Method for determining applicability of power system frequency linear analysis model based on LPV theory

Abstract

The application discloses a power system frequency linear analysis model applicability determination method based on LPV theory, relates to the technical field of power system safe operation determination, and comprises the following steps: Taylor expansion calculation linearization matrix and high-order term matrix of a state space equation based on an original model of equipment, singular value and an eigenvector of the model are obtained through singular value decomposition, induced norms of key quantities such as singular value and gain operator are calculated, then the maximum structure direction is determined through singular value decomposition, an error mapping is constructed by establishing a gain operator, and the dynamic response error in the worst case is quantified. The application can give the dynamic response error in the worst case, and then the dynamic response error in the worst case is converted into a frequency error upper limit, so that it is clear whether the model approximation will change the frequency stability conclusion, thereby providing a reliable linearizable region boundary for large-scale wind power grid simulation.

Description

Technical Field

[0001] This invention relates to the field of power system safety operation judgment technology, specifically a method for determining the applicability of power system frequency linear analysis models based on LPV theory. Background Technology

[0002] The frequency of a power system reflects the supply and demand balance of active power across the entire system, and its dynamic characteristics directly affect system security and frequency regulation capabilities. In frequency analysis, commonly used average frequency models, system frequency response models, and some spatiotemporal frequency distribution models typically employ linearized generator models. These methods can significantly improve computational efficiency in synchronous machine-dominated systems while maintaining a certain level of accuracy. Furthermore, by introducing a limiting element, they can partially characterize the operating boundaries of the speed governor.

[0003] With the increasing proportion of new energy equipment connected to the grid, the applicability of linearized models faces challenges. On the one hand, traditional frequency models have already linearized the equipment during construction, making it difficult to directly apply the original nonlinear models to frequency analysis. On the other hand, new energy equipment itself has high-order, nonlinear control characteristics, making it complex and inefficient to directly use its original models for analysis. Although existing research has proposed linearization modeling methods for different types of equipment, current work lacks a systematic method for determining the applicability of frequency linear analysis models; that is, how to quantify its analytical accuracy and applicability under different operating points and parameter conditions remains unresolved. Summary of the Invention

[0004] (a) Technical problems to be solved This invention provides a method for determining the applicability of power system frequency linear analysis models based on LPV theory. For the applicability determination of power system frequency linear analysis models, a quantitative evaluation index that can combine higher-order nonlinear effects and parameter changes is established.

[0005] (II) Technical Solution To achieve the above objectives, the present invention provides the following technical solution: a method for determining the applicability of a power system frequency linearity analysis model based on LPV theory, comprising the following execution steps: The electromagnetic, mechanical, and operating parameters of the power system are obtained and transformed into voltage and current equations in Park coordinates. Combined with the electromagnetic torque expression and mechanical motion equations of the power system, a nonlinear state-space function is constructed to characterize the dynamic behavior of the power system. Under the specified operating conditions of the power system, the steady-state operating point is solved based on the nonlinear state-space function. At each of the steady-state operating points, the first-order partial derivatives of the nonlinear state-space function with respect to the state variables and input variables are calculated to construct a benchmark linear model. Based on each of the aforementioned steady-state operating points, a higher-order Taylor expansion is performed on the nonlinear state-space function to obtain the higher-order state action matrix, the higher-order state-input coupling matrix, and the higher-order input action matrix, respectively. When the operating parameters change, a time-varying parameter matrix is ​​obtained by taking the partial derivative of the nonlinear state-space function with respect to the operating parameters. The time-varying parameter matrix is ​​used to characterize the offset of the linear state matrix and the input matrix relative to the reference linear model when the operating parameters change. Singular value decomposition is performed on the time-varying matrices of each parameter to obtain the maximum singular value and its corresponding left and right singular vectors. The upper bound of the perturbation amplitude of the baseline linear model caused by the change of the running parameters is obtained by calculating the induced norm. In the baseline linear model, the output response increment caused by the state higher-order action matrix and the input higher-order action matrix, as well as the output response increment caused by their coupling, are calculated respectively to construct a mapping operator from the perturbation of the baseline linear model to the output error; by calculating the induced norm of the mapping operator, the dynamic error amplification factor of the model structure perturbation on the output error is obtained. The dynamic error amplification factor, higher-order nonlinear error, and parameter perturbation error are superimposed to construct an expression for the upper bound of the total error. When the upper bound of the total error satisfies the allowable error range of the engineering, the corresponding range of operating parameter changes is solved to determine the linearizable index.

