Transient synchronous stability analysis method for multiple parallel grid-following type converters

Abstract

The invention relates to a transient synchronous stability analysis method for multiple parallel grid-following type converters, which comprises the following steps of: firstly, constructing a grid-following type converter grid-connected system model considering an inner loop current and phase-locked loop coupling effect under a multi-machine system framework; then, through a Newton-Raphson method and an equal area criterion, quantitative evaluation of the existence of the balance point and visual analysis of a transient synchronization behavior are respectively realized; the method comprises the steps that firstly, transient synchronization stability criteria with physical significance are strictly deduced through extended application of a Lyapunov direct method, key factors prone to causing transient synchronization instability and influence rules of the key factors are disclosed, and finally time domain simulation is conducted through MATLAB / SIMULIN so as to verify the accuracy of the method. The transient synchronous instability characteristic description method and the risk quantitative evaluation system have important theoretical guiding significance for parameter optimization design of a power electronic system.

Description

Technical Field

[0001] The present invention belongs to the technical field of synchronous control of power electronic systems, and particularly relates to a transient synchronous stability analysis method for multi-parallel grid-following converters. Background Art

[0002] In recent years, new energy systems represented by solar energy and wind energy have been massively connected to the modern power grid through power electronic devices such as grid-connected voltage source converters (VSCs). As the core device for new energy grid connection, the mainstream control strategy of VSCs is the grid-following mode and relies on a phase-locked loop to achieve grid synchronization. The dynamic characteristics of the phase-locked loop directly dominate the transient synchronous stability of the grid-following converter, which makes the VSC synchronization mechanism based on the phase-locked loop a key research object.

[0003] Traditional transient synchronous stability analysis is usually based on the assumption of ideal inner-loop current, only focuses on the dynamic characteristics of the phase-locked loop, and good results have been achieved in the research on the transient synchronous stability of grid-following converters based on the phase-locked loop. However, ignoring the interaction between the inner-loop current and the phase-locked loop may lead to an overly optimistic conclusion in the analysis of transient synchronous stability. Chen Junru et al. established a system model of a single-voltage-source converter with current transient and phase-locked loop, and verified the influence of the current transient of the converter on the dynamic characteristics of the phase-locked loop. Professor Wu Chao et al. further derived a fourth-order model of a single-voltage-source converter system considering inner-loop current control, proposed an improved equal-area method to reveal the necessity of considering inner-loop current control in transient synchronous stability analysis, and pointed out that when the inner-loop current control bandwidth is insufficient, the traditional second-order phase-locked loop model will lead to inaccurate stability evaluation results. Therefore, especially when the inner-loop current control bandwidth is small, the dynamic characteristics of the inner-loop current will significantly affect the transient synchronous stability of the grid-connected system of the grid-following converter, and there are problems such as complex transient models, unclear instability mechanisms, and lack of transient synchronous stability criteria, which urgently need further in-depth research.

[0004] In view of this, it is necessary to construct a description method for the intuitive transient synchronous instability characteristics and a transient synchronous instability quantitative evaluation criterion for a multi-parallel grid-following converter grid-connected system considering the coupling effect of the inner-loop current and the phase-locked loop. Summary of the Invention

[0005] 1. Object of the Invention

[0006] The present invention aims to solve the problem of ignoring the coupling effect between the inner-loop current and the phase-locked loop in the existing transient synchronous stability analysis of grid-following converters, and proposes A Method for Analyzing the Transient Synchronization Stability of a Multi-Parallel Grid-Following Inverter to achieve a comprehensive and accurate characterization of the nonlinear dynamic characteristics of a multi-parallel grid-following converter grid-connected system.

[0007] 2. Technical Solution

[0008] To achieve the above objectives, the following steps are mainly included:

[0009] Step 1: Construction of a multi-parallel VSCs grid-connected model considering the inner-loop current and PLL coupling effect;

[0010] Step 2: Analysis of the transient synchronization instability mechanism of a multi-parallel VSCs grid-connected system considering the inner-loop current;

[0011] Step 3: Construction of a transient synchronization stability criterion based on the approximate Lyapunov direct method;

[0012] Step 4: Case study and time-domain simulation.

