Harmonic balance based constant amplitude oscillation solving method, device, equipment and medium

Abstract

The application relates to a harmonic balance-based equal-amplitude oscillation solving method, device, equipment and medium, which comprises the following steps: determining the highest order of the oscillation component in a to-be-solved vector, performing small-signal stability analysis on an original vector without the oscillation component based on a small-signal stability analysis result, generating a nonlinear equation according to the analysis result and an iteration strategy when an equal-amplitude oscillation starting condition is met, solving the nonlinear equation based on a Newton method solving strategy and the highest order of the oscillation component in the to-be-solved vector to obtain an initial solving result, and judging whether single clipping is triggered according to the solving result, wherein when the single clipping is not triggered, a final equal-amplitude oscillation solving result is obtained according to the initial solving result. Therefore, the problems that the equal-amplitude oscillation solving method in the related art has defects in principle, the solving result is incomplete, and the method cannot be applied to more profound stability analysis are solved, and the accuracy of equal-amplitude oscillation solving of a VSC system and the reliability of a solving process are improved.

Description

Technical Field

[0001] This application relates to the field of steady-state analysis technology for power systems, and in particular to a method, apparatus, equipment, and medium for solving constant-amplitude oscillations based on harmonic balance. Background Technology

[0002] With my country's vigorous promotion of the construction of new power systems, more and more new energy sources are being connected to the system via VSC (Voltage Source Converter). VSCs are typical high-order nonlinear components, and their steady-state and dynamic performance is influenced by both the modulation method and the control loop, significantly increasing the difficulty of system analysis and control. A typical example is the broadband oscillation (small-signal stability) problem caused by the interaction of various factors, including the volatility of new energy sources, changes in system operation modes, and multi-timescale control of the converter.

[0003] In related technologies, most studies on broadband oscillations first assume system stability, then linearize the state space at its equilibrium point or develop a dedicated transfer function containing closed-loop system modes for stability analysis and control. However, although this approach has guided the stable operation of actual power grids and accumulated some experience for operators and manufacturers, actual accidents involving broadband oscillations still occur frequently, indirectly reflecting the incompleteness of the underlying mechanisms of related solutions.

[0004] A novel perspective is to directly assess the actual operating state of the system at the moment the oscillation suppression strategy is implemented: it is generally believed that the existence of a negative damping mode in the system is the fundamental cause of broadband oscillations. Suppressing constant-amplitude or negatively damped oscillations is a practical requirement for new power systems. Accurately calculating the amplitude and frequency of each harmonic of the constant-amplitude oscillation, and obtaining a linearized model based on the calculation results, is essential to provide a rigorous theoretical basis for the design of oscillation suppression. Furthermore, the limiting element is a crucial part of the VSC controller design, as it can limit the controller's output to an upper or lower limit.

[0005] In related technologies, research has shown that when a system reaches a constant-amplitude oscillation state, the amplitude limiting element may or may not be triggered. For solving constant-amplitude oscillations, the describing function method is used. The describing function method is defined as the complex ratio (amplitude ratio, phase difference) between the fundamental component and the input sinusoidal signal in the steady-state sinusoidal response of a nonlinear element. This method was first proposed by P.J. Daniel in 1940 and is mainly used to analyze the stability and self-oscillation problems of a class of nonlinear systems when no input signal is applied. The describing function method is an equivalent linearization method for studying the stability of nonlinear control systems from the perspective of the frequency domain. In Soviet literature, this method is often referred to as the harmonic balance method. Furthermore, the describing function method can only be used to study the frequency response characteristics of a system and cannot provide precise information about the time response.

[0006] However, while this method is unaffected by the system's order, it does offer some approximations. Furthermore, the solution method for VSC constant-amplitude oscillations has inherent limitations in principle, and the results are incomplete, making it unsuitable for more in-depth stability analysis and other problems, which urgently need to be addressed. Summary of the Invention

[0007] This application provides a method, apparatus, device, and medium for solving constant-amplitude oscillations based on harmonic balance, to address the shortcomings in the principle of related technologies for solving constant-amplitude oscillations of VSCs, the incompleteness of the solution results, and the inability to apply them to more in-depth stability analysis. It provides a method for small-signal stability analysis and oscillation suppression in new power systems, improving the accuracy of solving constant-amplitude oscillations in systems containing VSCs and the reliability of the solution process.

[0008] The first aspect of this application provides a method for solving constant-amplitude oscillations based on harmonic balance, comprising the following steps:

[0009] The highest order of the oscillating component in the vector to be determined is determined; based on a preset small-signal stability analysis strategy, a small-signal stability analysis is performed on the original vector without oscillating components, and the analysis results are used to determine whether the preset constant-amplitude oscillation initiation condition is met; if the preset constant-amplitude oscillation initiation condition is met, at least one nonlinear equation that satisfies the preset condition is generated based on the analysis results and a preset iterative strategy, and the at least one nonlinear equation is solved based on a preset Newton-Laurel method solution strategy and the highest order of the oscillating component in the vector to be determined to obtain an initial solution result, and the solution result is used to determine whether a single amplitude limit is triggered, and if the single amplitude limit is not triggered, the final constant-amplitude oscillation solution result is obtained based on the initial solution result.

[0010] Optionally, in some embodiments, after determining whether the single limit is triggered based on the solution result, the method further includes: if the single limit is triggered, then generating at least one new nonlinear equation that satisfies the preset conditions based on the analysis result and the preset iterative strategy, until the single limit is not triggered.

[0011] Optionally, in some embodiments, after determining whether the preset constant amplitude oscillation initiation condition is met based on the analysis results, the method further includes: if the preset constant amplitude oscillation initiation condition is not met, then constant amplitude oscillation solution is not performed.

[0012] Optionally, in some embodiments, before solving the at least one nonlinear equation based on the preset Newton-Laurel method solution strategy and the highest order of the oscillating component in the vector to be solved, the method further includes: performing a parameter scan on the at least one nonlinear equation to obtain the initial values ​​of the at least one nonlinear equation.

[0013] Optionally, in some embodiments, the step of solving the at least one nonlinear equation based on the preset Newton-Lager method solution strategy and the highest order of the oscillating component in the vector to be solved to obtain the initial solution result further includes: determining whether the amplitude of the first-order oscillating component in the vector to be solved to the amplitude of the current-order oscillating component in the vector to be solved is not equal to a preset threshold; if the amplitude of the first-order oscillating component in the vector to be solved to the amplitude of the current-order oscillating component in the vector to be solved is not equal to the preset threshold, then determining whether the current order is less than the highest order; if the current order is less than the highest order, then incrementing the current order by 1 until the current order is greater than or equal to the highest order, thereby obtaining the initial solution result.

[0014] A second aspect of this application provides a constant-amplitude oscillation solver based on harmonic balance, comprising:

[0015] The system comprises a determination module for determining the highest order of the oscillating component in the vector to be determined; an analysis module for performing small-signal stability analysis on the original vector without oscillating components based on a preset small-signal stability analysis strategy, and determining whether a preset constant-amplitude oscillation initiation condition is met based on the analysis results; and a solution module for generating at least one nonlinear equation that satisfies the preset condition based on the analysis results and a preset iterative strategy when the preset constant-amplitude oscillation initiation condition is met, solving the at least one nonlinear equation based on a preset Newton-Laurel method solution strategy and the highest order of the oscillating component in the vector to be determined, obtaining an initial solution result, determining whether a single amplitude limit is triggered based on the solution result, and obtaining the final constant-amplitude oscillation solution result based on the initial solution result when the single amplitude limit is not triggered.

[0016] Optionally, in some embodiments, after determining whether the single limit has been triggered based on the solution result, the solution module is further configured to: when the single limit is triggered, regenerate at least one new nonlinear equation that satisfies the preset conditions based on the analysis result and the preset iterative strategy, until the single limit is not triggered.

[0017] Optionally, in some embodiments, after determining whether the preset constant amplitude oscillation initiation condition is met based on the analysis results, the analysis module is further configured to: not perform constant amplitude oscillation solution when the preset constant amplitude oscillation initiation condition is not met.

[0018] Optionally, in some embodiments, before solving the at least one nonlinear equation based on the preset Newton-Lager method solution strategy and the highest order of the oscillating component in the vector to be solved, the solution module is further configured to: perform parameter scanning on the at least one nonlinear equation to obtain the initial values ​​of the at least one nonlinear equation.

[0019] Optionally, in some embodiments, the solving module is further configured to: determine whether the amplitude of the first-order oscillation component in the vector to be solved to the amplitude of the current-order oscillation component in the vector to be solved are all not equal to a preset threshold; when the amplitude of the first-order oscillation component in the vector to be solved to the amplitude of the current-order oscillation component in the vector to be solved are all not equal to the preset threshold, determine whether the current order is less than the highest order; when the current order is less than the highest order, increment the current order by 1 until the current order is greater than or equal to the highest order, and obtain the initial solution result.

