Time-Series Prediction of Electricity Load for Charging Piles in a Region of China Based on Broad Learning System

Abstract

This paper introduces a novel electricity load time-series prediction model, utilizing a broad learning system to tackle the challenge of low prediction accuracy caused by the unpredictable nature of electricity load sequences in a specific region of China. First, a correlation analysis with mutual information is utilized to identify the key factors affecting the electricity load. Second, variational mode decomposition is employed to obtain different mode information, and then a broad learning system is utilized to build a prediction model with different mode information. Finally, particle swarm optimization is used to fuse the prediction models under different modes. Simulation experiments using real data validate the efficiency of the proposed method, demonstrating that it offers higher accuracy compared to advanced modeling techniques and can assist in optimal electricity-load scheduling decision-making. Additionally, the $R^2$ of the proposed model is 0.9831, the $P_{RMSE}$ is 21.8502, the $P_{MAE}$ is 17.0097, and the $P_{MAPE}$ is 2.6468.

1. Introduction

Load forecasting plays a critical role in ensuring the reliable operation and efficient management of smart grids. The precise prediction of electricity demand is a crucial step in enhancing the safety and cost-effectiveness of charging stations. Additionally, it serves as a key factor in supporting smart grid security monitoring, cost management, regulatory compliance, and decision-making processes.

Currently, traditional load forecasting methods include wavelet analysis, regression analysis, neural network, support vector machine, and Monte Carlo simulation [1,2,3]. For example, Buzna et al. [4] proposed a hierarchical probabilistic load prediction method, which utilizes a hierarchical approach to decompose the problem into sub-problems in low-level areas, combines principal component analysis to reduce the dimensionality of the sub-problems, and then solves the problem by standard probabilistic models, such as gradient-enhanced regression tree, quantile regression forest method, and quantile regression neural network, so that the total load of high-level geographic areas can be predicted. Yi et al. [5] employed Monte Carlo modeling and Markov decision making to determine drivers’ travel paths, and the results show that changes in the travel chain lead to significant differences in the distribution of charging loads between weekdays and weekends, and that high temperatures and traffic congestion cause an increase in the amplitude of charging loads. Tang et al. [6] introduced a probabilistic model for node charging demand by considering spatiotemporal dynamics within graph-theoretic integrated systems. Zhu et al. [7] proposed a charging load prediction method for electric vehicles based on the long short-term memory method, and the simulation results based on the actual data show that this method has better performance. Rao et al. [8] proposed a new class of nonlinear functional regression models. The above methods are able to achieve an effective prediction of electricity loads to a certain extent, but they still suffer from problems such as underfitting and the tendency to fall into local optimization during training.

In addition, electricity load is subject to randomness and uncertainty due to a variety of random factors such as traffic, weather, etc., which makes it difficult to choose an appropriate method. In recent years, a variety of deep learning algorithms have been utilized in power system forecasting. For example, Wang et al. [9] established a photovoltaic power prediction model based on wavelet transform and convolutional neural network. Chang et al. [10] proposed a prediction model for photovoltaic output day ahead based on deep belief network. Yi et al. [11] proposed a time-series prediction of monthly commercial electric vehicle charging loads based on a deep sequence-to-sequence model. Compared with single-layer networks, deep neural networks show a higher prediction accuracy, but their complex network structure also brings problems such as time-consuming training and difficulty in tuning parameters. For this reason, a new neural network structure, i.e., a broad learning system (BLS), has been proposed [12,13]. And some scholars have applied it to multi-view classification [14], harmonic detection [15], and energy consumption prediction [16], which have all achieved computational results with high accuracy while ensuring computational efficiency. However, no study in the literature has addressed the application of BLS in electricity load prediction. Therefore, the application of BLS to electricity load prediction in this paper is of great research significance.

This paper introduces a new electricity load time-series prediction model based on BLS, with the goal of addressing issues related to redundant, periodic, and stochastic electricity load data, as well as reducing the time spent training deep neural networks as seen in previous research. Initially, the data correlation analysis method is employed to identify the primary factors that have a direct impact on the electricity load. Then, the variational modal decomposition [17] is used to obtain the data under different modes; and the BLS-based electricity load prediction models are established for different modes, respectively. Finally, a multi-model fusion strategy based on particle swarm optimization (PSO) [18,19] is proposed in order to fuse the prediction models in different modes. At the same time, experiments are being carried out using actual electricity load data to confirm the efficacy of the proposed method.

This paper is structured as follows. Section 2 analyzes the electricity load data characteristics. Section 3 shows in detail how to build an electricity load time-series prediction model. Simulation experiments based on actual data are given in Section 4. Section 5 provides a summary of the paper and outlines potential future research directions.

