A stability analysis method of load frequency control system considering participation of electric vehicles

Abstract

The application discloses a stability analysis method of a load frequency control system considering participation of an electric vehicle, constructs a new Lyapunov functional based on a time delay interval, establishes different sub-functionals in different time delay intervals, and integrates a time delay product type item to capture more time delay information; and then by combining the proposed functional with a zero equation method, the application discards a mode of using a single and fixed functional in the whole time delay interval in the traditional way, reduces the conservativeness of a criterion, and obtains a maximum time delay margin, so as to provide less conservative stability analysis and control design basis for the load frequency control system containing the electric vehicle.

Description

Technical Field

[0001] This invention belongs to the field of load frequency control stability analysis technology, and in particular, relates to a stability analysis method for a load frequency control system that takes into account the participation of electric vehicles. Background Technology

[0002] Load frequency control is a crucial element in maintaining frequency stability and power balance in power systems. With the widespread application of communication networks in cyber-physical power systems, time delays are inevitably introduced into the control loop, which weakens system dynamic performance and threatens stability. Especially in power systems containing electric vehicles, frequent charging and discharging further exacerbates the system's nonlinear and time-varying characteristics, making time-delay-dependent stability analysis more complex.

[0003] Existing research mainly employs two types of methods: one is frequency domain analysis based on eigenvalues, which is suitable for constant delays but struggles to handle time-varying delays; the other is time domain analysis based on Lyapunov functionals. To reduce conservatism, researchers have proposed methods such as augmented functionals, delay-decomposition functionals, and delay-product functionals. However, these methods typically use a single Lyapunov functional across the entire time delay interval, making it difficult to reflect the dynamic changes in the time delay across different intervals.

[0004] A patent application with publication number CN119448328A discloses a stability analysis method for time-delay load frequency control considering electric vehicle access. This method falls under the technical field of load frequency control stability analysis, including: establishing a nominal load frequency control model for the load frequency control system using model reconstruction based on the generation characteristics of electric vehicles and distributed power sources; establishing a deterministic load frequency control model for the load frequency control system based on the fluctuations in the state of charge of electric vehicles; introducing auxiliary polynomials to construct two types of improved relaxation matrix functions; combining multiple integrals; and using the Lyapunov stability analysis method and the Bessel-Legendre inequality to obtain the asymptotic stability criterion for the nominal load frequency control system and the robust stability criterion for the uncertain load frequency control system, respectively; and completing the stability analysis of the time-delay load frequency control. This patent analyzes the entire time delay interval based on a unified Lyapunov functional structure, implicitly assuming that the system has consistent dynamic characteristics across different delay intervals, making it difficult to characterize the differences in the system's dynamic behavior when the time delay function changes across different intervals. Summary of the Invention

[0005] This invention addresses the issue that existing stability analysis methods typically employ a single Lyapunov functional across the entire time delay interval, making it difficult to reflect the dynamic changes in time delay across different intervals. Therefore, this invention proposes a stability analysis method for a load frequency control system that considers the participation of electric vehicles.

[0006] A stability analysis method for a load frequency control system considering electric vehicle participation includes the following steps:

[0007] S1. Construct a dynamic model of a single-area load frequency control system that includes electric vehicles, wind power, and time-varying time delays. The dynamic model includes the output power difference of electric vehicles, the output power variation of wind power, and the time-varying time delay function. ; S2. Based on the preset upper bound of time delay and interval division parameters, establish a piecewise Lyapunov functional based on time delay intervals, and use the corresponding Lyapunov matrix for different time delay sub-intervals. S3. Using the zero-equality method combined with the piecewise Lyapunov functional, estimate the integral term that appears after differentiating the Lyapunov functional and combining it with the integral inequality, and derive the time-delay-related stability criterion; the stability criterion is expressed in the feasible form of a set of linear matrix inequalities; S4. Solve the linear matrix inequalities to verify the satisfaction of the stability criterion. If satisfied, the load frequency control system containing electric vehicles is determined to be stable, and the maximum time-delay margin of the system is calculated and output; S5. Analyze the relationship between PI controller parameters and time delay margin through simulation.

[0008] Further, in step S1, constructing the dynamic model of the single-region load frequency control system includes the following steps: S11. Define the state vector of the control system, with the following expression: ,in, For frequency deviation, For the mechanical power output of the generator, For valve opening, This indicates the output power deviation of electric vehicles. For state variables; S12. Construct the system state equations, the expression of which is: ;in, Represents a known real matrix. Represents a known real matrix. Indicates the system status. Represents a known real matrix. Indicates a disturbance input; S13. Introduce time-varying delay function The time-varying time-delay function satisfy , , All are constants; S14. The system state equation is reconstructed, and the reconstructed expression is:

[0009] In the formula, , , All are block matrices.

