A capacity matching method for grid-forming and grid-following converters based on semidefinite programming

Abstract

The present invention discloses a method for capacity ratioing of grid-type and grid-following converters based on semidefinite programming, comprising the following steps: S1, establishing state space models of the grid-type converter and the grid-following converter respectively, and constructing an overall state space model of the station containing the grid-type and grid-following converters through coordinate transformation; S2, describing the stability index using semidefinite programming based on the Lyapunov equation, and deriving an analytical expression for the stability sensitivity of the grid-type converter ratio based on the primal variable and dual variable of the semidefinite programming; S3, initializing the grid-type converter ratio, and iterating with the stability sensitivity of the grid-type converter ratio as the gradient until the objective function value reaches the stability index limit or the difference between the objective function values ​​of two iterations is less than the convergence gap, thereby obtaining the optimal grid-type converter ratio. The present invention can accurately and quickly solve the optimal grid-type converter ratio, thereby ensuring the stability and economy of the new energy power generation system.

Description

Technical Field

[0001] The present invention relates to the technical field of operation control of a new energy power generation system, and in particular to a capacity ratio matching method for grid-forming and grid-following converters based on semi-definite programming. Background Art

[0002] In renewable energy power generation systems, the control strategies for connecting renewable energy sources like wind power and photovoltaics to the AC grid via converters are categorized into two main types: grid-following control and grid-forming control. Grid-following control is the most widely used. Grid-following converters offer excellent stability in strong grids with high short-circuit rates. However, as the "dual-high" characteristics of new power systems deepen, renewable energy sources gradually replace traditional synchronous power sources, reducing grid strength. Grid-following converters are prone to small-disturbance stability issues such as subsynchronous and supersynchronous oscillations.

[0003] By simulating the control methods of synchronous generators, grid-type converters exhibit characteristics that are dual to those of grid-following converters, demonstrating excellent weak-grid adaptability. Existing research has shown that grid-type converters exhibit "positive resistance" circuit characteristics, which can mitigate the negative damping introduced by grid-following converters, significantly improving system oscillation damping and enhancing system stability.

[0004] However, while grid-based control improves stability, it also sacrifices economic efficiency. On the one hand, grid-based equipment is expensive, and on the other hand, its active support control requires active power reserve, which reduces operational economics. Therefore, an accurate and fast grid-based converter ratio optimization method is urgently needed to determine the minimum grid-based converter ratio that meets stability requirements. Summary of the Invention

[0005] The purpose of the present invention is to provide a capacity ratio matching method for grid-forming and grid-following converters based on semi-definite programming, which can accurately and quickly solve the optimal grid-forming converter ratio and ensure the stability and economy of the new energy power generation system.

[0006] To achieve the above objectives, the present invention provides a method for capacity matching of grid-forming and grid-following converters based on semidefinite programming, comprising the following steps:

[0007] S1. Establish state space models of grid-forming converters and grid-following converters respectively, and construct the overall state space model of the station including grid-forming converters and grid-following converters through coordinate transformation;

[0008] S2. Use the semidefinite programming based on the Lyapunov equation to describe the stability index, and derive the stability sensitivity analytical expression of the grid-type converter ratio according to the primal and dual variables of the semidefinite programming;

[0009] S3. Initialize the grid-type converter ratio, and iterate with the stability sensitivity of the grid-type converter ratio as the gradient until the objective function value reaches the stability index limit or the difference between the objective function values ​​of two iterations is less than the convergence gap, and then stop, to obtain the optimal grid-type converter ratio.

[0010] Preferably, in S1, the wind and solar power stations include Grid-type converter and The grid-type converter uses the equivalent impedance Z c connected to the power grid;

[0011] The construction process of the overall state space model of the station is as follows:

[0012] The mathematical model of each control link and filter circuit of the grid-following converter is:

[0013] ;

[0014] Where, 、 They are the phase angle and frequency obtained by the grid-type phase-locked loop, 、 They are the phase-locked loop PI parameters, 、 are the power outer loop PI parameters, 、 are the current inner loop PI parameters, 、 are the active and reactive power output by the converter respectively, represents the reference value, is the filter inductor, 、 are the filter outlet current and voltage respectively, represents the reference value, is the converter output voltage.

