas in a large power system network. Since HVDC allows power transmission between unsynchronized AC distribution systems, it helps increase system stability by preventing cascading failures from propagating from one part of a wider power transmission grid to another. Changes in load, for example, that cause portions of the AC network to become unsynchronized would not similarly affect the DC link. Likewise, the power flow through the DC link would tend to stabilize the AC network [15]. In addition, adding suitable supplementary controllers to the HVDC has been shown to improve the transient stability of the power system [16], [17]. The disadvantages of using HVDC technology are both the cost and complexity of the equipment. Another approach to damp power system oscillations in the transmission path is to use FACTS devices. For the past two decades, FACTS devices have been progressively developing for better control of power flow in a power transmission system, improving the transient stability, damping power system oscillations, and providing voltage stability by using high power semiconductors technology [18]. Various types of FACTS devices have been developed and employed in power systems including Satic VAr Compensator (SVC), Static Synchronous Compensator (STATCOM), Unified Power Flow Controller (UPFC), and Thyristor Controlled Series Compensator (TCSC). Since FACTS devices are already starting to be employed in power system grids to compensate for reactive power and provide voltage support, they can potentially be used for damping the oscillations of the power system and also work in coordination with the Power System Stabilizer (PSS) on

27 4 the generator to improve the overall power system stability [19]. This method will be discussed in detail in Chapter Damping at the generator location Generator supplementary control, commonly referred to as PSS, has been studied extensively as an effective means of damping oscillations. PSS acts through the excitation system of a generator to increase the damping of electromechanical oscillations by generating a component of electrical torque proportional to the input signals. Several types of supplementary signals, such as speed and frequency deviations, and accelerating power, have been successfully utilized as PSS inputs. Most of the generating units nowadays are equipped with PSSs to improve transient stability and provide effective damping [1], [20], [21]. 1.3 Power System Stabilizers (PSSs) Application of the PSS in damping power system oscillations is an important technique employed by electrical power utilities to overcome the oscillatory stability problems via the excitation system. As mentioned in the previous section, the PSS is designed to aid system stability by introducing a supplementary control signal into the Automatic Voltage Regulator (AVR) of a generator along with the voltage error [22]. Nonetheless, this traditional concept of applying the output of the PSS at the AVR can be slightly modified, based on the location of the PSS. The PSS can be applied to FACTS devices installed in the transmission path, such as SVC or STATCOM, and consequently modulate the output of the FACTS device in order to supply a proper signal to damp the system oscillations [18].

28 5 The main approaches, reported in the literature for designing the PSSs are discussed in detail in the following two subsections Conventional Power System Stabilizer (CPSS) The Conventional Power System Stabilizer (CPSS) is considered one of the first generations of power system stabilizers that have been widely used in electrical power plants. Indeed, it has been used for many years to damp electromechanical oscillations and enhance power system dynamic stability [23], [24]. CPSS is a fixed parameter, lead-lag type compensator and its design is based on classical control theory, using transfer functions. By appropriately tuning the parameters of the CPSS, it improves its system damping capability [25]. However, power systems nowadays are increasingly complex and their operating conditions vary over a wide range due to different types of disturbances, load changes and changes in the topological configuration of the grid. Hence, a fixed-parameter controller will not be able to maintain its desired performance level under system parameter variations. The following setbacks are associated with applying a CPSS to power systems [18], [26]: Choosing a proper transfer function for a CPSS that gives a suitable supplementary stabilizing signal, covering all frequency ranges of interest Effective tuning of the CPSS parameters Tracking the variation of the system operating conditions. The inherent aforementioned drawbacks limit the application of the CPSS and make it less than optimal for use in power systems. Regardless of these disadvantages, it is standard practice to equip all generating units with either a conventional Proportional-

29 6 Integral or lead-lag compensator (CPSS), to aid in damping power system oscillations. Therefore, using any type of conventional controller, as a point of reference in order to test and evaluate the performance of new adaptive or advanced control algorithms, is a wellestablished common practice. On the other hand, adaptive control theory can provide an adequate solution in coping with the parameter changes of systems, making it a candidate for use in replacing fixed parameter controllers [27] Adaptive Power System Stabilizer (APSS) Because of the nonlinearity, time varying and stochastic nature of the power systems, the application of adaptive control techniques to power systems has attracted attention. Many adaptive control algorithms have been proposed to be incorporated in the design of PSSs. Researchers, over the years, have investigated different adaptive control schemes based on PSS designs, including self-tuning adaptive control, the use of artificial intelligence in identifying and controlling the system and other classes of adaptive control schemes [8], [11], [28]. Some of these adaptive techniques are discussed broadly in the following subsections Self-Tuning Control Based Adaptive PSS Self-tuning adaptive control is one of the most successful and well-established adaptive techniques. It comprises of an on-line identification algorithm to identify the dynamic behavior of the system and a control algorithm obtained based on the identified model [29]. The structure of the self-tuning control is illustrated in Fig.1.1. Different types of identification algorithms, used to identify the parameters of the system, have been investigated and reported in the literature. The most commonly used are

30 7 Kalman Filter (KF) and Recursive Least Squares (RLS) algorithms, which have been successfully applied and have demonstrated excellent results in tracking the system parameters [29], [30]. Once the system parameters are correctly identified, a control algorithm is used to calculate the required control signal to damp system oscillations. Various control algorithms, including Minimum Variance (MV), Pole Assignment (PA) and Pole Shift (PS) have been proposed to compute the control signal [30], [31]. All the control algorithms compute the control signals based on the assumption that the on-line identified model correctly resembles the actual plant as closely as possible. Plant y(t) u(t) Controller (t) ^ On-line Parameter Identification Self-tuning Adaptive Power System Stabilizer Figure 1.1 General scheme of self-tuning adaptive control Artificial Intelligence Based Adaptive PSS The methodology of Artificial Intelligence (AI) has successfully contributed to the design of adaptive control systems for various types of applications. The merits of high speed, robustness and generalization ability are of considerable advantage in power system

31 8 applications [32], [33]. Different types of AI techniques such as Artificial Neural Networks (ANNs), Fuzzy Logic (FL) and Neuro-Fuzzy (NF) are successfully employed in the design of adaptive PSSs to damp power system oscillations. The ability of ANNs to learn the system dynamics by adapting their weights, make them suitable candidate for adaptive control applications [33]. ANNs based PSSs have been widely investigated and shown to provide good performance in terms of system identification and designing a control system. Various kinds of network topologies have been developed to construct different types of ANNs [34], [35], [36], [37], [38]. One of the most well-known network topologies is the use of an indirect adaptive control scheme where two networks are employed, one as a neuro-identifier and the other as a neurocontroller. The neuro-identifier is used to estimate the dynamic model of the plant and provide an update to the neuro-controller. The neuro-controller is used to produce a supplementary control signal. The neuro-identifier and neuro-controller parameters are updated on-line using a back-propagation algorithm. In this manner, the controller can adapt to the changes in the system configuration and operating conditions [39]. One of the disadvantages of this control scheme, however, is the lack of description because ANN is treated as a black box. Indeed, it is difficult for the outside user to understand the control process which, in turn, discourages the application of the control scheme that might have given a satisfactory output. In addition, for higher order models, the number of layers and neurons in the neural network might significantly increase which leads to a complex network and, as a result, heavy computation time [33]. Another approach to design an adaptive PSS utilizing artificial intelligence is the use of FL technique. FL has been proven effective for complex, nonlinear and imprecisely

32 9 defined systems. A Fuzzy Logic Controller (FLC) provides a feasible alternative to capture the approximate and qualitative aspects of human reasoning and a decision making process to control a system [26], [40]. The basic FLC design steps are [41]: Fuzzification: transfers the crisp input variables to corresponding fuzzy variables Rule Definition: contains the meaning of the linguistic value of the process state and control output variable Inference: used to obtain strength of each rule according to the membership value Defuzzification: used to convert the set of modified control output values into nonfuzzy (crisp) control values. Various techniques have been applied to make the FLC adaptive [42], [43], [44], [45]. The main concept of these techniques is to modify the parameters of the FLCs in order to change their behavior. By modifying the membership functions and the consequent parameters of the controller, the FLC is considered adaptive [46]. Although FLCs are considered robust and have a relatively low computation requirement which decreases the development time and cost, using FLCs with a poorly defined nonlinear system may not be suitable due to the following limitations [4]: Difficulties in constructing a rule-base for FLC Difficulty with selection of membership functions and tuning the parameters Combining fuzzy logic and neural networks has played a significant role in improving controller performance [40]. By fusing both techniques, the learning algorithms used to train the neural networks can be applied to adapt fuzzy systems. This methodology provides an adaptive network that is suitable for the adaptation of the control system parameters through the application of learning algorithms, used in the area of the neural

