Point-interval electricity price prediction method based on weight combination mechanism and deep learning
By adopting a weight combination mechanism and deep learning method in electricity price prediction, combined with improved data decomposition and prediction models, the shortcomings of the existing electricity price prediction model under high-frequency and high-volatility data are solved, and more accurate and reliable electricity price prediction is achieved, providing electricity price fluctuation range and uncertainty analysis is provided, and more scientific market decision-making and regulation are supported.
Patent Information
- Application Number
- CN202411584980.4
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2024-11-07
- Publication Date
- 2025-03-07
AI Technical Summary
The existing electricity price prediction model is difficult to provide accurate predictions under high-frequency and high-volatility data, and lacks a reasonable description of the range and uncertainty of electricity price fluctuations, making it difficult to meet the decision-making needs of market participants and regulations makers.
Using a point-interval electricity price prediction method based on weight combination mechanism and deep learning, the data decomposition of improved adaptive noise-fully integrated empirical mode decomposition model, electricity price prediction of bidirectional time convolution network and bidirectional long and short-term memory network, and the interval prediction of multi-objective optimized mucosmic algorithm and kernel density estimation, provides a more comprehensive analysis of electricity price fluctuation range and uncertainty.
It significantly improves the accuracy and reliability of electricity price predictions, provides specific point prediction values and range predictions for electricity price fluctuations, helping market participants and regulations makers to formulate strategies and market regulation measures more scientifically, reduce economic losses and improve the overall operating efficiency of the power market.
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Figure CN119579224A_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the technical field of electricity price prediction, and in particular to a point-interval electricity price prediction method based on a weight combination mechanism and deep learning. Background Art
[0002] With the liberalization and deregulation of the power industry, the power market has become increasingly dynamic and highly uncertain. The fluctuation of electricity prices is not only affected by the relationship between supply and demand, but also by a variety of external factors such as meteorological conditions and energy prices. Therefore, establishing an accurate electricity price forecasting model has always been an important issue for power market participants to avoid risks and maximize economic benefits. Accurate electricity price forecasting not only helps power production enterprises and consumers to arrange resource allocation reasonably, but also helps rule makers to formulate more scientific market regulation measures.
[0003] However, existing electricity price forecasting models often fail to meet the demand for accurate forecasting under high-frequency and highly volatile data. Traditional statistical methods or single deep learning models are insufficient in coping with the complex dynamic changes in electricity prices, and it is difficult to balance the accuracy of forecasts with uncertainty analysis. Existing methods can usually only provide specific point forecast values and lack a reasonable description of the range and uncertainty of electricity price fluctuations. The lack of interval forecasts makes it difficult for market participants to fully assess risks when formulating strategies, which may lead to market decision-making errors and even greater economic losses when electricity prices fluctuate violently.
[0004] The present invention aims to overcome the above-mentioned shortcomings and provide a point-interval electricity price prediction method based on weight combination mechanism and deep learning, provide a more comprehensive range of electricity price fluctuations, and provide more reliable decision support for market participants and regulation makers. Summary of the invention
[0005] The present invention provides a point-to-interval electricity price prediction method based on a weight combination mechanism and deep learning. A point-to-interval electricity price prediction method based on a weight combination mechanism and deep learning comprises the following steps:
[0006] S1, data collection and processing: collect electricity price data from the power market, sort out the factors affecting electricity price fluctuations, including meteorological factors, crude oil prices, and coal prices, obtain raw data, and clean the raw data;
[0007] S2, data decomposition: the original data is decomposed using the improved adaptive noise fully integrated empirical mode decomposition model;
[0008] S3, electricity value prediction: Based on the results of the original data decomposition, a bidirectional temporal convolutional network BiTCN model and a bidirectional long short-term memory network BiLSTM model are constructed. According to the prediction results of the BiTCN model and the BiLSTM model, the multi-objective optimization slime mold MOSMA algorithm is used to determine the weights to obtain the final electricity price prediction value;
[0009] S4, interval prediction based on kernel density estimation: After obtaining the final electricity price forecast value, the distribution of the forecast error is calculated by the kernel density estimation method, and the confidence interval of the electricity price is calculated according to the set confidence level.
