Electric power system saddle node bifurcation point solving method based on hybrid pre-estimation-correction and secant method

Abstract

The invention discloses an electric power system saddle node bifurcation point solving method based on a hybrid estimation-correction and secant method, and belongs to the technical field of electric power system stability analysis. The method comprises the following steps: establishing an extended power flow equation set; obtaining an effective initial value through a pre-estimation-correction method; dynamically processing a network equation for a line fault scene; judging a bifurcation point interval based on the linear relation and the sensitivity; and combining a secant method with self-adaptive step length control to realize accurate positioning of the bifurcation point. According to the method, the defect that the numerical value is unstable near the bifurcation point in the traditional method is overcome, the calculation efficiency and robustness are improved, the method is suitable for a complex power system containing faults and new energy, and effective support is provided for stability analysis.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of power system stability analysis, and particularly relates to a power system saddle node bifurcation point solving method based on a hybrid prediction-correction and secant method. BACKGROUND

[0002] In a power system, a saddle node bifurcation point is a key node marking the transition of the system from stable operation to unstable operation, and its accurate solution has irreplaceable significance for power system stability analysis, operation optimization and fault prevention. With the expansion of the scale of the power system, the increase of the proportion of new energy grid connection and the increase of complex fault scenarios, higher requirements are put forward for the accuracy, efficiency and robustness of the solution of the saddle node bifurcation point.

[0003] In the prior art, the traditional solution method (such as the Newton-Raphson method) is prone to solution divergence when approaching the saddle node bifurcation point, because the Jacobian matrix of the system is close to singular, resulting in failure of bifurcation point positioning. Although some methods attempt to optimize the solution logic, they lack an effective step length control mechanism, and have the defects of low calculation efficiency and insufficient accuracy. At the same time, for fault scenarios such as line removal, existing methods are difficult to quickly adapt to fault states, and are prone to solution process interruption or solution reliability decline, which cannot meet the needs of post-fault stability analysis of actual power systems. In addition, some methods cannot accurately determine whether the solution is on the upper half branch, the lower half branch or the bifurcation point itself, and are prone to failure in solving the bifurcation point due to solution to the lower half branch.

[0004] Therefore, there is an urgent need for a saddle node bifurcation point solving method that can avoid the singularity problem of the Jacobian matrix, has effective step length control, adapts to fault scenarios and accurately determines the position of the solution, in order to solve many pain points of the prior art and provide reliable technical support for the stable operation of complex power systems. SUMMARY

[0005] The technical problem to be solved by the present application is to provide a power system saddle node bifurcation point solving method based on a hybrid prediction-correction and secant method, which can accurately and quickly solve the saddle node bifurcation point in a complex power system, thereby providing strong support for stability analysis and operation optimization of the power system, and solving the technical problems of poor numerical stability, low calculation efficiency and lack of effective step length control in the prior art power system saddle node bifurcation point solving method.

[0006] The application adopts the following technical solutions: The power system saddle node bifurcation point solving method based on a hybrid prediction-correction and secant method comprises the following steps: S1, obtaining network parameters, generator parameters and load parameters of a power system, establishing a power flow equation of the power system based on the above parameters, and introducing parameters to form an extended power flow equation set by the continuation method; S2. Solve the extended power flow equations obtained in step S1 using the prediction-correction method. First, obtain the voltage estimate through a linear method and use it as an approximate solution. Then, perform vertical and horizontal corrections on the approximate solution to handle the possibility that the conventional power flow problem may not converge, and obtain effective initial values. S3. For the scenario of line disconnection under network fault, the network equation is first transformed into a specific form that adapts to the fault scenario. Then, based on the effective initial value obtained in step S2, the transformed network equation is processed by combining vertical and horizontal correction processes to ensure the continuity of the solution process under the fault scenario and output the horizontally corrected equation set. S4. Based on the horizontally corrected equation set output in step S3, and combined with the effective initial values ​​obtained in step S2, determine the range of feasible solutions. First, approximate the relevant parameters of the saddle-node bifurcation point near the feasible solution through linear relationships. Then, determine whether the current solution is in the upper half, lower half, or bifurcation point based on the sensitivity sign. At the same time, record the last effective solution of the upper half and the first effective solution of the lower half to determine the initial solution interval where the saddle-node bifurcation point is located. S5. Based on the initial solution interval determined in step S4, the saddle-node bifurcation point is searched by combining the secant method and adaptive step size control. If the solution does not converge at the current step size, the process returns to step S4 and the step size is adjusted and recalculated until the saddle-node bifurcation point is accurately located, thus completing the entire solution process.

[0007] Preferably, in step S2, the process of obtaining the voltage estimate using a linear method is as follows: Given a step size, a linear relationship is established between the voltage change and the parameter change mentioned in step S1 to obtain a voltage estimate, which is then used as an approximation of the solution to the extended power flow equations.

[0008] Preferably, in step S2, the vertical correction specifically involves solving the equation. The aforementioned level correction specifically involves solving the equation. ,in, For the first A column vector consisting of the variables from each iteration. For the first The margin parameters obtained from the round, To correct the step size, For the first The margin parameters obtained from the round, For the first The variable vector of the round of iteration One tracking parameter, For the first The predicted results of each tracking parameter.

[0009] Preferably, in step S3, transforming the network equations into a specific form adapted to the fault scenario specifically includes: When a line is removed from the AC system, the network equation is written in a form related to the line state parameters, in which the admittance matrix varies with the line state.

[0010] Preferably, the processing of the transformed network equation by combining the vertical correction and horizontal correction processes specifically comprises: In the vertical correction, given the fixed value of the parameter in step S1, the network mismatch is re-solved by forming the admittance matrix depending on the parameter value; In the horizontal correction, the parameter in step S1 is taken as the to-be-solved quantity, the admittance matrix is updated with the parameter when calculating the network mismatch, and the partial derivative of the function with respect to the parameter is updated in the iteration using the Jacobian matrix.

[0011] Preferably, in step S4, the approximate solution of the parameter related to the saddle node bifurcation point by the linear relationship specifically comprises: According to the Jacobian matrix and the voltage component of the current solution, the rate of change of the parameter λ is calculated to obtain the approximate value of the saddle node bifurcation point.

[0012] Preferably, the judging the position of the current solution according to the sign of the sensitivity specifically comprises: The sensitivity value is calculated, when the sensitivity is positive, the current solution is in the upper half branch; when the sensitivity is negative, the current solution is in the lower half branch; when the sensitivity is zero, the current solution is at the bifurcation point.

