Power system transient stability analysis method based on spherical geometry

Abstract

The invention relates to a power system transient stability analysis method based on spherical geometry, and the method comprises the following steps: 1, carrying out the dimension reduction processing of a high-dimensional multi-machine power system, and obtaining the projection of a global manifold in a unit projection sphere of a three-dimensional phase space through a contraction projection transformation method; and step 2, two groups of spherical coordinates SAIs which are symmetrical about the center of the sphere can be obtained by constructing and observing a geometric distribution situation graph of the SAIs of the high-dimensional multi-machine power system on the sphere, and a spherical triangle area formed by three SAIs on the same hemisphere is used for defining a spherical surface curved surface triangle area SAST as an SAIs evaluation parameter. The problem that the stability degree cannot be effectively distinguished due to the fact that the indexes are the same when different faults exist in an existing power system transient stability analysis method is solved, simulation results show that evaluation parameters provided by the method have the advantage of being higher in accuracy, and the transient stability degree of the system can be effectively judged.

Description

Technical Field

[0001] The present invention relates to the field of power system stability, and particularly to a transient stability analysis method for power systems based on spherical geometry. Background Art

[0002] With the continuous growth of power load and the access of large-scale intermittent energy sources such as wind energy and photovoltaic power generation, modern power systems exhibit significant uncertainty and low inertia characteristics, resulting in significant changes in the system stability mechanism. Such changes have triggered a series of new stability problems, posing challenges to traditional stability analysis methods. Therefore, there is an urgent need to develop more effective stability analysis theories and mathematical methods.

[0003] When studying the transient stability of a power system, it is essential to comprehensively understand the dynamic characteristics of the system. Studying the solution trajectories of the equations can obtain the topological space of the system. When analyzing the global topological structure of the system, it is necessary to study the boundary of the complete stability region of the system. When studying the stability region of a certain system, it is necessary to find the set composed of the solutions that satisfy the operating equations of the system. All trajectories starting from within this set will converge to a specified state space. At this time, when characterizing the boundary of this specified state space, the stability of the system can be evaluated, that is, the maximum range deviating from the stable operating state can be determined.

[0004] Therefore, when studying the dynamic characteristics of the system, it is necessary to start with accurately characterizing the boundary of the stability region (Boundary of the Stability Region, BSR). Currently, the methods for characterizing BSR mainly include the energy function method, the manifold estimation method, numerical methods, and topological and dynamic characteristic analysis methods, etc. The first two analysis methods are more commonly used. The core of the energy function method is to approximate the stability region of the system using the convex boundary of the Lyapunov function. This method is simple and direct, and can avoid complex numerical integration calculations of the trajectories. However, the construction methods of energy functions are diverse and have different forms, lacking generality. The manifold estimation method combines the advantages of the energy function method and develops a calculation method for the stable manifold, more efficiently realizing the characterization of BSR. At the beginning of the 19th century, Hilbert studied two-dimensional topological manifolds and first defined the plane through neighborhoods as an ideal tool for describing manifolds, thus laying the foundation for manifold geometry. In 1993, Guckenheimer et al. introduced the concept of geodesic level sets to locally approximate the stable manifold of saddle points, initially realizing the visualization of the manifold geometric structure. Since then, a large number of research results on manifold calculation methods have emerged, such as the trajectory arc length method, the adaptive orbit extension method, the simplex method, and the geometric perturbation method, etc., providing a theoretical basis and research ideas for the BSR characterization of dynamic systems.

[0005] Numerous subsequent studies have shown that the size of the stability region reflects the ability of the system to maintain stability. Based on this, transient stability studies such as the critical clearing angle and stability margin analysis of power systems have been realized. This indicates that if a comprehensive topological analysis of the manifold can be completed and the dynamic characteristics of the system at infinity can be provided, it will be of great significance for the establishment and application of the complete BSR theory.

[0006] The topological space of a manifold is used to describe the properties of a space with local Euclidean Geometry (EG). For each manifold, there exists a neighborhood that can be homeomorphic to an open set in Euclidean space, ensuring that the manifold has a smooth structure. By studying the topological structure, the metric structure and natural coordinate system on the manifold can be obtained. When describing the topological structure of a manifold, spherical geometry (SG) is used as a specific type of manifold. Although EG is very practical on a plane, SG is more suitable when dealing with curved surfaces, the Earth's surface, and other positively curved spaces. The key difference between EG and SG can be regarded as the difference between a plane and a sphere. The sphere has positive curvature while the plane has zero curvature, which means that the straight lines on the sphere are great circles, while on the plane they are straight lines, making the geodesics on the sphere more natural and in line with intuitive perception. At the same time, the calculation of the area of a fixed region on the sphere is also closer to the actual geographical area and more intuitive. These characteristics have made the research in many fields focus on SG: in addition to being frequently used in astronomy, geophysics, and satellite communication fields, the research on VR panoramic images, surround sound, and robot joints in spherical space has gradually attracted the attention of scholars in various fields.

[0007] Currently, the main methods for power system transient stability analysis include time-domain simulation method, energy function method, and extended equal-area method, etc. Among them, the time-domain simulation method is more commonly used. It can intuitively display the electromechanical transient process of the system, provide the time response of each variable in the system, and has wide model adaptability. The energy function method mainly includes the UEP method, PEBS method, and BCU method. Its core is to analyze the stability of the first swing of the system power angle based on the energy function. The core of the extended equal-area method is to divide multiple generators in the system into critical units and non-critical units, and then equivalent them to a two-machine system, and use the "equal-area" criterion to judge the transient stability of the system and conduct quantitative analysis.

[0008] However, the SAIs evaluation parameter Q proposed based on EG in the current technical literature "Ma Meiling, Wang Jie, Li Penghan, etc. Global Manifold Analysis on the Power Angle Stability Region and Boundary of Power System [J]. Proceedings of the CSEE, 2020, 40(18): 5865-5875." 2There are defects, that is, different fault conditions will result in the same indicators, making it impossible to effectively distinguish the severity of their stability levels. Summary of the Invention

[0009] Regarding the problem that the SAIs evaluation parameter Q2 proposed based on EG in the literature has the same indicators under different fault conditions, with certain precision defects and thus unable to effectively distinguish the severity of its stability level, a transient stability analysis method for power systems based on spherical geometry is proposed. The present invention will, on the basis of existing transient stability analysis methods, combine manifolds and their characteristics in the SG space to construct an index for the transient stability level of power systems. In order to expand the application scope of power system stability theory and master the topological structure of the stability domain, the present invention studies the distribution characteristics of manifolds and singularities at infinity (SAIs) on the BSR, improves the BSR structure of the power system, performs corresponding equivalent changes on the multi-machine power system through the contraction projection transformation method, and constructs an index for the transient stability level of the power system based on the correlation characteristics between SAIs and the projection space.

