Electrical Power and Energy Systems 63 (2014) 1023–1029 Contents lists available at ScienceDirect Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes Implementation of supervisory controller for solar PV microgrid system using adaptive neural model Ho Pham Huy Anh ⇑ FEEE, DCSELAB, Ho Chi Minh City University of Technology, VNU-HCM, Viet Nam a r t i c l e i n f o Article history: Received 20 July 2013 Received in revised form 21 June 2014 Accepted 23 June 2014 Available online 30 July 2014 Keywords: Solar photovoltaics (solar PV) Solar PV microgrid system Back Propagation learning algorithm (BP) Adaptive neural-based supervisory controller Modeling and identification a b s t r a c t This paper investigates a novel forward adaptive neural model which is applied for modeling and implementing of the supervisory controller of the solar PV microgrid system The nonlinear features of the solar PV microgrid system were thoroughly modeled based on the adaptive identification process using experimental input–output training data This paper proposes the novel use of a back propagation (BP) algorithm to generate the adaptive neural-based supervisory controller for the solar PV microgrid system The simulation results show that the proposed adaptive neural-based supervisory controller trained by Back Propagation learning algorithm yields outstanding performance and perfect accuracy Ó 2014 Elsevier Ltd All rights reserved Introduction Hybrid renewable energy systems can be classified into two main types: grid-connected and standalone The renewable energy sources can be PV or wind generators (or both), according to the availability of solar radiation or wind velocity (or both) at the system site Batteries are often used as a backup source to supply the system when the renewable energy source is unavailable Other backup sources can be used with or without batteries such as fuel cells (e.g electrolysers, supercapacitors and flywheel energy storage) Diesel generators could be used as secondary sources of renewable energy The standalone system might provide dc power, ac power, or both dc and ac power [1–3] The grid-connected systems can work on standalone mode when the utility grid is unavailable For the most part, fuel cells and diesel generators are not used with such grid-connected systems The supervisory controllers manage the power according to the type and different components of the system The supervisory controllers could be divided generally to two kinds; conventional-based and artificial intelligence-based methods A small-scale hybrid PV-wind generation system with batteries works only in standalone mode as proposed in [4] The supervisory controller with a fault ride through strategy is explained in Ref [5] ⇑ Tel.: +84 08 39490415 E-mail address: hphanh@hcmut.edu.vn http://dx.doi.org/10.1016/j.ijepes.2014.06.068 0142-0615/Ó 2014 Elsevier Ltd All rights reserved The supervisory controller of a hybrid wind-PV-fuel cell (FC) energy system is proposed in [6–8] Every source is connected to the ac bus bar via an inverter to supply the load The FC– electrolyzer combination is used as a backup and long-term storage system The battery bank is used in the system as a shorttime backup to supply the transient power At any given time, the supervisory controller controls any excess wind-PV-generated power to be supplied to the electrolyser The hydrogen, which is delivered to the hydrogen storage tanks by a gas compressor, is consequently generated If the generated power is less than the load demand, the FC stack begins to produce energy for the load using hydrogen from the storage tanks A steady state model was used in the papers with no dynamical results This study demonstrates that the low voltage distribution network is supervised to optimize energy flow and control power quality [9] This kind of system is supplied by renewable energy sources, diesel generators, and energy storage backups The system is controlled, according to international power quality standards The algorithm is universal and adapts its control variables A supervisory controller for the hybrid PV-wind system with batteries is proposed in [10] The PV is directly connected in parallel with the batteries to supply the ac load through a three phase inverter which is connected from the other side to a wind generator The power management strategy is simplified in this configuration as the batteries act as a constant voltage load line which charges both ways by the PV and the wind generators A dump load can be switched