Power load combination forecasting based on triangular fuzzy discrete difference equation forecasting model and PSO-SVR

Abstract

In this paper, we develop a new triangular fuzzy series combination forecasting method based on triangular fuzzy discrete difference equation forecasting model and PSO-SVR, and use the developed forecasting method to power load forecasting. First, we propose a triangular fuzzy discrete difference equation (TFDDE) forecasting model to predict the triangular fuzzy series, which can accurately predict the fluctuating trend and is suitable for small sample data. Then, the support vector regression optimized by particle swarm optimization (PSO-SVR) is adopted to further improve the forecast result of TFDDE forecasting model, in which the parameters of support vector regression are optimally obtained by particle swarm optimization algorithm so as to avoid the blindness of artificial selection. Finally, the practical example of load forecasting of US PJM power market is employed to illustrate the proposed forecasting method. The experimental results show that the proposed forecasting method produces much better forecasting performance than some existing triangular fuzzy series models. The proposed combination forecasting method, which fully capitalizes on the time series forecasting model and intelligent algorithm, makes the triangular fuzzy series prediction more accurate than before and has good applicability. This is the first attempt of employing discrete difference equation theory for the triangular fuzzy series forecasting.

Keywords

Power load forecasting triangular fuzzy discrete difference equation support vector regression particle swarm optimization

1 Introduction

Power load forecasting plays an important role in the management of modern power systems. Therefore, improving the accuracy and efficiency of power load forecasting has become an important research field in the operation and management of modern power systems [1]. Load forecasting studies can be categorized based on forecasting period into three categories: very short term load forecast, short term load forecast and medium-long term load forecast [2]. Compared with very short term load forecast and short term load forecast, medium-long term load forecast is more significant in distribution power system planning and analysis [3]. Thus, this article focus on improving the accuracy of medium-long term load forecast.

The technical literature displays a wide range of methodologies and models for load forecasting, among them, the classical power load forecasting methods includes exponential smoothing [4], ARMA [5], Box-Jenkins ARIMA [6], regression analysis [7], and transfer function (dynamic regression) [8], expert systems [9], and other traditional mathematical statistics methods [10, 11]. Although the forecasting results of these methods usually show good performances, their forecasting precisions still need to be improved.

In recent years, with the development of artificial intelligence technology, artificial intelligence has been gradually introduced into power load forecasting, which includes artificial neural network [12], radial basis function neural network [13], particle swarm optimization [14], general simulation (GSIM) theory [15], support vector regression (SVR) [16], least squares support vector machine and fruit fly optimized algorithm [17], genetic algorithm optimized cyclic support vector machine [18] and differential evolution algorithm optimized support vector regression [19]. AlRashidi and El-Naggar used the Egyptian networks and particle swarm optimization to forecast the long term power load peak values [20]. Additionally, Hu et al developed a short-term power load forecasting model based on the generalized regression neural network with decreasing step fruit fly optimization algorithm [21]. However, there are still shortcomings among them, such as complex procedures, low precision, and poor stability [22].

Load forecasting is vital to the whole power industry. However, it is a difficult task. Firstly, because the load time series exhibit multiple levels of seasonality [23]. Secondly, because there are many exogenous variables that may be considered, with highly volatile with occasional large spikes [24], so using fuzzy numbers to represent power load data is more reasonable than precise numbers [25]. But previous studies conducted on precise numbers are not applicable to these data. In the fuzzy environment, triangular fuzzy numbers are a common expression form for uncertain information, which compensates for the lack of precise numbers and interval numbers. Thus, it is natural to develop a triangular fuzzy number series forecasting method for the power load forecasting.

Zeng et al. [25] proposed a triangular fuzzy series forecasting based on the grey model and neural network. However, the essence of grey model is matching the raw series with an exponential type curve, and its prediction curve is a smooth curve which gives expression to the development trend of the series, so the grey model cannot predict the fluctuating trend of the series effectively. Recently, Chen et al. [26] provided an efficient gradient forecasting search method utilizing the discrete difference equation model. Their work shows that the discrete difference equation forecasting method is a potential time series forecasting tool for the time series with inflection points. Thus, an interesting and important issue to be solved is how to develop a triangular fuzzy discrete difference equation prediction model. Up to now, there has been no research about this issue.

In this paper, we propose a power load combination forecasting approach based on triangular fuzzy series discrete difference equation forecasting model and PSO-SVR. A triangular fuzzy discrete difference equation (TFDDE) prediction model is developed, which can predict the fluctuating trend of the triangular fuzzy series effectively. Moreover, in order to improve the accuracy of power load forecasting, we use the support vector regression optimized by particle swarm optimization algorithms (SVR-PSO) to adjust the forecasting result of TFDDE model. Finally, an illustrated example for forecasting power load is used to verify the effectiveness of our proposed method. The proposed method provides a new triangular fuzzy combination forecasting framework in enrich the power load forecasting. This is also the first attempt of employing discrete difference equation for the triangular fuzzy series forecasting.

The rest of the paper is organized as follows. Section 2 introduces the basic concepts of triangular fuzzy numbers and discrete difference equation prediction model. Triangular fuzzy discrete difference equation forecasting model is constructed in Section 3. In Section 4, we develop a triangular fuzzy series combination forecasting framework based on discrete difference equation forecasting model and PSO-SVR. Section 5 presents the power load forecasting practical example. Finally, conclusions are drawn in Section 6.

2 Preliminaries

2.1 Triangular fuzzy numbers

Zadeh [27] proposed the concept of fuzzy sets, which are widely used in uncertain environment. Triangular fuzzy number is one of the major components, which is defined as follows.

Definition 1. [28] If a fuzzy number $\tilde{x}$ on the real domain R, its membership function $μ_{\tilde{x}} (x)$ is $μ_{\tilde{x}} (x) = {\begin{matrix} \frac{x - x_{l}}{x_{m} - x_{l}}, & x_{l} ⩽ x ⩽ x_{m}, \\ \frac{x_{r} - x}{x_{r} - x_{m}}, & x_{m} ⩽ x ⩽ x_{r}, \\ 0, & else . \end{matrix}$ (1)

Then $\tilde{x}$ is called a triangular fuzzy number, denoted as $\tilde{x} = [x_{l}, x_{m}, x_{r}]$ , where 0 < x_l ⩽ x_m ⩽ x_r. When x_l = x_m = x_r, $\tilde{x}$ degenerates into a precise number.

2.2 The discrete difference equation prediction model

The discrete difference equation (DDE) prediction model was developed by Chen and Lee [26], which has higher prediction accuracy than the grey prediction model GM for dynamic time series with inflection points. The DDE prediction model can be detailed as follows.

