A method and system for stability analysis of power systems with time delay

Abstract

The present application belongs to the technical field of power system, and particularly relates to a time-delay power system stability analysis method and system. The method firstly establishes a load frequency control system model including time delay and nonlinear load disturbance, wherein the nonlinear load disturbance is a nonlinear disturbance taking unknown exogenous load disturbance as current and delayed state variable; then an augmented Lyapunov-Krasovskii functional is constructed, and an analysis method based on time-delay dependent matrix free matrix integral inequality and anti-convex combination inequality is adopted to analyze the stability of the load frequency control system, so as to obtain the stability criterion of the load frequency control system. The present application plays an important role in improving the calculation accuracy and time delay upper limit, and is helpful to expand the stable operation area of the power system.

Description

Technical Field

[0001] This invention belongs to the field of power system technology, specifically relating to a method and system for stability analysis of time-delay power systems. Background Technology

[0002] The construction of smart grids, the large-scale grid connection of distributed renewable energy, and the advancement of ultra-high voltage transmission projects have led to the development of modern power systems towards greater scale and complexity, with advantages such as cross-regional compensation, grid peak shifting, and mutual backup between generating units becoming more pronounced. However, the gradual expansion of the power grid, its increasingly complex operating characteristics, and the limitations of existing control measures can cause dynamic instability in regional power grids, even leading to large-scale blackouts and causing extremely serious social impacts. Therefore, for the power system to operate stably and safely, it is essential to ensure that the grid frequency is maintained at a certain fixed value or fluctuates within a small range around it. Load Frequency Control (LFC) technology is one of the most commonly used methods to achieve this goal. With the emergence of numerous dedicated networks and the use of synchronous phasor measurement technology, information in load frequency control systems is mostly transmitted through open communication networks. These open communication networks can achieve large-scale, large-volume information exchange, but due to network bandwidth limitations, there are many challenging problems during transmission, such as network congestion, waiting, discontinuous data transmission, and data packet loss. Therefore, time delays inevitably occur during communication, and the magnitude of these time delays poses a potential threat to the stable operation of the power system. Therefore, studying the stability problem of time-delay power systems based on load frequency control has significant practical implications.

[0003] LFC systems with communication channels are typical time-delay systems, which can be analyzed using Lyapunov functional methods combined with linear matrix inequalities (LMI) techniques. Since these methods provide sufficient conditions for system stability, they exhibit a degree of conservatism. Therefore, to continuously improve system performance and reduce the conservatism of the results, research has focused on two aspects: constructing suitable Lyapunov-Krasovskii functionals and selecting more accurate derivative estimation techniques. Many effective research methods have been proposed, such as Lyapunov functionals containing triple and quadruple integral terms, time-delay partitioning methods, Wirtinger integral inequalities, free-matrix-based integral inequalities, Bessel-Legendre integral inequalities, auxiliary-function-based integral inequalities, relaxed integral inequalities, and anti-convex combination methods. However, the stability criteria obtained by these methods are highly conservative, which is not conducive to expanding the stable operating region of power systems. Summary of the Invention

[0004] The purpose of this invention is to provide a method and system for stability analysis of time-delay power systems, in order to solve the problem that existing methods are not conducive to expanding the stable operating range of power systems.

[0005] To address the aforementioned technical problems, this invention provides a method for stability analysis of time-delay power systems, comprising the following steps:

[0006] 1) Establish a load frequency control system model that includes time delay and nonlinear load disturbance, wherein the nonlinear load disturbance is a nonlinear disturbance that treats unknown exogenous load disturbance as current and delay state variables;

[0007] 2) Construct an augmented Lyapunov-Krasovskii functional, and use the analysis methods of free matrix integral inequalities based on time delay dependency matrices and anti-convex combination inequalities based on time delay dependency matrices to analyze the stability of the load frequency control system, thereby obtaining the stability criterion of the load frequency control system.

[0008] Wherein, the free matrix integral inequality based on the time delay dependency matrix is ​​obtained by applying the constant matrix N in the free matrix integral inequality. i Set as the time-delay dependency matrix that introduces time delay The anticonvex combinatorial inequality based on the time delay dependency matrix is ​​defined by setting the constant matrix S in the anticonvex combinatorial inequality as the time delay dependency matrix that introduces the time delay. h(t)∈{h1,h2}, where h1 and h2 are given constants and 0≤h1≤h2. μ is a given constant and μ < 1.

