A modeling method for power systems, a calculation method for power flow models, and electronic equipment.

Abstract

This invention provides a modeling method for power systems, a calculation method for power flow models, and electronic equipment. By introducing a combination of an ideal phase-shifting transformer and ordinary transmission lines, and adding an AD node, this invention successfully solves the non-convergence problem that may occur in the classical fully embedded method for power flow calculation. By splitting the phase-shifting transformer and introducing it into the AD node as a node with zero power, the stability and convergence of the power flow calculation are ensured. Based on this, the reconstructed power flow model includes multiple constraints, such as the complex power balance equations of the PQ and AD nodes, the active power balance equations of the PV nodes, and voltage amplitude constraint equations, further improving the calculation accuracy and reliability. This method can effectively cope with the complex situations in large-scale power systems, significantly improve the convergence of power flow calculations, and provide a strong guarantee for the stable operation of power systems.

Description

Technical Field

[0001] This invention relates to the technical field of power systems, specifically to a power system modeling method, a power flow model calculation method, and electronic equipment. Background Technology

[0002] Power flow calculation in power systems is a fundamental analytical method for determining the steady-state operating parameters of a power grid. Its goal is to solve for the voltage magnitude, voltage phase angle, and the distribution of active and reactive power in each branch of the system, given the network topology, generator output, load distribution, and control parameters.

[0003] The existing fully embedded method (HEM) suffers from convergence failure in power systems with large phase shift angles. Summary of the Invention

[0004] The purpose of this invention is to overcome the above-mentioned technical deficiencies and provide a power system modeling method, a power flow model calculation method, and an electronic device to solve the technical problem that the existing fully embedded method (HEM) fails to converge in power systems with large phase shift angles.

[0005] To achieve the above-mentioned technical objectives, the present invention adopts the following technical solution: In a first aspect, the present invention provides a modeling method for a power system, comprising: Identify the target factors that cause the holomorphic embedding method to fail to converge; Based on the aforementioned target factors, the phase-shifting transformers carried in the target power system are decomposed into a combination of ideal phase-shifting transformers and ordinary transmission lines, and new nodes are introduced as AD nodes to construct a power model; wherein, the AD node is a power node with injected power that is always zero. Based on the power model, the power flow model is reconstructed to obtain the reconstructed power flow model; wherein, the reconstructed power flow model includes the complex power balance equations of PQ nodes and AD nodes, the active power balance equations and voltage amplitude constraint equations of PV nodes, the voltage amplitude constraint equations of the slack nodes, and the phase-shift constraint equations of the ideal phase-shifting transformer.

[0006] Furthermore, the step of determining the target factors that cause the holomorphic embedding method to fail to converge includes: A power deviation-based detection method is used to identify the target factors that cause the fully embedded method to fail to converge; wherein, the target factors include numerical instability caused when the phase shift angle of the phase-shifting transformer exceeds a first threshold.

[0007] Furthermore, the step of determining the target factors causing the fully pure embedding method to fail to converge using a power bias-based detection method includes: Perform power flow calculations using the fully embedded method on the target power system, record the voltage update value of each node during the power series recursion process, and calculate the voltage oscillation amplitude. Calculate the power deviation value for each node; wherein the power deviation value is the difference between the actual injected power and the theoretical power flow power of the node; If a pair of nodes simultaneously meets the following conditions, it is determined to be an abnormal node connected to the phase-shifting transformer: the power deviation value exceeds the second threshold, the voltage oscillation amplitude exceeds the third threshold, and the fluctuation trend is synchronized. The pair of nodes is determined to be connected by a phase-shifting transformer, and the phase shift angle of the phase-shifting transformer exceeding a first threshold is taken as the target factor.

[0008] Furthermore, the complex power balance equations for the PQ node and the AD node are expressed as follows:

[0009] In the formula, Represented as the total number of nodes in the system. This represents the element in the i-th row and k-th column of the nodal admittance matrix. Let be the complex voltage at node k. For nodes Injection complex power The conjugate value, The complex power flowing from node i to node m The conjugate value of , where im represents the ideal phase-shifting transformer connected between nodes i and m. This represents a set of ideal phase-shifting transformers. Represents the set of PQ nodes. Represents the set of AD nodes. This indicates the calculation of the conjugate value of a complex number. Represented as the complex voltage at node i The conjugate value, ; The active power balance equation and voltage amplitude constraint equation of the PV node are expressed as follows:

[0010] In the formula, Represented as the complex voltage at node i The conjugate value, This represents the element in the i-th row and k-th column of the nodal admittance matrix. Let be the complex voltage at node k. This is represented by the injected active power at PV node i. It is represented as the sum of the active power of all ideal phase-shifting transformers connected to node i. This represents the real-time active power consumed by the ideal phase-shifting transformer itself. This indicates the voltage amplitude specified for a PV node or a ballast node. Represented as a set of PV nodes, This indicates the calculation of the real part of a complex number; This indicates the calculation of the absolute value of a complex number.

[0011] Furthermore, the voltage magnitude constraint equation for the balancing node is expressed as:

[0012] In the formula, Represented as a set of balanced nodes; The phase-shifting constraint equation of the ideal phase-shifting transformer is expressed as:

[0013] In the formula, This represents a set of ideal phase-shifting transformers. This represents an ideal phase-shifting transformer connected between nodes i and m; This represents the voltage magnitude at the new node m. θ represents the transformer turns ratio. shift This indicates the phase shift angle of the transformer.

[0014] Secondly, the present invention provides a method for calculating a power flow model of a power system, comprising: For the reconstructed power flow model, complex variables are embedded, and a power flow model in the form of fully embedded variables is constructed; wherein, the reconstructed power flow model is the reconstructed power flow model described above. The fully embedded power flow model is solved using a preset solution algorithm until the convergence accuracy is met, and the result parameters of each node in the target power system are obtained.

[0015] Furthermore, the fully embedded power flow model includes: complex power balance equations for PQ and AD nodes in a fully embedded form, active power balance equations and voltage magnitude constraint equations for PV nodes in a fully embedded form, voltage magnitude constraint equations for slack nodes in a fully embedded form, phase-shift constraint equations for the ideal phase-shifting transformer in a fully embedded form, and fully embedded functions of active and reactive power flowing through the ideal phase-shifting transformer.

[0016] Furthermore, the complex power balance equations for the PQ node and AD node in the purely embedded form are expressed as follows:

[0017] In the formula, ; The active power balance equation and voltage amplitude constraint equation for a purely embedded PV node are expressed as follows:

[0018] In the formula, ; The voltage magnitude constraint equation for a purely embedded balancing node is expressed as:

[0019] The phase-shifting constraint equation for the ideal phase-shifting transformer in its pure embedded form is expressed as:

[0020] In the formula, ; The holomorphic functions of active and reactive power flowing through an ideal phase-shifting transformer are expressed as:

[0021] In the formula, Represented as embedded complex variables, This represents the element in the i-th row and k-th column of the cascaded admittance matrix; This represents the parallel admittance at node i. It is represented as a holomorphic function of the complex voltage at node k. This represents the conjugate of the complex power injected into node i. It is expressed as a holomorphic function of the active power flowing through an ideal phase-shifting transformer. It is expressed as a holomorphic function of the reactive power flowing through an ideal phase-shifting transformer. This is expressed as the parallel admittance at node i. It is represented as a holomorphic function of the complex voltage at node i. Represented as a set of PQ nodes, Represented as a set of AD nodes, It can be represented as a holomorphic function of the complex voltage conjugate at node i. This is represented by the injected active power at PV node i. The imaginary unit, Let i be a holomorphic function of the reactive power of PV node i. Let be the set of PV nodes, where n represents the order of the power series; s is the embedding factor of the complex variable. n This represents the embedding factor of the complex variable of order n.

