A method for determining the boundary of the theoretical operating region of an LC type converter based on the homotopy method

Abstract

This invention relates to a method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method, belonging to the field of power systems. The method includes: establishing modulation voltage equations based on a steady-state power model; establishing voltage limit boundary equations based on converter modulation ratio constraints; evaluating and marking feasible regions through parameter space preprocessing, and automatically generating initial seed points for boundary tracing; and employing a prediction-correction framework based on the homotopy continuation method to bidirectionally trace from the seed points and completely solve the voltage amplitude boundary. This method, combining analytical calculation and numerical tracing, determines the theoretical feasible operating domain of a grid-connected converter that simultaneously satisfies power flow solvability and physical amplitude constraints. It can rapidly advance along the boundary curve, significantly reducing computational dimensionality and memory overhead, providing a theoretical boundary basis for the design, tuning, and system stability analysis of grid-connected converters, and contributing to improving the operational reliability of high-proportion renewable energy power systems.

Description

Technical Field

[0001] This invention relates to a method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method, belonging to the field of power system technology. Background Technology

[0002] With the accelerated construction of a new power system based on new energy sources, the integration of a high proportion of renewable energy into the grid has become an inevitable trend. Grid-connected converters, with their ability to actively construct grid voltage and frequency and provide synchronization support, have become core equipment for ensuring the safe and stable operation of the power system.

[0003] However, due to the random fluctuations in renewable energy output and load demand, as well as potential grid fault interference, grid-connected converters often operate under complex, time-varying conditions, characterized by large-scale dynamic changes in the grid's equivalent impedance. To define the stable operating boundary of the converter from a physical perspective, it is necessary to plot the theoretical operating domain of the grid-connected converter. The concept of the theoretical operating domain of the grid-connected converter reveals the inherent power transmission limit of the converter under different grid impedance variations, thus providing a theoretical benchmark for its optimized control and stability improvement in complex weak grid environments.

[0004] Traditional methods for plotting the theoretical operating domain of grid-type converters typically employ a high-density, full-plane discrete scanning method. This discretization approach not only suffers from significant computational redundancy, resulting in excessive computational load, but also often requires storing massive amounts of intermediate data to ensure plotting accuracy, leading to high memory consumption and making it difficult to meet the timeliness requirements of power system analysis. Summary of the Invention

[0005] To address the shortcomings of existing technologies, this invention provides a method for determining the theoretical operating domain boundary of an LC-type converter based on the homotopy method. Utilizing the continuous differentiability of the theoretical operating domain boundary, the homotopy continuation method transforms the static equation solving into dynamic path tracing. This method can rapidly advance along the boundary curve, significantly reducing computational dimensionality and memory overhead, thereby achieving automatic, fast, and computationally efficient accurate mapping of the theoretical operating domain of a network-type converter.

[0006] The present invention adopts the following technical solution:

[0007] A method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method includes the following steps:

[0008] Step 1: Based on the single-phase equivalent model of the grid-type converter, calculate the modulation voltage equation, and according to the maximum transmission power limit constraint, obtain the discriminant based on the grid impedance, and establish the solvable boundary equation of the power flow.

[0009] Step 2: Based on the converter modulation ratio limit, establish the voltage over-limit boundary equation based on the grid impedance;

[0010] Step 3: Based on the target grid impedance parameters, perform spatial preprocessing, evaluate and mark the feasible region, and automatically calculate the seed point for initiating high-precision boundary tracing;

[0011] Step 4: Based on the prediction-correction framework of the homotopy continuation method, bidirectional tracking is performed along the contour lines of the power flow solution boundary equation and the voltage over-limit boundary equation. Through adaptive arc length stepping and local Newton correction, the theoretical operating boundary of the grid converter is obtained.

[0012] This invention, based on the converter modulation voltage expression, derives the power flow solvable boundary equations analytically using the square root expression. Based on the converter modulation voltage expression and the converter modulation ratio constraint, it establishes a voltage limit boundary equation based on grid impedance. Based on these two boundary equations, a robust tracking algorithm integrating global preprocessing and adaptive homotopy continuation is proposed to accurately and completely map this nonlinear boundary. Ultimately, the two boundaries together define the feasible operating region that simultaneously satisfies the power flow solvable constraint and the modulation ratio constraint, achieving a reliable characterization of the theoretical operating region of the grid-connected converter. This reveals the inherent power transmission limit of the converter under different grid impedance variations, thus providing a theoretical benchmark for its optimized control and stability improvement in complex weak grid environments.

