Method for transient stability boundary description of multi-inverter hybrid system based on reverse integration

Abstract

The application provides a multi-inverter hybrid system transient stability boundary delineation method based on reverse integration, and belongs to the field of electrical engineering. The delineation method combines a circuit model of a grid-constructed inverter, a grid-following inverter and an infinite grid hybrid system and a virtual synchronous machine mathematical model of the grid-constructed inverter, derives a second-order differential-algebraic equation set of the hybrid system, further obtains system steady-state equilibrium points by solving the equation set and classifies, and reversely integrates from the type I unstable equilibrium point to obtain a transient stability boundary of the hybrid system. The application is simple and intuitive, has a solid theoretical basis, can quickly and accurately identify the transient stability boundary of the grid-connected inverter system, and has important significance for analyzing the transient stability of the multi-grid-connected inverter hybrid system.

Description

A method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration. Technical Field

[0001] This invention relates to a method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration, belonging to the field of electrical engineering. Background Technology

[0002] In multi-inverter hybrid systems, especially in systems with multiple grid-connected and grid-linked inverters, finding an effective, fast, simple, and accurate method to identify the transient synchronization stability boundary of the system is a highly valuable problem and is of great significance for further guiding the design of the system's steady-state operating point.

[0003] The article titled "A Review of Transient Synchronous Stability Research in New Energy Power Systems" (Geng Hua, He Changjun, Liu Yushuang, et al. A Review of Transient Synchronous Stability Research in New Energy Power Systems [J]. High Voltage Engineering, 2022, 48(09): 3367-3383.) delves into the analytical methods for transient synchronous stability of single-unit and multi-unit hybrid systems of new energy equipment. It summarizes the advantages and disadvantages of the equal area criterion, phase plane method, Lyapunov direct method, and impedance method. It also points out that existing research has conducted rich studies on transient synchronous instability mechanisms, analytical methods, and stabilization methods, but has not formed a systematic research method similar to that of traditional power grids. In particular, research on transient synchronous stability at the level of multi-unit and hybrid systems is still insufficient.

[0004] Titled "Transient Stability Analysis and Enhancement of an IslandedMicrogrid with Grid-forming Converters and Grid-following Converters" (Ding Y, Gao F, Hou K, et al. The paper, "Transient Stability Analysis and Enhancement of Islanded Microgrids with Grid-Connected and Grid-Connected Converters" (Ding Yawen, Gao Fei, Hou Kai, et al. Transient Stability Analysis and Enhancement of Islanded Microgrids with Grid-Connected and Grid-Connected Converters [C] / / 2024 4th Power System and Green Energy Conference. Shanghai, China: Institute of Electrical and Electronics Engineers, 2024, 1362-1367), considers the dynamics of the phase-locked loop and current loop of the grid-connected converter. It establishes the state-space equations of an 8th-order nonlinear system for multi-machine hybrid systems of grid-connected and grid-connected converters and estimates the system's attraction domain using the Takagi-Sugeno method (TS method). However, the TS method requires solving linear matrix inequalities. For higher-order models containing multiple nonlinear terms, the computational cost increases significantly as the microgrid scale expands. Furthermore, the TS method, based on the idea of ​​local linearization, may not be able to fully capture the global nonlinear behavior of the system when faced with large disturbances far from the equilibrium point, thus leading to misjudgment of stability.

[0005] As the above analysis shows, the stability of power systems with high proportions of power electronic grid-connected interfaces has received widespread attention and research, especially the transient stability of hybrid power electronic equipment systems with different control methods. Some studies use the derivation of complete state-space models or reduced-order models of the system to analyze stability. These methods can comprehensively and relatively accurately show the stable state of the system, but the derivation process is relatively complex and requires high real-time computing power. If a simple and effective transient stability analysis method could be proposed, requiring only the solution of the system's steady-state operating point and a single inverse integration to obtain the system's transient stability boundary, and thus guiding the action time of relay protection equipment, this would not only enrich and improve the transient stability theory of grid-connected inverters but also have significant implications for practical engineering applications.

[0006] In summary, existing research on transient stability analysis methods for multi-machine hybrid systems still has the following problems:

[0007] 1. Existing research mostly focuses on the analysis of instability mechanisms at the equipment level. The integration of a large number of new energy equipment will bring about system-level stability problems. There is very little knowledge about the stability analysis and instability mechanisms at the system level, and the interaction between multiple pieces of equipment and between equipment and the system needs to be clarified.