on with batteries fully charged but the batteries are later disconnected to prevent 1024 H.P.H Anh / Electrical Power and Energy Systems 63 (2014) 1023–1029 overcharging One of the drawbacks is that there is no ability in this scheme to provide PV or wind generators control Furthermore, the batteries’ charging and discharging is not fully controlled Recently, authors in [11,12] introduced the supervisory control system for hybrid wind diesel microgrid Up to now, there are many researches focus on artificial intelligence-based methods applied to supervisory control of hybrid microgrid systems A standalone system with hybrid PV-diesel power generators and flywheel backup energy storage system is proposed in [13] A pump is used as an auxiliary load to absorb the extra power from the system A fuzzy logic supervisory controller is proposed to manage the power from the generators to the load According to the generated PV power and the rotor speed of the flywheel, the fuzzy controller adjusts the references for the diesel generator output power and the pump demand A fuzzy logic supervisor is proposed also in Ref [14] for a grid-connected wind generated system The supervisory controller regulates the power of the wind generator according to the change in the grid frequency The pitch angle is controlled to match the reference power generated by the supervisory controller Hong et al in [15] developed of intelligent MPPT (maximum power point tracking) control for a grid-connected hybrid power generation system In a microgrid system [16], the PV generators could be used to remove frequency deviations using fuzzy supervisory controller This controller increases or decreases the PV output power to match a high frequency or a low frequency respectively In Ref [17], the fuzzy supervisor controls the pitch angle of a fixed speed wind generator and the reactive power output of the static VAR compensator to smooth the wind generator output power and regulate the grid voltage respectively A neural networks-based supervisory controller manages the power in a PV standalone system with batteries Two neural networks are used: one neural network for direct control and the second to adapt the first one to optimize the system’s operation [18] Other applications of neural and fuzzy techniques in supervisory control of microgrid systems were investigated in [19–22] Artificial bee colony-based approach was applied in [23] as to solve the problem of capacitor placement for net saving maximization and system stability enhancement in distribution networks The drawback of these researches is that the proposed intelligent supervisory controllers were unable to adaptively generate the switching control outputs Unfortunately, up to now, the use of adaptive neural network-based supervisory controller for the microgrid systems has not yet been adequately studied To overcome this gap, this paper proposes the novel use of adaptive neural MIMO model to generate the supervisory controller for the solar PV microgrid systems The Back Propagation (BP) learning algorithm is used to process the experimental input–output data that is measured from the optimal desired operation of the solar PV microgrid systems as to optimize all nonlinear and dynamic features of this system Thus, the BP algorithm optimally generates the appropriate neural weightings to perfectly characterize the features of the supervisory controller for the solar PV microgrid systems These good obtained results are due to proposed adaptive neural MIMO model combines the extraordinary approximating capability of the neural system with the powerful predictive and adaptive potentiality of the nonlinear ARX structure that is implied in the proposed adaptive neural-based model Consequently, the proposed method of the generation of the adaptive supervisory controller for the solar PV microgrid systems has successfully modeled the nonlinear features of the desired operation of the solar PV microgrid system with good performance The rest of the paper is organized as follows Section ‘Implementation of supervisory control for the solar PV microgrid system’ introduces the implementation of supervisory controller in solar PV microgrid systems Section ‘Adaptive neural MIMO model for supervisory control of the solar PV microgrid system’ presents the novel adaptive neural MIMO model using for the implementation of supervisory controller in solar PV microgrid systems The results from the proposed adaptive neural-based supervisory controller are presented in