[0010] The technical solution of the present invention is as follows:

[0011] A transient stability analysis method for power systems based on spherical geometry, comprising the following steps:

[0012] Step 1: Perform dimensionality reduction on the high-dimensional multi-machine power system. After obtaining the projection of the global manifold in the unit projection sphere in the three-dimensional phase space through the contraction projection transformation method and obtaining the geometric distribution of the singularities at infinity SAIs of the high-dimensional multi-machine power system on the spherical surface of the unit projection sphere, then conduct observation and analysis;

[0013] Step 2: By constructing and observing the geometric distribution diagram of SAIs of the high-dimensional multi-machine power system on the spherical surface, two sets of SAIs can be obtained and wherein, and are two sets of spherical coordinates symmetric about the center of the sphere obtained through the mapping relationship between any SAIs in the high-dimensional multi-machine power system and the surface of the unit sphere in the phase plane;

[0014] Step 3: Construct an SAIs evaluation parameter based on spherical geometry SG to measure the transient stability level of the power system: Define the surface area SAST of the spherical triangle formed by 3 SAIs on the same hemisphere as the SAIs evaluation parameter for measuring the surface area of the spherical triangle on the spherical surface.

[0015] Furthermore, the expression form of SAST is as follows:

[0016] SAST = R 2 (α + β + γ - π) × 100% (17)

[0017]

[0018] Wherein, R is the radius of the sphere, α, β, and γ are the interior angles of the spherical triangle, and a, b, and c are the spherical distances between three points.

[0019] Furthermore, the contraction projection transformation method in step 1 is specifically implemented as follows:

[0020] Any point (x1, x2, x3) in the high-dimensional multi-machine power system can form a mapping relationship with the corresponding point (y1, y2, y3) in a certain finite space, and the corresponding coordinate mapping is as follows:

[0021]

[0022] Wherein, After the transformation, when a certain point in the high-dimensional multi-machine power system approaches infinity, the trajectory will be mapped to a new spherical surface:

[0023]

[0024] When x i → ∞, h also tends to infinity, and correspondingly

[0025] The distribution of SAIs can be observed through six coordinate ranges in the finite space S. These six coordinate ranges are tangent to the points S1(1, 0, 0), S2(0, 1, 0), S3(0, 0, 1), S′1(-1, 0, 0), S′2(0, -1, 0), and S′3(0, 0, -1) on the finite space S respectively;

[0026] The finite space S is a unit sphere, its coordinate system is (y1, y2, y3), and it satisfies the spherical equation S1, S′1, S2, S′2, S3, S′3 are the points on S that intersect with the coordinate system (y1, y2, y3) of the unit sphere. Taking these six points as the center points respectively, six corresponding phase planes can be constructed;

[0027] Let the coordinate system of the space outside the unit sphere S be (x1, x2, x3). Any point in this system can form a certain mapping relationship with S through these 6 phase planes. When the point in the system is a SAIs, it will be contracted and projected onto the surface of the unit sphere, and other points will be contracted and projected into the sphere. Taking the phase plane where S1(1, 0, 0) is located as an example, for any point N(x1, x2, x3) in the high-dimensional multi-machine power system, assuming this point is a SAIs, then this point will be contracted and projected onto the surface of the unit sphere. Through the collinear relationship between this point and the point N * (1, u1, u2) and the point N′(y1, y2, y3) on the sphere surface, the specific coordinate relationship of the contraction projection transformation can be determined; according to the collinear relationship between N(x1, x2, x3) and N * (1, u1, u2), we can get:

[0028]

[0029] Let u3 = 1 / x1 and take the partial derivative of u, then the definition of the contraction projection transformation can be obtained:

[0030]

[0031] Suppose the equation has a solution of After determining and the values, the specific coordinates of the point (1, u1, u2) on the phase plane can be determined as which is the SAIs coordinates mapped on this phase plane;

[0032] According to the collinear relationship between N′(y1, y2, y3) and N * (1, u1, u2), there must exist an arbitrary scalar such that the following relational expression holds:

[0033] (y1, y2, y3) = d(1, u1, u2) (8)

[0034] Combined with the spherical equation the value of d can be determined:

[0035]

[0036] From this, the coordinates of the point on the phase plane mapped on the sphere surface can be determined as:

[0037]

[0038] Combined with the coordinates of the SAIs in the high-dimensional multi-machine power system mapped onto the phase plane obtained in formula (7) The coordinates of the SAIs in the high-dimensional multi-machine power system mapped on the spherical surface can be obtained as follows:

[0039]

[0040] According to the symmetry of the sphere, the remaining SAIs can be obtained in other phase planes.

[0041] Furthermore, the process of obtaining the two sets of spherical coordinates symmetric about the center of the sphere in step 2 is as follows:

[0042] When the high-dimensional multi-machine power system is disturbed, it is first necessary to identify the corresponding coherent machine groups according to the characteristics of the generator swing curves; when a fault occurs in the system, there are two groups of clustering situations: one group is the leading machine group S with severe disturbance and rapidly changing power angle curves; the other group is the remaining machine group A with relatively stable power angle curves. The rotor motion equations of the two groups of machine groups are added respectively to obtain

[0043]

[0044] For the leading machine group S and the remaining machine group A, where k S and k A are the ratios of the damping coefficient D to the inertia time constant M of the two groups of machine groups respectively; m and n are the numbers of generators in the two groups of machine groups; δ S and δ A are the equivalent rotor angle centers of the two groups of machine groups respectively; ω S and ω A are the equivalent electrical angular velocities of the two groups of machine groups respectively; P ms and P ma are the equivalent mechanical powers of the two groups of machine groups respectively; P es and P ea are the equivalent electromagnetic powers of the two groups of machine groups respectively; and represent their derivatives with respect to time t respectively; in order to visually display the position distribution of the SAIs in the three-dimensional phase space, assuming that there is no oscillation phenomenon between the two groups of machine groups, let Δδ = δ S -δ A , and the third-order equation of formula (12) is obtained as follows:

[0045]

[0046] According to the definition of the contraction projection transformation mentioned in formula (7), the state variables (Δδ, ω S , ω A ) in the above multi-machine power system in formula (13) and the point N on the phase plane S1(1,0,0) *There is always a collinear relationship among (1, u1, u2): 1 / Δδ = u1 / ω S = u2 / ω A ; Let u3 = 1 / Δδ, and taking the derivative of u gives:

[0047]

[0048] where and respectively represent the derivative of u with respect to time t;

[0049] Let Solving the equation, three sets of solutions are obtained on the phase plane S1(1, 0, 0), that is, three sets of singular points A V (1, 0, 0), B V (1, 0, k A ) and C V (1, -k S , 0);

[0050] Through the mapping relationship between any SAIs in the high - dimensional multi - machine power system in Equation (11) between the phase plane and the surface of the unit sphere, two sets of spherical coordinates symmetric about the center of the sphere can be obtained; the specific coordinates of the 6 SAIs mapped on the surface of the sphere are:

[0051]

[0052] where After the coordinate transformation of A V (1, 0, 0), two fixed points projected on the unit sphere surface and After the coordinate transformation of B V (1, 0, k A ) to obtain the coordinates projected on the unit sphere surface and are symmetric about the center of the sphere and are distributed on the circumferential line of the sphere surface; after the coordinate transformation of C V (1, -k S , 0) to obtain the coordinates projected on the unit sphere surface and are symmetric about the center of the sphere and are distributed on the circumferential line of the sphere surface.

