A Robust Stability Analysis Method for a Grid-Connected Rectifier under a Weak Grid

Abstract

The present invention discloses a method for analyzing the robust stability of a grid-connected rectifier under a weak grid, which includes the following steps: S1. Establish a differential equation for the grid-connected rectifier connected to the weak grid to describe the dynamic behavior of the system; S2. Use the structural perturbation theory to obtain the perturbation model of the system; S3. Under different perturbation levels, use the bifurcation theory to solve the Hopf bifurcation point of the system and construct a bifurcation diagram; S4. For the Hopf bifurcation points under different perturbation levels, adopt the singular perturbation theory to track the evolution law of the limit cycle; S5. By analyzing the bifurcation diagram and the phase diagram corresponding to the limit cycle, visually display the instability path of the grid-connected rectifier under the weak grid. The present invention can accurately analyze the instability mechanism of the grid-connected rectifier under weak grid conditions and provide theoretical support for system design and operation.

Description

Technical Field

[0001] The present invention relates to the technical field of distributed energy grid connection, and particularly to a method for analyzing the robust stability of a grid-connected rectifier under a weak grid. Background Art

[0002] With the continuous increase in the penetration rate of new energy, the access of intermittent energy sources such as wind energy and photovoltaic energy has significantly changed the dynamic characteristics of traditional power grids, making the power grid exhibit more significant "weakening" characteristics, including problems such as high impedance, low short-circuit ratio, and low inertia. Under weak grid conditions, there are complex nonlinear interactions between the grid-connected rectifier and the power grid, which may lead to system instability. Therefore, studying the robust stability and instability path of the grid-connected rectifier under a weak grid can provide effective guidance for system parameter adjustment.

[0003] In existing research, to simplify the analysis process, the dynamic characteristics of the current loop of the grid-connected rectifier are usually ignored, and mainly linearization analysis methods are used, such as small-signal analysis and impedance analysis. However, these methods fail to fully reveal the nonlinear characteristics in the system, resulting in limitations in the assessment of system stability and difficulty in accurately identifying the formation mechanism of the instability path. The limitations of the above analysis methods significantly restrict the effectiveness of system design and operation optimization, and it is difficult to meet the requirements for the stability and reliability assessment of grid-connected rectifiers under weak grid conditions. Summary of the Invention

[0004] To solve the problem of assessing the stability and reliability of grid-connected rectifiers under weak grid conditions, the present invention proposes a method for analyzing the robust stability of a grid-connected rectifier under a weak grid to solve the above problems.

[0005] The present application discloses a method for analyzing the robust stability of a grid-connected rectifier under a weak grid, including the following steps:

[0006] S1. Establish a differential equation for the grid-connected rectifier accessing the weak grid to describe the dynamic behavior of the system;

[0007] S2. Based on the differential equation of the grid-connected rectifier accessing the weak grid, use the structural perturbation theory to obtain the perturbation model of the system;

[0008] S3. Under different perturbation levels, use the bifurcation theory to solve the Hopf bifurcation point of the system and construct a bifurcation diagram;

[0009] S4. For the Hopf bifurcation points under different perturbation levels, use the singular perturbation theory to track the evolution law of the limit cycle;

[0010] S5. By analyzing the bifurcation diagram and the phase diagram corresponding to the limit cycle, visually display the instability path of the grid-connected rectifier under the weak grid and directly determine the robustness of the system.

[0011] Preferably, the S1 includes the following steps:

[0012] S11. Construct a model of the phase-locked loop:

[0013]

[0014] wherein, , and are state variables of the phase-locked loop, and are PI control coefficients of the phase-locked loop, is the q-axis component of the PCC point voltage, is the AC grid frequency, is the ideal value of the AC grid frequency;

[0015] S12. Construct a model of the voltage loop:

[0016]

[0017] wherein, is the DC voltage on the rectifier side, is the reference value of the voltage , is the state variable of the voltage controller, is the d-axis component of the voltage on the rectifier side, is the q-axis component of the voltage on the rectifier side, is the d-axis component of the current flowing through the filter inductor, is the q-axis component of the current flowing through the filter inductor, is the DC bus capacitor of the grid-connected rectifier, is the DC bus load resistance of the grid-connected rectifier;

