A Region Sensitivity Analysis Method Based on Improved 3D Adaptive Sparse Mesh Interpolation

Abstract

This invention belongs to the field of power transmission capacity sensitivity analysis technology, specifically relating to a regional sensitivity analysis method for static maximum transmission capacity based on an improved 3D adaptive sparse grid interpolation method. The method includes: importing system parameters to obtain a deterministic MSTC calculation model; establishing a probabilistic model of uncertain sources in the power system and converting it into the required random input variables; correcting the boundaries of the random input variables to obtain a surrogate model for MSTC; generating MCS samples of random inputs to obtain the probability results of MSTC; calculating the CSM and CSV numerical results related to the random inputs of MSTC; selecting the top-ranking important random input variables; combining the corresponding CSM and CSV numerical results; analyzing the graphical expressions of the important random input variables; and determining adjustment strategies. This method efficiently identifies important random variables, thereby providing effective strategies for their adjustment and MSTC control, thus improving the stability and economic operation of the system.

Description

Technical Field

[0001] This invention belongs to the field of power transmission capacity sensitivity analysis technology, specifically relating to a regional sensitivity analysis method for static maximum power transmission capacity based on an improved dimensional adaptive sparse grid interpolation method. Technical Background

[0002] The grid connection of renewable energy sources such as wind power introduces numerous uncertainties into the power system (referred to as random input variables in probabilistic analysis). Furthermore, user electricity consumption behavior is also a typical source of uncertainty in modern power systems. These uncertainties make the determination of the Maximum Static Transfer Capacity (MSTC) a stochastic process, significantly impacting the security and stability of power transmission and increasing the decision-making difficulty for transmission operators.

[0003] To overcome this challenge, it is necessary to propose corresponding strategies to control the impact of uncertainty on the power system to a certain extent, ensuring the safe and stable transmission of electricity and maximizing the mean of the MSTC while minimizing its variance. As a fundamental step in achieving this goal, it is necessary to quantify the impact of random input variables on the MSTC in advance.

[0004] Sensitivity analysis is an effective method for quantifying the impact of random variables on target output and serves as the basis for adjusting the target output. Both Global Sensitivity Analysis (GSA) and Regional Sensitivity Analysis (RSA) can be used to analyze the degree of influence of input variables on target output variables. In practical applications, a method is needed to efficiently identify these important random variables, thereby providing effective strategies for their adjustment and MSTC control, which is beneficial for improving the stability and economical operation of the system.

[0005] While GSA (Guided Analytical Stability) indicators can determine the importance ranking of input variables (here, importance is generally ranked according to the degree of influence of the input variables on a certain feature of the target output variable, which can be variance, probability density, etc.), schedulers can only take some countermeasures to roughly control the most important random inputs for a specific purpose. This reflects GSA's lack of ability to explicitly identify the exact distribution range of so-called influential random input variables, thus making it impossible to formulate accurate and detailed adjustment strategies. RSA (Responsive Analysis) is a method that quantifies the importance of the probability distribution (or moments) of random input variables to the target output within a region. As proposed in references 13 J. Sinclair, “Response to the PSACOIN level Sexercise. In: PSACOIN level Sintercomparison,” Nuclear Energy Agency, Organization for Economic Cooperation and Development, 1993. and S. Tarantola et al., “Sensitivity analysis using contribution to sample variance plot: Application to a water hammer model,” Reliab. Eng. Syst. Saf., vol. 99, pp. 62-73, Mar. 2012., the Contribution to Sample Mean (CSM) and Contribution to Sample Variance (CSV) indices measure the impact of the distribution intervals of independent random input variables on the output mean and variance, and can generally be analyzed in detail using graphical expressions. However, under the original definitions of CSM and CSV, ignoring the correlation between random input variables can lead to significant errors. Similarly, the literature J. Lei et al., “A Concise Transformation Combined With Adaptive Kriging Model for Efficiently Estimating Regional Sensitivity on Failure Probability,” IEEE Access, vol. 7, pp. 135457-135471, 2019, proposed a failure probability contribution index as a regional sensitivity index, and used the corresponding graphical expression to quantify the impact of random input variables on the failure probability of roof truss structures in any distributed region.As a typical application of RSA results, the literature P. Wei et al., “Regional sensitivity analysis using revised mean and variance ratio functions,” Reliab. Eng. Syst. Saf., vol. 121, pp. 121-135, Jan. 2014, reveals the significant advantage of RSA (compared to GSA) by adjusting the boundaries of the critical distribution intervals of random variables marked by RSA results to demonstrate the impact on the output mean and variance: RSA can explicitly identify regions of important random input variables that need adjustment, thereby achieving control over the target output distribution (or certain moments). However, this technique has not yet been applied to the control of mean and variance in probabilistic analyses of power systems, such as probabilistic MSTC.

[0006] However, relying solely on the RSA algorithm to identify critical regions among all random input variables requires significant computational or analytical costs. In existing literature, such as P. Wei et al., “Moment-independent regional sensitivity analysis: Application to an environmental model,” Environ. Modell. Softw., vol. 47, pp. 55-63, Sep. 2013, after probability calculations, it is necessary to re-establish the bivariate Copula function between each input variable and the target output, and then use numerical integration on the Copula function to calculate the failure probability contribution index—a process with extremely high computational demands. For CSM and CSV, calculating and comparing their graphical expressions among all random variables, especially in the case of high-dimensional random variables, also requires very tedious analytical work. Therefore, pre-selecting important random variables before RSA is a more effective way to solve this problem. P. Wei et al. proposed combining GSA and RSA, allowing analysts to first identify important random variables using GSA, and then perform the RSA process using the same computational sample to analyze the important distribution intervals of the identified important random input variables. On the one hand, in existing literature such as P. Wei et al., “Moment-independent sensitivity analysis using copula,” Risk Anal., vol. 34, issue 2, pp. 210-222, Feb. 2014 and P. Wei et al., “Moment-independent regional sensitivity analysis: Application to an environmental model,” Environ. Modell. Softw., vol. 47, pp. 55-63, Sep. 2013.0, GSA and RSA based on the Borgonovo index require the establishment of a joint distribution function between each output and input random variable, which greatly limits their application in the case of high-dimensional random input variables. On the other hand, no research has yet proposed a GSA index to complement CSM and CSV in order to achieve the purpose of joint application of GSA and RSA.

[0007] Furthermore, Monte Carlo simulation (MCS) is considered the only available method for calculating the RSA index. However, the significant time consumption of MCS in handling probabilistic problems is widely acknowledged in the field. To address this challenge, existing techniques employ two approaches: firstly, using Sobol sequences or Latin hypercube samples to improve convergence and achieve higher computational accuracy with fewer samples; secondly, using regular low-rank approximation, basis adaptive sparse polynomial chaotic expansion, and dimension adaptive sparse grid interpolation (DASGI) to establish surrogate models, thereby reducing the computation time per sample and improving the overall computational efficiency of the MCS process. Notably, these two methods can be further combined to further enhance computational efficiency.

[0008] For surrogate model-based methods, the first two methods require all random variables to follow the same marginal distribution, and then determine the basis functions to be used based on the type of marginal distribution. However, this increases the computational cost of building the surrogate model or reduces its accuracy because the original model becomes more complex when combined with Copula or Nataf transforms to ensure that random variables have a uniform marginal distribution. In contrast, DASGI is more flexible. For information about the input random variables, it only needs the distribution range of each random input variable, so it can also be used to approximate only some intermediate parts of the original model. However, in multi-output applications, DASGI is slightly cumbersome because, compared to the single-output case, it must consider all outputs to select the placement point, or it can only build a surrogate model for each output separately. However, MSTC computation is a typical single-output model, which matches the advantages of the DASGI method well. Summary of the Invention

[0009] The present invention aims to propose a regional sensitivity analysis method based on an improved dimensional adaptive sparse grid interpolation method, which can efficiently identify important random variables, thereby providing effective strategies for their adjustment and MSTC control, so as to improve the stability and economic operation of the system.

