A Comparison of MLP, RNN and ESN in Determining Harmonic Contributions from Nonlinear Loads

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1 A Comparison of MLP, RNN and ESN in Determining Harmonic Contributions from Nonlinear Loads Jing Dai, Pinjia Zhang, Joy Mazumdar, Ronald G Harley and G K Venayagamoorthy 3 School of Electrical and Computer Engineering Georgia Institute of Technology Atlanta, GA 333, USA jingdai@gatechedu {pzhang, rharley}@ecegatechedu Power Conversion Division Siemens Energy and Automation Alpharetta, GA 35, USA joymazumdar@siemenscom 3 Department of Electrical Engineering Missouri University of Science and Technology Rolla, MO 6549 USA gkumar@ieeeorg Abstract- With the wide use of power electronics devices, harmonic currents are being injected into the power system, known as harmonic pollution Although IEEE standards [][] have required the utilities and customers to limit the amount of harmonic current and voltage, the practical evaluation is complicated, as it is difficult to separate the contributions from the utilities and customers A neuralnetwork-based harmonic current prediction scheme was previously proposed by the authors to estimate the true harmonic current attributed to the nonlinearity of the load, instead of the distorted power supply To test the feasibility of different types of neural networks in this application, this paper compares the performances and computational effort of three types of neural networks: Multilayer perceptron networks (MLP), simple recurrent network (RNN) and echo state network (ESN) I INTRODUCTION With the wide use of power electronics devices, harmonic currents are being injected into the power system, known as harmonic pollution IEEE standards 59 [][] and the IEC--3 [3] limit the amount of harmonic current and voltage generated by utilities and customers If the harmonics generated by each customer, especially large industrial plant, has serious negative effects on the power quality, then this can be identified by this proposed method, and the utility can adopt appropriate methods to penalize these offending customers Since the interactions between loads and sources are complex, to identify accurate harmonics generated by an individual nonlinear load becomes a challenge Figure shows a typical network structure When the voltage at the point of common coupling (PCC) is sinusoidal, the harmonic current generated by the nonlinear load is defined as the true harmonic current contributions from this load However, there are often other loads connected to the PCC If the voltage at the PCC is distorted, these other loads will have harmonic currents to the system even if the loads are linear As a result, simply measuring the harmonic currents of each individual load is not sufficiently accurate since the harmonic current is caused by both the load itself as well as the distorted PCC voltage Researchers have proposed various tools based on conventional power system analysis methods to address this problem The harmonic active method [4] and critical impedance measurement method [5] can achieve a certain degree of accuracy; however, these methods are based on some fundamental assumptions such as prior knowledge of the source impedance The authors have previously [6] presented a novel load modeling approach based on neural networks to identify the nonlinear characteristics of a load Since neural networks are known as effective tools of function approximation, the relationship between the input voltage and output current of the load can be learnt by a neural network after proper training; and then, if fed with a sinusoidal input voltage, the output current of the network should be the true distorted current of the load Therefore, the properly trained neural network can serve as an accurate model of the nonlinear load to predict the true distorted current attributed to the load is() t = i () t + ih() t Nonlinear Load RS Supply Network LS vt () PCC Other Loads Fig A simplified power distribution system The approximation capability of different types of neural networks may vary dramatically In order to find out which type of neural network is suitable for this particular application, this paper investigates the feasibility of three types of neural network: Feed-forward Multilayer perceptron networks (MLP), simple recurrent network (RNN) and echo state network (ESN) In this paper, the neural network-based load modeling scheme is introduced briefly in section II; the structures and training algorithms of MLP, RNN and ESN are described in section III; the experimental setup and the experimental results, which are used to validate the proposed scheme for different types of neural network, are discussed in section IV The training, testing results, and harmonic current prediction results of the three types of neural networks are shown in section V The comparison of these neural networks in performances, required calculation efforts and the convergence properties is shown in section VI Authorized licensed use limited to: Missouri University of Science and Technology Downloaded on March 4, at :33:4 EST from IEEE Xplore Restrictions apply

