Nonlinear Modeling Method of Synchronous Rotating Coordinate System Phase-Locked Loop Based on Sector Interval Method

Abstract

The present invention discloses a nonlinear modeling method for a synchronous rotating coordinate system phase-locked loop based on the sector interval method, belonging to the technical field of nonlinear modeling of grid-connected converters. Based on the nonlinear model of the synchronous rotating coordinate system phase-locked loop, its operating point is converted to the coordinate origin, and the sine nonlinear link in the phase-locked loop control is selected as the fuzzy object; the initial sector interval is designed according to the selected fuzzy object; the phase-locked loop control state matrix is determined according to the designed sector interval; the linear matrix inequality of the phase-locked loop control system is written based on the state matrix, the common positive definite matrix P is solved, and the size of the sector interval of the phase-locked loop fuzzy system is adjusted by the presence or absence of the matrix P, that is, the maximum universe of discourse of the state variables in the fuzzy object is obtained; the membership function and fuzzy rules of the phase-locked loop control system are determined according to the maximum universe of discourse, and the equivalent model of the system is constructed by the membership function and fuzzy rules. The present invention can achieve accurate modeling with fewer fuzzy rules.

Description

Technical Field

[0001] The present invention relates to the technical field of non - linear modeling of grid - connected converters, and particularly to a non - linear modeling method for a phase - locked loop in a synchronous rotating coordinate system based on the sector - interval method. Background Art

[0002] As non - renewable energy, fossil energy is becoming increasingly scarce in the development of human society. Therefore, people's attention has shifted to renewable energy (photovoltaic, wind power). Thus, new - energy power generation will be favored in the future. However, as the interface between new energy and the power system, the stability of grid - connected converters plays a crucial role.

[0003] Nowadays, the research on the stability of grid - connected converters focuses on small - signal disturbance stability analysis, and the research on the transient large - signal stability of grid - connected converters is still in the preliminary exploration stage. In the analysis of the transient large - signal stability of grid - connected converters, due to coordinate transformation, the grid - connected system exhibits non - linear characteristics, making its control process complex. Currently, there is a lack of rigorous analysis based on Lyapunov's direct method for the transient synchronous stability research of grid - connected converters. The main difficulty lies in the fact that the original system has strong non - linear characteristics, which is not convenient for the selection of Lyapunov equations. Therefore, it is necessary to determine its linear equivalent model to simplify the difficulty of finding Lyapunov equations. For this reason, a non - linear modeling method for a phase - locked loop in a synchronous rotating coordinate system based on the sector - interval method is proposed. Summary of the Invention

[0004] The technical problem to be solved by the present invention is: how to achieve accurate modeling of the system with fewer fuzzy rules. The present invention provides a non - linear modeling method for a phase - locked loop in a synchronous rotating coordinate system based on the sector - interval method, accurately describes the non - linear characteristics of the phase - locked loop, and uses the LMI toolbox to solve the Lyapunov function of the system, which is of great significance for the large - signal stability analysis of the phase - locked loop.

[0005] The present invention solves the above - mentioned technical problems through the following technical solutions. The present invention includes the following steps:

[0006] S1: According to the non - linear model of the phase - locked loop in the traditional synchronous rotating coordinate system, transfer the static operating point to the origin by variable substitution, and select the non - linear term as the fuzzy object;

[0007] S2: Select the sector interval according to the fuzzy object selected in step S1, and select the sector - interval dividing line;

[0008] S3: Determine the first state matrix and the second state matrix according to the sector interval determined in step S2;

[0009] S4: Write out the linear matrix inequality group according to the first state matrix and the second state matrix determined in step S3, and use the LMI toolbox to solve the common positive definite matrix P. If a positive definite matrix P can be found, then reduce the value of the lower bound slope k of the sector interval. If a common positive definite matrix cannot be found, then increase the value of the lower bound slope k of the sector interval. If a positive definite matrix P cannot be found all the time, it means that this modeling method is not applicable. The finally obtained value is the final value of the lower bound slope k of the sector interval. 2 2 2

[0010] S5: According to the finally determined lower bound slope k of the sector interval in step S4, obtain the maximum universe of discourse of the state variable x. 2 1

[0011] S6: Determine the membership function and fuzzy rules according to the maximum universe of discourse of the state variable x determined in step S5. 1

[0012] S7: Determine the equivalent model of the synchronous rotating coordinate system phase-locked loop based on the sector interval method according to the membership function and fuzzy rules determined in step S6.

