Electric power spot market electricity price difference prediction method based on probability density modeling

Abstract

The invention provides an electricity spot market electricity price difference prediction method based on probability density modeling, and belongs to the technical field of electricity market electricity price prediction and transaction strategy optimization. According to the method, a three-level technical framework of marginal distribution prediction, joint distribution fusion and price difference distribution derivation is constructed. Firstly, a skewed DeepAR model is adopted to predict probability density of a day-ahead electricity price and probability density of a real-time electricity price; secondly, constructing a joint probability density function of the two prices based on t-Copula; and finally projecting the joint probability density to a price difference space through a numerical integration method, and outputting a probability density prediction result of the price difference. The method can accurately describe the skewness of the electricity price, the peak characteristic and the dependency relationship among the prices, and provides a probability basis for arbitrage decision and risk management of electricity market participants.

Description

Technical Field

[0001] This invention relates to the field of electricity market price forecasting and trading strategy optimization, specifically a method for predicting the price difference in the electricity spot market based on probability density modeling. Background Technology

[0002] With a high proportion of renewable energy (wind power and photovoltaic) being integrated into the power system, the intermittent and random nature of its output has led to increasingly drastic fluctuations in the day-ahead (DA) and real-time (RT) electricity prices in the spot market. The price difference between the two has become a core arbitrage basis for market participants (such as power generation companies and electricity sales companies). However, the formation mechanism of the price difference is affected by multiple factors such as load characteristics, renewable energy output, and grid constraints, exhibiting three core characteristics: skewness, peaking, and time-series dependence. Traditional point prediction methods (such as ARIMA and conventional neural networks) are unable to characterize its probability distribution characteristics, resulting in the inability to quantify the uncertainty of the prediction results and making it difficult to support risky trading decisions.

[0003] Existing technologies suffer from two key shortcomings: First, they primarily focus on independent forecasting of electricity prices in a single market (DA or RT), failing to specifically model the price spread—a directly related indicator of returns—and ignoring the inherent dependency between day-ahead and real-time electricity prices (especially tail-end linkage under extreme price conditions). Second, they mostly output deterministic forecast values, failing to provide probability distribution information on the price spread (such as forecast ranges and probabilities of extreme price spreads), making it difficult for market participants to assess arbitrage risks. Therefore, there is an urgent need for a systematic forecasting method that can accurately characterize the probability distribution of the price spread and integrate the dependency on electricity prices, providing quantitative risk support for electricity spot market transactions. Summary of the Invention

[0004] This invention aims to solve the problems of neglecting the dependence of electricity prices and lacking probability distribution output in existing electricity price spread prediction. It provides a whole-process method from "marginal probability density prediction → joint distribution construction → price spread distribution calculation" to achieve accurate output of price spread probability density and prediction interval, providing a scientific basis for market participants' arbitrage decisions and risk management.

[0005] The electricity spot market price difference prediction method proposed in this invention is based on a three-level technical framework: marginal distribution prediction → joint distribution fusion → price difference distribution quantification. This method includes the following steps: 1. Price spread probability density distribution modeling In this invention, the price difference is calculated by subtracting the real-time price from the day-ahead electricity price, as shown below: To model the probability density distribution of the price spread, we first need to model the probability distributions of day-ahead and real-time electricity prices. In this invention, the Jones-Faddy skewed t-distribution is used to characterize the marginal probability density of day-ahead electricity prices, which can be expressed as: in in B () is the beta function.

[0006] The probability density modeling of real-time electricity prices follows the same principle; therefore, it can be obtained through distribution parameter estimation. , Integrating yields the cumulative distributions of each component: in, Position parameters It is a scale parameter. It is the left tail weight parameter. This is the right-tailed heavy parameter, which is subsequently predicted using a neural network. Based on Sklar's theorem, the joint probability distribution of day-ahead and real-time electricity prices can be decomposed into their respective marginal distributions and a Copula function describing the dependency structure between variables. Specifically, the joint probability density function of day-ahead and real-time electricity prices can be expressed as: in For the skewed day-ahead and real-time electricity price distributions, there are in and These are the correlation coefficient and the degrees of freedom, respectively.

