Study on stabilization and optimization of a large - Scale system applying for power systems

traints. The author applied analytical methods to propose a novel control strategy using modified Riccati equations in dealing with the transient stability problem of the three-machine electric power system. The author also suggested a hybrid control scheme applying a PD-like fuzzy logic controller and superconducting magnetic energy storage devices to tackle the load- frequency control of a five-machine interconnected power network. Although the research results are obtained by numerical simulation demonstrations, this study is completely significant in practical applications. c, Research methods - Theoretical research: Analyze the theory, build theoretical basis for control problems under uncertainties with a number of given laws 3 and constraints. The goal is to maintain the system frequency at the nominal value and bring the stability back to the power grid. - Tools: Lyapunov’s stability theory, Riccati equation, linear algebra, fuzzy logic control, etc. - Based on the above tools, this thesis presented a control strategy applying modified Riccati equations to find an optimal feedback control law with a state gain vector. The goal of this control scheme is to damp instantaneous fluctuations caused by noises, thereby ensuring the stability of the system. The next part of the study focuses on designing an intelligent control strategy applying a PD-type fuzzy logic controller and superconducting magnetic energy storage (SMES) to maintain the network frequency at the nominal value. The control performances obtained in this study are verified through numerical simulations using MATLAB/Simulink package. It is supposed that these research achievements are able to be a fundamental study for the further studies of large-scale systems under uncertainties, then they can be applied for the reality. 3. Scientific and practical significances - Reaffirm the correctness of several theories which have been applied for studies on the control engineering of optimization and fuzzy logic. - Propose a simplest and best method for calculation to obtain good control performances satisfying acceptable tolerances. Then, this method has been applied for two typical control problems of the large-scale power system: transient stabilization and load-frequency control problems. 4 - Practical significances: There is no doubt that the power system plays an extremely important role for a nation, thus the stability and reliability of power transmission need to be seriously considered. The control strategies proposed in this study are able to enhance the control performances and ensure the stability and economy of the power network. The main contents of the thesis The thesis includes four chapters: Chapter I. Overview of large-scale systems: generally analyze large-scale systems with domestic and foreign studies which were reported in the past. Chapter II. The theory of distributed systems and decentralized control. This chapter mainly focused on building a mathematical model for the large-scale power networks and evaluating the stability of the systems. The author proposed two control problems, i.e. transient stability and load-frequency control for whole the multi- machine electric power plant. Chapter III. Study on an effective decentralized control strategy to stabilize a large-scale power grid. The proposed control method using improved Riccati equations has been applied for a three-machine power system. Simulation results implemented in MATLAB/Simulink environment were presented to demonstrate control quality of the proposed strategy. Chapter IV. Study on load-frequency control of an interconnected power system. In this chapter, the author presented a novel control scheme to maintain the network frequency at a nominal value (50Hz). The proposed control method is a hybrid integration of 5 PD-like fuzzy logic controllers and SMES devices. Good simulation achievements clarified the feasibility of the control strategy studied in this chapter. Conclusion and discussion: Conclude studied results and present discussions for future work. CHAPTER 1: OVERVIEW 1.1 Characteristics of large-scale systems 1.2. Related work 1.2.1. Domestic studies 1.2.2. Foreign studies 1.3. Brief conclusions relating to the thesis 1.4. Conclusion for chapter I - Study on stability of the large-scale systems under uncertainties is a new research in Vietnam and in the world. In fact, the electric power grid in Vietnam is considered to be a typical example of the large-scale systems, so that this study plays an important role in reality. - There have been a number of studies on the large-scale systems, especially the interconnected power networks. In dealing with the stability of a multi-machine electric power grid, there exist a lot of control strategies, such as PID, fuzzy logic, neural network, optimal and adaptive control methods. However, each control strategy still needs to be further improved to design the best control method in dealing with the stability of a large-scale power system. 