Weighted Fuzzy Time Series Model for Load Forecasting

NCITPA 25 Weighted Fuzzy Time Series Model for Load Forecasting Yao-Lin Huang * Department of Computer and Communication Engineering, De Lin Institute of Technology yaolinhuang@gmail.com * Abstract Electric load forecasting is one of the most critical issues in power system operation and planning. In this paper, we propose a weighted fuzzy time series model (WFTS) on short-term load to improve prediction accuracy. The proposed model uses weighted fuzzy method to calculate the forecasting value in the defuzzification process. The empirical data of load in Chongqing China is used to assess the model s effectiveness. The experimental results prove that the proposed WFTS model is applicable and performs better than other listing methods. Keywords: Weighted fuzzy time series, Load forecasting, Defuzzification. Introduction Load forecasting plays important role in electric energy consumption. Accuracy prediction of short-term load can help power utilities to make efficient planning and management power system []. However, the complicated nonlinear behavior of load time series influences the forecasting techniques. During the past decades, a wide variety of approaches have been developed for load forecasting to improve the estimating accuracy. For example, P.A. Mastorocostas et al. [2] presented a hybrid fuzzy modeling method by employing the orthogonal least squares method to create the fuzzy model and a constrained optimization algorithm to perform the parameter learning for short-term load forecasting. S.E. Papadakis [3] suggested fuzzy models for short-term load forecasting. The model building process includes fuzzy C-regression method, genetic algorithm and a hybrid genetic/least squares algorithm. Yang and Stenzel [4] presented a new regression tree method for short-term load forecasting Both increment and non-increment tree are built according to the historical data to provide the data space partition and input variable selection. Based on state space and Kalman filter approach, H.M. Al-Hamadi [] presented a novel time-varying weather and load model for solving the short-load forecasting problem. Based on the idea of simulative forecasting, Chongqing Kang et al. [5] introduced an integrated model which can synthesize individual forecasting model results. Based on Bayesian clustering by dynamics (BCD) and support vector regression (SVR), Fan et al. [6] proposed a novel forecasting model for day ahead electricity load forecasting. Several other typical approaches for short-term load forecasting can be found in [7-4]. However, load forecasting problems encounter nonlinear data and large fluctuations, and thus it is difficult to make accurate prediction for power utilities. Traditional forecasting methods can only deal with numerical data and fail to solve the 8

NCITPA 25 problems in which the historical data are linguistic values. In recent years, as a good nonlinear model, fuzzy time series have great progress in various applications. Fuzzy time series possess a capability to yield the nonlinear relationships between input and output. Fuzzy set theory was firstly introduced by Zadeh [5-7] to deal with linguistic values. Song and Chissom [8] successfully applied the concept of fuzzy sets in the time-invariant and time-variant fuzzy time series models [9-2] to forecast enrollments. To reduce the complexity of the forecasting procedure and improve forecasting accuracy, Chen [22] introduced the first-order fuzzy logical relationships (FLRs) and IF-THEN rules based on fuzzy theory. Huarng [23, 24] presented computational and heuristic schemes by properly partitioning the lengths of intervals. Wong et al. [25] presented automatically adaptive method which can analyze the window size of fuzzy time series according to the prediction accuracy in the training phase and heuristic rules to yield forecasted values in the testing phase. Yolcu et al. Yu [26] presented the refined approach in the formulation of FLRs to apply the FLRs more effectively. Hwang et al. [27] presented the time-variant fuzzy model using the variations in fuzzy relationship matrix. [28] applied the constrained optimization which used a single-variable to calculate the ratio of the interval lengths. Kuo et al. [29, 3] and Huang et al. [3, 32] presented the hybrid forecasting models which integrated fuzzy methods and particle swarm optimization (PSO) to increase forecast accuracy. Adebiyi et al. [33] presented a hybridized approach in which artificial neural networks was used to increase forecasting accuracy in stock index. Sadaei et al. [34] proposed an integrated model, which can synthesize individual forecasting model results a sophisticated exponentially weighted fuzzy algorithm that is aligned with an enhanced harmony search. Chen et al. [35, 36] presented a new model for fuzzy forecasting based on two-factors second-order fuzzy-trend logical relationship groups and the probabilities of trends of FLRs. From literature review, fuzzy time series are applicable to solve load forecasting problems. For fixed lengths of intervals, the forecasting rules are key issues impacting prediction accuracy. In this paper, a weighted fuzzy time series (WFTS) model is applied to improve forecasting accuracy. The empirical example of short-term load in Chongqing of China [37] is used to evaluate the effectiveness of the proposed forecasting model. The indicators of mean squared error (MSE) and mean absolute percentage error (MAPE) are applied to estimate the forecasting accuracy. In WFTS model, Chen s method [8] is applied to construct fuzzy logical relationships (FLRs) and the weighted method is used to calculate defuzzify values in the training and testing phase. The experimental results show that the WFTS model outperforms other existing models. The remaining content of this paper is organized as follows. Section 2 introduces basic concepts of fuzzy time series. Section 3 illustrates weighted fuzzy time series and the detail procedures using the empirical case of hourly load. Section 4 evaluates the forecasting accuracy of the proposed model by compared with the existing methods. Finally, the paper is concluded in section 5. 2. Literature review Fuzzy theory was first introduced by Zadeh [5]. Later, the concept and application of Fuzzy time series were developed by Song and Chissom [9]. Some basic definitions and 82