[0013] Preferably, in step 1, based on the single-phase equivalent model of the grid-type converter, the relationship between its output voltage and line resistance is established. and inductor Filter parameters and Grid voltage angular frequency and given power command , Based on the mathematical relationship between them, the complex expression for the modulation voltage is obtained as follows:

[0014] (1)

[0015] In the formula These are intermediate substitution parameters introduced to separate the real and imaginary parts and simplify the complex number expression when deriving the analytical model. They represent intermediate parameter one, intermediate parameter two, intermediate parameter three, and intermediate parameter four, respectively. Their specific expressions are obtained from equation (2):

[0016] (2)

[0017] In equation (2), For line resistance, For line inductance, This is the grid voltage. To output active power, To output reactive power, The angular frequency of the power grid. For filtering inductors, For filtering capacitors, The discriminant based on grid impedance is obtained from equation (3):

[0018] (3)

[0019] Based on the necessary and sufficient condition for the existence of a real steady-state solution in the system, the key discriminant regarding the grid impedance is extracted from the output voltage equation, which is: Discriminant This is a mathematical premise that the system has a power flow solution; in equation (3) From equation (4), we get:

[0020] (4)

[0021] The boundary equation for the power flow of a grid-connected converter based on grid impedance is obtained from equation (5):

[0022] (5).

[0023] Preferably, in step 2, the peak value of the converter modulation voltage is affected by its DC-side voltage. The limitation is that, when using sinusoidal pulse width modulation (SPWM), the maximum output fundamental voltage amplitude is [value missing]. Based on the modulation voltage equation established in step 1, the amplitude of the modulation voltage is calculated. Setting this amplitude equal to the aforementioned physical threshold, we obtain a voltage over-limit boundary equation regarding the grid impedance:

[0024] (6)

[0025] In the formula, F ( R s , L s ) represents the voltage limit boundary equation. The amplitude of the modulated voltage for the converter; The upper limit threshold of the voltage amplitude is obtained from equation (7):

[0026] (7)

[0027] In the formula, It is a DC voltage.

[0028] The voltage limit boundary equation defines the operating limit boundary of the converter under the premise of meeting its own physical voltage output capability. It is a nonlinear implicit equation that cannot be explicitly expressed as a single variable function and needs to be solved and tracked by specialized numerical methods.

[0029] Preferably, in step 3, the process of automatically calculating the boundary tracking seed point using spatial preprocessing of the target grid impedance parameters is as follows:

[0030] (3.1) Define the parameter space and construct the computational grid:

[0031] Let the range of values ​​for the resistance of the target power grid line be... The inductance value range is Then the two-dimensional impedance parameter space is defined as:

[0032] (8)

[0033] Construct a uniform, regular computational grid covering the entire space. Assume the resistance direction is divided into m equal parts and the inductance direction into n equal parts. Then the grid node coordinates are:

[0034] (9)

[0035] in:

[0036] (10)

[0037] (3.2) Node status assessment and classification:

[0038] For each grid node Perform the following judgments in sequence:

[0039] First, a mathematical solvability check is performed: the solvable boundary equation of the power flow from step 1 is called. If the solvability condition is met, it is marked as a solvable node and the process proceeds to the next step; otherwise, it is marked as a node in the "unsolvable region".

[0040] Then, voltage limiting constraint judgment is performed: for solvable nodes, the voltage over-limit boundary equation from step 2 is called to check whether the output voltage meets the requirements.

[0041] (11)

[0042] in V represents the number of output ports. k V represents the modulated voltage amplitude of the converter under the corresponding grid node, which is obtained by substituting the impedance parameter of the node into equation (1) to calculate the modulus; max This indicates the maximum allowable voltage output capability of the converter in the linear modulation region, determined by the DC-side voltage U. dc Decision, i.e., V max=U dc / 2;V min This indicates the minimum voltage limit to ensure stable operation of the converter, which is preset according to the actual operating standards of the project;

[0043] If equation (11) is satisfied, that is, all voltage constraints are satisfied, then it is marked as a "feasible zone" node; otherwise, it is marked as a "voltage limit exceeded zone" node.

[0044] (3.3) Identify the core feasible region:

[0045] Collect all marked "feasible region" nodes to form a feasible node set. :

[0046] (12)

[0047] Based on feasible node set Based on the spatial distribution of the nodes, a connected component analysis algorithm is used to identify the connected component with the most nodes as the core feasible region, denoted as . Connectivity analysis algorithms are mature existing technologies, typically used to classify adjacent elements with the same attributes in grid data.