33 10 network while, maintaining the rule-base structure of fuzzy logic systems [47]. Several network structures combining fuzzy systems (FSs) and neural networks have been introduced in the literature [11], [48], [49], [50]. Among them, the most well-known network is the Adaptive Neuro-Fuzzy Inference System (ANFIS) [11], [51], [52]. ANFIS architecture is typically represented by a five-layer network with two inputs and one output. Two sets of parameters need to be adjusted to make the network adaptive. One set contains the membership functions and the other contains the consequent parameters. The control methodology of using the adaptive neuro-fuzzy controllers to damp power system oscillations is well-known and widely discussed in the literature [11], [49], [50], [53], [54]. The control algorithm can be successfully applied, as a supplementary controller, to the excitation system located at the generator or the SVC device installed on the transmission lines. A comparison of two adaptive controllers applied to the excitation system of a generator to damp low frequency oscillations and improve the performance of the power system dynamics is given in [11]. The adaptive neuro-fuzzy controller proposed by the authors was designed based on ANFIS Sugeno network with two inputs. The results demonstrated that the neuro-fuzzy controller provided superior performance, compared to the proportional integral derivative based genetic algorithm control system. The time response, overshoot and settling time are significantly improved by using the adaptive neuro-fuzzy controller. A similar neuro-fuzzy controller was also applied to an SVC device to aid in damping power system oscillations. The neuro-fuzzy controller proposed in [53] was applied to an SVC device to damp the low frequency oscillations. Based on a comparison of the system performance with the proportional-integral, fuzzy logic and the proposed

34 11 controller it is concluded that the best damping is achieved through the neuro-fuzzy controller. A similar conclusion is provided by [54] when a neuro-fuzzy controller is applied to an SVC device installed in a multi-machine system However, the complexity of the control system, mentioned in [11], [53] and [54], leads to a heavy computational burden and has large memory space requirements, especially when applying the control algorithm to a real-time system. Therefore, it is desirable to design a simplified version of the adaptive neuro-fuzzy controller in which the performance is not compromised and the computational time is reduced. This is one of the main objectives and contributions of this thesis. The performance of the simplified version of the adaptive neuro-fuzzy controller will be compared to the typical adaptive neuro-fuzzy controller and the results will be investigated in detail. 1.4 Statement of the Problem It is mentioned in section 1.2.1, that the FACTS devices that are already installed in the transmission path can be also used to provide damping to power system oscillations. By introducing a supplementary control signal to the voltage control loop of the FACTS device, damping of system oscillations can be achieved. The focus of this dissertation is to develop an adaptive control system that can provide a supplementary control signal to a FACTS device to damp power system oscillations and enhance system stability. The adaptive controller developed in this thesis is used to provide a supplementary control signal to a Static VAr Compensator (SVC). The general control systems structure for designing an adaptive controller for an SVC device has two steps: estimation of the system parameters by an on-line identification method and calculation of the control parameters using an adaptation technique. If the

35 12 system parameters vary, the identifier will provide an estimate of these parameters and the adaptation mechanism will subsequently tune the controller. The commonly used input signal to the controller is either the generator speed deviation or the generator electrical power deviation, while the input signals to the identifier are the control signal and the speed or power deviations. Considering the supplementary controller structure mentioned above, two problems arise when using the generator speed deviation or the electrical power deviation as input signals to the identifier and the controller employed at the SVC. First, since the SVC is usually located far away from the generating unit, any delay or interruption in the communication system that transmits these signals from the sending end, where the generator is located, to the identifier and controller, might negatively affect the performance of both systems [55]. Second, in a complex system such as a multi-machine system, it is not always clear which generating unit should be selected to use its signals as inputs to the identifier and controller. To overcome the aforementioned problems, a few recent studies have proposed the use of voltage or power of the bus, where the SVC is installed, as an input signal [56]. Moreover, some studies discussed the effect of the delayed speed deviation signals on the performance of the controller [55]. A new simplified adaptive Neuro-Fuzzy (NF) control system, employing ANFIS network structure, is proposed in this dissertation. The power deviation at the SVC device bus is used as an input signal to the identifier and controller. In this way, both the identifier and controller will receive their input signals locally. Using this control system structure will eliminate the problem of sending the speed deviation or the electrical power deviation

36 13 signals from the generating stations to the FACTS device. In addition, in a multi-machine system, it is more suitable and accurate to use the power signal at the SVC device bus as an input signal to the identifier and controller instead of using the speed deviation of one of the generators in the network. The proposed simplified NF controller demonstrates similar performance compared to the typical adaptive NF controller, but with a smaller number of parameters to optimize, reducing the computation time for real-time application. 1.5 Dissertation Objectives and Contributions The objective of this dissertation is to focus on designing a new simplified version of an adaptive NF controller applied to a FACTS device, namely SVC to damp power oscillations. The implementation of the proposed adaptive control system in a real-time physical model of the power system is also investigated. The main objectives and contributions of this dissertation are summarized as follows: 1. An Adaptive Simplified Neuro-Fuzzy Controller (ASNFC) applied to an SVC device to damp power system oscillations and enhance system stability is investigated. The proposed adaptive controller is constructed employing an ANFIS network. It consists of a reduced number of input membership functions and consequent parameters. Unlike the common practice of using the generator speed or the power angle deviations as inputs to the controller, the input to the ASNFC is the power deviation at the bus where the SVC is installed. 2. An Adaptive Neuro-Identifier (ANI) is constructed, employing a Multilayer Perceptron (MLP) network, to represent the nonlinear plant. The ANI is first trained offline, under various operating conditions, to make the estimated model output follow the output of the actual plant by minimizing a certain cost function. After

37 14 the offline training, the weights of ANI are updated on-line to predict the plant output one step ahead and provide an update to the proposed ASNFC. 3. The ASNFC, Adaptive Neuro-Fuzzy Controller (ANFC) and CPSS are applied to an SVC device located in the middle of a Single Machine Infinite Bus (SMIB) system. The performance of the ASNFC is examined over a range of operating conditions. In addition, the behaviour of the ASNFC is also investigated in a multimachine power system simulation environment under various operating conditions. 4. To ensure the practicality of the proposed ASNFC and verify its application in a real-time power system, as a proof of concept the ASNFC is implemented and tested on a physical model of a real power system. The ASNFC is applied as a supplementary controller to the Automatic Voltage Regulator (AVR) of a generating unit. The performance of the proposed controller is examined under various types of operating conditions to verify its effectiveness. 1.6 Outlines of the Dissertation Following the Introduction, this dissertation is organized as follows: Chapter 2 presents an introduction to the SVC device and its contribution in enhancing power system stability and damping oscillations. The schematic diagram, the mathematical model, and the control structure of the SVC device considered in the simulation studies are introduced. In Chapter 3, the basic concept of the direct and indirect adaptive control schemes is given. The artificial neural network, and its application in the field of system identification, is also discussed. Different neural network topologies and the backpropagation learning algorithm used to adapt their parameters, are explained in detail.

38 15 Finally, the proposed adaptive neuro-identifier used to track the behaviour of the plant is introduced. In Chapter 4, the structure of the adaptive neuro-fuzzy controller when merging the neural network with fuzzy logic, is described. Since an overview of the neural networks and their architectures were introduced in Chapter 3, an overview of fuzzy logic systems and their basic structures is explained in this chapter. The fuzzy logic based neural network and its various network architectures, the basic structure of the neuro-fuzzy controller and the on-line adaptation technique used to adjust the parameters of the neuro-fuzzy controller, are also given. Finally, the design of the proposed adaptive simplified neuro-fuzzy controller is discussed. In Chapter 5, the application of the proposed controller to a single machine infinite bus system and a multi-machine power system for damping power system oscillations, is investigated. The simulation results are obtained under different types of disturbances and operating conditions. In Chapter 6, the implementation of the proposed ASNFC and experimental tests on a physical power system is demonstrated. The behavior of the ASNFC, ANFC and CPSS in the actual physical power system is observed. The performance of the simplified and detailed adaptive neuro-fuzzy controllers is evaluated using the Time-Integral Performance Criteria. Finally, conclusions and suggested future research are given in Chapter 7.

