Explicit unified modeling method for static voltage stability boundary of new energy weak sending end system

Abstract

The application discloses an explicit unified modeling method for static voltage stability boundary of a new energy weak sending end system, and belongs to the technical field of static voltage stability analysis and probability risk assessment of a new energy power system. The application aims to solve the problems of the static voltage stability boundary of the new energy weak sending end system, such as difficult unified description of the static voltage stability boundary, difficult identification of boundary formation mechanism and large calculation amount of probability evaluation. The application establishes a non-smooth unified complementary steady-state model considering device constraint switching, constructs a unified static voltage stability critical boundary augmented equation, establishes a deterministic nonlinear equation group about polynomial spectral coefficients, solves the equation group to obtain an explicit spectral mapping, and finally realizes the explicit unified modeling and probability evaluation of the static voltage stability boundary of the new energy weak sending end system through the calculation result of the explicit spectral mapping. The application provides a theoretical basis for new energy sending checking and stability margin configuration.

Description

Technical Field

[0001] This invention belongs to the technical field of static voltage stability analysis and probabilistic risk assessment of new energy power systems. Background Technology

[0002] With the large-scale centralized development of new energy sources such as wind power and photovoltaics, and the continuous expansion of long-distance transmission, the weak-end systems of new energy sources are gradually showing characteristics such as insufficient synchronous support, weak AC grid structure, and enhanced coupling among multiple power plants. After the mixed access of grid-connected and grid-built new energy equipment, the reactive power support capacity, control method, and operating limits of the equipment will significantly affect the static voltage stability limit of the system. When new energy equipment reaches operating limits such as reactive power, capacity, or current, the original control target may be released or switched, causing the system steady-state equation and stability boundary formation mechanism to change accordingly. At this time, the static voltage stability boundary may be formed by saddle-node bifurcation corresponding to Jacobian matrix degeneration, or by limit-induced bifurcation caused by equipment limit triggering. Existing continuous power flow and static stability boundary calculation methods are usually based on fixed control modes or single equipment limits, making it difficult to uniformly describe the switching between multiple types of equipment constraints and different boundary mechanisms. In addition, the output of new energy sources has randomness and multi-power plant correlation, making the location of stability boundary, boundary point state, and stability margin exhibit probabilistic characteristics. Existing methods such as Monte Carlo simulation, point estimation, and surrogate models typically require repeated power flow calculations and stability limit searches for a large number of stochastic scenarios, resulting in significant computational costs. Existing multinomial chaos methods are mainly used for conventional power flow states or stability margin analysis, and struggle to simultaneously establish explicit mappings between stochastic outputs and boundary coordinates, boundary point states, and equipment constraint states. Therefore, it is necessary to propose a method that can uniformly describe equipment constraint switching and different static voltage stability boundary mechanisms, and efficiently conduct probabilistic risk assessments. Summary of the Invention

[0003] The purpose of this invention is to provide an explicit unified modeling method for the static voltage stability boundary of a new energy weak-transmission system, addressing the challenges of unifying the description of the static voltage stability boundary, identifying the boundary formation mechanism, and consuming a large amount of probability assessment calculation when equipment operating limit triggering, control state switching, and the randomness of new energy output all work together in the new energy weak-transmission system.

[0004] The steps of this invention are: S1. Incorporate network power balance, equipment steady-state external characteristics, equipment control conditions and equipment operating limits into a unified steady-state model, and then obtain a non-smooth unified complementary steady-state model that takes into account equipment constraint switching by the complementary relationship between equipment limit margin and control release variables. S2. The smooth Fischer-Burmeister function is used to process the non-smooth unified complementary steady-state model to obtain the smooth unified steady-state equation. The smooth unified steady-state equation, Jacobi degeneracy condition and critical vector normalization condition are combined to construct the unified static voltage stability critical boundary augmented equation. S3. Random boundary parameters obtained through random mapping relationships Substituting into the augmented equation of the unified static voltage stability critical boundary, we obtain the static voltage stability boundary coordinates, the unified steady-state variables of the boundary points, and the critical vector. S4. Polynomial chaotic expansion is performed on the static voltage stability boundary coordinates, boundary point unified steady-state variables and critical vectors through orthogonal polynomial basis functions, and a deterministic nonlinear equation system about the polynomial spectral coefficients is established by intrusive PCE-Galerkin projection. After solving, an explicit spectral mapping is obtained. S5. Calculate the static voltage stability boundary coordinates, critical power transmission, boundary point physical state, and equipment constraint state corresponding to different new energy random outputs based on explicit spectral mapping, and calculate the external risk of the operating point crossing the static voltage stability boundary. Quantile stability margin at different quantile probability levels Thick probability boundaries formed between different conditional quantile boundaries This enables explicit unified modeling and probabilistic evaluation of the static voltage stability boundary of new energy weak-sending systems.

[0005] The non-smooth unified complementary steady-state model of this invention is as follows: (10) In the formula, For steady-state algebraic variables, As a subordinate advancing coordinate, For independent parameter vectors, It is a non-negative vector. For an algebraic equation that always holds true under a fixed control mode, This is the vector form of the reactive power control release equation. For the limit margin function, " "Indicates a complementary relationship, for the first..." There are several limits. and .

[0006] The unified static voltage stability critical boundary augmented equation of this invention is as follows: (18) In the formula, To unify steady-state variables, It is a non-zero critical vector. For smoothing parameters, For the transpose of a given reference direction vector, To account for the smooth uniform equation of constraint switching, To account for the smooth uniform equations regarding constraint switching The transpose of the Jacobian matrix.

[0007] The random boundary parameters of this invention The process of obtaining it is as follows: For new energy power stations Let the predicted active power output be... The actual contribution was Then predict the residuals Written as: (20) It is constructed through marginal distribution, correlation model, or empirical samples. Random mapping relationship with ξ The relationship between new energy output and independent random variables was obtained: (twenty one) In the formula, To make predictions and contribute efforts, for The random mapping relationship with ξ; The active power output of each station is recorded as an independent parameter. Then, by selecting and transforming parameters, a mapping is established from the random output of new energy sources to the independent parameter space of the static voltage stability boundary: (twenty two) In the formula, For the physical domain of the boundary parameters, For the mapping from standard random variables to the boundary parameter space, To select operators.

[0008] This invention addresses random boundary parameters. When substituting into the augmented equation for the unified static voltage stability critical boundary, if λ increases monotonically from the stable operating point towards the critical boundary, and when the augmented equation for the unified static voltage stability critical boundary has multiple candidate critical solutions under given independent parameters, the coordinates of the first candidate critical boundary encountered are selected as the effective static voltage stability boundary coordinates according to the given advancement direction from the stable operating point along the subordinate advancement coordinates. Then, the static stability domain under the smoothing parameter τ is expressed as: (25) In the formula, For the set of real numbers, Boundary mapping for subordinate advancing coordinate λ; The coordinates of the system's current or planned operating point in the direction of propulsion. Then, the static stability margin for a given q is: (26) In the formula, The coordinates of the system's current or planned operating point in the direction of propulsion; when When the running point is within the static stability region; when When the running point is at the critical boundary; when At that time, the running point had already crossed the static stability boundary; When the independent parameters of the weak-end system of new energy are When λ is also a power component of a single renewable energy power station, the critical power transmission capacity is further given: (27) In the formula, Let q be an s-dimensional column vector of all 1s, where s is the total number of new energy power stations included in the independent parameter q; Corresponding external delivery stability margin for: (28) In the formula: for The power is transmitted from the operating point below.

[0009] The polynomial chaotic expansion of this invention yields: (43) In the formula, To obtain the unified steady-state variable approximation for boundary points using the PCE representation truncated by the total order K, The approximate boundary coordinates are obtained using the PCE representation truncated with the total order K. This is the approximate value of the critical vector obtained by using the PCE representation truncated with the total order K. , These are the boundary coordinate spectral coefficients of the unified steady-state variable and the critical vector at the boundary point, respectively. for 3D real vector space, For index The corresponding polynomial chaotic basis functions, This is the first multi-index. To correspond to the total order of PCE truncation A set of multiple indicators.