[0013] Furthermore, in the said step S1, the traditional non-linear model of the phase-locked loop in the synchronous rotating coordinate system is:

[0014]

[0015] where, α is the output of the integrator in the phase-locked loop PI controller, θ pll is the output phase angle of the phase-locked loop, θ g is the grid phase angle, ω g is the grid angular velocity.

[0016] Furthermore, in the said step S1, shifting the static operating point of the traditional non-linear model of the phase-locked loop in the synchronous rotating coordinate system to the origin is:

[0017]

[0018] where, x 1 = θ pll - θ g is the difference between the output phase angle of the phase-locked loop and the output phase angle when the system is stable, where θ pll is the output phase angle of the phase-locked loop, θ g is the grid phase angle, x 2 = α - α 0 , where α is the output of the integrator in the phase-locked loop PI controller, α 0 is its steady-state value; k p and k iare the proportional coefficient and integral coefficient in the PI controller, respectively, I c is the amplitude of the grid-connected current, L g is the grid line inductance, V g is the amplitude of the grid voltage, is the phase angle caused by the grid impedance, ω g is the grid angular velocity; select the non-linear term as the fuzzy object.

[0019] Furthermore, in the step S2, select and as the sector interval demarcation lines, where k 1 = 1, 0 ≤ k 2 <1, the initial value of k 2 is selected as any value in this range, and the final value is to be determined in step S4.

[0020] Furthermore, in the step S3, the first state matrix A 1 、the second state matrix A 2 are expressed as follows:

[0021]

[0022]

[0023] Furthermore, in the step S4, the linear matrix inequality group is shown as follows:

[0024]

[0025] Furthermore, in the step S5, x 1 ∈(-a, a), where a satisfies k 2 a - sina = 0.

[0026] Furthermore, in the step S6, the membership functions ω 1 and ω 2 are respectively as follows:

[0027]

[0028]

[0029] where, ω 1 + ω 2 = 1;

[0030] The fuzzy rules are:

[0031] If x 1 = 0, then the state matrix A = A 1 ;

[0032] If x 1 ≠ 0, then the state matrix A = A 2 .

[0033] Furthermore, in the step S7, the fuzzy object is replaced by the following formula:

[0034]

[0035] The equivalent model of the synchronous rotating coordinate system phase-locked loop based on the sector interval method is:

[0036]

[0037] wherein,

[0038] The present invention has the following advantages compared with the prior art: The nonlinear modeling method of the synchronous rotating coordinate system phase-locked loop based on the sector interval method can achieve accurate modeling of the system with fewer fuzzy rules, greatly reducing the difficulty of solving the Lyapunov function of the system and facilitating the large-signal stability analysis of the phase-locked loop by using the direct Lyapunov method. Description of the Drawings

[0039] Figure 1 is a schematic diagram of the principle of the synchronous rotating coordinate system phase-locked loop in an embodiment of the present invention;

[0040] Figure 2 is a schematic diagram of the maximum sector interval of the fuzzy object in an embodiment of the present invention;

[0041] Figure 3 is a schematic diagram of the maximum stable domain of the large signal of the grid-connected synchronous control system under the maximum universe of discourse in an embodiment of the present invention;

[0042] Figure 4 is a schematic diagram of the membership function of the large-signal fuzzy model of the synchronous rotating coordinate system phase-locked loop in an embodiment of the present invention;

