Robust Neural Controllers for Power System Based on New Reduced Models

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE Robust Neural Controllers for Power System Based on New Reduced Models Wissem BAHLOUL1, Mohamed CHTOUROU 2, Mohsen BEN AMMAR 2, Hsan HADJABDALLAH 2 1Biomedical Department, Higher Institute of Biotechnology of Sfax, University of Sfax, Street Soukra, 3029 Sfax, Tunisia 2Department of Electrical Engineering, National Engineering School of Sfax, University of Sfax, Street Soukra, 3029 Sfax, Tunisia [email protected], mohamed.ch[email protected]u.tn, mohsen.b[email protected], hsan.ha[email protected]u.tn DOI: 10.15598/aeee.v21i2.4690 Article history: Received Sep 01, 2022; Revised Jan 14, 2023; Accepted Mar 27, 2023; Published Jun 30, 2023. This is an open access article under the BY-CC license. Abstract. This paper presents an advanced control method for the stabilization of Electric power systems. This method is a decentralized control strategy based on a set of neural controllers. Essentially, the largescale power system is decomposed into a set of subsystems in which each one is constituted by a single machine connected to a variable bus. For each subsystem, a neural controller is designed to respond to a performance index. The neural controller is a feed-forward multi-layered one. Its training method is accomplished for different rates of desired terminal voltage and is based on the perturbed electrical power system model. For a single machine, the synaptic weights of corresponding neural controller are adjusted to force the machine outputs to converge into expected one obtained by the load flow program. To evaluate the performance and effectiveness of the proposed control method, it has been applied to the WSCC power system under severe operating conditions. The obtained results compared to the ones of conventional controllers proved the high quality of the proposed controller in terms of transient stability and voltage regulation of the considered electrical power system. Keywords Decentralized control, mathematic model reduction, neural controller, power system. 1. Introduction To maintain the interconnected power system stability under several disturbances, robust excitation controllers are used for the different generators [1] and [2]. These controllers require the entire power system state variables to be available at one central computing station in order to make centralized controller law [3] and [4]. However, several researchers have pointed out that the implementation of a centralized controller possesses certain difficulties, particularly when the complexity of the interconnected area increases [5], [6], [7] and [8]. More importantly, the centralized control poses a heavy risk of instability if a loss or retard of information occurs in their communication system. In this respect, a decentralized control approach is used [9], [10], [11], [12], [13], [14], [15], [16], [17] and [18]. The advantage of the decentralized control applications does reduce the complexity of the system model and allows more feasible control implementation. Several design approaches have been proposed and successfully applied to improve the transient power system stability. In [9], based on the LMI approach, a decentralized controller is designed. In [10], a H∞controller for Single Machine Infinite Bus (SMIB) system is proposed. A decentralized controller is designed for a varying system model as presented in [11]. However, most of these works just employ the linearized power system model for designing these controllers. Generally, the SMIB system is simplified and linearized as a single machine connected to an infinite bus model under normal operating conditions. It is obvious that ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 107 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE when major or inter-zone faults occur, the behaviour of the power system may change significantly. Therefore, the conventional linear controllers are not sufficient to guarantee the system stability under these circumstances. In recent years, in order to improve the transient stability, much more attention has been paid to power system control, using the latest developed nonlinear control theory [19], [20] and [21]. Taking