[0013] Preferably, in step S5, the searching of the saddle node bifurcation point by combining the secant method and the adaptive step control specifically comprises: Based on the last valid solution of the upper half branch and the first valid solution of the lower half branch, the secant method is used to iteratively solve the point where the sensitivity is zero; in the iteration process, if the current step does not converge, return to the last step and reduce the step size to recalculate until convergence.

[0014] Preferably, the adjusting the step size specifically comprises: According to the sensitivity obtained by the solution, the step size is dynamically adjusted by using the linear approximation method to bypass the saddle node bifurcation point and ensure the calculation accuracy.

[0015] In the second aspect, the embodiments of the present application provide a power system saddle node bifurcation point solving system combining the prediction-correction method and the secant method, comprising: An equation module obtains network parameters, generator parameters and load parameters of a power system, establishes a power flow equation of the power system based on the above parameters, and forms an extended power flow equation set by introducing parameters through the continuation method; The solving module solves the extended power flow equation set obtained by the equation module by using a prediction-correction method, obtains a voltage estimation value by a linear method as an approximate solution, and performs vertical correction and horizontal correction on the approximate solution to process a situation where a conventional power flow problem may not converge, and obtains an effective initial value; The fault module, for the scenario of line disconnection under network fault, first converts the network equation into a specific form suitable for the fault scenario, and then processes the converted network equation based on the effective initial value obtained by the solving module and the vertical correction and horizontal correction process, to ensure the continuity of the solving process under the fault scenario, and outputs the horizontal corrected equation set; The interval module determines a feasible solution range based on the horizontal corrected equation set output by the fault module and the effective initial value obtained by the solving module, approximates the solution of the saddle node bifurcation point related parameters near the feasible solution by a linear relationship, and determines the initial solution interval of the saddle node bifurcation point according to the sensitivity sign to judge whether the current solution is in the upper half branch, the lower half branch or the bifurcation point, and records the last effective solution of the upper half branch and the first effective solution of the lower half branch. The positioning module searches for the saddle node bifurcation point based on the initial solution interval determined by the interval module and the secant method and adaptive step control, and if the solution does not converge under the current step, returns to the interval module and adjusts the step to recalculate until the saddle node bifurcation point is accurately positioned, and the entire solving process is completed.

[0016] In a third aspect, a computer device includes a memory, a processor, and a computer program stored in the memory and executable on the processor, and the processor implements the steps of the power system saddle node bifurcation point solving method of the mixed prediction-correction and secant method.

[0017] In a fourth aspect, an embodiment of the present application provides a computer readable storage medium including a computer program, and the computer program implements the steps of the power system saddle node bifurcation point solving method of the mixed prediction-correction and secant method when executed by a processor.

[0018] In a fifth aspect, a chip includes a memory, a processor, and a computer program stored in the memory and executable on the processor, and the processor implements the steps of the power system saddle node bifurcation point solving method of the mixed prediction-correction and secant method.

[0019] In a sixth aspect, an embodiment of the present application provides an electronic device including a computer program, and the computer program implements the steps of the power system saddle node bifurcation point solving method of the mixed prediction-correction and secant method when executed by the electronic device.

[0020] Compared with the prior art, the present application has at least the following beneficial effects: The application discloses a power system saddle node bifurcation point solving method combining a mixed prediction-correction and secant method, constructs an extended power flow equation set by introducing a continuation parameter, realizes accurate positioning of a bifurcation point by combining linear prediction, vertical / horizontal correction and the secant method, and avoids the problem of singularity of a Jacobian matrix near a saddle node bifurcation point in a traditional method by extending the power flow equation through the continuation method; the prediction-correction double correction mechanism ensures acquisition of an effective initial value, lays a foundation for subsequent calculation; a fault processing module is adaptively matched with a line cutout scene, and continuity of solving is ensured; interval determination, the secant method and adaptive step control are combined, the search range is determined, and the precision is dynamically adjusted, and the core pain points of poor numerical stability, low efficiency and no effective step control in the prior art are solved. The whole process is progressive, realizes full-link optimization from a basic model to accurate positioning, can be adapted to a large-scale power system containing faults and new energy grid connection, and is wide in application range and strong in robustness.

[0021] Further, the difficulty of initial value solving is simplified and the accuracy of an approximate solution is improved. By establishing a linear correlation between the voltage variation and the continuation parameter variation under a given step, the approximate solution can be quickly obtained without complex iteration, and the initial value solving efficiency is greatly improved. The application of the linear relationship not only conforms to the local characteristics of the power system power flow equation, but also quickly locks the approximate range of the solution, provides a high-quality starting point for subsequent vertical correction and horizontal correction, and reduces the iteration number of the correction link. Compared with the traditional blind trial calculation or complex iteration method for obtaining the initial value, the method has small calculation amount and fast response, effectively avoids the convergence difficulty of subsequent solving caused by the initial value deviation, and provides a key support for the efficiency of the whole method.

[0022] Further, the vertical correction focuses on the convergence problem of the power flow equation under the fixed continuation parameter, and the horizontal correction optimizes the coupling relationship between the parameter and the voltage, and the double correction mechanism forms a closed loop. By clearly defining the physical meaning and mathematical correlation of each parameter (such as the core variable of the extended power flow equation set and the to-be-solved variable), the correction process is ensured to be logical and reproducible. Compared with a single correction method, the scheme not only solves the pain point of non-convergence of the conventional power flow problem, but also accurately adjusts the deviation of the solution, so that the effective initial value is closer to the true solution, provides a high-reliability foundation for subsequent fault processing and bifurcation point solving, and improves the numerical stability of the whole technical scheme.

[0023] Further, by converting the network equation into a form related to the line state parameter, the admittance matrix can be dynamically adjusted with the line outage state, and the physical characteristics of the fault scene are accurately matched. Compared with the traditional method, the direct use of the original network equation in the fault scene leads to the problem of solution interruption or accuracy reduction. This scheme realizes the seamless connection of the fault state and the solving model. Without reconstructing the overall solving framework, the solving process is ensured to be continuous by equation form adaptation, which reduces the complexity of fault handling and ensures the effectiveness of the solution in the fault scene, expanding the practical application scenarios of the method and providing reliable support for stability analysis of power systems after faults.

