Modelling of a nonlinear switched

MODELLING OF A NONLINEAR SWITCHED RELUCTANCE DRIVE BASED ON ARTIFICIAL NEURAL NETWORKS Ç. Elmas * , Ş. Sağıroğlu + , İ.Çolak * , G. Bal * * Technical Education Faculty, Gazi University, Ankara, Turkey. + Engineering Faculty, Erciyes University, Kayseri, Turkey. Abstract - Switched Reluctance Motors (SRMs) are increasingly popular machines in electrical drives, whose performances are directly related to their operating condition. Their dynamic characteristics vary as condition change. Recently, several methods of modelling of the magnetic saturation of SRMs have been proposed. However, the SRM is nonlinear and cannot be adequately described by such models. Artificial Neural Networks (ANNs) may be used to overcome this problem. This paper presents a method which uses backpropagation algorithm to handle one of the modelling problems in an switched reluctance motor. The simulated waveforms of a phase current are compared with those obtained from a real switched reluctance commercial motor. Experimental results have validated the applicability of the proposed method. INTRODUCTION An important characteristic of the SRM drive is its inherent nonlinearity. The inductance of the magnetic circuit is a nonlinear function of both phase current and rotor position. In addition, the system handles energy most efficiently when the energy conversion cycles are made as square as possible, maximising the ratio of energy converted to energy input [12, 13]. This leads a particularly difficult problem because of their complicated magnetic circuit, which operates at varying levels of saturation under operating conditions. Square energy conversion cycles are created by driving the motor into magnetic saturation and bring the energy handling requirements of inverter into closer alignment with the energy conversion characteristics of motor [12, 13]. This can results in reduced switch requirements and energy savings. The recirculated energy in a drive with an applied voltage requires current flow and acts to increase the inverter and motor losses that accompany the current flow. Stephenson and Corda [1] proposed a quite successful method to model the flux linkage as a function of current and rotor position. This method has been modified by several others [2, 3, 4]. Torrey and Lang [5] have also proposed a method to provide analytical expressions for the flux linkage and current for every rotor position within a single summary equation. In contrast to the above methods, there have been many attempts to generate the necessary static magnetisation curves by Finite Element Analysis (FEA) [6]. Recently, the authors have reported an application of ANN for modelling of the magnetic nonlinearity of the magnetisation curves [14]. Artificial Neural Network (ANN) techniques have grown rapidly in recent years. Extensive research has been carried out on the application of artificial intelligence. Artificial Neural Network technology has the potential to accommodate an improved method of determining nonlinear model which is complementary to conventional techniques. NN are alone nonlinear and actual algorithmic relevant set of training examples is required which can be derived from operating plant data. This paper investigates the use of ANNs for the modelling of the magnetic nonlinearity of the SRM. Since this method does not require any prior information regarding the SRM system apart from the input and output signals, it is quite simple and cost effective. The modelling method in this paper departs significantly from previous modelling method by the authors, in which the magnetisation curves are represented by functions of flux linkage against rotor position, rather than current. In the paper, first, magnetic nonlinearity of the SRM is presented, then ANN approach to the modelling of the SRM is presented. ANN training requirements are discussed next and finally, the models are verified through comparisons with experimentally measured results. MAGNETIC NONLINEARITIES OF THE SRM The first step in modelling the nonlinearities of the SRM is predicting (ψ/θ/i) curves for a given motor. For the experimental motor, these curves are shown in Fig. 1. Although the construction of the SRM is quite simple, it is very difficult to derive a comprehensive mathematical model for the behaviour of the machine. Many attempts have been made by different researchers to overcome this problem. The structure of Stephenson and Corda method is that flux linkage is modelled as a function of current, with rotor position as a parameter. This method is based on storing the (ψ/θ/i) information in a look up table. Elmas and Zelaya de la parra [4] described a similar method which applies Least Squares curve fitting methods to produce a representation of the measured magnetic data as a series of polynomials. Pulle [3] has investigated the merits of representing the magnetising curve of an SRM by customised cubic splines and storing the coefficients in a new data base with the aim of improving the method suggested by Stephenson and Corda. Miller and McGilp [1] have also adapted the Stephenson and Corda method. Their