Industrial load demand response rebound effect identification method based on Bayesian intermediary analysis

Abstract

The invention discloses an industrial load demand response rebound effect identification method based on Bayesian intermediary analysis, and belongs to the technical field of industrial load demand response rebound effect identification. According to the method, a Bayesian intermediary analysis framework, systematized rebound effect modeling, parameter estimation based on MCMC sampling, quantification of an equipment interaction effect and an environment regulation effect, a complete algorithm implementation process, an innovative effect decomposition and synthesis method and a strict model comparison mechanism are introduced; accuracy and reliability of identification of the industrial load demand response rebound effect are remarkably improved, and the method has important theoretical value and application prospect.

Description

Technical Field

[0001] The invention discloses an industrial load demand response rebound effect identification method based on Bayesian mediation analysis, belonging to the technical field of industrial load demand response rebound effect identification. Background Art

[0002] With the deepening of the reform of the electricity market, demand response, as an important means of demand-side management, has been widely applied. The core idea of demand response is to encourage users to adjust their electricity consumption behaviors to cope with the problem of power supply-demand imbalance, thereby improving the stability and efficiency of the power grid. However, after industrial users implement demand response, load rebound often occurs. Load rebound refers to the situation where, after the demand response ends, the electricity load of users quickly returns to or even exceeds the level before the response. This phenomenon not only reduces the effect of demand response but also may have an adverse impact on the safe and stable operation of the power grid. Specifically, load rebound may lead to drastic fluctuations in the power grid load, increasing the difficulty of power grid dispatching and even triggering stability problems in the power system.

[0003] Currently, the research on the demand response rebound effect has the following main problems: lack of in-depth analysis of the internal mechanism of the rebound effect: existing methods mostly focus on the overall performance of the rebound effect, lacking in-depth analysis of its internal mechanism. Traditional methods are difficult to distinguish the contributions of direct and indirect effects and cannot provide a basis for refined management; the uncertainty characteristics have not been effectively described: the rebound effect is affected by various factors and has obvious uncertainty characteristics. However, most existing studies use deterministic methods and are difficult to effectively describe and quantify this uncertainty; lack of effective mathematical tools: there is a complex mediation relationship between the equipment state changes of industrial loads and the rebound effect, but there is a lack of effective mathematical tools to characterize this relationship.

[0004] Therefore, there is an urgent need for a new technical solution in the existing technologies to solve this problem. Summary of the Invention

[0005] Aiming at the above problems existing in the identification of the industrial load demand response rebound effect, the purpose of the present invention is to propose an industrial load demand response rebound effect identification method based on Bayesian mediation analysis. This method first establishes a system model including direct effect, equipment state effect, environmental regulation effect, and equipment interaction effect, and quantitatively analyzes each effect through a Bayesian inference framework. The present invention uses MCMC sampling to achieve parameter estimation and verifies the reliability of the results through model comparison and stability analysis. This method can accurately identify and quantify the load rebound effect after demand response and provide decision-making support for industrial users to participate in demand response.

[0006] To achieve the above object, the present invention adopts the following technical solutions: An industrial load demand response rebound effect identification method based on Bayesian mediation analysis, comprising the following steps:

[0007] Step S1: System model construction

[0008] Decompose the rebound effect amplitude R(t) into direct effect D(t), equipment state effect S(t), environmental regulation effect E(t), equipment interaction effect I(t), and system random disturbance term ε(t), and construct an equipment state transition equation;

[0009] Step S2: Construction of Bayesian inference framework

[0010] Set the prior distributions of the direct effect coefficient α, equipment state influence coefficient β, environmental regulation coefficient γ, equipment interaction effect coefficient δ, and system noise parameter σ ∈ and construct an equipment state equation and a likelihood function of the rebound effect;

[0011] Step S3: Effect decomposition and quantitative analysis

[0012] Calculate the direct effect D(t), single-equipment indirect effect, environmental regulation effect E(t), and equipment interaction effect I(t), and synthesize the total effect. The total effect:

[0013]

[0014] Posterior probability of the total effect:

[0015]

[0016] In the formula, represents the posterior mean estimate of the direct effect; IE i (t) is the indirect effect of the i-th equipment; CE ij (t) is the interaction effect between the i-th equipment and the j-th equipment; is the posterior mean estimate of the environmental regulation effect; M is the length of the MCMC sampling chain; TE (m) (t) is the total effect obtained from the m-th sampling; I is the indicator function; n is the total number of equipment;

[0017] Step S4: Preprocess the industrial load data, use the preprocessed data for model training, validation, and testing, and use the trained model to identify and quantify the load rebound effect after demand response.

[0018] Furthermore, in the industrial load demand response rebound effect identification method based on Bayesian mediation analysis, Step S1 includes:

[0019] 1) Rebound effect modeling

[0020] Let the amplitude of the rebound effect at time point t be R(t), and its mathematical expression is:

[0021] R(t) = D(t) + S(t) + E(t) + I(t) + ε(t);

[0022] In the formula, D(t) represents the direct effect; S(t) represents the equipment state effect; E(t) represents the environmental regulation effect; I(t) represents the equipment interaction effect; ε(t) represents the system random disturbance term;

[0023] Among them;

[0024] (1) The mathematical expression of the direct effect D(t) is as follows:

[0025] D(t) = α·DR(t);

[0026] In the formula, DR(t) represents the intensity of demand response; α is the direct effect coefficient;

[0027] (2) The mathematical expression of the equipment state effect S(t) is as follows:

[0028]

[0029] Among them, ES i (t) represents the state of the i-th equipment; β i is the influence coefficient of the i-th equipment state on the rebound effect, simply referred to as the equipment state influence coefficient β; n is the total number of equipment;

[0030] (3) The mathematical expression of the environmental regulation effect E(t) is as follows:

[0031] E(t) = γ·T(t)·DR(t);

[0032] Among them, T(t) represents the environmental temperature; γ is the environmental regulation coefficient; DR(t) represents the intensity of demand response;

[0033] (4) The mathematical expression of the equipment interaction effect I(t) is as follows:

[0034]

[0035] Among them, δ ij represents the interaction coefficient between equipment i and j, simply referred to as the equipment interaction effect coefficient δ; ES i (t) represents the state of the i-th equipment; ES j (t) represents the state of the j-th equipment;

[0036] (5) The system random disturbance term ε(t) follows a normal distribution:

[0037] ε(t)~N(0,σ 2 );

[0038] where N(0,σ 2 ) represents a normal distribution with a mean of 0 and a variance of σ 2 ;

[0039] 2) State transition equation modeling

[0040] (1) Equipment state equation

[0041] For equipment i, its state transition equation is:

[0042] ES i (t) = f i (DR(t), T(t), ES i-1 (t)) + ξ i (t);

[0043] where f i (DR(t), T(t), ES i-1 (t)) = η i ·DR(t) + λ i ·T(t) + μ i ·ES i-1 (t) + v i , η i is the influence coefficient of demand response on the state of equipment i; λ i is the influence coefficient of temperature; μ i is the influence coefficient of upstream equipment; ν i is the basic operating parameter; ξ i (t) is the random disturbance term of equipment state;

[0044] (2) Determine the interaction relationship between equipment, and the interaction relationship between equipment is represented by a state transition matrix:

[0045]

[0046] In the formula, A is the state transition matrix, B is the input matrix, and Ξ(t) is the system noise vector.