[0015] Preferably, the mathematical models of each control link and filter circuit of the grid-type converter are:

[0016] ;

[0017] Where, The frequency obtained by network-type virtual synchronous control is D 、 J are the damping coefficient and inertia of the virtual synchronous control loop, E is the internal potential obtained by voltage droop control, is the voltage droop coefficient, k pv 、 k ivare the voltage outer loop PI parameters, k pc 、 k ic are the current inner loop PI parameters, P 、 Q are the active and reactive power output by the converter respectively, represents the reference value, L f is the filter inductor, 、 are the filter outlet current and voltage respectively, represents the reference value, is the converter output voltage.

[0018] Preferably, the mathematical models of the grid-type converter, the grid-forming converter, and the AC network are combined by coordinate transformation and linearized at the equilibrium point to obtain the overall state space model of the station:

[0019] ;

[0020] Where, is the system state variable, the matrix A is the system state matrix, the small signal stability of the system is related to the matrix A is related to the eigenvalues ​​of .

[0021] Preferably, in S2, the Lyapunov equation is used to describe the stability of the system. Lyapunov's second method states that the system is stable if and only if there exists a real symmetric positive definite matrix satisfy , the system is asymptotically stable, and the related Lyapunov function is , is any given negative real number;

[0022] The stability index is further designed as the solution to the following semidefinite programming problem:

[0023] ;

[0024] The Lyapunov equation requires , so set a constant that tends to 0 Make , which is consistent with is equivalent to setting , if the system is stable, the stability index obtained by semidefinite programming is will be negative;

[0025] Stability index The absolute value of is the lower limit of the convergence rate after perturbation, and from the above semidefinite programming problem we get:

[0026] ;

[0027] Define an analytical formula for accurately calculating stability sensitivity, and derive the sensitivity of Jacobian matrix elements based on the convexity of semidefinite programming. The analytical expression of the stability sensitivity is obtained by using the chain rule. .

[0028] Preferably, stability sensitivity The derivation method is as follows:

[0029] First, change the original form of the semidefinite programming problem above into the standard form:

[0030] ;

[0031] in, ,vector Including stability indicators and the symmetric matrix The lower triangular elements of , , ; Consider all the constraints of the original semidefinite program and define ,in , , Indicates all n A set of dimensional real symmetric matrices; corresponding to the constraints in the aforementioned original semidefinite programming, , , , , , , , , , represent n The basis of the symmetric matrix, ;

[0032] Next, write the standard form of the dual problem of the above standard form semidefinite programming problem:

[0033] ;

[0034] Among them, the symmetric matrix Z Represents the original variable The dual variable of

[0035] Then, according to Slater's condition, there exists a strictly feasible solution to the original problem , making , and there exists a strictly feasible solution to the dual problem that satisfies and , ; This condition ensures the strong duality of the semidefinite programming problem, at the optimal value The duality gap at is 0: ; and because and ,have to:

[0036] ;

[0037] Since the proposed semidefinite programming problem is based on matrix inequalities, while the standard convex problem is based on scalar inequalities, in order to conveniently derive parameter sensitivities using dual variables, the "svec" calculation is used to vectorize the matrix in the semidefinite programming problem, resulting in a standard convex optimization form. The parameter sensitivity formula for the semidefinite programming problem is then established in the same way as for the traditional convex problem.

[0038] In order to facilitate the representation and matrix operations, define the function "svec": Mapped to , ; Then, the relationship between matrix and vector calculation is established through the "svec" function and inner product:

[0039] ;

[0040] ;

[0041] make , then the above formula is expressed as:

[0042] ;

[0043] Since the duality gap is 0, we have:

[0044] ;

[0045] Using symmetric Kronecker products To simplify the expression, it is defined as dimensional symmetric matrix X ,exist M , N ∈ , such that:

[0046] ;

[0047] therefore Expressed as:

[0048] ;

[0049] Then, by combining and , define the function G :

[0050] ;

[0051] in, , given the parameters The optimal value of Department In essence, G It can be regarded as a variant of the KKT condition under the matrix inequality, since c and are all continuously differentiable, and the derivative of the function is expressed as:

[0052] ;

[0053] At the optimal value is non-singular, parameter sensitivity The calculation formula is:

[0054] ;

[0055] Among them, the derivative Expressed as:

[0056] ;

[0057] vector The first element is the stability index , for the sake of simplicity, we define Vector , so the stability index Too The first element of

[0058] Finally, by applying the chain rule, we obtain the stability index Sensitivity to the controllable variable, i.e., the grid-type converter ratio:

[0059] ;

[0060] in, is the Jacobian matrix element pair containing The partial derivative of .