aim was to represent flux linkage as a function of rotor position, with current as an undetermined parameter rather than position. In addition, an original work was published by Torrey and Lang [5]. The goal of their method was to provide complete analytic expressions of the flux linkage current information for every rotor position within a single summary equation. The general equation of SRM for only one phase is as follows: VRi d dt =+ ψθ(,) i (1) For the solution of the Eq. (1), it is necessary to model magnetic nonlinearities in the form of i(ψ,θ) rather than ψ (i,θ) form as shown in Fig. 1. This is because after each integration step the solution to Eq. 1 yields a value for the flux which can be used to find the corresponding current value for the next integration step. The variation of flux- linkage with current is the same for the remaining phases except for the angular dependence, which takes 1 A 2 A 3 A 4 A 5 A 6 A Fig. 1 The variation of flux-linkage with current. into account the physical interpolar spacing. The experimental motor has a total of eight symmetrically located stator poles used by a total of four phases (two poles per phase). NEURAL NETWORK MODELLING OF THE SRM Ability and adaptability to learn, generalisation, less information requirement, fast real-time operation and ease of implementation have made ANNs popular in the last few years. ANNs have been applied in many areas [7, 11]. Dynamic system modelling, identification and control using ANNs are particularly very promising [7, 8, 10, 11]. As a result of that, the modelling of SRM has been employed using the Backpropagation (BP) [9], which is the most popular algorithm in the arena of neural networks. Backpropagation. The standard backpropagation by Rumelhart and McClelland has been demonstrated on various problems [7, 8, 9, 10, 11]. The reasons for using this algorithm are that its structure is well understood and its recent successful applications encourage the applicants. This algorithm consists of a number of propagation errors (PEs), a transfer function for each PE in the layers, number of connections between layers (at least three layers; an input, a hidden layer, an output layer) and an algorithm or learning rule which is the generalised delta rule. This rule is simple and give a prescription for changing the weights (w ij ) in any feedforward network to map the input-output pairs. This change is based on gradient descent and relies on propagating an error occurred from an output PE backwards, towards input layer through the PEs in the hidden layers according to the errors. A simple operation for calculating the output takes place when a set of input is entered to the input layer. The calculation direction is from input layer towards the output layer via hidden layers. The number of PEs in the input layer is equal to the number of inputs in each input pattern, and each of these PEs receives one of the inputs. The number of PE in the output layer is the output of the network. The number of PEs in the hidden layer depends on the discretion of the network designer. However, there is no given clear explanation implemented. Fig. 2 shows the topology of NN with biases. Generally, the weights between the layers are initialised with small random values. This ensures that the networks train and function easier. So it is important that the weights do not start with the same value, thus nonsymmetric weights can be obtained for internal representations. During training the feedforward computation and the adjustment to the weights based upon the error are determined. During recall only the feedforward computation takes place as mentioned earlier. The algorithm is simple and relies on propagating an error signal from the output layer backwards towards input layer through hidden layers. The operation of calculating the output takes place when the input signal is entered to input layer. The calculation direction is from input layer towards the output layer via hidden layers. The feedforward computation and the weights' adjustments based upon the error are determined during training. The feedforward computation only takes place in recall. +1 Bias +1 Bias O u t p u t s Output Layer Hidden Layer Input Layer More Layers Weights I n p u t s Fig. 2. Topology of backpropagation neural network. A simple training and recall chart is given in Fig. 3. It shows the sequence of training and recalling procedure of backpropagation. For some applications more than one hidden layer are used. Initialise weights randomly Start training Present input set to input layer Calculate output throughout PEs Adjusting weights using gradient descent Start testing Present either training or testing Set to input layer of network An actual output from the network error ? set completed? Stop Calculating output throughtout PEs not acceptable acceptable no yes Fig. 3 Training and recall flow chart of backpropagation algorithm. In the feedforward computation an input set passes onto the hidden layer from input layer. The output of each PE in the layer is calculated a weighted sum of its input, then passes the sum through its activation function