Evaluation and regulation method for electro-mechanical integrated energy system based on bootstrap sampling

Abstract

The present application relates to the technical field of electricity-gas integrated energy system, and more particularly to a kind of electricity-gas integrated energy system evaluation control method based on bootstrap sampling, comprising: S1, obtain electric historical prediction error data, set multiple sample quantities different historical prediction error subsets, respectively execute steps S2 to S4;S2, initialization K=1, generate the Kth Bootstrap sample set with Bootstrap sampling method;S3, execute iteration process;S4, form the probability box corresponding to current sample quantity;S5, select the best sample quantity;S6, generate multiple prediction error samples based on the probability box corresponding to the best sample quantity;S7, add prediction error sample to the predicted value of electricity-gas integrated energy system, obtain corresponding actual operation sample, input the steady-state analysis model of the system.The present application can accurately depict the uncertainty range of multi-source prediction error in electricity-gas integrated energy system in the actual scene of limited historical data, unknown distribution characteristics and cognitive uncertainty.

Description

Technical Field

[0001] This invention relates to the field of integrated electric-gas energy system technology, and in particular to an evaluation and control method for integrated electric-gas energy systems based on bootstrap sampling. Background Technology

[0002] Energy plays an irreplaceable role in global development. Traditionally, power and natural gas systems have been operated by different companies with independent mechanisms, leading to low efficiency in cross-energy coordination and redundant resource allocation. With the advancement of dual-carbon goals and the large-scale integration of renewable energy, gas-fired power generation, due to its excellent peak-shaving performance, low carbon emission intensity, and good complementarity with renewable energy, is widely used in modern power systems. Against this backdrop, the Integrated Electricity-Gas System (IEGS) has emerged, significantly improving the absorption capacity of renewable energy sources (RES) and overall energy utilization efficiency by deeply coupling the power grid and natural gas network at both the physical and operational levels.

[0003] Steady-state modeling and analysis of IEGS (Integrated Energy Generation Systems) are fundamental to its planning, scheduling, and risk assessment. However, IEGS operation faces multiple uncertainties, primarily including the intermittency and volatility of renewable energy output, and the time-varying characteristics of electricity and natural gas loads. These uncertainties are highly non-Gaussian and non-stationary, causing frequent shifts in system operating points, thus posing challenges to safety, economy, and reliability. Therefore, accurately quantifying and effectively characterizing the impact of these uncertainties on IEGS becomes a crucial prerequisite for supporting its safe and efficient operation.

[0004] In uncertainty modeling, existing research often employs probability distribution assumptions (such as Gaussian and Weibull distributions) or relies on large-scale historical data to construct empirical models to characterize the randomness of RES output and load. While these methods are mathematically concise, their core assumptions often fail to capture the complex statistical characteristics of real-world data, easily introducing significant model errors. More importantly, in real-world scenarios, missing measurements, sparse data, or incomplete information (e.g., lack of observable data for user-side distributed photovoltaic systems) often result in prediction errors containing not only random uncertainty but also cognitive uncertainty stemming from insufficient knowledge. This type of cognitive uncertainty cannot be adequately captured by traditional probabilistic models.

[0005] To address cognitive uncertainty, some studies have attempted to introduce methods such as evidence theory, fuzzy sets, or Bayesian inference. However, these methods generally suffer from high computational complexity, strong dependence on prior information, and difficulty in compatibility with existing optimization scheduling frameworks, limiting their engineering applicability in large-scale IEGS. Especially when sample sizes are limited or data quality is low, constructing an uncertainty representation model that reflects both the inherent variability of data and accommodates cognitive uncertainty without relying on strong distribution assumptions has become a current research challenge.

[0006] The core issue is that traditional methods struggle to balance accuracy, robustness, and computational feasibility in uncertainty modeling under limited and non-ideal data conditions. On one hand, over-reliance on parametric distributions ignores the complex structure of real data; on the other hand, nonparametric or high-dimensional uncertainty modeling methods often suffer from the "curse of dimensionality" or poor convergence. Furthermore, in the context of multi-energy coupling in IEGS, the propagation and interaction of uncertainties between electro-aerobic subsystems further exacerbate the modeling difficulty.

[0007] Therefore, how to accurately characterize the uncertainty range of multi-source prediction errors in a real-world scenario with limited historical data, unknown distribution characteristics, and cognitive uncertainties, in order to support subsequent reliable risk assessment and scheduling decisions, has become an urgent problem to be solved. Summary of the Invention

[0008] To address the aforementioned shortcomings of existing technologies, the present invention aims to provide a bootstrap sampling-based evaluation and control method for integrated electric-gas energy systems. This method can accurately characterize the uncertainty range of multi-source prediction errors in integrated electric-gas energy systems in real-world scenarios where historical data is limited, distribution characteristics are unknown, and cognitive uncertainties exist, thereby supporting subsequent reliable risk assessment and scheduling decisions.