2. Analysis of Electricity Load Characteristics

This section briefly analyzes the characteristics of electricity load.

2.1. Load Characteristics

Load characteristics in power systems include the following:

• Periodicity: Electricity load can be based on time-divided cycles to complete electricity load prediction. Periodicity shows that the change trend of electricity load is roughly the same in a continuous period of time and is roughly the same as a change law measured in days.
• Conditionality: The methods of load prediction in the power system all need specific conditions, which need to be satisfied after the conditions in order to minimize the likelihood of distortion of the results.
• Randomness; Electricity load can reflect randomness in cyclical loads, where the load of electricity used on any given day can change randomly. In real life, such randomly changing loads are more difficult to predict.

These characteristics must be taken into account for an accurate time-series prediction model of electricity load.

2.2. Analysis of Influencing Factors of Electricity Load

The electricity load prediction system is a multivariate input nonlinear system. The electricity load is affected by a variety of variables, and the correlation between the variables and the electricity load in the prediction system of different regions shows different correlations. Therefore, if the system wants to better predict the changes of the electricity load, and to design a high-precision and high-performance electricity load prediction model, it must first comprehend the factors that influence these fluctuations.

• Climatic factor: Climatic factors include temperature, humidity, and rainfall. Among them, temperature is one of the many influencing factors that have the greatest impact on the change in electricity load. Typically, there is a relationship between temperature and electricity demand, meaning that fluctuations in temperature impact shifts in electricity usage.
• Date factor: Electricity load is subject to change with time; the normal electricity load change will not jump in a split second and a complete daily load in different moments of the load value is not the same. The selection of date factors will directly affect the effect of electricity load prediction.
• Random factor: The factors affecting electricity load prediction are not limited to the above, but include the influence of a variety of factors. The main random factors are holidays, policies, etc. Therefore, it is necessary to select appropriate influencing factors according to the actual electricity consumption, and to analyze the selection of influencing factors through a prediction error of electricity load prediction.

3. Time-Series Prediction Model of Electricity Load Using BLS

This section describes how to build a time-series prediction model of electricity load for charging piles using BLS in a region of China.

3.1. Design of Electricity Load Time-Series Modeling Structure

Due to the large number and complexity of influencing factors affecting electricity load, it is essential to take these complex characteristics into account in order to establish an accurate time-series prediction model for electricity load. A novel modeling structure is then proposed, as depicted in Figure 1.

From Figure 1, the correlation analysis of the electricity load data is firstly carried out to determine the main factors affecting the electricity load. Due to the complexity and variability in electricity load data, we consider the use of variational modal decomposition to process the data to obtain different feature components. Based on these feature components, the electricity load prediction models based on the broad learning system are built, respectively. However, the prediction performance of different models varies, so we propose a multi-model fusion strategy based on PSO to improve the performance of the electricity load time-series prediction model.

3.2. Correlation Analysis and Modal Decomposition

Based on the analysis of electricity load-related characteristics, it is clear that the factors related to electricity load need to be obtained for modeling. There may be redundancies in these factors, which can affect the accuracy of electricity load prediction. In order to avoid this problem, the mutual information method [20] is adopted in this paper for the selection of the main affecting factors. The mutual information value between the electricity load and related factors is calculated as

$$I(X,Y)=\sum _{x\in X}\sum _{y\in Y}p(x_o,y_o)log\frac{p(x_o,y_o)}{p\left(x_o\right)p\left(y_o\right)},$$

(1)

where $X=\left[x_1,x_2,\dots ,x_o,\dots ,x_N\right]$ and $Y=\left[y_1,y_2,\dots ,y_o,\dots ,y_N\right]$ are related factors and electricity load, respectively. Moreover, a higher value of mutual information indicates a higher degree of correlation between the two parameters.

On this basis, it is also necessary to process the historical information of electricity load to extract its potential cyclical features, which will enhance the model training process. The variational modal decomposition method utilized in this paper is able to separate the original signal into several individual FM–AM signals with varying central frequencies. While the empirical mode decomposition can decompose the signal from high frequency to low frequency, it is plagued by end-point effects and mode overlapping, potentially causing the resulting modes to inadequately capture the intrinsic properties of the original signal. The variational modal decomposition method utilizes Wiener filtering, Hilbert transform, and frequency mixing techniques to convert the signal decomposition challenge into a variational constraint-solving problem. This approach successfully addresses the issues of end-point effects and mode aliasing encountered in empirical modal decomposition.

