LMI-based two-level VSC-HVDC minimum DC capacitance calculation method

Abstract

The invention relates to the technical field of power electronics and direct current transmission, in particular to a two-level VSC-HVDC minimum direct current capacitance calculation method based on LMI. The method comprises the following steps: establishing a state equation comprising a main circuit and a controller dynamic model, converting a three-phase alternating current quantity into a direct current quantity under a synchronous rotating coordinate system by utilizing Park transformation, and deducing a system state equation in combination with a Kirchhoff's law; performing Taylor expansion and linearization on the nonlinear equation, and constructing a small signal model based on a Lyapunov stability theory; by separating a diagonal matrix and defining a stability criterion in an LMI form, minimum direct-current capacitance solving is converted into a convex optimization problem, a critical value is iteratively searched by adopting a dichotomy method to determine minimum capacitance, and meanwhile, a tolerance mechanism is introduced to improve the robustness of an algorithm. Through combination of mathematical modeling and an optimization algorithm, the calculation efficiency and accuracy are effectively balanced, and a theoretical basis is provided for reducing the cost and the size of the converter.

Description

Technical Field

[0001] The present invention relates to the technical field of power electronics and HVDC transmission, and particularly relates to a method for calculating the minimum DC capacitor of a two-level VSC-HVDC based on LMI. Background Art

[0002] VSC-HVDC (voltage source converter based high voltage direct current) has become a key technology for building a new power system due to its ability to independently control active and reactive power and its characteristic of no commutation failure; there have been many scientific and technological demonstration projects of VSC-HVDC technology in various application scenarios such as new energy grid connection, long-distance large-capacity power transmission, grid interconnection, DC grid construction, large city power supply, and power supply for passive networks; VSC-HVDC can quickly control the AC voltage and has good application prospects in scenarios of power supply to passive networks.

[0003] The DC capacitor is an important component on the DC side of the converter, and its main functions are to stabilize the DC voltage, buffer power fluctuations, and suppress harmonics; the size of the DC capacitor directly affects the dynamic performance, cost, and volume of the system; a small DC capacitor can significantly reduce the cost and space while meeting the system performance requirements, which is very necessary for VSC-HVDC in special scenarios such as offshore platforms; regarding the calculation of the minimum DC capacitor of VSC-HVDC, the key issue is the selection of the stability criterion; in the small-signal stability analysis of VSC-HVDC, the existing stability criteria include the generalized Nyquist criterion based on the impedance model and the eigenvalue method based on the state-space model; the generalized Nyquist criterion determines the stability of the system by analyzing the Nyquist plot of the transfer function matrix and finds its minimum value by adjusting the DC capacitor value; the eigenvalue method considers the role of the controller, based on the small-signal model, and determines the stability of the system by calculating the eigenvalues of the Jacobian matrix, and then gradually tries to calculate the minimum value of the DC capacitor.

[0004] Zhang Xu, Hao Zhiguo, Li Yujun, etc. proposed a sufficient condition for the stability criterion with a DC capacitor variable in the literature titled "Assessment of DC Voltage Oscillation Stability of VSC-HVDC System Based on Electrical Torque Method" (Power System Technology, 2024, 48(10): 4306-4316), from which the minimum value of the DC capacitor can be deduced.

[0005] Regarding the problem of solving the minimum value of the DC capacitor in a two-level VSC-HVDC system, the generalized Nyquist criterion requires frequent plotting and analysis of the Nyquist diagram, resulting in low computational efficiency when applied to high-dimensional VSC-HVDC systems. The eigenvalue method is generally considered a relatively direct and efficient stability criterion. However, when applied to a closed-loop system considering the dynamics of the controller, due to the high-dimensional and asymmetric characteristics of the Jacobian matrix, the computational complexity of eigenvalues increases, thus reducing the efficiency of solving the minimum value of the DC capacitor. It can be seen that the existing technologies have the problem of low computational efficiency. In addition, although the stability criterion of the electrical torque method with a DC capacitor gives an analytical expression, it is only a sufficient condition for small-signal stability and has strong conservatism, and cannot accurately obtain the accurate minimum DC capacitor value. Summary of the Invention

[0006] The object of the present invention is to propose a method for calculating the minimum DC capacitor of a two-level VSC-HVDC based on LMI in view of the problems in the background technology.

