A method for calculating the minimum DC capacitance of two-level VSC-HVDC based on LMI

Abstract

The present invention relates to the field of power electronics and direct current transmission technology, specifically a method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI. The method comprises the following steps: establishing a state equation including a dynamic model of a main circuit and a controller, converting a three-phase AC quantity into a DC quantity in a synchronous rotating coordinate system using Park transformation, and deriving the system state equation in combination with Kirchhoff's law; performing Taylor expansion and linearization on the nonlinear equation, and constructing a small signal model based on Lyapunov stability theory; converting the minimum DC capacitance solution into a convex optimization problem by separating the diagonal matrix and defining a stability criterion in the form of LMI, and adopting a bisection method to iteratively search for a critical value to determine the minimum capacitance, while introducing a tolerance mechanism to improve the robustness of the algorithm. The present invention effectively balances computational efficiency and accuracy by combining mathematical modeling with an optimization algorithm, providing a theoretical basis for reducing the cost and volume of converters.

Description

Technical Field

[0001] The present invention relates to the technical field of power electronics and direct current transmission, and in particular to a method for calculating the minimum direct current capacitance 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 new power systems due to its independent control of active and reactive power and its lack of commutation failure. VSC-HVDC technology has been demonstrated in numerous scientific and technological projects in various application scenarios, including renewable energy grid integration, long-distance large-capacity transmission, grid interconnection, DC grid construction, large-scale urban power supply, and passive network power supply. VSC-HVDC can quickly control AC voltage and has good application prospects in scenarios where power is supplied to passive networks.

[0003] DC capacitors are important components on the DC side of the converter. Their main functions are to stabilize 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. Small DC capacitors can significantly reduce cost and space while meeting system performance requirements, making them very necessary in VSC-HVDC systems in special scenarios such as offshore platforms. The key issue in calculating the minimum DC capacitor of VSC-HVDC is the selection of stability criteria. In the small-signal stability analysis of VSC-HVDC, 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 finding its minimum value by adjusting the DC capacitor value. The eigenvalue method considers the role of the controller and, based on the small-signal model, determines the stability of the system by calculating the eigenvalues ​​of the Jacobian matrix, and then gradually calculates the minimum value of the DC capacitor.

[0004] Zhang Xu, Hao Zhiguo, Li Yujun, et al. proposed a sufficient condition for stability criterion with DC capacitance variable in the paper titled "Evaluation of DC voltage oscillation stability of VSC-HVDC system based on electrical torque method" (Power System Technology, 2024, 48(10): 4306-4316), which can infer the minimum value of DC capacitance.

[0005] Aiming at the problem of solving the minimum DC capacitance of two-level VSC-HVDC; the generalized Nyquist criterion requires frequent drawing and analysis of Nyquist diagrams, resulting in low computational efficiency when applied to high-dimensional VSC-HVDC systems; the eigenvalue method is generally considered to be a more direct and relatively efficient stability criterion, but when applied to a closed-loop system after considering the controller dynamics, the high-dimensional and asymmetric characteristics of the Jacobian matrix will increase the computational complexity of the eigenvalue, thereby reducing the efficiency of solving the minimum DC capacitance; it can be seen that the existing technology has the problem of low computational efficiency; in addition, although the DC capacitance stability criterion of the electrical torque method gives an analytical expression, it is only a sufficient condition for small signal stability and is highly conservative, and cannot accurately derive the minimum DC capacitance value. Summary of the Invention

[0006] The present invention aims to solve the problems existing in the background technology and propose a method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI.

[0007] The technical solution of the present invention is a method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI, comprising the following specific implementation steps:

[0008] S1. Establish the main circuit and controller state equations for the two-level VSC-HVDC system: Based on the symmetric harmonicity assumption, use the Park transformation to convert the three-phase AC quantities into DC quantities in a synchronously rotating coordinate system. Combined with Kirchhoff's law, 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 equations to output the VSC-HVDC system state equation.