[0024] Further, in the vertical correction, the continuation parameter is fixed and the admittance matrix is reconstructed to ensure that the mismatch quantity focuses on the static load flow characteristics after the fault; in the horizontal correction, the admittance matrix and the Jacobian matrix partial derivative are dynamically updated to adapt to the coupling relationship between the parameters and the fault state. This scene-based processing method takes into account the core objectives of different correction stages and ensures the accuracy of the mathematical model through dynamic matrix updating. Compared with the unified processing method in fault handling, this scheme effectively avoids the interference of fault factors on correction accuracy, ensures the high reliability of the equation set after horizontal correction in the fault scene, provides accurate input for subsequent bifurcation point approximation solving and interval determination, and further improves the robustness of the method in the fault scene.

[0025] Further, the rate of change of the continuation parameter is calculated using the Jacobian matrix and voltage component of the current solution, which fully utilizes the existing calculation data, eliminates the need for additional complex modeling, and significantly reduces the computational load. The approximation solving process is consistent with the mathematical properties of the saddle-node bifurcation point of the power system, quickly locks the approximate range of the bifurcation point, and provides quantitative support for subsequent sensitivity judgment and interval determination. Compared with the complex iterative method of directly solving the bifurcation point, this scheme uses a "approximate solving - precise positioning" step-by-step strategy to improve efficiency, ensuring the reliability of the approximate solution and narrowing the range for subsequent secant method search, achieving a balance between efficiency and accuracy.

[0026] Further, by explicitly defining the solution positions corresponding to positive, negative, and zero sensitivities (upper half branch, lower half branch, and bifurcation point), a direct correlation between quantitative indicators and physical states is established, avoiding the subjectivity of traditional methods that rely on empirical judgment or complex curve fitting. This judgment rule can quickly distinguish the attribution interval of the solution and avoid the problem of bifurcation point solution failure caused by solving to the lower half branch, providing clear evidence for recording key solution values and determining the initial solution interval. The judgment process only requires calculating the sensitivity sign, which has small computational load and fast response, does not affect the overall solving efficiency, and ensures the accuracy of the initial solution interval, laying a foundation for subsequent precise positioning.

[0027] Further, based on the key solution value of the upper half branch and the lower half branch, a search interval is constructed, the secant method uses two-point linear interpolation to quickly approach the bifurcation point, compared with the Newton-Raphson method, the operation amount is smaller and the convergence is more stable without calculating the Jacobian matrix. The mechanism of returning to the last step and reducing the step size when not converging effectively avoids the numerical oscillation problem near the bifurcation point and ensures that the solving process does not interrupt. The scheme not only takes advantage of the high efficiency of the secant method, but also compensates for the convergence defect near the singular point through step size adjustment, realizes the balance of fast search and stable convergence, greatly improves the positioning accuracy and efficiency of the saddle-node bifurcation point, and solves the pain points of the traditional method in the convergence difficulty and insufficient accuracy near the bifurcation point.

[0028] Further, by dynamically adjusting the step size through linear approximation of sensitivity, the step size is accurately matched with the distance from the bifurcation point: a larger step size can be used to improve efficiency when far away from the bifurcation point, and the step size is automatically reduced to ensure accuracy when approaching the bifurcation point. Compared with the fixed step size method, this scheme not only avoids the positioning deviation caused by large step size, but also prevents the low efficiency caused by small step size, realizes the intelligent optimization of step size, and ensures that the singular region near the bifurcation point is bypassed, further improving the stability and reliability of the solution, and providing key technical support for accurately positioning the saddle-node bifurcation point.

[0029] It can be understood that the beneficial effects of the above-mentioned second aspect to the sixth aspect can be referred to the related description in the first aspect, which will not be repeated here.

[0030] In summary, the present application overcomes the numerical defects of the traditional method in solving the bifurcation point through hybrid algorithm design and adaptive control mechanism, significantly improves the calculation efficiency, stability and scene adaptability, and provides key technical support for safe operation of power system.

[0031] The technical solutions of the present application will be further described in detail below with the help of the drawings and examples. BRIEF DESCRIPTION OF DRAWINGS

[0032] Figure 1 The flowchart of the present application; Figure 2 The pre-estimation-correction schematic diagram; Figure 3 The IEEE-39 system block diagram; Figure 4 The schematic diagram of the computer equipment provided by an embodiment of the present application; Figure 5 The block diagram of a chip provided by the present application according to an embodiment.

[0033] Wherein, 60. Computer device; 61. Processor; 62. Memory; 63. Computer program; 600. Electronic device; 610. Processing unit; 620. Storage unit; 6201. Random access storage unit; 6202. Cache storage unit; 6203. Read-only storage unit; 6204. Program / utility; 6205. Program module; 630. Bus; 640. Display unit; 650. Input / output interface; 660. Network adapter; 700. External device. DETAILED DESCRIPTION

[0034] The technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some of the embodiments of the present application, rather than all the embodiments of the present application. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work fall within the scope of protection of the present application.

[0035] In the description of the present application, it should be understood that the terms "include" and "contain" indicate the existence of described features, whole, steps, operations, elements and / or components, but do not exclude the existence or addition of one or more other features, whole, steps, operations, elements, components and / or sets thereof.

[0036] It should also be understood that the terms used in the present application specification are only for the purpose of describing specific embodiments and are not intended to limit the present application. As used in the present application specification and the appended claims, unless otherwise clear from the context, the singular forms "a", "an" and "the" are intended to include the plural forms.

[0037] It should be further understood that the term "and / or" used in the present application specification and the appended claims means any combination of one or more of the associated listed items and all possible combinations, and includes these combinations, for example, A and / or B can represent three cases of A alone, A and B together, and B alone. In addition, the character " / " in the present application generally represents an "or" relationship between the front and rear associated objects.

[0038] It should be understood that although the terms first, second, third, etc. may be used in the embodiments of the present application to describe the preset ranges, etc., these preset ranges should not be limited to these terms. These terms are only used to distinguish the preset ranges from each other. For example, the first preset range can also be referred to as the second preset range, and similarly, the second preset range can also be referred to as the first preset range without departing from the scope of the embodiments of the present application.

[0039] Depending on the context, the word "if" as used herein can be interpreted to mean "when" or "while" or "in response to determining" or "in response to detecting." Similarly, the phrase "if it is determined" or "if [a stated condition or event] is detected" can be interpreted to mean "when it is determined" or "in response to determining" or "when [a stated condition or event] is detected" or "in response to detecting [a stated condition or event]."