[0020] A third aspect of this application provides an electronic device, including: a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the program to implement the constant-amplitude oscillation solution method based on harmonic balance as described in the above embodiments.

[0021] A fourth aspect of this application provides a computer-readable storage medium having a computer program stored thereon, which is executed by a processor to implement the constant-amplitude oscillation solution method based on harmonic balance as described in the above embodiments.

[0022] Therefore, this application determines the highest order of the oscillating component in the vector to be determined, and performs small-signal stability analysis on the original vector without oscillating components based on a small-signal stability analysis strategy. When the constant-amplitude oscillation initiation condition is met, a nonlinear equation is generated based on the analysis results and iterative strategy. The nonlinear equation is then solved based on the Newton-Laurel method and the highest order of the oscillating component in the vector to be determined, yielding an initial solution. The solution result is then used to determine whether a single amplitude limit has been triggered. If the single amplitude limit has not been triggered, the final constant-amplitude oscillation solution is obtained based on the initial solution result. This solves the problems in related technologies where the solution methods for constant-amplitude oscillations of VSCs are theoretically insufficient and the solution results are incomplete, making them unsuitable for deeper stability analysis. It provides a small-signal stability analysis and oscillation suppression method for novel power systems, improving the accuracy and reliability of the solution process for constant-amplitude oscillations in systems containing VSCs.

[0023] Additional aspects and advantages of this application will be set forth in part in the description which follows, and in part will be obvious from the description, or may be learned by practice of this application. Attached Figure Description

[0024] The above and / or additional aspects and advantages of this application will become apparent and readily understood from the following description of the embodiments taken in conjunction with the accompanying drawings, wherein:

[0025] Figure 1This is a flowchart of the constant-amplitude oscillation solution method based on harmonic balance provided in the embodiments of this application;

[0026] Figure 2 This is a schematic diagram of a standard test system for a constant amplitude oscillation calculation method according to an embodiment of this application;

[0027] Figure 3 This is a schematic diagram illustrating how bilateral (asymmetric) clipping results in continuous input and segmented output according to an embodiment of this application.

[0028] Figure 4 This is a schematic diagram illustrating the implementation process of a constant-amplitude oscillation algorithm according to an embodiment of this application;

[0029] Figure 5 This is a schematic diagram of a PSCAD simulation waveform provided according to an embodiment of this application;

[0030] Figure 6 This is a schematic diagram of a harmonic balance-based constant-amplitude oscillation solver provided according to an embodiment of this application;

[0031] Figure 7 This is a schematic diagram of the structure of an electronic device provided according to an embodiment of this application. Detailed Implementation

[0032] The embodiments of this application are described in detail below. Examples of these embodiments are shown in the accompanying drawings, wherein the same or similar reference numerals denote the same or similar elements or elements having the same or similar functions throughout. The embodiments described below with reference to the accompanying drawings are exemplary and intended to explain this application, and should not be construed as limiting this application.

[0033] The following describes, with reference to the accompanying drawings, a method, apparatus, device, and medium for solving constant-amplitude oscillations based on harmonic balance according to embodiments of this application. Addressing the shortcomings of the aforementioned constant-amplitude oscillation solving methods in the background art—namely, incomplete solution results that cannot be applied to deeper stability analysis—this application provides a constant-amplitude oscillation solving method based on harmonic balance. In this method, the highest order of the oscillation component in the vector to be solved is determined; based on a preset small-signal stability analysis strategy, small-signal stability analysis is performed on the original vector without oscillation components, and the analysis results are used to determine whether a preset constant-amplitude oscillation initiation condition is met; if the preset constant-amplitude oscillation initiation condition is met, at least one nonlinear equation satisfying the preset condition is generated based on the analysis results and a preset iterative strategy; and based on a preset Newton-Laurent method solution strategy and the highest order of the oscillation component in the vector to be solved, at least one nonlinear equation is solved to obtain an initial solution result. The solution result is then used to determine whether a single amplitude limit is triggered, and if the single amplitude limit is not triggered, the final constant-amplitude oscillation solution result is obtained based on the initial solution result. Therefore, this addresses the shortcomings in the underlying principles of related technologies for solving constant-amplitude oscillations of VSC, as well as the incompleteness of the solution results, which prevents their application to more in-depth stability analysis. It provides a method for small-signal stability analysis and oscillation suppression in novel power systems, improving the accuracy of solving constant-amplitude oscillations in systems containing VSC and the reliability of the solution process.

[0034] Before introducing the embodiments of this application, let's first introduce the description function method in related technologies.

[0035] It should be noted that the presence of negative damping modes in power electronic converter systems is generally considered the root cause of broadband oscillations. When harmonics of a specific frequency are excited, they participate in the closed-loop control of the system, thereby dynamically altering the system's operating trajectory. If the system's control and protection logic is designed reasonably, the aforementioned negative damping divergence process usually evolves into constant-amplitude oscillations, deteriorating the system's power quality and exciting harmonic power flows. This constant-amplitude oscillation state is generally considered to contradict the principle of safe and stable system operation. Therefore, suppressing constant-amplitude or negative damping oscillation processes is a practical requirement for new power systems. Accurately calculating the amplitude and frequency of each harmonic of the constant-amplitude oscillation, and obtaining a linearized model based on the calculation results, is essential to providing a rigorous theoretical basis for the design of oscillation suppression.

[0036] The describing function method in related technologies decomposes the closed loop into a linear part and a nonlinear part (which can be single-input single-output or multiple-input multiple-output), and uses the generalized Nyquist criterion "graphically" or constructs a system of nonlinear equations to numerically solve for the harmonic oscillation amplitude / frequency.

[0037] However, applying the describing function method to solve for VSC constant-amplitude oscillations has the following obvious shortcomings in principle:

[0038] (1) The describing function method can only be applied to constant-amplitude oscillations triggered by the limiting element, but not to constant-amplitude oscillations not triggered by the limiting element, because the latter does not have the condition for separating the limiting nonlinearity. This inherent defect of the describing function method has even led many studies to believe that the limiting element is a sufficient condition for triggering constant-amplitude oscillations.

[0039] (2) The describing function method usually requires the artificial extraction of a nonlinear element (such as a limiting element). The linear part generally refers to the transfer function containing several linear controllers (which should have low-pass characteristics). In fact, it completely ignores nonlinear elements such as phase-locked loops (PLLs) and there is no theoretical basis to explain the rationality of simplification or the necessity of low-pass characteristics. This similarity to the idea and form of small-signal stability has even led some studies to regard the solution of constant amplitude oscillation as a stability analysis problem, which obviously confuses the steady-state and dynamic analysis of the system.

[0040] (3) Theoretically, every controller has a limiting element, but the describing function method can generally only consider limiting at a single point or two structurally symmetrical points (such as the d-axis inner loop current control and q-axis inner loop current control in the dq coordinate system control of a converter), and can only solve for variables related to the input and output of the limiting element. From the perspective of the observability of the transfer function with respect to the system mode, it can be understood as "partial observability". It is obviously impossible to solve all variables using a single loop, but subsequent stability analysis usually requires a relatively "complete" (AC and DC) trajectory of the system. Therefore, the describing function method has the problem that the solution is not complete enough and cannot be applied to more in-depth stability analysis.

[0041] (4) Most importantly, since amplitude limiting is not a sufficient condition for constant amplitude oscillation, simulation must be used to confirm which amplitude limiting element is triggered before the describing function method can be applied. If an iterative algorithm is used, its initial value generally depends on the simulation results, which obviously deviates from the goal of steady-state analysis (simulation is only used to observe phenomena and verify results, and should not be used as the starting point for the specific implementation of the method).

[0042] In summary, the describing function method is only an approximate solution for VSC constant-amplitude oscillations and the relevant mechanism is incomplete. Its reliability in practical applications cannot be guaranteed. Therefore, there is an urgent need to develop new and more accurate methods for calculating constant-amplitude oscillations.

[0043] To address the aforementioned issues, this application provides a method for solving constant-amplitude oscillations based on harmonic balance. This application shifts the research perspective, treating the power electronic converter system in a constant-amplitude oscillation state as a nonlinear time-varying periodic system. From the perspective of input-output energy balance, not only is the fundamental source-load energy balanced, but the harmonic energy is also in source-load balance; this application collectively refers to this as harmonic balance. Furthermore, strictly speaking, the describing function method is also based on harmonic balance, but it artificially defines a linear system; however, this so-called "linear system" is essentially strongly nonlinear.