[0013] Among them, the specific process of the above Step 1 includes: in an N-parallel VSCs grid-connected system, N PLLs are defined with N dq-axis reference frames. In addition, the DQ axis is defined as the reference frame, with the grid voltage vector U g as the reference, θ g and ω g representing the phase angle and angular velocity of U g Taking the k-th grid-following converter VSC k (k = 1,..., N) as an example, the expression of the terminal voltage vector U k of VSC t,k in the DQ frame is,

[0014]

[0015] where the subscript j represents the j-th VSC and j ≠ k, I j representing the output current vector of VSC j I k representing the output current vector of VSC k Z g = R g + jX g and Z linek = R line,k + jX line,k representing the grid-side impedance and line impedance respectively, R g and R line,k representing the grid-side resistance and line resistance respectively, X g and X line,k representing the grid-side reactance and line reactance respectively.

[0016] Decompose U t,k into d-axis and q-axis components, and its expression is,

[0017]

[0018] where the power angle δ k= θ PLL,k -θ g denotes that U t,k leads U g by the formed angle difference, δ kj = δ k -δ j denotes the VSC k and the VSC j by the formed power angle difference. is the injection angle of the current vector I k θ. I,k Denotes the current injection angle, θ Zlinek and θ Zg respectively denote the impedance angles of Z linek and Z g .

[0019] From Kirchhoff's voltage law and the inner-loop current control block diagram, the control equation of the inner-loop current of the VSC k is as follows,

[0020]

[0021] where the superscript "·" is the derivative with respect to time t, and respectively denote the active current and reactive current deviations. x dC,k and x qC,k are the state variables of the current-loop control, e d,k and e q,k are the control variables of the current-loop control, K pC and K iC respectively denote the proportional gain and integral gain of the inner-loop current PI regulator, and L f denotes the filter inductor on the AC output side of the grid-connected converter.

[0022] According to the phase-locked loop control block diagram, the state equation of the phase-locked loop in the VSC k can be derived as follows,

[0023]

[0024] where K pPLL and K iPLL respectively denote the proportional gain and integral gain of the phase-locked loop PI regulator, and △ω k is defined as the angular frequency difference between the output angular frequency ω k of the phase-locked loop and ω g .

[0025] Furthermore, by combining Equation (2), Equation (4), and Equation (5), the sixth-order nonlinear dynamic model of the VSC k is obtained.

[0026]

[0027] Among them, M k is the equivalent inertia, D eq,k is the equivalent damping, P m,k and P e,k are the equivalent mechanical power and the equivalent electromagnetic power respectively, y d,k and y q,k The detailed expressions are as follows:

[0028]

[0029] Among them, L g and L line,k represent the grid-side inductor and the line inductor respectively. Obviously, the sixth-order nonlinear model has the characteristics of nonlinearity, strong coupling and high order. This model fully considers the coupling effect between the inner-loop current dynamic characteristics and the phase-locked loop, providing a theoretical basis for subsequent transient synchronization stability analysis.

[0030] Among them, the specific process of step 2 includes: for the sixth-order nonlinear model constructed in step 1, analyze its transient instability mechanism from two aspects: the existence of the equilibrium point and the transient synchronization behavior.