[0040] Various structural diagrams according to the disclosed embodiments of the present application are shown in the drawings. These diagrams are not drawn to scale, in which certain details are exaggerated for clarity of presentation and may be omitted, for example. The shapes and relative sizes of the various regions, layers, and the relative positions of these regions / layers shown in the drawings are merely examples and may deviate in practice due to manufacturing tolerances or technical limitations, and regions / layers with different shapes, sizes, and relative positions can be additionally designed according to actual needs by those skilled in the art.

[0041] The present application provides a power system saddle node bifurcation point solving method combining prediction-correction and secant method, first obtains network parameters, generator parameters, load parameters and the like of a power system, establishes power flow equations of the power system; then adopts a prediction-correction framework, obtains initial values and processes line faults through vertical correction and horizontal correction; then under horizontal correction, approximates a saddle node bifurcation point by using a linear relationship, and judges whether a current solution is located on an upper half branch, a lower half branch or a bifurcation point according to sensitivity signs; finally, through secant method and adaptive step control, precisely locates the saddle node bifurcation point. The method can realize fast and robust saddle node bifurcation point solving in a large-scale power system containing line faults and new energy grid connection, provides strong support for power system stability analysis and operation optimization, and can judge whether a current solution is located on an upper half branch or a lower half branch according to a value of , avoids the problem of saddle node bifurcation point solving failure caused by solving to the lower half branch, has a step control link, and can return to the last step and calculate by using a smaller step in the case of non-convergence, so that the accuracy of calculation can be ensured.

[0042] Please refer to Figure 1 , the present application is a power system saddle node bifurcation point solving method combining prediction-correction and secant method, comprising the following steps: S1, establishing power system equations, obtaining network parameters, generator parameters, load parameters and the like of a power system, and establishing power flow equations of the power system; According to various parameters in the power system, a parameter is introduced by using the continuation method, and then the following equation group is formed: (1) In the vicinity of a certain feasible solution, there are: (2) Right now: (3) At the saddle junction Where, there is Therefore This is unusual. Conventional power flow methods fail to converge. Therefore, we employ the extended power flow prediction-correction approach for solution.

[0043] S2. The prediction-correction method is adopted. The prediction method is used to obtain the initial value better. Since the power flow problem does not necessarily converge, two correction methods, vertical correction and horizontal correction, are adopted. For the forecasting stage, such as Figure 2 In part AB, given step size Below, the voltage estimate is obtained. . The estimated value can be approximated using a linear method. Equation (3) has already formed the... and The linear relationship between them leads to: (4) right ,use As a solution Approximate to .

[0044] The calibration process is divided into vertical calibration and horizontal calibration.

[0045] The approximate solution obtained using the prediction method, for vertical correction, such as Figure 2 The middle BC part, i.e., the solution: (5) For the horizontal correction process, such as Figure 2 The DE part, i.e., the solution: (6) S3. For fault handling, in the case of disconnecting lines under network faults, the network equations are written in a specific form and handled separately in vertical and horizontal corrections. For fault handling, in the event of a network fault, the line is disconnected. If the disconnection occurs in an AC system, the network equations are written as follows: (7) Throughout the system, regarding The variables are only reflected in the network equations.

[0046] Note the network equation Therefore, we get: (8) At this point, for vertical correction, Given the network equations, we only need to address the mismatch. When solving for the mismatch, we first form... Dependent admittance matrix Then, the network mismatch is recalculated. For horizontal correction, Given. At this point, note that... Since it is a quantity to be determined, it should not only be included in the calculation of mismatch. The updated admittance matrix also requires updating the function's relationship with respect to the Jacobian matrix during iteration. The partial derivatives of . Therefore, for level correction, the Jacobian matrix should be increased by one dimension: (9) here express exist The position in the middle.

[0047] S4. Approximate solution of the bifurcation point of the saddle knot: For the system of equations under horizontal correction, the solution can be obtained through linear relationship near a certain feasible solution, and it can be determined whether the current solution is in the upper half branch, the lower half branch, or the bifurcation point. Regarding the solution for the bifurcation point of the saddle-node, for the system of equations under horizontal correction, we have: (10) in, The voltage at a given node.

[0048] In the vicinity of some feasible solution, we have: (11) Right now: (12) Therefore, we can obtain: (13) Therefore, if a solution converges, it can be obtained during level correction using the method described above. Then, by using equation (14), we can determine whether the current solution is in the upper branch, the lower branch, or a bifurcation point.

[0049] (14) S5. The saddle-node bifurcation point is further solved using the secant method and step size control.

[0050] First, the running point of the upper branch is obtained through vertical correction. Then, by continuously applying horizontal correction, the running point of the first lower branch is obtained. and record the last point on the upper branch In the process of horizontal correction, it is judged to be in the upper branch or lower branch by solving equation (12), and the step length is controlled by the absolute value of

[0051] In the interval , according to the size of , the secant method is used to solve the point =0, and the saddle node bifurcation point is searched. If the solved saddle node bifurcation point meets the requirement that =0, stop. Otherwise, use the newly obtained solution to update , and repeat this step.As for the secant method for solving the point =0, the specific process is as follows: Let , if and are obtained, the secant method has the following when solving: (15) Therefore: (16) It is obtained that: (17) In the entire process of solving the saddle node bifurcation point, attention must be paid to the control of the step length. When the saddle node bifurcation point is quickly solved, according to obtained by solving at this step, the linear approximation method is used, and the arc lengths of and are equal to the step length , that is: (18) It is obtained that: (19)That is, there is a larger voltage drop at the saddle node bifurcation point. The calculation around the saddle node bifurcation point is bypassed, and if the step does not converge, it is returned to the previous step, and the half is used to solve until convergence is achieved.

[0052] In another embodiment of the present application, a hybrid power system saddle node bifurcation point solving system combining the prediction-correction method and the secant method is provided, which can be used to realize the hybrid power system saddle node bifurcation point solving method combining the prediction-correction method and the secant method. Specifically, the hybrid power system saddle node bifurcation point solving system combining the prediction-correction method and the secant method comprises an equation module, a solving module, a fault module, an interval module and a positioning module.

