A Risk Analysis Method for Unbalanced Distribution Networks Based on Multi-Level Sparse Polynomials

Abstract

This invention discloses a risk analysis method for unbalanced distribution networks based on multi-level sparse polynomials, belonging to the technical field of distribution network risk analysis. The method includes: establishing a distribution network input-output model; approximating the distribution network input-output model as a linear regression model; calculating the ANCOVA exponent and p-value of the linear regression model to sparsify the distribution network input-output model, obtaining a preliminary sparsified distribution network input-output model; suppressing higher-order cross terms of the preliminary sparsified distribution network input-output model according to a preset truncation rule; shrinking the coefficients of the preliminary sparsified distribution network input-output model according to optimized polynomial chaotic expansion coefficients; using a preset residual Akaike information content criterion as the model balance criterion to obtain a final sparsified distribution network input-output model; and obtaining the distribution network risk analysis results based on the final sparsified distribution network input-output model. This invention can provide quantitative basis for the operation and control of distribution networks.

Description

Technical Field

[0001] This invention relates to the field of distribution network risk analysis technology, and in particular to a method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials. Background Technology

[0002] Due to the volatility and randomness of distributed power sources (such as wind power and photovoltaics), the safe operation and control of the distribution network are posed significant challenges. The integration of these distributed power sources into the distribution network makes it difficult for traditional distribution network risk analysis methods to handle uncertainties and calculate power flow.

[0003] Traditional distribution network risk analysis is mostly based on the probabilistic power flow (PLF) method, which quantifies uncertainties. However, this method requires a large amount of sample data to ensure the accuracy of the calculation. Although the probabilistic power flow method can provide the results of uncertainty quantification, its computational efficiency is low because each quantitative analysis requires complex power flow calculations, and the computational cost is huge when the sample size is large. Summary of the Invention

[0004] The purpose of this invention is to overcome the shortcomings of the prior art and provide a risk analysis method for unbalanced distribution networks based on multi-level sparse polynomials. This method can identify and remove polynomial terms that have no significant impact on the output, thereby optimizing the model, improving the reliability of distribution network risk analysis results, and providing quantitative basis for the operation and control of distribution networks.

[0005] To achieve the above objectives, the present invention is implemented using the following technical solution:

[0006] This invention provides a risk analysis method for unbalanced distribution networks based on multi-level sparse polynomials, including:

[0007] A power distribution network input-output model is established using polynomial chaos expansion theory.

[0008] Solve the polynomial chaotic expansion coefficients of the power distribution network input-output model to obtain the power distribution network input-output data pairs. Approximate the power distribution network input-output model as a linear regression model. Calculate the regression coefficients of the linear regression model based on the power distribution network input-output data pairs.

[0009] Calculate the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model. Based on the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model, calculate the ANCOVA index and p-value of the linear regression model.

[0010] Based on the ANCOVA index and P-value, a preliminary sparse distribution network input-output model is obtained.

[0011] The higher-order cross terms of the preliminary sparse distribution network input-output model are suppressed according to the preset truncation rule. The coefficients of the preliminary sparse distribution network input-output model are shrunk according to the optimized polynomial chaotic expansion coefficients. The preset residual Akaike information content criterion is used as the model balance criterion to obtain the final sparse distribution network input-output model.

[0012] Based on the final sparse distribution network input-output model, the node voltage random quantities are defined, and the distribution network risk analysis indicators are calculated based on the node voltage random quantities to obtain the distribution network risk analysis results.

[0013] Optionally, the power distribution network input-output model is represented as follows:

[0014] ;

[0015] in, Represents the input variables of the distribution network The corresponding output response; Indicates the multi-order subscript of a polynomial The expansion coefficients; Represents the input variables of the distribution network Corresponding polynomial multi-order subscript basis functions; A set representing multiple levels of subscripts; , , These represent the zero-order undetermined coefficients, the first-order undetermined coefficients of the i-th distribution network input variable, and the second-order undetermined coefficients of the interaction terms of the i-th and j-th distribution network input variables, respectively. Indicates the dimension of the input variables of the distribution network; , Represent the i-th distribution network input variables respectively First-order Hermite orthogonal polynomial, i-th distribution network input variable and the j-th distribution network input variable The second-order Hermite orthogonal polynomial; Indicates the number of input variables in the distribution network; Let i represent the i-th distribution network input variable, j-th distribution network input variable, ..., j-th distribution network input variable. Undetermined coefficients of multi-order interaction terms of input variables of a power distribution network; Represents the i-th distribution network input variable The j-th distribution network input variable ,…,No. Individual power distribution network input variables The multi-order Hermite orthogonal polynomial.

[0016] Optionally, the polynomial chaotic expansion coefficients are expressed as:

[0017] ;

[0018] ;

[0019] in, Represents the coefficients of the polynomial chaotic expansion; Represents the collocation point polynomial matrix; Indicates the output response of the distribution network; Indicates matrix transpose; , , These represent the first distribution network input variable in the first column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions; , , These represent the first distribution network input variable in the second column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions; , , These represent the first distribution network input variable in the (P-1)th column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions.