Section ‘Identification and implementation of the adaptive neural MIMO model for supervisory control of the solar PV microgrid system’ Finally, Section ‘Conclusions’ contains the concluding remarks Implementation of supervisory control for the solar PV microgrid system We consider an implementation of a supervisory controller for the solar PV microgrid systems illustrated in Fig This scheme introduces the novel use of adaptive neural MIMO model to generate the supervisory controller for the solar PV microgrid systems Fig illustrates the working principle of proposed supervisory controller for the solar PV microgrid systems The proposed neural NARX-based supervisory controller is designed through two phases: offline training phase and then online operating phase In the training phase, based on the experimental input–output data measured from the optimal desired operation of the solar PV microgrid system, the Back Propagation (BP) learning algorithm is applied to optimally generate the appropriate neural weightings which perfectly characterize the features of the supervisory controller for the solar PV microgrid systems Then, in the operating phase, the neural networks-based supervisory controller will optimally manage the power in a PV grid-connected system This neural networks-based supervisory controller adapt well the input variables including PV power available and required load power to optimize the system’s operation via appropriate switching outputs S1, S2, S3 This controller is concerned with the utility grid not with controlling the local generators The grid-connected systems can work on standalone mode when the utility grid is unavailable and in grid-connected systems, the utility grid is a secondary source Four modes of switching operation of the proposed neural-based supervisory controller for the solar PV microgrid systems were tabulated in Table as follows Adaptive neural MIMO model for supervisory control of the solar PV microgrid system The adaptive forward neural MIMO controller used in this paper is a combination between the Multi-Layer Perceptron Neural Networks (MLPNN) structure and the Auto-Regressive with eXogenous input (ARX) model Due to this combination, adaptive forward neural MIMO model possesses both of powerful universal approximating feature from MLPNN structure and strong predictive feature from nonlinear ARX model A fully connected 3-layer feed-forward MLP-network with n inputs, q hidden units (also called ‘‘nodes’’ or ‘‘neurons’’), and m outputs units is shown in Fig In Fig 2, w10, ., wq0 and W10, .,Wm0 are weighting values of Bias neurons of Input Layer and Hidden Layer respectively Forwardly we consider an Auto-Regressive with eXogenous input (ARX) model with noisy input, which can be described as AðqÀ1 ÞyðtÞ ẳ Bq1 ịut Tị ỵ Cq1 ịetị with Aq1 ị ẳ ỵ a1 q1 ỵ a2 q2 Bq1 ị ẳ b1 ỵ b2 q1 Cq1 ị ẳ c1 þ c2 qÀ1 þ c3 qÀ2 ð1Þ H.P.H Anh / Electrical Power and Energy Systems 63 (2014) 1023–1029 1025 Fig Schematic of a supervisory controller for the solar PV microgrid systems Table Four modes of switching operation of the proposed adaptive neural-based supervisory controller for the solar PV microgrid systems Mode S1 S2 S3 Conditions OFF ON ON OFF ON ON OFF OFF OFF OFF ON ON (Solar (Solar (Solar (Solar power) power) power) power) PW % PW < PL (Consumed load power) PW > PL (Consumed load power) PW > PL (PL = 0) where e(t) is the white noise sequence with zero mean and unit variance; u(t) and y(t) are input and output of system respectively; q is the shift operator and T is the time delay From Eq (2), not considering the noise component e(t), we have the general form of the discrete ARX model in z-domain (with the time delay T = nk = 1) À1 À1 À2 Ànb b1 z ỵ b2 z ỵ ỵ bnb z yz ị ẳ uz1 ị ỵ a1 z1 ỵ a2 z2 ỵ ỵ ana zÀna ð2Þ in which na and nb are the order of output y(zÀ1) and input u(zÀ1) respectively We investigate the potentiality of various simple adaptive neural MIMO models in order to exploit them in modeling, identification and control as well The adaptive neural-based supervisory controller of the solar PV microgrid system is investigated Thus, by embedding a 3-layer MLPNN (with number of neurons of hidden layer equal 5) in a 1st order ARX model with its characteristic equation induced from (3) as follows: s1hat kị ẳ b11 pS kị ỵ b12 pL ðkÞ À a11 s1 ðk À 1Þ À a12 s2 ðk À 1Þ À a13 s3 ðk À 1ị s2hat kị ẳ b21 pS kị ỵ b22 pL ðkÞ À a21 s1 ðk À 1Þ À a22 s2 ðk À 1Þ À a23 s3 ðk À 1Þ s3hat kị ẳ b31 pS kị ỵ b32 pL kị a31 s1 ðk À 1Þ À a32 s2 ðk À 1Þ À a33 s3 