LTP system feature vector calculation method for power system stability analysis and application

Abstract

The invention discloses an LTP system feature vector calculation method for power system stability analysis and application, and belongs to the field of linear periodic time-varying system analysis in a power system.The LTP system feature vector calculation method comprises the steps that dynamic differential equations of a left feature vector and a right feature vector of LTP system feature values are obtained; establishing a relation among the LTP system characteristic value, the left characteristic vector, the right characteristic vector and the state transition matrix, and deducing to obtain solutions of the left characteristic vector and the right characteristic vector at 0 and T moments; based on the differential equation of the right feature vector, deriving a solution of the right feature vector in a period T through a forward modeling form; and based on the state transition matrix and the differential equation of the left eigenvector, deriving in a time inversion form to obtain a solution of the left eigenvector of the eigenvalue in the LTP system in a period T. Through the LTP system feature vector calculation method disclosed by the invention, the requirement for ill-conditioned STM inversion is avoided, and more reliable left and right feature vectors are obtained.

Description

Technical Field

[0001] The present invention relates to the field of linear periodic time-varying system analysis in power systems, and more specifically to a method for calculating an LTP system eigenvector for power system stability analysis and its application. Background Art

[0002] The widespread use of power electronics has led to a shift from traditional synchronous generator-dominated power systems to power electronics-dominated power systems (PEPS). The switching dynamics and negative-sequence control of power electronics introduce inter-sequence dynamic coupling, resulting in complex, periodic, and time-varying characteristics. This poses significant challenges to the stable operation of PEPS, with persistent broadband oscillations becoming a major concern.

[0003] Modal analysis, which assesses damping, oscillation frequencies, participation factors, and sensitivities, has long been considered a systematic approach to addressing these oscillation problems. Much research has focused on modal analysis based on linear time-invariant (LTI) theory, where time-varying characteristics are eliminated to obtain a transformed LTI model. However, LTI modeling suffers from a difficult trade-off between accuracy and computational efficiency, particularly in large-scale PEPS systems, which limits its applicability. In contrast, recent advances in linear time-periodic (LTP) systems offer superior accuracy and efficiency: these methods directly linearize the system to obtain the LTP system while preserving the time-varying characteristics and system order. Consequently, modal analysis of LTP systems, with eigenvalues ​​and eigenvectors forming the cornerstone, has become increasingly important in PEPS.

[0004] However, unlike the LTI eigenvector, the LTP eigenvector is periodically time-varying, and existing methods usually derive the LTP eigenvector using the forward form starting from time 0. This results in the left eigenvector and the right eigenvector in the LTP system depending on the inverse matrix Φ of the state transition matrix (STM) respectively. -1 (t, 0) and the matrix Φ(t, 0). However, in PEPS, high-dimensional STMs are generally ill-conditioned, making the solution of their inverse matrix unreliable, thus limiting the accurate quantitative calculation of the LTP left eigenvector. Therefore, it is particularly necessary to find a method to accurately calculate the left eigenvector of the LTP system. Summary of the Invention

[0005] In response to the above problems, the present invention provides a method and application for calculating the LTP system eigenvector for power system stability analysis. Its purpose is to calculate the left eigenvector of the LTP system eigenvalue by time reversal, thereby avoiding the need for inverting the pathological STM and obtaining more reliable left eigenvector and right eigenvector of the LTP system eigenvalue.

[0006] To achieve the above object, the present invention provides a method for calculating the LTP system eigenvector for power system stability analysis, comprising the following steps:

[0007] S1: Based on the Floquet-Lyapunov method, the dynamic differential equations for obtaining the left and right eigenvectors of the LTP system eigenvalues ​​are derived;

[0008] S2: Based on the Floquet-Lyapunov method, the relationship between the eigenvalues, left eigenvectors, right eigenvectors and state transition matrix of the LTP system is established, and the solutions of the left eigenvector and right eigenvector at time 0 and time T are derived, where T is the period of the LTP system;

[0009] S3: Based on the state transfer matrix and the differential equation of the LTP right eigenvector obtained in step S1, the solution of the right eigenvector of the LTP system eigenvalue within a period T is derived by forward formalism;

[0010] S4: Based on the state transfer matrix and the differential equation of the left eigenvector of the eigenvalue in the LTP system obtained in step S1, a solution of the left eigenvector of the eigenvalue in the LTP system within a period T is derived through time reversal formalism.

[0011] The present invention also discloses a stability analysis method for a power system, comprising:

[0012] The left eigenvector and the right eigenvector of the LTP system eigenvalue obtained by using the above-mentioned LTP system eigenvector calculation method for power system stability analysis;

[0013] The left eigenvector and the right eigenvector of the LTP system eigenvalue are used to perform stability analysis of the power system.