[0010] Optionally, the data decomposition in S2 includes:
[0011] S21, constructing a first residual signal: constructing a residual signal, and calculating the first residual signal through an EMD model, which is expressed as:
[0012] X i (t) = X (t) + ε0E1 (W i (t));
[0013] r1(t)= <MX i (t)>;
[0014] Where M(·) represents the local mean of the signal, and <·> means taking the mean;
[0015] S22, calculating the first-order modal component: calculating the first-order modal component according to the first residual signal, expressed as:
[0016] IMF1 n (t) = X(t) - r1(t);
[0017] S23, estimate the second-order residual signal: estimate the second residual signal as a series of r1(t)=ε1E2W i The mean of (t) is used to obtain the second-order modal component, which is expressed as:
[0018]
[0019] S24, repeat the iteration to obtain higher-order modal components: repeatedly calculate the first-order modal components and estimate the second-order residual signal to obtain the j-th residual signal and the j-th modal component, which can be expressed as:
[0020] r j (t)= <M(r j-1 (t)+ε j-1 E j W i (t))>;
[0021]
[0022] Optionally, the electricity value prediction in S3 includes:
[0023] S31, BiTCN model prediction: The time series data is processed through convolution operations, including forward learning and backward learning. The forward convolution layer and the backward convolution layer learn the time series data through two forward TCN residual blocks and two backward TCN residual blocks respectively, and integrate the output of the residual blocks to obtain the preliminary electricity price prediction results;
[0024] S32, BiLSTM model prediction: Using the bidirectional long short-term memory network BiLSTM model, the time series data is processed through the forward layer and the reverse layer to obtain another preliminary electricity price prediction result;
[0025] S33, electricity price forecast value output: Use the multi-objective optimization slime mold MOSMA algorithm to optimize the prediction results of the BiTCN model and the BiLSTM model. The optimization targets are the mean square error and standard deviation to obtain the final electricity price forecast value.
[0026] Optionally, the BiTCN model prediction in S31 includes:
[0027] S311, convolution operation: Let the convolution kernel be K = (k1, k2, ..., k m ), the input time series is a=(h1,h2,...,h n ), then a n The causal convolution of is represented as:
[0028]
[0029] Among them, P n-m+1 For input data;
[0030] S312, calculation of dilated convolution: reduce the number of convolution neurons between two adjacent layers by the value of the dilation coefficient l, then a n The dilated convolution is expressed as:
[0031]
[0032] Where d is the expansion factor;
[0033] S313, residual connection: The residual connection in the TCN model consists of dilated causal convolution and nonlinear mapping, expressed as:
[0034] K(a i ) = ω2[Relu(ω1α1)];
[0035] Among them, ω1 and ω2 are the weights of the convolutional layer, and Relu is the activation function to activate the output of the residual block;
[0036] S314, one-dimensional full convolution operation: The residual block also includes a one-dimensional full convolution operation. Assuming that the input and output sequences have the same length, the output is expressed as:
[0037] P(α i )=K(α i )+H i ;
[0038] P i+1 =Relu[P(α i )];
[0039] Among them, P i+1 It is the full convolution output result;
[0040] S315, final result integration of the forward and backward residual blocks: In the forward TCN and backward TCN residual blocks, the last time step obtained by the last residual block is taken out, and the output results of the two time steps are averaged to obtain the final prediction result.
[0041] Optionally, the BiLSTM model in S32 is expressed as:
[0042] h i =σ t (W1f i +W3h i-1 );
[0043] r i =σ t (W2f i +W5r i+1 );
[0044] Y i =σ t (W4h i +W6r i );
[0045]
[0046] Among them, h i Represents the output of the forward layer in the bidirectional long short-term memory network, r i is the output result of the reverse layer, f i Represents the i-th data of the input, Y i represents the output of the BiLSTM output layer, σ t (x) is the activation function, and W is the weight matrix of different input and output layers.
[0047] Optionally, the output of the electricity price forecast value in S33 includes:
[0048] S331, select data set: select the data set for MOSMA optimization;
[0049] S332, define objective function: define a multi-objective function for MOSMA optimization, including mean square error and standard deviation as optimization targets;
[0050] S333, set MOSMA parameters: set MOSMA parameters, including population size N, maximum number of iterations t max , population N pop , two objective functions f1 and f2, population search lower limit L b And the population search upper limit U b ;
[0051] S334, perform multi-objective optimization: use MOSMA to optimize until the Pareto optimal solution is reached;
[0052] S335, output the optimal parameter combination: output the optimal parameter combination obtained by MOSMA optimization, reconstruct it with the prediction results of the BiLSTM model and the BiTCN model, and obtain the final electricity price prediction value.