[0010] The regional sensitivity analysis method based on the improved dimensional adaptive sparse grid interpolation method in this invention includes:

[0011] Step S1: Import the system parameters to obtain the deterministic MSTC calculation model of the system;

[0012] Step S2: Establish a probabilistic model of the uncertain sources in the power system and transform it into the required random input variables;

[0013] Step S3: Correct the boundaries of the random input variables to obtain the surrogate model of MSTC, i.e., g. D (X);

[0014] Step S4: Obtain the β-index based on the numerical boundaries of the corrected random input variables;

[0015] Step S5: Generate randomly input MCS samples and input them into the surrogate model to obtain the probability results of MSTC. Calculate the CSM and CSV numerical results related to the random input of MSTC.

[0016] Step S6: Calculate the TCSM and TCSV values ​​of all variables, and sort the random variables from largest to smallest according to the corresponding TCSM and TCSV values, and select the important random input variables with the highest ranking.

[0017] Step S7: For important random input variables, combine the corresponding CSM and CSV numerical results to draw their graphical expressions and obtain their derivative results;

[0018] Step S8: Based on the graphical expressions of the important random input variables obtained in Step S7, analyze the CSM, CSV, and DPE of these random inputs to determine the adjustment strategy;

[0019] Specifically, the probability result of MSTC is updated after adjusting the target random input variable, and the mean of MSTC is increased based on the TCSM and CSM values ​​or its variance is decreased based on the TCSV and CSV values.

[0020] Furthermore, step S1 includes:

[0021] S11: In MSTC calculations, uncertainties such as random loads and wind speeds in wind farms should be treated as random variables, represented as m-dimensional random vectors ξ. The joint distribution function of ξ can be modeled using Gaussian connection functions, as shown in equation (1) below.

[0022] F(ξ)=C(F1(ξ1),F2(ξ2),...,F m (ξ m ))

[0023] Where F(ξ) is the cumulative distribution function (CDF) containing m variables ξ; F k (ξ k ) is the k-th ξ(ξ) k The CDF of ξ, where m represents the dimension of ξ, i.e., the number of random variables;

[0024] Where ξ r and ξ k The correlation coefficient between them is denoted as ρS,(r,k) Calculate using the following formula (2):

[0025]

[0026] Cov and Var are mathematical operations for calculating covariance and variance;

[0027] The random variable is transformed into injection power in the following way:

[0028] X = h(ξ)

[0029] Where X represents the transformed m-dimensional variable random input, including random load and wind power;

[0030] S12: Determine the limiting state of the system using the solution of the following AC-OPF model:

[0031]

[0032] st

[0033]

[0034] Among them, R Sink and R Source Let R be the set of nodes in the receiving network and the sending network, respectively. Sink and R Source Union of; P L,i and Q L,i P represents the active and reactive loads at node i, respectively. G,i and Q G,i V represents the active and reactive power outputs of the generator at node i, respectively. i and V j The voltage magnitudes at nodes i and j, respectively, G ij and B ij Let θ be the real and imaginary parts of the elements in the i-th row and j-th column of the nodal admittance matrix. ij S is the voltage phase difference between node i and node j. ij This represents the apparent power flowing from node i to node j. The superscripts "max" and "min" indicate the upper and lower limits of the corresponding variables.

[0035] S13: Determine the branch corresponding to the tie line, and then (5) determine the active power flowing from the sending network to the receiving network on the tie line, wherein, according to the optimal solution of the AC-OPF model, the active power transmitted by each branch is determined by the following formula:

[0036] P ij =V i 2 G ij -V i Vj (G ij cosθ ij +B ij sinθ ij )

[0037] Where P ij Let be the active power flowing from node i to node j;

[0038] S14: Calculate MSTC from the sum of active power on the tie line, denoted as P. MSTC The calculation process is as follows:

[0039] P MSTC =∑P TL

[0040] Among them, P TL This represents the active power on the connection line from the sending network to the receiving network.

[0041] Furthermore, in step S2, the MSTC model is established as follows:

[0042] P MSTC =g(X),

[0043] Where g(X) represents P with random variable X as input. MSTC calculate.

[0044] Furthermore, step S3 includes:

[0045] S31: Selection of Configuration Points

[0046] The number of Chebyshev Gauss-Lobatto (CGL) nodes is determined by the following formula (8):

[0047]

[0048] Where n is the number of CGL nodes, and d is a positive integer representing the interpolation level;

[0049] The specific CGL node is determined by the following formula:

[0050]

[0051] in It is the j-th CGL node at interpolation level d;

[0052] For the case of m multivariate inputs, the m-dimensional CGL nodes are determined by their corresponding interpolation levels d1, d2, ..., dn. m The combination of all CGL nodes in each dimension forms the structure, where the number of CGL nodes in each dimension is denoted as n1, n2, ..., n. m ;

[0053] For each CGL node, the r-th element of the placement point of the random input variable X in the MSTC model expression (7) is obtained through the linear transformation shown in (10), that is, the r-th random variable X. r The value obtained at this configuration point is represented by x. col,r express:

[0054]

[0055] in and For X r The upper and lower boundaries, and Estimated using MCS samples. Therefore, (x) can be formed. col,1 ,…,x col,r ,…,x col,m Configuration points;

[0056] The conventional boundaries of the placement points are estimated from the MCS samples in the following manner:

[0057]

[0058] in and X is the traditional upper and lower bounds of the r-th random input; MCS,r Let r represent the MCS sample of the r-th random input variable;

[0059] The modified upper and lower boundaries are obtained in the following way:

[0060]

[0061] in and X represents the upper and lower boundaries after the modification of the r-th random input; MCS,r (1-α) and X MCS,r (α) represents the MCS sample value of the r-th random input at its lower 1-α and α quantiles, respectively;

[0062] S32: Full-grid interpolation

[0063] The full-grid interpolation of g(X) can be represented by tensor product as follows:

[0064]

[0065] in For a general symbol, it represents the full-grid interpolation function of g(X) constructed from the full tensor product of m univariate interpolation functions; It is the tensor product; For the m-th input variable at interpolation level dm The jth m Interpolation basis functions for each configuration point; It is a location point of X, obtained by linear transformation of the full combination of all CGL nodes in each dimension;

[0066] S33: Sparse mesh interpolation

[0067] Using the Smolyak algorithm, in d r At interpolation levels ≥ 1, for the r-th univariate interpolation function Define an interpolation increment as follows.

[0068]

[0069] The interpolation function of g(X) is used through interpolation increments. It is expressed as follows:

[0070]

[0071] Where A q,m (g) is the interpolation function of g(X) based on the Smolyak algorithm; q is the given interpolation depth, satisfying q≥m;

[0072] Will be by A q,m (g) represents the tensor product of a univariate interpolation function for a subset of loci points as follows:

[0073]

[0074] Where A q,m (g) is the m-dimensional sparse grid interpolation function at interpolation depth q, where |d| is the overall index of the interpolation level, and |d| = d1 + ... + d m ;

[0075] S34: 3D Adaptive Sparse Mesh Interpolation

[0076] The interpolation function A of g(X) based on the Smolyak algorithm q,m (g) Rewritten as follows:

[0077]

[0078] Where ΔA q,m (g) can be represented as follows:

[0079]

[0080] Where p is an m-dimensional composite index (p1,…,p) mEach element in each dimension refers to the number of new CGL nodes added in the corresponding dimension when the value changes from |d|≤q-1 to |d|≤q. For the corresponding newly added configuration points, yes The corresponding interpolation basis function, For stratified surplus, let l = |d| - m. When g(X) is continuous, The following conditions must be met:

[0081]

[0082] In A q,m In the calculation of (g), the interpolation depth q is gradually increased, and the convergence accuracy is adjusted according to the preset value. Make a judgment to terminate A. q,m The calculation of (g) is then used to adaptively establish the DASGI surrogate model g(X). D (X).

[0083] Furthermore, step S4 includes:

[0084] When constructing the MSTC agent model, when q = m + 1, there are 2m + 1 indispensable configuration points, denoted as... The expression is as follows:

[0085]

[0086] Furthermore, when r = 1, 2, ..., m, Replace the rth term with and To construct and The r-th column is denoted as and

[0087] Based on these configuration points, the index β is defined for the r-th random input using the following formula. r :

[0088]

[0089] Using β r The β-index quantifies the impact of the r-th random input on the MSTC value.

