Small interference stability analysis method suitable for flexible direct current asymmetric working condition

Abstract

The invention relates to the technical field of power systems and automation thereof, in particular to a small-interference stability analysis method suitable for a flexible direct current asymmetric working condition, and the method comprises the following steps: S1, building an MMC-HVDC nonlinear periodic time-varying state space model; s2, solving a steady-state periodic trajectory of the MMC-HVDC system and calculating a Jacobian matrix time sequence; and S3, solving the state transition matrix based on the Floquet theorem and carrying out eigenvalue decomposition. The technical problems of high model order and large calculation amount in MMC-HVDC system small interference stability analysis under the asymmetric working condition are solved.

Description

Technical Field

[0001] This invention relates to the field of power systems and their automation technology, specifically to a small-disturbance stability analysis method applicable to flexible DC asymmetric operating conditions. Background Technology

[0002] Modular multilevel converters (MMCs), as an advanced voltage source converter, possess significant advantages such as high output waveform quality, low switching losses, strong fault handling capabilities, and high scalability, making them a core component of high voltage direct current (HVDC) transmission systems. With the increasing demand for renewable energy grid integration and the intelligent upgrading of power systems, MMC-HVDC technology is widely used in inter-regional grid interconnection and centralized renewable energy transmission. Its voltage levels and capacities are continuously increasing, placing higher demands on system stability control.

[0003] However, the MMC-HVDC system faces severe stability challenges in actual operation. Due to its high degree of power electronics, complex control circuitry, and increased integration of renewable energy sources, the system exhibits complex dynamic characteristics such as wideband resonance and multimodal coupling. High-frequency resonance incidents have been observed multiple times in engineering practice: the State Grid Chongqing-Hubei DC project experienced harmonic resonances of 700Hz and 1800Hz; the world's first flexible DC grid, the Zhangbei project, experienced high-frequency oscillations of around 1500Hz during AC-side charging at the Kangbao station; and the Southern Power Grid's Luxi back-to-back flexible DC project experienced 1200Hz resonance when connected to a weak AC system. These phenomena not only affect power quality but, in severe cases, can lead to converter lockout and system shutdown, posing a significant threat to the safe and stable operation of the power grid.

[0004] For stability analysis of MMC-HVDC systems, existing technologies have developed relatively mature solutions under symmetrical operating conditions. A typical method is based on the harmonic balance principle, which transforms a system with periodic time-varying solutions into an equivalent linear time-invariant system, and then uses eigenvalue analysis to achieve small-disturbance stability assessment. This method effectively simplifies the analysis process under symmetrical power grid conditions. However, in actual engineering, non-ideal operating conditions such as three-phase power grid voltage asymmetry, line impedance parameter imbalance, and differences in bridge arm inductance are common. Under asymmetrical operating conditions, the MMC converter generates abundant harmonic components, and strong coupling effects exist between harmonics of different frequencies. In this case, the traditional method based on the harmonic balance principle needs to consider multiple harmonic frequency components, causing the system model order to increase by a factor of 2n+1 with the number of harmonics n, leading to the "curse of dimensionality." This modeling approach is not only cumbersome and lacks clear physical meaning, but also significantly increases computational complexity and hardware costs, making it difficult to meet the needs of rapid analysis and real-time control in engineering applications.

[0005] Therefore, developing an efficient small-disturbance stability analysis method suitable for asymmetric operating conditions, and avoiding the problem of model order expansion in traditional techniques, is of great theoretical significance and engineering value for improving the safe and stable operation of MMC-HVDC systems. Summary of the Invention

[0006] To address the aforementioned issues, a small-disturbance stability analysis method suitable for flexible DC asymmetric operating conditions is provided, solving the technical problems of high model order and large computational load in the small-disturbance stability analysis of MMC-HVDC systems under asymmetric operating conditions.

[0007] To address the problems of existing technologies, this invention provides a small-disturbance stability analysis method suitable for flexible DC asymmetric operating conditions, comprising the following steps: Step S1: Establish the MMC-HVDC nonlinear periodic time-varying state-space model; A nonlinear periodic time-varying state-space model for a modular multilevel converter-HVDC system suitable for asymmetric operating conditions is constructed to clarify the relationship between system state variables and input variables. Step S2: Solve for the steady-state periodic trajectory of the MMC-HVDC system and calculate the Jacobian matrix time series; The Newton-Krylov algorithm is used to solve the steady-state periodic trajectory of the system, and the periodic time series of the Jacobian matrix is ​​obtained at this trajectory; Step S3: Solve the state transition matrix based on Floquet's theorem and perform eigenvalue decomposition; Using the fundamental solution set as initial conditions, the state transition matrix is ​​solved by numerical integration, and Floquet multipliers are obtained by eigenvalue decomposition to determine the system stability.