[0009] Its beneficial effects are as follows: Based on the traditional free matrix integral inequality, this invention introduces a time-delay dependency matrix. Replacing the previous constant matrix with a time delay not only fully utilizes the upper and lower bounds of the time delay and their rate of change, but also introduces more free matrices, providing greater degrees of freedom during scaling and reducing the conservatism of the results. Furthermore, to further reduce the conservatism of the results, a time delay dependency matrix is ​​introduced based on the anti-convex combination inequality. This new estimation method breaks through the limitations of existing anti-convex combination inequalities, expanding its application scope and allowing the lower limit of time delay to be no longer confined to zero. This new method plays a crucial role in improving computational accuracy and the upper limit of time delay, contributing to obtaining a larger upper limit of time delay and thus expanding the stable operating range of power systems.

[0010] Furthermore, the free matrix integral inequality based on the time-delay dependency matrix mentioned in step 2) is:

[0011] For any positive definite matrix and Arbitrary constant matrix and and all differentiable functions Represents an n×n dimensional matrix space over the real number field. Represents a 3n×3n dimensional matrix space over the real number field. Describe a 3n×n dimensional matrix space over the real number field if the following conditions are met:

[0012]

[0013] Then the following inequalities hold:

[0014]

[0015] In the formula, Sym{···} represents the sum of the matrix within the parentheses and the transpose of the matrix within the parentheses. h 12 =h2-h1; And N ij Let be a constant matrix, i = 1, 2, j = 1, 2, 3, 4; the upper right corner T of the parameter indicates the transpose of the parameter, and the top · of the parameter indicates the first derivative of the parameter.

[0016] Furthermore, the anti-convex combinatorial inequality based on the time-delay dependency matrix in step 2) is:

[0017] For any vectors ω1, ω2, there exist constant matrices X1, X2 and a time-delay dependency matrix. Real scalars a≥0, b≥0, and satisfying Given a + b = 1, the following inequalities hold:

[0018]

[0019] In the formula, h 12 =h2-h1, and S i It is a constant matrix, i = 1, 2, 3, 4; the upper right corner T of the parameter indicates the transpose of the parameter; It represents the n×n dimensional matrix space over the real number field.

[0020] Furthermore, the augmented Lyapunov-Krasovskii functional constructed in step 2) is:

[0021]

[0022]

[0023]

[0024]

[0025]

[0026] In the formula, col{…} represents the column vector of elements within the parentheses, x(t) is the state variable, θ is the integral term, and P is a positive definite matrix. Q1, Q2, and Q3 are all positive definite matrices. η2(s) is an augmented vector; W1 and W2 are both positive definite matrices. h 12 =h2-h1; M1, M2, M3, and M4 are all positive definite matrices, and λ is the integral term; the superscript T of the parameter indicates the transpose of the parameter, and the dot on the parameter indicates the first derivative of the parameter.

[0027] Its beneficial effects are: in V1(x t A single integral term is introduced into the augmented variable of ). and and double integral terms and This establishes more connections between different cross vectors, compared to the single integral terms constructed in existing techniques. This invention is in V2(x t The form of ) was introduced in The integral term, of which It depends on the instantaneous state variable x(t) and its derivative. The augmented term further strengthens the relationship between the LK functional and the state variables. Furthermore, in V3(x t ) and V4(x t The paper introduces integral functionals in the form of double and triple integrals, which make full use of the boundary information of the system time delay, and helps to obtain a larger upper bound on the time delay and reduce the conservatism of the conclusions.

[0028] Furthermore, the load frequency control system model established in step 1) is as follows:

[0029]

[0030] In the formula, x(t) is the state variable; D is the generator's moment of inertia, T ch The inertial time constant of the steam turbine. T is the speed drop coefficient of the speed governor. g K is the inertial time constant, β is the frequency deviation factor, and K is the frequency deviation factor. P and K I These represent the proportional gain and integral gain of the PI controller, respectively, and M is the generator's moment of inertia. ΔPd φ(t) is the power grid load variable; φ(t) is a continuous initial phasor function on t∈[-h2,0].

[0031] Furthermore, in step 2), during the stability analysis, if the disturbance received by the load frequency control system is less than the set value, the FΔP value in the load frequency control system will be adjusted. d If (t) is set to 0, i.e., σ = ο = 0, where σ and ο are given nonnegative scalars, then the following corollary is obtained:

[0032] Given constants 0 ≤ h1 ≤ h2 and μ < 1, if there exists a positive definite matrix... and Symmetric matrix and arbitrary matrices Represents an n×n dimensional matrix space over the real number field. Represents a 3n×3n dimensional matrix space over the real number field. Represents a 2n×2n dimensional matrix space over the real number field. Represents a 5n×5n dimensional matrix space over the real number field. Represent a 3n×n dimensional matrix space over the real number field, for any matrix satisfying the constraints... If the following matrix inequalities hold, then FΔP satisfies... d The load frequency control system with t = 0 is asymptotically stable.

