Microsoft Word - 18-2970_sx2_ETASR_V9_N5_pp4673-4678_2 Engineering, Technology & Applied Science Research Vol. 9, No. 5, 2019, 4673-4678 4673 www.etasr.com Nechadi: Adaptive Fuzzy Type-2 Synergetic Control Based on Bat Optimization for Multi-Machine … Adaptive Fuzzy Type-2 Synergetic Control Based on Bat Optimization for Multi-Machine Power System Stabilizers Emira Nechadi Ferhat Abbas Setif 1 University Setif, Algeria emira.nechadi@univ-setif.dz Abstract—A new, adaptive, fuzzy type-2 fast terminal, synergetic multi-machine power system stabilizer is proposed in this study, based on the Bat algorithm. The time spent to reach the equilibrium point, from any initial state, is guaranteed to be finite. The adaptive fuzzy type-2 design is applied to estimate the unknown functions of a multi-machine power system. The parameters of the fast terminal synergetic control are optimized, using bat metaheuristic method. In order to test the robustness of the proposed stabilizer, three load conditions, of the multi- machine power system are studied. A comparison of the proposed adaptive fuzzy type-2 synergetic power system stabilizer with bat conventional approach is presented, indicating improved performance. The control system stability is assessed by the second theorem of Lyapunov and is proven to be asymptotically stable. Keywords-adaptive fuzzy type-2 design; fast terminal synergetic control; bat algorithm; Lyapunov stability; power system stabilizer I. INTRODUCTION A power system must remain stable and capable of withstanding a wide range of disturbances, in order to provide secure and reliable services. In a power system, the active power depends on the phase angle between the sending and receiving-end voltages, whereas the reactive power depends on the voltage magnitudes. A dynamic model of the system can be described, by the relationships between active and reactive powers and the bus voltage and frequency [1]. In a stable power system, when synchronous generators are subjected to a disturbance, they either return quickly to their original state or to a new stable operating point. Disturbances cause mechanical oscillations, which must be damped [2]. Power systems are complex nonlinear systems that often exhibit low frequency oscillations, due to insufficient damping caused by adverse operating conditions, which can lead the underlying machine to lose synchronism [3]. Power System Stabilizers (PSS) are designed to suppress these oscillations and improve overall stability by applying supplementary control through the excitation controller (AVR) [4]. Conventional PSS, consisting of cascade connected lead–lag compensators derived from a linearized model of the power system around a certain operating point have long been used to damp oscillations, regardless of the varying loading conditions or disturbances. However PSS control strategies based on linear models often fail to provide satisfactory results over a wide range of operating conditions [5]. Authors in [4, 5] presented a comprehensive approach for tuning the conventional PSS parameters and their effect on the dynamic performance of the power system. However PSS designed to damp one single oscillation mode can produce adverse effects in other modes. Several PSS design techniques have been reported [6, 7]. Pole placement or eigenvalues methods are used in [8-11]. Classical optimization techniques failed to provide optimum PSS parameters [12]. Heuristic techniques, such as Genetic Algorithms (GA), have already been applied to PSS design [13]. A particle swarm optimization (PSO) algorithm has been used in [14], for optimizing PSS parameters. Research has been conducted in optimization using Bat algorithm in [15, 16]. Optimization of PSS parameters, based on Bat algorithm, has been also reported [17-19]. Recently, a new synergistic control scheme, which combines control theory with heuristic optimization and computational intelligence methods, has emerged [20-22]. This study proposes an adaptive fuzzy type-2 fast terminal synergetic power system stabilizer. Fast terminal’s synergetic control parameters are determined, using the Bat optimization method, as shown in [23, 24]. Adaptive fuzzy type-2 design is used to approximate unknown functions in the multi machine power system model. II. FAST TERMINAL SYNERGETIC CONTROL In this section, the Fast Terminal Synergetic (FTSYN) controller is developed