[0048] (3.4) Determine the initial point for the homotopy continuation method:

[0049] In the core feasible area In this process, the initial seed point (i.e., the robust initial point) for initiating the homotopy continuation method is selected using the geometric center method. :

[0050] (13)

[0051] in, This represents the number of nodes in the set. This preprocessing step, through grid scanning, classification evaluation, and region analysis, calculates a homotopy continuation method for tracing the starting point with good numerical properties, avoiding manual trial and error and improving the robustness of subsequent feasible region boundary calculations.

[0052] Preferably, in step 4, the prediction-correction framework of the homotopy continuation method is used to perform bidirectional tracking along the contour lines of the power flow solution boundary equation and the voltage limit boundary equation. Through adaptive arc length stepping and local Newton correction, the theoretical operating boundary of the grid converter is obtained. The process is as follows:

[0053] (4.1) Establishing the homotopy equation and parameterization path:

[0054] The solvable boundary equation of the power flow D(R) s ,L s ) and voltage limit boundary equation F(R) s ,L sAs the target boundary function H to be tracked, the boundary curve to be determined is represented in parametric form using the arc length parameterization method:

[0055] (14)

[0056] Where s is the arc length parameter along the curve; the tracing problem is transformed into solving for... Continuous paths;

[0057] (4.2) Prediction step:

[0058] At the current boundary point At this point, calculate the gradient of the function. ; Tangent direction of the boundary curve at the current boundary point Orthogonal to the gradient, i.e., tangent direction for:

[0059] (15)

[0060] H H represents the target boundary function to be tracked; when tracking the voltage limit boundary, H is the voltage limit equation F(R) in step 2. s ,L s ), corresponding to formula (6); when there is a solution boundary for tracking the power flow, H corresponds to the discriminant equation D(R) described in step 1. s ,L s ), corresponding to formula (5).

[0061] Normalize the tangent and determine its sign based on its consistency with the tangent direction from the previous step. Perform Euler prediction along the tangent direction:

[0062] (16)

[0063] In the formula This is the current adaptive arc length step size;

[0064] (4.3) Calibration step:

[0065] With prediction points Using the initial values, the equation is solved using Newton's iterative method. The iteration format is:

[0066] (17)

[0067] In the formula It is the normal unit vector perpendicular to the tangent;

[0068] Iterate until the residual , ε To preset the tolerance, the convergence point is recorded as the new boundary point. ;

[0069] (4.4) Adaptive arc length step control:

[0070] The arc length step size is dynamically adjusted based on the convergence behavior of the correction process. If the Newton iteration converges within 10 iterations, it is considered a smooth path, and the current arc length step size is adjusted accordingly. Multiply by a scaling factor greater than 1; if the number of iterations exceeds a set threshold or diverges, reduce the step size and restart from the previous step. The system performs prediction and correction at points; the step size is limited to a preset minimum and maximum value to ensure the stability and efficiency of tracking.

[0071] (4.5) Two-way tracking:

[0072] Starting from the seed point, prediction-correction tracking is performed independently along the positive and negative tangent directions, and terminated when the termination condition is met.

[0073] (4.6) Boundary curve synthesis and feasible region definition:

[0074] Connect the discrete point sequences obtained by tracing in two directions in spatial order to form a smooth, closed boundary curve. The region enclosed by the boundary curve is the theoretical operating domain of the grid-type converter that simultaneously satisfies the power flow solution and modulation ratio constraint.

[0075] Preferably, in step (4.5), the termination condition is that the process terminates when one of the following three conditions is met:

[0076] Scenario 1: Variables are outside the physically feasible range:

[0077] (18)

[0078] Case 2: Tracing into the region where the discriminant is less than zero, i.e., the region with no solution:

[0079] (19)

[0080] Case 3: The cumulative arc length reaches the preset maximum value or the step size is lower than the minimum threshold.

[0081] For any details not covered in this invention, please refer to the prior art.