39 16 CHAPTER 2 SVC BASED-POWER SYSTEM STABILIZER 2.1 Introduction For the past two decades, Flexible AC Transmission Systems (FACTS) devices have been implemented in the power systems to enhance their capacity, stability, security and quality. Since FACTS devices are designed based on advanced power electronics technology, they are capable of providing control action at high speed [57]. Several types of FACTS devices have been developed for application in power systems [58]. The function of these devices is primarily to control the power flow through the transmission lines and regulate the voltage level where they are installed. FACTS devices can generally be classified as below [17]: shunt connected controllers such as static VAr compensator (SVC) and static synchronous compensator (STATCOM) series connected controllers such as static synchronous series compensator (SSSC), series capacitive reactance compensator (SCRC), thyristor-switched series capacitor (TCSC), and thyristor-controlled series reactor (TCSR) a combination of shunt and series connected controllers such as unified power flow controller (UPFC), thyristor-controlled phase shifting transformer (TCPST).

40 17 A general review of the SVC and its contribution to enhancing system stability and power capacity is given in this chapter. A description of the SVC mathematical model used in this thesis is also presented. 2.2 Static VAr Compensator (SVC) SVC is considered as the first generation of shunt connected FACTS devices that have been implemented in power systems to provide fast-acting reactive power and voltage support to the power grid. By incorporating inductive and capacitive branches, SVC is able to regulate the voltage at a chosen bus by supplying or absorbing reactive power. The advantages of simplicity, low losses, low harmonics production and low cost have made SVC to be used extensively compared to other shunt FACTS devices [59]. In fact, many SVCs have been installed at power plants around the world and are considered attractive elements to enhance the performance of power systems. SVC installations, for instance, can be found in Finland and Norway where SVCs have been commissioned to damp interarea oscillations and enable a power transfer increase across a limited interface [60]. An SVC is installed in the Kangasala substation, south of Finland, to improve the transmission capability of the Finnish transmission system. The SVC operates under power oscillation damping control mode to damp inter-area mode of oscillations. Parameters of the Power Oscillation Damping (POD) control, along with the PSS, are tuned and specified to damp 0.3 Hz inter-area oscillations. A typical structure of the SVC that consists of a Thyristor-Controlled Reactor (TCR), a Thyristor-Switched Capacitor (TSC) and a harmonic filter used to filter the harmonics generated by the TCR is illustrated in Fig. 2.1 [61].

41 18 Figure 2.1 Schematic diagram of an SVC Location of the SVC in the power system network is important to dictate its effectiveness. Since the voltage drop along an uncompensated transmission line is the largest at the midpoint, it is ideal to connect the SVC at the middle of the transmission line [18], [58]. An SVC connected at the middle of a transmission line of a single machine infinite bus (SMIB) system is illustrated in Fig. 2.2 [58]. The transmission line reactance is denoted as jx1 and jx2. V s, Vm m and V 0 r represent the voltage at the sending end, middle bus voltage and voltage at the receiving end, respectively. The currents passing through the first segment and second segment of the transmission line are denoted as I1, I2, respectively. The current going to the middle bus from the SVC is denoted as Isvc.

42 19 V s Vm m V r 0 jx 1 jx 2 I I svc 1 I 2 SVC Figure 2.2 A Transmission line with SVC connected at the middle of a SMIB system 2.3 Steady State Power Transfer Capacity An SVC can be employed to enhance the power transfer capacity of a transmission line [58]. As shown in Fig. 2.2, the SVC splits the transmission line into two parts. The first segment transmits power from the generator to the midpoint while the second part carries power from the midpoint to the receiving bus. The power across the first half line section in connecting the generator with the middle bus and the power transfer in the second half line section connecting the middle bus with the receiving bus, assuming, X1= X2 can be described as [18]: P = V s,r V m X/2 sin δ 2 (2.1) where X = X 1 + X 2 (2.2) For simplicity, if Vs, Vm and Vr are equal and represented by V, Eqn. (2.1) can be written as: P = 2V2 X sin δ 2 (2.3)

43 20 It is clear from Eqn. 2.3 that the maximum transmittable power across the line has increased compared to the maximum power transmitted in the uncompensated case. In other words, the SVC located at the middle bus almost doubles the steady-state power limit and increases the stable angular difference between the sending bus and the receiving bus from 90 o to 180 o. Fig. 2.3 shows the effect of the SVC compensation on a power-angle diagram. Figure 2.3 Effect of an SVC on P-δ diagram 2.4 Enhancement of Transient Stability An SVC can significantly improve the transient stability of the power system. The enhancement is achieved primarily through voltage control exercised by the SVC at the interconnected bus. This can be demonstrated by a well-known concept in power system stability, named Equal Area Criterion [62]. The power-angle curves for (a) an

44 21 uncompensated system and (b) midpoint SVC compensated system for a SMIB system are illustrated in Fig Assume both systems are transmitting the same level of power and are subject to the same fault at the generator terminals for an equal length of time. The initial operating points for both systems, which are indicated by rotor angles δ1 and δp1, are correspond to the intersections between the power-angle curves and the mechanical input line Pm, which is the same in both cases. When a fault occurs at the generator terminals, the active power from the generator will be reduced. Since the mechanical input remains the same, the generator accelerates and the rotor angle will reach the values of δ2 and δp2. Moreover, the accelerating energy, A1 and Ap1, will accumulate in both the uncompensated and compensated systems, respectively. When the fault is cleared, the electrical power exceeds the mechanical power and thus the generator begins to decelerate. However, the rotor angle continues to increase to δ3 and δp3 due to the stored kinetic energy in the rotor. The decline in the rotor angle begins only when the decelerating energies, represented by A2 and Ap2, become equal to the accelerating energies, A1 and Ap1, respectively. The power system returns to stable operation if the post-fault angular swing, δ3 and δp3, does not exceed the maximum limits of δcrit and δp Crit, respectively. The further the angular over-swing from its maximum limit, the more transient stability in the system. The transient stability margins in a SMIB system is shown in Fig It is evident that Apmargin significantly exceeds Amargin, and, therefore, the system-transient stability is greatly enhanced by the installation of an SVC [18].

45 22 Figure 2.4 P-δ Diagram curves illustrating transient stability margins in a SMIB system: (a) without SVC (b) with SVC 2.5 Mathematical Model of an SVC Auxiliary Control As mentioned in section 2.1, the main objective of FACTS devices including SVC is to support bus voltage and reactive power in the power grid. However, FACTS devices are also capable of enhancing the transient stability and damping power system oscillations [55], [57], [63]. By adding a suitable supplementary control signal to the summing junction of the voltage control loop of an SVC, damping power system oscillations using SVC can be successfully achieved. Since the signal addition causes the bus voltage to vary, an accelerating or decelerating torque on the generator rotor will be introduced [29]. The SVC can be represented by a susceptance BSVC that varies within limits depending on the control provided by the voltage regulator of the SVC [63], [64]. The variable susceptance modifies the admittance of the transmission line and, consequently, varies the voltage of the bus where the SVC is installed. Positive value of BSVC represents a capacitive mode of operation while negative value indicates an inductive mode.

46 23 The voltage of the controlled bus is measured and compared to a reference value. The voltage error is amplified and used to modify the susceptance at the controlled bus. When the bus voltage drops below the reference value, the BSVC value will be positive to inject reactive power to the system. On the other hand, when the bus voltage increases beyond the reference value, BSVC value will be negative to absorb reactive power from the system. The block diagram of an SVC controller with a supplementary control signal, represented by a first order model is shown in Fig Vm - AVR Vref Controller SVC K svc T B max - + B SVC 1 s. svc u(t) B min Supplementary Control Figure 2.5 Block diagram of SVC with a supplementary controller The dynamic equation representing the variation of the susceptance in the SVC is given by: B SVC = 1 T SVC [K SVC ( V ref V m + u(t)) B SVC ] (2.4) The block diagram, shown in Fig. 2.5, consists of three main parts. The first part is an AVR that is used to adjust the output of the SVC to control the bus voltage. The second part is the SVC device which represents the TSC, TCR and firing angles α. The third part is a supplementary controller that is used to generate a damping signal. The input signal to

47 24 the supplementary controller, in this thesis, is chosen to be the active power deviation of the bus where the SVC is located. From Fig. 2.2, the middle bus voltage Vm can be expressed in terms of BSVC as: V m = V r + j X 2 I 1 + B SVC V m X 2 (2.5) Simplifying Eqn. 2.5, the middle bus voltage Vm can be written as: V m = V r + j X 2 I 1 1 B SVC X 2 (2.6) The parameters of the SVC controller used in this thesis are provided in Appendix A. It is important to mention that the net power at the middle bus, Psvc, zero since the input power to the bus is equal to the output power. Power at the middle bus, used as the SVC input signal, is derived from a measurement of the voltage at the middle bus and the current flow in to the middle bus. It can be seen from Fig. 2.2 that knowing the voltage at the middle bus Vm and the current flowing into the transmission lines I1, the power at the middle bus can be obtained using Eqn. 2.7 P svc = V m I 1 cos (δ m θ) (2.7) where δ m and θ are the angle of the voltage, V m, at the middle bus and the angle of the current, I 1, respectively. 2.6 Conclusions A general introduction about FACTS devices is provided in this chapter. The electronic structure of the SVC is described, and the functions of the SVC in increasing the power transfer capacity and enhancing power system stability are explained. The mathematical model of the SVC with a supplementary control signal is also introduced.