[0020] Optionally, the power distribution network input-output model is approximated as a linear regression model. Based on the power distribution network input-output data pairs, the regression coefficients of the linear regression model are calculated, including:

[0021] The linear regression model is expressed as:

[0022] ;

[0023] Set the input and output data pairs of the distribution network Construct as a design matrix According to the design matrix The regression coefficients of the linear regression model are obtained and expressed as follows:

[0024] ;

[0025] in, Indicates the output response of the distribution network; , , These represent the intercept term, the regression coefficient of the i-th distribution network input variable, and the regression coefficient of the interaction term between the i-th and j-th distribution network input variables, respectively. , , Let i, j, and j represent the i-th distribution network input variable, respectively. One power distribution network input variable; Represents the residuals of a linear regression model; Indicates the dimension of the input variables of the distribution network; This represents the k-th power distribution network input variable; This represents the output response corresponding to the k-th power distribution network input variable; Indicates the number of input variables in the distribution network; Represents the regression coefficients of a linear regression model; Let i represent the i-th distribution network input variable, j-th distribution network input variable, ..., j-th distribution network input variable. The regression coefficients of the interaction terms of the input variables of the power distribution network.

[0026] Optionally, the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model can be calculated using the following formula:

[0027] ;

[0028] ;

[0029] ;

[0030] in, , , Let Si represent the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model, respectively. , , These represent the output response corresponding to the k-th distribution network input variable, the average output of the linear regression model, and the output response of the k-th distribution network predicted by the linear regression model, respectively. This indicates the number of input variables in the power distribution network.

[0031] Optionally, based on the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model, calculate the ANCOVA index and p-value of the linear regression model, using the following formula:

[0032] ;

[0033] ;

[0034] ;

[0035] ;

[0036] in, This represents the sum of squares of the regressions in a linear regression model that includes only the j-th power distribution network input variable. This represents the sum of squares of the regression in a linear regression model that includes all input variables from the power distribution network. This represents the sum of squares of the regression in the linear regression model, excluding the j-th distribution network input variable. Represents the ANCOVA exponent of the j-th distribution network input variable; This represents the sum of squares in a linear regression model; The F-statistic represents the j-th input variable of the power distribution network; This represents the mean square error associated with the j-th distribution network input variable; This represents the mean square error of the input variables in the distribution network. This represents the degrees of freedom of the j-th power distribution network input variable; This represents the sum of squared residuals in a linear regression model. Indicates the number of input variables in the distribution network; This indicates the number of degrees of freedom in a linear regression model; This represents the P-value corresponding to the F-statistic of the j-th power distribution network input variable; The F-statistic represents the input variables of the power distribution network; This represents a probability function.

[0037] Optionally, based on the ANCOVA index and P-value, a preliminary sparse distribution network input-output model is obtained, including:

[0038] By removing distribution network input variables whose ANCOVA exponent is less than the first threshold or whose P value is less than the second threshold from the distribution network input-output model, a preliminary sparsed distribution network input-output model is obtained.

[0039] Optionally, the preset truncation rule is expressed as:

[0040] ;

[0041] in, Represents a set of d-dimensional non-negative integer vectors; Multinomial subscripts representing input variables of the power distribution network; The multinomial subscript represents the i-th power distribution network input variable. Indicates hyperparameters; Indicates the upper limit of the given total order;

[0042] Higher-order cross terms in the preliminary sparse distribution network input-output model are suppressed according to a preset truncation rule, including:

[0043] According to the preset truncation rules The polynomial multi-order subscripts in the initial sparse distribution network input-output model are retained. The sum is not greater than the given upper limit of total order. All higher-order cross terms, or remove non-zero polynomial multi-order subscripts from the initial sparsified distribution network input-output model. and polynomial multi-order subscripts Higher-order cross terms that are greater than the preset order value.

[0044] Optionally, the optimized polynomial chaotic expansion coefficients are expressed as:

[0045] ;

[0046] The preset residual Akaike information content criterion is expressed as follows:

[0047] ;

[0048] in, Indicates the number of input variables in the distribution network; Indicates the output response of the distribution network; Represents the collocation point polynomial matrix; Represents the coefficients of the polynomial chaotic expansion; This represents the regularization parameter for Lasso regression; Indicates the adaptive LASSO regression weights; Indicates the multi-order subscript of a polynomial The expansion coefficients; The second norm of a vector; The polynomial chaotic expansion coefficients that minimize the expression within the parentheses. The value; This represents the number of non-zero polynomial chaotic expansion coefficients;

[0049] Based on the optimized polynomial chaotic expansion coefficients, the coefficients of the initial sparse distribution network input-output model are shrunk. The preset residual Akaike information content criterion is used as the model balance criterion to obtain the final sparse distribution network input-output model, including:

[0050] Optimize the polynomial chaos expansion coefficients Items with a coefficient less than the preset value will be deleted;

[0051] Minimize the preset residual Akaike information content criterion This yields the final sparsed power distribution network input-output model that achieves the best balance between model complexity and goodness of fit.