[0082] The beneficial effects of this invention are as follows:

[0083] This invention proposes a method for determining the theoretical operating domain boundary of LC converters based on the homotopy method. By combining two boundary equations with pre-processed homotopy continuation tracing, this method can completely determine the power flow solution boundary and voltage amplitude boundary of the grid-connected converter on the grid impedance plane, thus clearly delineating its steady-state feasible operating domain. This method is computationally efficient and reliable, providing clear boundary criteria for the parameter design and capacity configuration of grid-connected converters, and offering an effective analytical tool for evaluating the stability margin and safety risks of converters under specific grid conditions. Attached Figure Description

[0084] The accompanying drawings, which form part of this application, are used to provide a further understanding of this application. The illustrative embodiments of this application and their descriptions are used to explain this application and do not constitute an undue limitation of this application.

[0085] Figure 1 This is a schematic diagram of the method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method according to the present invention.

[0086] Figure 2 This is a three-phase topology diagram of the main circuit of a grid-type converter;

[0087] Figure 3 The single-phase equivalent circuit diagram of a grid-type converter;

[0088] Figure 4 This is a theoretical operating domain partitioning diagram of a grid-type converter obtained using the traditional global grid scanning method;

[0089] Figure 5 This is the theoretical operating domain boundary diagram obtained using the homotopy method proposed in this invention. Detailed Implementation

[0090] To enable those skilled in the art to better understand the technical solutions in this specification, the technical solutions in the embodiments of this invention will be clearly and completely described below with reference to the accompanying drawings. However, this is not the only description; all aspects not described in detail herein are based on conventional techniques in the art.

[0091] Example 1

[0092] A method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method, such as... Figure 1 As shown, it includes the following steps:

[0093] Step 1: Based on the single-phase equivalent model of the grid-type converter, calculate the modulation voltage equation, and according to the maximum transmission power limit constraint, obtain the discriminant based on the grid impedance, and establish the solvable boundary equation of the power flow.

[0094] Step 2: Based on the converter modulation ratio limit, establish the voltage over-limit boundary equation based on the grid impedance;

[0095] Step 3: Based on the target grid impedance parameters, perform spatial preprocessing, evaluate and mark the feasible region, and automatically calculate the seed point for initiating high-precision boundary tracing;

[0096] Step 4: Based on the prediction-correction framework of the homotopy continuation method, bidirectional tracking is performed along the contour lines of the power flow solution boundary equation and the voltage over-limit boundary equation. Through adaptive arc length stepping and local Newton correction, the theoretical operating boundary of the grid converter is obtained.

[0097] In this embodiment, firstly, the modulation voltage equation is established based on the single-phase equivalent model of the grid-type converter. Secondly, the discriminant is obtained using the necessary and sufficient condition for the existence of real solutions in the system, and the solvable boundary equation for power flow is established; the voltage limit boundary equation is established based on the converter modulation ratio constraint. Then, through parameter space preprocessing, feasible regions are evaluated and marked, and initial seed points for boundary tracking are automatically generated. Finally, a prediction-correction framework based on homotopy continuation is used to bidirectionally track from the seed points to completely solve the voltage amplitude boundary. This method, by combining analytical calculation and numerical tracking, determines the theoretical feasible operating region of the grid-type converter that simultaneously satisfies power flow solvability and physical amplitude limiting constraints, providing a theoretical boundary basis for the design, tuning, and system stability analysis of the grid-type converter, and helping to improve the operational reliability of high-proportion renewable energy power systems.

[0098] Example 2

[0099] A method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method is shown in Example 1. The difference is that in step 1, the output voltage and line resistance are established based on the single-phase equivalent model of the grid-type converter. and inductor Filter parameters and Grid voltage angular frequency and given power command , Based on the mathematical relationship between them, the complex expression for the modulation voltage is obtained as follows:

[0100] (1)

[0101] In the formula These are intermediate substitution parameters introduced to separate the real and imaginary parts and simplify the complex number expression when deriving the analytical model. They represent intermediate parameter one, intermediate parameter two, intermediate parameter three, and intermediate parameter four, respectively. Their specific expressions are obtained from equation (2):

[0102] (2)

[0103] In equation (2), For line resistance, For line inductance, This is the grid voltage. To output active power, To output reactive power, The angular frequency of the power grid. For filtering inductors, For filtering capacitors, The discriminant based on grid impedance is obtained from equation (3):

[0104] (3)

[0105] Based on the necessary and sufficient condition for the existence of a real steady-state solution in the system, the key discriminant regarding the grid impedance is extracted from the output voltage equation, which is: Discriminant This is a mathematical premise that the system has a power flow solution; in equation (3) From equation (4), we get:

[0106] (4)

[0107] The boundary equation for the power flow of a grid-connected converter based on grid impedance is obtained from equation (5):

[0108] (5).