[0009] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0010] The evaluation and control method for integrated electric-gas energy systems based on bootstrap sampling includes the following steps:

[0011] S1. Obtain historical prediction error data of renewable energy generation, electricity load and natural gas load in the integrated electricity-gas energy system and preprocess it, and set up multiple historical prediction error subsets with different sample sizes; for each historical prediction error subset with different sample sizes, execute steps S2 to S4 respectively.

[0012] S2. Based on the historical prediction error subset of the current sample size, initialize the number of samplings K=1, and use the Bootstrap sampling method with replacement to generate the Kth Bootstrap sample set;

[0013] S3. After generating a new Bootstrap sample set, perform the following iterative process:

[0014] S31. Calculate the empirical cumulative distribution function of each Bootstrap sample set, and calculate the Wasserstein distance between each empirical cumulative distribution function and the empirical cumulative distribution function of the historical prediction error subset of the current sample size;

[0015] S32. Based on the Wasserstein distance sequences corresponding to K Bootstrap sample sets, calculate the variance coefficient β of the sequence. k ;

[0016] S33. Determine the variance coefficient β of S32. k Is it less than or equal to the preset convergence threshold β? If yes, stop sampling, determine the current sampling number K as the final sampling number, and go to S4; if no, let K=K+1, and return to step S2 to generate the Kth Bootstrap sample set.

[0017] S4. Based on the empirical cumulative distribution function of all Bootstrap sample sets corresponding to the final sampling number K, take the maximum and minimum values ​​at all points to construct the upper and lower bounds of the cumulative distribution function, forming a probability box corresponding to the current sample size.

[0018] S5. Construct a true cumulative distribution function based on the complete historical dataset, and calculate the average Kullback-Leibler divergence between the probability box corresponding to each sample size and the true cumulative distribution function in order of increasing sample size; select the smallest sample size that makes the change in the average Kullback-Leibler divergence between consecutive X adjacent sample sizes less than a preset threshold as the optimal sample size.

[0019] S6. A two-layer sampling strategy is adopted to generate multiple prediction error samples based on the probability boxes corresponding to the optimal sample size.

[0020] S7. For each prediction error sample generated in step S6... The predicted value added to the electric-gas integrated energy system Above, obtain the corresponding actual running samples. The data is then input into the steady-state analysis model of the system for risk assessment or optimized scheduling decisions.

[0021] Compared with the prior art, the present invention has the following advantages:

[0022] 1. Effectively characterizes the boundary of cognitive uncertainty. Traditional methods typically rely on pre-defined probability distributions (such as Gaussian or Weibull distributions) to describe prediction errors, which are insufficient to reflect cognitive uncertainty caused by missing data or lack of knowledge. This approach generates a family of empirical cumulative distribution functions through Bootstrap resampling and constructs probability boxes based on their upper and lower bounds. Without any distributional assumptions, it can explicitly characterize the range of cognitive uncertainty caused by finite samples, significantly improving the robustness of uncertainty modeling.

[0023] 2. Achieve adaptive convergence control in the sampling process. Existing nonparametric methods often use a fixed number of samplings, which can easily lead to computational redundancy or insufficient convergence. This scheme introduces a dynamic convergence criterion based on the variance coefficient of the Wasserstein distance sequence, which can automatically determine the required number of Bootstrap samplings based on the stability of the current sample set, thus avoiding unnecessary computational overhead while ensuring modeling accuracy.

[0024] 3. Addressing the challenge of determining "how much historical data is sufficient," this solution constructs a true reference distribution using complete historical data and identifies information saturation points by analyzing the changing trend of the average Kullback-Leibler divergence, thereby selecting the minimum sample size that meets the accuracy requirements. Compared to blindly using all data or arbitrarily selecting subsets, this strategy achieves a better balance between data utilization efficiency and modeling reliability.

[0025] 4. Supports efficient and highly compatible subsequent decision-making applications. The generated probability boxes can be directly used for two-level sampling, producing prediction error samples that conform to the uncertainty boundary, seamlessly integrating with the IEGS steady-state analysis model. Compared to complex frameworks such as evidence theory or Bayesian methods, this scheme has a clear computational structure, is easy to embed into existing optimization scheduling processes, and balances engineering practicality with theoretical rigor.