48 25 CHAPTER 3 ADAPTIVE NEURO-IDENTIFIER 3.1 Introduction Control systems have played an important role in the advance of modern life and technology. They are found in various applications such as space-vehicle systems, power systems, manufacturing, industrial process, robotic systems and others. The basic concept of a control system is to maintain the conditions of a system at determined values and counteract random disturbances caused by external forces. This can be achieved via a feedback system where a controller senses the operation of a system, compares it against a desired behaviour, computes corrective actions and actuates the system to obtain the desired response [65]. Nonetheless, if the parameters of the controlled system vary over a wide range of operating conditions and are subject to disturbances, the performance of the conventional controller, with constant parameters, cannot provide effective control and its performance will deteriorate. Therefore, it is desirable to develop a controller that has the capability to adjust its parameters according to the environment in which it works to provide satisfactory control performance. An adaptive control system provides an effective solution for designing controllers applied to systems whose parameters change continuously during operation and no prior knowledge can be obtained of when these changes will take place. By using this type of

49 control scheme, the controller can constantly adapt itself to the current behaviour of the system. The adaptive control techniques can be classified as [66], [67]: 26 Direct Adaptive Technique Indirect Adaptive Technique. In the direct adaptive approach, known as Model Reference Adaptive Control (MRAC), the difference between the output of the plant and the output of the model, known as plant model error, is used to directly adjust the parameters of the controller in real-time. The adjusting mechanism continues until the plant model error reaches zero. The performance of this technique relies on the determination of a suitable reference model and the derivation of an appropriate learning mechanism. This type of adaptation scheme is described in Fig Figure 3.1 Direct adaptive control scheme

50 27 The basic idea of the indirect adaptive approach, shown in Fig. 3.2, is that a suitable controller can be designed if a model of the plant is estimated on-line from available inputoutput measurements. The adaptation of the parameters is done in two stages: the first stage is to identify the plant parameters on-line while the second stage is to compute the controller parameters on-line, based on the identified model. The indirect adaptive control scheme offers various combinations of control algorithms and parameter estimation techniques [68]. Desired Performance Controller Design ŷ p Plant Model Estimation u(t) y p Desired output Controller Control Plant Plant output Feed-back Figure 3.2 Indirect adaptive control scheme In this chapter, a description of system identification using Neural Networks (NNs) will be introduced. The architectures for different types of neural networks including Multilayer Perceptron (MLP), are described. The adaptation technique used to adapt its parameters and track the behaviour of the plant is explained. Finally, the proposed adaptive neuro-identifier used in this thesis is also introduced.

51 System Identification and Neural Networks The goal of system identification is to obtain a knowledge of the properties of a system based on its input and output data. It provides a mathematical representation of a physical system, called a model, which describes the static and dynamic behavior of that system in a sufficiently accurate manner [69]. System identification has attracted the interest of scientists and engineers for many decades since it is widely used in many applications. For instance, the key to successful application of an indirect adaptive controller to practical power systems is the identified model of the power system dynamics. If the power system dynamics are accurately identified, a suitable adaptive controller can be successfully designed. Models, in general can be classified as simple, such as linear and time invariant, or complex, such as nonlinear and time-varying models. Linear models can be used to produce accurate identification of a system s behavior, given the system is operating within a narrow region. However, if the system, being modeled, has many operating regions to cover, a nonlinear model may be required [70]. A large variety of identification methods have been proposed in the literature. Theoretically, any identification method can be used to obtain the identified model of the controlled system. However, for certain applications, some identification methods are more suitable than others. For example, a real-time identification technique is used in the power system to identify its parameters since it is considered a time-varying system. For other applications, a non-real time identification method might be sufficient. Many algorithms, used for on-line identification to estimate the dynamics of the power system, have been discussed in the literature. Recursive Least Squares (RLS) and

52 29 Kalmen Filter (KF) have been successfully applied to identify the parameters of the power system and update the adaptive controller used as a supplementary controller for an SVC [29], [30], [71], [72]. By using these algorithms, the system is constructed as a 3 rd order Auto-regressive Moving Average (ARMA) linear discrete model. It is assumed that the system is working around a certain operating point for a certain period of time, which enables the estimated coefficients of the linear model to converge to the actual values. However, proper design of an RLS algorithm for the purpose of parameter identification is critical, and extra care should be taken to ensure stability, especially under large disturbances [73]. Fuzzy logic has also been used in system identification applied to power systems [8], [74]. A fuzzy identifier has been used to track the parameters of the power system and update the adaptive controller. Based on the knowledge of the plant, the input signals to the fuzzy system are fuzzified, the rule table is constructed and the output signals are finally defuzzified. Parameters of the fuzzy identifier are updated in real-time by minimizing a defined cost function using the gradient descent method [8]. Another successful tool used in power system identification is the use of neural networks. The powerful approximation abilities make neural networks attractive for being utilized in many nonlinear applications where traditional analytical approaches cannot be handled easily. Since power systems are highly nonlinear systems, using a neural network is a suitable approach to identify their parameters. Different kinds of neural network structures such as Radial Basis Function (RBF) [75], Cerebellar Model Articulation Controller (CMAC) [76] and MLP [39] have been successfully used in system identification.

53 30 The basic elements of artificial neural networks, their common network topologies and the backpropogation learning algorithm will be discussed in section and its related subsections Artificial Neural Networks Artificial Neural Networks (ANNs) are biologically inspired computational models that consist of processing elements called neurons with connections between them that constitute the network structure. They are essentially nonlinear function approximators that utilize process inputs to estimate process outputs. An important feature of the ANNs is the ability to adjust their connections through an adaptive learning process called Learning. Learning can be accomplished using a series of examples and patterns. Information obtained through learning is retained and represented by a set of connection weights within the neural network structure [33], [77]. As mentioned before, an ANN comprises of many neurons that are connected to form a network. A simple neuron model consists of two main parts: a linear combiner and a nonlinear activation function. Typically, the neuron has more than one input and can be mathematically modeled as shown in Fig The input signals, x1, x2 xn are weighted by scalers w2, w2 wn and added up together to produce the net input to the activation function. The output signal of the neuron, y, can be expressed as: n y = f ( w k x k + b) (3.1) k=1

54 31 Input signals x 1 x w1 w2 Activation function f (Φ) y Output signal x n wn b Bias Figure 3.3 Simple model of an artificial neuron It is worth mentioning that the weights are the most important parameters in determining the output of the neural network. They are used to adjust the relative importance of the connection between the neurons, according to a modification rule. It can be noted from Eqn. 3.1 that the effect of the bias, b, is to increase or decrease the input to the activation function. The activation function is utilized to transform the activity level of the neuron into the output signal. Many activation functions such as a hard-limit, sigmoid, Gaussian and hyperbolic tangent functions have successfully been used to build neural networks [4], [33], [78], [79]. The choice of the activation function relies on the applications where the neural network is used. The most common activation functions used in multi-layer networks are the sigmoid and hyperbolic tangent functions [26], [80]. The outputs of sigmoid and hyperbolic tangent functions are described in Eqns. 3.2 and 3.3, respectively. f(x) = 1 (3.2) 1 + e x

55 Neural Network Topologies 32 f(x) = ex e x e x (3.3) + e x The neurons themselves are not very powerful in terms of computation or representation. However, their interconnection allows one to encode relations between the variables and gives different powerful processing capabilities. The way the neurons are connected within the neural network and the type of activation function used to construct the network, yield to different network architectures. In general, three different types of network architectures can be identified [39]. 1. Single-layer Feed-forward Architecture A feed-forward network has a layered structure. A single-layer network, shown in Fig. 3.4, consists of multi-input and multi-output signals. The input signals are connected to each of the neurons in the network. The sum of the products of the weights and the inputs is calculated in each node. The input layer is not accounted for since no computation has taken place there. The limitation of this type of network is, regardless of how many neurons the network has or what kind of activation function is chosen, that it can only approximate a linear function. The common approach of approximating a nonlinear function can be obtained by using a multi-layer perceptron.