ðk À 1Þ ð3Þ We will design the proposed adaptive neural–based supervisory controller of the solar PV microgrid system (with na = 1, nb = 1, nk = 1) with inputs (including two input values pw(k), pl(k) and three recurrent delayed output values s1(k À 1), s2(k À 1), Fig Structure of feed-forward MLPNN 1026 H.P.H Anh / Electrical Power and Energy Systems 63 (2014) 1023–1029 s3(k À 1)) and three output values s1hat(k), s2hat(k) and s3hat(k) We remember that two input values pw(k), pl(k), representing the two power inputs [MW] of the solar PV and the load, respectively and the three output values s1hat(k), s2hat(k) and s3hat(k) representing the responding switching output of the adaptive neural-based supervisory controller Its structure is shown in Fig By this way, the fifteen parameters a11, a12, a13, b11, b12, a21, a22, a23, b21, b22, a31, a32, a33, b31, b32 of the ARX structure of three switching output variables s1hat(t), s2hat(t) and s3hat(t), respectively, now become adaptively nonlinear and will be determined from the weighting values Wij and wjl of the proposed adaptive neural MIMO NARX model This feature makes adaptive neural MIMO NARX model very powerful in modeling, identification and in model-based advanced control as well The prediction error approach, which is the strategy applied here, is based on the introduction of a measure of closeness in terms of a mean sum of square error (MSSE) criterion: EN h; Z N ị ẳ N X ^t jhịT ẵytị y ^t jhị ẵytị y 2N tẳ1 4ị Based on the conventional error Back-Propagation (BP) training algorithms, the weighting value is calculated as follows: Wðk ỵ 1ị ẳ Wkị k @EWkịị @Wkị 5ị with k is kth iterative step of calculation and k is learning rate which is often chosen as a small constant value Concretely, the weights Wij and wjl of neural MIMO NARX are then updated as: W ij k ỵ 1ị ẳ W ij kị ỵ DW ij k ỵ 1ị DW ij k ỵ 1ị ẳ k di Oj ^i ð1 À y ^i Þðyi À y ^i Þ di ¼ y ð6Þ with di is search direction value of ith neuron of output layer (i = [1 ? m]); Oj is the output value of jth neuron of hidden layer ^i are truly real output and predicted output (j = [1 ? q]); yi and y of ith neuron of output layer (i = [1 ? m]), and wjl k ỵ 1ị ẳ wjl kị ỵ Dwjl k ỵ 1ị Dwjl k ỵ 1ị ẳ k dj Á ul m X dj ¼ Oj ð1 À Oj Þ di W ij ð7Þ Identification and implementation of the adaptive neural MIMO model for supervisory control of the solar PV microgrid system In general, the procedure which must be executed when attempting to identify a dynamical system consists of four basic steps STEP STEP STEP STEP (Getting Training Data) (Select Model Structure) (Estimate Model) (Validate Model) In Step 1, the identification procedure is based on experimental input–output data values measured from the desired input–output of the adaptive neural–based supervisory controller of the solar PV microgrid system The two input values pw(k), pl(k), representing the two power inputs [MW] of the solar PV and the load and the three desired referential output values s1hat(k), s2hat(k) and s3hat(k) representing the responding switching output of the adaptive neural–based supervisory controller Fig 4a presents the collected input–output data composes of the two input signals pw(k), pl(k) applied to the neural–based supervisory controller of the solar PV microgrid system and Fig 4b introduces the referential output values s1hat(k), s2hat(k) and s3hat(k) Back Propagation (BP) learning algorithm based on the error between the (s1, s2, s3) reference switching outputs and the responding (s1hat, s2hat, s3hat) switching outputs of adaptive neural MIMO NARX model to update the weights of proposed neural MIMO NARX supervisory controller Fig illustrates identification scheme of the neural MIMO NARX supervisory controller using proposed neural MIMO NARX model for solar PV microgrid system The second step relates to selecting the model structure The block diagram in Fig illustrates the identification scheme of the proposed intelligent model The proposed adaptive neural MIMO NARX model structure was attempted Its model structure was presented in Fig The third step estimates values for the trained adaptive Neural NARX model The optimal fitness value to use for the BP-based optimization and identification process is calculated The estimation result is presented in Fig This figure represents the fitness convergence values of the proposed neural-based supervisory controller which correspond to adaptive neural NARX identified and optimized with Back Propagation (BP) learning algorithm The fitness value of the