Method for analyzing interaction of multiple grid-connected inverters based on matrix norm theory

Abstract

The application discloses a method for analyzing the interaction of multiple grid-connected inverters based on matrix norm theory, and aims at the interaction between grid-connected inverters, takes a power distribution system containing two grid-connected inverters as an analysis object, establishes an equivalent grid-connected model containing two inverters based on the Norton theorem, derives a transfer function matrix of a control system, and quantitatively analyzes the relationship between the interaction between inverters and the system frequency, equivalent grid-connected inductance and controller combination mode based on the spectral norm and condition number of the transfer function matrix, so as to provide a reference for the coordinated control of the inverters. A simulation model is built in Matlab / Simulink for analysis. The application provides a simple and effective method for analyzing the interaction of inverters, and provides a theoretical method for the coordinated control of the inverters.

Description

Technical Field

[0001] This invention relates to the field of multi-grid-connected inverter control technology, and specifically to a method for analyzing the interaction effects of multi-grid-connected inverters based on matrix norm theory. Background Technology

[0002] With the escalating global energy crisis and increased environmental awareness, new energy power generation, represented by wind and solar power, has attracted significant attention due to its cleanliness and renewability. The installed capacity of wind and solar power has increased significantly in recent years. As a crucial interface for grid connection of new energy sources, a large number of inverters are installed in the distribution network. Interactions exist between controllers with similar structures and functions within these inverters. Furthermore, the long-distance transmission of wind and solar power, coupled with line impedance, can cause the new energy grid connection system to exhibit a weak grid state. This exacerbates the interaction between inverters, reducing power quality and even disrupting the stable operation of the power system. Therefore, the stable operation of inverters is crucial for the absorption of new energy sources such as wind power.

[0003] Current research focuses on the interaction effects between FACTS devices, with limited research on the interaction effects between grid-connected inverters. Furthermore, the analysis methods for inverter interaction effects often employ relative gain matrices; however, relative gain matrices based on steady-state operating points sometimes lead to incorrect assessments. In summary, current research on the interaction effects between inverters is insufficient, making it even more difficult to achieve coordinated control of multi-inverter grid-connected systems. Summary of the Invention

[0004] To address the interaction problem among multiple grid-connected inverters, this invention provides a method for analyzing the interaction effects of multiple grid-connected inverters based on matrix norm theory. This method uses the spectral norm and condition number of the system transfer function matrix as evaluation indicators to analyze the relationship between interaction effects and frequency, equivalent grid-connected inductance, and the combination mode of the inverter outer loop controller, providing a theoretical basis for achieving coordinated control of multiple grid-connected inverters.

[0005] The technical solution adopted in this invention is as follows:

[0006] The method for analyzing the interaction effects of multiple grid-connected inverters based on matrix norm theory includes the following steps:

[0007] Step 1: Construct a grid-connected model for a single inverter, where the inverter is an LCL type inverter, and a dual-loop control strategy is adopted to suppress resonance spikes;

[0008] Step 2: Obtain the system control block diagram from Step 1, simplify it according to Mason's formula, and obtain the equivalent circuit of a single inverter according to Norton's theorem;

[0009] Step 3: Study the interaction between the two inverters. Replace the inverters with Norton equivalent circuits. Based on Kirchhoff's current law, the transfer function matrix of the grid-connected system with the two inverters can be obtained.

[0010] Step 4: Based on the transfer function matrix of the grid-connected system of two inverters obtained in Step 3, derive the evaluation index of inverter interaction influence according to matrix norm theory.

[0011] Step 5 also includes, for a grid-connected system with two inverters, three control scenarios are established to explore the impact of different factors on the interaction between inverters:

[0012] ① To investigate the relationship between the interaction between inverters and frequency, while keeping other parameters constant, harmonic currents at frequencies of 250Hz, 650Hz and 850Hz were injected into the inverter reference current, and the spectral norm and condition number of the transfer function matrix of the control system at the corresponding frequencies were calculated.