[0053] Optionally, the multi-objective function in S332 is expressed as:
[0054]
[0055] Among them, N is the number of prediction samples, y i is the true value, is the predicted value, and std is the standard deviation function.
[0056] Optionally, the kernel density estimation method in S4 is expressed as:
[0057]
[0058] Among them, n represents the total number of samples, h represents the window width, and X i is the specified sample, and K(.) is the selected Gaussian kernel function.
[0059] Beneficial effects of the present invention:
[0060] The present invention significantly improves the accuracy of electricity price forecasting by realizing two modes, namely point forecasting and interval forecasting. Point forecasting provides specific electricity price forecasting values for rule makers, power producers, consumers, etc., so as to more intuitively understand the possible trend of future electricity prices. Interval forecasting provides the possible range of electricity prices and their degree of uncertainty, helping all parties to make decisions within a wider price range while taking into account the uncertainty of the forecast. This accurate forecasting has important guiding significance for formulating power production plans, power consumption strategies, and power price risk management, and helps all parties in the power market to participate in market competition more fairly, reduce information asymmetry and speculative behavior in the market, and improve the overall operating efficiency of the power market.
[0061] The present invention, through accurate electricity price forecasting, allows rule makers to better understand market dynamics, predict future supply and demand relationships and price trends in the electricity market, and thus formulate more scientific and reasonable electricity market regulations and price control measures. When abnormal conditions occur in the electricity market or price fluctuations are large, rule makers can use the electricity price forecast results to conduct timely market intervention and regulation to prevent excessive market fluctuations or imbalances. This not only helps maintain the stability and healthy development of the electricity market, but also promotes energy transformation and sustainable development, promotes the green transformation of the electricity market and reduces pollution and damage to the environment. BRIEF DESCRIPTION OF THE DRAWINGS
[0062] In order to more clearly illustrate the technical solutions in the present invention or the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings in the following description are only for the present invention. For ordinary technicians in this field, other drawings can be obtained based on these drawings without paying creative work.
[0063] Figure 1 Schematic diagram of the prediction method flow in an embodiment of the present invention. DETAILED DESCRIPTION
[0064] The present invention is described in detail below in conjunction with the accompanying drawings and specific embodiments. At the same time, it is explained here that in order to make the embodiments more detailed, the following embodiments are the best and preferred embodiments, and those skilled in the art may also adopt other alternatives to implement some known technologies; and the accompanying drawings are only for more specific description of the embodiments, and are not intended to specifically limit the present invention.
[0065] It should be noted that the references to "one embodiment", "an embodiment", "an exemplary embodiment", "some embodiments" and the like in the specification indicate that the embodiments described may include specific features, structures or characteristics, but not every embodiment may include the specific features, structures or characteristics. In addition, when a specific feature, structure or characteristic is described in conjunction with an embodiment, it should be within the knowledge of a person skilled in the art to implement such feature, structure or characteristic in conjunction with other embodiments (whether or not explicitly described).
[0066] In general, a term can be understood, at least in part, from its use in context. For example, depending, at least in part, on the context, the term "one or more" as used herein can be used to describe any feature, structure, or characteristic in the singular sense, or can be used to describe a combination of features, structures, or characteristics in the plural sense. Additionally, the term "based on" can be understood as not necessarily intended to convey an exclusive set of factors, but can instead, depending, at least in part, on the context, allow for the presence of other factors that are not necessarily explicitly described.
[0067] like Figure 1 As shown, a point-interval electricity price prediction method based on weight combination mechanism and deep learning includes the following steps:
[0068] (1) Data collection and processing: Collect electricity price data from the power market, sort out influencing factors such as meteorological factors, crude oil prices, coal prices, etc., obtain raw data, and clean the obtained raw data.