[0008] 2. Numerical integration can be used to simulate and analyze complex new energy systems, and the system stability can be determined based on the simulation trajectory. However, this method is not a necessary and sufficient condition for estimating stability, and it is difficult to characterize the true stability boundary of the system, which is not conducive to guiding the design of control parameters and the operation of relay protection equipment. Summary of the Invention

[0009] The technical problem to be solved by the present invention is to overcome the limitations of the above-mentioned technical solutions. In view of the two problems mentioned above, a method for characterizing the transient stability boundary of a multi-inverter hybrid system is provided.

[0010] The objective of this invention is achieved as follows: This invention provides a method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration. The multi-inverter hybrid system involves m grid-connected inverters and n grid-connected inverters, and all m grid-connected inverters and n grid-connected inverters are connected to the same infinite power grid via a common bus. The characterization method includes the following steps:

[0011] Step 1: For grid-connected inverters, phase-locked loop synchronous control is adopted, and for grid-connected inverters, virtual synchronous machine control is adopted. A circuit model for analysis is established, including the expression of the grid connection point voltage, the steady-state equation of the phase-locked loop of the grid-connected inverter, and the swing equation of the virtual synchronous machine of the grid-connected inverter.

[0012] Step 2: Solve for the steady-state equilibrium point of the multi-inverter hybrid system using the steady-state equations of the phase-locked loop of the grid-connected inverter and the swing equations of the virtual synchronous machine of the grid-connected inverter. Denote any one of these steady-state equilibrium points as the system equilibrium point x. i i = 1, 2, 3…;

[0013] Step 3: Calculate the system equilibrium point x one by one. i The Jacobian matrix J of the system i Eigenvalues, and determine the equilibrium point x of each system based on their signs. i The types include Type I unstable equilibrium points. and stable equilibrium point

[0014] Step 4, establish δ CFM x-axis, ω GFM Let y be a plane coordinate system; find any Type I unstable equilibrium point. The stable unit eigenvector y of the Jacobian matrix at point is i And with this type I unstable equilibrium point Let be the center of a circle, and let be the radius of the circle. Construct a circle A on the plane coordinate system with radius ∈ . Circle A is connected to the stable unit eigenvector y of the Jacobian matrix. i Intersection produces intersection point and intersection

[0015] Step 5: From the two intersection points obtained in Step 4, the unstable equilibrium point is obtained using the swing equations of the forward integral grid-connected inverter and the reverse integral grid-connected inverter. The trajectory of the stable manifold is obtained, and the transient stability boundary of the multi-inverter hybrid system is obtained through this trajectory.

[0016] Preferably, the implementation process of step 1 is as follows:

[0017] Ignoring current loop dynamics and phase-locked loop dynamics, the voltage source characteristics and current source characteristics exhibited by (m+n) grid-connected inverters with different control methods are simplified to ideal voltage sources. Ideal Current Source Simplifying an infinite power grid into an ideal power grid voltage source, namely the grid voltage U0, where U GFM For the terminal voltage amplitude of a grid-connected inverter, I GFL To determine the output current amplitude of the grid-connected inverter, δ GFM δ represents the phase difference between the terminal voltage of the grid-connected inverter and the grid voltage. GFL The phase difference between the output phase of the phase-locked loop of the grid-connected inverter and the grid voltage is given by j, where j is the imaginary unit and e is the natural constant.

[0018] For grid-connected inverters, phase-locked loop synchronous control is adopted, and for grid-connected inverters, virtual synchronous machine control is adopted. A circuit model for analysis is established, including the expression of the grid connection point voltage, the steady-state equation of the phase-locked loop of the grid-connected inverter, and the swing equation of the virtual synchronous machine of the grid-connected inverter.

[0019] The expression for the grid connection point voltage is:

[0020]

[0021] Among them, U g δ is the grid connection point voltage. g K1 is the phase of the grid voltage, K2 is the first impedance ratio, and K1 = X. g1 / (X g1 +X g2 K2 = X g2 / (X g1 +X g2 ), X g1 X is the line impedance from the grid side to the grid connection point. g2 X is the line impedance from the port of the grid-connected inverter to the grid connection point. g4 For parallel impedance, X g4 =X g1 / / X g2 " / / " represents parallel operations;

[0022] The grid-connected inverter uses phase-locked loop synchronous control. The steady-state equation of the phase-locked loop of the grid-connected inverter is as follows:

[0023] U q =K1U GFM sin(δ GFM -δ GFL )-K2U0sinδ GFL +X g5 I d，GFL =0

[0024] Among them, U q To match the q-axis component of the terminal voltage of the grid-connected inverter, X g5 For series impedance, X g5 =X g4 +X g3 X g3 To determine the line impedance from the grid-connected inverter port to the grid connection point, I d，GFL This refers to the d-axis component of the output current of a grid-connected inverter.