[0047] Furthermore, the industrial load demand response rebound effect identification method based on Bayesian mediation analysis, step S2 includes:

[0048] 1) Prior distribution setting

[0049] (1) Set the following for the direct effect coefficient and the equipment state influence coefficient:

[0050]

[0051] where α is the direct effect coefficient; Represents the initial uncertainty of the direct effect size;

[0052]

[0053] where β i is the influence coefficient of the i-th device state on the rebound effect, simply referred to as the device state influence coefficient β; Represents the prior uncertainty of the degree of device influence;

[0054] (2) The environmental adjustment coefficient is set as follows:

[0055] γ~N(0,0.5 2 );

[0056] γ is the environmental adjustment coefficient;

[0057] (3) The interaction effect coefficient is set as follows:

[0058] The interaction coefficient between devices adopts the following prior:

[0059]

[0060] where, δ ij represents the interaction coefficient between devices i and j, simply referred to as the device interaction effect coefficient δ;

[0061]

[0062] (4) The system noise parameter σ ∈ is set as follows: The system noise parameter σ ∈ adopts a half-Cauchy distribution as the prior distribution;

[0063] 2) Likelihood function construction

[0064] (1) The likelihood function of the device state equation

[0065] For device i, the likelihood function of its state observation is:

[0066]

[0067] where f i (DR(t),T(t),ES i-1 (t))=η i ·DR(t)+λ i ·T(t)+μ i ·ES i-1 (t)+v i ,η i is the influence coefficient of the demand response on the state of device i; λ i is the influence coefficient of temperature; μi is the influence coefficient of the upstream equipment; ν i is the basic operating parameter; ξ i (t) is the random disturbance term of the equipment state; θ represents the set of all relevant parameters in the model, and θ specifically includes the direct effect coefficient α; the influence coefficient β of the state of the i-th equipment on the rebound effect i , i = 1, 2, …, n; the environmental regulation coefficient γ; the interaction coefficient δ between equipment i and j ij , i = 1, 2, …, n, j = 1, 2, …, n, and j ≠ i; the influence coefficient η of the demand response on the state of equipment i i , the temperature influence coefficient λ i , the upstream equipment influence coefficient μ i , the basic operating parameter v i , the variance σ of the system random disturbance term 2 and the variance of the observation noise of each equipment

[0068] (2) The likelihood function of the rebound effect

[0069] For the amplitude R(t) of the rebound effect observed at the time point t, its likelihood function is:

[0070]

[0071] where: μ R (t) = D(t) + S(t) + E(t) + I(t) represents the expected value of the rebound effect, is the variance of the observed value of the rebound effect, which characterizes the magnitude of the random uncertainty in the observation process.

[0072] Furthermore, in the method for identifying the rebound effect of industrial load demand response based on Bayesian mediation analysis, in step S3,

[0073] The calculation process of the direct effect D(t) is as follows:

[0074] Through the Bayesian mediation analysis framework, the direct effect is separated from the complex system; the expression is:

[0075] D(t) = αDR(t);

[0076] where α is the direct effect coefficient, which characterizes the direct impact of the unit DR intensity on the load; this coefficient obtains the posterior distribution {α (1) , α (2) , …, α (M)} through MCMC sampling under the Bayesian inference framework, where α (M) represents the value of the direct effect coefficient obtained by the M-th MCMC sampling, and then calculates its mean and confidence interval:

[0077]

[0078] CI 95％ (D(t)) = [q 0.025 (α·DR(t)), q 0.975 (α·DR(t))];

[0079] where, represents the posterior mean estimate of the direct effect and is the best point estimate of the true direct effect; m represents the m-th sampling index in the MCMC sampling sequence; M represents the total number of MCMC samplings, CI 95％ (D(t)) represents the 95% confidence interval of the direct effect, which is used to quantify the uncertainty range of the direct effect estimate; q 0.025 represents the 2.5% quantile of the posterior distribution, i.e., the lower bound value; q 0.975 represents the 97.5% quantile of the posterior distribution, i.e., the upper bound value;

[0080] The calculation process of the indirect effect of a single device is as follows:

[0081] The indirect effect of the i-th device is generated by the DR instruction through its state transfer:

[0082] IE i (t) = η i ·β i ·DR(t);

[0083] The calculation process of the device interaction effect is as follows:

[0084] The interaction between devices is modeled by the state transition matrix A, and the transfer effect of device i on j is:

[0085] CE ij (t) = μ ij ·β j ·DR(t);

[0086] The calculation process of the environmental regulation effect is as follows:

[0087] E(t) = γ·T(t)·DR(t);

[0088] where, η i is the influence coefficient of the demand response on the state of device i, indicating the direct influence degree of a unit-intensity demand response signal on the state change of device i; β i is the influence coefficient of the state of the i-th device on the rebound effect, indicating the contribution degree of the state change of device i to the overall rebound effect, β j is the influence coefficient of the state of the j-th device on the rebound effect, indicating the contribution degree of the state change of device j to the overall rebound effect; μij is the transfer coefficient from device i to j in the state transition matrix A; γ is the environmental adjustment coefficient; T(t) is the environmental temperature.

[0089] Furthermore, in the industrial load demand response rebound effect identification method based on Bayesian mediation analysis, the data preprocessing method includes: standardization processing, time series smoothing processing, and quartile method outlier correction processing.

[0090] Furthermore, in the industrial load demand response rebound effect identification method based on Bayesian mediation analysis, in step S4, a three-layer verification system is adopted:

[0091] 1) Model comparison:

[0092]

[0093] In the formula, Δ TE represents the difference in the total effect, which is used to quantify the difference degree between the simplified model R(t) = α0 + α1DR(t) + ∈

[0094] and the complete model in terms of predicting the total effect; represents the total effect value estimated by the simplified model, that is, the predicted value considering only the direct linear impact of demand response; represents the total effect value estimated by the complete model, including all components of the direct effect, device state effect, environmental adjustment effect, and device interaction effect;

[0095] Among them: R(t) represents the amplitude of the rebound effect at time point t; α0 represents the baseline constant term of the rebound effect, reflecting the natural rebound level of the system without demand response signal; α1 represents the linear impact coefficient of demand response intensity, reflecting the direct impact of unit demand response intensity on the rebound effect; DR(t) represents the demand response intensity at time point t; ∈ represents the random error term, which follows a normal distribution with a mean of 0 and a constant variance, and is used to capture the random fluctuations not explained by the model;

[0096] 2) Stability test:

[0097] Verify the stability of the model through the Bootstrap resampling method; specifically as follows:

[0098] Calculate the coefficient of variation of the effect estimate:

[0099]

[0100] In the formula, CV represents the coefficient of variation, which is the ratio of the standard deviation to the mean, and is used to measure the relative dispersion degree of the estimated value; σ bootrepresents the standard deviation of the effect estimate obtained by Bootstrap resampling, reflecting the degree of fluctuation of the estimate; μ boot represents the mean of the effect estimate obtained by Bootstrap resampling, reflecting the central tendency of the estimate; a coefficient of variation less than 0.2 is an empirical threshold;

[0101] 3) Sensitivity analysis:

[0102] Calculate the total effect change rate:

[0103]

[0104] In the formula, S θ represents the sensitivity index of parameter θ, reflecting the response degree of the model output to the change of this parameter; when S θ < 2, it indicates that the sensitivity of the model to this parameter is within a reasonable range and the model has good stability; on the contrary, if S θ ≥ 2, it indicates that the model is overly sensitive to this parameter and the model structure needs to be further optimized or the parameter estimation method needs to be re-evaluated.

