A Multi-Time-Scale Load Forecasting Method for Electric Vehicle Charging Stations Based on Improved LSTM Network and Error Correction

CN117474039BActive Publication Date: 2026-09-01STATE GRID HUNAN ELECTRIC POWER COMPANY LIMITED +1
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Patent Information

Application Number
CN202311371029.6
Authority / Receiving Office
CN · China
Patent Type
Patents(China)
Current Assignee / Owner
Filing Date
2023-10-23
Publication Date
2026-09-01
Estimated Expiration
2043-10-23

AI Technical Summary

Technical Problem

LSTM网络能够解决电动汽车充电站的负荷预测问题,但其挖掘非连续数据间蕴含的有效信息和潜在关系的能力一般,可能导致预测精度不足

Benefits of technology

[0037]本发明利用负荷预测先验知识,将历史负荷数据、气象信息、日期信息、电价信息等负荷影响因素作为输入参数代入CNN-LSTM神经网络模型中进行训练,能够解决现有方法不适用于以充电站为单位的小范围区域的负荷预测问题,实现以充电站为基本单位的电动汽车多时间尺度充电负荷预测,从而能够很好地拟合以月、周、日和小时为代表的电动汽车多时间尺度充电负荷曲线。

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Abstract

This invention discloses a multi-timescale load forecasting method for electric vehicle charging stations based on an improved LSTM network and error correction. The method includes: collecting and preprocessing historical load data to obtain several training samples for load forecasting at different timescales (monthly, weekly, daily, and hourly); each training sample for load forecasting at each timescale includes historical load data and meteorological, date, and electricity price information for the forecast time; building multiple CNN-LSTM models for load forecasting at different timescales, and training them using the training samples to obtain load forecasting models with various patterns; for the time when the load of the electric vehicle charging station is to be predicted, obtaining input samples for each pattern, and using their respective load forecasting models to make predictions, obtaining load forecast values ​​for multiple timescales, and then using the Random Forest (RF) algorithm for error correction. This invention improves the accuracy of load forecasting for electric vehicle charging stations.
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Description

Technical Field

[0001] This invention belongs to the field of load forecasting technology, specifically relating to a multi-timescale load forecasting method for electric vehicle charging stations based on an improved LSTM network and error correction. Background Technology

[0002] Electric vehicles (EVs), as mobile energy storage devices, offer strong technical support for peak shaving and valley filling in power distribution networks, renewable energy absorption, and frequency regulation due to their highly flexible dispatchability. Currently, China has built the world's largest EV charging network in terms of the number of facilities, coverage area, and vehicle coverage. The large-scale growth in the number of electric vehicles and charging facilities places further demands on charging load forecasting technology. Unlike traditional loads, EV charging loads exhibit both temporal and nonlinear characteristics, and the uncertainty brought about by random charging by vehicle owners exacerbates the difficulty of accurate load forecasting.

[0003] Most existing EV charging load forecasting methods predict the medium- to long-term number of various types of electric vehicles in a city, such as private cars, buses, and taxis, and then use statistical methods, such as Monte Carlo simulations, to establish a medium- to long-term charging demand model for urban electric vehicles. This method is suitable for cities or large-scale local areas. However, in real-world scenarios, EV charging demand response and charging infrastructure planning and construction are mostly based on charging stations as the basic unit. Currently, there is a severe lack of load forecasting methods for charging stations, which cannot support practical applications based on charging stations as the basic unit.

[0004] Long Short-Term Memory (LSTM) networks, as a more advanced nonlinear recurrent neural network, take into account both the temporal and nonlinear relationships of load data, avoiding the "vanishing gradient" problem in model training. They have been widely used in load forecasting in recent years. While LSTM networks can solve the load forecasting problem for electric vehicle charging stations, their ability to extract effective information and potential relationships between discontinuous data is generally limited, potentially leading to insufficient prediction accuracy. Summary of the Invention

[0005] This invention provides a multi-timescale load forecasting method for electric vehicle charging stations based on an improved LSTM network and error correction. It improves the LSTM model by leveraging the advantage of CNN in effectively extracting data features, considers multiple factors for load forecasting of electric vehicle charging stations, and performs error correction to improve the accuracy of load forecasting for electric vehicle charging stations.