[0053] The equation module obtains network parameters, generator parameters and load parameters of the power system, establishes power flow equations of the power system based on the above parameters, and forms an extended power flow equation set by introducing parameters through the continuation method. The solving module solves the extended power flow equation set obtained by the equation module by using a prediction-correction method, obtains a voltage estimated value as an approximate solution through a linear method, and performs vertical correction and horizontal correction on the approximate solution to handle a situation where a conventional power flow problem may not converge, to obtain an effective initial value. The fault module, for the scenario of line removal under network fault, first converts the network equation into a specific form suitable for the fault scenario, and then based on the effective initial value obtained by the solving module, combines the vertical correction and horizontal correction processes, and processes the converted network equation respectively to ensure the continuity of the solving process under the fault scenario, and outputs the horizontal corrected equation set. The interval module determines a feasible solution range based on the horizontal corrected equation set output by the fault module and the effective initial value obtained by the solving module, approximates the solution of the saddle node bifurcation point related parameters through linear relationship near the feasible solution, and determines the initial solution interval of the saddle node bifurcation point according to the sensitivity sign to judge whether the current solution is in the upper half branch, the lower half branch or the bifurcation point, and records the last effective solution of the upper half branch and the first effective solution of the lower half branch to determine the initial solution interval of the saddle node bifurcation point. The positioning module searches for the saddle node bifurcation point based on the initial solution interval determined by the interval module and the secant method and adaptive step control. If the solution does not converge under the current step, return to the interval module and adjust the step to recalculate until the saddle node bifurcation point is accurately positioned, and the entire solving process is completed.

[0054] The application provides a terminal device, which comprises a processor and a memory for storing a computer program, wherein the computer program comprises program instructions, and the processor is used for executing the program instructions stored in the computer storage medium. The processor can be a central processing unit (CPU), and can also be other general-purpose processors, graphics processing units (GPUs), tensor processing units (TPUs), digital signal processors (DSPs), application specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs) or other programmable logic devices, discrete gates or transistor logic devices, discrete hardware components and the like, which are the computing core and control core of the terminal, and are suitable for implementing one or more instructions, and are specifically suitable for loading and executing one or more instructions to implement a corresponding method flow or a corresponding function; the processor in the embodiments of the application can be used for the operation of the saddle-node bifurcation point solving method of the hybrid prediction-correction and secant method of the power system, comprising: The network parameters, generator parameters and load parameters of the power system are acquired, the power flow equation of the power system is established based on the above parameters, and the parameter is introduced to form an extended power flow equation set by the continuation method; the extended power flow equation set obtained by using the prediction-correction method is solved, the voltage estimation value is obtained by a linear method and is taken as an approximate solution, and the vertical correction and the horizontal correction are performed on the approximate solution to process the situation that the conventional power flow problem may not converge, so that an effective initial value is obtained; for the scene of line removal under network fault, the network equation is converted into a specific form suitable for the fault scene, and then based on the obtained effective initial value, the converted network equation is processed in combination with the vertical correction and the horizontal correction process, so that the continuity of the solving process under the fault scene is ensured, and the equation set after the horizontal correction is output; based on the output equation set after the horizontal correction, the feasible solution range is determined in combination with the obtained effective initial value, the parameters related to the saddle-node bifurcation point are approximately solved by a linear relationship near the feasible solution, and then the current solution is determined to be in the upper half branch, the lower half branch or the bifurcation point according to the sensitivity sign, and the last effective solution of the upper half branch and the first effective solution of the lower half branch are recorded to determine the initial solution interval of the saddle-node bifurcation point; based on the determined initial solution interval, the saddle-node bifurcation point is searched in combination with the secant method and the adaptive step control, if the solving under the current step length does not converge, the step length is returned to be adjusted and recalculated until the saddle-node bifurcation point is accurately positioned, and the whole solving process is completed.

[0055] Please refer toFigure 4 In the embodiment, the terminal device is a computer device, and the computer device 60 includes a processor 61, a memory 62, and a computer program 63 stored in the memory 62 and executable on the processor 61. When the computer program 63 is executed by the processor 61, the power system saddle node bifurcation point solving method based on the hybrid predictor-corrector and secant method is implemented. To avoid repetition, the functions of the computer program 63 are not described herein. Alternatively, when the computer program 63 is executed by the processor 61, the functions of each model / unit in the power system saddle node bifurcation point solving system based on the hybrid predictor-corrector and secant method are implemented. To avoid repetition, the functions of the computer program 63 are not described herein.

[0056] The computer device 60 can be a desktop computer, a notebook computer, a palm computer, a cloud server, or the like. The computer device 60 can include, but is not limited to, the processor 61 and the memory 62. Those skilled in the art can understand that the computer device 60 can include more or fewer components, or some components can be combined, or different components can be included, for example, the computer device can further include an input / output device, a network access device, a bus, and the like. Figure 4 The computer device 60 is merely an example and does not limit the computer device 60. The computer device 60 can include more or fewer components, or some components can be combined, or different components can be included, for example, the computer device can further include an input / output device, a network access device, a bus, and the like.

[0057] The processor 61 can be a central processing unit (CPU), and can also be other general-purpose processors, graphics processing units (GPUs), tensor processing units (TPUs), digital signal processors (DSPs), application specific integrated circuits (ASICs), field-programmable gate arrays (FPGAs) or other programmable logic devices, discrete gates or transistor logic devices, discrete hardware components, or the like. The general-purpose processor can be a microprocessor or can be any conventional processor.

[0058] The memory 62 can be an internal storage unit of the computer device 60, for example, a hard disk or a memory of the computer device 60. The memory 62 can also be an external storage device of the computer device 60, for example, a plug-in hard disk, a smart media card (SMC), a secure digital (SD) card, a flash card, or the like.

[0059] Furthermore, the memory 62 may include both internal storage units of the computer device 60 and external storage devices. The memory 62 is used to store computer programs and other programs and data required by the computer device. The memory 62 can also be used to temporarily store data that has been output or will be output.

[0060] Please see Figure 5 The terminal device is an electronic device 600, which is manifested in the form of a general-purpose computing device. The components of the electronic device may include, but are not limited to: at least one processing unit 610, at least one storage unit 620, a bus 630 connecting different platform components (including storage unit 620 and processing unit 610), a display unit 640, etc.

[0061] The storage unit stores program code, which can be executed by the processing unit 610 to perform the steps described in the method section of this specification according to various exemplary embodiments of the present invention. For example, the processing unit 610 can perform actions such as... Figure 1 The steps are shown in the figure.