[0105] Through the above design scheme, compared with the prior art, the present invention has the following advantages:

[0106] 1. The Bayesian mediation analysis framework is introduced for the first time

[0107] The present invention applies Bayesian mediation analysis to the research of the rebound effect of industrial load demand response for the first time, solving the problem that it is difficult for traditional methods to distinguish direct effects and indirect effects. Most of the prior arts adopt deterministic models or simple regression analyses and cannot effectively decompose the various components of the rebound effect. The present invention realizes the precise decomposition and quantification of the rebound effect by constructing a system model including direct effects, equipment state effects, environmental regulation effects, and equipment interaction effects. The Bayesian mediation analysis framework can not only identify the contributions of various effects but also quantify their uncertainties, providing more refined decision-making support for industrial users.

[0108] 2. Systematic modeling of the rebound effect

[0109] The present invention proposes a systematic rebound effect model, decomposing the rebound effect into four main components: direct effect, equipment state effect, environmental regulation effect, and equipment interaction effect, and describing the dynamic evolution process between equipment states through state transition equations. The prior arts usually regard the rebound effect as a whole and lack in-depth analysis of its internal mechanism. The model of the present invention not only considers the direct impact of demand response on load but also captures the complex interactions between equipment through equipment state effects and equipment interaction effects, as well as the regulatory effect of environmental factors on the rebound effect. This systematic modeling method can more comprehensively reflect the formation mechanism of the rebound effect.

[0110] 3. Parameter Estimation and Uncertainty Quantification Based on MCMC Sampling

[0111] The present invention uses the Markov Chain Monte Carlo (MCMC) sampling method for parameter estimation, which can effectively handle the uncertainties in the model. Most of the existing technologies use point estimation methods and cannot quantify the uncertainties of parameters. By obtaining the posterior distribution of parameters through MCMC sampling, the present invention can not only provide point estimates of parameters but also calculate their credible intervals, thus more accurately reflecting the uncertainties of parameters.

[0112] 4. Quantification of Equipment Interaction Effects and Environmental Regulation Effects

[0113] The present invention first proposes a method for quantifying equipment interaction effects and environmental regulation effects. Existing technologies usually ignore the cascading relationships between devices and the regulatory effects of environmental factors, resulting in biases in the estimation of rebound effects. The present invention quantifies the interaction effects between devices through path analysis methods and captures the influence of external factors such as temperature on rebound effects through environmental regulation coefficients. This refined quantification method can more accurately predict the magnitude and duration of rebound effects.

[0114] 5. Model Comparison and Consistency Testing Mechanism

[0115] The present invention designs a strict model comparison and consistency testing mechanism, which can verify the consistency between complex models and simple models. Existing technologies usually lack effective model comparison methods, resulting in greater arbitrariness in model selection. By calculating the total effect differences and constructing consistency indicators, the present invention can objectively evaluate the consistency between different models, ensuring the rationality of model selection. This model comparison mechanism not only improves the reliability of the model but also provides a basis for subsequent model optimization.

[0116] In summary, through the introduction of the Bayesian mediation analysis framework, systematic rebound effect modeling, parameter estimation based on MCMC sampling, quantification of equipment interaction effects and environmental regulation effects, complete algorithm implementation processes, innovative effect decomposition and synthesis methods, and strict model comparison mechanisms, the present invention significantly improves the accuracy and reliability of identifying industrial load demand response rebound effects, and has important theoretical value and application prospects. Description of the Drawings

[0117] The description of the drawings here is used to provide a further understanding of the present invention and constitutes a part of this application for the present invention. The schematic embodiments of the present invention and their descriptions are used to understand the present invention and do not constitute an improper limitation of the present invention. In the drawings:

[0118] Figure 1This is the overall algorithm flowchart of the industrial load demand response rebound effect identification method system based on Bayesian mediation analysis proposed by the present invention;

[0119] Figure 2 This is the Bayesian network structure diagram in the embodiment of the present invention;

[0120] Figure 3 This is the posterior distribution diagram of the direct effect of the driving force in the embodiment of the present invention;

[0121] Figure 4 This is the posterior distribution diagram of the indirect effect of device 1 in the embodiment of the present invention;

[0122] Figure 5 This is the posterior distribution diagram of the indirect effect of device 2 in the embodiment of the present invention;

[0123] Figure 6 This is the posterior distribution diagram of the temperature regulation effect in the embodiment of the present invention;

[0124] Figure 7 This is the posterior distribution diagram of the device interaction effect in the embodiment of the present invention;

[0125] Figure 8 This is the posterior distribution diagram of the total effect in the embodiment of the present invention. Detailed implementation manners

[0126] To make the objectives, technical solutions and advantages of the present invention clearer, the technical solutions in 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, rather than all of the embodiments.

[0127] Figure 1 It shows the overall algorithm process of the system for the industrial load demand response rebound effect identification method based on Bayesian mediation analysis. Starting from the theoretical basis, this process covers the citation of the basic theoretical framework, the construction of the system model, the design of the Bayesian inference framework, and the effect decomposition and quantitative analysis. Subsequently, through the algorithm implementation process, data preprocessing, model training and result analysis are realized, with particular emphasis on the application of the MCMC sampling technique. In the technical implementation part, a verification mechanism such as model comparison and stability analysis is mainly constructed to ensure the reliability of the results. Finally, in the application verification stage, through algorithm verification, effect analysis, exception handling and performance evaluation, the effectiveness and reliability of the entire identification method are ensured. This process fully reflects the complete transformation from theory to practice, aiming to promote the intelligent and automated scheduling of power demand response and improve the economy and reliability.

[0128] The industrial load demand response rebound effect identification method based on Bayesian mediation analysis includes the following steps:

[0129] Step S1: System model construction

[0130] In industrial load demand response, the rebound effect refers to the phenomenon that after the implementation of demand response (such as peak shaving and valley filling, load adjustment, etc.), the load rebounds within a certain period of time. This rebound phenomenon not only affects the effect of demand response, but also may pose challenges to the safety and stability of the power grid. Therefore, accurately describing and quantifying the rebound effect is an important issue in demand response research.

[0131] To deeply analyze the rebound effect, the present invention proposes a systematic model, which comprehensively depicts the mechanism of rebound by decomposing multiple components of the rebound effect. These components include the direct effect, the equipment state effect, the environmental regulation effect, the equipment interaction effect, and the random disturbance term. Each component has its independent physical background and mathematical description, and they are interrelated through specific mathematical formulas and jointly act on the total rebound effect of the system.

