A power distribution network variational iteration method stability analysis method

Abstract

The application discloses a power distribution network variational iteration method stability analysis method in the field of efficient numerical simulation algorithm of distributed photovoltaic system, which comprises the following steps: firstly, constructing a photovoltaic mathematical model based on algebraic equations; then, judging whether each parameter of the physical system and the control system meets the numerical stability condition; then, calculating the fixed step length updating expression, and performing numerical stability analysis on the fixed step length updating expression, calculating the error iteration format, and expanding the maximum equivalent step length of the variational iteration method; finally, introducing the event trigger related theory, taking the norm of the error vector as the trigger judgment condition, obtaining the error threshold, and realizing the partial state quantity lock value through the trigger condition. The method is based on the approximate analytical algorithm, and compared with the traditional numerical integral method, allows a larger simulation step length; the precision and efficiency in the simulation process can be considered, the simulation efficiency of the photovoltaic power station is improved, and then the simulation efficiency of the modern power distribution network is improved.

Description

Technical Field

[0001] This invention belongs to the research field of efficient numerical simulation algorithms for distributed photovoltaic systems, specifically involving a variational iterative method for stability analysis of distribution networks. Background Technology

[0002] Modern active power distribution systems with high-density distributed photovoltaic (PV) access are multidimensional, high-order, complex, nonlinear rigid systems. These systems contain numerous differential, integral, and algebraic equations, with intricate relationships between state variables, making unified solutions difficult. Dynamic numerical algorithms are commonly used to solve such models, and research on these algorithms has always been a key aspect of electrical simulation. Dynamic methods must meet both accuracy and speed requirements, which is crucial for real-time tracking and control of modern active power distribution networks with high-density distributed power access, where dynamic characteristics are uncertain and state changes are random. Currently, the common solution method for differential equation systems (ODEs) is numerical methods, which discretize the differential equations and sequentially obtain the function values ​​for each time step using initial values. These algorithms struggle to balance accuracy and speed. Considering the greater differences in the characteristics of state variables in modern power distribution networks; the interleaving of multiple time scales and the significant rigidity issues; and the high-order, high-dimensional, and highly nonlinear characteristics of the system model, the dynamic simulation of power distribution systems with high-density distributed power access faces severe challenges. Therefore, while ensuring the accuracy of the simulation, it is necessary to propose simulation algorithms that consume fewer resources, are simpler to compute, and are faster to meet the needs of large-scale systems.

[0003] For calculating state variables, the most accurate and fastest method is explicit analytical solutions. This only requires providing the simulation time step; the instantaneous value at that moment can be calculated using functional relationships. Compared to numerical integration, this avoids resource-intensive iterative calculations. However, in practical applications, various systems often lack direct analytical solutions. In recent years, the Variational Iteration Method (VIM) has been widely used in approximate analytical calculations due to its high computational accuracy. When applied to distributed generation simulation in power distribution networks, this method improves computational efficiency compared to traditional implicit algorithms. Considering that this method essentially involves obtaining the variable parameter calculation formulas for each state variable, it requires updating the state variable expressions at each simulation time step, leading to unnecessary computational resource consumption. Efficiency can still be optimized by reducing the computational load. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a variational iterative stability analysis method for power distribution networks, thereby resolving the problems mentioned in the background section.

[0005] The objective of this invention can be achieved through the following technical solutions:

[0006] A variational iterative stability analysis method for power distribution networks includes the following steps:

[0007] First, the variational iteration method is applied to obtain the approximate analytical solutions corresponding to each differential equation in photovoltaics, and a photovoltaic mathematical model based on algebraic equations is constructed.

[0008] Then, based on the construction of a photovoltaic mathematical model based on algebraic equations, the state equation corresponding to the photovoltaic model is obtained. By calculating the spectral radius of the state matrix, the maximum allowable step size under numerical stability is derived. Based on the numerical stability based on spectral analysis, it is determined whether the parameters of the physical system and the control system meet the numerical stability conditions.

[0009] Next, the fixed-step update expression is calculated, and numerical stability analysis is performed on the fixed-step update expression. Its error iteration scheme is calculated, and the maximum equivalent step size of the extended variational method is determined.

