Adjustable load response capability probability modeling method based on Gaussian mixture model

Abstract

The invention discloses an adjustable load response capability probability modeling method based on a Gaussian mixture model. The method comprises the following steps: S1, collecting historical response data of an adjustable load; s2, selecting the electricity consumption of the previous day, the temperature of the current day and the declaration response power of the current day to form a vector X of an input variable, and then taking the actual response power of the current day as another vector Y; s3, based on historical data of X and Y, establishing a Gaussian mixture model, and fitting joint probability distribution of [X, Y]; s4, for the future response condition, after X is given, the conditional distribution of Y is obtained according to the joint distribution of [X, Y]; and S5, solving a mean value, a standard deviation and a quantile of each probability to serve as a probability prediction model of the adjustable load response capability. According to the method, the diversity of the load response can be comprehensively described, and different probability distributions are generated under different conditions, so that the prediction of the load response is more flexible, high in adaptability and high in accuracy.

Description

Technical Field

[0001] The present invention relates to the technical field of power system operation control, and in particular to a probabilistic modeling method for adjustable load response capability based on a Gaussian mixture model. Background Art

[0002] In the dispatching of modern power systems, load response capability plays an important role as one of the key factors in power grid dispatching. Load demand response refers to the behavior of power users to flexibly adjust their power load according to the needs, price signals or other incentives of power operators under power market conditions. This approach can not only improve the reliability of the power system, but also effectively reduce power costs.

[0003] However, the prediction of load response capability faces many challenges. Traditional load response prediction methods usually rely on historical data and make predictions through statistical analysis or simple time series models, but these methods have certain limitations. Especially when faced with complex load patterns and changing dispatch requirements, traditional methods often find it difficult to accurately reflect the actual changing trend of load response.

[0004] As the demand for flexible dispatch and precise load response in power systems increases, how to use more accurate modeling methods to effectively predict load response capabilities has become an important issue in the field of power dispatch. Therefore, a new load response capability probabilistic modeling method is urgently needed, which can accurately predict the probability distribution of future load response capabilities based on historical data and provide strong support for power system dispatch. Summary of the invention

[0005] The purpose of the present invention is to provide a probabilistic modeling method for adjustable load response capability based on a Gaussian mixture model to solve the above defects.

[0006] In order to achieve the above object, the present invention provides the following technical solutions:

[0007] The probabilistic modeling method of adjustable load response capacity based on Gaussian mixture model includes the following steps:

[0008] S1. Collect historical response data of adjustable loads;

[0009] S2, select the power consumption of the previous day, the temperature of the day, and the response power reported on the day to form a vector X of input variables, and then take the actual response power of the adjustable load on the day as another vector Y;

[0010] S3. Based on the historical data of X and Y, a Gaussian mixture model is established and the joint probability distribution of [X, Y] is fitted;

[0011] S4. For future responses, given X, the conditional distribution of Y can be obtained based on the joint distribution of [X, Y].

[0012] S5. For the conditional distribution of Y, the mean, standard deviation and quantiles of each probability are calculated as the probabilistic prediction model of the adjustable load response capacity.

[0013] Preferably, in step S1, the historical response data includes data of each time period in a day when the adjustable load makes response adjustment and power consumption of the previous day; the data of each time period includes: temperature of each time period, declared response power of each time period and actual response power of each time period.

[0014] Preferably, in step S2, the vector X=[E,T,P decl ]; where E represents the electricity consumption of the previous day; T = [T1, T2, ..., T i ,...,T n ] contains the temperature of each period, T i represents the temperature in the ith period; P decl =[P decl,1 ,P decl,2 ,...,P decl,i ,...,P decl,n ] includes the declared response power for each period, P decl,i represents the declared response power in the ith period;

[0015] The vector Y=P resp , where P resp =[P resp,1 ,P resp,2 ,...,P resp,i ,...,P resp,n ], including the actual response power of each time period, P resp,i represents the declared response power in the ith period;

[0016] Assuming there are n response periods, the dimension of vector x is 2n+1, and the vector y The dimension of is n.