[0014] The present invention discloses a device for calculating an LTP system eigenvector, comprising:

[0015] Differential equation acquisition module: used to derive the dynamic differential equations of the LTP left eigenvector and right eigenvector to obtain the eigenvalues ​​of the LTP system based on the Floquet-Lyapunov method;

[0016] First eigenvector solution module: Based on the Floquet-Lyapunov method, the relationship between the LTP system eigenvalue, left eigenvector, right eigenvector and state transition matrix is ​​established, and the solutions of the left eigenvector and right eigenvector at time 0 and T are derived;

[0017] The second eigenvector solving module: Based on the state transfer matrix and the differential equation of the LTP right eigenvector obtained in the differential equation obtaining module, the solution of the LTP right eigenvector within a period T is derived through forward formalism;

[0018] The third eigenvector solving module: Based on the state transfer matrix and the differential equation of the LTP left eigenvector obtained in the differential equation obtaining module, the solution of the LTP left eigenvector within a period T is derived through time reversal form.

[0019] The present invention discloses an electronic device, comprising a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, the steps of the method for calculating the characteristic vector of the LTP system are realized.

[0020] The present invention discloses a computer-readable storage medium on which a computer program is stored. When the computer program is executed by a processor, the steps of the above-mentioned method for calculating the characteristic vector of the LTP system are realized.

[0021] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects:

[0022] The LTP system eigenvector calculation method for power system stability analysis provided by the present invention calculates the LTP left eigenvector by time inversion without relying on the need to invert the pathological STM, making the obtained LTP left and right eigenvectors more reliable and laying the foundation for modal analysis of the LTP system. BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Figure 1 A flowchart of a method for calculating an LTP system eigenvector for power system stability analysis provided according to an embodiment of the present invention;

[0024] like Figure 2 Schematic diagram comparing the solution of the left eigenvector of the LTP system based on time reversal according to an embodiment of the present invention and the forward solution of the left eigenvector in the prior art. DETAILED DESCRIPTION

[0025] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only intended to illustrate the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0026] Example 1:

[0027] Figure 1 FIG2 is a diagram showing a method for calculating an LTP system eigenvector for power system stability analysis according to a first embodiment of the present invention, which includes the following steps:

[0028] S1: Based on the Floquet-Lyapunov method, the dynamic differential equations of the LTP left eigenvector and right eigenvector for obtaining the eigenvalues ​​of the LTP system are derived. In an optional embodiment, the dynamic equations of the LTP system with general periodic coefficients (for example, the power frequency period in the power system) can be expressed as follows:

[0029]

[0030] Where x(t)=[x1(t),x2(t),...,x n (t)] T represents the column vector consisting of n state variables of the LTP system at time t, represents the time derivative of x(t), A(t) represents the system matrix, and T represents the fundamental period of the system.

[0031] Furthermore, according to the Floquet-Lyapunov method, there exists a periodic transformation matrix P(t) that can transform the LTP system into an LTI system, and the systems before and after the transformation have the same stability. The LTI system satisfies:

[0032]

[0033] Among them, v(t) represents a column vector consisting of a set of new state variables obtained based on the transformation of x(t), represents the time differential of v(t), Q is a time-invariant n-order square matrix, and P(t) is an n-order square matrix with a period of T, representing a special coordinate transformation that transforms the time-invariant system matrix A(t) into a constant matrix Q. The transformation matrix P(t) satisfies:

[0034]

[0035] in, represents the differential of P(t) with respect to time, P -1(t) represents the inverse of P(t), const represents a constant, and the constant matrix Q is further decomposed to obtain the eigenvalues ​​of the LTP system:

[0036] VQU=Λ=diag(λ1,λ2,...,λ n ) (4)

[0037] Where V and U represent the left and right eigenvectors of the constant matrix Q, respectively, and the diagonal matrix Λ represents the Floquet characteristic index λ of the LTP system. i (i=1,2,...,n), which is the eigenvalue of LTP.

[0038] Furthermore, by combining the transformation matrix P(t) with the right eigenvector U of the Q matrix, we can construct the coordinate transformation R(t)=P(t)U. The transformation matrix R(t) realizes that the system matrix A(t) remains unchanged and is converted into the eigenvalue matrix Λ of the LTP system:

[0039]

[0040] Where L(t) = R -1 (t), represents the left eigenvector matrix of the LTP system, R(t) represents the right eigenvector matrix of the LTP system, Represents the time derivative of the right eigenvector matrix R(t).