[0022] Furthermore, the step of using a preset solution algorithm to solve the fully embedded power flow model until the convergence accuracy is met, and obtaining the result parameters of each node in the target power system, includes: For the fully embedded power flow model, a corresponding recursive equation is constructed; wherein, the recursive equation establishes a recursive relationship between the parameters to be solved by extracting the relationship between the coefficients of higher-order power series and the coefficients of lower-order power series. For the recursive equation, recursive calculations are performed to obtain the result parameters for each node in the target power system; wherein, the recursive equation includes: The recursive equations for nodes PQ and AD are expressed as follows:

[0023] In the formula, The current recursion order is... For the nth power series coefficients of the voltage at node k, ... These are the coefficients of the nth power series of active power losses in a phase-shifting transformer. The coefficients of the nth power series of reactive power loss in a phase-shifting transformer are given. For node i, the complex voltage V i The conjugate of the reciprocal term, The reciprocal of the voltage at node i The Conjugate of order coefficients, The lower-order recursion sequence number. This means that nodes with AD are positive and nodes without AD are negative. The recurrence equations for PV nodes and equilibrium nodes are expressed as follows:

[0024]

[0025] In the formula, Represented as the Kronecker delta function, Let the coefficients of the nth power series of reactive power at PV node i be represented. ; The recursive equation for a phase-shifting transformer is expressed as:

[0026] In the formula, This represents the nth-order coefficient of the voltage at primary node i of the phase-shifting transformer. This represents the nth-order coefficient of the voltage at the newly added node m of the phase-shifting transformer. ; The unified recursive equation is expressed as:

[0027] In the formula, A is the recursive matrix required for recursive calculation, which remains constant during the recursive calculation process; x[n] is the coefficient of the nth power series to be calculated. This is the right-hand side term of the recursive equation, which is calculated from the coefficients of a power series of order less than n.

[0028] Thirdly, the present invention provides an electronic device, comprising: a memory, and one or more processors communicatively connected to the memory; the memory stores instructions executable by the one or more processors, the instructions being executed by the one or more processors to cause the one or more processors to implement the method described above.

[0029] Beneficial effects: This invention successfully solves the non-convergence problem that may occur in the classical fully embedded method for power flow calculation by introducing a combination of an ideal phase-shifting transformer and a conventional transmission line, and adding an AD node. By splitting the phase-shifting transformer and introducing it as an AD node with zero power, the stability and convergence of the power flow calculation are ensured. Based on this, the reconstructed power flow model includes multiple constraints, such as the complex power balance equations of the PQ and AD nodes, the active power balance equations of the PV nodes, and voltage amplitude constraint equations, further improving the calculation accuracy and reliability. This method can effectively handle the complex situations in large-scale power systems, significantly improve the convergence of power flow calculation, and provide a strong guarantee for the stable operation of power systems. Attached Figure Description

[0030] Figure 1 This is a flowchart illustrating a power system modeling method provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of the improved power flow calculation model for a phase-shifting transformer provided in an embodiment of the present invention; Figure 3 This is a schematic diagram of the node voltage magnitude update results of the case1888rte system under the classic fully pure embedding method provided in the embodiments of the present invention; Figure 4 This is a schematic diagram of a comparison table of two classic fully pure embedding forms provided in the embodiments of the present invention; Figure 5 This is a schematic diagram of the comparison table of the first six power series coefficients of node 7 of the three methods provided in the embodiments of the present invention (phase shift angle is 2°); Figure 6 This is a schematic diagram of a comparison table of the number of phase-shifting transformers and the phase-shifting angle of multiple example systems provided in the embodiments of the present invention; Figure 7 This is a schematic diagram comparing the computational efficiency and convergence of the proposed method and the NR method provided in the embodiments of the present invention. Detailed Implementation

[0031] To enable those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present application, and not all embodiments. Based on the embodiments in the present application, all other embodiments obtained by those of ordinary skill in the art without creative effort should fall within the scope of protection of the present application.

[0032] In related technologies, with the transformation of the global energy structure and the development of a low-carbon economy, modern power systems have begun to integrate new energy sources, DC transmission, and various flexible control devices (such as phase-shifting transformers and static synchronous compensators) on a large scale. These emerging devices optimize power transmission and distribution through efficient power electronics technology, enhancing the flexibility and stability of the power grid. However, with the widespread integration of these devices, the complexity of the power system has increased significantly, especially in the face of the volatility and uncertainty of large-scale renewable energy, making the need for precise analysis of the power system increasingly urgent.

[0033] In this environment, traditional power flow calculation methods, such as the Newton-Raphson method (NR method), are inefficient due to their sensitivity to initial conditions and the need to recalculate the Jacobian matrix in each iteration, making them unable to meet the dual requirements of real-time performance and accuracy in modern power grids. In contrast, the fully embedded method (HEM), as a non-iterative approach, theoretically avoids dependence on initial conditions by utilizing the analytical extension of complex variables and supports parallel recursive computation. Therefore, it is considered an important development direction for next-generation power flow calculation methods.

[0034] Holomorphic embedding (HEM) transforms the power flow equations of a power system into analytic functions on the complex plane and recursively approximates the steady-state solution using power series expansion. Theoretically, this method can stably solve the power flow problem of a power system without depending on the initial conditions. However, large-scale system tests in practice show that the traditional classical holomorphic embedding (CHEM) method suffers from severe convergence failures when dealing with power systems containing large phase shifts.

[0035] For example, in the 1888 node power system in Europe, the convergence rate of CHEM decreases when the phase shift angle exceeds a certain degree, and even convergence failure occurs. That is, in the standard system (without a phase shift transformer), HEM can achieve a 100% convergence rate when the phase shift angle is 0 degrees, but once a phase shift transformer is introduced, especially when the phase shift angle is greater than a certain value, the convergence of CHEM drops sharply.

[0036] The root cause of this convergence problem lies in the inadequacy of the traditional phase-shifting transformer model. The presence of the phase shift angle significantly increases the value of the right-hand side of the matrix, which is equivalent to connecting a much larger admittance value at the node than the actual situation, thus deviating from the true power grid operating state of the system and leading to a deterioration in convergence.

[0037] In this embodiment, the target factor that causes the pure embedding method to fail to converge can be the numerical instability caused when the phase shift angle of the phase-shifting transformer exceeds the first threshold.

[0038] like Figure 1 , Figure 2 and Figure 3 As shown, to address the aforementioned technical problems, this embodiment provides a power system modeling method, including: Step S12: Identify the target factors that cause the holomorphic embedding method to fail to converge.

[0039] In this embodiment, a power deviation-based detection method can be used to determine the target factors that cause the fully embedded method to fail to converge; wherein, the target factors include numerical instability caused when the phase shift angle of the phase-shifting transformer exceeds a first threshold.

[0040] It should be noted that in power systems, the update process of node voltage amplitude is usually a smooth and convergent process. However, in the case1888rte system, the voltage amplitude update process of nodes 422 and 982 exhibited violent oscillations. Figure 3 The results show that the voltage amplitude fluctuations at the two nodes are almost identical, exhibiting clear synchronization characteristics. Typically, excessively large voltage amplitude fluctuations can be a signal of non-convergence in power flow calculations and warrant special attention.

[0041] In power systems, physically adjacent nodes often influence each other, but their voltage fluctuations are usually not perfectly synchronized without a direct electrical connection. The highly synchronized voltage fluctuations between node 422 and node 982 indicate some kind of direct electrical connection between these two nodes. According to the system configuration, nodes 422 and 982 are connected via a phase-shifting transformer (PST). A PST controls the direction of power flow by adjusting the phase difference between voltages, thus affecting grid stability. In this power system, the PST is the only device connecting these two nodes. Therefore, the abnormal fluctuations in node voltage amplitude are closely related to the connection between these two nodes via the PST.

[0042] Understandably, the working principle of a phase-shifting transformer is primarily to change the voltage phase difference by adjusting the phase shift angle. This process is highly nonlinear, and the power-voltage equation of a phase-shifting transformer includes complex exponential terms. These nonlinear terms make the model highly sensitive during numerical calculations. If the phase-shifting transformer model fails to accurately reflect these nonlinear characteristics during power flow calculations, convergence oscillations may occur. This instability directly leads to drastic fluctuations in voltage amplitude, ultimately causing the power flow calculation to fail to converge.

[0043] Holo-embedded power flow (HEM) is a non-iterative power flow calculation method that expands the voltage function using complex variables and often relies on the convergence of power series for the solution. In HEM, the voltage update process is adjusted by continuously calculating the power series coefficients. For linear models, the power series coefficients typically decay rapidly, and the convergence process is relatively smooth. However, if strong nonlinear terms (complex exponential terms) such as those of phase-shifting transformers are introduced into the model, the decay of the power series coefficients becomes very slow, or even diverges, manifesting as oscillations in higher-order coefficients. These oscillations become apparent during the voltage update process, manifesting as synchronous fluctuations in the node voltage amplitude.