[0006] In some feasible embodiments, a doubly-fed wind turbine is selected as the power system, and the input variable matrix of the power system is: The state variable matrix is The output matrix is The integrated nonlinear state-space function in Park coordinates is as follows: ; in, and This represents the functional relationship in a first-order differential equation.

[0007] In some feasible embodiments, the steady-state operating point is obtained using the nonlinear state-space function at selected operating points with rated wind speed and rated power. The equation is: ; Then calculate the first derivative: ; ; ; Integrated output: ; This is used as the baseline linear model.

[0008] In some feasible embodiments, during the execution of the higher-order Taylor expansion, the... Perform second-order partial derivatives: ; ; ; Then, construct the state-space equations after Taylor expansion: ; in, ; ; In the above formula , The modulation effect of higher-order terms on the coefficients of first-order terms was characterized; the changes in operating parameters were obtained. Calculate the time-varying matrix of parameters and : ; ; Then solve: ; ; in, and This represents the offset relative to the baseline linear model after the operating parameters change.

[0009] In some feasible embodiments, under zero initial conditions of the power system, its output expression is: ; The difference between it and the linear time-invariant model based on the steady-state operating point is as follows: ; in, For the gain operator, its expression is: ; The difference between the gain operators of the two is... The expression is: ; in, and These are the time-varying matrices of the parameters obtained.

[0010] In some feasible embodiments, the output response increment caused by the higher-order action matrix of the state is calculated respectively: ; The output response increment caused by the input higher-order action matrix: ; And the output response increment caused by their coupling: ; in, As the baseline matrix, This is the output matrix.

[0011] In some feasible embodiments, the time-varying matrix varies with different parameters. Below, by introducing a state transition matrix Calculate the dynamic error amplification of the output error by the perturbation in the power system, where the setting is... The expression is: , ; in Define the corresponding expression for the dynamic error amplification degree, using the reference matrix. for: ; Differentiating it, we get: ; in , Solving the equation, we get: ; Used to indicate exist Disturbances generated at any time propagate through the power system to The magnification of time, substitute The expression, and the perturbation theory is used to... The expression is simplified to: .

[0012] In some feasible embodiments, the reference matrix is... Perform singular value decomposition: ; Among them, singular values satisfy: ; Before extraction Left singular vector in each main channel and right singular vector : ; ; ; The principal singular values ​​of the matrix are: ; Time-varying matrix of parameters The primary singular value is: ; because exist Disturbances generated at any time propagate through the power system to The magnification of time satisfies: ; Seeking to obtain Singular values ​​of a matrix and parameter time-varying matrix singular values The following relationship exists: ; in: ; For all The original main channel output is The output corresponding to the disturbance is The following inequalities are satisfied: .

[0013] In some feasible embodiments, the parameter time-varying matrix is ​​used. Substitute into From: ; Let the above formula be: ; Its induced norm is: ; According to the definition of the induced norm, the following inequality always holds: ; in That is 2-norm || || 2 .

[0014] In some feasible embodiments, the time-varying matrix varies with different parameters. and In the case of, calculate respectively and The value, Based on the relative error definition of the gain operator, the linearizable index e is defined as follows: , ; Then, an error threshold range is set, and the linearizable index... If the error is within the aforementioned error threshold range, it means that the overall dynamic response error of the power system is also controlled within the error threshold range.

[0015] (III) Beneficial Effects: Compared with the prior art, this invention has the following beneficial effects: This invention establishes a quantifiable transition mechanism from a nonlinear model to a linear model through higher-order Taylor expansion and parameter perturbation decomposition.

[0016] By introducing a time-varying parameter matrix, the changes in operating point and matrix changes in the power system are decoupled, the influence of different operating conditions on the system structure is clarified, and then the maximum structural direction is determined by singular value decomposition. A gain operator is constructed to establish an error mapping, thereby quantifying the dynamic response error under the worst case.

