A defense method against zero-dynamic attacks on wind power generation systems

Abstract

The present invention belongs to the field of network security and discloses a defense method for zero-dynamic attacks on wind power generation systems, comprising: constructing a mathematical model of the wind power generation system; introducing a gain-scheduled PI controller and electronically adjustable passive components into the wind power generation system, wherein the electronically adjustable passive components are controllable inductance modules and controllable capacitance modules, which function to adjust the inductance and capacitance of the system; designing a zero-dynamic attack signal, injecting it into the system, and analyzing the destructive factors of the zero-dynamic attack; implementing defense against the designed zero-dynamic attack signal, dynamically adjusting and optimizing the parameter values ​​of the component adjustment coefficients to adjust the inductance and capacitance, converting unstable zero points into stable zero points, weakening the destructiveness of the zero-dynamic attack, and achieving defense. This method enables the system to maintain stable operation in complex environments, overcomes the dependence limitations of existing technologies, and enhances the robustness of the defense.

Description

Technical Field

[0001] The present invention belongs to the field of network security, and in particular relates to a defense method for zero-dynamic attacks on a wind power generation system. Background Art

[0002] Zero-Dynamics Attack (ZDA), as a new attack method in the field of network security, has caused huge losses to wind power generation systems due to its unpredictability and concealment.

[0003] Existing defenses against these attacks fall into two main categories: one is to modify the system model to make it difficult for attackers to obtain precise system parameters; the other is to introduce external perturbations or modify control variables to interfere with the attack. However, these methods still have limitations in their application to multi-input, multi-output systems.

[0004] A generalized keeper (GH) is a holding device proposed to deal with zero-dynamics attack (ZDA). It controls the system zero points by designing a specific holding function so that all zero points are located within the unit circle, thereby weakening or eliminating the impact of ZDA on the system.

[0005] A generalized sampler (GS) is a sampling device that can flexibly adjust the zero-point characteristics of a sampling system, aiming to increase the degree of freedom in zero-point configuration of a sampled data system. The goal is to place the zero point of the sampled data system within the unit circle, thereby stabilizing the zero dynamics and avoiding internal state divergence caused by ZDA.

[0006] However, the implementation of generalized keepers and generalized samplers relies on precise control of the system zero point, which typically requires complex numerical optimization algorithms and high-precision modeling. However, in practical control systems, due to system parameter uncertainties such as device aging and environmental changes, the design of GH and GS may fail. This is especially true in dynamic environments, where frequent adjustments to the keepers or samplers' parameters are required to ensure zero-point stability. This significantly increases the difficulty and cost of system implementation.

[0007] Furthermore, existing GH and GS methods mostly assume that system parameters are constant or change slowly. However, in some industrial scenarios, system parameters may fluctuate significantly due to changes in operating conditions. For example, in a wind power generation system, changes in wind speed directly affect the system's dynamic characteristics, and GH and GS cannot adapt to these changes in real time. Implementing generalized retainers and samplers in existing technologies in such situations requires careful design of the retaining function or precise calculation of sampling weights, which greatly increases the design difficulty. Summary of the Invention

[0008] In order to solve this technical problem, the present invention provides a defense method against zero-dynamic attacks on wind power generation systems. This method effectively reduces the destructive impact of zero-dynamic attacks, effectively reduces costs and improves reliability, enables the system to maintain stable operation in complex environments, overcomes the dependence limitations of existing technologies, and enhances the robustness of defense.

[0009] In order to achieve the above object, the present invention is achieved through the following technical solutions:

[0010] The present invention is a defense method for zero-dynamic attacks on a wind power generation system, which specifically includes the following steps:

[0011] Step 1: Construct a mathematical model of the wind power generation system and obtain the transfer function of the power generation system;

[0012] Step 2: Introduce a gain-scheduled PI controller and electronically adjustable passive components into the wind power generation system obtained in step 1, wherein the gain-scheduled PI controller gain scheduling includes a proportional gain k p and integral gain k i The electronically adjustable passive components are specifically a controllable inductance module and a controllable capacitance module, and the function of the controllable inductance module and the controllable capacitance module is to adjust the inductance L and the capacitance C of the wind power generation system;

[0013] Step 3: design a zero-dynamic attack signal, inject the designed zero-dynamic attack signal into the wind power generation system, and analyze the destructive factors of the zero-dynamic attack;

[0014] Step 4: Implement defense against the zero dynamic attack signal designed in step 3, and dynamically adjust and optimize the component adjustment coefficient k l And the adjustment coefficient k of the component to the capacitance c The parameter values ​​are selected to adjust the inductor L and capacitor C, converting the unstable zero point into a stable zero point, weakening the destructiveness of the zero dynamic attack and achieving defense.

[0015] A further improvement of the present invention is that: the step 1 specifically includes the following steps:

[0016] Step 1.1: Assume that the electromotive force of the wind power generation system is E, which can be expressed as:

[0017] E=K g ω g

[0018] Among them, K g is the electromagnetic torque coefficient, ω g is the generator speed;

[0019] The voltage balance equation in the entire wind power generation system is:

[0020]

[0021] Where U is the motor terminal voltage, is the voltage drop across the resistor, U L is the voltage drop across the inductor, i L is the current through the inductor, I is the motor armature current, R1 and R2 are the resistances in the system, and the torque balance equation is:

[0022]

[0023] Among them, K t is the potential constant, the total moment of inertia J=J d +J m +J h , J d is the moment of inertia of the wind turbine, J m is the moment of inertia of the direct-drive permanent magnet synchronous generator (D-PMSG), J h For the gearbox with equivalent moment of inertia, combining the characteristics of capacitance and inductance, the above formula is combined to obtain

[0024]

[0025] in,

[0026]

[0027] Perform Laplace transform,

[0028] U(s)=K g ((a0s 3 +a1s 2 +a2s+1)ω g (s));

[0029] Where L and C are the inductance and capacitance of the wind power system before the controllable elements are deployed;

[0030] Step 1.2, the transfer function G1(s) of the wind power generation system:

[0031]

[0032] make Then we have:

[0033]

[0034] A further improvement of the present invention is that in step 2, a gain-scheduled PI controller is introduced into the power generation system obtained in step 1, specifically:

[0035] Step 2.1: Set the proportional gain k p and integral gain ki The values ​​of are dynamically associated with the real-time inductance L(t) and real-time capacitance C(t), respectively, so that the gain-scheduled PI controller can adapt to the changes in wind power system parameters. The association relationship is defined as follows:

[0036] K p (t)=αL(t)+βC(t)

[0037] K i (t) = γL(t) + δC(t)

[0038] Among them, the sampling time t = kT, T is the sampling period, α, β, γ, δ are the internal adjustment parameters of the gain-scheduled PI controller, which is designed and adjusted according to the needs of the wind power generation system, and flexibly responds to the real-time changes of the real-time inductance L(t) and real-time capacitance C(t). The gain-scheduled PI controller automatically adjusts K according to the changes in the system state. p (t) and K i (t).