[0010] Finally, the theory of event triggering is introduced, and the norm of the error vector is used as the triggering condition to obtain the error threshold. The state variables are locked through the triggering condition.

[0011] Preferably, the photovoltaic mathematical model is established as follows:

[0012] First, the differential equations in distributed photovoltaic systems are transformed into algebraic equations using a variational algorithm, and the corrected functional is constructed as shown below:

[0013]

[0014] Then, a mathematical model is constructed by combining algebraic and differential equations for each module in the distributed photovoltaic system, including the photovoltaic array, boost circuit, and inverter circuit.

[0015] Finally, variational methods were used to transform all the differential equations in the photovoltaic system into a system of algebraic equations.

[0016] Preferably, the mathematical model corresponding to the photovoltaic array is as follows:

[0017]

[0018]

[0019] The mathematical model for the Boost circuit is as follows:

[0020]

[0021]

[0022]

[0023] The mathematical model corresponding to the inverter circuit is as follows:

[0024]

[0025]

[0026]

[0027]

[0028]

[0029]

[0030] .

[0031] Preferably, the mathematical model of the grid-connected inverter after converting the differential equation into an algebraic equation is first integrated. The controller adopts single-loop constant power control, and the variational method uses a second-order scheme, resulting in the following corresponding algebraic model:

[0032]

[0033] Then, for each simulation time step, an alternating solution calculation is performed, using the results of interactive differential equations and algebraic equations to determine the modulated voltage in the controller. u md1 , u md2 The integral equation has no other state variables coupled, so the Euler method is directly applied to discretize it. Other variables are solved using variational iteration to obtain an approximate analytical solution, taking the output current as the input. i d , i q， Integral part of the modulated wave voltage u md1 , u mq1 Once the state vector is formed, the corresponding state-space expression is as follows:

[0034] .

[0035] Preferably, according to the convergence principle of discrete systems, when the spectral radius of the state matrix of the discrete system is less than 1, the system satisfies the numerical stability condition. Based on the parameters in the known matrix, it is determined whether the system can be stable. When other undetermined parameters are known, the maximum allowable simulation step size is calculated based on the spectral radius being less than 1.

[0036] Preferably, based on the principle of fixed-step update, a locking condition is applied in the k-th simulation step, with a locking condition of n steps. The original state equation then becomes as follows:

[0037]

[0038] To obtain the state-space equation corresponding to the fixed-step update, since each state variable is locked for n steps, the equivalent step size in the simulation process is expanded to nΔ. t By analyzing the state matrix A’ The spectral radius is used to determine whether the system is numerically stable.

[0039] Preferably, the state-space expression is simplified to the following form:

[0040]

[0041] And assume that the state vector is when the steady state is reached. There is an error between the state vector at the start of the lock-up period and the state vector at the steady state. e 0. After locking the value, the state matrix changes to... A 1. Then the new state-space expression for the system is:

[0042]

[0043] remember A 1= A +Δ A Substituting the above equation and changing it, we get the following:

[0044]

[0045] Subtract from both sides of the equation and A + C 1. The iterative equation for the error is as follows:

[0046]

[0047] Then, the error iteration equation in the above equation is transformed into a continuous form using a continuous method. The equation is then transformed from continuous to discrete form:

[0048]

[0049] By applying the inequality constraints of norm multiplication, we obtain the following equation:

[0050]

[0051] The above equation is a basic first-order differential equation. We can calculate it using the general formula, and by calculating the range of values ​​for the error vector norm in this differential inequality, we can derive the boundary conditions for the event triggering, which can then be written in inequality form:

[0052]

[0053] Taking the value corresponding to t-t0=Δt as its threshold, the final constraint conditions are as follows:

[0054] .

[0055] The beneficial effects of this invention are:

[0056] 1. The method of this invention is based on an approximate analytical algorithm, which is applied to a modern power distribution system with a large number of photovoltaic systems connected. It transforms the system into a system of algebraic equations and provides a numerical stability verification method, which allows for a larger simulation step size compared to traditional numerical integration methods.