[0090] Furthermore, step S5 includes:

[0091] Calculate the values ​​of CSM and CSV in the following way:

[0092]

[0093] Where f(x) is the joint distribution function of X, CSM r (α) and CSV r (α) Numerical calculations are performed using the MCS method in the following manner:

[0094]

[0095] Where N s The number of samples in the MCS. Not greater than αN s The largest integer, It is the t-th MCS sample after sorting the MCS samples in ascending order of the value of the r-th random variable, where Δα is the numerical step size of α, and the u-th value of α is denoted as α(u), which can be determined in the following way:

[0096]

[0097] Furthermore, step S6 includes:

[0098] S61: For the r-th random variable, its TCSM and TCSV are calculated in the following manner:

[0099]

[0100] Random variables are ranked by importance from largest to smallest based on their TCSM or TCSV values. The larger the TCSM or TCSV value, the more important the random variable is to the mean or variance of the target output. Multiple random inputs with high importance are selected in sequence for further RSA.

[0101] S62: Based on the β-indices of each random input, exclude several spurious important random variables among the selected random variables whose β-indice values ​​are very small.

[0102] Furthermore, step S7 includes:

[0103] Based on α and its corresponding series of CSM / CSV discrete point values, plot the α-CSM for each random variable. r (α) curve or α-CSV r (α) curve, and then α-CSM. r (α) or α-CSV r (α) Plot: The horizontal axis of the graph represents the quantiles of the random variable, and the vertical axis represents the degree of accumulation of the mean or variance of the output variable when the random variable accumulates from the lower bound of the distribution to the quantile α.

[0104] The α-CSM is obtained in the following way. r (α) or α-CSV r(α) The derivative of the plot expression (DPE):

[0105]

[0106] Wherein, the DPE when u=1 is defined as equal to the DPE when u=2;

[0107] According to DPE, the regional sensitivity of a random variable is described as a precise region. If DPE is greater than 1, the random variable distributed in that region contributes more to the mean or variance of the target output and is defined as a sensitive region. Conversely, if DPE is less than 1, the definition of an insensitive region is specified. The larger the value of DPE, the more sensitive it is to the mean or variance of the target output, and vice versa.

[0108] Furthermore, the strategies that can be formulated in step S8 include:

[0109] Strategy 1: Increase the value of the random variable when it is in a region of small values;

[0110] Strategy 2: When the random variable falls into the region of large values, reduce it;

[0111] Strategy 3, a combination of strategy 1 and strategy 2;

[0112] Adjusted random variable W ad It can be calculated in the following ways;

[0113]

[0114] Among them, W ad (α) is W ad The value at α, W0(α) is the value of the original random variable at α; ΔW1 and ΔW2 represent the adjustment amounts for the target regions α1 and α2 corresponding to strategy 1 and strategy 2, respectively. Note that for strategy 1, For strategy 2, For strategy 3, α1 and α2 are both non-empty sets; the adjustment amounts ΔW1 and ΔW2 are non-negative numbers.

[0115] This invention presents a regional sensitivity analysis method based on an improved dimensional adaptive sparse grid interpolation method. It employs the DASGI method to construct a surrogate model, thereby improving the computational efficiency of MSTC in probabilistic analysis. Simultaneously, it improves the determination of the boundary points for the DASGI method, further accelerating the probabilistic and sensitivity analysis processes of MSTC. Accordingly, it proposes Total CSM (TCSM) and Total CSV (TCSV) as GSA indices for pre-analysis of RSA, to pre-screen important random input variables. Finally, based on the RSA results, the selected important random input variables are adjusted to adjust the mean or variance of MSTC, thereby improving the system's stability and economical operation. Attached Figure Description

[0116] Figure 1 This is a flowchart of a regional sensitivity analysis method based on an improved dimensional adaptive sparse grid interpolation method in an embodiment of the present invention.

[0117] Figure 2 This is a diagram showing the α-CSMr(α) and α-CSVr(α) graphs and their derivatives in an embodiment of the present invention.

[0118] Figure 3 The following are schematic diagrams illustrating the adjustment of the distribution of random variables in an embodiment of the present invention: (a) and (b) are schematic diagrams illustrating the control of the mean and variance, respectively.

[0119] Figure 4 This is a schematic diagram of the modified IEEE 118 node system used in the calculation example section of this embodiment of the invention.

[0120] Figure 5 This is a schematic diagram of the probability density distribution of the probability MSTC results in the calculation example section of this embodiment of the invention.

[0121] Figure 6 This is a schematic diagram showing the values ​​and importance ranking of different GSA results in an embodiment of the present invention.

[0122] Figure 7 This is a schematic diagram of the distribution of α-CSM110(α), α-CSV110(α), and their DPEs after adjustment at the 110th random input in the calculation example section of this embodiment of the invention.

[0123] Figure 8 This is a schematic diagram of the RSA and adjustment results of the 110th, 111th, 112th and 77th random inputs in the calculation example section of this embodiment of the invention.

[0124] Figure 9This is a schematic diagram of the randomly input α-CSVr(α) plot and the corresponding variance reduction adjustment strategy selected in the calculation example section of this embodiment of the invention.

[0125] Figure 10 This is a schematic diagram of the variance results of MSTC obtained by adjusting G1 and G2 in the calculation example section of this embodiment of the invention.

[0126] Figure 11 This is a schematic diagram showing the results of probability calculation and sensitivity analysis of MSTC in the calculation example section of this embodiment of the invention.

[0127] Figure 12 This is a schematic diagram illustrating the variance of MSTC obtained under different adjustment combinations in the calculation example section of this embodiment of the invention. Detailed Implementation

[0128] The flowchart of the region sensitivity analysis method based on the improved dimensional adaptive sparse grid interpolation method in this embodiment is basically as follows: Figure 1 As shown, including

[0129] Step S1: Import the system parameters to obtain the deterministic MSTC calculation model of the system;

[0130] In some embodiments, this step specifically includes:

[0131] S11: In MSTC calculations, uncertainties such as random loads and wind speeds in wind farms should be treated as random variables, represented as an m-dimensional random vector ξ. The joint distribution function of ξ can be modeled using a Gaussian connection function, as shown in formula (1).

[0132] F(ξ)=C(F1(ξ1),F2(ξ2),...,F m (ξ m )) (1)

[0133] Where F(ξ) is the cumulative distribution function (CDF) containing m variables ξ; F k (ξ k ) is the k-th ξ(ξ) k The CDF of ξ; m represents the dimension of ξ, i.e., the number of random variables.

[0134] The correlation between ξ can be described by the Spearman correlation coefficient, where ξ r and ξ k The correlation coefficient between them can be calculated using formula (2), denoted as ρ. S,(r,k) .

[0135]

[0136] Cov and Var are mathematical operations used to calculate covariance and variance.

[0137] When performing power flow or optimal power flow (OPF) calculations, these random variables need to be transformed into injected power and then input into the power flow or optimal power flow model. That is, wind speed is transformed into wind turbine output, without the need for additional operations on uncertain loads. This process can be simply represented as (3), where X represents the transformed m-dimensional variable random input, including random loads and wind power.

[0138] X=h(ξ) (3)

[0139] S12: In practical applications, tie lines are lines that transmit electrical energy from the sending-end network to the receiving-end network. MSTC can be defined as: the sum of active power transmitted through all tie lines when the power system is operating in the scenario where the power shortage in the receiving-end grid is the greatest, by minimizing the power generation of the receiving-end network (i.e., satisfying the receiving-end load as much as possible with the generators of the sending-end network). When considering a series of static constraints, the limit state of the system is determined by the solution of the AC-OPF model shown in formula (4).

[0140]

[0141] Among them, R Sink and R Source Let R be the set of nodes in the receiving network and the sending network, respectively. Sink and R Source The union of P. L,i and Q L,i P represents the active and reactive loads at node i, respectively. G,i and Q G,i V represents the active and reactive power outputs of the generator at node i, respectively. i and V j The voltage magnitudes at nodes i and j, respectively, G ij and B ij Let θ be the real and imaginary parts of the elements in the i-th row and j-th column of the nodal admittance matrix. ij S represents the voltage phase difference between node i and node j. ij Let represent the apparent power flowing from node i to node j. The superscripts "max" and "min" indicate the upper and lower limits of the corresponding variables.

[0142] S13: Based on the results of the OPF model in formula (4), the active power transmitted by the branch can be determined by formula (5).

[0143] P ij =V i 2 G ij -Vi V j (G ij cosθ ij +B ij sinθ ij (5)

[0144] Where P ij Let be the active power flowing from node i to node j.