[0007] Based on the definition of price spread, the cumulative distribution of price spread can be obtained as follows: thereby obtaining According to the transformed Perform variable substitution Therefore, we obtain Based on the relationship between CDF and PDF, the PDF of the price difference can be obtained as follows: Its joint probability density relationship with day-ahead electricity price and real-time electricity price is as follows: That is Based on the derivation of the day-ahead and real-time electricity price joint probability density function, the expression for the price difference probability density function is obtained as follows: By numerical summation approximation, the probability density distribution of the price difference can be expressed as: 2. Day-ahead / real-time electricity price marginal probability density prediction based on Skewed DeepAR To address the skewed and peak characteristics of DA and real-time electricity prices, a Jones-Faddy skewed t-distribution-driven DeepAR neural network (Skewed DeepAR) is designed to output the marginal probability density functions (PDFs) of both.

[0008] The model input features include: historical day-ahead electricity price series Historical real-time electricity price series Time-series covariates (load forecast values) Date coding, new energy output forecast For numerical features, Z-score standardization is used; for date features, trigonometric function encoding (such as sin(2)) is used. πt / 24) Depicting intraday periodicity.

[0009] The prediction for the future from time 1 to time T can be expressed as: The Jones-Faddy skewed t-distribution parameter output model uses a recurrent neural network (RNN) as its basic architecture, through hidden states... h t The formula for updating the hidden state to pass on temporal dependency information is: in, f For RNN activation functions, X t Let θ be the covariate at time t, and Θ be the RNN parameters. The model output layer consists of four fully connected layers, outputting the skewed t-distribution parameters of the day-ahead / real-time electricity price: Day-ahead electricity price forecast distribution parameters: location parameters Scale parameters Left tail weight parameters Right tail weight parameters ; RT electricity price parameters: location parameters Scale parameters Left tail weight parameters Right tail weight parameters ; Based on the Jones-Faddy skewed t-distribution parameters, the marginal PDFs of day-ahead and real-time electricity prices can be constructed separately (taking day-ahead electricity prices as an example, and the same applies to real-time electricity prices), as shown in the following expressions: Therefore, the prediction problem can be expressed as: The marginal CDF is obtained based on the prediction results: To optimize the marginal distribution parameters of Skewed DeepAR and ensure its accurate fit to the probabilistic characteristics of historical electricity prices, a negative log-likelihood (NLL) loss function is used for model training, based on the historical day-ahead electricity price series. With real-time electricity price series Its joint likelihood function is the product of the marginal likelihoods at each time step (simplified based on the assumption of temporal independence): The parameter update logic is as follows: To avoid numerical underflow and to adapt to the optimization logic of deep learning, the natural logarithm of the joint likelihood is taken and then negative, resulting in the negative log-likelihood loss.

[0010] 3. Construction of the joint probability density of day-ahead and real-time electricity prices based on t-Copula To capture the dependence between DA and real-time electricity prices (especially the tail linkage under extreme prices), a two-dimensional joint PDF is constructed by fusing the marginal distributions of the two based on Sklar's theorem and using the t-Copula function.

[0011] 3.1 t-Copula parameter estimation Using historical day-ahead / real-time electricity price data, the t-Copula parameter (correlation coefficient) is estimated in two steps using the inference function method (IFM). r and degrees of freedom n ).

[0012] Step 1: Based on historical data, use the trained Skewed DeepAR model to convert historical electricity prices into uniformly distributed samples: in, u t , v t ∈[0,1], satisfying a uniform distribution (property integral transformation property); Step 2: For uniformly distributed samples {( u t , v t) The density function parameters of t-Copula are optimized using maximum likelihood estimation (MLE). in, It is the t-Copula density function.

[0013] 3.2 Constructing the joint PDF based on the marginal PDF and the estimated parameters of the t-Copula density function, and deriving the day-ahead-real-time electricity price joint PDF using Sklar's theorem: in Differentiation yields then The core advantage of this modeling method is that it preserves the skewness and peak characteristics of day-ahead and real-time electricity prices (determined by the marginal PDF) while using the t-Copula density function to characterize the dependency between the two (especially the tail linkage under extreme prices).

[0014] 4. Derivation of Price Difference Probability Density and Prediction Interval Output Based on Numerical Integration According to the definition of price difference According to market rules, the day-ahead and real-time electricity price combined PDF is projected into a price difference PDF through numerical integration, and the forecast range is output.