proposed adaptive neural-based supervisory i¼1 pl(k) ps(k) s3hat(k) s3(k-1) TWO POWER INPUT VALUES OF TRAINING DATA 50 POWER of SOLAR PV [kW] in which dj is search direction value of jth neuron of hidden layer (j = [1 ? q]); Oj is the output value of jth neuron of hidden layer (j = [1 ? q]) ; ul is input of lth neuron of input layer (l = [1 ? n]) pl(k) 30 20 10 0 10 15 20 10 15 20 45 s2hat(k) s2(k-1) pl(k) s1hat(k) ps(k) s1(k-1) Fig Model structure of the adaptive neural-based supervisory controller of the solar PV microgrid system POWER of LOAD [kW] ps(k) 40 40 35 30 25 20 time [hour] Fig 4a Two power input signals pw(k), pl(k) of training data for identification process 1027 H.P.H Anh / Electrical Power and Energy Systems 63 (2014) 1023–1029 THREE OUTPUT VALUES OF TRAINING DATA S1 0.5 0 10 15 20 10 15 20 10 15 20 S2 0.5 S3 0.5 time [hour] Fig 4b Three switching output signals of training data for identification process Fig Identification scheme of the neural-based supervisory controller using proposed adaptive neural MIMO NARX model adaptive neural-based solar PV microgrid supervisory controller are implied in the three responding output switching signals (s1, s2, s3) from two power input values (pw(k), pl(k)) The last step relates to validating the resulting nonlinear adaptive models Applying the same training diagram in Fig 5, a good validating result demonstrates the performance of the resulting forward neural MIMO NARX (FNMN) model presented in Fig The error which is optimized nearly zero between the real reference output switching signals (s1, s2, s3) and the forward neural MIMO NARX model responding output signals (sh1, sh2, sh3) asserts the very good performance of proposed neural MIMO NARX controller Furthermore, as for consolidating the performance of the trained neural-based supervisory controller, we have applied another set of daily input power Pw, Pl values (see Table 2) for testing the adaptive performance of the trained neural-based supervisory controller The output switching S1, S2, S3 results precisely illustrated in Fig 8a and 8b once more confirms the efficiency of the proposed adaptive neural-based supervisory controller for the solar PV microgrid system Finally, Fig illustrates the auto-tuning variation of adaptive ARX parameters of proposed forward neural MIMO NARX Model of the adaptive neural-based supervisory controller Concretely, the fifteen parameters a11, a12, a13, b11, b12, a21, a22, a23, b21, b22 and a31, a32, a33, b31, b32 of the two 1st order ARX structure integrated in proposed FNMN11 model were adaptively auto-tuning as illustrated in Fig These results show that the parameters of the ARX structure integrated in proposed FNMN models now become adaptively nonlinear and will be adaptively determined from the optimized weighting values Wij and wjl of the forward neural MIMO NARX model This feature once more proves that the proposed adaptive forward neural MIMO NARX (FNMN) supervisory controller is very powerful and adaptive in identification and in model-based advanced control as well Table tabulates the optimized weighting values of the proposed forward neural MIMO NARX model The final structures of proposed neural MIMO NARX model identified and optimized by BP learning algorithm are shown in Fig FITNESS CONVERGENCE OF ADAPTIVE NEURAL MIMO NARX MODEL IDENTIFICATION 10 VALIDATION OF ADAPTIVE NEURAL MIMO NARX MODEL IDENTIFICATION -2 S1 10 -4 -6 -2 10 10 error ERROR S1 ref S1 neural model 0.5 -8 x 10 10 10 15 20 10 15 20 -3 S2 -10 10 S2 ref S2 neural model 0.5 200 300 400 500 600 700 800 900 ITERATIONS Fig Fitness convergence of proposed adaptive neural-based supervisory controller identification controller identification produces an excellent global optimal value (equal to 0.00000000086) These good results are due to how the proposed model combines the extraordinary approximating capability of the neural system with the powerful predictive and adaptive potentiality of the nonlinear NARX structure that is implied in the adaptive neural MIMO NARX model Consequently, the complex features of the 10 15 10 15 20 0.5 -0.5 S3 100 error error -12 10 20 S3 ref S3 neural model 0.5 0 10 15 20 10 15 20 0.5 -0.5 -1 time [hour] Fig Validation of proposed adaptive neural-based supervisory controller identification 1028 H.P.H Anh / Electrical Power and Energy Systems 63 (2014) 1023–1029 Table Set of daily power Pw, Pl input values for testing the adaptive performance of the trained neural-based supervisory controller 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 0 0 20 73 80 154 179 187 179 160 104 38 0 0 0 33 37 39 40 43 58 60 70 75 68 60 50 62 52 54 45 40 62 80 73 60 48 40 32 À33 À37 À39 À40 À43 À58 À60 À50 À2 12 94 129 125 146 106 59 À2 À62 À80 À73 À60 À48 À40 À32 S1 Putility (MW) ON ? 