[0025] The grid-connected inverter uses virtual synchronous machine control, and the swing equation of the virtual synchronous machine of the grid-connected inverter is as follows:

[0026]

[0027] Where ω0 is the per-unit value of the fundamental frequency angular velocity, ω b T is the actual value of the fundamental frequency angular velocity. J D is the inertia time constant. g ω is the damping coefficient. GFM For the output angular frequency of the grid-connected inverter, P m For grid-connected inverters, the output active power command value is P. e This refers to the actual value of the active power output of a grid-connected inverter.

[0028] Preferably, the implementation process of step 2 is as follows:

[0029] Given the following constraints:

[0030]

[0031] Solve for the system equilibrium point x i , in, The equilibrium point for grid-connected inverters, The balance point for grid-connected inverters.

[0032] Preferably, the system equilibrium point x in step 3 i The Jacobian matrix J of the system i for:

[0033]

[0034] Where a is the first constant coefficient, a = (c1cosδ) i GFM0 -c2sin(δ i GFM0 -δ i GFL0 )) / T J c1 is the second constant coefficient, c1 = U GFM U0 / (X g1 +X g2 c2 is the third constant coefficient, c2 = X g1 U GFM I GFL / (X g1 +X g2 );

[0035] Determine the system equilibrium point x based on its sign. i The implementation process of the type is as follows:

[0036] Operations on determinants |λI-J i |=0 to obtain the first eigenvalue λ1 and the second eigenvalue λ2 of the Jacobian matrix, where I is the identity matrix and λ is the eigenvalue of the system Jacobian matrix J, and the following judgment is made:

[0037] If λ1 and λ2 are both negative, then the equilibrium point x of the system will be... i This is denoted as the stable equilibrium point. h = 1, 2, 3, ...;

[0038] If one of λ1 and λ2 is positive and the other is negative, then the equilibrium point x of the system will be... i This is denoted as a Type I unstable equilibrium point. And j≠h;

[0039] For each system equilibrium point x obtained in step 2 i After performing type determination, if the type I unstable equilibrium point is not found. Or a stable equilibrium point has not been found. If it is determined that the multi-inverter hybrid system does not have a practically meaningful stability boundary, this characterization method ends; otherwise, proceed to step 4.

[0040] Preferably, the determinant operation |λI-J i |=0 expands as follows:

[0041]

[0042] Preferably, step 4 involves finding any type I unstable equilibrium point. The stable unit eigenvector y of the Jacobian matrix at point is i The process is as follows:

[0043] Take any type I unstable equilibrium point By matrix operations (λI-J) i )y=0 yields the unstable equilibrium point of type I. The two right eigenvectors y of the Jacobian matrix at point . i1 and y i2 Among them, the eigenvectors corresponding to negative eigenvalues ​​are the stable eigenvectors. Normalizing these stable eigenvectors yields the Type I unstable equilibrium points. The stable unit eigenvector y of the Jacobian matrix at point i .

[0044] Preferably, the implementation process of step 5 is as follows:

[0045] Based on the actual engineering application requirements, only the system dynamics within one cycle are considered, and the grid-connected inverter is designed to output only positive active power. Therefore, the system balance point is subject to the following constraints:

[0046] δ GFM ∈[-π, π]

[0047] δ GFL ∈[0, 1.5π]

[0048] In this case, only one stable equilibrium point and one type I unstable equilibrium point can be solved;

[0049] The trajectory is obtained from the swing equation of the forward integral grid-connected inverter at the two intersection points obtained in step 4. and trajectory Then, the following judgment is made:

[0050] State 1, if trajectory trajectory If the points remain within the chosen circle A, then the two intersection points obtained in step 4... Starting from the point where the reverse integral grid-connected inverter begins its swing equation, the two new trajectories obtained constitute the unstable equilibrium point. The stable manifold, i.e., the stable equilibrium point. The stable boundary is determined, and the combination of two new trajectories is denoted as the trajectory.

[0051] State 2, if trajectory If we move beyond the chosen circle A, we reduce the radius ∈ and redraw the circle. Specifically, the center remains unchanged, and we take a new radius ∈' = k∈ (0 < k < 1). We then draw a circle B with radius ∈ on the phase plane coordinate system. This circle B is related to the stable unit eigenvector y of the Jacobian matrix. i The intersection produces intersection points A' and B'. The swing equation for the grid-connected inverter is then constructed again using forward integration at intersection points A' and B'. If the forward integration trajectory still deviates from circle B, the value of k is reduced until it no longer deviates. Then, the swing equation for the grid-connected inverter is constructed again using reverse integration. The two new trajectories obtained constitute the unstable equilibrium point. The stable manifold, i.e., the stable equilibrium point. The stable boundary is determined, and the combination of two new trajectories is denoted as the trajectory.