[0057] 2. The method of this invention is based on the event triggering principle. Under certain conditions, it locks the values ​​of the computationally complex state variables and uses fixed step size update or update when the error vector norm meets certain conditions. This can balance the accuracy and efficiency in the simulation process. By improving the simulation efficiency of photovoltaic power plants, it can improve the simulation efficiency of modern power distribution networks and create conditions for large-scale power distribution system simulation. Attached Figure Description

[0058] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0059] Figure 1 This is a flowchart of the method of the present invention;

[0060] Figure 2 This is a simulation diagram of the grid-connected three-phase voltage source inverter used in this invention;

[0061] Figure 3 This is a comparison diagram of active power waveforms calculated using the variational method with different timing lengths in this invention;

[0062] Figure 4 This is a simulation comparison chart of active power using the Euler method, trapezoidal method, and variational method in this invention with a step size of 0.1ms.

[0063] Figure 5 This is a comparison chart of the active power of the inverter under time-triggered, fixed-step-size-triggered, and event-triggered modes in this invention.

[0064] Figure 6 This is a comparison diagram of inverter voltage under time-triggered, fixed-step-size-triggered, and event-triggered conditions in this invention;

[0065] Figure 7This is a structural diagram of the IEEE-33 node test system with distributed photovoltaic system used in this invention;

[0066] Figure 8 This is a comparison diagram of the active power waveforms of bus 8 of the test system under different simulation methods in this invention;

[0067] Figure 9 This is a comparison diagram of reactive power waveforms of bus 8 of the test system under different simulation methods in this invention;

[0068] Figure 10 This is a comparison diagram of voltage waveforms of bus 8 of the test system under different simulation methods in this invention. Detailed Implementation

[0069] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0070] Please see Figure 1 As shown, this invention proposes a variational iterative method for stability analysis of distribution networks, comprising the following steps:

[0071] Step 1: Apply variational iteration method to obtain approximate analytical solutions to the differential equations in photovoltaics, construct a photovoltaic mathematical model based on algebraic equations, and then proceed to Step 2.

[0072] In the process of establishing the mathematical model of the distributed photovoltaic system, based on the iterative format of the variational algorithm, the mathematical model of the distributed photovoltaic system, including modules such as photovoltaic arrays, boost circuits, and inverter circuits, is as follows:

[0073] First, a variational algorithm is used to transform the differential equations in distributed photovoltaic systems into algebraic equations, with the original format as follows:

[0074] (1)

[0075] in, y ( t ) is the function corresponding to the state variable. 、 These are linear operators and nonlinear operators, respectively. Based on variational theory, the corrected functional of equation (1) can be constructed as follows:

[0076] (2)

[0077] in, nThis indicates the number of iterations for the variational scheme. λ (τ) denotes the Lagrange multiplier. for The restricted variation means that the variational result of the expression is 0.

[0078] Then, a mathematical model is constructed for each module in the distributed photovoltaic system, including the photovoltaic array, boost circuit, inverter circuit, etc., by combining algebraic and differential equations. The mathematical models corresponding to each module are as follows:

[0079] Photovoltaic array:

[0080] (3)

[0081] (4)

[0082] in, u PV , i PV These represent the output voltage and current of the photovoltaic array, respectively. i Boost,L This represents the input current of the Boost circuit. C PV This represents the DC equivalent capacitance of a photovoltaic cell.

[0083] Boost circuit:

[0084] (5)

[0085] (6)

[0086] (7)

[0087] in, u Boost Indicates the chopper output voltage. i Boost Indicates the chopper output current. D Indicates the duty cycle of the PWM signal. L Boost , C Boost These are the inductor and capacitor in the Boost circuit. u M The output voltage of the photovoltaic array corresponding to the maximum power point. k p1 , k i1 These are the controller parameters in the Boost circuit.