[0006] To achieve the above technical objectives, the present invention adopts the following technical solution:

[0007] A multi-timescale load forecasting method for electric vehicle charging stations based on an improved LSTM network and error correction includes:

[0008] Step 1: Collect historical load data of electric vehicle charging station meters and preprocess it to obtain several training samples for load prediction at different time scales.

[0009] The various load forecasts at different time scales include load forecasts based on months, weeks, days, and hours; and load forecasts based on months, weeks, days, and hours with forecast periods of 1 hour, 1 hour, 15 minutes, and 15 minutes, respectively.

[0010] The training samples for load forecasting at each time scale include historical load data for that time scale several times prior to the forecast time, as well as meteorological information, date information, and electricity price information for the forecast time.

[0011] Step 2: Build multiple CNN-LSTM neural network models corresponding to load prediction at different time scales, and train the corresponding CNN-LSTM neural networks using training samples to obtain load prediction models of various modes.

[0012] Step 3: For the load forecasting time of the electric vehicle charging station, obtain input samples for load forecasting at various time scales using the same method as in Step 1; use different load forecasting models to forecast the load within the time scale starting from the forecasting time based on the corresponding input samples, and obtain load forecast values ​​for multiple time scales.

[0013] Step 4: For the load forecast values ​​at each time scale, the Random Forest (RF) algorithm is used to correct the errors.

[0014] Furthermore, the preprocessing of historical load data includes: using the mean of the preceding and following data to fill in some missing data, and then using the min-max standardization rule to perform a linear transformation on the load dataset after data filling, limiting the load data to the range of [-1,1].

[0015] Furthermore, the prediction period corresponding to the time scale of month and week is 1 hour, and the prediction period corresponding to the time scale of day and hour is 15 minutes.

[0016] Furthermore, meteorological information refers to the season to which the forecast time belongs: spring, summer, autumn, and winter, represented by numbers 1-4 respectively.

[0017] Date information includes: (1) whether the predicted time belongs to a holiday, workday or weekend. If it does, use 1 to indicate it; otherwise, use 0 to indicate it. (2) the weekday of the predicted time, using numbers 1-7 to indicate Monday to Sunday. (3) the time period of the predicted time, setting 5 minutes per period, dividing the 24 hours of a day into periods 0-96.

[0018] Electricity price information refers to the time-of-use electricity price at the predicted time, using numbers 1-3 to represent the off-peak, normal, and peak electricity prices.

[0019] Furthermore, the CNN-LSTM neural network model includes a CNN network and an LSTM network;

[0020] The CNN network includes three convolutional layers with 256, 32 and 1 convolutional kernels respectively. Each convolutional kernel is 3*3 in size, and the input feature dimension of the CNN network is 8.

[0021] The LSTM network has 4 stacked layers, 1 input feature dimension, and 128 hidden layer nodes.

[0022] The input samples are first processed by a CNN network to extract features, compressing the multidimensional input data into a 1-dimensional structure before being fed into an LSTM network. The first LSTM stack layer of the LSTM network receives the output of the CNN network, and the second to fourth LSTM stack layers each receive the output of the previous LSTM stack layer and perform calculations. Finally, the load prediction value is output through a fully connected layer.

[0023] Furthermore, the calculation formula for each LSTM stack layer is as follows:

[0024] i t =σ(w ix x t +w ih h t-1 +b i )

[0025] f t =σ(w fx x t +w fh h t-1 +b f )

[0026] o t =σ(w ox x t +w oh h t-1 +b o )

[0027] c t =f tc t-1 +i t tanh(w cx x t +w ch h t-1 +b c )

[0028] h t =o t tanhc t

[0029] In the formula, i t ,f t ,o t ,c t ,h t These represent the input gate, forget gate, output gate, memory unit update, and current time output, respectively; w ix ,w ih ,w fx ,w fh ,w ox ,w oh ,w cx ,w ch The matrix weights are the products of the corresponding gate, the input, and the output from the previous time step; x t Input for the current time, h t-1 The output from the previous moment; b i ,b f ,b o ,b c This is the bias term for the corresponding gate.