Let x⁽⁰⁾ = {x⁽⁰⁾ (1) , x⁽⁰⁾ (2) , ⋯ , x⁽⁰⁾ (n)}, n ∈ Z, be the original time series. Then, we can get an increasing sequence x⁽¹⁾ = {x⁽¹⁾ (1) , x⁽¹⁾ (2) , … x⁽¹⁾ (n)} based on accumulated generating operation (AGO) as Equation (2). $x^{(1)} (i) = \sum_{k = 1}^{i} x^{(0)} (k), i = 1, 2, \dots, n .$ (2)

Then, a second-order discrete difference equation is constructed as follows: $\begin{matrix} x^{(1)} (p + 2) + α x^{(1)} (p + 1) + β x^{(1)} (p) = 0, \\ p = 1, 2, \dots, n - 2 . \end{matrix}$ (3)

Among them, α and β are the parameters to be estimated, and p take the general integer. Linear least squares estimation [29] is applied to estimate α and β. Then, the discrete difference Equation (3) can be solved by invoking the initial conditions [26].

Finally, we can use the inverse accumulative generation operation to restore the AGO operation and obtain the predicted values, which is as follows: ${\hat{x}}^{(0)} (i) = {\hat{x}}^{(1)} (i) - {\hat{x}}^{(1)} (i - 1), i = 1, 2, \dots, n .$ (4)

Thus, we can get the final predicted results of the original sequence ${\hat{x}}^{(0)} = {{\hat{x}}^{(0)} (1), {\hat{x}}^{(0)} (2), \dots, {\hat{x}}^{(0)} (n)}$ .

3 Triangular fuzzy discrete difference equation forecasting model

In order to forecast triangular fuzzy time series, we will develop a triangular fuzzy discrete difference equation (TFDDE) forecasting model. The operation of TFDDE forecasting model can be detailed as follows:

(1) Gather original time series data.

Let ${\tilde{x}}^{(0)} = {{\tilde{x}}^{(0)} (1), {\tilde{x}}^{(0)} (2), \dots, {\tilde{x}}^{(0)} (n)}$ be the original triangular fuzzy time series, where ${\tilde{x}}^{(0)} (i) = [x_{l}^{(0)} (i), x_{m}^{(0)} (i), x_{r}^{(0)} (i)]$ , i = 1, 2, ⋯ , n.

(2) Triangular fuzzy sliding average (TFSA)

Due to the irregular variation of the original triangular fuzzy time series, sliding average is applied to eliminate the accidental factors and find out the development trend of fuzzy time series and improve the prediction accuracy. The calculation formula of the sliding average can be written as: ${\tilde{y}}^{(0)} (i) = {\begin{matrix} \frac{1}{3} [2 {\tilde{x}}^{(0)} (1) \oplus {\tilde{x}}^{(0)} (2)], i = 1; \\ \frac{1}{3} [{\tilde{x}}^{(0)} (i - 1) \oplus {\tilde{x}}^{(0)} (i) \oplus {\tilde{x}}^{(0)} (i + 1)], i = 2, \dots, n - 1 \\ \frac{1}{3} [{\tilde{x}}^{(0)} (n - 1) \oplus 2 {\tilde{x}}^{(0)} (n)], i = n . \end{matrix}$ (5)

In which, ${\tilde{y}}^{(0)} (i) = [y_{l}^{(0)} (i), y_{m}^{(0)} (i), y_{r}^{(0)} (i)]$ is the modified triangular fuzzy time series.

(3) Triangular fuzzy accumulated generating operation (TFAGO)

The increasing TFAGO series of ${\tilde{y}}^{(0)}$ is ${\tilde{y}}^{(1)} = {{\tilde{y}}^{(1)} (1), {\tilde{y}}^{(1)} (2), \dots, {\tilde{y}}^{(1)} (n)}$ n ∈ Z, which can be obtained by Equation (6).

$\begin{matrix} {\tilde{y}}^{(1)} (i) & = & \oplus_{k = 1}^{i} {\tilde{y}}^{(0)} (k) \\ = & [\sum_{k = 1}^{i} y_{l}^{(0)} (k), \sum_{k = 1}^{i} y_{m}^{(0)} (k), \sum_{k = 1}^{i} y_{r}^{(0)} (k)] \end{matrix}$ (6)

(4) Build triangular fuzzy discrete difference equation (TFDDE) model

We build triangular fuzzy second-order discrete difference equation model for TFAGO series as Equation (7). ${\tilde{y}}^{(1)} (p + 2) + α {\tilde{y}}^{(1)} (p + 1) + β {\tilde{y}}^{(1)} (p) = 0$ (7)

In Equation (7), the integral development coefficients, α and β represent the integral development tendency of the triangular fuzzy series, not the tendency of a single boundary point. And can be estimated by ordinary least square method.

(5) Solve of triangular fuzzy discrete difference equation.

According to the ordinary least square method, we have $\begin{matrix} (\begin{matrix} α_{l} \\ β_{l} \end{matrix}) = {(A^{T} A)}^{- 1} A^{T} y_{l}^{(1)}, (\begin{matrix} α_{m} \\ β_{m} \end{matrix}) = {(B^{T} B)}^{- 1} B^{T} y_{m}^{(1)}, \\ (\begin{matrix} α_{r} \\ β_{r} \end{matrix}) = {(C^{T} C)}^{- 1} C^{T} y_{r}^{(1)} . \end{matrix}$ (8) where $\begin{matrix} A & = & [\begin{matrix} - y_{l}^{(1)} (2) & - y_{l}^{(1)} (1) \\ - y_{l}^{(1)} (3) & - y_{l}^{(1)} (2) \\ ⋮ & ⋮ \\ - y_{l}^{(1)} (n - 1) & - y_{l}^{(1)} (n - 2) \end{matrix}], y_{l}^{(1)} = [\begin{matrix} y_{l}^{(1)} (3) \\ y_{l}^{(1)} (4) \\ \dots \\ y_{l}^{(1)} (n) \end{matrix}], \\ B & = & [\begin{matrix} - y_{m}^{(1)} (2) & - y_{m}^{(1)} (1) \\ - y_{m}^{(1)} (3) & - y_{m}^{(1)} (2) \\ ⋮ & ⋮ \\ - y_{m}^{(1)} (n - 1) & - y_{m}^{(1)} (n - 2) \end{matrix}], y_{m}^{(1)} = [\begin{matrix} y_{m}^{(1)} (3) \\ y_{m}^{(1)} (4) \\ \dots \\ y_{m}^{(1)} (n) \end{matrix}], \\ C & = & [\begin{matrix} - y_{r}^{(1)} (2) & - y_{r}^{(1)} (1) \\ - y_{r}^{(1)} (3) & - y_{r}^{(1)} (2) \\ ⋮ & ⋮ \\ - y_{r}^{(1)} (n - 1) & - y_{r}^{(1)} (n - 2) \end{matrix}], y_{r}^{(1)} = [\begin{matrix} y_{r}^{(1)} (3) \\ y_{r}^{(1)} (4) \\ \dots \\ y_{r}^{(1)} (n) \end{matrix}] . \end{matrix}$

According to Equation (8), we can get the development coefficients of three boundary points respectively. And, the integral development coefficients can be obtained by aggregating development coefficients of the three boundary points of the triangular fuzzy series based on Equation (9). $α = a α_{l} + b α_{m} + c α_{r}, β = a β_{l} + b β_{m} + c β_{r},$ (9)

Where a, b, c > 0 and a + b + c = 1. Additionally, since the mid boundary point plays the leading role of the three boundary points, thus b > a and b > c.