[0052] trajectory This is the transient stability boundary of the hybrid system.

[0053] Compared with the prior art, the beneficial effects of the present invention include:

[0054] 1. The inverse integration method proposed in this invention naturally takes into account system damping. Compared with traditional transient stability analysis methods such as the equal area method and energy function method, it is less conservative and can obtain more accurate transient stability boundaries, which greatly expands the operating domain of hybrid systems.

[0055] 2. Compared with other stability analysis methods, the inverse integration method proposed in this invention requires less computing power, is conducive to real-time calculation, can timely and effectively evaluate the fault-crossing capability of the hybrid system, and provide the critical clearing time, providing a reference for the operation of relay protection equipment. Attached Figure Description

[0056] Figure 1 is a topology diagram of the multi-inverter hybrid system involved in this invention connected to an infinite power grid;

[0057] Figure 2 is a schematic diagram of the analysis circuit model in an embodiment of the present invention.

[0058] Figure 3 shows the transient stability boundary results of the hybrid system under normal operating conditions obtained in the embodiment of the present invention;

[0059] Figure 4 shows the transient stability boundary and critical clearing time of the hybrid system during a fault in an embodiment of the present invention.

[0060] Figure 5 is a schematic diagram of step 5 in an embodiment of the present invention. Detailed Implementation

[0061] The present invention will be further described below with reference to the accompanying drawings and specific embodiments.

[0062] Figure 1 is a topology diagram of the multi-inverter hybrid system involved in this invention connected to an infinite grid. As shown in Figure 1, the hybrid system involved in this invention includes m grid-connected inverters and n follow-grid inverters. The voltage source inverters employ a virtual synchronous machine control strategy, and there is one filter inductor L between the port and the grid connection point. GFMa1 1 line inductor L GFMα2 (α = 1, 2, ..., m), the filtered output three-phase voltage u is collected. abc Three-phase current i abc The power supply outer loop is a virtual synchronous machine, and the amplitude u of the external reference voltage of the output filter is... dqref and phase θ PSC The voltage and current are fed into the inner loop, ultimately generating a PWM wave. The current source inverter employs a single synchronous phase-locked loop control strategy, with one filter inductor L between the port and the grid connection point. GFLβ1 1 line inductor L GFLβ2 (β=1, 2, …, n), the filtered output three-phase voltage u is collected. abc The supply phase-locked loop performs phase locking and outputs a tracking phase θ. PLL Simultaneously, the three-phase current i outside the filter is collected.abc and voltage u abc The inner current loop is supplied, ultimately generating a PWM wave. The inductance of the mains line is L. g .

[0063] The relevant electrical parameters for implementing this invention are set as follows: reference angular frequency ω b =100π, base capacity S N =10MVA, reference voltage U N =690V. Voltage source inverter port to grid connection point line reactance X GFM =0.26, Line reactance X from the current source inverter port to the grid connection point GFL =0.3, power grid line reactance X g =0.2. Voltage source inverter terminal voltage U GFM =1, Current source inverter terminal current I GFL =1. Virtual synchronizer inertial time constant T J =6, damping coefficient D g =10, mechanical power P m =1 (during normal operation), P m =0.6 (in case of fault), grid voltage U0=1 (in normal operation), U0=0.5 (in case of fault).

[0064] This invention provides a method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration. The multi-inverter hybrid system involves m grid-connected inverters and n grid-connected inverters, and all m grid-connected inverters and n grid-connected inverters are connected to the same infinite power grid via a common bus. The method includes the following steps:

[0065] Step 1: For grid-connected inverters, phase-locked loop synchronous control is adopted, and for grid-connected inverters, virtual synchronous machine control is adopted. A circuit model for analysis is established, including the expression of the grid connection point voltage, the steady-state equation of the phase-locked loop of the grid-connected inverter, and the swing equation of the virtual synchronous machine of the grid-connected inverter.

[0066] In this embodiment, the implementation process of step 1 is as follows:

[0067] Ignoring current loop dynamics and phase-locked loop dynamics, the voltage source characteristics and current source characteristics exhibited by (m+n) grid-connected inverters with different control methods are simplified to ideal voltage sources. Ideal Current Source Simplifying an infinite power grid into an ideal power grid voltage source, namely the grid voltage U0, where U GFM For the terminal voltage amplitude of a grid-connected inverter, I GFL To determine the output current amplitude of the grid-connected inverter, δ GFMδ represents the phase difference between the terminal voltage of the grid-connected inverter and the grid voltage. GFL The phase difference between the output phase of the phase-locked loop of the grid-connected inverter and the grid voltage is given by j, where j is the imaginary unit and e is the natural constant.