[0026] In summary, this method can accurately characterize the uncertainty range of multi-source prediction errors in a real-world scenario with limited historical data, unknown distribution characteristics, and cognitive uncertainty, thereby supporting reliable risk assessment and scheduling decisions, and effectively balancing computational efficiency and model accuracy.

[0027] Preferably, in step S31, the Wasserstein distance W corresponding to the k-th Bootstrap sample set is... k The formula for calculation is:

[0028] ;

[0029] In the formula, This represents the empirical cumulative distribution function corresponding to the k-th Bootstrap sample set; This represents the empirical cumulative distribution function of the current subset of historical prediction errors; This indicates the prediction error.

[0030] Compared to traditional convergence criteria based on sample mean or variance, this approach allows the Wasserstein distance to capture a more comprehensive picture of the distribution shape, tail features, and overall offset, making it particularly suitable for non-Gaussian, heavy-tailed, or asymmetric prediction error distributions. Therefore, this method more accurately reflects whether the Bootstrap sample set sufficiently approximates the true distribution structure of the original data, thus avoiding premature stopping or redundant sampling and enhancing the stability and reliability of the iteration process.

[0031] Preferably, in step S32, the variance coefficient β k The formula for calculation is:

[0032] ;

[0033] ;

[0034] Where Var(·) represents variance operation; This indicates the average distance.

[0035] This setup allows for adaptive assessment of the stability of Bootstrap sampling. Traditional methods often rely on a fixed number of samplings or simple statistics (such as the rate of change of the mean) to determine convergence, which is insufficient to handle the dynamic characteristics of distribution differences across different sample sets. This approach introduces a dimensionless variance coefficient β. k This effectively reflects the relative volatility level of the Wasserstein distance sequence. When β k When the value tends to stabilize and falls below the preset threshold, it indicates that the contribution of new samples to the overall distribution estimation has become saturated, thus enabling a more scientific and flexible convergence determination and avoiding waste of computing resources or insufficient modeling.

[0036] Preferably, the preset convergence threshold β is 0.05.

[0037] Preferably, in step S4, the upper bound of the cumulative distribution function is... and the lower world The method for determining it is as follows:

[0038] ;

[0039] ;

[0040] In the formula, This represents the empirical cumulative distribution function corresponding to the k-th Bootstrap sample set; This indicates the prediction error.

[0041] This setup, through the construction of probability boxes using extreme value aggregation strategies, enables a non-parametric, conservative, and information-rich representation of prediction error uncertainty under limited historical data, providing a flexible and reliable tool for quantifying the uncertainty of complex energy systems.

[0042] Preferably, S6 includes:

[0043] S61. For each prediction error variable Generate the first random number The inverse function value of the cumulative distribution function of the lower bound of the probability box is calculated based on the first random number. The inverse function value of the upper bound cumulative distribution function To determine the range of values ​​for this variable under cognitive uncertainty;

[0044] S62. Generate a second random number. The specific value of the variable is calculated by linear interpolation within the range of the second random number. , which is the j-th sampled element of this variable;

[0045] S63. Repeat steps S61 and S62 until the j-th sampled element of all prediction error variables is obtained, thus forming the j-th prediction error sample. , where the subscript u represents the number of prediction error variables;

[0046] S64. Repeat steps S61 to S63 until a preset number of prediction error samples are obtained.

[0047] This setup achieves two key advantages: 1. Effective sampling and propagation of cognitive uncertainty. Traditional Monte Carlo methods rely on a single probability distribution, which struggles to reflect the ambiguity of distribution boundaries caused by insufficient data or knowledge gaps. This method introduces an "outer layer" of random numbers to determine the uncertainty boundary (i.e., the upper and lower bounds of the probability box), and then uses an "inner layer" of random numbers to sample within the boundary, thus achieving hierarchical modeling and sampling of cognitive uncertainty. This two-layer structure preserves the statistical characteristics of the original data while reflecting a conservative estimate of the unknown distribution, significantly improving the rationality and robustness of the sampling results in the propagation of uncertainty.

[0048] 2. It balances computational efficiency with sampling diversity. Compared to directly using complex nonparametric distributions or focal element combinations in evidence theory, this method employs a simple linear interpolation strategy. While ensuring that sampling points uniformly cover the probability box, it avoids high-dimensional integration or complex inference operations, exhibiting good computational scalability. It is particularly suitable for large-scale, multi-source uncertainty scenarios such as integrated electric and gas energy systems, and can efficiently generate diverse operational samples with limited computational resources.

[0049] Preferably, in step S61, the first random number The second random number is between 0 and 1; in step S62, the second random number A random number between 0 and 1.