[0030] Furthermore, step 4 uses the random forest algorithm to correct the error of the load forecast values ​​for each time scale, specifically as follows:

[0031] P2′(t)=P2(t)+e1(t)

[0032] e1(t)=f[e(tT),e(t-2T),…,e(t-nT)]

[0033]

[0034] e(t) = P1(t) - P2(t)

[0035] In the formula, P2′(t) is the corrected load forecast value at time t, P2(t) is the original load forecast value at time t, i.e., the load forecast value at time t obtained in step 4, e1(t) is the error forecast value at time t, T is the forecast period, n is the number of forecast periods included in the current time scale, e is the load forecast error, P1(t) is the actual load value at time t, and f i () represents the prediction result of the i-th decision tree, Mtree is the number of decision trees in the random forest.

[0036] Beneficial effects

[0037] This invention utilizes prior knowledge of load forecasting, and uses historical load data, meteorological information, date information, electricity price information, and other load-influencing factors as input parameters to train a CNN-LSTM neural network model. This can solve the problem that existing methods are not suitable for load forecasting in small areas with charging stations as the unit, and realize multi-timescale charging load forecasting of electric vehicles with charging stations as the basic unit. Thus, it can fit the multi-timescale charging load curves of electric vehicles represented by months, weeks, days, and hours very well.

[0038] Furthermore, considering the highly random nature of charging load, this invention employs the RF algorithm to further refine the output of the trained prediction model. Compared to the traditional LSTM prediction model and the unrefined CNN-LSTM prediction model, it achieves higher prediction accuracy in multi-timescale load prediction for charging stations. This invention's prediction method can help mitigate the impact on the power grid caused by the large-scale integration of electric vehicles and facilitates efficient management and control of large-scale electric vehicles. Attached Figure Description

[0039] Figure 1 This is a model diagram of the prediction method of the present invention;

[0040] Figure 2 This is a graph showing the October charging load forecast results for the EV charging station of this invention.

[0041] Figure 3 This is a graph showing the predicted charging load of the EV charging station for the first week of October according to the present invention.

[0042] Figure 4 This is a graph showing the predicted charging load of the EV charging station for October 5th according to the present invention.

[0043] Figure 5 This is a graph showing the predicted charging load of the EV charging station of this invention at 0:00 on September 5th. Detailed Implementation

[0044] To enable those skilled in the art to better understand the technical solution of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0045] The example uses smart meter load data from a charging station in Changsha, Hunan Province as the dataset. This dataset contains over 30,000 records, representing the charging load of the charging station every 15 minutes from January to December 2022. Load data from January to August is used as the training set, and load data from September to December is used as the test set.

[0046] like Figure 1 As shown, this embodiment of the invention provides a multi-timescale load forecasting method for electric vehicle charging stations based on an improved LSTM network and error correction, specifically including the following steps:

[0047] I. Constructing training samples.

[0048] Since this invention requires the realization of load prediction for electric vehicle charging stations at multiple time scales, different training samples need to be constructed for load prediction at different time scales.

[0049] In this embodiment, since the load forecasting needs to be divided into medium-term load forecasting based on the time scale, short-term load forecasting based on the week, short-term load forecasting based on the day, and ultra-short-term load forecasting based on the hour, it is necessary to build CNN-LSTM neural network models for the above four different time scale load forecasting in step 2 described later, corresponding to modes 1-4, and requiring different input variables respectively.

[0050] For the input variables of load forecasting at each time scale, there are 8-dimensional data features, with historical load information constituting a 1-dimensional data feature, meteorological information constituting a 1-dimensional data feature, date information constituting a 5-dimensional data feature, and electricity price information constituting a 1-dimensional data feature.

[0051] 1. Historical Load Information. First, historical load data from smart meters at electric vehicle charging stations are collected as the original dataset, specifically historical load data with a sampling period of 15 minutes. Then, the mean of the preceding and following data is used to fill in some missing data to reduce its impact on the accuracy of the prediction model, that is, the load mean of the previous and following time points is used to fill in the missing values. Then, the min-max standardization rule is used to linearly transform the original dataset, and its size is limited to the range of [-1, 1], as shown in Equation (1).

[0052]

[0053] In equation (1) x * For the normalized data, x max and x min These are the maximum and minimum values ​​of the original dataset after the missing data has been filled in.