[0010] This invention is for solving polynomial spectral coefficients, in By substituting randomness into the augmented equation of the unified static voltage stability critical boundary and simultaneously solving the polynomial chaotic expansion, the explicit approximate expression is obtained, yielding the residual function in the sense of truncated PCE: (44) In the formula, For the mapping from standard random variables to the boundary parameter space, To unify the augmented equation for the static voltage stability critical boundary, c is a column vector formed by stacking all spectral coefficients in a fixed order. It is a standard independent random vector; The Galerkin projection requires that the residual be orthogonal to every basis function in the chosen polynomial space, i.e.: (45) In the formula, residual function Galerkin projection residual in the direction of the β-th polynomial chaotic basis function Let ξ be the joint probability measure of a standard independent random vector. For joint probability measure Defined inner product, Here, β is the polynomial chaotic basis function corresponding to the multi-index β; β is the second multi-index. Use and The numerical approximation of the discrete Galerkin equations for matching: (46) In the formula, To unify the augmented equations for the static voltage stability critical boundary, The set of quadrature node indices used for numerical integration; , For the first Each quadrature node and its corresponding weight; Therefore, for all The core solution equation is obtained as follows: (47) In the formula, This is the residual vector of the overall nonlinear equation system formed by stacking all the corresponding Galerkin projection residuals in a fixed order; Solving for the spectral coefficients yields the corresponding vector. Extract the corresponding spectral coefficients of each explicit approximation expression to obtain the explicit static stability boundary in the random parameter space: (48) In the formula, For an explicit statically stable boundary in the random parameter space, Let ξ be the range of values ​​for the standard random vector. when If it is invertible within the parameter region, then an approximate explicit relation in the physical boundary parameter space is further given: (49) In the formula, Let be the explicit function of the static voltage stability boundary coordinates in the physical boundary parameter space obtained by the K-order PCE approximation. Random mapping for contributing to new energy The inverse mapping of .

[0011] The extraterritorial risk of the operating point of this invention crossing the static voltage stability boundary Represented as: (29) In the formula, The probability of an event occurring. This is an indicator function for events outside the domain. Let q be the integral measure of the probability measure of the random boundary parameter vector q in the physical parameter space Q; Quantile stability margin at different quantile probability levels Represented as: (32) In the formula, Let τ be the value of the static stability margin random variable under the smoothing parameter τ. Let be the cumulative distribution function of the static stability margin random variable; Thick probability boundaries formed between different conditional quantile boundaries Represented as: (38) In the formula, As a baseline independent parameter vector, For parameter points in the extended boundary parameter space, which consists of a reference independent parameter vector and subordinate propagation coordinates, Reference independent parameter vector The physical domain, , These are the quantile probability levels corresponding to a given baseline independent parameter vector. , The static voltage stability boundary coordinate quantiles, , These are two given quantile probability levels.

[0012] An explicit unified modeling system for the static voltage stability boundary of a new energy weak-sending-end system includes: The unified steady-state modeling module is used to incorporate network power balance, device steady-state external characteristics, device control conditions and device operating limits into a unified steady-state model, construct device limit margin functions and control release variables, and characterize the control state switching before and after device limit triggering through the complementary relationship between device limit margin and control release variables, thus obtaining a non-smooth unified complementary steady-state model that takes into account device constraint switching. The critical boundary construction module is used to process the complementary relationship in the non-smooth unified complementary steady-state model using the smooth Fischer-Burmeister function, establish the smooth unified steady-state equation, and, based on the Jacobi degeneracy condition of the smooth unified steady-state equation with respect to the unified steady-state variables, introduce the non-zero critical vector and the critical vector normalization condition to construct the unified static voltage stability critical boundary augmented equation. The random parameter mapping module is used to establish a random mapping between standard independent random vectors and random output of new energy sources based on the probability distribution of the prediction error of new energy power output and the correlation between multiple power stations. It also obtains random boundary parameters through parameter selection and transformation, and substitutes the random boundary parameters into the unified static voltage stability critical boundary augmented equation so that the static voltage stability boundary coordinates, boundary point unified steady-state variables and critical vectors form a random response. The explicit spectral mapping solution module is used to select orthogonal polynomial basis functions that are compatible with the probability measure of standard independent random vectors, perform truncated polynomial chaotic expansion on the static voltage stability boundary coordinates, unified steady-state variables of boundary points, and critical vectors, substitute the obtained expansion into the augmented equation of the unified static voltage stability critical boundary to form a residual function, and use Galerkin projection to make the residual function orthogonal to each basis function in the selected polynomial space, construct a deterministic nonlinear equation system about the polynomial spectral coefficients, and solve it to obtain the explicit spectral mapping between the random power output of new energy and the static voltage stability boundary; The probability assessment module is used to calculate the static voltage stability boundary coordinates, critical power transmission, physical state of boundary points and equipment constraint state corresponding to different new energy random outputs based on the explicit spectrum mapping, and to calculate the extra-domain risk of the operating point crossing the static voltage stability boundary, the quantile stability margin under different quantile probability levels and the probability thick boundary formed between different conditional quantile boundaries. The result output module is used to output one or more of the following: static voltage stability boundary coordinates, critical power transmission, boundary point physical state, equipment constraint state, external risk, quantile stability margin, and probabilistic thick boundary. Each module is implemented by at least one processor executing a computer program stored in memory.

[0013] The present invention provides a computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, it implements the explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending system as described in any one of claims 1 to 5.

[0014] This invention maintains a low approximation error within the validated principal probability region and avoids solving non-smooth boundary equations for a large number of random samples one by one in probability assessment. Furthermore, it can obtain stability margins at different quantile levels, providing a theoretical basis for the verification and stability margin configuration of new energy transmission. Attached Figure Description

[0015] Figure 1 This is a schematic diagram of multiple critical candidate points in the case of boundary intersection; Figure 2 This is a modified structural diagram of the CSEE-RAS new energy weak-end test system; Figure 3 This is the original non-smooth static voltage stability boundary diagram, where (a) is a fixed P W3 The static voltage stability boundary surface under the condition, (b) is a fixed P W1 and P W3 The switching boundary between SNB and LIB dominant mechanisms under certain conditions; Figure 4 It is a smooth explicit approximate static voltage stability boundary map obtained by using the intrusive PCE-Galerkin method; Figure 5 This is a diagram showing the explicit approximate boundary coordinate error distribution under a given cross section, where (a), (b), and (c) represent P, respectively. W3 =490 MW (RMSE=0.0993MW), P W3 =505 MW (RMSE=0.0772MW) and P W3 Boundary coordinate error distribution under the condition of 520 MW (RMSE=0.0894MW); Figure 6 It is a statistical diagram of the nodal voltage distribution and its approximate error at the static voltage stability boundary point; Figure 7 This is a comparison chart of the consistency of reactive power, capacity and current limit utilization rates of station W4; Figure 8 The modified CSEE-RAS system critical coordinates and critical total power transmission probability density and external risk map, where (a) is the probability density and external risk of the critical active power output of station W4, and (b) is the probability density and external risk of the critical total power transmission. Figure 9These are spatial distribution diagrams of the W4 critical output and conditional probability thick boundary of the station at different quantile levels, where (a) is the W4 critical output boundary surface of the station at different quantile levels, and (b) is the spatial distribution of the thickness of the conditional probability thick boundary. Figure 10 This is an equivalent system topology diagram of a new energy base in Northwest China; Figure 11 This is a distribution diagram of the explicit approximate static voltage stability boundary and dominant mechanism under representative parameter cross-sections of a new energy base in Northwest China. Figure 12 This is a boundary coordinate error distribution map under different representative parameter sections of a new energy base in Northwest China. (a), (b) and (c) are the boundary coordinate error distributions under the conditions of ξ4=1.5 (RMSE=0.237MW), ξ4=0 (RMSE=0.180MW), and ξ4=-1.5 (RMSE=0.392MW), respectively. Figure 13 This is a probability density and external risk map of the critical coordinates and critical total power transmitted to a new energy base in Northwest China. (a) represents the probability density and external risk of the critical active power output of the W5 power station, and (b) represents the probability density and external risk of the critical total power transmitted to the power station. Detailed Implementation

[0016] This invention discloses an explicit unified modeling and probabilistic evaluation method for the static voltage stability boundary of a renewable energy weak-end system, taking into account constraint switching. Based on the equipment steady-state control equations, power balance, equipment external characteristics, and various limits are incorporated into a unified complementary steady-state model. A smoothed Fischer-Burmeister function is used to handle non-smooth relationships at points of change in active constraints. A critical augmented equation is constructed using a unified Jacobian matrix degradation condition, enabling the description of saddle-node bifurcation (SNB), limit-induced bifurcation (LIB), and their boundary switching under different equipment limits within the same model framework. Then, an invasive PCE-Galerkin projection is applied to the boundary coordinates, boundary point physical states, and critical vectors to establish an explicit spectral mapping between the renewable energy random output and the static voltage stability boundary. The boundary formation mechanism is explained based on equipment limit margins, control release states, and Jacobian degradation characteristics. Furthermore, the joint probability distribution of renewable energy output is mapped to the boundary parameter space to calculate external risks, quantile stability margins, and probabilistic thick boundaries, achieving a unified analysis of deterministic physical boundaries and probabilistic risks. Finally, based on the improved CSEE-RAS weak-end system and the equivalent system of a new energy base in Northwest China, the boundary approximation accuracy, physical constraint consistency and probabilistic energy efficiency of the proposed method were verified.