[0088] Inverter circuit (using constant power control, retaining only the current loop, and performing Parker transformation with the d-axis in phase with A):

[0089] (8)

[0090] (9)

[0091] (10)

[0092] (11)

[0093] (12)

[0094] (13)

[0095] (14)

[0096] in, u md and u mq , u id and u iq , and , and These represent the dq components of the SPWM modulation voltage, inverter circuit voltage, inverter input voltage, and inverter circuit output current, respectively. k 0 indicates the modulation ratio. k p2 , k i2 These are the PI parameters for constant power control. When the inverter uses sinusoidal pulse width modulation, k 0 take .assumed u boost It is 800V. U AC If the voltage is 380V, then it can be calculated that... m =2.227.

[0097] Finally, the variational method is used to transform all the differential equations in the photovoltaic system into a system of algebraic equations. Taking equation (8) as an example, the general steps for finding an approximate analytical solution using the variational method are given. Combining equation (2), its iterative format is as follows:

[0098] (15)

[0099] Considering the alternating solution process used in DAEs, for differential equations,u sd , u id , L f All are known quantities obtained from algebraic equations, which can be represented by parameters. A Perform a substitution. Applying variational operations to equation (15), we have:

[0100] (16)

[0101] According to variational correlation theory, as the iteration proceeds, the left side of equation (16) gradually approaches 0, from which it can be deduced that:

[0102] (17)

[0103] According to the above formula, we can solve for... λ =-1, substituting the Lagrange operator into equation (15) yields the following result. i d The corrected functional can be obtained by using any initial function and iterative calculation to obtain the approximate analytical solutions of each order of the state variable. The same method can be used to calculate the corrected functionals of other state variables.

[0104] Step 2: Obtain the state equation corresponding to the photovoltaic model, derive the maximum allowable step size under numerical stability by calculating the spectral radius of the state matrix, and determine whether the parameters of the physical system and control system meet the numerical stability conditions based on the numerical stability method of spectral analysis, and then proceed to Step 3.

[0105] First, integrate the mathematical model of the grid-connected inverter after converting the differential equation into an algebraic equation. The controller adopts single-loop constant power control, and the variational method uses a second-order scheme. The corresponding algebraic model is as follows:

[0106] (18)

[0107] In equation (18), the subscript 0 indicates that the variable is the initial value after the calculation formula is updated once.

[0108] Finally, for each simulation time step, an alternating solution calculation is performed, achieved through the results of interactive differential equations and algebraic equations. The modulation voltage in the controller is taken into consideration. u md1 , u md2 Since there are no other state variables coupled in the integral equation, the Euler method is directly applied for discretization, and the other variables are solved using variational iteration to obtain an approximate analytical solution. The output current is taken as an example. i d , i q，Integral part of the modulated wave voltage u md1 , u mq1 Once the state vector is formed, the corresponding state-space expression is as follows:

[0109] (19)

[0110] Where, Δ t For simulating step size, C This represents a constant matrix, whose element values ​​can be calculated from the inverter's system parameters. M Indicates the output of the inverter u i With modulated wave u m The ratio of [the parameters]. According to the convergence principle of discrete systems, when the spectral radius of the state matrix of a discrete system is less than 1, the system satisfies the numerical stability condition. Based on the parameters in this known matrix, it can be determined whether the system is stable. When other undetermined parameters are known, the maximum allowable simulation step size can be calculated based on the spectral radius being less than 1.

[0111] Step 3: Calculate the fixed-step update expression, perform numerical stability analysis on the fixed-step update expression, calculate its error iteration scheme, expand the maximum equivalent step size of the variational method, and then proceed to Step 4.

[0112] Based on the principle of fixed-step update, assuming that locking is performed in the k-th simulation step for n steps, the original state equation will become as follows:

[0113] (20)

[0114] Obtain the state-space equation corresponding to the fixed-step update. Since each state variable is locked for n steps, this method expands the equivalent step size in the simulation process to nΔ. t For this form, it is still possible to analyze the state matrix. A’ The spectral radius is used to determine whether the system is numerically stable.

[0115] Step 4: Introduce event triggering theory, use the norm of the error vector as the triggering condition to obtain the error threshold, and lock some state variables through the triggering condition to save simulation computing resources, improve the simulation speed of the original variational iterative method, and further improve the simulation efficiency of the entire system.