[0044] This application takes a harmonic balance perspective and characterizes several common nonlinear elements of VSC (including PLL, VSC modulation, AC / DC power balance of the VSC body, and (potential) limiting nonlinearity) in the frequency domain rather than the time domain. A set of nonlinear equations is listed modularly and systematically. Since these nonlinear equations are not differential equations, they have the potential for expansion, such as considering delay nonlinearity. This application adopts an iterative method to solve the amplitude and frequency of constant-amplitude oscillations with high precision, including the 1st, 2nd, ..., nth order components of the oscillation. A process for obtaining the initial values ​​of the iteration is specially designed.

[0045] Furthermore, if we examine the solution process of the aforementioned constant-amplitude oscillation problem from the perspective of nonlinear dynamics, it is essentially a process of solving a "stable limit cycle." Therefore, theoretically, it can also be extended to the solution of unstable limit cycles or semi-stable limit cycles, and the results can be used to guide the assessment of system power quality. Since the solution process fully utilizes the concept of "harmonics," which is of particular concern in power systems, the solution method of this application is expected to provide a basis for solving the steady state of general nonlinear dynamic systems.

[0046] The method for solving constant-amplitude oscillations based on harmonic balance of this application will be described in detail below with reference to specific embodiments.

[0047] Specifically, Figure 1 This is a flowchart illustrating a method for solving constant-amplitude oscillations based on harmonic balance, provided in an embodiment of this application.

[0048] like Figure 1 As shown, the method for solving constant-amplitude oscillations based on harmonic balance includes the following steps:

[0049] In step S101, the highest order of the oscillating component in the vector to be determined is determined.

[0050] It should be noted that the constant amplitude oscillation solution method based on harmonic balance in this application requires first dividing the vector to be solved and the intermediate vector, determining the corresponding AC / DC vector type, and then generating the corresponding vector and Toeplitz matrix according to the rules.

[0051] Specifically, Figure 2This is a schematic diagram of a standard test system for the constant amplitude oscillation calculation method provided in this application embodiment. This application embodiment considers, for example... Figure 2 The method shown here for solving the constant-amplitude oscillation generated by a three-phase two-level VSC connected to a weak power grid assumes that the system is three-phase symmetrical and that switching frequency harmonics are completely ignored. The DC side of this system uses an ideal current source to simulate the output of new energy sources (such as wind turbines and photovoltaics), which is a commonly used equivalent method for new energy power plants. The VSC control loop uses typical dq coordinate system decoupled control, with the d-axis outer loop controlling the DC bus voltage and the q-axis outer loop reference value set to 0. Each PI (proportional integral) controller has a limiting element. Although the method in this embodiment can consider any (specifically to) Figure 2 This refers to the situation where the limiting circuit (using four PI controllers plus an external PWM (Pulse Width Modulation) modulation stage) is triggered. However, this application manually sets the limiting values, only considering the possibility of the outer loop limiting on the d-axis being triggered. Furthermore, the above setup fully preserves the typical nonlinear dynamics of the VSC, thus it can be selected as a standard system for studying the calculation method of constant-amplitude oscillations in the VSC.

[0052] It is understandable that the time-domain variables used to write nonlinear equations can be represented by their frequency-domain Fourier coefficients as two main categories: "DC signal vector" and "AC signal vector". The meanings of some vectors in the embodiments of this application will be explained below.

[0053] In this embodiment, the "DC signal vector" includes the DC voltage u dc The potential component within the d-axis is u md The potential component within the q-axis is u mq The voltage signal component at the d-axis grid connection point is u. ld The q-axis grid connection point voltage signal component is u lq The d-axis injected grid-connected current signal component is i ld The q-axis injected grid-connected current signal component is i lq The phase θ of the grid connection point output from the phase-locked loop, and the signal i of the reference value signal of the inner d-axis loop (i.e., the outer d-axis loop output) before being limited. p*ld The reference value signal of the d-axis inner loop (i.e., the output of the d-axis outer loop) is limited to the signal i. *ld wait.

[0054] In this embodiment, the "AC signal vector" includes the voltage u at the grid connection point of phase a. la Injecting current i at the grid connection point of phase a la a-phase modulation ratio signal m ia In addition, it also includes L(cos(θ(t))) and L(sin(θ(t))) for the Park transform.

[0055] In addition, the dominant (first-order) oscillation frequency f s It must be a variable to be determined (not an FFT signal), or it can be a vector f composed of the frequency shifts of the fundamental wave and harmonics.

[0056] When constant amplitude oscillation occurs, it is assumed that the DC signal has a frequency of f. s The dominant harmonic component and a DC component (considered to have a frequency of 0) are present. The AC signal cosθ contains a fundamental component with frequency f1 and f1±f s Harmonic components. The above AC / DC harmonic components are collectively referred to as first-order oscillatory components. Therefore, when a DC signal in the dq coordinate system is transformed by the inverse Park transform to obtain an AC signal in the abc coordinate system, it will generate a frequency of f1, f1±f. s and f1±2f s The signal, and simultaneously the AC signal in the abc coordinate system, after Park transformation to generate the DC signal in the dq coordinate system, contains a frequency of 0, f. s and 2f s The amount.

[0057] Repeating the above process, since the harmonic components of both AC and DC signals are theoretically infinite, the calculations must be manually truncated to the nth-order oscillation component. Because passive devices such as inductors and capacitors have high high-frequency impedance and controllers typically have low-pass properties, the amplitude of the high-frequency component (or high-frequency energy coupled nonlinearly from the low-frequency component) is usually much smaller than that of the low-frequency component. Therefore, considering that second- or third-order components are generally sufficient to describe the characteristics of a constant-amplitude oscillation state, the AC and DC vectors are uniformly represented in the following form:

[0058] g = [g -1 g0 g1] [3×(2×2n+1)]×1 T

[0059] g k =[0 n×1 g _k-n×s L g _k-1×s g _k g _k+1×s L g _k+n×s 0 n×1 ] (2×2n+1)×1 T k = -1, 0, 1

[0060] In the above formula, g includes g -1 The three sub-vectors g0 and g1 (all column vectors) have different values ​​for AC signals. For AC signals, g0 is a zero vector, while for DC signals, g0 is a zero vector. -1g1 is a zero vector. The order of each sub-vector is 2×2n+1. In this embodiment, n is to be 0, 1, 2, 3. Therefore, the sub-vector also contains a large number of zero elements (the number can be even greater when solving for higher-order components). That is, the higher-order harmonic components are artificially set to 0 to prevent the AC / DC higher-order components generated by the coupling of AC lower-order components from being mistakenly identified as DC / AC lower-order components. In other words, it is similar to the computer's memory needing to be large enough, or requesting a large enough memory space during the operation.

[0061] Furthermore, since this application embodiment initially considers a three-phase symmetrical system, a single-phase (default a-phase) AC signal can be used to replace the three-phase AC signal for subsequent equation writing and numerical calculation. Subvectors with indices 1 / 0 / -1 are represented as "generalized positive-sequence / negative-sequence / zero-sequence subvectors," where the generalized positive-sequence and negative-sequence vectors satisfy a "symmetric conjugate relationship," meaning that elements with symmetrical positions are conjugates, such as g. 1+s =g (-1-s) * .

[0062] Taking the generalized positive-order subvector as an example, only the component with index 1 satisfies the traditional positive and negative order definition of "phase a leads phase b and phase c by 120 degrees each", while the components with other indices satisfy different phase shift relationships according to the frequency domain (obviously not 120 degrees). This further highlights the necessity of using a modular approach to write equations to solve the stability limit cycle.

[0063] It should be noted that each component in the vector is described by two undetermined coefficients, which can be either rectangular coordinates (Re(.)+jIm(.)) or polar coordinates (Mag(.)∠Ang(.)). The former is theoretically easier to write equations for, while the latter is also used when dealing with specific nonlinearities.

[0064] Optionally, based on the time-frequency conversion relationship, the time-domain signal multiplication operation can be transformed into a frequency-domain signal convolution operation. The frequency-domain signal convolution operation can then be transformed into the product of the Toeplitz matrix (T(.)) generated from the aforementioned higher-order vectors and a column vector to obtain the Fourier coefficients in the frequency domain. All three operations satisfy the commutative law of multiplication. Therefore, based on the above operational principles, the unknown components in each vector to be determined can be set as undetermined coefficients. Sufficient harmonic balance relationships can be sought to formulate equations such that the real and imaginary parts of the equations are both 0, forming a system of nonlinear equations. Furthermore, the above modular generation method of variables and equations can be achieved using symbolic computation in Matlab.

[0065] Therefore, this application can determine the most suitable unknown vector for characterizing the power / internal potential product nonlinearity, trigonometric function expansion nonlinearity, and amplitude-limited piecewise nonlinearity in constant-amplitude oscillations, the variable setting method considering different orders of oscillation components, and the polar / rectangular coordinate representation of the variables, and generate the corresponding vector and Toeplitz matrix to facilitate the extraction of equations from the corresponding positions in subsequent matrix operations, thus ensuring the scalability of the method.