[0017] This invention includes the following steps: S1. Obtain the network topology parameters, new energy equipment parameters, equipment control parameters and equipment operating limits of the new energy weak transmission end system. Incorporate network power balance, equipment steady-state external characteristics, equipment control conditions and equipment operating limits into a unified steady-state model. Characterize the triggering of equipment operating limits and the corresponding control state switching through the complementary relationship between equipment limit margin and control release variables. S2. The complementary relationship is processed by the smooth Fischer-Burmeister function, and the unified static voltage stability critical boundary equation is constructed by combining the degeneration condition of the Jacobian matrix of the smooth unified steady-state equation with respect to the unified steady-state variables. The static voltage stability boundary coordinates and corresponding boundary point states under different equipment constraint states are obtained. S3. Establish a stochastic mapping relationship between the random output of new energy sources and the independent parameters of the static voltage stability boundary. Substitute the stochastic boundary parameters obtained by mapping the random output of new energy sources into the unified static voltage stability critical boundary augmented equation, so that the static voltage stability boundary coordinates, boundary point unified steady-state variables and critical vectors form a stochastic response in the stochastic parameter space. S4. Select orthogonal polynomial basis functions that are compatible with the probability distribution of random power output from new energy sources, perform polynomial chaotic expansion on the static voltage stability boundary coordinates, unified steady-state variables at boundary points, and critical vectors, and establish a deterministic nonlinear equation system about the polynomial spectral coefficients using intrusive PCE-Galerkin projection; by solving the deterministic nonlinear equation system, obtain the explicit spectral mapping between the random power output from new energy sources and the static voltage stability boundary coordinates and boundary point states. S5. Based on the explicit spectral mapping, calculate the static voltage stability boundary coordinates, critical external power, boundary point physical state and equipment constraint state corresponding to different new energy random outputs, and calculate the external risk of the operating point crossing the static voltage stability boundary, the quantile stability margin under different confidence levels and the probabilistic thick boundary formed between different conditional quantile boundaries, so as to realize the explicit unified modeling and probabilistic evaluation of the static voltage stability boundary of the new energy weak transmission end system.

[0018] See appendix Figure 1-13 This invention proposes an explicit unified modeling and probabilistic evaluation method for the static voltage stability boundary of a new energy weak-sending system, taking into account equipment constraint switching. The specific implementation steps are as follows: 1. The steady-state algebraic variables used to uniformly represent node voltage magnitude, node voltage phase angle, equipment output power, and equipment control mode are denoted as... The static voltage stability boundary analysis parameters are denoted as... .in: It is an independent parameter vector used to characterize external given quantities such as operating mode, injected power, or control settings; As a subordinate propulsion coordinate, it is used to characterize the critical level reached along a specified direction of transmission, load growth, or power transfer. , and They represent dimension, 3D real vector space and the set of real numbers; This defines the physical domain of the boundary parameters. Where: and They are respectively and The dimension of . Before the equipment operating limits are triggered, the always-held network power balance equation, the equipment steady-state external characteristics, and the fixed control conditions are expressed as: (1).

[0019] For the releaseable control conditions satisfied when the equipment is in the normal control range, set the control residual: (2).

[0020] This constitutes the steady-state model of the system under a fixed control mode: (3).

[0021] For a given independent parameter vector The system steady-state model follows the subordinate coordinate progression. A steady-state solution branch is formed; when the system reaches the critical state of static voltage stability along the steady-state solution branch, the corresponding critical propagation coordinates are: Thus forming a As independent parameters, with For the static voltage stability boundary of the subordinate propulsion coordinate: (4).

[0022] Further construct an equipment limit margin function to characterize the equipment's reactive power, voltage, current, capacity, and control saturation operation limits. have: (5).

[0023] It meets the following requirements within the specified feasible operating area: (6).

[0024] Furthermore, nonnegative vectors are introduced. This describes the release of the control residual after the limit is triggered. And the limit margin... With release variables The relationship should be complementary: (7) When the limit is not triggered, And there must be ;when There will always be a time when... This indicates that the equipment has reached the corresponding limit, and the original control residual has been released.

[0025] Then the first includes reactive power, capacity and current limits. The reactive power control release equation for this device is: (8) in, and The first Release variables when a device reaches its upper or lower reactive power limits; When the equipment reaches the capacity limit, along the capacity limit face Release variables in the normal direction; When the device triggers the current limit, along the current limit face Release variables in the normal direction.

[0026] Furthermore, the release effect vector of the equipment limit trigger on the control residual is expressed as: (9) In the formula: To control the release of the correlation matrix; Indicates the first The limit release variable is for the first... The effect coefficient of the control residual. If the two are unrelated, then If the two are related, The value is determined by the definition of the control residual and the limit margin.

[0027] Specifically, regarding the reactive power limit Its limit release direction is Similarly, for the lower limit of reactive power, its limit release direction is -1. From the above definition of the limit release direction, we can obtain that if the first... The limit value is the first If there is an upper limit constraint on the controlled variable, then When a lower bound constraint is applied... For nonlinear limit relationships such as capacity constraints and current margins, the first... Capacity margin relationship of each controlled quantity For example, its limit release direction is... Related to, for When the first The limit value is the first When considering the capacity margin relationship of a controlled quantity As for the first Current limit Its limit release direction is also The above variables , The first The reactive power output and maximum reactive power output of the equipment; For the first Active power output of the equipment; , The first The rated capacity and upper limit of each piece of equipment; For the first The node voltage amplitude of the equipment; For the first The upper limit of the allowable current amplitude for each device. Based on this, a non-smooth unified complementary steady-state model considering device constraint switching is constructed: (10).

[0028] In the aforementioned unified complementary steady-state model, the network power balance equation and the steady-state external characteristics of the equipment are retained before and after the switching of equipment control modes. The control release variable only participates in the release of control residuals when the corresponding equipment limit is triggered. Therefore, when the equipment has not reached the operating limit, the model maintains the original control mode; when the equipment reaches operating limits such as reactive power, capacity, or current, the control target is released by changing the activity state of the corresponding complementary variable. This allows the steady-state operating state before and after the limit trigger to be continuously described in the same mathematical model, without the need to establish independent steady-state equations for different equipment constraint states.

[0029] 2. The non-smooth complementary relationship between the equipment limit margin function and the control release variable is transformed into a continuously differentiable smooth relationship using the smooth Fischer-Burmeister function. The smoothing function corresponding to each complementary relationship is: (11) In the formula: The smoothing parameter is used to adjust how well the smoothed Fischer-Burmeister function approximates the original non-smooth complementary relation. The smaller the value, the closer the smooth equation is to the original complementary condition.

[0030] Write all smooth complementary relationships in vector form: (12).

[0031] Let the unknown unified steady-state variable be . Then the smooth uniform equation taking into account constraint switching can be written as: (13) The smooth and unified steady-state equation is used to describe, within the same model framework, the states of a device not reaching its limit, a single device triggering its limit, multiple devices triggering their limits simultaneously, and the corresponding control mode switching states.

[0032] Furthermore, the smooth unified steady-state equation is given with respect to the unified steady-state variables. Jacobian matrix: (14).

[0033] Then smoothing parameter Corresponding unified static stability boundary for: (15).