and presents the activation value to the output layer. This simply explains how a PE works. At this stage of training, X represents the input vector, Position and Flux, and C represents the desired output vector, Current. BP is briefly reviewed here. If the network contains n inputs and m outputs, X and C are given by: (2) Xfxxx x Cfccc c n m = = (, , , ) (, , , , ) 123 123 If C net is the output vector of the neural network, the aim of BP is to minimise the error values between the output of the system (C) and the output of the network (C net ). This error is considered as a function of the connection weights. When the examples X and C are presented to the net, an output of j-th PE in the kth layer is calculated as; First, the inputs are multiplied by related weights and then they are summed as; (3) net x j k ji i n i = = ∑ ω 0 ) Second, the output of the j-th PE in the k-th layer is calculated as a function of net j as; (4) cf netj k = ( net j where f is a transfer or threshold function. The transfer function used in training of this work is given as follows; fc cT cT cT cT k netj netj netj netj netj () exp( ) exp( ) (exp( ) exp( )) = +− − + ++ − + (5) where T is a threshold and c is the sum of the weighted inputs for the j-th PE in the k-th layer. The activation function is a 'smoothed' form of the threshold function. The function used in the backpropagation network should be monotonically increasing and continuously differentiable such as hyperbolic tangent. It should be noted that not all of the nets used the hyperbolic tangent as given in Eq.(5). The input and output layer uses linear activation function. netj The output obtained is used to feed the PEs in the further layer as the inputs to it/them. This process continues till reaching the output layer. When feedforward process has completed, the backpropagating starts. A backpropagation net learns by making changes in its weights in a direction to minimise the error between the a desired value and its prediction. The changes have been done using the steepest decent or generalised delta rule. Assume that there are s input/output pairs, x and c, available for training the network. After presentation of a pair of s, the weights are changed as follows: (6) ωω ji s ji s ji s() ( ) () =+ −1 ∆ω with given by three equations when two hidden layers are considered: ∆ω ji s() hidden to output weights: ∆ω ji sk k ij k netj k ji sk fnetcc ()() ( ) () () ( )() ()[ ] =−+ − η 1 − α∆ω 1 ) − 1 ∆ω ) −− 12 ∆ω ) (7) where j represents the number of PE in the output layer i represents the number of PEs in the second hidden layer. hidden to hidden weights: ∆ω ji sk k i k ji sk f net error ()() ( ) () ()( () −− − − =+ 12 1 1 ηα (8) where; error f net c c kk ji k neti k ij sk i m ( ) ( ) () () ( )() '()[ ] −− − = =−       ∑ 11 1 1 ω (9) where i represents the number of PE in the output layer j represents the number of PEs in the second hidden layer. m is equal to the number of PEs in the output layer. hidden to input weights: ∆ω ji sk k j k i m ij ks i s ji sk f net error x ()() () ()()()() ()( '()[ −− − = − =       + ∑ 22 1 1 1 ηωα (10) where i represents the number of PE in the second layer. j represents the number of PEs in the first hidden layer. m is equal to the number of PEs in the second layer. where : learning coefficient, η : momentum coefficient, α : delta weights from the i-th to the j-th unit in the k-th layer ∆ω ji s() : previous delta weights of the k-th layer. ∆ω ji s(−1 After the first pair, the rest of the input set is applied to the network. The weights of the network have been set of randomised with a set of values which are distributed uniformly between -0.1 and +0.1. The selected seed was 1. Training and testing of neural network. The first and usually longest step in this work was to collect data from the system. The data set used in training was obtained from the SRM machine experimentally. Obtaining accurate data has taken an important place to train the networks more accurately. Generally, this is the most critical to prospective success. It must be possible to gather an adequate sample of characteristics data so that networks can learn efficiently, otherwise, it may be hard or infeasible to train a neural network. During training a reasonable strategy is to start with a few hidden nodes and increase the number while monitoring generalisation by testing at each epoch. The most common index of generalisation for BP is mean squared error, calculated by squaring each error, summing the squares, then averaging the sum by number of outputs and data patterns. A good technique for preventing overtraining is to stop training when the improvement of the mean squared error is stop. After a successful training the neural network model is replaced with the SRM system. The neural network is here a part of a larger application, within which it acts like a callable function: the application passes a set of input values to the neural network model that the model produces phase current. Backpropagation network used in modelling. The backpropagation network used in modelling is shown in Fig. 4 with a block diagram. This structure