[0050] This setup, using uniformly distributed random numbers in the [0,1] interval as input, naturally matches the domain of the cumulative distribution function, ensuring that the samples generated by the inverse transformation method are statistically representative. Especially for non-parametric boundaries, this standardized input avoids systematic errors introduced by biases in the distribution assumptions, thus improving the reliability of the sampling results.

[0051] Preferably, in S5, the average Kullback-Leibler divergence The calculation formula is:

[0052] ;

[0053] In the formula, N x The number of discretization points; This represents the value of the i-th error variable at the j-th discrete point; and These are the upper and lower bounds of the probability box, respectively, and their cumulative distribution functions. This is the true cumulative distribution function obtained based on the complete historical dataset; This indicates the width of the discretization interval.

[0054] Preferably, in step S1, the historical prediction error data comes from the day-ahead prediction data of the integrated electric-gas energy system, with a time resolution of 15 minutes, and the historical prediction error subset consists of the difference between the prediction data and the actual operating data.

[0055] This setup, where day-ahead forecast data is typically used to guide next-day scheduling decisions, allows the error characteristics to accurately reflect the system's uncertainty under typical operating conditions. Employing a 15-minute time resolution not only preserves the short-term fluctuations in load and renewable energy output but also fully captures the dynamic response process within the electro-pneumatic coupling system. This makes subsequent bootstrap sampling and probability box construction more closely resemble actual operating scenarios, significantly enhancing the model's ability to characterize real-world operational uncertainties.

[0056] Preferably, in S5, the preset threshold is greater than or equal to 0.005 and less than or equal to 0.01.

[0057] This setting effectively avoids over-reliance on large-scale historical data. In practical engineering, historical data is often limited and expensive to obtain. Using too small a threshold might lead to higher precision requirements for the algorithm, necessitating more samples to meet convergence conditions and wasting computational resources. Conversely, too large a threshold might prematurely terminate the iteration, preventing the model from fully capturing data features. Setting the threshold within the range of [0.005, 0.01] ensures robust modeling while reasonably limiting the data volume requirement, significantly improving the method's practicality in data-scarce scenarios. Attached Figure Description

[0058] To make the objectives, technical solutions, and advantages of the invention clearer, the invention will now be described in further detail with reference to the accompanying drawings, wherein:

[0059] Figure 1 This is a flowchart of the method;

[0060] Figure 2 This is a flowchart illustrating the process of determining the final number of samples K in Example 1.

[0061] Figure 3 This is a schematic diagram of the uncertainty quantification framework based on bootstrap sampling in Example 1;

[0062] Figure 4 This is a schematic diagram of the prediction error probability box model under different sample sizes in Example 2.

[0063] Figure 5 The prediction error KLD under different sample sizes in Example 2 avg Schematic diagram. Detailed Implementation

[0064] The present invention will be further described below with reference to the accompanying drawings and embodiments.

[0065] To make the objectives, technical solutions, and advantages of the embodiments of the present invention clearer, the technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only a part of the embodiments of the present invention, not all of them. Therefore, the following detailed description of the embodiments of the present invention provided in the accompanying drawings is not intended to limit the scope of the claimed invention, but merely to represent selected embodiments of the invention. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without inventive effort are within the scope of protection of the present invention.

[0066] It should be noted that similar reference numerals and letters in the following figures indicate similar items. Therefore, once an item is defined in one figure, it does not need to be further defined and explained in subsequent figures. In the description of this invention, it should be noted that the terms "center," "upper," "lower," "left," "right," "vertical," "horizontal," "inner," and "outer," etc., indicate the orientation or positional relationship based on the orientation or positional relationship shown in the figures, or the orientation or positional relationship commonly used when the product is in use. They are only for the convenience of describing the invention and simplifying the description, and do not indicate or imply that the device or element referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on the invention. Furthermore, the terms "first," "second," and "third," etc., are only used to distinguish descriptions and should not be construed as indicating or implying relative importance. In addition, the terms "horizontal," "vertical," etc., do not indicate that the component is required to be absolutely horizontal or suspended, but can be slightly tilted. For example, "horizontal" simply means that its direction is more horizontal than "vertical," and does not mean that the structure must be completely horizontal, but can be slightly tilted. In the description of this invention, it should also be noted that, unless otherwise explicitly specified and limited, the terms "set," "install," "connect," and "link" should be interpreted broadly. For example, they can refer to a fixed connection, a detachable connection, or an integral connection; they can refer to a mechanical connection or an electrical connection; they can refer to a direct connection or an indirect connection through an intermediate medium; and they can refer to the internal connection of two components. Those skilled in the art can understand the specific meaning of the above terms in this invention based on the specific circumstances.