[0018] S13. Construct a model of the current loop:

[0019]

[0020]

[0021] wherein, and are state variables of the current controller, is the d-axis component of the current, is the q-axis component of the current, is the reference value of is the reference value of , and is the PI control coefficient for voltage control, is the voltage on the rectifier side reference value, is the voltage on the rectifier side reference value, and is the PI control coefficient for voltage control;

[0022] S14. Construct the dynamic models of the delay module, LCL filter, and grid impedance:

[0023]

[0024]

[0025]

[0026]

[0027] Among them, is the delay period, is the filter inductor, is the filter capacitor, is stray resistance, is the d-axis component of the voltage of the filter capacitor and stray resistance, is the q-axis component of the voltage of the filter capacitor and stray resistance, is the d-axis component of the voltage of the filter capacitor, is the q-axis component of the voltage of the filter capacitor, and are the line inductors, is corresponding resistance, is corresponding resistance, is the d-axis component of the weak AC grid voltage, is the q-axis component of the weak AC grid voltage;

[0028] S15. Obtain the differential equation of the grid-connected rectifier connected to the weak grid by synthesizing S11 - S14, where the state vector is:

[0029]

[0030] The differential equation of the grid-connected rectifier connected to the weak grid is:

[0031]

[0032] Among them, is the state matrix of the system.

[0033] Preferably, the perturbation model of the system is:

[0034]

[0035] where the matrix is the structured perturbation of the differential equation of the grid-connected rectifier connected to a weak grid, and the matrices and both reflect the structural information of the perturbation, and the matrix represents the unknown perturbation;

[0036] If there are multiple perturbations in the system, then the perturbation model of the system is:

[0037]

[0038] If , , , then the perturbation models shown in Equation and Equation are in the same form.

[0039] Preferably, the S3 includes the following steps:

[0040] Assume is the measurement index of the perturbation degree of the parameter, and use Equation or Equation to construct the perturbation model of the grid-connected rectifier connected to a weak grid at the , and perturbation levels;

[0041] For the perturbation models at the , and perturbation levels, use the bifurcation theory to detect the location of the Hopf bifurcation point, and construct the bifurcation diagram of the system accordingly.

[0042] Preferably, the step of using the bifurcation theory to detect the location of the Hopf bifurcation point includes the following steps:

[0043] Express Equation (10) as , where represents the bifurcation parameter. When , if there is a pair of pure imaginary eigenvalues in and , then the system undergoes a Hopf bifurcation at the point , represents the eigenvalue, and is the imaginary part of the eigenvalue.

[0044] Preferably, S4 includes the following steps:

[0045] Starting from the Hopf bifurcation point, the singular perturbation theory is used to analyze the evolution process of the limit cycle of the system under parameter changes, and further reveal the dynamic characteristics of the limit cycle.

[0046] Preferably, the process of using the singular perturbation theory to analyze the evolution process of the limit cycle of the system under parameter changes is shown as follows:

[0047]

[0048] where, is the amplitude of the limit cycle, and are the partial derivatives of the time scale, is a small parameter, .

[0049] Preferably, S5 includes the following steps:

[0050] For a determined perturbation level, combined with the change trend of the limit cycle generated by the adjoint Hopf bifurcation and the stability of the limit cycle, reveal the instability path of the system;

[0051] By comparing and analyzing the bifurcation diagrams of the system under different perturbation levels, further judge the robust stability of the system, so as to provide effective guidance for the adjustment of system parameters.

[0052] Advantages of the present invention:

[0053] (1) The present invention fully considers the interaction between dynamic links when the grid-connected rectifier is connected to the weak grid system, combines the bifurcation theory and the structural perturbation theory, systematically studies the robust stability characteristics of the system, clarifies the influence of parameters on the system stability, and provides a comprehensive and specific guiding basis for optimizing the system parameter adjustment.