[0068] For grid-connected inverters, phase-locked loop synchronous control is adopted, and for grid-connected inverters, virtual synchronous machine control is adopted. A circuit model for analysis is established, including the expression of the grid connection point voltage, the steady-state equation of the phase-locked loop of the grid-connected inverter, and the swing equation of the virtual synchronous machine of the grid-connected inverter.

[0069] The expression for the grid connection point voltage is:

[0070]

[0071] Among them, U g δ is the grid connection point voltage. g K1 is the phase of the grid voltage, K2 is the first impedance ratio, and K1 = X. g1 / (X g1 +X g2 K2 = X g2 / (X g1 +X g2 ), X g1 X is the line impedance from the grid side to the grid connection point. g2 X is the line impedance from the port of the grid-connected inverter to the grid connection point. g4 For parallel impedance, X g4 =X g1 / / X g2 " / / " represents parallel operations;

[0072] The grid-connected inverter uses phase-locked loop synchronous control. The steady-state equation of the phase-locked loop of the grid-connected inverter is as follows:

[0073] U q =K1U GFM sin(δ GFM -δ GFL )-K2U0sinδ GFL +X g5 I d，GFL =0

[0074] Among them, U q To match the q-axis component of the terminal voltage of the grid-connected inverter, X g5 For series impedance, X g5 =X g4 +X g3 X g3 To determine the line impedance from the grid-connected inverter port to the grid connection point, I d，GFLThis refers to the d-axis component of the output current of a grid-connected inverter.

[0075] The grid-connected inverter uses virtual synchronous machine control, and the swing equation of the virtual synchronous machine of the grid-connected inverter is as follows:

[0076]

[0077] Where ω0 is the per-unit value of the fundamental frequency angular velocity, ω b T is the actual value of the fundamental frequency angular velocity. J D is the inertia time constant. g ω is the damping coefficient. GFM For the output angular frequency of the grid-connected inverter, P m For grid-connected inverters, the output active power command value is P. e This refers to the actual value of the active power output of a grid-connected inverter.

[0078] Figure 2 is a schematic diagram of the analysis circuit model in an embodiment of the present invention.

[0079] Step 2: Solve for the steady-state equilibrium point of the multi-inverter hybrid system using the steady-state equations of the phase-locked loop of the grid-connected inverter and the swing equations of the virtual synchronous machine of the grid-connected inverter. Denote any one of these steady-state equilibrium points as the system equilibrium point x. i , i = 1, 2, 3...

[0080] In this embodiment, the implementation process of step 2 is as follows:

[0081] Given the following constraints:

[0082]

[0083] Solve for the system equilibrium point x i , in, The equilibrium point for grid-connected inverters, The balance point for grid-connected inverters.

[0084] Step 3: Calculate the system equilibrium point x one by one. i The Jacobian matrix J of the system i Eigenvalues, and determine the equilibrium point x of each system based on their signs. i The types include Type I unstable equilibrium points. and stable equilibrium point

[0085] In this embodiment, the implementation process of step 3 is as follows: Step 3, the system equilibrium point x i The Jacobian matrix J of the systemi for:

[0086]

[0087] Where a is the first constant coefficient, a = (c1cosδ) i GFMo -c2sin(δ i GFM0 -δ i GFL0 )) / T J c1 is the second constant coefficient, c1 = U GFM U0 / (X g1 +X g2 c2 is the third constant coefficient, c2 = X g1 U GFM I GFL / (X g1 +X g2 );

[0088] Determine the system equilibrium point x based on its sign. i The implementation process of the type is as follows:

[0089] Operations on determinants |λI-J i |=0 to obtain the first eigenvalue λ1 and the second eigenvalue λ2 of the Jacobian matrix, where I is the identity matrix and λ is the eigenvalue of the system Jacobian matrix J, and the following judgment is made:

[0090] If λ1 and λ2 are both negative, then the equilibrium point x of the system will be... i This is denoted as the stable equilibrium point. h = 1, 2, 3, ...;

[0091] If one of λ1 and λ2 is positive and the other is negative, then the equilibrium point x of the system will be... i This is denoted as a Type I unstable equilibrium point. And j≠h;

[0092] For each system equilibrium point x obtained in step 2 i After performing type determination, if the type I unstable equilibrium point is not found. Or a stable equilibrium point has not been found. If it is determined that the multi-inverter hybrid system does not have a practically meaningful stability boundary, this characterization method ends; otherwise, proceed to step 4.