[0008] As a specific embodiment of the present invention, step S1 includes: Step S1.1: Define the MMC main loop topology; Define the main components and parameters of the system based on the MMC power circuit topology; Step S1.2: Derive the mathematical model equations of the MMC-HVDC system; Based on Kirchhoff's laws and the working principle of the converter, the nonlinear periodic time-varying state-space equation of the MMC-HVDC system is established: (1); Equation (1) above can be expressed as: (2);

[0009] In the formula: Represents a vector consisting of system state variables. This represents a vector consisting of system inputs. Represents a vector field.

[0010] As a specific embodiment of the present invention, step S2 includes: Step S2.1: Apply the Newton-Krylov algorithm to solve for the steady-state periodic trajectory; Step S2.2: Calculate the periodic time series of the Jacobian matrix.

[0011] As a specific embodiment of the present invention, step S2.1 includes: obtaining the steady-state periodic trajectory of equation (1) using the Newton-Krylov algorithm. .

[0012] As a specific embodiment of the present invention, step S2.2 includes: linearizing at the steady-state periodic trajectory to obtain the linear periodic time-varying system equation characterizing the stability of the system under small disturbances: (3); In the formula: This represents a vector composed of system state variables, since It has periodic and time-varying properties, therefore It is a periodic variable coefficient matrix, and for the MMC system, its period T = 0.02s.

[0013] As a specific embodiment of the present invention, step S3 includes: Step S3.1: Establish the fundamental solution matrix differential equation; Step S3.2: Solve for the state transition matrix using numerical integration; Step S3.3: Eigenvalue decomposition and stability criterion.

[0014] As a specific embodiment of the present invention, in step S3.1, according to Floquet's theorem, the fundamental solution matrix of equation (2) is obtained: (4); In the formula: The fundamental solution matrix of the system, For the period is The function, It is a non-singular constant matrix. It is an identity matrix.

[0015] As a specific embodiment of the present invention, in step S3.2, the above-mentioned system of differential equations is processed over one period. Numerical integration is performed within the interval to obtain... State transition matrix at time step: (5).

[0016] As a specific embodiment of the present invention, in step S3.3, the equation (5) is... The following system of differential equations is solved by numerical integration: (6); definition One of the characteristic values ​​is , For matrix eigenvalues, where are called eigenvalues. For Floquet multipliers, For Floquet index; for Eigenvalue decomposition yields n values. and Where n is the system order; when any one >0 or When the value is greater than 1, the system is unstable; when any one of them is greater than 1, the system is unstable. ≤0 or When ≤1, the system is stable or critically stable; therefore, the system stability criterion can be obtained as: (7); The stability of the LTP system was analyzed by calculating the Floquet multipliers.

[0017] A data processing device, comprising: Memory, used to store computer programs; A processor is used to implement a small-disturbance stability analysis method suitable for flexible DC asymmetric operating conditions when executing the computer program.

[0018] A computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps of a small-disturbance stability analysis method applicable to flexible DC asymmetric operating conditions.

[0019] The advantages of this invention compared to the prior art are: 1. Significantly reduces model complexity and avoids the "curse of dimensionality". This invention directly processes periodic time-varying systems based on Floquet's theorem, eliminating the need to convert them into time-invariant systems through the harmonic balance principle. This fundamentally avoids the problem in traditional methods where the model order increases by a factor of 2n+1 with the number of harmonics n. By establishing the original periodic time-varying state-space model, while preserving the physical essence of the system, the model dimension is controlled at a level comparable to the original system, effectively solving the technical bottleneck of cumbersome modeling steps and ambiguous physical meaning under asymmetric conditions.

[0020] 2. Improve the efficiency of stability analysis calculations and reduce engineering application costs. The Newton-Krylov algorithm is used to solve the steady-state periodic trajectory, and the state transition matrix is ​​calculated by combining it with the numerical integration method, thus avoiding the huge computational cost of eigenvalue decomposition of high-dimensional matrices.

[0021] 3. Enhance the universality of the method to cover a variety of asymmetric scenarios. This invention eliminates the need to adjust the modeling method for different asymmetry types (voltage asymmetry, impedance asymmetry, parameter asymmetry, etc.), and can be directly applied to various non-ideal operating conditions. By analyzing system stability using Floquet multipliers, it can capture the dynamic characteristics of periodic time-varying systems, overcoming the shortcomings of traditional methods in terms of insufficient analysis accuracy under strongly coupled harmonic environments, and providing a unified solution for stability assessment under complex operating conditions.