[0015] 4.1 Integral Expression for Price Difference PDF According to the probability integral transformation, the price difference PDI The PDF can be represented as a single-variable integral of the day-ahead electricity price, as follows: Perform variable substitution get Differentiation yields in, The value represents the real-time electricity price. Because the joint PDF contains nonlinear terms, this integral has no analytical solution and needs to be approximated using numerical methods.

[0016] 4.2 Numerical Integration Methods A three-step method of "truncating invalid regions → discretizing the grid → linear interpolation" is adopted to reduce the computational load while ensuring accuracy. Step 1: Truncate the invalid interval. Based on the marginal PDF of day-ahead / real-time electricity prices, determine the valid interval of the probability density set: in, For the minimum threshold (e.g.) This ensures that the cumulative probability within the effective interval is ≥99.9%; Step 2: Discretize the day-ahead electricity price range. Divided into M equally spaced discrete points Discrete spacing ; Step 3: Linear interpolation for non-discrete points. For each day-ahead discrete electricity price point... Calculate the corresponding real-time electricity price : like p RT It falls exactly on the discrete grid of the real-time electricity price. (), directly obtain the corresponding joint PDF value ; like p RT If a point does not fall on a discrete grid, find its two nearest discrete points. , (satisfy ≤ ≤ The joint PDF value is calculated using linear interpolation: Step 4: Discretize and sum to approximate the integral. Transform the continuous integral into a summation of discrete points to obtain an approximate value for the price difference PDF: 4.3 Normalization Correction and Prediction Interval Calculation Normalization correction: Because truncating invalid intervals and linear interpolation may introduce errors, The total probability may deviate from 1, and normalization is needed to ensure that it conforms to the basic properties of PDF: in, Let be the set of discrete points representing the price difference. The discrete interval of the price difference; Prediction interval calculation: Integrate the normalized price spread PDF to obtain the cumulative distribution function (CDF). Use quantiles as the prediction interval (e.g., 90% confidence interval is...). ),in It is the inverse function of CDF (quantile function).

[0017] The beneficial effects of this invention are as follows: This invention constructs a systematic solution for predicting the probability density of electricity price differences through "joint modeling, skewness capture, and numerical solution," fundamentally improving the theoretical rigor and practical value of probability prediction. Specifically, this is reflected in: 1. A price spread prediction framework based on joint probability density modeling is proposed, which solves the theoretical defect of traditional methods that ignore price correlation. By deriving the integral relationship between the price spread probability density and the day-ahead-real-time electricity price joint probability density, and introducing Copula theory to construct the joint distribution, the problem of correlation modeling in the "prediction-difference" strategy is overcome, providing a rigorous theoretical basis for price spread probability prediction.

[0018] 2. A DeepAR probability density prediction model guided by a skewed t-distribution was designed to effectively capture the asymmetric distribution characteristics of electricity prices. By introducing the Jones-Faddy skewed t-distribution into the DeepAR framework and using four neural networks to output distribution parameters, the skewness, kurtosis, and heavy-tailed characteristics of electricity prices are modeled simultaneously, significantly improving the accuracy of marginal probability density prediction.

[0019] 3. A numerical conditional density projection method is proposed, solving the numerical computation problem of complex integrals without closed-form solutions. By discretizing the integration region, performing linear interpolation, and normalization correction, an accurate projection of the joint probability density onto the price difference space is achieved, providing an efficient and feasible solution for the numerical computation of probability density.

[0020] 4. By organically integrating the aforementioned core technologies through a probabilistic framework, a complete probability density prediction system for electricity price spreads has been constructed. This system forms a complete closed loop from marginal distribution prediction and joint distribution modeling to numerical integration solutions, significantly improving the accuracy, reliability, and practicality of electricity price spread probability prediction, and providing a scientific probabilistic basis for risk quantification and arbitrage decision-making for electricity market participants. Detailed Implementation

[0021] The specific implementation of this invention includes the following steps: 1. Predicting the marginal probability density of day-ahead and real-time electricity prices; 2. Constructing the joint probability density of day-ahead and real-time electricity prices; 3. Solving for the probability density and prediction interval of the price difference using a numerical integration method, with specific details as follows: 1. Marginal probability density prediction of day-ahead / real-time electricity prices based on Skewed DeepAR To address the skewed and peak characteristics of day-ahead and real-time electricity prices, a Jones-Faddy skewed t-distribution-driven DeepAR neural network (Skewed DeepAR) was designed to predict their marginal probability density functions (PDFs).