After parameter estimation, the values of α and β are substituted into Equation (10) to obtain the predicted value of ${\tilde{y}}^{(1)} (i)$ , which are denoted as ${\hat{\tilde{y}}}^{(1)} (i)$ , i = 1, 2, ⋯ , n. ${\begin{matrix} {\hat{\tilde{y}}}^{(1)} (1) = {\tilde{x}}^{(0)} (1), {\hat{\tilde{y}}}^{(1)} (2) = {\tilde{x}}^{(0)} (1) + {\tilde{x}}^{(0)} (2), \\ {\hat{\tilde{y}}}^{(1)} (i) = - α {\hat{\tilde{y}}}^{(1)} (i - 1) - β {\hat{\tilde{y}}}^{(1)} (i - 2), i = 3, 4, \dots, n . \end{matrix}$ (10)

(6) Apply inverse triangular fuzzy accumulated generating operation (I-TFAGO)

Finally, we can obtain the predicted value of original triangular fuzzy time series, ${\hat{\tilde{y}}}^{(0)} (i) = [{\hat{y}}_{l}^{(0)} (i), {\hat{y}}_{m}^{(0)} (i), {\hat{y}}_{r}^{(0)} (i)]$ , i = 3, 4, ⋯ , n, based on the inverse triangular fuzzy accumulated generating operation (I-TFAGO), where

$\begin{matrix} {\hat{y}}_{l}^{(0)} (i) = {\hat{y}}_{l}^{(1)} (i) - {\hat{y}}_{l}^{(1)} (i - 1), {\hat{y}}_{m}^{(0)} (i) = {\hat{y}}_{m}^{(1)} (i) - {\hat{y}}_{m}^{(1)} (i - 1), \\ {\hat{y}}_{r}^{(0)} (i) = {\hat{y}}_{r}^{(1)} (i) - {\hat{y}}_{r}^{(1)} (i - 1) \end{matrix}$ (11)

Remark 1. When $x_{l}^{(0)} (i)$ i = 1, 2, ⋯ , n, ${\tilde{x}}^{(0)} (i)$ degenerates a precise number. At the same time, if triangular fuzzy sliding average is skipped over, then Equation (9) degenerates into the Equation (3). Thus, the DDE prediction model is a special case of the developed TFDDE forecasting model.

4 Combination forecasting based on TFDDE and PSO-SVR

4.1 Transformation of the triangular fuzzy series

In order to measure the residual between the original data ${\tilde{x}}^{(0)}$ and the predicted value of TFDDE forecasting model ${\hat{\tilde{y}}}^{(0)}$ , the original triangular fuzzy number need to be transformed. The detailed transformation process, which can ensure the triangle fuzzy number boundary and sequence integrity, is as follows:

For the triangular fuzzy number series $\tilde{x} = {\tilde{x} (1), \tilde{x} (2), \dots, \tilde{x} (n)}$ , $\tilde{x} (i) = [x_{l} (i), x_{m} (i), x_{r} (i)]$ (i = 1, 2, ⋯ , n) can be transformed into s (i), f (i) and m (i). The conversion process is as follows: ${\begin{matrix} s (i) = \frac{x_{r} (i) - x_{l} (i)}{2}, \\ f (i) = \frac{x_{l} (i) + x_{m} (i) + x_{r} (i)}{3}, \\ m (i) = x_{m} (i) . \end{matrix}$ (12)

In which, s (i), f (i) and m (i) are called the cover area, gravity center and mid boundary point of the membership function, respectively.

Conversely, the three boundary points of $\tilde{x} (i)$ can be restored from s (i), f (i) and m (i) as follows: ${\begin{matrix} x_{l} (i) = \frac{3 f (i) - m (i) - 2 s (i)}{2}, \\ x_{m} (i) = m (i), \\ x_{r} (i) = \frac{3 f (i) - m (i) + 2 s (i)}{2} . \end{matrix}$ (13)

According to Equation (12), original data ${\tilde{x}}^{(0)} (i)$ can be transformed into s (i), f (i) and m (i). Similarly, the predicted value of the TFDDE ${\hat{\tilde{y}}}^{(0)} (i)$ , can be transformed into $\hat{s} (i)$ , $\hat{f} (i)$ and $\hat{m} (i)$ . Then, the forecasting residual of the TFDDE model can be calculated as follows

$\begin{matrix} r_{s} (i) & = & s (i) - \hat{s} (i), r_{m} (i) = m (i) - \hat{m} (i), \\ r_{f} (i) & = & f (i) - \hat{f} (i) \end{matrix}$ (14)

It can be seen that the three transformation sequences are constrained by the three boundary points of the triangular fuzzy number at the same time, which not only guarantees the integrity of the fuzzy number, but also weakens the jumping degree of some boundary points of the fuzzy number and increases the smoothness of the modeling sequence.

4.2 PSO-SVR

To improve the forecast result of TFDDE forecasting model, the support vector regression (SVR) is employed to predict the forecasting residual of the TFDDE model, i.e., r_s (i), r_f (i) and r_m (i).

The support vector regression (SVR), an extension of support vector machine, is a machine learning method based on statistical leaning theory [30, 31] and can be used to time series forecasting. The basic idea of SVR is to map the input vector x_i (i = 1, 2, ⋯ , n) to the high dimensional feature space by the nonlinear mapping function φ (x), and perform linear regression. The regression function of SVR in the high dimensional feature space can be described as $f (x) = ω^{T} φ (x) + b,$ (15)

Where φ (x) is a nonlinear mapping function, ω is the weight factor, and b is the threshold value. The empirical risk can be defined as follows: $R_{emp} = \frac{1}{n} \sum_{i = 1}^{n} L_{ɛ} (y_{i}, f (x_{i}))$ (16)

Where L_ɛ (y_i, f (x_i)) is called ɛ-intensive loss function. L_ɛ (y_i, f (x_i)) is as follows: $L_{ɛ} (y_{i}, f (x_{i})) = {\begin{matrix} | f (x_{i}) - y_{i} | - ɛ, & if | f (x_{i}) - y_{i} | ⩾ ɛ; \\ 0, & otherwise . \end{matrix}$ (17)

By introducing two non-negative slack variables ξ_i and $ξ_{i}^{*}$ , the coefficients of Equation (15) can be obtained by the following optimization problem: $\begin{matrix} min \frac{1}{2} ω^{T} ω + C \sum_{i = 1}^{n} (ξ_{i} + ξ_{i}^{*}) \\ s . t . {\begin{matrix} y_{i} - (ω^{T} φ (x) + b) ⩽ ɛ + ξ_{i}, i = 1, 2, \dots, n, \\ (ω^{T} φ (x) + b) - y_{i} ⩽ ɛ + ξ_{i}^{*}, i = 1, 2, \dots, n, \\ ξ_{i}, ξ_{i}^{*} ⩾ 0, i = 1, 2, \dots, n . \end{matrix} \end{matrix}$ (18)