[0007] The technical solution of the present invention: A method for calculating the minimum DC capacitor of a two-level VSC-HVDC based on LMI includes the following specific implementation steps: S1. Establish the main circuit and controller state equations of the two-level VSC-HVDC system: Based on the symmetric harmonic-free assumption, use the Park transformation to convert three-phase AC quantities into DC quantities in the synchronous rotating coordinate system, and combine Kirchhoff's law to derive the main circuit state equation. Subsequently, establish the PI controller state equations for the sending-end constant DC voltage-zero reactive power control and the receiving-end droop control. Finally, substitute the controller output variables into the main circuit equation to output the VSC-HVDC system state equation; S2. Perform Taylor series expansion on the VSC-HVDC system state equation, retain the first-order term, and construct a small-signal model; S3. Based on the small-signal model, use the Lyapunov stability theory of linear time-invariant systems to construct an LMI stability criterion; Preferably, the construction of the LMI stability criterion is specifically as follows: Introduce a diagonal matrix to adjust the criterion structure, and verify the stability of the system with a given capacitor value through inequality constraint conditions; The decision variable P is a symmetric positive definite matrix. Combine parameters and tolerances to ensure the positive definite margin, and output the LMI stability criterion; S4. According to the LMI-form stability criterion, use the bisection method to solve the minimum value of the DC capacitor; S5. Build a two-level VSC-HVDC simulation for supplying power to a passive network on a simulation platform, conduct simulation verification, and obtain the equilibrium point of the system; S6. Output the minimum value of the DC capacitor.

[0008] Preferably, the establishment of the main circuit and controller state equations includes: S21. Define the reference direction, and use Kirchhoff's voltage and current laws to obtain the circuit equation set of the system in the three-phase stationary coordinate system, and transform it to the synchronous rotating coordinate system by Park transformation; S22. By establishing the PI controller state equation of the rectifier constant DC voltage - zero reactive power control and the inverter droop control equation, define the intermediate state variables and generate reference values based on the instantaneous power theory, and combine the current loop PI parameter adjustment to dynamically track the error to generate variables that control the main circuit; S23. Substitute the variables that control the main circuit output by the controller into the main circuit state equation to obtain the controller output: ; where, k pi1 is the rectifier side proportionality coefficient; k pi2 is the inverter side proportionality coefficient; U s1d and U s1q are the d-axis and q-axis components of the sending-end AC system voltage; U c1d and U c1q are the d-axis and q-axis components of the sending-end filter capacitor voltage; i 1d and i 1q are the d-axis and q-axis components of the sending-end current; w1 is the sending-end system angular frequency, w1 = 2πf1; f1 is the power grid frequency; L1 is the equivalent inductance of the sending-end AC side; L2 is the inductance of the connection part between the inverter and the AC system; w2 is the receiving-end system angular frequency; i 2d and i 2q are the components of the inverter AC side current in the dq coordinate system; i 1dr and i 1qr are the reference currents of the rectifier in the dq coordinate system, the d-axis is the active component, and the q-axis is the reactive component; m2, m3 are the intermediate state variables in the rectifier control system; m5, m7 are the intermediate state variables; U s2d and U s2q are the components of the voltage at the common connection point of the receiving-end AC system in the dq coordinate system; S24. Obtain the state equation of the two-level VSC-HVDC system: Define the state variable x: x = [i line , i ld , i 1q , U dc1 , m1, m2, m3, i 2d , i 2q , U s2d , U s2q , U dc2, m4, m5, m6, m7, i ld , i lq T ; where T is the matrix transpose operation; i line is the DC line current; U dc1 and U dc2 are the DC voltages; m1, m2, m3 are intermediate state variables in the rectifier control system; m4, m5, m6, m7 are intermediate state variables; U s2dr and U s2qr are the reference values of the AC voltages on the inverter side: d-axis and q-axis components; Define the input vector u = [U dcr , i 1qr , w n , Q n , U n , U s2qr T ; where U dcr is the reference voltage on the DC side; w n is the rated frequency of the system; Q n is the rated reactive power; Calculate to obtain: the state equation of the two-level VSC-HVDC system ; ; In the formula, the vector h(x) is the non-linear part of the system, and A, B, C are constant matrices; M(C) is a diagonal matrix; M r (C) = diag(L1, L2, C, 1, 1, 1, 1, 1); M i (C) = diag(L2, L2, C f , C, 1, 1, 1, 1, L l , L l ); G is the non-linear gain matrix.