[0145] Once the branch corresponding to the tie line is determined, the active power flowing from the sending end network to the receiving end network on the tie line can be determined by formula (5).

[0146] S14: MSTC can be calculated from the sum of active power on the tie line as shown in formula (6), denoted as P. MSTC .

[0147] P MSTC =∑P TL (6)

[0148] Among them, P TL This represents the active power on the connection line from the sending network to the receiving network.

[0149] Step S2: Establish a probabilistic model of the uncertain sources in the power system and transform it into the required random input variables.

[0150] In some embodiments, this step specifically includes:

[0151] In probabilistic scenarios, random loads, wind power, etc., can be treated as random input variables, and P... MSTC The target output is considered as the MSTC model. Therefore, the MSTC model can be simply expressed as Equation (7).

[0152] P MSTC =g(X), (7)

[0153] Where g(X) is the symbolic expression of formulas (4) and (5).

[0154] Step S3: Based on the theory of DASGI and combined with the random input variable boundary correction method proposed in this study, the surrogate model of MSTC is obtained, namely g. D (X).

[0155] In some embodiments, this step specifically includes:

[0156] S31: Selection of Configuration Points

[0157] Chebyshev Gauss-Lobatto (CGL) nodes are an excellent choice for generating placement points for interpolation. For the univariate case, the number of CGL nodes can be determined by formula (8).

[0158]

[0159] Where n is the number of CGL nodes, and d is a positive integer representing the interpolation level. The specific number of CGL nodes can be determined by formula (9).

[0160]

[0161] in It is the j-th CGL node at interpolation level d. Obviously, the CGL nodes at interpolation level d are a subset of the CGL nodes at interpolation level d+1.

[0162] For the case of m multivariate inputs, the m-dimensional CGL nodes are determined by their corresponding interpolation levels d1, d2, ..., dn. m The combination of all CGL nodes in each dimension forms the structure, where the number of CGL nodes in each dimension is denoted as n1, n2, ..., n. m .

[0163] For the CGL node, the r-th element of the placement point of the random input variable X in formula (7) can be obtained through the linear transformation shown in (10), that is, the r-th random variable X. r The value obtained at this configuration point is represented by x. col,r express.

[0164]

[0165] in and For X r The upper and lower boundaries, and It can be estimated using MCS samples. Therefore, (x) can be formed. col,1 ,…,x col,r ,…,x col,m ) configuration points.

[0166] The traditional method of directly estimating the boundary of the placement point based on the MCS sample is shown in Equation (11), where the r-th random input is used to determine the upper and lower boundaries. and For example:

[0167]

[0168] in and X represents the upper and lower bounds of the r-th random input in the traditional method; MCS,r Represents the MCS sample of the r-th random input variable.

[0169] To determine a narrower range of configuration points, the improved upper and lower boundaries are obtained as shown in formula (12):

[0170]

[0171] in and X represents the upper and lower boundaries after the modification of the r-th random input; MCS,r (1-α) and X MCS,r (α) represents the MCS sample value of the r-th random input at its 1-α and α quantiles, respectively.

[0172] S32: Full-grid interpolation

[0173] The full-grid interpolation of a general deterministic model g(X) with m-dimensional input can be expressed as the tensor product shown in equation (13):

[0174]

[0175] in It can be considered as a general symbolic identifier, representing the full-grid interpolation function of g(X) constructed from the full tensor product of m univariate interpolation functions; It is the tensor product; For the m-th input variable at interpolation level d m The jth m The specific expression for the interpolation basis function for each configuration point can be found in the literature WAKlimke, “Uncertainty modeling using fuzzy arithmetic and sparse grids,” Ph.D. dissertation, mathematics, Univ. Stuttgart, Stuttgart, Germany, 2005; It is a configuration point of X, which can be obtained by linear transformation of the full combination of all CGL nodes in each dimension.

[0176] S33: Sparse mesh interpolation

[0177] Using the Smolyak algorithm, in d r At interpolation levels ≥ 1, for the r-th univariate interpolation function Define an interpolation increment As shown in (14):

[0178]

[0179] The interpolation function of g(X) can be obtained through interpolation increments. Represented as formula (15):

[0180]

[0181] Where A q,m (g) is the interpolation function of g(X) based on the Smolyak algorithm; q is the given interpolation depth, satisfying q≥m.

[0182] Furthermore, from formula (15), the tensor product of the univariate interpolation function shown in formula (16) can be obtained, which can be calculated from the subset of placement points used in formula (13):

[0183]

[0184] Where A q,m (g) is the m-dimensional sparse grid interpolation function at interpolation depth q, where |d| is the overall index of the interpolation level, and |d| = d1 + ... + d m .

[0185] The computational cost of formula (16) depends on the full combination of all interpolation depths in each dimension, where each parameter satisfies q-m+1≤|d|≤q.

[0186] S34: 3D Adaptive Sparse Mesh Interpolation

[0187] Based on the above derivation, A q,m (g) can be determined using formula (16). However, the computational efficiency of this method is fixed when the interpolation depth is given, and the accuracy of the model cannot be predicted to a certain extent before the expression of the interpolation function is fully determined. Therefore, in application scenarios, its performance depends only on the predetermined q. In formula (15), the summation condition can be decomposed into |d|≤q-1 and |d|=q, where the sum when |d|≤q-1 is exactly equal to A. q-1,m (g). Therefore, formula (15) can also be rewritten as formula (17):

[0188]

[0189] It should be noted that among the nested placement points generated by (9) and (10), the placement point of |d|≤q-1 is completely contained in the placement point of |d|≤q. Therefore, only the placement point of |d|=q is newly generated when |d|≤q-1 changes to |d|≤q.

[0190] In formula (17), ΔA q,m (g) can be expressed as formula (18), and some of its expressions are set as stratified surplus.

[0191]

[0192] Where p is an m-dimensional composite index (p1,…,p) m Each element in each dimension refers to the number of new CGL nodes added in the corresponding dimension when the value changes from |d|≤q-1 to |d|≤q. For the corresponding newly added configuration points, yes The corresponding interpolation basis function, let l = |d| - m, when g(X) is continuous, Satisfies formula (19):

[0193]

[0194] By gradually increasing the interpolation depth q, the convergence accuracy can be adjusted according to a preset value. Make a judgment to terminate A. q,m The calculation of (g). Therefore, the DASGI of g(X) can be adaptively established, i.e., g D (X).

[0195] Step S4: Obtain the β-index based on the numerical boundaries of the random input variables during the DASGI implementation process.

[0196] In some embodiments, the step includes:

[0197] When constructing the MSTC agent model, when q = m + 1, there are 2m + 1 indispensable configuration points, denoted as... The expression is shown in formula (20):

[0198]

[0199] Furthermore, when r = 1, 2, ..., m, Replace the rth term with and To construct and The r-th column is denoted as and

[0200] Based on these configuration points, a new index can be defined for the r-th random input using formula (21):

[0201]

[0202] β can be used r The β-index is used to quantify the impact of the r-th random input on the MSTC value.

[0203] Step S5: Generate randomly input MCS samples and input them into the surrogate model to obtain the probability results of MSTC. Calculate the CSM and CSV numerical results related to the random input of MSTC.

[0204] In practical applications of this step in some embodiments, the correlation between random inputs has a significant impact on the probability results. Therefore, it is necessary to redefine CSM and CSV for the relevant random input variables. In this example, CSM and CSV can be expressed as Equation (22).

[0205]

[0206] Where f(x) is the joint distribution function of X, under this assumption, CSM r (α) and CSV r (α) can all be numerically calculated using only the MCS method and formula (23), which is one of the most important reasons for choosing DASGI to accelerate the solution of the deterministic MSTC model:

[0207]

[0208] Where N s The number of samples in the MCS. Not greater than αN s The largest integer. It is the t-th MCS sample after sorting the MCS samples by the value of the r-th random variable from smallest to largest. Δα is the numerical step size of α, which can be set to 0.01. The u-th value of α is denoted as α(u), which can be determined by formula (24).

[0209]

[0210] Notice:

[0211] (1) The CSM and CSV of some intermediate variables can also be obtained through formula (23);

[0212] (2) Only N needs to be performed s The deterministic MSTC calculation can be performed by reordering the current MCS samples multiple times and then combining them with formula (23) to obtain the CSM and CSV numerical results of all random input variables.