[0009] S2. Perform Taylor series expansion on the VSC-HVDC system state equation, retain the linear terms, and construct a small signal model.

[0010] S3. Based on the small signal model, the LMI stability criterion is constructed using the Lyapunov stability theory of linear steady-state systems;

[0011] Preferably, the LMI stability criterion is constructed as follows:

[0012] A diagonal matrix is ​​introduced to adjust the criterion structure, and the system stability of a given capacitance value is verified through inequality constraints.

[0013] The decision variable P is a symmetric positive definite matrix, which is combined with parameters and tolerance to ensure positive definiteness margin and output the LMI stability criterion;

[0014] S4. Based on the stability criterion of LMI form, use the bisection method to solve the minimum value of DC capacitance;

[0015] S5. Build a two-level VSC-HVDC simulation for powering the passive network on the simulation platform, perform simulation verification, and obtain the system balance point;

[0016] S6, minimum output DC capacitance.

[0017] Preferably, the establishment of the main circuit and controller state equations includes:

[0018] S21. Define the reference direction and use Kirchhoff's voltage and current laws to derive the circuit equations for the system in the three-phase stationary coordinate system. Use Park transformation to transform to the synchronously rotating coordinate system.

[0019] S22. Establishing the PI controller state equation for the rectifier constant DC voltage-zero reactive power control and the inverter droop control equation, defining intermediate state variables and generating reference values ​​based on instantaneous power theory, and adjusting the dynamic tracking error in combination with the current loop PI parameters to generate variables that control the main circuit;

[0020] S23. Substitute the controller output variable that controls the main circuit into the main circuit state equation to obtain the controller output: ;

[0021] Among them, k pi1 is the proportional coefficient of the rectifier side; k pi2 is the inverter side proportional coefficient; U s1d and U s1q are the d-axis and q-axis components of the AC system voltage at the sending end; 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 is the component of the sending-end current on the d-axis and q-axis; 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 between the inverter and the AC system; w2 is the receiving-end system angular frequency; i 2d and i 2q is the component of the inverter AC side current in the dq coordinate system; i 1dr and i 1qr is the reference current of the rectifier in the dq coordinate system, the d-axis is the active component, and the q-axis is the reactive component; m2 and m3 are the intermediate state variables in the rectifier control system; m5 and m7 are intermediate state variables; U s2d and U s2q is the component of the voltage at the common connection point of the receiving AC system in the dq coordinate system;

[0022] S24. Obtain the state equation of the two-level VSC-HVDC system:

[0023] Define the state variable x:

[0024] 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 ;

[0025] Where T is the matrix transpose operation; i line is the DC line current; U dc1 and U dc2 is the DC voltage; m1, m2, m3 are the intermediate state variables in the rectifier control system; m4, m5, m6, m7 are the intermediate state variables; U s2dr and U s2qr is the reference value of the inverter side AC voltage: d-axis and q-axis components;

[0026] Define the input vector u=[U dcr ,i 1qr ,w n ,Q n ,U n ,U s2qr ] T ;

[0027] Among them, U dcr is the reference voltage on the DC side; w n is the system rated frequency; Q n is the rated reactive power;

[0028] Calculated: Two-level VSC-HVDC system state equation ;

[0029] ;

[0030] In the formula, vector h(x) is the nonlinear part of the system, 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 nonlinear gain matrix.

