Multi-direct-drive fan system order reduction method based on balanced truncation theory

Abstract

The invention discloses a multi-direct-drive fan system order reduction method based on a balance truncation theory, which can cover a multi-machine system simplification method of an aggregation model, and realizes ordered truncation of main dynamic energy channels of the system through collaborative analysis of controllability and observability. Firstly, linearization is carried out near a system balance point, and a standard state space model is constructed; secondly, directly performing balance transformation on the full-order small signal model without depending on a physical structure and manual partitioning, so that the controllability and the observability are uniformly distributed in a state space; then, the state variables are sorted according to Hankel singular values, the part with remarkable dynamic contribution in an input and output channel is automatically recognized, and information loss caused by improper structure division is avoided; and finally, by cutting off the redundant state, a reduced-order model of which the dynamic characteristic is highly close to that of an original system is constructed. A simulation result shows that when parameters are consistent, the method is naturally degraded into a traditional aggregation model; and when the parameters are inconsistent, the performance of the reduced-order model is obviously superior to that of an aggregation model, and stronger robustness is shown. The method provides an effective new way for simplifying a multi-direct-drive wind power plant system model in a different-wind-speed scene.

Description

Technical Field

[0001] This invention belongs to the field of power technology, and relates to new energy technology, and in particular to a method for reducing the order of a multi-direct-drive wind turbine system based on the balance truncation theory. 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 method for reducing the order of a multi-direct-drive wind turbine system based on the equilibrium truncation theory, so as to solve the problem that the nonlinear dynamic equations of new energy units are complex and that existing technologies lack effective methods for reducing the order.

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

[0007] A method for order reduction in multi-direct-drive wind turbine systems based on balanced truncation theory is proposed. First, taking a DDWF system connected to a weak AC power grid as an example, the operating range of the system's full-order model is analyzed. Then, linearization points are randomly selected. Next, small-signal models under different operating conditions are established at these linearization points. The order of these small-signal models is reduced using singular perturbation theory, resulting in a set of reduced-order small-signal models under different operating conditions. Finally, a large-scale nonlinear order reduction mathematical model for the DDWF system is established by fitting the nonlinear relationship between the operating parameters and the reduced-order models through polynomial regression.

[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 This diagram shows the structure of a multi-direct-drive wind turbine grid connection 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 technical approach for reducing the order of a multi-direct-drive fan system based on the balanced truncation theory.

[0015] Figure 3 This is a Hankel singularity distribution diagram for five operating conditions.

[0016] Figure 4 The graph shows the results of the participation factor calculation under five operating conditions.

[0017] Figure 5 This is a comparison chart of damping characteristics before and after the order of decrease for five different operating conditions. Detailed Implementation

[0018] This invention provides a method for order reduction in a multi-direct-drive wind turbine system based on the balanced truncation theory. 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, in conjunction with the accompanying drawings, further details the specific implementation schemes of this invention. Figure 2 The embodiments are described in detail below.

[0019] Step 1: Linearize the system near its equilibrium point to construct a standard state-space model;

[0020] 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;

[0021]

[0022] A: n×n state matrix; B: n×m input matrix; C: p×n output matrix.

[0023] Step 2: Directly perform a balance transformation on the full-order small-signal model to achieve a unified distribution of controllability and observability in the state space; sort the state variables according to the Hankel singular values ​​to automatically identify the parts with significant dynamic contributions in the input and output channels, avoiding information loss caused by improper structural partitioning;

[0024] For the system shown in equation (1), its controllability Gramian P matrix and observability Gramian Q matrix satisfy the following Lyapunov equation:

[0025]

[0026] P measures the controllability of a system, satisfying the condition that for any initial state x(0), there exists an input u(t) such that the system can transition to the origin in finite time; Q measures the ability to determine the initial state x(0) from the system output y(t); since the original system

[0027] Since P≠Q, the importance of state variables is difficult to compare directly; therefore, a similarity transformation is introduced to make the state variables of the new system symmetric in terms of controllability and observability, thereby achieving equilibrium.