[0043] Figure 5 is the synchronous rotating coordinate system phase-locked loop in an embodiment of the present invention Comparison diagram of the dynamic responses of the system before and after fuzzy;

[0044] Figure 6 is the synchronous rotating coordinate system phase-locked loop in an embodiment of the present invention Comparison diagram of the dynamic responses of the system before and after fuzzy;

[0045] Figure 7 is a schematic diagram of the process of the nonlinear modeling method of the synchronous rotating coordinate system phase-locked loop in an embodiment of the present invention. Detailed Embodiments

[0046] The embodiments of the present invention will be described in detail below. These embodiments are implemented on the premise of the technical solution of the present invention, and detailed implementation manners and specific operation processes are given. However, the protection scope of the present invention is not limited to the following embodiments.

[0047] This embodiment provides a technical solution: a nonlinear modeling method for a synchronous rotating coordinate system phase-locked loop based on the sector interval method. Based on the principle block diagram of the phase-locked loop in the traditional synchronous coordinate system, the synchronous control differential equation of the system is established, and the sector interval method is used to model the nonlinear system of the synchronous rotating coordinate system phase-locked loop, which specifically includes the following steps (see Figure 7 ):

[0048] Step 1: According to the nonlinear model of the phase-locked loop in the traditional synchronous rotating coordinate system, the static operating point is transferred to the origin by using the method of variable substitution, and the nonlinear term is selected as the fuzzy object, that is, is used as the fuzzy object;

[0049] Step 2: Select the sector interval according to the fuzzy object selected in Step 1, and select and as the sector interval boundary;

[0050] Step 3: Determine the state matrices A 1 and A 2 according to the sector interval determined in Step 2;

[0051] Step 4: Write out the linear matrix inequality group according to the state matrices A 1 and A 2 determined in Step 3, and use the LMI toolbox to solve the common positive definite matrix P. If a positive definite matrix P can be found, then reduce the value of the lower bound slope k 2 of the sector interval. If a common positive definite matrix cannot be found, then appropriately increase the value of k 2 . If a positive definite matrix P cannot be found all the time, it means that this modeling method is not applicable; the finally obtained value is the final value of k 2 ;

[0052] Step 5: According to the finally determined lower bound slope k 2 of the sector interval in Step 4, obtain the maximum universe of discourse of the state variable x 1 , x 1 ∈(-a,a);

[0053] Step 6: Determine the membership function and fuzzy rules according to the maximum universe of discourse of the state variable x 1 determined in Step 5;

[0054] Step 7: Finally determine the equivalent model of the synchronous rotating coordinate system phase-locked loop based on the sector interval method according to the membership function and fuzzy rules determined in Step 6.

[0055] In this embodiment, the structural schematic diagram of the synchronous rotating coordinate system phase-locked loop is shown as Figure 1 shown. In grid-connected synchronous control, the grid-connected converter is equivalent to a current source. I c is the reference current amplitude, θ pll is the phase angle output by the phase-locked loop, R g is the grid parasitic resistance, L g is the grid parasitic inductance, V g is the grid amplitude, θ g is the grid phase angle.

[0056] In this embodiment, the reference value I c of the grid-connected current of the system is 3.93 A, the reference current phase angle is the phase angle output by the phase-locked loop, the amplitude V g of the grid voltage is 120 V, the grid frequency is 50 Hz, the grid parasitic resistance Z g is 0.5 Ω, the grid parasitic inductance L g is 0.001 H, the proportional coefficient k p is 1.457, the integral coefficient k i is 129.4813, and the influence phase angle of the line impedance on the phase-locked loop output

[0057] According to Figure 1 it is obtained that the non-linear model of the phase-locked loop under the traditional synchronous rotating coordinate system is:

[0058]

[0059] where α is the output of the integrator in the phase-locked loop PI controller, θ pll is the phase angle output by the phase-locked loop, θ g is the grid phase angle, ω g is the grid angular velocity.