6 CHAPTER 2: DISTRIBUTED SYSTEMS AND DECENTRALIZED CONTROL FOR LARGE-SCALE POWER SYSTEMS 2.1. Introduction A multi-machine power grid is considered to be a typical example of large-scale systems. Such a power system normally consists of several generation areas; each area is considered to be a control area which includes three basic components: a governor, a turbine and a synchronous machine with electrical load. Figure 2.1. A large-scale power system model consists of N interconnected areas In this study, the author proposes two stability control problems for a multi-machine power system. The first one is a transient stability problem, and the second one is load-frequency control problem which is the key for automatic generation control (AGC) of a power grid. 2.2. Mathematical model of a multi-machine power system Consider a n-machine electric power system with an equation describing rotor motion of the ith generator as follows: i i i i mi eiM D P P      , i = 1, 2,, n (2.1) 7 Where: ij ij1 cos( ) n ei i j i jj P E E Y        In (2.1), ,i miM P và iE are constant for each synchronous generator. Let: i i D M  , i = 1, 2,, n (2.2) Let a vector 2( 1)nx R  be: 0 0 1 2 1, 1 1 1, 1,( , ,..., , ,..., ) T n n n n n n n n n nx            (2.3) where in in n    , ,in i n i i        and 0ij are solved from the power balance: 0 ij( )ei miP P  với i = 1, 2,, n – 1 (2.4) The Lur’e-Postnikov function as: ( ) T x Ax Bf y y C x      (2.5) Equation (2.5) can be rewritten as:   0 1 ( ) h ( ) 1 0 0 0 1 i i i i i i i x x y x y x                        , i = 1, 2,, n – 1 (2.10) The system given by (2.10) presenting interconnected Lur’e – Postnikov subsystems. These subsystems are given by: 1 ( ) ; , 1, 2,..., s T i i i i i i ij ij i i i j x A x b e h c x i s         2.2.2. Analysis of subsystems Each subsystem can be modelled by the following equation: 8 1 0 1 ( ) 1 0 0 [0 1] i i i i i x x y y x                       (2.13) Based on Walker and McClamroch (1967), we select a Lyapunov candidate as: 0 ( ) ( ) T i ic x T i i i i i i i i iV x x H x y dy    (2.18) Where Hi is a constant matrix, and i is a scalar number. According to [20], the Popov constraint as:  1 1Re (1 j )c (A j I) 0Ti i i ib         (2.19) . 21/2 (7.15)( ) ( ) g T i i i i i i i iV x y x       (2.20) Now we carry out stable area as * 0.ix  Based on a procedure proposed by Walker and McClamroch, let 1 0i   and consider the following condition:   2 22 2 2 ( 1) 0 0i i i                (2.22) An approximation i of the stable region i ( )i i  . We get:  0: ( ) , 1,2,..., -1i i i i ix V x V i n    (2.36) where 0 iV is determined from (2.35). 2.2.3. Stable region With each given value i , we can select a positive number 0 ij(0 os )i i c    given in (2.30) ' ' 0 0sin( ) sin , k = 1, 2ik ik in iny y     1i i i    (2.52) Where, ( )m i i iG   is combined with (2.51), one can be obtained below: 9 1 1 1 0 1 0 0 1 ( ) 2 cos sin sin( ) ( ) n i M i i n i in in in i ij ij ij ji m i j i H M M A M A H                             i = 1, 2,, n – 1 (2.53) From (2.53) it is possible to recognize that the smaller Aịj is, the easier the value of εi can be selected. This means that the decomposition of the power system model should be performed in such a way that the resulting subsystems are weakly coupled. This is also the general principle to analyze a large-scale electric power system. 2.3. Control strategies to stabilize a power system 2.3.1. Introduction 2.3.2. Transient stability problem Recall a N-machine electric power system with the following mathematical [18-20]: 0 0 0 ( ) ( ) ( ) ( , ( )), ( ) , 1,2,3,..., i i i i i i i i x t A x t B u t f t x t t t x t x i N         (2.60) In dealing with the nonlinearities of the system, the author suggested using a two-step method based on Riccati equations to establish the linear decentralized control law as follows. Step 1: Establish the modified algebraic Riccati equations in a form as follows: 10   1 1, 1, 0 N T T T i i i i i ij ij ij i i i i i i j j i N T T ij i i ji ji i j j i A P PA P p G G P PB R B P p W W W W Q                    (2.64) where Ri > 0 and Qi(ni x