[0033]

[0034]

[0035] In the formula, Sym{…} represents the sum of the matrix within the parentheses and the transpose of the matrix within the parentheses; α is a real scalar; the superscript T in the parameter indicates the transpose of the parameter; diag{···} represents a diagonal matrix consisting of the elements in parentheses.

[0036] Further, in step 1), the nonlinear load disturbance is a norm-bounded nonlinear function of the current and delay state variables, i.e.:

[0037] FΔP d (t)=f(x(t),x(th(t)))

[0038] The corresponding norm boundedness condition that needs to be satisfied is:

[0039] f T (x(t),x(th(t)))f(x(t),x(th(t)))≤σ 2 x T (t)U T Ux(t)+ο 2 x T (th(t))V T Vx(th(t))

[0040] In the formula, σ and ο are given nonnegative scalars; U and V are constant matrices.

[0041] To address the aforementioned technical problems, the present invention also provides a time-delay power system stability analysis system. This system includes a memory and a processor, wherein the processor executes computer instructions stored in the memory to implement the time-delay power system stability analysis method described above, and achieves the same beneficial effects as the method. Attached Figure Description

[0042] Figure 1 This is a block diagram of the single-region LFC system considering time delay used in this invention;

[0043] Figure 2 This is a flowchart of the time-delay power system stability analysis method of the present invention;

[0044] Figure 3 This is a frequency response curve under the influence of steady time delay according to the present invention;

[0045] Figure 4 This is a frequency response curve diagram under the influence of time-varying time delay of the present invention;

[0046] Figure 5 This is a structural diagram of the time-delay power system stability analysis system of the present invention. Detailed Implementation

[0047] This invention first fully considers the impact of the time delay rate of change on the system's stable region. By integrating multi-dimensional information such as time delay, time delay rate of change, and system state in a load frequency regulation networked system, an augmented Lyapunov-Krasovskii functional is constructed. Then, when processing the integral term in the functional's derivative, methods such as the free matrix integral inequality and anti-convex combination, based on the time delay dependency matrix method, are employed to fully consider the upper and lower bounds of the time delay and its rate of change, resulting in a stability criterion with less conservatism. Finally, numerical examples are used to analyze the relationship between the PI controller gain and the system's time delay stability margin under different time delay types and disturbance conditions. Simulation results demonstrate the effectiveness and superiority of the proposed method.

[0048] The following sections will first introduce the free matrix integral inequalities based on time-delay dependency matrices and the anti-convex combinatorial inequalities based on time-delay dependency matrices. In the following formulas, and Let them represent the n-dimensional vector space and the n×m-dimensional matrix space over the real number field, respectively; R represents the set of n-order positive symmetric matrices; R T I represents the transpose of matrix R; P > 0 indicates that matrix P is symmetric and positive definite; n×n and 0 n×n Let X represent the identity matrix and the zero matrix of order n×n, respectively; col{…} represents the column vector of the elements in the parentheses, and diag{···} represents the diagonal matrix composed of the elements in the parentheses; Sym{X}=X+X T ; "*" indicates a symmetric term in the matrix; e i (i = 1, 2, ..., 14) represents the block identity matrix; η2(s) and These represent the augmented vector and its transpose, respectively. The remaining parameters will be explained in detail where they appear.

[0049] Lemma 1: For any positive definite matrix and arbitrary constant matrix and all differentiable functions If the following conditions are met:

[0050]

[0051] Then the following inequalities hold:

[0052]

[0053] In the formula,

[0054] Improved Lemma 1 (based on the free matrix integral inequality of time-delay dependent matrices): For any positive definite matrix and arbitrary constant matrix and all differentiable functions If the following conditions are met:

[0055]

[0056] Then the following inequalities hold:

[0057]

[0058] In the formula, And N ij (i = 1, 2, j = 1, 2, 3, 4) is a constant matrix.

[0059] Lemma 2: For any vectors ω1, ω2, there exist constant matrices X1, X2 and an arbitrary constant matrix S, with real scalars a ≥ 0, b ≥ 0, and satisfying Given a + b = 1, the following inequalities hold:

[0060]

[0061] Improved Lemma 2 (based on the anticonvex combinatorial inequality of the time-delay dependency matrix): For any vectors ω1, ω2, there exist constant matrices X1, X2 and a time-delay dependency matrix. Real scalars a≥0, b≥0, and satisfying Given a + b = 1, the following inequalities hold:

[0062]

[0063] In the formula, h 12 =h2-h1, and S i (i = 1, 2, 3, 4) is a constant matrix.

[0064] Based on this, the present invention will be further described in detail below with reference to the accompanying drawings and embodiments.