56 33 Input Layer Output Layer Figure 3.4 Single-layer feed-forward network 2. Multi-layer Feed-forward Architecture In this type of network, two or more single-layer networks are connected together to form one network. Each layer consists of neurons which receive their inputs from the neurons located in the layer directly before them and send their outputs to the neurons located in the subsequent layer. The layer whose output is the network output is called an output layer, while other layers are called hidden layers. A multi-layer network, often known as an MLP network, with one hidden and one output layer is shown in Fig Input Layer Hidden Layer Output Layer Figure 3.5 Multi-layer perceptron network

57 34 3. Recurrent Networks A Recurrent Neural Network (RNN) is another class of neural network which contains feedback connections between the outputs and inputs of the network. Using this kind of network structure will allow signal flow in both forward and backward directions, providing the network with a dynamic memory, useful to mimic dynamic systems [81]. However, the drawback of using the RNN is the difficulty to train such a network, compared with the MLP, due to the feedback connections. The configuration of an RNN is shown in Fig Input Layer Output Layer Figure 3.6 Recurrent Neural Network The Back-propagation Learning Algorithm As previously mentioned, by adjusting the weights of the neural network, the output of the network will be altered. The weight modifications can be achieved by applying a suitable learning algorithm, which leads the network to converge to a desired value. The back-propagation is a learning algorithm, which has been widely utilized to train the multilayer neural networks [82], [83]. The learning technique is based on the gradient method

58 35 which minimizes the accumulated mean squared error between the actual and desired outputs, using the derivatives of the error. By passing the derivative of the error from the output layer of the network back toward the input layer, the weights of the network can be adjusted in such a manner. Assuming the error function E, for a given training data set p, can be described in the form of a well-known error function called the sum of squared errors: E = 1 2 ( (d pk y pk ) 2 ) (3.4) p k where d pk is the desired output at instant time k for pattern p and y pk is the actual output at instant time k for pattern p. The objective is to minimize the error function E, to a value close to zero so the output of the network is closed to the desired value. It is assumed that by minimizing the error of each pattern individually, E will also be minimized. Therefore, notation p can be neglected assuming there is only one pattern to be considered. A three-layer network, shown in Fig. 3.7, is considered as an example in order to explain the back-propagation algorithm. The output of neuron j, located in the hidden layer, is given as: O j = φ (net j ) (3.5) where φ represents the activation function used for the output of neuron j. net j is called the local field and it is given by: net j = W ji i O i (3.6) where Wji and Oi are the weight and the input signal to neuron j, respectively.

59 36 Similarly, the output of neuron k, which represents the output of the neural network, can be expressed as: O k = φ (net k ) (3.7) The local field net k is given by: net k = W kj j O j (3.8) where Wkj is the weight related to neuron k and Oj is the output of neuron j. net j net k X i O i O j O k W ji W kj i j k Figure 3.7 Three-layer neural network The network, shown in Fig. 3.7, is capable of calculating the total error E, for a given training set. Typically, the weights are the only parameters of the network that can be iteratively modified to make the error function as small as possible. The relation between the error function E, and the weights of the network can be defined as a quadratic function,

60 37 illustrated in Fig If the slope is positive, the weights should be decreased by a small amount to lower the error. On the contrary, if the slope is negative, the weights of the network should be increased. Error Weights Figure 3.8 Relationship between the error function and weights of the network By applying the Chain Rule method, the partial derivative of the error function E, with regard to the weights W kj, leading to an output unit change, can be calculated as: E W kj = E O k O k net k net k W kj (3.9) The weights can be updated using the gradient descent method as follows: W kj (k + 1) = W kj (k) + η E W kj (3.10) where η is the learning rate of the back-propagation algorithm. It can be noted from Fig. 3.7 that changing W kj will only affect the output of neuron k, while changing W ji will affect

61 38 the output of neurons j and k. Therefore, the change in E with regard to W ji can be written as the sum of the changes to each of the output units. The adaptation of weights between hidden j, and input i, layers can be expressed as follows: E W ji = E O j O j net j net j W ji (3.11) The term E O j can be calculated as: E O j = E O k O k net k net k O j (3.12) Hence, the adaptation of the hidden layer weights can be written as: E W ji = E O k k O k net k net k O j O j net j net j W ji (3.13) W ji (k + 1) = W ji (k) + η E W ji (3.14) The back-propagation algorithm requires a large number of training examples in order to provide an acceptable level of accuracy. It is also important to carefully select the learning rate to ensure the convergence of the network, as a large value of η might lead to network instability and a small value will cause a very slow convergence Proposed Adaptive Neuro-Identifier (ANI) A Multilayer Perceptron network is employed to represent the dynamics of the plant. The structure of the MLP is illustrated in Fig The proposed network has 6 inputs, one hidden layer of 9 neurons with hyperbolic tangent functions, and an output layer with one neuron having linear node characteristics. The overall structure of the plant with the ANI is shown in Fig. 3.10

62 39 The output of the ANI is given by: ΔP svc (k + 1) = f [ΔP svc (k), ΔP svc (k 1), ΔP svc (k 2), u(k), u(k 1), u(k 2)] (3.15) ΔPsvc(k) ΔPsvc(k-1) ΔPsvc(k-2) u(k) u(k-1) ΔP svc (k+1) u(k-2) Figure 3.9 Adaptive neuro-identifier architecture Figure 3.10 Overall plant structure with the ANI where ΔPsvc(k) is the power deviation at the SVC bus and u(k) is the control signal, both at time step k. The output of the ANI is the predicted ΔP svc at time step (k+1). The inputs to

63 40 the ANI are scaled before being applied to the network to take a value in the range of [+1,- 1]. To properly derive the network output at the (k+1) time step, the identifier is first trained to make the estimated model output,δp svc (k), follow the actual plant output, ΔP svc (k), by minimizing the following cost function: J i (k) = 1 e 2 i 2 (k) = 1 [ΔP 2 svc(k) ΔP svc (k)] 2 (3.16) The weights of the identifier are updated on-line using the gradient descent method as follows [39]: W(k) = W(k 1) η W J i (k) (3.17) where W(k) is the weights matrix at time k, η is the network learning rate, and W J i (k) is the gradient of J i (k) with respect to the weight matrix W(k). The gradient is calculated by: W J i (k) = [ΔP svc (k) ΔP svc (k)] ΔP svc (k) W(k) (3.18) 3.3 Conclusions The direct and indirect adaptive approaches used in the design of adaptive control systems are introduced in this chapter. The use of different kinds of algorithms to identify the parameters of the plant, including RLS, KF, fuzzy logic and artificial neural networks, was reviewed. Since, ANNs are powerful tools that have useful features, such as the capability to learn and approximate nonlinear functions, they have been used in system identification to estimate the parameters of the plant. A description of different neural network architectures has been introduced. The back-propagation algorithm employed to train the network is also explained. Finally, the proposed adaptive neuro-identifier used in this research is described.

64 41 CHAPTER 4 ADAPTIVE NEURO-FUZZY CONTROL SYSTEM 4.1 Introduction Neuro-Fuzzy Control (NFC) has been widely used in many control system applications [84], [85], [86], [87], [88]. It represents a control approach where fuzzy logic and artificial neural networks are combined. The basic idea of a neuro-fuzzy system is to model a fuzzy logic system by a neural network and apply the learning algorithms developed in the field of neural networks to adapt the parameters of the fuzzy system. An NFC can be defined as a multi-layer network that has the elements and functions of typical fuzzy logic control systems, with additional capability to adjust its parameters via learning techniques [89]. The motive of combining fuzzy logic with neural networks is to take advantage of their strengths and overcome shortcomings. In fact, fuzzy logic systems and neural networks can be considered complementary technologies. In a neuro-fuzzy system, the fuzzy system can be provided by an automatic tuning mechanism without altering its functionality. A background about fuzzy logic, fuzzy logic based neural network and its various network architectures, the basic structure of a NFC, and the on-line adaptation technique used to adjust the parameters of the NFC controller, will be described in this chapter. Furthermore, a simplified version of the NFC will be introduced. The methodology applied