[0031] (1) Analysis of the existence of the equilibrium point: The sufficient condition for the existence of the system equilibrium point U tq,k (x 1 , x 2 ,..., x N ) = 0, k = 1, 2,..., N. This is a highly nonlinear system of equations. Due to the influence of cross-coupling terms and current-loop control, it is difficult to directly solve analytically. Among them, the vector x k =(δ k , △I d,k , △I q,k ), k = 1, 2,..., N represents the state variables of the VSC k , including the power angle δ k and the current deviation

[0032] △I d,k , △I q,k . The Newton-Raphson method is used to perform numerical calculations on the nonlinear system of equations: (a) Initial value selection: Initialize the state variable matrix x (i) . Set the differential terms on the left side of the sixth-order nonlinear model to zero, and the initial steady-state values of the state variables can be accurately calculated, and the steady-state values before the fault are selected as the initial values; (b) Taylor expansion: Expand the matrix U tq (x 1 , x 2 ,..., x N) Perform Taylor expansion at the initial estimation point and retain the first-order term to obtain the linearized approximation The superscript i is the current iteration number, [J (i) ] represents the Jacobian matrix of the current iteration point, and the matrix x is obtained (i) ; (c) Iterative solution: Update the estimated value, and use the x obtained from (b) (i+1) As the initial value for the next iteration. Repeat (a) and (b) until all elements in the offset matrix are less than the convergence threshold 10 7 , the iterative calculation stops.

[0033] (2) Transient synchronization behavior analysis: The inner loop current dynamics will be directly reflected in P m,k , which is refined into and They are defined as the equivalent mechanical power when the inner loop current characteristics are ignored and considered respectively. The expressions are as follows:

[0034]

[0035] The equal-area criterion method is used to intuitively analyze the transient synchronization behavior of the sixth-order nonlinear model, revealing the influence of the dynamic characteristics of the inner-loop current on the transient synchronization behavior of the grid-connected system with multiple parallel grid-following converters.

[0036] The specific process of step 3 includes: (1) function construction and verification: ignoring the nonlinear damping effect, combining the first integral method and the extended invariance principle to construct a candidate Lyapunov function,

[0037]

[0038] In the formula, δ s,k Represents VSC k The stable equilibrium point is obtained by the Newton-Raphson method in step 2, λ k is an undetermined parameter that can adjust the weight of the mixed term without changing the equilibrium point. The derivative of the candidate Lyapunov function The expression is,

[0039]

[0040] In the formula, H k The detailed expression of the sub-block is:

[0041]

[0042] Formula (11) consists of a quadratic term plus the last term, which is obtained by choosing the parameter λ k Make the quadratic term a positive definite term, then is negatively semidefinite and satisfies the principle of extension invariance.k When the inequality (12) is satisfied, the candidate Lyapunov function V can be used for the study of transient synchronization stability.

[0043]

[0044] (2) Critical energy value calculation: Use the constructed V above to derive the critical energy value V considering non - linear damping dissipation at the unstable equilibrium point cr , and its expression is as follows.

[0045] V cr =V k (△ω k ) + V p (δ u , △ω u ) + ∫D eq △ωdδ (12)

[0046] where δ u represents the unstable equilibrium point, △ω u represents the angular frequency deviation at the unstable equilibrium point, D eq △ω corresponds to the frictional force during the transient process, and the dissipation of the damping D eq is related to the integral path of the frictional force. Since the integral term of the frictional force is difficult to calculate accurately, the triangular area approximation method is used to estimate the damping dissipation. Specifically, assume that the integral path of the frictional force of the VSC k is approximately the triangular area. The base of this triangle corresponds to the change range of the system path from the initial point δ 0 to the maximum phase angle δ max,k , and the height corresponds to the maximum angular velocity deviation △ω max,k . The expression for the maximum phase angle is as follows.

[0047]

[0048] where the value range of δ max,k is (δ s,k , δ u,k , and its value equation is determined by V p,k (δ 0 ) = V p,k (δ k ). Define W da,k as the damping dissipation energy of the system running from δ 0 to δ s,k . At the SEP, it has the minimum transient potential energy V pmin = 0 and the maximum transient kinetic energy V kmax,k . V kmax,k is determined by subtracting the damping dissipation from the initial transient energy of the system.