2 II HARMONIC CURRENT ESTIMATION SCHEME The harmonic current estimation scheme [6] is shown in Fig The distorted power supply voltage v abc and the resultant distorted line current i abc are measured to identify nonlinear characteristic of the load The identification neural network (ANN) is trained to learn the nonlinear characteristics The estimation neural network (ANN), which is an exact replica of ANN with trained and fixed weights, is used to predict the true current harmonics The identification neural network (ANN) is trained with the distorted voltage v abc as input and the distorted load current i abc as output After training, a mathematically generated sine wave is supplied to the estimation neural network ANN, to predict the load current harmonics when the load is supplied by a clean sinusoidal voltage source Ideally, ANN can learn the exact nonlinear characteristics of the load, and therefore the distortion present in the output of ANN, iˆabc distorted, can be considered as the exact current harmonics attributed to the nonlinearity of the load Due to the nature of the sigmoidal transfer function, the outputs of the neurons in the hidden layer are limited to values between and Therefore, the inputs to both ANN and ANN are scaled to [- ] v s III RS LS v sine i abc iˆabc distorted iˆabc i abc v abc Fig Proposed harmonic current estimation scheme NEURAL NETWORK STRUCTURE AND TRAINING A MLP neural networks The MLP network is one of the most popular topologies in use nowadays and its structure is shown in Fig 3 This network consists of a set of input neurons, output neurons and one or more layers of hidden neurons The MLP network is one kind of a family of Feed-forward neural networks, where data flows into the network through the input layer, passes through the hidden layer and finally flows out of the network through the output layer, without any direct data-flow loop The network therefore has a simple interpretation as a form of input-output model, with network weights as free parameters [7] bias x(k) x(k-) x(k-) Input Layer Hidden Layer Output Layer W V ^ y(k) Fig 3 MLP neural network architecture The online training algorithm of an MLP consists of two main steps: ) Forward propagation At any sample step k, the input vector x is fed via the corresponding weights in the input matrix W to each neuron in the hidden layer The active vector a is determined using the following equation: a= Wx () Each of the hidden layer neuron activation in a is then passed through a sigmoidal function to determine the hidden layer decision vector d, which is d =, i {,,, m}, () i ai + e where m is the number of neurons in the hidden layer The decision vector is then fed to the corresponding weights in the output weight matrix V The output of the MLP network is computed as y = ( Vd) T (3) ) Error Back-Propagation The output error e is calculated as e = y y (4) where y is the desired value Then e is back propagated through the network to determine the errors e d and e ai, which are computed as ed = Ve, (5) eai = di ( di ) e (6) di The changes of input and output weight matrices at sample step k are computed according to the following equations: T Δ W( k) = γmδw( k ) + γ geax (7) T Δ V( k) = γmδv( k ) + γ ged (8) where γ and m γ g [,] are the momentum gain and learning gain The final step of the training is to update the input and output weight matrices W and V at sample step k as: W( k) = W( k ) +Δ W( k) (9) V( k) = V( k ) +Δ V( k) () B Simple Recurrent Neural Network A simple recurrent neural network is also known as the Elman network [8] The structure of an RNN is shown in Authorized licensed use limited to: Missouri University of Science and Technology Downloaded on March 4, at :33:4 EST from IEEE Xplore Restrictions apply

3 Fig 4 At each sample step k, the outputs of the hidden neurons at sample step (k-) of the network are used as the additional inputs to the network Adding feedback from previous step introduces a kind of memory to the network, which increases the capability of the network to learn the dynamic characteristics of the system The training algorithm of RNNs is known as the Backpropagation through time method [9] The RNN is unfolded through time, and then back-propagation training is used to update weights as in an MLP bias v(k) v(k-) v(k-) Input Layer Hidden Layer Output Layer W V Fig 4 RNN architecture i(k) C Echo State Neural Network (ESN) The ESN []-[4] is a special form of recurrent neural network proposed in recent years for complex dynamic systems modeling A large (eg hidden neurons) RNN is used as a dynamic reservoir, which can be excited by suitably presented input and/or feedback output The architecture of the ESN used