[0022] The model input features include: historical 14-day day-ahead electricity price series. Historical 14-day real-time electricity price series Time-series covariates (load forecast values) for the target date Date coding, new energy output forecast For numerical features, Z-score standardization is used; for date features, trigonometric function encoding (such as sin(2)) is used. πt / 24) Depicting intraday periodicity.

[0023] The prediction for the future from time 1 to time T can be expressed as: The Jones-Faddy skewed t-distribution parameter output model uses a recurrent neural network (RNN) as its basic architecture, through hidden states... h t The formula for updating the hidden state to pass on temporal dependency information is: in, f For RNN activation functions, X t Let Θ be the covariate at time t, and Θ be the RNN parameters. The model output layer consists of four fully connected layers, outputting the distribution parameters of day-ahead / real-time electricity prices. Day-ahead electricity price distribution parameters: location parameters Scale parameters Left tail weight parameters Right tail weight parameters ; Real-time electricity price parameters: Location parameters Scale parameters Left tail weight parameters Right tail weight parameters ; Based on the Jones-Faddy skewed t-distribution parameters, the marginal PDF of day-ahead / real-time electricity prices can be constructed (taking day-ahead electricity prices as an example, and the same applies to real-time electricity prices): Therefore, the prediction problem can be expressed as: Thus, the marginal CDF is obtained: To optimize the marginal distribution parameters of Skewed DeepAR and ensure its accurate fit to the probabilistic characteristics of historical electricity prices, a negative log-likelihood (NLL) loss function is used for model training, based on the historical day-ahead electricity price series. With real-time electricity price series Its joint likelihood function is the product of the marginal likelihoods at each time step: The parameter update logic is as follows: To avoid numerical underflow and to adapt to the optimization logic of deep learning, the natural logarithm of the joint likelihood is taken and then negative, resulting in the negative log-likelihood loss.

[0024] 2. Construction of the joint probability density of day-ahead and real-time electricity prices based on t-Copula To capture the dependency between day-ahead and real-time electricity prices (especially tail-end linkage under extreme price conditions), based on Sklar's theorem, a t-Copula function is used to fuse the marginal distributions of the two, constructing a two-dimensional joint PDF: 2.1 t-Copula parameter estimation Using historical day-ahead / real-time electricity price data, the t-Copula parameter (correlation coefficient) is estimated in two steps using the inference function method (IFM). r and degrees of freedom n ).

[0025] Step 1: Based on historical data, use the trained Skewed DeepAR model to convert historical electricity prices into uniformly distributed samples: in, u t , v t ∈[0,1], satisfying a uniform distribution (property integral transformation property); Step 2: For uniformly distributed samples {( u t , v t) The density function parameters of t-Copula are optimized using maximum likelihood estimation (MLE). in, c t-Copula The t-Copula density function is expressed as: t ν (⋅) represents the degrees of freedom. n The t-distribution CDF, It is its inverse function (used to map uniform samples back to the t-distribution space).

[0026] 2.2 Joint PDF Construction Based on the marginal PDF and the t-Copula density function with estimated parameters, the joint PDF of day-ahead and real-time electricity prices is derived using Sklar's theorem: in Differentiation yields then The core advantage of this formula is that it retains the skewness and peak characteristics of day-ahead and real-time electricity prices (determined by the marginal PDF) while also characterizing their dependence on each other through the t-Copula density function (especially the tail linkage under extreme prices).

[0027] 3. Derivation of Price Difference Probability Density and Prediction Interval Output Based on Numerical Integration According to the definition of price difference According to market rules, the day-ahead and real-time electricity price combined PDF is projected into a price difference PDF through numerical integration, and the forecast range is output.

[0028] 3.1 Integral Expression for Price Difference PDF According to the probability integral transformation, the price difference P DI The PDF can be represented as a single-variable integral of the day-ahead electricity price, as follows: Perform the following variable substitutions get in, I The value represents the real-time electricity price. Because the joint PDF contains nonlinear terms, this integral has no analytical solution and needs to be approximated using numerical methods.

