Power system stability analysis method considering communication time delay and periodic sampling

Abstract

The invention discloses a power system stability analysis method considering communication time delay and periodic sampling, an augmented bilateral closed-loop functional is constructed, a functional correlation matrix in the functional does not need to meet positive definite definition, and only a functional structure needs to meet the requirements of the closed-loop functional; then, dynamic information of a power system model is fully utilized, and some zero equations containing free matrixes are established; a second-order Bessel-Legendre integral inequality is combined to obtain a power system stability criterion; finally, results of other literatures are compared in case simulation, and the stability criterion obtained through the method has superiority and effectiveness.

Description

Technical Field

[0001] This invention belongs to the field of power system stability analysis technology, and in particular, relates to a power system stability analysis method that considers communication delay and periodic sampling. Background Technology

[0002] With the increasing automation and informatization of power systems, their dynamic stability has become a major concern. Modern power grids are expanding their coverage, and transmission and transformation capacities and voltage levels are constantly improving. Regional power grid interconnection, as a key development trend, enables mutual support between hydropower and thermal power resources, cross-regional load compensation and regulation, and effectively leverages core advantages such as peak shaving and valley filling, shared unit reserves, supply and demand balance, and emergency support.

[0003] In modern power systems, communication networks are ubiquitous, and system state information and control signals are often transmitted through communication links. This inevitably introduces factors such as communication delay and periodic sampling. These factors significantly affect the dynamic performance and stability of the system, and may even lead to system instability. Therefore, power system stability analysis methods that consider communication delay and periodic sampling characteristics have become one of the key research directions. In existing studies on power system stability, the stability of time-delay systems usually relies on the Lyapunov-Krasovskii functional method. This method transforms the system stability problem into a matrix inequality problem by constructing a suitable functional structure. To reduce the conservatism of the criterion, scholars have proposed various improvement strategies, including the piecewise time-delay interval method, the integral inequality method, and functional construction methods based on polynomial approximation. However, these methods generally have some shortcomings, such as the matrices introduced in the functional often needing to satisfy positive definiteness constraints, thus limiting the flexibility of the functional structure; and insufficient utilization of system dynamic information, failing to fully explore the intrinsic constraint relationships between state variables. Therefore, how to further utilize the system's dynamic information to construct effective zero inequalities containing free matrices remains a key issue.

[0004] A patent application with publication number CN110456768A discloses a stability determination method for a periodic sampling control system with communication delay. The specific steps of this method include: establishing a sampling control system model, constructing a Lyapunov functional, differentiating the constructed functional, establishing sufficient conditions to guarantee the stability of the sampling control system based on the free matrix integral inequality method, and then solving the linear matrix inequalities that satisfy the corresponding conditions to obtain the maximum permissible sampling period to guarantee system stability under a certain communication delay. Compared with other existing methods, the stability analysis method proposed in this patent yields results with less conservatism, thus reducing the hardware implementation cost of the sampling control system design. However, this patent does not fully utilize the dynamic information of the system to construct an effective zero-equality containing free matrices. Summary of the Invention

[0005] This invention primarily addresses the issue that the stability of existing time-delay systems typically relies on the Lyapunov-Krasovskii functional method. This method transforms the system stability problem into a matrix inequality problem by constructing a suitable functional structure. To reduce the conservatism of the criterion, scholars have proposed various improvement strategies, including the piecewise time-delay interval method, the integral inequality method, and functional construction methods based on polynomial approximation. However, these methods generally have some shortcomings, such as the matrices introduced in the functional often needing to satisfy positive definiteness constraints, thus limiting the flexibility of the functional structure; and insufficient utilization of system dynamic information, failing to fully explore the intrinsic constraints between state variables. Therefore, this invention proposes a power system stability analysis method that considers communication delay and periodic sampling.

[0006] A power system stability analysis method considering communication delay and periodic sampling includes the following steps: S1. Establish a power system model that considers communication delay and periodic sampling, wherein the sampling period of the periodic sampling satisfies ,in The power system model includes communication delay. The state variables of the power system model are defined as follows: The output variable is defined as The expression for the power system model is:

[0007] In the formula, Indicates the sampling time. Indicates load disturbance. This represents the change in system frequency; S2. Construct an augmented two-sided closed-loop functional. The functional correlation matrix does not need to satisfy the positive definite constraint; only the structure of the augmented two-sided closed-loop functional needs to meet the requirements of a closed-loop functional. The expression of the augmented two-sided closed-loop functional is as follows: ; S3. Using the dynamic information of the power system model, establish a zero equation containing free matrices; S4. Combining the second-order Bessel-Legend inequality, differentiate the augmented two-sided closed-loop functional constructed in S2 and process its integral terms to derive the stability basis of the linear matrix inequality for the asymptotic stability of the power system. S5. Based on the stability criteria in step S4, calculate the maximum value of the sampling interval that the power system can tolerate under different time delays.