in this paper is shown in Fig 5 u(n) x(n) y(n) Fig 5 ESN architecture The training algorithm of an ESN consists of the following steps: ) Generate a recurrent neural network randomly Three weight matrices are needed They are the input weight matrix W in, internal weight matrix W and output feedback matrix W back Once they have been generated, they will not change during the entire training process Only when a recurrent neural network has the echo state property [9] can it be used for dynamic system modeling Intuitively, the echo state property can be described as follows: if the network has been run for a very long time (from minus infinity time in the definition), the current network state is uniquely determined by the history of the input and the (teacher-forced) output The echo state property is related to the algebraic properties of the weight matrix W; however, there is no known necessary and sufficient algebraic condition which allows one to decide whether the network has the echo state property, given W in, W and W back But there are certain conditions to increase the possibility of the RNN having the echo state property Usually W is generated following the principles described as follows: Generate a sparse matrix W and make sure the mean value of all the weights in it is about zero Normalize W to a matrix W with unit spectral radius as: W W =, () λmax where λ max is the spectral radius of W Scale W to W, as W = αw, () where α < ) Feed the teacher input and teacher output data (training data) to the ESN When the training data are fed to the ESN, they will activate dynamics within the dynamic reservoir At each sample step, compute the internal dynamic reservoir states according to the following equation: in back x( n+ ) = tanh( W u( n+ ) + Wx( n) + W y( n)) (3) where u is the input vector, x is the internal units states vector and y is the output vector Since at time n=, x() and y() are not defined, the following initial conditions are used: Initialize the network state arbitrarily, eg to zero state x()=; Set y()= Then, at each sample step, collect the input u(n) and x(n) together as a new row of a matrix M In other words, the n-th row of M is the concatenated vector of u(n) and x(n) Meanwhile, collect the teacher output data into another matrix T 3) Wash out the initial memory of the dynamic reservoir Since the arbitrarily generated network states (eg zero state) contain an initial memory which is not caused by the input, the information from times n=, n, is discarded (n is different for different systems and the length of input sequence) By the time n, it is assumed that the effects of the arbitrary starting state have died out and the network states are a pure reflection of the teacher-forced input and output So only keep a part of matrix M and T, to obtain two new matrices denoted as M forget and T forget 4) Compute output weights Multiply the pseudoinverse of M forget by T forget, to obtain out T matrix ( W ) : out T forget ( W ) = pseudoinverse( M ) T (4) Authorized licensed use limited to: Missouri University of Science and Technology Downloaded on March 4, at :33:4 EST from IEEE Xplore Restrictions apply

4 Now the output yn ˆ( ) of the ESN approximates the actual system output yn ( ) by the equation, L out yˆ( n) y() n = Wi d() n (5) i= where d(n) contains the values in the dynamic reservoir More specifically, the output weights are computed such that the mean squared training error (MSE) is minimized r r L out MSE= ( yn ( ) yn ˆ( )) = ( yn ( ) Wi dn ( )) (6) r n= r n= i= where r is the length of the Input/Output sequence used for training When the training of ESN is completed, all the weight matrices are fixed, and the network is fed with new input and the estimated output is computed according to the following equations: in back x( n+ ) = tanh( W u( n+ ) + Wx( n) + W y( n)) (7) out yn ( + ) = W ( un ( + ), xn ( + )) (8) IV EXPERIMENTAL RESULTS The proposed scheme is validated by experiment using a drive-connected induction motor as the nonlinear load As shown in Fig 6 The variable speed drive (ABB ACS 5) is supplied from the utility source, which is a distorted power supply For validation purposes, a California Instruments 5 ix harmonic generator, which is capable of providing voltages with programmable distortion levels and zero internal impedance, is used as clean power pure sine wave source to measure the true current harmonics taken by the VSD Utility Supply Network v s Clean Power Supply S v pcc v pcc Data Acquisition ( LABVIEW ) v sine i a Weights ACS5 (VSD) i a error ˆa dist i iˆa Induction Motor Fig 6 Experimental setup The three-phase voltages and currents are measured using LEM LV 5-P voltage transducers and the LEM LAH 5- NP current transducers, respectively The measured data are then acquired and stored using a National Instruments data acquisition system with an Analog-to-Digital resolution of 6 bits, at a sampling frequency