[0132] Specifically, the direct effect is the direct impact of demand response intervention on the load, which is determined by the intensity of demand response and a coefficient; the equipment state effect reflects the indirect impact brought about by changes in equipment state, and the state of each equipment has different degrees of influence on the rebound effect; the environmental regulation effect considers external factors, such as temperature, etc., on the regulation effect of demand response; the equipment interaction effect further affects the rebound effect through the interaction between equipment, especially important in scenarios where multiple devices work together; finally, the random disturbance term describes the unforeseen fluctuations in the system, which considers possible external disturbances or internal uncertainties of the system.

[0133] The construction of this system model not only helps to understand the load change process after demand response, but also provides a theoretical framework and calculation basis for further effect analysis and prediction. By accurately modeling each effect, the internal mechanism of the rebound effect can be revealed, and then data support and decision-making basis can be provided for formulating more refined demand response strategies.

[0134] The construction of the system model includes the following two steps:

[0135] I. Modeling of the rebound effect

[0136] Let the amplitude of the rebound effect at time point t be R(t), and its mathematical expression is:

[0137] R(t) = D(t) + S(t) + E(t) + I(t) + ε(t);

[0138] In the formula, D(t) represents the direct effect; S(t) represents the equipment state effect; E(t) represents the environmental regulation effect; I(t) represents the equipment interaction effect; ε(t) represents the system random disturbance term. The physical meaning and mathematical expressions of each component are as follows:

[0139] (2) Direct effect D(t)

[0140] The direct effect describes the direct impact of demand response intervention on the load:

[0141] D(t) = α·DR(t);

[0142] In the formula, DR(t) represents the intensity of demand response; α is the direct effect coefficient.

[0143] (3) Equipment status effect S(t)

[0144] The equipment status effect reflects the indirect impact generated by changes in equipment status:

[0145]

[0146] In the formula, ES i (t) represents the status of the i-th equipment; β i is the influence coefficient of the i-th equipment status on the rebound effect, abbreviated as: equipment status influence coefficient β; n is the total number of equipment.

[0147] (4) Environmental regulation effect E(t)

[0148] The environmental regulation effect takes into account the regulatory role of external factors such as temperature:

[0149] E(t) = γ·T(t)·DR(t);

[0150] In the formula, T(t) represents the environmental temperature; γ is the environmental regulation coefficient; DR(t) represents the intensity of demand response.

[0151] (5) Equipment interaction effect I(t)

[0152] The equipment interaction effect describes the interaction between equipment:

[0153]

[0154] In the formula, δ ij represents the interaction coefficient between equipment i and j, abbreviated as: equipment interaction effect coefficient δ; ES i (t) represents the status of the i-th equipment, ES j (t) represents the status of the j-th equipment.

[0155] (6) System random disturbance term ε(t)

[0156] Represents the random fluctuations in the system, following a normal distribution:

[0157] ε(t) ~ N(0, σ 2 );

[0158] Among them, N(0, σ 2 ) represents a normal distribution with a mean of 0 and a variance of σ 2 .

[0159] II. State Transition Equation Modeling

[0160] Describe the dynamic evolution process of the device state effect and the device interaction effect, and use the state transition matrix to represent the interaction relationship between devices. Among them, the state transition equation modeling includes:

[0161] (1) Device State Equation

[0162] For device i, its state transition equation is:

[0163] ES i (t) = f i (DR(t), T(t), ES i-1 (t)) + ξ i (t);

[0164] Among them, f i (DR(t), T(t), ES i-1 (t)) = η i ·DR(t) + λ i ·T(t) + μ i ·ES i-1 (t) + v i , η i is the influence coefficient of demand response on the state of device i, simply referred to as the demand response influence coefficient η; λ i is the influence coefficient of temperature; μ i is the influence coefficient of the upstream device; ν i is the basic operating parameter; ξ i (t) is the device state random disturbance term.

[0165] (2) Description of Device Interaction Effect

[0166] The interaction relationship between devices is represented by the state transition matrix:

[0167]

[0168] In the formula, A is the state transition matrix, B is the input matrix, and Ξ(t) is the system noise vector.

[0169] Step S2: Construction of Bayesian Inference Framework

[0170] The Bayesian inference framework plays an important role in the analysis of the rebound effect, especially in the face of uncertainty. The Bayesian inference framework relies on setting the prior distribution of parameters. The prior distribution reflects the expectation or uncertainty of the parameter values before the observation data, when there is no observation data. In the framework of the present invention, the setting of the prior distribution takes into account the physical characteristics of the various components of the rebound effect and the relevant background knowledge. For example, for the direct effect coefficient and the device state influence coefficient, the prior distribution is set as a normal distribution to reflect the initial assumptions about the possible values of these coefficients. For the environmental regulation coefficient and the interaction effect coefficient, a distribution form that conforms to the actual physical phenomena is also adopted to ensure that the model is more in line with the actual situation.

[0171] By constructing a likelihood function, the parameters in the model are related to the observed data. The role of the likelihood function is to represent the probability of the occurrence of the observed data given the parameters. In the analysis of the rebound effect, the likelihood function not only reflects the evolution process of the device state, but also describes how factors such as the demand response intensity and the environmental temperature affect the observed value of the rebound effect. By comparing these variables with the actual observed data, the estimation of each parameter can be gradually corrected.

[0172] Through the above steps, the Bayesian inference framework can combine prior knowledge and observed data to provide the optimal estimate of the parameters through the posterior distribution. This process can not only effectively quantify the influence degree of each effect, but also capture the uncertainty existing in the system, and thus provide strong support for the further analysis and decision-making of the load rebound effect.

[0173] The construction of the Bayesian inference framework includes the following two steps:

[0174] I. Setting the prior distribution

[0175] Set the prior distribution for each effect coefficient and noise parameter, reflecting the assumptions about the initial uncertainty of the parameters.

[0176] (1) For the coefficients of the direct effect and the device state effect, considering their physical meanings, the following is set:

[0177]

[0178] where α is the direct effect coefficient; represents the initial uncertainty about the magnitude of the direct effect.

[0179] For the device state influence coefficient:

[0180]

[0181] where β i is the influence coefficient of the i-th device state on the rebound effect; Represents the prior uncertainty of the impact degree on the device.

[0182] (2) Environmental adjustment coefficient

[0183] The adjustment effect of environmental factors such as temperature is usually small:

[0184] γ ∼ N(0, 0.5 2 );

[0185] γ is the environmental adjustment coefficient, and choosing a smaller variance reflects this prior knowledge.

[0186] (3) Interaction effect coefficient

[0187] The interaction coefficient between devices adopts the following prior:

[0188]

[0189] where δ ij represents the interaction coefficient between devices i and j; reflects that the device interaction effect is usually less than the main effect. The "main effect" is the direct influence of a single dominant factor in the rebound effect model, mainly including the direct effect D(t) = α · DR(t) and the device state effect Device interaction effect describes the mutual influence and coupling effect between device states, which is a second-order or higher-order effect. The device interaction effect usually appears in complex industrial systems where multiple devices work together, but its intensity is generally less than the main effect. Therefore, a smaller variance value is used in the prior distribution setting to reflect this statistical characteristic.