[0049]

[0050] VSC k The system runs along the path from δ 0 to δ s,k The damping dissipation W da,k and the path from δ s,k runs to δ max,k The damping dissipation W dd,k is approximately estimated by the triangle area,

[0051]

[0052] Based on the above analysis, considering the approximate dissipation of D eq The critical energy value V of the dissipation cr can be expressed as,

[0053]

[0054] where γ = 8M k V p,k (δ 0 ) + ((δ s,k - δ 0 )D eq,k ) 2 . Based on the above constructed V and the critical energy value V cr , the analytical expression of the transient synchronization stability criterion is,

[0055]

[0056] where η is defined as the transient synchronization stability index, which is used to quantitatively evaluate the transient synchronization stability. When η ≥ 0, the system is in the transient synchronization stable state, and the larger the η value, the better the transient synchronization stability. On the contrary, the system is in the unstable state.

[0057] Among them, the specific process of step 4 includes: obtaining the transient synchronization stability criterion based on step 3, constructing a transient synchronization stability evaluation system for the multi - parallel grid - following converters, extracting the key factors that are prone to cause transient synchronization instability and analyzing their influence laws. Finally, through the MATLAB / SIMULIN platform for time - domain simulation, the accuracy and effectiveness of the proposed method are verified. It mainly covers the coordinate transformation block, the VSC model block in dq coordinates, the outer - loop voltage control block, the inner - loop current control block, and the phase - locked loop module.

[0058] Through the above steps, a transient synchronization stability analysis method for multi - parallel grid - following converters is given.

[0059] The excellent effects of the present invention are as follows: Under the framework of a multi-machine system, a sixth-order nonlinear model of the grid-following converter considering the coupling effect of the inner-loop current and the phase-locked loop is established, realizing a more accurate characterization of the transient synchronous stability of the system. Through the Newton-Raphson method and the equal-area criterion, the quantitative evaluation of the existence of the system equilibrium point and the visual analysis of the transient synchronous behavior are respectively realized, verifying the necessity of the inner-loop current in the transient synchronous stability analysis. Combining the first integral method and the extended invariance principle to construct a Lyapunov function, incorporating the nonlinear damping approximate dissipation into the critical energy, and strictly deriving a transient synchronous stability criterion with physical significance; Based on this criterion, a transient synchronous instability characteristic description method and a risk quantification evaluation system are proposed to reveal the key factors and their action mechanisms that are prone to cause transient synchronous instability, providing a reliable theoretical basis for the parameter optimization design of power electronic systems and having important guiding significance for ensuring the safe and stable operation of new power systems. Description of the Drawings

[0060] Figure 1 The figure shows a flow chart of a transient synchronous stability analysis method for a multi-parallel grid-following converter.

[0061] Figure 2a The figure shows the topological structure diagram of an N-VSC grid-connected system.

[0062] Figure 2b The figure shows a VSC k Grid-connected system control module block diagram

[0063] Figure 3a The figure shows the power angle diagram without considering the dynamic characteristics of the inner-loop current.

[0064] Figure 3b The figure shows the power angle diagram considering the dynamic characteristics of the inner-loop current.

[0065] Figure 4a The figure shows that the system path is from δ 0 to δ s,k Damping approximate dissipation diagram

[0066] Figure 4b The figure shows that the system path is from δ s,k to δ max,k Damping approximate dissipation diagram

[0067] Figure 5 The figure shows a simplified diagram of N parallel grid-following converters.

[0068] Figure 6 The figure shows the attraction domain estimation diagram under different SCR changes.

[0069] Figure 7 The figure shows the time-domain simulation diagram of scenario A.