[0060] Let the left side of Equation (1) be 0, that is, when working stably:

[0061]

[0062] It is obtained that α = α 0 , Let x 2 = α - α 0 , where is the phase angle caused by the grid impedance. After shifting the static operating point of the non-linear model of the phase-locked loop under the traditional synchronous rotating coordinate system to the origin, the model is shown as follows:

[0063]

[0064] According to Step 1, select one of them as the fuzzy object.

[0065] In Step 2, select the initial sector interval according to the fuzzy object, and its dividing lines are and

[0066] In Step 3, obtain the state matrices and

[0067] In Step 4, write the linear matrix inequality group according to the obtained state matrices as shown in the following formula:

[0068]

[0069] Using the LMI toolbox, a positive definite matrix P cannot be found. Increase k 2 , at this time k 2 = 0.2, and obtain the state matrices and A positive definite matrix can be found. Decrease k 2 , at this time k 2 = 0.15, and obtain the state matrices and A positive definite matrix P cannot be found. Increase k 2 , and so on. Finally, determine k 2 = 0.1826, The sector interval is as Figure 2 shown. The Lyapunov function V(x) = x T Px. Let to determine the maximum stable region of the system, as Figure 3 shown.

[0070] In Step 5, finally, according to k 2 a - sina = 0, obtain a = 0.84π, that is, the maximum universe of discourse of the state variable x 1 is (-a, a).

[0071] In Step 6, determine the membership function according to the maximum universe of discourse, as shown in the following formula:

[0072]

[0073]

[0074] where ω 1 + ω 2= 1;

[0075] The membership function is as Figure 4 shown, and the fuzzy rules are determined as:

[0076] 1) If x 1 = 0, then the state matrix A = A 1 ;

[0077] 2) If x 1 ≠ 0, then the state matrix A = A 2 ;

[0078] In step 7, according to the membership function and fuzzy rules determined in step 6, the fuzzy object can be replaced by the following formula:

[0079]

[0080] Finally, the equivalent model of the synchronous rotating coordinate system phase-locked loop based on the sector interval method is as follows:

[0081]

[0082] Among them, where

[0083] As Figure 5 and Figure 6 are for verifying the dynamic characteristics of the phase-locked loop in the synchronous rotating coordinate system. Output phase angle jumps of 0.2π and -0.7π are set at 0.05 s and 0.15 s respectively. It can be seen that the phase-locked loop model constructed by the sector interval method has the same dynamic characteristics as the traditional phase-locked loop model, that is, the phase-locked loop model constructed by the present invention is correct.

[0084] In summary, the above-mentioned non-linear modeling method of the synchronous rotating coordinate system phase-locked loop based on the sector interval method can achieve accurate modeling of the system with fewer fuzzy rules, greatly reducing the difficulty of solving the Lyapunov function of the system and facilitating the use of the Lyapunov direct method for large-signal stability analysis of the phase-locked loop.

[0085] Although the embodiments of the present invention have been shown and described above, it can be understood that the above embodiments are exemplary and should not be construed as limiting the present invention. Those of ordinary skill in the art can make changes, modifications, substitutions, and variations to the above embodiments within the scope of the present invention.

