Novel high-dimensional system simplification method

Abstract

The invention discloses a novel high-dimensional system simplification method, which utilizes the advantage of high numerical calculation speed of a simulation platform, combines a singular perturbation theory with a numerical analysis method, and replaces the complexity of full-order system nonlinear modeling with the complexity of polynomial parameters, thereby realizing order reduction of a DDWF system. The method comprises the following steps: firstly, analyzing an operation range of a full-order model of a DDWF incorporated weak alternating current system, secondly, randomly screening linearization points, and respectively establishing small signal models under different working conditions at the linearization points; then reducing the order of the small signal model according to a singular perturbation theory to obtain an order-reduced small signal model set under different working conditions; and fitting a non-linear relationship between the operation parameters and the reduced-order model through numerical analysis. The result shows that the order reduction method saves the calculation cost to a great extent while keeping the dynamic characteristics of the system, and is suitable for the modeling requirements of the complex system of the electromagnetic simulation platform.

Description

Technical Field

[0001] This invention belongs to the field of power technology, relates to new energy technology, and particularly to a novel method for simplifying high-dimensional systems. Background Technology

[0002] In recent years, due to energy shortages and worsening environmental problems, the utilization of new energy sources has become a trend. The nonlinear dynamic equations of new energy units are complex, and there is a lack of effective methods for order reduction. Relying on electromagnetic transient simulation analysis to analyze stability problems is insufficient to systematically reveal the stability mechanism of power grid operation. There is an urgent need for effective methods for reducing the order of nonlinear differential equations in order to analyze the stability mechanism of the entire system.

[0003] To address the aforementioned problems, numerous scholars have conducted research and derived corresponding solutions in the existing technology. Kokotovic et al. proposed the concept of slow manifolds for nonlinear singular perturbation systems as early as the 1980s and studied the time-scale decomposition and model order reduction of nonlinear systems. However, solving nonlinear partial differential equations to determine the integral manifold is overly complex. Furthermore, this method does not assess the stability of the system before and after order reduction, failing to meet the requirements for complex system modeling and stability analysis. Currently, singular perturbation methods are commonly used for order reduction calculations in power systems. That is, for systems with multi-time-scale characteristics, fast and slow variables are first decoupled. By treating fast variables as boundary layer systems, fast dynamic variables are ignored while large-scale dynamic processes are retained, thus reducing the requirements for stability analysis. When the system operates at the equilibrium point, this reduced-order model can well reproduce the transient response characteristics of the full-order system. However, when the system operating parameters vary over a wide range, the reduced-order model based on linearization at the equilibrium point becomes unapplicable.

[0004] Due to the inherent complexity of nonlinear systems, the aforementioned algorithms generally suffer from drawbacks such as relatively difficult implementation and limited application scope. When the system order is high, the "curse of dimensionality" easily arises. To resolve the contradiction between high-dimensional computation and computational cost, we take a DDWF system integrated into a weak power grid as an example and introduce a least-squares polynomial fitting method to perform large-scale order reduction operations on the nonlinear system. Least-squares polynomial fitting is a mathematical optimization technique used to find a polynomial function that minimizes the sum of squared errors between the function and given data points. By using least-squares polynomial fitting to replace the complexity of the nonlinear DDWF system with the complexity of polynomial coefficients, we not only effectively restore the order reduction effect of single-point singular perturbations but also significantly save computation time, making it suitable for the requirements of complex system modeling in electromagnetic simulation platforms. Summary of the Invention

[0005] The technical problem to be solved by this invention is to propose a novel method for simplifying high-dimensional systems, in order to address the issue that the nonlinear dynamic equations of new energy units are complex and that existing technologies lack effective methods for reducing their order.

[0006] This invention is achieved using the following technical solution:

[0007] A novel method for simplifying high-dimensional systems is proposed. First, taking a DDWF system integrated into a weak AC power grid as an example, the operating range of the full-order model of the system is analyzed. Then, linearization points are randomly selected. Next, small-signal models under different operating conditions are established at the linearization points. The order of the small-signal models is reduced according to singular perturbation theory to obtain a set of reduced-order small-signal models under different operating conditions. Finally, the nonlinear relationship between the operating parameters and the reduced-order models is fitted by polynomial regression to establish a mathematical model for large-scale nonlinear order reduction of the DDWF system.