[0034] At the critical boundary point, a non-zero critical left vector is introduced. This reflects the critical degradation direction of equation (13) as a whole. However, to avoid When the direction changes abruptly, a direction reference normalization condition is introduced to satisfy the following: (16) (17).

[0035] Combining the aforementioned smooth unified steady-state equation, Jacobi degradation condition, and critical vector normalization condition, an augmented equation for the unified static voltage stability critical boundary is constructed: (18) In the formula, To unify steady-state variables, It is a non-zero critical vector. For smoothing parameters, For the transpose of a given reference direction vector, To account for the smooth uniform equation of constraint switching, To account for the smooth uniform equations regarding constraint switching The transpose of the Jacobian matrix.

[0036] For a given independent parameter vector The augmented equation for the unified static voltage stability critical boundary may have multiple critical candidate solutions, especially when different boundary branches approach or intersect. Multiple critical points satisfying the augmented equation may be obtained simultaneously within the same parameter section. For example... Figure 1 As shown, this implementation method uses the subordinate propulsion coordinates from the stable operating point. Based on the given forward propagation process, the obtained critical candidate points are physically screened, and the effective critical point first reached along the propagation direction is taken as the current critical point. The corresponding static voltage stability physical boundary points can then be used to obtain the physical boundary mapping of the system: (19) This avoids misjudging non-first-encounter critical branches as actual stable boundaries by directly determining the boundary based solely on the mathematical roots of the augmented equation.

[0037] Based on the equipment limit margin, control release variable, and Jacobian matrix degradation characteristics in the augmented equation of the unified static voltage stability critical boundary, a posteriori discrimination is performed on the saddle-node bifurcation (SNB), the limit-induced bifurcation (LIB), and the switching of their dominant mechanisms corresponding to the critical boundary point.

[0038] 3. In probability space Set standard independent random vectors The probability distribution of the prediction error of new energy power output is as follows: ; , These are the sample space and the events on it. Algebra; , , Probability distribution and standard random vector, respectively. The range of values ​​and A real vector space of dimension , where This represents the number of independent, standardized random variables. For renewable energy power plants... Let the predicted active power output be... The actual contribution was Then predict the residuals It can be written as: (20).

[0039] Furthermore, by constructing marginal distributions, correlation models, or empirical samples, we can obtain... and random mapping relationship The relationship between new energy output and independent random variables can be obtained: (twenty one).

[0040] The active power output of each station is recorded as an independent parameter. Then, by selecting and transforming parameters, a mapping from the random output of new energy sources to the independent parameter space of the static voltage stability boundary can be established: (twenty two) in, To select operators.

[0041] For any operating region in the physical parameter space Boundary parameters The probability of falling into this area is: (twenty three) In the formula: and The scope is usually determined by the combined factors of new energy capacity, control constraints, operating mode, and engineering analysis requirements.

[0042] To describe the main landing region of a random run, consider the probabilistic principal mass region. and satisfy (twenty four) In the formula: For the probability principal mass region The probability quality covered, i.e. the confidence level of the region.

[0043] Therefore, the uncertainty of the actual output of new energy power plants is uniformly converted into boundary-independent parameters. The uncertainty. For each random sample All can be accessed through Determine the corresponding boundary parameter points so that the subsequent static voltage stability boundary coordinates, boundary point states, and critical vectors can all be considered as random variables. The response is as follows: When the prediction errors of different new energy power stations are independent, the mapping is directly constructed based on the marginal probability distribution of each station; when the prediction errors of different new energy power stations are correlated, the mapping is constructed after being transformed to an independent random coordinate space through a correlation model.

[0044] 4. Random boundary parameters Substitute into the unified static voltage stability critical boundary equation. If If the smoothing parameter τ increases monotonically from the stable operating point towards the critical boundary, then the static stability region under the smoothing parameter τ can be expressed as: (25).

[0045] The coordinates of the system's current or planned operating point in the direction of propulsion. Then given The static stability margin is: (26).

[0046] when When the running point is within the static stability region; when When the running point is at the critical boundary; when At that time, the running point had already crossed the static stability boundary.

[0047] When the independent parameters of the weak-end system of new energy are ,and When considering the power component of a single renewable energy power plant, the critical power transmission capacity can be further given: (27) In the formula: Independent parameters The total number of new energy power stations included.

[0048] Corresponding external delivery stability margin for: (28) In the formula: for The power is transmitted from the operating point below.

[0049] Therefore, this implementation not only treats the critical boundary coordinates as a random response, but also incorporates the critical total power transmitted and the stability margin from the current operating point to the critical boundary into a unified probabilistic representation, thereby enabling the description of the impact of random power output from new energy sources on static voltage stability from the perspectives of boundary position and operating margin.

[0050] 5. Map the joint probability distribution of random power output from new energy sources to the static voltage stability boundary, and calculate the out-of-domain risk when the operating point is outside the static voltage stability domain: (29).

[0051] Let the distribution function of the boundary coordinate random variable λ be... for: (30).

[0052] This leads to the quantiles of the lower boundary coordinates λ at confidence level α. for: (31).

[0053] Similarly, the static stability margin distribution function is obtained. Corresponding quantile margin Critical power distribution function quantile They are respectively: (32) (33).

[0054] Furthermore, under random operating parameters, the positional changes of the same stability boundary corresponding to different probability levels can be obtained, i.e., the probability thick boundary. The actual independent parameters... Represented as baseline parameters and random disturbances sum: (34).

[0055] In the given Below, boundary coordinates Let be a conditional random variable, and its distribution function be: (35).

[0056] quantile probability The quantiles are: (36).

[0057] According to equation (36) Termination Boundary : (37).

[0058] This yields the conditional probability thick boundary formed between the two given conditional quantile boundaries: (38) In the formula: , For two different quantiles, and have .

[0059] The corresponding boundary surface thickness is: (39) The external risk, quantile stability margin, critical power transmission quantile value, and probabilistic thick boundary are used to characterize the static voltage stability limit change and its probabilistic risk characteristics caused by random power output from new energy sources.

[0060] 6. For standard independent random vectors Select orthogonal polynomial basis functions that are compatible with its probability distribution. Where, corresponding to the total order of PCE truncation Multi-indicator set for: (40) In the formula: For all A set of multiple non-negative integer indices.

[0061] This gives the polynomial basis functions. The orthogonality relation is: (41) In the formula: Multiple indices for basis functions; It is the norm of the basis functions; This refers to the Kronecker function.

[0062] polynomial base for: (42).

[0063] In the orthogonal polynomial space, this embodiment simultaneously performs a polynomial chaotic expansion on the unified steady-state variables, boundary coordinates, and critical vectors of the static voltage stability boundary points to obtain: (43) in , These are the spectral coefficients of the unified steady-state variable at the boundary point, the boundary coordinates, and the critical vector, respectively. Used to restore the node voltage, device power, and constraint status of boundary points; Used to explicitly characterize the critical boundary positions corresponding to the random output of different new energy sources; Its function is to describe the critical degradation direction and ensure the closure of the augmented equation; To augment the critical solution coefficients A real vector space of dimension, and .

[0064] 7. To solve for the spectral coefficients, Substituting the augmented equation of the unified static voltage stability critical boundary, introducing stochasticity, and simultaneously solving the polynomial chaotic expansion, the explicit approximate expression obtained yields the residual function in the sense of truncated PCE: (44) In the formula: It is a column vector formed by stacking all spectral coefficients in a fixed order.

[0065] The Galerkin projection requires that the residual be orthogonal to every basis function in the chosen polynomial space, i.e.: (45).

[0066] In numerical computation, the use of and The numerical approximation of the discrete Galerkin equations for matching: (46) In the formula: The set of quadrature node indices used for numerical integration; , For the first Each quadrature node and its corresponding weight.

[0067] Therefore, for all The core solution equation can be obtained as follows: (47).

[0068] Solving for it yields the corresponding spectral coefficient vector. Furthermore, the corresponding spectral coefficients of each explicit approximation expression can be extracted from them. From this, the explicit static stability boundary in the random parameter space can be obtained: (48).

[0069] when If it is invertible within the parameter region, then an approximate explicit relation in the physical boundary parameter space can be further given: (49).