[0053] Furthermore, the calculation steps of the spherical triangle area in step 3 are as follows:

[0054] First, determine the radius of the sphere as R, and obtain the three vertices H, J and L of the spherical triangle; then calculate the side length of the spherical triangle, that is, the spherical distance between the three vertices, and use the spherical distance formula to obtain a = Rarccos (H·J), b = Rarccos (J·L), c = Rarccos (L·H), where "·" represents dot product calculation; after the spherical distance is determined, the spherical cosine theorem is used to describe the relationship between the side length of the spherical triangle and the corresponding internal angle, and the sizes of the three spherical internal angles α, β and γ can be further determined: cos (α) = (cosa-cosbcosc) / sinbsinc, cos (β) = (cosb-cosacosc) / sinasinc, cos (γ) = (cosc-cosacosb) / sinasinb; finally, the surface area S of the spherical triangle is determined by calculating the spherical angle excess E = α + β + γ - π 球Δ The calculation formula is as follows:

[0055] S 球Δ =R 2 (α+β+γ-π) (16)

[0056] For a three-dimensional sphere, any closed area distributed on the sphere is based on the spherical triangle. The area of ​​the spherical triangle is S 球Δ It is an important measure to describe the physical properties of a sphere.

[0057] The beneficial effects of the present invention are:

[0058] With the rapid development of large interconnected power grids and new energy sources in my country, the stability mechanism of new power systems has undergone tremendous changes, and the safe and stable operation of power grids faces severe challenges. The manifold and spherical geometry theory provides new ideas for the study of the dynamic characteristics of nonlinear power systems. From the perspective of the stable domain structure, the present invention elaborates in detail the role of infinite singularities in the manifold topology and the characterization of the boundaries of the complete stable domain, aiming to provide a more comprehensive theoretical support for the stable operation of the power grid. Based on the Poincare compactness theory, the coordinates of the infinite singularities of the multi-machine power system are derived, and the geometric distribution of the infinite singularities located on the (hyper) sphere is analyzed in combination with the concept of spherical geometry. Using the mapping relationship between the infinite singularities and the stable domain of the power system, an infinite singularity evaluation parameter based on the principle of spherical geometry is proposed to determine the transient stability of the power system. The simulation results show that the evaluation parameters proposed by the present invention have the advantage of higher accuracy and can effectively determine the transient stability of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0059] Figure 1 is a position distribution diagram of the stable manifold on the boundary of the present invention;

[0060] Figure 2Schematic diagram of six coordinate ranges of the present invention;

[0061] Figure 3 Geometric distribution diagram of SAIs of the multi-machine power system of the present invention;

[0062] Figure 4 Test system diagram of 16-machine 68-node of the present invention;

[0063] Figure 5 SAIs evaluation parameter Q proposed in the reference 2 Figure;;

[0064] Figure 6 SAIs evaluation parameter SAST diagram proposed by the present invention;

[0065] Figure 7 Power angle curve diagram of the test system under Fault 1 of the present invention;

[0066] Figure 8 Power angle curve diagram of the test system under Fault 4 of the present invention;

[0067] Figure 9 Power angle curve diagram of the test system under Fault 7 of the present invention. Detailed implementation method

[0068] The present invention will be described in detail below with reference to the accompanying drawings and specific embodiments. This embodiment is implemented on the premise of the technical solution of the present invention, and gives detailed implementation methods and specific operation processes, but the protection scope of the present invention is not limited to the following embodiments.

[0069] A power system transient stability analysis method based on spherical geometry, comprising the following steps:

[0070] (Basic Sections 1.2, 1.3) Calculation method of SAIs in three-dimensional space and how a multi-machine power system applies this SAIs calculation method. Through these two basic contents, SAIs are effectively mapped out, which is convenient for constructing relevant indicators according to the geometric distribution of SAIs later. Whether it is the indicators obtained from previous articles (the indicator constructed in previous articles here is SVI, and its name is changed here to Q 2 ), or the indicator SAST of this article, both require these two basic calculation methods to be realized. (Section 1.1 only briefly explains the necessity of SAIs research)

[0071] (Improved Implementation Section 2.2) Construct the indicator SAST. (Section 2.1 compares and explains the superiority of SG compared to EG)

[0072] (Improved Implementation Comparison and Verification) (Section 3.1) Through actual simulation verification, compare the two indicators Q 2and SAST. It is found that the previous metric Q 2 has defects, that is, the same metric will appear in different fault conditions, so it is impossible to effectively distinguish the severity of its stability. However, the metric SAST established in the present invention effectively avoids this defect, so the present invention summarizes that SAST has superiority in terms of accuracy.

[0073] (Section 3.2) At the same time, by comparing with the "critical stable time tcr under fault conditions", the effectiveness of the SAST metric is verified.

[0074] The specific implementation process includes the following steps:

[0075] 1 Calculation method of SAIs in power system

[0076] 1.1 Correlation analysis between SAI and complete BSR

[0077] For a nonlinear system, the core of studying its stability lies in analyzing the stability of the system's equilibrium point. When calculating the equilibrium point of a nonlinear system, consider a system in the following form:

[0078]

[0079] Any equation dx / dt = f(x) about the unknown x is set as a certain nonlinear system equation;

[0080] Among them, the equilibrium point of the system refers to the solution that satisfies the equation f(x s ) = 0, and the state transition process φ(t, x(0)) represents the solution trajectory starting from the moment t = 0. In the phase space, the set of all initial points that can converge to the equilibrium point x s is called the stable domain of this equilibrium point:

[0081] Ω(x s ) = {x(t)|limφ(t, x) = x s , t→+∞} (2)

[0082] When x = x s , f(x) = 0 can be obtained. At this time, x s is considered as the equilibrium point of the system f(x). Formula (2) is to let x start from the initial point and change with the increase of time t, and finally make x approach the equilibrium point x s we require. Through this formula, the x that can approach x s as time goes by is found, and these x constitute the stable domain of the equilibrium point x s ; Ω(x s ) is to find the x that meets the condition that x approaches x s as time goes by.Stored therein, representing all x that meet the requirements, i.e., the equilibrium point x s The stable region around it can also be called the stable boundary;