[0008] As a preferred embodiment of the present invention, the embodiment includes the following steps:

[0009] Step 1: First, randomly select multiple linearization points on the nonlinear perturbation trajectory of DMSG, and then establish a full-order linearization model at each linearization point;

[0010] Step 2: Based on the singular perturbation theory, the full-order linearization model at each linearization point is divided into fast and slow variables. By ignoring the fast state variable, the fast and slow variables are decoupled, and a set of single-point order reduction models under different operating conditions is established.

[0011] Step 3: The operating parameters of each linearization point are mapped one-to-one with the parameters of its singular perturbation reduced-order state equation. Numerical methods are used to fit the nonlinear relationship between the operating parameters and the parameters of the reduced-order model, thereby establishing a response-driven DDWF large-scale reduced-order numerical model.

[0012] The beneficial effects of this invention are as follows: First, the operating range of the full-order model of the DDWF system integrated into a weak AC system is analyzed; second, linearization points are randomly selected, and small-signal models under different operating conditions are established at these points; then, the order of the small-signal models is reduced according to singular perturbation theory to obtain a set of reduced-order small-signal models under different operating conditions; finally, the nonlinear relationship between the operating parameters and the reduced-order models is fitted through numerical analysis. This method utilizes the advantage of the high numerical computation speed of the simulation platform, combining singular perturbation theory with numerical analysis methods. It replaces the complexity of nonlinear modeling of the full-order system with the complexity of polynomial parameters, solving the problems of computational difficulty, loss of dynamic characteristics, and limitation to linear systems inherent in the singular perturbation method. Simulation analysis results of the DDWF system before and after a large-scale order reduction verify the effectiveness and accuracy of the method. The proposed order reduction method greatly saves computational costs while preserving the dynamic characteristics of the system, making it suitable for the requirements of complex system modeling on electromagnetic simulation platforms. Attached Figure Description

[0013] Figure 1 The diagram shows the topology of a DDWF-connected weak AC power grid system, used to verify the effectiveness of the high-dimensional system order reduction method described in this invention.

[0014] Figure 2 This invention provides a novel simplified method for high-dimensional systems.

[0015] Figure 3 A flowchart of the algorithm for achieving large-scale nonlinear order reduction when DDWF is integrated into a weak AC power grid system.

[0016] Figure 4 After establishing a set of multi-point linearized models, the effect diagram of fitting the nonlinear relationship between the state equation parameters and the operating parameter wind speed through univariate first-order, univariate second-order, and univariate third-order polynomial regression analysis is shown.

[0017] Figure 5 The graph shows the effect of fitting the nonlinear relationship between the state equation parameters and the running parameters v and Ugd using bivariate third-order polynomial regression analysis after establishing a multi-point linearized model set. Detailed Implementation

[0018] This invention provides a novel method for simplifying high-dimensional systems. To make the objectives, technical solutions, and effects of this invention clearer, the specific implementation schemes of this invention are described in detail below with reference to the accompanying drawings and examples. The specific examples described in this invention are only for explaining the invention and are not intended to limit the invention. The following description, with reference to the accompanying drawings, further details the specific implementation schemes of this invention. Figure 3 The embodiments are described in detail below.

[0019] Step 1: First, randomly select multiple linearization points on the nonlinear perturbation trajectory of DMSG, and then establish a full-order linearization model at each linearization point;

[0020] The direct-drive wind turbine integrated into the weak current grid system consists of a DDWF system and a weak current grid system, and its topology is as follows: Figure 1 As shown. The weak AC power grid system consists of transmission line resistor R1, inductor L1, grid connection point capacitor C1, 3kV / 35kV transformers, and 35kV / 330kV transformers. The DDWF system consists of wind turbines, permanent magnet synchronous generators (PMSG), machine-side converters (MSC), grid-side converters (GSC), phase-locked loops (PLL), and DC capacitors C1 and C2. DC It consists of a filter inductor L. Where C... DC For DC capacitors, L g For the wind turbine grid-side filter inductor, uDC The voltage of the DC capacitor is u. s i s These are the stator winding voltage and current, u t i g These represent the voltage and current at the GSC output, u g i1 is the voltage at the fan outlet, i1 is the current flowing through the AC transmission line, and u1 is the grid-connected voltage.