[0069] (2) Data decomposition based on improved adaptive noise fully integrated empirical mode decomposition:
[0070] Firstly, the original electricity price series is decomposed using the improved adaptive noise fully integrated empirical mode decomposition model. The specific steps are as follows:
[0071] The operator M(·) is defined to represent the local mean of the signal; <·> represents taking the mean.
[0072] 1) Construct X i (t) = X (t) + ε0E1 (W i (t)), the first residual signal is calculated by the EMD model as r1(t)= <MX i (t)>;
[0073] 2) Calculate the first-order modal component, which can be expressed as IMF1 n (t) = X(t) - r1(t);
[0074] 3) Estimate the second residual signal as a series of r1(t)=ε1E2W i (t), so the second-order modal component can be expressed as
[0075] 4) Repeat the above steps 2) and 3), then the jth residual signal and the jth order modal component can be expressed as r j (t)= <M(r j-1 (t)+ε j-1 E j W i (t))>,
[0076] (3) Constructing a bidirectional temporal convolutional network and bidirectional long short-term memory network electricity price prediction model based on multi-objective optimization slime mold algorithm optimization:
[0077] After obtaining the decomposition results of electricity prices, two prediction models are constructed, namely, a bidirectional temporal convolutional network model and a bidirectional long short-term memory network model. According to the prediction results of the two models, the multi-objective optimization slime mold algorithm is used to optimize the weights.
[0078] 1) The specific steps of BiTCN prediction model construction are as follows:
[0079] The core of BiTCN is the convolution operation, which processes time series data through convolution layers. It is divided into forward learning and backward learning. It consists of two forward TCN residual blocks and two backward TCN residual blocks. The final result is obtained by integrating the outputs of these residual blocks. Its basic structure is the same, and the content of the forward convolution calculation of the input sequence is as follows:
[0080] ① Assume that the convolution kernel is K = (k1, k2, ..., k m ), the input time series is a=(h1,h2,...,h n ), then a n The causal convolution is expressed as P n-m+1 For input data.
[0081] ② By reducing the value of the expansion coefficient l, the number of convolutional neurons between two adjacent layers is reduced. n The dilated convolution can be expressed as d is the expansion factor, which grows exponentially with 2.
[0082] ③ Residual connections are usually used in TCN. The residual block is composed of a dilated causal convolution and a nonlinear mapping, which can be expressed as K(a i )=ω2[Relu(ω1α1)], ω1 and ω2 are the weights of the convolution layer; Relu is the activation function, which activates the residual block output. The other part is a one-dimensional full convolution operation, which makes the input and output sequences have the same length. The output can be expressed as P(α i )=K(α i )+H i , Pi+1 =Relu[P(α i )],P i+1 This is the full convolution output result.
[0083] ④ In the forward TCN and backward TCN residual blocks, the last time step obtained by the last residual block is taken out, and then the two time steps are averaged to obtain the final prediction result.
[0084] 2) The specific expression of BiLSTM prediction model construction is as follows:
[0085] h i =σ t (W1f i +W3h i-1 ) (1)
[0086] r i =σ t (W2f i +W5r i+1 ) (2)
[0087] Y i =σ t (W4h i +W6r i ) (3)
[0088]
[0089] Where: h i Represents the output of the forward layer in the bidirectional long short-term memory network; r i is the output result of the reverse layer; f i Represents the i-th data of the input; Y i represents the output of the BiLSTM output layer; σ t (x) is the activation function; W is the weight matrix of different input and output layers.
[0090] 3) After obtaining the prediction results of BiLSTM and BiTCN, the multi-objective optimization slime mold algorithm (MOSMA) is used to determine the weights. The multi-objective function optimized by MOSMA uses mean square error and standard deviation respectively, and the formula is as follows:
[0091]
[0092] Among them, N is the number of prediction samples, y i is the true value, is the predicted value, and std is the standard deviation function.
[0093] The specific process is described as follows:
[0094] Step 1: Select the dataset for MOSMA optimization;
[0095] Step 2: Define objective functions f1 and f2 and establish a training model;
[0096] Step 3: Set MOSMA parameters, including population size N, maximum number of iterations t max , population N pop , two objective functions f1 and f2, population search lower limit L b And the population search upper limit U b wait;
[0097] Step 4: Use MOSMA to optimize until the Pareto optimal solution is reached;
[0098] Step 5: Output the optimal parameter combination of the model and reconstruct it with the prediction results of BiLSTM and BiTCN to obtain the final prediction result.