[0083] According to the above definition, it is not difficult to find that on the boundary of Ω(x s ), there are critical stable equilibrium points everywhere, and for such equilibrium points (denoted as x u ), the Jacobian matrix has both eigenvalues with positive real parts and eigenvalues with negative real parts. This means that within the neighborhood of the equilibrium point x u , there are both an unstable manifold W u (x u ) = {x|φ(t, x) → x u , t → -∞} and a stable manifold W s (x u ) = {x|φ(t, x) → x u , t → +∞}. Among them, W s (x u ) starts from afar and finally converges to x u , and W u (x u ) starts from this point and gradually moves away from x u . The union of the stable manifolds of these specific equilibrium points on the boundary constitutes the BSR of the stable equilibrium point x s , that is

[0084]

[0085] where i = 1, 2,..., m, and m is the number of equilibrium points.

[0086] It can be seen that to determine the BSR of an equilibrium point x s , it is not necessary to calculate all the trajectory sets that can converge to this point, but only to calculate the union of the stable manifolds on the boundary.

[0087] For an equilibrium point, the manifold approaching this point is called the stable manifold, and the manifold moving away from this point is called the unstable manifold. Figure 1 The schematic diagram of the position of the stable manifold on the boundary is given. At this time, there is a situation where when the number of equilibrium points on the boundary is small, the local manifolds within the neighborhood of the equilibrium point are not sufficient to form a closed - form BSR. As Figure 1 shows, the stable equilibrium point is surrounded by unstable equilibrium points; this region is not closed, and it is found that there are no closed curves at both ends. To obtain a complete closed - form BSR, the addition of SAIs is required. SAIs start or end at infinity and end or start at the unstable equilibrium points in the figure respectively, jointly forming a closed - form BSR.

[0088] From the perspective of topological structure, only the local information of the BSR is not sufficient to reflect the complete dynamic characteristics of the nonlinear system. At this time, it is necessary to further analyze whether the manifolds on the boundary of the stable region can curl inward to form a closed BSR. To obtain the global structure of the BSR, the contraction projection transformation method is usually used to project the system into a finite space range, and observe the topological structure of the stable region of the system based on SAIs and its complete BSR in the projection space.

[0089] 1.2 Calculation of SAIs in Three-Dimensional Space

[0090] Taking a 3D nonlinear system as an example, the contraction projection transformation method and the forms and distributions of SAIs in three-dimensional space are specifically described below. Any point (x1, x2, x3) in the original system can form a mapping relationship with the corresponding point (y1, y2, y3) in a certain finite space, and the corresponding coordinate mapping is as follows:

[0091]

[0092] where After the transformation, when a point in the original system approaches infinity, the trajectory will be mapped to a new spherical surface:

[0093]

[0094] When x i →∞, h also tends to infinity, and the corresponding Specifically, all trajectories in the original system will be contraction-projected into the interior of a closed finite space S, and the trajectories at infinity will be contraction-projected onto the surface of the finite space S. If there are SAIs in the original system at infinity, they must be distributed on the surface of this finite space S.

[0095] The distribution of SAIs can be observed through six coordinate ranges in the finite space S, and these six coordinate ranges are tangent to the points S1(1, 0, 0), S2(0, 1, 0), S3(0, 0, 1), S′1(-1, 0, 0), S′2(0, -1, 0), and S′3(0, 0, -1) on the finite space S respectively.

[0096] As Figure 2 shown, the finite space S is a unit sphere with the coordinate system (y1, y2, y3), and it satisfies the spherical equation S1, S′1, S2, S′2, S3, S′3 are the points on S that intersect with the coordinate system (y1, y2, y3) of the unit sphere. Taking these six points as the center points respectively, six corresponding phase planes can be constructed. These six phase planes can form a cube that is circumscribed to the unit sphere. Through these six phase planes, the global structure of this unit sphere can be effectively observed. Taking the three vertices S1, S2, and S3 as examples respectively to illustrate the process of establishing the phase plane: At S1(1, 0, 0), a coordinate system is established with the u1-axis parallel to the y2-axis and the u2-axis parallel to the y3-axis; at S2(0, 1, 0), a coordinate system is established with the v1-axis parallel to the y1-axis and the v2-axis parallel to the y3-axis; at S3(0, 0, 1), a coordinate system is established with the w1-axis parallel to the y1-axis and the w2-axis parallel to the y2-axis.

[0097] Let the coordinate system of the space outside the unit sphere S be (x1, x2, x3). Any point in this system can form a certain mapping relationship with S through these 6 phase planes. When the point in the system is a SAIs, it will be contracted and projected onto the surface of the unit sphere, and other points will be contracted and projected into the sphere. Taking the phase plane where S1(1, 0, 0) is located as an example, for any point N(x1, x2, x3) in the original system, assuming this point is a SAIs, then this point will be contracted and projected onto the surface of the unit sphere. Through the collinear relationship between this point and the point N * (1, u1, u2) on the phase plane and the point N′(y1, y2, y3) on the sphere surface, the specific coordinate relationship of the contraction and projection transformation can be determined. According to the collinear relationship between N(x1, x2, x3) and N * (1, u1, u2), we can get:

[0098]

[0099] Let u3 = 1 / x1 and take the partial derivative of u, then the definition of the contraction and projection transformation can be obtained:

[0100]

[0101] Suppose the equation has a solution of After determining and the values, the specific coordinates of the point (1, u1, u2) on the phase plane can be determined as which is the SAIs coordinates mapped on this phase plane.

[0102] According to the collinear relationship between N′(y1, y2, y3) and N * (1, u1, u2), there must exist an arbitrary scalar such that the following relational expression holds:

[0103] (y1,y2,y3) = d(1,u1,u2) (8)

[0104] Combined with the spherical equation the value range of d can be determined:

[0105]

[0106] From this, the coordinates of the points on the phase plane mapped on the spherical surface can be determined as:

[0107]

[0108] Combined with the coordinates of the SAIs in the original system obtained in Equation (7) mapped onto the phase plane the coordinates of the SAIs in the original system mapped on the spherical surface can be obtained as:

[0109]

[0110] The above is the calculation process of shrinking and projecting a certain SAIs on the spherical surface in the phase plane where S1(1,0,0) of the original system is located. According to the symmetry of the sphere, the remaining SAIs can be obtained in other phase planes.