the other view into consideration, the modelling of the whole Electric Power System (EPS) for the decentralized control represents a challenge for many recent researchers [22], [23], [24] and [25]. In order to enhance the stability of the interconnected nonlinear power systems within the whole operating region, the design of the decentralized nonlinear controllers still remains a challenging task for researchers in this area of study. Different techniques of nonlinear controller are proposed in the literature and the learning capacities of the nonlinear neural controller [26], [27], [28], [29] and [30] for a system with complex nonlinear behaviour as well as for an extended domain of operation is a potential solution for designing a new controller. Accordingly, a neural control, which is proposed by some researchers in the last 40 years has been continuously evolving until today. In 1990 [26], Nguyen and Widrow introduced a new schema of neural control with a full state feedback and an emulator for designing a dynamic system. The backpropagation algorithm and the specialized training method are operated in this paper. However, authors in [27] suggested another schema of neural control with output feedback. They investigated two types of training methods; generalized and specialized. Jadlovsa published a paper in 2000 [28] that focuses on the control structure for target and trajectory tracking by neuro-controller, which drives a dynamical system from an initial condition to given finite states. In some other papers [29], [30], [31], [32] and [33], a novel intelligent system with adaptive control architecture for power system is designed. It mainly contains two neural networks: one as a controller and another as an identifier. The last system is trained with extensive test data offline in addition to an adjusted online one. In the current investigation, a new robust decentralized neural controller is proposed and developed, which uses only local information for a large interconnected nonlinear power system. Accordingly, two stages are actually to pursue: in the first, the whole interconnected power system is decomposed into independent sub-systems founded on load flow program results and new reduction method. Then, the multimachine power system model is transformed into independent reduced models designed for each generator separately. Each model consists of a single generator connected to a variable Bus. The last Bus is designed by three new parameters (Ri,Xiand Vsi). By adapting the parameters values in the healthy and faulty system states, the proposed independent dynamic model for each generator makes it possible to design three topologies for each sub-system states (before fault, during fault and after fault). In the second stage, each sub-system designed with reduced model is controlled using Decentralized Neural Controller (DNC) techniques. The considered neural controller has been nominated as robust one because it has been trained for different fault types occurring in the power system. For these different faults, different mathematical models representing the faulted multi-machine power system have been considered for the training of the neural controller. Seizing the performances of the neural controller offered by authors in [27], the terminal voltage of each generating buses is oriented to this aimed value. For the training of the DNC, the specialized learning architecture is used to specifically learn in the region of interest. In this case, no stability problems for the control system are noticed [26] and [27]. This paper is structurally written as follows. Section 2. details the reduction method of the EPS. The DNC design is introduced in Sec. 3. In order to validate the proposed controller, simulation results are provided in Sec. 4. Lastly, the conclusion is given in Sec. 5. 2. Reduction Method of the Interconnected Power System Model The fame multi-machine model of interconnected power system is complex and multivariable [1] and [2]. A new reduction method is proposed in order to synthesize a robust decentralized neural controller consisting in simplifying the complexity of the model and reducing the number of numerous variables. 