[0052] Optionally, the formula for the node voltage random quantity is:

[0053] ;

[0054] in, express node The phase voltage is a random quantity; Represents a set of multi-level subscripts after sparsification; Indicates the multi-order subscript of a polynomial The expansion coefficients; Represents the input variables of the distribution network Corresponding polynomial multi-order subscript basis functions;

[0055] Based on the random quantities of node voltage, risk analysis indicators for three-phase unbalanced distribution networks are calculated, and the results of distribution network risk analysis are obtained, including:

[0056] Risk analysis indicators for calculating the probability of node phase voltage exceeding limits, the severity of node phase voltage, and the node voltage imbalance based on the random quantity of node voltage are given by:

[0057] ;

[0058] ;

[0059] ;

[0060] in, , They represent node The probability of phase voltage exceeding the upper limit node The probability of phase voltage exceeding the lower limit; express node The phase voltage is a random quantity; , They represent node Phase upper limit, node Phase lower limit; Indicates the probability of occurrence; , express node The severity of phase voltage exceeding the upper limit node Severity of phase voltage exceeding the lower limit; Indicates the portion exceeding the limit; Represents the expectation operator; express node Phase voltage imbalance risk analysis indicators; express The average value of the three-phase voltage at the node; Indicates the voltage imbalance threshold;

[0061] The system-level voltage risk aggregation analysis index is calculated based on the node-phase voltage exceedance probability and the node-phase voltage severity, using the following formula:

[0062] ;

[0063] ;

[0064] in, express node Phase voltage risk; This represents the aggregated analysis index of system-level voltage risk; Indicates the weighting coefficient; This represents the voltage risk vector for all phase nodes; , Let L1 norm and L∞ norm be represented respectively;

[0065] The formula for calculating system-level power flow risk analysis indicators is as follows:

[0066] ;

[0067] ;

[0068] in, express branch road The trend risk of phase, express A branch formed by two nodes; express branch road The trend of the times; express branch road The upper limit of the current trend; Indicates system-level power flow risk analysis indicators; This represents the power flow risk vector for all branches and phases;

[0069] Based on the node voltage imbalance risk analysis index, the system-level voltage risk aggregation analysis index, and the system-level power flow risk analysis index, the distribution network risk analysis results are obtained.

[0070] Compared with the prior art, the beneficial effects achieved by the present invention are as follows:

[0071] This invention combines ANCOVA exponential sensitivity analysis with a truncated, coefficient- and criterion-based hierarchical sparsified distribution network input-output model. It effectively solves the problem of characterizing and quantifying power flow uncertainty in distribution networks with a high proportion of renewable energy. It can identify and remove polynomial terms that have no significant impact on the output, thereby optimizing the model. Using the sparsified model, it analyzes the voltage random quantities of the coupled distribution network, ultimately quantifying the risk analysis indicators of the distribution network. It can quantitatively analyze the operational risks of the distribution network from multiple dimensions and obtain the overall risk level through system-level aggregation methods, improving the reliability of risk analysis results for three-phase unbalanced distribution networks and providing a quantitative basis for the operation and control of distribution networks. Attached Figure Description

[0072] Figure 1 The diagram shown is a flowchart of one embodiment of the unbalanced distribution network risk analysis method based on multi-level sparse polynomials of the present invention.

[0073] Figure 2 The figure shows the voltage PDF and CDF curves of node 33 in the improved IEEE-118 node power distribution system under the influence of uncertain factors.

[0074] Figure 3 The figure shows the active power PDF and CDF curves of the branch under the influence of uncertain factors in the improved IEEE-118 node distribution system of this invention. Detailed Implementation

[0075] The technical solution of the present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. It should be understood that the embodiments of the present invention and the specific features in the embodiments are detailed descriptions of the technical solution of the present invention, rather than limitations thereof. In the absence of conflict, the embodiments of the present invention and the technical features in the embodiments can be combined with each other.

[0076] The term "and / or" simply describes the relationship between related objects, indicating that three relationships can exist. For example, A and / or B can represent: A alone, A and B simultaneously, or B alone. Additionally, the character " / " generally indicates that the preceding and following related objects have an "or" relationship.

[0077] Example 1

[0078] like Figure 1 As shown in the figure, this embodiment introduces a risk analysis method for unbalanced distribution networks based on multi-level sparse polynomials, including:

[0079] Step 1: Establish the input and output model of the distribution network, specifically as follows:

[0080] Based on the theory of polynomial chaos expansions (PCE), the input and output relationships of the distribution network are modeled. The output responses of the distribution network, such as node voltages, are expanded, and the relationships between input variables, such as loads, and these output responses are established, i.e.:

[0081] ;

[0082] in, Represents the input variables of the distribution network The corresponding output response specifically refers to key operating parameters of the distribution network, such as node voltages or branch power flows. It relates to uncertain input variables of the distribution network. The function; Indicates the multi-order subscript of a polynomial The expansion coefficients are obtained by linear regression methods such as least squares; Indicates multi-level subscripts, where Input variables for the distribution network dimensionality For uncertain input variables of distribution networks that follow a standard normal distribution, such as distributed photovoltaic power output and load fluctuations; Represents the input variables of the distribution network Corresponding polynomial multi-order subscript basis functions; A set representing multiple levels of subscripts;

[0083] The basis functions are orthogonal, that is:

[0084] ;

[0085] in, Let it be the expected function; Represents the input variables of the distribution network Corresponding polynomial multi-order subscript basis functions; For Kroneck symbol, when The value is 1 if it is true, and 0 otherwise.

[0086] For distribution network input variables that follow a standard normal distribution The original random variable can be obtained in the following ways The result of the conversion is:

[0087] ;

[0088] in, For the original first in the distribution network Each power distribution network input variable (such as photovoltaic output, load, etc.). For variables The cumulative distribution function; It is the inverse function of the cumulative distribution function of the standard normal distribution; For the first Each power distribution network input variable.