[0010] Further, in step S2, candidate Lyapunov functionals are constructed: , The piecewise Lyapunov functional based on time delay intervals is expressed as follows:

[0011] In the formula, It is a time-varying time-delay function. and For the divided time-delay sub-intervals, The upper limit of the time lag, For additional integral functional terms.

[0012] Furthermore, the aforementioned The expressions are as follows:

[0013]

[0014]

[0015] In the formula, ; , , , ( =1,2,3) are all positive definite matrices. Represents a state combination vector. This represents a given computation function. Represents a state combination vector. Indicates the system at time 10:00 s The state vector; The The expression is: .

[0016] In the formula, , It is a symmetric positive definite matrix. These represent time and integration variables, respectively.

[0017] Furthermore, step S1 also includes a dynamic model of the electric vehicle system and the wind turbine generator, expressed as:

[0018]

[0019] In the formula, and These correspond to the time constant and gain of the electric vehicle system, respectively. Let be the time constant of the wind power system. and These represent the changes in output power of wind turbine generator sets and wind power, respectively.

[0020] Furthermore, in step S3, the stability criterion is derived using the zero equality method combined with piecewise Lyapunov functional derivation, specifically including: for a given constant... There exist symmetric positive definite matrices and arbitrary matrices. , The symmetric positive definite matrix includes: , , , , ,

[0021] Conditions met:

[0022] And matrix inequalities Established; In the formula, , , This represents a block matrix.

[0023] Furthermore, in step S5, the simulated system parameters satisfy: , , , , , , Participation factors of governor and turbine And the participation factors of electric vehicles ,satisfy This ensures that the total control quantity is reasonably distributed between these two parts; the time-varying delay parameter satisfies... .

[0024] Furthermore, the expression for the PI controller is:

[0025] In the formula, Indicates control input, Represents the frequency deviation coefficient. This is the adjustment coefficient. This is the load damping coefficient. .

[0026] Furthermore, the integral inequality is the Bessel-Legendre inequality or the Wirtinger inequality, used to estimate the integral terms in the functional derivative.

[0027] A load frequency control system employs the stability analysis method described above for a load frequency control system considering electric vehicle participation. Compared with the prior art, the beneficial effects of the present invention are as follows: 1. This invention captures more time-delay information by constructing different Lyapunov functionals within different time-delay intervals, and incorporating time-delay product terms into these functionals, thus eliminating the need for a single Lyapunov functional (LKF) covering the entire time-delay interval as in the past. 2. By combining Lyapunov functionals with the zero-equation method, a novel time-delay-related stability criterion suitable for single-region load frequency control systems for electric vehicles is derived. The method of this invention abandons the traditional approach of using a single, fixed functional across the entire time-delay interval, reducing the conservatism of the criterion and obtaining maximum time-delay margin, providing a less conservative basis for stability analysis and control design of load frequency control systems containing electric vehicles.

[0028] 2. This invention combines the inverse convex combination lemma and second-order integral inequalities (such as the second-order Bessel-Legendre inequality) to handle the cross terms and integral terms in the derivative of a functional. This allows for a more precise estimation of the integral term after the functional is differentiated, thereby reducing the conservatism of the criterion and enhancing its applicability to time-varying delay conditions. It is suitable for load frequency control systems where time delay fluctuations are caused by the charging and discharging of electric vehicles. Attached Figure Description

[0029] Figure 1 This is a flowchart of the present invention; Figure 2 This is a schematic diagram of the dynamic model of the single-area load frequency control system of the present invention; Figure 3 Frequency deviation of the LFC system of this invention The dynamic response curve. Detailed Implementation

[0030] 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.