[0067] Example 1

[0068] like Figure 1 As shown, this invention provides a method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling, comprising the following steps:

[0069] S1. Obtain historical prediction error data of renewable energy generation, electricity load and natural gas load in the integrated electricity-gas energy system and preprocess it, and set multiple historical prediction error subsets with different sample sizes.

[0070] In practice, the historical prediction error data originates from the day-ahead prediction data of the integrated electricity-gas energy system, with a time resolution of 15 minutes. The subset of historical prediction errors consists of the difference between the predicted data and the actual operating data. Since day-ahead prediction data is typically used to guide the next day's scheduling decisions, its error characteristics can accurately reflect the uncertainty of the system under typical operating conditions. Using a 15-minute time resolution not only preserves the short-term fluctuation characteristics of load and renewable energy output but also fully captures the dynamic response process in the electricity-gas coupled system. This makes subsequent bootstrap sampling and probability box construction more closely resemble actual operating scenarios, significantly enhancing the model's ability to characterize real-world operational uncertainties.

[0071] For each subset of historical prediction errors for each sample size, steps S2 to S4 are executed respectively.

[0072] S2. Based on the historical prediction error subset of the current sample size, initialize the number of samplings K=1, and use the Bootstrap sampling method with replacement to generate the Kth Bootstrap sample set;

[0073] S3. After generating a new Bootstrap sample set, perform the following iterative process:

[0074] S31. Calculate the empirical cumulative distribution function of each Bootstrap sample set, and calculate the Wasserstein distance between each empirical cumulative distribution function and the empirical cumulative distribution function of the historical prediction error subset of the current sample size;

[0075] In practice, the Wasserstein distance W corresponding to the k-th Bootstrap sample set is... k The formula for calculation is:

[0076] ;

[0077] In the formula, This represents the empirical cumulative distribution function corresponding to the k-th Bootstrap sample set; This represents the empirical cumulative distribution function of the current subset of historical prediction errors; This indicates the prediction error.

[0078] Compared to traditional convergence criteria based on sample mean or variance, the Wasserstein distance can more comprehensively capture distribution shape, tail features, and overall offset information, making it particularly suitable for non-Gaussian, heavy-tailed, or asymmetric prediction error distributions. Therefore, this method can more accurately reflect whether the Bootstrap sample set has sufficiently approximated the true distribution structure of the original data, thus avoiding premature stopping or redundant sampling and enhancing the stability and reliability of the iterative process.

[0079] S32. Based on the Wasserstein distance sequences corresponding to K Bootstrap sample sets, calculate the variance coefficient β of the sequence. k ;

[0080] In specific implementation, the variance coefficient β k The formula for calculation is:

[0081] ;

[0082] ;

[0083] Where Var(·) represents variance operation; This indicates the average distance.

[0084] Traditional methods often rely on fixed sampling times or simple statistics (such as the rate of change of the mean) to determine convergence, which is insufficient to handle the dynamic characteristics of distribution differences across different sample sets. This scheme introduces a dimensionless variance coefficient β. k This effectively reflects the relative volatility level of the Wasserstein distance sequence. When β k When the value tends to stabilize and falls below the preset threshold, it indicates that the contribution of new samples to the overall distribution estimation has become saturated, thus enabling a more scientific and flexible convergence determination and avoiding waste of computing resources or insufficient modeling.

[0085] S33. Determine the variance coefficient β of S32. k If the sample size is less than or equal to the preset convergence threshold β, then stop sampling, determine the current sampling number K as the final sampling number, and go to S4; otherwise, let K=K+1, and return to step S2 to generate the Kth Bootstrap sample set; in specific implementation, the preset convergence threshold β is 0.05.

[0086] The value of K affects the accuracy of the probability box. A larger K value results in more bootstrap sampling, leading to a more accurate probability box model. Some studies suggest K should be greater than 20, others greater than 80, and still others even greater than 1000. These recommendations are ambiguous for finite sample sets of varying sizes. Therefore, this invention proposes a bootstrap sampling method with a termination condition to determine a suitable K value. The final number of samplings K is as follows: Figure 2 As shown.

[0087] S4. Based on the empirical cumulative distribution function of all Bootstrap sample sets corresponding to the final sampling number K, take the maximum and minimum values ​​at all points to construct the upper and lower bounds of the cumulative distribution function, forming a probability box corresponding to the current sample size.