[0021] The GSC employs a DC voltage outer loop and a grid-connected dq-axis current inner loop control strategy to achieve the functions of controlling DC voltage stability and grid-connected reactive power; the MSC employs a speed outer loop and a stator dq-axis current inner loop control strategy to achieve the functions of maximum power tracking of the wind turbine and minimizing generator losses.

[0022] The established DDWF is incorporated into the dynamic mathematical model of the weak AC system. At the steady-state operating point, it is linearized to establish a small-signal model as shown in equation (1).

[0023]

[0024] A: 18×18 state matrix;

[0025] B: 18×2 input matrix;

[0026] C: 10×18 level output matrix;

[0027] ΔX: Linearized state variables, detailed state variables are shown in equation (2);

[0028] ΔY: Linearized output variable, detailed output variables are shown in equation (3);

[0029] ΔU: Linearized input variable, detailed input variables are shown in equation (4).

[0030]

[0031] ΔY=[Δu DC ,Δω,Δi sd , Δi sq ,Δu sd ,Δu sq , Δi gd , Δi gq ,Δu gd ,Δu gq , Δθ pll ] T (3)

[0032] ΔU=[Δu DCref ,Δω sref ] T (4)

[0033] The small-signal model of the full-order DDWF integrated into the weak AC power grid system has a total of 18 orders, of which the DDWF part has 14 orders and the AC system part has 4 orders.

[0034] Step 2: Based on the singular perturbation theory, divide the full-order linearized model at each linearized point into fast and slow variables. By ignoring the fast state variable, the fast and slow variables are decoupled, and a set of single-point reduced-order models under different operating conditions is established.

[0035] The singular perturbation method, also known as the small parameter method, is an approximate analytical method for solving differential equations. For systems with multi-timescale characteristics, the singular perturbation method is used to decouple fast and slow variables. By treating fast variables as boundary layers, it ignores fast dynamic variables in the DDWF system while retaining large-timescale dynamic processes, thereby reducing the order of the DDWF model used for stability analysis. The derivation of the singular perturbation method is as follows:

[0036] Nonlinear dual-time-scale systems are generally described using the following model:

[0037]

[0038] y(t)=C1x1(t)+C2x2(t) (7)

[0039] In the formula, State variables with slow time scales; For fast time-scaled state variables; u(t)∈R m is the input variable for the state-space equation; μ is the singular perturbation parameter, and 0 < μ << 1; A ij B i (i,j=1,2) are the system's state matrix and input matrix, respectively. Due to the nonlinear characteristics of the system, A ij B i (i,j=1,2) has a nonlinear functional relationship with the system state variables.

[0040] The basic premise of the singular perturbation method is that when a fast variable undergoes its transient process, we assume that the variable remains in a constant state. When a slow variable exhibits a significant change, we consider that the transient process of the fast variable has ended and it has reached its corresponding quasi-steady state. If A22 is a non-singular matrix and the variable x2(t) is asymptotically stable, when μ→0, equations (5)-(7) degenerate into the following form:

[0041]

[0042] 0 = A 21 x 1s (t)+A 22x 2s (t)+B2u s (t) (9)

[0043] y s (t)=C1x 1s (t)+C2x 1s (t) (10)

[0044] From equation (9), the singular perturbation form of the fast variable can be obtained as follows:

[0045]

[0046] Substituting equation (11) into equation (8), we obtain the following singular perturbation reduced-order equation for the original linear system:

[0047]

[0048] x 1s (0)=x 10 (13)

[0049] In the formula, Where x 2s To find the quasi-steady-state value of the fast variable x2, we can obtain an approximate solution x1 by solving the system of equations (12). 1s .