[0041] Arrange equation (5) and express the eigenvalue matrix Λ as diag(λ1,λ2,...,λ n ), the right eigenvector matrix and the left eigenvector matrix are expressed as R(t)=[r1(t),r2(t),...,r n (t)] and L(t)=[l1(t); l2(t);...; l n (t)], where r i (t) and l i (t) are the eigenvalues ​​λ i The right eigenvector (column vector) and left eigenvector (row vector) of , and then the LTP eigenvalue λ can be obtained i The dynamic differential equation of the LTP eigenvector is:

[0042]

[0043] in, is the differential of the right eigenvector with respect to time; is the time derivative of the left eigenvector.

[0044] S2: Based on the Floquet-Lyapunov method, the relationship between the eigenvalues, left eigenvectors, right eigenvectors and state transition matrix of the LTP system is established, and the solutions of the left eigenvector and right eigenvector at time 0 and T are derived. Specifically, according to the Floquet-Lyapunov method, the form of the LTP system solution can be expressed as:

[0045] x(t)=P(t)e Qt x(0) (8)

[0046] Among them, e Qt Represents Qt's matrix exponential.

[0047] According to equation (1) in step S1, the dynamic stability of the LTP system is characterized by the corresponding matrix equation, which is obtained:

[0048]

[0049] Where I is the identity matrix, the solution of Equation (9) is Φ(t, 0), and Φ(t, 0) is the standard fundamental solution matrix of Equation (1), also known as the STM of the system, which satisfies the following relationship x(t) = Φ(t, 0)x(0).

[0050] According to formula (8), the state transfer matrix can be further expressed as:

[0051] Φ(t,0)=P(t)e Qt (10)

[0052] Among them, under the initial value setting of the standard basic solution matrix Φ(0,0)=I, Φ(0,0) represents the solution at time 0, and P(0) at this time is also the unit matrix.

[0053] Furthermore, Equation (10) can be rewritten using the LTP system eigenvalue, left eigenvector, and right eigenvector representations:

[0054] Φ(t,0)=R(t)e Λt L(0) (11)

[0055] Among them, Φ(t,0) represents the transition of the state variable x(t) from time 0 to time t. According to the periodicity of the LTP eigenvector, the right eigenvector satisfies R(T) = R(0). Therefore, the state transition matrix Φ(T,0) within one period has the following form:

[0056] Φ(T,0)=R(T)e ΛT L(0)=R(0)e ΛT L(0) (12)

[0057] Based on formula (12), the values ​​of the eigenvalue Λ and the eigenvector at time 0 and time T can be obtained by performing eigendecomposition on Φ(T,0); Φ(T,0) can be obtained by solving the periodic coefficient linear differential equation system of formula (9). In an optional embodiment of the present invention, the solution is achieved by calling a differential equation solver in Matlab, and other methods can also be used.

[0058] S3: Based on the state transfer matrix and the differential equation (6) of the right eigenvector of the eigenvalue of the LTP system obtained in step S1, the solution of the LTP right eigenvector in one period T is obtained by forward derivation. Forward derivation means that the solution of the LTP right eigenvector is calculated in time sequence within one time period. Equation (5) is rewritten as:

[0059]

[0060] Subtract λ from both sides of the equation (13) i I, we can get (A(t)–λ i I) in the form of characteristic decomposition:

[0061]

[0062] From Equation (14), we can see that the left eigenvector L(t) and the right eigenvector R(t) of A(t) are also the system matrix (A(t)–λ i I) left and right eigenvectors, and the eigenvalue matrix of the latter needs to subtract λ compared to the former i I, that is (Λ–λ i I). Therefore, (A(t)–λ i I) The state transition matrix Φ corresponding to 0 to t (t is greater than or equal to 0) r (t,0) can be expressed as:

[0063]

[0064] According to formula (11), the numerator on the right side of the equal sign in (15) can be expressed as Φ(t,0):

[0065]

[0066] Therefore, based on the initial value r of the right eigenvector i (0) and (A(t)–λ i I) state transition matrix Φ r (t,0), the solution r of the right eigenvector of formula (6) i (t) can be expressed as:

[0067]

[0068] Thus, the right eigenvector r of the LTP system is obtained i (t).

[0069] S4: Based on the state transfer matrix and the differential equation of the left eigenvector of the eigenvalue in the LTP system obtained in step S1, the solution of the left eigenvector of the eigenvalue in the LTP system within a period T is obtained by time reversal derivation. The inverse derivation means that the solution of the LTP left eigenvector is calculated in reverse time order within a time period. The reverse time order means derivation from T to 0. Specifically, the solution of the left eigenvector of equation (7) is similar to the solution of the right eigenvector. It is necessary to consider the system matrix (λ i I–A T (t))’s characteristic structure.