[0213] Step S6: Calculate the TCSM and TCSV of all variables and sort the random variables according to their values, and select the important random input variables with the highest ranking (largest TCSM or TCSV value).

[0214] In some embodiments, the step includes:

[0215] S61: For the r-th random variable, let its TCSM (denoted as TCSM) r ) for uniform contribution lines and α-CSM r The region enclosed by the (α) line. Therefore, it quantifies the α-CSM. r The total deviation of the (α) feature can be used as a GSA index to measure the general influence of the r-th random input variable on the target output mean. It can be numerically calculated according to formula (25) and the TCSV under the same definition can be determined. r :

[0216]

[0217] Using formulas (24) and (25), the importance ranking of random variables can be obtained based on the results of TCSM or TCSV. A larger TCSM or TCSV value indicates that the random variable is more important to the mean or variance of the target output. Therefore, the most important random input can be selected for further RSA to reduce analysis costs.

[0218] S62: Using the β-index calculated in formula (21), for the selected highly correlated important random variables, spurious important random variables with very small β-index values ​​are excluded.

[0219] Step S7: For important random input variables, combine the corresponding CSM and CSV numerical results to draw their graphical expressions and obtain their derivative results.

[0220] In some embodiments, this step is specifically as follows:

[0221] The regional sensitivity of a specific random variable over its entire distribution range can be visually described using graphical expressions of CSM and CSV, known as α-CSM. r (α) diagram and α-CSV r (α) diagram.

[0222] Using the mathematical model y = ln(x1) 2 +2x2 2 Taking +1) as an example, assuming x1 and x2 follow independent standard normal distributions, a series of discrete point pairs of α and CSM / CSV can be obtained through the steps shown in formulas (23) and (24), and then these point pairs are plotted as follows Figure 2 The curves shown in sections (a) and (b) correspond to the aforementioned α-CSM. r (α) diagram and α-CSV r (α) Graph. The horizontal axis of the graph represents the quantiles of the random variable, and the vertical axis represents the degree of accumulation of the mean or variance of the output variable as it accumulates from the lower bound of the distribution to the quantile α.

[0223] Within a specific range or region, the slope of a point or interval on a graphical representation corresponds to the magnitude of the influence of the portion of the random variable at that location on the mean or variance of the target output. The slope of the graphical representation can be represented using α-CSM. r (α) or α-CSV r (α) The derivative of the plot is used to describe the plot, denoted as the derivative of the plot expression (DPE), which can be numerically obtained by formula (26):

[0224]

[0225] Here, the DPE when u=1 is defined as equal to the DPE when u=2.

[0226] when At that time, the DPE curve is as follows Figure 2 As shown in sections (c) and (d), the derivative of the uniform contribution line is clearly a constant with a value of 1, highlighted by the black dashed line.

[0227] According to the Degree of Sensitivity (DPE), the regional sensitivity of a random variable can be described as a precise region. If the DPE is greater than 1, the portion of the random variable distributed within that region contributes more to the mean or variance of the target output, and this region is defined as sensitive. Conversely, a lower DPE specifies an insensitive region. A larger DPE value indicates greater sensitivity to the mean or variance of the target output, and vice versa.

[0228] Step S8: Based on the graphical representation of the important random inputs in step S7, analyze the CSM, CSV, and DPE of these random inputs to determine the adjustment strategy; after adjusting the target random input, update the probability results of MSTC, either by increasing the mean of MSTC (based on TCSM and CSM) or decreasing its variance (based on TCSV and CSV).

[0229] In some embodiments, the step includes:

[0230] Select important random variables based on TCSM and TCSV, using α-CSM. r (α) diagram and α-CSV r (α) The analysis of the region and its DPEs (Distributed Physical Parameters) on the importance of the region to the MSTC mean and variance reveals the region with the greatest impact on the adjustment of the MSTC mean and variance. Figure 2 Taking CSM and CSV in x2 as an example, from Figure 3As can be seen, the region highlighted by the blue triangle has a significantly larger DPE, and this region is judged to be a sensitive region; while the region marked by the red square has a much smaller DPE, i.e., an insensitive region. Numerically speaking, the core operation of adjusting a random variable is to move these samples from the sensitive region to the adjacent insensitive region, thereby reducing the mean or variance of the MSTC, and vice versa. Figure 3 As shown, the solid black arrows and dashed black arrows represent increases and decreases, respectively.

[0231] Therefore from Figure 3 It can be seen that there are three basic adjustment strategies:

[0232] Strategy 1: When a random variable is in a region of low value, increase its value, for example, by providing some extra power for wind power generation or increasing the consumption of probabilistic loads;

[0233] Strategy 2, when the random variable falls into the large value region, reduces it, which is the opposite of Strategy 1;

[0234] Strategy 3 is a combination of strategy 1 and strategy 2.

[0235] Adjusted random variable W ad It can be calculated using formula (27).

[0236]

[0237] Among them, W ad (α) is W ad The value at α, W0(α) is the value of the original random variable at α; ΔW1 and ΔW2 represent the adjustment amounts for the target regions α1 and α2 corresponding to strategy 1 and strategy 2, respectively. Note that for strategy 1, For strategy 2, For strategy 3, α1 and α2 are both non-empty sets; the adjustment amounts ΔW1 and ΔW2 are non-negative numbers.

[0238] Calculation Example

[0239] This example validates the effectiveness of the proposed method on an improved IEEE 118 system. By comparing the probabilistic results with the MSTC model based on Direct Current Optimal Power Flow (DC-OPF) (DC-OPF-MCS), the efficiency and accuracy of the DASGI-based surrogate model are verified. A comparison between the traditional DASGI method (TDASGI-MCS) and the improved DASGI method of this technique (MDASGI-MCS, α = 0.05, ε = 0.1) demonstrates the effectiveness of the proposed method for correcting the boundary conditions of the DASGI method. Furthermore, since DC-OPF is already an acceleration and approximation of the AC-OPF model, the combination of the DC-OPF model and the DASGI method is no longer used as a comparison method.

[0240] By comparing the proposed sensitivity analysis method with the traditional variance-based Sobol sensitivity index, i.e., the total effective ST index, the superiority of the proposed method is demonstrated. Let ST be the ST index result of the r-th random variable on the target output. r It can be calculated using (28):

[0241]

[0242] Where E[·] represents the mathematical operation for calculating the mean; X ～r Excluding X r A vector of random input variables. It is worth noting that ST r The larger the value, the stronger X. r The greater the impact on MSTC variance, the better. Meanwhile, with 10... 5 The MCS method for a simple random sample is used as an accuracy reference, denoted as MCS-ref (i.e., AC-OPF-MCS).

[0243] 1. Introduction to the test cases

[0244] Test calculation example Figure 4 As shown, 14 wind farms were connected in the standard IEEE 118 system. Since the line capacity data for this example was not available, the transmission line capacity was set to three times the apparent power of the branch in the standard example results.

[0245] For uncertain sources, in this example, active power load and wind speed at the wind farm are treated as random variables, denoted as L for the sending and receiving end networks, respectively. Source W Source ,L Sink and W Sink Furthermore, in probabilistic calculations, reactive load is obtained by assuming the corresponding load power factor is the same as in the original example. Let L...Source and L Sink It follows a normal distribution, with a mean equal to the original work load and a standard deviation equal to 5% of its mean. W Source It follows a Weibull distribution with shape and scale parameters of 1.837 and 7.218, respectively; W Sink The random variables follow a log-normal distribution, with a log-mean and standard deviation of 1.731 and 0.441, respectively. The rank correlation coefficients within and between these random variable groups are shown in Table 1. The random variable groups are ordered as L. Source L Sink W Source and W Sink The random variables in each group are sorted in ascending order according to their corresponding node numbers. Based on the above settings, there are a total of 113 random variables, including 99 active loads and 14 wind speeds (i.e., 14 wind power outputs).

[0246] The power output model for the wind turbine can be found in the literature M.Aien, M. Rashidinejad, and M.F. Firuz-Abad, “Probabilistic optimal power flow in correlated hybrid wind-PVpower systems: A review and anew approach,” Renew. Sustain. Energy Rev., vol.41, no.4, pp.1437–1446, Feb. 2015., with cut-in, rated, and cut-out wind speeds of 3 m / s, 10 m / s, and 25 m / s, respectively. Each wind turbine has a rated power of 2 MW and a reactive power output of -0.002 MVar.