[0022] 4. Provides a precise theoretical basis for resonance suppression strategies. This invention utilizes Floquet multipliers obtained through eigenvalue decomposition to not only determine system stability but also quantitatively analyze the oscillation frequencies and damping characteristics of each mode. Combined with sensitivity analysis of the state transition matrix, key influencing parameters of the system can be located, providing quantitative guidance for developing targeted resonance suppression strategies (such as control parameter optimization and additional damping control design), thereby improving the system's fault ride-through capability and operational reliability. Attached Figure Description

[0023] Figure 1 This is a flowchart of a small-disturbance stability analysis method applicable to flexible DC asymmetric operating conditions according to the present invention.

[0024] Figure 2 This is an MMC power circuit diagram.

[0025] Figure 3 This invention relates to a small-disturbance stability analysis method for flexible DC asymmetric operating conditions, which is based on Floquet's theorem and includes an MMC-HVDC stability analysis process.

[0026] Figure 4 This is a block diagram of a typical control link in a flexible DC power transmission system for an asynchronous interconnected network project.

[0027] Figure 5 This is a time-domain waveform diagram of the active power of a flexible DC power system in an asynchronous interconnected network project.

[0028] Figure 6 This is the active power waveform spectrum diagram of a flexible DC power system in an asynchronous interconnection project.

[0029] Figure 7 It is the characteristic trajectory of the inner loop proportional coefficient of the current in a flexible DC engineering project of an asynchronous interconnection project. Detailed Implementation

[0030] To further understand the features, technical means, and specific objectives and functions achieved by the present invention, the present invention will be described in further detail below with reference to the accompanying drawings and specific embodiments.

[0031] Reference Figure 1 As shown, a small-disturbance stability analysis method suitable for flexible DC asymmetric operating conditions includes the following steps: Step S1: Establish a nonlinear periodic time-varying state-space model for MMC-HVDC; construct a nonlinear periodic time-varying state-space model for a modular multilevel converter-HVDC system suitable for asymmetric operating conditions, and clarify the relationship between system state variables and input variables; Step S2: Solve for the steady-state periodic trajectory of the MMC-HVDC system and calculate the Jacobian matrix time series; use the Newton-Krylov algorithm to solve for the steady-state periodic trajectory of the system, and obtain the periodic time series of the Jacobian matrix at this trajectory; Step S3: Solve the state transition matrix based on Floquet's theorem and perform eigenvalue decomposition; using the fundamental solution set as initial conditions, solve the state transition matrix through numerical integration, perform eigenvalue decomposition to obtain Floquet multipliers, and determine the system stability.

[0032] As a specific embodiment of the present invention, step S1 includes: Step S1.1: Define the MMC main loop topology; like Figure 2 The MMC power circuit shown in the figure: and These represent the AC side voltage and current, respectively. and These represent the currents of the upper and lower bridge arms, respectively. and These represent the voltages of the upper and lower bridge arms, respectively. and These represent the sum of the capacitor voltages of the upper and lower bridge arms, respectively. and They represent the first Switching functions for each submodule; x and They represent the first The capacitor voltage of each submodule; and These represent the DC-side current and voltage, respectively. and These represent the modulation ratios of the upper and lower bridge arm submodules, respectively; above, .

[0033] Define the main components and parameters of the system based on the MMC power circuit topology; Step S1.2: Derive the mathematical model equations of the MMC-HVDC system; Based on Kirchhoff's laws and the working principle of the converter, the nonlinear periodic time-varying state-space equation of the MMC-HVDC system is established: (1); In the formula: and These are the bridge arm resistor and the bridge arm inductance, respectively. , It is the leakage inductance of the converter transformer; , It is a submodule capacitor. This is the number of bridge arm submodules; , , and These represent the sum of the bridge arm current, voltage, capacitor voltage, and the three-phase common-mode value of the modulation ratio, respectively. , , and These represent the sum of the bridge arm current, voltage, capacitor voltage, and the three-phase differential mode value of the modulation ratio, respectively.

[0034] Equation (1) above can be expressed as: (2); In the formula: Represents a vector consisting of system state variables. This represents a vector consisting of system inputs. Represents a vector field.

[0035] As a specific embodiment of the present invention, step S2 includes: Step S2.1: Apply the Newton-Krylov algorithm to solve for the steady-state periodic trajectory; Step S2.2: Calculate the periodic time series of the Jacobian matrix.