In Model (18), (x₁, y₁), (x₂, y₂), ⋯, (x_n, y_n) are pair of input and output vectors, C is the trade-off parameter between the first and the second terms of the objective function, ξ_i and $ξ_{i}^{*}$ are the upper training error and lower training error subject to ɛ-insensitive loss factor respectively. The Model (18) can be transformed into the dual problem and its solution is given by Equation (19). $f (x) = \sum_{i = 1}^{n} (λ_{i}^{*} - λ_{i}) K (x_{i}, x_{j}) + b$ (19)

Where $λ_{i}^{*}$ and λ_i are the Lagrange multipliers that can be got by solving the dual problem, K (x_i, x_j) is the Kernel function. In the feature space, the kernel function must satisfy Mercer’s theorem [32] and can be described as K (x_i, x_j) = φ (x_i) φ (x_j). Among the four typical kernel functions, the Gaussian RBF is the most widely used kernel function, which can be expressed as $K (x_{i}, x_{j}) = exp {- \frac{{∥ x_{i} - x_{j} ∥}^{2}}{δ^{2}}}$ (20)

The main motivations for which we consider the SVR are its good properties, such as (i) good generalization capability; (ii) ability to handle high dimensional input spaces also when few training samples are available; and (iii) relatively limited computational load in the training phase [33 –39].

For the SVR forecasting model, the parameters, i.e., ɛ, C and δ have great influence on the forecasting accuracy, the particle swam optimization (PSO) is employed to search the optimal parameters. The advantage of using PSO [40] is that it is simple and easy to implement, no gradient information is needed, and there are few parameters, especially its natural real-coded characteristics are particularly suitable for handling real optimization problems.

4.3 Triangular fuzzy series combination forecasting based on TFDDE and PSO-SVR

Based on the developed TFDDE forecasting model and PSO-SVR, we develop a novel triangular fuzzy series combination forecasting method, which is shown in Fig. 1.

Fig.1

The process of the triangular fuzzy series combination forecasting method based on TFDDE and PSO-SVR.

The detailed Steps can be listed as follows:

Step 1. The original triangular fuzzy time series ${\tilde{x}}^{(0)}$ is pre-processed using sliding average based on Equation (5). Then, we get a new triangular fuzzy series ${\tilde{y}}^{(0)}$ ;

Step 2. Build a triangular fuzzy discrete difference equation (TFDDE) forecasting model, see Equation (7); And obtain the TFDDE model predicted value ${\hat{\tilde{y}}}^{(0)}$ for original time series;

Step 3. According to Equation (12), original data ${\tilde{x}}^{(0)} (i)$ can be transformed into s (i), f (i) and m (i). Similarly, the predicted value of the TFDDE, ${\hat{\tilde{y}}}^{(0)} (i)$ , can be transformed into $\hat{s} (i)$ , $\hat{f} (i)$ and $\hat{m} (i)$ ;

Step 4. Calculate the residual series, r_s (i), r_f (i) and r_m (i), by Equation (14).

Step 5. Let r_s (i - k), r_s (i - k + 1), ⋯, r_s (i - 1) be the input data and r_s (i) be the output data. The PSO-SVR is employed to predict the residual series r_s (i). By analogy, r_f (i) and r_m (i) can also be predicted in the same way. The predicted residual series are denoted as ${\hat{r}}_{s} (i)$ , ${\hat{r}}_{f} (i)$ and ${\hat{r}}_{m} (i)$ .

Step 6. Aggregate the predicted results of TFDDE and PSO-SVR, as follows:

$\begin{matrix} {\hat{s}}^{*} (i) & = & \hat{s} (i) + {\hat{r}}_{s} (i), {\hat{f}}^{*} (i) = \hat{f} (i) + {\hat{r}}_{f} (i), \\ {\hat{m}}^{*} (i) & = & \hat{m} (i) + {\hat{r}}_{m} (i), i = 1, 2, \dots, n . \end{matrix}$ (21)

Step 7. The final predicted results of the original triangular fuzzy series are restored from ${\hat{s}}^{*} (i)$ , ${\hat{f}}^{*} (i)$ and ${\hat{m}}^{*} (i)$ based on Equation (13). And can be denoted as ${\hat{\tilde{x}}}^{(0), *} (i) = [{\hat{x}}_{l}^{(0), *} (i), {\hat{x}}_{m}^{(0), *} (i), {\hat{x}}_{r}^{(0), *} (i)]$ , i = 1, 2, ⋯ , n.

Step 8. The relative error and the mean relative error (MRE) were selected as evaluation criteria, which can be calculated as follows: $\begin{matrix} e_{l} (i) & = & \frac{x_{l}^{(0)} (i) - {\hat{x}}_{l}^{(0), *} (i)}{x_{l}^{(0)} (i)}, \\ e_{m} (i) & = & \frac{x_{m}^{(0)} (i) - {\hat{x}}_{m}^{(0), *} (i)}{x_{m}^{(0)} (i)}, \end{matrix}$ (22) $e_{r} (i) = \frac{x_{r}^{(0)} (i) - {\hat{x}}_{r}^{(0), *} (i)}{x_{r}^{(0)} (i)} .$ $\begin{matrix} MRE & = & \frac{1}{3 n} \sum_{i = 1}^{n} (\frac{| x_{l}^{(0)} (i) - {\hat{x}}_{l}^{(0), *} (i) |}{x_{l}^{(0)} (i)} + \frac{| x_{m}^{(0)} (i) - {\hat{x}}_{m}^{(0), *} (i) |}{x_{m}^{(0)} (i)} \\ + \frac{| x_{r}^{(0)} (i) - {\hat{x}}_{r}^{(0), *} (i) |}{x_{r}^{(0)} (i)}) . \end{matrix}$ (23)

5 Practical example

5.1 The predicted results of the TFDDE forecasting model

In this section, we use the historical load data of the US PJM power market from January 2013 to May 2018 (a total of 65 issues) as an example. Among them, the mean value of each month is taken as the mid boundary of triangular fuzzy number. Meanwhile, the minimum and maximum of each month are taken as the left and right boundary of triangular fuzzy number, respectively. Firstly, the sliding average is employed to correct the singularity of the original triangular fuzzy time series, so as to eliminate the accidental factors and obtain a new triangular fuzzy time series. Then, the ordinary least square method is used to estimate the corresponding parameters of the TFDDE forecasting model. The integral expansion coefficients can be obtained by Equation (9), where a : b : c = 1 : 2 :1. Moreover, the integral expansion coefficients are substituted into Equation (7) to obtain the predicted value of TFAGO series. Finally, we get the predicted value of the original data based on I-TFAGO. The mean relative error (MRE) of the developed TFDDE prediction model is 3.95%.

Furthermore, we compare the TFDDE forecasting model with the triangular fuzzy series forecasting method based on grey model (TFGM) [25]. We use the TFGM to predict the power load, and obtained the MRE of TFGM is 7.27%. As can be seen, the prediction accuracy using our proposed TFDDE method are much better than those under the TFGM.