[0247] Table 1 Rank correlation coefficients of random variables

[0248]

[0249] 2. Performance of the DASGI-based MSTC model and the GSA-based model for most important random input recognition

[0250] In this section, the performance of MDASGI-MCS in probability MSTC calculation and sensitivity analysis is demonstrated through the probability density plot of MSTC and the results of importance ranking of random inputs.

[0251] The probability density of MSTC can be obtained by using MDASGI-MCS, TDASGI-MCS, DC-OPF-MCS, and MCS-ref. Figure 5 As shown, the mean / variance of MSTC and its execution time are shown in Table II. Note that MCS-ref uses 10... 5A simple random sample, while the other three methods use 10 4 Probability calculations are performed on a simple random sample.

[0252] from Figure 5 As can be seen from the table, both MDASGI-MCS and TDASGI-MCS can accurately obtain the probability distribution of MSTC. The overlap area of ​​the probability density map between MDASGI-MCS (TDASGI-MCS) and MCS-ref is 0.9818 (0.9805). That is to say, after modifying the placement point boundary of the DASGI method, its accuracy is slightly improved. As can be seen from Table 2, it saves more than 15% of the computation time compared with TDASGI-MCS. Note that the computation time of 23.3826s (27.6090s) for MDASGI-MCS (TDASGI-MCS) includes 23.2420s (27.4033s) for building a surrogate model with 259 (299) placement points, and 0.1406s (0.2057s) for reusing the deterministic MSTC calculation 104 times.

[0253] Table 2 shows the variation of the mean and variance of MSTC over the probability calculation time.

[0254]

[0255] Furthermore, the probability distribution, mean, and variance of MSTC obtained through DC-OPF-MCS are compared with... Figure 5 There are significant deviations from the corresponding reference data in Table 2. Furthermore, the time taken by DC-OPF-MCS is nearly six times that of MDASI-MCS, indicating that DC-OPF-MCS is not the preferred method for MSTC probability calculation.

[0256] Regarding the performance of these methods in handling GSA, Table 3 shows the five most important random inputs obtained by MDASGI-MCS, TDASGI-MCS, DC-OPF-MCS, and MCS-ref for the three GSA metrics (TCSM, TCSV, and ST). Furthermore, the numerical values ​​of these GSA metrics for the five most important random inputs are as follows: Figure 6 As shown.

[0257] Table 3 shows the five most important random inputs for TCSM, TCSV, and ST selection.

[0258]

[0259]

[0260] √: The result is the same as the random input result corresponding to MCS-ref.

[0261] The results in Table 3 show that the five most important random input choices selected by MDASGI-MCS and TDASGI-MCS are exactly the same as those selected by MCS-ref. Meanwhile, according to... Figure 6 The two DASGI methods yielded almost identical GSA index values, which verifies the accuracy of the proposed MDASGI-MCS and allows it to be further applied to the sensitivity analysis of MSTC.

[0262] However, although DC-OPF-MCS can serve as another method to accelerate probability calculations and sensitivity analysis, according to TCSV, three of the five most important random input quantities or their order obtained by DC-OPF-MCS differ, and its GSA index value deviates significantly from the index value obtained by MDASGI-MCS for the same importance, especially... Figure 6 The first, second, and third important indicators in (b) and (c).

[0263] As can be seen from all the results in this section, the surrogate model based on the DASGI method provides an efficient and highly accurate method for the probability calculation and sensitivity analysis of MSTC. By modifying the boundary of the configuration points, the performance of this DASGI method (i.e., MDASGI-MCS) in probability calculation is further improved, with the total computation time saving more than 15% compared to TDASGI-MCS. The method based on the DCOPF model (i.e., DC-OPF-MCS) can alleviate some of the computational burden (but still requires nearly 6 times the computation time compared to MDASGI-MCS), at the cost of a loss of accuracy in the probability distribution and sensitivity results of MSTC. Therefore, it is unnecessary to further combine the DC-OPF model with the DASGI method.

[0264] For TCSM, TCSV, and ST, TCSM and TCSV require only one probability calculation for MSTC, while ST requires m+1 probability calculations for MSTC. Clearly, the newly proposed indices, TCSM and TCSV, are more efficient in this application. Furthermore, there are RSA indices for TCSM and TCSV, namely CSM and CSV, which can be further analyzed to provide adjustment strategies for improving the mean and reducing the variance of MSTC, a capability lacking in the ST index.

[0265] Therefore, MDASGI-MCS is the best choice for further analysis, which can be used to adjust the mean and variance of MSTC based on the RSA indicators, namely CSM and CSV, using appropriate adjustment strategies.

[0266] 3. RSA single random input and its adjustment

[0267] This section uses CSM and CSV to perform RSA of the most important random inputs' MSTC and further provides explicit adjustment strategies to improve the mean or reduce the variance of the MSTC. Here, the adjustment amount for wind power is set to 20% of its capacity.

[0268] α-CSM of the 110th random input 110 (α) plots and α-CSV110(α) plots and their DPEs are as follows Figure 7 As shown in (a) and (b), and the schematic diagram illustrating the MSTC adjustments for increasing the mean and decreasing the variance. Figure 7 The gray dashed lines in (a) and (b) are the derivatives of the uniform contribution lines.

[0269] like Figure 7 As shown in (a), α-CSM 110 The (α) plot is very close to the uniform contribution line, indicating that the 110th random input makes a nearly uniform contribution to the MSTC mean. Even so, CSM can be observed. 110 (α) The DPE of the graph, i.e., dα-CSM 110 (α)dα shows a decreasing trend, where dα-CSM 110 (α)dα is significantly less than 1 in the region exceeding 20%. Therefore, to improve the mean of MSTC, strategy 2 can be used to adjust the 110th random input variable in the region exceeding 20%. The adjusted distribution of wind power generation at the corresponding bus is as follows. Figure 7 As shown in (c).

[0270] Similarly, for Figure 7 α-CSV in (b) r In graph (α), nearly 20% of the area above and below the graph shows significant variation; therefore, strategy 3 can be used for adjustment. The adjusted distribution of the random input is as follows. Figure 7 As shown in (d). It must be emphasized that, Figure 7 (c) and Figure 7 The adjustments in (d) are mutually exclusive, with the aim of achieving targeted adjustments to the output mean or variance based on CSM or CSV, respectively.

[0271] After adjustment, the mean and variance of MSTC and their corresponding original values ​​are shown in Table 4.

[0272] Table 4 shows the mean and variance of MSTC adjusted from the 110th random input.

[0273]

[0274] As shown in Table 4, both CSM-based and CSV-based adjustments successfully achieved their objectives, increasing the mean by 0.6832% and reducing the variance by 21.20%. However, each adjustment strategy affects the mean and variance of the MSTC. CSM-based adjustments reduced the variance by 13.20%, while CSV-based adjustments increased the mean by 0.1337%. Even so, random variations in the mean or variance should not exceed the target adjustment.

[0275] Furthermore, compared to the adjustability of the variance in this case, whether it is the minimum value of the GSA index (TCSM) 110 =0.0144, much smaller than Figure 7 TCSV 110 =0.0592), or Figure 7 The lack of significant sensitivity differences in the 110th random input region in (a) reflects the poor adjustability of the MSTC mean, where the adjustment ratio of the mean is much smaller than the adjustment ratio of the variance seen in Table 4.

[0276] For the 110th, 111th, and 112th most important random inputs selected from Table 3 for TCSM and TCSV in the previous section, their α-CSM r (α) and α-CSV r (α) The graph and the adjusted MSTC results, as well as the results of the 77th important random input selected by ST, are shown below. Figure 8 As shown.

[0277] like Figure 8 As shown in (a) and (b), the more significant graphs of random inputs are located further outside the uniform contribution line. Clearly, the α-CSV of the three wind power types... r The (α) plots show similar trends, therefore the same adjustment strategy was used for variance control of MSTC, based on α-CSM. r The adjustment of the (α) diagram is also related to Figure 7 The adjustment for the 110th random input is the same. Therefore, the wind power adjustment strategy is the same as... Figure 7 Similarly, for the load, i.e., the 77th random input, Strategy 1 is used, adjusting the lower 10% region to ΔW1, equal to 10% of its mean. The adjusted MSTC is as follows: Figure 8 As shown in (c) and (d), it can be observed that adjustments to the more important random inputs typically have a greater impact on the mean or variance of the MSTC.