[0070] Figure 8 Shown are the estimated attraction regions under different Z line1,i changes

[0071] Figure 9 Shown is the time-domain simulation diagram of Scenario B

[0072] Figure 10a Shown is the three-dimensional graph of η under different changes in the PLL PI gain

[0073] Figure 10b Shown is the stable region in the PLL PI gain parameter space

[0074] Figure 11 Shown is the time-domain simulation diagram of Scenario C

[0075] Figure 12a Shown are different R g and changes

[0076] Figure 12b Shown is R g and parameter space

[0077] Figure 13 Shown is the time-domain simulation diagram of Scenario D

[0078] Figure 14a Shown are different X line,k and changes

[0079] Figure 14b Shown is X line,k and parameter space

[0080] Figure 15 Shown is the time-domain simulation diagram of Scenario E Specific implementation manner

[0081] To clearly understand the objectives, features, and advantages of the present invention, the following further detailed description is provided in conjunction with the accompanying drawings and specific implementation examples:

[0082] Figure 1 Schematic diagram of the process for transient synchronization stability analysis of a multi-parallel grid-following converter. As Figure 1 shown, it includes the following steps:

[0083] Step 1: Taking the case of N parallel grid-connected converters connected to the grid as an example, its typical topological structure and control module are shown in Figure 2. A multi-parallel grid-connected converter model considering the coupling effect of the inner-loop current and the phase-locked loop is constructed, and the model equations are not elaborated here. The main system parameters are set as shown in Table 1. In this example, the designed bandwidth of the inner-loop current is about 80 times that of the phase-locked loop design bandwidth.

[0084] Table 1 Main System Parameters

[0085]

[0086] Step 2: Analyze the transient instability mechanism of the model from the aspects of the existence of the equilibrium point and the transient synchronization behavior. First, through the Newton-Raphson method, determine whether there is an equilibrium point in the system during the transient period. If there is no equilibrium point, the system is unstable. Second, use the equal-area method to visually analyze the transient synchronization behavior of the model. As shown in Figure 3, S a and S d represent the accelerating and decelerating regions respectively, and curves I and II represent the changes of and respectively, verifying that the inner-loop current will cause changes in the equilibrium point and the decelerating area, ultimately affecting the transient synchronization behavior.

[0087] Step 3: First, combine the first integral method and the extended invariance principle to construct a candidate Lyapunov function V. In this example, λ k (k = 1,..., N) takes 0.06 to make satisfy semi-negativity, ensuring that V is applicable to the analysis of transient synchronization stability. Use V to derive the critical energy value V cr at the unstable equilibrium point considering nonlinear damping dissipation, where the nonlinear damping dissipation is approximated by the triangular area method (as shown in Figure 4). Then, based on W and W cr at the unstable equilibrium point, calculate the transient synchronization stability index η.

[0088] Step 4: Extract the key factors that may affect the transient synchronization stability of the nonlinear system through the transient synchronization stability criterion constructed in Step 3, such as the short-circuit ratio SCR (a basic index quantifying the grid connection strength, SCR ≈ 1X g ), line impedance, PI gain of the phase-locked loop, line reactance, grid-side resistance, current reference value, and reveal their specific impacts on the transient synchronization stability. This experiment uses N (N = 4) parallel VSCs scenarios as shown in Figure 5 for simulation. Based on the circuit models in Figure 2 and Figure 6 , it is established in MATLAB / SIMULINK. When the simulation scenario is t = 0.5 s, the grid voltage U g drops sharply to 0.3 p.u. Unless otherwise specified, the parameter settings remain unchanged.

[0089] The region of attraction is an approximate estimate value on the contour plot of the proposed Lyapunov function W at a fixed level W cr above. Figure 6 It shows that a larger SCR will increase the region of attraction, that is, a larger SCR will enhance the transient synchronization stability of the system. Table 2 gives the results of the transient synchronization stability analysis for Case A. As the SCR decreases, the stability index η decreases. Under conditions A1 and A2, the calculated results of η are both greater than 0, and the system under A1 is more stable than that under A2 (A1: η = 0.8747 is greater than A2: η = 0.3053). Figure 7 Through time-domain simulation, it shows that the waveform oscillation amplitude under condition A2 is greater than that under condition A1, which is consistent with the theoretical results. Under condition A3, further reducing the value of SCR, the system becomes unstable (η < 0), which is consistent with the simulation results, and the simulation results verify the accuracy of the theoretical analysis.