[0062] Storage unit 620 may include a readable medium in the form of a volatile storage unit, such as random access memory (RAM) 6201 and / or cache memory 6202, and may further include a read-only memory (ROM) 6203.

[0063] Storage unit 620 may also include a program / utility 6204 having a set (at least one) program module 6205, such program module 6205 including but not limited to: operating system, one or more application programs, other program modules and program data, each or some combination of these examples may include an implementation of a network environment.

[0064] Bus 630 can represent one or more of several types of bus structures, including a memory cell bus or memory cell controller, a peripheral bus, a graphics acceleration port, a processing unit, or a local bus using any of the multiple bus structures.

[0065] The electronic device 600 can also communicate with one or more external devices 700 such as a keyboard or pointing device, a Bluetooth device, or a database, and / or one or more devices that enable a user to interact with the electronic device 600 and / or one or more devices (e.g., routers, modems) that enable the electronic device 600 to communicate with one or more other computing devices. Such communication can occur via an input / output interface 650. Still yet, the electronic device 600 can communicate with one or more networks, such as a local area network (LAN), a general wide area network (WAN), and / or the Internet, through a network adapter 660. The network adapter 660 can be any of a plurality of different types of adapters suitable for interfacing the electronic device 600 to various networking schemes. It should be appreciated that, while not shown explicitly, other hardware and / or software modules could be used in conjunction with the electronic device 600, including but not limited to microcode, device drivers, redundant processing units, external disk drive arrays, RAID systems, tape drives, and data archival storage systems, etc.

[0066] Embodiment 4 The present application also provides a storage medium, specifically a computer readable storage medium, which is a memory device in the terminal device, and is used to store programs and data. It should be understood that the computer readable storage medium herein can include the built-in storage medium in the terminal device, and of course can also include the expansion storage medium supported by the terminal device, and can be any tangible medium containing or storing programs, which can be used by or in conjunction with an instruction execution system, device or apparatus. The computer readable storage medium provides a storage space, which stores the operating system of the terminal. In addition, one or more instructions suitable for being loaded and executed by the processor are also stored in the storage space, and the instructions can be one or more computer programs (including program codes). It should be noted that more specific examples of the computer readable storage medium include an electrical connection having one or more wires, a portable disk, a hard disk, a random access memory, a read-only memory, an erasable programmable read-only memory, an optical fiber, a portable compact disk read-only memory, an optical storage device, a magnetic storage device, or any suitable combination of the above.

[0067] Computer readable storage media further includes data signals transported through a carrier wave and a communications medium, including or excluding wired storage media.

[0068] The program code can be executed by using any combination of one or more programming languages including an object oriented programming language such as Java, C++ or the like, and a conventional procedural programming language such as a "C" language or similar programming languages. The program code can execute entirely on the user's computing device, partly on the user's computing device, as a stand-alone software package, partly on the user's computing device and partly on a remote computing device or entirely on the remote computing device or server. In the latter scenario, the remote computing device can be connected to the user's computing device through any type of network, including a local area network ("LAN"), or a wide area network ("WAN"), or the connection can be made to an external computing device, for example, through the Internet using an Internet Service Provider ("ISP").

[0069] The one or more instructions stored in the computer readable storage medium can be loaded and executed by a processor to implement the corresponding steps of the power system saddle node bifurcation point solving method with hybrid prediction-correction and secant method in the above embodiments; the one or more instructions stored in the computer readable storage medium are loaded and executed by the processor to implement the following steps: The network parameters, generator parameters and load parameters of the power system are acquired, the power flow equation of the power system is established based on the parameters, the parameter is introduced by the continuation method to form an extended power flow equation set, the extended power flow equation set obtained by the prediction-correction method is solved, the voltage estimation value is obtained by a linear method and is taken as an approximate solution, and the approximate solution is subjected to vertical correction and horizontal correction respectively to process the situation that the conventional power flow problem may not converge, and an effective initial value is obtained; for the scene of line removal under network fault, the network equation is converted into a specific form suitable for the fault scene, the effective initial value obtained is combined with the vertical correction and horizontal correction processes, and the converted network equation is processed respectively to ensure the continuity of the solving process under the fault scene, and an equation set after horizontal correction is output; based on the output equation set after horizontal correction, the effective initial value is combined to determine the feasible solution range, the parameters related to the saddle node bifurcation point are approximately solved by a linear relationship near the feasible solution, the current solution is determined to be in the upper half branch, the lower half branch or the bifurcation point according to the sensitivity sign, and the last effective solution of the upper half branch and the first lower half branch effective solution are recorded to determine the initial solution interval of the saddle node bifurcation point; based on the determined initial solution interval, the secant method and adaptive step control are combined to search for the saddle node bifurcation point, if the solution does not converge under the current step, the step is adjusted and recalculated until the saddle node bifurcation point is accurately located, and the whole solving process is completed.

[0070] The database involved in each embodiment provided in the present application can include at least one of a relational database and a non-relational database. The non-relational database can include a distributed database based on a blockchain, and the like, without being limited thereto. The processor involved in each embodiment provided in the present application can be a general-purpose processor, a central processing unit, a graphics processing unit, a digital signal processor, a programmable logic device, a data processing logic device based on quantum computing, and the like, without being limited thereto.

[0071] To make the objectives, technical solutions, and advantages of the embodiments of the present application clearer, the technical solutions in the embodiments of the present application will be described clearly and completely below with reference to the drawings in the embodiments of the present application. Obviously, the described embodiments are some embodiments of the present application, rather than all the embodiments. The components of the embodiments of the present application described and shown in the drawings herein can be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present application provided in the drawings is not intended to limit the scope of the claimed present application, but only represents selected embodiments of the present application. All other embodiments obtained by a person of ordinary skill in the art without creative labor based on the embodiments in the present application belong to the scope of protection of the present application.

[0072] In order to verify the power system saddle node bifurcation point solving method of the mixed prediction-correction and secant method proposed in the application, the inventors verified the accuracy of the method on an IEEE-39 node system. The schematic diagram of the IEEE-39 node system is shown in Figure 3 , the reference capacity of the generator is set , the line break fault is set for the system respectively, and the corresponding saddle node bifurcation point of the system at this time is solved respectively . The corresponding solving result is shown in Table 1.