[0278] For the adjustment of the 77th random input, the increase in the mean of MSTC was slightly greater than that for the adjustment of the 112th random input. This anomaly could be due to differences in adjustment strategy, adjustment range size, and uncertainty source type. Nevertheless, the adjustment to the mean is negligible. Furthermore, the decrease in variance of MSTC was significantly smaller than that of the other three methods. Clearly, in Table 3, the second most important random input selected by the ST index, such as... Figure 8 The adjustment results of CSM(CSV) and TCSM(TCSV) shown severely contradict the importance ranking obtained by ST. Therefore, due to the lack of a corresponding RSA indicator, the further application of ST in random input adjustment is severely limited.

[0279] 4. Adjust for multiple random inputs, and reject false results for important random variables.

[0280] Since the mean remains almost unchanged during the adjustment process, only CSV-based adjustment is used to reduce the variance of MSTC in this section. MSTC variance control is achieved by adjusting multiple random inputs selected by TCSV, rather than a single random input. Furthermore, to demonstrate the effectiveness of the β-index in rejecting false positives for important random variables, the correlation coefficient between the 60th random variable (active load at the 114th bus) and the 110th random variable (wind speed at the 82nd node) is set to 0.95, while all other correlation coefficients are zero. The original variance of MSTC obtained by the reference method is 1753MW. 2 Based on the proxy model described above, the variance of MSTC is 1748MW. 2 With a deviation of only 0.3173% from the reference value, the result can be considered very accurate, which once again demonstrates the applicability of the surrogate model.

[0281] In addition, TCSV selected six of the most important random inputs, as shown in the β-indices in Table 5. Their α-CSV... r (α) diagram Figure 9 As shown.

[0282] Table 5. Sensitivity results and variance after random input adjustment.

[0283]

[0284] As shown in Table 5, the β-indices of the 60th random input are much smaller than those of the other random inputs. This indicates that the 60th random input has no significant impact on the MSTC value compared to the other five random inputs. Therefore, considering the 60th random input as a significant source of uncertainty is an incorrect choice.

[0285] from Figure 9It can be seen that all selected random inputs contribute significantly to the MSTC variance. Therefore, Strategy 3 is adopted for adjustment, and the specific wind power and load parameters are shown in Table 6.

[0286] For two sets of target random inputs: 1) the top 5 most important random variables in Table 5, denoted as G1; 2) all random input variables in Table 5 except for the 60th random variable, denoted as G2. G1 represents the result of direct TCSV selection without spurious rejections, and G2 represents the random input group adjusted by the β-index. As the number of adjusted random inputs increases, from the most important inputs to those with less influence, the variance of MSTC is as follows: Figure 10 As shown.

[0287] Table 6 shows the wind power and load adjustment parameters corresponding to the selected random inputs.

[0288]

[0289] *:P WP is the rated capacity of the wind farm;

[0290] ^:P L is the mean value of the uncertain load.

[0291] Depend on Figure 10 It can be seen that: 1) As shown by the decreasing red and blue lines, the variance of MSTC tends to decrease with the increase of adjusted random inputs; 2) The almost invisible blue bars when adjusting with two random inputs indicate that adjusting for spurious acceptances has no meaningful effect on reducing the variance of MSTC. The blue line shows almost no change in variance compared to using only one random input, suggesting the necessity of using a β-index to reject spurious acceptances; 3) By rejecting erroneous acceptances, each newly considered random input has a significant effect on reducing the variance of MSTC. Furthermore, the more important the newly involved random input, the greater the adjustment to the variance, which is evident from… Figure 10 The degree of decline in the red bars reflects this.

[0292] 5. Scenarios with higher wind power integration

[0293] Based on the original test cases, the comprehensive wind power of each wind farm was increased to 1.5 times that of the original test cases to verify the performance of the proposed method in scenarios with higher wind power integration. In addition, other configuration points remained unchanged.

[0294] The probability density and RSA results of the proposed method (MDASGI-MCS) and the reference method (MCS-ref) are as follows: Figure 11 As shown.

[0295] like Figure 11 As shown in (a) and (b), the area of ​​overlap between the probability density obtained by MDASGI-MCS and the probability density obtained by MCS-ref is 0.9868, which is very close to 1. Figure 12 Based on the GSA results in (b), it can be concluded that the proposed method can also accurately estimate the probability density of MSTC and select the correct important random inputs when the wind power integration is greater.

[0296] and Figure 5 and Figure 6 (b) Comparing the corresponding results under the original scenario, the distribution range of MSTC is significantly wider and the TCSV index value is significantly larger, indicating that as the integration of wind power increases (i.e., uncertainty increases), the fluctuation of MSTC during operation tends to be larger, thus highlighting the importance of taking some actions to control transmission uncertainty.

[0297] The TCSV method selects three important random inputs, whose α-CSV r (α) diagram Figure 11 As shown in (c). Corresponding to Figure 12 DPEs in (d). Figure 11 As shown in (c), the selected random inputs have almost the same sensitivity range as the original scenario. Therefore, the same adjustment strategy can be used to control them. However, with the increase of the overall wind power, α-CSV r The maximum DPE value in the (α) plot shows a significant increase, among which Figure 11 The maximum DPF value of the 110th random input in (d) is approximately 2.5, which is achieved through... Figure 8 (b) Its maximum value is estimated to be 2.0. This change indicates that the expansion of the uncertainty source size may have a more significant impact on transmission uncertainty, illustrating the urgent need for the technology proposed in this paper.

[0298] All combinations of the identified key random inputs were adjusted, and the adjusted MSTC variance is as follows: Figure 12 As shown. From Figure 12 As can be seen, all adjustments to the random input reduced the variance of MSTC to varying degrees. (Compared to...) Figure 8 Similarly, adjusting for a more important random input reduces the variance of the MSTC. Furthermore, the variance of the MSTC obtained by adjusting a set of random inputs is smaller than the variance obtained by adjusting for any suitable subset of random inputs; specifically, the variance obtained by adjusting the 110th and 112th random inputs is 3495.3 MW. 2 The variance obtained by adjusting only one of the random inputs is 3948.4MW.2 and 4820.7MW 2 When all three important random inputs are intentionally controlled, we can obtain... Figure 12 The minimum variance is 2783.3 MW. 2 Similar to the conclusions of the previous section, the more random inputs adjusted based on RSA results, the higher the adjustment cost, and the smaller the MSTC variance. Therefore, a balance should be considered between the variance change and economic cost of the corresponding adjustment strategy to propose an accurate adjustment strategy.

Claims

1. A regional sensitivity analysis method based on an improved 3D adaptive sparse grid interpolation method, characterized in that, include: Step S1: Import the system parameters to obtain the deterministic MSTC calculation model of the system; Step S2: Establish a probabilistic model of the uncertain sources in the power system and transform it into the required random input variables; Step S3: Correct the boundaries of the random input variables to obtain the surrogate model of MSTC, i.e. ,include: S31: Selection of Configuration Points The number of Chebyshev Gauss-Lobatto (CGL) nodes is determined by the following formula: ， Where n is the number of CGL nodes, and d is a positive integer representing the interpolation level; The specific CGL node is determined by the following formula: ， in It is the j-th CGL node at interpolation level d; For the case of m multivariate inputs, the m-dimensional CGL nodes are determined by the corresponding interpolation levels d1, d2, …, d m The total number of CGL nodes in each dimension is formed by the combination of all CGL nodes in each dimension, where the number of CGL nodes in each dimension is denoted as n1, n2, …, n. m ; For each CGL node, the following linear transformation is used to obtain the r-th element of the placement point of the random input variable X in the MSTC model expression, i.e., the r-th random variable X. r The value obtained at this configuration point is represented by x. col,r express: ， in and For X r The upper and lower boundaries, and By estimating the MCS samples, (x) is formed. col,1 , …,x col,r , …, x col,m Configuration points; The conventional boundaries of the placement points are estimated from the MCS samples in the following manner: ， in and X is the traditional upper and lower bounds of the r-th random input; MCS,r Let r represent the MCS sample of the r-th random input variable; The modified upper and lower boundaries are obtained in the following way: ， in and X represents the upper and lower boundaries after the modification of the r-th random input; MCS,r (1-α) and X MCS,r (α) represents the MCS sample value of the r-th random input at its lower 1-α and α quantiles, respectively; Step S4: Obtain the β-index based on the numerical boundaries of the corrected random input variables; Step S5: Generate randomly input MCS samples and input them into the surrogate model to obtain the probability results of MSTC. Calculate the CSM and CSV numerical results related to the random input of MSTC. Step S6: Calculate the TCSM and TCSV values ​​of all variables, and sort the random variables from largest to smallest according to the corresponding TCSM and TCSV values, and select the important random input variables with the highest ranking. Step S7: For important random input variables, combine the corresponding CSM and CSV numerical results to draw their graphical expressions and obtain their derivative results; Step S8: Based on the graphical expressions of the important random input variables obtained in Step S7, analyze the CSM, CSV, and DPE of these random inputs to determine the adjustment strategy; Specifically, the probability result of MSTC is updated after adjusting the target random input variable, and the mean of MSTC is increased based on the TCSM and CSM values ​​or its variance is decreased based on the TCSV and CSV values.