[0061] Preferably, in S3, the initialization grid-type converter ratio is the upper limit of the site grid-type converter ratio , set the stability index limit to , the optimization algorithm process is as follows:

[0062] S31. Set the initial value of the optimized grid-type converter ratio parameter ,make ;

[0063] S32, calculate the objective function value according to the method in step S2 Sensitivity and stability ;

[0064] S33, determining the termination condition;

[0065] S34, iterative update of grid-type converter ratio :

[0066] ;

[0067] in, is the iteration step, and the updated Substitute into step S32, let .

[0068] Preferably, in S33, specifically:

[0069] Termination condition 1: When When , the difference between the objective function values ​​of the two iterations satisfies the convergence gap ε, that is:

[0070] ;

[0071] Termination condition 2: When the stability index is within the limit Range Within, that is:

[0072] ;

[0073] If any of the termination conditions is met, the algorithm stops and outputs the optimization result. If none of the termination conditions are met, the algorithm proceeds to step S34 to iteratively update the grid-type converter matching parameters.

[0074] Therefore, the present invention adopts the above-mentioned method of capacity ratio of grid-type and grid-following converters based on semidefinite programming, which can use the strong duality of semidefinite programming to analytically obtain stability sensitivity, and further obtain the optimal grid-type converter ratio through iteration, avoiding the process of traditional numerical perturbation method to solve approximate sensitivity through multiple optimizations, significantly improving calculation efficiency and accuracy, and realizing accurate and rapid solution of the optimal grid-type converter ratio, thereby ensuring the stability and economy of the new energy power generation system.

[0075] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0076] Figure 1 This is a flow chart of an embodiment of a method for capacity ratio matching of grid-forming and grid-following converters based on semidefinite programming according to the present invention;

[0077] Figure 2 Schematic diagram of the wind and solar power station topology and network control method targeted by the present invention;

[0078] Figure 3 Schematic diagram of the overall state space model of the station according to the present invention;

[0079] Figure 4 This is a flow chart of the grid-type converter ratio optimization algorithm of the present invention;

[0080] Figure 5 This is a comparison chart of the errors between the stability sensitivity analytical method and the numerical perturbation method in the present invention;

[0081] Figure 6 This is a diagram verifying the effectiveness of a method for capacity matching of grid-forming and grid-following converters based on semidefinite programming according to the present invention. DETAILED DESCRIPTION

[0082] The technical solution of the present invention is further described below with reference to the accompanying drawings and embodiments.

[0083] Unless otherwise defined, the technical or scientific terms used in the present invention shall have the usual meanings understood by persons of ordinary skill in the field to which the present invention belongs. The words "first", "second" and similar terms used in the present invention do not indicate any order, quantity or importance, but are only used to distinguish different components. Words such as "include" or "comprise" mean that the elements or objects preceding the word include the elements or objects listed after the word and their equivalents, without excluding other elements or objects. Words such as "connect" or "connected" are not limited to physical or mechanical connections, but may include electrical connections, whether direct or indirect. "Up", "down", "left", "right" and the like are only used to indicate relative positional relationships. When the absolute position of the object being described changes, the relative positional relationship may also change accordingly.

[0084] Matrix inequality notation is used in linear algebra and matrix analysis to express specific relationships between matrices, where A 0 means that the matrix A is a positive definite matrix, A 0 means that the matrix A is a positive semidefinite matrix, A B means AB is a negative semidefinite matrix.

[0085] Example 1

[0086] like Figure 1 As shown, the present invention provides a method for capacity matching of grid-forming and grid-following converters based on semidefinite programming, comprising the following steps:

[0087] S1. Establish the state space models of the grid-forming converter and the grid-following converter respectively, and construct the overall state space model of the station including the grid-forming converter and the grid-following converter through coordinate transformation.

[0088] Wind and solar station topology and network control methods are as follows Figure 2 As shown, the wind and solar stations include Grid-type converter and The grid-type converter uses the equivalent impedance Z c connected to the power grid;

[0089] The construction process of the overall state space model of the station is as follows:

[0090] The mathematical model of each control link and filter circuit of the grid-following converter is:

[0091] ;

[0092] Where, 、 They are the phase angle and frequency obtained by the grid-type phase-locked loop, 、 They are the phase-locked loop PI parameters, 、 are the power outer loop PI parameters, 、 are the current inner loop PI parameters, 、 are the active and reactive power output of the converter respectively ( ref represents reference value), is the filter inductor, 、 They are the filter outlet current and voltage ( represents reference value), is the converter output voltage.