[0044] The Newton-Raphson (NR) method is a commonly used iterative power flow calculation method. However, when encountering convergence problems, this method often diverges directly, failing to provide data on intermediate oscillations. Therefore, in the NR method, if the calculation fails to converge, the system will directly report an error or fail to obtain a solution, without showing the oscillation process. In contrast, in the fully embedded method, due to its recursive nature, the voltage amplitude update process is more explicit, exposing any instability, especially when the coefficients of higher-order power series cannot decay, making the oscillation phenomenon more pronounced.

[0045] Therefore, inaccurate modeling of the phase-shifting transformer is the root cause of non-convergence in power flow calculations. The nonlinear characteristics of the phase-shifting transformer closely match the sensitivity of the pure embedding method. Incorrect handling of nonlinear terms in phase-shifting transformer modeling often leads to severe oscillations in voltage amplitude, thus preventing power flow calculations from converging. By comparing the performance of the NR method and the pure embedding method, it can be seen that the pure embedding method has a unique advantage in the explicit representation of oscillation phenomena, which provides strong evidence for identifying potential problems in phase-shifting transformer modeling in this implementation method.

[0046] Therefore, it is understandable that the abnormal voltage amplitude fluctuations between nodes 422 and 982, and the characteristic of these two nodes being connected via a phase-shifting transformer, indicate that phase-shifting transformer modeling may be the cause of non-convergence in the power flow calculation. The interaction between the nonlinear terms in the phase-shifting transformer model and the convergence requirements of the fully embedded method further exacerbates the voltage amplitude fluctuations, leading to computational instability.

[0047] Step S14: Based on the target factors, the phase-shifting transformer carried in the target power system is split into a combination of an ideal phase-shifting transformer and a normal transmission line, and a new node is introduced as an AD node to construct a power model; wherein, the AD node is a power node with injected power that is always zero.

[0048] like Figure 2 As shown, in this embodiment, a new node m can first be introduced between the original connected node i of the phase-shifting transformer and the target node n. The function of node m is as a virtual node to isolate the nonlinear effects of the phase-shifting transformer, providing a simplified power system model for subsequent calculations. The introduction of the new node m not only helps to separate the function of the phase-shifting transformer but also effectively reduces unstable factors in the calculation.

[0049] Then, the phase-shifting transformer that originally connected node i and node n can be split into two devices: an ideal phase-shifting transformer and a conventional transmission line. Specifically, the ideal phase-shifting transformer can connect node i and the newly added node m, specifically describing the voltage transformation relationship. In this model, the ideal phase-shifting transformer can be used to adjust the voltage phase difference without involving the line's resistance and reactance. The conventional transmission line can connect the newly added node m and the target node n, retaining the resistance, reactance, and susceptance parameters of the original line where the phase-shifting transformer was located, ensuring that the original system's electrical characteristics are preserved.

[0050] In this embodiment, the newly added node can be defined as an AD node (Augmented Device Node). Specifically, the active and reactive power of this node are both zero, and the voltage amplitude and voltage phase angle of this node are variables to be determined. It should be noted that this newly added node can be classified as a PQ node, but unlike a conventional PQ node, the power of the AD node is always zero, and its purpose is to provide a voltage reference point for the system without participating in actual power flow.

[0051] In this embodiment, it is understood that the introduction of the AD node does not affect the power balance of the entire power system; it serves only as a virtual node to supplement the nonlinear effects of the phase-shifting transformer, thereby helping the power system to achieve better stable convergence during power flow calculations. In this way, numerical instability that may have been caused by the nonlinear characteristics of the phase-shifting transformer in the original system is effectively isolated, avoiding excessive voltage amplitude fluctuations and convergence failures.

[0052] Step S16: Reconstruct the power flow model based on the power model to obtain the reconstructed power flow model; wherein, the reconstructed power flow model includes the complex power balance equations of PQ nodes and AD nodes, the active power balance equations and voltage amplitude constraint equations of PV nodes, the voltage amplitude constraint equations of the slack nodes, and the phase-shift constraint equations of the ideal phase-shifting transformer.

[0053] In this embodiment, the injected active and reactive power of the PQ node are known quantities, while the voltage amplitude and voltage phase angle are unknown quantities. The PQ node can be represented as a load node, which can be a residential load.

[0054] In this embodiment, the PV node can be a voltage control node, which can be used to characterize a power supply with automatic voltage regulation capability. The injected active power and voltage amplitude of the PV node can be known quantities, while its injected reactive power and voltage phase angle are unknown quantities.

[0055] In this embodiment, the voltage amplitude and voltage phase angle of the balancing node are known quantities, while the injected active power and reactive power are unknown quantities.

[0056] In this embodiment, the AD node, as a newly added virtual node, functions to ensure that the nonlinear effects of the phase-shifting transformer are correctly reflected. At this node, the injected power is set to zero, and the voltage amplitude and phase angle are the variables to be solved. This node, combined with the complex power balance equations of other nodes, ensures the balance of the system current flow, avoiding the instability caused by the phase-shifting transformer.

[0057] This embodiment successfully addresses the non-convergence problem that may occur in the classical fully embedded method for power flow calculation by introducing a combination of an ideal phase-shifting transformer and a conventional transmission line, and adding an AD node. By splitting the phase-shifting transformer and introducing it as a node with zero power in the AD node, the stability and convergence of the power flow calculation are ensured. Based on this, the reconstructed power flow model includes multiple constraints, such as the complex power balance equations of the PQ and AD nodes, the active power balance equations of the PV nodes, and voltage amplitude constraint equations, further improving the calculation accuracy and reliability. This method can effectively handle the complex situations in large-scale power systems, significantly improve the convergence of power flow calculation, and provide strong support for the stable operation of power systems.

[0058] In some implementations, the step of determining the target factor that causes the holomorphic embedding method to fail to converge includes: Step S122: Use a power deviation-based detection method to determine the target factors that cause the fully embedded method to fail to converge; wherein, the target factors include numerical instability caused when the phase shift angle of the phase-shifting transformer exceeds a first threshold.

[0059] In this embodiment, the first threshold can be represented as a critical point discovered during simulation using a power deviation detection method. This threshold indicates that when the phase shift angle of the phase-shifting transformer exceeds this value, the power flow calculation of the power system will exhibit numerical instability. Specifically, after the phase shift angle exceeds the first threshold, the power deviation at each node in the power system will suddenly jump from a normal level to an abnormal level. This abrupt change indicates that the power flow of the power system begins to fluctuate significantly and cannot maintain a normal stable state. Simultaneously, the voltage amplitude changes exhibit significant synchronous oscillations, meaning that the voltage amplitude fluctuations at multiple nodes in the system become synchronized and their amplitudes increase significantly. This indicates that the system's stability is compromised, the normal voltage distribution is disrupted, and the fully embedded method (HEM) cannot converge.

[0060] Understandably, the first threshold is a trend inflection point identified during simulation based on the actual system performance. By analyzing the changing patterns of power deviation and voltage fluctuations in the system, it can be found that when the phase shift angle exceeds a certain specific value, the computational stability of the system undergoes a significant change; this specific value is the first threshold.

[0061] It is also understandable that the specific value of the first threshold is not fixed, but varies with the characteristics of different power systems (e.g., system configuration, number of nodes, phase-shifting transformer parameters, etc.).

[0062] This implementation method effectively identifies the target factors causing non-convergence of the fully embedded method by introducing a power deviation-based detection method, particularly the numerical instability caused when the phase shift angle of the phase-shifting transformer exceeds a threshold. This detection method allows for real-time monitoring of power deviation changes during power flow calculations, thereby accurately pinpointing the key factors causing non-convergence. Especially in complex power systems, it can promptly detect abnormal voltage amplitude fluctuations caused by the nonlinear characteristics of the phase-shifting transformer, providing a clear basis for subsequent optimization adjustments. This method improves the stability and accuracy of power flow calculations, ensures the reliability of the fully embedded method in practical applications, and provides more effective technical support for the safety analysis and dispatch of large-scale power grids.

[0063] In some implementations, the step of using a power bias-based detection method to determine the target factors causing the fully pure embedding method to fail to converge includes: Step S1222: Perform power flow calculation using the fully embedded method on the target power system, record the voltage update value of each node during the power series recursion process, and calculate the voltage oscillation amplitude.