[0093] In this embodiment, the determinant operation |λI-J i |=0 expands as follows:

[0094]

[0095] Step 4, establish δ GFM x-axis, ω GFM Let y be a plane coordinate system; find any Type I unstable equilibrium point. The stable unit eigenvector y of the Jacobian matrix at point is i And with this type I unstable equilibrium point Let be the center of a circle, and let be the radius of the circle. Construct a circle A on the plane coordinate system with radius ∈ . Circle A is connected to the stable unit eigenvector y of the Jacobian matrix. i Intersection produces intersection point and intersection

[0096] In this embodiment, step 4 involves finding any type I unstable equilibrium point. The stable unit eigenvector y of the Jacobian matrix at point is i The process is as follows:

[0097] Take any type I unstable equilibrium point By matrix operations (λI-J) i )y=0 yields the unstable equilibrium point of type I. The two right eigenvectors y of the Jacobian matrix at point . i1 and y i2 Among them, the eigenvectors corresponding to negative eigenvalues ​​are the stable eigenvectors. Normalizing these stable eigenvectors yields the Type I unstable equilibrium points. The stable unit eigenvector y of the Jacobian matrix at point i .

[0098] Step 5: From the two intersection points obtained in Step 4, the unstable equilibrium point is obtained using the swing equations of the forward integral grid-connected inverter and the reverse integral grid-connected inverter. The trajectory of the stable manifold is obtained, and the transient stability boundary of the multi-inverter hybrid system is obtained through this trajectory.

[0099] In this embodiment, the implementation process of step 5 is as follows:

[0100] Based on the actual engineering application requirements, only the system dynamics within one cycle are considered, and the grid-connected inverter is designed to output only positive active power. Therefore, the system balance point is subject to the following constraints:

[0101] δ GFM ∈[-π, π]

[0102] δ GFL ∈[0, 1.5π]

[0103] In this case, only one stable equilibrium point and one type I unstable equilibrium point can be solved;

[0104] The trajectory is obtained from the swing equation of the forward integral grid-connected inverter at the two intersection points obtained in step 4. and trajectory Then, the following judgment is made:

[0105] State 1, if trajectory trajectory If the points remain within the chosen circle A, then the two intersection points obtained in step 4... Starting from the point where the reverse integral grid-connected inverter begins its swing equation, the two new trajectories obtained constitute the unstable equilibrium point. The stable manifold, i.e., the stable equilibrium point. The stable boundary is determined, and the combination of two new trajectories is denoted as the trajectory.

[0106] State 2, if trajectory If we move beyond the chosen circle A, we reduce the radius ∈ and redraw the circle. Specifically, the center remains unchanged, and we take a new radius ∈' = k∈ (0 < k < 1). We then draw a circle B with radius ∈ on the phase plane coordinate system. This circle B is related to the stable unit eigenvector y of the Jacobian matrix. i The intersection produces intersection points A' and B'. The swing equation for the grid-connected inverter is then constructed again using forward integration at intersection points A' and B'. If the forward integration trajectory still deviates from circle B, the value of k is reduced until it no longer deviates. Then, the swing equation for the grid-connected inverter is constructed again using reverse integration. The two new trajectories obtained constitute the unstable equilibrium point. The stable manifold, i.e., the stable equilibrium point. The stable boundary is determined, and the combination of two new trajectories is denoted as the trajectory.

[0107] trajectory This is the transient stability boundary of the hybrid system.

[0108] Figure 5 is a schematic diagram of step 5 in an embodiment of the present invention.

[0109] To demonstrate the beneficial effects of the present invention, simulations were performed.

[0110] In the simulation, two scenarios were implemented: normal operation of the hybrid system and current limiting during a fault. The specific results are shown in Figures 3 and 4.

[0111] During normal operation, positive integrals are taken simultaneously at initial state points (-0.6, 0.98), (0, 0.99), and (-1.3, 1) inside and outside the system attraction domain, and at a point on the boundary (0.52312, 1.00078) to verify the accuracy of the boundary. Figure 3 shows the results. As can be seen from Figure 3, the starting points inside and on the boundary are eventually attracted by the stable equilibrium point, indicating that the system has successfully stabilized. However, the starting points outside the boundary are not attracted by the stable equilibrium point. Although the system may be attracted by the stable equilibrium point in the next cycle, this would cause slippage and loss of synchronism in the actual system, which may lead to serious faults such as relay protection failure, power reversal, overvoltage, and voltage oscillation, and is therefore considered unstable. This result successfully verifies that the trajectory is the transient stability boundary of the hybrid system. It is worth noting that the model established in this embodiment does not include type II unstable equilibrium points, so the trajectory is not closed. When a fault occurs, the model parameters are adjusted to simulate the power grid fault, and the fault trajectory and critical clearing time t are calculated. CCT They are shown together in Figure 4.