[0031] Preferably, the conversion process to the synchronously rotating coordinate system using Park transformation is as follows:

[0032] The Park transformation matrix is: ;

[0033] Where 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 the three-phase AC quantity into the DC quantity in the synchronous rotating coordinate system;

[0034] The state equation of the sending-end AC system is: ;

[0035] Where, L1 is the equivalent inductance on the AC side of the sending end; R1 is the equivalent resistance on the AC side of the sending end; and is the time derivative of the d-axis and q-axis current components; i 1d and i 1q is the component of the sending-end current on the d-axis and q-axis; w1 is the sending-end system angular frequency, w1=2πf1; f1 is the grid frequency; U s1d and U s1q are the d-axis and q-axis components of the AC system voltage at the sending end; U c1d and U c1q are the d-axis and q-axis components of the sending-end filter capacitor voltage;

[0036] The state equation of the DC circuit is: ;

[0037] Where, C is the DC side capacitance; U dc1 and U dc2 is the DC voltage; i 1d and i 1q is the component of the rectifier AC side current in the dq coordinate system; i 2d and i 2q is the component of the inverter AC side current in the dq coordinate system; U cld and U c1q is the component of the rectifier AC side voltage in the dq coordinate system; U c2d and U c2q is 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; R dc is the DC line resistance;

[0038] ;

[0039] Where U s2d and Us2q is the component of the voltage at the common connection point of the receiving AC system in the dq coordinate system; i ld and i 1q is the component of the AC load current in the dq coordinate system; L2 and R2 are the inductance and resistance of the connection 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.

[0040] Preferably, the adjustment process of adjusting the dynamic tracking error is as follows:

[0041] The state equation for constructing the rectifier control system is:

[0042] ;

[0043] ;

[0044] 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 DC voltage actually measured on the rectifier side; i 1dr and i 1qr is the reference current 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 is the proportional coefficient and integral coefficient of the PI controller; i 1d and i 1q is the component of the actual current on the AC side of the rectifier in the dq coordinate system;

[0045] The state equation for constructing the inverter control system is:

[0046] ;

[0047] ;

[0048] Where m4, m5, m6, and m7 are intermediate state variables; U s2dr and U s2qr is the reference value of the AC voltage on the inverter side: d-axis and q-axis components; i 2dr and i 2qr is the reference value of the inverter current: d-axis is the active component, q-axis is the reactive component; k pv2 and k iv2 are the proportional coefficient and integral coefficient of the current loop PI controller;

[0049] The inverter control method belongs to grid-type control, and the droop control link is:

[0050] ;

[0051] According to the instantaneous reactive power theory: ;

[0052] Where w n is the system rated frequency; P s and P n is 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 is the actual reactive power and the rated reactive power.

[0053] Preferably, the linearized small signal model is: ;

[0054] Where Δx is a small perturbation, Δx=xx e ;x e is the equilibrium state; H is the matrix, ;

[0055] 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 does not contain equilibrium states; all non-zero elements in A1 and A2 contain equilibrium states.

[0056] Preferably, the construction process of the LMI stability criterion is as follows:

[0057] S71, separate the diagonal matrix M(C), from M(C) -1 =M2(C)M1;

[0058] Among them, M2(C)=diag(1,1,1,1 / C,1,1,1,1,1,1,1,1 / C,1,1,1,1,1,1,1);

[0059] M1=diag(1 / (L dc ),1 / (L1),1 / (L1),1,1,1,1,1 / (L2),1 / (L2),1 / (C f ),1 / (Cf ),1,1,1,1,1,1 / (L1),1 / (L1));

[0060] S72. Define the matrix R = M1H, where M>0 and M<0 represent positive definite and negative definite, respectively.

[0061] When the value of the scalar variable C is given When , according to the Lyapunov stability theory of linear time-invariant systems, there is a stability criterion in the form of LMI: ;

[0062] Where, 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.

[0063] Preferably, the simulation verification includes:

[0064] In dynamic simulation, the DC capacitance value is gradually reduced, and the DC voltage, receiving-end bus voltage, and power fluctuations are monitored;

[0065] The robustness of the calculated results is verified by comparing the critical capacitance value of 0.35 pu with the instability threshold of 0.34 pu.

[0066] Preferably, the dichotomy solution process includes:

[0067] The initial search interval is set to 0 to the rated capacitance value, and the interval is narrowed through iteration until the tolerance requirement is met;

[0068] In each iteration, the convex optimization tool is called to solve the LMI feasibility problem and the search interval is updated based on the results.