[0073] Table 1: Values of the corresponding saddle node bifurcation points solved under different line break faults of the IEEE-39 node system

[0074] It can be obtained from the table that the saddle node bifurcation point solving method described in the application can accurately solve the saddle node bifurcation point of the curve under the line break fault .

[0075] Finally, it needs to be pointed out that the above description is only for the specific embodiments of the application, and those skilled in the art can make various similar expressions in the complex power system containing new energy, direct current transmission and the like under the inspiration of the application without violating the purpose and claims of the application, as long as the relevant parameters of the new energy and direct current transmission are read in step S1; the same operation is performed subsequently to solve the corresponding saddle node bifurcation point , and such transformation falls within the protection scope of the application.

[0076] In summary, the power system saddle node bifurcation point solving method of the mixed prediction-correction and secant method of the application effectively solves the numerical instability problem of the traditional Newton-Raphson method near the saddle node bifurcation point due to the singularity of the Jacobian matrix by introducing the continuation parameter to construct the extended power flow equation, combining the linear prediction, vertical / horizontal correction and secant method. The simulation verification of the method in the IEEE-39 node system shows that it can accurately solve the saddle node bifurcation point under different line break faults, has strong numerical stability and high calculation efficiency. In addition, through the sensitivity judgment and adaptive step control, invalid iteration is avoided, the adaptability to complex power system scenarios is improved, and a reliable technical means is provided for system stability analysis and operation optimization.

[0077] ​Those skilled in the art can clearly understand that, for the convenience and brevity of description, only the above-mentioned division of each functional unit and module is exemplified, and in actual application, the above-mentioned functions can be completed by different functional units and modules according to needs, that is, the internal structure of the device is divided into different functional units or modules to complete all or part of the functions described above. Each functional unit and module in the embodiment can be integrated in one processing unit, or each unit can be physically present separately, or two or more units can be integrated in one unit. The above-mentioned integrated unit can be realized in the form of hardware or software. In addition, the specific names of each functional unit and module are only for the convenience of mutual distinction, and do not limit the protection scope of the present application. The specific working process of the units and modules in the above system can refer to the corresponding process in the foregoing method embodiments, which will not be described here.

[0078] In the above embodiments, the description of each embodiment has its own emphasis, and the parts not described or recorded in detail in a certain embodiment can be referred to the related description of other embodiments.

[0079] Those of ordinary skill in the art can realize that the units and algorithm steps of each example described in combination with the embodiments disclosed in the present application can be realized by electronic hardware or a combination of computer software and electronic hardware. Whether the functions are executed in hardware or software depends on the specific application and design constraints of the technical solution. Professional technicians can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the present application.

[0080] In the embodiments provided by the present application, it should be understood that the disclosed apparatus / terminal and method can be implemented by other ways. For example, the above-mentioned apparatus / terminal embodiments are only schematic, and the division of the modules or units is only a logical function division, and there can be another division way in actual implementation, for example, a plurality of units or components can be combined or integrated into another system, or some features can be ignored or not executed. In addition, the displayed or discussed mutual coupling or direct coupling or communication connection can be indirect coupling or communication connection through some interface, device or unit, and can be electrical, mechanical or other forms.

[0081] The units described as separate components can or can not be physically separated, and the components shown as units can or can not be physical units, that is, they can be located in one place, or can be distributed on multiple network units. Part or all of the units can be selected according to actual needs to achieve the purpose of the embodiment.

[0082] In addition, each functional unit in each embodiment of the present application can be integrated in one processing unit, or each unit can be physically present separately, or two or more units can be integrated in one unit. The integrated unit can be realized in the form of hardware or in the form of a software functional unit.

[0083] The integrated module / unit, if realized in the form of a software functional unit and sold or used as an independent product, can be stored in a computer-readable storage medium. Based on such understanding, all or part of the processes in the above-mentioned embodiment methods can also be completed by a computer program instructing related hardware, and the computer program can be stored in a computer-readable storage medium. The computer program can implement the steps of each method embodiment when executed by a processor. The computer program includes computer program code, which can be in the form of source code, object code, executable files, or some intermediate forms. The computer-readable medium can include any entity or device capable of carrying the computer program code, recording medium, U disk, mobile hard disk, magnetic disk, optical disk, computer memory, read-only memory (ROM), random access memory (RAM), electrical carrier signal, telecommunication signal, and software distribution medium, etc. It should be noted that the computer-readable medium can include or exclude contents according to the requirements of legislation and patent practice in the jurisdiction, for example, in some jurisdictions, according to legislation and patent practice, the computer-readable medium does not include electrical carrier signals and telecommunication signals.

[0084] The present application is described with reference to flowcharts and / or block diagrams according to the methods, devices, and computer program products of embodiments of the present application. It should be understood that each flow and / or block in the flowcharts and / or block diagrams, and the combination of flows and / or blocks in the flowcharts and / or block diagrams can be implemented by computer program instructions. These computer program instructions can be provided to a general-purpose computer, a special-purpose computer, an embedded processor, or other programmable data processing devices to produce a machine, so that the instructions executed by the computer or other programmable data processing devices produce a device that implements the functions specified in the flowcharts and / or block diagrams. Figure 1 The functions specified in one flow or multiple flows and / or blocks Figure 1 The device that implements the functions specified in one block or multiple blocks.

[0085] These computer program instructions can also be stored in a computer-readable storage medium that can guide the computer or other programmable data processing devices to work in a specific manner, so that the instructions stored in the computer-readable storage medium produce a manufactured product including instruction devices that implement the functions specified in the flowcharts and / or block diagrams. Figure 1one or more processes and / or blocks Figure 1 the function specified in the one or more blocks.

[0086] These computer program instructions can also be loaded into computer or other programmable data processing devices, so that a series of operational steps are performed on the computer or other programmable data processing devices to generate a computer-implemented process, so that the instructions executed on the computer or other programmable data processing devices provide a process for implementing the flow Figure 1 one or more processes and / or blocks Figure 1 the function specified in the one or more blocks.

[0087] The above is only to illustrate the technical idea of the present application, and cannot limit the protection scope of the present application. Any modification made according to the technical idea of the present application on the basis of the technical scheme falls within the protection scope of the claims of the present application.