[0064] In this embodiment, the fully embedded method can be applied to the target power system for power flow calculation. The fully embedded method updates the voltage of each node through a power-series recursion. In each iteration, the updated value of the node voltage is calculated, and the voltage oscillation amplitude is calculated based on this value. The voltage oscillation amplitude can be represented as the magnitude of the node voltage change during the recursive process, and can manifest as rapid voltage fluctuations or unstable behavior. By recording the updated voltage values ​​and oscillation amplitudes, the stability of the power system can be monitored, providing fundamental data for subsequent analysis. That is, during the power flow calculation, the voltage update status of each node can be recorded, and the voltage values ​​of each node can be compared to determine the oscillation amplitude (i.e., the fluctuation range of the voltage value) during the voltage update process.

[0065] Step S1224: Calculate the power deviation value of each node; wherein the power deviation value is the difference between the actual injected power of the node and the theoretical power flow.

[0066] In this embodiment, the power deviation value of each node can be calculated. The power deviation can be expressed as the difference between the actual injected power and the theoretical power flow at the node. The actual injected power is calculated based on the power system operating conditions and voltage, while the theoretical power flow is calculated based on the system's power flow model. The power deviation reflects the difference in power flow at the nodes and is an important indicator of whether the power flow calculation has converged. That is, for each node, its actual injected power is compared with its theoretical power flow, and the difference between them, i.e., the power deviation, is calculated.

[0067] Step S1226: If a pair of nodes simultaneously meet the following conditions, they are determined to be abnormal nodes connected to the phase-shifting transformer: the power deviation value exceeds the second threshold, the voltage oscillation amplitude exceeds the third threshold, and the fluctuation trend is synchronized.

[0068] In this embodiment, all nodes can be screened, and nodes whose power deviation exceeds a second threshold and whose voltage oscillation amplitude exceeds a third threshold can be identified. If the voltage oscillation trends among these nodes are synchronized, these nodes can be identified as abnormal nodes. It should be noted that it is also possible to further determine whether these abnormal nodes are connected through a phase-shifting transformer. It is understood that a phase-shifting transformer is an important device in a power system, used to regulate the phase difference of voltage; however, the nonlinear characteristics of a phase-shifting transformer can cause abnormalities in power flow and voltage distribution within the system.

[0069] In this embodiment, the second threshold can be represented as a critical value for the power deviation. When the power deviation of a node exceeds this threshold, it indicates that there is abnormal power flow or calculation error at that node. This second threshold can be a value determined based on simulation analysis and historical data, representing the critical point at which the power deviation in the power system abnormally increases. If the power deviation exceeds the second threshold, it indicates abnormal power flow at that node, which may be due to instability caused by the nonlinear effect of the phase-shifting transformer or other external factors.

[0070] In this embodiment, the third threshold can be represented as a critical value for the voltage oscillation amplitude, indicating that when the node voltage oscillation amplitude exceeds this threshold, the stability of the power system is threatened, which may lead to non-convergence of power flow calculations. Similarly, this third threshold can also be a value determined based on simulation analysis and historical data.

[0071] Step S1228: Determine that the paired nodes are connected through a phase-shifting transformer, and take the phase shift angle of the phase-shifting transformer exceeding the first threshold as the target factor.

[0072] In this embodiment, once a pair of nodes is identified as being connected through a phase-shifting transformer, the phase shift angle of the phase-shifting transformer can be detected. If the phase shift angle exceeds a first threshold, it can be determined that the phase-shifting transformer is a key factor causing numerical instability in the power system.

[0073] This implementation method effectively identifies the target factors causing the non-convergence of the fully embedded method by combining power deviation detection and voltage oscillation analysis. Specifically, by recording the voltage update values ​​and oscillation amplitudes of nodes during the power series recursion process, abnormal fluctuations caused by the nonlinear effect of phase-shifting transformers in the power system can be accurately captured. When the power deviation value of a pair of nodes in the power system exceeds the second threshold, and the voltage oscillation amplitude exceeds the third threshold and the fluctuation trend is synchronized, it can be promptly determined that these nodes are abnormal nodes connected by phase-shifting transformers. At this time, it is further confirmed that the phase shift angle of the phase-shifting transformer exceeds the first threshold, which is the root cause of system instability. Through this method, the key factors affecting the convergence of the fully embedded method can be accurately identified, thereby providing a scientific basis for system optimization, improving the stability and accuracy of power flow calculation, and ensuring the reliability and accuracy of the power system under complex conditions.

[0074] In some implementations, the complex power balance equations for the PQ node and the AD node are expressed as follows:

[0075] In the formula, Represented as the total number of nodes in the system. This represents the element in the i-th row and k-th column of the nodal admittance matrix. Let be the complex voltage at node k. For nodes Injection complex power The conjugate value, The complex power flowing from node i to node m The conjugate value of , where im represents the ideal phase-shifting transformer connected between nodes i and m. This represents a set of ideal phase-shifting transformers. Represents the set of PQ nodes. Represents the set of AD nodes. This indicates the calculation of the conjugate value of a complex number. Represented as the complex voltage at node i The conjugate value, ; The active power balance equation and voltage amplitude constraint equation of the PV node are expressed as follows:

[0076] In the formula, Represented as the complex voltage at node i The conjugate value, This represents the element in the i-th row and k-th column of the nodal admittance matrix. Let be the complex voltage at node k. This is represented by the injected active power at PV node i. It is represented as the sum of the active power of all ideal phase-shifting transformers connected to node i. This represents the real-time active power consumed by the ideal phase-shifting transformer itself. This indicates the voltage amplitude specified for a PV node or a ballast node. Represented as a set of PV nodes, This indicates the calculation of the real part of a complex number; This indicates the calculation of the absolute value of a complex number.

[0077] In this implementation, the power flow and voltage changes between nodes are accurately described using complex power balance equations, reducing calculation errors. Through reasonable voltage amplitude constraints and phase-shift constraint equations, the system can converge better under complex conditions, avoiding oscillations and divergences common in traditional methods. This implementation can handle complex equipment such as phase-shifting transformers in large-scale power systems, maintaining good stability and convergence performance, especially when the system contains multiple nonlinear components. Through these improvements, the accuracy and stability of power flow calculations are significantly enhanced, particularly when dealing with large-scale power grids and complex equipment, providing more reliable calculation results for system optimization and real-time scheduling.

[0078] In some implementations, the voltage magnitude constraint equation of the balancing node is expressed as:

[0079] In the formula, Represented as a set of balanced nodes; The phase-shifting constraint equation of the ideal phase-shifting transformer is expressed as:

[0080] In the formula, This represents a set of ideal phase-shifting transformers. This represents an ideal phase-shifting transformer connected between nodes i and m; This represents the voltage magnitude at the new node m. θ represents the transformer turns ratio. shift This indicates the phase shift angle of the transformer.

[0081] In this implementation, on the one hand, by introducing an ideal phase-shifting transformer and voltage amplitude constraints, computational oscillations and instability caused by the nonlinear characteristics of the phase-shifting transformer are avoided, thus improving system stability. On the other hand, by precisely constraining the node voltage amplitude and phase angle, the voltage updates in power flow calculations are ensured to be more accurate, contributing to refined control of system operation. Furthermore, this implementation can effectively handle complex power systems containing nonlinear components such as phase-shifting transformers, especially in systems with multiple nodes and multiple devices, maintaining high computational accuracy and convergence performance.

[0082] This embodiment provides a method for calculating a power flow model of a power system, including: Step S22: For the reconstructed power flow model, embed complex variables and construct a power flow model in a fully embedded form; wherein, the reconstructed power flow model is the reconstructed power flow model provided in the above embodiments; Step S24: Use a preset solution algorithm to solve the fully embedded power flow model until the convergence accuracy is met, and obtain the result parameters of each node in the target power system.

[0083] In this embodiment, a power series recursive algorithm can be used to solve the power flow model of the fully embedded form until the convergence accuracy is met, so as to obtain the result parameters of each node in the target power system.

[0084] In this embodiment, the result parameters may include voltage amplitude, voltage phase angle, power flow, or voltage deviation, etc.