The variational modal decomposition method breaks down the initial data into a series of eigenmode functions that vary in center frequencies based on a predetermined number of decompositions. The method takes the optimal center frequency and power spectrum center of each eigenmode function as the target, and the sum of the decomposed signals is equal to the original signal as the constraint.

$$\left\{\begin{array}{c}\underset{\left\{u_k\right\},\left\{{\omega }_k\right\}}{min}\left\{{\displaystyle \sum _k^K}{∥{\partial }_t[(\delta \left(t\right)+\frac{j}{\pi t})\ast u_k\left(t\right)]e^{-j{\omega }_{k}t}∥}_2^2\right\}\hfill \\ \mathrm{s}.\mathrm{t}.f\left(t\right)={\displaystyle \sum _k^K}{u}_k\left(t\right)\hfill \end{array}\right.$$

(2)

where $u_k\left(t\right)$ and ${\omega }_k$ are, respectively, the eigenmode function obtained from the decomposition of electricity load data and the corresponding center frequency, $f\left(t\right)$ is the original signal, $\delta \left(t\right)$ is the impulse function, and K is the artificially set decomposition quantity.

In order to solve the above constraint equation, it is transformed into an unconstrained problem by introducing the Lagrange multiplier $\lambda \left(t\right)$ and the quadratic penalty term $\alpha$. The unconstrained equations’ saddle points are determined through the alternating direction multiplier method, with updates for the intrinsic modal function $u_k\left(t\right)$, center frequency ${\omega }_k$, and the frequency domain of the Lagrange multiplier $\lambda \left(t\right)$ as outlined in the following equations.

$${\widehat{u}}^{n+1}\left(\omega \right)=\frac{\widehat{f}\left(\omega \right)-{\displaystyle \sum _{i\ne k}}{\widehat{u}}_i\left(\omega \right)+\frac{\widehat{\lambda }\left(\omega \right)}{2}}{1+2\alpha {(\omega -{\omega }_k)}^2},$$

(3)

$${\omega }_k^{n+1}=\frac{{\int }_0^{\infty }\omega {\left|{\widehat{u}}_k\left(\omega \right)\right|}^{2}d\omega }{{\int }_0^{\infty }{\left|{\widehat{u}}_k\left(\omega \right)\right|}^{2}d\omega },$$

(4)

$${\widehat{\lambda }}^{n+1}\left(\omega \right)={\widehat{\lambda }}^n\left(\omega \right)+\tau \left[\widehat{f}\left(\omega \right)-\sum _k{\widehat{u}}_k^{n+1}\left(\omega \right)\right],$$

(5)

$$\sum _k\frac{{∥{\widehat{u}}_k^{n+1}-{\widehat{u}}_k^n∥}_2^2}{{∥{\widehat{u}}_k^n∥}_2^2}\le \epsilon .$$

(6)

After performing continuous iterations, the inverse Fourier transform of the decomposition results is conducted to obtain K sets of eigenmode functions.

3.3. Electricity Load Time-Series Modeling Based on Broad Learning System

Based on the structure of the BLS in Figure 1, the features extracted from the input data in the BLS are mapped into feature nodes, which are then subject to nonlinear transformation to produce enhancement nodes. These nodes, along with the feature nodes, are connected directly to the output layer. Unlike deep learning, which improves fitting ability by increasing the number of neural network layers, BLS enhances network performance by expanding feature nodes and enhancement nodes in width. This flat network structure is characterized by fast training.

The input data X are first obtained by feature extraction to obtain a series of feature nodes, where the i-th group of feature mapping is obtained by the following equation:

$$F_i=\psi \left(X{W}_{e_i}+b_{e_i}\right),i=1,2,\dots ,m,$$

(7)

where $b_{e_i}$ and $W_{e_i}$ are the bias and the weight associated with the feature node, respectively, and $\psi$ is the feature mapping incentive function.

The combination of all feature nodes is denoted as

$$F^m=\left[F_1,F_2,\dots ,F_m\right].$$

(8)

There is an extension of all feature nodes to the enhancement layer via a nonlinear excitation function. The j-th enhancement node is represented as

$$E_j=\xi \left(F^m{W}_{e_j}+b_{e_j}\right),j=1,2,\dots ,n,$$

(9)

where $W_{e_j}$ and $b_{e_j}$ are the weight and bias associated with the enhancement node, respectively, and $\xi$ is the nonlinear excitation function.

$$\xi \left(x\right)=\frac{1-e^{-2x}}{1+e^{-2x}}.$$

(10)

In addition, all augmented nodes are described as

$$E^n=\left[E_1,E_2,\dots ,E_n\right].$$

(11)

The feature node layer and the augmentation node layer are connected to each other through a weight matrix, which ultimately forms the mathematical expression of the BLS.

$$\begin{array}{c}y=\left[F_1,F_2,\dots ,F_m|E_1,E_2,\dots ,E_n\right]W,\hfill \\ =\left[F^m|E^n\right]W,\hfill \end{array}$$

(12)

where W is the weight matrix and y is the output of BLS, i.e., the prediction value of electricity load.