[0066] In step S102, based on the preset small-signal stability analysis strategy, the original vector without oscillation components is subjected to small-signal stability analysis, and the preset constant-amplitude oscillation initiation condition is determined according to the analysis results.

[0067] This application includes a pre-defined small-signal stability analysis strategy. This strategy is used to perform small-signal stability analysis on the original vector (which does not contain oscillating components) obtained in step S101, thereby obtaining the stability analysis result of the original vector. The pre-defined constant-amplitude oscillation initiation condition in this embodiment is whether a negative damping mode exists in the small-signal stability analysis result. If a negative damping mode exists, the vector to be determined is deemed to meet the constant-amplitude oscillation initiation condition.

[0068] Specifically, as analyzed above, the order of each sub-vector in this embodiment is 2×2n+1. In this embodiment, n is chosen to be 0, 1, 2, or 3. Therefore, the sub-vector also contains a large number of 0 elements (the number can be even greater when solving for higher-order components). That is, higher-order harmonic components are artificially set to 0 to prevent AC / DC higher-order components generated by the coupling of AC lower-order components from being mistakenly identified as DC / AC lower-order components. Therefore, in this embodiment, n=0, small-signal stability analysis is performed on the original vector without oscillating components, and the condition for constant-amplitude oscillation initiation is determined based on the analysis results.

[0069] This application assumes that the system does not have harmonic components. Small-signal stability analysis determines whether it is necessary to calculate constant-amplitude oscillations and extracts initial value information for iteration. This step belongs to the traditional view of small-signal stability analysis. Traditional methods for small-signal stability analysis assume that the system does not have harmonics at non-set oscillation frequencies. Strictly speaking, this only determines the system's oscillation initiation conditions, i.e., the starting point information for the system's gradual transition to a constant-amplitude oscillation state. Therefore, it is very meaningful for solving the amplitude and phase of constant-amplitude oscillations using iterative methods.

[0070] Optionally, in some embodiments, after determining whether the preset constant amplitude oscillation condition is met based on the analysis results, the method further includes: if the preset constant amplitude oscillation condition is not met, then the constant amplitude oscillation solution is not performed.

[0071] Specifically, if the small-signal stability analysis result of the oscillating component in the vector to be determined does not meet the preset constant-amplitude oscillation condition, that is, there is no negative damping mode, then the vector to be determined will not be solved for constant-amplitude oscillation and the process will end.

[0072] In step S103, if the preset constant amplitude oscillation condition is met, at least one nonlinear equation that satisfies the preset condition is generated based on the analysis results and the preset iterative strategy. Based on the preset Newton-Laurent method solution strategy and the highest order of the oscillation component in the vector to be solved, at least one nonlinear equation is solved to obtain the initial solution result. Based on the solution result, it is determined whether the single amplitude limit is triggered. If the single amplitude limit is not triggered, the final constant amplitude oscillation solution result is obtained based on the initial solution result.

[0073] Specifically, if the vector to be determined has a negative damping mode, then the minimum number of nonlinear equations are generated based on the order of the desired highest oscillation component. Then, this application uses the Newton-Laurel equation to solve the linear equation to obtain the initial solution result. Based on the solution result, it is determined whether amplitude limiting is triggered, and the constant amplitude oscillation is recalculated based on the existing results.

[0074] The following embodiments detail the method for characterizing the typical nonlinear environment of VSC under constant amplitude oscillation conditions provided in this application.

[0075] Understandably, to reduce the computational load and improve the iteration speed, it is necessary to carefully select the category and representation (polar coordinates, rectangular coordinates) of the vector to be calculated. This embodiment determines u as... la m ia u dc ,θ,i p*ld Let f be the variables (vectors) to be determined, where θ and i are... p*sd Using polar coordinates, u la m ia u dc Cartesian coordinates are used. Furthermore, of the four key nonlinear components mentioned above, the handling of the two product nonlinearities in the equation formulation is intuitive; the handling of the remaining two nonlinear components requires special explanation:

[0076] 1) Sine / cosine functions of DC signal vectors – soft nonlinear, continuously differentiable.

[0077] The time-domain form of a DC signal vector g is g(t) (n=3):

[0078] g(t) = ω1t + θ0 + m s cos(ω s t+θ s )+m 2s cos(2ω s t+θ 2s )+m3s cos(3ω s t+θ 3s )

[0079] According to the Jacobi-Anger expansion formula:

[0080]

[0081]

[0082]

[0083]

[0084] The sine and cosine vectors of the DC signal required by combining the Park transform and its inverse transform can be approximated as:

[0085] cos(g(t))=cos[ω1t+θ0+m s cos(ω s t+θ s )+m 2s cos(2ω s t+θ 2s )+m 3s cos(3ω s t+θ 3s )]

[0086] =c1c s c 2s c 3s -c1c s s 2s s 3s -c1s s s 2s c 3s -c1s s c 2s s 3s -s1s s c 2s c 3p +s1s s s 2s c 3s -s1s s c 2s s 3s -s1s s s 2s c 3s

[0087] ≈c1c s c 2s -c1s s s 2s -s1s sc 2s -s1s s s 2s

[0088] sin(g(t))=sin[ω1t+θ0+m s cos(ω s t+θ s )+m 2s cos(2ω s t+θ 2s )+m 3s cos(3ω s t+θ 3s )]

[0089] =s1c s c 2s c 3s -s1c s s 2s s 3s -s1s s s 2s c 3s -s1s s c 2s s 3s +c1s s c 2s c 3s -c1s s s 2s s 3s +c1s s c 2s s 3s +c1s s s 2s c 3s

[0090] ≈s1c s c 2s -s1s s s 2s +c1s s c 2s +c1s s s 2s

[0091] Among them:

[0092] c1=cos(ω1t+θ0)

[0093] c s =cos[m s cos(ω s t+θ s )]≈B(0,m s )-2B(2,m s )cos(2(ωs t+θ s ))+2B(4,m s )cos(4(ω s t+θ s ))

[0094] ≈B(0,m s )-2B(2,m s )cos(2(ω s t+θ s ))

[0095] c 2s =cos[m 2s cos(2ω s t+θ 2s )]≈B(0,m 2s )-2B(2,m 2s )cos(2(2ω s t+θ 2s ))+2B(4,m 2s )cos(4(2ω s t+θ 2s ))

[0096] ≈B(0,m 2s )-2B(2,m 2s )cos(2(2ω s t+θ 2s ))

[0097] c 3s =cos[m 3s cos(3ω s t+θ 3s )]≈B(0,m 3s )-2B(2,m 3s )cos(2(3ω s t+θ 3s ))+2B(4,m 4s )cos(4(3ω s t+θ 3s ))≈1

[0098] s1=sin(ω1t+θ0)

[0099] s s =sin[m s cos(ω s t+θ s )]≈2B(1,m s )cos(ω s t+θ s )-2B(3,m s )cos[3(ωs t+θ s )]≈2B(1,m s cos(ω) s t+θ s )

[0100] s 2s =sin[m 2s cos(ω 2s t+θ 2s )]≈2B(1,m 2s cos(2ω) s t+θ 2s )-2B(3,m 2s )cos[3(2ω s t+θ 2s )]≈2B(1,m 2s cos(2ω) s t+θ 2s )

[0101] s 3s =sin[m 3s cos(ω 3s t+θ 3s )]≈2B(1,m 3s cos(3ω) s t+θ 3s )-2B(3,m 3s )cos[3(3ω s t+θ 2s )]≈2B(1,m 3s cos(3ω) s t+θ 3s )≈0

[0102] It can be seen that the Jacobi-Anger method completely ignores the third-order oscillatory component of the phase. This is partly because the relative and absolute values ​​of the amplitude of the third-order oscillatory component are usually much smaller than those of the second and first orders. Furthermore, considering the third-order component would lead to "over-limiting" in multiplication operations; that is, if the third-order phase component is to be considered, the matrix order should be increased. In addition, since the Bessel function (B(.)) uses the amplitudes of each harmonic component of the phase, the rationale for using polar coordinates to represent the phase vector can be confirmed. Finally, using the aforementioned principle of "multiplying a Toeplitz matrix by a vector," the frequency domain Fourier coefficients of the time-domain signals cos(g(t)) and sin(g(t)) are obtained.