[0116] First, the state-space expression for each step of equation (19) can be simplified as follows:

[0117] (twenty one)

[0118] And assume that the state vector is when the steady state is reached. There is an error between the state vector at the start of the lock-up period and the state vector at the steady state. e 0. After locking the value, the state matrix changes to... A 1. Then the new state-space expression for the system is:

[0119] (twenty two)

[0120] remember A 1= A +Δ A The substitution formula (22) can be modified, namely:

[0121] (twenty three)

[0122] You can eliminate both sides, subtracting from both sides of the equation simultaneously. and A + C The iterative equation for the error can be obtained as follows:

[0123] (twenty four)

[0124] Finally, the error iteration equation of equation (24) above is transformed into a continuous form using the continuous method, so that it can be subjected to inequality transformation. From the transformation equation between continuous and discrete forms, we can derive:

[0125] (25)

[0126] Finding the norm of this expression, and applying the inequality constraints of norm multiplication, we obtain:

[0127] (26)

[0128] Equation (26) is the basic first-order differential equation, which can be calculated using the general formula. By calculating the range of values ​​for the norm of the error vector in this differential inequality, the boundary conditions for the event triggering can be derived, which can be written in inequality form as follows:

[0129] (27)

[0130] Considering that the right side of equation (27) is a monotonically increasing function, for a discrete-time system, the left side should be less than or equal to the minimum value of the right side, that is, the value corresponding to t-t0=Δt is taken as its threshold. The final constraint conditions are as follows:

[0131] (28)

[0132] Applying the above-designed technical solution to practice, the simulation system is as follows: Figure 2 As shown, the DC-side circuit on the left side of the converter is connected to an 800V DC input, and the right side of the converter is connected to a filter inductor and then grid-connected via a line. The grid line voltage is 380V RMS and the frequency is 50Hz. (Filter inductor...) L f The line resistance is 4 mH. R line The inductance is 0.1 Ω. L line The power reference value is 1mH. The system simulation time is 5s, the converter uses PQ control, and the power reference value is... P ref It is 20kW, of which it drops to 15kW in 2-3 seconds. Q ref 20kvar. Controller parameter settings. k p =1.5, k i =30. According to an embodiment of the present invention, a distributed photovoltaic grid-connected inverter is designed based on the stability analysis and parameter sensitivity study of the variational iterative method for distribution networks. A simulation model is built on the MATLAB platform to simulate the stability of each algorithm and the effect of event triggering, verifying the feasibility of the method of the present invention.

[0133] like Figure 3 The image shows a comparison of the active power waveforms calculated using the variational method with different step sizes in this embodiment. The simulation step sizes are 1 ms, 1.7 ms, 1.73 ms, and 1.74 ms, respectively, for comparison. The inverter output active power is shown below. Figure 3 As shown in the figure, when the simulation step size is less than 1.73 ms, the system can converge and oscillate around the steady-state value at 1.73 ms. However, when the step size is 1.74 ms, the system gradually diverges from the start of the calculation. The simulation results are consistent with the theoretical maximum simulation step size.

[0134] like Figure 4 The figure shows a simulation comparison of active power using the Euler method, trapezoidal method, and variational method with a step size of 0.1ms. At the start of operation, the reference active power of each inverter is 20kW; after 2 seconds, the power decreases to 15kW; and after 3 seconds, it recovers to 20kW. The simulation results are as follows... Figure 4 As shown, the horizontal axis represents time in seconds, and the vertical axis represents active power in kilowatts. The waveform comparison shows that, using the same 0.1ms simulation step size, the variational method achieves accuracy close to the trapezoidal method, and both methods outperform the Euler method. Table 2 compares the total time taken for 100 five-second simulations using each simulation algorithm. Because the variational method allows for larger step sizes in simulation calculations, it significantly improves simulation efficiency.