[0013] ② To investigate the interaction between inverters and the equivalent grid-connected inductance L g Given the relationship between the two, keeping other factors constant, the equivalent grid-connected inductance is set to 1mH, 2mH and 3mH respectively, and the spectral norm and condition number of the transfer function matrix of the control system under the corresponding equivalent grid-connected inductance are calculated.

[0014] ③ To explore the relationship between the interaction between grid-connected inverters and the combination mode of the current outer loop controller, while keeping other factors constant, the inverter outer loop controller was set to three combination modes: PI+PI, PI+quasi-PR, and quasi-PR+quasi-PR. The spectral norm and condition number of the control system transfer function matrix were calculated for different controller combination modes.

[0015] Step 5 sets up different control scenarios and analyzes the relationship between the interaction between inverters and frequency, equivalent grid-connected inductance, and the combination mode of the inverter outer loop controller, providing a method to suppress the interaction between inverters.

[0016] Step 6: Build a grid-connected model of two inverters in Matlab / simulink, and compare the simulation results with the results calculated based on matrix spectral norm and condition number to verify the correctness of the interactive analysis method proposed in this invention.

[0017] In step 1, the inverter grid connection model is as follows: Figure 1 As shown, where U g L is the system voltage. g As the equivalent inductance of a weak power grid, L1, C, and L2 form an LCL-type filter, H i1 H is the inner loop capacitor current feedback coefficient. i2 This represents the feedback coefficient of the outer loop grid-connected current. refFor reference input current, G i (s) is the transfer function of the PI-type outer loop current controller. The inverter adopts a single-pole frequency-doubling sine pulse width modulation (SPWM) control strategy. inv (s) is the voltage transfer function from the modulation wave side to the converter valve side.

[0018] In step 2, the inverter control block diagram is simplified using Mason blotting. The simplified control block diagram is as follows: Figure 2 As shown,

[0019] in:

[0020]

[0021]

[0022] The system loop gain T(s) is:

[0023] T(s) = G x1(s) G x2(s) H i2 (3)

[0024] In step 2, the equivalent circuit of a single inverter is shown in equation (4):

[0025]

[0026] Where: i gi (s) represents the inverter output current; G i i represents the control coefficient of the controlled source in the Norton equivalent circuit; gref (s) represents the inverter reference output current; Y i For the Norton circuit's equivalent admittance; U PCC (s) represents the inverter's grid connection point voltage.

[0027] In step 3, the transfer function matrix of the grid-connected system with two inverters is shown in equation (5):

[0028]

[0029] Where: i g1 and i g2 These are the output currents of the two inverters, i gref1(s) and i gref2(s) These are the reference output currents of the two inverters, respectively; G1 is the controlled source control coefficient in the equivalent Norton circuit of inverter 1, and G2 is the controlled source control coefficient in the equivalent Norton circuit of inverter 2; Y1 is the equivalent admittance in the equivalent Norton circuit of inverter 1, and Y2 is the equivalent admittance in the equivalent Norton circuit of inverter 2; Y g For equivalent grid-connected admittance; ug This is the grid voltage.

[0030] In step 3, Kirchhoff's current law is used to obtain the transfer function matrix of the grid-connected system of two inverters. From the transfer function matrix, it can be seen that the output of any inverter is not only related to its own reference current, but also affected by the reference current of other inverters and the grid voltage. This theoretically confirms that there is an interactive influence between inverters.

[0031] In step 4, the inverter interaction impact evaluation index is specifically as follows:

[0032] (1) Spectral norm of the transfer function matrix:

[0033] Let the transfer function matrix of the control system be G(s), then the eigenvalues ​​λ of G(s) can be obtained by equation (6).

[0034] det[λI-G(s)]=0 (6);

[0035] Where: I is the identity matrix.

[0036] Based on the eigenvalues ​​of the transfer function matrix, the spectral norm of the transfer function matrix is:

[0037]

[0038] Where: λ max Let G(s) be the largest eigenvalue; G(s) T This is the transpose of G(s). The closer the spectral norm of the transfer function matrix of the control system is to the value "1", the smaller the interaction between the input and output channels of the system.