[0069] Compared with the prior art, the above technical solution of the present invention has the following beneficial technical effects:

[0070] The present invention designs a two-level VSC-HVDC minimum DC capacitance calculation method based on LMI. Through a 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 VSC-HVDC DC capacitance minimization calculation; its beneficial technical effects are mainly reflected in the transformation of nonlinear stability analysis into a convex optimization problem through system modeling, small signal linearization and LMI stability criterion construction, and the combination of the bisection iterative algorithm significantly improves the calculation efficiency and accuracy, and can accurately reduce the DC capacitance to 0.35pu while ensuring system stability, reducing computing resource consumption 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, ensuring that it can still be solved reliably in high-dimensional complex systems, providing theoretical support for reducing the cost and volume of the converter. At the same time, simulation verification shows that this method has significant advantages in DC voltage, receiving bus voltage and power stability, and is suitable for actual engineering scenarios such as new energy grid connection. BRIEF DESCRIPTION OF THE DRAWINGS

[0071] Figure 1 A schematic flow chart of a method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to an embodiment of the present invention;

[0072] Figure 2 A single-line diagram of a system for supplying power from an infinite grid to a passive network via a two-level VSC-HVDC according to an embodiment of the present invention;

[0073] Figure 3 A block diagram of a constant DC voltage-zero reactive power control system for a sending-end converter station according to an embodiment of the present invention;

[0074] Figure 4 A block diagram of droop control for a receiving-end converter station provided in one embodiment of the present invention;

[0075] Figure 5 A diagram showing the effect of a drop in DC capacitance on DC voltage according to an embodiment of the present invention;

[0076] Figure 6 A diagram showing the effect of a drop in DC capacitance on the receiving-end bus phase voltage provided by an embodiment of the present invention;

[0077] Figure 7 This is a diagram showing the impact of a drop in DC capacitance on the output power of a receiving-end converter station provided by one embodiment of the present invention. DETAILED DESCRIPTION

[0078] The present invention proposes a method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI, such as Figure 1 As shown, the specific implementation steps include the following:

[0079] S1. Based on the symmetric harmonicity assumption, the state equation of the two-level VSC-HVDC system is established as follows:

[0080] S11. Establish the state equation 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. The reference direction is defined as follows: Figure 2 As shown, Kirchhoff's voltage / current law is used to derive the circuit equations of the system in the three-phase stationary coordinate system, and Park transformation is used to transform it into the synchronous rotating coordinate system;

[0081] Subscripts 1 and 2 are used to represent the sending end system and the receiving end system, respectively. Subscripts d and q are used to represent the d-axis component and q-axis component of the variable, respectively.

[0082] The Park transformation matrix is: ;

[0083] Where θ is the electrical angle of the d-axis (with the a-axis as a reference); T abc / dq0 It is the transformation matrix from the three-phase stationary coordinate system (abc) to the synchronous rotating coordinate system (dq0), which converts the three-phase AC quantity (including but not limited to voltage and current) into the DC quantity in the synchronous rotating coordinate system;

[0084] The state equation of the sending-end AC system is: ;

[0085] Where L1 is the equivalent inductance on the AC side of the sending end, which represents the electromagnetic inertia of the sending end circuit and affects the dynamic response of the current; R1 is the equivalent resistance on the AC side of the sending end, which represents the active power loss of the circuit and affects the steady-state relationship between voltage and current. and is the time derivative of the d-axis and q-axis current components; i 1d and i 1q is the component of the sending-end current on the d-axis and q-axis; w1 is the sending-end system angular frequency, w1=2πf1 (f1 is the grid frequency); U s1d and U s1q are the d-axis and q-axis components of the AC system voltage at the sending end; U c1d and U c1q are the d-axis and q-axis components of the sending-end filter capacitor voltage;