[0028] The equilibrium transformation matrix T is constructed using the following steps:

[0029] 1) Perform singular value decomposition on P and Q:

[0030]

[0031] Among them, U C U O They are orthogonal matrices, ∑ C , ∑ O It is a diagonal matrix;

[0032] 2) Construct the intermediate matrix:

[0033]

[0034] Perform singular value decomposition on it:

[0035] H = U H Σ H V H T (5)

[0036] 3) Construct the equilibrium transformation matrix:

[0037]

[0038] Using this transformation matrix to perform state transformation on the original system, we can obtain the equilibrium system model:

[0039] x b =A b x b +B b u

[0040] y = C b x b +Du (7)

[0041] Where, x b =Tx, where A is the state variable in the equilibrium coordinates;b =TAT -1 B b =TB,C b =CT -1 At this point, the controllable and observable Gramian of the equilibrium system are all diagonal matrices:

[0042] P b =Q b =Σ=diag(σ1,σ2,…,σ n (8)

[0043] Where σ i These are the Hankel singular values, reflecting the combined energy contribution of the i-th state to the system's input-output transfer, and are arranged in descending order; that is:

[0044] σ1≥σ2≥…≥σ n >0 (9)

[0045] Where σ i This is the Hankel singular value.

[0046] Step 3: By truncating redundant states, a reduced-order model with dynamic characteristics highly similar to the original system is constructed; the details are as follows:

[0047] Hankel singular values ​​reflect the global contribution of equilibrium state variables to the dynamic characteristics of the system; therefore, after truncation, only the first r state variables are retained, i.e., σ1 to σ2 are selected. r Discard the state components corresponding to the smaller singular values; let:

[0048]

[0049] Where x1∈R r Denotes the reserved subset of states, x2∈R n-r The truncated part; discarding x2 and keeping x1, we obtain the reduced-order system:

[0050]

[0051] This reduced-order system not only retains the most significant energy channels, but also has a strict upper bound on the error:

[0052]

[0053] Where G(s) is the transfer function of the full-order system, G r (s) is the transfer function of the reduced-order system;

[0054] Regarding the choice of order, the cutoff point is determined by normalizing the Hankel singular values ​​and calculating their cumulative contribution; for example, if the total energy contribution target is 99%, then the following condition must be met: (13)

[0055]

[0056] This allows us to determine a reasonable value for r and construct the corresponding reduced-order system accordingly.

[0057] Using a three-DDWF system as an example, the effectiveness of the balanced truncation order reduction method is verified from the following three aspects:

[0058] 1) The time-domain transient response characteristics of the full-order system and the reduced-order system should match as closely as possible;

[0059] 2) A multi-DDWF order reduction system should contain all SSO modes;

[0060] When applied to a reduced-order system, the influence mechanism of DDWF control parameters on each SSO mode should remain as unchanged as possible.

[0061] To verify the applicability of the proposed reduced-order model to different wind speed conditions, time-domain and frequency-domain simulations were conducted under five wind speed conditions, focusing on the three aspects mentioned above. First, the Hankel singular value distribution of the system was calculated and compared; the calculation results are shown below. Figure 3 .according to Figure 3 It was found that in the five operating conditions, the first 18 singular values ​​accounted for the vast majority of the energy, while the singular values ​​after the 18th order decayed rapidly. Therefore, the 18th order can be regarded as the dividing point where the system's dynamics are dominated, and the states corresponding to subsequent orders have a relatively small impact on the overall system response.

[0062] To explore the applicability of the reduced-order system model to the damping characteristic analysis of multi-DDWF systems, the evolution of the oscillation frequency versus damping ratio curves under SSO mode was examined by adjusting key controller parameters. This allowed for the evaluation of the effectiveness and reliability of the reduced-order system model in damping characteristic analysis under different scenarios. First, the state variable participation factors of the full-order system under the SSO mode were calculated for five operating conditions, and the results are as follows: Figure 4 . Figure 4 This demonstrates the degree of participation of different state variables in the SSO mode under five test conditions. For each SSO mode, the state variable with the participation factor ranking in the top two and having a participation factor greater than 0.1 is called the dominant state variable of that SSO mode.