[0039] The internal adjustment parameters α, β, γ, and δ of the gain-scheduled PI controller are set to fixed values. The transfer function of the wind power generation system becomes:

[0040]

[0041] Where G1 is the transfer function of the wind power system, and G2 is the transfer function of the gain-scheduled PI controller;

[0042] Step 2.2: Convert the wind power system transfer function from the frequency domain to the state space table in the time domain and select the state variable Among them, ω g is the angular velocity of the generator, and the high-order derivatives represent the corresponding rate of change. The state variables are used to describe the dynamic behavior of the wind power generation system. The state space expression is:

[0043]

[0044] y(t)=Cx(t)

[0045] The system matrix is ​​expressed as:

[0046]

[0047] B=[Kk i Kk p 0 0] T

[0048] C=[1 0 0 0]

[0049] Step 2.3: To remotely operate a continuous-time system, a gain-scheduled PI controller is used to perform this task, which is to remotely operate a continuous-time wind power generation system. The input and output of the gain-scheduled PI controller are processed at discrete time points. The discretized state equation is expressed as:

[0050]

[0051] For ease of expression, the above formula is simplified to:

[0052] x(k+1)=Gx(k)+Hu(tk)

[0053] y(k)=Cx(k)

[0054] Wherein, sampling time t=kT, where T is the sampling period;

[0055] In the discretization process, the zero-order hold method (ZOH) is used for discretization, and the discretization matrix of the wind power generation system is:

[0056]

[0057]

[0058] A further improvement of the present invention is that in step 2, the electronically adjustable passive components are introduced into the wind power generation system obtained in step 1, specifically:

[0059] The controllable inductor module realizes real-time dynamic adjustment of the total inductance by connecting the adjustable inductor array in parallel with the original inductance L of the wind power generation system. The adjustment formula is:

[0060] L ′ =Lk l

[0061] Among them, L ′ is the equivalent inductance after adjustment of the controllable inductance module, L is the original inductance of the wind power generation system, k l k is the adjustment coefficient of the component to the inductance, by adjusting l , effectively controlling the inductance and ensuring that its regulation meets the actual needs of the power system;

[0062] The controllable capacitor module is connected in parallel with the original capacitance of the wind power generation system to the adjustable capacitor array to achieve real-time dynamic adjustment of the total capacitance. The adjustment formula is:

[0063] C ′ =Ck c

[0064] Among them, C ′ is the equivalent capacitance after adjustment by the controllable capacitor module, C is the original capacitance of the wind power generation system, k cis the adjustment coefficient of the component to the capacitance;

[0065] In order to prevent the parameter adjustment from affecting the normal operation of the original wind power generation system, the adjustment coefficient k of the component to the inductance is set to l And the adjustment coefficient k of the component to the capacitance c Configure as a continuously adjustable value within a specified parameter range, defining the parameter adjustment range:

[0066] k l,min ≤k l ≤k l,max

[0067] k c,min ≤k c ≤k c,max

[0068] Equivalent inductance L after adjustment by the controllable inductance module ′ And the equivalent capacitance C after adjustment by the controllable capacitance module ′ It is a linear function of the inductance L and the capacitance C. As long as the boundary values ​​meet the required conditions, all intermediate values ​​meet the same criteria.

[0069] A further improvement of the present invention is that in step 3, the zero dynamic attack signal is specifically designed as follows:

[0070] Step 3.1: Based on the Euclidean algorithm, the transfer function of the wind power system in step 2.2 is rewritten to represent the transfer function of the feedback loop:

[0071]

[0072] Where Q(s) is the quotient and R(s) is the remainder;

[0073] Step 3.2: Convert the wind power system into Byrnes-Isidori form

[0074] η(t+1)=G0η(t)+H0C c ξ(t)

[0075] ξ(t+1)=(G c +H c λ T )ξ(t)+H c b m (u(t)-C0η(t))

[0076] y(t)=C c ξ(t)

[0077] in,

[0078]

[0079] λ T ξ(t)=y(t)+b m C0η(t)-b m u(t)

[0080]

[0081] C c =[1 0 0]

[0082] Among them, b m is the maximum power numerator coefficient of the Byrnes-Isidori form transfer function; η(t) and ξ(t) represent the internal and external states of the wind power system, G0, H0, and C0 are the minimum realizations of the transfer function of the feedback path R(s) / E(s). Applying the Euclidean algorithm, the transfer function of the equation in step 3.2 is derived and expressed after sampling as

[0083] η[k+1]=G d η[k]+H d C c ξ[k]

[0084] ξ[k+1]=(G c +H c λ T )ξ[k]+H c b m (u[k]-C d η[k])

[0085] y[k]=C c ξ[k]

[0086] Step 3.3: The goal of the zero-dynamic attack signal is to cut off the connection between the internal and external states of the wind power generation system. Assume that the attacker injects an attack signal a[k] into the wind power generation system that introduces a gain-scheduled PI controller, i.e., the closed-loop system. The attack signal is designed as:

[0087] z[k+1]=G d z[k]

[0088] a[k]=C d z[k]

[0089] After the attack is launched, the sampling is converted into

[0090] z[k+1]=G d z[k]

[0091] ξ[k+1]=(G c +H c λ T )ξ[k]+H c b m u[k]

[0092] y[k]=C c ξ[k]

[0093] A further improvement of the present invention is that in step 3, the destructive factors of the zero dynamic attack are analyzed, specifically:

[0094] For the discretized wind power generation x[k+1]=Gx[k]+Hu[k];y[k]=Cx[k], design zero dynamic attack z[k+1]=G d z[k], the impact of zero dynamic attack on the internal state of the wind power system depends on G d , assuming G d is an n×n matrix with eigenvalues ​​λ1,λ2,…,λ n , the initial state z[0] is replaced by G d The eigenvector of is represented as follows:

[0095] z[0]=c1v1+c2v2+…+c n v n

[0096] Among them, v i is the eigenvalue λ i The corresponding eigenvector, c i It is v i Coefficient, according to the state equation of the open-loop system, that is, the wind power generation system without the introduction of the gain scheduling PI controller, is:

[0097]

[0098] Because G d Decompose into G d =VΛV -1 , where Λ is the diagonal matrix of eigenvalues ​​and V is the matrix of eigenvectors, we get:

[0099] z[k]=VΛ k V -1 z[0]

[0100] Substituting z[0] into the above equation, we can obtain and satisfy the following conditions:

[0101]

[0102] Analyzing the eigenvalue λ i The modulus |λ i |The impact on the state vector z[k] is as follows:

[0103] If there exists an eigenvalue λ i So that |λ i |>1, then It grows exponentially with k, so the state vector z[k] will also become unbounded, causing z[k] to be unstable;

[0104] If all eigenvalues ​​satisfy |λ i |≤1, then It does not grow unbounded with k, but remains bounded or converges to zero as k increases, thus ensuring that z[k] remains bounded or converges to a stable value;

[0105] The impact of zero dynamic attack on the internal state of wind power generation system depends on G d , and meet the following conditions:

[0106]

[0107] in, For existence, Indicates any one.

[0108] A further improvement of the present invention is that: in step 4, the defense method for the zero dynamic attack signal designed in step 3 is to offset the unstable zero point of the wind power generation system after sampling. There are two types of unstable zero points of the wind power generation system after sampling. One is the unstable zero point of the original system after sampling mapping, and the other is additionally introduced due to sampling. By adjusting k l and k c , converting unstable zero points into stable zero points, specifically including:

[0109] Step 4.1. Consider the designed gain-scheduled PI controller. The expression of the zero point is:

[0110]

[0111] Due to the deployment of electronically adjustable passive components, the above expression is transformed into:

[0112]

[0113] It can be seen that the adjustment of the zero point mainly depends on k l and k c The choice of k l and k c The choice of is constrained to ensure that the primary functionality of the system is not compromised. Therefore, not all systems can implement defenses against zero-dynamic attacks.

[0114] Step 4.2: Under the condition that the sampling period of the continuous-time system is T, the obtained discrete system is expressed by the transfer function

[0115]

[0116] Among them, D d(z) is the characteristic polynomial of the discrete system. For a wind power generation system with a relative degree greater than 2, the degree of the characteristic polynomial of the discrete system increases after sampling, thereby introducing additional zeros. These additional zeros are sampling-induced zeros.

[0117] The degree of discretization of the characteristic polynomial increases, which is expressed as

[0118] D d (z) = b0 + b1z -1 +b2z -2 +…+b n z -n

[0119] Among them, the coefficient b i It is a function of the continuous system coefficients a0, a1, a2. According to the parameter definition in the system, a0, a1, a2 are directly related to the inductance L and capacitance C, so b i It is also a function of L and C, recorded as:

[0120] b i =f(a0,a1,a2,…)=g(L,C)

[0121] The change of the inductor L and capacitor C values ​​will affect the position of the root of the characteristic polynomial Dd(z) of the discrete system. The characteristic polynomial D d The (z) roots correspond to the zeros in the discrete system.

[0122] After sampling the system, the composition of the zero point becomes more complicated. In order to deal with the unstable zero introduced by sampling, it is necessary to select appropriate values ​​within the parameter range.

[0123] A further improvement of the present invention is that in step 4, the parameter k l and k c The optimization is achieved by particle swarm optimization (PSO) algorithm, which includes the following steps:

[0124] Step 4.1.1. Initialize the positions and velocities of particles in the wind power system, where each particle represents a candidate solution.

[0125] Step 4.1.2. In each iteration, the transfer characteristics of the wind power system are evaluated by calculating the inductance and capacitance corresponding to each particle to determine whether the wind power system meets the stability condition, that is, |P| < 1;

[0126] Step 4.1.3: Update the particle's historical best position p according to its performance best and the global optimal position g best ;

[0127] Step 4.1.4: Adjust the speed and position of the particles, iterate and find the optimal solution, and finally output the optimal inductance k. l and capacitance k c parameter.

[0128] The beneficial effects of the present invention are:

[0129] 1. Reduce implementation difficulty and cost

[0130] This invention uses electronically adjustable passive components, eliminating the complex design and numerical optimization required for zero-point configuration in traditional GH and GS systems. These components stabilize the system's zero-point position by automatically and in real time adjusting the system's inductance and capacitance, eliminating the need for precise numerical optimization and high-precision models. This eliminates the need for frequent parameter adjustments and costly numerical calculations, enabling a simpler control approach and significantly reducing the difficulty and cost of system implementation.

[0131] Because they no longer rely on complex numerical calculations and high-precision modeling, electronically tunable passive components effectively reduce costs and improve reliability by simplifying the implementation process, enabling the system to maintain stable operation in complex environments.

[0132] 2. Enhance robustness in dynamic environments

[0133] The electronically adjustable passive components designed in this invention, combined with a gain-scheduled PI controller, monitor the system's operating status and automatically adapt to real-time changes in environmental conditions, effectively addressing uncertainties in dynamic environments. In applications such as wind power generation, the electronically adjustable passive components can adjust the system zero point in real time based on changes in external factors such as wind speed, thereby ensuring system stability and improving protection against ZDA.

[0134] Since the present invention can dynamically adjust key parameters, it can automatically adapt to changes in system characteristics even in industrial scenarios such as wind speed fluctuations, ensuring the stability of the system zero point, thereby improving the defense effect and the overall robustness of the system.

[0135] 3. Overcoming the limitations of system zero-point configuration's reliance on feedback

[0136] In traditional methods, system zero-point configuration relies on system status or output feedback. Incomplete system status information or invalid feedback information can limit the control effectiveness of GH and GS. This invention enhances system adaptability by directly adjusting physical parameters, avoiding feedback dependency and ensuring stable system operation even in the presence of invalid feedback or incomplete information.

[0137] Since the present invention is based on the design of physical parameter adjustment, it can work effectively under conditions of incomplete system status feedback, ensuring the continuous stability of the system in complex environments such as network attacks, overcoming the dependence limitations of existing technologies, and enhancing the robustness of defense. BRIEF DESCRIPTION OF THE DRAWINGS

[0138] Figure 1 It is a structural diagram of the wind power generation system of the present invention.