[0086] The state equation of the DC circuit is: ;

[0087] Where, C is the DC side capacitor (to buffer power fluctuations and maintain voltage stability); U dc1 and U dc2 is the DC voltage (in this embodiment, the voltage on the rectifier side and the inverter side); i 1d and i 1qis the component of the rectifier AC side current in the dq coordinate system; i 2d and i 2q is the component of the inverter AC side current in the dq coordinate system; U cld and U c1q is the component of the rectifier AC side voltage in the dq coordinate system; U c2d and U c2q is 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 (characterizing line loss);

[0088] ;

[0089] Where U s2d and U s2q is the component of the voltage at the common connection point of the receiving AC system in the dq coordinate system; i ld and i 1q is the component of the AC load current in the dq coordinate system; L2 and R2 are the inductance and resistance of the connection between the inverter and the AC system; C f is the AC side filter capacitor (filters high-frequency harmonics and improves power quality); w2 is the receiving end system angular frequency (2π×50 / 60rad / s in this embodiment);

[0090] S12. Establish the state equation of the rectifier control system and the inverter control system in the synchronous rotating coordinate system. The basic controller of the converter is the PI controller, such as Figure 3 As shown in Figure 1, to clamp the voltage of the DC transmission line, at least one converter station in the VSC-HVDC system must be responsible for regulating the DC voltage. Therefore, the control method at the sending end is determined to be constant DC voltage control. However, controlling the DC voltage can only regulate active power, but not reactive power.

[0091] Therefore, in order to avoid increasing losses by transmitting reactive power, the control method at the sending end also combines zero reactive power control. In addition, Figure 4 As shown in Figure 2, since the converter station at the receiving end is connected to a passive network, its control target is set as adjusting the frequency and voltage, and droop control is used to achieve this.

[0092] in accordance with Figure 3 , we can get the state equation of the rectifier control system as:

[0093] ;

[0094] ;

[0095] Where m1, m2, and m3 are intermediate state variables in the rectifier control system, which are used to describe the dynamic regulation process; U dcr is the reference voltage on the DC side; U dc1 is the DC voltage actually measured on the rectifier side; i 1dr and i 1qr is the reference current of the rectifier in the dq coordinate system (d-axis is the active component, q-axis is the reactive component); k pv1 and k iv1 is the proportional coefficient and integral coefficient of the PI controller; i 1d and i 1q is the component of the actual current on the AC side of the rectifier in the dq coordinate system;

[0096] in accordance with Figure 4 , we can get the state equation of the inverter control system as:

[0097] ;

[0098] ;

[0099] Where m4, m5, m6, and m7 are intermediate state variables used to describe the dynamic adjustment process of the control system (such as integrator output); U s2dr and U s2qr is the reference value of the inverter side AC voltage (d-axis and q-axis components), generated by droop control; i 2dr and i 2qr is the reference value of the inverter current (d-axis is the active component, q-axis is the reactive component); k pv2 and k iv2 are the proportional coefficient and integral coefficient of the current loop PI controller, which are used to adjust the current tracking error;

[0100] At the same time, the control mode of the inverter belongs to grid-type control, and the droop control link is:

[0101] ;

[0102] According to the instantaneous reactive power theory: ;

[0103] Where w n is the system rated frequency (reference value); P s and P n is 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, which defines the slope of voltage changing with reactive power; Q s and Q nis the actual reactive power and the rated reactive power;

[0104] S13, substitute the variables output by the controller that control the main circuit into the main circuit state equation, according to Figure 3 and Figure 4 , the controller output is:

[0105] ;

[0106] Where k pi1 k is the proportional coefficient on the rectifier side, which is used to adjust the dynamic tracking error of the rectifier side current; pi2 is the inverter side proportional coefficient, which is used to adjust the dynamic tracking error of the inverter side current;

[0107] S14. Obtain the state equation of the two-level VSC-HVDC system:

[0108] Define the state variable x:

[0109] 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 ;

[0110] Define the input vector u=[U dcr ,i 1qr ,w n ,Q n ,U n ,U s2qr ] T ;

[0111] Calculated: Two-level VSC-HVDC system state equation ;

[0112] ;

[0113] In the formula, vector h(x) is the nonlinear part of the system, 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 ,Ll ); G is the nonlinear gain matrix.