Besides, Fig. 2 gives the curves of predicted results of TFDDE forecasting model and TFGM. We can see that the prediction curve of TFGM is smooth, which indicates the general development tendency of the raw series. But the fluctuation of the raw series is not shown in the curve. Compared with TFGM, the TFDDE forecasting model has better prediction effect at the inflection point than the gray model, which can reflect the trend of original data well and greatly improve the accuracy of prediction.

Fig.2

The curves of predicted results of TFDDE and TFGM.

5.2 The adjustment of predicted result based on PSO-SVR

Next, the PSO-SVR is adopted to adjust the predicted values of TFDDE. Firstly, According to Equation (14), we obtain the residual sequence, and use PSO-SVR to predict the residuals. Then, we can obtain the predicted residuals. We use the particle swarm optimization algorithm to find the best parameters for SVR, i.e., C = 4.0373 and ɛ = 73.4674.

The predicted results of TFDDE and PSO-SVR according to Equations (13) and (23) and get the final predicted results of our developed combination forecasting method, which are shown in Table 1. And it is evident that the combination forecasting model tends to fit closer to the original value with a smaller forecasting error. The errors of TFDDE model are almost higher than the combination forecasting method, which means that our developed combination forecasting method is more accurate and practical. Therefore, the results show that the developed combination forecasting method can efficiently improve the forecasting accuracy of theTFDDE.

Table 1
The final predicted result of the combination forecasting method

Month Raw values Predicted values Relative errors

4 [10243.40 13976.89 20353.80] [10157.18 14000.68 20338.90] 0.00842 –0.00170 0.00073

5 [9842.70 13779.07 20278.10] [9901.85 13803.43 20268.53] –0.00601 –0.00177 0.00047

6 [10143.20 15008.65 21436.60] [10152.38 14983.73 21377.56] –0.00091 0.00166 0.00275

7 [10658.70 15909.16 22841.10] [10598.74 15884.1 22849.13] 0.00563 0.00158 –0.00035

8 [10229.10 15305.14 21097.60] [10220.98 15329.6 21157.14] 0.00079 –0.00160 –0.00282

9 [9818.30 14073.89 22762.30] [9828.88 14048.97 22702.74] –0.00108 0.00177 0.00262

10 [9754.10 13567.11 17239.30] [9769.84 13542.64 17322.94] –0.00161 0.0018 –0.00485

11 [10806.40 14900.93 19677.30] [10734.57 14898.77 19674.83] 0.00665 0.00014 0.00013

12 [10688.70 16001.12 20558.80] [10705.29 15976.11 20642.97] –0.00155 0.00156 –0.00409

13 [13269.20 18452.56 24433.30] [13210.05 18427.81 24442.52] 0.00446 0.00134 –0.00038

14 [13079.30 17410.49 22306.80] [13071.05 17434.86 22367.50] 0.00063 –0.00140 –0.00272

15 [11996.90 15829.24 21536.30] [12005.42 15805.04 21475.30] –0.00071 0.00153 0.00283

16 [9834.90 13415.2 18167.40] [9894.07 13440.42 18157.66] –0.00602 –0.00188 0.00054

17 [9751.30 13608.3 18800.20] [9810.03 13633.53 18791.70] –0.00602 –0.00185 0.00045

18 [9861.20 15124.43 21257.10] [9844.65 15149.93 21171.65] 0.00168 –0.00169 0.00402

19 [9778.30 14926.98 21418.70] [9827.54 14951.27 21422.78] –0.00504 –0.00163 –0.00019

20 [10523.60 15284.27 20870.00] [10465.67 15259.2 20878.51] 0.0055 0.00164 –0.00041

21 [9875.30 14019.46 21349.40] [9863.91 14043.56 21409.41] 0.00115 –0.00172 –0.00281

22 [10347.70 13267.94 16199.90] [10339.29 13292.73 16261.38] 0.00081 –0.00187 –0.00380

23 [11509.50 15264.3 21252.20] [11518.84 15240.3 21192.78] –0.00081 0.00157 0.00280

24 [11145.50 15544.74 19447.20] [11136.09 15568.67 19507.34] 0.00084 –0.00154 –0.00309

25 [11603.80 17316.17 23636.00] [11613.83 17290.73 23578.36] –0.00086 0.00147 0.00244

26 [12347.90 18385.58 24723.60] [12357.45 18361.08 24664.54] –0.00077 0.00133 0.00239

27 [11327.70 15066.55 21785.50] [11307.70 15091.04 21708.41] 0.00177 –0.00162 0.00354

28 [9739.60 13042.48 16446.20] [9730.34 13067.21 16506.18] 0.00095 –0.00190 –0.00365

29 [9775.30 13739.55 18574.30] [9831.70 13764.21 18565.44] –0.00577 –0.00179 0.00048

30 [10197.20 15115.47 20969.70] [10206.53 15090.75 20911.03] –0.00092 0.00164 0.00280

31 [9899.40 15448.57 21759.60] [9956.76 15474.29 21749.70] –0.00579 –0.00166 0.00046

32 [10525.20 15176.21 20734.10] [10541.93 15151.39 20819.90] –0.00159 0.00164 –0.00414

33 [9882.90 14441.23 20886.30] [9893.83 14415.94 20826.35] –0.00111 0.00175 0.00287

34 [9971.30 13055.29 16402.80] [9961.99 13080.12 16461.78] 0.00093 –0.00190 –0.00360

35 [10441.20 13470.46 18057.30] [10524.19 13445.88 18071.95] –0.00795 0.00183 –0.00081

36 [9114.80 14049.02 18859.20] [9123.92 14023.96 18800.01] –0.00100 0.00178 0.00314

37 [12026.90 17268.24 22804.40] [12036.13 17243.17 22746.56] –0.00077 0.00145 0.00254

38 [11566.80 16513.33 21842.20] [11482.98 16537.87 21827.72] 0.00725 –0.00149 0.00066

39 [10426.10 14102.06 19345.10] [10486.91 14125.87 19335.15] –0.00583 –0.00169 0.00051

40 [10141.90 13740.01 17703.30] [10129.51 13765.12 17762.81] 0.00122 –0.00183 –0.00336

41 [9793.80 13274.56 19106.30] [9851.24 13299.34 19095.51] –0.00587 –0.00187 0.00056

42 [10578.70 15384.32 21069.40] [10586.70 15360.15 21008.33] –0.00076 0.00157 0.00290

43 [10126.60 16134.75 22427.50] [10211.40 16110.5 22444.67] –0.00837 0.0015 –0.00077

44 [11106.50 16581.53 22231.70] [11060.44 16557.02 22253.55] 0.00415 0.00148 –0.00098

45 [9900.00 14850.81 21880.60] [9813.77 14876.29 21863.63] 0.00871 –0.00172 0.00078

46 [9569.70 12745.74 15987.60] [9630.26 12770.43 15978.40] –0.00633 –0.00194 0.00058