[0111] 1.3 Calculation of SAIs for Multi - machine Power System

[0112] The motion equation of the multi - machine power system has a relatively high dimension, and directly calculating the high - dimensional manifold is somewhat challenging. In order to apply the coordinate transformation method mentioned above to the multi - machine power system, it is necessary to reduce the dimension of the high - dimensional multi - machine system. Then, through the shrinking projection transformation, the projection of the global manifold in the three - dimensional phase space is obtained, and then the observation and analysis are carried out.

[0113] When the multi - machine power system is disturbed, it is first necessary to identify the corresponding homologous machine groups according to the characteristics of the generator swing curve. When a fault occurs in a certain part of the system, there are two component - group situations: one is the leading machine group S with serious disturbance and rapidly changing power - angle curve; the other is the remaining machine group A with relatively stable power - angle curves for the rest. The rotor motion equations of the two groups of machine groups are added respectively to obtain

[0114]

[0115] For the leading machine group S and the remaining machine group A, where k S and k A are the damping coefficients of the two groups of machine groups respectively D and the ratio of the inertia time constant M. m and n are the number of generators in the two groups of machine groups respectively. δ S and δ A are the equivalent rotor - angle centers of the two groups of machine groups respectively. ω S and ωA The equivalent electrical angular velocities of two groups of clusters, respectively. P ms and P ma The equivalent mechanical powers of two groups of clusters, respectively. P es and P ea The equivalent electromagnetic powers of two groups of clusters, respectively, and are respectively represented as their derivatives with respect to time t. In order to visually display the position distribution of SAIs in three-dimensional phase space, it is assumed that there is no oscillation phenomenon between the two groups of clusters, and let Δδ = δ S -δ A , and the third-order equation of Equation (12) is obtained as follows:

[0116]

[0117] According to the definition of the contraction projection transformation mentioned in Equation (7), the state variables (Δδ, ω S , ω A ) in the above multi-machine power system and the point N * (1, u1, u2) on the phase plane S1(1, 0, 0) always have a collinear relationship: 1 / Δδ = u1 / ω S = u2 / ω A . Let u3 = 1 / Δδ, and the derivative of u can be obtained as:

[0118]

[0119] Let Solve the equation, and three sets of solutions are obtained on the phase plane S1(1, 0, 0), that is, three sets of singular points A V (1, 0, 0), B V (1, 0, k A ) and C V (1, -k S , 0).

[0120] Through the mapping relationship between any SAIs in the original system in Equation (11) and the surface of the unit sphere through the phase plane, two sets of spherical coordinates symmetric about the center of the sphere can be obtained. Their position distribution is as Figure 3 shown, and the specific coordinates of the 6 SAIs mapped on the surface of the sphere are:

[0121]

[0122] where

[0123] After the coordinate transformation of A V (1, 0, 0), two fixed points projected on the unit sphere are obtained and after passing through B V (1, 0, k A ) coordinate transformation to obtain the coordinates projected on the unit sphere and are symmetric about the center of the sphere and are distributed on the green circular line on the sphere surface. The collinear relationship between the two points is represented by a green line segment with an arrow; after passing through C V (1, -k S , 0) coordinate transformation to obtain the coordinates projected on the unit sphere and are symmetric about the center of the sphere and are distributed on the yellow circular line on the sphere surface. The collinear relationship between the two points is represented by a yellow line segment with an arrow.

[0124] 2SG and the construction method of transient stability degree index

[0125] In the actual transient analysis process of a multi - machine power system, the grid structure of the system after a fault is determined at the moment of fault clearing, and the corresponding system equivalent parameters are also determined. That is to say, it is not necessary to wait until the actual power angle runs to infinity. At the moment of fault clearing, the position distribution and evaluation parameters of SAIs can be obtained, so as to effectively measure the transient stability of the system. When constructing the evaluation parameters for the change in the position distribution of SAIs, the SAIs projected on the sphere can be observed and analyzed from different geometric fields. Different geometric fields have different representation methods and advantages. Next, this section will compare the characteristics of the EG space and the SG space in detail, and construct the evaluation parameters of SAIs based on the characteristics of the SG space, and its superiority will be proved below.

[0126] By observing the geometric distribution of SAIs of the above - mentioned multi - machine system on the sphere, two groups of SAIs can be obtained, and these two groups of SAIs and are respectively symmetric about the center of the unit sphere. According to their position distribution on the sphere, two inscribed plane triangles symmetric about the center of the sphere can be constructed based on EG. The literature (Ma Meiling, Wang Jie, Li Penghan, etc. Global manifold analysis of power angle stability region and boundary in power system [J]. Proceedings of the CSEE, 2020, 40(18): 5865 - 5875.) takes a certain group of SAIs (taking as an example) on the same hemisphere as an example, and takes the area of the inscribed plane triangle formed by them as the evaluation parameter Q 2 of the change in the position distribution of SAIs, and conducts the stability degree analysis of the multi - machine system based on this evaluation parameter, as Figure 5 shown.

[0127] Corresponding to the planar case of EG, if the coordinate positions of these two groups of SAIs are analyzed from the perspective of SG, theoretically, more accurate singularity change evaluation parameters can be obtained. Next, the present invention will prove this from the perspective of SG and construct SAIs evaluation parameters based on SG to measure the transient stability degree of the power system.

[0128] 2.1 SG and Spherical Triangles

[0129] SG is a field of geometry that involves the properties of points, curves, and angles on a three-dimensional sphere. There are many differences between SG and the more commonly used planar EG. Regarding the relationship change from the plane to the sphere, the main differences between the two are shown in Table 1.

[0130] Table 1 Comparison of Euclidean Geometry with Spherical Geometry

[0131] TABLE 1 Comparison of Euclidean Geometry with Spherical Geometry

[0132]

[0133] For SG, the characteristics of a certain area on the sphere can be effectively observed from any direction. As the value changes, the geometric figures shown on the sphere are also different. In SG, the simplest geometric figure is the spherical triangle, which is composed of three vertices on the sphere and the great circular arc segments between them. When analyzing the characteristics of the spherical triangle, the area of the spherical triangle is an important indicator. The steps to calculate the area of the spherical triangle are as follows:

[0134] First, determine that the radius of the sphere is R and obtain the three vertices H, J, and L of the spherical triangle. Next, calculate the side lengths of the spherical triangle, that is, the spherical distances between the three vertices. Using the spherical distance formula, they are respectively obtained as a = R arccos(H·J), b = R arccos(J·L), c = R arccos(L·H), where "·" represents the dot product calculation. After the spherical distances are determined, the relationship between the side lengths of the spherical triangle and the corresponding interior angles can be described by the spherical cosine theorem, and the sizes of the three spherical interior angles α, β, and γ can be further determined: cos(α) = (cos a - cos b cos c) / sin b sin c, cos(β) = (cos b - cos a cos c) / sin a sin c, cos(γ) = (cos c - cos a cos b) / sin a sin b. Finally, by calculating the spherical excess E = α + β + γ - π, the surface area S of the spherical triangle is determined. 球Δ The calculation formula is as follows:

[0135] S 球Δ = R2 (α + β + γ - π)(16)

[0136] For a three - dimensional sphere, any closed region distributed on the sphere is based on spherical triangles, and the area S of a spherical triangle 球Δ is an important measure to describe the physical characteristics of the sphere. As Figure 3 shown, after the contraction projection transformation of a multi - machine power system, its SAIs are usually distributed on the surface of the three - dimensional projection space (unit sphere). As the BSR changes, the SAIs will move along the sphere. As the endpoints of the BSR, the stability and position of the SAIs depend on the structural characteristics of the post - fault system. With the supplement of SAIs, on the one hand, it helps to outline the complete structure of the BSR, and on the other hand, the change of the BSR can also be reflected through the distribution characteristics of SAIs, thus reflecting the stability degree of the system.