[0139] Figure 2 It is the equivalent circuit diagram of the power generation link of the present invention.

[0140] Figure 3 It is a schematic diagram of the wind power generation system of the present invention being subjected to a zero-power attack.

[0141] Figure 4 It is a schematic diagram of the changes in the internal and external states of the system when the present invention attacks the unstable zero point introduced after sampling.

[0142] Figure 5 It is a schematic diagram of the changes in the state variables of the system when the present invention attacks the unstable zero point introduced after sampling.

[0143] Figure 6 It is a schematic diagram of the changes in the internal and external states of the system after the defense method of the present invention is deployed.

[0144] Figure 7 It is a schematic diagram of the changes in system state variables after the defense method of the present invention is deployed. DETAILED DESCRIPTION

[0145] The following diagrams illustrate embodiments of the present invention. For clarity, many practical details are included in the following description. However, it should be understood that these practical details are not intended to limit the present invention. In other words, in some embodiments of the present invention, these practical details are not essential.

[0146] The present invention proposes a defense method for zero dynamic attacks on wind power generation systems. Compared with the complex design of zero point configuration, complex numerical optimization algorithms and high-precision system modeling of GH and GS, the introduction of electronically adjustable passive components simplifies the zero point adjustment process. By monitoring the changes in system parameters in real time and automatically adjusting the values ​​of inductance and capacitance to maintain the stability of the system zero point, complex numerical optimization and high-precision modeling are no longer required, thereby reducing the difficulty and cost of system implementation. In addition, electronically adjustable passive components can dynamically adjust key parameters according to real-time changes in system operating conditions (such as load, environmental conditions, etc.). This capability enables the system to effectively combat ZDA when facing uncertain factors. In addition, combined with a gain-scheduled PI controller, the present invention can achieve real-time adjustment of the PI control gain, ensuring the zero point stability of the system under different conditions, thereby improving the robustness of the system.

[0147] Specifically, the defense method of the present invention includes the following steps:

[0148] Step 1: Construct a mathematical model of the wind power generation system and obtain the transfer function of the power generation system;

[0149] The wind power generation system of the present invention is composed as follows: 1. Wind turbine: including blades, hub and rotor; 2. Transmission system: including main shaft and gearbox; 3. Generator: mainly permanent magnet synchronous generator (PMSG); 4. Power conversion system: including rectifier, inverter and filter.

[0150] Wind turbines capture the kinetic energy of wind and convert it into mechanical energy using blades, hubs, and rotors. The mechanical energy is transmitted to the generator through the main shaft and gearbox of the transmission system. The generator converts the mechanical energy into electrical energy. Finally, the power conversion system adjusts the AC power output by the generator into stable DC power or AC power that meets grid connection requirements through rectifiers, inverters, and filters, thereby achieving efficient conversion and output of wind energy into electrical energy.

[0151] The power generated by the generator is regulated by the converter to meet the needs of the power grid, stabilize the voltage and frequency, and ensure the reliable grid-connected operation of the system. The monitoring and data acquisition system tracks the operating status and key parameters in real time, and ensures the efficient operation of the equipment and timely problem handling through data analysis, thereby extending the service life and improving power generation efficiency. Figure 1 shown.

[0152] From the system structure, it can be seen that the power generation link of the wind power generation system is at risk of being attacked through the input signal. Interference with the input signal may damage the power generation link, affecting power generation efficiency and system stability. Zero dynamic attack achieves the purpose of attack by manipulating the input signal to damage the power generation link and affect the normal operation of the system. The circuits involved are as follows: Figure 2 shown.

[0153] Assume that the electromotive force of the wind power generation system is E, which can be expressed as:

[0154] E=K g ω g

[0155] Among them, K g is the electromagnetic torque coefficient, ω g is the generator speed.

[0156] According to Kirchhoff's voltage law, the voltage balance equation in the entire wind power generation system can be derived as follows:

[0157]

[0158] Where U is the motor terminal voltage, is the voltage drop across the resistor, U L is the voltage drop across the inductor, i L is the current through the inductor, I is the motor armature current, R1 and R2 are the resistances in the system, and the torque balance equation is:

[0159]

[0160] Among them, K t is the potential constant, the total moment of inertia J=J d +J m +J h , J d is the moment of inertia of the wind turbine, J m is the moment of inertia of the direct-drive permanent magnet synchronous generator (D-PMSG), J h For the gearbox with equivalent moment of inertia, combining the characteristics of capacitance and inductance, the above formula is combined to obtain

[0161]

[0162] in,

[0163]

[0164] Perform Laplace transform,

[0165] U(s)=K g ((a0s 3 +a1s 2 +a2s+1)ω g (s));

[0166] Where L and C are the inductance and capacitance of the wind power system before the controllable elements are deployed;

[0167] The transfer function G1(s) of the wind power generation system is:

[0168]

[0169] make Then we have:

[0170]

[0171] Step 2: Introduce a gain-scheduled PI controller and electronically adjustable passive components into the wind power generation system obtained in step 1, wherein the gain scheduling includes a proportional gain k p and integral gain k i The electronically adjustable passive components are specifically a controllable inductance module and a controllable capacitance module. The function of the controllable inductance module and the controllable capacitance module is to adjust the inductance L and capacitance C of the wind power generation system.

[0172] A common PI controller is:

[0173]

[0174] To further optimize the performance of the controller and ensure its stability and robustness under various operating conditions, the proportional gain k p and integral gain k i The values ​​of are dynamically associated with the real-time inductance L(t) and real-time capacitance C(t), respectively, so that the gain-scheduled PI controller can adapt to the changes in wind power system parameters. The association relationship is defined as follows:

[0175] K p (t)=αL(t)+βC(t)

[0176] K i (t) = γL(t) + δC(t)

[0177] Among them, the sampling time t = kT, T is the sampling period, α, β, γ, δ are the internal adjustment parameters of the gain-scheduled PI controller, which is designed and adjusted according to the needs of the wind power generation system, and flexibly responds to the real-time changes of the real-time inductance L(t) and real-time capacitance C(t), ensuring the efficient and stable operation of the control system under different working conditions. The gain-scheduled PI controller automatically adjusts K according to the changes in the system state. p (t) and K i (t). This mechanism improves the system’s adaptability in complex environments and makes the controller design more versatile and practical.