47 [10517.40 13733.62 18024.80] [10507.21 13759.02 18084.86] 0.00097 –0.00185 –0.00333

48 [10729.20 15754.71 21323.90] [10740.49 15729.91 21265.01] –0.00105 0.00157 0.00276

49 [10843.50 15570.55 21792.60] [10851.59 15545.48 21733.43] –0.00075 0.00161 0.00272

50 [10432.70 14734.11 19917.70] [10492.22 14758.55 19909.47] –0.00571 –0.00166 0.00041

51 [9906.00 14673.9 20913.60] [9916.13 14648.99 20855] –0.00102 0.0017 0.00280

52 [9513.40 12874.73 16528.10] [9504.43 12898.34 16587.12] 0.00094 –0.00183 –0.00357

53 [9746.70 12998.52 18569.70] [9736.34 13023.38 18627.82] 0.00106 –0.00191 –0.00313

54 [10191.30 14598.06 20484.50] [10198.93 14574.3 20425.67] –0.00075 0.00163 0.00287

55 [10217.10 15591.42 21533.70] [10227.39 15566.38 21473.84] –0.00101 0.00161 0.00278

56 [9841.30 14977.4 21017.20] [9827.15 14999.96 20933.98] 0.00144 –0.00151 0.00396

57 [9811.70 13921.42 20727.80] [9796.78 13946.67 20641.60] 0.00152 –0.00181 0.00416

58 [9828.00 13221.58 17129.60] [9818.77 13246.94 17187.96] 0.00094 –0.00192 –0.00341

59 [10004.50 14256.61 17820.20] [9995.85 14281.4 17879.53] 0.00086 –0.00174 –0.00333

60 [11744.60 16034.42 20830.10] [11687.02 16009.46 20839.60] 0.0049 0.00156 –0.00046

61 [12204.90 17632.07 22853.80] [12216.05 17607.18 22793.95] –0.00091 0.00141 0.00262

62 [10762.30 15310.59 19913.40] [10819.68 15335.81 19902.92] –0.00533 –0.00165 0.00053

63 [11505.90 15157.28 18793.70] [11446.32 15133.2 18802.96] 0.00518 0.00159 –0.00049

64 [10366.00 13681.94 17674.10] [10359.04 13706.45 17733.35] 0.00067 –0.00179 –0.00335

65 [9803.80 13427.8 18114.70] [10002.93 13707.13 17497.22] –0.02031 –0.02080 0.03409

MRE 0.26%

Month	Raw values	Predicted values	Relative errors
4	[10243.40 13976.89 20353.80]	[10157.18 14000.68 20338.90]	0.00842 –0.00170 0.00073
5	[9842.70 13779.07 20278.10]	[9901.85 13803.43 20268.53]	–0.00601 –0.00177 0.00047
6	[10143.20 15008.65 21436.60]	[10152.38 14983.73 21377.56]	–0.00091 0.00166 0.00275
7	[10658.70 15909.16 22841.10]	[10598.74 15884.1 22849.13]	0.00563 0.00158 –0.00035
8	[10229.10 15305.14 21097.60]	[10220.98 15329.6 21157.14]	0.00079 –0.00160 –0.00282
9	[9818.30 14073.89 22762.30]	[9828.88 14048.97 22702.74]	–0.00108 0.00177 0.00262
10	[9754.10 13567.11 17239.30]	[9769.84 13542.64 17322.94]	–0.00161 0.0018 –0.00485
11	[10806.40 14900.93 19677.30]	[10734.57 14898.77 19674.83]	0.00665 0.00014 0.00013
12	[10688.70 16001.12 20558.80]	[10705.29 15976.11 20642.97]	–0.00155 0.00156 –0.00409
13	[13269.20 18452.56 24433.30]	[13210.05 18427.81 24442.52]	0.00446 0.00134 –0.00038
14	[13079.30 17410.49 22306.80]	[13071.05 17434.86 22367.50]	0.00063 –0.00140 –0.00272
15	[11996.90 15829.24 21536.30]	[12005.42 15805.04 21475.30]	–0.00071 0.00153 0.00283
16	[9834.90 13415.2 18167.40]	[9894.07 13440.42 18157.66]	–0.00602 –0.00188 0.00054
17	[9751.30 13608.3 18800.20]	[9810.03 13633.53 18791.70]	–0.00602 –0.00185 0.00045
18	[9861.20 15124.43 21257.10]	[9844.65 15149.93 21171.65]	0.00168 –0.00169 0.00402
19	[9778.30 14926.98 21418.70]	[9827.54 14951.27 21422.78]	–0.00504 –0.00163 –0.00019
20	[10523.60 15284.27 20870.00]	[10465.67 15259.2 20878.51]	0.0055 0.00164 –0.00041
21	[9875.30 14019.46 21349.40]	[9863.91 14043.56 21409.41]	0.00115 –0.00172 –0.00281
22	[10347.70 13267.94 16199.90]	[10339.29 13292.73 16261.38]	0.00081 –0.00187 –0.00380
23	[11509.50 15264.3 21252.20]	[11518.84 15240.3 21192.78]	–0.00081 0.00157 0.00280
24	[11145.50 15544.74 19447.20]	[11136.09 15568.67 19507.34]	0.00084 –0.00154 –0.00309
25	[11603.80 17316.17 23636.00]	[11613.83 17290.73 23578.36]	–0.00086 0.00147 0.00244
26	[12347.90 18385.58 24723.60]	[12357.45 18361.08 24664.54]	–0.00077 0.00133 0.00239
27	[11327.70 15066.55 21785.50]	[11307.70 15091.04 21708.41]	0.00177 –0.00162 0.00354
28	[9739.60 13042.48 16446.20]	[9730.34 13067.21 16506.18]	0.00095 –0.00190 –0.00365
29	[9775.30 13739.55 18574.30]	[9831.70 13764.21 18565.44]	–0.00577 –0.00179 0.00048
30	[10197.20 15115.47 20969.70]	[10206.53 15090.75 20911.03]	–0.00092 0.00164 0.00280
31	[9899.40 15448.57 21759.60]	[9956.76 15474.29 21749.70]	–0.00579 –0.00166 0.00046
32	[10525.20 15176.21 20734.10]	[10541.93 15151.39 20819.90]	–0.00159 0.00164 –0.00414
33	[9882.90 14441.23 20886.30]	[9893.83 14415.94 20826.35]	–0.00111 0.00175 0.00287
34	[9971.30 13055.29 16402.80]	[9961.99 13080.12 16461.78]	0.00093 –0.00190 –0.00360
35	[10441.20 13470.46 18057.30]	[10524.19 13445.88 18071.95]	–0.00795 0.00183 –0.00081
36	[9114.80 14049.02 18859.20]	[9123.92 14023.96 18800.01]	–0.00100 0.00178 0.00314
37	[12026.90 17268.24 22804.40]	[12036.13 17243.17 22746.56]	–0.00077 0.00145 0.00254
38	[11566.80 16513.33 21842.20]	[11482.98 16537.87 21827.72]	0.00725 –0.00149 0.00066
39	[10426.10 14102.06 19345.10]	[10486.91 14125.87 19335.15]	–0.00583 –0.00169 0.00051
40	[10141.90 13740.01 17703.30]	[10129.51 13765.12 17762.81]	0.00122 –0.00183 –0.00336
41	[9793.80 13274.56 19106.30]	[9851.24 13299.34 19095.51]	–0.00587 –0.00187 0.00056
42	[10578.70 15384.32 21069.40]	[10586.70 15360.15 21008.33]	–0.00076 0.00157 0.00290
43	[10126.60 16134.75 22427.50]	[10211.40 16110.5 22444.67]	–0.00837 0.0015 –0.00077
44	[11106.50 16581.53 22231.70]	[11060.44 16557.02 22253.55]	0.00415 0.00148 –0.00098
45	[9900.00 14850.81 21880.60]	[9813.77 14876.29 21863.63]	0.00871 –0.00172 0.00078
46	[9569.70 12745.74 15987.60]	[9630.26 12770.43 15978.40]	–0.00633 –0.00194 0.00058
47	[10517.40 13733.62 18024.80]	[10507.21 13759.02 18084.86]	0.00097 –0.00185 –0.00333
48	[10729.20 15754.71 21323.90]	[10740.49 15729.91 21265.01]	–0.00105 0.00157 0.00276
49	[10843.50 15570.55 21792.60]	[10851.59 15545.48 21733.43]	–0.00075 0.00161 0.00272
50	[10432.70 14734.11 19917.70]	[10492.22 14758.55 19909.47]	–0.00571 –0.00166 0.00041
51	[9906.00 14673.9 20913.60]	[9916.13 14648.99 20855]	–0.00102 0.0017 0.00280
52	[9513.40 12874.73 16528.10]	[9504.43 12898.34 16587.12]	0.00094 –0.00183 –0.00357
53	[9746.70 12998.52 18569.70]	[9736.34 13023.38 18627.82]	0.00106 –0.00191 –0.00313
54	[10191.30 14598.06 20484.50]	[10198.93 14574.3 20425.67]	–0.00075 0.00163 0.00287
55	[10217.10 15591.42 21533.70]	[10227.39 15566.38 21473.84]	–0.00101 0.00161 0.00278
56	[9841.30 14977.4 21017.20]	[9827.15 14999.96 20933.98]	0.00144 –0.00151 0.00396
57	[9811.70 13921.42 20727.80]	[9796.78 13946.67 20641.60]	0.00152 –0.00181 0.00416
58	[9828.00 13221.58 17129.60]	[9818.77 13246.94 17187.96]	0.00094 –0.00192 –0.00341
59	[10004.50 14256.61 17820.20]	[9995.85 14281.4 17879.53]	0.00086 –0.00174 –0.00333
60	[11744.60 16034.42 20830.10]	[11687.02 16009.46 20839.60]	0.0049 0.00156 –0.00046
61	[12204.90 17632.07 22853.80]	[12216.05 17607.18 22793.95]	–0.00091 0.00141 0.00262
62	[10762.30 15310.59 19913.40]	[10819.68 15335.81 19902.92]	–0.00533 –0.00165 0.00053
63	[11505.90 15157.28 18793.70]	[11446.32 15133.2 18802.96]	0.00518 0.00159 –0.00049
64	[10366.00 13681.94 17674.10]	[10359.04 13706.45 17733.35]	0.00067 –0.00179 –0.00335
65	[9803.80 13427.8 18114.70]	[10002.93 13707.13 17497.22]	–0.02031 –0.02080 0.03409
MRE		0.26%