[0054] In this embodiment, the load forecast input variable for medium-term load forecasting (monthly) corresponds to Mode 1, and its historical load information is the load data of the 744 moments before the time to be measured (prediction period of 1 hour, i.e., the previous 31 days); the load forecast input variable for short-term load forecasting (weekly) corresponds to the load data of the 168 moments before the time to be measured (prediction period of 1 hour, i.e., the previous 7 days); the load forecast input variable for short-term load forecasting (dayly) corresponds to Mode 3, and its historical load information is the load data of the 96 moments before the time to be measured (prediction period of 15 minutes, i.e., the previous 1 day); the load forecast input variable for ultra-short-term load forecasting (hourly) corresponds to Mode 4, and its historical load information is the load data of the 16 moments before the time to be measured (prediction period of 15 minutes, i.e., the previous 4 hours).

[0055] 2. Meteorological information refers to the season at the time of forecast. The meteorological information for all models is spring, summer, autumn, and winter, represented by numbers 1-4.

[0056] 3. Date information. The date information for all modes is divided into: (1) Whether the day to which the forecast time belongs is a holiday, workday or weekend, represented by 3-dimensional data. If it belongs, use 1; if it does not, use 0. (2) The weekday of the day to which the forecast time belongs, using numbers 1-7 to represent Monday to Sunday. (3) The time period of the forecast time, that is, the forecast cycle number of the forecast time in 1 day. In the load forecast of Mode 1 and Mode 2, numbers 1-24 are used to represent 24 time periods of 1 day; in the load forecast of Mode 3 and Mode 4, numbers 1-96 are used to represent 96 time periods of 1 day.

[0057] 4. Electricity price information refers to the time-of-use electricity price at the predicted time. All modes use numbers 1-3 to represent the off-peak, normal, and peak electricity prices.

[0058] II. Building the CNN-LSTM model.

[0059] Multiple CNN-LSTM neural network models for load prediction at different time scales were built, and the corresponding CNN-LSTM neural networks were trained using training samples to obtain load prediction models with various modes.

[0060] Each CNN-LSTM neural network model includes both a CNN network and an LSTM network.

[0061] The CNN network consists of three convolutional layers with 256, 32, and 1 kernels respectively. Each kernel is 3*3 in size, and the input feature dimension of the CNN network is 8.

[0062] The LSTM network has 4 stacked layers, 1 input feature dimension, and 128 hidden layer nodes. Its calculation formula is as follows:

[0063] i t =σ(w ix x t +w ih h t-1 +b i (2)

[0064] f t =σ(w fx x t +w fh h t-1 +b f (3)

[0065] o t =σ(w ox x t +w oh h t-1 +b o (4)

[0066] c t =f t c t-1 +i t tanh(w cx x t +w ch h t-1 +b c (5)

[0067] h t =o t tanhc t (6)

[0068] In equations (2)-(6), i t ,f t ,o t ,c t ,h t These represent the input gate, forget gate, output gate, memory unit update, and current time output, respectively. ix ,w ih ,w fx ,w fh ,w ox ,w oh ,w cx ,w ch This represents the matrix weights corresponding to the gate multiplied by the input and the output from the previous time step. t Input for the current time, h t-1 This is the output from the previous moment. (b) i ,b f,b o ,b c This is the bias term for the corresponding gate.

[0069] The input samples are first processed by a CNN network to extract features, compressing the multidimensional input data into a 1-dimensional structure before being fed into an LSTM network. The first LSTM stack layer of the LSTM network receives the output of the CNN network, and the second to fourth LSTM stack layers all receive the output of the previous LSTM stack layer and perform calculations. Finally, the load prediction value is output through a fully connected (Dense) layer.

[0070] III. Actual load forecast.

[0071] For the load forecasting time of electric vehicle charging stations, the method of obtaining training samples in step 1 is used to obtain input samples for load forecasting at various time scales; different load forecasting models are used to forecast the load within the time scale starting from the forecasting time according to the corresponding input samples, so as to obtain load forecast values ​​at multiple time scales.

[0072] The medium-term load forecasting model, corresponding to Mode 1, outputs load values ​​for 744 time points, including the time to be forecasted and subsequent time points, with a batch size of 16. The short-term load forecasting model, corresponding to Mode 2, outputs load values ​​for 168 time points, including the time to be forecasted and subsequent time points, with a batch size of 128. The short-term load forecasting model, corresponding to Mode 2, outputs load values ​​for 96 time points, including the time to be forecasted and subsequent time points, with a batch size of 128. The short-term load forecasting model, corresponding to Mode 4, outputs load values ​​for 8 time points, including the time to be forecasted and subsequent time points, with a batch size of 128.