[0114] S2. Perform Taylor series expansion on the VSC-HVDC system state equation and retain the first-order term. The linearized small signal model is: ;

[0115] Where Δx is a small perturbation, Δx=xx e ;x e is the equilibrium state; H is the matrix, ;

[0116] 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 does not contain equilibrium states; all non-zero elements in A1 and A2 contain equilibrium states.

[0117] S3. Based on the small signal model, the LMI stability criterion is constructed using the Lyapunov stability theory of linear steady-state systems. Specifically:

[0118] S31, separate the diagonal matrix M(C), from M(C) -1 =M2(C)M1;

[0119] Among them, M2(C)=diag(1,1,1,1 / C,1,1,1,1,1,1,1,1 / C,1,1,1,1,1,1,1);

[0120] 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));

[0121] S32. Define the matrix R = M1H, where the matrices M>0 and M<0 represent positive definite and negative definite, respectively;

[0122] When the value of the scalar variable C is given When , according to the Lyapunov stability theory of linear time-invariant systems, there is a stability criterion in the form of LMI: ;

[0123] Where, 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.

[0124] S4. Based on the stability criterion of LMI form, an algorithm for solving the minimum value of DC capacitance is constructed based on convex optimization theory;

[0125] Convex feasibility problem ~CVX: find P subject to P T ,P>0,~LMI;

[0126] Assume the DC capacitor rating is C r =400 uF, then the minimum DC capacitance is solved by Algorithm 1:

[0127] Algorithm 1, the algorithm for solving the minimum DC capacitance based on the bisection method:

[0128] The input parameters of the algorithm include the initial lower limit value l=0, the initial upper limit value u=400uF and the allowable error The final output is the minimum DC capacitance C that meets the accuracy requirements. ∗ , its core process is as follows:

[0129] A1. Calculate the median value ;

[0130] A2. Solve convex feasibility problems using convex optimization tools ;

[0131] A3. If the problem is feasible, update the current upper limit to Otherwise, adjust the lower limit to ;

[0132] A4. Repeat steps A1 to A3 until the interval width Less than or equal to the preset tolerance .

[0133] S5. Build a two-level VSC-HVDC simulation for powering the passive network on the MATLAB / Simulink simulation platform, run the simulation, obtain the system equilibrium point, and substitute it into step S2. In this embodiment, there are 20 state variables that need to be determined in the equilibrium state.

[0134] S6. After the equilibrium point is obtained in step S5, the minimum DC capacitance value can be calculated using CVXtool in MATLAB according to the algorithm designed in step S4.

[0135] It should be noted that the numerical stability of the bisection method is poor. Once a wrong judgment is made at a certain time, the credibility of the solution will be greatly reduced. At the same time, it involves the positive definiteness judgment of the matrix, so special caution is required when using this algorithm, and a tolerance needs to be set. Specifically, as long as the solution with the weakest positive definiteness satisfies the positive definiteness condition, the remaining feasible solutions are all satisfied. Therefore, first, the parameter γ can be introduced to make P>γI, and the weakest positive definite solution can be found among all feasible solutions by minimizing γ. Then, let γ>1e-5 instead of γ>0, and sacrifice conservatism by using the tolerance tol=(1e-5)-0 to provide a positive definite margin. However, this approach also has disadvantages. When the tolerance is set too large, it will cause excessive conservatism, and when it is too small, the solution may exceed the actual stable interval. Therefore, it is necessary to rely on multiple calculations and adjust the tolerance tol to ensure the accuracy of the solution.