[0031] Example 1 like Figure 1 As shown, a stability analysis method for a load frequency control system considering electric vehicle participation includes the following steps: S1. Construct a dynamic model of a single-area load frequency control system that includes electric vehicles, wind power, and time-varying time delays. The dynamic model includes the output power difference of electric vehicles, the output power variation of wind power, and the time-varying time delay function. ; S2. Based on the preset upper bound of time delay and interval division parameters, establish a piecewise Lyapunov functional based on time delay intervals, and use the corresponding Lyapunov matrix for different time delay sub-intervals. S3. Using the zero-equality method combined with the piecewise Lyapunov functional, estimate the integral term that appears after differentiating the Lyapunov functional and combining it with the integral inequality, and derive the time-delay-related stability criterion; the stability criterion is expressed in the feasible form of a set of linear matrix inequalities; S4. Solve the linear matrix inequalities to verify the satisfaction of the stability criterion. If satisfied, the load frequency control system containing electric vehicles is determined to be stable, and the maximum time-delay margin of the system is calculated and output; S5. Analyze the relationship between PI controller parameters and time delay margin through simulation.

[0032] In this embodiment, express Vieuxiliary space. Indicates all A set of real matrices of order 1. (separately, ) denotes the set of symmetric matrices (specifically, the set of positive definite matrices). The set of positive integers. Superscript and These represent the inverse and transpose of a matrix or vector, respectively. (Symbols) , and These represent the symmetric terms, column vectors, and diagonal matrix of the block matrix, respectively.

[0033] A schematic diagram of a single-area load frequency control system is shown below. Figure 2 As shown. Among them, , , and These represent frequency deviation, generator mechanical power output, valve opening, and load disturbance, respectively. , and These correspond to the inertia coefficient, turbine time constant, and governor constant, respectively. ACE is calculated from the frequency deviation and tie-line power deviation, although the tie-line power deviation is zero in a single-zone load frequency control system. The control error in this zone is... Processing steps. The ACE input is sent to the PI controller to generate a control signal. , This represents the time-varying delay introduced by signal transmission. The dynamic models of electric vehicle systems and wind turbine generators can be described as follows: (1) (2) in, and These correspond to the time constant and gain of the electric vehicle system, respectively. Used to indicate the output power deviation of electric vehicles. and These refer to the changes in output power of the wind turbine generator and the wind power itself, respectively. The designed PI controller is as follows: (3) in, Indicates control input, Represents the frequency deviation coefficient. This is the adjustment coefficient. This is the load damping coefficient. .

[0034] Figure 1 Medium parameters and These represent the participation factors of the governor and turbine, and the participation factor of the electric vehicle, respectively.

[0035] In this embodiment, step S1, which involves constructing the dynamic model of the single-region load frequency control system, further includes the following steps: S11. Define the state vector of the control system, and its expression is: ,in, For frequency deviation, For the mechanical power output of the generator, For valve opening, This indicates the output power deviation of electric vehicles. For state variables; S12. Construct the system state equations, the expression of which is: (5) in, Represents a known real matrix. Represents a known real matrix. Indicates the system status. Represents a known real matrix. Indicates a disturbance input; Specifically,

[0036]

[0037]

[0038] .

[0039] S13. Introduce time-varying delay function The time-varying time-delay function satisfy , , All are constants; S14. The system state equation is reconstructed, and the reconstructed expression is: (6) In the formula, , , All are block matrices. Specifically: , , , , , .

[0040] External disturbances This does not affect the internal stability of the system. Therefore, the stability analysis of system expression (6) will only consider... .

[0041] In this embodiment, Lemma 1 and Lemma 2 are referenced. Lemma 1: Given any scalar... and and There exists a matrix ,as well as , so that: (7) (8) In the above formula, Represents the integral vector. Represents a vector;

[0042]

[0043]

[0044]

[0045] .

[0046] Lemma 2: Given a random variable , , And satisfy , , so that: (9) In the above formula, Represents a symmetric matrix. Represents any matrix, Represents a symmetric matrix. Represents an arbitrary matrix; .

[0047] In this embodiment, before obtaining the improved time-delay-related stability criterion, the following notation is introduced for simplicity: , , , , , , , , ,

[0048] ; In the above formula, Indicates the system state within the time interval Integral vector within, Indicates the system state within the time interval The vector obtained by weighted integration within the vector. Represents the integral variable. Both represent state combination vectors. Represents the augmented state combination vector. This represents a given computation function. Represents any matrix, Represents a diagonal matrix. Represents a symmetric matrix. It represents any positive integer.

[0049] In step S3, the stability criterion is derived using the zero equality method combined with piecewise Lyapunov functional derivation, specifically including: given a constant , , If a symmetric matrix exists , , and , , , , , , , , , , , , , , and arbitrary matrices , , , .

[0050] (10) (11) (12) (13) In the above formula, Each represents a defined matrix block. Let each represent an arbitrary matrix. This represents the defined matrix block; , , , , , , , , , , ,

[0051] , , , , , , , , , , , , , .