2.1. Aggregation of the Power System Producer Nodes The EPS is designed for n nodes, which are classified into two types: the nodes producing Ngnumbers and the others consuming NcNumbers. It is worth noticing that the nodes total number is equal to N=Ng+Nc. The expression of the injected current ¯ Iiin each node of the power system is expressed by the following expression: ¯ Ii= N X j=1 ¯ YNij ¯ Vj.(1) ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 108 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE Thus, the matrix writing of the current is given by: ¯ I=¯ YN¯ V , (2) where ¯ YNdenotes the nodal matrix of dimension (N×N). By decomposing the vector ¯ Iinto two subones, each is corresponding to the currents injected in the load nodes and to the number Nc. The currents injected in the generator nodes correspond to the number Ng. Accordingly, the matrices ¯ YNand ¯ Vwill be decomposed by the same manner. In this respect, the last matrix Eq. (2) turns up into the following form: ¯ I=  −¯ Icc . . . . . . . . . . . . . . . . . . ¯ Ig−¯ Icp  =     ¯ Ycc . . .¯ Ycp . . . . . . . . . . . . . . . . . . ¯ YT cp . . .¯ Ypp      ¯ Vc . . . . . . . . . ¯ Yp  ↓Nc − − ↑Ng . (3) The load current is expressed as a function of bus voltage ¯ Yi, real and reactive power (Pci and Qci) obtained by Load Flow Program (LFP): ¯ Ici =¯ Yci ¯ Vi,(4) where ¯ Yci =Pci +jQci V2 i . Thus, the last current can be written in the following matrix form: ¯ Ic=¯ Yc¯ V . (5) Similarly to the above decomposition, the vectors ¯ Ic is expressed as follows: ¯ Ic=  ¯ Icc . . . . . . . . . ¯ Icp  = =    ¯ Yc . . .0 . . . . . . . . . . . . . . . . . . 0. . .¯ Yp     ·  ¯ Vc . . . . . . . . . ¯ Yt , (6) where the sub-vectors ¯ Vcand ¯ Vtdefine the voltages at the consuming nodes and at the terminals of the generators connected to the EPS, respectively. Indeed, the sum of the two vectors of the injected currents ¯ I given by Eq. (3) and the load current ¯ Icgiven by Eq. (6) provides the following Eq. (7):   0 . . . . . . . . . ¯ Ig = =    ¯ Ycc . . .¯ Ycp . . . . . . . . . . . . . . . . . . ¯ YT cp . . .¯ Ypp     ·  ¯ Vc . . . . . . . . . ¯ Yt , (7) which makes it possible to express the generator nodes current vector ¯ Igby the following matrix equation: ¯ Ig=¯ Yp¯ Vt,(8) where ¯ Yp=¯ Y′ pp −¯ YT cp(−¯ Y′ cc)−1¯ Ycp, which is the reduced admittance matrix. Its dimension has been reduced from (N×N) to (Ng×Ng). This procedure, therefore, is helpful to switch from a grid of N=Ng+Ncnodes to another reduced one of Ngproducer nodes. Hence, the interconnected power system aggregated by the Ngproducer nodes is schematized in Fig. 1. Fig. 1: The interconnected power system aggregated by the 𝑁𝑔 producer nodes. Fig. 2: Reduced model for power system used for decoupling of the ith generator. Fig. 3: Blondel diagram of one machine connected to the power system . Fig. 4: The control structure scheme corresponding to each generator of the EPS. 𝛿 𝑖 𝐼𝑔𝑖 𝐼 𝑞𝑖 𝑉𝑞𝑖 𝑗𝑋 𝑖 𝐼 𝑔𝑖 𝑗𝑋 𝑑𝑖 ′𝐼 𝑔𝑖 𝑗𝑋 𝑞𝑖 𝐼 𝑔𝑖 𝐸𝑞𝑖 𝑉 ത 𝑡𝑖 𝑉 ത 𝑠𝑖 𝐸 𝑖 ′ 𝑅 𝑖 𝐼 𝑔𝑖 𝑋 𝑞𝑖 𝐼 𝑑𝑖 𝑋𝑑𝑖 ′𝐼𝑑𝑖 𝑋𝑖𝐼𝑑𝑖 𝐸 𝐼𝑖 (𝑋 𝑑𝑖 − 𝑋 𝑞𝑖 )𝐼 𝑑𝑖 𝑉 𝑡𝑞𝑖 𝑉𝑑𝑖 𝐼 𝑑𝑖 𝐸 𝑞𝑖 ′ 𝑅 𝑖 𝐼 𝑞 𝐼ҧ 𝑔1 𝐼ҧ 𝑔𝑁𝑔 Transmission system [Yp] 𝑉𝑡𝑖 𝐼ҧ 𝑔2 𝐺1 𝐺2 𝐺𝑁𝑔 . . . 𝐼𝑔𝑖 𝑋𝑖 𝑅𝑖 𝑉𝑠𝑖 𝐺𝑖 𝑉𝑡𝑖 Rest of the EPS 𝑃 𝑒𝑖 𝑄𝑒𝑖 G1 NC1 G2 NC2 G3 NC3 SS3 SS2 SS1 Fig. 1: The interconnected power system aggregated by the Ng producer nodes. 2.2. Decoupling of Each Generator as Single Machine Sub-system For decoupling each generator in independent subsystem, the matrix expression (Eq. (8)) can be inverted to formulate the vector of the voltages at the generating nodes: ¯ Vt=¯ Zp¯ Ig,(9) where ¯ Zp=¯ Y−1 p. Based on Eq. (9), the terminal voltage of the ith node generator can be transformed regarding impedances and injected currents by: ¯ Vti = Ng X j=1 ¯ Zij ¯ Igj =¯ Zii ¯ Igi + Ng X j=1 j=i ¯ Zij ¯ Igj.(10) The injected current of each generator node ¯ Igj considering the equation is possibly expressed as follows: ¯ Igj = Ng X k=1 ¯ Yjk ¯ Vtk.