[0089] Distribution network output response Regarding input variables of the distribution network The polynomial can be expressed as:

[0090] ;

[0091] in, , , These represent the zero-order undetermined coefficients, the first-order undetermined coefficients of the i-th distribution network input, and the second-order interaction term undetermined coefficients of the i-th and j-th distribution network input variables, respectively. Indicates the dimension of the input variables of the distribution network; , Represent the i-th distribution network input variables respectively First-order Hermite orthogonal polynomial, i-th distribution network input variable and the j-th distribution network input variable The second-order Hermite orthogonal polynomial; Let i represent the i-th distribution network input variable, j-th distribution network input variable, ..., j-th distribution network input variable. Undetermined coefficients of multi-order interaction terms of input variables of a power distribution network; Represents the i-th distribution network input variable The j-th distribution network input variable ,…,No. Individual power distribution network input variables The multi-order Hermite orthogonal polynomials; where the omitted elements are combinations of higher orders.

[0092] Locus of points polynomial matrix Represented as:

[0093] ;

[0094] in, , , These represent the first distribution network input variable in the first column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions; , , These represent the first distribution network input variable in the second column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions; , , These represent the first distribution network input variable in the (P-1)th column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions.

[0095] Solving for polynomial chaotic expansion coefficients using the least squares method , is represented as:

[0096] ;

[0097] in, Indicates the output response of the distribution network; This indicates the matrix transpose.

[0098] The Latin hypercube sampling method is used to generate Input variables of power distribution network ,in .

[0099] Input variables for the k-th distribution network Running the power distribution network input-output model To obtain the corresponding distribution network output response A set of power distribution network input and output data pairs was obtained. .

[0100] Step 2: Approximate the power distribution network input-output model as a linear regression model, specifically:

[0101] Sensitivity analysis was performed using the Analysis of Covariance (ANCOVA) index to determine the output response of the distribution network. This can be achieved through an input variable related to the distribution network. We can approximate this with a linear regression model, which can include main effects (the influence of individual variables) and interaction effects (the influence of combinations of variables). The approximate linear regression model is expressed as:

[0102] ;

[0103] in, Indicates the output response of the distribution network; , , These represent the intercept term, the regression coefficient of the i-th distribution network input variable, and the regression coefficient of the interaction term between the i-th and j-th distribution network input variables, respectively. , , Let i, j, and j represent the i-th distribution network input variable, respectively. One power distribution network input variable; Represents the residuals of a linear regression model; Let i represent the i-th distribution network input variable, j-th distribution network input variable, ..., j-th distribution network input variable. The regression coefficients of the interaction terms of the power distribution network input variables; where the omitted terms are higher-order combinations.

[0104] The least squares method is used to fit a linear regression model to calculate the regression coefficients. The input and output data of the distribution network are compared. Construct as a design matrix Design matrix Each row corresponds to a power distribution network input variable, such as voltage, or an interaction term between power distribution network input variables. These interaction terms are used to capture the mutual influence between input variables. Each column corresponds to the category of input variables in each row. For example, if the first number in each row is voltage, then that column represents voltage. In this case, the linear regression model can be transformed into... The regression coefficients of the linear regression model are expressed as:

[0105] .

[0106] Step 3: Calculate the ANCOVA index and p-value of the linear regression model, specifically:

[0107] The next step is to calculate the ANCOVA index, which focuses on the contribution of each regression term (input variable or interaction term) to the variance of the linear regression model output, as well as the statistical significance of these contributions.

[0108] Variance decomposition (based on regression): The total sum of squares (SST) of a linear regression model can be decomposed into the regression sum of squares (SSR) and the residual sum of squares (SSE), and all sub-terms are orthogonal.

[0109] Sum of squares in a linear regression model Represented as:

[0110] ;

[0111] Sum of squared residuals of a linear regression model Represented as:

[0112] ;

[0113] Sum of squares in linear regression model Represented as:

[0114] ;

[0115] in, , , These represent the output response corresponding to the k-th distribution network input variable, the average output of the linear regression model, and the output response of the k-th distribution network predicted by the linear regression model, respectively. This indicates the number of input variables in the power distribution network.

[0116] The relationship between the total sum of squares, the regression sum of squares, and the residual sum of squares in a linear regression model is as follows:

[0117] ;

[0118] The statistic measures the proportion of variance explained by a linear regression model:

[0119] ;

[0120] ANCOVA Variance Contribution (Partial Sum of Squares): The core of ANCOVA lies in calculating the partial sum of squares, which quantifies the additional contribution of a specific variable or interaction term to the total variance after considering the effects of other variables. This is typically done by comparing the regression sums of squares of models that include and do not include the specific term. For the j-th distribution network input variable... (As part of the main effect or interaction term), its unique variance contribution after considering all other variables can be expressed as the regression sum of squares of a linear regression model that only includes the j-th distribution network input variable. ,Right now:

[0121] ;

[0122] in, This represents the sum of squares of the regression in a linear regression model that includes all input variables from the power distribution network. This represents the sum of squares of the regression in the linear regression model, excluding the j-th distribution network input variable.