[0052] In the above formula, , , This represents the defined matrix. Represents a constant. , , , , , , , , , , , , , , This represents the defined vector.

[0053] In this embodiment, in step S2, a candidate Lyapunov functional is constructed: (14) The piecewise Lyapunov functional based on time delay intervals is expressed as follows: (15) In the formula, It is a time-varying time-delay function. and For the divided time-delay sub-intervals, The upper limit of the time lag, For additional integral functional terms.

[0054] The The expressions are as follows:

[0055]

[0056]

[0057] In the formula, ; , , , ( =1,2,3) are all positive definite matrices. Represents a state combination vector. This represents a given computation function. Represents a state combination vector. Indicates the system at time 10:00 s The state vector; The The expression is: .

[0058] In the formula, , It is a symmetric positive definite matrix. These represent time and integration variables, respectively.

[0059] exist At that time, for Lyapunov functionals Taking the derivative, we get: (16) (17) In the above formula, , , , , .

[0061] Therefore, applying Lemma 1, when And when conditions (10) to (11) are true, we can obtain:

[0062]

[0063] (18) In the formula, .

[0064] By applying Lemma 2 to the last two terms in equation (18), we obtain:

[0065]

[0066] Furthermore, the zero equation is given below: (19) Therefore, we can conclude that: (20) Therefore, if And if conditions (12) to (13) are true, then we can obtain ,in For a sufficiently small scalar. Following steps (16)-(20) above, when At that time, if And since conditions (10)-(11) are true, we can obtain: (twenty one) Therefore, if And if conditions (12)-(13) are true, then we can obtain ,in It is a sufficiently small scalar.

[0067] In summary, if And since conditions (11)-(12) are true, we can finally obtain ,in .

[0068] This embodiment reduces the conservatism of the stability criterion by using a piecewise Lyapunov functional, and the time delay margin is significantly better than existing methods, which can ensure the stability of the system frequency under electric vehicle charging and discharging and wind power fluctuations.

[0069] Example 2 like Figure 1 As shown, a stability analysis method for a load frequency control system considering electric vehicle participation includes the following steps: S1. Construct a dynamic model of a single-area load frequency control system that includes electric vehicles, wind power, and time-varying time delays. The dynamic model includes the output power difference of electric vehicles, the output power variation of wind power, and the time-varying time delay function. ; S2. Based on the preset upper bound of time delay and interval division parameters, establish a piecewise Lyapunov functional based on time delay intervals, and use the corresponding Lyapunov matrix for different time delay sub-intervals. S3. Using the zero-equality method combined with the piecewise Lyapunov functional, estimate the integral term that appears after differentiating the Lyapunov functional and combining it with the integral inequality, and derive the time-delay-related stability criterion; the stability criterion is expressed in the feasible form of a set of linear matrix inequalities; S4. Solve the linear matrix inequalities to verify the satisfaction of the stability criterion. If satisfied, the load frequency control system containing electric vehicles is determined to be stable, and the maximum time-delay margin of the system is calculated and output; S5. Analyze the relationship between PI controller parameters and time delay margin through simulation.

[0070] In this embodiment, a case study was conducted to verify a single-area load frequency control system. The system parameters were set as follows: , , , , , , , .

[0071] Given Including PI controller ( , The time delay margin of the single-area load frequency control system is shown in Table 1. Table 1 When Time delay margin

[0072] All time delay margins obtained by this invention are superior to those in the existing literature [1], clearly demonstrating that the proposed time delay-related stability criterion has lower conservatism.

[0073] Literature [1] is Zeng, HB, Zhou, SJ, Zhang, XM, and Wang, W. (2022). Delay-dependent stability analysis of load frequency control systems with electric vehicles. IEEE Transactions on Cybernetics, 52(12), 13645-13653. Example 3 To verify the effectiveness of the proposed criterion, simulation tests were conducted. Given... , And the time-varying time-delay function is The time delay satisfies the upper bound. With rate of change constraints, a time-domain simulation of the closed-loop system is performed to obtain the dynamic response of the system's frequency deviation, as shown below. Figure 3 As shown. Figure 3 The simulation results show that the frequency deviation of the load frequency control system converges, validating the effectiveness of the proposed method. The simulation results also show that the frequency deviation... The system can quickly converge to zero after the disturbance, which directly verifies that the system is stable under this time delay, thus confirming the effectiveness of the stability judgment of the present invention.