(11) ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 109 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE Replacing the expression ¯ Igj in Eq. (10), the terminal voltage of the ith node generator is obtained by: ¯ Vti =¯ Zii ¯ Igi + Ng X j=1 j=i ¯ Zij Ng X k=1 ¯ Yjk ¯ Vtk = =¯ Zii ¯ Igi + Ng X j=1 j=i ¯ Zij Ng X k=1 k=i ¯ Yjk ¯ Vtk +¯ Vti Ng X j==1 j=i ¯ Zij ¯ Yji, (12) ¯ Vti   1−Ng j=1 j=i ¯ Zij ¯ Yji  =¯ Zii ¯ Igi + Ng X j=1 j=i ¯ Zij Ng X k=1 k=i ¯ Yjk ¯ Vtk, (13) we pose: µi=1 1− Ng X j=1 j=i ¯zij ¯γji , then ¯ Vti =µi¯ Zii ¯ Igi +µi Ng X j=1 j=i ¯ Zij Ng X k=1 k=i ¯ Yjk ¯ Vtk. Basing on Eq. (12) and Eq. (13), the ith generator node terminal voltage is solely represented with an equivalent impedance and voltage, as follow: ¯ Vti =¯ Zi¯ Igi +¯ Vsi,(14) with: ¯ Vsi =µi Ng X j=1 j=i ¯ Zij Ng X k=1 k=i ¯ Yjk ¯ Vtk. In the same context, we pose: Vsi=|¯ Vsi|,(15) Ri=real(¯ Ri),(16) Xi=Imag(¯ Zi).(17) Consequently, using Eq. (14) allows us to model the power system as an independent Nggenerator connected to the rest of the grid. Hence, the rest of the EPS is assimilated by an impedance ¯ Ziin series with a voltage source ¯ Vsi as shown in Fig. 2 which represents new mathematic reduction model for power system. The parameters rates (Ri,Xiand Vsi) for any steady state are calculated by Eq. (15), Eq. (16) and Eq. (17): this means that each generator has an aggregate view of the remaining Ng−1generators equivalent to a variable parameter bus (see Fig. 2). For the study and control of the transient regime, the generator Giwill be modeled by its adequate dynamic model. Fig. 1: The interconnected power system aggregated by the 𝑁𝑔producer nodes. Fig. 2: Reduced model for power system used for decoupling of the ith generator. Fig. 3: Blondel diagram of one machine connected to the power system. Fig. 4: The control structure scheme corresponding to each generator of the EPS. 𝛿 𝑖 𝐼𝑔𝑖 𝐼 𝑞𝑖 𝑉𝑞𝑖 𝑗𝑋 𝑖 𝐼 𝑔𝑖 𝑗𝑋 𝑑𝑖 ′𝐼 𝑔𝑖 𝑗𝑋 𝑞𝑖 𝐼 𝑔𝑖 𝐸𝑞𝑖 𝑉 ത 𝑡𝑖 𝑉 ത 𝑠𝑖 𝐸 𝑖 ′ 𝑅 𝑖 𝐼 𝑔𝑖 𝑋 𝑞𝑖 𝐼 𝑑𝑖 𝑋𝑑𝑖 ′𝐼𝑑𝑖 𝑋𝑖𝐼𝑑𝑖 𝐸 𝐼𝑖 (𝑋 𝑑𝑖 − 𝑋 𝑞𝑖 )𝐼 𝑑𝑖 𝑉 𝑡𝑞𝑖 𝑉𝑑𝑖 𝐼 𝑑𝑖 𝐸 𝑞𝑖 ′ 𝑅 𝑖 𝐼 𝑞 𝐼ҧ 𝑔1 𝐼ҧ 𝑔𝑁𝑔 Transmission system [Yp] 𝑉𝑡𝑖 𝐼ҧ 𝑔2 𝐺1 𝐺2 𝐺𝑁𝑔 . . . 𝐼𝑔𝑖 𝑋𝑖 𝑅𝑖 𝑉𝑠𝑖 𝐺𝑖 𝑉𝑡𝑖 Rest of the EPS 𝑃 𝑒𝑖 𝑄𝑒𝑖 G1 NC1 G2 NC2 G3 NC3 SS3 SS2 SS1 Fig. 2: Reduced model for power system used for decoupling of the ith generator. 2.3. Dynamic Model of Synchronous Generator The detailed nonlinear model of a synchronous generator is a seventh-order model while the popular thirdorder model is of crucial interest for studying control systems of the generator as well as their stability analysis [1] and [2]. Therefore, the detailed nonlinear model is usually reduced to a generalized nonlinear thirdorder model [1]. It is expressed per unit as follows:          dδi(t) dt =ωi(t), dωi(t) dt =−KDi 2Hiωi(t) + ωs 2HI(Pmi(t)−Pei(t)) , dE′ qi(t) dt =1 T′ d0i(t)Efdi(t)−E′ qi + (Xdi −X′ di)Idi, (18) where δiis the generator rotor angle, ωiis the difference between the generator angular speed and the synchronous angular speed, E′ qi is the transient EMF in quadrature axis qand Pei is the electrical power. Pmi and Efdi are the two inputs of the system corresponding to the mechanical power and the excitation control voltage, respectively. KDi,Hiand T′ d0iare the dumping constant, the inertia constant and the excitation circuit time constant, respectively. The Blondel diagram corresponding to the synchronous generator using to calculate the algebraic electrical equations is presented by Fig. 3 [1]. In this diagram, the terminal voltage of the ith generator is expressed by the sum vector: ¯ Vti=¯ Vsi +Ri¯ Igi +jXi¯ Igi. Referring to the Blondel diagram, the algebraic electrical equations could be calculated as: Pei =EqiIqi,(19) Eqi =E′ qi + ∆XdiIdi,(20) Idi =XqsiE′ qi −(XqsiVqi +RiVdi) R′ xi ,(21) Iqi =RiE′ qi +Xdsi′Vdi −RiVqi) R′ xi ,(22) where Vqi =Vsi cos(δi);Vdi =Vsi sin(δi);X′ dsi =X′ di + Xi;Xdsi =Xdi +Xi;Xqsi =Xqi +Xi;Rxi′=R2 i+ X′ dsiXqsi;Rxi=R2 i+XdsiXqsi and ∆Xdi =Xdi −X′ di. ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 110 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE Fig. 1: The interconnected power system aggregated by the 𝑁𝑔producer nodes. Fig. 2: Reduced model for power system used for decoupling of the ith generator. Fig. 3: Blondel diagram of one machine connected to the power system. Fig. 4: The control structure scheme corresponding to each generator of the EPS. 𝛿 𝑖 𝐼𝑔𝑖 𝐼 𝑞𝑖 𝑉𝑞𝑖 𝑗𝑋 𝑖 𝐼 𝑔𝑖 𝑗𝑋 𝑑𝑖 ′𝐼 𝑔𝑖 𝑗𝑋 𝑞𝑖 𝐼 𝑔𝑖 𝐸𝑞𝑖 𝑉 ത 𝑡𝑖 𝑉 ത 𝑠𝑖 𝐸 𝑖 ′ 𝑅 𝑖 𝐼 𝑔𝑖 𝑋 𝑞𝑖 𝐼 𝑑𝑖 𝑋𝑑𝑖 ′𝐼𝑑𝑖 𝑋𝑖𝐼𝑑𝑖 𝐸 𝐼𝑖 (𝑋 𝑑𝑖 − 𝑋 𝑞𝑖 )𝐼 𝑑𝑖 𝑉 𝑡𝑞𝑖 𝑉𝑑𝑖 𝐼 𝑑𝑖 𝐸 𝑞𝑖 ′ 𝑅 𝑖 𝐼 𝑞 𝐼ҧ 𝑔1 𝐼ҧ 𝑔𝑁𝑔 Transmission system [Yp] 𝑉𝑡𝑖 𝐼ҧ 𝑔2 𝐺1 𝐺2 𝐺𝑁𝑔 . . . 