[0054] (2) Through bifurcation analysis and limit cycle tracking, the present invention intuitively shows the instability path of the grid-connected rectifier under weak grid conditions, clearly reveals the dynamic evolution law during the system instability process, and provides a scientific and reliable theoretical support for the optimal control design of the system and the in-depth understanding of the instability mechanism. Description of the Drawings

[0055] Figure 1 is the flow chart of the method for analyzing the robust stability of the grid-connected rectifier under weak grid in the embodiment of the present invention;

[0056] Figure 2 is the structural diagram of the grid-connected rectifier connected to the weak grid system in the embodiment of the present invention;

[0057] Figure 3System bifurcation diagram of the embodiment of the present invention;

[0058] Figure 4 Bifurcation diagrams under different perturbation levels of the embodiment of the present invention;

[0059] Figure 5 Phase diagram of the system limit cycle of the embodiment of the present invention;

[0060] Figure 6 Phase diagram of the stable region of the embodiment of the present invention;

[0061] Figure 7 Time-domain diagram of the stable region of the embodiment of the present invention;

[0062] Figure 8 Phase diagram of the unstable limit cycle region of the embodiment of the present invention. Detailed implementation manners

[0063] To make the objectives, technical solutions and advantages of the present application clearer and more understandable, the following examples are given with reference to the accompanying drawings to further elaborate on the present application in detail.

[0064] The embodiment of the present invention discloses a method for robust stability analysis of a grid-connected rectifier under a weak grid. The structure of the grid-connected rectifier connected to the weak grid system is as Figure 2 shown, Figure 2 in is the AC grid voltage, is the impedance, is the voltage at the PCC point, is and voltage, is the damping resistor. Analyze the system according to the steps shown in Figure 1 as follows. The specific steps are as follows:

[0065] S1. System modeling: Establish a differential equation for the grid-connected rectifier connected to the weak grid to describe the dynamic behavior of the system.

[0066] S11. Construct the model of the phase-locked loop:

[0067]

[0068] where, , and are the state variables of the phase-locked loop, and are the PI control coefficients of the phase-locked loop, is the q-axis component of the voltage at the PCC point, is the AC grid frequency, is the ideal value of the AC grid frequency.

[0069] S12. Construct the model of the voltage loop:

[0070]

[0071] where, is the DC voltage on the rectifier side, is the voltage reference value, is the state variable of the voltage controller, is the d-axis component of the voltage on the rectifier side, is the q-axis component of the voltage on the rectifier side, is the d-axis component of the current flowing through the filter inductor, is the q-axis component of the current flowing through the filter inductor, is the DC bus capacitor of the grid-connected rectifier, is the DC bus load resistance of the grid-connected rectifier.

[0072] S13. Construct the model of the current loop:

[0073]

[0074]

[0075] where, and are the state variables of the current controller, is d-axis component of the current, is q-axis component of the current, is reference value, is reference value, , and are the PI control coefficients of the voltage control, is the reference value of the voltage on the rectifier side, is the reference value of the voltage on the rectifier side, and are the PI control coefficients of the voltage control.

[0076] S14. Construct the dynamic models of the delay module, LCL filter and grid impedance:

[0077]

[0078]

[0079]

[0080]

[0081] Among them, is the delay period, is the filter inductor, is the filter capacitor, is the stray resistance of is the d-axis component of the voltage of the filter capacitor and the stray resistance, is the q-axis component of the voltage of the filter capacitor and the stray resistance, is the d-axis component of the voltage of the filter capacitor, is the q-axis component of the voltage of the filter capacitor, and are the line inductors, is the corresponding resistance, is the corresponding resistance, is the d-axis component of the weak AC grid voltage, is the q-axis component of the weak AC grid voltage.

[0082] S15. By synthesizing S11 - S14, the differential equation for the grid-connected rectifier connected to the weak grid is obtained, where the state vector is:

[0083]

[0084] The differential equation for the grid-connected rectifier connected to the weak grid is:

[0085]

[0086] Among them, is the state matrix of the system.