[0136] The minimum DC capacitance of this embodiment is 0.35 pu. To verify the accuracy of the obtained result, the following simulation test is performed. The impact of the DC capacitance drop on the system stability is tested. The process is as follows: the system is started at 0 seconds, and the DC capacitance is set to the rated value of 400 uF; at 0.4 seconds, the DC capacitance drops to 0.35 pu; at 1 second, the DC capacitance drops again to 0.34 pu; Figure 5 、 Figure 6 and Figure 7 It can be seen that between 0 seconds and 0.4 seconds, that is, when the DC capacitance is at its rated value, the system is stable, and the DC voltage, receiving-end bus voltage, and receiving-end converter output power are all at the preset operating point; at 0.4 seconds, the DC capacitance drops to 0.35 pu. Although the DC voltage oscillation and receiving-end bus voltage ripple increase, they do not undermine the stability of the system; however, after the first second, as the DC capacitance drops again to 0.34 pu, although the output power of the receiving-end converter does not change significantly, the DC voltage oscillates significantly, indicating that the DC current is unstable and has flipped; at the same time, the receiving-end bus voltage exhibits severe harmonics, indicating that the system has lost the ability to maintain stable operation at the original operating point, that is, it has become unstable.

[0137] The embodiments of the present invention are described in detail above with reference to the accompanying drawings, but the present invention is not limited thereto. Various changes can be made within the scope of knowledge possessed by those skilled in the art without departing from the spirit of the present invention.

Claims

1. A method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI, characterized in that: The specific implementation steps include the following: S1. Establish the main circuit and controller state equations for the two-level VSC-HVDC system: Based on the symmetric harmonicity assumption, use the Park transformation to convert the three-phase AC quantities into DC quantities in a synchronously rotating coordinate system. Combined with Kirchhoff's law, 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 equations to output the VSC-HVDC system state equation. The establishment of the main circuit and controller state equations includes: A1. Define the reference direction and use Kirchhoff's voltage and current laws to derive the circuit equations for the system in a three-phase stationary coordinate system. Use Park transformation to convert the equations to a synchronously rotating coordinate system. A2. By establishing the PI controller state equation for the rectifier's constant DC voltage-zero reactive power control and the inverter's droop control equation, intermediate state variables are defined and reference values ​​are generated based on instantaneous power theory. Dynamic tracking error is adjusted using the current loop PI parameters to generate variables that control the main circuit. A3. Substitute the controller output variables that control the main circuit into the main circuit state equation to obtain the controller output: ； Among them, k pi1 is the proportional coefficient of the rectifier side; k pi2 is the inverter side proportional coefficient; U s1d and U s1q are the d-axis and q-axis components of the AC system voltage at the sending end; 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 is the component of the sending-end current on the d-axis and q-axis; 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 between the inverter and the AC system; w2 is the receiving-end system angular frequency; i 2d and i 2q is the component of the inverter AC side current in the dq coordinate system; i 1dr and i 1qr is the reference current of the rectifier in the dq coordinate system, the d-axis is the active component, and the q-axis is the reactive component; m2 and m3 are the intermediate state variables in the rectifier control system; m5 and m7 are intermediate state variables; U s2d and U s2q is the component of the voltage at the common connection point of the receiving AC system in the dq coordinate system; A4. Derive 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 is the DC voltage; m1, m2, m3 are the intermediate state variables in the rectifier control system; m4, m5, m6, m7 are the intermediate state variables; U s2dr and U s2qr is the reference value of the inverter side AC voltage: 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 system rated frequency; Q n is the rated reactive power; Calculated: Two-level VSC-HVDC system state equation ; ； In the formula, vector h(x) is the nonlinear part of the system, 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 nonlinear gain matrix; S2. Perform Taylor series expansion on the VSC-HVDC system state equation, retain the linear terms, and construct a small signal model. S3. Based on the small signal model, the LMI stability criterion is constructed using the Lyapunov stability theory of linear steady-state systems; The construction of LMI stability criterion is as follows: A diagonal matrix is ​​introduced to adjust the criterion structure, and the system stability of a given capacitance value is verified through inequality constraints. The decision variable P is a symmetric positive definite matrix, which is combined with parameters and tolerance to ensure positive definiteness margin and output the LMI stability criterion; S4. Based on the stability criterion of LMI form, use the bisection method to solve the minimum value of DC capacitance; S5. Build a two-level VSC-HVDC simulation for powering the passive network on the simulation platform, perform simulation verification, and obtain the system balance point; S6, minimum output DC capacitance.