[0036] As a specific embodiment of the present invention, step S2.1 includes: obtaining the steady-state periodic trajectory of equation (1) using the Newton-Krylov algorithm. .

[0037] As a specific embodiment of the present invention, step S2.2 includes: linearizing at the steady-state periodic trajectory to obtain the linear-time-period (LTP) system equation characterizing the stability of the system under small disturbances: (3); In the formula: This represents a vector composed of system state variables, since It has periodic and time-varying properties, therefore It is a periodic variable coefficient matrix, and for the MMC system, its period T = 0.02s.

[0038] As a specific embodiment of the present invention, step S3 includes: Step S3.1: Establish the fundamental solution matrix differential equation; Step S3.2: Solve for the state transition matrix using numerical integration; Step S3.3: Eigenvalue decomposition and stability criterion.

[0039] As a specific embodiment of the present invention, in step S3.1, according to Floquet's theorem, the fundamental solution matrix of equation (2) is obtained: (4); In the formula: The fundamental solution matrix of the system, For the period is The function, It is a non-singular constant matrix. It is an identity matrix.

[0040] As a specific embodiment of the present invention, since The system exhibits periodic and time-varying properties, and the stability of the solution depends on the exponential part of equation (4), i.e., the matrix. The eigenvalues. Therefore, in step S3.2, the above system of differential equations is processed over one period. Numerical integration is performed within the interval to obtain... State transition matrix at time step: (5).

[0041] As a specific embodiment of the present invention, the system matrix is ​​actually solved directly. It is difficult. Therefore, in step S3.3, for equation (5) The following system of differential equations is solved by numerical integration: (6); definition One of the characteristic values ​​is , For matrix eigenvalues, where are called eigenvalues. For Floquet multipliers, For Floquet index; for Eigenvalue decomposition yields n values. and Where n is the system order; when any one >0 or When the value is greater than 1, the system is unstable; when any one of them is greater than 1, the system is unstable. ≤0 or When ≤1, the system is stable or critically stable; therefore, the system stability criterion can be obtained as: (7); The stability of the LTP system is analyzed by calculating the Floquet multipliers. The specific process is as follows: Figure 3 As shown.

[0042] A data processing device, comprising: Memory, used to store computer programs; A processor is used to implement a small-disturbance stability analysis method suitable for flexible DC asymmetric operating conditions when executing the computer program.

[0043] A computer-readable storage medium storing a computer program, which, when executed by a processor, implements the steps of a small-disturbance stability analysis method applicable to flexible DC asymmetric operating conditions. Specific Implementation

[0044] A model from an asynchronous interconnected power grid project is used for verification. The typical control elements of a flexible DC power grid project are as follows: Figure 4 As shown in the figure, , These are the PI transfer functions for the current loop and the power loop, respectively. These are the decoupling control coefficients for the current loop; This is the transfer function of the voltage feedforward stage, which is 1 when no filter is added to the voltage feedforward stage. The transfer function for the positive / negative order separation stage; This is the output value of the phase-locked loop. This is the transfer function for the PI element of the phase-locked loop.

[0045] In the PSCAD / EMTDC simulation model, the AC side is set to generate a time of 2 seconds. During a phase short-circuit fault, due to the three-phase asymmetry of the system, a 100Hz harmonic generated by the negative sequence component appears in the active power waveform. Increasing the inner current loop proportional coefficient from 1 to 1.48 at 2.5s shows that harmonic oscillations occur in the system at this time, and the active power waveform is as follows: Figure 5 As shown.

[0046] right Figure 5 The waveform was subjected to Fourier analysis, and the analysis results are as follows: Figure 6 As shown, the oscillation frequency of the system at this time is 162Hz.

[0047] Plot the characteristic trajectory of the proportional coefficient of the inner current loop, such as... Figure 7 As shown, it can be seen that as the parameters increase, a pair of characteristic exponents of the system gradually approach and cross the unit circle, that is, the system becomes unstable.

[0048] Depend on Figure 7 The predictable oscillation frequency of the system is (16.2 + 50n) Hz, where n can take any integer value. Figure 6 The simulation results shown are consistent.

[0049] This embodiment analyzes the critical parameters for system stability by plotting the characteristic trajectory of the system under the change of the proportional coefficient of the inner current loop, and compares and verifies the results with the results of electromagnetic transient simulation.

[0050] The above embodiments only illustrate one or more implementations of the present invention, and their descriptions are relatively specific and detailed, but they should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of the present invention should be determined by the appended claims.