[0093] The mathematical models of each control link and filter circuit of the grid-type converter are as follows:

[0094] ;

[0095] Where, The frequency obtained by network-type virtual synchronous control is D 、 J are the damping coefficient and inertia of the virtual synchronous control loop, E is the internal potential obtained by voltage droop control, k Q is the voltage droop coefficient, k pv 、 k ivare the voltage outer loop PI parameters, k pc 、 k ic are the current inner loop PI parameters, P 、 Q are the active and reactive power output of the converter respectively ( ref represents reference value), L f is the filter inductor, i odq 、 u odq They are the filter outlet current and voltage ( represents reference value), u idq is the converter output voltage.

[0096] For the above system, according to Figure 3 The modeling method is to combine the mathematical models of the grid-type converter, the grid-forming converter and the AC network through coordinate transformation, and linearize them at the equilibrium point. The overall state space model of the station can be obtained as follows:

[0097] ;

[0098] Where, is the system state variable, the matrix A is the system state matrix, the small signal stability of the system is related to the matrix A is related to the eigenvalues ​​of .

[0099] S2. Use the semidefinite programming based on the Lyapunov equation to describe the stability index, and derive the analytical expression of the stability sensitivity of the grid-type converter ratio according to the primal and dual variables of the semidefinite programming.

[0100] Using the Lyapunov equation to describe the stability of the system, Lyapunov's second method points out that if and only if there exists a real symmetric positive definite matrix satisfy , the system is asymptotically stable, and the related Lyapunov function is , is any given negative real number.

[0101] In order to quantify the dynamic behavior of the system, the stability index is further designed as the solution of the following semidefinite programming problem:

[0102] ;

[0103] The Lyapunov equation requires , so set a constant that tends to 0 Make , which is consistent with are equivalent. At the same time, such a transformation makes the feasible domain from an open set to a closed set, ensuring the existence of extreme points in the feasible domain. In addition, in order to prevent the objective function value of this problem from being negative infinity, set If the system is stable, the stability index obtained by semidefinite programming is Will be negative.

[0104] Stability index The absolute value of is the lower limit of the convergence rate after perturbation, because from the above semidefinite programming problem we can get:

[0105] ;

[0106] The method based on Lyapunov function provides an overall evaluation of stability, that is, whether the system is stable and the convergence speed. In this embodiment, the stability index based on Lyapunov function is selected. , because this indicator is more operational. Different from eigenvalue analysis, stability index This is given by a convex semidefinite programming problem with strong duality. In this example, an efficient method is designed to exploit this strong duality to calculate stability sensitivity, significantly reducing the computational burden while ensuring analytical accuracy. This method is suitable for determining stability and assessing convergence speed, but the study of oscillation modes is beyond the scope of this discussion.

[0107] The stability constraint can be described as: , is a controllable variable that can be used to enhance the stability index. In this embodiment Indicates the capacity ratio of grid-type converters, In order to make the stability index meet the constraints, it is necessary to adjust the corresponding sensitivity by calculating the stability index. But due to and The functional relationship is implicit, and it is very difficult to obtain the sensitivity. The mainstream sensitivity analysis method is the numerical perturbation method. The stability sensitivity of is estimated as follows:

[0108] ;

[0109] However, this estimate strongly depends on the perturbation Therefore, the numerical perturbation method is inaccurate, and the stability index needs to be calculated twice to obtain a single variable The sensitivity of the calculation is very high.

[0110] In this embodiment, an analytical formula for accurately calculating the stability sensitivity is defined. Starting from the convexity of semidefinite programming, the sensitivity of the Jacobian matrix elements is derived. The analytical expression of the stability sensitivity is obtained by using the chain rule. , specifically:

[0111] First, change the original form of the semidefinite programming problem above into the standard form:

[0112] ;

[0113] in, ,vector Including stability indicators and the symmetric matrix The lower triangular elements of , , Consider all the constraints in the original semidefinite program and define ,in , , represents the set of all n-dimensional real symmetric matrices. Corresponding to the constraints in the original semidefinite programming, , , , , , , , , , represents the basis of n-dimensional symmetric matrices, .

[0114] Next, write the standard form of the dual problem of the above standard form semidefinite programming problem:

[0115] ;

[0116] Among them, the symmetric matrix Z represents the original variable The dual variable of .