[0070] After the spectral coefficients are solved, for new random power output samples of new energy sources, there is no need to solve the original critical boundary augmented equations sample by sample again. It is only necessary to calculate the corresponding orthogonal polynomial basis functions and substitute them into the explicit spectral expression shown in equation (43) to obtain the corresponding boundary coordinates and boundary point states. This transforms the nonlinear critical boundary search that is repeatedly performed in a large number of random scenarios into explicit polynomial substitution calculation.

[0071] 8. To evaluate the effectiveness of the invasive PCE-Galerkin approximation explicit boundary solution, based on independent validation samples... By combining the criteria of continuous advancement and the first encounter with the terminal event, the boundary reference solutions are obtained respectively. and and from Extracting the node voltage magnitude vector The following discrimination indices are constructed. The absolute errors between the approximate solutions and the reference solutions for the boundary coordinates and nodal voltages are as follows: (50).

[0072] The root mean square error of both can be uniformly expressed as: (51) In the formula: It is a set of indices for independent validation samples, and each element corresponds to a different validation sample number.

[0073] In addition, to evaluate the ability of the explicit model to correctly recover the constraint activity state and residual constraint margin of the critical boundary points, the first definition is used. The reactive power, capacity, and current limit utilization rates of the new energy power stations are as follows: (52) (53) (54).

[0074] In the above formulas (50)-(54), the boundary coordinate error is used to evaluate the accuracy of the explicit spectral model in restoring the geometric position of the static voltage stability boundary, and the node voltage error is used to evaluate the accuracy of the explicit spectral model in restoring the system state at the boundary point; the reactive power, capacity, and current limit utilization rates are used to determine whether the boundary points obtained by the explicit spectral model have consistent equipment constraint activity states with the original non-smooth unified complementary model. Therefore, the above verification not only examines the position error of the explicit boundary, but also examines the consistency of the physical state of the boundary point and the equipment constraint mechanism.

[0075] 9. The numerical solution, verification, and probabilistic evaluation of the explicit boundary model include: Based on the given order of polynomial chaotic expansion and the probability model of random input, generate orthogonal polynomial basis functions, numerical quadrature nodes and their weights that match the probability measure of random variables, and use the deterministic critical boundary solution or the result of low-order polynomial chaotic expansion as the initial value for solving the spectral coefficients. The deterministic nonlinear equations formed by the intrusive PCE-Galerkin projection are solved iteratively. When the smoothing parameter decreases, the boundary mechanism is switched, or the convergence of the equations decreases, at least one of the following methods is used to improve the convergence of the spectral coefficient solution: Newton iteration, damped Newton iteration, line search, smoothing parameter homotopy extension, or order-increasing polynomial expansion. The spectral coefficients corresponding to the static voltage stability boundary coordinates, the unified steady-state variables at the boundary points, and the critical vector are obtained. Substitute the obtained spectral coefficients into the explicit spectral expression, calculate the static voltage stability boundary coordinates and boundary point states for the independent verification samples, and compare them with the first critical boundary reference solution obtained by the original non-smooth unified complementary steady-state model along the given advancement direction; when the boundary coordinate error, node voltage error or equipment limit utilization state does not meet the preset accuracy requirements, increase the order of the polynomial chaotic expansion, adjust the smoothing parameters, or use the original non-smooth boundary solution to re-verify the corresponding parameter region. Within the parameter region verified by the boundary approximation accuracy and physical constraint consistency, the static voltage stability boundary coordinates, boundary point physical state, static voltage stability margin, and critical power transmission for the new energy random sample are directly calculated using the explicit spectral expression. For samples in the edge region or low probability tail region of the joint probability distribution, the original non-smooth unified complementary steady-state model is used for back substitution, and the explicit boundary results are verified based on the first critical event encountered along the given propagation direction. Final output static voltage stability explicit boundary approximation expression Explicit approximation expression for boundary point states and risks outside the territory Boundary coordinate quantiles Quantile stability margin Critical power transmission quantile and probability thick boundary .

[0076] Case Analysis To verify the effectiveness of the proposed explicit unified modeling and probabilistic evaluation method for static voltage stability boundary considering constraint switching, the modified CSEE-RAS weak-end test system was first used as a theoretical verification test.

[0077] The modified CSEE-RAS system A area sender structure is as follows: Figure 2 As shown, B01 is the common grid connection point, and the external power grid on the B11 side adopts the Thevenin equivalent, whose equivalent electromotive force is... The equivalent impedance is The system baseline capacity is 1000 MVA, and the baseline voltage is 500 kV. Wind farms W1-W4 are connected to nodes B03, B04, B06, and B05, respectively. Basic parameters for each wind farm are detailed in Table 1. Wind power data is sourced from the Australian Energy Market Operator (AEMO) and has been scaled to match the power scale of the example system using capacity equivalence scaling. This paper uses the wind power prediction residual as a stochastic source of uncertainty in renewable energy output and employs a Beta distribution to construct the output power of W1-W3, i.e., independent parameters. The probabilistic bounded model is used; the correlation between multiple stations is described by Gaussian Copula, and the generated random power samples are mapped to the static stable boundary parameter space. The correlation coefficient matrix is: (55).

[0078] W4 output of GFM type As a boundary propulsion power variable, i.e. This is used to characterize the static stability limit of the weak-end system under the influence of random fluctuations in wind farm output. The probabilistic model parameters for the output of each GFL unit are detailed in Table 2. The line parameters adopt the standard parameters of the CSEE-RAS system, and equivalent processing is performed on multiple parallel lines.

[0079] Table 1 Basic parameters of wind farm

[0080] Table 2. Probabilistic Model Parameters for Active Power Output of GFL-type Wind Farms

[0081] To examine the true state of the stability boundary of the simulation system and obtain physical reference results independent of the PCE approximation, the given equation (10) is applied. When taking 490MW, 505MW and 520MW respectively, and along and The direction is added to solve for the first statically stable critical coordinates reached by the system. Based on this, three sets of non-smooth reference boundary surfaces were generated, such as... Figure 3 As shown in (a), the statistical results of the number of boundary mechanism points and the boundary coordinate range of each surface are shown in Table 3.

[0082] Table 3 Different P W3 Statistical results of the boundary zone points below

[0083] Depend on Figure 3 As can be seen in (a) and Table 3, the critical boundary coordinates Follow and The overall trend of the increase is decreasing, indicating that after the active power output of the GFL-type wind farm increases, the critical active power space for wind farm W4 to continue increasing is correspondingly reduced, thus compressing the static voltage stability margin available for W4 to continue increasing active power output. Although the boundary coordinates remain continuous throughout the parameter region, there are still local alternations and narrow switching mechanisms between the SNB and LIB dominant regions, indicating that the unified physical boundary can be formed by the first encounter points of different critical branches. When the SNB branch intersects or approaches the LIB branch induced by the equipment limit in the parameter space, the dominant condition determining the system stability limit can switch, while the critical boundary coordinates themselves can still remain continuous, but their local gradients may undergo abrupt changes or significant changes. Specifically, with The capacity increased from 490MW to 520MW, and the proportion of SNB points increased from 24.33% to 65.33%, indicating a gradual shift in the boundary-dominant mechanism from LIB to SNB. It should be noted that within the aforementioned verification area, the first-time device constraint encountered at all 498 LIB reference points was the W4 current limit.

[0084] To visually verify the local structure of the dominant mechanism switching, further fixation , and along Adding directions to construct stable boundary slices, such as Figure 3 As shown in Figure (b), the switching mechanism between SNB and LIB under this operating condition is located at [488.949, 491.324] MW. Within a narrow range of variation, the LIB branch, dominated by the W4 current limit, and the SNB branch approach each other and compete. Even minor changes can trigger a switch in the dominant mechanism, indicating that the location of the mechanism switch is highly sensitive to operating parameters.

[0085] Based on the numerical settings shown in Tables 1 and 2, the random parameter mapping is substituted into the unified boundary equation of equation (18), and then... A third-order invasive PCE-Galerkin projection is performed using a homotopy method to obtain the third-order PCE spectral coefficients (K=3). Then, according to equation (49), the following is obtained: When taking 490MW, 505MW and 520MW respectively, and along and The third-order explicit approximation boundary with added direction is as follows: Figure 4 As shown.