[0065] Method Implementation Examples:

[0066] An embodiment of the stability analysis method for time-delay power systems according to the present invention, the overall process of which is as follows: Figure 2 As shown, the process is as follows:

[0067] Step 1: Establish a load frequency control system model that includes time delay and nonlinear load disturbance. The nonlinear load disturbance is a nonlinear disturbance that treats unknown exogenous load disturbances as current and delay state variables.

[0068] In the stability analysis of time-delay power systems, high-order complex systems are often studied by analyzing simple, low-order linear systems. This embodiment will use a simplified load frequency control model, the control structure block diagram of which is shown below. Figure 1 As shown.

[0069] Figure 1 In this context, β is the frequency deviation factor, Δf is the change in system frequency, and ΔP is the frequency deviation factor. m It is the change in mechanical power, ΔP v It is the change in the opening degree of the automotive control valve, ΔP d M is the change in grid load, D is the generator's moment of inertia, and D is the generator's damping coefficient. It is the speed drop coefficient of the speed governor, T ch It is the turbine's inertial time constant, T. g It is the inertial time constant. Figure 1 The control system shown is based on the principle that when the grid load changes by ΔP... d This causes a frequency deviation Δf in the system, which generates a feedback signal ΔP used to regulate the prime mover. v This causes the output power increment ΔP of the prime mover to increase. m The change in load is compensated for, thereby bringing the system frequency back to a given value. An exponential function is typically used. Let h represent the time delay, where h + This is the upper bound of the time lag.

[0070] Area Control Error (ACE) only considers frequency variations, load variations, and switching power variations, which are caused by deviations in equipment and dispatching. For a single-area LFC system, ACE is calculated as ACE = βΔf. The following key state and output variables are selected: The following system state-space model can be obtained:

[0071]

[0072] In the formula, ω(t)=ΔP d .

[0073] Meanwhile, using ACE as the input to the designed controller, the output obtained using the PI controller can be:

[0074] u(t) = -K P ACE-K I ∫ACE (2)

[0075] In the formula, K P and K IThese are the proportional and integral gains, respectively.

[0076] Therefore, the following variables are defined:

[0077]

[0078] Combining equation (3), equation (2) can be further written as:

[0079] u(t)=-Ky(th(t)) (4)

[0080] To simplify the analysis, the PI control problem can be transformed into a static output feedback control problem. The following virtual state variables are defined:

[0081] x(t)=[Δf ΔP m ΔP v ∫ACE] T (5)

[0082] Substituting the PI controller (4) containing time-varying time-delay components into system (1), we can obtain the general form of the LFC system containing time-varying time-delay components:

[0083]

[0084] In the formula, and φ(t) is the state variable of the power system; φ(t) is a continuous initial phasor function on t∈[-h2,0]; h(t) is a time-varying time-delay function that satisfies:

[0085]

[0086] In the formula, h1, h2 and μ<1 are constants.

[0087] To more accurately estimate the intensity of load disturbances to the power system, unknown exogenous load disturbances can be treated as nonlinear disturbances of current and delay state variables:

[0088] FΔP d (t)=f(x(t),x(th(t))) (8)

[0089] The following norm boundedness conditions must be met:

[0090]

[0091] In the formula, U and V are constant matrices of appropriate dimensions, and σ and ο are given nonnegative scalars.

[0092] Step two: Introduce relevant propositions and lemmas.

[0093] Proposition 1: For any positive definite matrix and arbitrary constant matrix and all differentiable functions If the following conditions are met:

[0094]

[0095] Then the following inequalities hold:

[0096]

[0097] In the formula, And N ij (i = 1, 2, j = 1, 2, 3, 4) is a constant matrix.

[0098] To obtain a stability criterion with less conservatism, this invention proposes the inequality in Proposition 1, termed the improved free matrix integral inequality based on the time-delay dependency matrix method. Compared to the traditional free matrix integral inequality, this inequality introduces a time-delay dependency matrix... Replacing the previous constant matrix with a more flexible matrix not only fully utilizes the upper and lower bounds of the time delay and their rate of change, but also introduces more free matrices, providing greater degrees of freedom during scaling and reducing the conservatism of the results. When this condition is met, Proposition 1 can be simplified to the traditional free matrix integral inequality. Therefore, this new estimation method plays an important role in improving computational accuracy and the upper bound of time delay, and helps to expand the stable operating range of the power system.

[0099] Proposition 2: For any vectors ω1, ω2, there exist constant matrices X1, X2 and a time-delay dependency matrix. Real scalars a≥0, b≥0, and satisfying Given a + b = 1, the following inequalities hold:

[0100]

[0101] In the formula, h 12 =h2-h1, and S i (i = 1, 2, 3, 4) is a constant matrix.