[0137] It can be seen from this that there is a certain connection between the change of SAIs and the stability degree of the system. In the following, taking the SAIs distribution parameters as the benchmark, a transient stability degree index of the power system is constructed.

[0138] 2.2 Construction of the transient stability degree index of the power system

[0139] Projecting the position distribution of SAIs on the sphere onto the inscribed plane triangle inside the sphere, and using the plane image to represent the spherical image, the SAIs evaluation parameter constructed based on this is not accurate. Although this EG - based representation form can achieve the measurement goal to a certain extent, it will inevitably have the situation that the data correlation does not match the actual situation, resulting in some error situations. Specifically, the SAIs evaluation parameter Q proposed based on EG 2 It can be determined that when the position distribution of SAIs in a multi - machine power system changes, different areas of the inscribed plane triangle will be generated. The three vertices of this plane triangle are the 3 SAIs mapped on the sphere of the same hemisphere. At this time, there may be a situation where the areas of two different inscribed plane triangles are the same, and the evaluation parameter Q 2 will not be able to measure the difference between the two situations.

[0140] When the above problems occur, the SAIs evaluation parameter constructed based on the spherical triangle area of SG has better performance and also has certain advantages in numerical calculation. The present invention uses the area of the spherical triangle (Surface area of a spherical triangle, SAST) formed by 3 SAIs on the same hemisphere to define the area of the spherical surface triangle as the SAIs evaluation parameter. According to the analysis of SG and spherical triangles in the previous text, the expression form of SAST can be obtained as follows:[[]]

[0141] SAST = R 2 (α + β + γ - π) × 100% (17)

[0142]

[0143] Wherein, R is the radius of the sphere, α, β, and γ are the interior angles of the spherical triangle, and a, b, and c are the spherical distances between three points. This evaluation parameter reflects the change trend of the equilibrium point at infinity, reflects the expansion and contraction of the stable region, and can also measure the change in the transient stability degree of the system.

[0144] 3 Simulation Analysis and Verification

[0145] The transient stability assessment of the power system needs to consider the structural stability of the system and the state change after being disturbed. To verify the effectiveness of this evaluation parameter in large power grids, this section conducts simulations on the IEEE 16-machine 68-bus system model, as Figure 4 shown, analyzes the transient power angle stability degree of the system after different fault lines are removed. Explore the distribution characteristics of SAIs after the multi-machine system undergoes the contraction projection transformation, compare the differences between the two SAIs evaluation parameters Q 2 and SAST, and verify the superiority of SAST compared to Q 2 . At the same time, verify the effectiveness of SAST in measuring the transient power angle stability degree of the power system. The specific steps are as follows:

[0146] 1) Remove different fault lines, simplify the multi-machine system model into an equivalent model according to the grouping situation of the generating units, and calculate the corresponding equivalent parameters;

[0147] 2) Obtain the SAIs distribution according to the contraction projection transformation method given in Section 1.2;

[0148] 3) Calculate the SAIs evaluation parameter as the stability degree change index under different fault line conditions according to the content in Section 2.2.

[0149] This section selects the following three-phase short-circuit fault locations: Fault 1 (Line 28 - 26), Fault 2 (Line 24 - 68), Fault 3 (Line 67 - 66), Fault 4 (Line 15 - 42), Fault 5 (Line 19 - 68), Fault 6 (Line 10 - 31), and Fault 7 (Line 13 - 17), which are marked in Figure 4 . The fault duration is set to 300 ms. The grouping situations of the system under different fault conditions are also completely different. After the fault occurs, when the leading machine group S is identified, the remaining machine groups are all classified as the remaining machine group A. After equivalent processing, the equivalent power angles δ S and δ A of the two groups of machine groups and the corresponding damping coefficient k Sand k A According to the damping coefficient k S and k A the SAI evaluation parameters SAST and Q can be further calculated 2 to analyze the transient stability degree under different fault conditions.

[0150] 3.1 Comparison of Different SAI Evaluation Parameters

[0151] As shown in Table 2, (the fault locations here are randomly selected for convenience of analyzing the superiority of the coordinate SAST proposed in the present invention) Fault 1 means setting a three-phase short-circuit fault at Node 31 and clearing the fault line 28 - 26 after 300 ms; Fault 2 means setting a three-phase short-circuit fault at Node 26 and clearing the fault line 24 - 68 after 300 ms; Fault 3 means setting a three-phase short-circuit fault at Node 81 and clearing the fault line 67 - 66 after 300 ms; Fault 4 means setting a three-phase short-circuit fault at Node 15 and clearing the fault line 15 - 42 after 300 ms. Fault 5 means setting a three-phase short-circuit fault at Node 20 and clearing the fault line 19 - 68 after 300 ms; Fault 6 means setting a three-phase short-circuit fault at Node 10 and clearing the fault line 10 - 31 after 300 ms; Fault 7 means setting a three-phase short-circuit fault at Node 13 and clearing the fault line 13 - 17 after 300 ms; It is not difficult to find that there are two groups of faults (Fault 1 and 4, Fault 2 and 3) in the system where the Q 2 indexes are equal, but the SAI evaluation parameter SAST values of the two faults are not equal. This situation indicates that the evaluation parameter SAST is more accurate than Q 2 in terms of precision.

[0152] Among them, G i' {i'∈(1,16)} represents the 16 generators in this system, and different leading generator groups will be generated under different fault conditions; t cr is the critical stability time of the system under fault conditions, and its length is positively correlated with the transient stability degree of the power system. (The explanation of t cr exists in Section 3.2 for using SAST to judge the transient power angle stability degree of a multi-machine system, that is, this t is defined for the node by the power angle difference being greater than 180°, cr and the effectiveness of SAST is verified by comparing the critical clearing time with the SAST index.)