[0074] 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 stability analysis method for a load frequency control system considering electric vehicle participation, characterized in that, Includes the following steps: S1. Construct a dynamic model of a single-area load frequency control system that includes electric vehicles, wind power, and time-varying time delays. The dynamic model includes the output power difference of electric vehicles, the output power variation of wind power, and the time-varying time delay function. ; S2. Based on the preset upper bound of time delay and interval division parameters, establish a piecewise Lyapunov functional based on time delay intervals, and use the corresponding Lyapunov matrix for different time delay sub-intervals. S3. Using the zero-equality method in combination with the piecewise Lyapunov functional, the integral term that appears after differentiating the Lyapunov functional and combining it with the integral inequality is estimated, and the time-delay-related stability criterion is derived; the stability criterion is expressed in the feasibility form of a set of linear matrix inequalities. S4. Solve the linear matrix inequality to verify the satisfaction of the stability criterion. If satisfied, determine that the load frequency control system containing electric vehicles is stable, and calculate and output the maximum time delay margin of the system. S5. Analyze the relationship between PI controller parameters and time delay margin through simulation.

2. The stability analysis method for a load frequency control system considering electric vehicle participation according to claim 1, characterized in that, In step S1, constructing the dynamic model of the single-area load frequency control system includes the following steps: S11. Define the state vector of the control system, with the following expression: ,in, For frequency deviation, For the mechanical power output of the generator, For valve opening, This indicates the output power deviation of electric vehicles. For state variables; S12. Construct the system state equations, the expression of which is: ;in, Represents a known real matrix. Represents a known real matrix. Indicates the system status. Represents a known real matrix. Indicates a disturbance input; S13. Introduce time-varying delay function The time-varying time-delay function satisfy , , All are constants; S14. The system state equation is reconstructed, and the reconstructed expression is: In the formula, , , All are block matrices.

3. The stability analysis method for a load frequency control system considering electric vehicle participation according to claim 1, characterized in that, In step S2, candidate Lyapunov functionals are constructed: , The piecewise Lyapunov functional based on time delay intervals is expressed as follows: In the formula, It is a time-varying time-delay function. and For the divided time-delay sub-intervals, The upper limit of the time lag, For additional integral functional terms.

4. The stability analysis method for a load frequency control system considering electric vehicle participation according to claim 3, characterized in that, The The expressions are as follows: In the formula, ; , , , ( =1,2,3) are all positive definite matrices. Represents a state combination vector. This represents a given computation function. Represents a state combination vector. Indicates the system at time 10:00 The state vector; The The expression is: 。 In the formula, , It is a symmetric positive definite matrix. These represent time and integration variables, respectively.

5. The stability analysis method for a load frequency control system considering electric vehicle participation according to claim 1, characterized in that, Step S1 also includes a dynamic model of the electric vehicle system and the wind turbine generator, expressed as: In the formula, and These correspond to the time constant and gain of the electric vehicle system, respectively. Let be the time constant of the wind power system. and These represent the changes in output power of wind turbine generator sets and wind power, respectively.

6. The stability analysis method for a load frequency control system considering electric vehicle participation according to claim 1, characterized in that, In step S3, the stability criterion is derived using the zero equality method combined with piecewise Lyapunov functional derivation, specifically including: for a given constant There exist symmetric positive definite matrices and arbitrary matrices. , The symmetric positive definite matrix includes: 、 、 、 、 、 Conditions met: And matrix inequalities Established; In the formula, , , This represents a block matrix.

7. The stability analysis method for a load frequency control system considering electric vehicle participation according to claim 1, characterized in that, In step S5, the simulated system parameters satisfy: , , , , , , The participation factors of governor and turbine And the participation factors of electric vehicles ,satisfy The time-varying time-delay parameters satisfy .

8. The stability analysis method for a load frequency control system considering electric vehicle participation according to claim 1, characterized in that, The expression for the PI controller is: In the formula, Indicates control input, Represents the frequency deviation coefficient. This is the adjustment coefficient. This is the load damping coefficient. .

9. The stability analysis method for a load frequency control system considering electric vehicle participation according to claim 1, characterized in that, The integral inequality is called the Bessel-Legendre inequality or the Wirtinger inequality, and it is used to estimate the integral terms in the derivative of a functional.

10. A load frequency control system, characterized in that, The stability analysis method for a load frequency control system considering electric vehicle participation, as described in any one of claims 1 to 9, is adopted.

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