[0086] To evaluate the accuracy of the explicit approximate statically stable boundary, respectively in Figure 3 (a) and Figure 4 On the three curved surfaces shown, 300 sampling boundary points were taken from each surface according to an equally spaced grid, and the distribution error of the boundary coordinates on the corresponding curved surfaces was calculated. The results are as follows. Figure 5 As shown. It can be seen that there are 3 fixed... Boundary coordinates of the cross section The absolute error distributions are generally similar, with the absolute error in most verification areas below 0.2MW, and the maximum value occurring at coordinates (570, 472). At that time, the larger absolute errors were mostly concentrated at a few edges of the verification window, with a maximum value of 0.7281MW, but the high error region was relatively small; as As the magnitude of the error increases, the maximum absolute error decreases, while the area covered by contour lines with moderate error relatively increases. Therefore, it can be seen that... The increase in magnitude did not lead to a monotonically increasing local maximum absolute error, but rather transformed it from a concentrated deviation at a few edge points into a relatively wider distribution of smaller amplitude local deviations. Meanwhile, relative to the rated capacity of each wind farm shown in Table 1, 10 -2 The order-of-magnitude RMSE error also demonstrates that the third-order explicit boundary maintains a small and stable approximation accuracy throughout the region of the physical parameters under investigation.

[0087] Only verify boundary coordinates The explicit approximation model cannot fully demonstrate that it effectively preserves the physical meaning of the unified boundary equation shown in equation (18). Therefore, node voltages and device limit utilization rates are further extracted from the PCE state spectrum and compared with the actual non-smooth reference boundary to verify the approximation accuracy of the proposed method for the original non-smooth boundary. First, the node voltage distributions of the proposed method at the 50th percentile and the actual non-smooth model, as well as the statistical results of the 50th percentile absolute error and RMSE error between the node voltage approximation values ​​and the non-smooth reference values ​​at the selected 900 comparison grid points, are as follows: Figure 6 As shown in the figure, the node voltage curves of the proposed method at the 50th percentile and the actual non-smooth model basically coincide, and the absolute error and RMSE error between the two calculated by equations (50) and (51) are only within 10. -3 and 10-4 The magnitude of the data verified the boundary approximation accuracy of the proposed method. In comparison, due to the presence of structured network equipment at node B05, its reactive power / voltage support capability is stronger, its node voltage is closer to the rated value, and it is the least sensitive to boundary approximation, thus achieving the highest approximation accuracy. Overall, the proposed explicit model can accurately recover the statically stable boundary voltage state of different nodes within the investigated parameter region, without exhibiting a significant amplification of voltage errors at individual nodes.

[0088] In addition, according to equations (52)-(54), the reactive power, capacity, and current limit utilization rates of the new energy power station W4 obtained by the PCE-Galerkin method at the above 900 verification points are calculated for the explicit approximation model and the non-smooth reference model, respectively, and then the ratio of the two is obtained as shown in the figure. Figure 7 The dashed line in the figure represents the 45° consistency discrimination curve. As can be seen from the figure, the ratios of reactive power, capacity, and current limit utilization of W4 are all highly consistent with the 45° line, indicating that the explicit model can not only approximate the physical distribution of critical boundary coordinates and voltage, but also maintain a residual constraint margin consistent with the actual reference at its approximately static stable boundary points.

[0089] To quantify the risks of renewable energy transmission, based on the Beta margin distribution and Gaussian Copula correlation analysis of each station shown in Table 2, 10 independent standard Gaussian spaces were generated. 5 The W4 critical active power of the third-order invasive PCE-Galerkin explicit boundary corresponding to each sample is directly calculated using Equation (49). The critical total power to be transmitted can be obtained according to equation (27). For the active power output of W4, its risk of transmission outside the region can be obtained from equation (29). The statistical results of the above variables are as follows: Figure 8 As shown. By Figure 8 As can be seen in (a), The probability density of the power transmission capacity exhibits a unimodal distribution. Within the region with higher power output (940-960 MW), the risk curve rises rapidly, meaning that even small changes in output within this region can cause significant changes in external risk. At the two extremes of the distribution, the risk curve changes relatively slowly due to the lower probability of the corresponding random states. Specifically, when the output of W4 is 950.38 MW, the risk of static instability in transmission is approximately 50%; when the output of W4 increases to 976.64 MW, the risk probability reaches 95%. Similarly, the critical total power transmission capacity... The probability density and risk curves are as follows: Figure 8 As shown in (b), with Figure 8Compared to (a), the latter has a more concentrated probability distribution and a narrower static stability range for power transmission. When the total power transmitted reaches 2550.34MW, the risk of static instability is about 50%, and when it reaches 2553.09MW, the risk of power transmission instability reaches 90%.

[0090] To analyze the impact of the randomness of new energy output on the static stability boundary, under a given... and The following analysis and consideration Spatial differences in boundary coordinates λ under boundary randomness. The power output of station W3 is calculated using equation (49). exist The boundary coordinates λ corresponding to the boundary quantiles of 50% and 95%, i.e., the boundary surface of W4 output force. like Figure 9 As shown in (a). As can be seen from the figure, with... , As the value increases, all three quantile surfaces show a decreasing trend, indicating that when the outputs of W1 and W2 are large, the available critical active space of W4 is limited. Furthermore, in the case of... - Within the region, the area between the 5% and 95% conditional quantile surfaces constitutes the central 90% probability boundary band. The corresponding conditional probability thick boundary can be obtained from equations (38) and (39), and its boundary thickness can be calculated. The spatial distribution of the boundary thickness is as follows: Figure 9 As shown in (b).

[0091] Figure 9 Figure (b) shows that within the investigated region, the central 90% condition thickness is between 27.826 MW and 30.065 MW, indicating that the randomness of new energy sources extends the deterministic static voltage stability limit into a critical region with a probabilistic width. In contrast, the boundary thickness variation in the example is relatively small, only 2.24 MW, which suggests that within the investigated region, the... Conditional randomness causes The dispersion of the critical position is generally stable, remaining at around 30MW.

[0092] Furthermore, in terms of computational efficiency, the proposed explicit approximation model was compared with the non-smooth unified boundary pointwise method and the extended CPF commonly used in static stability analysis to verify the advantages of the proposed method in terms of computational performance. The three methods were used to solve 10,000 identical Monte Carlo samples under the same computational environment, and the statistical results of their respective computational time are shown in Table 4.

[0093] Table 4 Comparison of computation time

[0094] In Table 4, the initial modeling time refers to the one-time computation time required to solve the explicit approximation expression. The non-smooth unified boundary pointwise method and the extended CPF do not require pre-solving the explicit boundary expression, therefore only the sample computation time is included. As shown in Table 4, due to the initial modeling time, the computation time for the first 100 points of the proposed model is higher than the two comparative methods; however, as the computational scale increases, the computational time of the comparative methods increases significantly. The total time for the proposed method to solve 10,000 sample points is 429.8510 s, which is approximately 18.2 times faster than the non-smooth unified boundary pointwise method and the extended CPF method, respectively. The time spent solving sample points accounts for a very small percentage, only 12.56 parts per million. The results show that the computational cost of the proposed explicit model is mainly concentrated in the one-time solution of the explicit approximation boundary expression. As the sample size increases or the number of explicit model calls increases, the proportion of the one-time modeling cost in the total computation gradually decreases, verifying the computational efficiency advantage of the proposed method in large-scale probability assessment and multi-scenario repeated analysis.

[0095] Taking a new energy base in a region lacking synchronous power supply as a real-world system example, its equivalent topology is as follows: Figure 10 As shown. Wind farms W1-W5 are connected to nodes 1-5 respectively. GFL type W1-W4 use PQ control, with rated active power outputs of 500, 300, 500, and 500MW respectively; GFM type W5 uses V / Q droop control, with a rated active power output of 400MW, and its reactive power, capacity, and current upper limits are 90Mvar, 440MVA, and 1.16pu respectively. Each wind farm cluster is collected at 330kV and stepped up to 750kV, then connected to the external synchronous grid via a long-distance transmission channel. The unit impedance of the 330kV and 750kV transmission lines is 0.05+j0.31Ω / km and 0.01167+j0.27Ω / km respectively. This section still uses the wind power output data published by AEMO, which is processed as necessary and applied to the numerical example verification. To better describe the state of the approximate explicit boundary in space, this invention adopts the fourth-order invasive PCE-Galerkin method. Based on this, the active power output of the GFL-type cluster wind farm is selected as the independent boundary parameter, i.e. The edge distribution parameters of its active power output are shown in Table 5. The Gaussian Copula correlation matrix of each cluster wind farm under the edge distribution is as follows: (56).