[0088] In specific implementation, the upper bound of the cumulative distribution function and the lower world The method for determining it is as follows:

[0089] ;

[0090] ;

[0091] In the formula, This represents the empirical cumulative distribution function corresponding to the k-th Bootstrap sample set; This indicates the prediction error.

[0092] This method first defines a method for calculating the probability box for quantifying mixed uncertainty, namely, using the cumulative distribution function obtained from all K bootstrap sets to calculate the upper and lower bounds of the probability box.

[0093] This invention utilizes bootstrap sampling to construct an uncertainty quantification model. Bootstrap sampling is a sampling strategy used to estimate the distribution of random variables. It involves sampling with replacement from the original finite sample set to create multiple "new sample sets," known as bootstrap sets. The size of the bootstrap sets is typically the same as the original sample set. This method processes a small set of samples without any prior knowledge.

[0094] Figure 3This paper describes a framework for uncertainty quantification based on bootstrap sampling. Starting with the original sample set, the bootstrap sampling process generates K bootstrap sets by randomly resampling with replacement from the original dataset. Each bootstrap set is the same size as the original dataset but may contain duplicate samples. From these K bootstrap sets, K cumulative distribution functions (CDFs) are obtained, and they are aggregated to construct a probability box that quantifies uncertainty by providing upper and lower bounds on the CDFs across the bootstrap sets.

[0095] S5. Construct a true cumulative distribution function based on the complete historical dataset, and calculate the average Kullback-Leibler divergence between the probability boxes corresponding to each sample size and the true cumulative distribution function in ascending order of sample size. Select the smallest sample size that ensures the change in the average Kullback-Leibler divergence between X consecutive adjacent sample sizes is less than a preset threshold as the optimal sample size. In specific implementation, X is 3, and the preset threshold is greater than or equal to 0.005 and less than or equal to 0.01.

[0096] Wherein, the average Kullback-Leibler divergence The calculation formula is:

[0097] ;

[0098] In the formula, N x The number of discretization points; This represents the value of the i-th error variable at the j-th discrete point; and These are the upper and lower bounds of the probability box, respectively, and their cumulative distribution functions. This is the true cumulative distribution function obtained based on the complete historical dataset; This indicates the width of the discretization interval.

[0099] S6. A two-layer sampling strategy is adopted to generate multiple prediction error samples based on the probability boxes corresponding to the optimal sample size.

[0100] In specific implementation, S6 includes:

[0101] S61. For each prediction error variable Generate the first random number The inverse function value of the cumulative distribution function of the lower bound of the probability box is calculated based on the first random number. The inverse function value of the upper bound cumulative distribution function To determine the range of values ​​for the variable under cognitive uncertainty; wherein, the first random number A random number between 0 and 1;

[0102] S62. Generate a second random number. The specific value of the variable is calculated by linear interpolation within the range of the second random number. , as the j-th sampling element of the variable; where, the second random number A random number between 0 and 1;

[0103] S63. Repeat steps S61 and S62 until the j-th sampled element of all prediction error variables is obtained, thus forming the j-th prediction error sample. , where the subscript u represents the number of prediction error variables;

[0104] S64. Repeat steps S61 to S63 until a preset number of prediction error samples are obtained.

[0105] Traditional Monte Carlo methods rely on a single probability distribution, making it difficult to reflect the ambiguity of distribution boundaries caused by insufficient data or lack of knowledge. This method introduces an "outer layer" of random numbers to determine the uncertainty boundary (i.e., the upper and lower bounds of the probability box), and then uses an "inner layer" of random numbers to sample within the boundary, achieving hierarchical modeling and sampling of cognitive uncertainty. This two-layer structure preserves the statistical characteristics of the original data while reflecting a conservative estimate of the unknown distribution, significantly improving the rationality and robustness of the sampling results in uncertainty propagation. Furthermore, compared to directly using complex nonparametric distributions or focal element combinations in evidence theory, this method employs a simple linear interpolation strategy, ensuring uniform coverage of the probability box by sampling points while avoiding high-dimensional integration or complex inference operations, resulting in good computational scalability. It is particularly suitable for large-scale, multi-source uncertainty scenarios such as integrated electric and gas energy systems, and can efficiently generate diverse operational samples with limited computational resources.

[0106] S7. For each prediction error sample generated in step S6... The predicted value added to the electric-gas integrated energy system Above, obtain the corresponding actual running samples. The data is then input into the steady-state analysis model of the system for risk assessment or optimized scheduling decisions.

[0107] In practical implementation, the existing model can be used directly for the steady-state analysis model of the system. The structure or parameters of the model are not the inventive technical content of this invention and will not be described in detail here.