[0050] Step 3: Map the operating parameters of each linearization point to the parameters of its singular perturbation reduced-order state equation, and use numerical methods to fit the nonlinear relationship between the operating parameters and the reduced-order model parameters, thereby establishing a response-driven DDWF large-scale reduced-order numerical model. The specific steps are as follows:

[0051] For a given set of data points (x) i ,y i ), where i = 1, 2, ..., N, we want to find a polynomial function that minimizes the sum of the squares of the perpendicular distances (i.e., errors) between all data points and the polynomial. Let its fitting function be as shown in equation (14):

[0052]

[0053] In the formula, m is the order of the fitted function, and N is the number of sampling points. According to the principle of minimizing error using the least squares method, the following formula should be minimized:

[0054]

[0055] By the extremum condition of a multivariable function, let have to

[0056]

[0057] Introduce inner products for arbitrary functions u(x) and v(x).

[0058]

[0059] Then equation (18) can be expressed as

[0060]

[0061] Represented in matrix form as

[0062]

[0063] Let {1, x, ..., x} m Using} as the basis functions, a polynomial fit is performed, and the above normal equation system is transformed into:

[0064]

[0065] Calculate the values ​​of each element in the coefficient matrix and the right-hand side of the system of equations one by one, thus solving for a0, a1…a n Finally, the fitting polynomial is obtained:

[0066]

[0067] The effectiveness of the proposed DDWF large-scale order reduction method is verified from the following three aspects.

[0068] 1) The DDWF reduced-order system should contain all the SSO modes that need to be retained, and the eigenvalues ​​and frequencies of the same mode should be as similar as possible;

[0069] 2) Before and after the replacement, the singular perturbation reduced state space equation is consistent with the least squares parameter fitting results.

[0070] 3) The calculation speed should be effectively improved before and after order reduction.

[0071] The verification process is as follows:

[0072] Taking a single DDWF connected to a weak power grid system as an example, the selected test conditions are shown in Table 2.

[0073] Wind speed (m / s) Grid connection point voltage Ugd (kV) Operating Condition 1 7.5 2.175 Operating Condition 2 11.25 2.2125 Operating Condition 3 3.75 2.1375

[0074] The results of retaining the SSO mode for the above three test conditions are as follows:

[0075]

[0076] As shown in the table, when the operating parameters change continuously within a certain range, the large-scale reduced-order system model based on polynomial fitting can cover all SSO modes of the full-order system model. The calculated results of the characteristic roots and frequencies of the same mode are basically consistent with those of the full-order model, and the characteristic root variation patterns of the full-order model and the reduced-order model are the same.

[0077]

[0078] As shown in the table, compared to the single-point singular perturbation order reduction method, the polynomial fitting algorithm can significantly improve the computation speed. For an 18th-order single DDWF connected to a weak grid system, when wind speed and grid connection voltage change continuously, the bivariate fourth-order polynomial fitting yields the best results, but has the longest computation time; the bivariate second-order polynomial fitting yields the worst results, but has the shortest computation time. Considering both the requirements for computational accuracy and time, since the bivariate third-order fitting already meets the accuracy requirements and has a shorter computation time than the fourth-order polynomial, we ultimately choose the bivariate third-order polynomial fitting to obtain the singular perturbation order reduction model for a large-scale DDWF connected to a weak grid system. Therefore, when the system order is small, using multivariate high-order fitting can ensure computational accuracy; when the system order is too high, computational efficiency can be improved by sacrificing some accuracy, i.e., appropriately reducing the polynomial order to save computational costs.

[0079] In summary, the novel high-dimensional system simplification method described in this invention leverages the high numerical computation speed of simulation platforms, combining singular perturbation theory with numerical analysis methods. It replaces the complexity of nonlinear modeling of full-order systems with the complexity of polynomial parameters, thereby achieving order reduction for DDWF systems. Verification results show that the proposed order reduction method significantly reduces computational costs while preserving the system's dynamic characteristics, making it suitable for the requirements of complex system modeling on electromagnetic simulation platforms.

[0080] Finally, it should be noted that the above examples of the present invention are merely illustrative and not intended to limit the implementation of the invention. Although the applicant has described the present invention in detail with reference to preferred embodiments, those skilled in the art can make other variations and modifications based on the above description. It is impossible to exhaustively list all possible implementations here. All obvious variations or modifications derived from the technical solutions of the present invention are still within the scope of protection of the present invention.