of 8 khz The scheme has to be applied to each phase individually The phase A is used as an example to demonstrate the proposed method The measured phase A voltage and current with distorted and clean power supplies, are shown in Fig7 (a) (b) and Fig 7 (c) (d), respectively The total harmonic distortion (THD) of the distorted utility voltage is 45%, and the THD of the CI 5 ix voltage is only % The use of the clean power source is only for validation purposes Fig 7 Measured voltage and current with distorted and clean power supplies V LOAD MODELING USING MLP, RNN AND ESN The feasibility of three different types of neural networks: MLP, RNN and ESN with different numbers of hidden neurons in this application are tested, and the performances are compared Due to page limitations, only the training, testing and prediction results of networks containing 5 neurons are shown in this section A Training, Testing and Prediction Data Construction To test the feasibility of using a MLP, RNN and ESN in this application, the measured data are separated into 3 parts: training data, testing data, and prediction data The training data are the first samples of the measured voltage and current with the distorted power supply For the ESN, the first samples are not used in training, which is called the forget period The testing data are 6 samples after the training data, with the distorted power supply, to test the training results Then 3 k samples of measured voltage and current with clean power supply are used to test the harmonic current prediction results of the three neural networks For the ESN, the learning of load characteristics does not only lie in the trained weights, but also lies in the hidden neuron values, or dynamic reservoir, which can be considered as the memory of recurrent neural networks Therefore the prediction data and the testing data are concatenated for harmonic current prediction, as shown in Fig8 Authorized licensed use limited to: Missouri University of Science and Technology Downloaded on March 4, at :33:4 EST from IEEE Xplore Restrictions apply

5 5-5 - Fig 8 Structure of training, testing and prediction data B Neural Network Training Results The training results of the MLP, RNN and ESN are shown in Fig9 (a), (b), and (c), respectively The Mean Square Errors (MSEs) of the three neural networks are shown in Fig 9 (d) The training of the ESN is only a one-step calculation of output weights by pseudo inverse, so there is no step-bystep MSE available Fig 9 shows that the MSEs of the MLP and the RNN converge to a similar value, while the MSE of the ESN is much smaller This is understandable as the training of the ESN can be considered optimal, as pseudo inverse is used C Neural Network Testing Results To test the learning capability of the three neural networks, a set of voltage and current data with distorted power supply is fed to the neural networks, as shown in Fig 8, and the neural network outputs are compared with the actual outputs in each case The testing results are shown in Fig Fig (a) (b) and (c) show the comparison of neural network outputs and the actual current for the MLP, RNN and ESN respectively This shows that the RNN performs slightly better than the MLP, while the ESN gives a much better testing result From the testing results of the MLP and RNN, it can be concluded that the training of the MLP and RNN is insufficient; while the ESN is already sufficiently trained with the same training data However, it should also be noted that the performance of the ESN is highly dependent on the randomly pre-determined weights of the hidden neurons, which reflects the dynamic feature of each hidden neuron in the dynamic reservoir If the randomly selected weights can accurately represent all the dynamic characteristics of the system, then the ESN can perform accurately as a system simulator Fig shows the testing results of sufficiently trained MLP and RNN For sufficient training, training data with 5 k samples, instead of k samples, are used This figure shows that the RNN performs better than the MLP network, which is caused by its recurrent feature: the outputs are both determined by the inputs and the previous states of the network This recurrent feature makes recurrent neural network a better tool for dynamic system modeling D Harmonic Current Prediction Results After testing the learning capability of the neural networks, a pure sinusoidal wave is supplied to each type of neural network to predict the actual distorted current attributed to the nonlinearity of the load The prediction results are shown in Fig Fig (a), (b), (c) show the predicted currents by MLP, RNN and ESN, respectively, and the actually measured current with a clean power supply The prediction results of sufficiently trained MLP and RNN are shown in Fig 3 It can be observed in MLPNN Ouput (a) MLP SRN RNN Output (b) RNN 8 6 ESN Output - MLPNN RNN ESN 4 - MSE (c) ESN Fig 9 Neural Network Training of MLP, RNN, and ESN (d) MSE Authorized licensed use limited to: Missouri University of Science and Technology Downloaded on March 4, at :33:4 EST from IEEE Xplore Restrictions apply