[0108] Traditional methods typically rely on pre-defined probability distributions (such as Gaussian or Weibull distributions) to describe prediction errors, which struggle to reflect cognitive uncertainty caused by missing data or insufficient knowledge. This approach generates a family of empirical cumulative distribution functions through bootstrap resampling and constructs probability boxes based on their upper and lower bounds. Without any distributional assumptions, it explicitly characterizes the range of cognitive uncertainty caused by finite samples, significantly improving the robustness of uncertainty modeling. Furthermore, existing nonparametric methods often use a fixed number of samplings, which can lead to computational redundancy or insufficient convergence. This approach introduces a dynamic convergence criterion based on the variance coefficient of the Wasserstein distance sequence, automatically determining the required number of bootstrap samplings based on the stability of the current sample set, ensuring modeling accuracy while avoiding unnecessary computational overhead.

[0109] Addressing the challenge of determining the adequacy of historical data, this solution constructs a true reference distribution using complete historical data and identifies information saturation points by analyzing the changing trend of the average Kullback-Leibler divergence, thereby selecting the minimum sample size required to meet accuracy requirements. Compared to blindly using all data or arbitrarily selecting subsets, this strategy achieves a better balance between data utilization efficiency and modeling reliability. Furthermore, the probability boxes generated by this method can be directly used for two-level sampling, producing prediction error samples that conform to the uncertainty boundary, seamlessly integrating with the IEGS steady-state analysis model. Compared to complex frameworks such as evidence theory or Bayesian methods, this solution has a clear computational structure, is easily embedded into existing optimization scheduling processes, and balances engineering practicality with theoretical rigor.

[0110] This method can accurately characterize the uncertainty range of multi-source prediction errors in a real-world scenario with limited historical data, unknown distribution characteristics, and cognitive uncertainty, thereby supporting subsequent reliable risk assessment and scheduling decisions, and effectively balancing computational efficiency and model accuracy.

[0111] Example 2

[0112] To better illustrate the effectiveness of this method, the following simulation experiment is conducted.

[0113] In this simulation experiment, the hybrid uncertainty quantification method of this approach was tested under given sample conditions. Renewable energy and electricity load forecasting error data were obtained from open-access day-ahead forecasts of load and onshore wind power in Country B from October 31 to November 30, 2024, with a resolution of 15 minutes. Natural gas load forecasting data also ensured a consistent 15-minute resolution.

[0114] The performance of the proposed hybrid uncertainty quantification method was evaluated. Figure 4The prediction error probability box model is shown under different sample sizes. The true CDF curve is calculated using complete sample data. RES power generation: (a) 10 samples, (b) 50 samples, (c) 200 samples; Electricity load: (d) 10 samples, (e) 50 samples, (f) 200 samples; Natural gas load: (g) 10 samples, (h) 50 samples, (i) 200 samples.

[0115] In each subplot, the x-axis represents the relative prediction error and is fixed within the range of [-0.5, 0.5] for standardized visualization; the y-axis displays the cumulative probability. The true CDF is obtained from the complete prediction error dataset derived from the references. This shows that when data is insufficient, cognitive uncertainty has a significant impact on the quantification of mixed uncertainty; while with sufficient data, cognitive uncertainty is negligible, and random uncertainty becomes the dominant factor. Therefore, a sufficient sample size can reduce cognitive uncertainty, making the upper and lower bounds of the CDF estimated by the probability box model closer to reality.

[0116] KLD avg The deviation between the constructed probability box model and the true CDF was quantified. Higher KLD avg The value reflects the stronger impact of cognitive uncertainty. Figure 5 Results for 30 probabilistic box models built for RES, electricity, and gas loads are shown, where (a) RES generation, (b) electricity load, and (c) gas load. The sample sizes used in these models range from 10 to 300, with each KLD... avg The values ​​were obtained through an average of 10 independent bootstrap sampling runs. It can be seen that further increasing the sample size only leads to a decrease in KLD. avg Negligible changes occurred. This result indicates that the cognitive uncertainty caused by data scarcity has been largely eliminated. Therefore, in KLD... avg The sample size was chosen when the condition was stable because further increases would only bring negligible improvements. Based on this analysis, a sample size of 200 was used in the subsequent case study because KLD... avg All types of uncertainty have been stabilized.

[0117] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention and not to limit the technical solutions. Those skilled in the art should understand that any modifications or equivalent substitutions to the technical solutions of the present invention without departing from the spirit and scope of the present invention should be covered within the scope of the claims of the present invention.