Claims

1. Nonlinear Modeling Method of Synchronous Rotating Coordinate System Phase Locked Loop Based on Sector Interval Method Characterized in that: It includes the following steps: S1: According to the nonlinear model of the phase locked loop in the traditional synchronous rotating coordinate system, the static operating point is transferred to the origin by variable substitution, and the nonlinear term is selected as the fuzzy object; In the step S1, transferring the static operating point of the nonlinear model of the phase locked loop in the traditional synchronous rotating coordinate system to the origin is: where x 1 = θ pll - θ g is the difference between the output phase angle of the phase-locked loop and the output phase angle at system stability, where θ pll is the output phase angle of the phase-locked loop, and θ g is the grid phase angle, and x 2 = α - α 0 , where α is the output of the integrator in the PI controller of the phase-locked loop, and α 0 is its steady-state value; k p and k i are the proportional coefficient and integral coefficient in the PI controller, respectively. I c is the amplitude of the grid-connected current, and L g is the inductance of the power grid line, and V g is the amplitude of the grid voltage, is the phase angle caused by the grid impedance, ω g is the angular velocity of the power grid; the non-linear term is selected as the fuzzy object; S2: Select the sector interval according to the fuzzy object selected in the step S1, and select the sector interval dividing line; In the step S2, select and as the dividing lines of the sector intervals, where k 1 = 1, 0 ≤ k 2 < 1, the initial value of k 2 is selected as any value within this range, and the final value is to be determined in step S4; S3: Determine the first state matrix and the second state matrix according to the sector interval determined in the step S2; S4: Write out a set of linear matrix inequalities based on the first state matrix and the second state matrix determined in step S3, and use the LMI toolbox to solve for the common positive definite matrix P. If a positive definite matrix P can be found, then decrease the value of the lower bound slope k of the sector interval. If a common positive definite matrix cannot be found, then increase the value of the lower bound slope k of the sector interval. If a positive definite matrix P cannot be found all the time, it means that this modeling method is not applicable; the finally obtained value is the final value of the lower bound slope k of the sector interval. 2 If a positive definite matrix P can be found, then decrease the value of the lower bound slope k of the sector interval. If a common positive definite matrix cannot be found, then increase the value of the lower bound slope k of the sector interval. 2 If a positive definite matrix P cannot be found all the time, it means that this modeling method is not applicable; the finally obtained value is the final value of the lower bound slope k of the sector interval. 2 ​ S5: Obtain the maximum universe of discourse of the state variable x according to the lower bound slope k of the sector interval finally determined in step S4 2 , 1 ; S6: Determine the membership function and fuzzy rules according to the maximum universe of discourse of the state variable x determined in step S5 1 ; S7: Determine the equivalent model of the synchronous rotating coordinate system phase locked loop based on the sector interval method according to the membership function and fuzzy rules determined in the step S6.

2. The nonlinear modeling method of the synchronous rotating coordinate system phase locked loop based on the sector interval method according to claim 1, Characterized in that: In the step S1, the nonlinear model of the phase locked loop in the traditional synchronous rotating coordinate system is: where α is the output of the integrator in the PLL PI controller, and θ pll is the output phase angle of the PLL, θ g is the grid phase angle, and ω g is the grid angular velocity.

3. The nonlinear modeling method of the synchronous rotating coordinate system phase locked loop based on the sector interval method according to claim 1, Characterized in that: In the step S3, the first state matrix A 1 and the second state matrix A 2 are expressed as follows:

4. The nonlinear modeling method of the synchronous rotating coordinate system phase locked loop based on the sector interval method according to claim 3, Characterized in that: In the step S4, the linear matrix inequality group is as follows:

5. The nonlinear modeling method of the synchronous rotating coordinate system phase locked loop based on the sector interval method according to claim 4, Characterized in that: In the step S5, x 1 ∈ (-a, a), where a satisfies k 2 a - sina = 0.

6. The nonlinear modeling method of the synchronous rotating coordinate system phase locked loop based on the sector interval method according to claim 5, Characterized in that: In the step S6, the membership functions ω 1 and ω 2 are respectively as follows: where ω 1 + ω 2 = 1; The fuzzy rule is: If x 1 = 0, then the state matrix A = A 1 ; If x 1 ≠ 0, then the state matrix A = A 2 .

7. The nonlinear modeling method of the synchronous rotating coordinate system phase locked loop based on the sector interval method according to claim 6, Characterized in that: In the step S7, the fuzzy object is replaced by the following formula: The equivalent model of the synchronous rotating coordinate system phase locked loop based on the sector interval method is: Among them,

Images