[0190] (4) System noise parameter

[0191] The system noise parameter adopts a half-Cauchy distribution as the prior. This distribution selection ensures the non-negativity of the noise parameter, while allowing for large tail uncertainties, enabling the model to adapt to possible abnormal observations and large fluctuations. Compared with the normal distribution or uniform distribution, the half-Cauchy distribution is more suitable as the prior distribution of the variance parameter in Bayesian inference:

[0192] σ ∈ ~HalfCauchy(1);

[0193] where the half-Cauchy distribution (HalfCauchy) refers to the probability distribution defined on the positive real axis, and its probability density function is:

[0194]

[0195] II. Likelihood function construction

[0196] Based on the state space model, construct the state equation of the device and the likelihood function of the rebound effect for subsequent parameter estimation of the fuzzy set. Based on the framework of the state space model, construct the following likelihood function:

[0197] (1) Likelihood function of the device state equation

[0198] For each device i, the likelihood function of its state observation is:

[0199]

[0200] where f i (DR(t), T(t), ES i-1 (t)) = η i ·DR(t) + λ i ·T(t) + μ i ·ES i-1 (t) + v i , η i is the influence coefficient of the demand response on the state of device i; λ i is the influence coefficient of temperature; μ i is the influence coefficient of the upstream device; ν i is the basic operating parameter; ξ i (t) is the device state random disturbance term; θ represents the set of all relevant parameters in the model, and θ specifically includes the direct effect coefficient α; the influence coefficient β i of the state of the i-th device on the rebound effect, i = 1, 2,..., n; the environmental adjustment coefficient γ; the interaction coefficient δ ij between device i and j, i = 1, 2,..., n, j = 1, 2,..., n, and j ≠ i; the influence coefficient η i of the demand response on the state of device i, the temperature influence coefficient λ i , the upstream device influence coefficient μ i , the basic operating parameter v i , the variance σ 2 of the system random disturbance term, and the variance

[0201] (2) Likelihood function of the rebound effect

[0202] For the magnitude R(t) of the rebound effect at the observed time point t, its likelihood function is:

[0203]

[0204] where: μ R (t) = D(t) + S(t) + E(t) + I(t) represents the expected value of the rebound effect, It is the variance of the rebound effect observation value, representing the magnitude of random uncertainty during the observation process.

[0205] Step S3: Effect decomposition and quantification

[0206] Effect decomposition and quantification are the core links to achieve the characterization of the feasible region of industrial load operation. By precisely analyzing the immediate and secondary impacts of demand response (DR) instructions on industrial loads, it can provide a scientific basis for subsequent optimization decisions. The following is a detailed description of the implementation process of effect decomposition and quantification in specific implementations:

[0207] The direct effect reflects the immediate regulation effect of demand response (DR) instructions on industrial loads and is a core component of the rebound effect. In industrial scenarios, DR instructions may achieve load reduction by directly reducing equipment power or adjusting the production rhythm, but its effect may vary due to factors such as equipment type and operating status. To precisely quantify this effect, the present invention separates the direct effect from the complex system through a Bayesian mediation analysis framework. The expression is:

[0208] D(t) = αDR(t);

[0209] In the formula, α is the direct effect coefficient, representing the direct impact of unit DR intensity on the load. This coefficient obtains the posterior distribution {α (1) , α (2) , …, α (M)} through MCMC sampling under the Bayesian inference framework, where α (M) represents the value of the direct effect coefficient obtained from the M - th MCMC sampling, and then calculates its mean and confidence interval:

[0210]

[0211] CI 95％ (D(t)) = [q 0.025 (α·DR(t)), q 0.975 (α·DR(t))];

[0212] In the formula, represents the posterior mean estimate of the direct effect and is the best point estimate of the true direct effect; m represents the m - th sampling index in the MCMC sampling sequence; M represents the total number of MCMC samplings, usually set to a sufficiently large value (such as 1000 or 10000) to ensure the convergence of the sampling distribution; CI 95％ (D(t)) represents the 95% confidence interval of the direct effect, used to quantify the uncertainty range of the direct effect estimate; q 0.025 represents the 2.5% quantile of the posterior distribution, that is, the lower bound value; q 0.975Represents the 97.5% quantile of the posterior distribution, i.e., the upper bound value. The quantile-based confidence interval construction method can effectively capture the uncertainty of parameters and provide reasonable uncertainty quantification even in the case of asymmetric distributions.

[0213] Indirect effects describe the secondary impacts generated by DR instructions through device state changes, environmental factor regulation, and interactions among devices. For example, DR may temporarily shut down a device, resulting in higher energy consumption when it restarts due to backlogged production demands, or affect the operating efficiency of other devices through temperature changes. To capture such complex relationships, the present invention proposes a hierarchical quantification method.

[0214] Indirect effect of a single device:

[0215] The indirect effect of the i-th device is generated by the DR instruction through its state transfer:

[0216] IE i (t) = η i ·β i ·DR(t);

[0217] Interaction effect between devices:

[0218] The interaction between devices is modeled by the state transition matrix A, and the transfer effect of device i on j is:

[0219] CE ij (t) = μ ij ·β j ·DR(t);

[0220] Environmental regulation effect:

[0221] The regulation effect of the environmental temperature T(t) on the DR effect is:

[0222] E(t) = γ·T(t)·DR(t);

[0223] In the formula, η i is the influence coefficient of the demand response on the state of device i, representing the direct influence degree of a demand response signal with unit strength on the state change of device i; β i is the influence coefficient of the state of the i-th device on the rebound effect, representing the contribution degree of the state change of device i to the overall rebound effect, β j is the influence coefficient of the state of the j-th device on the rebound effect, representing the contribution degree of the state change of device j to the overall rebound effect; μ ij is the transfer coefficient of device i to j in the state transition matrix A; γ is the environmental regulation coefficient; T(t) is the environmental temperature.

[0224] The total effect is the comprehensive manifestation of the direct effect and various indirect effects, and its significance evaluation can determine whether the DR afterload rebound is statistically significant. Through posterior probability calculation, the present invention quantifies the effect direction and intensity, providing a scientific basis for decision-making.

[0225] Total effect synthesis:

[0226]

[0227] Significance evaluation:

[0228] Calculate the posterior probability that the total effect is greater than zero:

[0229]

[0230] In the formula, represents the posterior mean estimate of the direct effect D(t); IE i (t) is the indirect effect of the i-th device; CE ij (t) is the interaction effect between the i-th device and the j-th device; is the posterior mean estimate of the environmental regulation effect; M is the length of the MCMC sampling chain; TE (m) (t) is the total effect obtained from the m-th sampling; I is the indicator function.

[0231] Figure 1 Among them, NUTS (No-U-Turn Sampler) is a Markov chain Monte Carlo (MCMC) sampling algorithm for Bayesian inference. NUTS is an extension of Hamiltonian Monte Carlo (HMC), aiming to automatically adjust key parameters (such as step size and path length) in the sampling process, thereby improving sampling efficiency and avoiding the complexity of manual parameter tuning.