The weight matrix W can be computed by a pseudoinverse algorithm.

$$W={\left[F^m|E^n\right]}^{+}y.$$

(13)

Let

$$\Omega =\left[F^m|E^n\right].$$

(14)

The weight matrix W can be approximately calculated as

$$W={\left(\Omega {\Omega }^T+\eta I\right)}^{-1}{\Omega }^{T}y,$$

(15)

$${\Omega }^{+}=\underset{\eta \to 0}{lim}{\left(\Omega {\Omega }^T+\eta I\right)}^{-1}{\Omega }^T,$$

(16)

where $\eta$ represents the ridge coefficient.

It can be seen that BLS training does not require a learning algorithm based on gradient descent, and it is faster and less likely to fall into local optimization. Using BLS with a single hidden layer structure and expanding the number of nodes in width as a prediction method can fit the historical data better and ensure the computational efficiency of the model.

3.4. PSO-Based Fusion of Independent Output Result

Based on the BLS model established in different modes, the final electricity load prediction model is established:

$$Y\left(t\right)=\sum _{o=1}^{K+1}{a}_{o}y{\left(t\right)}_o,$$

(17)

where $a_o$$(0<a_o<1)$ is the linear fusion parameter.

To choose the most appropriate $a_o$, the PSO method is used here. The optimization objective is defined as

$$e=\frac{1}{M}\sum _{t=1}^M\left|Y_{\mathrm{real}}\left(t\right)-\sum _{o=1}^{K+1}{a}_{o}y{\left(t\right)}_o\right|,$$

(18)

where $Y_{\mathrm{real}}\left(t\right)$ is the real value of electricity load data and M is the number of testing data.

The position and velocity update equations for the PSO algorithm are shown below.

$$V_h^{g+1}=\varpi V_h^g+c_1{r}_1(P_h-L_h^g)+c_2{r}_2(G_h-L_h^g),$$

(19)

$$L_h^{g+1}=L_h^g+V_h^{g+1},$$

(20)

where $h=1\dots H$; H denotes the total number of particles; $V_h^g$ denotes the velocity of the particle at the gth iteration; $c_1$ and $c_2$ are the learning factors; $r_1$ and $r_2$ are random numbers between [0, 1]; $P_h$ is the particle’s individual extreme value; $G_h$ is the particle’s global optimum; $L_h^g$ is the particle’s position at the gth iteration; and $\varpi$ is the inertia factor.

4. Experiments and Discussion

In this section, we experimentally verify the effectiveness of the proposed method using actual electricity load data from a region in Wuhan, China.

4.1. Performance Metrics

To assess the performance of the proposed model, we utilize the following performance metrics for testing its effectiveness. Here, we consider the correlation coefficient ($R^2$), root-mean-squared error ($P_{RMSE}$), mean absolute error ($P_{MAE}$), and mean absolute percentage error ($P_{MAPE}$).

$$R^2=1-\frac{{\sum }_{t=1}^M{\left[Y\left(t\right)-{\overline{Y}}_{\mathrm{real}}\left(t\right)\right]}^2}{{\sum }_{t=1}^M{\left[Y_{\mathrm{real}}\left(t\right)-{\overline{Y}}_{\mathrm{real}}\left(t\right)\right]}^2},$$

(21)

$$P_{RMSE}=\sqrt{\frac{{\sum }_{t=1}^M{\left[Y\left(t\right)-Y_{\mathrm{real}}\left(t\right)\right]}^2}{M}},$$

(22)

$$P_{MAE}=\frac{1}{M}{\sum }_{t=1}^M\left|Y\left(t\right)-Y_{\mathrm{real}}\left(t\right)\right|,$$

(23)

$$P_{MAPE}=\frac{100}{M}{\sum }_{t=1}^M\left|\frac{Y\left(t\right)-Y_{\mathrm{real}}\left(t\right)}{Y_{\mathrm{real}}\left(t\right)}\right|,$$

(24)

where ${\overline{Y}}_{\mathrm{real}}\left(t\right)$ is the mean value for electricity load data in testing data and $Y\left(t\right)$ is the prediction value.