65 42 in the design and the structure of the simplified neuro-fuzzy controller will also be explained. 4.2 Fuzzy Logic Systems In the real world, there are a lot of imprecise conditions that defy a simple True or False statement as a description of their state. A computer system and its binary logic are incapable of adequately representing these vague, yet understandable, states and conditions. Fuzzy logic, which was developed in the mid 1960 s by L.A Zadeh, is a branch of mathematics that deals with vague and linguistic representations of data that mimic human understanding or intuition [90]. It expands the reach of traditional binary logic by allowing for the use of analog values as inputs and outputs in logic calculations. Fuzzy logic was developed based on the concept of Fuzzy Set Theory. It is considered a valuable tool, which can be used to solve highly complex problems where a mathematical model is too difficult, or impossible, to create. The applications of fuzzy logic can be found in many engineering and scientific works. Fuzzy logic has been successfully used in numerous applications, such as control systems engineering, image processing, power system engineering, industrial automation, robotics, consumer electronics, optimization, medical diagnosis and treatment plans, and stock trading [47] Fuzzy Set Theory Fuzzy set can be defined by changing the usual definition of the characteristic function of a crisp set, to introduce degree of membership. A fuzzy set A in a reference set X, called the universe of discourse, is defined by a mapping function, called the Membership Function (MF), that takes values in the range between 0 and 1. This can

66 43 mathematically be written as: μa: X [0, 1]. The MF is a curve that defines how each point in the input space is mapped to a membership value between 0 and 1. The higher the membership X has in the fuzzy set A, the truer that X is A [91]. Many membership functions, such as triangular, trapezoidal, bell and Gaussian, are used in fuzzy logic; however, triangular and trapezoidal are the most common ones. Unfortunately, there are no general rules or guidelines for selecting the appropriate shape of the membership functions. The fact that trapezoidal and triangular shapes are the ones most used in the literature is because they produce good results for most input variables, in various applications. Fuzzy logic has operators defined in a similar way to the classical Boolean logic. The AND operator can be evaluated, for example, using min while the max operator represents the OR operator and the NOT is replaced by 1-A in fuzzy logic. Let X be a fuzzy set, and A and B are two fuzzy sets with the membership functions μa(x) and μb(x), respectively. Then the union, intersection and complement of fuzzy sets can be respectively defined as: μ A μ B (x) = max ( μ A (x), μ B (x)) (4.1) μ A μ B (x) = min ( μ A (x), μ B (x)) (4.2) μ A(x) = 1 μ A (x) (4.3) There are some other definitions in the literature. For instance, the intersection operator, also known as the T-norm operator, could also be described as the algebraic product of two fuzzy sets: μ A μ B (x) = μ A (x) μ B (x) (4.4) The choice of the fuzzy operator, in the end, depends on the expert knowledge and implementation feasibility [92].

67 Linguistic Variable Since fuzzy logic deals with events and situations with subjectively defined attributes, a proposition in fuzzy logic does not have to be either true or false. For example, a room temperature can be described as cold, cool, comfortable, warm or hot; as opposed to only cold or hot. The descriptions mentioned previously are known as the linguistic variables in fuzzy logic terminology. The range of possible values of a linguistic variable is called the universe of discourse. In the case of a temperature example, the universe of discourse can be within the interval [10 C, 35 C]. However, for simplicity, a common practice is to normalize or scale the values to be in the range of [-1, +1] [47] Fuzzy IF-THEN Rules A single fuzzy IF-THEN statement can be explained as follows: IF (x is A) THEN (y is B) where x and y are the input and output variables, respectively. A and B are the linguistic values defined by fuzzy sets on the ranges x and y. The IF part is called the antecedent while THEN part is called the consequent. The IF-THEN rule can be interpreted in a way such that, if the antecedent is a fuzzy statement that is true to some degree of membership, then the consequent is also true to that same degree Structure of a Fuzzy Logic System The basic structure of a fuzzy logic system is illustrated in Fig It can be seen that in order to design a fuzzy logic system, four steps are required to be considered [92]:

68 45 Knowledge Base Input Fuzzification Interface Defuzzification Interface Output Inference Engine Fuzzification Figure 4.1 Structure of a fuzzy system Fuzzification is the process of mapping the input data into corresponding universes of discourses and converting the input into suitable linguistic values. The task of the fuzzification process can be summarized as follows: Measure the values of input variables Map the values of input variables to a corresponding universe of discourse Convert the input data into appropriate linguist values Knowledge Base The purpose of the knowledge base or rule-base step is to provide definitions that express the relation between the input and output fuzzy variables. It defines the control goals by means of a set of linguistic control rules. The rule-base is often expressed in the form of IF-THEN rule.

69 Fuzzy Inference The fuzzy inference is considered the core of any fuzzy logic system. The fuzzy inference mechanism is a process by which the input values for each of the fuzzy variables in the antecedent are matched with all rules in the fuzzy rule-base and an inferred fuzzy set is obtained. The membership values obtained in the fuzzification step are combined through a specific fuzzy operator to obtain the firing strength of each rule. Based on the firing strength, the consequent part of each qualified rule is produced. There are two methods that are used in fuzzy inference known as Mamdani and Sugeno inference systems [26], [47], [93]. The difference between the two methods resides in the consequent part. Mamdani fuzzy inference expects the output membership function to be fuzzy sets while Sugeno fuzzy inference method treats the consequent parts as either linear polynomials or constants in the form of single spikes. The output of each rule is weighted by the firing strength of the rule and the final output is the weighted average of all rule outputs Defuzzification The process by which a non-fuzzy output is attained from the fuzzy set is called defuzzification. There are several defuzzification methods, such as Center of Area (COA), Mean of Maximum (MOM), Smallest of Maximum (SOM) and Largest of Maximum (LOM), used in the defuzzification process of a fuzzy system. The two approaches most commonly used are the COA and MOM methods. The COA, also known as the Center of Gravity (COG), calculates the center of gravity of the distribution of the membership degrees under the curve. This method can be expressed, in the discrete form, as:

70 47 COA = n k=1 x. μ A (x) n μ A (x) k=1 (4.5) where μa (x) is the membership function of a fuzzy set A defined in the universe x, and n is the number of quantization levels of the output. The MOM method calculates the output value by averaging only the part of the inferred fuzzy set whose membership functions reach the maximum. The output of using this method can be described, in the discrete form, as: MOM = x i l l i=1 (4.6) where l is the number of elements, xi, with membership equal to the maximum value. Fig. 4.2 shows different types of defuzzification methods for a fuzzy function. μ A Smallest of Max Mean of Max x Largest of Max Center of Area Figure 4.2 Defuzzification methods for a fuzzy function 4.3 Neuro-Fuzzy Systems As mentioned in section 4.1, the neuro-fuzzy system is an artificial intelligence approach, resulting from the merging of a fuzzy logic system and a neural network structure. The integrated system is called neuro-fuzzy, and has the advantage of tuning the

71 48 rules of the fuzzy system using learning algorithms applied to neural networks. In return, the neural network can improve the transparency by having the rule-based fuzzy reasoning considered in its construction [94]. Many neuro-fuzzy network structures have been presented in the literature. Some of these networks are Fuzzy Adaptive Learning Control Network (FALCON) [95], Adaptive Neuro-Fuzzy Inference System (ANFIS) [96], Fuzzy Net (FUN) [97] and others [98]. However, one of the most well-known networks that has been reported in many publications and applied to various applications is the ANFIS [26] Adaptive Neuro-Fuzzy Inference System (ANFIS) ANFIS was first introduced by Takagi and Sugeno in 1985 and further developed by Jang [95]. The network is built to have the capability of ANNs in adapting and learning, together with the merit of approximate reasoning offered by fuzzy logic. Unlike neural networks, the weights of the connections between nodes located in one layer and the nodes in the subsequent layer are constant and have values of one. There are two main ANFIS structures known as the first-order or zero-order Sugeno models. A typical rule in the first-order Sugeno model has the form of: IF input 1 = x1 and input 2 = x2 THEN output is y = ax1 + bx2 + c where {a,b,c} is a parameter set. For the zero-order Sugeno model, the output y is considered a constant and does not depend on the inputs to the network. The fuzzy IF-THEN rule for this type can be written as: IF input 1 = x1 and input 2 = x2 THEN output is y = c where a=b=0.

72 The basic structure of a first-order ANFIS with two inputs and one output is depicted in Fig. 4.3 [49]. 49 A1 x1 x2 x1 A2 Π N x2 B1 Π N B2 x1 x2 Figure 4.3 ANFIS structure with two inputs and one output The function of each layer can be described as follows [92]: Layer 1: Input Membership Layer The first layer represents the MFs and contains adaptive nodes. The membership value specifying the degree to which an input value belongs to a fuzzy set, is determined in this layer. The output of the nodes in this layer can be defined by: O 1,i = μ Ai (x 1 ) for i = 1,2 O 1,i = μ Bi 2 (x 2 ) for i = 3,4 (4.7) Assuming that the membership functions are triangular functions, the output from node Ai can be given as: O 1,i = μ Ai (x 1 ) = max (min ( x 1 a, c x 1 b a c b ), 0) for i = 1,2 (4.8) where x1 is the input of node i, Ai is linguistic label associated with this node, and {a, b, c} is the parameter set of the triangular membership function.