NCITPA 25 principles of time series are used to cope with the ambiguity and chaotic of fuzzy forecasting problems. Let U be the universe of discourse, where U = {u, u2,, un}. A fuzzy set A in U can be defined as: A / A( u) / u A( u2) / u2 A( un ) un () where A is the membership function of the fuzzy set A, A : U [, ] and A (u i ), i n, denotes the grade of membership of u i in U and A (u i ) [, ]. In [8-2, 22], some definitions of the fuzzy time series are described as follows. Definition : Let Y(t)(t =,,, 2, 3, 4, ), a subset of real numbers, be the universe of discourse by which fuzzy sets i (t) are defined. If F(t) is a collection of (t), 2 (t),, then F(t) is the fuzzy time series defined as Y(t). Definition 2: When F(t) and F(t ) are fuzzy sets, if there exists a FLR, R(t, t), such that F(t) = F(t ) R(t, t ), where the symbol denotes an operation, then F(t) is said to be caused by F(t ). If F(t) is caused by F(t ) only, the first-order FLR is represented by F(t ) F(t), where F(t ) is the current state and F(t) is the next state, respectively. Definition 3: Let F(t n), F(t n + ),, F(t ), and F(t) be the fuzzy sets in a time series. If F(t) is caused by F(t n), F(t n + ), and F(t ), then the nth-order FLR is defined as follows: F(t n), F(t n + ),, F(t ) F(t), where F(t n), F(t n + ),, F(t ) denotes to the current state and F(t) denotes to the next state. Definition 4: All FLRs can be grouped together according to the same current state. Let A i, A j, and A k be fuzzy sets. By Definition 2, the FLRs are A i A j and A i A k. These two FLRs can be grouped into one fuzzy logical group (FLG) as A i A j, A k. Table Actual load, fuzzy set, and midpoint Hour Actual load Fuzzy set Midpoint 2269.5 A 8 225 2 23.6 A 6 25 3 782.3 A 3 75 4 684.5 A 2 65 45 2695.2 A 2 265 46 2643.4 A 2 265 47 2543 A 255 3. WFTS forecasting model In WFTS forecasting model, a weighted defuzzify method is used to improve fuzzy approach. The key issue of the weighted fuzzy time series (WFTS) is based on the adapting weights between output values of fuzzy rules and the cyclic time series values of previous experience. The empirical example of day-type 24-hour load [37] for WFTS model is 83

NCITPA 25 presented as follows. Step : Define the universe of discourse U as U = [4, 29]. Step 2: Divide U into ten intervals as U = { u, u 2, u 3, u 4, u 5, u 6, u 7, u 8, u 9, u, u, u 2, u 3, u 4, u 5 }. (2) 2 5 Step 3: Define the fuzzy sets as = 2 3 5 6 7 8 9 2 3 5 = 2 3 5 6 7 8 9 2 3 5 = 2 3 5 6 7 8 9 2 3 5 Step 4: Fuzzify historical data as showed in Table in which Columns 2 lists actual load and Column 3 shows fuzzy sets. Table 2 The st-order fuzzy logical relationships for hourly load with 47 training data The st-order FLRs A, A 2 A 2, A 2 2, A 2 3 A 3 2, A 3 4, A 3 6 A 2, A 2 2 A 3 2 Step 5: Generate all fuzzy logical relationships. According to Definitions 2 and 3, Table 2 shows all st-order FLRs. Table 3 The st-order fuzzy logical groups for hourly load 47 training data The st-order FLGs A, A 2 A 8 6, A 8, A 9 A 2, A 2, A 3 A 9 7, A, A A 3 2, A 4, A 6 A 7 A 4 2, A 6 A, A, A 3 A 6 3, A 4, A 6, A 7, A 8 A 2, A 2 A 7 6, A 7, A 8, A 9 A 3 2 84

