A new method for short-term load forecasting based on chaotic time series and neural network

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1 A new method for short-term load forecasting based on chaotic time series and neural network Sajjad Kouhi*, Navid Taghizadegan Electrical Engineering Department, Azarbaijan Shahid Madani University, Tabriz, Iran. *Corresponding author. Tel.: address: (S. Kouhi). Tel.: ; Abstract This paper illustrates application of neural network to chaotic time series prediction. Electricity load time series is modeled as chaotic time series and predicted by using MLP neural network. For the sake of training NN, LM training algorithm is used that is one of the most efficient learning mechanisms for the prediction. The LM method trains a NN 1-1 times faster than the gradient descent back propagation (GDBP) algorithm. Proposed method is examined in New York electricity market with different forecasting horizons. Keywords: Chaotic time series, Neural network, Short term load forecast 1. Introduction Time series forecasting involves the study of the past and present behavior of the system to prediction of the future. Chaotic time series are a kind of between regular and random systems. Chaos theory is used to scrutiny the behavior of the dynamical systems that are extremely sensitive to initial conditions such as noise and error [1], [2]. The chaotic time series prediction has been used in wide range of applications such as: in finance [3], signal processing [4], power load [5], weather forecast [6], hydrological prediction [7] and Sunspot prediction [8]. Short-term load forecasting (STLF) plays a key role in operation of both traditional and deregulated power systems. In deregulated electricity market, STLF is a useful tool for economic and reliable operation of power system. Many operating decisions are based on load forecast such as: dispatch scheduling of generating production, reliability and security analysis and maintenance plan for generators [9]. Therefore, load forecasts are vital for the market players in competitive electricity market [1]. Hence, improving the accuracy of STLF can increase the appropriateness of planning and scheduling and reduce operational costs of power systems. Load forecasting algorithms are includes traditional methods and modern intelligent methods [11]. Traditional methods based on mathematical statistics including regression analysis method [12], autoregressive integrated moving average (ARIMA) [13], state space model [14], exponential soothing [15], and etc. These methods have the advantage of mature technology and simple algorithm, but these are based on linear analysis and none of them can forecast the nonlinear load series accurately [11]. The modern intelligent forecasting methods have shown better performance for non-linearity of the time series. Also, they do not require any complex mathematical formulations or quantitative correlation between inputs and outputs. Effective utilization of intelligent algorithms in the context of ill-defined processes (such as load time series), have led to their wide application in STLF [9]. The intelligent algorithms are based on artificial neural network (ANN) based methods [16], [17]. In this paper, a neural network with Multi-layer perception (MLP) structure is used for modeling and prediction of a chaotic electricity load. Levenberg-Marquardt (LM) training algorithm is the one of the 227

2 most efficient learning mechanism [18], which trains a NN 1-1 times faster than the gradient descent back propagation (GDBP) algorithm [19]. Therefore, LM learning algorithm is used to trains NN. The remaining parts of the paper are organized as follows. Chaotic time series prediction method is described in section 2. In the third section, the numerical results of the proposed method are illustrated. Finally conclusion is discussed in section Chaotic time series prediction method A) Phase space reconstruction Phase space reconstruction is the process of finding a space in which the dynamics are smooth and no intersections or overlaps occur in the orbits of the attractor. Embedding theorem of the Taken provides the conditions under which a chaotic time-series can be reconstructed into a M-dimensional vector with two conditions: the embedding dimension and the time delay [2]. Given an chaotic time series x ( t), t 1,2,, N (load time-series), selecting the embedding dimension M and the delay-time t, the phase space can be explained as follows: [ x(1), x(1 t ),, x(1 ( M 1) * t ) 2 1 [ x(2), x(2 t ),, x(2 ( M 1) * t ) L [ x( L), x( L t ),, x( L ( M 1) * t ) Where, L N ( M 1) * t is the length of the reconstructed phase space and is a point or vector in t the construction phase space. There is a determinism map f(x) (mapping the t+1 to the t ) meeting the following equation: t 1 f ( t ) (2) The f(x) is so called the chaotic time series predicting model. Neural network with Multi-layer perception (MLP) structure is used to do mapping procedure. (1) B) Multi-layer perception NN For the forecaster blocks, we are used Multi-layer perception (MLP) structure with Levenberg-Marquardt (LM) learning algorithm. LM training algorithm is the one of the most efficient learning mechanism for the prediction[18]. The LM method trains a NN 1-1 times faster than the gradient descent back propagation (GDBP) algorithm [19]. Mathematical details of the LM algorithm are discussed in [21]. In [22] Kolmogorov s theorem express, a problem can be solved with MLP by using one hidden layer, provided it has the proper number of neurons. So, we are used one hidden layer in the MLP in structure NN. By accounting chaos feature selection and MLP-NN, The proposed forecast method consists of three adjustable parameters: neurons of neural network s hidden layer, embedding dimension, and delay-time. In order to fine-tune the adjustable parameters, this paper has been incorporated a kind of cross-validation technique. Selection of the validation set, which is unseen for the forecaster influence efficiency of the cross-validation technique [23]. The day before the forecast day is considered as validation set and the 39 days before considered as training set. To fine-tune the adjustable parameter we are used to step procedure. At first step, the N H is assumed constant and Training phase of the forecaster with different values for the chaos feature selection parameters is executed and minimum of the validation set error is selected as the optimal point. Then, the feature selection parameters are assumed constant and previous step executed for N H. The validation error is also used to avoid the overtraining problem of the forecaster. 228

3 3. Simulations The proposed chaotic time series prediction has been tested for load forecast of New York electricity market in 24, which is widely known in the US. Historical data of New York electricity market has been presented in their website [24]. Forecast time step is assumed one hour. By using cross-validation technique the adjustable parameters are determined as: delay-time 45 min (.75hour), embedding dimension is 12 and number of hidden neurons is 13. Data of the 25 hours ago are using to forecast. Data of Previous day is select as validation set and 24 other are used as training data. The forecast accuracy is in term of mean absolute percentage error (MAPE). MAPE is defined as follows: N 1 LActual ( k ) LForecasted ( k ) MAPE (3) N k 1 LActual ( k ) Where, N is the forecast horizon, L Actual (k) is the actual load of hour k, and L Forecasted (k) is the load forecast of hour k. Actual signal and predicted load curve for one-day (24 hour), three-day (72 hour) and one-week (168 hour) are shown in figures 1, 2, and 3, respectively.prediction error for figures 1, 2, and 3 are.8169%, %, and %, respectively. As seen in these figures, the forecasted curve accurately follows actual load. Also in fig. 4, the predicted values are plotted with respect to observed values which show a high correlation coefficient between predicted and observed values. Fig. 1. Actual and predicted load for one day (24 hour) Fig. 2 Actual and predicted load for three day (72 hour) 229

4 Fig. 3 Actual and predicted load for 7 day (168 hour) Fig. 4 Correlation function between predicted and actual values for load prediction of New York electricity market 4. Conclusion In this this paper, electricity load is modeled as chaotic time series. A neural network with Multi-layer perception (MLP) structure is used is used for forecasting of chaotic time series of electricity load. LM training algorithm is the one of the most efficient learning mechanism that trains a NN 1-1 times faster than the gradient descent back propagation (GDBP) algorithm. The prediction results show high correlation coefficients with observed data. This method can be adapted easily to any system by tuning adjustable parameters. References [1] E. N. Lorenz, Deterministic nonperiodic flow, Journal of the atmospheric sciences, vol. 2, no. 2, pp ,

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