[0070] like Figure 2 The figure is a comparison diagram of the solution of the left eigenvector of the LTP system based on time reversal according to an embodiment of the present invention and the forward solution of the left eigenvector in the prior art. Specifically, the traditional forward solution is based on the properties of the dual matrix, (λ i I–A T The eigenvalue of (t)) is (A(t)–λ i I) The opposite of the eigenvalue, i.e. (λ i I–Λ), and (λ i I–A T The left eigenvector of (t)) is R T (t), the right eigenvector is L T (t). Therefore, (λ i I–A T The state transition matrix Φ corresponding to (t)) from 0 to t (t is greater than or equal to 0) l (t,0) is expressed as:

[0071]

[0072] Therefore, based on the initial value of the left eigenvector and the state transfer matrix Φ of formula (7) l (t,0), the solution of the left eigenvector is expressed as:

[0073]

[0074] Transpose both sides of the above equation to get the left eigenvector l i (t) form:

[0075]

[0076] According to formula (20), the left eigenvector l i (t) and Φ(t,0) satisfy the algebraic equation relationship, the left eigenvector li The dynamics of (t) will depend on the inverse matrix Φ of the state transition matrix Φ(t,0) -1 (t, 0). However, due to the ill-conditioned nature of Φ(t, 0) in PEPS, inverting Φ(t, 0) usually leads to divergence of the calculated results over time, which limits the effectiveness of the forward-based algorithm for solving the left eigenvector of LTP in Equation (20).

[0077] Based on this, in the embodiment of the present invention, the LTP left eigenvector is derived by time reversal. Specifically, the forward derivation of the left eigenvector in formula (20) is converted into its inverse process, and the derivation l i The initial condition of (t) is l i (0)Change to l i (T), that is, from l i (T) Find l i (t):

[0078]

[0079] Among them, time reversal makes the inverse of Φ(t,0) in Equation (20) eliminated, Φ -1 (t,0) is replaced by Φ(T,t) matrix multiplication to avoid the inverse operation, where Φ(T,t) represents the transfer of the state variable x(t) from time t to time T.

[0080] The left eigenvector solved by the LTP system eigenvector calculation method for power system stability analysis disclosed in the present invention does not need to rely on the need for inverting the ill-conditioned STM, making the obtained left eigenvector and right eigenvector of the LTP system more reliable, laying the foundation for the modal analysis of the LTP system.

[0081] Example 2:

[0082] The present invention also discloses a method for analyzing the stability of a power system, which includes a method for analyzing the stability of a power system using the eigenvectors of the LTP system solved by the method in Example 1, namely the left eigenvector and the right eigenvector.

[0083] Example 3:

[0084] The present invention discloses a device for calculating an LTP system eigenvector, comprising:

[0085] Differential equation acquisition module: used to derive the dynamic differential equations of the LTP left eigenvector and right eigenvector to obtain the eigenvalues ​​of the LTP system based on the Floquet-Lyapunov method;

[0086] First eigenvector solution module: Based on the Floquet-Lyapunov method, the relationship between the LTP system eigenvalue, left eigenvector, right eigenvector and state transition matrix is ​​established, and the solutions of the left eigenvector and right eigenvector at time 0 and T are derived;

[0087] The second eigenvector solving module: Based on the state transfer matrix and the differential equation of the LTP right eigenvector obtained in the differential equation obtaining module, the solution of the LTP right eigenvector within a period T is derived through forward formalism;

[0088] The third eigenvector solving module: Based on the state transfer matrix and the differential equation of the LTP left eigenvector obtained in the differential equation obtaining module, the solution of the LTP left eigenvector within a period T is derived through time reversal form.

[0089] The specific working process of the above modules is described in Example 1 and will not be described in detail to avoid redundancy.

[0090] Example 4:

[0091] The present invention also discloses an electronic device, including a memory and a processor. The memory stores a computer program. When the processor executes the computer program, the steps of the method for calculating the LTP system characteristic vector in embodiment 1 are implemented.

[0092] Example 5:

[0093] The present invention also discloses a computer-readable storage medium on which a computer program is stored. When the computer program is executed by a processor, the steps of the method for calculating the LTP system characteristic vector in embodiment 1 are implemented.

[0094] In summary, the technical solution conceived by the present invention calculates the left eigenvector of the LTP system through time reversal, eliminating the need to invert the ill-conditioned STM. This makes the obtained left and right eigenvectors of the LTP system more reliable, laying the foundation for modal analysis of the LTP system. This further improves the accuracy of power system stability analysis.