Claims

1. A method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration, wherein the multi-inverter hybrid system comprises m grid-connected inverters and n grid-connected inverters, and the m grid-connected inverters and n grid-connected inverters are all connected to the same infinite power grid via a common bus, characterized in that... The characterization method includes the following steps: Step 1, using phase-locked loop synchronous control for grid-connected inverters and virtual synchronous machine control for grid-connected inverters, establish an analysis circuit model, including the expression for the grid connection point voltage, the steady-state equation of the phase-locked loop for grid-connected inverters, and the swing equation of the virtual synchronous machine for grid-connected inverters; Step 2, solving for the steady-state equilibrium point of the multi-inverter hybrid system using the steady-state equation of the phase-locked loop for grid-connected inverters and the swing equation of the virtual synchronous machine for grid-connected inverters, and denoteing any one of these steady-state equilibrium points as the system equilibrium point x. i i = 1, 2, 3...; Step 3, calculate the system equilibrium point x one by one. i The Jacobian matrix J of the system i Eigenvalues, and determine the equilibrium point x of each system based on their signs. i The types include Type I unstable equilibrium points. and stable equilibrium point Step 4, establish δ GFM x-axis, ω GFM Let y be the phase plane coordinate system; find any type I unstable equilibrium point. The stable unit eigenvector y of the Jacobian matrix at point is i And with this type I unstable equilibrium point Let be the center of a circle, and let be the radius of the circle. Construct a circle A on the plane coordinate system with radius ∈ . Circle A is connected to the stable unit eigenvector y of the Jacobian matrix. i Intersection produces intersection point and intersection Step 5: From the two intersection points obtained in Step 4, find the appropriate radius of circle A ∈ using the swing equation of the forward integral grid-connected inverter; then, obtain the unstable equilibrium point using the swing equation of the reverse integral grid-connected inverter. The trajectory of the stable manifold is obtained, and the transient stability boundary of the multi-inverter hybrid system is obtained through this trajectory.

2. The method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration according to claim 1, characterized in that, The implementation process of step 1 is as follows: Ignoring the dynamics of the current loop and the phase-locked loop, the voltage source characteristics and current source characteristics exhibited by (m+n) grid-connected inverters with different control methods are simplified to ideal voltage sources. Ideal Current Source Simplifying an infinite power grid into an ideal power grid voltage source, namely the grid voltage U0, where U GFM For the terminal voltage amplitude of a grid-connected inverter, I GFL To determine the output current amplitude of the grid-connected inverter, δ GFM δ represents the phase difference between the terminal voltage of the grid-connected inverter and the grid voltage. GFL To represent the phase difference between the output phase of the phase-locked loop (PLL) of the grid-connected inverter and the grid voltage, where j is the imaginary unit and e is the natural constant, a PLL synchronous control is used for the grid-connected inverter, and a virtual synchronous machine control is used for the grid-connected inverter. An analytical circuit model is established, including the expression for the grid connection point voltage, the steady-state equation of the PLL for the grid-connected inverter, and the swing equation of the virtual synchronous machine for the grid-connected inverter. The expression for the grid connection point voltage is: Among them, U g δ is the grid connection point voltage. g K1 is the phase of the grid voltage, K2 is the first impedance ratio, and K1 = X. g1 / (X g1 +X g2 K2 = X g2 / (X g1 +X g2 ), X g1 X is the line impedance from the grid side to the grid connection point. g2 X is the line impedance from the port of the grid-connected inverter to the grid connection point. g4 For parallel impedance, X g4 =X g1 / / X g2 " / / " represents parallel operation; the grid-connected inverter adopts phase-locked loop synchronous control, and the steady-state equation of the phase-locked loop of the grid-connected inverter is: U q =K1U GFM sin(δ GFM -δ GFL )-K2U0sinδ GFL +X g5 I d，GFL Among them, U q To match the q-axis component of the terminal voltage of the grid-connected inverter, X g5 For series impedance, X g5 =X g4 +X g3 X g3 To determine the line impedance from the grid-connected inverter port to the grid connection point, I d，GFL The output current d-axis component of the grid-connected inverter is given. The grid-connected inverter uses virtual synchronous machine control, and the swing equation of the virtual synchronous machine of the grid-connected inverter is as follows: Where ω0 is the per-unit value of the fundamental frequency angular velocity, ω b T is the actual value of the fundamental frequency angular velocity. J D is the inertia time constant. g ω is the damping coefficient. GFM For the output angular frequency of the grid-connected inverter, P m For grid-connected inverters, the output active power command value is P. e This refers to the actual value of the active power output of a grid-connected inverter.