[0029] 3.2 Numerical Integral Approximation A three-step method of "truncating invalid regions → discretizing the grid → linear interpolation" is adopted to reduce the computational load while ensuring accuracy. Step 1: Truncate the invalid interval. Based on the marginal PDF of day-ahead / real-time electricity prices, determine the valid interval of the probability density set: in, For the minimum threshold (e.g.) This ensures that the cumulative probability within the effective interval is ≥99.9%; Step 2: Discretize the day-ahead electricity price range. Divided into M equally spaced discrete points Discrete spacing ; Step 3: Linear interpolation for non-discrete points. For each day-ahead discrete electricity price point... Calculate the corresponding real-time electricity price ,like p RT It falls precisely on the discrete grid of the real-time electricity price, allowing direct acquisition of the corresponding joint PDF value: like p RT If a point does not fall on a discrete grid, find its two nearest discrete points. , (satisfy ≤ ≤ The joint PDF value is calculated using linear interpolation: Step 4: Discretize and sum to approximate the integral. Transform the continuous integral into a summation of discrete points to obtain an approximate value for the price difference PDF: 3.3 Normalization Correction and Prediction Interval Calculation Because truncating invalid intervals and linear interpolation may introduce errors. The total probability (discrete summation × price difference interval) may deviate from 1, and needs to be normalized to ensure that it conforms to the basic property of PDF, that is, the total probability sum is 1. The calculation method is as follows: in, Let Δ be the set of discrete points of the price difference. p DI The discrete interval of the price difference; Integrating the normalized price spread PDF yields the cumulative distribution function (CDF). Use quantiles as the prediction interval (e.g., 90% confidence interval is...). ),in It is the inverse function of CDF (quantile function).

Claims

1. A method for predicting electricity spot market price spreads based on probability density modeling, characterized in that, The steps include: (1) Based on the skewed DeepAR model, predict the probability density functions of day-ahead electricity price and real-time electricity price respectively; (2) Based on the t-Copula function, the probability density functions of the day-ahead electricity price and the real-time electricity price are fused to construct a joint probability density function of the two prices; (3) Project the joint probability density function onto the price difference space using the numerical integration method to solve for the probability density function of the price difference.

2. The method for predicting electricity spot market price spreads based on probability density modeling as described in claim 1, characterized in that, The price difference Defined as day-ahead electricity price Subtract real-time electricity price The following is an example: .

3. The method for predicting electricity spot market price spreads based on probability density modeling as described in claim 2, characterized in that, In step (1), the probability density function of electricity price is characterized by the Jones-Faddy skewed t-distribution, and its function expression is as follows: ； in: B () is the beta function; By integrating the probability density function, the corresponding cumulative distribution function is obtained: in, Position parameters It is a scale parameter. It is the left tail weight parameter. It is the right tail weight parameter.

4. The method for predicting electricity spot market price spreads based on probability density modeling according to claim 3, characterized in that, Based on Sklar's theorem, the joint probability distribution of day-ahead electricity prices and real-time electricity prices is decomposed into a marginal distribution and a Copula function, whose joint probability density function is expressed as: ； in: , is the t-Copula density function; For the partially distributed day-ahead and real-time electricity price distributions, there are in and These are the correlation coefficient and degrees of freedom, respectively. It is the inverse cumulative distribution function of a t-distribution with v degrees of freedom.

5. The method for predicting electricity spot market price spreads based on probability density modeling according to claim 4, characterized in that, The cumulative distribution function of the price difference is defined as: And thus obtain After transformation, we obtain: Perform variable substitution ,get: Based on the relationship between CDF and PDF, the PDF representing the price difference is: Its joint probability density relationship with day-ahead electricity price and real-time electricity price is as follows: That is: Based on the derivation of the day-ahead and real-time electricity price joint probability density function, the expression for the price difference probability density function is obtained as follows: By numerical summation approximation, the probability density distribution of the price difference is expressed as: in, For discretized day-ahead electricity price points, The interval is discrete.