[0017] Preferably, the S3 step includes the following specific steps:

[0018] S31. Use equations (1) to (4) to construct a basic Gaussian mixture model:

[0019]

[0020] ω k >0 (2),

[0021]

[0022] Where PDF x represents the probability density function when the random variable vector X takes X = x, N k (x; μ k ,Σ k ) represents the probability density function of the multidimensional normal distribution of the kth component, K and N represent the normal distribution component and dimension of the Gaussian mixture model respectively, ω k , μ k ,Σ k They represent the weight coefficient, mean vector and covariance matrix of the kth normal distribution component respectively. T represents the transpose of the matrix, |Σ k | represents the determinant of the kth covariance matrix;

[0023] S32. Use equations (5) to (7) to construct the probability density function of the joint probability distribution of [X, Y]:

[0024]

[0025]

[0026] Formula (5) represents the probability density function of the joint probability distribution of [X, Y] when X = x, Y = y; in formula (6), μ k is the mean vector of the kth normally distributed component of the joint probability distribution of [X,Y]; in formula (7), Σ k is the covariance matrix of the kth normally distributed component of the joint probability distribution of [X,Y], is the covariance matrix of the kth normally distributed component of vector y, is the covariance matrix of the kth normally distributed component of vector y and vector x, is the covariance matrix of the kth normally distributed component of vector x and vector y, is the covariance matrix of the kth normally distributed component of vector x;

[0027] S33. Based on the collected historical data, the expectation maximization algorithm is used to fit the parameters of the joint probability distribution of [X,Y]. [X,Y] is a new vector formed by merging vector x and vector y. The expectation step and the maximization step are iterated alternately.

[0028] Preferably, the step S33 is specifically as follows:

[0029] S331, initialize Gaussian mixture model parameters ω k ,μ k ,Σ k ;

[0030] S332. In the expectation step, using formula (8), according to the existing Gaussian mixture model parameters, the probability of each sample point belonging to each component is estimated:

[0031]

[0032] In formula (8), p ik Indicates the probability that each sample point belongs to each component, [x, y] i represents the i-th sample of the joint distribution [X,Y], ω k Represents the weight coefficient of the kth normal distribution component;

[0033] S333. In the maximization step, the parameters of the Gaussian mixture model are optimized using equations (9) to (13) to maximize the likelihood objective function:

[0034]

[0035] Formula (9) is the likelihood objective function, where M is the total number of samples; Formula (10) is the value of the objective function; Formula (11) is the update formula of the component weight coefficient; Formula (12) is the component mean update formula, where is to convert [x,y] i The vector formed by transposition; Formula (13) is the component covariance matrix update formula;

[0036] S334. Use formula (14) to construct the iterative convergence criterion:

[0037] |Obj-Obj′|<ε (14),

[0038] In formula (14), Obj, Obj′ are the likelihood objective function values ​​calculated this time and last time according to formula (10), respectively; ε is the convergence criterion. If formula (14) is satisfied, the iteration is terminated and the parameter value is output; otherwise, return to step S332 to continue the iteration.

[0039] Preferably, the S4 step comprises the following specific steps:

[0040] S41. Define the Gaussian mixture model of the joint distribution [Y, X] using equation (15):

[0041]

[0042] S42. Define the probability density function of the conditional Gaussian mixture model using formula (16):

[0043]

[0044] In formula (16), PDF(y|x) is the probability density function of the Gaussian mixture model of Y given X = x;

[0045] S43. Use formula (17) to construct a conditional Gaussian mixture model of the variable vector Y:

[0046]

[0047] In formula (17), ω k ′ is the weight coefficient of the kth normal distribution component of the constructed conditional Gaussian mixture model, is the mean vector of the kth normally distributed component of Y given X = x, It is the covariance matrix of the kth normally distributed component of Y given X=x.

[0048] Preferably, the specific steps of step S5 are as follows:

[0049] S51. Calculate the parameter ω′ of the conditional Gaussian mixture model constructed by equation (17) using equation (18)-equation (20): k ,

[0050]

[0051] S52. Use equations (21) to (23) to construct a linear combination of Y to represent the J-th dimension vector of Y:

[0052] Y J =c T Y (21),

[0053] c T =[c1,c2,...,c j ,...c n ] T (twenty two),

[0054]

[0055] Where Y J represents the J-th dimension variable of vector Y, c T is a coefficient vector;

[0056] S53, based on the Gaussian mixture model, the quantile solution algorithm of a certain probability is constructed by using equations (24)-(25) to solve the univariate nonlinear equation y J , we get Y J The probability α quantile Qt Y J |The value of α:

[0057]

[0058] QUR J |α=yJ (25),

[0059] In formula (24), CDF y J |x represents y J The cumulative distribution function under the condition of X = x; Φ represents the cumulative probability density function of the standard normal distribution; Formula (25) represents the quantile Qt Y J |The value of α is equal to y J The solution.