[0109] Example 3

[0110] A method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method is shown in Example 2. The difference is that in step 2, the peak value of the converter modulation voltage is affected by its DC side voltage. The limitation is that, when using sinusoidal pulse width modulation (SPWM), the maximum output fundamental voltage amplitude is [value missing]. Based on the modulation voltage equation established in step 1, the amplitude of the modulation voltage is calculated. Setting this amplitude equal to the aforementioned physical threshold, we obtain a voltage over-limit boundary equation regarding the grid impedance:

[0111] (6)

[0112] In the formula, F ( R s , L s ) represents the voltage limit boundary equation. The amplitude of the modulated voltage for the converter; The upper limit threshold of the voltage amplitude is obtained from equation (7):

[0113] (7)

[0114] In the formula, It is a DC voltage.

[0115] The voltage limit boundary equation defines the operating limit boundary of the converter under the premise of meeting its own physical voltage output capability. It is a nonlinear implicit equation that cannot be explicitly expressed as a single variable function and needs to be solved and tracked by specialized numerical methods.

[0116] Example 4

[0117] A method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method is shown in Example 3. The difference is that in step 3, the process of automatically calculating the boundary tracking seed point using spatial preprocessing of the target grid impedance parameters is as follows:

[0118] (3.1) Define the parameter space and construct the computational grid:

[0119] Let the range of values ​​for the resistance of the target power grid line be... The inductance value range is Then the two-dimensional impedance parameter space is defined as:

[0120] (8)

[0121] Construct a uniform, regular computational grid covering the entire space. Assume the resistance direction is divided into m equal parts and the inductance direction into n equal parts. Then the grid node coordinates are:

[0122] (9)

[0123] in:

[0124] (10)

[0125] (3.2) Node status assessment and classification:

[0126] For each grid node Perform the following judgments in sequence:

[0127] First, a mathematical solvability check is performed: the solvable boundary equation of the power flow from step 1 is called. If the solvability condition is met, it is marked as a solvable node and the process proceeds to the next step; otherwise, it is marked as a node in the "unsolvable region".

[0128] Then, voltage limiting constraint judgment is performed: for solvable nodes, the voltage over-limit boundary equation from step 2 is called to check whether the output voltage meets the requirements.

[0129] (11)

[0130] in V represents the number of output ports. k V represents the modulated voltage amplitude of the converter under the corresponding grid node, which is obtained by substituting the impedance parameter of the node into equation (1) to calculate the modulus; max This indicates the maximum allowable voltage output capability of the converter in the linear modulation region, determined by the DC-side voltage U. dc Decision, i.e., V max =U dc / 2;V min This indicates the minimum voltage limit to ensure stable operation of the converter, which is preset according to the actual operating standards of the project;

[0131] If equation (11) is satisfied, that is, all voltage constraints are satisfied, then it is marked as a "feasible zone" node; otherwise, it is marked as a "voltage limit exceeded zone" node.

[0132] (3.3) Identify the core feasible region:

[0133] Collect all marked "feasible region" nodes to form a feasible node set. :

[0134] (12)

[0135] Based on feasible node set Based on the spatial distribution of the nodes, a connected component analysis algorithm is used to identify the connected component with the most nodes as the core feasible region, denoted as . Connectivity analysis algorithms are mature existing technologies, typically used to classify adjacent elements with the same attributes in grid data.

[0136] (3.4) Determine the initial point for the homotopy continuation method:

[0137] In the core feasible area In this process, the initial seed point (i.e., the robust initial point) for initiating the homotopy continuation method is selected using the geometric center method. :

[0138] (13)

[0139] in, This represents the number of nodes in the set. This preprocessing step, through grid scanning, classification evaluation, and region analysis, calculates a homotopy continuation method for tracing the starting point with good numerical properties, avoiding manual trial and error and improving the robustness of subsequent feasible region boundary calculations.