[0087] S2. Disturbance modeling: Based on the differential equation of the grid-connected rectifier connected to the weak grid, the disturbance model of the system is obtained using the structural disturbance theory as follows:

[0088]

[0089] Among them, the matrix is the structural disturbance of the differential equation of the grid-connected rectifier connected to the weak grid, the matrices and both reflect the structural information of the perturbation, and the matrix represents the unknown disturbance;

[0090] If there are multiple disturbances in the system, then the disturbance model of the system, i.e., Equation (11), can be expressed as:

[0091]

[0092] If , , ,then the perturbation models shown in Equation and Equation are in the same form.

[0093] S3. Solve the bifurcation diagram based on bifurcation theory: At different perturbation levels, use bifurcation theory to solve the Hopf bifurcation points of the system and construct the bifurcation diagram.

[0094] Assume is the measurement index of the perturbation degree of the parameter. Use Equation or Equation to construct the perturbation models of the grid-connected rectifier connected to the weak grid at 、 and perturbation levels;

[0095] For the perturbation models at 、 and perturbation levels, use bifurcation theory to detect the positions of the Hopf bifurcation points and construct the bifurcation diagram of the system accordingly. Detecting the positions of the Hopf bifurcation points using bifurcation theory includes the following steps:

[0096] Express Equation (10) as , where represents the bifurcation parameter. When , there is a pair of pure imaginary eigenvalues in and , then the system undergoes a Hopf bifurcation at the point . represents the eigenvalue, and is the imaginary part of the eigenvalue. In this embodiment, the system bifurcation diagram drawn by combining S1, S2, and S3 is as shown in Figure 3 、 Figure 4 .

[0097] S4. Limit cycle evolution law: For the Hopf bifurcation points at different perturbation levels, use singular perturbation theory to trace the evolution law of the limit cycle.

[0098] Starting from the Hopf bifurcation point, use singular perturbation theory to analyze the evolution process of the limit cycle of the system under parameter changes, and further reveal the dynamic characteristics of the limit cycle. The evolution process of the limit cycle is shown in the following equation:

[0099]

[0100] where is the amplitude of the limit cycle, and is the partial derivative of the time scale, is a small parameter, .

[0101] Based on the system bifurcation diagram plotted in S3, the limit cycle traced using the singular perturbation theory is obtained as Figure 5 shown.

[0102] S5. Analyze the robust stability and instability path: By analyzing the bifurcation diagram and the phase diagram corresponding to the limit cycle, visually display the instability path of the grid-connected rectifier under a weak grid, and directly determine the robustness of the system. For a determined perturbation level, combine the change trend of the limit cycle generated by the adjoint Hopf bifurcation and the stability of the limit cycle to reveal the instability path of the system; then, by comparing and analyzing the bifurcation diagrams of the system under different perturbation levels, further judge the robust stability of the system, thereby providing effective guidance for the adjustment of system parameters. The specific steps are as follows:

[0103] Bifurcation analysis and stability study: In the AC grid connected to the grid-connected system, the line inductance will weaken the stability of the AC grid. To analyze its influence, in this embodiment, the line inductance is used as the key control variable for bifurcation analysis, and the results are as Figure 3 shown:

[0104] When , the system will undergo a Hopf bifurcation;

[0105] When , the system will undergo a Saddle-node bifurcation;

[0106] The above bifurcation behaviors reveal the significant influence of parameter changes on the system stability and dynamic behavior, providing a theoretical basis for optimizing the operation of the grid-connected rectifier.

[0107] Robust stability analysis: Based on the structural perturbation theory, in this embodiment, for the line inductance at , and three perturbation levels, the corresponding perturbation models are constructed, and the system bifurcation diagrams as Figure 4 shown are plotted. It can be seen from Figure 4 that:

[0108] As the perturbation level increases, the stable range of the DC bus load resistance of the grid-connected rectifier gradually decreases. In particular, when the perturbation level reaches When this occurs, two new Hopf bifurcation points are added to the system, significantly complicating the bifurcation characteristics of the system. This indicates that the sensitivity of the system to structural perturbations significantly affects its stability and dynamic behavior. To improve the robustness of the system, the analysis results in Figure 3 and Figure 4 can be combined to optimize and adjust the parameters of the system so that it operates in a stable region, thereby achieving the coordination of stability and robustness.