Claims

1. A method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling, characterized in that, Includes the following steps: S1. Obtain historical prediction error data of renewable energy generation, electricity load and natural gas load in the integrated electricity-gas energy system and preprocess it, and set up multiple historical prediction error subsets with different sample sizes; for each historical prediction error subset with different sample sizes, execute steps S2 to S4 respectively. S2. Based on the historical prediction error subset of the current sample size, initialize the number of samplings K=1, and use the Bootstrap sampling method with replacement to generate the Kth Bootstrap sample set; S3. After generating a new Bootstrap sample set, perform the following iterative process: S31. Calculate the empirical cumulative distribution function of each Bootstrap sample set, and calculate the Wasserstein distance between each empirical cumulative distribution function and the empirical cumulative distribution function of the historical prediction error subset of the current sample size; S32. Based on the Wasserstein distance sequences corresponding to K Bootstrap sample sets, calculate the variance coefficient β of the sequence. k ; S33. Determine the variance coefficient β of S32. k Is it less than or equal to the preset convergence threshold β? If yes, stop sampling, determine the current sampling number K as the final sampling number, and go to S4; if no, let K=K+1, and return to step S2 to generate the Kth Bootstrap sample set. S4. Based on the empirical cumulative distribution function of all Bootstrap sample sets corresponding to the final sampling number K, take the maximum and minimum values ​​at all points to construct the upper and lower bounds of the cumulative distribution function, forming a probability box corresponding to the current sample size. S5. Construct a true cumulative distribution function based on the complete historical dataset, and calculate the average Kullback-Leibler divergence between the probability box corresponding to each sample size and the true cumulative distribution function in order of increasing sample size; select the smallest sample size that makes the change in the average Kullback-Leibler divergence between consecutive X adjacent sample sizes less than a preset threshold as the optimal sample size. S6. A two-layer sampling strategy is adopted to generate multiple prediction error samples based on the probability boxes corresponding to the optimal sample size. S7. For each prediction error sample generated in step S6... The predicted value added to the electric-gas integrated energy system Above, obtain the corresponding actual running samples. The data is then input into the steady-state analysis model of the system for risk assessment or optimized scheduling decisions.

2. The method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling according to claim 1, characterized in that: In step S31, the Wasserstein distance W corresponding to the k-th Bootstrap sample set is... k The formula for calculation is: ； In the formula, This represents the empirical cumulative distribution function corresponding to the k-th Bootstrap sample set; This represents the empirical cumulative distribution function of the current subset of historical prediction errors; This indicates the prediction error.

3. The method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling according to claim 2, characterized in that: In step S32, the variance coefficient β k The formula for calculation is: ； ； Where Var(·) represents variance operation; This indicates the average distance.

4. The method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling according to claim 3, characterized in that: The preset convergence threshold β is 0.

05.

5. The method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling according to claim 1, characterized in that: In step S4, the upper bound of the cumulative distribution function and the lower world The method for determining it is as follows: ； ； In the formula, This represents the empirical cumulative distribution function corresponding to the k-th Bootstrap sample set; This indicates the prediction error.

6. The method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling according to claim 5, characterized in that, S6 include: S61. For each prediction error variable Generate the first random number The inverse function value of the cumulative distribution function of the lower bound of the probability box is calculated based on the first random number. The inverse function value of the upper bound cumulative distribution function To determine the range of values ​​for this variable under cognitive uncertainty; S62. Generate a second random number. The specific value of the variable is calculated by linear interpolation within the range of the second random number. , which is the j-th sampled element of this variable; S63. Repeat steps S61 and S62 until the j-th sampled element of all prediction error variables is obtained, thus forming the j-th prediction error sample. , where the subscript u represents the number of prediction error variables; S64. Repeat steps S61 to S63 until a preset number of prediction error samples are obtained.

7. The method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling according to claim 6, characterized in that: In step S61, the first random number A random number between 0 and 1; In step S62, the second random number A random number between 0 and 1.

8. The method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling according to claim 1, characterized in that: In S5, the average Kullback-Leibler divergence The calculation formula is: ； In the formula, N x The number of discretization points; This represents the value of the i-th error variable at the j-th discrete point; and These are the upper and lower bounds of the probability box, respectively, and their cumulative distribution functions. This is the true cumulative distribution function obtained based on the complete historical dataset; This indicates the width of the discretization interval.

9. The method for evaluating and controlling an integrated electric-gas energy system based on bootstrap sampling according to claim 1, characterized in that: In step S1, the historical prediction error data comes from the day-ahead prediction data of the integrated electric-gas energy system, with a time resolution of 15 minutes. The historical prediction error subset consists of the difference between the prediction data and the actual operating data.

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