[0009] Preferably, the transformation process of converting to the synchronous rotating coordinate system by Park transformation is as follows: The Park transformation matrix is: ; In the formula, θ is the electrical angle of the d-axis, with the a-axis as the reference; T abc / dq0 is the transformation matrix from the three-phase stationary coordinate system abc to the synchronous rotating coordinate system dq0, which converts three-phase AC quantities into DC quantities in the synchronous rotating coordinate system; The state equation of the sending-end AC system is: ; In the formula, L1 is the equivalent inductance on the sending-end AC side; R1 is the equivalent resistance on the sending-end AC side; and​​ are the time derivatives of the d-axis and q-axis current components; i 1d and i 1q are the d-axis and q-axis components of the sending-end current; w1 is the angular frequency of the sending-end system, w1 = 2πf1; f1 is the grid frequency; U s1d and U s1q are the d-axis and q-axis components of the sending-end AC system voltage; U c1d and U c1q are the d-axis and q-axis components of the sending-end filter capacitor voltage; The state equation of the DC loop is: ; Where C is the DC-side capacitor; U dc1 and U dc2 are the DC voltages; i 1d and i 1q are the components of the rectifier AC-side current in the dq coordinate system; i 2d and i 2q are the components of the inverter AC-side current in the dq coordinate system; U cld and U c1q are the components of the rectifier AC-side voltage in the dq coordinate system; U c2d and U c2q are the components of the inverter AC-side voltage in the dq coordinate system; i line is the DC line current; L dc is the DC line inductance; R dc is the DC line resistance; ; Where U s2d and U s2q are the components of the receiving-end AC system common connection point voltage in the dq coordinate system; i ld and i 1q are the components of the AC load current in the dq coordinate system; L2 and R2 are the inductance and resistance of the connection part between the inverter and the AC system; C f is the AC-side filter capacitor; w2 is the angular frequency of the receiving-end system.

[0010] Preferably, the adjustment process of the dynamic tracking error is as follows: The state equation of the rectifier control system is constructed as: ; ; Where m1, m2, m3 are intermediate state variables in the rectifier control system; U dcr is the reference voltage on the DC side; U dc1 is the actually measured DC voltage on the rectifier side; i 1dr and i1qr is the reference current of the rectifier in the dq coordinate system, where the d-axis is the active component and the q-axis is the reactive component; k pv1 and k iv1 are the proportional coefficient and integral coefficient of the PI controller; i 1d and i 1q are the components of the actual current on the AC side of the rectifier in the dq coordinate system; The state equation for constructing the inverter control system is: ; ; where m4, m5, m6, m7 are intermediate state variables; U s2dr and U s2qr are the reference values of the AC voltage on the inverter side: d-axis and q-axis components; i 2dr and i 2qr are the reference values of the inverter current: the d-axis is the active component and the q-axis is the reactive component; k pv2 and k iv2 are the proportional coefficient and integral coefficient of the current-loop PI controller; The control mode of the inverter belongs to network-forming control, and the droop control link is: ; According to the instantaneous reactive power theory: ; where w n is the rated frequency of the system; P s and P n are the actual active power and the rated active power; U n is the rated voltage; k q is the voltage-reactive power droop coefficient; Q s and Q n are the actual reactive power and the rated reactive power.

[0011] Preferably, the linearized small-signal model is: ; where Δx is a small perturbation, Δx = x - x e ; x e is the equilibrium state; H is a matrix, ; where A r,dc = -(A dc,r ) T = -[0, 0, 1, 0, 0, 0] T ; A i,dc = -(A dc,i ) T = -[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0]T ; H r = A r + A1; H i = A i + A1; A r and A i and A do not contain the equilibrium state; the non - zero elements in A1 and A2 contain the equilibrium state.

[0012] Preferably, the construction process of the LMI stability criterion is as follows: S71. Separate the diagonal matrix M(C), from M(C) -1 = M2(C)M1; where, M2(C) = diag(1, 1, 1, 1 / C, 1, 1, 1, 1, 1, 1, 1, 1 / C, 1, 1, 1, 1, 1, 1); M1 = diag(1 / (L dc ), 1 / (L1), 1 / (L1), 1, 1, 1, 1, 1 / (L2), 1 / (L2), 1 / (C f ), 1 / (C f ), 1, 1, 1, 1, 1, 1 / (L1), 1 / (L1)); S72. Define the matrix R = M1H, the matrices M>0 and M<0 represent positive definite and negative definite respectively; When the value of the scalar variable C is given , according to the Lyapunov stability theory of linear time - invariant systems, there is a stability criterion in the form of LMI ~ LMI: ; In the formula, the diagonal matrix N = diag(1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1); P is a symmetric positive definite matrix and is a decision variable.

[0013] Preferably, the simulation verification includes: Gradually reduce the DC capacitance value in the dynamic simulation, and monitor the DC voltage, the receiving - end bus voltage and the power fluctuation; Verify the robustness of the calculation results by comparing the critical capacitance value of 0.35 p.u. with the instability threshold of 0.34 p.u.

[0014] Preferably, the bisection method solving process includes: Set the initial search interval to be from 0 to the rated capacitance value, and gradually reduce the interval through iteration until the tolerance requirement is met; Call the convex optimization tool to solve the LMI feasibility problem in each iteration, and update the search interval according to the result.