𝐼𝑔𝑖 𝑋𝑖 𝑅𝑖 𝑉𝑠𝑖 𝐺𝑖 𝑉𝑡𝑖 Rest of the EPS 𝑃 𝑒𝑖 𝑄𝑒𝑖 G1 NC1 G2 NC2 G3 NC3 SS3 SS2 SS1 Fig. 3: Blondel diagram of one machine connected to the power system. By referring to Eq. (19), Eq. (20), Eq. (21) and Eq. (22) and by exploiting the coefficients aji given in Tab. 1, the electric power expression is given by: Pei(t) = (a1icos(δi) + a2isin(δi) + a3iE′ qi)E′ qi+ +a4isin2(δi) + a5isin(2δi) + a0i. (23) Tab. 1: Coefficients of electrical power equation. Coefficients Quantity a0i RiXqsW iV 2 si∆Xdi R′2 x a1i −RiVsi(Rxi +Xqsi∆Xdi) R′2 xi a2i Vsi(RxiXdsi′−R2 i∆Xdi) R′2 xi a3i RiRx R′2 x a4i −Ri∆XdiV2 si(Xqsi +Xdsi′) Ri a5i ∆XdiV2 si(R2 i−Xdsi′Xqsi) 2R′2 xi Replacing Pei and Idi by their expressions, the dynamic model of the synchronous machine connected to the rest of the grid is given by the state model presented by Eq. (24):                dδi dt =ωi, dωi dt =−KDi 2Hiω−ωs 2HiPmi +a0i+E′ qi (a1icos δi+ +a2isin δi+a3iE′ qi+a4isin2δi+a5isin 2δi, dE′ qi dt =1 T′ d0ia6iE′ qi +a7isin δi+a8icos δi+Efdi, (24) where a6i=−Rxi R′ xi ;a7i=RiVsi R′ xi ∆Xdi and a8i=XqsiVsi R′ xi ∆Xdi. From another perspective, the generator terminal voltage Vti represents the measurable output of the generator. According to Blondel’s diagram, the direct and quadrature components of Vti are illustrated by: Vtdi =−XqiIqi,(25) Vtqi =E′ qi −X′ diIdi,(26) Vti =qV2 tdi +V2 tqi.(27) By replacing Idi and Iqi by their expressions in Eq. (21) and Eq. (22), we obtain the new expression for the generator terminal voltage: Vti(t) = 1 R′ xi b0i+E′ qi b1isin δi+b2icos δi+b3iE′ qi+ +b4isin2δi+b5isin 2δi0.5, (28) where the coefficients bji are given in Tab. 2 below. Tab. 2: Coefficients of terminal voltage equation. Coefficients Quantity b0iV2 si(R2 iX2 qi +X2 qsiX′2 di) b1i2VsiRi(X′ dsiX2 qi −X′2 diXqsi +R′ xiX′ di) b2i2Vsi(R′ xiX′ diXqsi −R2 iX2 qi −X2 qsiX′2 di) b3iR′2 xi + (R2 iX2 qi +X2 qsiX′2 di −2R′ xiX′ diXqsi) b4iV2 si(X2 qiX′2 dsi +R2 i(X′2 di −X2 qi) + X2 qsiX′2 di) b5iV2 siRi(X′2 diXqsi −X′ dsiX2 qi) The quantities available to act on the behavior of the power system are the mechanical input power applied to the machine shaft and the rotor excitation signal. In this design, we assume that the mechanical input power Pmi is slowly changing compared to the excitation control voltage. Thus, let Pmi =Pmi0be a positive constant and the only control input signal is u(t) = Efdi(t). The state space model of the power system presented by Eq. (24) has a nonlinear behavior, which can be given by: (˙ X=A(X(t), θ(t)) ·X(t) + B·u(t), Y(t) = h(X(t), θ(t)),(29) where X(t)∈R3denotes the vectors of the 3 states, Y(t)∈R2is column that represents the 2 outputs and θ(t)represents vector ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 111 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE variable parameters: X(t) = δi(t)ωi(t)E′ qi(t)T, Y(t)=[ωi(t)Vti(t)]T,θ(t)=[Ri(t)Xi(t)Vsi(t)]T, A=      0 1 ωs 2Hiδia4isin2δi+ a5isin 2δi−KDi 2Hi 1 T′ d0iδia7isin δi+ a8icos δi0 0 ωs 2Hia1icos δi+ a2isin δi+a3iE′ qi  a6i T′ d0i      , B=     0 0 Pm+a0 2Hi 0 01 T′ d0i      ,0 Efd(t)and hrepresents the observation equation: h= [ωi(t)Vti(t)]T. In this study, the machine model has been used in its discrete form according to Euler’s method with a sampling time Te: (X(k+ 1) = (I+Te·A)X(k) + Te·B·U, Y(k) = h(X(k), θ(k)).(30) 3. Proposed Decentralized Neural Control Strategy Besides, the nonlinearity and the complexity of the model above (Eq. (30)), the EPS can be the site of different default types in disturbance states. For this reason and exploiting the neural networks capacity of learning different functioning conditions, a neural control will be considered for the excitation command of the generators connected to the EPS. The neural control strategy inspired by [27] have shown noticeable performances and use only outputs as feedback signal. 3.1. The Neural Control Scheme The neural control scheme consists in associating a Neural Controller (NCi) with each machine, which maintains its terminal voltage round a desired one and mainly to enhance the transient stability even in the presence of severe perturbations in the EPS. For illustrative reasons, the suggested control structure for an EPS composed by three generators (Ng= 3) is presented by Fig. 4. The suggested control strategy is a decentralized one where a decentralized neural controller is trained for each single machine and different operating conditions. Two types of faults are considered: local and interzone one. Strategy requires only direct measurements from individual generators, thereby making implementation simple in practice. Fig. 1: The interconnected power system aggregated by the 𝑁𝑔producer nodes. Fig. 2: Reduced model for power system used for decoupling of the ith generator. Fig. 3: Blondel diagram of one machine connected to the power system. Fig. 4: The control structure scheme corresponding to each generator of the EPS. 