[0008] Furthermore, in step S3, the zero equation containing the free matrix is ​​derived by processing the terms and related terms in the augmented two-sided closed-loop functional using integration by parts, and the dimension of the free matrix is ​​adapted to the dimension of the state variables of the power system model.

[0009] Furthermore, the linear matrix inequality satisfies, given a scalar , and There exists a matrix , , , , , , The free matrix includes , , , , , , (j=1, 2, 3, 4, 5,6, 7; k=j, 8).

[0010] Furthermore, for The expression for the linear matrix inequality is:

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[0013] In the formula, Let be a matrix constructed using system parameters; when the above linear matrix inequalities hold, the power system is asymptotically stable.

[0014] Furthermore, the matrix satisfy:

[0015] In the above formula, the , The expressions are as follows:

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[0017] The matrix The expression is:

[0018] The matrix The expression is: .

[0019] Furthermore, in step S3, the construction interval of the zero equation containing the free matrix includes the sampling period. interval, The interval, and when the communication delay When the sampling period is less than The zero equation, by exploring the inherent constraints of power system state variables within each interval, supplements system dynamic information to reduce the conservatism of stability criteria.

[0020] Furthermore, in step S2, the augmented bilateral closed-loop functional includes multiple subfunctionals, the expressions of which are as follows:

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[0028] In the formula, , , The expression is as follows:

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[0049] Furthermore, differentiating the constructed functional yields the following integral term:

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[0054] .

[0055] Furthermore, the integral term is estimated, specifically by providing a positive definite matrix. With any matrix For any differentiable function , , sum vector The following inequalities hold: ; Calculation yields ;in,

[0056] .

[0057] Furthermore, the verification of the linear matrix inequality employs Schur's complement theorem, when... and That is, the derivative of the augmented two-sided closed-loop functional. .

[0058] Compared with the prior art, the beneficial effects of the present invention are as follows: 1. The augmented two-sided closed-loop functional constructed by this invention only needs to meet the closed-loop structure requirements and does not require positive definite matrix, which greatly improves the flexibility of functional design. The functional structure can be adjusted according to the dynamic characteristics of different power systems (such as single-area / multi-area load frequency control systems), avoiding the problem of insufficient functional adaptability caused by positive definite constraints.

[0059] 2. This invention employs the second-order Bessel-Legendé integral inequality, and uses it in conjunction with the zero inequality containing free matrices, to achieve high-precision boundary estimation of the integral term; simultaneously, it overcomes the limitation that the matrix must be positive definite, requiring only... This reduces the stringency of matrix constraints and avoids estimation bias caused by positive definiteness requirements, ensuring that the stability criterion can truly reflect the actual stability boundary of the power system. Attached Figure Description

[0060] Figure 1 This is a flowchart illustrating the principle of the present invention. Figure 2 This is a load frequency control diagram for a single-area power system according to the present invention; Figure 3 This is a system state response diagram of the present invention. Detailed Implementation

[0061] To clearly illustrate the technical features of the present invention, the invention will be described in detail below through specific embodiments and in conjunction with the accompanying drawings. Many specific details are set forth in the following description to provide a thorough understanding of the invention; however, the invention may be implemented in other ways different from those described herein, and therefore, the scope of protection of the invention is not limited to the specific embodiments disclosed below. In the present invention, unless otherwise expressly specified and limited, the first feature "on" or "below" the second feature may mean that the first and second features are in direct contact, or that the first and second features are in indirect contact through an intermediate medium. In the description of this specification, references to terms such as "an embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that the specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Moreover, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0062] Example 1 like Figure 1 As shown, a power system stability analysis method considering communication delay and periodic sampling includes the following steps: S1. Establish a power system model that considers communication delay and periodic sampling, wherein the sampling period of the periodic sampling satisfies ,in The power system model includes communication delay. The state variables of the power system model are defined as follows: The output variable is defined as The expression for the power system model is:

[0063] In the formula, Indicates the sampling time. Indicates load disturbance. This represents the change in system frequency; S2. Construct an augmented two-sided closed-loop functional. The functional correlation matrix does not need to satisfy the positive definite constraint; only the structure of the augmented two-sided closed-loop functional needs to meet the requirements of a closed-loop functional. The expression of the augmented two-sided closed-loop functional is as follows: ; The constructed closed-loop functional introduces and Two terms. Furthermore, by using integration by parts to handle these two terms, two zero equations containing free matrices are introduced to further reduce the conservatism of the system stability analysis.