2. The method according to claim 1, characterized in that, Step S1 includes: S11: In MSTC calculations, the uncertainties of random loads and wind speeds from wind farms are treated as random variables, represented as m-dimensional random vectors ξ. The joint distribution function of ξ is modeled using a Gaussian connection function, as shown in the following equation. ， Where F(ξ) is the cumulative distribution function CDF containing m variables ξ; F k (ξ k ) is the k-th ξ, i.e. ξ k The CDF, where m represents the dimension of ξ, i.e. the number of random variables; Where ξ r and ξ k The correlation coefficient between them is denoted as Calculate using the following formula: ， Cov and Var are mathematical operations for calculating covariance and variance; The random variable is transformed into injection power in the following way: ， Where X represents the transformed m-dimensional variable random input, including random load and wind power; S12: Determine the limiting state of the system using the solution of the following AC-OPF model: ， Among them, R Sink and R Source Let R be the set of nodes in the receiving network and the sending network, respectively. Sink and R Source Union of; P L,i and Q L,i P represents the active and reactive loads at node i, respectively. G,i and Q G,i V represents the active and reactive power outputs of the generator at node i, respectively. i and V j The voltage magnitudes at nodes i and j, respectively, G ij and B ij Let θ be the real and imaginary parts of the elements in the i-th row and j-th column of the nodal admittance matrix. ij S is the voltage phase difference between node i and node j. ij Let $\mathbf{i}$ be the apparent power flowing from node $i$ to node $j$; the superscripts "max" and "min" indicate the upper and lower limits of the corresponding variables. S13: Determine the branch corresponding to the tie line, and then determine the active power flowing from the sending-end network to the receiving-end network on the tie line. The active power transmitted in each branch is determined using the following formula based on the optimal solution of the AC-OPF model: ， Where P ij Let be the active power flowing from node i to node j; S14: Calculate MSTC from the sum of active power on the tie line, denoted as P. MSTC The calculation process is as follows: ， Among them, P TL This represents the active power on the connection line from the sending network to the receiving network.

3. The method according to claim 2, characterized in that, In step S2, the MSTC model is established as follows: ， Where g(X) represents P with random variable X as input. MSTC Calculation process.

4. The method according to claim 3, characterized in that, Step S3 also includes: S32: Full-grid interpolation The full-grid interpolation of g(X) can be represented by tensor product as follows: ， in For a general symbol, it represents the full-grid interpolation function of g(X) constructed from the full tensor product of m univariate interpolation functions; It is the tensor product; For the m-th input variable at interpolation level d m The jth m Interpolation basis functions for each configuration point; It is a location point of X, obtained by linear transformation of the full combination of all CGL nodes in each dimension; S33: Sparse mesh interpolation Using the Smolyak algorithm, in d r At interpolation levels ≥ 1, for the r-th univariate interpolation function Define an interpolation increment as follows: ， The interpolation function of g(X) is used through interpolation increments. It is expressed as follows: ， Where A q,m (g) is the interpolation function of g(X) based on the Smolyak algorithm; q is the given interpolation depth, satisfying q≥m; Will be by A q,m (g) represents the tensor product of a univariate interpolation function for a subset of loci points as follows: ， Where A q,m (g) is the m-dimensional sparse grid interpolation function at interpolation depth q, where |d| is the overall index of the interpolation level, and |d| = d1 + … + d m ; S34: 3D Adaptive Sparse Mesh Interpolation The interpolation function A of g(X) based on the Smolyak algorithm q,m (g) Rewritten as follows: ， Where ΔA q,m (g) means the following: ， Where p is an m-dimensional composite index (p1, …, p) m Each element in each dimension refers to the number of new CGL nodes added in the corresponding dimension when the value changes from |d|≤q-1 to |d|≤q. For the corresponding newly added configuration points, yes The corresponding interpolation basis function, For tiered surplus, let When g(X) is continuous, The following conditions must be met: ， In A q,m In the calculation of (g), the interpolation depth q is gradually increased, and the convergence accuracy is adjusted according to the preset value. Make a judgment to terminate A. q,m The calculation of (g) is then used to adaptively establish a DASGI surrogate model for g(X). .

5. The method according to claim 4, characterized in that, Step S4 includes: When constructing the MSTC agent model, when q = m+1, there are 2m+1 indispensable configuration points, denoted as... = [ , , ]; The expression is as follows: ， Furthermore, when r = 1, 2, …, m, The rth term is replaced with and To construct and The r-th column is denoted as and ; Based on these configuration points, the index β is defined for the r-th random input using the following formula. r : ， Using β r The β-index quantifies the impact of the r-th random input on the MSTC value.

6. The method according to claim 5, characterized in that, Step S5 includes: The values ​​of CSM and CSV are calculated using the following method, with numerical calculations performed via the MCS method: ， Where N s The number of samples in the MCS. Not greater than αN s The largest integer, It is the t-th MCS sample after sorting the MCS samples in ascending order of the value of the r-th random variable, where Δα is the numerical step size of α, and the u-th value of α is denoted as α(u), which is determined in the following way: 。 7. The method according to claim 6, characterized in that, Step S6 includes: S61: For the r-th random variable, its TCSM and TCSV are calculated in the following manner: ， Random variables are ranked by importance from largest to smallest based on their TCSM or TCSV values. The larger the TCSM or TCSV value, the more important the random variable is to the mean or variance of the target output. Multiple random inputs with high importance are selected in sequence for further RSA. S62: Based on the β-index of each random input, exclude several spurious important random variables with small β-index values ​​from the selected random variables.

8. The method according to claim 7, characterized in that, Step S7 includes: Based on α and its corresponding series of CSM / CSV discrete point values, plot the α-CSM for each random variable. r (α) curve or α-CSV r (α) curve, and then α-CSM. r (α) or α-CSV r (α) Plot: The horizontal axis of the graph represents the quantiles of the random variable, and the vertical axis represents the degree of accumulation of the mean or variance of the output variable when the random variable accumulates from the lower bound of the distribution to the quantile α. The α-CSM is obtained in the following way. r (α) or α-CSV r (α) The derivative of the graphical representation of the graph, DPE: ， Wherein, the DPE when u = 1 is defined as equal to the DPE when u = 2; According to DPE, the regional sensitivity of a random variable is described as a precise region. If DPE is greater than 1, the random variable distributed in that region contributes more to the mean or variance of the target output and is defined as a sensitive region. Conversely, if DPE is less than 1, the definition of an insensitive region is specified. The larger the value of DPE, the more sensitive it is to the mean or variance of the target output, and vice versa.

9. The method according to claim 8, characterized in that, The strategies defined in step S8 include: Strategy 1: Increase the value of the random variable when it is in a region of small values; Strategy 2: When the random variable falls into the region of large values, reduce it; Strategy 3, a combination of strategy 1 and strategy 2; Adjusted random variable Calculated in the following manner; ， in, for The value at α Let be the value of the original random variable at α; ΔW1 and ΔW2 represent the adjustment amounts for the target regions α1 and α2 corresponding to strategy 1 and strategy 2, respectively; for strategy 1, α2 = For strategy 2, α1 = For strategy 3, α1 and α2 are both non-empty sets; the adjustment amounts ΔW1 and ΔW2 are non-negative numbers.

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