6 Fig and Fig 3 that, like the testing results, the performances of the RNN and ESN in harmonic current prediction are better than MLP, which is caused by their stronger capability of dynamic system modeling Again, like the testing results, the performance of the ESN is highly dependent on the randomly selected weights of the hidden neurons VI COMPARISON OF MLP, RNN AND ESN A Comparisons of Neural Network Performances The MSEs of MLP, RNN and ESN with 5 and 3 hidden neurons in training, testing and prediction are compared in Table I In this table, the MLP and RNN are both trained with sufficient training data; while a much shorter training data is used for training the ESN During training, the MSE 8 MLPNN Output 8 SRN Output 8 ESN Output Fig Neural Network Testing of MLP, RNN, and ESN 8 MLPNN Output 8 RNN Output Fig Testing results of MLP, RNN after sufficient training 8 MLPNN Output 8 SRN RNN Output 8 ESN Output (a) MLP (b) RNN (c) ESN Fig Distorted current prediction of MLP, RNN, and ESN 8 MLPNN Output 8 RNN Output Fig 3 Distorted current prediction of MLP, RNN after sufficient training Authorized licensed use limited to: Missouri University of Science and Technology Downloaded on March 4, at :33:4 EST from IEEE Xplore Restrictions apply

7 of the ESN is much smaller than the MSEs of the MLP and RNN, which are quite close As explained in Section VB, this is because of the fundamental difference between the training algorithms In testing and prediction, the MSE of the MLP is larger than the MSEs of the RNN and the ESN, which are still quite close This implies that the learning capabilities of the ESN and the RNN are higher than the MLP in nonlinear load modeling However, the performance of the ESN is more dependent on the randomly generated initial weights than the MLP and RNN, as some of the weights in the ESN are not updated during training For the MLP and RNN, the 5-neuron network and 3- neuron network have similar performances, which indicate that the increase of hidden neurons is not improving the learning capability with a certain length of training data The performance of the 5-neuron ESN is even better than a 3-neuron ESN It can be concluded that a 5-neuron ESN is already sufficient to learn the complexity of the system, so an increase of hidden neurons does not imply a better performance Besides, the increase of hidden neurons in the ESN may greatly change the dynamic characteristics of the dynamic reservoir, which may not enhance the learning capability B Calculation effort comparison The computational effort of each neural network is compared in Table II A PC with Pentium 4 CPU working at 34 GHz is used for testing In training, the computation time of the ESN is shorter than for the MLP and RNN, which is again the result of different training algorithms For the ESN, the training is simply some matrix computations without iteration, while for the MLP and RNN, the back-propagation algorithm requires more computational effort In contrast, during testing and prediction, the computational effort required by the ESN is larger than for the MLP and RNN, which is because the hidden neurons are sparsely connected, and more computation is required for every iteration step Because of the same reason, the increase of hidden neurons for the ESN dramatically increases the computational time C Convergence property comparison The convergence of the MLP and RNN seems much easier than for the ESN, because of the dependence of the ESN on the initial weights For the MLP and RNN, the backpropagation algorithm can provide a reliable convergence property, although more training data are needed However, for the ESN, when the initially generated weights can not represent the dynamics of the system, although the pseudoinverse calculation in training is still giving an accurate training result considering its optimal nature, during the testing, the ESN can easily become unstable, as the error is dramatically increased after iterations D Current Harmonics Components Estimation The overall current harmonic prediction results are shown in table III The errors of the estimated current total harmonic distortions (THDs) of the RNN and ESN are smaller than for the MLP; while the performances of the 5- neuron neural networks are better than the 3-neuron networks The measured THD with distorted power supply is 839%; while the measured current THD with clean power supply is 7379% This results show that the neural networks are capable of giving reliable estimation of the true current harmonics attributed to the nonlinear load VII CONCLUSIONS A neural network-based harmonic current prediction method was proposed in the authors previous work to estimate the true current harmonics