[0153] Table 2 SAI evaluation parameters (Q 2 and SAST)

[0154] TABLE 2 SAI evaluation parameters (Q 2and SAST) under different fault conditions

[0155]

[0156] 1) The SAIs evaluation parameter Q for Fault 1 and Fault 4 2 is the same, both being 0.2817. However, the SAIs evaluation parameter SAST for Fault 1 is 37.0086, while that for Fault 4 is 32.8762, and the two are not the same;

[0157] 2) The SAIs evaluation parameter Q for Fault 2 and Fault 3 2 is the same, both being 0.2614. However, the SAIs evaluation parameter SAST for Fault 2 is 34.118, and that for Fault 3 is 33.9754.

[0158] Both of the above two situations indicate that the evaluation parameter Q 2 has certain precision defects when measuring the transient stability degree of the power system. To clearly show the geometric distribution of the two types of SAIs evaluation parameters, the following takes Fault 1 and Fault 4 as examples for comparative analysis. Table 3 gives the system equivalent parameters under the two fault conditions and the specific coordinate distributions of the corresponding two groups of SAIs.

[0159] Under the two fault conditions, the parameter geometric distribution diagrams of the system SAIs evaluation parameter Q 2 corresponding to SAST are as shown in Figure 5 and Figure 6 shown. Among them, is a group of SAIs under the condition of Fault 1, and the blue shaded part is the geometric distribution of the SAIs evaluation parameter under the condition of Fault 1. is a group of SAIs under the condition of Fault 4, and the red shaded part is the geometric distribution of the SAIs evaluation parameter under the condition of Fault 4. For the same two fault types, Figure 5 the areas of the two inscribed plane triangles represented by the evaluation parameter Q 2 are equal, and no correct judgment result is given. While Figure 6 the areas of the spherical surface curved triangles represented by the evaluation parameter SAST under the two fault conditions are not equal, and the transient stability degree of the power system can be well distinguished. SAST can more accurately determine the transient stability degree of the system under different fault conditions than Q 2 can.

[0160] Table 3 System equivalent parameters corresponding to different faults

[0161] TABLE 3 System equivalent parameters corresponding to different faults

[0162]

[0163] 3.2 Using SAST to judge the transient power angle stability degree of a multi-machine system

[0164] Another example is to select Fault 1, Fault 4, and Fault 7 in Table 2 to verify the effectiveness of the evaluation parameter SAST for transient power angle stability analysis. Under different fault conditions, the 16 generators in the system are divided into the leading machine group S and the remaining machine group A according to the coherence of the power angle curves. After Fault 1, Fault 4, and Fault 7 occur separately, the generator power angle curves of the simulation system are respectively as Figure 7 、 Figure 8 and Figure 9 shown.

[0165] According to the curve change trend, it can be observed that the system is torn into a two-group machine group operation mode, reflecting the change trend of the power angles of the two groups of machine groups. As Figure 7 shown, after Fault 1, the system is divided into: the leading machine group S = {2, 3, 4, 5, 6, 7, 8, 9} and the remaining machine group A = {1, 10, 11, 12, 13, 14, 15, 16}, and the equivalent power angle δ S of the leading machine group is represented by a thick black dashed line in the figure, and the equivalent power angle δ A of the remaining machine group is represented by a thick red dashed line in the figure. Then it is pointed out in turn that, as Figure 8 shown, after Fault 4, the system is divided into: the leading machine group S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16} and the remaining machine group A = {15}, δ S is represented by a thick black dashed line, and δ A is represented by a thick red dashed line. As Figure 9 shown, after Fault 7, the system is divided into: the leading machine group S = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} and the remaining machine group A = {13, 14, 15, 16}, δ S is represented by a thick black dashed line, and δ A is represented by a thick red dashed line.

[0166] After determining the equivalent power angles of the two groups of machine groups, set the difference between the equivalent power angles of the two groups of machine groups as the boundary of 180°. When the difference between the equivalent power angles of the two groups of machine groups is greater than 180°, the multi-machine system loses stability, and the corresponding time t is the critical stability time t cr under this fault condition, and the length of this time is positively correlated with the transient stability degree of the power system. When tcr When t is longer, the system has more time to adapt to the fault, thus slowing down the steady-state process, which means that the transient stability of the system is better and the frequency, voltage and power can be better maintained, thus reducing the risk of system instability. cr If the fault is too short, the system may not be able to adapt to the fault, resulting in severe fluctuations in frequency and voltage, which may lead to system collapse, which is detrimental to the transient stability of the system.

[0167] By comparing Figure 7 , Figure 8 and Figure 9 The system power angle curve can be intuitively obtained:

[0168] 1) System t after Fault 1 cr is 1038ms. After fault 4, the system t cr The comparison between Fault 1 and Fault 4 shows that after Fault 1, the system has a longer time to adapt to the occurrence of the fault, and the transient stability of the system after Fault 1 is stronger than that after Fault 4.

[0169] 2) System t after fault 7 cr It is 605ms. The comparison between fault 7 and fault 4 shows that the critical clearing time of the system after fault 7 is shorter than that of fault 4, and the transient stability of the system after fault 7 is weaker than that of fault 4.

[0170] It can be judged that the transient stability of the system after fault 1 is stronger than that after fault 4, and the transient stability after fault 4 is stronger than that after fault 7. Next, the SAIs evaluation parameter SAST established by the present invention is compared with the critical removal time t cr Comparative analysis is performed to verify the effectiveness of SAST.

[0171] After shrinking and projecting the equivalent model, the SAI evaluation parameter SAST based on SG can be further obtained. Different faults correspond to different SAST and critical removal time t cr By comparing the change degree of SAST with t cr The degree of change can verify the effectiveness of SAST for transient power angle stability. As shown in Table 2, the SAI evaluation parameter SAST of faults 1 to 7 gradually decreases from 37.0086 for fault 1 to 24.6918 for fault 7, and its critical fault removal time t cr It also gradually decreases from 1038ms for fault 1 to 605ms for fault 7. The corresponding decreasing relationship between the two verifies the effectiveness of the evaluation parameter SAST in measuring the transient stability of the power system.

[0172] 4 Summary

[0173] The present invention explores the importance of the stability domain and the manifold in the nonlinear dynamic system, and emphasizes the role of SAI in the boundary of the complete stability domain. By introducing the contraction projection transformation method and the SAI calculation method in the three-dimensional space, through the derivation of the mathematical model and equations, the coordinate mapping relationship between them is elaborated, and the equivalent calculation of the multi-machine power system is carried out to obtain the projection space topological structure based on SAI. Combining the concepts of spherical geometry and spherical triangle, with the stability of the system and the state change after being perturbed as a reference, the SAI evaluation parameters based on EG are optimized to a certain extent, and an SAI evaluation parameter based on SG is proposed. The superiority of this parameter in terms of accuracy is verified, and its effectiveness in measuring the transient stability degree of the power system is also verified. Overall, the present invention provides a new perspective and theoretical support for the transient stability analysis of the power system, and provides useful ideas and methods for expanding the application scope of the power system stability theory.