[0135] Table 1

[0136]

[0137] like Figure 5 , Figure 6 The figure shows the simulation results comparing the grid-connected inverters in this embodiment under time-triggered, fixed-step-triggered, and event-triggered conditions. At startup, the reference active power of each inverter is 20kW; after 2 seconds, the power decreases to 15kW; and after 3 seconds, it recovers to 20kW. The simulation results are as follows: Figure 5 and Figure 6 As shown, where, Figure 5 This is a comparison chart of the active power output of the inverter under different triggering methods. The horizontal axis represents time in seconds, and the vertical axis represents active power in kilowatts. Figure 6 This graph compares the output voltage of the inverter under different triggering methods. The horizontal axis represents time (seconds), and the vertical axis represents active power (volts). Figure 5 It is known that when using steady-state triggering, since the current value is still the previous steady-state value at the instant the current reference value is updated, substituting the difference into the modulation wave calculation formula will cause large fluctuations, which is not allowed in simulation. The accuracy and convergence speed of fixed-step triggering are not as good as event triggering, while event triggering can basically achieve the same effect as the original method.

[0138] like Figure 7 The diagram shows the structure of the IEEE-33 node test system with distributed photovoltaic (PV) power generation used in this embodiment. The distributed PV power generation system is connected to load nodes 8, 13, 25, and 33. The total system load is 5084.26 + j2547.32 kVA. The simulation step size is 10 μs. During normal operation, the initial environmental conditions of the PV power generation system are 1000 W / m². 2 Both are at 25℃ and operate at unity power factor. Line parameters: C PV =10 -4 F, L Boost =50 mH, C Boost =5 mF, L f =5 mH. Controller parameters: k p1 = k p2 =1.5, k i1 = k i2 =30. The total simulation time is 5 seconds. When the system runs for 2.5 seconds, the photovoltaic irradiance drops to 800W / m².2 It recovers to 1000 W / m in 3 seconds. 2 .

[0139] like Figures 8 to 10 The figures show the simulation results of the improved variational method based on event triggering, the variational method without triggering locking, and the trapezoidal integral method in this embodiment. The per-unit values ​​of the active power, reactive power, and effective voltage of node 8 are compared. The simulation results are as follows: Figures 8 to 10 As shown, where, Figure 8 The waveform of active power at node 8 is shown. The horizontal axis represents time in seconds, and the vertical axis represents active power in pu. Figure 9 The waveform diagram for reactive power at node 8 is shown. The horizontal axis represents time (in seconds), and the vertical axis represents reactive power (in pu). Figure 10 The graph shows the effective voltage waveform at node 8, with the horizontal axis representing time (seconds) and the vertical axis representing voltage (pu). The comparison shows that the basic variational method and the trapezoidal method have similar accuracy, while the improved variational method has an error of approximately 0.1% when calculating the bus voltage. This error is small and basically meets the accuracy requirements. The simulation time for each algorithm is shown in Table 2. The simulation time mainly consists of distributed photovoltaic calculations and power flow calculations. Table 2 shows that the improved variational method reduces the photovoltaic calculation time by approximately 29% compared to the trapezoidal method.

[0140] Table 2

[0141]

[0142] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0143] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention.