[0103] 2) Controller limiting – hard nonlinearity, continuously non-differentiable (piecewise points)

[0104] In related technologies, when using the describing function method to solve for constant-amplitude oscillations, the limiting is represented as a function (N(Mag)) of the controller output and the dominant component of the controller input, related to the input amplitude Mag. Depending on whether the coupling effect of non-dominant components is considered, the function value may be a real number (i.e., the input does not undergo phase shift after the limiting nonlinearity) or a complex number (i.e., the input undergoes phase shift after the limiting nonlinearity). Its main drawback is that it defaults to "symmetric limiting" (not considering the DC component of the input) or only considers unilateral limiting (which is still not the most generalized conclusion); in addition, research in related technologies excessively pursues the analytical form of the describing function expression, but with the consideration of more practical factors, the analytical form is often difficult to write, and corresponding methods must be developed to test and ensure the accuracy of the numerical results of the controller limiting nonlinearity.

[0105] Since the input to the limiting circuit considered in this application is a DC signal, the input-output relationship during limiting triggering can be transformed into the following problem: Assuming the time-domain expression of the limiting input is g(t), and the upper and lower limits of the limiting circuit are l... u With l p At this time, in each dominant oscillation T p The output is a piecewise function (5 segments), such as... Figure 3 As shown. Although the input, after being subjected to amplitude-limited nonlinearity, may generate higher-order components, since the calculation artificially truncates it to no more than the third order, the Fourier series of the piecewise function can be directly calculated to obtain the coefficients of the output signal (this is actually one of the calculation processes of the describing function). To simplify the calculation of the Fourier series, it is necessary to determine the dividing time of the piecewise function; considering that the amplitude of higher-order components is usually relatively smaller than that of lower-order components, and that the analytical form of the piecewise function at the turning point needs to be substituted into the integral operation required to calculate the Fourier series, at the turning point (angle), the higher-order oscillatory components of the input are ignored and only the first-order oscillatory component is considered, which is set to l. u or l p This allows us to calculate the analytical form of the turning point (equivalent to an angle). Since calculating the angle requires arcsine or arccosine operations (which introduces a constraint to subsequent iterative operations, namely that the asin function must guarantee that the input is [-1,1]), we can reaffirm the rationality of expressing the amplitude-limited input in polar coordinates.

[0106] Furthermore, since the numerical calculation of Fourier coefficients is easily implemented using MATLAB, and by changing the limiting value, the bilateral asymmetric limiting can be degenerated into a bilateral symmetric limiting or a unilateral asymmetric limiting, the describing function can be accurately derived using MATLAB numerical computation based on the existing definition of the describing function. Therefore, this application also simplifies the derivation of the describing function. Thus, this application completely avoids manually deriving the analytical form of the simple describing function and then substituting it into the numerical calculation of the function value, possessing stronger scalability and higher computational accuracy.

[0107] Furthermore, if we assume that the input signal g is limited... p If the output is g, then the above process can be described as g = L(g p ,l u ,l p This nonlinear relationship can be modified by considering l. u ,l p Provide a unified description of whether the event is triggered or not.

[0108] If we consider the modulation limiting of VSC, the signal of the input limiting stage is a signal dominated by the fundamental frequency component. We can still use the above method to treat the output signal as a piecewise function. However, the computational complexity of calculating the Fourier series of the output signal increases rapidly (about 4 times that of input / output DC signals). Therefore, we can consider making reasonable simplifications to the corresponding frequency components, such as ignoring the higher-order components of the modulation voltage.

[0109] Optionally, in some embodiments, before solving at least one nonlinear equation based on a preset Newtonian solution strategy and the highest order of the oscillating component in the vector to be solved, the method further includes: performing a parameter scan on at least one nonlinear equation to obtain initial values ​​for at least one nonlinear equation.

[0110] It should be noted that time-varying periodic systems, represented by constant-amplitude oscillations, can necessarily be solved using a sufficient number of nonlinear equations to determine the same number of unknowns. Non-constant-amplitude oscillations can be considered a generalization of this conclusion (n=0). The embodiments of this application reduce the number of equations through reasonable analysis. For example, in the case of non-limited-triggered constant-amplitude oscillations, since the limiting nonlinearity is not triggered, the corresponding equations can be removed to reduce the computational load. This ensures that each equation is a nonlinear equation and is continuously differentiable with respect to the unknowns.

[0111] Specifically, this application found through experiments that the iterative results obtained solely based on small-signal analysis results, with all harmonic components set to 0 (some variables cannot be set to 0 to meet differentiability requirements), easily converge to the iterative solution assuming small-signal stability at n=0, deviating from the target of solving constant-amplitude oscillations. Therefore, this application addresses the aforementioned deficiency by successively increasing the order of the considered target oscillation components and employing a partial parameter sweep method (after reasonable analysis) to obtain initial values ​​capable of solving nonlinear equations.

[0112] Optionally, in some embodiments, based on a preset Newton-Laurel method solution strategy and the highest order of the oscillating component in the vector to be solved, at least one nonlinear equation is solved to obtain an initial solution result. The solution further includes: determining whether the amplitude of the first-order oscillating component in the vector to be solved to the amplitude of the current-order oscillating component in the vector to be solved is not equal to a preset threshold; if the amplitude of the first-order oscillating component in the vector to be solved to the amplitude of the current-order oscillating component in the vector to be solved is not equal to the preset threshold, then determining whether the current order is less than the highest order; if the current order is less than the highest order, then incrementing the current order by 1 until the current order is greater than or equal to the highest order, thereby obtaining the initial solution result.

[0113] In this embodiment, the preset threshold for the amplitude of the current order oscillation component in the vector to be solved is 0. When the amplitude of the current order oscillation component in each vector is 0, step S102 is re-executed to perform small-signal stability analysis on the oscillation component in the vector to be solved, and to determine or supplement some initial values ​​for iteration. After obtaining reasonable initial values ​​and ensuring that the number of equations is the same as the number of unknowns, the mature Newton-Laurent method is used to solve for each harmonic component, and the remaining equations are substituted to solve for the intermediate vector.

[0114] The following embodiments will provide a detailed description of the method for formulating nonlinear equations and obtaining iterative initial values ​​for solving constant-amplitude oscillations provided in this application:

[0115] Based on the above assumptions, u sa m ia u dc ,θ, f and are vectors to be solved, and can be considered as "known quantities" when writing equations. Several "intermediate variables" are generated using the following method (u in the following description). ga_1 L g L f C d u dc_1 i lq_1 i dc_0 k p_udc k i_udc k p_cc k i_cc k p_pll k i_pll (As known):

[0116] ai la :i la =(u la -u la ). / (2πfX g (Based on Ohm's law and the fact that passive devices do not have frequency coupling effects, X) g =jL g i saU represents the injected current vector at the grid connection point of phase a—an AC vector. ga (The vector representing the infinite bus voltage – an AC vector);

[0117] bP ac :P ac =3T ila ×(u la +2πfX f (Based on Ohm's law and the basic principles of power calculation, X) f =jL f P ac (Indicates the power injected into the AC side of the VSC port);

[0118] cP dc :P dc =T udc ×i dci i dci =i dc -T udc ×(2πfX c (Based on Ohm's law and the basic principles of power calculation, i) dc The vector i represents the DC vector formed by ideal current sources on the DC side. dci X represents the current vector injected into the DC side of VSC via capacitor shunt—the DC vector. c =jC d );

[0119] d. (Based on Park transformation, Indicates the DC voltage reference value The vectors formed are DC vectors, L(cos(θ(t))) and L(sin(θ(t))) representing the frequency domain vectors of two trigonometric functions—AC vectors, u. md / u mq This represents the potential components within the d / q axes—the DC vector. Similarly, u ld / u lq and i ld / i lq It can also be derived from u la and i la List;

[0120] e. PI Controller: Based on the principle that linear controllers do not introduce frequency coupling effects and harmonic balance, and to facilitate the differentiation of variables in subsequent equation writing, Est(.) is used to represent the output result—all of which are DC vectors. Specifically, it consists of the following 4 equations:

[0121] 1. The outer loop PI control of the d-axis voltage generates the reference value for the inner loop control of the d-axis:

[0122]

[0123] 2. The d-axis current inner loop PI control generates the d-axis internal potential component:

[0124]

[0125] 3. The q-axis current inner loop PI control generates the q-axis internal potential component:

[0126]

[0127] 4. PLL (Phase-locked loops) generate the grid connection point phase:

[0128] Est(θ)=(u lq ).×(k p_pll +k i_pll . / (2πf))

[0129] It should be noted that in the above PI controller operation, 2πf is in the denominator, and one component of the generalized zero-sequence vector in f is 0. This inevitably leads to a non-zero input-output result of NAN, which physically means that the PI controller has infinite gain for the DC signal. However, this relationship cannot be used to write an equation, indirectly indicating that if the system is in steady state, the DC bias in the input PI controller signal (the component with the subscript _0, not other harmonic vectors) must be 0. Therefore, an alternative equation can be written. In addition, in the matrix-vector multiplication operation, redundant data at undesired vector positions can be cleaned up in a timely manner to ensure that the components of the AC / DC vectors are distributed according to their set method.