[0060] The beneficial effects of the present invention are:

[0061] (1) The present invention provides a probabilistic modeling method for adjustable load response capability based on a Gaussian mixture model. By introducing a conditional Gaussian mixture model and utilizing the multi-component characteristics of the Gaussian mixture model, the method can comprehensively characterize the diversity of load responses and generate different probability distributions under different conditions, thereby making the prediction of load response more flexible and adaptable.

[0062] (2) The adjustable load response capability probability modeling method based on the Gaussian mixture model of the present invention makes full use of historical load response data to fit the model parameters. By learning a large amount of data, it can effectively improve the accuracy of the load response probability model and form response capability quantiles corresponding to different probabilities, thereby providing more reliable information for the dispatch of the power system and helping the power grid dispatchers to rationally plan the load response, thereby improving the economy and safety of the power system operation. BRIEF DESCRIPTION OF THE DRAWINGS

[0063] Figure 1 is a flow chart of the method of the present invention;

[0064] Figure 2 It is a probability density function diagram of the Gaussian mixture model in the method of the present invention. DETAILED DESCRIPTION

[0065] The present invention is further described below in conjunction with the embodiments. It should be noted that these are merely examples and illustrations of the concept of the present invention. Those skilled in the art may make various modifications or additions to the described specific embodiments or replace them in a similar manner. As long as they do not deviate from the concept of the invention or exceed the scope defined by the claims, they should be deemed to fall within the protection scope of the present invention.

[0066] Embodiment 1:

[0067] In this embodiment, the method for probabilistic modeling of adjustable load response capability based on Gaussian mixture model is to use conditional Gaussian mixture model to model the probability distribution of adjustable load response capability, such as Figure 1 As shown, the following steps are included:

[0068] S1. Collect historical response data of adjustable loads.

[0069] The historical response data include the data of each time period during the day when the adjustable load responds and adjusts, as well as the power consumption of the previous day; and the data of each time period include: the temperature of each time period, the declared response power of each time period, and the actual response power of each time period.

[0070] S2. Select the power consumption of the previous day, the temperature of the day, and the response power reported on the day to form a vector X of input variables, and then use the actual response power of the adjustable load on the day as another vector Y.

[0071] Vector X = [E, T, P decl ], E represents the electricity consumption of the previous day; T=[T1,T2,...,T i ,...,T n ] contains the temperature of each period, T i represents the temperature in the ith period; P decl =[P decl,1 ,P decl,2 ,...,P decl,i ,...,P decl,n ] includes the declared response power for each period, P decl,i Represents the declared response power in the i-th time period.

[0072] Vector Y = P resp , where P resp =[P resp,1 ,P resp,2 ,...,P resp,i ,...,P resp,n ], including the actual response power of each time period, P resp,i Represents the declared response power in the i-th time period.

[0073] Assuming there are n response periods, the dimension of vector x is 2n+1, and the vector y The dimension of is n.

[0074] S3. Based on the historical data of X and Y, a Gaussian mixture model is established. Each component of the Gaussian mixture model is a normal distribution, and the probability density function is a convex combination of the probability density functions of all normal distribution components. For a three-component Gaussian mixture model, which contains three normal distribution components, when the weight coefficient of the component is set to [0.3, 0.6, 0.1], its probability density function shows a trend as follows: Figure 2 The Gaussian mixture model probability density function is shown, and the expectation maximization algorithm is used to fit the parameters of the joint probability distribution of [X, Y]. [X, Y] is a new vector formed by merging vector x and vector y. The specific steps are as follows:

[0075] S31. Use equations (1) to (4) to construct a basic Gaussian mixture model:

[0076]

[0077] ω k >0 (2),

[0078]

[0079] Where PDF x represents the probability density function when the random variable vector X takes X = x, N k (x; μ k ,Σ k ) represents the probability density function of the multidimensional normal distribution of the kth component, K and N represent the normal distribution component and dimension of the Gaussian mixture model respectively, ω k , μ k ,Σ k Respectively represent the weight coefficient, mean vector and covariance matrix of the kth normal distribution component, T represents the transpose of the matrix, |Σ k | represents the determinant of the kth covariance matrix.