[0099] (4) Interval prediction based on kernel density estimation:
[0100] After obtaining the prediction results, the absolute error between the true value and the predicted value is calculated, and the error is estimated by kernel density. The formula is: Where n represents the total number of samples; h represents the window width; X i is the specified sample; K(.) is the selected Gaussian kernel function. According to the kernel density estimate of the error, different confidence levels are set to calculate the confidence interval. For example, for a 95% confidence interval, it is necessary to find the upper and lower quantiles of the probability density function, denoted as q0.025 and q0.975. The final interval prediction is in is the predicted value at time t.
[0101] (5) Example analysis:
[0102] By collecting price data from a certain electricity market, we conduct case analysis. In order to evaluate the prediction performance of the model, this paper selects mean absolute error (MAE), mean square error (MSE), root mean square error (RMSE), and mean absolute percentile error (MAPE) as evaluation indicators. The calculation formula is as follows:
[0103]
[0104] Among them, y i represents the true value, represents the predicted value of the true value, i = 1, 2, ..., n, n is the number of samples. The results are shown in Table 1. As can be seen from Table 1, the model proposed by this method has the highest accuracy.
[0105] Table 1 Comparison of model results
[0106] Evaluation indicators This method BiTCN BiLSTM LSTM MAE 20.01 45.31 34.41 54.21 RMSE 34.14 67.41 72.76 84.12 MAPE 18.43% 24.21% 23.98% 31.07%
[0107] The interval prediction indicators are the coverage of the prediction interval (PICP) and the average bandwidth of the prediction interval (PINAW), and the calculation formula is as follows:
[0108]
[0109] Table 2 Interval prediction results of this method
[0110] Confidence 95% 90% 85% PICP 0.9631 0.9164 0.8685 PINAW 0.54 0.42 0.36
[0111] The present invention covers any substitution, modification, equivalent method and scheme made on the essence and scope of the present invention. In order to make the public have a thorough understanding of the present invention, specific details are described in detail in the following preferred embodiments of the present invention, but those skilled in the art can fully understand the present invention without the description of these details. In addition, in order to avoid unnecessary confusion about the essence of the present invention, well-known methods, processes, procedures, components and circuits are not described in detail.
[0112] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as the scope of protection of the present invention.
Claims
1. A point-to-interval electricity price prediction method based on weight combination mechanism and deep learning, characterized in that: The following steps are involved: S1, data collection and processing: collect electricity price data from the power market, sort out the factors affecting electricity price fluctuations, including meteorological factors, crude oil prices, and coal prices, obtain raw data, and clean the raw data; S2, data decomposition: the original data is decomposed using the improved adaptive noise fully integrated empirical mode decomposition model; S3, electricity value prediction: Based on the results of the original data decomposition, a bidirectional temporal convolutional network BiTCN model and a bidirectional long short-term memory network BiLSTM model are constructed. According to the prediction results of the BiTCN model and the BiLSTM model, the multi-objective optimization slime mold MOSMA algorithm is used to determine the weights to obtain the final electricity price prediction value; S4, interval prediction based on kernel density estimation: After obtaining the final electricity price forecast value, the distribution of the forecast error is calculated by the kernel density estimation method, and the confidence interval of the electricity price is calculated according to the set confidence level.