[0096] Table 5 Edge distribution parameters of active power output in GFL-type wind farms

[0097] The random variable vector of each wind power cluster is ; using the active power output of W5 as the subordinate boundary coordinate, i.e. The critical total power transmission capacity is defined by the active power flow of the 750kV transmission section. .

[0098] Since this example contains four independent standard normal coordinates — This corresponds to a hypersurface in the four-dimensional input space. If two physical power variables are directly fixed, after Gaussian Copula correlation mapping, the physical cross-section is... In space, the coordinate plane is usually no longer a regular coordinate plane, which increases the complexity of inverse mapping and verification point construction. To observe the local morphology of the explicit boundary in three-dimensional space, this invention fixes the boundary in an independent probability coordinate space. and The independent variable, that is, let Under the conditions, construct Corresponding to - Two-dimensional input space cross section such as Figure 11 As shown, its physical meaning is given. and corresponding and The 50th conditional quantile value. As can be seen from the graph, , The variation ranges are [242.37, 306.33]MW and [143.10, 186.63]MW, respectively, with corresponding critical coordinates. The variation range is [289.93, 372.06] MW, indicating that the output variation of multiple wind farms will significantly change the critical output upper limit of W5 along the active power propulsion direction.

[0099] In addition, the 100 boundary points of this section include 57 SNB points and 43 LIB points, of which 42 LIB points are triggered by the reactive power limit of W5, and another one is triggered by the current limit of W5. Although Figure 11 The locally closed or intersecting boundary segments reflect the switching between different device constraints and SNB mechanisms in the parameter space, but the change in the dominant mechanism did not cause the critical coordinates. The fracture further validates the ability of the explicit boundary on this representative section to approximate the local mechanism switching structure.

[0100] To compare the approximate accuracy of different random boundary coordinates, in Under the conditions, respectively corresponding to The 93.32%, 50%, and 6.68% quantiles were taken as follows: The values ​​are 0 and -1.5. Then, 100 sampling boundary points are taken at equal intervals on each of the three representative sections, and the spatial distribution of the error at each point is statistically analyzed. Figure 12 As shown. Figure 12 (a) corresponds to The error of the W4 condition high quantile section mainly follows the path of A localized banded distribution was formed, with a maximum absolute error of 0.507 MW. This section contains 82 SNB points and 18 LIB points, with the boundary dominated by SNB, indicating that the increase in local error does not only occur in the LIB or equipment constraint switching area; Figure 12 (b) corresponds to The W4 condition median section, except for the low ,Low Apart from isolated large errors near the corner points, most other areas maintained a low error level. Although the maximum absolute error of this section reached 0.733MW, its 95th percentile error was only 0.216MW, indicating that the maximum error was mainly caused by a few local points and did not represent the approximate level of the entire section. Figure 12 (c) corresponds to The W4 condition low quantile section has larger errors mainly concentrated in the low quantile section. On one side and its adjacent local region, the spatial non-uniformity is significantly stronger than on the other two sections. Therefore, the approximation error of the explicit boundary does not increase uniformly across the entire parametric section, but is mainly concentrated in local regions outside the probability center. The conditional median section has the highest overall approximation accuracy, while the approximation of the conditional low quantile section is relatively more difficult. This result demonstrates that the PCE-Galerkin model can cover the main probability region well, but also indicates that it is still necessary to combine deterministic non-smooth boundary solutions for verification when conducting extreme risk analysis in low-probability marginal regions.

[0101] In summary, the proposed explicit model can approximate the critical boundary formed by the SNB and LIB surfaces well, and maintains high accuracy within the main region of the joint probability. Error comparisons across different representative cross sections show that the approximation error increases with the degree to which the random input deviates from the probability center, but larger errors are mainly concentrated in local edge regions and do not appear universally across the entire parameter cross section.

[0102] Based on the aforementioned verification of the accuracy of the explicit boundary approximation, further research was conducted on the critical output. With critical total output power The probability assessment. First, generate 10 in an independent standard Gaussian space. 7 The correlated active power outputs of the Monte Carlo samples were obtained by performing Gaussian Copula correlation transformation and inverse Beta transformation on samples W1–W4. These were then substituted into the PCE-Galerkin explicit boundary to obtain the critical boundary coordinates for each sample. And calculate the critical total power transmitted based on the boundary point states. . and The probability statistics are shown in Table 6, and their probability density and cumulative distribution are as follows: Figure 13 As shown.

[0103] Table 6 and Probability statistics results

[0104] critical The 5th, 50th, and 95th percentiles are 286.75MW, 332.77MW, and 375.83MW, respectively. Combined with the current operating point... =219.46MW, with quantile stability margins of 67.29MW, 113.31MW, and 156.37MW, respectively. Significant differences exist among the three quantile margins, with the 5%–95% quantile margin spanning 89.08 MW, indicating that stochastic output significantly alters the critical margin of W5 along the active power propulsion direction. compared to, Figure 13 Critical total power transmitted in (b) The distribution is more concentrated. The 5%, 50%, and 95% stability margins are 85.42 MW, 95.39 MW, and 103.57 MW, respectively, with a range of 18.15 MW between the 5% and 95% quantiles, indicating that random power output has little impact on the stability margin of the critical power output.

[0105] In summary, the PCE–Galerkin explicit boundary can maintain a low approximation error within the validated principal probability region and avoids solving non-smooth boundary equations for a large number of random samples one by one in probability assessment. Furthermore, it can obtain stability margins at different quantile levels, providing a theoretical basis for the verification and configuration of stability margins for new energy transmission.

Claims

1. An explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending-end system, characterized in that: The steps are as follows: S1. Incorporate network power balance, equipment steady-state external characteristics, equipment control conditions and equipment operating limits into a unified steady-state model, and then obtain a non-smooth unified complementary steady-state model that takes into account equipment constraint switching by the complementary relationship between equipment limit margin and control release variables. S2. The smooth Fischer-Burmeister function is used to process the non-smooth unified complementary steady-state model to obtain the smooth unified steady-state equation. The smooth unified steady-state equation, Jacobi degeneracy condition and critical vector normalization condition are combined to construct the unified static voltage stability critical boundary augmented equation. S3. Random boundary parameters obtained through random mapping relationships Substituting into the augmented equation of the unified static voltage stability critical boundary, we obtain the static voltage stability boundary coordinates, the unified steady-state variables of the boundary points, and the critical vector. S4. Polynomial chaotic expansion is performed on the static voltage stability boundary coordinates, boundary point unified steady-state variables and critical vectors through orthogonal polynomial basis functions, and a deterministic nonlinear equation system about the polynomial spectral coefficients is established by intrusive PCE-Galerkin projection. After solving, an explicit spectral mapping is obtained. S5. Calculate the static voltage stability boundary coordinates, critical power transmission, boundary point physical state, and equipment constraint state corresponding to different new energy random outputs based on explicit spectral mapping, and calculate the external risk of the operating point crossing the static voltage stability boundary. Quantile stability margin at different quantile probability levels Thick probability boundaries formed between different conditional quantile boundaries This enables explicit unified modeling and probabilistic evaluation of the static voltage stability boundary of new energy weak-sending systems.

2. The explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending-end system according to claim 1, characterized in that: The nonsmooth unified complementary steady-state model is as follows: （10） In the formula, For steady-state algebraic variables, As a subordinate advancing coordinate, For independent parameter vectors, It is a non-negative vector. For an algebraic equation that always holds true under a fixed control mode, This is the vector form of the reactive power control release equation. For the limit margin function, "" indicates a complementary relationship, and for the j-th limit value, we have and .

3. The explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending-end system according to claim 1, characterized in that: The unified static voltage stability critical boundary augmented equation is: （18） In the formula, To unify steady-state variables, It is a non-zero critical vector. For smoothing parameters, For the transpose of a given reference direction vector, To account for the smooth uniform equation of constraint switching, To account for the smooth uniform equations regarding constraint switching The transpose of the Jacobian matrix.