Claims

1. A novel method for simplifying high-dimensional systems, characterized in that, Includes the following steps: Step 1: First, randomly select multiple linearization points on the nonlinear perturbation trajectory of the high-dimensional system, and then establish a full-order linearization model at each linearization point. Step 2: Based on the singular perturbation theory, the full-order linearization model at each linearization point is divided into fast and slow variables. By ignoring the fast state variable, the fast and slow variables are decoupled, and a set of single-point order reduction models under different operating conditions is established. Step 3: The operating parameters of each linearized point are mapped one-to-one with the parameters of its singular perturbation reduced-order state equation. Numerical methods are used to fit the nonlinear relationship between the operating parameters and the parameters of the reduced-order model, thereby establishing a simplified model of the response-driven high-dimensional nonlinear system.

2. The method as described in claim 1, characterized in that, Linearization of high-dimensional nonlinear systems at sample points includes: Based on the nonlinear dynamic mathematical model of the high-dimensional system, it is linearized at the sample points to establish a linearized model at the equilibrium point as shown in equation (1); the linearized models of multiple sample points constitute a set of multi-point linearized models; where m is the number of input variables, n is the order of the state equation, and p is the number of output variables; A: An n×n state matrix; B is an n×m order input matrix; C is a p×n output matrix.

3. The method as described in claim 1, characterized in that, A set of single-point order reduction models under different operating conditions is established using a singular perturbation algorithm, including: For systems with multi-timescale characteristics, the singular perturbation method is used to decouple fast and slow variables. By treating fast variables as boundary layer systems, fast dynamic variables in high-dimensional systems are ignored while large-timescale dynamic processes are retained, thereby reducing the order of high-dimensional systems used for stability analysis. The derivation of the singular perturbation method is as follows: Nonlinear dual-time-scale systems are generally described using the following model: y(t)=C1x1(t)+C2x2(t) (4) In the formula, x1(t)∈R n1 For a slow-time scale state variable; for a fast-time scale state variable; u(t)∈R m The input variables for the state-space equations are μ; for odd x2(t)∈R n2 Different perturbation parameters, and 0 < μ < < 1; A ij B i (i,j=1,2) are the system's state matrix and input matrix, respectively; due to the nonlinear characteristics of the system, A ij B i (i,j=1,2) has a nonlinear functional relationship with the system state variables; If A 22 Since the matrix is ​​non-singular and the variable x2(t) is asymptotically stable, when μ→0, equations (2)-(4) degenerate into the following form: 0=A 21 x 1s (t)+A 22 x 2s (t)+B2u s (t) (6) y s (t)=C1x 1s (t)+C2x 1s (t) (7) From equations (5) to (7), the singular perturbation form of the fast variable can be obtained as follows: Substituting equation (8) into equations (5) and (6), we obtain the following singular perturbation reduced-order equations for the original linear system: x 1s (0)=x 10 (11) In the formula, Where x 2s To find the quasi-steady-state value of the fast variable x2, solving the system of equations () yields an approximate solution x1. 1s .

4. The method as described in claim 1, characterized in that, The nonlinear relationship between the operating parameters and the parameters of the reduced-order state equation is calculated using least-squares polynomial fitting, including: For a given set of data points (x) i ,y i ), where i = 1, 2, ..., N, we want to find a polynomial function that minimizes the sum of squares of the perpendicular distances (i.e., errors) between all data points and the polynomial; let its fitting function be as shown in equation (12): In the formula, m is the order of the fitting function, and N is the number of sampling points; According to the principle of minimizing error in the least squares method, the following expression should be minimized: By the extremum condition of a multivariable function, let have to Introduce inner products for arbitrary functions u(x) and v(x). Then equation (14) can be expressed as Represented in matrix form as Let {1, x, ..., x} m Using} as the basis functions, a polynomial fit is performed, and the above normal equation system is transformed into: Calculate the values ​​of each element in the coefficient matrix and the right-hand side of the system of equations one by one, thus solving for a0, a1…a n Finally, the fitting polynomial is obtained: At this point, the large-scale order reduction model of the high-dimensional nonlinear system has been completed.

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