[0102] To further reduce the conservatism of the results, Proposition 2 is a newly improved integral inequality that breaks through the limitations of existing anticonvex combinations based on time-delay-dependent matrix methods, expanding its applicability by making its lower time-delay limit not strictly zero. Therefore, Proposition 2 is more universally applicable.

[0103] Lemma 1: For any positive definite matrix and all differentiable functions And a < b, such that the following inequality holds:

[0104]

[0105] In the formula, χ1=ω(b)-ω(a),

[0106] Lemma 2: For any positive definite matrix Real scalars a and b satisfy phasor-valued functions And a < b, such that the following inequality holds:

[0107]

[0108]

[0109] In the formula,

[0110] Lemma 3: For phasor-valued functions Time-varying delay d(t)∈[0,h], symmetric matrix R>0, and any matrix S1 satisfying Where R1 = diag{R, 3R}, the following inequality holds:

[0111]

[0112] in,

[0113] Step 3: Construct an augmented Lyapunov-Krasovskii functional. Based on the content introduced in Step 2, analyze the stability of the load frequency control system using the free matrix integral inequality based on the time-delay dependency matrix and the anti-convex combination inequality based on the time-delay dependency matrix, thereby obtaining the stability criterion for the load frequency control system. That is, by constructing a suitable Lyapunov-Krasovskii functional and applying relatively novel analytical methods such as Proposition 1 and Proposition 2, the stability criterion for the time-delay LFC system is obtained. For simplicity, the following vectors and matrices are defined:

[0114]

[0115]

[0116]

[0117] Theorem 1: For given constants 0 ≤ h1 ≤ h2, μ < 1 and λ > 0, if there exists a positive definite matrix and Symmetric matrix and any matrix For satisfying the constraints If the following matrix inequalities hold, then system (6) is asymptotically stable:

[0118]

[0119]

[0120] In the formula, Ξ=Θ1+Θ2+Θ 31 +Θ 41 +Θ 42 +Θ 43 ,

[0121] h 12 =h2-h1, e i =[0 n×(i-1)n I n×n 0 n×(14-i)n ] T ,i=1,2,...,14.

[0122] Proof: Choose the following Lyapunov-Krasovskii functional:

[0123]

[0124] In the formula,

[0125] As is well known, a suitable Lyapunov-Krasovskii functional is crucial for reducing the conservatism of the results. In V1(x t A single integral term is introduced into the augmented variable of ). and double integral terms This establishes more connections between different cross vectors, compared to the single integral terms constructed in existing techniques. This invention is in V2(x t The form of ) was introduced in The integral term, of which It depends on the instantaneous state variable x(t) and its derivative. The augmented term further strengthens the relationship between the LK functional and the state variables. Furthermore, in V3(x t ) and V4(x t The paper introduces integral functionals in the form of double and triple integrals, which make full use of the boundary information of the system time delay, and helps to obtain a larger upper bound on the time delay and reduce the conservatism of the conclusions.

[0126] Along the solution trajectory of system (6) for V(x) t Differentiation yields:

[0127]

[0128]

[0129]

[0130]

[0131] In the formula,

[0132] From proposition 1 to equation (21) An estimation yields the following:

[0133]

[0134] Lemma 1 and Proposition 2 relate to equation (21) It can be estimated that:

[0135]

[0136] Substituting equations (23) and (24) into equation (21), we get:

[0137]

[0138] By dealing with the integral terms in equation (22) using Lemma 2 and Lemma 3, we can obtain:

[0139]

[0140]

[0141]

[0142]

[0143]

[0144]

[0145] Y3+Y6≤-ξ T (t)ψ2ξ(t) (32)

[0146] Substituting equations (26) to (32) into equation (22), we get:

[0147]

[0148] In summary, from equations (19), (20), (25), and (33), we can obtain:

[0149]

[0150] Based on the Lyapunov stability principle, when At that time, system (6) is asymptotically stable, that is:

[0151]

[0152] By further applying Schur's complement lemma to equation (35), we can obtain equation (10), thus completing the proof.

[0153] In the When processing the integral term, the improved free matrix integral inequality proposed in Proposition 1 is applied to... The estimation process introduces multiple free matrices, effectively utilizing more time delay and derivative information. For the single-integral term... When performing analytical analysis, the single integration interval is divided into two sub-intervals: [th(t), t-h1] and [t-h2, th(t)]. The estimation is performed using the auxiliary function single integral inequality in Lemma 1 and the improved anti-convex combination technique in Proposition 2, which greatly improves the estimation accuracy of the integral term and helps to obtain a larger upper bound on the time delay.