[0015] Compared with the prior art, the above - mentioned technical solution of the present invention has the following beneficial technical effects: The present invention designs a method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI. Through the method based on linear matrix inequality and convex optimization, it effectively solves the technical problems of low efficiency, high complexity and strong conservatism in the existing calculation of minimizing the DC capacitance of VSC-HVDC. Its beneficial technical effects are mainly reflected in that through system modeling, small-signal linearization and the construction of LMI stability criterion, the non-linear stability analysis is transformed into a convex optimization problem. Combining with the bisection iterative algorithm significantly improves the calculation efficiency and accuracy. It can accurately reduce the DC capacitance to 0.35 p.u. on the premise of ensuring system stability, reducing the consumption of computing resources and avoiding conservative errors compared with traditional methods. In addition, the robustness of the algorithm is optimized by separating the diagonal matrix and introducing a tolerance mechanism to ensure reliable solution in high-dimensional complex systems, providing theoretical support for reducing the cost and volume of converters. At the same time, simulation verification shows that this method has significant advantages in terms of DC voltage, receiving-end bus voltage and power stability, and is applicable to actual engineering scenarios such as new energy grid connection. BRIEF DESCRIPTION OF THE DRAWINGS

[0016] Figure 1 FIG. is a schematic flow chart of a method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI provided by an embodiment of the present invention; Figure 2 FIG. is a single-line diagram of a system in which an infinite power grid supplies power to a passive network through a two-level VSC-HVDC provided by an embodiment of the present invention; Figure 3 FIG. is a control block diagram of constant DC voltage-zero reactive power for a sending-end converter station provided by an embodiment of the present invention; Figure 4 FIG. is a droop control block diagram for a receiving-end converter station provided by an embodiment of the present invention; Figure 5 FIG. is a diagram showing the influence of the decrease of DC capacitance on DC voltage provided by an embodiment of the present invention; Figure 6 FIG. is a diagram showing the influence of the decrease of DC capacitance on the receiving-end bus phase voltage provided by an embodiment of the present invention; Figure 7 FIG. is a diagram showing the influence of the decrease of DC capacitance on the output power of the receiving-end converter station provided by an embodiment of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0017] A method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI proposed by the present invention, as Figure 1 shown, includes the following specific implementation steps: S1. Establish the state equation of the two-level VSC-HVDC system based on the symmetric harmonic-free assumption, specifically: S11. Establish the state equations of the two-level VSC-HVDC main circuit in the synchronous rotating coordinate system. The research object of the embodiment is a three-phase symmetrical system, and the variables are approximately represented by the corresponding fundamental frequency components. Define the reference directions as follows Figure 2 As shown, using Kirchhoff's voltage / current law, obtain the circuit equations of the system in the three-phase stationary coordinate system, and transform them to the synchronous rotating coordinate system using the Park transformation; Use subscripts 1 and 2 to represent the sending-end system and the receiving-end system respectively, and use subscripts d and q to represent the d-axis component and q-axis component of the variable; The Park transformation matrix is: ; In the formula, θ is the electrical angle of the d-axis (referenced to the a-axis); T abc / dq0 is the transformation matrix from the three-phase stationary coordinate system (abc) to the synchronous rotating coordinate system (dq0), which converts three-phase alternating quantities (including but not limited to voltage and current) into direct current quantities in the synchronous rotating coordinate system; The state equation of the sending-end AC system is: ; In the formula, L1 is the equivalent inductance of the sending-end AC side, which represents the electromagnetic inertia of the sending-end circuit and affects the current dynamic response; R1 is the equivalent resistance of the sending-end AC side, which represents the active power loss of the circuit and affects the steady-state relationship between voltage and current; and are the time derivatives of the d-axis and q-axis current components; i 1d and i 1q are the d-axis and q-axis components of the sending-end current; w1 is the angular frequency of the sending-end system, w1 = 2πf1 (f1 is the power grid frequency); U s1d and U s1q are the d-axis and q-axis components of the sending-end AC system voltage; U c1d and U c1q are the d-axis and q-axis components of the sending-end filter capacitor voltage; The state equation of the DC loop is: ; In the formula, C is the DC side capacitor (buffering power fluctuations and maintaining voltage stability); U dc1 and U dc2 are the DC voltages (the rectifier side and inverter side voltages in this embodiment); i 1d and i 1q are the components of the rectifier AC side current in the dq coordinate system; i 2d and i 2q are the components of the inverter AC side current in the dq coordinate system; U cld and U c1q are the components of the rectifier AC side voltage in the dq coordinate system; U c2d and U c2qis the component of the inverter AC-side voltage in the dq coordinate system; i line is the DC line current; L dc is the DC line inductance (to suppress current mutation); R dc is the DC line resistance (to represent line losses); ; In the formula, U s2d and U s2q are the components of the voltage at the common connection point of the receiving-end AC system in the dq coordinate system; i ld and i 1q are the components of the AC load current in the dq coordinate system; L2 and R2 are the inductance and resistance of the connection part between the inverter and the AC system; C f is the AC-side filter capacitor (to filter out high-frequency harmonics and improve power quality); w2 is the angular frequency of the receiving-end system (in this embodiment, it is 2π×50 / 60 rad / s); S12. Establish the state equations of the rectifier control system and the inverter control system in the synchronous rotating coordinate system. The basic controller of the converter is a PI controller. As Figure 3 shown, in order to clamp the voltage of the DC transmission line, at least one converter station in the VSC-HVDC must undertake the task of regulating the DC voltage. Therefore, the control mode of the sending end is determined as constant DC voltage control; however, controlling the DC voltage can only regulate the active power and cannot regulate the reactive power; Therefore, in order to avoid increasing losses by transmitting reactive power, the control mode of the sending end also combines zero reactive power control. In addition, as Figure 4 shown, since the converter station at the receiving end is connected to a passive network, its control objective is set to adjust the frequency and voltage here, and droop control is used to achieve it; According to Figure 3 , the state equation of the rectifier control system can be obtained as: ; ; In the formula, m1, m2, and m3 are intermediate state variables in the rectifier control system, used to describe the dynamic regulation process; U dcr is the reference voltage on the DC side; U dc1 is the actually measured DC voltage on the rectifier side; i 1dr and i 1qr are the reference currents of the rectifier in the dq coordinate system (the d-axis is the active component, and the q-axis is the reactive component); k pv1 and k iv1 are the proportional coefficient and integral coefficient of the PI controller; i 1d and i 1q are the components of the actual current on the AC side of the rectifier in the dq coordinate system; According to Figure 4 , the state equation of the inverter control system can be obtained as follows: ; ; where m4, m5, m6, and m7 are intermediate state variables used to describe the dynamic regulation process of the control system (such as the output of the integrator); U s2dr and U s2qr are the reference values of the AC voltage on the inverter side (d-axis and q-axis components), generated by droop control; i 2dr and i 2qr are the reference values of the inverter current (the d-axis is the active component, and the q-axis is the reactive component); k pv2 and k iv2 are the proportional coefficient and integral coefficient of the current-loop PI controller, used to adjust the current tracking error; At the same time, the control mode of the inverter belongs to the network-forming control, and the droop control link is: ; According to the instantaneous reactive power theory, we have: ; where w n is the rated frequency of the system (reference value); P s and P n are the actual active power and the rated active power; U n is the rated voltage (reference value); k q is the voltage-reactive power droop coefficient, defining the slope of the voltage change with respect to the reactive power; Q s and Q n are the actual reactive power and the rated reactive power; S13. Substitute the variables output by the controller that control the main circuit into the state equation of the main circuit. According to Figure 3 and Figure 4 , the output of the controller is obtained as: ; where k pi1 is the rectifier-side proportional coefficient, used to adjust the dynamic tracking error of the rectifier-side current; k pi2 is the inverter-side proportional coefficient, used to adjust the dynamic tracking error of the inverter-side current; S14. Obtain the state equation of the two-level VSC-HVDC system: Define the state variable x: x = [i line , i ld , i 1q , U dc1 , m1, m2, m3, i2d , i 2q , U s2d , U s2q , U dc2 , m4, m5, m6, m7, i ld , i lq ) T ; Define the input vector u = [U dcr , i 1qr , w n , Q n , U n , U s2qr ) T ; Calculate to obtain: The state equation of the two-level VSC-HVDC system ; ; In the formula, the vector h(x) is the nonlinear part of the system, and A, B, and C are constant matrices; M(C) is a diagonal matrix; M r (C) = diag(L1, L2, C, 1, 1, 1, 1, 1); M i (C) = diag(L2, L2, C f , C, 1, 1, 1, 1, L l , L l ); G is the nonlinear gain matrix.