ni) and Pi(ni x ni) are defined matrices. Step 2: Solve the above Riccati equations to find the control law as follows: 1 ( ) ( )i i i T i i i i u t K x t K R B P     (2.65) The feedback control law mentioned in (2.65) is capable of recovering the stability of a large-scale system after presence of disturbances. Hence, it is also applied to an interconnected electric power system, particularly the three-machine network. In this perspective, the corresponding control law can be given below:       0 0 0 1 ( ) ( ) ( ) ( ) ( ) ( ) i i i i i i i i Pi mi mi Xi ei ei T i i i i u t K x t K t K t K P t P K X t X K R B P                     (2.66) The effectiveness of this control law will be specifically demonstrated in the following section through a number of numerical simulations using MATLAB/Simulink package. 11 2.3.3. Fuzzy logic applied for load-frequency control 2.3.3.1 Definition of fuzzy logic control 2.3.3.2. Principles and steps to design a fuzzy logic controller 2.3.4. Fuzzy logic controllers 2.4. Conclusions for Chapter 2 The main objective of this study is to analyze the large-scale nonlinear and uncertain systems with a typical example of interconnected power networks. The author also presented an approach to evaluate the stability region for such a power system in a case of considering it as a set of weakly interconnected subsystems by applying Lyapunov theory. For the stability issue of a large-scale power system, the author proposed two problems: transient stability and load-frequency control. To each control problem, the author presented a particular control strategy which will be discussed in the following chapters. Chapter 3: Decentralized control strategy to stabilize a power system 3.1. Introduction 3.2. Structure of a multi-machine electric power system Figure 3.1: Typical structure of a power system 12 3.3. Mathematical model of a power system Figure 3.2. Three-machine electric power system model 3.4. A control strategy to stabilize an interconnected power system Assuming that this model consists of N subsystems, its corresponding mathematical representation is as follows1: 0 0 0 ( ) ( ) ( ) ( , ( )), ( ) , 1,2,3,..., i i i i i i i i x t A x t B u t f t x t t t x t x i N         (3.23) where, fi(t, x(t)) = fi(x) denotes the interconnected components which represent all nonlinear characteristics of the ith subsystem. Such terms should satisfy the Lipshitz conditions as: ( ) , ( ) ( ) i n i i f x c x x y f x f y h x y        (3.24) 13 1, ( ) ( , ), 1,2,3,..., . N i ij ij i j j j i f x G g x x i N     (3.25) Where gij(xi, xj) is a nonlinear function, which must satisfy the following constraint: ( , ) W (t) (t) , , ji nn ij i j i i ij j i jg x x x W x x x      (3.26) ( )ix t ( ) . ( )i i iu t K x t *( ) 0ix t  Figure 3.3. Feedback control model for the control area #i 3.5. Simulation and discussion In this chapter, the author presents three simulation cases to demonstrate the effectiveness of the proposed control strategy. First, three steps need to be implemented as follows : Step 1: Design of the control plant, a typical example of an interconnected electric power grid. Using the simulation parameters given in the two reports, the following equations can be obtained for the first and second machines: 1, ( ) ( ) ( ) ( , ), 1,2,3,..., . N i i i i i ij ij ij i j j j i x t A x t B u t p G g x x i N       (3.30) 14 Step 2: Find the solution of the algebraic Riccati equations given in (3.23). Using MATLAB environment, solving the Riccati equations presented in (2.64) with the given simulation parameters, the control law, especially the following two gain vectors (computed from (3.26)), can be obtained: 1 1 1 1 2 2 2 2 1 2 [ ] [ 174.7398 42.2510 10.7218 5.2562] . [ ] [ 174.2508 29.2102 10.7003 5.4231] P X P X K K K K K K K K K K                    (3.31) Step 3: Carry out the necessary numerical simulations to demonstrate the feasibility of the proposed control method. In this step, we give two simulation cases with the initial conditions indicated in Table. Figure 3.4: Dynamic responses of the first and second machines in the simulation case #1 0 1 2 0 0.5 1   1 ,2 ( ra d ) time (s) (a) Machine 1 Machine 2 0 1 2 -2 0 2 4   1 ,2 ( ra d /s ) time (s) (b) Machine 1 Machine 2 0 1 2 -0.5 0 0.5  P m 1 ,2 ( p .u .) time (s) (c) Machine 1 Machine 2 0 1 2 -0.5 0 0.5 1  X e 1 ,2 ( p .u .) time (s) (d) Machine 1 Machine 2 15 Figure 3.5: Dynamic responses of the first and second machines in the simulation case #2 Figure 5: Comparative results of settling times for four state parameters 0 1 2 0 0.5 1 time (s) (a)   1 ,2 ( ra d ) Machine 1 