[0085] This implementation significantly improves the accuracy and convergence of power flow calculations by embedding complex variables into the power flow model of the power system and transforming it into a fully embedded form, combined with a pre-defined solution algorithm. First, the reconstructed power flow model, by introducing complex variables, can more accurately describe the voltage and power flow in the power system, especially when dealing with complex equipment such as phase-shifting transformers, avoiding nonlinear errors in traditional methods. Second, by using a fully embedded form, this method fully utilizes complex analysis theory, ensuring the stable convergence of power series during the power flow calculation process, thereby reducing oscillations and instabilities in the calculation. By employing a pre-defined solution algorithm, the system can efficiently solve the fully embedded power flow model and obtain accurate result parameters for each node in the target power system while meeting convergence accuracy requirements. This method effectively improves computational efficiency, ensuring that key parameters such as voltage amplitude and phase angle can be obtained quickly and accurately in large-scale power systems, ensuring stable operation and optimized scheduling of the power system. This implementation, through accurate modeling and efficient solving, solves the convergence problem in traditional power flow calculations, especially when dealing with complex power systems, providing more reliable and faster calculation results, and has significant engineering application value.

[0086] In some implementations, the fully embedded power flow model includes: complex power balance equations for PQ and AD nodes in a fully embedded form, active power balance equations and voltage magnitude constraint equations for PV nodes in a fully embedded form, voltage magnitude constraint equations for slack nodes in a fully embedded form, phase-shift constraint equations for the ideal phase-shifting transformer in a fully embedded form, and fully embedded functions of active and reactive power flowing through the ideal phase-shifting transformer.

[0087] This implementation significantly improves the stability and accuracy of calculations by applying a fully virtually embedded form to the power flow model of the power system. By introducing complex power balance equations, voltage magnitude constraint equations, and phase-shift constraint equations, it ensures accurate modeling of the effects of various nodes (e.g., PQ nodes, PV nodes, slack nodes) and ideal phase-shifting transformers, effectively avoiding numerical instability problems caused by nonlinear devices in traditional calculations. Furthermore, using a fully virtually embedded form to handle the power flowing through the phase-shifting transformer makes the system calculations more accurate, especially when dealing with large-scale and complex power systems, while still maintaining high convergence and computational efficiency. This method improves the reliability of power flow calculations and provides a more accurate basis for system optimization and scheduling.

[0088] In some implementations, the complex power balance equations for the PQ node and AD node in the purely embedded form are expressed as:

[0089] In the formula, ; The active power balance equation and voltage amplitude constraint equation for a purely embedded PV node are expressed as follows:

[0090] In the formula, ; The voltage magnitude constraint equation for a purely embedded balancing node is expressed as:

[0091] The phase-shifting constraint equation for the ideal phase-shifting transformer in its pure embedded form is expressed as:

[0092] In the formula, ; The holomorphic functions of active and reactive power flowing through an ideal phase-shifting transformer are expressed as:

[0093] In the formula, Represented as embedded complex variables, This represents the element in the i-th row and k-th column of the cascaded admittance matrix; This represents the parallel admittance at node i. It is represented as a holomorphic function of the complex voltage at node k. This represents the conjugate of the complex power injected into node i. It is expressed as a holomorphic function of the active power flowing through an ideal phase-shifting transformer. It is expressed as a holomorphic function of the reactive power flowing through an ideal phase-shifting transformer. This is expressed as the parallel admittance at node i. It is represented as a holomorphic function of the complex voltage at node i. Represented as a set of PQ nodes, Represented as a set of AD nodes, It can be represented as a holomorphic function of the complex voltage conjugate at node i. This is represented by the injected active power at PV node i. The imaginary unit, Let i be a holomorphic function of the reactive power of PV node i. Let be the set of PV nodes, where n represents the order of the power series; s is the embedding factor of the complex variable. n This represents the embedding factor of the complex variable of order n.

[0094] In some implementations, the step of solving the fully embedded power flow model using a preset solution algorithm until the convergence accuracy is met, and obtaining the result parameters of each node in the target power system, includes: For the fully embedded power flow model, a corresponding recursive equation is constructed; wherein, the recursive equation establishes a recursive relationship between the parameters to be solved by extracting the relationship between the coefficients of higher-order power series and the coefficients of lower-order power series. For the recursive equation, recursive calculations are performed to obtain the result parameters for each node in the target power system; wherein, the recursive equation includes: The recursive equations for nodes PQ and AD are expressed as follows:

[0095] In the formula, The current recursion order is... For the nth power series coefficients of the voltage at node k, ... These are the coefficients of the nth power series of active power losses in a phase-shifting transformer. The coefficients of the nth power series of reactive power loss in a phase-shifting transformer are given. For node i, the complex voltage V i The conjugate of the reciprocal term, The reciprocal of the voltage at node i The Conjugate of order coefficients, The lower-order recursion sequence number. This means that nodes with AD are positive and nodes without AD are negative. The recurrence equations for PV nodes and equilibrium nodes are expressed as follows:

[0096]

[0097] In the formula, Represented as the Kronecker delta function, Let the coefficients of the nth power series of reactive power at PV node i be represented. ; The recursive equation for a phase-shifting transformer is expressed as:

[0098] In the formula, This represents the nth-order coefficient of the voltage at primary node i of the phase-shifting transformer. This represents the nth-order coefficient of the voltage at the newly added node m of the phase-shifting transformer. ; The unified recursive equation is expressed as:

[0099] In the formula, A is the recursive matrix required for recursive calculation, which remains constant during the recursive calculation process; x[n] is the coefficient of the nth power series to be calculated. This is the right-hand side term of the recursive equation, which is calculated from the coefficients of a power series of order less than n.

[0100] In one specific implementation, a fully pure embedding method based on an improved phase-shifting transformer model is provided. This method first introduces a power deviation detection method to identify the key factors causing the classical fully pure embedding method to fail to converge. Based on this, an improved phase-shifting transformer model and its corresponding fully pure embedding form are constructed, and the calculation steps for the initial solution and higher-order power series are derived.

[0101] Terminology Explanation: Holomorphic Embedding Method (HEM) is a non-iterative algorithm for power flow calculation in power systems.

[0102] Taylor expansion (TE) is a mathematical method that approximates a function using a power series form based on the derivative information at a certain point.

[0103] Power Flow Calculation (PFC) is a numerical calculation process that, based on known node voltages and load conditions in a power system, determines the voltage magnitude and phase angle at each node, as well as the power flow and distribution along lines. This calculation is based on a steady-state power system model and is one of the fundamental tools for power grid planning, dispatching, and analysis.

[0104] The Newton-Raphson algorithm (NR) is an iterative algorithm for solving nonlinear equations.

[0105] Per unit (pu) is a numerical notation method commonly used in power systems and engineering calculations. It represents the relative value of various physical quantities and parameters and is dimensionless.

[0106] Power flow calculation is a fundamental tool for power system planning and operation analysis. Essentially, it involves solving high-dimensional nonlinear equations. Currently, the most commonly used power flow calculation method in industry is the iterative Newton-Raphson (NR) method. However, this method has the following shortcomings: 1) It is sensitive to the selection of initial values; unreasonable initial values ​​may lead to slow convergence or even non-convergence; 2) Each iteration requires regenerating the Jacobian matrix and calculating correction equations, resulting in a slow computation speed for the NR method; 3) When power flow fails to converge, it is impossible to determine whether the flow itself has no solution or whether a solution exists but has not been found. With the expansion of power system scale and the large-scale integration of power electronic equipment, the operation mode of power systems is becoming increasingly complex and variable, and the non-convergence of power flow is becoming more serious, challenging the reliability of power system safety and stability analysis. Therefore, there is an urgent need to propose more reliable power flow calculation methods.

[0107] To address the aforementioned problems of traditional iterative power flow calculation methods, a recursive power flow calculation method, namely the Holomorphic Embedding Method (HEM), is proposed. This method, based on complex analysis theory, embeds complex variables into the power flow equations and constructs the variables to be solved as holomorphic functions. Then, based on the expansion properties of holomorphic functions, the power flow solution is transformed into recursively obtaining the power series coefficients of the holomorphic functions. This method requires no initial values ​​or the formation of a Jacobian matrix. The holomorphic embedding method guarantees convergence to a workable solution when a power flow exists and provides a clear numerical oscillation signal when there is no solution.