[0178] To simplify the system, the internal adjustment parameters α, β, γ, and δ of the gain-scheduled PI controller are set to fixed values. The transfer function of the wind power generation system becomes:

[0179]

[0180] Among them, G1 is the transfer function of the wind power generation system, and G2 is the transfer function of the gain-scheduled PI controller.

[0181] To facilitate subsequent system analysis, the system is converted from the transfer function form in the frequency domain to the state space expression in the time domain. Select the state variable Among them, ω g is the angular velocity of the generator, and the higher-order derivatives represent the corresponding rate of change. The state variables are used to describe the dynamic behavior of the wind power generation system. Based on this choice, the state space expression is:

[0182]

[0183] y(t)=Cx(t)

[0184] The system matrix is ​​expressed as:

[0185]

[0186] B=[Kk i Kk p 0 0] T

[0187] C=[1 0 0 0]

[0188] To remotely operate a continuous-time system, we assume that a gain-scheduled PI controller is used to perform this task. The gain-scheduled PI controller can implement a variety of control strategies. Discretization is used to process continuous-time signals in the gain-scheduled PI controller so that they can adapt to discrete sampling periods. Sampling time t = kT, where T is the sampling period, is set. At this time, the input and output of the gain-scheduled PI controller are processed at discrete time points. The discretized state equation is expressed as:

[0189]

[0190] For ease of expression, the above formula is simplified to:

[0191] x(k+1)=Gx(k)+Hu(tk)

[0192] y(k)=Cx(k)

[0193] During the discretization process, the zero-order hold (ZOH) method is used to adjust the system's state equation to reflect the dynamic characteristic changes brought about by discretization while keeping the static output equation of the system unchanged. Specifically, using ZOH for discretization, the discretization matrix of the wind power generation system is:

[0194]

[0195] Electronically adjustable passive components are a type of dynamically adjustable passive components, specially designed for use in the fields of power electronics and automatic control, providing flexible and dynamic response capabilities in circuits.

[0196] The controllable inductor module is connected in parallel with the existing inductor to form an adjustable inductor array or use an adjustable inductor module based on power electronics. By using power electronic switches, these small inductors can be quickly controlled to be connected or disconnected, thereby achieving real-time dynamic adjustment of the total inductance.

[0197] In order to facilitate analysis and application, a simplified adjustment formula is introduced, which is expressed as:

[0198] L ′ =Lk l

[0199] Among them, L ′ is the equivalent inductance after adjustment of the controllable inductance module, L is the original inductance of the wind power generation system, k l k is the adjustment coefficient of the component to the inductance, by adjusting l , effectively controlling the inductance and ensuring that its regulation meets the actual needs of the power system;

[0200] Similarly, the controllable capacitor module and the original capacitance of the wind power generation system are connected in parallel with the adjustable capacitor array to achieve real-time dynamic adjustment of the total capacitance. The adjustment formula is:

[0201] C ′ =Ck c

[0202] Among them, C ′ is the equivalent capacitance after adjustment by the controllable capacitor module, C is the original capacitance of the wind power generation system, k c is the adjustment coefficient of the component to the capacitance;

[0203] In order to prevent the parameter adjustment from affecting the normal operation of the original wind power generation system, the adjustment coefficient k of the component to the inductance is set to l And the adjustment coefficient k of the component to the capacitance c Configure as a continuously adjustable value within a specified parameter range, defining the parameter adjustment range:

[0204] k l,min ≤k l ≤k l,max

[0205] k c,min ≤k c ≤k c,max

[0206] Equivalent inductance L after adjustment by the controllable inductance module ′ And the equivalent capacitance C after adjustment by the controllable capacitance module ′It is a linear function of the inductance L and the capacitance C. As long as the boundary values ​​meet the required conditions, all intermediate values ​​meet the same criteria.

[0207] Step 3: Design a zero-dynamic attack signal, inject the designed zero-dynamic attack signal into the wind power generation system, and analyze the destructive factors of the zero-dynamic attack.

[0208] Based on the Euclidean algorithm, the transfer function of the wind power system is rewritten to represent the transfer function of the feedback loop:

[0209]

[0210] Here, Q(s) is the quotient and R(s) is the remainder; this form can be interpreted as the transfer function representation of the feedback loop.

[0211] Converting wind power generation systems to Byrnes-Isidori form

[0212] η(t+1)=G0η(t)+H0C c ξ(t)

[0213] ξ(t+1)=(G c +H c λ T )ξ(t)+H c b m (u(t)-C0η(t))

[0214] y(t)=C c ξ(t)

[0215] in,

[0216]

[0217] λ T ξ(t)=y(t)+b m C0η(t)-b m u(t)

[0218]

[0219] C c =[1 0 0]

[0220] Among them, b m is the maximum power numerator coefficient of the Byrnes-Isidori form transfer function; η(t) and ξ(t) represent the internal and external states of the wind power generation system, G0, H0 and C0 are the minimum realizations of the feedback path R(s) / E(s) transfer function. Applying the Euclidean algorithm, the transfer function of the above equation is derived and expressed after sampling as

[0221] η[k+1]=G d η[k]+H d C c ξ[k]

[0222] ξ[k+1]=(G c +H c λ T )ξ[k]+H c b m (u[k]-C d η[k])

[0223] y[k]=C c ξ[k]

[0224] The construction of the system matrix can refer to the discretization matrix of the wind power generation system:

[0225]

[0226] The zero-dynamic characteristic is described by a transfer function or state-space model, which manifests as a situation where the system's output remains unchanged under certain inputs, but its internal state deviates significantly. Attackers exploit this characteristic by designing signals so that the system's output appears normal, while its internal state gradually deviates from the expected trajectory, ultimately leading to system failure. The goal of the zero-dynamic attack signal is to cut off the connection between the internal and external states of the wind power generation system. Suppose the attacker injects an attack signal a[k] into the wind power generation system that introduces a gain-scheduled PI controller, i.e., a closed-loop system, as shown in the following example: Figure 3 To achieve this goal, the attack signal is designed to be:

[0227] z[k+1]=G d z[k]

[0228] a[k]=C d z[k]

[0229] After launching the attack, the sampling expression formula is transformed into

[0230] z[k+1]=G d z[k]

[0231] ξ[k+1]=(G c +H c λ T )ξ[k]+H c b m u[k]

[0232] y[k]=C c ξ[k]

[0233] From the above formula, we can see that the internal state of the system is replaced by z[k], and the state space expression of the system is And y(t) = Cx(t), constructing the attack signal needs to meet the following conditions:

[0234]

[0235] When the state after the attack converges locally, the zero-power attack is invalid, indicating a failure. Therefore, to construct a more effective attack, the condition ||G0||>1 must be satisfied. In this case, the state continues to diverge, which can be considered a successful and effective attack.