[0039] (2) The condition number k of the transfer function matrix:

[0040] The condition number of the transfer function matrix is ​​obtained from equation (8):

[0041] k = ||G(s)||·||G(s) -1 ||(8)

[0042] Where: k is the condition number of the matrix; G(s) -1 Let k denote the inverse matrix of G(s); ||·|| is an arbitrary norm of the matrix. When the condition number is too large, the system will be in an "ill-conditioned" state, so k should be as small as possible.

[0043] By combining the spectral norm and condition number of the transfer function matrix of the inverter grid-connected system, it is possible to quantitatively analyze the degree of interaction between inverters.

[0044] In step 4, the spectral norm of the transfer function matrix characterizes the dynamic characteristics of the system, while the condition number of the matrix characterizes the steady-state characteristics. Combining the spectral norm and condition number of the control system's transfer function matrix can more fully reflect the degree of interaction between inverters. Based on matrix norm theory, an analytical index for the interaction between inverters is proposed, avoiding the erroneous analysis that sometimes occurs when based on the relative gain matrix at the steady-state operating point, thus making the analysis of interaction more accurate.

[0045] This invention provides a method for analyzing the interactive effects of multiple grid-connected inverters based on matrix norm theory. The technical advantages are as follows:

[0046] 1) Regarding the evaluation index of interaction effects, this invention addresses the shortcomings of traditional relative gain matrix analysis methods based on steady-state operating points. It proposes an evaluation index based on matrix norm theory, which comprehensively considers the dynamic and steady-state performance of the system, using both spectral norm and condition number. This index is simple to calculate and can quickly and efficiently assess the degree of interaction effects between inverters.

[0047] 2) Regarding the influencing factors of interaction, this invention analyzes the relationship between interaction and frequency, equivalent grid-connected inductance, and the combination mode of the inverter outer loop controller, providing a theoretical method for suppressing interaction and achieving coordinated control among inverters. Attached Figure Description

[0048] Figure 1 This is a structural diagram of a single inverter according to the present invention.

[0049] Figure 2 This is a simplified control block diagram of the inverter described in this invention.

[0050] Figure 3 This is the equivalent Norton circuit diagram of the inverter described in this invention.

[0051] Figure 4 This is a diagram of the equivalent grid-connected model of the multi-inverter system described in this invention.

[0052] Figure 5 The THD diagram shows the output current of inverter 1 when it is connected to the grid alone according to the present invention.

[0053] Figure 6 The THD diagram of the output current of inverter 1 when the inverter described in this invention is simultaneously connected to the grid is shown.

[0054] Figure 7(a) shows the L described in this invention. g THD diagram of inverter 1 output current when = 2mH;

[0055] Figure 7(b) shows the L described in this invention. g THD diagram of inverter 1 output current when =3mH.

[0056] Figure 8(a) shows the THD of the output current of inverter 1 when the inverter outer loop controller combination mode is quasi-PR+quasi-PR as described in this invention.

[0057] Figure 8(b) shows the THD of the output current of inverter 1 when the inverter outer loop controller combination mode is PI+quasi-PR as described in this invention.

[0058] Figure 9 This is a flowchart of the present invention. Detailed Implementation

[0059] This paper presents a method for analyzing the interaction effects of multiple grid-connected inverters based on matrix norm theory. Taking a power distribution system with two grid-connected inverters as the analysis object, an equivalent grid-connected model with two inverters is established based on Norton's theorem. The transfer function matrix of the control system is derived from this model. Based on matrix norm theory, the spectral norm and condition number of the transfer function matrix are used as indicators to quantitatively analyze the relationship between the interaction effects between inverters and the system frequency, equivalent grid-connected inductance, and controller combination, providing a reference for coordinated inverter control. A simulation model is built in Matlab / Simulink for analysis. The specific flowchart is shown below. Figure 9 As shown, it includes the following steps:

[0060] Step 1: Construct a grid-connected model for a single inverter. The inverter used is an LCL type, and a dual-loop control strategy is employed to suppress resonance spikes. The inverter structure is as follows: Figure 1 As shown.