Figure 3 reflects the predicted results raw values and our developed combination forecasting method. Obviously, our developed combination forecasting method gives acceptable predictions most of the time, and the prediction curve of our developed combination forecasting method is better than the TFDDE model.

Fig.3

The power load curves of the developed combination forecasting.

5.3 Comparative analysis

For the purpose of comparison, the developed combination forecasting method in this paper and NNTFGM in [25] were used, and the predicted results are obtained respectively. It can be concluded that the MRE for our combination forecasting method is 0.26%, and the MRE for NNTFGM is 1.69%. Therefore, our developed method improves the accuracy of the medium-long term load forecasting significantly. Besides, to make a clearer comparison between the developed combination forecasting method and NNTFGM, the comparison between these methods and the real load is made as shown in Figs. 3 and 4. It can be concluded that the combination method in this paper is mostly close to the real load.

Fig.4

The power load curves of NNTFGM.

In view of the model effectiveness and efficiency on the whole, we can conclude that the proposed model is quite competitive against the compared model. To sum up, we can make the conclusions as follows.

Firstly, this is the first attempt of employing discrete difference equation theory for the triangular fuzzy series forecasting. It complements the theoretical absence of discrete difference equations in fuzzy environments. And in some practical applications, the original data obtained are not precise numbers and the representation of these data by fuzzy numbers will be more reasonable, so our developed forecasting framework has a broad prospect for application. Secondly, the proposed TFDDE prediction model has a better prediction effect at the inflection point, which can well reflect the fluctuant trend of the triangular fuzzy series. Thirdly, SVR is shown to be very resistant to the over learning problem and has been widely and successfully applied to many forecasting problems. Besides, in order to gain optimization parameters of SVR, PSO is implemented to automatically perform the parameter selection in SVR modeling, which can avoid the blindness of artificial selection effectively and more reasonable. Overall, our proposed combination forecasting framework not only achieves better results, but also the models developed in this paper are effective and applicable, and can enhance forecasting ability.

6 Conclusion

In this paper, we have developed a triangular fuzzy series combination forecasting method based on triangular fuzzy discrete difference equation (TFDDE) prediction model and PSO-SVR. In which, we first propose a triangular fuzzy discrete difference equation (TFDDE) forecasting model to predict the triangular fuzzy series, and it has a better prediction effect at the inflection point. Furthermore, we use PSO-SVR to improve the prediction accuracy of TFDDE. Moreover, a practical example of load forecasting is employed to illustrate the proposed forecasting method. The prediction performance of our developed combination forecasting method surpasses that of both standard TFGM and NNTFGM with their smallest MRE index under the same training and test conditions. In conclusion, the proposed combination forecasting method is applicable and practical in accurately predicting the medium-long term power load. In our future works, we will introduce other influencing factors that affect the power load into the hybrid model to further improve the prediction accuracy.

Footnotes

Acknowledgments

The work was supported by National Natural Science Foundation of China (Nos. 71501002, 61502003, 71871001, 71771001, and 71701001), Anhui Provincial Natural Science Foundation (Nos. 1608085QF133, 1508085QG149).

References

Qiu

, Suganthan

P.N.

, Amaratunga

G.A.J.

, Ensemble incremental learning random vector functional link network for short-term electric load forecasting, Knowledge-Based Systems 145 (2018), 182–196.

Dangha

T.H.

, Bianchi

F.M.

, Olsson

, Local short term electricity load forecasting: Automatic approaches, Filippo Maria Bianchi 22 (2017), 4267–4274.