[0090] Table 2 Results of transient synchronization stability analysis under different SCR values

[0091] Case A <![CDATA[X g > η Results of the Proposed Method Simulation Results A1 0.1818 (SCR = 5.5) 0.8747(>0) Stable Stable A2 0.3333 (SCR = 3.0) 0.3053(>0) Stable Stable A4 0.6667 (SCR = 1.5) -0.1429(<0) Unstable Unstable

[0092] Keep the VSC 2-4 impedance value unchanged, and select four different Z 1 of the VSC line1,i (i = 1, 2, 3, 4) (Table 3). Z line1,1 , Z line2,1 and Z line3,1 have almost the same impedance angle but different amplitudes. Z line3,1 and Z line4,1 have almost the same magnitude but different impedance angles. Figure 8 It shows that a smaller impedance magnitude or impedance angle will increase the region of attraction, that is, a smaller Z line1,i will enhance the transient synchronization stability of the system. Table 3 gives the transient synchronization stability results for Case B, Figure 9 and the accuracy of the conclusion of the proposed method is verified through time-domain simulation.

[0093] Table 3 Results of transient synchronization stability analysis under different Z line1,i values

[0094] Case B <![CDATA[Z line1,i > η Results of the Proposed Method Simulation Results B1 0.19+j0.065 0.4894(>0) Stable Stable B2 0.1046+j0.3217 -0.0245(<0) Unstable Unstable B3 0.2508+j0.7726 -0.1698(<0) Unstable Unstable B4 0.6599+j0.4739 0.3266(>0) Stable Stable

[0095] Figure 10 shows that a larger proportional gain K pPLL or a smaller integral gain K iPLL can significantly increase η, thereby improving the transient synchronization stability. Table 4 gives the transient synchronization stability results for Case C, Figure 11 and the accuracy of the conclusion of the proposed method is verified through time-domain simulation.

[0096] Table 4 Different K iPLL and K pPLL Results of transient synchronous stability under different values

[0097] Case C <![CDATA[K iPLL > <![CDATA[K pPLL > η Results of the Proposed Method Simulation Results C1 0.15 10 0.1074 Stable Stable C2 0.15 12 -0.0927 Unstable Unstable C3 0.50 12 0.0850 Stable Stable

[0098] In Case D, the grid voltage drops sharply to 0.25 p.u. Figure 12 shows that a larger R g or can significantly increase η, thus improving transient synchronous stability. And when taking the minimum values of both R g and (point a in Figure 12(a)), the system is most prone to instability risk. Table 5 gives the transient synchronous stability results of Case D, Figure 13 and the accuracy of the conclusions of the proposed method is verified through time-domain simulation.

[0099] Table 5 Results of transient synchronous stability under different R g and values

[0100]

[0101] Figure 14 shows that a smaller X line,k or can significantly increase η, thus improving transient synchronous stability. And when taking the maximum values of both X line,k and (point b in Figure 14(a)), the system is most prone to instability risk. Table 5 gives the transient synchronous stability results of Case D, Figure 15 and the accuracy of the conclusions of the proposed method is verified through time-domain simulation.

[0102] Table 6 Results of transient synchronous stability under different X line,k and values

[0103]

[0104] This specification describes each case in a progressive structure, highlighting the unique features of each case. For the same or similar parts, reference can be made to each other. Among them, Case A is taken as an example for detailed analysis, focusing on the analysis method and related conclusions of transient synchronous stability. The analysis methods of other cases (such as B, C, D, and E) are similar to that of Case A, so the description is relatively brief, only highlighting their key differences and results. Through specific cases, this article systematically expounds the principle and implementation mode of the present invention, aiming to help understand the core idea of the method. For those skilled in the art, without departing from the overall concept of the present invention, modifications or equivalent replacements can be made to the specific implementation modes. At the same time, according to the idea of the present invention, adaptive adjustments can also be made in the implementation modes and application scopes. In summary, the content of this specification should not be construed as a limitation to the present invention.