NCITPA 25 Step 6: Create FLR groups. Based on Definition 3, all FLRs with the same current state are taken together to yield a FLR group. By grouping Table 2, Table 3 shows FLR groups Step 8: Forecast and defuzzify. The WFTS model is used in each forecasting. Suppose a forecast from time t to time t is denoted as A i A j where subscripts i and j denote interval labels. Let Y j, Y (j+), Y (j+2),, Y (j+ π) be the historical data in interval j, m j, m j2,, and m jk be the midpoints of A j, A j2,, and A jk, respectively. For a forecast of time t, F c (t) are forecasted values calculated by Chen s method [22]. The WFTS model calculates the forecasted value F c (t) according to two rules as follows Rule -. If a th-order FLR group is to one, i.e., A i, A i( ),, A i2, A i A j, then the maximum membership value of A j occurs at interval u j. Therefore the forecasted value is the IFC of u j as follows: F(t) =.5 F c (t) +.5 Y(t 24). (3) Rule -2. If a th-order FLR group is to many, i.e., A i, A i( ),, A i2, A i A j, A j2,, A jk, then the maximum membership value of A j occurs at interval u j ( k). Therefore the forecasted value is the average of the IFCs of u j, u j2,, u jk as follows: k ( ) ( ).5 F t c j F t.5 Y ( t 24) (4) k If a next state is not existed, i.e., Equation (7), then F(t) is conducted by the following rules: Rule 2-. If the current state of a FLR exists in the current state of trained FLR groups, then Rules - and -2 are applied to forecast. Rule 2-2. If the current state of the FLR does not exist in the current state of trained FLR groups and F(t ) is the forecasted value of time t, then F(t) = F(t ). Step 9: Compute forecasting accuracy. The measure indicators of MSE and MAPE are used to estimate the forecasting accuracy. Let N be the number of forecasted data. The estimated indicators of MSE and MAPE are defined as follows: N 2 ( F i Y i i v ( ) ( )) MSE (5) N MAPE N N i Fv ( i) Y( i) % Y( i) (6) 85

Electric load NCITPA 25 3 28 Actual Load 26 24 22 2 8 6 4 3 5 7 9 3 5 7 9 2 23 25 27 29 3 33 35 37 39 4 43 45 47 Hours Fig.. Time series data of electric load 4. Experimental Results The actual data of hourly load in Chongqing of China [37] are listed in Table for model validation. As shown in Fig., the historical data are separated into two parts: in-sample data from the first -24 hours for training and out-of-sample data from the next 25-48 hours for testing. Table 4 Comparison of 24-hour load forecasting in the testing phase Time Actual 2-order 3-order Day Proposed Naïve FTS [22] Hour Load MA MA Ahead WFTS 272.4 299.2 239.3 2396. 2269.5 22 26.8 2 863.2 272.4 235.8 223.3 23.6 983.33 882.82 3 626. 863.2 967.8 244.9 782.3 25 867.25 4 624.9 626. 744.7 853.9 684.5 7 696.2 5 553. 624.9 625.5 74.7 692.4 7 66.45 6 556.2 553. 589 6.4 622.9 65 65 22 2643.4 2695.2 2723.8 2667.6 2569.6 265 265 23 2543 2643.4 2669.3 2697 249.4 265 2424.6 24 2222.8 2543 2593.2 2627.2 299.2 2583.33 2327.87 MSE 2429.7 4733.6 689.69 683.93 34.55 433.5 MAPE 5.95% 8.3% 9.83% 5.52% 6.96% 4.7% In the testing phase, hourly data of electric load are used to verify the forecasting accuracy. A comparison of electric load among naïve method, 2-order and 3-order move average, 24-hour ahead method, FTS method [22] and WFTS model is shown in Table 4. For proposed WFTS model, the MSE value is 433.5 and the MAPE value is 4.7% for st-order FLRs. Fig. 3 and Fig. 4 show the trend of forecasting error rates which are MSE and MAPE values, respectively. From the graph of time series plot revealed in Fig. 2-4, we conclude that the 86

MAPE MSE Load NCITPA 25 proposed WFTS model can provide valuable information used in future planning and management for power utilities. 3 28 26 24 Real Naïve 2-order MA 3-order MA DayAhead FTS WFTS 22 2 8 6 4 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 2 2 22 23 24 Hours Fig. 2. Comparison of load forecasting in the testing phase 7 6 5 4 3 2 Naïve 2-order MA 3-order MA Day Ahead Forecasting Models FTS WFTS Fig. 3. Comparison of MSE in different models % % 9% 8% 7% 6% 5% 4% 3% Naïve 2-order MA 3-order MA Day Ahead FTS Forecasting Models WFTS Fig. 4. Comparison of MAPE in different models 87

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