73 50 Layer 2: Firing Strength Layer The output of each node in this layer is the product of all incoming signals and it represents the firing strength of a rule. Each node in this layer is a fixed node which performs the fuzzy AND operation using the algebraic product. The output of each node in this layer is given by: O 2,i = w i = μ Ai (x 1 ) μ Bi (x 2 ) for i = 1,2 (4.9) Layer 3: Normalized Firing Strength Layer The nodes in this layer are fixed nodes and they calculate the normalized firing strength for each rule which is given by: O 3,i = w i = w i n i=1 w i for i = 1,2 (4.10) Layer 4: Consequent Layer The output of each node in this layer is adaptive and represents the weighted consequent part of the rule table. The output of each node can be expressed as: O 4,i = f i = w i (p i x 1 + q i x 2 + r i ) for i = 1,2 (4.11) The parameter sets {pi, qi, ri} is called Consequent Parameters (CPs). Layer 5: Defuzzification Layer This layer is the output layer and it acts as a defuzzifier. The single node in this layer is a fixed node which computes the overall output as the summation of all incoming signals. The output of this layer is given by: n O 5,i = y = f i i=1 (4.12)

74 51 Like any other neural networks, a set of parameters in ANFIS, is required to be updated in order for the network to be adaptive. These parameters are the MFs represented by layer 1 and the CPs represented by layer 4. The common adaptation technique is based on a gradient descent method [96] Structure of the Neuro-Fuzzy Controller (NFC) The structure of a typical NFC is shown in Fig The two inputs to the controller are usually the error signal and change-of-error. The error signal e(t) represents the difference between the actual output of the plant and a desired set-point while the changeof-error e(t) is the difference between the error e(t) and the previous error value e(t-1). A negative sign of e(t) means that the output of the plant y(t) has a value above the desired value yd since e(t)= yd y(t), while a positive sign of e(t) indicates that output of the plant is below the desired value. Furthermore, a negative sign of e(t) suggests that the plant output has increased when compared to its previous value y(t-1) while a positive e(t) means the opposite. μ A w e(t) K1 f K3 u(t) Δe(t) K2 Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Figure 4.4 Architecture of an adaptive neuro-fuzzy controller

75 52 The input scaling factors, K1 and K2 are typically used to map the real input to the normalized input space in which the membership functions are defined. In general, the normalization range can be in the range of [-1, +1] for the universe of discourse. It is noted that the input scaling factors influence the sensitivity of the NFC and affect its performance [46], [99]. On the other hand, the output scaling factor K3 is used to map the output of the fuzzy inference system to the real output. The value of K3 should be appropriately selected so the output range of the NFC will not exceed a certain boundary, where a physical limitation is violated. It is clear that the output scaling factor has the most influence on system stability and oscillation tendency [46]. Considering the membership functions, located in the first layer of the NFC, have triangular shapes, there will be seven triangular MFs associated with fuzzy linguistic sets used for each input to the controller. The input MFs of the NFC is depicted in Fig As shown in Fig. 4.5, the centers of the membership functions are distributed evenly along the normalized input space, which is a common technique used in most fuzzy control applications. Given that the peak value of a MF is equal to 1, the cross point between two MFs is at 0.5. The linguistic terms used for the membership functions are large positive (LP), medium positive (MP), small positive (SP), zero (ZO), small negative (SN), medium negative (MN) and large negative (LN). Since the NFC has two inputs of which each has seven MFs, the rule-base table associated with the controller will contain 49 rules. The Sugeno-Type rule-base table with 49 rules is shown in Table. 4.1.

76 53 NB NM NS ZO PS PM PB Membership Degree Normalized Input Variable Figure 4.5 Triangular membership functions of the inputs Table 4.1 Sugeno-type rule-base table with 49 rules e e PB PM PS ZO NS NM NB NB NM NS ZO PS PM PB

77 On-line Adaptation Technique In order to make the NFC adaptive, two set of parameters are required to be adjusted. This can be achieved through network learning of which the membership functions of linguistic terms and the consequent parameters can be modified, using certain adaptation techniques [100]. One of these adaptation schemes is the back-propagation algorithm described in Chapter 3. The back-propagation algorithm can be employed to adjust the centers of the MFs and the CPs of the NFC. Given the cost function, described in Eqn. 4.13, the centers of the MFs and CPs can be updated on-line using the gradient descent method [49], [101]. J c (k) = 1 2 e c[(k + 1)] 2 (4.13) where e c (k + 1) is the error signal between the estimated output, provided by the identifier, and the desired system output at time step (k + 1). Assuming θ is an arbitrary parameter in the NFC, updating the MFs and CPs can be achieved using the gradient optimization method given as [55]: θ(k + 1) = θ(k) η J c θ (4.14) where θ J c (k) = [e c (k + 1) e c(k + 1) ] [ u(k) u(k) θ ] (4.15) u(k) θ = u(k) O O ε S O θ (4.16)

78 η and u(k) are the learning rate and output of the NFC, respectively. S and O* are the set of nodes whose outputs depend on θ and the output of nodes belonging to S, respectively. For the output node, u(k) is given by: O For an internal node, u(k) is given as: O 55 u(k) O = K 3 (4.17) u(k) = K 3 u(k) O i l P l+1 O n=1 n O n l+1 O i l (4.18) where O l i is the output of the i th node of the l th layer and P is the number of nodes in the (l+1) layer. The updating of the membership centers and consequent parameters of the neuro-fuzzy controller takes place every sampling period. The number of neurons, shown Fig. 4.4, in the adaptive neuro-fuzzy controller are 14, 49, 49, 7 and 1 for layers L1, L2, L3, L4 and L5, respectively. This means that fourteen center points in the membership functions (seven for each input) and seven consequent parameters need to be updated on-line. A network with this number of parameters to update is considered relatively complex and computationally expensive, especially for real-time applications [45]. As stated in Chapter 1, the purpose of the adaptive controller, in this thesis, is to be applied to an SVC device, described in Chapter 2, in order to damp power system oscillations and enhance system stability. Since the SVC is an electronic device, designed based on high-speed power electronic components, it is desirable to design a controller that possesses the characteristic of having a fast response time. Therefore, the objective is to

79 56 design a simplified version of the ANFC, described in section 4.3.2, that requires less computation time, and apply it to the SVC device. It is necessary for the simplified controller to provide similar performance as compared to the ANFC Adaptive Simplified Neuro-Fuzzy Controller (ASNFC) The essence of designing an NFC is to be able to transform a Fuzzy Logic Controller (FLC) and represent it in a neural network structure. This implies that it is imperative that the FLC is the main part taken into consideration when designing an NFC. If an FLC is constructed, an NFC can be designed and represented by a neural network, accordingly. Therefore, in order to develop a simplified version of an NFC, a simplified FLC is first required. In general, it is always desirable to design the simplest control system that performs the expected task it is built for, as long as its accuracy is not compromised. In fact, in designing FLCs, interpretability and accuracy are the most important aspects to be considered [102]. Although, accuracy and interpretability represent contradictory objectives, an optimum fuzzy control system design should satisfy both criteria, to a certain degree. For instance, a complex FLC might successfully control a high-order nonlinear system accurately; nonetheless, the drawback would be the difficulty in expressing the behaviour of the controller in an understandable way. On the contrary, a simple FLC can be easily understood but, its performance is not satisfactory. Therefore, a trade-off between the interpretability and accuracy should always be considered Simplification of the Rule-Base Structure There have been several publications proposing the design of simplified fuzzy logic controllers using different simplification approaches. One of these techniques, is reducing

80 57 the size of the fuzzy rule tables [45], [103], [104]. The principle of designing the proposed ASNFC is based on the concept of reducing the fuzzy rule-base table, illustrated in Table 4.1. It can be seen from Table 4.1 that the rule-base table can be viewed as a Toeplitz structure with zero diagonal line. Having such a structure provides the advantage of using the symmetrical property of the table to construct a one-dimensional fuzzy-rule table. A new variable, called the signed distance, can be introduced to build a Simplified Fuzzy Logic Controller (SFLC). This new variable represents the distance, d, to an actual state from the main diagonal line, called the switching line. The distance can be positive or negative, depending on the position of the actual state in the rule-base table, illustrated in Fig. 4.6 [105]. e NB NM NS ZO PS PM PB e PB PM PS ZO NS NM NB + d - d Switching Line Figure 4.6 The distance, d, between the switching line and actual state Looking at Fig. 4.6, it should be noted that a control action can be proportionally related to the perpendicular distance from any consequent in the table to the switching line. Three distances, d1, d2 and d3, can be obtained in both the upper half-plane and the lower half-plane. These distances will have negative signs if they are located in the upper half-