2. The method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to claim 1, characterized in that: The conversion process to the synchronous rotating coordinate system using Park transformation is as follows: The Park transformation matrix is: ； Where 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 the three-phase AC quantity into the DC quantity in the synchronous rotating coordinate system; The state equation of the sending-end AC system is: ; Where, L1 is the equivalent inductance on the AC side of the sending end; R1 is the equivalent resistance on the AC side of the sending end; and is the time derivative of the d-axis and q-axis current components; i 1d and i 1q is the component of the sending-end current on the d-axis and q-axis; w1 is the sending-end system angular frequency, w1=2πf1; f1 is the grid frequency; U s1d and U s1q are the d-axis and q-axis components of the AC system voltage at the sending end; 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 circuit is: ; Where, C is the DC side capacitance; U dc1 and U dc2 is the DC voltage; i 1d and i 1q is the component of the rectifier AC side current in the dq coordinate system; i 2d and i 2q is the component of the inverter AC side current in the dq coordinate system; U cld and U c1q is the component of the rectifier AC side voltage in the dq coordinate system; U c2d and U c2q is 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; R dc is the DC line resistance; ； Where U s2d and U s2q is the component of the voltage at the common connection point of the receiving AC system in the dq coordinate system; i ld and i 1q is the component of the AC load current in the dq coordinate system; L2 and R2 are the inductance and resistance of the connection 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.

3. The method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to claim 2, characterized in that: The adjustment process of dynamic tracking error is as follows: The state equation for constructing the rectifier control system is: ； ； 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 DC voltage actually measured on the rectifier side; i 1dr and i 1qr is the reference current 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 is the proportional coefficient and integral coefficient of the PI controller; i 1d and i 1q is the component 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, and m7 are intermediate state variables; U s2dr and U s2qr is the reference value of the AC voltage on the inverter side: d-axis and q-axis components; i 2dr and i 2qr is the reference value of the inverter current: d-axis is the active component, 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 inverter control method belongs to grid-type control, and the droop control link is: ； According to the instantaneous reactive power theory: ; Where w n is the system rated frequency; P s and P n is 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 is the actual reactive power and the rated reactive power.

4. The method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to claim 3, characterized in that: The linearized small signal model is: ; Where Δx is a small perturbation, Δx=xx e ;x e is the equilibrium state; H is the 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 does not contain equilibrium states; all non-zero elements in A1 and A2 contain equilibrium states.

5. The 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: Separate the diagonal matrix M(C), from 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,1,1,1,1,1,1,1); 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)); Define the matrix R=M1H, where the matrices M>0 and M<0 represent positive definite and negative definite respectively; When the value of the scalar variable C is given When , according to the Lyapunov stability theory of linear time-invariant systems, there is a stability criterion in the form of LMI: ; Where, 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.

6. The method for calculating the minimum DC capacitance of a two-level VSC-HVDC based on LMI according to claim 1, characterized in that: Simulation verification includes: In dynamic simulation, the DC capacitance value is gradually reduced, and the DC voltage, receiving-end bus voltage, and power fluctuations are monitored; The robustness of the calculated results is verified by comparing the critical capacitance value of 0.35 pu with the instability threshold of 0.34 pu.

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

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