1, OFF ? S1 S2 S3 0 0 0 1 1 1 1 1 0 0 0 1 1 1 1 0 0 0 0 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 POWER of SOLAR PV [MW] S2 ref S2 neural model 10 15 20 25 S3 ref S3 neural model 0.5 10 15 20 25 Time [hour] Fig 8b Consolidation testing of proposed adaptive neural-based supervisory controller Adaptive NARX parameters' auto-tuning of proposed neural MIMO NARX model 20 10 -10 -20 -30 -40 -50 -60 -70 -80 POWER of LOAD [MW] 25 50 20 20 0.5 100 15 15 150 10 10 200 5 TWO POWER INPUT VALUES OF CONSOLIDATION TESTING S1 ref S1 neural model 0.5 S2 PL (MW) S3 PW (MW) Adaptive NARX parameters' Values t (hour) THREE SWITCHING OUTPUT VALUES of CONSOLIDATION TESTING 25 10 15 20 25 30 35 40 45 time (samples X 30minutes) 80 Fig Adaptive NARX parameters’ auto-tuning of proposed neural MIMO NARX supervisory controller 60 40 20 10 15 20 conventional supervisory controller using relays and digital logic control circuits which required high hardware cost, maintaining fee, unable with variable input values (of solar power and load power) and other disadvantages On the contrary, the novel adaptive neural MIMO NARX-based supervisory controller using adaptively switching soft-computing control which required low software cost, maintaining free, highly adaptive performance with variable input values (from solar power and load power) and other 25 Time [hour] Fig 8a Two power input signals pw(k), pl(k) of data for consolidation testing In summary, for comparing between the results respectively obtained using the novel neural-based controller and the conventional supervisory controller, it convincingly shows that the Table Optimized weights of forward neural MIMO-NARX controller – total number of weighting values = 68 j wji – weights of Input Layer i Wj0 – weight of Bias Input layer Wkj – weights of Hidden layer i k=1 k=2 k=3 0.1669 0.0075 0.0251 À0.0032 À0.0254 2.5321 À13.865 41.918 À5.7831 À45.182 2.568 À51.631 3.3732 À31.643 À28.857 À0.003 À0.019 À0.004 À0.027 0.006 À1.0561 0.1474 À0.2834 0.0142 À0.3565 1.023 À0.096 0.311 À0.109 0.329 0.0816 0.0091 0.0196 0.0047 À0.019 À1.056 0.147 À0.283 0.0136 À0.356 1.023 À0.095 0.311 À0.109 0.329 À0.167 0.0027 0.02635 À0.0043 À0.0115 À0.699 0.0386 0.0182 0.0349 À0.017 0.492 À0.037 0.038 À0.016 À0.013 0.016 0.00338 0.00848 À0.0105 0.00306 Wk0 – weight of Bias Hidden layer À5.134 Wkj – weights of Hidden layer Wk0 – weight of Bias Hidden layer À12.923 Wkj – weights of Hidden layer Wk0 – weight of Bias Hidden layer À6.1125 H.P.H Anh / Electrical Power and Energy Systems 63 (2014) 1023–1029 advantages Furthermore, in comparison with the other intelligent supervisory controllers introduced in [16,17,19,22,23], the proposed adaptive neural-based controller can online adaptively generate the switching control outputs, which seems unable with the other intelligent supervisory controllers previously suggested Conclusions In this paper a new approach of forward neural MIMO NARX model firstly utilized in modeling and identification of the adaptive supervisory controller applied in the solar PV microgrid system Training and testing results showed that the newly proposed adaptive neural MIMO NARX model presented in this paper can be used in online control with better dynamic property and strong robustness This proposed intelligent neural MIMO NARX model is quite suitable to be applied for the modeling, identification and control of various hybrid PV-wind-fuel cell microgrid systems, including linear and nonlinear processes without concerns of large change in external environments Acknowledgement This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 107.04-2012.23 References [1] Khan M Supervisory hybrid control of a wind energy conversion and battery storage system Graduate Department of Electrical and Computer Engineering, University of Toronto, PhD thesis; 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Back-Propagation (BP) training algorithms, the weighting value is calculated as follows: Wk ỵ 1ị ẳ Wkị k @EWkịị @Wkị 5ị with k is kth iterative step of calculation and k is learning rate which is often... s1hat(k), s2hat(k) and s3hat(k) Back Propagation (BP) learning algorithm based on the error between the (s1, s2, s3) reference switching outputs and the responding (s1hat, s2hat, s3hat) switching