[0130] Furthermore, nonlinear equations can be generated or extracted using intermediate variables:

[0131] a. Internal potential balance relationship: u la +2πfX f .×i la =0.5×T mia ×u dc Since the internal potential is an AC vector, the 2×(2×n+1) equation can be extracted from the generalized positive sequence vector.

[0132] b. VSC AC / DC side power balance relationship: P ac =P dc Since power is a DC vector, the (2×n+1) equation can be extracted from the generalized zero-sequence vector.

[0133] c. Related to Park transform: Est(θ) = θ; Est(u md )=u md ;Est(u mq)=u mq Since the relevant vectors are all DC vectors, (2×n+1) equations can be extracted from the generalized zero-sequence vectors. According to the previous analysis, when n=3, two equations need to be subtracted to avoid the AC vectors from exceeding the limit.

[0134] d. With potential amplitude-limiting nonlinearity Related: Since power is a DC vector, the (2×n+1) equation can be extracted from the generalized zero-sequence vector.

[0135] Combining the aforementioned substitution equations and the Jacobi-Anger expansion variable truncation principle, the number of equations can be appropriately replaced and reduced. For example, if the infinite bus voltage phase is chosen to be 0, then based on the power balance relationship (no active power loss), it can be confirmed that the imaginary part of the grid connection point voltage satisfies... This reduces the number of power balance equations by one (although it lacks universality, it reduces the amount of iterative computation). Furthermore, θ can be fixed when writing the equations. 1+s The phase is 0 because the infinite bus voltage contains only the fundamental component. The PLL is essentially only used to synchronize the frequency of the PCC point phase with the fundamental frequency. When the constant amplitude oscillation actually occurs, the absolute value of the harmonic phase is not fixed but the relative value is fixed. That is, the system essentially has infinitely many stable limit cycles, but the amplitude and frequency of the obtained constant amplitude oscillation are determined, which is also in line with the solution objective of the method developed in this patent.

[0136] Ultimately, ensuring that all equations are nonlinear and the number of equations equals the number of unknowns, the Newton-Raphson method can be used for iterative solution. However, it is well known that the results of the Newton-Raphson method are highly sensitive to initial conditions, especially for the constant-amplitude oscillation solution involved in this application: since this application considers the oscillation caused by the presence of a negatively damped mode in the system, when solving for constant-amplitude oscillations, simply setting the amplitude of each variable to 0 will solve for the oscillations. p If we take the frequency corresponding to the negative damping mode as the small-signal stability analysis (such as the state-space method) to calculate, then we will naturally get a solution to the nonlinear equation system. However, this solution does not actually exist, nor is it the expected constant amplitude oscillation state. Therefore, we must develop a feasible initial value selection algorithm.

[0137] Optionally, in some embodiments, after determining whether a single limit has been triggered based on the solution results, the method further includes: if a single limit has been triggered, then generating at least one new nonlinear equation that satisfies the preset conditions based on the analysis results and the preset iterative strategy, until the single limit is not triggered.

[0138] It should be noted that, based on the above analysis, amplitude limiting is not a sufficient condition for constant-amplitude oscillations. Therefore, it is necessary to determine whether a single amplitude limit has been triggered based on the calculation results excluding amplitude limiting, and to consider adding equations for re-iteration to ultimately solve for the constant-amplitude oscillations accurately. It should be noted that to ensure that some nonlinear functions of the nonlinear equation (mainly the arcsine and arccosine functions at the piecewise moments of the piecewise function) are always confined within the default domain during iteration, it is necessary to check and manually adjust the results of each iteration to ensure that the iteration can always run. However, ideally, the result should be obtained in fewer than 5 iterations.

[0139] Specifically, since the constant-amplitude oscillation triggered by the amplitude limit can be regarded as a transition from the constant-amplitude oscillation without the amplitude limit, the solution of the constant-amplitude oscillation without the amplitude limit can be used as the initial value of the constant-amplitude oscillation triggered by the amplitude limit. Finally, the problem is focused on the selection of the initial value of the constant-amplitude oscillation without the amplitude limit.

[0140] The initial value selection method for the amplitude limiting non-triggered constant amplitude oscillation in this application embodiment includes the following steps:

[0141] (1) Assuming that the desired solution is to the third-order constant amplitude oscillation component, the solution should start from the first-order constant amplitude oscillation component (i.e., according to the above principle of nonlinear equation generation, the number of equations and variables to be solved are reduced accordingly).

[0142] (2) Using traditional methods for small-signal stability analysis of the system, assuming no oscillations occur, the intermediate elements of the generalized positive and negative sequence vectors and the initial oscillation frequency can be solved through another simple (harmonic-insensitive) nonlinear iteration, which can be used as partial initial values ​​for solving the first-order constant-amplitude oscillation. In addition, based on the conclusions of the small-signal stability analysis (e.g., comparing the amplitude of the frequency impedance model), the dominance of the AC / DC harmonic growth rate during the oscillation stage is predicted. A parameter scanning method is adopted, with a step size of 1% of the rated DC / AC voltage, and the real part of the first-order harmonic AC / DC oscillation amplitude is substituted. The scanning range is 0-40% of the rated value (because exceeding this range usually triggers VSC modulation oversaturation or DC voltage amplitude too low and collapses); the phase of the first-order phase oscillation of the phase-locked loop is handled in the same way (note that this variable cannot be 0); the initial values ​​of the remaining variables are all set to 0; the nonlinear equations are solved step by step according to different combinations of the two parameter scanning methods; when it is confirmed that the solution of each first-order harmonic component is not 0, the iteration is stopped, and the solution process of the first-order oscillation component is considered to be finished.

[0143] (3) Use the calculation results of the first-order oscillation iteration process as the initial value of the second-order oscillation nonlinear iteration, and expand the corresponding equations and unknowns; add the second-order oscillation phase (note that this variable cannot be 0) as the variable to be scanned, and the scanning range is 0-40% of the amplitude of the first-order oscillation component obtained in step b (changing by 1% step size), and the amplitude of the other newly added variables can be selected as 0; when it is confirmed that the solution of each first-order and second-order harmonic components is not 0, stop the iteration and consider the solution process of the second-order oscillation component to be completed.

[0144] (4) Use the calculation results of the second-order oscillatory iteration process as the initial value of the third-order oscillatory nonlinear iteration to expand the corresponding equations and unknowns. According to the previous description, the angle component usually only needs to be taken up to the second order, so there is no need to add new scanning variables, and the third-order oscillatory component should be able to be solved quickly.

[0145] Therefore, considering that there are many instances of constant-amplitude oscillations originating from negative damping in power systems based on new energy sources in related technologies, this application focuses on this specific scenario and returns to the basic law of closed-loop system operation—harmonic (energy) balance. It directly solves the amplitude and frequency of constant-amplitude oscillations in the frequency domain, which solves many shortcomings of related technologies based on the describing function method, such as excessive reliance on simulation results, unattainable assumptions, incomplete solution results, or unclear physical mechanisms. It develops a limit cycle solution theory, which has strong scalability and can be used for more fundamental oscillation analysis and control.

[0146] The following examples, in conjunction with the accompanying drawings, illustrate the implementation process of the constant amplitude oscillation algorithm provided in this application.

[0147] Specifically, Figure 4 This is a schematic diagram illustrating the implementation process of the constant-amplitude oscillation algorithm provided in the embodiments of this application, such as... Figure 4 As shown, the constant-amplitude oscillation algorithm includes the following steps:

[0148] Step S401: Divide the vector to be determined into the intermediate vector, determine the corresponding AC / DC vector type, and determine the highest order N of the oscillation component in the vector to be determined.

[0149] Step S402: Set the order n = 0 and perform small-signal stability analysis.

[0150] Step S403: Determine whether a negative damping mode exists, i.e., determine whether it is necessary to calculate constant amplitude oscillation. If the conditions for constant amplitude oscillation are met, proceed to step S404; otherwise, end the process.

[0151] Step S404: Execute n = n + 1.

[0152] Step S405: Determine or supplement the nonlinear equations.

[0153] Step S406: Determine or supplement some initial values ​​for iteration based on small signal stability analysis or the results of the previous iteration.

[0154] Step S407: Parameter scan to determine the initial values ​​for the remaining iterations (optional).

[0155] Step S408: Solve the nonlinear equations using the Newton-Lambert method.

[0156] Step S409: Determine whether the amplitude of each vector's 1-n order oscillation component is not 0. If the amplitude of the oscillation component is not 0, proceed to step S410; otherwise, proceed to step S406.

[0157] Step S410: Determine if n≥N. If n≥N is true, proceed to step S404; otherwise, proceed to step S411.