Claims

1. A variational iterative stability analysis method for distribution networks, characterized in that, The method includes the following steps: Step 1: Apply variational iteration method to obtain approximate analytical solutions to the differential equations in photovoltaics, and construct a photovoltaic mathematical model based on algebraic equations; The process of establishing the photovoltaic mathematical model is as follows: First, the differential equations in distributed photovoltaic systems are transformed into algebraic equations using a variational algorithm, and the corrected functional is constructed as shown below: In the formula, n This indicates the number of iterations of the variational scheme. λ (τ) denotes the Lagrange multiplier. for The restricted variation means that the variational result of the expression is 0. 、 These are linear operators and nonlinear operators, respectively. Then, a mathematical model is constructed by combining algebraic and differential equations for each module in the distributed photovoltaic system, including the photovoltaic array, boost circuit, and inverter circuit. Finally, variational methods were used to transform all the differential equations in the photovoltaic system into a system of algebraic equations. The mathematical model corresponding to the photovoltaic array is as follows: In the formula, u PV , i PV These represent the output voltage and current of the photovoltaic array, respectively. i Boost,L This represents the input current of the Boost circuit. C PV This represents the DC equivalent capacitance of a photovoltaic cell; The mathematical model for the Boost circuit is as follows: In the formula, u Boost Indicates the chopper output voltage. i Boost Indicates the chopper output current. D Indicates the duty cycle of the PWM signal. L Boost , C Boost These are the inductor and capacitor in the Boost circuit. u M The output voltage of the photovoltaic array corresponding to the maximum power point. k p1 , k i1 These are the controller parameters in the Boost circuit; The mathematical model corresponding to the inverter circuit is as follows: In the formula, u md and u mq , u id and u iq , and , and These represent the dq components of the SPWM modulation voltage, inverter circuit voltage, inverter input voltage, and inverter circuit output current, respectively. k 0 indicates the modulation ratio. k p2 , k i2 These are the PI parameters for constant power control. m Indicates the output of the inverter u i With modulated wave u m The ratio; Step 2: Obtain the state equation corresponding to the photovoltaic model by constructing a photovoltaic mathematical model based on algebraic equations. Derive the maximum allowable step size under numerical stability by calculating the spectral radius of the state matrix. Determine whether the parameters of the physical system and the control system meet the numerical stability conditions based on the numerical stability method of spectral analysis. First, the mathematical model of the grid-connected inverter after converting the differential equation into an algebraic equation is integrated. The controller adopts single-loop constant power control, and the variational method uses a second-order scheme, resulting in the following corresponding algebraic model: In the formula, the subscript 0 indicates that the variable is the initial value after the calculation formula is updated once; Then, for each simulation time step, an alternating solution calculation is performed, using the results of interactive differential equations and algebraic equations to determine the modulated voltage in the controller. u md1 , u md2 The integral equation has no other state variables coupled, so the Euler method is directly applied to discretize it. Other variables are solved using variational iteration to obtain an approximate analytical solution, taking the output current as the input. i d , i q， Integral part of the modulated wave voltage u md1 , u mq1 Once the state vector is formed, the corresponding state-space expression is as follows: Where, Δ t For simulating step size, C This represents a constant matrix, whose element values ​​can be calculated from the inverter's system parameters; According to the convergence principle of discrete systems, when the spectral radius of the state matrix of a discrete system is less than 1, the system satisfies the numerical stability condition. Based on the parameters in the known matrix, it is determined whether the system can be stable. When other undetermined parameters are known, the maximum allowable simulation step size is calculated based on the spectral radius being less than 1. Step 3: Calculate the fixed-step update expression, perform numerical stability analysis on the fixed-step update expression, calculate its error iteration scheme, and determine the maximum equivalent step size of the extended variational method; Step 4: Introduce event triggering theory, use the norm of the error vector as the triggering condition, obtain the error threshold, and lock some state variables through the triggering condition.

2. The stability analysis method for distribution networks using variational iterative method according to claim 1, characterized in that, Based on the principle of fixed-step update, a locking condition is applied in the k-th simulation step, with a locking condition of n steps. The original state equation then becomes as follows: To obtain the state-space equation corresponding to the fixed-step update, since each state variable is locked for n steps, the equivalent step size in the simulation process is expanded to nΔ. t By analyzing the state matrix A’ The spectral radius is used to determine whether the system is numerically stable.

3. The stability analysis method for distribution networks using variational iterative method according to claim 1, characterized in that, The state-space expression can be simplified to the following form: And assume that the state vector is when the steady state is reached. There is an error between the state vector at the start of the lock-up period and the state vector at the steady state. e 0. After locking the value, the state matrix changes to... A 1. Then the new state-space expression for the system is: remember A 1= A +Δ A Substituting the above equation and changing it, we get the following: Subtract from both sides of the equation and A + C 1. The iterative equation for the error is as follows: Then, the error iteration equation in the above equation is transformed into a continuous form using a continuous method. The equation is then transformed from continuous to discrete form: By applying the inequality constraints of norm multiplication, we obtain the following equation: The above equation is a basic first-order differential equation. We can calculate it using the general formula, and by calculating the range of values ​​for the error vector norm in the differential inequality, we can derive the boundary conditions for the event triggering, which can then be written in inequality form: Taking the value corresponding to t-t0=Δt as its threshold, the final constraint conditions are as follows: 。

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