81 58 plane and positive signs if they are in the lower half-plane. This can clearly be illustrated in Fig The switching line, shown in Fig. 4.7, can be represented by a general straight line equation given by: A e + B Δe + C = 0 (4.19) where A, B and C are constants and they are equal to A=B=-1, C=0. The perpendicular distance between the switching line and a given point P (e, e), located on the phase-plane, can be shown in Fig. 4.8 and expressed as [106]: d = Ae + BΔe + C A 2 + B 2 (4.20) Figure 4.7 Illustration of d1, d2 and d3 between the switching line and actual states in the phase-plane Substituting the constant values, A, B and C into Eqn. 4.20, yields:

82 59 (e + Δe) d = f 1 (e, Δe) = 2 (4.21) Four symmetrical triangular MFs with 50% overlap in the range of [0, +1] are chosen to be the input to the SFLC. This range is considered, as opposed to the range of [- 1,+1] used in the typical fuzzy logic controller, since d1 can be located in the upper halfplane or the lower half-plane, Fig. 4.7, with negative or positive signs, respectively. Therefore, a calculation of only one distance from the switching line is required and a positive or negative sign can be associated, based on the location of the actual state in the phase-plane. e P (e, e) d e A e +B e + C=0 Figure 4.8 Perpendicular distance between point P (e, e) and the switching line in the phase-plane

83 Membership Degree 60 The values of d1, d2 and d3 for the upper and lower half-plane represent the centers of the triangular membership functions. The rule-base table is reduced to one-dimensional with the fuzzy linguistic terms, zero (ZO), small (S), medium (M) and big (B) for the distance, d, and fuzzy singletons for the control signal. The control signal for any point in the phase-plane is given by [45]: u = K 3 S u u us (4.22) where K3 is the output scaling factor, uus is the unsigned control action and Su is given by: 1 when e + Δe 0 S u = f 2 (e, Δe) = { 1 otherwise (4.23) The reduced rule-base table and input membership functions are shown in Table 4.2 and Fig. 4.9, respectively. Table 4.2 Reduced fuzzy rule-base table Normalized Input Variable Figure 4.9 Input membership functions for the SFLC

84 Control System Design of the proposed ASNFC The reduced rule-base table shown in Table 4.2 is used for the design of the proposed ASNFC. Since the objective is to design a simple NFC with a reduced number of parameters to update, a zero-order Sugeno-type fuzzy controller based ANFIS, is employed to construct the proposed controller. The overall structure of the proposed ASNFC is illustrated in Fig D + - K 1 K 2 f 1(.) f 2(.) d n S u ANFIS u us K 3 u(k) Plant ΔP SVC (k) ASNFC - D ei + D ANI Identifier D D D ec + - ΔP d (k+1) Figure 4.10 Overall control system structure of ASNFC The ASNFC comprises of an ANFIS network, with a reduced number of layers and nodes, and two function blocks, f 1 and f 2, given by Eqns and 4.23, respectively. As shown in Fig. 4.10, the input to the ANFIS network is the distance in the phase-plane, d, while the output is the unsigned control action, uus. The cost function considered to update the centers of the MFs and the CPs of the proposed controller is defined in Eqn

85 62 J(k) = 1 2 e c(k + 1) 2 = 1 2 [ΔP svc (k + 1) ΔP d (k + 1) ] 2 (4.24) where ΔP svc (k + 1) and ΔP d (k + 1) are the estimated power deviation and the desired value at time step k+1, respectively. Since the desired value at time step k+1 is always zero, Eqn can be written as: J(k) = 1 2 e c(k + 1) 2 = 1 2 [ΔP svc (k + 1)] 2 (4.25) The update of the MFs centers and CPs is taking place at every sampling time, employing Eqns and Eqn can be expressed in terms of ΔP svc (k + 1) and the control signal u(k) as: ΔP svc (k+1) u(k) θ J c (k) = [ΔP svc (k + 1) ΔP svc (k + 1) ] [ u(k) u(k) θ ] (4.26) Three terms need to be calculated in (4.26). The terms ΔP svc (k + 1) and are the estimated output and the Jacobian of the plant, respectively. They can be obtained from the neuro-identifier. The term u(k) θ can be calculated using the backpropagation algorithm, as expressed in Eqn The new ANFIS network consists of 4 layers and has 4, 4, 4 and 1 neurons for L1, L2, L3 and L4, respectively. Since the rule-base table is reduced to a one-dimensional ruletable instead of a two-dimensional rule-table, the second layer described in section will not be applicable in the proposed design. It is obvious that the number of layers and neurons of the new ANFIS network is significantly reduced which yields to a simple type of ANFIS network. In addition, the control parameters required to be updated on-line are reduced from twenty one (fourteen center points in the MFs plus seven CPs) to eight (four

86 63 center points in the MFs plus four CPs). This will reduce the overall computation time of the controller. 4.4 Conclusions In this chapter, the adaptive neuro-fuzzy control system has been described. The fundamentals of the fuzzy logic system and its structure are explained in detail. The architecture of ANFIS network, employed to construct the adaptive neuro-fuzzy controller and the function of each layer, has been introduced. Furthermore, the learning technique used to update the MFs and CPs of the ANFC, at each sampling instant, has been demonstrated. Finally, the principle and the methodology of designing the proposed ASNFC, based on simplifying the fuzzy rule-base table and its control system structure, have been presented.

87 64 CHAPTER 5 APPLICATION OF AN ADAPTIVE SIMPLIFIED NEURO- FUZZY CONTROLLER IN POWER SYSTEM 5.1 Introduction Power systems are nonlinear systems and operate over a wide range. They are subject to unpredictable load changes and faults that introduce power system oscillations. These transient oscillations require damping; otherwise, maintaining stability and a good dynamics of the power network will be critical and difficult to achieve. It is desirable to develop a controller that has the ability to adjust its own parameters on-line, according to the environment it works in, to provide satisfactory control performance. For successful use of a controller in power systems, the flexibility of an adaptive controller is a major advantage as it determines its applicability to different conditions. It is also desirable that dependence on outside interference in its execution be kept to a minimum. As a rule, the larger the number of controller coefficients that need to be tuned manually, the more difficult it is to apply to practical situations. The performance of the proposed ASNFC applied to an SVC device, connected at the middle of the transmission line of a single machine infinite bus system, is investigated in this chapter. Moreover, the performance of the proposed ASNFC when installed on a multi-machine system that exhibits multi-modal oscillations is also tested. Various case

88 65 studies are presented for different operating conditions to demonstrate the effectiveness of the ASNFC. A neuro-identifier is used to track the system and update the parameters of the ASNFC on-line. Eight parameters are required to be updated in order for the proposed controller to be adaptive. As mentioned in Chapter 4, four input MFs and four CPs are considered in the design of the ASNFC. The centers of the input MFs and CPs are updated every sampling interval. In addition, a comparison between a traditional ANFC, with twenty one parameters to update, and the proposed ASNFC in damping power system oscillations, is carried out. The conventional PSS is also used to design an SVC Conventional Power System Stabilizer (SCPSS) applied as a supplementary controller to the SVC device, to damp power system oscillations. A comparison of the damping effectiveness of the proposed ASNFC, ANFC and the SCPSS is conducted under different operating conditions. 5.2 SMIB System Configuration The system model considered for studying the performance of the proposed adaptive controller is depicted in Fig The simulation study has been performed on a single machine connected to a constant voltage bus through a long AC transmission line, with the SVC at the middle of the line. As shown in Fig. 2.5 in Chapter 2, the SVC is modeled as a susceptance that varies within a limit, depending on the control provided by the voltage regulator of the SVC.

89 66 Figure 5.1 ASNFC for an SVC device in an SMIB system The proposed ASNFC is connected to the summing junction of the voltage regulator of the SVC. The variable susceptance Bsvc is used to update the admittance matrix. The voltage of the middle bus is measured and compared with its reference value. The voltage error is amplified and used to change the susceptance of a reactor unit at the controlling bus. The general control system structure, shown in Fig. 4.10, for the ASNFC applied to the SVC device to damp power system oscillations, is given in the previous chapter, section The control structure has two steps: estimation of the system parameters online, using ANI and calculation of the controller parameters using the gradient descent method. If the system parameters vary, the identifier will provide an estimate of these parameters and the adaptation mechanism will subsequently tune the controller parameters.