Claims

1. A method for finding saddle-node bifurcation points in power systems by hybrid predictor-corrector and secant method, characterized in that, The method comprises the following steps: S1, obtaining network parameters, generator parameters and load parameters of a power system, establishing a power flow equation of the power system based on the parameters, and introducing parameters to form an extended power flow equation set by using the continuation method; S2, solving the extended power flow equation set obtained in step S1 by using a prediction-correction method, obtaining a voltage estimation value by using a linear method as an approximate solution, and performing vertical correction and horizontal correction on the approximate solution to handle a situation that a conventional power flow problem may not converge, and obtaining an effective initial value; S3, for a scenario of line removal under network failure, first converting a network equation into a specific form suitable for the fault scenario, then based on the effective initial value obtained in step S2, combining the vertical correction and horizontal correction processes, and processing the converted network equation respectively to ensure the continuity of the solving process under the fault scenario, and outputting a horizontally corrected equation set; S4, based on the horizontally corrected equation set output in step S3, combining the effective initial value obtained in step S2 to determine a feasible solution range, first approximately solving the parameters related to the saddle node bifurcation point by using a linear relationship near the feasible solution, then judging the position of the current solution according to the sensitivity sign, recording the last effective solution of the upper half branch and the first effective solution of the lower half branch to determine the initial solution interval of the saddle node bifurcation point; S5, based on the initial solution interval determined in step S4, combining the secant method and adaptive step control to search for the saddle node bifurcation point, if the solution does not converge under the current step, returning to step S4 and adjusting the step to recalculate until the saddle node bifurcation point is accurately located, and completing the entire solving process.

2. The hybrid predictor-corrector and secant method based power system saddle node bifurcation point solving method according to claim 1, wherein, In step S2, the process of obtaining the voltage estimation value by using the linear method is as follows: Under a given step, the voltage variation and the parameter variation in step S1 are associated by using a linear relationship to obtain the voltage estimation value, and the voltage estimation value is taken as an approximate value of the solution of the extended power flow equation set.

3. The hybrid predictor-corrector and secant method based power system saddle node bifurcation point solving method according to claim 1, wherein, In step S2, the vertical correction specifically involves solving the equation. The aforementioned level correction specifically involves solving the equation. ,in, For the first A column vector consisting of the variables from each iteration. For the first The margin parameters obtained from the round, To correct the step size, For the first The margin parameters obtained from the round, For the first The variable vector of the round of iteration One tracking parameter, For the first The predicted results of each tracking parameter.

4. The hybrid predictor-corrector and secant method based power system saddle node bifurcation point solving method according to claim 1, wherein, In step S3, the conversion of the network equation into a specific form suitable for the fault scenario specifically includes: When the line is removed in an alternating current system, the network equation is written in a form related to the line state parameter, and the admittance matrix changes with the line state.

5. The hybrid predictor-corrector and secant method based power system saddle node bifurcation point solving method according to claim 4, wherein, The processing of the converted network equation by combining the vertical correction and horizontal correction processes specifically includes: In the vertical correction, the fixed value of the parameter in step S1 is given, the admittance matrix depending on the parameter value is formed, and the network mismatch is recalculated; In the horizontal correction, the parameter in step S1 is taken as a to-be-solved quantity, the admittance matrix is updated with the parameter when the network mismatch is calculated, and the partial derivative of the function with respect to the parameter is updated in the iteration using the Jacobian matrix.

6. The hybrid predictor-corrector and secant method based power system saddle node bifurcation point solving method according to claim 1, wherein, In step S4, the approximate solution of the parameters related to the saddle node bifurcation point by using the linear relationship specifically includes: According to the Jacobian matrix and the voltage component of the current solution, the rate of change of the parameter λ is calculated to obtain the approximate value of the saddle node bifurcation point.

7. The hybrid predictor-corrector and secant method based power system saddle node bifurcation point solving method according to claim 6, wherein, The position of the current solution according to the sensitivity sign specifically includes: The sensitivity value is calculated, when the sensitivity is positive, the current solution is in the upper half branch; when the sensitivity is negative, the current solution is in the lower half branch; and when the sensitivity is zero, the current solution is at the bifurcation point.

8. The hybrid predictor-corrector and secant method based power system saddle node bifurcation point solving method according to claim 1, wherein, In step S5, the search for the saddle-node bifurcation point by combining the secant method and adaptive step control specifically includes: Based on the last valid solution of the upper branch and the first valid solution of the lower branch, the point with zero sensitivity is iteratively solved by using the secant method; during the iteration process, if the current step does not converge, the last step is returned and the step size is reduced for recalculation until convergence.

9. The hybrid predictor-corrector and secant method based power system saddle node bifurcation point solving method according to claim 8, wherein, The step size adjustment specifically includes: According to the sensitivity obtained by the step, the step size is dynamically adjusted by using the linear approximation method to bypass the saddle-node bifurcation point and ensure the calculation accuracy.

10. A power system saddle node bifurcation point solving system mixing predictor-corrector and secant method, characterized in that, It includes: The equation module obtains the network parameters, generator parameters and load parameters of the power system, establishes the power flow equation of the power system based on the above parameters, and forms an extended power flow equation set by introducing parameters through the continuation method; The solving module solves the extended power flow equation set obtained by the equation module by using the predictor-corrector method, first obtains the voltage estimate value by linear method and takes it as the approximate solution, and then performs vertical correction and horizontal correction on the approximate solution to handle the situation that the conventional power flow problem may not converge, to obtain the effective initial value; The fault module, for the scenario of line removal under network fault, first converts the network equation into a specific form suitable for the fault scenario, and then based on the effective initial value obtained by the solving module, combines the vertical correction and horizontal correction process, and processes the converted network equation respectively, to ensure the continuity of the solving process under the fault scenario, and outputs the horizontal corrected equation set; The interval module, based on the horizontal corrected equation set output by the fault module, determines the feasible solution range in combination with the effective initial value obtained by the solving module, first approximates the solution of the saddle-node bifurcation point related parameters by linear relationship near the feasible solution, and then determines whether the current solution is in the upper branch, lower branch or bifurcation point according to the sensitivity sign, and records the last valid solution of the upper branch and the first valid solution of the lower branch to determine the initial solution interval of the saddle-node bifurcation point; The positioning module, based on the initial solution interval determined by the interval module, searches for the saddle-node bifurcation point by combining the secant method and adaptive step control, if the solution does not converge under the current step size, return to the interval module and adjust the step size for recalculation until the saddle-node bifurcation point is accurately positioned, and the entire solving process is completed.

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