[0236] For the discretized wind power generation x[k+1]=Gx[k]+Hu[k];y[k]=Cx[k], design zero dynamic attack z[k+1]=G d z[k], the impact of zero dynamic attack on the internal state of the wind power system depends on G d , assuming G d is an n×n matrix with eigenvalues ​​λ1,λ2,…,λ n , the initial state z[0] is replaced by G d The eigenvector of is represented as follows:

[0237] z[0]=c1v1+c2v2+…+c n v n

[0238] Among them, v i is the eigenvalue λ i The corresponding eigenvector, c i It is v i Coefficient, according to the state equation of the open-loop system, that is, the wind power generation system without the introduction of the gain scheduling PI controller, is:

[0239]

[0240] Because G d Decompose into G d =VΛV -1 , where Λ is the diagonal matrix of eigenvalues ​​and V is the matrix of eigenvectors, we get:

[0241] z[k]=VΛ k V -1 z[0]

[0242] Substituting z[0] into the above equation, we can obtain and satisfy the following conditions:

[0243]

[0244] Analyzing the eigenvalue λ i The modulus |λ i |The impact on the state vector z[k] is as follows:

[0245] If there exists an eigenvalue λ i So that |λ i |>1, then It grows exponentially with k, so the state vector z[k] will also become unbounded, causing z[k] to be unstable;

[0246] If all eigenvalues ​​satisfy |λ i |≤1, then It does not grow unbounded with k, but remains bounded or converges to zero as k increases, thus ensuring that z[k] remains bounded or converges to a stable value;

[0247] The impact of zero dynamic attack on the internal state of wind power generation system depends on G d , and meet the following conditions:

[0248]

[0249] in, For existence, Indicates any one.

[0250] Step 4: Implement defense against the zero dynamic attack signal designed in step 3, and dynamically adjust and optimize the component adjustment coefficient k l And the adjustment coefficient k of the component to the capacitance c The parameter values ​​are selected to adjust the inductor L and capacitor C, converting the unstable zero point into a stable zero point, weakening the destructiveness of the zero dynamic attack and achieving defense.

[0251] The defense method for the zero dynamic attack signal designed in step 3 is to offset the unstable zero point of the wind power generation system after sampling. There are two types of unstable zero points of the wind power generation system after sampling. One is the unstable zero point of the original system after sampling mapping, and the other is additionally introduced due to sampling. By adjusting k l and k c , converting unstable zeros into stable zeros.

[0252] Considering the designed gain-scheduled PI controller, the expression of the zero point is:

[0253]

[0254] Due to the deployment of electronically adjustable passive components, the above expression is transformed into:

[0255]

[0256] It can be seen that the adjustment of the zero point mainly depends on k l and k c The choice of k l and kc The choice of is constrained to ensure that the primary functionality of the system is not compromised. Therefore, not all systems can implement defenses against zero-dynamic attacks.

[0257] Under the condition that the sampling period of the continuous-time system is T, the discrete system obtained is expressed by the transfer function

[0258]

[0259] Among them, D d (z) is the characteristic polynomial of the discrete system. For a wind power generation system with a relative degree greater than 2, the degree of the characteristic polynomial of the discrete system increases after sampling, thereby introducing additional zeros. These additional zeros are sampling-induced zeros.

[0260] The characteristic polynomial Dd(z) of the sampled discrete system is derived from the characteristic polynomial D(s) of the continuous system. The degree of the discretization of the characteristic polynomial increases, which is expressed as

[0261] D d (z) = b0 + b1z -1 +b2z -2 +…+b n z -n

[0262] Among them, the coefficient b i It is a function of the continuous system coefficients a0, a1, a2. According to the parameter definition in the system, a0, a1, a2 are directly related to the inductance L and capacitance C, so b i It is also a function of L and C, recorded as:

[0263] b i =f(a0,a1,a2,…)=g(L,C)

[0264] The change of the inductor L and capacitor C values ​​will affect the position of the root of the characteristic polynomial Dd(z) of the discrete system. The characteristic polynomial D d The (z) roots correspond to the zeros in the discrete system.

[0265] After sampling the system, the composition of zeros becomes more complex. In order to deal with the unstable zeros introduced by sampling, it is necessary to select appropriate values ​​within the parameter range. In order to successfully implement the defense strategy, a particle algorithm based on k is designed to select the appropriate value. l and k c value.

[0266]

[0267] The main function of the algorithm is to calculate the parameter k l and k cOptimize and find the combination that minimizes the maximum zero of the system. When complete defense cannot be achieved, the destructive impact of zero-power attacks can be reduced.

[0268] To verify the present invention, we injected an attack and deployed defenses to verify performance, as follows:

[0269] Table 1 System parameters

[0270] parameter Numerical <![CDATA[Rotor inertia (J d )]]> <![CDATA[2.13×10 -5 kg·m 2 ]]> <![CDATA[Generator inertia (J m )]]> <![CDATA[4×10 -6 kg·m 2 ]]> <![CDATA[Gearbox inertia (J h )]]> <![CDATA[6.5×10 -7 kg·m 2 ]]> <![CDATA[Resistor R1]]> 21Ω <![CDATA[Resistor R2]]> 1.2Ω Inductor L 0.7H Capacitor C 5F <![CDATA[Gearbox ratio n g > 2 <![CDATA[Tip speed ratio λ o > 1.38

[0271] The simulation parameters are set as shown in Table 1, which reasonably demonstrates the destruction process of zero-dynamic attack and the effectiveness of the designed defense strategy.

[0272]

[0273] Assume that the constant wind speed V = 15m / s and the sampling period T = 1.5s. According to the information in Table 1, the complete system model can be derived, where λ0 represents the tip speed ratio, n g Indicates the gearbox speed ratio, which determines the system input w g is (35).