[0108] Traditional iterative power flow calculation methods, when encountering convergence issues, often produce voltage results that significantly deviate from the actual operating state of the system, making it difficult to provide effective reference information for subsequent power flow convergence adjustments. As a recursive power flow calculation method, the fully virtuous embedding method overcomes the initial value dependency problem to some extent and has demonstrated excellent convergence performance in numerous existing studies. However, limited by factors such as numerical calculation accuracy, embedding form construction, and system characteristics, the fully virtuous embedding method may still fail to converge in practical applications. For example, in some system examples with a large number of nodes (such as case 1888rte and case 13659pegase), the classic fully virtuous embedding method fails to converge. Currently, to improve the convergence performance of the fully virtuous embedding method, various improvement schemes have been proposed, such as flexible fully virtuous embedding, parameterized path fully virtuous embedding, and restarted fully virtuous embedding. Although these methods improve the convergence of the fully virtuous embedding method, they still have shortcomings in revealing the reasons for the convergence failure of the classic fully virtuous embedding method in certain power system examples. Therefore, further exploring its convergence mechanism and constructing a better form of fully pure embedding remains a key challenge that needs to be overcome in this field.

[0109] Existing holomorphic embedding methods have improved the convergence of holomorphic embedding to some extent, but they still have shortcomings in revealing the reasons for the convergence failure of classical holomorphic embedding in some power system examples. Among the many problems that may affect the convergence of power flow, such as model parameters and injected data, how to accurately locate the key factors and make targeted adjustments remains a weak link in current research.

[0110] Therefore, this implementation scheme takes the failure of the classical fully pure embedding method to converge in some complex power grids as its starting point. First, based on the simulation results of the example, a power deviation detection method is introduced to analyze the key factors that cause the classical fully pure embedding method to fail to converge. Then, the modeling method of the phase-shifting transformer is improved, the recursive equation of the proposed fully pure embedding form and the corresponding solution process are derived, and the convergence performance of the proposed method is compared with that of NR and the classical fully pure embedding. This implementation scheme provides a fully pure embedding method based on the improved modeling of the phase-shifting transformer.

[0111] The technical solution adopted by this implementation plan to solve its technical problems is as follows: 1. Power Deviation Detection Method First, a power deviation detection method is introduced to identify the key factors causing the classical fully embedded method to fail to converge, and further improvement strategies are proposed to enhance the convergence of power flow calculations. The specific steps of the power deviation detection method will be explained in detail in the case 1888rte example. This example system contains 1888 nodes and 2531 lines, with an active load of 59111MW and a reactive load of 2271MVar.

[0112] In the case1888rte system, the power deviation of some nodes (node ​​422 and node 982) is significantly greater than that of other nodes, with a difference of several orders of magnitude, and exhibits an approximately symmetrical distribution along the horizontal axis.

[0113] Further analysis revealed that nodes with larger power deviations also exhibited abnormal behavior with large fluctuations in their voltage amplitude update process. Figure 3 The voltage amplitude updates at nodes 422 and 982 are shown, both exhibiting severe oscillations with almost identical amplitudes, indicating an anomaly between these two nodes. Further observation reveals that nodes 422 and 982 are connected via a phase-shifting transformer, suggesting that phase-shifting transformer modeling may be the cause of the power flow calculation non-convergence. It should be noted that this power deviation and voltage fluctuation anomaly is observable using the pure embedding method but not using the NR rule.

[0114] 2. Construct a power flow calculation model including an improved phase-shifting transformer. Next, the power flow calculation model with improved phase-shifting transformer is introduced.

[0115] like Figure 2 As shown, the phase-shifting transformer connecting node i and node n is replaced by a model consisting of an ideal phase-shifting transformer proposed in this embodiment and a conventional transmission line. This model includes an ideal phase-shifting transformer from node i to the newly introduced node m, and a conventional transmission line connecting node m and node n. The ideal phase-shifting transformer model is used to describe the voltage and power transmission relationships between node i and node m. Figure 2 S im P represents the complex power flowing from node i to node m. im Q represents the active power flowing from node i to node m. im This represents the reactive power flowing from node i to node m.

[0116] Based on this, the power flow equations after introducing an ideal phase-shifting transformer are constructed below.

[0117] Suppose an N-node system containing PQ nodes, PV nodes, and slack nodes. The specific meanings of the three types of nodes are as follows: PQ node: P and Q are given, and the voltage phase angle and magnitude are variables to be determined; PV node: P and voltage magnitude are given, and Q and voltage phase angle are variables to be determined; Slack node: There is only one slack node, and its voltage magnitude and phase angle are given, while P and Q are variables to be determined.

[0118] The corresponding set of nodes is denoted as follows: , and If a phase-shifting transformer exists in the system, according to Figure 2 The modeling approach introduces a new node m. This newly introduced node is treated as a PQ node with zero injected power and voltage as the variable to be determined. The set of newly added nodes to which it belongs is denoted as m. The total number of newly added nodes is denoted as NAD. After a new node is introduced, the system node set is updated to... The total number of system nodes will be expanded accordingly. ( ).

[0119] After introducing the new node, the power flow model is adjusted as shown in equations (1)-(4). The power flow equations of the original PQ node and PV node need to be modified accordingly. The complex power Sim injected and discharged by the ideal phase-shifting transformer should be considered in the construction of the power flow model, as shown in equations (1) and (2). The voltage amplitude constraint of the slack node remains unchanged, corresponding to equation (3).

[0120] (1) (2) (3) In the formula: This represents the element in the i-th row and k-th column of the nodal admittance matrix; The complex voltage at node k ( ,in Voltage amplitude, (Voltage phase angle); Inject complex power into node i ( j is the imaginary part unit. To inject active power; (for injecting reactive power). Specify the voltage amplitude for PV nodes or ballast nodes; This indicates the calculation of the conjugate value of a complex number; This indicates the calculation of the real part of a complex number; This indicates the calculation of the absolute value of a complex number. .

[0121] N represents the number of nodes, and the asterisk in the upper right corner represents the deconjugate value; S im Let represent the complex power flowing from node i to node m. This is contained in equation (1). The value of the sign is determined by the set to which the node belongs: if the node is a newly added node, then... If positive, use a positive sign; otherwise, use a negative sign.

[0122] An ideal phase-shifting transformer constrains the voltage amplitude and phase relationship at its two ends, and the corresponding constraint equation is shown in (4): (4) In the formula: V represents the set of ideal phase-shifting transformers; im represents the ideal phase-shifting transformer connected between nodes i and m; m τ represents the voltage amplitude at the new node m; τ represents the transformer turns ratio; θ shift This indicates the phase shift angle of the transformer.

[0123] In the revised power flow equations, the changes in the number of variables and equations are as follows: If NAD ideal phase-shifting transformers are added to the system, then 4NAD equations are added accordingly (of which, equations (1) and (4) each increase by 2NAD). Simultaneously, 2NAD complex power variables are introduced. (i.e., NAD each for active and reactive power), and 2NAD node voltage variables (i.e., NAD each for the real and imaginary parts of the voltage). Therefore, the number of new equations is equal to the number of new variables, thus ensuring the solvability of the modified power flow equations in terms of dimensionality.

[0124] 3. Construct a fully embedded form with an improved phase-shifting transformer. The fully embedded form of the power flow equations for PQ nodes, PV nodes, slack nodes, and ideal phase-shifting transformers is constructed as shown in equations (5)-(8).

[0125] (5) (6) (7) (8) In the formula: Represents embedded complex variables; This represents the element in the i-th row and k-th column of the cascaded admittance matrix; This represents the parallel admittance at node i. It is important to note that, to ensure the holomorphic properties of the embedding form, The corresponding embedding format is Instead This is because the former follows the Cauchy-Riemann (Cauchy-Riemann) model. The Riemann condition is not followed by the latter. and The holomorphic functions representing the active and reactive power flowing through an ideal phase-shifting transformer can be expressed as a power series summation as shown in equation (9).

[0126] (9) In the formula: n represents the order of the power series; s is the embedding factor of the complex variable, s n represents the embedding factor of the nth-order complex variable. im represents the phase-shifting transformer number connecting node i and node m.

[0127] 4. Derive the initial power series (initial solution) The constructed holomorphic embedding form retains the advantage of the classical holomorphic embedding form having a definite initial solution, and the initial solutions of each variable are shown in equation (10).

[0128] (10) 5. Derive higher-order power series (recursive equations) By extracting coefficients of the same power series, factors are embedded on both sides of the equation. The coefficients of the nth power series are equal. By shifting the coefficients of the higher power series to the left and the coefficients of the lower power series to the right, the following recursive equations can be obtained as shown in (11)-(14).

[0129] (11) (12) (13) (14) In the formula: The complex voltage at node i The reciprocal of .