[0117] Then, according to Slater's condition, there exists a strictly feasible solution to the original problem , making , and there exists a strictly feasible solution to the dual problem that satisfies and , This condition ensures the strong duality of the semidefinite programming problem, and the optimal value The duality gap at is 0: . And because and , we can get:

[0118] ;

[0119] Since the proposed semidefinite programming problem is based on matrix inequalities, while the standard convex problem is based on scalar inequalities, in order to conveniently derive parameter sensitivities using dual variables, the "svec" calculation is used to vectorize the matrix in the semidefinite programming problem, resulting in a standard convex optimization form. The parameter sensitivity formula for the semidefinite programming problem is then established using the traditional convex problem method.

[0120] In order to facilitate the representation and matrix operations, define the function "svec": Mapped to , Then, we establish the relationship between matrix and vector calculations through the "svec" function and inner product:

[0121] ;

[0122] ;

[0123] , then the above formula is expressed as:

[0124] ;

[0125] Since the duality gap is 0, we have:

[0126] ;

[0127] Using symmetric Kronecker products To simplify the expression, it is defined as for any n-dimensional symmetric matrix X, there exists M, N∈ , such that:

[0128] ;

[0129] therefore Expressed as:

[0130] ;

[0131] Then, by combining and , define the function G:

[0132] ;

[0133] in, , given the parameters The optimal value of Department In essence, G is considered as a variant of the KKT condition under the matrix inequality. are all continuously differentiable, and the derivative of the function is expressed as:

[0134] ;

[0135] At the optimal value is non-singular, parameter sensitivity The calculation formula is:

[0136] ;

[0137] Among them, the derivative Expressed as:

[0138] ;

[0139] Need explanation, vector The first element is the stability index , for the sake of simplicity, we define Vector , so the stability index Too The first element of .

[0140] Finally, by applying the chain rule, we obtain the stability index Sensitivity to the controllable variable, i.e., the grid-type converter ratio:

[0141] ;

[0142] in, is the Jacobian matrix element pair containing d The partial derivatives of Since it is an explicit function of , the partial derivative can be derived conveniently.

[0143] S3. Initialize the grid-type converter ratio, and iterate with the stability sensitivity of the grid-type converter ratio as the gradient until the objective function value reaches the stability index limit or the difference between the objective function values ​​of two iterations is less than the convergence gap, and then stop, to obtain the optimal grid-type converter ratio.

[0144] Initialize the grid-type converter ratio to the upper limit of the station grid-type converter ratio , set the stability index limit to , the optimization algorithm process is as follows Figure 4 As shown, specifically:

[0145] S31. Set the initial value of the optimized grid-type converter ratio parameter ,make .

[0146] S32, calculate the objective function value according to the method in step S2 Sensitivity and stability .

[0147] S33, determine the termination conditions, specifically:

[0148] Termination condition 1: When When , the difference between the objective function values ​​of the two iterations satisfies the convergence gap ε, that is:

[0149] ;

[0150] Termination condition 2: When the stability index is within the limit Range Within, that is:

[0151] ;

[0152] If any of the termination conditions is met, the algorithm stops and outputs the optimization result. If none of the termination conditions are met, the algorithm proceeds to step S34 to iteratively update the grid-type converter matching parameters.

[0153] S34, iterative update of grid-type converter ratio d :

[0154] ;

[0155] in, is the iteration step, and the updated Substitute into step S32, let .

[0156] The method of the present invention is illustrated below using a specific example. Table 1 shows the main parameters of the new energy station, grid-forming type, and grid-following type converters used in the examples of the present invention. The grid-forming type ratio is optimized using a weak grid operating condition with a short-circuit ratio of 1.2 at the station as an example.

[0157] Table 1 Main parameters of new energy station, grid-forming and grid-following converters

[0158] ;

[0159] In order to verify the accuracy and speed of the sensitivity analysis formula in step S2, the above scenario is applied to perform stability sensitivity calculation test.

[0160] The test was conducted on a computing platform equipped with an Intel i5-12450H CPU processor and 16GB of memory. 100 grid-type converter ratios were randomly generated to calculate the sensitivity and record the cumulative calculation time. In order to illustrate the significant improvement in calculation accuracy and efficiency, three perturbation values ​​were selected. 、 、 , the sensitivity is calculated using the numerical perturbation method for comparison.