[0017] This invention provides a reliable linearizable region boundary for large-scale wind power grid simulation by providing the worst-case dynamic response error and converting it into an upper bound for the frequency error. Attached Figure Description

[0018] Figure 1 This is a flowchart illustrating the applicability determination method for a power system frequency linearity analysis model based on LPV theory provided in an embodiment of the present invention. Figure 2 In the method for determining the applicability of a power system frequency linear analysis model based on LPV theory provided in this embodiment of the invention, the derivative of the state variable in the nonlinear system has a functional relationship with itself. Figure 3 In the method for determining the applicability of the power system frequency linear analysis model based on LPV theory provided in the embodiments of the present invention, a schematic diagram of the line connection of the doubly fed wind turbine generator of the power system under study is selected. Figure 4In the applicability determination method of the power system frequency linear analysis model based on LPV theory provided in the embodiments of the present invention, a comparative schematic diagram of the nonlinear model and the linear model of the doubly fed wind turbine under disturbance is provided under the conditions that the proportional coefficient of the phase-locked loop is 0.3 and the integral parameter is 0.7. Figure 5 In the applicability determination method of the power system frequency linear analysis model based on LPV theory provided in the embodiments of the present invention, a comparison diagram of the nonlinear model and the linear model of the doubly fed wind turbine under disturbance is shown when the proportional parameter and integral parameter of the phase-locked loop are 40 and 1600, respectively. Detailed Implementation

[0019] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the present invention without creative effort are within the scope of protection of the present invention.

[0020] It should be noted that, where there is no conflict, the features in the embodiments of the present invention can be combined with each other.

[0021] Combination Figures 1 to 5 The applicability determination method of the power system frequency linear analysis model based on LPV theory shown is used to replace the original mathematical model of the nonlinear doubly fed wind turbine (DFIG) with the LPV linear model within a certain operating range, quantify the maximum dynamic error between the two, and determine whether it affects the frequency analysis.

[0022] For reference here Figure 1 First, it is necessary to establish the nonlinear state-space function of DFIG. For DFIG, its dynamics consist of three parts: 0. Stator and rotor voltage equations (Parker coordinate system); 1. Expression for electromagnetic torque; 2. Equations of motion for machines.

[0023] Under the Park transformation (dq coordinate system), there exist terms for the product of flux linkage and current, and a coupling term between rotational speed and flux linkage. The nonlinear state-space function is: ; The input variable matrix is ​​as follows: For reference here Figure 3 It generally includes rotor voltage and mechanical power; The state variable matrix is ​​x, and the variables are generally rotor current, flux linkage, and rotational speed, etc. The output matrix is ​​y, which is generally electromagnetic power or power grid frequency coupling quantity.

[0024] Among them and Representing the functional relationship in a first-order differential equation, for a power system, the output quantity y is generally not directly related to the input quantity u, that is, y is only a function of x.

[0025] Considering that power grid frequency analysis is essentially small-disturbance analysis, and small-disturbance analysis must be conducted near the steady-state operating point, the following steps first determine the steady-state operating point at the selected operating point with rated wind speed and rated power (grid voltage) using the aforementioned constructed nonlinear state-space function. :

[0026] ; The steady-state operating point can be understood as the point at which the power system reaches a steady state, and all state variables are in equilibrium.

[0027] Then, at each steady-state operating point obtained above, partial derivatives are taken with respect to the state variables and the input to obtain the Jacobian matrix: ; ; ; Among them This represents the linearized relationship between states; This represents the linearized relationship between the input and the state.

[0028] Integrated output: ; This is used as a baseline linear model for small-signal stability analysis.

[0029] Furthermore, the above linear state-space equations can be expressed using the following Taylor expansion: ; In this matrix, A represents the relationship between the state variable x and its derivative, and B represents the relationship between the input u and the differential of the state variable. This represents the matrix relationship between state variables and output quantities, which has been shown here. Its processing is for a time-invariant form.

[0030] The superscript T represents matrix transpose, H represents the relationship between higher-order terms of Taylor expansion and the differentials of state variables or output quantities, and the superscript f represents the relationship between higher-order terms and the differentials of state variables.

[0031] Regarding why higher-order expansion is needed, it is important to understand that the linear model obtained above ignores the higher-order effects (second-order terms) of parameter coupling. These terms will become apparent when the operating range expands (wind speed changes).