[0073] IV. Error Correction.

[0074] For the load forecast values ​​at each time scale, the random forest algorithm is used for error correction. The specific calculation formula is as follows:

[0075] P2′(t)=P2(t)+e1(t) (7)

[0076] e1(t)=f[e(tT),e(t-2T),…,e(t-nT)] (8)

[0077]

[0078] e(t)=P1(t)-P2(t) (10)

[0079] In the formula, P2′(t) is the corrected load forecast value at time t, P2(t) is the original load forecast value at time t, i.e., the load forecast value at time t obtained in step 4, e1(t) is the error forecast value at time t, T is the forecast period, n is the number of forecast periods included in the current time scale, e is the load forecast error, P1(t) is the actual load value at time t, and f i () represents the prediction result of the i-th decision tree, M tree is the number of decision trees in the random forest.

[0080] V. Predictive Performance Evaluation

[0081] Select root mean square error e RMSE Mean absolute error e MAE The coefficient of determination R² is used as a model evaluation metric to measure the predictive performance of the CNN-LSTM neural network model after error correction.

[0082]

[0083]

[0084]

[0085] In equations (11)-(13), x = {x1, x2, x3, ..., x n Let y = {y1, y2, y3, ..., y} be the load forecast value. n} represents the actual load value, n represents the number of predictions, and y represents the mean of the actual values.

[0086] According to the prediction method of the present invention, the predicted load values ​​of electric vehicle charging stations for each month, week, day, and hour from September to December 2022 are obtained.

[0087] The medium-term load forecast results (Mode 1) obtained by the forecast method of this invention, taking the electric vehicle charging station load forecast values ​​at 744 time points in October 2022 as an example, are compared with the LSTM single model and the uncorrected CNN-LSTM model as follows: Figure 2 As shown in Table 1, the performance of the three methods in medium-term load forecasting is evaluated. The results show that the CNN network slightly improves the prediction accuracy of the LSTM model, while the prediction error of the RF-corrected CNN-LSTM model is significantly reduced by 84.37% (RMSE) and 87.28% (MAE), respectively, verifying the good effect of the method of the present invention in charging load forecasting.

[0088] Table 1 Comparison of Medium-Term Load Forecast Results

[0089] LSTM 572.80 314.52 0.86 CNN-LSTM 539.84 297.83 0.90 RF-CNN-LSTM 84.36 37.89 0.99

[0090] The short-term load forecast results (mode 2 and mode 3) obtained by the forecast method of this invention are compared with the electric vehicle charging station load forecast values ​​(mode 2) for 168 time points in the first week of October 2022 and the electric vehicle charging station load forecast values ​​(mode 3) for 96 time points on October 5, 2022, using the method of this invention as examples. The results are as follows: Figure 3 and Figure 4 As shown in Table 2, the performance of the three methods in short-term load forecasting is evaluated. The results show that the LSTM model significantly lags behind the CNN-LSTM model in predicting daily loads with greater randomness. This is because the CNN network helps the model extract effective input features better. The CNN-LSTM model with RF correction shows a significant improvement in prediction accuracy compared to the uncorrected model.

[0091] Table 2 Comparison of Short-Term Load Forecast Results

[0092]

[0093]

[0094] The ultra-short-term load forecast results (mode 4) obtained by the method of this invention, taking the load forecast values ​​of electric vehicle charging stations at 00:00 on September 5, 2022, for a total of 8 time points as an example, are compared with the LSTM single model and the uncorrected CNN-LSTM model as follows: Figure 5 As shown in Table 3, the performance of the three methods in ultra-short-term load forecasting is evaluated. The results show that the volatility of the load curve increases significantly in a short period. When the load pattern changes drastically, the LSTM single model and the CNN-LSTM model without RF correction are not sensitive to the more random short-term load changes, resulting in poor prediction performance. However, the prediction model with RF correction can accurately capture the inflection point of load changes. Compared with the model before correction, RMSE and MAE are reduced by 16.71% and 46.08% respectively, and R² is improved by 6.67%, achieving an improvement in the accuracy of charging load forecasting.