[0161] The analytical method of this paper is used respectively. Numerical perturbation method, Numerical perturbation method, The numerical perturbation method is used to calculate the stability sensitivity under these 100 grid-type converter ratios, and the relative error diagram of the calculation results compared with the analytical benchmark values ​​under these 100 scenarios is obtained, as shown in the figure below. Figure 5 The maximum and average relative errors in 100 scenarios were calculated, and the cumulative calculation time was recorded, as shown in Table 2. By comparison, the sensitivity analysis formula in step S2 can accurately analyze the stability sensitivity. Compared with the numerical perturbation method, the cumulative calculation time is shortened to less than half, which significantly improves the calculation speed.

[0162] Table 2 Calculation accuracy and efficiency of the numerical perturbation method and the method proposed in this invention

[0163] ;

[0164] To verify the effectiveness of the optimization algorithm in step S3, the stability index limit is set to ,scope , the optimal grid-type converter ratio in the above scenario is 9.0%. Under each ratio, the active power command is set to increase the disturbance of 0.05pu at 5s, and the output active power curve is compared as shown in the figure below. Figure 6 As shown in the figure, when the grid-type converter ratio is 9.0%, the stability index is -0.559, which meets the stability index limit constraint. However, when the grid-type converter ratio is 8.5% and 8.0%, the stability index is -0.230 and 0.102, respectively, which do not meet the stability index limit constraint. Therefore, the optimization algorithm in step S3 can effectively obtain the optimal grid-type converter ratio that meets the stability index limit constraint.

[0165] Therefore, the present invention adopts the above-mentioned method for capacity ratio matching of grid-forming and grid-following converters based on semidefinite programming, which can use the duality of semidefinite programming to analytically obtain stability sensitivity, thereby achieving accurate and rapid solution to the optimal grid-forming converter ratio.

[0166] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A method for capacity matching of grid-forming and grid-following converters based on semidefinite programming, characterized by: The following steps are involved: S1. Establish state space models of grid-forming converters and grid-following converters respectively, and construct the overall state space model of the station including grid-forming converters and grid-following converters through coordinate transformation; S2. Use the semidefinite programming based on the Lyapunov equation to describe the stability index, and derive the stability sensitivity analytical expression of the grid-type converter ratio according to the primal and dual variables of the semidefinite programming; S3. Initialize the grid-type converter ratio, and iterate with the stability sensitivity of the grid-type converter ratio as the gradient until the objective function value reaches the stability index limit or the difference between the objective function values ​​of two iterations is less than the convergence gap, and then stop, to obtain the optimal grid-type converter ratio.

2. The method for capacity matching of grid-forming and grid-following converters based on semidefinite programming according to claim 1, characterized in that: In S1, wind and solar stations include Grid-type converter and The grid-type converter uses the equivalent impedance Z c connected to the power grid; The construction process of the overall state space model of the station is as follows: The mathematical model of each control link and filter circuit of the grid-following converter is: ； Where, 、 They are the phase angle and frequency obtained by the grid-type phase-locked loop, 、 They are the phase-locked loop PI parameters, 、 are the power outer loop PI parameters, 、 are the current inner loop PI parameters, 、 are the active and reactive power output by the converter respectively, represents the reference value, is the filter inductor, 、 are the filter outlet current and voltage respectively, represents the reference value, is the converter output voltage.

3. The method for capacity matching of grid-forming and grid-following converters based on semidefinite programming according to claim 2, characterized in that: The mathematical models of each control link and filter circuit of the grid-type converter are as follows: ； Where, The frequency obtained by network-type virtual synchronous control is D 、 J are the damping coefficient and inertia of the virtual synchronous control loop, E is the internal potential obtained by voltage droop control, is the voltage droop coefficient, k pv 、 k iv are the voltage outer loop PI parameters, k pc 、 k ic are the current inner loop PI parameters, P 、 are the active and reactive power output by the converter respectively, represents the reference value, L f is the filter inductor, 、 are the filter outlet current and voltage respectively, represents the reference value, is the converter output voltage.

4. The method for capacity matching of grid-forming and grid-following converters based on semidefinite programming according to claim 3, characterized in that: By combining the mathematical models of the grid-following converter, grid-forming converter, and AC network through coordinate transformation and linearizing them at the equilibrium point, the overall state space model of the station is obtained as follows: ； Where, is the system state variable, the matrix A is the system state matrix, the small signal stability of the system is related to the matrix A is related to the eigenvalues ​​of .