[0032] Therefore, a second-order expansion of the nonlinear function is required to quantify the rate of nonlinear growth when the state deviates from the steady-state operating point. In the DFIG power system example, these errors mainly originate from the product of rotational speed and flux linkage, or the product of current and flux linkage.

[0033] Specifically, in some embodiments of the present invention, during the execution of higher-order Taylor expansion, the following steps are taken: Perform second-order partial derivatives: ; ; ; Then, construct the state-space equations after Taylor expansion: ; in, ; ; In the above formula , This characterizes the modulation effect of higher-order terms on the coefficients of first-order terms; during the dynamic process, the values ​​of the state variable x and the input variable u will inevitably change, and correspondingly, and A change in the value of will result in a change in the coefficients of the state-space equations.

[0034] In DFIG, taking wind speed as an example, changes in wind speed cause changes in slip, which in turn causes equivalent changes in electromagnetic parameters. Therefore, by obtaining the changes in operating parameters... Calculate the time-varying matrix of parameters and It is used to represent the structural offset caused by changes in operating parameters:

[0035] ; ; Then solve: ; ; in, and This represents the offset relative to the baseline linear model obtained above after the operating parameters change.

[0036] Based on conservative considerations, as long as the difference between the linearized model corresponding to the maximum deviation and the linearized model based on the steady-state point is within an acceptable range, it can be determined whether the mode can be linearized to the first order.

[0037] Under zero initial conditions of the power system, its output expression is: ; The difference between this model and the linear time-invariant (LTI) model based on the steady-state point is as follows: ; in, For the gain operator, its expression is: ; The difference between the gain operators of the two is... The expression is: ; It characterizes the difference between two LTI (linear time-invariant) systems, where, and These are the time-varying matrices of the parameters obtained.

[0038] As can be seen from the above formula, by comparing the differences between the two gain operators, we can determine the difference between the output of the system and the original LTI system when the deviation from the original system is most severe, and thus determine whether the system can be linearized.

[0039] The difference in the gain operator is mainly caused by changes in matrices A and B. Therefore, by decomposing the expression of the gain operator, the output difference caused by matrix A, i.e., the expression for the output response increment, is: ; The output response increment caused by the input higher-order action matrix: ; And the output response increment caused by their coupling: ; in, As the baseline matrix, This is the output matrix.

[0040] Based on its basic characteristics, h(t) consists of three parts: input B, gain A, and output C0. Therefore, the influence of matrix A is achieved through the input, gain, and output, while the change of matrix B only goes through the gain and output parts and is no longer reflected in the input.

[0041] Given that the values ​​of each element in matrix A are often unknown, we aim to separate them from the known A0 in order to explore their value under different conditions. The error profile of the system is then analyzed to determine whether it is suitable for linearization.

[0042] Specifically, the time-varying matrix with different parameters Below, by introducing a state transition matrix The calculation determines the degree of dynamic error amplification of the output error due to perturbations in the power system.

[0043] Why is a state transition matrix introduced in the embodiments of this invention? It is important to understand here that, such as Figure 1 As shown, in the subsequent singular value calculations, singular values ​​are only used to explain the magnitude variables of the matrix. However, this power system is a dynamic system, and structural changes in a dynamic system will be amplified or attenuated through a time propagation mechanism. Therefore, in the embodiments of this invention, this propagation mechanism is addressed by the introduced state transition matrix. To express.

[0044] Among them, setting The expression is: , ; in Define the corresponding expression for the dynamic error amplification degree, using the reference matrix. for: ; Differentiating it, we get: ; in .

[0045] Solving the equation, we get: ; Used to indicate exist Disturbances generated at any time propagate through the power system to The magnification of time, substitute The expression yields ; Then, using perturbation theory to... After simplifying the expression, we get: ; The simplified version above Apply this to subsequent calculations and processing.