[0109] Analysis of the instability path: In Figure 5 , in this embodiment, the limit cycle of the Hopf bifurcation point shown in Figure 3 is traced and analyzed. The following results are obtained:

[0110] The stable limit cycle of the system extends into the unstable region. After extending to a certain region, the system undergoes period-doubling bifurcation, and the stable limit cycle transforms into an unstable limit cycle. Finally, after two fold bifurcations, the limit cycle of the system completely disappears. Combining the analysis in Figure 3 and Figure 5 can intuitively reveal the instability path of the system and its dynamic evolution characteristics, providing an important basis for the optimization of system operation.

[0111] Verification of typical scenarios: Figure 6 , Figure 7 and Figure 8 correspond to three typical cases in the bifurcation diagram in Figure 3 respectively, specifically:

[0112] Figure 6 (L s = 7.7 mH): The system is in a stable state;

[0113] Figure 7 (L s = 8.5 mH): The system converges to a stable limit cycle after a certain oscillation;

[0114] Figure 8 (L s = 10.3 mH): The system loses stability after oscillation.

[0115] Through the verification of the analysis results of the above typical scenarios, the accuracy and effectiveness of the bifurcation analysis in Figure 3 are further proven, thus supporting the comprehensive description of the dynamic characteristics of the system in this application.

[0116] In summary, the present application fully considers the interaction between dynamic links when a grid-connected rectifier is connected to a weak grid system. Combining bifurcation theory and structural perturbation theory, the robust stability characteristics of the system are systematically studied. Through analysis, the influence of parameters on the system stability is clarified, providing a comprehensive and specific guiding basis for optimizing system parameter adjustment. At the same time, through bifurcation analysis and limit cycle tracking, the present application intuitively shows the instability path of the grid-connected rectifier under weak grid conditions, clearly revealing the dynamic evolution law during the system instability process, providing a scientific and reliable theoretical support for the optimal control design of the system and the in-depth understanding of the instability mechanism.

[0117] The foregoing has shown and described the basic principles, main features and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited by the above embodiments, and what is described in the above embodiments and the specification is only to illustrate the principle of the present invention. Without departing from the spirit and scope of the present invention, the present invention will have various changes and improvements, and all these changes and improvements fall within the scope of the present invention claimed. The scope of the present invention claimed is defined by the appended claims and their equivalents.

Claims

1. A method for analyzing the robust stability of a grid-connected rectifier under a weak grid, characterized in that, It includes the following steps: S1. Establish the differential equation of the grid-connected rectifier connected to a weak grid to describe the dynamic behavior of the system; S2. Based on the differential equation of the grid-connected rectifier connected to a weak grid, use the structural perturbation theory to obtain the perturbation model of the system; S3. Under different perturbation levels, use the bifurcation theory to solve the Hopf bifurcation point of the system and construct a bifurcation diagram; S4. For the Hopf bifurcation points under different perturbation levels, adopt the singular perturbation theory to track the evolution law of the limit cycle; S5. By analyzing the bifurcation diagram and the phase diagram corresponding to the limit cycle, visually display the instability path of the grid-connected rectifier under a weak grid, and directly determine the robustness of the system.