2. A point-to-interval electricity price prediction method based on weight combination mechanism and deep learning according to claim 1, characterized in that: The data decomposition in S2 includes: S21, constructing a first residual signal: constructing a residual signal, and calculating the first residual signal through an EMD model, which is expressed as: X i (t)=X(t)+ε0E1(W i (t)); r1(t)=<MX i (t)>; Where M(·) represents the local mean of the signal, and <·> means taking the mean; S22, calculating the first-order modal component: calculating the first-order modal component according to the first residual signal, expressed as: S23, estimate the second-order residual signal: estimate the second residual signal as a series of r1(t)=ε1E2W i The mean of (t) is used to obtain the second-order modal component, which is expressed as: S24, repeat the iteration to obtain higher-order modal components: repeatedly calculate the first-order modal components and estimate the second-order residual signal to obtain the j-th residual signal and the j-th modal component, which can be expressed as: r j (t)=<M(r j-1 (t)+ε j-1 E j W i (t))>; 3. A point-to-interval electricity price prediction method based on weight combination mechanism and deep learning according to claim 2, characterized in that: The electricity value prediction in S3 includes: S31, BiTCN model prediction: The time series data is processed through convolution operations, including forward learning and backward learning. The forward convolution layer and the backward convolution layer learn the time series data through two forward TCN residual blocks and two backward TCN residual blocks respectively, and integrate the output of the residual blocks to obtain the preliminary electricity price prediction results; S32, BiLSTM model prediction: Using the bidirectional long short-term memory network BiLSTM model, the time series data is processed through the forward layer and the reverse layer to obtain another preliminary electricity price prediction result; S33, electricity price forecast value output: Use the multi-objective optimization slime mold MOSMA algorithm to optimize the prediction results of the BiTCN model and the BiLSTM model. The optimization targets are the mean square error and standard deviation to obtain the final electricity price forecast value.
4. The point-to-interval electricity price prediction method based on weight combination mechanism and deep learning according to claim 3 is characterized in that: The BiTCN model prediction in S31 includes: S311, convolution operation: Let the convolution kernel be K = (k1, k2, ..., k m ), the input time series is a=(h1,h2,...,h n ), then a n The causal convolution of is represented as: Among them, P n-m+1 For input data; S312, calculation of dilated convolution: reduce the number of convolution neurons between two adjacent layers by the value of the dilation coefficient l, then a n The dilated convolution is expressed as: Where d is the expansion factor; S313, residual connection: The residual connection in the TCN model consists of dilated causal convolution and nonlinear mapping, expressed as: K(a i )=ω2[Relu(ω1α1)]; Among them, ω1 and ω2 are the weights of the convolutional layer, and Relu is the activation function to activate the output of the residual block; S314, one-dimensional full convolution operation: The residual block also includes a one-dimensional full convolution operation. Assuming that the input and output sequences have the same length, the output is expressed as: P(a i )=K(a i )+H i ; P i+1 =Relu[P(α i )]: Among them, P i+1 It is the full convolution output result; S315, final result integration of the forward and backward residual blocks: In the forward TCN and backward TCN residual blocks, the last time step obtained by the last residual block is taken out, and the output results of the two time steps are averaged to obtain the final prediction result.
5. A point-to-interval electricity price prediction method based on weight combination mechanism and deep learning according to claim 4, characterized in that: The BiLSTM model in S32 is represented as: h i =σ t (W1f i +W3h i-1 ); r i =σ t (W2f i +W5r i+1 ); AND i =σ t (W4h i +W6r i ); Among them, h i Represents the output of the forward layer in the bidirectional long short-term memory network, r i is the output result of the reverse layer, f i Represents the i-th data of the input, Y i represents the output of the BiLSTM output layer, σ t (x) is the activation function, and W is the weight matrix of different input and output layers.
6. A point-to-interval electricity price prediction method based on weight combination mechanism and deep learning according to claim 5, characterized in that: The output of the electricity price forecast value in S33 includes: S331, select data set: select the data set for MOSMA optimization; S332, define objective function: define a multi-objective function for MOSMA optimization, including mean square error and standard deviation as optimization targets; S333, set MOSMA parameters: set MOSMA parameters, including population size N, maximum number of iterations t max , population N pop , two objective functions f1 and f2, population search lower limit L b And the population search upper limit U b ; S334, perform multi-objective optimization: use MOSMA to optimize until the Pareto optimal solution is reached; S335, output the optimal parameter combination: output the optimal parameter combination obtained by MOSMA optimization, reconstruct it with the prediction results of the BiLSTM model and the BiTCN model, and obtain the final electricity price prediction value.
7. A point-to-interval electricity price prediction method based on weight combination mechanism and deep learning according to claim 6, characterized in that: The multi-objective function in S332 is expressed as: Among them, N is the number of prediction samples, y i is the true value, is the predicted value, and std is the standard deviation function.
8. The point-to-interval electricity price prediction method based on weight combination mechanism and deep learning according to claim 7, characterized in that: The kernel density estimation method in S4 is expressed as: Among them, n represents the total number of samples, h represents the window width, and X i is the specified sample, and K(.) is the selected Gaussian kernel function.