attributed to the nonlinearity of the load The proposed method has been validated by experiment using a variable speed driveconnected induction motor as the nonlinear load In this paper, to test the feasibility of different kinds of TABLE I MSE COMPARISON FOR TRAINING, TESTING AND PREDICTION Number of Hidden Neurons 5 3 Types of Neural Networks MLP RNN ESN MLP RNN ESN Training MSE min e e-4 Testing MSE Prediction MSE TABLE II CALCULATION TIME COMPARISON FOR TRAINING, TESTING AND PREDICTION Number of Hidden Neurons 5 3 Types of Neural Networks MLP RNN ESN MLP RNN ESN Training Time (s) Testing Time (s) Prediction Time (s) TABLE III COMPARISON OF THE ACTUAL CURRENT HARMONICS AND THE ESTIMATED HARMONICS Harmonic 5 Hidden Neurons 3 Hidden Neurons Current with Current with Clean Power Distorted Power Components MLP RNN ESN MLP RNN ESN Supply Supply Fundamental % % % % % % % % 5-th 56% 494% 496% 587% 5% 486% 4883% 55% 7-th 447% 384% 397% 43% 48% 3773% 3847% 433% -th 85% 78% 835% 89% 36% 49% 635% 37% THD 756% 733% 74% 776% 79% 758% 7379% 839% Authorized licensed use limited to: Missouri University of Science and Technology Downloaded on March 4, at :33:4 EST from IEEE Xplore Restrictions apply

8 neural networks in this application, a performance comparison of three types of neural networks: MLP, RNN and ESN, with different numbers of hidden neurons, is shown in detail The training results, testing results and harmonic current prediction results are compared The computational effort of each neural network is also discussed It is shown that the MLP and RNN require a much larger size of training set than ESN The RNN and ESN give more accurate load modeling results than the MLP, but require more computational effort The choice between RNN and ESN is a tradeoff between convergence property and the size of necessary the training data In other words, when the training data is not sufficient, which is quite possible in practice, the ESN can give better system approximation results than the MLP and RNN The importance of this work lies in its feasible and reliable nature Only the voltages and the currents of a nonlinear load connected to the utility power supply need to be measured; the proposed method can provide an accurate and reliable nonlinear model of the load, which can estimate the true current harmonics attributed to the nonlinearity of the load This provides a reliable and convenient tool for the utilities to evaluate the harmonics attributed to each customer [] H Jaeger, The echo state approach to analyzing and training recurrent neural networks, GMD Report 48, German National Research Center for Information Technology, [3] H Jaeger, Short term memory in echo state networks, GMD Report 5, German National Research Center for Information Technology, [4] H Jaeger, Tutorial on Training Recurrent Neural Networks, Covering BPTT, RTRL, EKF and the Echo State Network approach, GMD Report 59 German National Research Center for Information Technology, ACKNOWLEDGMENT This work is financially supported by the National Science Foundation (NSF), USA, under Grant ECS 65 REFERENCES [] IEEE Standard 59-99, IEEE Recommended Practices and Requirements for Harmonic Control in Electric Power Systems [] M McGranaghan, Overview of the Guide for Applying Harmonic Limits on Power Systems IEEE 59A, in Proceedings of the IEEE International Conference on Harmonics and Quality of Power, Vol, pp , Oct 998 [3] Electromagnetic Compatibility (EMC)-Part 3: Limits-Section 6:Assessment of Emission Limits for Distorting Loads in MV and HV Power Systems, IEC -3-6,996 [4] W Xu, X Liu and Y Liu, An investigation on the validity of power-direction method for harmonic source determination, IEEE Transactions on Power Delivery, Vol 8, Issue, pp 4 9, Jan 3 [5] Chun Li, Wilsun Xu, and T Tayjasanant, A Critical Impedance Based Method for Identifying Harmonic Sources, IEEE Transactions on Power Delivery, Vol 9, Issue, pp , April 4 [6] J Mazumdar, R Harley, F Lambert and G K Venayagamoorthy, Neural Network Based Method for Predicting Nonlinear Load Harmonics, IEEE Transactions on Power Electronics, Vol, Issue 3, pp36-49, May, 7 [7] Simon Haykin, Neural Networks A comprehensive foundation, nd ed Englewood Cliffs, NJ: Prentice Hall, 998 [8] Elman, J L Finding structure in time, Cognitive Science, 4, 99, pp79- [9] Paul J Werbos, Backpropagation through time: What to do and how to do it, in Proc IEEE, Vol 78, no, pp 55-56, 99 [] H Jaeger, Reservoir Riddles: Suggestions for Echo State Network Research, in Proceedings of the IEEE International Conference on Neural Networks, IJCNN 5, Vol, pp 95 9, July/Aug 5 [] D Prokhorov, Echo State Networks: Appeal and Challenges, in Proceedings of the IEEE International Conference on Neural Networks, IJCNN 5, Vol, pp 95 9, July/Aug 5 Authorized licensed use limited to: Missouri University of Science and Technology Downloaded on March 4, at :33:4 EST from IEEE Xplore Restrictions apply