3. The method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration according to claim 2, characterized in that, The implementation process of step 2 is as follows: given the following constraints: Solve for the system equilibrium point x i , in, The equilibrium point for grid-connected inverters, The balance point for grid-connected inverters.

4. The method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration according to claim 1, characterized in that, The system equilibrium point x mentioned in step 3 i The Jacobian matrix J of the system i for: Where a is the first constant coefficient. c1 is the second constant coefficient, c1 = U GFM U0 / (X g1 +X g2 c2 is the third constant coefficient, c2 = X g1 U GFM I GFL / (X g1 +X g2 Determine the system equilibrium point x based on its sign. i The implementation process of the type is as follows: by determinant operation |λI-J i |=0 to obtain the first eigenvalue λ1 and the second eigenvalue λ2 of the Jacobian matrix, where I is the identity matrix and λ is the eigenvalue of the system Jacobian matrix J. The following judgment is made: if both λ1 and λ2 are negative, then the system equilibrium point x is... i This is denoted as the stable equilibrium point. h = 1, 2, 3, ...; If one of λ1 and λ2 is positive and the other is negative, then the equilibrium point x of the system will be... i This is denoted as a Type I unstable equilibrium point. And j≠h; for each system equilibrium point x obtained in step 2 i After performing type determination, if the type I unstable equilibrium point is not found. Or a stable equilibrium point has not been found. If it is determined that the multi-inverter hybrid system does not have a practically meaningful stability boundary, this characterization method ends; otherwise, proceed to step 4.

5. The method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration according to claim 4, characterized in that, The determinant operation |λI-J i |=0 expands as follows:

6. The method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration according to claim 4, characterized in that, Step 4 involves finding any type I unstable equilibrium point. The stable unit eigenvector y of the Jacobian matrix at point is i The process is as follows: Take any type I unstable equilibrium point By matrix operations (λI-J) i )y=0 yields the unstable equilibrium point of type I. The two right eigenvectors y of the Jacobian matrix at point . i1 and y i2 y is a type I unstable equilibrium point. The right eigenvector of the Jacobian matrix at point A is given by the eigenvectors corresponding to negative eigenvalues. The stable eigenvectors are obtained by normalizing these stable eigenvectors, which yields the Type I unstable equilibrium point. The stable unit eigenvector y of the Jacobian matrix at point i .

7. The method for characterizing the transient stability boundary of a multi-inverter hybrid system based on inverse integration according to claim 6, characterized in that, The implementation process of step 5 is as follows: Based on the actual engineering application requirements, only the system dynamics within one cycle are considered, and the grid-connected inverter is made to output only positive active power. Therefore, the system balance point is restricted as follows: δ GFM ∈[-π, π]δ GFL In the case of [0, 1.5π], only one stable equilibrium point and one type I unstable equilibrium point can be solved; the swing equation of the grid-connected inverter with forward integration at the two intersection points obtained in step 4 is used to obtain the trajectory. and trajectory Then the following judgment is made: State 1, if the trajectory trajectory If the points remain within the chosen circle A, then the two intersection points obtained in step 4... Starting from the point where the reverse integral grid-connected inverter begins its swing equation, the two new trajectories obtained constitute the unstable equilibrium point. The stable manifold, i.e., the stable equilibrium point. The stable boundary is determined, and the combination of two new trajectories is denoted as the trajectory. State 2, if trajectory If we move beyond the chosen circle A, we reduce the radius ∈ and redraw the circle. Specifically, the center remains unchanged, and we take a new radius ∈' = k∈ (0 < k < 1). We then draw a circle B with radius ∈ on the phase plane coordinate system. This circle B is related to the stable unit eigenvector y of the Jacobian matrix. i The intersection produces intersection points A' and B'. The swing equation for the grid-connected inverter is then constructed again using forward integration at intersection points A' and B'. If the forward integration trajectory still deviates from circle B, the value of k is reduced until it no longer deviates. Then, the swing equation for the grid-connected inverter is constructed again using reverse integration. The two new trajectories obtained constitute the unstable equilibrium point. The stable manifold, i.e., the stable equilibrium point. The stable boundary is determined, and the combination of two new trajectories is denoted as the trajectory. trajectory This is the transient stability boundary of the hybrid system.

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