[0158] Step S411: Determine whether the amplitude limiting element has been triggered (once). If the amplitude limiting is triggered, proceed to step S405 to iterate again and finally accurately solve the constant amplitude oscillation. Otherwise, end the process.

[0159] Therefore, this application, through simulation reflecting the fundamental principle of VSC steady-state operation in a physical linear combination, clarifies that the key to constant-amplitude oscillation lies in accurately solving the constant-amplitude oscillation process under the condition of amplitude limiting without triggering. Furthermore, the algorithm exhibits strong scalability, unifying the fundamental and harmonic components by reasonably defining generalized positive / negative / zero-sequence components, fully leveraging the high efficiency of matrix computation through real-frequency domain transformation relationships, and perfecting the iterative algorithm solution process for nonlinear equations through a unique initial value selection method. Ultimately, amplitude limiting triggering / non-triggering and the asymmetry of amplitude limiting triggering positions no longer become factors restricting the solvability or accuracy of constant-amplitude oscillation. The constant-amplitude oscillation solution algorithm of this application improves the solution of typical VSC nonlinear constant-amplitude oscillation / limit cycle algorithm / harmonic power flow, enabling users to gain a more fundamental understanding of the target object of stability control, namely, that steady-state analysis can better serve stability analysis.

[0160] To enable those skilled in the art to further understand the constant amplitude oscillation solution method based on harmonic balance of the present application embodiments, the following description is provided in conjunction with the best embodiment of a specific application.

[0161] This embodiment sets the hardware environment to a computer, and the software configuration to Windows 10, Matlab language environment software, and the electromagnetic state simulation software PSCAD. This embodiment selects... Figure 2 The system shown is considered as follows: Figure 5 The four scenarios are shown.

[0162] a) During stage t1, the system first oscillates to a constant amplitude oscillation without triggering any amplitude limiting.

[0163] b) At the start of stage t2, the upper limit of the d-axis outer loop controller limit is set to 200 to simulate the case of "single-sided weak limiting".

[0164] c) Increase the grid-side inductance L at the start of stage t3. g The simulation simulates a weakening power grid, simulating a "one-sided strong limiting" scenario.

[0165] d) At the start of the t4 stage, the lower limit of the d-axis outer loop controller limit is set to 0 to simulate the case of "bilateral (asymmetric) limit".

[0166] The iteration results of the four sets of tests are shown in Tables 1-4. Table 1 compares the calculation results of Test1 with the actual test results when n=1, 2, 3 and using the proposed initial value acquisition method. Table 2 compares the calculation results of Test2 (n=3) with the actual test results. Table 3 compares the calculation results of Test3 (n=3) with the actual test results. Table 4 compares the calculation results of Test4 (n=3) with the actual test results.

[0167] Table 1

[0168]

[0169] Table 2

[0170]

[0171]

[0172] Table 3

[0173]

[0174] Table 4

[0175]

[0176] Since it's impossible to obtain precise measurements of the dominant oscillation frequency during measurement (i.e., FFT resolution is finite), the Hann window function is used to eliminate spectral leakage, but it should still be used as a reference for calculated values, indirectly reflecting the importance of theoretically calculated limit cycles. Furthermore, another important conclusion is that taking n = N typically only guarantees the accuracy of the N-1th order harmonic component calculation, due to the frequency coupling effect of infinite orders of constant-amplitude oscillations. Therefore, to reflect the main characteristics of constant-amplitude oscillations, it is recommended that n ≥ 3 in practical applications.

[0177] It should be noted that the above embodiments are only used to illustrate the technical solutions of this application and not to limit them. Although this application has been described in detail with reference to the above embodiments, those skilled in the art should understand that modifications or equivalent substitutions can still be made to the specific implementation of this application. Any modifications or equivalent substitutions that do not depart from the spirit and scope of this invention should be covered within the scope of these claims.

[0178] In other words, the application of the constant amplitude oscillation solution method based on harmonic balance provided in this application includes, but is not limited to: other application objects (such as modular multilevel converters, single-phase inverters, synchronous generators, grid-type converters, and three-phase asymmetrical systems), other oscillation categories (such as forced oscillations in power systems, considering switching frequency components), and situations where multiple limiting links are triggered.

[0179] The constant-amplitude oscillation solution method based on harmonic balance proposed in this application determines the highest order of the oscillation component in the vector to be solved. Based on a small-signal stability analysis strategy, a small-signal stability analysis is performed on the original vector without oscillation components. When the constant-amplitude oscillation initiation condition is met, a nonlinear equation is generated based on the analysis results and an iterative strategy. The nonlinear equation is then solved based on the Newton-Laurel method and the highest order of the oscillation component in the vector to be solved, yielding an initial solution result. The solution result is then used to determine whether a single amplitude limit has been triggered. If the single amplitude limit has not been triggered, the final constant-amplitude oscillation solution result is obtained based on the initial solution result. This solves the problems in related technologies where the solution methods for constant-amplitude oscillations of VSCs are theoretically insufficient and the solution results are incomplete, making them unsuitable for deeper stability analysis. This provides a small-signal stability analysis and oscillation suppression method for novel power systems, improving the accuracy and reliability of the solution process for constant-amplitude oscillations in systems containing VSCs.

[0180] Next, referring to the accompanying drawings, a constant-amplitude oscillation solution device based on harmonic balance proposed according to an embodiment of this application is described.

[0181] Figure 6 This is a block diagram of the constant amplitude oscillation solution device based on harmonic balance according to an embodiment of this application.

[0182] like Figure 6 As shown, the constant amplitude oscillation solution device 10 based on harmonic balance includes: a determination module 100, an analysis module 200, and a solution module 300.

[0183] The system comprises: a determination module 100, used to determine the highest order of the oscillating component in the vector to be determined; an analysis module 200, used to perform small-signal stability analysis on the original vector without oscillating components based on a preset small-signal stability analysis strategy, and to determine whether the preset constant-amplitude oscillation initiation condition is met based on the analysis results; and a solution module 300, used to generate at least one nonlinear equation that satisfies the preset condition based on the analysis results and a preset iterative strategy when the preset constant-amplitude oscillation initiation condition is met, and to solve at least one nonlinear equation based on a preset Newton-Laurel method solution strategy and the highest order of the oscillating component in the vector to be determined, to obtain an initial solution result, and to determine whether a single amplitude limit is triggered based on the solution result, and to obtain the final constant-amplitude oscillation solution result based on the initial solution result when the single amplitude limit is not triggered.

[0184] Optionally, in some embodiments, after determining whether a single limit has been triggered based on the solution results, the solution module 300 is further configured to: when a single limit has been triggered, regenerate at least one new nonlinear equation that satisfies the preset conditions based on the analysis results and the preset iterative strategy, until the single limit has not been triggered.

[0185] Optionally, in some embodiments, after determining whether the preset constant amplitude oscillation start-up condition is met based on the analysis results, the analysis module 200 is further configured to: not perform constant amplitude oscillation solution when the preset constant amplitude oscillation start-up condition is not met.

[0186] Optionally, in some embodiments, before solving at least one nonlinear equation based on a preset Newtonian solution strategy and the highest order of the oscillating component in the vector to be solved, the solution module 300 is further configured to: perform a parameter scan on at least one nonlinear equation to obtain initial values ​​for at least one nonlinear equation.

[0187] Optionally, in some embodiments, the solving module 300 is further configured to: determine whether the amplitude of the first-order oscillation component in the vector to be solved to the amplitude of the current-order oscillation component in the vector to be solved are all not equal to a preset threshold; when the amplitude of the first-order oscillation component in the vector to be solved to the amplitude of the current-order oscillation component in the vector to be solved are all not equal to the preset threshold, determine whether the current order is less than the highest order; when the current order is less than the highest order, increment the current order by 1 until the current order is greater than or equal to the highest order, and obtain the initial solution result.

[0188] It should be noted that the foregoing explanation of the embodiment of the constant amplitude oscillation solution method based on harmonic balance also applies to the constant amplitude oscillation solution device based on harmonic balance in this embodiment, and will not be repeated here.

[0189] The constant-amplitude oscillation solution device based on harmonic balance proposed in this application determines the highest order of the oscillation component in the vector to be solved. Based on a small-signal stability analysis strategy, it performs small-signal stability analysis on the original vector without oscillation components. When the constant-amplitude oscillation initiation condition is met, a nonlinear equation is generated based on the analysis results and an iterative strategy. The nonlinear equation is then solved based on the Newton-Laurel method and the highest order of the oscillation component in the vector to be solved, yielding an initial solution result. The solution result is then used to determine whether a single amplitude limit is triggered. If the single amplitude limit is not triggered, the final constant-amplitude oscillation solution result is obtained based on the initial solution result. This solves the problems in related technologies where the solution methods for constant-amplitude oscillations of VSCs are theoretically insufficient and the solution results are incomplete, making them unsuitable for deeper stability analysis. It provides a small-signal stability analysis and oscillation suppression method for novel power systems, improving the accuracy and reliability of the solution process for constant-amplitude oscillations in systems containing VSCs.