[0061] Step 2: Obtain the system control block diagram from Step 1, and simplify it according to Mason's formula to obtain the simplified control block diagram, as shown below. Figure 2 As shown, the equivalent circuit of a single inverter is obtained according to Norton's theorem;

[0062] i gi (s)=G i i grefi (s)-Y i U PCC (s) (1)

[0063] Where: i gi (s) represents the inverter output current; G i i represents the control coefficient of the controlled source in the Norton equivalent circuit; gref (s) represents the inverter reference output current; Y i For the Norton circuit's equivalent admittance; U PCC (s) represents the inverter's grid connection point voltage.

[0064] Step 3: Replace the inverter with an equivalent Norton circuit to obtain the equivalent model of multi-inverter grid connection, such as... Figure 4As shown. This invention studies the interaction between two inverters. Based on Kirchhoff's current law, the transfer function matrix of the grid-connected system of two inverters can be obtained as shown in equation (2).

[0065]

[0066] Where: i g1 and i g2 These are the output currents of the two inverters, i gref1(s) and i gref2(s) These are the reference output currents of the two inverters; G1 and G2 are the control coefficients of the controlled source in the Norton circuit; Y1 and Y2 are the equivalent admittances in the Norton circuit; Y g For equivalent grid-connected admittance; u g This is the grid voltage.

[0067] Step 4: Using the transfer function matrix of the grid-connected system with two inverters obtained in Step 3, derive the evaluation index of inverter interaction effects based on matrix norm theory. Specifically:

[0068] (1) Spectral norm of the transfer function matrix:

[0069] Let the transfer function matrix of the control system be G(s), then the eigenvalue λ of G(s) can be obtained by equation (3).

[0070] det[λI-G(s)]=0 (3)

[0071] Where I is the identity matrix.

[0072] Based on the eigenvalues ​​of the transfer function matrix, the spectral norm of the transfer function matrix can be obtained as follows:

[0073]

[0074] Where: λ max Let G(s) be the largest eigenvalue; G(s) T This is the transpose of G(s). The closer the spectral norm of the transfer function matrix of the control system is to the value "1", the smaller the interaction between the input and output channels of the system.

[0075] (2) The condition number k of the transfer function matrix:

[0076] The condition number of a matrix can be obtained from equation (5):

[0077] k = ||G(s)||·||G(s) -1 || (5)

[0078] Where: k is the condition number of the matrix; G(s) -1Let k denote the inverse matrix of G(s); ||·|| is an arbitrary norm of the matrix. When the condition number is too large, the system will be in an "ill-conditioned" state, so k should be as small as possible.

[0079] The degree of interaction between inverters can be quantitatively analyzed by combining the spectral norm and condition number of the transfer function matrix of the inverter grid-connected system.

[0080] Step 5: For a grid-connected system with two inverters, in order to explore the degree of influence of different factors on the interaction between inverters, three control scenarios were set up, and the inverter parameters are shown in Table 1.

[0081] Table 1 Inverter Parameters

[0082]

[0083] The scenarios are as follows:

[0084] 1) To investigate the relationship between the interaction between inverters and frequency, keeping other parameters constant, harmonic currents of 250Hz, 650Hz, and 850Hz were injected into the inverter reference current. g Table 2 shows the calculation results of the spectral norm and condition number of the system transfer function matrix at different frequencies, with a set value of 1mH.

[0085] Table 2. Spectral norm and condition number at different frequencies

[0086]

[0087] As shown in Table 2, when the harmonic frequency is 250 Hz, the system spectral norm differs from "1" by only 0.0592, and the condition number is also relatively small, indicating that there is almost no interaction between inverters at this time. At 650 Hz, both the system spectral norm and the condition number increase, indicating that the interaction between inverters becomes more severe. When the frequency equals 850 Hz, both the system spectral norm and the condition number increase significantly, indicating that the interaction between inverters continues to intensify.

[0088] 2) To investigate the interaction between inverters and the equivalent grid-connected inductance L g With other factors remaining constant, the equivalent grid-connected inductance is set to 1mH, 2mH and 3mH respectively, and the corresponding spectral norm and condition number are shown in Table 3.