Machine 2 0 1 2 -2 0 2 4   1 ,2 ( ra d /s ) time (s) (b) Machine 1 Machine 2 0 1 2 -0.5 0 0.5  P m 1 ,2 ( p .u .) time (s) (c) Machine 1 Machine 2 0 1 2 -1 0 1 2  X e 1 ,2 ( p .u .) time (s) (d) Machine 1 Machine 2 Machine 1 Machine 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 (a) S e tt lin g t im e (s )   P m X e Machine 1 Machine 2 0 0.1 0.2 0.3 0.4 0.5 (b) S e tt lin g t im e (s )   P m X e 16 3.6. Conclusions for Chapter 3 This Chapter has studied an effective linear decentralized control strategy to find an optimal solution for the stabilization issue of the large-scale system. A three-machine electric interconnection model representing a typical case study of the large-scale systems has also been taken into account. First, this model is mathematically formulated then the linear decentralized control scheme is applied to recover the stability of the network after the presence of the disturbances. Numerical simulation results obtained have demonstrated the feasibility and superiority of the proposed control method. For future work, with the diversity and complexity of the practical electric power systems, modern control techniques, such as fuzzy logic and neural network should be considered to further improve the effectiveness of the studied regulation strategy. In this aspect, a robust control scheme can be built in an efficient integration with the decentralized control methodology proposed in the present paper. 17 CHAPTER 4: LOAD-FREQUENCY CONTROL OF A LARGE-SCALE POWER SYSTEM 4.1. Introduction 4.2. Mathematical model of a large-scale power system in LFC 4.2.1. Introduction to large-scale power systems 4.2.2. Mathematical model of a large-scale power system ,D iP ,tie iP ( )iF s , ( )T iG s, ( )G iG s , ( )P iG s , ( )G iP s , ( )Tnr iP s( )iU s Figure 4.2. A typical model of a generation area From Figure 4.2, a mathematical model of the above system can be described as follows: , , , ,( ) ( ) ( ) ( ) ( ) ,i P i Tnr i D i tie iF s G s P s P s P s       (4.1) , , ,( ) ( ). ( ),Tnr i Tnr i G iP s G s P s   (4.2) , , 1 ( ) ( ) ( ) ( ) .G i G i i i i P s G s U s F s R          (4.3) 1,, 2 ( ) ( ) ,areas # & # are connected directly ( ) . 0, otherwise n ij i j j j itie i T F s F s i j sP s                (4.4) 18 4.2.3. A model of superconducting magnetic energy storage - SMES Figurer 4.3. A model of SMES unit The output voltage of the converter is: , 1 2 0 0.cos ' .cos 'd i d d d dE U U V V     (4.11) where Vd0 and V’d0 , α and α’ denote the ideal no-load DC voltages and firing angles of converter R1 and R2, respectively. Assuming that such two 6-pulse AC/DC thyristor – based converters are completely symmetrical, the DC output of the 12-pulse converter can be calculated as follows , , 0, , 0 if rectifying mode 2 2 .cos with 0 if inverting mode 2 d i d i d i d i E E V E               (4.12) In terms of using the SMES as an efficient part of the LFC strategy, it is necessary to mathematically model such an SMES device. According to the tie-line bias control idea, ACE signals must be collected, and then they are taken to both the LFC regulator and the SMES. It means that each generating power should be equipped with an LFC controller and an SMES. The control idea is that the load changes can be compensated by charging or discharging of the inductor, thus the DC current Id,i should become a controlled 19 quantity. (a) ,D iP ( )iF s ( )jF s ,tie iP 1 iR , ( )G iP s , ( )T iP s , ( ) Calculation tie iP s SMES Model ( )iF s ,SM iP ,D mP ,D n P Tie-line mn Figure 4.7. SMES model in MATLAB/Simulink 4.3. LFC controllers 4.3.1 Conventional controllers 4.3.2. PD-like fuzzy logic controllers Control system using a PD-like fuzzy logic controller is shown in Figure 4.8. 20 ( )iF siU UK iuDefuzzi- fication Evaluation of control rules Fuzzifi- cation eK deK iace idace iACE idACE , Calculate ( )tie iP s Compute ACE &i idACE Rule base PD-BASED FUZZY LOGIC CONTROLLER Database , ( )L iP s ( )kF s ( )iF s CONTROL-AREA MODELi , ( )tie iP s , ( )tie iP s ( )iF s Figure 4.8. Structure of a PD-based FL controller for the area i. 4.4. Numerical simulation results and discussion - Case 1: Let load variation only appear randomly in the most important generating station (area 5) as shown in Figure 4.11(a). In this case, the step-load will increase at instants of 0(sec), 5(sec) and 10(sec) with the corresponding magnitudes of 0.005(p.u.), 0.01(p.u.) and 0.02(p.u.). Such step-load will decrease at instants of 15(sec) and 20(sec) to the amplitudes of 0.01(p.u.) and 0.05(p.u.), respectively. It would be consistent with an actual condition of daily load in a power plant. - Case 2: Let load variations occur at all generating areas in the power system model built earlier as shown in Figure 4.11(b). Load in the area #5 is the same of the first case, meanwhile loads in the other areas are embedded at different instants and amplitudes. 