Short-term load forecasting using SVR (support vector regression)-based radial basis function neural network with dual extended Kalman filter
Highlights
► Propose a prediction model integrating SVR, RBFNN, and DEKF for short-term load forecasting. ► SVR model is employed to determine both the structure and initial parameters of the RBFNN. ► DEKF is used as the learning algorithm to optimize the parameters of the RBFNN. ► Illustrate the performance of the proposed SVR–DEKF–RBFNN for predicting real-load data. ► Compare the simulation results with DEKF–RBFNN and GRD-RBFNN models.
Introduction
Load forecasting is an important task in the modern power system planning, operation and control [1], [2], [3], [4], [5], [6], [7], [8]. According to various kinds of lead time, the load forecasting can be generally divided into short-term, mid-term, and long-term. Short-term load forecasting, ranging from 1 h to several days ahead, is a key work for maintenance scheduling, power system security, economic dispatch, and market operation. Accurate load forecasting can improve the security of the power system and promote the economic efficiency of electric utilities.
Some techniques have been developed for short-term load forecasting during the last few years. These approaches found in the extant literature can mainly be divided into two categories: traditional approaches and artificial intelligence based methods. Traditional approaches are based on statistical models including linear regression methods [9], exponential smoothing [10], Box-Jenkins approaches [11], and Kalman filters [12]. Basically, most of the traditional approaches are based on linear analysis. However, the load series are usually nonlinear. These models have difficulties in solving the load forecasting. To copy with the nonlinearity, the artificial intelligence based techniques such as the neural network models [13], expert system models [14], fuzzy inference [15], and a chaotic ant swarm optimization method [16] have been used to obtain promising results.
ANNs (Artificial neural networks) have the capability for providing good solutions to construct complex and nonlinear relationships through a training process using historical data. The ANN models have been widely applied in the short-term load forecasting. An annealing clustering ANN [17], a BP (back propagation) neural network with rough set [18], a neural network combining wavelet transform and adaptive mutation particle swarm optimization [19], an ANN based on fuzzy logic methods [20], and Bayesian neural networks integrating Monte Carlo algorithm [21] have been proposed for short-term load forecasting. Among ANNs, the radial basis function neural network (RBFNN) has received considerable applications in many fields [22], [23], [24], [25]. Since RBFNN has only one hidden layer and has fast convergence speed. Besides, the RBFNN is often referred to as model-free estimators since they can be used to approximate the desired outputs without requiring a mathematical description of how the outputs functionally depend on the inputs [26], [27].
SVM (Support vector machine) is a powerful machine learning method based on statistical learning theory, which is proposed by Vapnik [28]. SVM implements the structural risk minimization principle instead of the usual empirical risk by minimizing the fitting error and reducing the upper bound of the generalization error at the same time, thus increasing the generalization ability of the model. Based on this principle, SVM provides a promising network structure. Moreover, the SVM will be equivalent to solving a linear constrained QP (quadratic programming) problem so that the solution of SVM is always unique and globally optimal. Originally, SVM has been developed for solving the pattern classification [29] and function estimation [30], [31]. With the introduction of Vapnik's ε-insensitive loss function, SVM has also been extended to solve the regression problems called SVR (support vector regression) [32]. Recently, SVR has been applied to short-term load forecasting and has shown good results [33], [34], [35], [36], [37], [38], [39].
The KF (Kalman filter) is an optimal recursive minimum-variance state estimator for linear dynamic systems. Through linearization, the EKF (extended Kalamn filter) has been widely adopted for state estimation of nonlinear system. The EKF-based training of neural networks, both multilayer perceptions and recurrent networks, has proven to be reliable and practical for many applications [40], [41]. Recently, EKF [42] and DEKF [43] have been also used for training RBFNNs. They have illustrated the ability of restraining noises in the training set, and the results have less error.
In this paper, a hybrid algorithm of prediction approach is proposed for short-term load forecasting. In the proposed algorithm, integrating the advantages of the SVR, RBFNN, and DEKF (dual extended Kalamn filter) to develop a prediction model. In the prediction model, first, SVR determines the initial parameters and the structures of the RBFNN. Then, DEKF is used as the learning technique to adjust the parameters of the RBFNN effectively. To evaluate the performance of the proposed method, training and testing on the real data of the historical electric load from Taipower Company is illustrated. The prediction results show that the proposed SVR–DEKF–RBFNN exhibits superior performance over DEKF–RBFNN and GRD-RBFNN algorithms.
Section snippets
Architecture of RBFNN
A RBFNN (radial basis function neural network) consists of three layers, the input layer, the hidden layer, and the output layer. The architecture of an RBFNN is shown in Fig. 1. When the Gaussian function is chosen as the radial basis function, the outputs of an RBFNN can be expressed in the form.where is the input vector, is the jth output, wij is the synaptic weight between the ith hidden neuron and
Review of SVR
Based on statistical and mathematical learning theory [28], SVR method approximates an unknown function by mapping input data into a high dimensional feature space through a nonlinear mapping function, and then a linear problem is constructed in this feature space. Consider a set of training data, such that is the input vector, is the corresponding output, and m denotes the number of data in the training set. The linear problem is formulated as
Kalman filter
The KF (Kalman filter) is an optimal dynamic estimator from available measurement data [44]. Originally, it was introduced in linear models, but Kalman filters also solve the nonlinear model through linearization. Consider the dynamic system modelwhere k is the time step, x(k) is the state variable, A(k) is the state transition matrix, y(k) is the measurement value, H(k) is the measurement matrix. Moreover, v(k) and q(k) are the zero-mean Gaussian process
Formulation of the DEKF for training the RBFNN
The essence of the dual Kalman framework [46] is to perform both the Kalman state and weight filter in parallel, simultaneously update the estimates of Kalman state and weight. At every step, an EKF state filter estimates the state using the current model estimate , while the EKF weight filter estimates the weights using the current state estimate . The brief structure of the dual estimation method is shown in Fig. 2.
The Kalman filter framework for designing a feed forward neural
Simulation results
The effectiveness of the proposed SVR–DEKF–RBFNN is evaluated using the real-load data from the Taipower Company in the year of 2007. Three cases are used to investigate the performance of the proposed model for load forecasting. Case 1 is to forecast one-day, three-day and seven-day ahead forecast of weekdays (Monday to Friday). Case 2 is to perform one-day, three-day and seven-day ahead forecast of weekends (Saturday). Case 3 is to investigate one-day and three-day ahead forecast of holidays
Conclusions
In this paper, an integrated prediction approach was proposed for short-term load forecasting. The proposed RBFNN is based on the combination of SVR and DEKF methods. After the initial parameters and structures of the RBFNN using SVR method, the DEKF is used to tune the parameters of RBFNNs to obtain an optimal model. Then, the optimal RBFNN is adopted to perform the short-term load prediction. The proposed method has been tested for three cases with different real-load data acquired from the
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