for the following nonlinear single- input/single-output (SISO) system: ( ) ( )   += = utxgtxfx xx ,, 2 21 � � (1) where [ ] 2 21 Rxxx T ∈= is the state vector, while ( )txf , and ( )txg , are unknown functions. In order to obtain the terminal convergence of the state variables, the following macro- variable is defined as a function of the state variables: 1 1 1 x x x λα βΨ = + +� (2) Corresponding author: Emira Nechadi Engineering, Technology & Applied Science Research Vol. 9, No. 5, 2019, 4673-4678 4674 www.etasr.com Nechadi: Adaptive Fuzzy Type-2 Synergetic Control Based on Bat Optimization for Multi-Machine … where α, β are positive constants. With a proper choice of λ, α, β and given an initial state ( ) 00 1 ≠x , the dynamics of the macro variable will reach the equilibrium point in a finite time. The exact time to reach zero s t , is determined by: ( ) ( ) 1 1 01 ln 1 s x t λ α β α λ β − + =    −   (3) and the equilibrium point at 0 is a terminal attractor. Introducing the typical constraint (4), the selected macro- variable is forced to evolve in a desired manner, despite the uncertainties and/or disturbances: 0,0 >=Ψ+Ψ ss TT � (4) where s T is a parameter to be chosen determining the rate of convergence to the attractor and can be made arbitrary small, considering only eventual control constraint. Using (2) and (4), the macro-variable derivative is given as: Ψ−=++ − s T xxxx 11 1212 λβλα�� (5) The fast terminal synergetic control is: ( ) ( )       Ψ+++−= −− s T xxxtxftxgu 1 ,, 1 122 1 λβλα (6) To prove the stability of fast terminal synergetic control, consider the following candidate Lyapunov function: ΨΨ= TV 2 1 (7) Therefore: ΨΨ= �� TV (8) ( ) ( )( )1 122 ,, −+++Ψ= λβλα xxxutxgtxf (9) 21 Ψ−= s T (10) Then: 0≤ i V� (11) III. DESIGN OF ADAPTIVE FUZZY TYPE 2 SYNERGETIC CONTROL Control law (6) ensures system stabilization and robustness, but it cannot be directly implemented, since functions ( )txf , and ( )txg , are not known. This can be overcome by approximating functions by two interval type-2 fuzzy adaptive systems. A fuzzy system that uses type-2 fuzzy sets and/or fuzzy logic and inference, is called a type-2 fuzzy system [25]. Based on the universal approximation theorem, unknown functions ( )txf , and ( )txg , can be approximated by: ( ) ( ) f T f xxf θξθ =,ˆ (12) ( ) ( ) g T g xxg θξθ =,ˆ (13) where [ ] m21 ,...,θ,θθθ = is the parameters’ vector, [ ]T m21 ,...,ξ,ξξξ = is the vector of Fuzzy Basis Functions (FBF), such that: [ ][ ] flfr T l T rf T θθξξθξ 2 1 = (14) [ ][ ] glgr T l T rg T θθξξθξ 2 1 = (15) where 1 2 T m l l l l ξ ξ ,ξ ,...,ξ =   , 1 2 T m r r r r ξ ξ ,ξ ,...,ξ =   , [ ]r 1r 2r mrθ θ ,θ ,...,θ= , and [ ]l 1l 2l mlθ θ ,θ ,...,θ= . This yields the minimum approximation error: u gf δδε += (16) where: ( ) ( ) * f T f xxf θξδ −= (17) ( ) ( ) * g T g xxg θξδ −= (18) and * f θ , * g θ are the optimal approximation parameters by letting: *~ fff θθθ −= (19) *~ ggg θθθ −= (20) The following control law: ( ) ( )       Ψ+++−= −− sT xxxtxftxgu 1 ,ˆ,ˆ 1 122 1 λβλα (21) under the adaptation laws: ( ) ff x θγξγθ 11 −Ψ=� (22) ( ) gg x θγξγθ 22 −Ψ=� (23) ensures the stability of the nonlinear system (1). The Lyapunov function is chosen as: g T gf T f V θθ γ θθ γ ~~ 2 1~~ 2 1 2 1 21 2 ++Ψ= (24) Therefore : ( ) ( ) 1 2 1 1 1 T T f g s T T f f g g V x x u T θ ξ θ ξ ε θ θ θ θ γ γ   = Ψ − − + − Ψ    + + � �� � � � � (25) Using (22) and (23): Ψ+−−Ψ−= εθθθθ g T gf T f sT V ��� ~~1 2 (26) and given the following inequalities being valid: 2 * 2 1~~ 2 1~ ff T ff T f θθθθθ +−≤− (27) 2 * 2 1~~ 2 1~ gg T gg T g θθθθθ +−≤− (28) Engineering, Technology & Applied Science Research Vol. 9, No. 5, 2019, 4673-4678 4675 www.etasr.com Nechadi: Adaptive Fuzzy Type-2 Synergetic Control Based on Bat Optimization for Multi-Machine … V� can be written as: 2 2 2 * * 1 1 1 2 2 1 1 2 2 T T f f g g s f g V T θ θ θ θ θ θ ε   ≤ − Ψ + + +    + + Ψ � � � �� (29) using: ( )222 2 1 22 1 εεε s s s s T T T T −Ψ−+Ψ=Ψ (30) then: 2 2 2 2* * 1 1 1 2 2 2 1 1 2 2 2 T T f f g g s s f g V T T θ θ θ θ θ θ ε   ≤ − Ψ + + +    + + � � � �� (31) and       +=       = 2 * 2 * 21 2 1 2 1 ,, 1 min gf sT θθµ γγα (32) Finally: 2 2 εµα s T VV ++−≤� (33) Integrating (33) from 0 to t, yields: ( ) ( ) ( ) ( )0 2 0 2 0 Vtd T dVtV t s t +++−≤ ∫∫ µττεττα (34) Terms ( ) ττα dV t ∫ 0 and ( ) ττε dT t s ∫ 0 2 2 are bounded. It can be concluded that Ψ and Ψ� are bounded ( ∞∈Ψ L and ∞∈Ψ L� ). Sinceε , f θ ~ and g θ ~ are bounded, hence )(tV is also bounded, guaranteeing the stability of the closed loop system. To optimize the synergetic parameters βα,, s T and λ, a fuzzy synergetic approach using Bat algorithm is employed. The typical values of the optimized parameters are taken as [ ]2.005.0 − for s T , [ ]300100− for α , [ ]15050− for β and [ ]101− for λ . IV. BAT ΟPTIMIZATION ΑLGORITHM Bat algorithm is a relatively new meta-heuristic optimization method [23]. This algorithm exploits the so-called echolocation of bats. Bats use sonar echoes to detect and avoid obstacles. They navigate by emitting high frequency sounds waves and detecting the time delay of the reflected waves. From the detected time delay, bats know how far away they are from the prey or the obstacle [25]. The algorithm, with the use of random walks (one solution is selected among the current best solutions and then the random walk is applied to generate a new solution for each bat), is presented in detail in [23, 24]. Applying the Bat optimization algorithm and after the optimization procedure, we find Ts=0.1000, α=240, β=140 and λ=3. V. POWER SYSTEM MODEL The nonlinear power system model considered in this paper represents a synchronous machine connected to an infinite bus via a double circuit transmission line. A nonlinear representation of the power system, considered during a transient period after a major disturbance has occurred, is given by [18, 19]: ( ) ( )   +=∆ ∆=∆ uxgxfMP MP � �ω (35) where ω∆ is the speed deviation, em PPP −=∆ the accelerating power, M is the inertia coefficient, Ru ∈ is the input, ( )xf and ( )xg are nonlinear functions and ( ) 0≠xg in the controllability region. The block diagram of a conventional lead-lag power system stabilizer is shown in Figure 1. Tw is the wash out time constant, T1i-T4i are the PSS time constants and Ki is the PSS parameter of generator i. The optimal parameters of the conventional PSS are obtained by BAT method and are listed in Table II [19]. Fig. 1. Conventional power system stabilizer VI. SIMULATION RESULTS To proof the robustness and effectiveness of the proposed optimal fuzzy synergetic PSS, simulations were carried out under different operating conditions of the multi-machine power system. To demonstrate the stability enhancement achieved with the proposed stabilizer, a three-phase fault test is applied at bus 7 of the multi-machine power system, with duration of 60ms before its clearance. Seven fuzzy sets were used for each variable of the proposed PSS. The fuzzy sets for [ ]55.0.5730,2−∈∆P and [ ]1.81430.1086,-∈Q are defined according to the membership functions shown in Figures 2 and 3 respectively. Three disturbance scenarios have been considered in the simulation, in order to test the robustness of the proposed control scheme. Table I describes these three cases. In each case, the proposed stabilizer is compared with a BAT CPSS and an AFT2 SYNPSS. Engineering, Technology & Applied Science Research Vol. 9, No. 5, 2019, 4673-4678 4676 www.etasr.com Nechadi: Adaptive Fuzzy Type-2 Synergetic Control Based on Bat Optimization for Multi-Machine … TABLE I. CASES OF LOADING CONDITIONS FOR THE SYSTEM (PU) Generator G1 G2 G3 Light case P 0.9649 1.00 0.45 Q 0.223 -0.1933 -0.2668 Normal case P 1.7164 1.630 0.85 Q 0.6205 0.0665 -0.1086 Heavy case P 3.5730 2.20 1.35 Q 1.8143 0.7127 0.4313 Fig. 2. Fuzzy sets for speed deviation Fig. 3. Fuzzy sets for accelerating power TABLE II. BAT CONVENTIONAL POWER SYSTEM STABILIZER PARAMETETERS K T1 T3 BATPSS1 46.6588 0.4153 0.2698 BATPSS2 8.4751 0.4756 0.1642 BATPSS3 4.2331 0.2513 0.1853 The three-machine test system, used to examine the inter- area oscillation control problem, is shown in Figure 4 for light load case, in Figure 5 for nominal load case and in Figure 6 for a heavy load case. (a) (b) (c) Fig. 4. Light load scenario (a) (b) (c) Fig. 5. Normal load scenario Engineering, Technology & Applied Science Research Vol. 9, No. 5, 2019, 4673-4678 4677 www.etasr.com Nechadi: Adaptive Fuzzy Type-2 Synergetic Control Based on Bat Optimization for Multi-Machine … (a) (b) (c) Fig. 6. Heavy load scenario The proposed stabilizer keeps the generator synchronized. Three studies have been performed to investigate the effect of the proposed BAT AFT2 SYNPSS. The results are compared with a BAT CPSS and an AFT2 SYNPSS. 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