[0123] The ANCOVA index can be defined as each input variable of the distribution network or its interaction term. The proportion of the total sum of squares, the ANCOVA index of the j-th distribution network input variable. Represented as:

[0124] ;

[0125] F-statistic of input variables through the distribution network The F-statistic of the power distribution network input variables is analyzed using p-values ​​to assess the statistical significance of each regression term. It measures whether the variance explained by a particular term is significantly greater than the random error. The F-statistic, representing the j-th input variable of the distribution network, is used to analyze the statistical significance of the j-th input variable of the distribution network. It is expressed as:

[0126] ;

[0127] in, This represents the mean square error associated with the j-th distribution network input variable; This represents the mean square error of the input variables in the distribution network. This represents the degrees of freedom of the j-th power distribution network input variable; This indicates the number of degrees of freedom in a linear regression model;

[0128] Calculate the F-statistic corresponding to the j-th distribution network input variable. value The p-value is used to analyze the statistical significance of the regression term. It represents the probability that, if the null hypothesis is true, the current sample result or a more extreme result will be obtained. value Represented as:

[0129] ;

[0130] Among them, the F-statistic of the input variables of the distribution network Used to measure whether the variance explained by a certain item is significantly greater than the random error; This represents a probability function.

[0131] Step 4: Preliminary sparsification of the power distribution network input-output model, specifically:

[0132] For sparsification of the distribution network input-output model, the statistically significant distribution network input variables and their interaction terms are identified based on the p-value (usually less than 0.05 or 0.01). The ANCOVA index provides the proportion of variance explained by each significant variable. Therefore, terms with insignificant p-values ​​or very small ANCOVA indices are removed from the linear regression model, i.e., removed from the distribution network input-output model, achieving sparsification. This effectively removes uncertain input variables or their interaction terms that have little impact on the distribution network's operating state, thereby optimizing the distribution network input-output model and obtaining a preliminary sparsified distribution network input-output model.

[0133] Step 5: Secondary sparse distribution network input-output model, specifically:

[0134] Hyperbolic truncation and truncation, coefficient, and criterion hierarchical sparsification methods are introduced to perform second-level sparsification on the undetermined coefficients of the preliminary sparsified distribution network input-output model. If the full-order exponent set is used directly, the number of polynomials in the preliminary sparsified distribution network input-output model will explode, which will lead to ill-conditioned regression and overfitting. Therefore, a two-layer combined sparsification is adopted, consisting of an outer hyperbolic hyperellipsoid and an inner adaptive coefficient and criterion hierarchical sparsification.

[0135] First, in the outer layer, higher-order cross terms in the preliminary sparsed distribution network input-output model are suppressed according to a preset truncation rule. Represented as:

[0136] ;

[0137] in, Represents a set of d-dimensional non-negative integer vectors; Represents the multi-order subscript of the polynomial for the i-th distribution network input; Indicates hyperparameters, ; Indicates the upper limit of the total order. ;

[0138] when When, the cutoff condition becomes This is the classic total order truncation, which means only retaining the multi-order subscripts in the initial sparsified distribution network input-output model. The sum is not greater than the given upper limit of total order. All higher-order cross terms.

[0139] when At this time, it forms a truncated "hyperellipsoid" shape, which, for higher-order cross terms, means deleting the non-zero polynomial multi-order subscripts in the initial sparsified distribution network input-output model. and polynomial multi-order subscripts Higher-order cross terms with a preset order value are penalized more severely for terms with non-zero multi-order subscripts or larger multi-order subscripts, making them easier to exclude.

[0140] In the inner layer, the coefficients of the initial sparse distribution network input-output model are shrunk based on the optimized polynomial chaotic expansion coefficients. Represented as:

[0141] ;

[0142] in, This represents the regularization parameter for Lasso regression; Indicates the adaptive LASSO regression weights; The second norm of a vector; The polynomial chaotic expansion coefficients that minimize the expression within the parentheses. The value; , Multinomial subscripts for ordinary least squares estimation The expansion coefficients are optimized to implement adaptive LASSO, prioritizing the retention of large coefficients and easily shrinking small coefficients to zero.

[0143] Optimized polynomial chaotic expansion coefficients There will be two constraints: the regularization parameter for Lasso regression. The larger the value, the heavier the penalty, and the sparser the initial sparse distribution network input-output model will be, with more coefficients being compressed to zero; for coefficients obtained by ordinary least squares... The larger the value, the higher the corresponding weight. The smaller the value, the smaller the L1 penalty applied to these coefficients, and therefore they are more likely to be retained in the initial sparsed distribution network input-output model;

[0144] To avoid the high time consumption caused by cross-validation, the pre-set residual Akaike information criterion (AIC) is used as the model balancing criterion to obtain the final sparse distribution network input-output model. Represented as:

[0145] ;

[0146] in, This represents the number of non-zero polynomial chaotic expansion coefficients. The complexity of the model, or effective degrees of freedom, is represented by the regularization parameter of Lasso regression. Decision made The larger the value, the more polynomial chaotic expansion coefficients are compressed to zero. The smaller the AIC, the better. By minimizing the AIC, a final sparse distribution network input-output model that achieves the best balance between model complexity and goodness of fit can be selected for risk analysis of the distribution network.