[0046] Considering that singular values ​​are a representation of energy transfer capability, the baseline matrix is ​​first analyzed. Perform singular value decomposition: ; Among them, singular values satisfy: ; Before extraction Left singular vector in each main channel and right singular vector : ; ; ; The principal singular values ​​of the matrix are: ; Since the main propagation channel has been selected, and within the scope of the problem being studied, The value of is generally relatively small, insufficient to significantly change its propagation path; therefore, similarly, the time-varying matrix of the parameters... The primary singular value is: ; Any vector in the main output subspace can be written as: ; because exist Disturbances generated at any time propagate through the power system to The magnification of time satisfies: ; Seeking to obtain Singular values ​​of a matrix and parameter time-varying matrix singular values The following relationship exists: ; in: ; For all The original main channel output is The output corresponding to the disturbance is The following inequalities are satisfied: ; The time-varying matrix of parameters will be used subsequently. Bring to From: ; Let the above formula be: ; Its induced norm is: ; According to the definition of the induced norm, the following inequality always holds: ; in That is 2-norm .

[0047] Similarly, we have: ; because and Since they are all constants, therefore: ; According to the aforementioned inequality, we have: ; Similarly, for the B matrix part, we have: ; For the part of the interaction between matrices AB, we have: ; That is, except and Apart from that, all the others are related to and The specific numerical value is irrelevant; that is, the linearized correlation index based on the steady-state point can be obtained by only calculating it once.

[0048] Finally, the time-varying matrix with different parameters... and In the case of, calculate respectively and The value of can be used to quickly determine the degree of linearization.

[0049] Specifically, the linearizable index e is defined according to the relative error of the gain operator: , ; Next, set the error threshold range, for example, at 10%, when the linearizable index... If the error is within 10% of the aforementioned error threshold, it means that the overall dynamic response error of the power system is also controlled within 10% of the error threshold.

[0050] For more details, please refer to the following: Figure 4 When the proportional and integral parameters of the phase-locked loop are 0.3 and 0.7 respectively: =0.727427, =0.910917; max=1.81999, max=10.0129, max=2.73656; Based on these data and the inequality relationships satisfied by the linearizability index, the following can be estimated: =1.32391, =9.12096, =1.81331; The overall linearizability index e = 12.2582.

[0051] For more details, please refer to the following: Figure 5 When the proportional and integral parameters of the phase-locked loop are 40 and 1600 respectively: =0.0047445, =0.0028940; max=0.829167, max=11.7938, max=0.446943.

[0052] Based on these data and the inequality relationships satisfied by the linearizability index, the following can be estimated: =0.00393, =0.03413, =6.1367×10 -6 .

[0053] The total linearizability index e = 0.03807.

[0054] As can be seen from the examples, the smaller the total linearizability index, the higher the linearity of the model, and the more reasonable it is to use a linearized model to describe the dynamics of the equipment.

[0055] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. The scope of protection of the present invention is defined by the claims. Similarly, any equivalent structural changes made based on the description and drawings of the present invention should also be included within the scope of protection of the present invention.

Claims

1. A method for determining the applicability of a power system frequency linearity analysis model based on LPV theory, characterized in that, Applicable to power systems used for power generation, including the following steps: The electromagnetic, mechanical, and operational parameters of the power system are obtained and transformed into state-space form. Combining voltage and current equations, electromagnetic torque, and mechanical motion equations, a nonlinear state-space function of the power system is constructed. A doubly-fed induction generator (DFIG) wind turbine is selected as the power system, and the matrix of its input variables is as follows: The state variable matrix is The output matrix is The integrated nonlinear state-space function in Park coordinates is as follows: ； in, and This represents the functional relationship in a first-order differential equation; Under specified operating conditions of the power system, the steady-state operating point is solved based on the nonlinear state-space function; at each steady-state operating point, the first-order partial derivatives of the nonlinear state-space function with respect to the state variables and input variables are calculated to construct a baseline linear model; at selected operating points with rated wind speed and rated power, the steady-state operating point is obtained using the nonlinear state-space function. The equation is: ； Then calculate the first derivative: ； ； ； Integrated output: ； Will As the benchmark linear model; Based on each of the aforementioned steady-state operating points, a higher-order Taylor expansion is performed on the nonlinear state-space function to obtain the higher-order state action matrix, the higher-order state-input coupling matrix, and the higher-order input action matrix, respectively; when the operating parameters change, the time-varying parameter matrix is ​​obtained by taking the partial derivative of the nonlinear state-space function; Singular value decomposition is performed on the time-varying matrices of each parameter to obtain the maximum singular value and its corresponding left and right singular vectors. The upper bound of the perturbation amplitude of the baseline linear model caused by the change of the running parameters is obtained by calculating the induced norm. The state higher-order action matrix, the input higher-order action matrix, and the output response increment caused by coupling are used to construct a mapping operator from the perturbation of the baseline linear model to the output error. The induced norm is calculated to obtain the dynamic error amplification factor, which reflects the potential error amplification of the frequency response caused by nonlinearity and parameter changes. The dynamic error amplification factor, higher-order nonlinear error, and parameter perturbation error are superimposed to construct an expression for the upper bound of the total error. When the upper bound of the total error is within the allowable error range of the engineering, the benchmark linear model is determined to be suitable for frequency linear analysis within the operating parameter range.