[0089] Table 3 Different L g Spectral norm and condition number at time

[0090]

[0091] As can be seen from Table 3, with L... g As L increases, both the system spectral norm and condition number increase. This indicates that the interaction effects between inverters increase with L. gIt becomes larger and stronger.

[0092] 3) To investigate the relationship between the interaction between grid-connected inverters and the combination mode of the current outer loop controller, keeping other factors constant, the inverter outer loop controller was set to three combination modes: "PI+PI", "PI+quasi-PR", and "quasi-PR+quasi-PR". L was taken as... g The system spectral norm and condition number corresponding to different combinations are shown in Table 4, with a value of 3mH.

[0093] Table 4. Spectral norm and condition number under different combinations

[0094]

[0095] As can be seen from Table 4, when using the same type of controller combination, the system's spectral norm and condition number are both large, indicating that the interaction between inverters is serious at this time; when using the "PI + quasi-PR" controller combination, the system's spectral norm is close to "1" and the condition number drops to 0.8542, indicating that the control system is in a good state and the interaction between inverters can be ignored.

[0096] Step 6: Build a grid-connected model of two inverters in Matlab / simulink, and compare the simulation results with the results calculated based on matrix spectral norm and condition number to verify the correctness of the interaction influence analysis method proposed in this invention.

[0097] Injecting 2.5A of the 5th harmonic, 5A of the 13th harmonic, and 4A of the 17th harmonic into the reference current of inverter 1, when inverter 1 is connected to the grid alone, its output current THD is as follows: Figure 5 As shown. Inverter 2 is then connected to the grid. After inverters 1 and 2 are simultaneously connected to the grid, the THD of the output current of inverter 1 is as follows: Figure 6 As shown.

[0098] from Figure 4 As can be seen, the output current of inverter 1 follows the reference value. (Comparison) Figure 5 and Figure 6 After inverter 2 is connected to the grid, the harmonic content of each order in the output current of inverter 1 increases. Specifically, the 5th harmonic increases by 1.72%, the 13th harmonic by 4.3%, and the 17th harmonic by 5.8%. This indicates an interaction between inverter 1 and inverter 2. At lower frequencies, the increase in harmonic current is small and its impact on the overall frequency is negligible; however, as the frequency increases, the rate of increase in harmonic current rises significantly. The simulation results are consistent with the analysis results based on matrix norm.

[0099] exist Figure 5 Based on L g Change them to 2mH and 3mH respectively, with different L gThe THD of the output current of inverter 1 is shown in Figures 7(a) and 7(b). The equivalent grid-connected inductance L... g The value of L is strongly correlated with the interaction between inverters. g When it is 2mH, compared to Figure 6 The growth rate of each harmonic increases; L g When the value is 3mH, the inverter output grid-connected current also generates harmonic currents not included in the reference current. With L... g As the value increases, the interaction between inverters is significantly enhanced, consistent with the conclusions in Table 3, further verifying the correctness of the matrix norm theory proposed in this paper for the analysis of interaction effects between inverters.

[0100] Finally, based on Figure 7(b), the inverter outer loop controller combination was replaced with quasi-PR+quasi-PR and PI+quasi-PR respectively, and the corresponding output current THD of inverter 1 is shown in Figures 8(a) and 8(b). From Figures 8(a), 8(b), and 7(b), it can be seen that when the inverter outer loop uses PI controllers, the inverter output current cannot accurately follow the reference current, the 13th and 17th harmonics in the high-frequency band are significantly higher than the reference value, and harmonic currents not included in the reference current are generated. When the inverters all use quasi-PR controllers, the harmonic current in the output current is suppressed, but the 13th and 17th harmonics in the output current are significantly lower than the reference value. When the inverter outer loop uses a PI+quasi-PR combination, not only are the harmonics not included in the reference current eliminated, but the output current also basically follows the reference value. The PI+quasi-PR controller pairing method avoids the complex coordination and tuning of multiple inverter control parameters and effectively suppresses the interaction between inverters.

[0101] In summary, without changing the control parameters of each inverter, the PI+quasi-PR controller pairing method can effectively suppress the interaction between inverters, providing a simple and effective method for suppressing inverter interaction.