[0232] Step S4: Algorithm process and model verification

[0233] The algorithm process and model verification are the key links to ensure the reliability and effectiveness of the industrial load operation feasible region characterization method. By designing a reasonable algorithm process and strict model verification steps, the generalization ability and stability of the model in complex industrial scenarios can be ensured. The following is a detailed description of the implementation process of the algorithm process and model verification:

[0234] Before the algorithm starts, it is necessary to preprocess the industrial load data to improve the robustness of the model. The preprocessing steps include:

[0235] Standardization processing:

[0236] Eliminate the dimension difference and avoid parameter estimation deviation:

[0237]

[0238] In the formula, x represents the original data variable, such as original observed values like demand response intensity, equipment status value, environmental temperature, etc.; μ x represents the mean of variable x, which is used to centralize the data and eliminate the offset effect; σ x represents the standard deviation of variable x, which is used to normalize the data and eliminate the influence of different dimensions; x std represents the standardized variable value, which is dimensionless, with a mean of 0 and a standard deviation of 1, and helps to improve the stability and convergence speed of model training.

[0239] Time series smoothing:

[0240] The moving window average method (window size w = 24 hours) is adopted to suppress high-frequency noise:

[0241]

[0242] In the formula, represents the smoothed status value of device i at time point t; ES i (t - k) represents the original status value of device i at time point t - k; w represents the size of the moving window, with the unit of hour, and is set to 24 hours in the present invention; k represents the time offset within the moving window, and its value range is from 0 to w - 1, which is used to traverse all time points within the window.

[0243] Outlier correction:

[0244] Based on the interquartile method (IQR = 1.5), outliers are identified and replaced with the local mean:

[0245]

[0246] In the formula, x(t) represents the original data value at time point t; x corrected (t) represents the data value after outlier correction at time point t; Q1 represents the first quartile of the data (i.e., the 25% quantile), representing the lower quartile point of the data distribution; Q3 represents the third quartile of the data (i.e., the 75% quantile), representing the upper quartile point of the data distribution; IQR represents the interquartile range, i.e., Q3 - Q1, which is used to measure the dispersion degree of the data; k′ represents the time offset during the calculation of the local mean, and its value range is from -2 to 2, which is used to calculate the average value of a total of 5 time points before and after the current time point. When the original data point x(t) is identified as an outlier (i.e., falling outside the interval [Q1 - 1.5IQR, Q3 + 1.5IQR]), it is replaced with the mean value of a total of 5 time points (including itself) before and after to reduce the negative impact of outliers on model training.

[0247] Obtain the posterior distribution of parameters using Bayesian inference and MCMC sampling, and adjust the device operation strategy through iterative optimization. Specifically as follows:

[0248] Compare the total effect difference between the simplified model (R(t) = α0 + α1DR(t) + ∈) and the complete model:

[0249] In the formula, R(t) represents the amplitude of the rebound effect at time point t; α0 represents the baseline constant term of the rebound effect, reflecting the natural rebound level of the system without a demand response signal; α1 represents the linear influence coefficient of the demand response intensity, reflecting the direct impact of unit demand response intensity on the rebound effect; DR(t) represents the demand response intensity at time point t; ∈ represents the random error term, which follows a normal distribution with a mean of 0 and a constant variance, and is used to capture the random fluctuations not explained by the model.

[0250]

[0251] In the formula, Δ TE represents the total effect difference, which is used to quantify the difference degree between the simplified model and the complete model in predicting the total effect; represents the total effect value estimated by the simplified model, that is, the predicted value considering only the direct linear influence of the demand response; represents the total effect value estimated by the complete model, including all components such as the direct effect, device state effect, environmental regulation effect, and device interaction effect. The smaller this difference value, the higher the consistency between the simplified model and the complete model; conversely, the larger the difference value, the more significant the prediction deviation caused by ignoring the complex effects.

[0252] Verify the stability of the model through the Bootstrap resampling method. Specifically as follows:

[0253] Perform Bootstrap resampling 1000 times and calculate the coefficient of variation of the effect estimate:

[0254]

[0255] In the formula, CV represents the coefficient of variation, which is the ratio of the standard deviation to the mean, and is used to measure the relative dispersion degree of the estimated value; σ boot represents the standard deviation of the effect estimate values obtained by Bootstrap resampling, reflecting the fluctuation degree of the estimated values; μ boot represents the mean of the effect estimate values obtained by Bootstrap resampling, reflecting the central tendency of the estimated values. A coefficient of variation less than 0.2 is an empirical threshold, indicating that the model estimate has sufficient stability and reliability.

[0256] By perturbing key parameters (such as direct effect coefficient, indirect effect coefficient, environmental regulation coefficient, etc.), the sensitivity of the model to parameter changes is evaluated. Specifically as follows:

[0257] Sensitivity analysis:

[0258] By perturbing the value of the direct effect coefficient α (such as α ± 10%, that is, fluctuating 10% up and down based on the current estimated value), calculate the total effect change rate:

[0259]

[0260] In the formula, S θ represents the sensitivity index of parameter θ, reflecting the response degree of the model output to the change of this parameter. When S θ < 2, it indicates that the sensitivity of the model to this parameter is within a reasonable range and the model stability is good; on the contrary, if S θ ≥ 2, it indicates that the model is overly sensitive to this parameter and it is necessary to further optimize the model structure or re-evaluate the parameter estimation method.

[0261] Example

[0262] This example aims to demonstrate the application of the method for identifying the rebound effect of industrial load demand response based on Bayesian mediation analysis described in the present invention. Through this example, the operation steps and expected effects of this method in actual industrial load demand response management can be specifically illustrated.

[0263] Taking the steel industry in a certain city as an example, historical electricity consumption data of the steel industry in this city is collected. The data includes the load fluctuation conditions of main equipment such as blast furnaces, converters, and electric arc furnaces, as well as records of relevant environmental temperatures and demand response measures. The data time span is one year to ensure that the load changes in different seasons and production conditions are covered.

[0264] Secondly, construct a Bayesian mediation analysis model, and the model structure refers to Figure 2The Bayesian network structure diagram shown. The model includes direct effects, device state effects, environmental regulation effects, device interaction effects, and a random disturbance term. Set the prior distributions of each effect coefficient and noise parameter to reflect the assumptions about the initial uncertainty of the parameters. Construct the likelihood functions of the device state equation and the rebound effect for subsequent parameter estimation. Use the Markov Chain Monte Carlo (MCMC) method to fit the model and obtain the posterior distribution of the parameters. Verify the reliability of the results through model comparison and stability analysis. Construct a two-way feedback model of Device 1 and Device 2 through 34 parameter nodes, where the device intercept term adopts a semi-Cauchy prior distribution (Device 1: scale parameter σ = 0.5; Device 2: μ = 1.2, σ = 0.3, and the path coefficient adopts a truncated normal distribution (-2 < β < 2). The system achieves dynamic coupling through 21 effect paths, specifically including a temperature-driving force interaction effect module (normal distribution, μ = 0.6 ± 0.2), a two-way device interaction effect chain (Device 1-2 and Device 2-1 double paths), and a three-level rebound effect conduction mechanism (including intercept term, path coefficient, and likelihood calculation module). Multilevel data integration is achieved through 5 types of interactions, where the width of the direct effect path is positively correlated with the effect size, and the indirect effect is realized through the conduction path of device → environment → device, and the model fitting degree reaches R 2 = 0.78, effectively characterizing the non-linear correlation characteristics of the dual-device system under complex working conditions.