[0140] Example 5

[0141] A method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method is shown in Example 4. The difference is that in step 4, the prediction-correction framework of the homotopy continuation method is used to perform bidirectional tracking along the contour lines of the power flow solution boundary equation and the voltage limit boundary equation. Through adaptive arc length stepping and local Newton correction, the theoretical operating boundary of the grid-type converter is obtained. The process is as follows:

[0142] (4.1) Establishing the homotopy equation and parameterization path:

[0143] The solvable boundary equation of the power flow D(R) s ,L s ) and voltage limit boundary equation F(R) s ,L s As the target boundary function H to be tracked, the boundary curve to be determined is represented in parametric form using the arc length parameterization method:

[0144] (14)

[0145] Where s is the arc length parameter along the curve; the tracing problem is transformed into solving for... Continuous paths;

[0146] (4.2) Prediction step:

[0147] At the current boundary point At this point, calculate the gradient of the function. ; Tangent direction of the boundary curve at the current boundary point Orthogonal to the gradient, i.e., tangent direction for:

[0148] (15)

[0149] H H represents the target boundary function to be tracked; when tracking the voltage limit boundary, H is the voltage limit equation F(R) in step 2. s ,L s ), corresponding to formula (6); when there is a solution boundary for tracking the power flow, H corresponds to the discriminant equation D(R) described in step 1. s ,L s ), corresponding to formula (5).

[0150] Normalize the tangent and determine its sign based on its consistency with the tangent direction from the previous step. Perform Euler prediction along the tangent direction:

[0151] (16)

[0152] In the formula This is the current adaptive arc length step size;

[0153] (4.3) Calibration step:

[0154] With prediction points Using the initial values, the equation is solved using Newton's iterative method. The iteration format is:

[0155] (17)

[0156] In the formula It is the normal unit vector perpendicular to the tangent;

[0157] Iterate until the residual , ε To preset the tolerance, the convergence point is recorded as the new boundary point. ;

[0158] (4.4) Adaptive arc length step control:

[0159] The arc length step size is dynamically adjusted based on the convergence behavior of the correction process. If the Newton iteration converges within 10 iterations, it is considered a smooth path, and the current arc length step size is adjusted accordingly. Multiply by a scaling factor greater than 1; if the number of iterations exceeds a set threshold or diverges, reduce the step size and restart from the previous step. The system performs prediction and correction at points; the step size is limited to a preset minimum and maximum value to ensure the stability and efficiency of tracking.

[0160] (4.5) Two-way tracking:

[0161] Starting from the seed point, prediction-correction tracking is performed independently along the positive and negative tangent directions, and terminated when the termination condition is met.

[0162] Termination occurs when one of the following three conditions is met:

[0163] Scenario 1: Variables are outside the physically feasible range:

[0164] (18)

[0165] Case 2: Tracing into the region where the discriminant is less than zero, i.e., the region with no solution:

[0166] (19)

[0167] Case 3: The cumulative arc length reaches the preset maximum value or the step size is lower than the minimum threshold.

[0168] (4.6) Boundary curve synthesis and feasible region definition:

[0169] Connect the discrete point sequences obtained by tracing in two directions in spatial order to form a smooth, closed boundary curve. The region enclosed by the boundary curve is the theoretical operating domain of the grid-type converter that simultaneously satisfies the power flow solution and modulation ratio constraint.

[0170] To verify the effectiveness and reliability of the method proposed in this invention, based on Figure 2 The network converter topology shown is simulated and verified. The specific circuit parameters used in the verification are shown in Table 1. Figure 2 In the middle parameters, T1~T6 represent the six power switching transistors of the three-phase full-bridge inverter circuit, C represents the supporting capacitor on the DC side of the inverter, and i La i Lb i Lc These represent the filter inductor currents of phases a, b, and c on the converter side, respectively. 1a u 1b u 1c i represents the modulated voltages of phases a, b, and c at the output of the converter. sa i sb i sc This represents the three-phase line current flowing into the grid side from phases a, b, and c, u. 2a u 2b u 2c These represent the grid voltages for phases a, b, and c, respectively.

[0171] Table 1 Main Circuit Parameters of Grid Converter

[0172]

[0173] In Table 1, the grid frequency f is used to determine the angular frequency ω = 2πf and the apparent power command. S out The power factor angle θ is used to decompose and calculate the active power output of the converter. P out and reactive power Q out The calculation formula is:

[0174] (20)

[0175] The obtained ω, P out , Q out Substituting the main circuit hardware parameters from Table 1 into the analytical equations of steps 1 and 2, we can obtain the result relating only to the grid impedance variable (R). s , L s The boundary function D(R) s ,L s ) and D(R s ,Ls This initiates subsequent preprocessing and tracking algorithms.

[0176] Figure 3 The single-phase equivalent circuit diagram of the grid-type converter is given. Based on this circuit, the modulation voltage is calculated, and the solvable boundary equation of the converter power flow based on the grid impedance is obtained. The voltage over-limit boundary equation is established based on the given DC side voltage.