2. The robust stability analysis method of the grid-connected rectifier according to claim 1 under a weak grid is characterized in that, The S1 includes the following steps: S11. Construct the model of the phase-locked loop: Among them, , and are the state variables of the phase-locked loop, and are the PI control coefficients of the phase-locked loop, is the q-axis component of the PCC point voltage, is the AC grid frequency, is the ideal value of the AC grid frequency; S12. Construct the model of the voltage loop: Among them, is the DC voltage on the rectifier side, is the reference value of the voltage , is the state variable of the voltage controller, is the d-axis component of the voltage on the rectifier side, is the q-axis component of the voltage on the rectifier side, is the d-axis component of the current flowing through the filter inductor, is the q-axis component of the current flowing through the filter inductor, is the DC bus capacitor of the grid-connected rectifier, is the DC bus load resistance of the grid-connected rectifier; S13. Construct the model of the current loop: Among them, and are the state variables of the current controller, is the d-axis component of the current, is the q-axis component of the current, is the reference value of is the reference value of , and are the PI control coefficients of the voltage control, is the reference value of the rectifier-side voltage , is the reference value of the rectifier-side voltage , and are the PI control coefficients of the voltage control; S14. Construct the dynamic models of the delay module, LCL filter, and grid impedance: Among them, is the delay period, is the filter inductor, is the filter capacitor, is the stray resistance of, is the d-axis component of the voltage of the filter capacitor and the stray resistance, is the q-axis component of the voltage of the filter capacitor and the stray resistance, is the d-axis component of the voltage of the filter capacitor, is the q-axis component of the voltage of the filter capacitor, and is the line inductor, is the corresponding resistance, is the corresponding resistance, is the d-axis component of the weak AC grid voltage, is the q-axis component of the weak AC grid voltage; S15. Synthesize S11 - S14 to obtain the differential equation of the grid-connected rectifier connected to a weak grid, where the state vector is: The differential equation of the grid-connected rectifier connected to a weak grid is: Among them, is the state matrix of the system.

3. The method for analyzing the robust stability of the grid-connected rectifier under a weak grid according to claim 2, characterized in that, The perturbation model of the system is: Among them, the matrix is the structured disturbance of the differential equation for the grid-connected rectifier connected to a weak grid. The matrices and both reflect the structural information of the perturbation. The matrix represents the unknown perturbation; If there are multiple disturbances in the system , then the disturbance model of the system is: If , , , then the perturbation models shown in Equation and Equation are in the same form.

4. The method for analyzing the robust stability of the grid-connected rectifier under a weak grid according to claim 3, wherein The S3 includes the following steps: Assumption is a measure index of the perturbation degree of the parameter. Using Equation or Equation to construct a perturbation model of the grid-connected rectifier connected to a weak grid under , and perturbation levels; For , and the perturbation models of the perturbation levels, the bifurcation theory is used to detect the position of the Hopf bifurcation point, and the bifurcation diagram of the system is constructed accordingly.

5. The method for analyzing the robust stability of the grid-connected rectifier under a weak grid according to claim 4, characterized in that, The steps for detecting the position of the Hopf bifurcation point using the bifurcation theory include the following steps: Express Equation (10) as , where denotes the bifurcation parameter, and when , it satisfies that there exists a pair of pure imaginary eigenvalues in and and , then the system undergoes a Hopf bifurcation at the point , denotes the eigenvalue, and is the imaginary part of the eigenvalue.

6. The method for analyzing the robust stability of the grid-connected rectifier under a weak grid according to claim 5, characterized in that, The S4 includes the following steps: Starting from the Hopf bifurcation point, use the singular perturbation theory to analyze the evolution process of the limit cycle under parameter changes, and further reveal the dynamic characteristics of the limit cycle.

7. The method for analyzing the robust stability of the grid-connected rectifier under a weak grid according to claim 6, wherein The analysis of the evolution process of the limit cycle under parameter changes using the singular perturbation theory is shown as follows: Among them, is the amplitude of the limit cycle, and is the partial derivative of the time scale, is a small parameter, .

8. The method for analyzing the robust stability of the grid-connected rectifier under a weak grid according to claim 7, characterized in that, The S5 includes the following steps: For a determined perturbation level, combine the change trend of the limit cycle generated by the associated Hopf bifurcation and the stability of the limit cycle to reveal the instability path of the system; By comparing and analyzing the bifurcation diagrams of the system under different perturbation levels, further judge the robust stability of the system, so as to provide effective guidance for the adjustment of system parameters.

Images