[0265] It should be noted that for the σ standard deviation parameter: In the present invention, σ represents the degree of dispersion of the probability distribution and is used to quantify the uncertainty of parameter estimation. In the semi-Cauchy prior distribution of Device 1: σ = 0.5. In the parameters of Device 2: σ = 0.3. For the μ mean parameter: In the present invention, μ represents the central position or expected value of the probability distribution. In the parameters of Device 2: μ = 1.2, indicating the central position of the distribution. In the temperature-driving force interaction effect module: μ = 0.6 ± 0.2, indicating the distribution mean of the environmental regulation coefficient γ. The device state influence coefficient β: In the invention, β represents the contribution degree of the device state change to the rebound effect. In the truncated normal distribution: -2 < β < 2, which limits the reasonable range of the influence coefficient

[0266] Figures 3 to 8 The influence of demand response on multiple key effect parameters of the industrial load rebound effect in this embodiment is intuitively shown through the posterior distribution diagram, Figure 3 is the posterior distribution diagram of the direct effect of the driving force; Figure 4 is the posterior distribution diagram of the indirect effect of Device 1; Figure 5 is the posterior distribution diagram of the indirect effect of Device 2; Figure 6 is the posterior distribution diagram of the temperature regulation effect; Figure 7 is the posterior distribution diagram of the device interaction effect; Figure 8This is the posterior distribution map of the total effect. The mean of the posterior distribution of the direct effect is 0.31, and the 95% highest density interval (HDI) is [0.12, 0.49]. Moreover, the probability that the effect is greater than zero is extremely high, approaching 100%, indicating that the demand response has a significant positive direct impact on the load. The means of indirect effect 1 (indirect_1) and indirect effect 2 (indirect_2) are 0.0086 and -0.044 respectively. Their 95% HDIs contain 0, and the probabilities that the effects are greater than zero are relatively low, 71.2% and 13.7% respectively, suggesting that these two indirect effects may not be statistically significant, and indirect effect 2 is more likely to have a negative impact. The means of the temperature influence effect (temp_effect) and the equipment interaction effect (interaction) are 0.011 and 0.0033 respectively. Similarly, their 95% HDIs also contain 0, and the probabilities that the effects are greater than zero are 54.9% and 51.4% respectively, indicating greater uncertainty for these two effects. Finally, the mean of the posterior distribution of the total effect (total) is 0.27, and the 95% HDI is [0.084, 0.49]. The probability that the effect is greater than zero is close to 100%, confirming that after considering all effects comprehensively, the overall impact of the demand response on the load is significant and positive. These results provide a scientific basis for industrial users to participate in demand response and contribute to optimizing the stability and economy of the power grid.

[0267] The above embodiments are only used to illustrate the present invention, rather than limiting the technical solutions described therein. Although the above embodiments are described in detail in this specification, the present invention is not limited to the above specific implementation manners. Therefore, any technical solutions and their improvements that modify the present invention or make equivalent substitutions without departing from the scope of the present invention are all covered by the scope of the claims of the present invention.

Claims

1. A method for identifying the rebound effect of industrial load demand response based on Bayesian mediation analysis, characterized in that, It includes the following steps: Step S1: System model construction Decompose the rebound effect amplitude R(t) into direct effect D(t), equipment state effect S(t), environmental regulation effect E(t), equipment interaction effect I(t), and system stochastic disturbance term ε(t), and construct an equipment state transition equation; Step S2: Bayesian inference framework construction The prior distribution settings of the direct effect coefficient α, the device state influence coefficient β, the environmental adjustment coefficient γ, the device interaction effect coefficient δ, and the system noise parameter σ ∈ and the construction of the device state equation and the likelihood function of the rebound effect; Step S3: Effect decomposition and quantitative analysis Calculate the direct effect D(t), single-equipment indirect effect, environmental regulation effect E(t), and equipment interaction effect I(t), and synthesize the total effect. The total effect: Posterior probability of the total effect: In the formula, represents the posterior mean estimate of the direct effect; IE i (t) is the indirect effect of the i-th device; CE ij (t) is the interaction effect between the i-th device and the j-th device; is the posterior mean estimate of the environmental adjustment effect; M is the length of the MCMC sampling chain; TE (m) (t) is the total effect obtained from the m-th sampling; I is an indicator function; n is the total number of equipment; Step S4: Preprocess the industrial load data, use the preprocessed data for model training, verification, and testing, and use the trained model to identify and quantify the load rebound effect after demand response.

2. The method for identifying the rebound effect of industrial load demand response based on Bayesian mediation analysis according to claim 1, wherein Step S1 includes: 1) Rebound effect modeling Let the rebound effect amplitude at time point t be R(t), and its mathematical expression is: R(t) = D(t) + S(t) + E(t) + I(t) + ε(t); In the formula, D(t) represents the direct effect; S(t) represents the equipment state effect; E(t) represents the environmental regulation effect; I(t) represents the equipment interaction effect; ε(t) represents the system stochastic disturbance term; Among them; (1) The mathematical expression of the direct effect D(t) is as follows: D(t) = α·DR(t); In the formula, DR(t) represents the intensity of demand response; α is the direct effect coefficient; (2) The mathematical expression of the equipment state effect S(t) is as follows: Among them, ES i (t) represents the state of the i-th device; β i is the influence coefficient of the i-th device state on the rebound effect, simply referred to as the device state influence coefficient β; n is the total number of devices; (3) The mathematical expression of the environmental regulation effect E(t) is as follows: E(t) = γ·T(t)·DR(t); Among them, T(t) represents the environmental temperature; γ is the environmental regulation coefficient; DR(t) represents the intensity of demand response; (4) The mathematical expression of the equipment interaction effect I(t) is as follows: Among them, δ ij represents the interaction coefficient between devices i and j, simply referred to as the device interaction effect coefficient δ; ES i (t) represents the state of the i-th device; ES j (t) represents the state of the j-th device; (5) The system stochastic disturbance term ε(t) follows a normal distribution: ε(t)~N(0,σ 2 ); Among them, N(0, σ 2 ) represents a normal distribution with a mean of 0 and a variance of σ 2 ; 2) State transition equation modeling (1) Equipment state equation For equipment i, its state transition equation is: ES i (t) = f i (DR(t), T(t), ES i-1 (t)) + ξ i (t); where f i (DR(t), T(t), ES i-1 (t)) = η i ·DR(t) + λ i ·T(t) + μ i ·ES i-1 (t) + v i , η i is the influence coefficient of the demand response on the state of device i; λ i is the influence coefficient of temperature; μ i is the influence coefficient of the upstream device; ν i is the basic operating parameter; ξ i (t) is the random disturbance term of the device state; (2) Determine the interaction relationship between equipment. The interaction relationship between equipment is represented by a state transition matrix: In the formula, A is the state transition matrix, B is the input matrix, and Ξ(t) is the system noise vector.