[0154] In processing V4(x) t When calculating the time derivative, the time delay segmentation technique is used to divide the double integral term. and The components are finely divided as shown in equation (22). The double integral terms Y1, Y2, Y4, and Y5 are estimated using the double integral inequality with auxiliary functions in Lemma 2, while the single integral term Y3+Y6 is handled using the relaxed integral inequality in Lemma 3. This fully utilizes the time delay and its derivative information, thereby further improving the estimation accuracy of the functional derivative and greatly reducing the scaling space on both sides of the inequality, effectively reducing the conservatism of the result.

[0155] When the LFC system is subjected to very small load disturbances or almost no disturbances, consider the FΔP in system (6). d Since (t) = 0, that is, σ = ο = 0, we can obtain the following corollary 1.

[0156] Corollary 1: For given constants 0 ≤ h1 ≤ h2 and μ < 1, if there exists a positive definite matrix and Symmetric matrix and any matrix For satisfying the constraints If the following matrix inequalities hold, then FΔP satisfies... d The system (6) with t = 0 is asymptotically stable:

[0157]

[0158]

[0159] In the formula, Π1,Π4,Π5,Π6,Γ,Θ 31 ,Θ 41 ,Θ 42 ,Θ 43 The parameters are the same as in Theorem 1.

[0160] Building upon Corollary 1, and in order to demonstrate the superiority of Propositions 1 and 2, Corollary 2 will present a stability criterion based on the traditional free matrix integral inequality and anticonvex combination method.

[0161] Corollary 2: For given constants 0 ≤ h1 ≤ h2 and μ < 1, if there exists a positive definite matrix and Symmetric matrix and any matrix For satisfying the constraints If the following matrix inequalities hold, then FΔP satisfies... d The system (6) with t = 0 is asymptotically stable:

[0162]

[0163]

[0164] in, The definitions of other symbols are the same as in Corollary 1.

[0165] Step four, we will now perform simulation analysis.

[0166] Simulation Example 1: A typical second-order system.

[0167] First, consider FΔP in the time-delay system (6). d Given (t) = 0, the following typical second-order system is used to verify whether the proposed method can effectively reduce the conservatism of the system criterion, where:

[0168]

[0169] Table 1 lists the maximum permissible upper limit of the system time delay h2 obtained using Corollary 1 and Corollary 2, given the lower limit h1 and the rate of change of time delay μ, respectively. When μ = 0.5 and h1 = 2, the maximum permissible upper limit h2 increases by 36.04%. When μ = 0.5 and h1 = 0, the maximum permissible upper limit h2 increases by 19.12%. It is evident that the upper limit of the time delay obtained using Corollary 1 is significantly better than the result obtained using Corollary 2; therefore, the method proposed in this invention significantly reduces the conservatism of the results.

[0170] Table 1 shows the upper bound of the maximum permissible time delay of the system when taking different h1 and μ values.

[0171]

[0172] When μ = 0.5, Table 2 provides the values ​​of the maximum permissible upper time delay bound h2 for different lower time delay bounds h1 when the system is stable. By comparing the simulation results with those of existing methods 1–6, Corollary 1 significantly improves the upper time delay bound of the system.

[0173] Table 2 shows the upper bound of the maximum permissible time delay of the system when μ = 0.5 and different h1 values ​​are used.

[0174]

[0175] When h1 = 0, Table 3 provides the values ​​of the maximum permissible upper bound of the time delay h2 for different time delay change rates μ when the system is stable. By comparing the simulation results with those of existing methods 7–13, Inference 1 significantly reduces the conservatism of the results. In summary, the analytical method used in this invention significantly improves the maximum permissible upper bound of the time delay, further verifying the superiority of the proposed method.

[0176] Table 3 shows the upper bound of the maximum permissible time delay of the system when taking different μ values.

[0177]

[0178]

[0179] Simulation Example 2: Single-area LFC system.

[0180] Considering the time-delay LFC system (6), the parameters of the PI type LFC system are shown in Table 4.

[0181] Table 4 System Parameters

[0182]

[0183] 1) Results analysis.

[0184] Table 5 shows the maximum allowable upper bound h2 of the LFC system obtained by adjusting the PI control gain under different time delay types (steady delay μ = 0 and time-varying delay μ = 0.5) and different disturbance conditions, using Theorem 1. The constant matrices U and V are both taken as 0.1I4. The simulation results in Table 5 show that the selection of the PI controller gain has a certain impact on the maximum allowable upper bound of the system's time delay, based on the same K... P and K I The larger the rate of change of time delay μ, the smaller the maximum allowable upper bound of time delay h2, and the time delay stability margin is larger under steady time delay conditions compared to time-varying time delay conditions. Furthermore, under the same control gain, as the disturbance variables σ and ο increase, the upper bound of time delay h2 decreases, demonstrating that the impact of load disturbances on the power system is very significant.