6. The method for predicting electricity spot market price spreads based on probability density modeling as described in claim 5, characterized in that, To address the skewness and peak characteristics of day-ahead and real-time electricity prices, a DeepAR neural network model driven by a Jones-Faddy skewed t-distribution is designed to output the marginal probability density functions (PDFs) of the two prices. Input features for the DeepAR neural network model include: historical day-ahead electricity price series. Historical real-time electricity price series Time-series covariates, including load forecasts. Date coding, new energy output forecast For numerical features, Z-score standardization is used, and for date features, trigonometric function encoding is used; the prediction for future times 1 to T is expressed as follows: The Jones-Faddy skewed t-distribution parameter output model uses a recurrent neural network (RNN) as its basic architecture, through hidden states. The formula for updating the hidden state to pass on temporal dependency information is: in, φ For RNN activation functions, Let Θ be the covariate at time t, and Θ be the RNN parameters. The DeepAR neural network model outputs the skewed t-distribution parameters of the day-ahead / real-time electricity price through four fully connected layers: Day-ahead electricity price forecast distribution parameters: location parameters Scale parameters Left tail weight parameters Right tail weight parameters ; RT electricity price parameters: location parameters Scale parameters Left tail weight parameters Right tail weight parameters ; Based on the Jones-Faddy skewed t-distribution parameters, the marginal PDFs of day-ahead and real-time electricity prices are constructed as follows: Therefore, the prediction problem is formulated as follows: The marginal CDF is obtained based on the prediction results: The DeepAR neural network model is trained using the negative log-likelihood (NLL) loss function for historical day-ahead electricity price series. With real-time electricity price series Its joint likelihood function is the product of the marginal likelihoods at each time step: The parameter update logic is as follows: Taking the negative of the natural logarithm of the joint likelihood yields the negative log-likelihood loss.

7. The method for predicting electricity spot market price spreads based on probability density modeling as described in claim 6, characterized in that, Also includes: Based on Sklar's theorem, a two-dimensional joint PDF is constructed by fusing the marginal distributions of day-ahead and real-time electricity prices using the t-Copula function.

8. The method for predicting electricity spot market price spreads based on probability density modeling according to claim 7, characterized in that, Also includes: t-Copula parameter estimation: Using historical day-ahead / real-time electricity price data, the t-Copula parameters, namely the correlation coefficient ρ and the degrees of freedom ν, are estimated in two steps using the inference function method (IFM). Step 1: Based on historical data, use the trained DeepAR neural network model to convert historical electricity prices into uniformly distributed samples: in, , satisfying uniform distribution; Step 2: For uniformly distributed samples The density function parameters of t-Copula are optimized by maximum likelihood estimation (MLE): in, The density function is t-Copula. A joint PDF is constructed based on the marginal PDF and the estimated parameters using the t-Copula density function. The joint PDF of day-ahead electricity price and real-time electricity price is derived using Sklar's theorem. in Differentiation yields then 。 9. The method for predicting electricity spot market price spreads based on probability density modeling as described in claim 8, characterized in that, According to the definition of price difference According to market rules, the day-ahead and real-time electricity price combined PDF is projected into a price spread PDF through numerical integration, and the forecast range is output as follows: The integral expression for the price difference PDF: According to the probability integral transformation, the PDF of the price difference PDI is represented as a univariate integral of the day-ahead electricity price, and the specific process is as follows: Perform variable substitution get Differentiation yields in, For real-time electricity prices; Numerical integration methods Step 1: Truncate invalid intervals. Based on the marginal PDF of day-ahead / real-time electricity prices, determine the valid intervals of probability density concentration: in, The threshold is used to ensure that the cumulative probability within the effective interval is ≥99.9%; Step 2: Discretize the day-ahead electricity price range. Divided into M equally spaced discrete points Discrete spacing ; Step 3: Linear interpolation is used to process non-discrete points for each day-ahead discrete electricity price point. Calculate the corresponding real-time electricity price : like It falls exactly on the discrete grid of the real-time electricity price. Directly obtain the corresponding joint PDF value ; like If a point does not fall on a discrete grid, find its two nearest discrete points. , ,satisfy ≤ ≤ The joint PDF value is calculated using linear interpolation: Step 4: Discretize and sum to approximate the integral. Transform the continuous integral into a sum of discrete points to obtain an approximate value of the price difference PDF. Normalization correction and prediction interval calculation: Normalization correction: in, Let be the set of discrete points representing the price difference. The discrete interval of the price difference; Prediction interval calculation: Integrate the normalized price spread PDF to obtain the cumulative distribution function CDF: Use the quantile as the prediction interval.

Images