[0018] Perform Taylor series expansion on the state equation of the VSC-HVDC system S2 and retain the first-order term, then the linearized small-signal model is: ; where, Δx is a small perturbation, Δx = x - x e ; x e is the equilibrium state; H is a matrix, ; In the formula, A r,dc = -(A dc,r ) T = -[0, 0, 1, 0, 0, 0] T ; A i,dc = -(A dc,i ) T = -[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0] T ; H r = A r + A1; H i = A i + A1; A r and A i do not contain the equilibrium state; the non-zero elements in A1 and A2 contain the equilibrium state.

[0019] S3. Based on the small-signal model, construct the LMI stability criterion by using the Lyapunov stability theory of linear time-invariant systems, specifically as follows: S31. Separate the diagonal matrix M(C), and from M(C) -1 = M2(C)M1; where M2(C) = diag(1, 1, 1, 1 / C, 1, 1, 1, 1, 1, 1, 1, 1 / C, 1, 1, 1, 1, 1, 1); M1 = diag(1 / (L dc ), 1 / (L1), 1 / (L1), 1, 1, 1, 1, 1 / (L2), 1 / (L2), 1 / (C f ), 1 / (C f ), 1, 1, 1, 1, 1, 1 / (L1), 1 / (L1)); S32. Define the matrix R = M1H, and matrices M > 0 and M < 0 represent positive definite and negative definite respectively; When the value of the scalar variable C is given , according to the Lyapunov stability theory of linear time-invariant systems, there is a stability criterion in the form of LMI ~ LMI: ; In the formula, the diagonal matrix N = diag(1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 1); P is a symmetric positive definite matrix and is a decision variable.