[0095] The above embodiments are intended only to illustrate the technical concepts and features of the present invention. Their purpose is to enable those skilled in the art to understand the contents of the present invention and implement them accordingly. They are not intended to limit the scope of protection of the present invention. Any equivalent changes or modifications based on the essence of the present invention are intended to be covered by the scope of protection of the present invention.

Claims

1. A method for calculating LTP system eigenvectors for power system stability analysis, characterized in that: These include: S1: Based on the Floquet-Lyapunov method, the dynamic differential equations for obtaining the left and right eigenvectors of the LTP system eigenvalues ​​are derived; S2: Based on the Floquet-Lyapunov method, the relationship between the eigenvalues, left eigenvectors, right eigenvectors and state transition matrix of the LTP system is established, and the solutions of the left eigenvector and right eigenvector at time 0 and time T are derived, where T is the period of the LTP system; S3: Based on the state transfer matrix and the differential equation of the LTP right eigenvector obtained in step S1, the solution of the right eigenvector of the LTP system eigenvalue within a period T is derived by forward formalism; S4: Based on the state transfer matrix and the differential equation of the left eigenvector of the eigenvalue in the LTP system obtained in step S1, a solution of the left eigenvector of the eigenvalue in the LTP system within a period T is derived through time reversal formalism.

2. The method for calculating the LTP system eigenvector for power system stability analysis according to claim 1, characterized in that: The dynamic differential equation in step S1 is as follows: Among them, λ i is the LTP system eigenvalue, r i (t) and l i (t) are the LTP system eigenvalues ​​λ i The right eigenvector and left eigenvector of , A(t) represents the system matrix, I is the identity matrix, is the differential of the right eigenvector with respect to time; is the time derivative of the left eigenvector.

3. The method for calculating the LTP system eigenvector for power system stability analysis according to claim 1, characterized in that: The values ​​of the left eigenvector and the right eigenvector of the eigenvalue at time 0 and time T in step S2 are solved based on the following formula: Φ(T,0)=R(T)e ΛT L(0)=R(0)e ΛT L(0) Where Φ(T,0) is the state transition matrix, the eigenvalue Λ and the eigenvector values ​​at time 0 and T are obtained by eigendecomposing Φ(T,0), the right eigenvector R(T) = R(0), and L(0) is the value of the left eigenvector at time 0.

4. The method for calculating the LTP system eigenvector for power system stability analysis according to claim 2, characterized in that: Based on the dynamic differential equation in step S1, the formula for calculating the right eigenvector of the eigenvalue is as follows: in, Φ(t,0) represents the transition of the state variable x(t) from 0 to time t, r i (0) is the initial value of the right eigenvector.

5. The method for calculating the LTP system eigenvector for power system stability analysis according to claim 2, characterized in that: Based on the dynamic differential equation in step S1, the formula for calculating the left eigenvector of the eigenvalue is as follows: Among them, Φ(T,t) represents the transfer of the state variable x(t) from time t to time T, l i (T) represents the value of the left eigenvector at time T.

6. A method for analyzing the stability of a power system, characterized in that: include: A left eigenvector and a right eigenvector of an LTP system eigenvalue obtained by using the LTP system eigenvector calculation method for power system stability analysis according to any one of claims 1 to 5; The left eigenvector and the right eigenvector of the LTP system eigenvalue are used to perform stability analysis of the power system.

7. A device for calculating an LTP system eigenvector, characterized in that: These include: Differential equation acquisition module: used to derive the dynamic differential equations of the LTP left eigenvector and right eigenvector to obtain the eigenvalues ​​of the LTP system based on the Floquet-Lyapunov method; First eigenvector solution module: Based on the Floquet-Lyapunov method, the relationship between the LTP system eigenvalue, left eigenvector, right eigenvector and state transition matrix is ​​established, and the solutions of the left eigenvector and right eigenvector at time 0 and T are derived; The second eigenvector solving module: Based on the state transfer matrix and the differential equation of the LTP right eigenvector obtained in the differential equation obtaining module, the solution of the LTP right eigenvector within a period T is derived through forward formalism; The third eigenvector solving module: Based on the state transfer matrix and the differential equation of the LTP left eigenvector obtained in the differential equation obtaining module, the solution of the LTP left eigenvector within a period T is derived through time reversal form.

8. An electronic device, characterized in that: The method comprises a memory and a processor, wherein the memory stores a computer program and the processor implements the steps of the method according to any one of claims 1 to 5 when executing the computer program.

9. A computer-readable storage medium, characterized in that A computer program is stored thereon, and when the computer program is executed by a processor, the steps of the method according to any one of claims 1 to 5 are implemented.

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