4.2. Experiments with Actual Data

A total of 460 sets of actual electricity load data were collected from a region in Wuhan, China. The first 368 datasets were utilized for model training and the remaining 92 datasets were employed for model testing. To validate the proposed model’s effectiveness, we compared it with the long short-term memory (LSTM) [21,22] model, nonlinear auto-regressive model with exogenous inputs (NARX) [23], BLS [24], and the extreme learning machine (ELM) [25] model. Model inputs included temperature, humidity, rainfall, charging time, charging power, policy, etc. The model output was electricity load.

The variational mode decomposition of electricity load data requires the determination of the number of decomposed modes. As the value was small, the data complexity could not be fully reduced. When the value is large, however, the signal will be over-decomposed, resulting in frequency aliasing. This paper uses the method proposed in [26] to determine the number of modes. The Figure 2 displays the outcome of the variational mode decomposition.

Each model is trained using the same electricity load data as well as the same data on factors influencing electricity load. The Figure 3, Figure 4, Figure 5, Figure 6 and Figure 7 display the outcomes of various models used to forecast electricity load. Here, the solid blue line with circle indicates the actual electricity load value. Among them, it can be deduced that the electricity load curve exhibits both periodic patterns and random variations based on the data. From Figure 3, the LSTM prediction error falls within the range of [−600, 600]. From Figure 4, the NARX has a prediction error between [−400, 300]. From Figure 5, the prediction error of BLS is between [−400, 600]. From Figure 6, the prediction error of ELM is between [−200, 400]. However, as shown in Figure 7, the prediction error of the proposed method is only ±60. These results show that although each prediction model can predict the trend of electricity load, the electricity load prediction model of the proposed method in this paper has a higher fit to the actual electricity load curve, and the proposed method is able to follow the actual electricity load values better.

The $R^2$, $P_{RMSE}$, $P_{MAE}$, and $P_{MAPE}$ results of the prediction performance of each comparative prediction method for electricity load prediction are shown in

Table 1. Prediction performance of different models.

Model	${\mathit{R}}^2$	${\mathit{P}}_{\mathit{RMSE}}$	${\mathit{P}}_{\mathit{MAE}}$	${\mathit{P}}_{\mathit{MAPE}}$
LSTM [21]	0.1398	199.6250	157.7817	24.4203
NARX [23]	0.1687	157.6158	125.4578	19.5021
BLS [24]	0.5974	139.0036	106.8798	17.1711
ELM [25]	0.4715	131.1854	108.5291	17.0097
Proposed	0.9831	21.8502	17.1680	2.6468

. From the results of these performance metrics, it is clear that LSTM has the worst prediction performance, indicating that LSTM may be suitable for modeling time-series data with many samples. In contrast, our proposed method is able to fully consider the effects of time-series, periodicity, and robustness of electricity load data. Compared with the LSTM model, the proposed method improves by 85.78%, 89.05%, 89.12%, and 89.16% on the $R^2$, $P_{RMSE}$, $P_{MAE}$, and $P_{MAPE}$ metrics, respectively. Compared with the NARX model, the proposed method improves by 82.84%, 86.14%, 86.32%, and 86.43% on the $R^2$, $P_{RMSE}$, $P_{MAE}$, and $P_{MAPE}$ metrics, respectively. Compared with the BLS model, the proposed method improves by 39.23%, 84.28%, 83.94%, and 84.59% on the $R^2$, $P_{RMSE}$, $P_{MAE}$, and $P_{MAPE}$ metrics, respectively. Compared with the ELM model, the proposed method improves by 52.04%, 83.34%, 84.18%, and 84.44% on the $R^2$, $P_{RMSE}$, $P_{MAE}$, and $P_{MAPE}$ metrics, respectively. Among these results, the prediction performance of BLS is better than that of LSTM, NARX, and ELM except for the proposed method. This is because BLS performs feature extraction and is able to fit the relationship between high-dimensional variables more accurately.

Based on the above discussion, the proposed model in this paper demonstrates high prediction accuracy and better prediction performance in electricity load time-series prediction.

5. Conclusions

To tackle the non-stationary nature of electricity load time-series data, which is influenced by various variables, a new modeling structure for electricity load time-series is suggested. Experiments comparing LSTM, NARX, BLS, and ELM methods are also performed. The experimental results based on actual data show that the proposed method can better tap the intrinsic change rule of electricity load sequence from back to front, which improves the prediction accuracy and prediction performance, and verifies the validity of the proposed method, which can be applied to large-scale electricity load prediction and provide a reference for the optimal scheduling of electricity load. In future research, we will consider using classification ideas to cluster the electricity load data and then perform individual modeling for different categories.