Hattab

, Ma’Itah

, Sweidan

, Rifai

, Momani

, Medium term load forecasting for jordan electric power system using particle swarm optimization algorithm based on least square regression methods, Journal of Power & Energy Engineering 5 (2017), 75–96.

Kanehira

, Takahashi

, Imai

, Funabiki

, A comparison of electric power smoothing control methods for distributed generation systems, Electrical Engineering in Japan 193 (2015), 49–57.

Niu

, Wei

, An improved short-term power load combined forecasting with arma-grach-ann- svm based on fhnn similar-day clustering, Journal of Software 8 (2013), 716–723.

Joshi

P.A.

, Patel

J.J.

, Computational analysis and intelligent control of load forecasting using time series method, Advances in Intelligent Systems and Computing 671 (2018), 297–306.

Cecati

, Kolbusz

, Różycki

, Siano

, Wilamowski

B.M.

, A novel rbf training algorithm for short-term electric load forecasting and comparative studies, IEEE Transactions on Industrial Electronics 62 (2015), 6519–6529.

Weron

, Modeling and forecasting electricity forward prices: A dsfm approach motivation, Hsc Books 20 (2006), 101–155.

Amaral

L.F.

, Souza

R.C.

, Stevenson

, A smooth transition periodic autoregressive (stpar) model for short-term load forecasting, International Journal of Forecasting 24 (2008), 603–615.

10.

Gross

, Galiana

F.D.

, Short-term load forecasting, Proceedings of the IEEE 75 (1987), 1558–1573.

11.

Zhang

, Fang

H.J.

, Jiang

T.Y.

, Fault detection for nonlinear networked control systems with stochastic interval delay characterisation, International Journal of Systems Science 43 (2012), 952–960.

12.

Aslan

, Yağan

Y.E.

, Artificial neural-network-based fault location for power distribution lines using the frequency spectra of fault data, Electrical Engineering 9 (2016), 301–311.

13.

Chang

W.Y.

, Short-term load forecasting using radial basis function neural network, Journal of Computer & Communications 3 (2016), 40–45.

14.

López

, Zhong

, Zheng

M.L.

, Short-term electric load forecasting based on wavelet neural network, particle swarm optimization and ensemble empirical mode decomposition, Energy Procedia 105 (2017), 3677–3682.

15.

Jia

N.X.

, Yokoyama

, Zhouand

Y.C.

and Gao

Z.Y.

, A flexible long-term load forecasting approach based on new dynamic simulation theory — gsim, International Journal of Electrical Power & Energy Systems 23 (2001), 549–556.

16.

Wang

, Wang

, A hybrid model of emd and pso-svr for short-term load forecasting in residential quarters, Mathematical Problems in Engineering 5 (2016), 1–10.

17.

Niu

, Ma

, Liu

, Power load forecasting by wavelet least squares support vector machine with improved fruit fly optimization algorithm, Journal of Combinatorial Optimization 33 (2016), 1–22.

18.

Wang

, Li

, Niu

, Tan

, An annual load forecasting model based on support vector regression with differential evolution algorithm, Applied Energy 94 (2012), 65–70.

19.

Pai

P.F.

, Hong

W.C.

, Support vector machines with simulated annealing algorithms in electricity load forecasting, Energy Conversion & Management 46 (2005), 2669–2688.

20.

Alrashidi

M.R.

, El-Naggar

K.M.

, Long term electric load forecasting based on particle swarm optimization, Applied Energy 87 (2010), 320–326.

21.

Zhao

, Guo

, An optimized grey model for annual power load forecasting, Energy 107 (2016), 272–286.

22.

Esteves

G.R.T.

, Bastos

B.Q.

, Cyrino

F.L.

, Calili

R.F.

, Souza

R.C.

, Long term electricity forecast: A systematic review, Procedia Computer Science 55 (2015), 549–558.

23.

Taylor

J.W.

, Triple seasonal methods for short-term electricity demand forecasting, European Journal of Operational Research 204 (2010), 139–152.

24.

Weron

, Misiorek

, Modeling and forecasting electricity loads: A comparison, Econometrics 20 (2005), 135–142.

25.

Zeng

X.Y.

, Shu

, Huang

G.M.

, Jiang

, Triangular fuzzy series forecasting based on grey model and neural network, Applied Mathematical Modelling 40 (2016), 1717–1727.

26.

Chen

C.M.

, Lee

H.M.

, An efficient gradient forecasting search method utilizing the discrete difference equation prediction model, Applied Intelligence 16 (2002), 43–58.

27.

Zadeh

L.A.

, Fuzzy sets, Inf Control 8 (1965), 338–353.

28.

Chen

S.M.

, Chen

J.H.

, Fuzzy risk analysis based on ranking generalized fuzzy numbers with different heights and different spreads, Expert Syst Appl 36 (2009), 6833–6842.

29.

Brunk

H.D.

, Bayes least squares linear estimation of densities, Communications in Statistics 13 (1984), 2253–2291.

30.

Kavousi-Fard

, Samet

, Marzbani

, A new hybrid modified firefly algorithm and support vector regression model for accurate short term load forecasting, Expert Systems with Applications 41 (2014), 6047–6056.

31.

Sanstad

A.H.

, Mcmenamin

, Sukenik

, Barbose

G.L.

, Goldman

C.A.

, Modeling an aggressive energy-efficiency scenario in long-range load forecasting for electric power transmission planning, Applied Energy 128 (2014), 265–276.

32.

Vapnik

V.N.

, The nature of statistical learning theory, IEEE Transactions on Neural Networks 10 (1995), 988–999.

33.

Amroune

, Bouktir

, Musirin

, Power system voltage stability assessment using a hybrid approach combining dragonfly optimization algorithm and support vector regression, Arabian Journal for Science & Engineering 43 (2018), 1–14.

34.

, Li

, Miao

, Chen

, Cao

, Shen

, Online prediction method of icing of overhead power lines based on support vector regression, International Transactions on Electrical Energy Systems 28 (2018), e2500.

35.

Gholami

, Ansari

H.R.

, Hosseini

, Prediction of crude oil refractive index through optimized support vector regression: A competition between optimization techniques, Journal of Petroleum Exploration & Production Technology 7 (2017), 195–204.

36.

Yaslan

, Bican

, Empirical mode decomposition based denoising method with support vector regression for time series prediction: A case study for electricity load forecasting, Measurement 103 (2017), 52–61.

37.

Amroune

, Bouktir

, Musirin

38.

, Li

, Miao

, Chen

, Cao

, Shen

, Online prediction method of icing of overhead power lines based on support vector regression, International Transactions on Electrical Energy Systems 28 (2018), e2500.

39.

Gholami

, Ansari

H.R.

, Hosseini

40.

Jiang

, Xia

, Tran

Q.A.

, Ha

Q.M.

, Tran

N.Q.

, Hu

, A new binary hybrid particle swarm optimization with wavelet mutation, Knowledge-Based Systems 130 (2017), 90–101.