[0130] The relationship of the recursive equations shown in (11)-(14) is unified so that all coefficients of the nth power series are on the left side of the equation, and all coefficients of power series of order lower than n are on the right side of the equation. The unified recursive equation can be expressed as (15): (15) In the formula, A is the recursive matrix required for recursive calculation, which remains constant during the recursive calculation process; x[n] is the coefficient of the nth power series to be calculated; This is the right-hand side term of the recursive equation, which is calculated from the coefficients of a power series of order less than n.

[0131] This implementation scheme verifies the effectiveness and accuracy of the proposed fully embedded method based on improved phase-shifting transformer modeling, and compares it with the CHEM-1, CHEM-2, and NR methods. All numerical examples were performed on a laptop equipped with an Intel Core i9 processor (2.60GHz) and 32GB of RAM. The programming was implemented using MATPOWER 4.1, and the convergence accuracy for power flow calculations was set to 1×10⁻⁶. 6 pu, with a per-unit power reference value of 100MW. CHEM-1 and CHEM-2 are two classic fully pure embedding methods. The table below lists the embedding method, admittance matrix processing, and other information for each method.

[0132] Figure 4 This table compares two classic holomorphic embedding forms. It should be noted that for classic holomorphic embedding form 1, the admittance matrix is ​​split into a series part and a parallel part, while for classic holomorphic embedding form 2, the admittance matrix is ​​split into a symmetric part and an asymmetric part.

[0133] First, the accuracy was verified by modifying the 9-node system. In the 9-node power system, one of the original transmission lines connecting nodes 7 and 8 was replaced with a phase-shifting transformer with the following parameters: resistance r = 0.00004 pu, reactance x = 0.0004 pu, turns ratio τ = 1, and the phase shift angle θshift changed from 1° to 5°.

[0134] As the phase shift angle θshift increases, the convergence performance of the two classical holomorphic embedding methods (CHEM-1 and CHEM-2) significantly decreases, and both fail to converge at θshift = 5°. In contrast, the method proposed in this embodiment exhibits consistent and stable convergence characteristics under different phase shift angle scenarios. The proposed method converges quickly within approximately 7 recursive calculations, verifying its superior and stable convergence performance. With increasing phase shift angle, the maximum value of the right-hand side of the admittance matrix increases significantly in the CHEM-1 and CHEM-2 methods, while the maximum value of the right-hand side of the admittance matrix remains unchanged in the method proposed in this embodiment.

[0135] Comprehensive analysis suggests that the two classic fully embedded methods, CHEM-1 and CHEM-2, have shortcomings when dealing with phase-shifting transformers, leading to a deterioration in the convergence performance of power flow calculations. This is because the presence of the phase shift angle significantly increases the value of the right-hand side of the matrix, equivalent to connecting admittance values ​​at nodes that are much larger than the actual values, thus deviating from the true operating state of the power grid and consequently causing convergence degradation. In contrast, the method proposed in this implementation scheme effectively avoids the above problems by introducing intermediate nodes and adopting an ideal phase-shifting transformer model, significantly improving the stability and convergence performance of the calculation.

[0136] Figure 5 The table presents a comparison of the coefficients V7[n] and their magnitudes |V7[n]| of the first six power series at node 7 for three methods (CHEM-1, CHEM-2, and the method proposed in this embodiment) when the phase shift angle θshift = 2°. As can be seen from the table, the magnitudes of the power series coefficients in the proposed method decrease rapidly with increasing power series order n, and the coefficients of the 6th power series are already less than 10. 4 pu. In comparison, the coefficients of power series of the CHEM-1 and CHEM-2 methods are larger than those of the proposed method, especially at orders n=1 and n=2, where the magnitudes are significantly larger than those of the proposed method in this implementation scheme. The proposed method achieves faster convergence.

[0137] Furthermore, the accuracy of the proposed method was compared with that of the NR method. Under different phase shift angles, the comparison results of the voltage amplitudes of each node calculated by the proposed method and the NR method in this implementation scheme are analyzed as follows: 1) When θshift=1°, 2°, and 3°, the results of the proposed method and the NR method on the voltage amplitudes of each node are consistent and almost overlap, indicating the accuracy of the proposed algorithm; 2) When the phase shift angle increases to 3.2°, the NR method shows obvious anomalies in the node voltage (the voltage of nodes 4-9 is less than 0.5 pu), indicating that the NR algorithm can no longer converge, while the proposed method can still converge, indicating that the proposed algorithm has better convergence performance.

[0138] To verify the effectiveness of the proposed method in large-scale systems, the convergence accuracy was set to 1×10⁻⁶. The system uses 6 PUs, with a maximum recursion count of 30. This implementation scheme selects 10 typical cases listed in Table 3 for testing. These cases all include a certain number of phase-shifting transformers. The table lists the number of phase-shifting transformers in each system, the phase-shifting angle of each transformer, the phase-shifting angle of each transformer, and the absolute value of the maximum phase-shifting angle. It can be seen that in cases 1888rte, 1951rte, 6468rte, and ACTIVSg10k, the absolute value of the maximum phase-shifting angle is relatively large (all exceeding 9°), while in other cases, the maximum phase-shifting angle does not exceed 4°.

[0139] Figure 6 A comparison table is presented showing the number of phase-shifting transformers and the magnitude of the phase-shifting angle in several example systems. Figure 6 As can be seen, the maximum power deviation corresponding to the maximum number of recursions was tested in 10 test systems using three methods (CHEM-1, CHEM-2, and the method proposed in this implementation). The results show that the proposed method consistently outperforms or is comparable to CHEM-1 and CHEM-2 in terms of power deviation, demonstrating superior convergence performance. Particularly in systems with large phase shift angles, such as the test systems case1888rte, case1951rte, case6468rte, and case_CTIVSg10k, the maximum power deviation of the proposed algorithm is significantly smaller than that of CHEM-1 and CHEM-2, demonstrating the significant advantages of the proposed method and further proving its effectiveness and reliability in complex power systems.

[0140] Furthermore, this implementation scheme compares the proposed method with the NR method in terms of convergence and computational efficiency. The NR method uses both flat start and non-flat start (using the results of two iterations of the fast decomposition method as initial values) for power flow calculation. Table 4 shows the computation time and the number of iterations (recursions) required for convergence for each example.

[0141] In the case_ACTIVSg10k system, the NR method failed to converge under flat-start conditions, while the method proposed in this implementation scheme converged successfully, fully demonstrating its superior convergence characteristics. Furthermore, in terms of computation time, the proposed method also exhibits a significant speed advantage over the non-flat-start NR method in both the case1354pegase and case_ACTIVSg10k systems. Comprehensive analysis further validates the application potential and engineering value of the proposed method in power flow analysis of complex power systems.

[0142] Figure 7A table comparing the computational efficiency and convergence of the proposed method with the NR method is presented.

[0143] This implementation plan takes the failure of the classical fully pure embedding method to converge in some complex power grids as a starting point. Based on the simulation results, a power deviation detection method is introduced, and the key factors that cause the classical fully pure embedding method to fail to converge are analyzed.

[0144] This implementation plan proposes an improved power flow calculation model for phase-shifting transformers, which adopts a modeling method combining ideal phase-shifting transformers with conventional transmission lines, replacing the traditional phase-shifting transformer model.

[0145] This implementation scheme proposes that the solution of the nonlinear equation system is non-iterative, has low dependence on initial values, can guarantee convergence when there is a solution, and can issue a signal when there is no solution and oscillation occurs.

[0146] This implementation scheme derives the recursive equation and corresponding solution process of the proposed holomorphic embedding form. The convergence performance of the proposed method is compared with that of NR and classical holomorphic embedding. The proposed method has better convergence performance in power flow calculation than the classical holomorphic embedding method, demonstrating good adaptability and engineering application potential.

[0147] This implementation scheme has higher reliability and lower dependence on the selection of initial values. It can issue a signal when the equation has no solution. This implementation scheme introduces a power deviation detection method to identify the key factors that cause the classical pure embedding method to fail to converge. This implementation scheme adopts a modeling method that combines an ideal phase-shifting transformer with a conventional transmission line, replacing the traditional phase-shifting transformer model. The pure embedding form constructed based on this method exhibits good convergence performance.

[0148] According to an embodiment of the present invention, an electronic device is provided. The electronic device in this embodiment may include one or more of the following components: a processor, a network interface, memory, non-volatile memory, and one or more application programs, wherein the one or more application programs may be stored in the non-volatile memory and configured to be executed by one or more processors, and the one or more programs are configured to perform the methods described in the foregoing method embodiments.