Claims

1. A transient synchronization stability analysis method for multiple parallel grid-following converters, characterized in that The following steps are involved: Step 1: Construction of a grid-connected model of multiple parallel VSCs considering the inner loop current and phase-locked loop coupling effects; In a grid-connected system with N parallel VSCs, N phase-locked loops are defined with N dq axis reference frames. In addition, the DQ axis is defined as the reference frame, with the grid voltage vector U g For reference, θ g and ω g Indicates U g The phase angle and angular velocity of the kth grid-connected converter VSC k (k=1,...,N) as an example, VSC under DQ framework k The terminal voltage vector U t,k The expression is, Where, subscript j represents the jth VSC and j≠k, I j Represents VSC j Output current vector, I k Represents VSC k Output current vector, Z g =R g +jX g and Z linek =R line,k +jX line,k Respectively represent the grid side impedance and line impedance, R g and R line,k Respectively represent the grid side resistance and line resistance, X g and X line,k They represent grid-side reactance and line reactance respectively. Will U t,k Decomposed into d-axis and q-axis components, the expression is, In the formula, the power angle δ k =θ PLL,k -θ g Indicates U t,k Ahead of U g The resulting angle difference, δ kj =δ k -δ j Represents VSC k and VSC j The power angle difference formed. is the current vector I k The injection angle θ I,k represents the current injection angle, θ Zlinek and θ Zg Respectively represent Z linek and Z g The impedance angle. According to Kirchhoff's voltage law and the inner loop current control block diagram, VSC k The control equation of the inner loop current is: Wherein, the superscript "·" is the derivative with respect to time t, and Represents the active current and reactive current deviation respectively. dC,k and x qC,k is the state variable of the current loop control, e d,k and e q,k is the control variable of the current loop control, K pC and K iC They represent the proportional gain and integral gain of the inner loop current PI regulator, L f Represents the filter inductance on the AC output side of the grid-connected converter. According to the phase-locked loop control block diagram, VSC k The state equation of the phase-locked loop can be derived as follows: Among them, K pPLL and K iPLL Respectively represent the proportional gain and integral gain of the phase-locked loop PI regulator, △ω k Defined as the output angular frequency ω of the phase-locked loop k With ω g angular frequency difference. Furthermore, by combining equations (2), (4), and (5), we can obtain VSC k The sixth-order nonlinear dynamic model of . Among them, M k is the equivalent inertia, D eq,k is the equivalent damping, P m,k and P e,k are the equivalent mechanical power and equivalent electromagnetic power respectively, y d,k and q,k The detailed expression is as follows: Among them, L g and L line,k They represent the grid-side inductance and line inductance respectively. The model fully considers the coupling effect between the dynamic characteristics of the inner loop current and the phase-locked loop. Subsequent transient synchronization stability analysis is based on this model. Step 2: Analysis of transient synchronous instability mechanism of multi-parallel VSCs grid-connected system considering inner loop current; For the sixth-order nonlinear dynamic model constructed in step 1, the Newton-Raphson method and the equal-area criterion are used to confirm the equilibrium point and analyze the transient synchronization behavior respectively. The method flow is as follows: First, confirm the existence of the equilibrium point: the sufficient condition for the existence of the system equilibrium point U tq,k (x1,x2,...,x N )=0,k=1,2,...,N, this is a highly nonlinear system of equations, where the vector x k =(δ k ,△I d,k ,△I q,k ), k=1,2,...,N represents VSC k The Newton-Raphson method is used to perform numerical calculations on the nonlinear equations. (a) Initial value selection: Initialize the state variable matrix x (i) , set the differential term on the left side of the sixth-order nonlinear model to zero, the initial steady-state value of the state variable can be accurately calculated, and the steady-state value before the fault can be selected As the initial value; (b) Taylor expansion: The matrix U tq (x1,x2,...,x N ) Perform Taylor expansion at the initial estimation point and retain the first-order term to obtain the linearized approximation The superscript i is the current iteration number, [J (i) ] represents the Jacobian matrix of the current iteration point, and the matrix x is obtained (i) ; (c) Iterative