[0185] Table 5 Upper bound of maximum permissible time delay of the system under different conditions

[0186]

[0187]

[0188] Table 6 shows different K values. P and K I The upper bound of the maximum permissible time delay of the system (μ = 0)

[0189]

[0190] Table 6 lists the upper bound of the maximum allowable time delay h2 of the system obtained using Corollary 1 when μ = 0 under different PI controller gains. It can be seen that, for the same time delay type, different controller gains have a significant impact on the system's time delay stability margin. When the proportional gain K... PWhen taking the same value, the upper bound of the time delay h2 changes with the integral gain K. I As K increases, it decreases, and K decreases. P The smaller the value, the more pronounced this trend becomes. However, the upper bound of the time delay h2 and K... P The relationship between them is quite complex. When K I When taking the same value, as K... P As the gain increases, the maximum allowable upper limit of time delay h2 first increases and then decreases. To design a high-performance controller, the relationships between controller gain and time delay stability margin, as described above, can serve as auxiliary conditions for selecting PI controller parameters, which is of great significance for enabling the power system to achieve a larger stable operating range.

[0191] 2) Result comparison.

[0192] Table 7 compares the simulation results with existing methods 14, 15, and 16 for different time delay types and given disturbance conditions. The simulation results show that the upper bound of the time delay obtained by Theorem 1 is significantly higher than the results obtained by other methods. Therefore, in the stability analysis process, compared with the Bessel-Legendre integral inequality in existing method 14, the auxiliary function integral inequality in existing method 15, and the Wirtinger-based integral inequality in existing method 16, the novel integral inequality improved by the time delay dependency matrix proposed in this invention not only significantly reduces the conservatism of the conclusions but also has a clear advantage in improving the time delay stability margin of practical networks.

[0193] Table 7 shows the upper bound of the maximum permissible time delay of the system obtained under different conditions using different methods.

[0194]

[0195] Table 8 shows the upper bound of the maximum permissible time delay of the system when h1 is taken in different ways.

[0196]

[0197] When K P =0.1,K I When h = 0.2, given a lower time delay bound h1, Table 8 lists the maximum permissible upper time delay bound h2 obtained using different methods. Comparing the results with existing methods 17-19, the h2 obtained by Corollary 1 is significantly larger, which is beneficial for expanding the stable operating region of the system. Therefore, the newly proposed integral inequality in this invention also applies to cases where the lower time delay bound is not zero.

[0198] 3) Simulation verification.

[0199] To verify the accuracy of the above theoretical results, we used MATLAB / Simulink to perform simulations based on the single-region LFC system model (6).

[0200] In the case of steady time delay: the load disturbance FΔP in the system model (6) d (t) is taken in the form of equation (9), where the matrix U = 0, V = 0.1I4, and the nonnegative scalars σ = 0, ο = 0.025. As shown in Table 5, when K P =0.1,K I When = 0.4, the upper bound of the maximum permissible time delay of the system is h. max = 3.32s. Based on the above conditions, Figure 3 The change in frequency deviation, Δf, is given. The curve in the figure shows that the frequency deviation eventually converges to zero, indicating that the grid frequency has returned to its rated value, meaning the system is approaching a stable state.

[0201] Time-varying and time-delay situations: Figure 4 The frequency response curves of system (6) with time-varying time delay and load disturbance are given. In this case, the load disturbance parameters are taken as U = V = 0.1I⁴, σ = ο = 0.025. When K... P =0.1,K I When = 0.6, the corresponding upper bound of the maximum permissible time delay is h. max =1.80s. From Figure 4 The simulation curves show that the system response time is quite significant, indicating that time delay has a substantial impact on the power system and must be considered. Ultimately, the frequency deviation converges to the equilibrium point, demonstrating asymptotic stability and confirming the accuracy of the theoretical results.

[0202] System Implementation Example:

[0203] An embodiment of the time-delay power system stability analysis system of the present invention, such as... Figure 5 As shown, the system includes a memory, a processor, and an internal bus. The processor and memory communicate and exchange data with each other via the internal bus. The memory includes at least one software function module stored in the memory. The processor executes various functional applications and data processing by running the software programs and modules stored in the memory, thereby implementing the time-delay power system stability analysis method described in the embodiments of the present invention.

[0204] The processor can be a microprocessor (MCU), a programmable logic device (FPGA), or other processing devices. The memory can be any type of memory that stores information using electrical energy, such as RAM and ROM; it can also be any type of memory that stores information using magnetic energy, such as hard disks, floppy disks, magnetic tapes, magnetic core memory, bubble memory, and USB flash drives; it can also be any type of memory that stores information using optical methods, such as CDs and DVDs; and of course, it can also be other types of memory, such as quantum memory and graphene memory.