[0147] Step Six: Conduct a risk analysis of the distribution network, specifically:

[0148] Using the final sparsed distribution network input-output model, a node voltage random quantity is defined, and its polynomial chaotic expansion formula is as follows:

[0149] ;

[0150] in, express node The phase voltage is a random quantity; The set of multi-order subscripts after sparsification is a set of multi-indexed subscripts after hyperbolic truncation and adaptive LASSO-AIC sparsification;

[0151] Risk analysis indicators for node-phase voltage exceedance probability, node-phase voltage severity, and node voltage imbalance are calculated based on random node voltage quantities. The formula is as follows:

[0152] ;

[0153] ;

[0154] ;

[0155] in, , They represent node The probability of phase voltage exceeding the upper limit node The probability of phase voltage exceeding the lower limit; express node The phase voltage is a random quantity; , They represent node Phase upper limit, node Phase lower limit; Indicates the probability of occurrence; , express node The severity of phase voltage exceeding the upper limit node Severity of phase voltage exceeding the lower limit; Indicates the portion exceeding the limit; Represents the expectation operator; express node Phase voltage imbalance risk analysis indicators; express The average value of the three-phase voltage at the node; Indicates the voltage imbalance threshold;

[0156] The system-level voltage risk aggregation analysis index is calculated based on the node phase voltage exceedance probability and node phase voltage severity. The formula is as follows:

[0157] ;

[0158] ;

[0159] in, express node Phase voltage risk; This represents a system-level voltage risk aggregation analysis index, which is an index that comprehensively considers the probability and severity of exceeding limits. This represents the weighting coefficient, used to balance the average risk and extreme risk of the system. This represents the voltage risk vector for all phase nodes; , Let L1 norm represent the overall risk level and L∞ norm represent the risk level under the worst-case scenario, respectively.

[0160] The formula for calculating system-level power flow risk analysis indicators is as follows:

[0161] ;

[0162] ;

[0163] in, express branch road The trend risk of phase, express A branch formed by two nodes; express branch road The trend of the times; express branch road The upper limit of the current trend; Indicates system-level power flow risk analysis indicators; This represents the power flow risk vector for all branch phases;

[0164] Based on the node voltage imbalance risk analysis index, the system-level voltage risk aggregation analysis index, and the system-level power flow risk analysis index, the distribution network risk analysis results are obtained.

[0165] This embodiment, through the above-mentioned analytical coupling, can quantitatively analyze the operational risks of the distribution network from three dimensions: voltage overrun, power flow overrun, and voltage imbalance. The overall risk level can be obtained through system-level aggregation methods. It can efficiently and accurately quantify the risk analysis indicators of the distribution network under uncertain conditions, and use the risk analysis indicators to provide a quantitative basis for the operation and control analysis of the distribution network.

[0166] Example 2

[0167] This embodiment presents an experimental example of a risk analysis method for unbalanced distribution networks based on multi-level sparse polynomials:

[0168] The unbalanced hyperbolic truncated LASSO-AIC and ANCOVA exponential combined sparse polynomial chaotic expansion (LASSO-AIC-ANCOVA-Polynomial Chaos Expansions, LAPCE) method is the method of this invention. This embodiment compares the results of the Polynomial Chaos Expansions (PCE), Monte Carlo Simulation (MCS), and LAPCE methods on the voltage magnitude probability density function (PDF) and cumulative distribution function (CDF) of distribution network nodes under the influence of uncertainties. Their voltage magnitude distribution curves almost overlap, indicating that the LAPCE method is very close to the traditional Monte Carlo method in terms of computational accuracy, while significantly improving computational efficiency. Figure 2 As shown.

[0169] The active power PDF and CDF curves of distribution network branches under the influence of uncertainties were compared between the MCS method and the LAPCE method. Their branch active power distribution curves almost overlapped, indicating that the LAPCE method is very close to the traditional Monte Carlo method in terms of calculation accuracy. Figure 3 As shown.

[0170] This embodiment uses the improved IEEE 33-node and IEEE 118-node systems as examples to verify the feasibility of the proposed method. Table 1 shows a comparison of the mean calculation results of node voltage and branch power flow using the LAPCE method and the MCS method in the IEEE 33-node system. Based on 10,000 samples, the table shows that the calculation results of the LAPCE method and the MCS method are almost identical, with very small differences in the mean values, proving that the LAPCE method has similar accuracy to the MCS method in voltage and power flow calculations. At the same time, the relative error in Table 1 is also very small, indicating that the LAPCE method not only maintains high calculation accuracy but also effectively reduces the amount of calculation and improves calculation efficiency. This result further verifies the efficiency and accuracy of the LAPCE method in distribution network voltage and power flow calculations, making it suitable for real-time risk analysis.

[0171] Table 1. Comparison of mean voltage and power flow results under different methods

[0172]

[0173] As shown in Table 2, in the IEEE 33-bus system, the relative errors of the standard deviations of node voltage and branch power flow were calculated using the LAPCE method and the MCS method. Based on 10,000 samples, the table shows that the calculated results of the standard deviations of node voltage and power flow are very close under different methods. The relative error of the LAPCE method is smaller and almost consistent with that of the MCS method.

[0174] Table 2 Comparison of voltage and power flow standard deviation results under different methods

[0175]

[0176] In summary, the IEEE 33-bus system section presents comparative results of the LAPCE and MCS methods in calculating nodal voltage and branch power flow. By comparing the calculated mean, standard deviation, and relative error, the LAPCE method's results are almost identical to the MCS method, regardless of whether wind power is integrated or not, demonstrating its efficiency and accuracy in voltage and power flow analysis. Furthermore, the LAPCE method achieves the same accuracy as traditional Monte Carlo methods with a smaller sample size, validating its application potential in real-time risk analysis.

[0177] As shown in Table 3, in the IEEE 118-node system, the relative errors of the mean and standard deviation of node voltage and branch power flow calculated using the LAPCE method and the MCS method are based on 10,000 samples. The table shows that the calculation results of the LAPCE method and the MCS method are almost consistent, and the relative errors of the mean and standard deviation of node voltage and branch power flow are very small.