Claims

1. A method for analyzing the stability of small disturbances under flexible DC asymmetric operating conditions, characterized in that, Includes the following steps: Step S1: Establish the MMC-HVDC nonlinear periodic time-varying state-space model; A nonlinear periodic time-varying state-space model for a modular multilevel converter-HVDC system suitable for asymmetric operating conditions is constructed to clarify the relationship between system state variables and input variables. Step S2: Solve for the steady-state periodic trajectory of the MMC-HVDC system and calculate the Jacobian matrix time series; The Newton-Krylov algorithm is used to solve the steady-state periodic trajectory of the system, and the periodic time series of the Jacobian matrix is ​​obtained at this trajectory; Step S3: Solve the state transition matrix based on Floquet's theorem and perform eigenvalue decomposition; Using the fundamental solution set as initial conditions, the state transition matrix is ​​solved by numerical integration, and Floquet multipliers are obtained by eigenvalue decomposition to determine the system stability.

2. The method for small-disturbance stability analysis applicable to flexible DC asymmetric operating conditions according to claim 1, characterized in that, Step S1 includes: Step S1.1: Define the MMC main loop topology; Define the main components and parameters of the system based on the MMC power circuit topology; Step S1.2: Derive the mathematical model equations of the MMC-HVDC system; Based on Kirchhoff's laws and the working principle of the converter, the nonlinear periodic time-varying state-space equation of the MMC-HVDC system is established: （1) Equation (1) above can be expressed as: （2）; In the formula: Represents a vector consisting of system state variables. This represents a vector consisting of system inputs. Represents a vector field.

3. The method for small-disturbance stability analysis applicable to flexible DC asymmetric operating conditions according to claim 1, characterized in that, Step S2 includes: Step S2.1: Apply the Newton-Krylov algorithm to solve for the steady-state periodic trajectory; Step S2.2: Calculate the periodic time series of the Jacobian matrix.

4. The method for small-disturbance stability analysis applicable to flexible DC asymmetric operating conditions according to claim 3, characterized in that, Step S2.1 includes: obtaining the steady-state periodic trajectory of equation (1) using the Newton-Krylov algorithm. .

5. The method for small-disturbance stability analysis applicable to flexible DC asymmetric operating conditions according to claim 4, characterized in that, Step S2.2 includes: linearizing at the steady-state periodic trajectory to obtain the linear periodic time-varying system equation characterizing the stability of the system under small disturbances. （3）； In the formula: This represents a vector composed of system state variables, since It has periodic and time-varying properties, therefore It is a periodic variable coefficient matrix, and for the MMC system, its period is T = 0.02s.

6. The method for small-disturbance stability analysis applicable to flexible DC asymmetric operating conditions according to claim 1, characterized in that, Step S3 includes: Step S3.1: Establish the fundamental solution matrix differential equation; Step S3.2: Solve for the state transition matrix using numerical integration; Step S3.3: Eigenvalue decomposition and stability criterion.

7. The method for small-disturbance stability analysis applicable to flexible DC asymmetric operating conditions according to claim 6, characterized in that, In step S3.1, according to Floquet's theorem, the fundamental solution matrix of equation (2) is obtained: （4）; In the formula: The fundamental solution matrix of the system, For the period is The function, It is a non-singular constant matrix. It is an identity matrix.

8. The method for small-disturbance stability analysis applicable to flexible DC asymmetric operating conditions according to claim 7, characterized in that, In step S3.2, the above system of differential equations is processed over one period. Numerical integration is performed within the interval to obtain... State transition matrix at time step: （5）； In step S3.3, for equation (5) The following system of differential equations is solved by numerical integration: （6）; definition One of the characteristic values ​​is , For matrix eigenvalues, where are called eigenvalues. For Floquet multipliers, For Floquet index; for Eigenvalue decomposition yields n values. and Where n is the system order; when any one >0 or When the value is greater than 1, the system is unstable; when any one of them is greater than 1, the system is unstable. ≤0 or When ≤1, the system is stable or critically stable; therefore, the system stability criterion can be obtained as: （7）; The stability of the LTP system was analyzed by calculating the Floquet multipliers.

9. A data processing device, characterized in that, include: Memory, used to store computer programs; A processor, configured to execute the computer program to implement the steps of a small-disturbance stability analysis method for flexible DC asymmetric operating conditions as described in any one of claims 1 to 8.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the steps of a small-disturbance stability analysis method for flexible DC asymmetric operating conditions as described in any one of claims 1 to 8.

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