3. The method for identifying the rebound effect of industrial load demand response based on Bayesian mediation analysis according to claim 2, wherein Step S2 includes: 1) Prior distribution setting (1) For the direct effect coefficient and equipment state influence coefficient, set as follows: where α is the direct effect coefficient; represents the initial uncertainty of the direct effect magnitude; Among them, β i is the influence coefficient of the i-th device state on the rebound effect, simply referred to as the device state influence coefficient β; represents the prior uncertainty of the degree of influence on the device; (2) For the environmental regulation coefficient, set as follows: γ ∼ N(0, 0.5 2 ); γ is the environmental regulation coefficient; (3) For the interaction effect coefficient, set as follows: The interaction coefficient between equipment adopts the following prior: Among them, δ ij represents the interaction coefficient between devices i and j, simply referred to as the device interaction effect coefficient δ; (4) System noise parameter σ ∈ It is set as follows: system noise parameter σ ∈ The semi-Cauchy distribution is used as the prior distribution; 2) Likelihood function construction (1) Likelihood function of the equipment state equation For equipment i, the likelihood function of its state observation is: where f i (DR(t), T(t), ES i-1 (t)) = η i ·DR(t) + λ i ·T(t) + μ i ·ES i-1 (t) + v i , η i is the influence coefficient of demand response on the state of device i; λ i is the influence coefficient of temperature; μ i is the influence coefficient of upstream devices; ν i is the basic operating parameter; ξ i (t) is the random disturbance term of device state; θ represents the set of all relevant parameters in the model, and θ specifically includes the direct effect coefficient α; the influence coefficient β i of the state of the i-th device on the rebound effect, i = 1, 2,..., n; the environmental adjustment coefficient γ; the interaction coefficient δ ij between device i and j, i = 1, 2,..., n, j = 1, 2,..., n, and j ≠ i; the influence coefficient η i of demand response on the state of device i i , the temperature influence coefficient λ i , the upstream device influence coefficient μ i , the basic operating parameter v 2 , the variance σ (2) Likelihood function of the rebound effect For the observed rebound effect amplitude R(t) at time point t, its likelihood function is: where: μ R (t) = D(t) + S(t) + E(t) + I(t) represents the expected value of the rebound effect, is the variance of the rebound effect observation value, characterizing the magnitude of random uncertainty during the observation process.

4. The method for identifying the rebound effect of industrial load demand response based on Bayesian mediation analysis according to claim 3, wherein In step S3, The calculation process of the direct effect D(t) is as follows: Separate the direct effect from the complex system through the Bayesian mediation analysis framework; the expression is: D(t) = αDR(t); Among them, α is the direct effect coefficient, which represents the direct impact of the unit DR intensity on the load; this coefficient obtains the posterior distribution {α (1) , α (2) , …, α (M)} through MCMC sampling under the Bayesian inference framework, where α (M) represents the value of the direct effect coefficient obtained from the M-th MCMC sampling, and then calculates its mean and confidence interval: In the formula, represents the posterior mean estimate of the direct effect, which is the best point estimate of the true direct effect; m represents the m-th sampling index in the MCMC sampling sequence; M represents the total number of MCMC samplings, and CI 95％ (D(t)) represents the 95% confidence interval of the direct effect, which is used to quantify the uncertainty range of the direct effect estimate; q 0.025 represents the 2.5% quantile of the posterior distribution, that is, the lower bound value; q 0.975 represents the 97.5% quantile of the posterior distribution, that is, the upper bound value; The calculation process of the single-equipment indirect effect is as follows: The indirect effect of the i-th equipment is generated by the DR instruction through its state transfer: IE i (t) = η i ·β i ·DR(t); The calculation process of the device interaction effect is as follows: The interaction between devices is modeled by the state transition matrix A, and the transfer effect of device i on j is: CE ij (t) = μ ij ·β j ·DR(t); The calculation process of the environmental regulation effect is as follows: E(t) = γ·T(t)·DR(t); where η i is the influence coefficient of the demand response on the state of device i, representing the direct influence degree of the demand response signal with unit strength on the state change of device i; β i is the influence coefficient of the state of the i-th device on the rebound effect, representing the contribution degree of the state change of device i to the overall rebound effect, β j is the influence coefficient of the state of the j-th device on the rebound effect, representing the contribution degree of the state change of device j to the overall rebound effect; μ ij is the transfer coefficient from device i to j in the state transition matrix A; γ is the environmental adjustment coefficient; T(t) is the environmental temperature.

5. The method for identifying the rebound effect of industrial load demand response based on Bayesian mediation analysis according to claim 4, wherein The data preprocessing methods include: normalization processing, time series smoothing processing, and quartile method outlier correction processing.

6. The method for identifying the rebound effect of industrial load demand response based on Bayesian mediation analysis according to claim 5, wherein In step S4, a three-layer verification system is adopted: 1) Model comparison: where, Δ TE represents the total effect difference, which is used to quantify the difference degree between the simplified model R(t) = α0 + α1DR(t) + ∈ and the complete model in predicting the total effect; represents the total effect value estimated by the simplified model, that is, the predicted value considering only the direct linear impact of demand response; represents the total effect value estimated by the complete model, including all components of the direct effect, equipment state effect, environmental regulation effect, and equipment interaction effect; Where: R(t) represents the amplitude of the rebound effect at time point t; α0 represents the baseline constant term of the rebound effect, reflecting the natural rebound level of the system without a demand response signal; α1 represents the linear influence coefficient of the demand response intensity, reflecting the direct influence of the unit demand response intensity on the rebound effect; DR(t) represents the demand response intensity at time point t; ∈ represents the random error term, which follows a normal distribution with a mean of 0 and a constant variance, and is used to capture the random variations that the model fails to explain; 2) Stability test: Verify the stability of the model through the Bootstrap resampling method; specifically as follows: Calculate the coefficient of variation of the effect estimate: In the formula, CV represents the coefficient of variation, which is the ratio of the standard deviation to the mean and is used to measure the relative dispersion of the estimated value; σ boot represents the standard deviation of the effect estimates obtained by Bootstrap resampling, reflecting the degree of fluctuation of the estimates; μ boot represents the mean of the effect estimates obtained by Bootstrap resampling, reflecting the central tendency of the estimates; a coefficient of variation less than 0.2 is an empirical threshold; 3) Sensitivity analysis: Calculate the total effect change rate: where S θ represents the sensitivity index of parameter θ, reflecting the response degree of the model output to the change of this parameter; when S θ < 2, it indicates that the sensitivity of the model to this parameter is within a reasonable range and the model has good stability; Conversely, if S θ ≥ 2, it indicates that the model is overly sensitive to this parameter, and it is necessary to further optimize the model structure or re-evaluate the parameter estimation method.

Images