Claims

1. A method for analyzing the interactive effects of multiple grid-connected inverters based on matrix norm theory, characterized in that... Includes the following steps: Step 1: Construct a grid-connected model for a single inverter; Step 2: Obtain the equivalent circuit of a single inverter according to Norton's theorem; Step 3: Replace all inverters with Norton equivalent circuits. Based on Kirchhoff's current law, the transfer function matrix of the inverter grid-connected system can be obtained. Step 4: Using the inverter grid-connected system transfer function matrix obtained in Step 3, derive the inverter interaction impact evaluation index based on matrix norm theory; Step 5: For a grid-connected system with two inverters, to explore the impact of different factors on the interaction between inverters, three control scenarios are established: ①. To investigate the relationship between the interaction between inverters and frequency, while keeping other parameters constant, harmonic currents were injected into the inverter reference current, and the spectral norm and condition number of the control system transfer function matrix at the corresponding frequency were calculated. ②. To investigate the interaction effects between inverters and the equivalent grid-connected inductance L g Given the relationship between the two factors, and keeping other factors constant, calculate the spectral norm and condition number of the transfer function matrix of the control system under the corresponding equivalent grid-connected inductance. ③. To explore the relationship between the interaction between grid-connected inverters and the combination mode of the current outer loop controller, keeping other factors constant, the inverter outer loop controller is set to three combination modes: PI+PI, PI+quasi-PR, and quasi-PR+quasi-PR. The spectral norm and condition number of the control system transfer function matrix are calculated for different controller combination modes.

2. The method for analyzing the interaction effects of multiple grid-connected inverters based on matrix norm theory according to claim 1, characterized in that: In step 4, the inverter interaction impact evaluation index is specifically as follows: (1) Spectral norm of the transfer function matrix: Let the transfer function matrix of the control system be G ( s ),but G ( s eigenvalues λ It can be obtained from equation (6); （6）； Where: I is the identity matrix; Based on the eigenvalues ​​of the transfer function matrix, the spectral norm of the transfer function matrix is: （7）； in: λ max for G ( s The largest eigenvalue; G ( s ) T for G ( s The transpose of ) (2) Condition number of the transfer function matrix k : The condition number of the transfer function matrix is ​​obtained from equation (8): （8）； in: k is the condition number of the matrix; G ( s ) -1 express G ( s The inverse matrix of ; ||·|| is any norm of the matrix; By combining the spectral norm and condition number of the transfer function matrix of the inverter grid-connected system, it is possible to quantitatively analyze the degree of interaction between inverters.

3. The method for analyzing the interaction effects of multiple grid-connected inverters based on matrix norm theory according to claim 1, characterized in that: The method also includes step 6, which involves building a grid-connected model of two inverters in Matlab / simulink and comparing the simulation results with the results calculated based on the matrix spectral norm and condition number to verify the correctness of the proposed interactive analysis method.

4. The method for analyzing the interaction effects of multiple grid-connected inverters based on matrix norm theory according to claim 1, characterized in that: In step 2, the equivalent circuit of a single inverter is shown in equation (4): （4）； in: This refers to the inverter output current. The control coefficients of the controlled sources in the Norton equivalent circuit; This is the inverter's reference output current. The equivalent admittance of the Norton circuit; This is the voltage at the inverter's grid connection point.

5. The method for analyzing the interaction effects of multiple grid-connected inverters based on matrix norm theory according to claim 1, characterized in that: In step 3, the transfer function matrix of the grid-connected system with two inverters is shown in equation (5): （5）； in: i g1 and i g2 These are the output currents of the two inverters, respectively. i gref1(s) and i gref2(s) These are the reference output currents of the two inverters, respectively. G 1 represents the control coefficient of the controlled source in the equivalent Norton circuit of inverter 1. G 2 represents the control coefficient of the controlled source in the equivalent Norton circuit of inverter 2; Y 1 represents the equivalent admittance of inverter 1 in the equivalent Norton circuit. Y 2 represents the equivalent admittance of the inverter 2 in the equivalent Norton circuit; Y g For equivalent grid-connected admittance; u g This is the grid voltage.

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