[0080] Formula (2) and formula (3) represent each normal distribution component ω k The weight coefficients are non-negative and sum to 1.

[0081] S32. Use equations (5) to (7) to construct the probability density function of the joint probability distribution of [X, Y]:

[0082]

[0083] Formula (5) represents the probability density function of the joint probability distribution of [X, Y] when X = x, Y = y; in formula (6), μ k is the mean vector of the kth normally distributed component of the joint probability distribution of [X,Y]; in formula (7), Σ k is the covariance matrix of the kth normally distributed component of the joint probability distribution of [X,Y], is the covariance matrix of the kth normally distributed component of vector y, is the covariance matrix of the kth normally distributed component of vector y and vector x, is the covariance matrix of the kth normally distributed component of vector x and vector y, is the covariance matrix of the kth normally distributed component of vector x.

[0084] S33. Based on the collected historical data, the expectation maximization algorithm is used to fit the parameters of the joint probability distribution of [X, Y]. [X, Y] is a new vector formed by merging vector x and vector y. The expectation step and the maximization step are iterated alternately, specifically including the following:

[0085] S331, initialize Gaussian mixture model parameters ω k ,μ k ,Σ k .

[0086] S332. In the expectation step, using formula (8), according to the existing Gaussian mixture model parameters, the probability of each sample point belonging to each component is estimated:

[0087]

[0088] In formula (8), p ik Indicates the probability that each sample point belongs to each component, [x, y] i represents the i-th sample of the joint distribution [X,Y], ω k Represents the weight coefficient of the kth normal distribution component.

[0089] S333. In the maximization step, the parameters of the Gaussian mixture model are optimized using equations (9) to (13) to maximize the likelihood objective function:

[0090]

[0091] Formula (9) is the likelihood objective function, where M is the total number of samples. This formula means that in a certain group ω k , μ k ,Σ k When the value of , the value of this expression reaches the maximum; Formula (10) is the value of the objective function; Formula (11) is the update formula of the component weight coefficient; Formula (12) is the update formula of the component mean, where is to convert [x,y] i The vector formed by transposition; Formula (13) is the update formula of the component covariance matrix. Update the parameter ω through formula (10)-formula (13) k , μ k ,Σ k The value of can make the likelihood objective function iterate towards the maximization direction.

[0092] S334. Use formula (14) to construct the iterative convergence criterion:

[0093] |Obj-Obj′|<ε (14),

[0094] In formula (14), Obj, Obj′ are the likelihood objective function values ​​calculated this time and last time according to formula (10), respectively; ε is the convergence criterion. If formula (14) is satisfied, the iteration is terminated and the parameter value is output; otherwise, return to step S332 to continue the iteration.

[0095] S4. For future responses, given X, the conditional distribution of Y can be obtained based on the joint distribution of [X, Y]. The specific steps are as follows:

[0096] S41. Define the Gaussian mixture model of the joint distribution [Y, X] using equation (15):

[0097]

[0098] In formula (15), for the Gaussian mixture model of the joint distribution [Y, X], it is necessary to rearrange the component mean vector μ according to the order of variables: k and the component covariance matrix Σ k , for example, for the vector X constructed in S2 = [E, T, P decl ] and Y=P resp , should be based on the variable P resp ,E,T,P decl The component mean vector μ is composed of their respective means. k and the component covariance matrix Σ consisting of their respective covariance matrices k .

[0099] S42. Define the probability density function of the conditional Gaussian mixture model using formula (16):

[0100]

[0101] In formula (16), PDF(y|x) is the probability density function of the Gaussian mixture model of Y given X=x.

[0102] S43. Use formula (17) to construct a conditional Gaussian mixture model of the variable vector Y:

[0103]

[0104] In formula (17), ω′ k is the weight coefficient of the kth normal distribution component of the constructed conditional Gaussian mixture model, is the mean vector of the kth normally distributed component of Y given X = x, It is the covariance matrix of the kth normally distributed component of Y given X=x.