[0095] Table 3 Comparison of Ultra-Short-Term Load Forecast Results

[0096] LSTM 624.05 466.25 0.40 CNN-LSTM 214.04 238.78 0.75 RF-CNN-LSTM 178.28 128.75 0.80

[0097] The above embodiments are preferred embodiments of this application. Those skilled in the art can make various changes or improvements based on them. Without departing from the overall concept of this application, these changes or improvements should fall within the scope of protection claimed in this application.

Claims

1. A multi-timescale load forecasting method for electric vehicle charging stations based on an improved LSTM network and error correction, characterized in that, include: Step 1: Collect historical load data of electric vehicle charging station meters and preprocess it to obtain several training samples for load prediction at different time scales. Among them, there are multiple load forecasts at different time scales, including load forecasts at monthly, weekly, daily, and hourly scales; and load forecasts at monthly, weekly, daily, and hourly scales with forecast periods of 1 hour, 1 hour, 15 minutes, and 15 minutes, respectively. The training samples for load forecasting at each time scale include historical load data for that time scale several times prior to the forecast time, as well as meteorological information, date information, and electricity price information for the forecast time. Step 2: Build multiple CNN-LSTM neural network models for load prediction at different time scales, and train the corresponding CNN-LSTM neural networks using training samples to obtain load prediction models with multiple modes. The CNN-LSTM neural network model includes a CNN network and an LSTM network; The CNN network includes three convolutional layers with 256, 32 and 1 convolutional kernels respectively. Each convolutional kernel is 3*3 in size, and the input feature dimension of the CNN network is 8. The LSTM network has 4 stacked layers, 1 input feature dimension, and 128 hidden layer nodes. The input sample is first processed by a CNN network to extract features, compressing the multidimensional input data into a 1-dimensional dimension before it is input into an LSTM network. The first LSTM stack layer of the LSTM network receives the output of the CNN network, and the second to fourth LSTM stack layers all receive the output of the previous LSTM stack layer and perform calculations. Finally, the load prediction value is output through a fully connected layer. The calculation formulas for each LSTM stack layer are as follows: ; In the formula, These represent the input gate, forget gate, output gate, memory unit update, and current time output, respectively. The matrix weights are the product of the corresponding gate, the input, and the output from the previous time step. Enter the current time. The output from the previous moment; This is the bias term for the corresponding gate; Step 3: For the load forecasting time of the electric vehicle charging station, obtain input samples for load forecasting at various time scales using the same method as in Step 1; use different load forecasting models to forecast the load within the time scale starting from the forecasting time based on the corresponding input samples, and obtain load forecast values ​​for multiple time scales. Step 4: For the load forecast values ​​at each time scale, the Random Forest (RF) algorithm is used for error correction. Step 4 uses the random forest algorithm to correct the error of the load forecast for each time scale. Specifically: ; ; ; ; In the formula, For the revised version Forecasted load values ​​at any time Before the correction The predicted load value at that time, i.e., the value obtained in step 4. Forecasted load values ​​at any time for Predicted time error value For the prediction cycle, For load forecasting error, for The actual load value at any given time. This refers to the number of forecast periods included in the current time period. For the first The prediction results of the decision tree is the number of decision trees in the random forest.

2. The load forecasting method according to claim 1, characterized in that, Preprocessing of historical load data includes: using the mean of previous and subsequent data to fill in some missing data, and then using the min-max standardization rule to perform a linear transformation on the load dataset after data filling, limiting the load data to the range of [-1,1].

3. The load forecasting method according to claim 1, characterized in that, The prediction period for time scales of month and week is 1 hour, and the prediction period for time scales of day and hour is 15 minutes.

4. The load forecasting method according to claim 1, characterized in that, Meteorological information refers to the season at which the forecast time belongs: spring, summer, autumn, and winter, represented by numbers 1-4 respectively. Date information includes: (1) whether the predicted time belongs to a holiday, workday or weekend. If it belongs, use 1 to indicate it, otherwise use 0 to indicate it; (2) the weekday of the predicted time, using numbers 1-7 to indicate Monday to Sunday; (3) the time period of the predicted time, setting 5 minutes per time period, dividing the 24 hours of a day into time periods 0-96. Electricity price information refers to the time-of-use electricity price at the predicted time, using numbers 1-3 to represent the off-peak, normal, and peak electricity prices.