[0190] Figure 7 A schematic diagram of the structure of an electronic device provided in an embodiment of this application. The electronic device may include:

[0191] The memory 701, the processor 702, and the computer program stored on the memory 701 and executable on the processor 702.

[0192] When the processor 702 executes the program, it implements the constant amplitude oscillation solution method based on harmonic balance provided in the above embodiments.

[0193] Furthermore, electronic devices also include:

[0194] Communication interface 703 is used for communication between memory 701 and processor 702.

[0195] The memory 701 is used to store computer programs that can run on the processor 702.

[0196] The memory 701 may include high-speed RAM (Random Access Memory) memory, and may also include non-volatile memory, such as at least one disk storage.

[0197] If the memory 701, processor 702, and communication interface 703 are implemented independently, then the communication interface 703, memory 701, and processor 702 can be interconnected via a bus to complete communication between them. The bus can be an ISA (Industry Standard Architecture) bus, a PCI (Peripheral Component Interconnect) bus, or an EISA (Extended Industry Standard Architecture) bus, etc. The bus can be divided into address bus, data bus, control bus, etc. For ease of representation, Figure 7 The bus is represented by a single thick line, but this does not mean that there is only one bus or one type of bus.

[0198] Optionally, in a specific implementation, if the memory 701, processor 702, and communication interface 703 are integrated on a single chip, then the memory 701, processor 702, and communication interface 703 can communicate with each other through an internal interface.

[0199] The processor 702 may be a CPU (Central Processing Unit), an ASIC (Application Specific Integrated Circuit), or one or more integrated circuits configured to implement the embodiments of this application.

[0200] This application also provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the above-described method for solving constant-amplitude oscillations based on harmonic balance.

[0201] In the description of this specification, the references to terms such as "one embodiment," "some embodiments," "example," "specific example," or "some examples," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of this application. In this specification, the illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples. Moreover, without contradiction, those skilled in the art can combine and integrate the different embodiments or examples described in this specification, as well as the features of different embodiments or examples.

[0202] Furthermore, the terms "first" and "second" are used for descriptive purposes only and should not be construed as indicating or implying relative importance or implicitly specifying the number of technical features indicated. Thus, a feature defined as "first" or "second" may explicitly or implicitly include at least one of that feature. In the description of this application, "N" means at least two, such as two, three, etc., unless otherwise explicitly specified.

[0203] Any process or method described in the flowchart or otherwise herein can be understood as representing a module, segment, or portion of code comprising one or N executable instructions for implementing custom logic functions or processes, and the scope of the preferred embodiments of this application includes additional implementations in which functions may be performed not in the order shown or discussed, including substantially simultaneously or in reverse order depending on the functions involved, as should be understood by those skilled in the art to which embodiments of this application pertain.

[0204] It should be understood that the various parts of this application can be implemented using hardware, software, firmware, or a combination thereof. In the above embodiments, the N steps or methods can be implemented using software or firmware stored in memory and executed by a suitable instruction execution system. For example, if implemented in hardware, as in another embodiment, it can be implemented using any one or a combination of the following techniques known in the art: discrete logic circuits having logic gates for implementing logical functions on data signals, application-specific integrated circuits (ASICs) having suitable combinational logic gates, programmable gate arrays (FPGAs), field-programmable gate arrays (FPGAs), etc.

[0205] Those skilled in the art will understand that all or part of the steps of the methods in the above embodiments can be implemented by a program instructing related hardware. The program can be stored in a computer-readable storage medium, and when executed, the program includes one or a combination of the steps of the method embodiments.

[0206] Although embodiments of this application have been shown and described above, it is understood that the above embodiments are exemplary and should not be construed as limiting this application. Those skilled in the art can make changes, modifications, substitutions and variations to the above embodiments within the scope of this application.

Claims

1. A method for solving constant-amplitude oscillations based on harmonic balance, characterized in that, Includes the following steps: Determine the highest order of the oscillatory component in the vector to be determined; Based on the preset small-signal stability analysis strategy, the original vector without oscillating components is subjected to small-signal stability analysis, and the results are used to determine whether the preset constant-amplitude oscillation initiation condition is met. as well as If the preset constant amplitude oscillation initiation condition is met, at least one nonlinear equation that satisfies the preset condition is generated based on the analysis results and the preset iterative strategy. Based on the preset Newton-Laurel method solution strategy and the highest order of the oscillation component in the vector to be solved, the at least one nonlinear equation is solved to obtain the initial solution result. Based on the solution result, it is determined whether the single amplitude limit is triggered. If the single amplitude limit is not triggered, the final constant amplitude oscillation solution result is obtained based on the initial solution result. After determining whether the single threshold has been triggered based on the solution result, the process further includes: If the single limit is triggered, at least one new nonlinear equation that satisfies the preset conditions is generated again based on the analysis results and the preset iterative strategy until the single limit is not triggered. The preset constant amplitude oscillation initiation condition is that the analysis result has a negative damping mode; the preset condition includes: the number of generated nonlinear equations is equal to the number of unknowns in the generated nonlinear equations, and each generated nonlinear equation is continuously differentiable with respect to the unknowns.

2. The method according to claim 1, characterized in that, After determining whether the preset constant-amplitude oscillation initiation condition is met based on the analysis results, the method further includes: If the preset constant amplitude oscillation initiation condition is not met, then constant amplitude oscillation solution will not be performed.

3. The method according to claim 1, characterized in that, Before solving the at least one nonlinear equation based on the preset Newton-Laurel method solution strategy and the highest order of the oscillating component in the vector to be solved, the method further includes: Perform a parameter scan on the at least one nonlinear equation to obtain the initial values ​​of the at least one nonlinear equation.

4. The method according to claim 1 or 3, characterized in that, The method of solving at least one nonlinear equation based on the preset Newton-Lager method solution strategy and the highest order of the oscillating component in the vector to be solved, to obtain the initial solution result, further includes: Determine whether the amplitude of the first-order oscillation component in the vector to be determined to the amplitude of the current-order oscillation component in the vector to be determined are all not equal to a preset threshold. If the amplitude of the first-order oscillation component in the vector to be determined is not equal to the preset threshold, then it is determined whether the current order is less than the highest order. If the current order is less than the highest order, the current order is incremented by 1 until the current order is greater than or equal to the highest order, thus obtaining the initial solution result.

5. A constant-amplitude oscillation solution device based on harmonic balance, characterized in that, For implementing the method as described in any one of claims 1-4, comprising: The determination module is used to determine the highest order of the oscillatory components in the vector to be determined. The analysis module is used to perform small-signal stability analysis on the original vector without oscillating components based on a preset small-signal stability analysis strategy, and to determine whether the preset constant-amplitude oscillation initiation condition is met based on the analysis results; and The solution module is used to generate at least one nonlinear equation that satisfies the preset conditions based on the analysis results and the preset iterative strategy when the preset constant amplitude oscillation initiation conditions are met. Based on the preset Newton-Laurel method solution strategy and the highest order of the oscillation component in the vector to be solved, the module solves the at least one nonlinear equation to obtain the initial solution result. Based on the solution result, the module determines whether the single amplitude limit is triggered. If the single amplitude limit is not triggered, the module obtains the final constant amplitude oscillation solution result based on the initial solution result. The preset constant amplitude oscillation initiation condition is that the analysis result has a negative damping mode; the preset condition includes: the number of generated nonlinear equations is equal to the number of unknowns in the generated nonlinear equations, and each generated nonlinear equation is continuously differentiable with respect to the unknowns; After determining whether the single amplitude limit has been triggered based on the solution result, the solution module is further configured to: When the single limit is triggered, at least one new nonlinear equation that satisfies the preset conditions is generated again based on the analysis results and the preset iterative strategy, until the single limit is not triggered.

6. The apparatus according to claim 5, characterized in that, After determining whether the preset constant-amplitude oscillation initiation condition is met based on the analysis results, the analysis module is further configured to: If the preset constant amplitude oscillation initiation condition is not met, constant amplitude oscillation solution will not be performed.

7. An electronic device, characterized in that, include: A memory, a processor, and a computer program stored in the memory and executable on the processor, the processor executing the program to implement the constant-amplitude oscillation solution method based on harmonic balance as described in any one of claims 1-4.

8. A computer-readable storage medium having a computer program stored thereon, characterized in that, The program is executed by the processor to implement the constant-amplitude oscillation solution method based on harmonic balance as described in any one of claims 1-4.

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