[0063] Figure 5 For different wind speed conditions, k i4 After increasing λ from 80 to 100 in steps of 1, 9,10 Curves showing the variation of mode oscillation frequency and damping ratio. Figure 5The results show that the SSO modes of the reduced-order system and the full-order system exhibit a high degree of similarity in the trends of oscillation frequency and damping ratio as a function of control parameters. As the control parameters gradually increase, the oscillation frequency of both models shows a monotonically increasing trend, while the absolute value of the damping ratio shows a monotonically decreasing characteristic. The results demonstrate that the reduced-order system model can accurately reproduce the damping characteristics of the full-order system model.

[0064] In summary, the method for reducing the order of a multi-direct-drive wind turbine system based on the balanced truncation theory described in this invention can more accurately preserve the key dynamic characteristics of the original system even when the wind turbine parameters are inconsistent, and effectively reflect the impact of changes in control parameters on damping characteristics.

[0065] 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 method for order reduction of a multi-direct-drive fan system based on balanced truncation theory, characterized in that, Includes the following steps: Step 1: Linearize the system near its equilibrium point to construct a standard state-space model; Step 2: Directly perform a balance transformation on the full-order small-signal model to achieve a unified distribution of controllability and observability in the state space; sort the state variables according to the Hankel singular values ​​to automatically identify the parts with significant dynamic contributions in the input and output channels, avoiding information loss caused by improper structural partitioning. Step 3: By truncating redundant states, a reduced-order model with dynamic characteristics that closely approximates the original system is constructed.

2. The method as described in claim 1, characterized in that, Linearization of multi-machine 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: n×n state matrix; B: n×m input matrix; C: p×n output matrix.

3. The method as described in claim 1, characterized in that, Using the equilibrium truncation theory, an equilibrium system is established based on dynamic characteristics, including: For the system shown in equation (1), its controllability Gramian P matrix and observability Gramian Q matrix satisfy the following Lyapunov equation: P measures the degree of controllability of a system, satisfying the condition that for any initial state x(0), there exists an input u(t) such that the system can transition to the origin in finite time; Q measures the ability to determine the initial state x(0) from the system output y(t); since the original system Since P≠Q, the importance of state variables is difficult to compare directly; therefore, a similarity transformation is introduced to make the state variables of the new system symmetric in terms of controllability and observability, thereby achieving equilibrium. The equilibrium transformation matrix T is constructed using the following steps: 1) Perform singular value decomposition on P and Q: Among them, U C U O They are orthogonal matrices, ∑ C , ∑ O It is a diagonal matrix; 2) Construct the intermediate matrix: Perform singular value decomposition on it: H=U H S H V H T (5) 3) Construct the equilibrium transformation matrix: Using this transformation matrix to perform state transformation on the original system, we can obtain the equilibrium system model: x b =A b x b +B b u y=C b x b +You (7) Where, x b =Tx, where A is the state variable in the equilibrium coordinates; b =TAT -1, B b =TB,C b =CT -1 ; At this point, the controllable and observable Gramian of the equilibrium system are all diagonal matrices: P b =Q b =Σ=diag(σ1,σ2,...,σ n ) (8) Where σ i These are the Hankel singular values, reflecting the combined energy contribution of the i-th state to the system's input-output transfer, and are arranged in descending order; that is: σ1≥σ2≥…≥σ n >0 (9) Where σ i This is the Hankel singular value.

4. The method as described in claim 1, characterized in that, Weakly correlated state variables are truncated based on Hankel singular values, including: Hankel singular values ​​reflect the global contribution of equilibrium state variables to the dynamic characteristics of the system; therefore, after truncation, only the first r state variables are retained, i.e., σ1 to σ2 are selected. r Discard the state components corresponding to the smaller singular values; let: Where x1∈R r Denotes the reserved subset of states, x2∈R n-r The truncated part; discarding x2 and keeping x1, we obtain the reduced-order system: This reduced-order system not only retains the most significant energy channels, but also has a strict upper bound on the error: Where G(s) is the transfer function of the full-order system, G r (s) is the transfer function of the reduced-order system; Regarding the choice of order, the cutoff point is determined by normalizing the Hankel singular values ​​and calculating their cumulative contribution; for example, if the total energy contribution target is 99%, then the following condition must be met: (13) This allows us to determine a reasonable value for r and construct the corresponding reduced-order system accordingly.

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