4. The explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending-end system according to claim 1, characterized in that: Random boundary parameters The process of obtaining it is as follows: For renewable energy power station i, let the predicted active power output be... The actual contribution was Then predict the residuals Written as: （20） It is constructed through marginal distribution, correlation model, or empirical samples. Random mapping relationship with ξ The relationship between new energy output and independent random variables was obtained: （21） In the formula, To make predictions and contribute efforts, for The random mapping relationship with ξ; The active power output of each station is recorded as an independent parameter. Then, by selecting and transforming parameters, a mapping is established from the random output of new energy sources to the independent parameter space of the static voltage stability boundary: （22） In the formula, For the physical domain of the boundary parameters, For the mapping from standard random variables to the boundary parameter space, To select operators.

5. The explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending-end system according to claim 1, characterized in that: In random boundary parameters When substituting into the augmented equation for the unified static voltage stability critical boundary, if λ increases monotonically from the stable operating point towards the critical boundary, and when the augmented equation for the unified static voltage stability critical boundary has multiple candidate critical solutions under given independent parameters, the coordinates of the first candidate critical boundary encountered are selected as the effective static voltage stability boundary coordinates according to the given advancing direction from the stable operating point along the subordinate advancing coordinates. Then, the static stability domain under the smoothing parameter τ is expressed as: （25） In the formula, For the set of real numbers, For the boundary mapping of the subordinate advancing coordinate λ; The coordinates of the system's current or planned operating point in the direction of propulsion. Then, the static stability margin for a given q is: （26） In the formula, The coordinates of the system's current or planned operating point in the direction of propulsion; when When the running point is within the static stability region; when When the running point is at the critical boundary; when At that time, the running point had already crossed the static stability boundary; When the independent parameters of the new energy weak transmission end system are When λ is also a power component of a single renewable energy power station, the critical power transmission capacity is further given: （27） In the formula, Let q be an s-dimensional column vector of all 1s, where s is the total number of new energy power stations included in the independent parameter q; Corresponding external delivery stability margin for: （28） In the formula: for The power is transmitted from the operating point below.

6. The explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending-end system according to claim 1, characterized in that: The polynomial chaotic expansion yields: （43） In the formula, To obtain the unified steady-state variable approximation for boundary points using the PCE representation truncated by the total order K, The approximate boundary coordinates are obtained using the PCE representation truncated with the total order K. This is the critical vector approximation obtained by using the PCE representation truncated with the total order K. , These are the spectral coefficients of the unified steady-state variable, critical vector, and boundary coordinates at the boundary point, respectively. for 3D real vector space, For index The corresponding polynomial chaotic basis functions, This is the first multi-index. To correspond to the total order of PCE truncation A set of multiple indicators.

7. The explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending-end system according to claim 1, characterized in that: To solve for the polynomial spectral coefficients, in By substituting randomness into the augmented equation of the unified static voltage stability critical boundary and simultaneously solving the polynomial chaotic expansion, the explicit approximate expression is obtained, yielding the residual function in the sense of truncated PCE: （44） In the formula, For the mapping from standard random variables to the boundary parameter space, To unify the augmented equation for the static voltage stability critical boundary, c is a column vector formed by stacking all spectral coefficients in a fixed order. It is a standard independent random vector; The Galerkin projection requires that the residual be orthogonal to every basis function in the chosen polynomial space, i.e.: （45） In the formula, residual function Galerkin projection residual in the direction of the β-th polynomial chaotic basis function Let ξ be the joint probability measure of a standard independent random vector. For joint probability measure Defined inner product, For the polynomial chaotic basis function corresponding to the multiple index β; β is the second multiple index; Use and The numerical approximation of the discrete Galerkin equations for matching: （46） In the formula, To unify the augmented equations for the static voltage stability critical boundary, The set of quadrature node indices used for numerical integration; , For the first Each quadrature node and its corresponding weight; Therefore, for all The core solution equation is obtained as follows: （47） In the formula, This is the residual vector of the overall nonlinear equation system formed by stacking all the corresponding Galerkin projection residuals in a fixed order; Solving for the spectral coefficients yields the corresponding vector. Extract the corresponding spectral coefficients of each explicit approximation expression to obtain the explicit static stability boundary in the random parameter space: （48） In the formula, For an explicit statically stable boundary in the random parameter space, Let ξ be the range of values ​​for the standard random vector. when If it is invertible within the parameter region, then an approximate explicit relation in the physical boundary parameter space is further given: （49） In the formula, Let be the explicit function of the static voltage stability boundary coordinates in the physical boundary parameter space obtained by the K-order PCE approximation. Random mapping for contributing to new energy The inverse mapping of .

8. The explicit unified modeling method for the static voltage stability boundary of a new energy weak-sending-end system according to claim 1, characterized in that: Off-domain risks of operating point crossing static voltage stability boundary Represented as: （29） In the formula, The probability of an event occurring. This is an indicator function for events outside the domain. Let q be the integral measure of the probability measure of the random boundary parameter vector q in the physical parameter space Q; Quantile stability margin at different quantile probability levels Represented as: （32） In the formula, Let τ be the value of the static stability margin random variable under the smoothing parameter τ. Let be the cumulative distribution function of the static stability margin random variable; Thick probability boundaries formed between different conditional quantile boundaries Represented as: （38） In the formula, As a baseline independent parameter vector, For parameter points in the extended boundary parameter space, which consists of a reference independent parameter vector and subordinate propagation coordinates, Reference independent parameter vector The physical domain, , These are the quantile probability levels corresponding to a given baseline independent parameter vector. , The static voltage stability boundary coordinate quantiles, , These are two given quantile probability levels.

9. A low-frequency voltage response mode attribution system for multi-active support equipment at the weak transmission end of a new energy source, used to implement the method of claims 1, 2, 3, 4, 5, 6, 7 or 8, characterized in that: include: The unified steady-state modeling module is used to incorporate network power balance, device steady-state external characteristics, device control conditions and device operating limits into a unified steady-state model, construct device limit margin functions and control release variables, and characterize the control state switching before and after device limit triggering through the complementary relationship between device limit margin and control release variables, thus obtaining a non-smooth unified complementary steady-state model that takes into account device constraint switching. The critical boundary construction module is used to process the complementary relationship in the non-smooth unified complementary steady-state model using the smooth Fischer-Burmeister function, establish the smooth unified steady-state equation, and, based on the Jacobi degeneracy condition of the smooth unified steady-state equation with respect to the unified steady-state variables, introduce the non-zero critical vector and the critical vector normalization condition to construct the unified static voltage stability critical boundary augmented equation. The random parameter mapping module is used to establish a random mapping between standard independent random vectors and random output of new energy sources based on the probability distribution of the prediction error of new energy power output and the correlation between multiple power stations. It also obtains random boundary parameters through parameter selection and transformation, and substitutes the random boundary parameters into the unified static voltage stability critical boundary augmented equation so that the static voltage stability boundary coordinates, boundary point unified steady-state variables and critical vectors form a random response. The explicit spectral mapping solution module is used to select orthogonal polynomial basis functions that are compatible with the probability measure of standard independent random vectors, perform truncated polynomial chaotic expansion on the static voltage stability boundary coordinates, unified steady-state variables of boundary points, and critical vectors, substitute the obtained expansion into the augmented equation of the unified static voltage stability critical boundary to form a residual function, and use Galerkin projection to make the residual function orthogonal to each basis function in the selected polynomial space, construct a deterministic nonlinear equation system about the polynomial spectral coefficients, and solve it to obtain the explicit spectral mapping between the random power output of new energy and the static voltage stability boundary; The probability assessment module is used to calculate the static voltage stability boundary coordinates, critical power transmission, physical state of boundary points and equipment constraint state corresponding to different new energy random outputs based on the explicit spectrum mapping, and to calculate the extra-domain risk of the operating point crossing the static voltage stability boundary, the quantile stability margin under different quantile probability levels and the probability thick boundary formed between different conditional quantile boundaries. The result output module is used to output one or more of the following: static voltage stability boundary coordinates, critical power transmission, boundary point physical state, equipment constraint state, external risk, quantile stability margin, and probabilistic thick boundary. Each module is implemented by at least one processor executing a computer program stored in memory.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the computer program is executed by the processor, it implements the explicit unified modeling method for the static voltage stability boundary of the new energy weak-sending end system as described in any one of claims 1 to 8.

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