𝛿 𝑖 𝐼𝑔𝑖 𝐼 𝑞𝑖 𝑉𝑞𝑖 𝑗𝑋 𝑖 𝐼 𝑔𝑖 𝑗𝑋 𝑑𝑖 ′𝐼 𝑔𝑖 𝑗𝑋 𝑞𝑖 𝐼 𝑔𝑖 𝐸𝑞𝑖 𝑉 ത 𝑡𝑖 𝑉 ത 𝑠𝑖 𝐸 𝑖 ′ 𝑅 𝑖 𝐼 𝑔𝑖 𝑋 𝑞𝑖 𝐼 𝑑𝑖 𝑋𝑑𝑖 ′𝐼𝑑𝑖 𝑋𝑖𝐼𝑑𝑖 𝐸 𝐼𝑖 (𝑋 𝑑𝑖 − 𝑋 𝑞𝑖 )𝐼 𝑑𝑖 𝑉 𝑡𝑞𝑖 𝑉𝑑𝑖 𝐼 𝑑𝑖 𝐸 𝑞𝑖 ′ 𝑅 𝑖 𝐼 𝑞 𝐼ҧ 𝑔1 𝐼ҧ 𝑔𝑁𝑔 Transmission system [Yp] 𝑉𝑡𝑖 𝐼ҧ 𝑔2 𝐺1 𝐺2 𝐺𝑁𝑔 . . . 𝐼𝑔𝑖 𝑋𝑖 𝑅𝑖 𝑉𝑠𝑖 𝐺𝑖 𝑉𝑡𝑖 Rest of the EPS 𝑃 𝑒𝑖 𝑄𝑒𝑖 G1 NC1 G2 NC2 G3 NC3 SS3 SS2 SS1 Fig. 4: The control structure scheme corresponding to each generator of the EPS. 3.2. Synthesis of the Neural Controllers In this section, the synthesis of the neural controller associated with each machine will be deeply analyzed. It is a feed forward multilayered network. Its input is constituted by the desired value of the terminal voltage at future instant (k+ 1), the actual terminal voltage and generator deviation speed. The output of the neural controller is the actual excitation voltage to be applied for the machine (see Fig. 5). Fig. 1: The three conditions steady state: before, during and after fault. Fig. 2: The training process of the neural network controller. Fig. 3: The WSCC-EPS with 3-machine and 9-bus; all impedances in p.u on a 100 MVA base. Input layer m = 0 Hidden layer m = 1 Output layer m = 2 𝜔11 0 1 𝑉 𝑡𝑖 (𝑘) 𝑉 𝑡𝑖 𝑑(𝑘 + 1) 𝜔11 1 . . . 𝑞1 1 1 𝜔𝑖(𝑘) 𝑞1 0 𝑞1 2 𝑢 𝑖 = 𝐸 𝑓𝑑𝑖 (𝑘) 𝑞2 1 𝑦𝑙(𝑘 + 1) 𝑦𝑙𝑑(𝑘 + 1) Reduced Model 𝑢(𝑘) 𝑌(𝑘) 𝑍−1 - + NCi 𝑋(𝑘 + 1) 𝑍−1 ℎ 𝑋(𝑘) G3 G1 G2 6 4 1 7 8 9 2 5 3 0.0085+j0.072 163 MW 6.7 Mvar 85 MW -10.9 Mvar 71.6 MW 27 Mvar 1.040 0.0° 1.025 9.3° 1.025 4.7° B/2=j0.0745 0.0119+j0.1008 0.010+j0.085 B/2=j0.1045 B/2=j0.088 j0.0586 j0.0625 Load A Load B Load C 0.032+j0.161 0.017+j0.092 0.039+j0.170 B/2=j0.079 B/2=j0.179 B/2=j0.153 j0.0576 Load A 125MW+j50Mvar B C 90MW+j30Mvar 100MW+j35Mvar Value Fig. 5: The three conditions steady state: before, during and after fault. 1) Neural Controller Architecture The neural controller is composed of one hidden layer and one output neuron. The numbers of hidden neurons will be fixed following learning trials. In this case, the architecture, which delivers the minimum error, is chosen. Furthermore, the most used activation function for hidden layer is that of sigmoid and for input and output layer is the linear one. The suggested 1 2 architecture of the neural controller is given by Fig. 5. ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 112 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE Referring to this architecture of the NC, the outputs of the neuron (a) at layer (m) are done by the following expression: qm a=fm a(X b ωm abqm−1 b),(31) where fm aare the activation functions. 2) Training Method In order to train an efficient neural controller, different types of eventual faults have been considered, local and inter-zone one [1]. Essentially, the equivalent circuit is specified by three different steps: before a fault, during the fault and after the elimination of the fault. It is worth noting that the values of the new parameters (Ri,Xiand Vsi) are variable and the faulted grid is considered in the steady state. For each steady state, the values of the new parameters are computed by a load flow program and reduction equations (Eq. (15), Eq. (16) and Eq. (17)) and then, updated in the single machine dynamic model which will be used in the training algorithm. The neural controller training has been accomplished based on the Fig. 6 which can be considered as a specialized training strategy [27]. The NCi is trained to find the actual SSi input u(k)that drives the system outputs Y to the desired Yd. This is accomplished by using the error between the desired and system outputs at future instant (k+ 1) to adjust the weights of the network using a descent procedure. Fig. 1: The three conditions steady state: before, during and after fault. Fig. 2: The training process of the neural network controller. Fig. 3: The WSCC-EPS with 3-machine and 9-bus; all impedances in p.u on a 100 MVA base. Input layer m = 0 Hidden layer m = 1 Output layer m = 2 𝜔11 0 1 𝑉 𝑡𝑖 (𝑘) 𝑉 𝑡𝑖 𝑑(𝑘 + 1) 𝜔11 1 . . . 