[0178] Table 3 Comparison of results using the MCS method (100,000 samples) and the LAPCE method (10,000 samples).

[0179]

[0180] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of a completely hardware embodiment, a completely software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product embodied on one or more computer-usable storage media (including, but not limited to, disk storage, CD-ROM, optical storage, etc.) containing computer-usable program code.

[0181] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus (systems), and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0182] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0183] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0184] The embodiments of the present invention have been described above with reference to the accompanying drawings. However, the present invention is not limited to the specific embodiments described above. The specific embodiments described above are merely illustrative and not restrictive. Those skilled in the art can make many other forms under the guidance of the present invention without departing from the spirit and scope of the claims. All of these forms are within the protection scope of the present invention.

Claims

1. A risk analysis method for unbalanced distribution networks based on multi-level sparse polynomials, characterized in that, include: A power distribution network input-output model is established using polynomial chaos expansion theory. Solve the polynomial chaotic expansion coefficients of the power distribution network input-output model to obtain the power distribution network input-output data pairs. Approximate the power distribution network input-output model as a linear regression model. Calculate the regression coefficients of the linear regression model based on the power distribution network input-output data pairs. Calculate the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model. Based on the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model, calculate the ANCOVA index and p-value of the linear regression model. Based on the ANCOVA index and P-value, a preliminary sparse distribution network input-output model is obtained. The higher-order cross terms of the preliminary sparse distribution network input-output model are suppressed according to the preset truncation rule. The coefficients of the preliminary sparse distribution network input-output model are shrunk according to the optimized polynomial chaotic expansion coefficients. The preset residual Akaike information content criterion is used as the model balance criterion to obtain the final sparse distribution network input-output model. Based on the final sparse distribution network input-output model, the node voltage random quantity is defined, and the distribution network risk analysis index is calculated based on the node voltage random quantity to obtain the distribution network risk analysis result. The power distribution network input / output model is represented as follows: ； in, Represents the input variables of the distribution network The corresponding output response; Indicates the multi-order subscript of a polynomial The expansion coefficients; Represents the input variables of the distribution network Corresponding polynomial multi-order subscript basis functions; A set representing multiple levels of subscripts; , , These represent the zero-order undetermined coefficients, the first-order undetermined coefficients of the i-th distribution network input variable, and the second-order undetermined coefficients of the interaction terms of the i-th and j-th distribution network input variables, respectively. Indicates the dimension of the input variables of the distribution network; , Represent the i-th distribution network input variables respectively First-order Hermite orthogonal polynomial, i-th distribution network input variable and the j-th distribution network input variable The second-order Hermite orthogonal polynomial; Indicates the number of input variables in the distribution network; Let i represent the i-th distribution network input variable, j-th distribution network input variable, ..., j-th distribution network input variable. Undetermined coefficients of multi-order interaction terms of input variables of a power distribution network; Represents the i-th distribution network input variable The j-th distribution network input variable ,…,No. Individual power distribution network input variables The multi-order Hermite orthogonal polynomial.

2. The method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials according to claim 1, characterized in that, The polynomial chaotic expansion coefficients are expressed as follows: ； ； in, Represents the coefficients of the polynomial chaotic expansion; Represents the collocation point polynomial matrix; Indicates the output response of the distribution network; Indicates matrix transpose; , , These represent the first distribution network input variable in the first column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions; , , These represent the first distribution network input variable in the second column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions; , , These represent the first distribution network input variable in the (P-1)th column of the configuration point polynomial matrix. The corresponding polynomial basis function, the second distribution network input variable The corresponding polynomial basis functions, the first Individual power distribution network input variables The corresponding polynomial basis functions.

3. The method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials according to claim 1, characterized in that, The power distribution network input-output model is approximated as a linear regression model. Based on the power distribution network input-output data pairs, the regression coefficients of the linear regression model are calculated, including: The linear regression model is expressed as: ； Set the input and output data pairs of the distribution network Construct as a design matrix According to the design matrix The regression coefficients of the linear regression model are obtained and expressed as follows: ； in, Indicates the output response of the distribution network; , , These represent the intercept term, the regression coefficient of the i-th distribution network input variable, and the regression coefficient of the interaction term between the i-th and j-th distribution network input variables, respectively. , , Let i, j, and j represent the i-th distribution network input variable, respectively. One power distribution network input variable; Represents the residuals of a linear regression model; Indicates the dimension of the input variables of the distribution network; This represents the k-th power distribution network input variable; This represents the output response corresponding to the k-th power distribution network input variable; Indicates the number of input variables in the distribution network; Represents the regression coefficients of a linear regression model; Let i represent the i-th distribution network input variable, j-th distribution network input variable, ..., j-th distribution network input variable. The regression coefficients of the interaction terms of the input variables of the power distribution network.

4. The method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials according to claim 1, characterized in that, The formulas for calculating the regression sum of squares, residual sum of squares, and total sum of squares of a linear regression model are as follows: ； ； ； in, , , Let Si represent the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model, respectively. , , These represent the output response corresponding to the k-th distribution network input variable, the average output of the linear regression model, and the output response of the k-th distribution network predicted by the linear regression model, respectively. This indicates the number of input variables in the power distribution network.