solution: Update the estimated value, and use the x obtained from (b) (i+1) As the initial value for the next iteration. Repeat (a) and (b) until all elements in the offset matrix are less than the convergence threshold 10 7 , the iterative calculation stops; Then, transient synchronous behavior analysis: the inner loop current dynamics will be directly reflected in P m,k , which is refined into and They are defined as the equivalent mechanical power when the inner loop current characteristics are ignored and considered respectively. The expressions are as follows: The equal-area criterion method is used to intuitively analyze the transient synchronization behavior of the sixth-order nonlinear dynamic model, thereby revealing the influence of the dynamic characteristics of the inner loop current on the transient synchronization behavior of the grid-connected system. Step 3: Construction of transient synchronization stability criterion based on approximate Lyapunov direct method For the sixth-order nonlinear dynamic model constructed in step 1, the transient synchronization stability criterion is strictly constructed based on the approximate Lyapunov direct method. The method flow is as follows: First, function construction and verification: Ignore the nonlinear damping effect, combine the first integral method and the extended invariance principle to construct a candidate Lyapunov function. In the formula, δ s,k Represents VSC k The stable equilibrium point is obtained by the Newton-Raphson method in step 2, λ k is an undetermined parameter that can adjust the weight of the mixed term without changing the equilibrium point. The derivative of the candidate Lyapunov function The expression is, In the formula, H k The detailed expression of the sub-block is: Formula (11) consists of a quadratic term plus the last term, which is obtained by choosing the parameter λ k Make the quadratic term a positive definite term, then is negatively semidefinite and satisfies the principle of extension invariance. k When inequality (11) is satisfied, V can be used to study transient synchronous stability. Secondly, the critical energy value is calculated: the critical energy value V considering nonlinear damping dissipation at the unstable equilibrium point is derived using the V constructed above cr , its expression is as follows, V cr =V k (△ω k )+V p (δ u ,△ω u )+∫D eq △ωdδ (12) Among them, δ u Indicates the unstable equilibrium point, represented by △ω u Denotes the angular frequency deviation at the unstable equilibrium point, D eq △ω corresponds to the friction force in the transient process, while the damping D eq The dissipation of is related to the integral path of friction. Since the integral term of friction is difficult to calculate accurately, the triangle area approximation method is used to estimate the damping dissipation. Specifically, assuming that VSC k The integral path of the friction force is approximated by the area of ​​a triangle whose base corresponds to the path of the system from the initial point δ0 to the maximum phase angle δ max,k The range of change, the height corresponds to the maximum angular velocity deviation △ω max,k The maximum phase angle expression is as follows, Among them, δ max,k The value range of (δ s,k ,δ u,k ], whose value equation is given by V p,k (δ0)=V p,k (δ k ) is determined. Define W da,k For the system to run from δ0 to δ s,k The damping dissipates energy. At SEP, it has the minimum transient potential energy V pmin = 0 and the maximum transient kinetic energy V kmax,k , V kmax,k Determined by the system's initial transient energy minus the damping dissipation, VSC k The grid-connected system runs along the path from δ0 to δ s,k The damping dissipation W da,k and the path from δ s,k Run to δ max,k The damping dissipation W dd,k The area of ​​the triangle is approximately estimated. Based on the above analysis, considering D eq The approximate critical energy value V of dissipation cr It can be expressed as, Where γ=8M k V p,k (δ0)+((δ s,k -δ0)D eq,k ) 2 Based on the above structure V and critical energy value V cr , the analytical expression of the transient synchronization stability criterion is, Among them, η is defined as the transient synchronization stability index, which is used to quantitatively evaluate the transient synchronization stability. η≥0 means that the system is in a transient synchronization stability state, and the larger the η value, the better the transient synchronization stability. Otherwise, the system is in an unstable state. Through the above steps, a transient synchronization stability analysis method for multiple parallel grid-following converters is given.

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