𝑞1 1 1 𝜔𝑖(𝑘) 𝑞1 0 𝑞1 2 𝑢 𝑖 = 𝐸 𝑓𝑑𝑖 (𝑘) 𝑞2 1 𝑦𝑙(𝑘 + 1) 𝑦𝑙𝑑(𝑘 + 1) Reduced Model 𝑢(𝑘) 𝑌(𝑘) 𝑍−1 - + NCi 𝑋(𝑘 + 1) 𝑍−1 ℎ 𝑋(𝑘) G3 G1 G2 6 4 1 7 8 9 2 5 3 0.0085+j0.072 163 MW 6.7 Mvar 85 MW -10.9 Mvar 71.6 MW 27 Mvar 1.040 0.0° 1.025 9.3° 1.025 4.7° B/2=j0.0745 0.0119+j0.1008 0.010+j0.085 B/2=j0.1045 B/2=j0.088 j0.0586 j0.0625 Load A Load B Load C 0.032+j0.161 0.017+j0.092 0.039+j0.170 B/2=j0.079 B/2=j0.179 B/2=j0.153 j0.0576 Load A 125MW+j50Mvar B C 90MW+j30Mvar 100MW+j35Mvar Value Fig. 6: The training process of the neural network controller. The weights of each NCi are updated to minimize the mean-square error. J(k) = 1 2 2 X l=1 αl(yd l(k+ 1) −yl(k+ 1))2,(32) where the constants αlare chosen by the designer to weigh the importance of each output error component in the control process. The final desired responses are: yd 1= 0 and yd 2=Vd t, where yl(k+ 1) denotes the predicted outputs as functions of future states X(k+ 1) and the input at time instant k. Taking each distinctive iterative process into consideration, the neural controller weights are adjusted using gradient method: ωm abnew =ωm abold −γ∂J(k) ∂ωm ab ,(33) where the learning rate value γis chosen in order to fix the convergence speed of the training algorithm. ∂J(k) ∂ωm ab =δm ab 2 X l=1 yd l(k+ 1) −yl(k+ 1)∂yl(k+ 1) ∂u(k), (34) where δm ab are obtained from Eq. (31) of NCi: δ1 a1=∂u(k) ∂ω1 a1 =u(k)(1 −u(k))q1 a,(35) δ0 ab =∂u(k) ∂ω0 ab =u(k)(1 −u(k))ω1 a1q1 b1−q1 bq0 a.(36) To determine the derivative of the SSi, outputs at future instant yl(k+ 1) with respect to his actual input u(k)are obtained by using the Jacobian of discrete model Eq. (30): ∂y1(k+ 1) ∂u(k)=T2 ek11iωs[k2icos (x1(k)) + +k3isin (x1(k)) + 2k4ix3(k)] , (37) ∂y2(k+ 1) ∂u(k)=1 2y2(k)Tek11i(C2isin (x1(k)) + +C3icos (x1(k)) + 2C4ix3(k)) . (38) 4. Validation of the DNC The WSCC system [34] shown in Fig. 7 is used as an exemplar system to test the performance of the proposed decentralized neural controllers. As it is presented in Fig. 4, each neural controller NCi(i= 1,2,3) is applied to each corresponding generator Giat the same time. Within a simulated process, the proposed robust neural excitation controller is compared to linear AVR+PSS excitation one [35]. Accordingly, several sequences are studied in order to show the effectiveness of the proposed nonlinear control. Indeed, a rich database was used for the learning phase of the NC. 4.1. Learning Phase Learning is carried out for three faulted states characterized by: a step change in load at the consumer nodes (node 2, 3 and 5), a three-phase short-circuit with two different periods at different lines (∆t= 30 ms ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 113 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE Fig. 1: The three conditions steady state: before, during and after fault. Fig. 2: The training process of the neural network controller. Fig. 3: The WSCC-EPS with 3-machine and 9-bus; all impedances in p.u on a 100 MVA base. Input layer m = 0 Hidden layer m = 1 Output layer m = 2 𝜔11 0 1 𝑉 𝑡𝑖 (𝑘) 𝑉 𝑡𝑖 𝑑(𝑘 + 1) 𝜔11 1 . . . 𝑞1 1 1 𝜔𝑖(𝑘) 𝑞1 0 𝑞1 2 𝑢 𝑖 = 𝐸 𝑓𝑑𝑖 (𝑘) 𝑞2 1 𝑦𝑙(𝑘 + 1) 𝑦𝑙𝑑(𝑘 + 1) Reduced Model 𝑢(𝑘) 𝑌(𝑘) 𝑍−1 - + NCi 𝑋(𝑘 + 1) 𝑍−1 ℎ 𝑋(𝑘) G3 G1 G2 6 4 1 7 8 9 2 5 3 0.0085+j0.072 163 MW 6.7 Mvar 85 MW -10.9 Mvar 71.6 MW 27 Mvar 1.040 0.0° 1.025 9.3° 1.025 4.7° B/2=j0.0745 0.0119+j0.1008 0.010+j0.085 B/2=j0.1045 B/2=j0.088 j0.0586 j0.0625 Load A Load B Load C 0.032+j0.161 0.017+j0.092 0.039+j0.170 B/2=j0.079 B/2=j0.179 B/2=j0.153 j0.0576 Load A 125MW+j50Mvar B C 90MW+j30Mvar 100MW+j35Mvar Value Fig. 7: The WSCC-EPS with 3-machine and 9-bus; all impedances in pu on a 100 MVA base. and 100 ms) and the change of grid topology by elimination of a transmission line. With these different types of faults, the three DNCs are trained. After training trials, the best choice is set at seven neurons in a hidden layer for each neural network. Furthermore, performance is measured in terms of mean squared error. The evolution of these mean square learning errors Ji(i= 1,2,3) obtained by the gradient descent algorithm is presented by Fig. 8. It is clear that the best training performance is in average 10−4at epoch 200. 