[0105] S5. For the conditional distribution of Y, obtain the component weight coefficient, mean, and standard deviation of the conditional Gaussian mixture model, and calculate the corresponding quantile for the required predicted probability as the probability prediction model for the adjustable load response capacity. The specific steps are as follows:

[0106] S51. Calculate the parameter ω′ of the conditional Gaussian mixture model constructed by equation (17) using equation (18)-equation (20): k ,

[0107]

[0108] S52. Use equations (21) to (23) to construct a linear combination of Y to represent the J-th dimension vector of Y:

[0109] Y J =c T Y (21),

[0110] c T =[c1,c2,...,c j ,...c n ] T (twenty two),

[0111]

[0112] Where Y J represents the J-th dimension variable of vector Y, c T is a coefficient vector.

[0113] S53, based on the Gaussian mixture model, the quantile solution algorithm of a certain probability is constructed by using equations (24)-(25) to solve the univariate nonlinear equation y J , we get Y J The probability α quantile Qt Y J |The value of α:

[0114]

[0115] QUR J |α=y J (25),

[0116] In formula (24), CDF y J |x represents y J The cumulative distribution function under the condition of X = x; Φ represents the cumulative probability density function of the standard normal distribution; Formula (25) represents the quantile Qt Y J |The value of α is equal to y J The solution.

[0117] The present invention is based on the probabilistic modeling method of adjustable load response capability of Gaussian mixture model. By introducing the conditional Gaussian mixture model and utilizing the multi-component characteristics of the Gaussian mixture model, the diversity of load responses can be comprehensively characterized and different probability distributions can be generated under different conditions, making the prediction of load response more flexible and adaptable.

[0118] The adjustable load response capability probability modeling method of the present invention is based on the Gaussian mixture model. It makes full use of historical load response data to fit the model parameters. By learning a large amount of data, it can effectively improve the accuracy of the load response probability model and form response capability quantiles corresponding to different probabilities, thereby providing more reliable information for the dispatch of the power system and helping the power grid dispatcher to reasonably plan the load response, thereby improving the economy and safety of the operation of the power system.

[0119] The above is an exemplary description of the invention. Obviously, the specific implementation of the present invention is not limited to the above-mentioned method. As long as such non-substantial improvements are made by adopting the method concept and technical solution of the present invention, or the concept and technical solution of the invention are directly applied to other occasions without improvement, they are all within the protection scope of the present invention.

Claims

1. A probabilistic modeling method for adjustable load response capability based on a Gaussian mixture model, characterized in that: The following steps are involved: S1. Collect historical response data of adjustable loads; S2, select the power consumption of the previous day, the temperature of the day, and the response power reported on the day to form a vector X of input variables, and then take the actual response power of the adjustable load on the day as another vector Y; S3. Based on the historical data of X and Y, a Gaussian mixture model is established and the joint probability distribution of [X, Y] is fitted; S4. For future responses, given X, the conditional distribution of Y can be obtained based on the joint distribution of [X, Y]. S5. For the conditional distribution of Y, the mean, standard deviation and quantiles of each probability are calculated as the probabilistic prediction model of the adjustable load response capacity.

2. The method for probabilistic modeling of adjustable load response capability based on Gaussian mixture model according to claim 1, characterized in that: In step S1, the historical response data includes data of each time period during a day when the adjustable load makes a response adjustment and the power consumption of the previous day; The data for each time period include: the temperature for each time period, the declared response power for each time period and the actual response power for each time period.

3. The method for probabilistic modeling of adjustable load response capability based on Gaussian mixture model according to claim 1, characterized in that: In step S2, the vector X=[E,T,P decl ]; where E represents the electricity consumption of the previous day; T = [T1, T2, ..., T i ,...,T n ] contains the temperature of each period, T i represents the temperature in the ith period; P decl =[P decl,1 ,P decl,2 ,...,P decl,i ,...,P decl,n ] includes the declared response power for each period, P decl,i represents the declared response power in the ith period; The vector Y=P resp , where P resp =[P resp,1 ,P resp,2 ,...,P resp,i ,...,P resp,n ], including the actual response power of each time period, P resp,i represents the declared response power in the ith period; Assuming there are n response periods, the dimension of vector x is 2n+1, and the dimension of vector y is n.