[0020] S4. Based on the stability criterion in the form of LMI, construct an algorithm for solving the minimum value of the DC capacitor based on convex optimization theory; Convex feasibility problem ~ CVX: find P subject to P T , P > 0, ~ LMI; Let the rated value of the DC capacitor C r = 400 uF, then the minimum value of the DC capacitor is solved by Algorithm 1: Algorithm 1, that is, an algorithm for solving the minimum value of the DC capacitor based on the bisection method: The input parameters of the algorithm include the initial lower limit value l = 0, the initial upper limit value u = 400 uF, and the allowable error , and the final output is the minimum value C of the DC capacitor that meets the accuracy requirements ∗ , and its core process is as follows: A1. Calculate the intermediate value ; A2. Solve the convex feasibility problem through the convex optimization tool ; A3. If the problem is feasible, update the current upper limit to ; otherwise, adjust the lower limit to ; A4. Iterate the above steps A1 - A3 repeatedly until the interval width is less than or equal to the preset allowable error .

[0021] S5. Build a two - level VSC - HVDC simulation for powering a passive network on the MATLAB / Simulink simulation platform, run the simulation, obtain the equilibrium point of the system, and substitute it into step S2. In this embodiment, there are 20 state variables to be determined in the equilibrium state.

[0022] S6. After obtaining the equilibrium point in step S5, the minimum value of the DC capacitor can be calculated using CVXtool in MATLAB according to the algorithm designed in step S4; It should be noted that the bisection method has poor numerical stability. Once a wrong judgment is made in a certain step, the credibility of the solution will be greatly reduced; at the same time, it involves the positive - definiteness judgment of matrices. Therefore, when using this algorithm, special caution is required and a tolerance needs to be set. Specifically, as long as the solution with the weakest positive - definiteness satisfies the positive - definite condition, then the remaining feasible solutions are also satisfied. Therefore, first, a parameter γ can be introduced to make P > γI and minimize γ to find the solution with the weakest positive - definiteness among all feasible solutions; then, let γ > 1e - 5 instead of γ > 0, and provide a positive - definite margin by sacrificing conservatism with a tolerance tol=(1e - 5)-0. However, this approach also has disadvantages. If the tolerance is set too large, it will cause excessive conservatism, and if it is too small, the solution may cross the actual stable interval. Therefore, multiple calculations are needed to adjust the tolerance tol to ensure the accuracy of the solution; The solution result of the minimum DC capacitor in this embodiment is 0.35 p.u.; to verify the accuracy of the obtained result, the following simulation tests are carried out; test the impact of the decrease in the DC capacitor on the system stability. The process is as follows: start the system at 0 seconds with the DC capacitor set to the rated value of 400 uF; at 0.4 seconds, the DC capacitor drops to 0.35 p.u.; at 1 second, the DC capacitor drops again to 0.34 p.u.; from Figure 5 , Figure 6 and Figure 7It can be seen that between 0 second and 0.4 seconds, that is, when the DC capacitor is at its rated value, the system is stable, and the DC voltage, the receiving-end bus voltage, and the output power of the receiving-end converter are all at the preset operating points; at 0.4 seconds, the DC capacitor drops to 0.35 p.u. Although the oscillation of the DC voltage and the ripple of the receiving-end bus voltage increase somewhat, it is not enough to disrupt the stability of the system; however, after the 1st second, that is, as the DC capacitor drops again, to 0.34 p.u., although the change in the output power of the receiving-end converter is not obvious, the DC voltage shows a large amplitude oscillation, indicating an unstable phenomenon of DC current reversal; at the same time, severe harmonics appear in the receiving-end bus voltage, indicating that the system has lost the ability to maintain stable operation at the original operating point, that is, it has become unstable.

[0023] The embodiments of the present invention have been described in detail above in conjunction with the accompanying drawings, but the present invention is not limited thereto. Various changes can be made without departing from the spirit of the present invention within the scope of knowledge possessed by those skilled in the art to which the present invention pertains.