0 20 40 60 80 100 120 140 160 180 200 Iteration 0 2 4 6 8 J 10-3 NC1 NC2 NC3 Fig. 8: Evolution of the mean square learning errors. 4.2. Simulation Results To illustrate the results of Synthesized DNCs, a new default, not used in training process, will be applied to the EPS. The evolution of the state variables will be visualized at the time when the machine is controlled by the corresponding DNC. After the learning phase is already held, the weights of the different neuronal networks are fixed, which will be used to generate control laws. Indeed, simulation is performed by three sequences, which are before fault, during fault and after fault. To characterize these sequences, the steps are as follows: Step 1: the system is in a healthy mode with an initial condition: X0= [δi0ωi0E′ qi0]. Step 2: a fault is applied at t1. Step 3: Elimination of the fault at t2. Step 4: the system is in a post fault state. In order to assess the effectiveness and robustness of the proposed DNCs, two types of faults in severe operating points of conditions are made during a simulating process: it firstly consists of step changing the load; and secondly, a three-phase short-circuit. 1) Step Change in Load A of Node 2 The applied fault to the power system is the step load changes at the consumer node 2 as addressed by: at t1= 10 s (Pc = 2 PC0and Qc = 2QC0). Both faults were selected in such a way that they were not part of the neural network trained database. Figure 9 and Fig. 10 illustrate the outputs and the states of generator G1 and G3 respectively. Based on Fig. 9 and according to the proposed strategy, the ϵscorresponding to the generator terminal voltage error, is much smaller than the one obtained by the classical controller (AVR+PSS). This criterion highlights the high performance of the proposed control in voltage regulation. Nevertheless, as it is presented in Fig. 10, it is possibly accentuated that the DNC gives high dynamic and static performances by compar- ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 114 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 21 |NUMBER: 2 |2023 |JUNE 0 10 20 30 40 50 60 70 t (s) 0.8 0.9 1 1.1 Vt1 (p.u.) NC1 AVR1+PSS1 desired value (a) 0 10 20 30 40 50 60 70 t (s) 0.7 0.8 0.9 1 1.1 1.2 1.3 Vt3 (p.u.) NC3 AVR3+PSS3 desired value (b) Fig. 9: Terminal voltage of the G1 (a) and G3 (b) following the step load change in Node 2. 0 10 20 30 40 50 60 70 0.6 0.8 1 delta1 (rad) NC1 AVR1+PSS1 desired value 0 10 20 30 40 50 60 70 -2 0 2 w1 (rad s) 0 10 20 30 40 50 60 70 t (s) 0.8 1 1.2 E'q1 (p.u.) (a) 0 10 20 30 40 50 60 70 -0.5 0 0.5 1 delta3 (rad) 0 10 20 30 40 50 60 70 -2 0 2 w3 (rad s) 0 10 20 30 40 50 60 70 t (s) 0.8 1 1.2 E'q3 (p.u .) NC3 AVR3+PSS3 desired value (b) Fig. 10: State variables response of the G1 (a) and G3 (b) towards the step load change at the consumer Node 2. ing it with the classical controller. The new controller eliminates the steady-state error for all variables (static error ϵs= 0). Moreover, the results exhibit the DNC satisfactory performances. They show more convenient characteristics than those of the AVR+PSS control techniques, particularly those related to the overshoot amplitude as well as the stabilizing time for all states (δi,ωiand E′ qi), as it is presented in Tab. 3. Finally, the control signals evolution (excitation voltage Efdi) is given in Fig. 11. It is clear to observe the use of saturation bloc (±5p.u.) to limit the voltage level for excitation coil protection. 2) Three Short Circuit A three short circuit is applied to the line connecting nodes: 1 to node 2 at t1= 5 s during ∆t= 500 ms. Later, the fault is eliminated by isolating the transmission line 1–2 at t2. With regard to Fig. 12 and Fig. 13, it is noted that by using the neural controller, the system returns to its equilibrium state after the elimination of the short circuit and the variation of its topology. However, with the classic AVR + PSS controller, the system loses its stability and basically ends up with divergence. In this respect, the improvement of the transient stability in Power Systems with Neural controller is easily noticeable with neural controller under different severe faults conditions. ©2023 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 115