5. The method for capacity matching of grid-forming and grid-following converters based on semidefinite programming according to claim 1, characterized in that: In S2, the Lyapunov equation is used to describe the stability of the system. Lyapunov's second method states that if and only if there exists a real symmetric positive definite matrix satisfy , the system is asymptotically stable, and the related Lyapunov function is , is any given negative real number; The stability index is further designed as the solution to the following semidefinite programming problem: ； The Lyapunov equation requires , so set a constant that tends to 0 Make , which is consistent with is equivalent to setting , if the system is stable, the stability index obtained by semidefinite programming is will be negative; Stability index The absolute value of is the lower limit of the convergence rate after perturbation, and from the above semidefinite programming problem we get: ； Define an analytical formula for accurately calculating stability sensitivity, and derive the sensitivity of Jacobian matrix elements based on the convexity of semidefinite programming. The analytical expression of the stability sensitivity is obtained by using the chain rule. .

6. The method for capacity matching of grid-forming and grid-following converters based on semidefinite programming according to claim 5, characterized in that: Stability sensitivity The derivation method is as follows: First, change the original form of the semidefinite programming problem above into the standard form: ； in, ,vector Including stability indicators and the symmetric matrix The lower triangular elements of , , ; Consider all the constraints of the original semidefinite program and define ,in , , Indicates all A set of dimensional real symmetric matrices; corresponding to the constraints in the aforementioned original semidefinite programming, , , , , , , , , , represent The basis of the symmetric matrix, ; Next, write the standard form of the dual problem of the above standard form semidefinite programming problem: ； Among them, the symmetric matrix Z Represents the original variable The dual variable of Then, according to Slater's condition, there exists a strictly feasible solution to the original problem , making , and there exists a strictly feasible solution to the dual problem that satisfies and , ; This condition ensures the strong duality of the semidefinite programming problem, at the optimal value The duality gap at is 0: ; and because and ,have to: ； Since the proposed semidefinite programming problem is based on matrix inequalities, while the standard convex problem is based on scalar inequalities, in order to conveniently derive parameter sensitivities using dual variables, the "svec" calculation is used to vectorize the matrix in the semidefinite programming problem, resulting in a standard convex optimization form. The parameter sensitivity formula for the semidefinite programming problem is then established in the same way as for the traditional convex problem. In order to facilitate the representation and matrix operations, define the function "svec": Mapped to , ; Then, the relationship between matrix and vector calculation is established through the "svec" function and inner product: ； ； make , then the above formula is expressed as: ； Since the duality gap is 0, we have: ； Using symmetric Kronecker products To simplify the expression, it is defined as dimensional symmetric matrix X ,exist M , N ∈ , such that: ； therefore Expressed as: ； Then, by combining and , define the function G : ； in, , given the parameters The optimal value of Department ;Will G It can be regarded as a variant of the KKT condition under the matrix inequality, since c and are all continuously differentiable, and the derivative of the function is expressed as: ； At the optimal value is non-singular, parameter sensitivity The calculation formula is: ； Among them, the derivative Expressed as: ； vector The first element is the stability index , for the sake of simplicity, we define Vector , so the stability index Too The first element of Finally, by applying the chain rule, we obtain the stability index Sensitivity to the controllable variable, i.e., the grid-type converter ratio: ； in, is the Jacobian matrix element pair containing d The partial derivative of .

7. The method for capacity matching of grid-forming and grid-following converters based on semidefinite programming according to claim 6, characterized in that: In S3, the initialization grid-type converter ratio is the upper limit of the site grid-type converter ratio. , set the stability index limit to , the optimization algorithm process is as follows: S31. Set the initial value of the optimized grid-type converter ratio parameter ,make ; S32, calculate the objective function value according to the method in step S2 Sensitivity and stability ; S33, determining the termination condition; S34, iterative update of grid-type converter ratio : ； in, is the iteration step, and the updated Substitute into step S32, let .

8. The method for capacity matching of grid-forming and grid-following converters based on semidefinite programming according to claim 7, characterized in that: In S33, specifically: Termination condition 1: When When , the difference between the objective function values ​​of the two iterations satisfies the convergence gap ε, that is: ； Termination condition 2: When the stability index is within the limit Range Within, that is: ； If any of the termination conditions is met, the algorithm stops and outputs the optimization result. If none of the termination conditions are met, the algorithm proceeds to step S34 to iteratively update the grid-type converter matching parameters.

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