[0274] Figure 4-5 As shown in Figure 2, when the parameters are α=1,β=-0.041,γ=0.87,δ=-0.022, the original system is no longer a minimum phase system. Similarly, the attack signal is injected after the system is running. After 100 seconds of injection of the attack signal, Figure 4 The system's internal state oscillates while the output remains almost unchanged, indicating that the attack is successful. Figure 5 Further inspection reveals that the system’s state variables also exhibit divergent oscillations after the attack, effectively demonstrating the destructive nature of the attack. Figure 5 As shown, the defense method is deployed in the designed system. Figure 6 It shows that after the attack signal is injected, the internal state η oscillates but gradually stabilizes. The results show that the proposed defense method effectively mitigates the destructive effects of zero-dynamic attacks. Figure 7 The changes in the system state variables are shown, indicating that although small oscillations occur after the attack, they eventually stabilize, further supporting the effectiveness of the proposed defense method.

[0275] The foregoing is merely an embodiment of the present invention and is not intended to limit the present invention. It will be apparent to those skilled in the art that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention are intended to be included within the scope of the claims of the present invention.

Claims

1. A defense method for zero-dynamic attacks on wind power generation systems, characterized by: The defense method specifically includes the following steps: Step 1: Construct a mathematical model of the wind power generation system and obtain the transfer function of the power generation system; Step 2: Introduce a gain-scheduled PI controller and electronically adjustable passive components into the wind power generation system obtained in step 1, wherein the gain-scheduled PI controller includes a proportional gain k p and integral gain k i The electronically adjustable passive components are specifically a controllable inductance module and a controllable capacitance module, and the function of the controllable inductance module and the controllable capacitance module is to adjust the inductance L and the capacitance C of the wind power generation system; Step 3: design a zero-dynamic attack signal, inject the designed zero-dynamic attack signal into the wind power generation system, and analyze the destructive factors of the zero-dynamic attack; Step 4: Implement defense against the zero dynamic attack signal designed in step 3, dynamically adjust and optimize the component's adjustment coefficient k for inductance l And the adjustment coefficient k of the component to the capacitance c The parameter values ​​of are used to adjust the inductor L and the capacitor C, converting the unstable zero point into a stable zero point, weakening the destructiveness of the zero dynamic attack, and achieving defense, where: In step 2, the electronically adjustable passive components are introduced into the wind power generation system obtained in step 1 as follows: The controllable inductor module realizes real-time dynamic adjustment of the total inductance by connecting the adjustable inductor array in parallel with the original inductance L of the wind power generation system. The adjustment formula is: L ′ =Luke l Among them, L ′ is the equivalent inductance after adjustment of the controllable inductance module, L is the original inductance of the wind power generation system, k l k is the adjustment coefficient of the component to the inductance, by adjusting l Control the inductance to ensure that its regulation meets the actual needs of the power system; The controllable capacitor module is connected in parallel with the original capacitance C of the wind power generation system to form an adjustable capacitor array to achieve real-time dynamic adjustment of the total capacitance. The adjustment formula is: C ′ =Ck c Among them, C ′ is the equivalent capacitance after adjustment by the controllable capacitor module, C is the original capacitance of the wind power generation system, k c is the adjustment coefficient of the component to the capacitance; In order to prevent the parameter adjustment from affecting the normal operation of the original wind power generation system, the adjustment coefficient k of the component to the inductance is set to l And the adjustment coefficient k of the component to the capacitance c Configured as a continuously adjustable value within the specified parameter range, define k l and k c Adjustment range: k l,min ≤k l ≤k l,max k c,min ≤k c ≤k c,max Equivalent inductance L after adjustment by the controllable inductance module ′ And the equivalent capacitance C after adjustment by the controllable capacitance module ′ It is a linear function of the inductance L and the capacitance C. As long as the boundary values ​​meet the required conditions, all intermediate values ​​meet the same criteria. In step 4, the zero dynamic attack signal designed in step 3 is defended, including: offsetting the unstable zero point of the wind power generation system after sampling. There are two types of unstable zero points of the wind power generation system after sampling. One is the unstable zero point of the original system mapped by sampling, and the other is additionally introduced due to sampling. The adjustment coefficient k of the inductance is adjusted by adjusting the element. l And the adjustment coefficient k of the component to the capacitance c , converting unstable zero points into stable zero points, specifically including: Step 4.

1. Consider the designed gain-scheduled PI controller. The expression of the zero point is: Due to the deployment of electronically adjustable passive components, the expression of the above zero point is transformed into: α, β, γ, and δ are the internal adjustment parameters of the gain-scheduled PI controller, which are designed and adjusted according to the needs of the wind power generation system. It can be seen that the adjustment of the zero point mainly depends on k l and k c choice; Step 4.2: Under the condition that the sampling period of the continuous-time system is T, the obtained discrete system is expressed by the transfer function Among them, D d (z) is the characteristic polynomial of the discrete system; The degree of discretization of the characteristic polynomial increases, which is expressed as D d (z)=b0+b1z -1 +b2z -2 +…+b n from -n Among them, the coefficient b i It is a function of the continuous system coefficients a0, a1, a2. According to the parameter definition in the wind power generation system, a0, a1, a2 are directly related to the inductance L and the capacitance C, so b i Denoted as: b i =f(a0,a1,a2,…)=g(L,C) The change of the inductor L and capacitor C values ​​will affect the position of the root of the characteristic polynomial Dd(z) of the discrete system. The characteristic polynomial D d The (z) roots correspond to the zeros in the discrete system.

2. The method for defending against zero-dynamic attacks on a wind power generation system according to claim 1, characterized in that: The step 1 specifically includes the following steps: Step 1.1: Assume that the electromotive force of the wind power generation system is E, which can be expressed as: E=K g oh g Among them, K g is the electromagnetic torque coefficient, ω g is the angular velocity of the generator; The voltage balance equation in the entire wind power generation system is: Where U is the motor terminal voltage, is the voltage drop across the resistor, U L is the voltage drop across the inductor, i L is the current through the inductor, I is the motor armature current, R1 and R2 are the resistances in the system, and the torque balance equation is: Among them, K t is the potential constant, the total moment of inertia J=J d +J m +J h , J d is the moment of inertia of the wind turbine, J m is the moment of inertia of the permanent magnet direct drive synchronous generator, J h In order to obtain the equivalent moment of inertia of the gearbox, the characteristics of capacitance and inductance are combined to obtain the voltage balance equation and torque balance equation. in, Perform Laplace transform, U(s)=K g ((a0s 3 +a1s 2 +a2s+1)ω g (s)); Where L and C are the inductance and capacitance of the wind power system before the controllable elements are deployed; Step 1.2, the transfer function G1(s) of the wind power generation system: make Then we have:

3. The method for defending against zero-dynamic attacks on a wind power generation system according to claim 2, characterized in that: In step 2, a gain-scheduled PI controller is introduced into the wind power generation system obtained in step 1, specifically: Step 2.1: Set the proportional gain k p and integral gain k i The values ​​of are dynamically associated with the real-time inductance L(t) and real-time capacitance C(t), respectively, so that the gain-scheduled PI controller can adapt to the changes in wind power system parameters. The association relationship is defined as follows: K p (t)=αL(t)+βC(t) K i (t)=γL(t)+δC(t) Among them, the sampling time t = kT, T is the sampling period, which flexibly responds to the real-time changes of the real-time inductance L(t) and real-time capacitance C(t). The gain-scheduled PI controller automatically adjusts K according to the system state changes. p (t) and K i (t); The internal adjustment parameters α, β, γ, and δ of the gain-scheduled PI controller are set to fixed values. The transfer function of the wind power generation system becomes: Where G1 is the transfer function of the wind power system, and G2 is the transfer function of the gain-scheduled PI controller; Step 2.2: Convert the wind power system transfer function from the frequency domain to the state space table in the time domain and select the state variable Among them, ω g is the angular velocity of the generator, and the high-order derivatives represent the corresponding rate of change. The state variables are used to describe the dynamic behavior of the wind power generation system. The state space expression is: y(t)=Cx(t) The wind power generation system matrix is ​​expressed as: B=[Kk i Kk p 0 0] T C＝[1 0 0 0]； Step 2.3: Use a gain-scheduled PI controller to remotely operate the continuous-time wind power generation system. The input and output of the gain-scheduled PI controller are processed at discrete time points. The discretized state equation is expressed as: For ease of expression, the above formula is simplified to: x(k+1)=Gx(k)+Hu(tk) y(k)=Cx(k) Wherein, sampling time t=kT, where T is the sampling period; In the discretization process, the zero-order hold method is used for discretization, and the discretization matrix of the wind power generation system is:

4. The method for defending against zero-dynamic attacks on a wind power generation system according to claim 3, characterized in that: In step 3, the zero dynamic attack signal design is specifically as follows: Step 3.1: Based on the Euclidean algorithm, the transfer function of the wind power system in step 2.2 is rewritten to represent the transfer function of the feedback loop: Where Q(s) is the quotient and R(s) is the remainder; Step 3.2: Convert the wind power system into the Byrnes-Isidori form: η(t+1)=G0η(t)+H0C c ξ(t) ξ(t+1)=(G c +H c λ T )ξ(t)+H c b m (u(t)-C0η(t)) y(t)=C c ξ(t) in, λ T ξ(t)=y(t)+b m C0η(t)-b m u(t) C c ＝[1 0 0] Among them, b m is the maximum power numerator coefficient of the transfer function; η(t) and ξ(t) represent the internal and external states of the wind power system. G0, H0, and C0 are the minimum realizations of the transfer function of the feedback path R(s) / E(s). Applying the Euclidean algorithm, the transfer function of the equation in step 3.2 is derived and expressed as η[k+1]=G after sampling. d η[k]+H d C c ξ[k] ξ[k+1]=(G c +H c l T )ξ[k]+H c b m (u[k]-C d the[k]) y[k]=C c ξ[k]; Step 3.3: The goal of the zero-dynamic attack signal is to cut off the connection between the internal and external states of the wind power generation system. Assume that the attacker injects an attack signal a[k] into the wind power generation system that introduces a gain-scheduled PI controller, i.e., the closed-loop system. The attack signal is designed as: z[k+1]=G d z[k] a[k]=C d z[k] After the attack is launched, the sampling is converted into z[k+1]=G d z[k] ξ[k+1]=(G c +H c l T )ξ[k]+H c b m u[k] y[k]=C c ξ[k] Among them, z[k] is the state vector.

5. The method for defending against zero-dynamic attacks on a wind power generation system according to claim 4, characterized in that: In step 3, the destructive factors of zero-dynamic attacks are analyzed, specifically: For the discretized wind power generation system x[k+1]=Gx[k]+Hu[k];y[k]=Cx[k], design zero dynamic attack z[k+1]=G d z[k], the impact of zero dynamic attack on the internal state of the wind power system depends on G d , assuming G d is an n×n matrix with eigenvalues ​​λ1,λ2,...,λ n , the initial state z[0] is replaced by G d The eigenvector of is represented as follows: z[0]=c1v1+c2v2+…+c n v n Among them, v i is the eigenvalue λ i The corresponding eigenvector, c i It is v i Coefficient, according to the state equation of the open-loop system, that is, the wind power generation system without the introduction of the gain scheduling PI controller, is: Because G d Decompose into G d =VΛV -1 , where Λ is the diagonal matrix of eigenvalues ​​and V is the matrix of eigenvectors, we get: z[k]=VΛ k IN -1 from[0] Substituting z[0] into the above equation, we can obtain and satisfy the following conditions: Analyzing the eigenvalue λ i The modulus |λ i |The impact on the state vector z[k] is as follows: If there exists an eigenvalue λ i So that |λ i |>1, then It grows exponentially with k, so the state vector z[k] will also become unbounded, resulting in instability of the state vector z[k]; If all eigenvalues ​​satisfy |λ i |≤1, then It does not grow unbounded with k, but remains bounded or converges to zero as k increases, thus ensuring that z[k] remains bounded or converges to a stable value; The impact of zero dynamic attack on the internal state of wind power generation system depends on G d , and meet the following conditions: in, For existence, Indicates any one.

6. The method for defending against zero-dynamic attacks on a wind power generation system according to claim 1, characterized in that: In step 4, the parameter k l and k c The optimization is achieved by particle swarm optimization (PSO) algorithm, which includes the following steps: Step 4.1.

1. Initialize the positions and velocities of particles in the wind power system, where each particle represents a candidate solution. Step 4.1.

2. In each iteration, the transfer characteristics of the wind power generation system are evaluated by calculating the inductance and capacitance corresponding to each particle to determine whether the wind power generation system meets the stability conditions. Step 4.1.3: Update the particle's historical best position p according to its performance best and the global optimal position g best ; Step 4.1.4: Adjust the speed and position of the particles, iterate and find the optimal solution, and finally output the optimal inductance k. l and the optimal capacitance k c parameter.

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