Claims

1. A method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI, characterized in that, It includes the following specific implementation steps: S1. Establish the state equations of the main circuit and controller of the two-level VSC-HVDC system: Based on the symmetric harmonic-free assumption, use the Park transformation to convert three-phase AC quantities into DC quantities in the synchronous rotating coordinate system, derive the state equations of the main circuit in combination with Kirchhoff's laws, then establish the state equations of the PI controllers for the sending-end constant DC voltage-zero reactive power control and the receiving-end droop control, and finally substitute the controller output variables into the main circuit equations to output the state equations of the VSC-HVDC system; S2. Perform Taylor series expansion on the state equations of the VSC-HVDC system, retain the first-order terms, and construct a small-signal model; S3. Based on the small-signal model, use the Lyapunov stability theory of linear time-invariant systems to construct an LMI stability criterion; The construction of the LMI stability criterion is specifically as follows: Introduce a diagonal matrix to adjust the criterion structure, and verify the stability of the system with a given capacitance value through inequality constraint conditions; The decision variable P is a symmetric positive definite matrix. Combine parameters and tolerances to ensure the positive definite margin, and output the LMI stability criterion; S4. According to the LMI-form stability criterion, use the bisection method to solve for the minimum value of the DC capacitance; S5. Build a two-level VSC-HVDC simulation for powering a passive network on the simulation platform, conduct simulation verification, and obtain the equilibrium point of the system; S6. Output the minimum value of the DC capacitance.

2. The method for calculating the minimum DC capacitor of a two-level VSC-HVDC based on LMI according to claim 1, characterized in that The establishment of the state equations of the main circuit and controller includes: S21. Define the reference direction, use Kirchhoff's voltage and current laws to obtain the circuit equations of the system in the three-phase stationary coordinate system, and transform them to the synchronous rotating coordinate system using the Park transformation; S22. By establishing the state equations of the PI controller for the rectifier constant DC voltage-zero reactive power control and the inverter droop control equation, define intermediate state variables and generate reference values based on the instantaneous power theory, and combine the current-loop PI parameters to adjust the dynamic tracking error to generate variables that control the main circuit; S23. Substitute the variable output by the controller and used to control the main circuit into the main circuit state equation to obtain the output of the controller: ; Among them, k pi1 is the rectifier side proportionality coefficient; k pi2 is the inverter side proportionality coefficient; U s1d and U s1q are the d-axis and q-axis components of the sending-end AC system voltage; U c1d and U c1q are the d-axis and q-axis components of the sending-end filter capacitor voltage; i 1d and i 1q are the d-axis and q-axis components of the sending-end current; w1 is the sending-end system angular frequency, w1 = 2πf1; f1 is the grid frequency; L1 is the equivalent inductance of the sending-end AC side; L2 is the inductance of the connection part between the inverter and the AC system; w2 is the receiving-end system angular frequency; i 2d and i 2q are the components of the inverter AC side current in the dq coordinate system; i 1dr and i 1qr are the reference currents of the rectifier in the dq coordinate system, the d-axis is the active component, and the q-axis is the reactive component; m2, m3 are the intermediate state variables in the rectifier control system; m5, m7 are the intermediate state variables; U s2d and U s2q are the components of the receiving-end AC system common connection point voltage in the dq coordinate system; S24. Obtain the state equations of the two-level VSC-HVDC system: Define the state variable x: x = [i line , i ld , i 1q , U dc1 , m1, m2, m3, i 2d , i 2q , U s2d , U s2q , U dc2 , m4, m5, m6, m7, i ld , i lq T ;​ where, T is the matrix transpose operation; i line is the DC line current; U dc1 and U dc2 are the DC voltages; m1, m2, m3 are intermediate state variables in the rectifier control system; m4, m5, m6, m7 are intermediate state variables; U s2dr and U s2qr are the reference values of the AC voltages on the inverter side: d-axis and q-axis components; Define the input vector u = [U dcr , i 1qr , w n , Q n , U n , U s2qr T ;​ Among them, U dcr is the reference voltage on the DC side; w n is the rated frequency of the system; Q n is the rated reactive power; Calculated: State equations of two-level VSC-HVDC system ; ； where the vector h(x) is the non-linear part of the system, and A, B, and C are constant matrices; M(C) is a diagonal matrix; M r (C) = diag(L1, L2, C, 1, 1, 1, 1, 1); M i (C) = diag(L2, L2, C f , C, 1, 1, 1, 1, L l , L l ); G is a non-linear gain matrix.