4.4.1. Capability of PD-based fuzzy logic controllers The maximum overshoots as well as the settling times of PD- based fuzzy logic controllers are much smaller than PI regulators. As a result, the control performances of the proposed controllers are much better than those of the conventional PI regulators for 21 conducting the maintenance of the network frequency against the load disturbances. Since load change only appears in the fifth area, the frequency of this station will be affected with the biggest deviation (the highest overshoot and the longest settling time). It means that the control indexes in this area are at the worst. 4.4.2. Effectiveness of the SMES devices Figure 4.11. Two load changes applied for simulation purpose 0 5 101520 30 40 50 0 0.005 0.01 0.015 0.02 0.025 Time (sec) (a) Lo ad v ar ia tio n (p .u .) A#5 0 5 101520 30 40 50 0 0.005 0.01 0.015 0.02 0.025 Time (sec) (b) Lo ad v ar ia tio n (p .u .) A#1 A#2 A#3 A#4 A#5 22 Figure 4.12. Frequency deviations in the areas #1, #4 and #5 in the first simulation case Figure 4.13. A comparison of maximum overshoots (absolute values) and settling times for all areas in the first simulation case. 0 10 20 30 40 50 -0.03 -0.02 -0.01 0 0.01 Time (sec) (a) F re q B ia s ( p .u .) A#5-PI A#5-FLC 0 10 20 30 40 50 -15 -10 -5 0 5 x 10 -3 Time (sec) (b) F re q B ia s ( p .u .) A#1-PI A#4-PI A#1-FLC A#4-FLC PI FLC 0 0.005 0.01 0.015 0.02 0.025 Type of Controller (a) M a x im u m o v e rs h o o ts ( p .u .) A#1 A#2 A#3 A#4 A#5 PI FLC 0 5 10 15 20 25 30 35 40 Type of Controller (b) S e tt lin g t im e s ( s e c ) A#1 A#2 A#3 A#4 A#5 23 4.5. Conclusion for Chapter 4 From this Chapter, it is well known the following conclusions can be drawn. - LFC is the core of an automatic generation control, playing an important role to ensure the stability and economy of a practical large-scale interconnected power plant. - PD-based fuzzy logic controllers can solve efficiently the issue of LFC, achieving the better control performances compared with conventional regulators, such as PI. - SMES units can be used in combination with efficient controllers to obtain the optimal control properties, such as the lowest overshoots and the smallest settling times. - Due to the complexity of a large and modern power system in practice, it is necessary to model more exactly such electric grid to deal with LFC problem by applying the improved control strategies as mentioned in this study. Finally, since loads of a power system, which depend only on customers, can vary randomly and continually over time, they should be measured to create the fully practical database for necessary control schemes. This might be of interest for our work in the future. CHAPTER 5: CONCLUSION AND DISCUSSION 5.1. Evaluation of research findings Study on large-scale systems under nonlinearities and uncertainties is a difficult task. Because of the high complexity and high order of the large-scale system, the calculation for analysis of the system rapidly increases. Moreover, the increasing effect of 24 uncertain and random disturbances make the design of control strategies more challenging. In this thesis, based on the existed research results for the large- scale systems, the author proposed a study focusing on stability solution for a multi-machine electric power grid, which is considered to be a typical example of the large-scale systems. To deal with this problem, the following tasks have been conducted: a) Modeling an interconnected power system under uncertainties. b) Proposed and designed a decentralized control using modified Riccati equations to stabilize the network, thereby the control method is able to quickly extinguish transient fluctuations caused by disturbances in the electric power grid. c) Presented a procedure to design a hybrid control scheme as an effective integration of PD-type fuzzy logic controllers and SMES devices for the load-frequency control. The random and continuous load variations in a power network strongly affect on the system frequency as well as the tie-line power flow. Applying the proposed control method in this study both the system frequency and the tie-line power flow can be maintained at the nominal and scheduled values, thus the stability and reliability of the grid can be guaranteed. With the proposed control s