was used for training and testing processes. After a couple of training, it was found that two layers network achieved the mapping task in high accuracy. The both learning and momentum coefficients were 0.018 and the number of epoch was 2000 for training. The most suitable network configuration found was 2x8x8x1. Current Position(s) Flux(s) (s) Fig. 4. Modelling the SRM using Artificial neural network. Calculation methods. The analysis is now proceeds by solving the characteristic differential equations for each topological mode by using the more accurate model for (ψ /θ/i) variations discussed above. Since four phases of the SRM are identical to each other only one model for (ψ/θ /i) is sufficient and is used for the other phases. Before going deep into calculation methods, it is necessary to explain the torque production mechanism for the SRM. The torque produced by a switched reluctance motor is proportional to the rate of change of coenergy as the rotor moves from one rotor position to another. The most general expression for instantaneous torque for one phase is: T W e cons t = ′ = ∂θ ∂θ (, ) tan i i (11) 150 where T e is torque, θ is rotor position and W' is the coenergy which is ′ = ζ W i ψθ (, ) i 0 d i (12) 50 The coenergy is a function of both rotor position and excitation current and hence, when evaluating the partial derivatives, it is necessary to keep the indicated variables constant. The method of calculation is as follows. Initially the values for phase flux (ψ), operating phase pole position (θ ) and phase current (i) are given as zero for the operating phase. Eq. (1) is now solved numerically using the Runge- Kutta numerical integration method. This yields a new value for ψ. The program now refers to the derived (ψ/θ/i) NN algorithm to find values for the phase current (i). Since the steady state conditions are assumed, the speed is constant. As the time constant of the mechanical system is much slower than the electrical time constant, the phase current and the phase flux can be accepted as a constant between two integration intervals. Since the speed is constant at a given value, the rotor position (θ) can replace time as the independent variable. The whole process is then repeated for the new values of θ and ψ. The accuracy of the results thus depends on the modelling of the flux current linkage. COMPARISON BETWEEN SIMULATION AND EXPERIMENTAL RESULTS To explore the effectiveness of this technique, both computer simulation and practical experimental work have been carried out. As indicated by Eqs. (11) and (12), the torque produced is based solely on the flux linkage/current relationship. This suggest that if the phase current is predicted correctly then the torque is also known. Thus, comparison between current waveforms from simulation and experiment should give enough evidence. Fig. 5 shows the variation of flux-linkage with current along with NN results. These results have also demonstrated the strong potential of the NN applied to the SRM. Fig. 6 illustrates simulation result and an actual measurement obtained by a data acquisition board. As seen from Fig. 6, there is generally good agreement between simulation and experimental results. Flux (mWb) 0 25 75 100 125 0 15 30 Position (degree) NN results Training data Fig. 5 The variation of flux-linkage with current along with NN results. Simulation result Experimental result (a) (b) Simulation result Experimental result Fig. 6 Current waveforms from simulation and experimental measurement, a) low speed, b) high speed. RESULTS Figs. 7-10 show simulation results for the SRM obtained by the proposed method. The motor was excited by a split DC source converter (SDCSC). The following values were used for the simulations: a DC link voltage of 200V, per phase resistance of 2.56Ω, input filter inductance of 240µH and two input filter capacitors of 1000µF (each). current torque flux Fig. 7 Phase current, flux and torque waveforms at 60 rad/s motor speed. Fig. 8 Coenergy at 60 rad/s motor speed. current torque flux Fig. 9 Phase current, flux and torque waveforms at 120 rad/s motor speed. Fig. 10 Coenergy at 120 rad/s motor speed. CONCLUSIONS Simulation results were verified through experimental results and ANN model was proven to be reasonably accurate. The advantages of the model developed here are that no a priori knowledge is required (model or equation), reduced mathematical complexity, and faster operation after training. However, it should be emphasised during the development, the collection of a data set is critical that the network can learn efficiently. The training period usually takes a long time. REFERENCES [1] Stephenson J. 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[12] Miller T.J.A., 1990, IEEE Trans. on. Ind. appl., Vol. IA-21, No. 5. [13] Stephenson M. and EL-Khazendar M.A., 1989, Proc. IEE, Vol. 136, Pt. B, No. 1. [14] Elmas Ç. Sağırolu Ş., Ç olak İ . and Bal G., 1994, MELECON'94, Part 2, 809-812. . achieved the mapping task in high accuracy. The both learning and momentum coefficients were 0. 018 and the number of epoch was 2000 for training. The most suitable network configuration found