5. The method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials according to claim 1, characterized in that, Based on the regression sum of squares, residual sum of squares, and total sum of squares of the linear regression model, calculate the ANCOVA index and p-value of the linear regression model using the following formula: ； ； ； ； in, This represents the sum of squares of the regressions in a linear regression model that includes only the j-th power distribution network input variable. This represents the sum of squares of the regression in a linear regression model that includes all input variables from the power distribution network. This represents the sum of squares of the regression in the linear regression model, excluding the j-th distribution network input variable. Represents the ANCOVA exponent of the j-th distribution network input variable; This represents the sum of squares in a linear regression model; The F-statistic represents the j-th input variable of the power distribution network; This represents the mean square error associated with the j-th distribution network input variable; This represents the mean square error of the input variables in the distribution network. This represents the degrees of freedom of the j-th power distribution network input variable; This represents the sum of squared residuals in a linear regression model. Indicates the number of input variables in the distribution network; This indicates the number of degrees of freedom in a linear regression model; This represents the P-value corresponding to the F-statistic of the j-th power distribution network input variable; The F-statistic represents the input variables of the power distribution network; This represents a probability function.

6. The method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials according to claim 1, characterized in that, Based on the ANCOVA index and P-value, a preliminary sparse distribution network input-output model is obtained, including: By removing distribution network input variables whose ANCOVA exponent is less than the first threshold or whose P value is less than the second threshold from the distribution network input-output model, a preliminary sparsed distribution network input-output model is obtained.

7. The method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials according to claim 1, characterized in that, The preset truncation rule is expressed as follows: ； in, Represents a set of d-dimensional non-negative integer vectors; Multinomial subscripts representing input variables of the power distribution network; The multinomial subscript represents the i-th power distribution network input variable. Indicates hyperparameters; Indicates the upper limit of the given total order; Higher-order cross terms in the preliminary sparse distribution network input-output model are suppressed according to a preset truncation rule, including: According to the preset truncation rules The polynomial multi-order subscripts in the initial sparse distribution network input-output model are retained. The sum is not greater than the given upper limit of total order. All higher-order cross terms, or remove non-zero polynomial multi-order subscripts from the initial sparsified distribution network input-output model. and polynomial multi-order subscripts Higher-order cross terms that are greater than the preset order value.

8. The method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials according to claim 1, characterized in that, The optimized polynomial chaotic expansion coefficients are expressed as follows: ； The preset residual Akaike information content criterion is expressed as follows: ； in, Indicates the number of input variables in the distribution network; Indicates the output response of the distribution network; Represents the collocation point polynomial matrix; Represents the coefficients of the polynomial chaotic expansion; This represents the regularization parameter for Lasso regression; Indicates the adaptive LASSO regression weights; Indicates the multi-order subscript of a polynomial The expansion coefficients; The second norm of a vector; The polynomial chaotic expansion coefficients that minimize the expression within the parentheses. The value; This represents the number of non-zero polynomial chaotic expansion coefficients; Based on the optimized polynomial chaotic expansion coefficients, the coefficients of the initial sparse distribution network input-output model are shrunk. The preset residual Akaike information content criterion is used as the model balance criterion to obtain the final sparse distribution network input-output model, including: Optimize the polynomial chaos expansion coefficients Items with a coefficient less than the preset value will be deleted; Minimize the preset residual Akaike information content criterion This yields the final sparsed power distribution network input-output model that achieves the best balance between model complexity and goodness of fit.

9. The method for risk analysis of unbalanced distribution networks based on multi-level sparse polynomials according to claim 1, characterized in that, The formula for the random quantity of the node voltage is: ； in, express node The phase voltage is a random quantity; Represents a set of multi-level subscripts after sparsification; Indicates the multi-order subscript of a polynomial The expansion coefficients; Represents the input variables of the distribution network Corresponding polynomial multi-order subscript basis functions; Based on the random quantities of node voltage, risk analysis indicators for three-phase unbalanced distribution networks are calculated, and the results of distribution network risk analysis are obtained, including: Risk analysis indicators for calculating the probability of node phase voltage exceeding limits, the severity of node phase voltage, and the node voltage imbalance based on the random quantity of node voltage are given by: ； ； ； in, , They represent node The probability of phase voltage exceeding the upper limit node The probability of phase voltage exceeding the lower limit; express node The phase voltage is a random quantity; , They represent node Phase upper limit, node Phase lower limit; Indicates the probability of occurrence; , express node The severity of phase voltage exceeding the upper limit node Severity of phase voltage exceeding the lower limit; Indicates the portion exceeding the limit; Represents the expectation operator; express node Phase voltage imbalance risk analysis indicators; express The average value of the three-phase voltage at the node; Indicates the voltage imbalance threshold; The system-level voltage risk aggregation analysis index is calculated based on the node-phase voltage exceedance probability and the node-phase voltage severity, using the following formula: ； ； in, express node Phase voltage risk; This represents the aggregated analysis index of system-level voltage risk; Indicates the weighting coefficient; This represents the voltage risk vector for all phase nodes; , Let L1 norm and L∞ norm be represented respectively; The formula for calculating system-level power flow risk analysis indicators is as follows: ； ； in, express branch road The trend risk of phase, express A branch formed by two nodes; express branch road The trend of the times; express branch road The upper limit of the current trend; Indicates system-level power flow risk analysis indicators; This represents the power flow risk vector for all branches and phases; Based on the node voltage imbalance risk analysis index, the system-level voltage risk aggregation analysis index, and the system-level power flow risk analysis index, the distribution network risk analysis results are obtained.

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