Claims

1. A stability analysis method for a time-delay power system, characterized in that, Includes the following steps: 1) Establish a load frequency control system model that includes time delay and nonlinear load disturbance, wherein the nonlinear load disturbance is a nonlinear disturbance that treats unknown exogenous load disturbance as current and delay state variables; 2) Construct an augmented Lyapunov-Krasovskii functional, and use the analysis methods of free matrix integral inequalities based on time delay dependency matrices and anti-convex combination inequalities based on time delay dependency matrices to analyze the stability of the load frequency control system, thereby obtaining the stability criterion of the load frequency control system. Wherein, the free matrix integral inequality based on the time delay dependency matrix is ​​the constant matrix in the free matrix integral inequality. Set as the time-delay dependency matrix that introduces time delay , The anticonvex combination inequality based on the time delay dependency matrix is ​​to convert the constant matrix in the anticonvex combination inequality into... Set as the time-delay dependency matrix that introduces time delay , , and All are given constants and , , Given a constant and .

2. The stability analysis method for time-delay power systems according to claim 1, characterized in that, The free matrix integral inequality based on the time delay dependency matrix mentioned in step 2) is: For any positive definite matrix and Arbitrary constant matrix and and all differentiable functions , Representing the real number field 3D matrix space, Representing the real number field 3D matrix space, Representing the real number field A 3D matrix space, if it satisfies: Then the following inequalities hold: In the formula, , , This represents the sum of the matrix within parentheses and the transpose of the matrix within parentheses. , , , , , ; ,and It is a constant matrix. The superscript T in the upper right corner of the parameter indicates the transpose of the parameter. It represents the first derivative of the parameter.

3. The stability analysis method for time-delay power systems according to claim 1, characterized in that, In step 2), the anti-convex combinatorial inequality based on the time-delay dependency matrix is: For any vector There exists a constant matrix. and time delay dependency matrix Real scalar and satisfy , The following inequalities hold: In the formula, , ,and It is a constant matrix. The "T" in the upper right corner of the parameter indicates the transpose of the parameter. Representing the real number field 3D matrix space.

4. The stability analysis method for time-delay power systems according to claim 1, characterized in that, The augmented Lyapunov-Krasovskii functional constructed in step 2) is: In the formula, , This represents a column vector containing the elements within the parentheses. For state variables, It is an integral term; It is a positive definite matrix. ; ; and All are positive definite matrices. , For augmented vectors; and All are positive definite matrices. ; ; All are positive definite matrices, and ; The term is the integral term; the superscript T on the parameter indicates the transpose of the parameter, and the dot on the parameter indicates the first derivative of the parameter.

5. The stability analysis method for time-delay power systems according to claim 1, characterized in that, The load frequency control system model established in step 1) is as follows: In the formula, For state variables; , , It is the generator's moment of inertia. The inertial time constant of the steam turbine. This is the speed drop coefficient of the speed controller. The inertial time constant, For frequency deviation factor, and These are the proportional gain and integral gain of the PI controller, respectively. The moment of inertia of the generator. , For power grid load variables; yes The initial phasor function is continuous on the upper surface.

6. The stability analysis method for time-delay power systems according to claim 5, characterized in that, In step 2), during the stability analysis, if the disturbance received by the load frequency control system is less than the set value, the load frequency control system will... Set to 0, that is ,in and Given a nonnegative scalar, the following corollary follows: For a given constant and If a positive definite matrix exists , , and symmetric matrix and any matrix , , , , Representing the real number field 3D matrix space, Representing the real number field 3D matrix space, Representing the real number field 3D matrix space, Representing the real number field 3D matrix space, Representing the real number field A 3D matrix space, for satisfying the constraints , If the following matrix inequalities hold, then satisfying... The load frequency control system is asymptotically stable. ， ， In the formula, , This represents the sum of the matrix within parentheses and the transpose of the matrix within parentheses; , , ; ; , , , , , , , , , , , , , , , , , , , , , , , , , , , , , It is a real scalar; the T in the upper right corner of the parameter indicates the transpose of the parameter; , , , This represents a diagonal matrix consisting of the elements in parentheses.

7. The stability analysis method for time-delay power systems according to claim 5, characterized in that, In step 1), the nonlinear load disturbance is a norm-bounded nonlinear function of the current and delay state variables, i.e.: The corresponding norm boundedness condition that needs to be satisfied is: In the formula, and Given a nonnegative scalar; and It is a constant matrix.

8. A stability analysis system for a time-delay power system, characterized in that, It includes a memory and a processor, the processor being used to execute computer instructions stored in the memory to implement the time-delay power system stability analysis method as described in any one of claims 1 to 7.

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