2. The method for determining the applicability of the power system frequency linearity analysis model based on LPV theory according to claim 1, characterized in that, During the execution of the higher-order Taylor expansion, Perform second-order partial derivatives: ； ； ； Then, construct the state-space equations after Taylor expansion: ； in, ； ； In the above formula , The modulation effect of higher-order terms on the coefficients of first-order terms was characterized; the changes in operating parameters were obtained. Calculate the time-varying matrix of parameters and : ； ； Then solve: ； ； in, and This represents the offset relative to the baseline linear model after the operating parameters change.

3. The method for determining the applicability of the power system frequency linearity analysis model based on LPV theory according to claim 1, characterized in that, Under zero initial conditions of the power system, its output expression is: ； The difference between it and the linear time-invariant model based on the steady-state operating point is as follows: ； in, For the gain operator, its expression is: ； ； The difference between the gain operators of the two is... The expression is: ； in, and These are the time-varying matrices of the parameters obtained.

4. The method for determining the applicability of the power system frequency linearity analysis model based on LPV theory according to claim 1, characterized in that, Calculate the output response increment caused by the higher-order action matrix of the state respectively: ； The output response increment caused by the input higher-order action matrix: ； And the output response increment caused by their coupling: ； in, As the baseline matrix, For the output matrix, and These are the time-varying matrices of the parameters obtained.

5. The method for determining the applicability of the power system frequency linearity analysis model based on LPV theory according to claim 1, characterized in that, Time-varying matrices with different parameters Below, by introducing a state transition matrix Calculate the dynamic error amplification of the output error by the perturbation in the power system, where the following is set: The expression is: ， ； in Define the corresponding expression for the dynamic error amplification degree, using the reference matrix. for: ； Differentiating it, we get: ； in , Solving the equation, we get: ； Used to indicate exist Disturbances generated at any time propagate through the power system to The magnification of time, substitute The expression, and the perturbation theory is used to... The expression is simplified to: 。 6. The method for determining the applicability of the power system frequency linearity analysis model based on LPV theory according to claim 5, characterized in that, For the reference matrix Perform singular value decomposition: ； Among them, singular values satisfy: ； Before extraction Left singular vector in each main channel and right singular vector : ； ； ； The principal singular values ​​of the matrix are: ； Time-varying matrix of parameters The primary singular value is: ； because exist Disturbances generated at any time propagate through the power system to The magnification of time satisfies: ； Seeking to obtain Singular values ​​of a matrix and parameter time-varying matrix The singular values ​​of the tactic elements have the following relationship: ； in: ； For all The original main channel output is The output corresponding to the disturbance is The following inequalities are satisfied: 。 7. The method for determining the applicability of the power system frequency linearity analysis model based on LPV theory according to claim 6, characterized in that, Time-varying matrix of parameters Substitute into From: ； Let the above formula be: ； Its induced norm is: ； According to the definition of the induced norm, the following inequality always holds: ； in That is 2-norm || ||2.

8. The method for determining the applicability of the power system frequency linearity analysis model based on LPV theory according to claim 7, characterized in that, Time-varying matrix with different parameters and In the case of, calculate respectively and The value, The linearizable index is defined based on the relative error of the gain operator. e : ， ； Then, an error threshold range is set, and the linearizable index... If the error is within the aforementioned error threshold range, it means that the overall dynamic response error of the power system is also controlled within the error threshold range.

Images