3. A method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to claim 2, characterized in that, The conversion process of transforming to the synchronous rotating coordinate system using the Park transformation is as follows: The Park transformation matrix is as follows: ; where θ is the electrical angle of the d-axis with reference to the a-axis; T abc / dq0 is the transformation matrix from the three-phase stationary coordinate system abc to the synchronous rotating coordinate system dq0, which converts three-phase alternating quantities into direct current quantities in the synchronous rotating coordinate system; The state equation of the sending-end AC system is as follows: ; Where, L1 is the equivalent inductance of the sending-end AC side; R1 is the equivalent resistance of the sending-end AC side; and are the time derivatives of the d-axis and q-axis current components; i 1d and i 1q are the d-axis and q-axis components of the sending-end current; w1 is the angular frequency of the sending-end system, w1 = 2πf1; f1 is the power grid frequency; U s1d and U s1q are the d-axis and q-axis components of the voltage of the sending-end AC system; U c1d and U c1q are the d-axis and q-axis components of the voltage of the sending-end filter capacitor; The state equation of the DC circuit is as follows: ; Where C is the DC-side capacitor; U dc1 and U dc2 are the DC voltages; i 1d and i 1q are the components of the rectifier AC-side current in the dq coordinate system; i 2d and i 2q are the components of the inverter AC-side current in the dq coordinate system; U cld and U c1q are the components of the rectifier AC-side voltage in the dq coordinate system; U c2d and U c2q are the components of the inverter AC-side voltage in the dq coordinate system; i line is the DC line current; L dc is the DC line inductance; R dc is the DC line resistance; ； where, U s2d and U s2q are the components of the voltage at the common connection point of the receiving-end AC system in the dq coordinate system; i ld and i 1q are the components of the AC load current in the dq coordinate system; L2 and R2 are the inductance and resistance of the connection part between the inverter and the AC system; C f is the filter capacitor on the AC side; w2 is the angular frequency of the receiving-end system.

4. A method for calculating the minimum DC capacitor of a two-level VSC-HVDC based on LMI according to claim 3, characterized in that The adjustment process of adjusting the dynamic tracking error is as follows: Construct the state equations of the rectifier control system as: ； ； where m1, m2, and m3 are intermediate state variables in the rectifier control system; U dcr is the reference voltage on the DC side; U dc1 is the actually measured DC voltage on the rectifier side; i 1dr and i 1qr are the reference currents of the rectifier in the dq coordinate system, the d-axis is the active component, and the q-axis is the reactive component; k pv1 and k iv1 are the proportional coefficient and integral coefficient of the PI controller; i 1d and i 1q are the components of the actual current on the AC side of the rectifier in the dq coordinate system; Construct the state equations of the inverter control system as: ； ； where m4, m5, m6, and m7 are intermediate state variables; U s2dr and U s2qr are the reference values of the AC voltage on the inverter side: d-axis and q-axis components; i 2dr and i 2qr are the reference values of the inverter current: the d-axis is the active component and the q-axis is the reactive component; k pv2 and k iv2 are the proportional coefficient and integral coefficient of the current loop PI controller; The control mode of the inverter belongs to the network-forming control, and the droop control link is: ； According to the instantaneous reactive power theory, there is: ; where, w n is the system rated frequency; P s and P n are the actual active power and the rated active power; U n is the rated voltage; k q is the voltage-reactive power droop coefficient; Q s and Q n are the actual reactive power and the rated reactive power.

5. A calculation method for the minimum DC capacitor of a two-level VSC-HVDC based on LMI according to claim 4, characterized in that, The linearized small-signal model is as follows: ; where, Δx is a small perturbation, Δx = x - x e ; x e is the equilibrium state; H is a matrix, ; where A r,dc = -(A dc,r ) T = -[0, 0, 1, 0, 0, 0] T ; A i,dc = -(A dc,i ) T = -[0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0] T ; H r = A r + A1; H i = A i + A1; A r and A i do not contain the equilibrium state; the non - zero elements in A1 and A2 all contain the equilibrium state.

6. A method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to claim 1, characterized in that The construction process of the LMI stability criterion is as follows: S61. Separate the diagonal matrix M(C) into M2(C)M1, where M(C) -1 = M2(C)M1; Among them, M2(C)=diag(1,1,1,1 / C,1,1,1,1,1,1,1,1 / C,1,1,1,1,1,1); M1 = diag(1 / (L dc ), 1 / (L1), 1 / (L1), 1, 1, 1, 1, 1 / (L2), 1 / (L2), 1 / (C f ), 1 / (C f ), 1, 1, 1, 1, 1, 1 / (L1), 1 / (L1)); S62. Define the matrix R = M1H, and the matrices M>0 and M<0 represent positive definite and negative definite respectively; When the value of the scalar variable C is given According to the Lyapunov stability theory of linear time-invariant systems, there is a stability criterion in the form of LMI~LMI: ; In the formula, the diagonal matrix N = diag(1,1,1,0,1,1,1,1,1,1,1,0,1,1,1,1,1,1); P is a symmetric positive definite matrix and is a decision variable.

7. A method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to claim 1, characterized in that, The simulation verification includes: Gradually reduce the DC capacitance value in the dynamic simulation, and monitor the DC voltage, receiving-end bus voltage, and power fluctuations; The robustness of the calculation results is verified by comparing the critical capacitance value of 0.35 p.u. with the instability threshold of 0.34 p.u.

8. A method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to claim 1, characterized in that The bisection method solving process includes: The initial search interval is set from 0 to the rated capacitance value, and the interval is reduced through iteration until the tolerance requirement is met; In each iteration, a convex optimization tool is called to solve the LMI feasibility problem, and the search interval is updated according to the results.

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