[0177] First, the operational domain is drawn using the traditional global grid scanning method. This method involves... A high-density grid scan is performed on the parameter plane to directly calculate the discriminant and modulation voltage amplitude at each point, and the region is colored based on the calculation results. The results are as follows: Figure 4 As shown in the figure, the white area represents the feasible region that simultaneously satisfies the power flow solvability and voltage limiting constraints; the black area represents the power flow unsolvable region where the discriminant is less than zero; and the gray area represents the voltage amplitude exceeding the limit region where a solution exists but the modulation voltage amplitude exceeds the threshold. The Short Circuit Ratio (SCR) is a core indicator for measuring the strength of a power grid. In this invention, it describes the short-circuit capacity S at the PCC point. ac With converter capacity S out The ratio between them:

[0178] (twenty one)

[0179] Among them, Z s Due to the line impedance R s and L s Z made the decision. s =R s +j·ωL s SCR is used here to quantify the strength of the power grid: the wider the area of ​​the operating domain boundary envelope, the stronger the converter's operational adaptability under a wide SCR range.

[0180] Figure 4 Boundary I is the power flow solution boundary of the grid-type converter, which is limited by the maximum transmission power limit. Its physical meaning is the maximum boundary of the grid impedance that the system can maintain static stable transmission under a given power reference. Boundary II is the voltage over-limit boundary of the grid-type converter, which is limited by the modulation ratio. Its physical meaning is the maximum boundary of the grid impedance that the converter can operate in the linear modulation region under a given power reference.

[0181] Secondly, the operating domain was drawn using the homotopy-based method for determining the theoretical operating domain boundary of LC-type converters proposed in this invention, and the results are as follows: Figure 5As shown in the figure, the light gray filled area represents the identified feasible region; the black dashed line represents the power flow solution boundary of the grid-type converter obtained using the homotopy method; and the black solid line represents the theoretical operating boundary drawn using the traditional global analytical calculation method.

[0182] contrast Figure 4 and Figure 5 It can be seen that the theoretical operating domain of the grid-type converter drawn based on the traditional global grid scanning method coincides with the theoretical operating domain drawn based on the preprocessing and homotopy continuation method, and its shape is continuous and complete. The boundary determination method based on the preprocessing and homotopy continuation method proposed in this invention can reliably and completely obtain the operating feasible domain boundary of the grid-type converter. This method avoids the disadvantage of computational density of the traditional grid scanning method. Through intelligent starting point selection and robust path tracing, it achieves results consistent with the analytical benchmark, verifying the correctness of the method of this invention.