[0149] According to embodiments of the present invention, a computer-readable storage medium is provided having a computer program stored thereon, which, when executed by a computer, causes the computer to perform the method described in any of the above embodiments.

[0150] According to embodiments of the present invention, a computer program product comprising instructions is also provided, which, when executed by a computer, cause the computer to perform a method in any of the above embodiments.

[0151] It should be noted that the terms "first," "second," etc., in the specification, claims, and accompanying drawings of this application are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments of this application described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" and "having," and any variations thereof, are intended to cover non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.

[0152] Optionally, specific examples in this embodiment can refer to the examples described in the above embodiments, and will not be repeated here.

[0153] The sequence numbers of the embodiments in this application are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.

[0154] In the above embodiments of this application, the descriptions of each embodiment have different focuses. For parts not described in detail in a certain embodiment, please refer to the relevant descriptions of other embodiments.

[0155] The above description is only a preferred embodiment of this application. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of this application, and these improvements and modifications should also be considered within the scope of protection of this application.

Claims

1. A modeling method for a power system, characterized in that, include: Identify the target factors that cause the holomorphic embedding method to fail to converge; Based on the aforementioned target factors, the phase-shifting transformers carried in the target power system are decomposed into a combination of ideal phase-shifting transformers and ordinary transmission lines, and new nodes are introduced as AD nodes to construct a power model; wherein, the AD node is a power node with injected power that is always zero. Based on the power model, the power flow model is reconstructed to obtain the reconstructed power flow model; wherein, the reconstructed power flow model includes the complex power balance equations of PQ nodes and AD nodes, the active power balance equations and voltage amplitude constraint equations of PV nodes, the voltage amplitude constraint equations of the slack nodes, and the phase-shift constraint equations of the ideal phase-shifting transformer. The complex power balance equations for the PQ node and the AD node are expressed as follows: In the formula, Represented as the total number of nodes in the system. This represents the element in the i-th row and k-th column of the nodal admittance matrix. Let be the complex voltage at node k. Inject complex power into node i The conjugate value, The complex power flowing from node i to node m The conjugate value of , where im represents the ideal phase-shifting transformer connected between nodes i and m. This represents a set of ideal phase-shifting transformers. Represents the set of PQ nodes. Represents the set of AD nodes. This indicates the calculation of the conjugate value of a complex number. Represented as the complex voltage at node i The conjugate value, ; The active power balance equation and voltage amplitude constraint equation of the PV node are expressed as follows: In the formula, This is represented by the injected active power at PV node i. It is represented as the sum of the active power of all ideal phase-shifting transformers connected to node i. This represents the real-time active power consumed by the ideal phase-shifting transformer itself. This indicates the voltage amplitude specified for a PV node or a ballast node. Represented as a set of PV nodes, This indicates the calculation of the real part of a complex number; This indicates the calculation of the absolute value of a complex number; The voltage magnitude constraint equation for the balancing node is expressed as: In the formula, Represented as a set of balanced nodes; The phase-shifting constraint equation of the ideal phase-shifting transformer is expressed as: In the formula, This represents the voltage magnitude at the new node m. θ represents the transformer turns ratio. shift This indicates the phase shift angle of the transformer.

2. The modeling method according to claim 1, characterized in that, The step of determining the target factors that cause the holomorphic embedding method to fail to converge includes: A power deviation-based detection method is used to identify the target factors that cause the fully embedded method to fail to converge; wherein, the target factors include numerical instability caused when the phase shift angle of the phase-shifting transformer exceeds a first threshold.

3. The modeling method according to claim 2, characterized in that, The step of determining the target factors causing the fully pure embedding method to fail to converge using a power deviation-based detection method includes: Perform power flow calculations using the fully embedded method on the target power system, record the voltage update value of each node during the power series recursion process, and calculate the voltage oscillation amplitude. Calculate the power deviation value for each node; wherein the power deviation value is the difference between the actual injected power and the theoretical power flow power of the node; If a pair of nodes simultaneously meets the following conditions, it is determined to be an abnormal node connected to the phase-shifting transformer: the power deviation value exceeds the second threshold, the voltage oscillation amplitude exceeds the third threshold, and the fluctuation trend is synchronized. The pair of nodes is determined to be connected by a phase-shifting transformer, and the phase shift angle of the phase-shifting transformer exceeding a first threshold is taken as the target factor.

4. A method for calculating a power flow model of a power system, characterized in that, include: For the reconstructed power flow model, complex variables are embedded and a power flow model in a fully embedded form is constructed; wherein, the reconstructed power flow model is the reconstructed power flow model as described in any one of claims 1-3; The fully embedded power flow model is solved using a preset solution algorithm until the convergence accuracy is met, and the result parameters of each node in the target power system are obtained.

5. The calculation method according to claim 4, characterized in that, The fully embedded power flow model includes: complex power balance equations for PQ and AD nodes in the fully embedded form, active power balance equations and voltage amplitude constraint equations for PV nodes in the fully embedded form, voltage amplitude constraint equations for slack nodes in the fully embedded form, phase-shift constraint equations for the ideal phase-shifting transformer in the fully embedded form, and fully embedded functions of active and reactive power flowing through the ideal phase-shifting transformer.

6. The calculation method according to claim 5, characterized in that, The complex power balance equations for the PQ node and AD node in the pure embedded form are expressed as follows: In the official ; The active power balance equation and voltage amplitude constraint equation for a purely embedded PV node are expressed as follows: In the official ; The voltage magnitude constraint equation for a purely embedded balancing node is expressed as: The phase-shifting constraint equation for the ideal phase-shifting transformer in its pure embedded form is expressed as: In the official ; The holomorphic functions of active and reactive power flowing through an ideal phase-shifting transformer are expressed as: In the formula, Represented as embedded complex variables, This represents the element in the i-th row and k-th column of the cascaded admittance matrix; This represents the parallel admittance at node i. It is represented as a holomorphic function of the complex voltage at node k. It is expressed as a holomorphic function of the active power flowing through an ideal phase-shifting transformer. It is expressed as a holomorphic function of the reactive power flowing through an ideal phase-shifting transformer. It is represented as the parallel admittance at node i. It is represented as a holomorphic function of the complex voltage at node i. It can be represented as a holomorphic function of the complex voltage conjugate at node i. The imaginary unit, Let be the holomorphic function of reactive power at PV node i, where n represents the order of the power series; s is the complex variable embedding factor, s n Denotes the embedding factor of the complex variable of order n. These are the nth-order power series coefficients of the active power of the phase-shifting transformer. The coefficients of the nth power series of reactive power of the phase-shifting transformer are given.

7. The calculation method according to claim 6, characterized in that, The step of solving the fully embedded power flow model using a preset solution algorithm until the convergence accuracy is met, and obtaining the result parameters of each node in the target power system, includes: For the fully embedded power flow model, a corresponding recursive equation is constructed; wherein, the recursive equation establishes a recursive relationship between the parameters to be solved by extracting the relationship between the coefficients of higher-order power series and the coefficients of lower-order power series. For the recursive equation, recursive calculations are performed to obtain the result parameters for each node in the target power system; wherein, the recursive equation includes: The recursive equations for nodes PQ and AD are expressed as follows: In the formula, The coefficients of the nth power series of the voltage at node k are... For node i, the complex voltage V i The conjugate of the reciprocal term, The reciprocal of the voltage at node i The Conjugate of order coefficients, The lower-order recursion sequence number. This means that nodes with AD are positive and nodes without AD are negative. The recurrence equations for PV nodes and equilibrium nodes are expressed as follows: In the formula, Represented as the Kronecker delta function, Let the coefficients of the nth power series of reactive power at PV node i be represented. ; The recursive equation for a phase-shifting transformer is expressed as: In the formula, The coefficients of the nth power series of the voltage at primary node i of the phase-shifting transformer are expressed as follows: This represents the nth power series coefficient of the voltage at the newly added node m of the phase-shifting transformer. ; The unified recursive equation is expressed as: In the formula, A is the recursive matrix required for recursive calculation, which remains constant during the recursive calculation process; x[n] is the coefficient of the nth power series to be calculated. This is the right-hand side term of the recursive equation, which is calculated from the coefficients of a power series of order less than n.

8. An electronic device, characterized in that, include: A memory, and one or more processors communicatively connected to the memory; The memory stores instructions that can be executed by the one or more processors to cause the one or more processors to implement the method as described in any one of claims 1 to 7.

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