4. The method for probabilistic modeling of adjustable load response capability based on Gaussian mixture model according to claim 1, characterized in that: The specific steps of step S3 are as follows: S31. Use equations (1) to (4) to construct a basic Gaussian mixture model: oh k >0 (2), Where PDF x represents the probability density function when the random variable vector X takes X = x, N k (x; μ k ,Σ k ) represents the probability density function of the multidimensional normal distribution of the kth component, K and N represent the normal distribution component and dimension of the Gaussian mixture model respectively, ω k , μ k ,Σ k Respectively represent the weight coefficient, mean vector and covariance matrix of the kth normal distribution component, T represents the transpose of the matrix, |Σ k | represents the determinant of the kth covariance matrix; S32. Use equations (5) to (7) to construct the probability density function of the joint probability distribution of [X, Y]: Formula (5) represents the probability density function of the joint probability distribution of [X, Y] when X = x, Y = y; in formula (6), μ k is the mean vector of the kth normally distributed component of the joint probability distribution of [X, Y]; in formula (7), Σ k is the covariance matrix of the kth normally distributed component of the joint probability distribution of [X,Y], is the covariance matrix of the kth normally distributed component of vector y, is the covariance matrix of the kth normally distributed component of vector y and vector x, is the covariance matrix of the kth normally distributed component of vector x and vector y, is the covariance matrix of the kth normally distributed component of vector x; S33. Based on the collected historical data, the expectation maximization algorithm is used to fit the parameters of the joint probability distribution of [X,Y]. [X,Y] is a new vector formed by merging vector x and vector y. The expectation step and the maximization step are iterated alternately.

5. The method for probabilistic modeling of adjustable load response capability based on Gaussian mixture model according to claim 4, characterized in that: The S33 step is specifically as follows: S331, initialize Gaussian mixture model parameters ω k ,μ k ,Σ k ; S332. In the expectation step, using formula (8), according to the existing Gaussian mixture model parameters, the probability of each sample point belonging to each component is estimated: In formula (8), p ik Indicates the probability that each sample point belongs to each component, [x, y] i represents the i-th sample of the joint distribution [X,Y], ω k Represents the weight coefficient of the kth normal distribution component; S333. In the maximization step, the parameters of the Gaussian mixture model are optimized using equations (9) to (13) to maximize the likelihood objective function: Formula (9) is the likelihood objective function, where M is the total number of samples; Formula (10) is the value of the objective function; Formula (11) is the update formula of the component weight coefficient; Formula (12) is the component mean update formula, where is to convert [x,y] i The vector formed by transposition; Formula (13) is the component covariance matrix update formula; S334. Use formula (14) to construct the iterative convergence criterion: |Obj-Obj′|<ε (14), In formula (14), Obj, Obj′ are the likelihood objective function values ​​calculated this time and last time according to formula (10), respectively; ε is the convergence criterion. If formula (14) is satisfied, the iteration is terminated and the parameter value is output; otherwise, return to step S332 to continue the iteration.

6. The method for probabilistic modeling of adjustable load response capability based on Gaussian mixture model according to claim 5, characterized in that: The specific steps of step S4 are as follows: S41. Define the Gaussian mixture model of the joint distribution [Y, X] using equation (15): S42. Define the probability density function of the conditional Gaussian mixture model using formula (16): In formula (16), PDF(y|x) is the probability density function of the Gaussian mixture model of Y given X = x; S43. Use formula (17) to construct a conditional Gaussian mixture model of the variable vector Y: In formula (17), ω k ′ is the weight coefficient of the kth normal distribution component of the constructed conditional Gaussian mixture model, is the mean vector of the kth normally distributed component of Y given X = x, It is the covariance matrix of the kth normally distributed component of Y given X=x.

7. The method for probabilistic modeling of adjustable load response capability based on Gaussian mixture model according to claim 6, characterized in that: The specific steps of step S5 are as follows: S51. Calculate the parameter ω′ of the conditional Gaussian mixture model constructed by equation (17) using equation (18)-equation (20): k , S52. Use equations (21) to (23) to construct a linear combination of Y to represent the J-th dimension vector of Y: AND J =c T And (21), c T =[c1,c2,...,c j ,...c n ] T (22), Where Y J represents the J-th dimension variable of vector Y, c T is a coefficient vector; S53, based on the Gaussian mixture model, the quantile solution algorithm of a certain probability is constructed by using equations (24)-(25) to solve the univariate nonlinear equation y J , we get Y J The probability α quantile Qt Y J |The value of α: Qt Y J |α=y J (25), In formula (24), CDF y J |x represents y J The cumulative distribution function under the condition of X = x; Φ represents the cumulative probability density function of the standard normal distribution; Formula (25) represents the quantile Qt Y J |The value of α is equal to y J The solution.

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