[0183] The above description represents the preferred embodiments of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for determining the boundary of the theoretical operating region of an LC type inverter based on the homotopy method, characterized in that, Includes the following steps: Step 1: Based on the single-phase equivalent model of the grid-type converter, calculate the modulation voltage equation, and according to the maximum transmission power limit constraint, obtain the discriminant based on the grid impedance, and establish the solvable boundary equation of the power flow. Step 2: Based on the converter modulation ratio limit, establish the voltage over-limit boundary equation based on the grid impedance; Step 3: Based on the target grid impedance parameters, perform spatial preprocessing, evaluate and mark the feasible region, and automatically calculate the seed point for initiating high-precision boundary tracing; Step 4: Based on the prediction-correction framework of the homotopy continuation method, bidirectional tracking is performed along the contour lines of the power flow solution boundary equation and the voltage over-limit boundary equation. Through adaptive arc length stepping and local Newton correction, the theoretical operating boundary of the grid converter is obtained. In step 3, the process of automatically calculating the boundary tracking seed point using spatial preprocessing of the target grid impedance parameters is as follows: (3.1) Define the parameter space and construct the computational grid: The range of the target grid line resistance is set as , the range of the inductance is set as , and the two-dimensional impedance parameter space is defined as （8） Construct a uniform, regular computational grid covering the entire space. Assume the resistance direction is divided into m equal parts and the inductance direction into n equal parts. Then the grid node coordinates are: （9） in: （10） (3.2) Node status assessment and classification: For each grid node The following determinations are made in sequence: First, a mathematical solvability check is performed: the power flow boundary equation from step 1 is called. If the solvability condition is met, it is marked as a solvable node and the process proceeds to the next step; otherwise, it is marked as a "no-solution zone" node. Then, voltage limiting constraint judgment is performed: for solvable nodes, the voltage over-limit boundary equation from step 2 is called to check whether the output voltage meets the requirements. （11） in V represents the number of output ports. k V represents the amplitude of the converter modulation voltage under the corresponding grid node. max V represents the maximum allowable voltage output capability of the converter in the linear modulation region. min This indicates the minimum voltage limit required to ensure stable operation of the converter; If equation (11) is satisfied, that is, all voltage constraints are satisfied, then it is marked as a "feasible zone" node; otherwise, it is marked as a "voltage limit exceeded zone" node. (3.3) Identify the core feasible region: Collect all the marked "feasible region" nodes to form a feasible node set : （12） Based on feasible node set Based on the spatial distribution of the nodes, a connected component analysis algorithm is used to identify the connected component with the most nodes as the core feasible region, denoted as . ; (3.4) Determine the initial point for the homotopy continuation method: In the core feasible area In the middle, the initial seed point for initiating the homotopy continuation method is selected according to the geometric center method. : （13） in, Indicates the number of nodes in the set; The implementation process of step 4 is as follows: (4.1) Establishing the homotopy equation and parameterization path: The solvable boundary equation of the power flow D(R) s ,L s ) and voltage limit boundary equation F(R) s ,L s As the target boundary function H to be tracked, the boundary curve to be determined is represented in parametric form using the arc length parameterization method: （14） Where s is the arc length parameter along the curve; the tracing problem is transformed into solving a problem that satisfies... Continuous paths; (4.2) Prediction step: At the current boundary point At this point, calculate the gradient of the function. ; Tangent direction of the boundary curve at the current boundary point Orthogonal to the gradient, i.e., tangent direction for: （15） H Represents the boundary function of the target to be tracked; Normalize the tangent and determine its sign based on its consistency with the tangent direction from the previous step. Perform Euler prediction along the tangent direction: （16） In the formula is the current adaptive arc length step; (4.3) Calibration step: The prediction point Newton iteration method is used to solve the equation The iteration format is: （17） In the formula is a normal unit vector perpendicular to the tangent line; Iterate until the residual , ε Take the convergence point as the new boundary point ; (4.4) Adaptive arc length step control: The arc length step size is dynamically adjusted based on the convergence behavior of the correction process. If the Newton iteration converges within 10 iterations, it is considered a smooth path, and the current arc length step size is adjusted accordingly. Multiply by a scaling factor greater than 1; if the number of iterations exceeds a set threshold or diverges, reduce the step size and restart from the previous step. Point-based prediction and correction; (4.5) Two-way tracking: Starting from the seed point, prediction-correction tracking is performed independently along the positive and negative tangent directions, and terminated when the termination condition is met. (4.6) Boundary curve synthesis and feasible region definition: Connect the discrete point sequences obtained by tracing in two directions in spatial order to form a smooth, closed boundary curve. The region enclosed by the boundary curve is the theoretical operating domain of the grid-type converter that simultaneously satisfies the power flow solution and modulation ratio constraint.

2. The method according to claim 1, wherein In step 1, based on the single-phase equivalent model of the grid-connected converter, the complex expression of the modulated voltage is obtained as （1） In the formula Let them represent intermediate parameter one, intermediate parameter two, intermediate parameter three, and intermediate parameter four, respectively. Their specific expressions are obtained from equation (2): （2） In equation (2), For line resistance, For line inductance, This is the grid voltage. To output active power, To output reactive power, The angular frequency of the power grid. For filtering inductors, For filtering capacitors, The discriminant based on grid impedance is obtained from equation (3): （3） Discriminant This is a mathematical premise that the system has a power flow solution; in equation (3) From equation (4), we get: （4） The boundary equation for the power flow of a grid-connected converter based on grid impedance is obtained from equation (5): （5）。 3. The method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method according to claim 2, characterized in that, In step 2, the voltage limit boundary equation is obtained from equation (6): （6） In the formula, F ( R s , L s ) represents the voltage limit boundary equation. The amplitude of the modulated voltage for the converter; The upper limit threshold of the voltage amplitude is obtained from equation (7): （7） In the formula, is a direct current voltage.

4. The method for determining the theoretical operating domain boundary of an LC converter based on the homotopy method according to claim 3, characterized in that, In step (4.5), the termination condition is that the process terminates when one of the following three conditions is met: Scenario 1: Variables are outside the physically feasible range: （18） Case 2: Tracing into the region where the discriminant is less than zero, i.e., the region with no solution: （19） Case 3: The cumulative arc length reaches the preset maximum value or the step size is lower than the minimum threshold.

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