[0064] S3. Using the dynamic information of the power system model, establish a zero equation containing free matrices; the construction interval and sampling period of the zero equation containing free matrices. interval, The interval, and when the communication delay When the sampling period is less than The zero equation, by exploring the inherent constraints of power system state variables within each interval, supplements system dynamic information to reduce the conservatism of the stability criterion; a free matrix is ​​defined, which includes... , , , , , , (j=1, 2, 3, 4, 5,6, 7; k=j, 8).

[0065] S4. Combining the second-order Bessel-Legend inequality, differentiate the augmented two-sided closed-loop functional constructed in S2 and process its integral terms to derive the stability basis of the linear matrix inequality for the asymptotic stability of the power system. S5. Based on the stability criteria in step S4, calculate the maximum value of the sampling interval that the power system can tolerate under different time delays.

[0066] In this embodiment, the following symbols are defined, where the superscript T and -1 represent the transpose and inverse of a matrix or vector, respectively. and Let represent the n-dimensional vector space and the n×m matrix space of the real number field, respectively. Representative matrix It is a symmetric positive definite matrix. (Symbol) and These represent the zero matrix and the identity matrix, respectively. In a block matrix, the symbols... It represents a symmetrical term. If it is a matrix, then express . Represents a block diagonal matrix. Denotes a set of column vectors, where Represents any matrix or vector. This is the speed drop coefficient of the speed controller. The inertial time constant, This represents the change in grid load. These are system control signals.

[0067] like Figure 2 As shown, based on the load frequency control block diagram of a single-region power system, the state variable is defined as follows: The output variable is .in, This represents the change in mechanical power. This indicates the change in the opening degree of the steam valve control valve. This represents the change in system frequency. Indicates the area control error. Given the inertial time constant of the steam turbine, the expression for the power system is derived as follows:

[0068] In the above formula, , , , , In the above formula, For the generator's moment of inertia, Indicates the generator damping coefficient. This represents the turbine time constant. This represents the time constant of the speed regulator. The frequency bias factor is represented by the PI controller. ,in, , Indicates controller gain. Sampling period. Because periodic sampling is being considered, therefore, Considering communication latency With the existence of , the power system can be represented in the following form:

[0069] in, .

[0070] Theorem 1: Given a positive definite matrix With any matrix For any differentiable function , , sum vector The following inequalities hold:

[0071] in,

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[0078] The following definition is given.

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[0102] In this embodiment, in step S2, an augmented two-sided closed-loop functional is constructed. The expression of the augmented two-sided closed-loop functional is: .

[0103] The augmented bilateral closed-loop functional includes multiple subfunctionals, the expressions of which are as follows:

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[0111] In the above formula, , , The expression is as follows:

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[0132] Differentiating the constructed subfunctional yields the following integral term:

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[0138] By estimating the above integral term using Theorem 1, we can obtain:

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[0145] Based on the following relationships between vectors

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[0152] ; Then the following zero equation containing free matrices holds:

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[0159] According to the above formula, we can obtain:

[0160] in:

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[0162] Applying Schur's complement theorem, the linear matrix inequality satisfies, given a scalar , and There exists a matrix , , , , , , .for The expression for the linear matrix inequality is:

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[0165] In the formula, Let be a matrix constructed using system parameters; the power system is asymptotically stable when the above linear matrix inequalities hold. The matrix... satisfy:

[0166] In the above formula, the , The expressions are as follows:

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[0168] The matrix The expression is:

[0169] The matrix The expression is: .

[0170] If the above linear matrix inequality holds, then we have and ,Right now Therefore, the sampling system is asymptotically stable.

[0171] Example 2 like Figure 1 and Figure 3 As shown, this embodiment is applied to a single-area load frequency control power system. This system includes core components such as a steam turbine, generator, speed governor, and PI controller, and uses fiber optic transmission for communication. A power system model is considered. ,in , , , , , .

[0172] For different time delays The maximum values ​​of the sampling intervals that power system stability can tolerate, derived from recent existing literature and the main conclusions of this application, are shown in Table 1. Table 1 shows that the results in this paper are an improvement over those in Literature 2 and Literature 3, implying that the asymptotic stability criterion for periodic sampling systems with communication delays obtained in this application is less conservative.

[0173] Table 1 Different delays Upper bound of the sampling interval below

[0174]

[0175] At that time, Yan and sampling period initial state The state-response diagram of the system is as follows: Figure 2 As shown, the system under consideration gradually approaches zero, which also demonstrates the effectiveness of the method proposed in this application.

[0176] Example 3 In this embodiment, to further utilize the dynamic information of the sampling system, the closed-loop functional constructed in this paper introduces... and Two terms. Furthermore, by using integration by parts to handle these two terms, two zero equations containing free matrices are introduced to further reduce the conservatism of the system stability analysis.

[0177] Obviously, the embodiments described above are merely examples for clearly illustrating the present invention and are not intended to limit the implementation of the present invention. Those skilled in the art can make other variations or modifications based on the above description. It is neither necessary nor possible to exhaustively list all possible implementations. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the claims of the present invention.

Claims

1. A power system stability analysis method considering communication delay and periodic sampling, characterized in that, Includes the following steps: S1. Establish a power system model that considers communication delay and periodic sampling, wherein the sampling period of the periodic sampling satisfies ,in The power system model includes communication delay. The state variables of the power system model are defined as follows: The output variable is defined as follows: The expression for the power system model is: In the formula, Indicates the sampling time. Indicates load disturbance. This represents the change in system frequency; S2. Construct an augmented two-sided closed-loop functional. The functional correlation matrix does not need to satisfy the positive definite constraint; only the structure of the augmented two-sided closed-loop functional needs to meet the requirements of a closed-loop functional. The expression of the augmented two-sided closed-loop functional is as follows: ； S3. Using the dynamic information of the power system model, establish a zero equation containing free matrices; S4. Combining the second-order Bessel-Legend inequality, differentiate the augmented two-sided closed-loop functional constructed in S2 and process its integral terms to derive the stability basis of the linear matrix inequality for the asymptotic stability of the power system. S5. Based on the stability criteria in step S4, calculate the maximum value of the sampling interval that the power system can tolerate under different time delays.

2. The power system stability analysis method considering communication delay and periodic sampling according to claim 1, characterized in that, In step S3, the zero equation containing the free matrix is ​​derived by processing the terms and related terms in the augmented two-sided closed-loop functional using integration by parts, and the dimension of the free matrix is ​​adapted to the dimension of the state variables of the power system model.

3. The power system stability analysis method considering communication delay and periodic sampling according to claim 1, characterized in that, The linear matrix inequality satisfies, given a scalar , and There exists a matrix , , , , , , The free matrix includes , , , , , , (j=1, 2, 3, 4, 5, 6, 7; k=j, 8).

4. The power system stability analysis method considering communication delay and periodic sampling according to claim 3, characterized in that, for The expression for the linear matrix inequality is: In the formula, Let be a matrix constructed using system parameters; when the above linear matrix inequalities hold, the power system is asymptotically stable.

5. The power system stability analysis method considering communication delay and periodic sampling according to claim 4, characterized in that, The matrix satisfy: In the above formula, the , The expressions are as follows: The matrix The expression is: The matrix The expression is: 。 6. The power system stability analysis method considering communication delay and periodic sampling according to claim 1, characterized in that, In step S3, the construction interval of the zero equation containing the free matrix includes the sampling period. interval, The interval, and when the communication delay When the sampling period is less than The zero equation, by exploring the inherent constraints of power system state variables within each interval, supplements system dynamic information to reduce the conservatism of stability criteria.

7. The power system stability analysis method considering communication delay and periodic sampling according to claim 1, characterized in that, In step S2, the augmented bilateral closed-loop functional includes multiple subfunctionals, the expressions of which are as follows: In the formula, , , The expression is as follows: 。 8. The power system stability analysis method considering communication delay and periodic sampling according to claim 7, characterized in that, Differentiating the constructed functional yields the following integral term: 。 9. A power system stability analysis method considering communication delay and periodic sampling according to claim 8, characterized in that, The integral term is estimated, specifically by providing a positive definite matrix. With any matrix For any differentiable function , , sum vector The following inequalities hold: ; Calculation yields ;in, 。 10. A power system stability analysis method considering communication delay and periodic sampling according to claim 9, characterized in that, The verification of the linear matrix inequality uses Schur's complement theorem, when... and That is, the derivative of the augmented two-sided closed-loop functional. .

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