[0174] The above-described embodiments merely represent one implementation manner of the present invention, and the description thereof is relatively specific and detailed, but it should not be construed as a limitation on the scope of the invention patent. It should be noted that for those of ordinary skill in the art, without departing from the concept of the present invention, several modifications and improvements can be made, and these all belong to the protection scope of the present invention. Therefore, the protection scope of the present invention patent shall be subject to the appended claims.

Claims

1. A transient stability analysis method for power systems based on spherical geometry, characterized in that It includes the following steps: Step 1: Perform dimensionality reduction on the high-dimensional multi-machine power system. After obtaining the projection of the global manifold in the unit projection sphere of the three-dimensional phase space through the contraction projection transformation method and getting the geometric distribution of the infinite singular points SAIs of the high-dimensional multi-machine power system on the spherical surface of the unit projection sphere, then conduct observation and analysis; Step 2: By constructing and observing the geometric distribution diagram of SAIs of the high-dimensional multi-machine power system on the sphere, two sets of SAIs can be obtained and wherein and are two sets of spherical coordinates symmetric about the center of the sphere obtained from the mapping relationship between any SAI in the high-dimensional multi-machine power system through the phase plane and the surface of the unit sphere Step 3: Construct SAIs evaluation parameters based on spherical geometry SG to measure the transient stability degree of the power system: Define the surface area SAST of the spherical triangle formed by 3 SAIs on the same hemisphere as the SAIs evaluation parameter.

2. The transient stability analysis method of the power system based on spherical geometry according to claim 1, characterized in that, The expression form of SAST is as follows: SAST = R 2 (α + β + γ - π) × 100% (17) Where, R is the radius of the sphere, α, β, γ are the interior angles of the spherical triangle, and a, b, c are the spherical distances between the three points.

3. The transient stability analysis method of a power system based on spherical geometry according to claim 1, characterized in that The specific implementation of the contraction projection transformation method in Step 1 is as follows: Any point (x1, x2, x3) in the high-dimensional multi-machine power system can form a mapping relationship with the corresponding point (y1, y2, y3) in a certain finite space, and the corresponding coordinate mapping is as follows: Among them, After transformation, when a certain point in the high-dimensional multi-machine power system approaches infinity, the trajectory will be mapped onto a new spherical surface: When x i → ∞, h also tends to infinity, and correspondingly The distribution of SAIs can be observed through six coordinate ranges in the finite space S, and these six coordinate ranges are tangent to the points S1(1, 0, 0), S2(0, 1, 0), S3(0, 0, 1), S′1(-1, 0, 0), S′2(0, -1, 0) and S′3(0, 0, -1) on the finite space S respectively; The finite space S is a unit sphere with a coordinate system (y1, y2, y3) and satisfies the spherical equation S1, S′1, S2, S′2, S3, S′3 are the points on S that intersect with the coordinate system (y1, y2, y3) of the unit sphere. Taking these six points as the center points respectively, six corresponding phase planes can be constructed; Let the coordinate system of the space outside the unit sphere S be (x1, x2, x3). Any point in this system can form a certain mapping relationship with S through these 6 phase planes. When the point in the system is a SAIs, it will be contracted and projected onto the surface of the unit sphere, and other points will be contracted and projected into the sphere. Taking the phase plane where S1(1, 0, 0) is located as an example, for any point N(x1, x2, x3) in the high-dimensional multi-machine power system, assuming this point is a SAIs, then this point will be contracted and projected onto the surface of the unit sphere. Through the collinear relationship between this point and the point N * (1, u1, u2) and the point N′(y1, y2, y3) on the sphere surface, the specific coordinate relationship of the contraction projection transformation can be determined; according to the collinear relationship between N(x1, x2, x3) and N * (1, u1, u2), we can obtain: Let u3 = 1 / x1, and take the partial derivative of u, then the definition of the contraction projection transformation can be obtained: Suppose the equation has a solution After determining and the values, the specific coordinates of the point (1, u1, u2) on the phase plane can be determined as which are the SAIs coordinates mapped on this phase plane; According to the collinear relationship between N′(y1, y2, y3) and N * (1, u1, u2), there must exist an arbitrary scalar such that the following relational expression holds: (y1, y2, y3) = d(1, u1, u2) (8) Combined with the spherical equation The value range of d can be determined as follows: Thus, the coordinates of the points mapped on the spherical surface in the phase plane can be determined as: Map the SAIs in the high-dimensional multi-machine power system obtained in Equation (7) to the coordinates on the phase plane It can be obtained that the coordinates of the SAIs coordinate mapping in the high-dimensional multi-machine power system on the spherical surface are as follows: According to the symmetry of the sphere, the remaining SAIs can be obtained in other phase planes.

4. The transient stability analysis method of the power system based on spherical geometry according to claim 1, wherein The obtaining process of the two sets of spherical coordinates symmetric about the center of the sphere in Step 2 is as follows: When the high-dimensional multi-machine power system is disturbed, it is first necessary to identify the corresponding homologous machine groups according to the characteristics of the generator swing curve; when a fault occurs in the system, there are two grouping situations: one group is the leading machine group S with serious disturbance and rapidly changing power angle curve; the other group is the remaining machine group A with relatively stable power angle curves of the rest. Add the rotor motion equations of the two groups of machine groups respectively to get For the leading fleet S and the remaining fleet A, where k S and k A are respectively the ratios of the damping coefficient D to the inertia time constant M of the two fleets; m and n are respectively the numbers of generators in the two fleets; δ S and δ A are respectively the equivalent rotor angle centers of the two fleets; ω S and ω A are respectively the equivalent electrical angular velocities of the two fleets; P ms and P ma are respectively the equivalent mechanical powers of the two fleets; P es and P ea are respectively the equivalent electromagnetic powers of the two fleets; and are respectively expressed as their derivatives with respect to time t; in order to be able to visually display the position distribution of SAIs in the three-dimensional phase space, assuming that there is no oscillation phenomenon between the two fleets, let Δδ = δ S -δ A , and the third-order equation of Equation (12) is obtained as follows: According to the definition of the contraction projection transformation mentioned in Equation (7), the state variables (Δδ, ω S , ω A ) of the above multi-machine power system and the point N * (1, u1, u2) on the phase plane S1(1, 0, 0) always have a collinear relationship: 1 / Δδ = u1 / ω S = u2 / ω A ; Let u3 = 1 / Δδ, and taking the derivative of u gives: Among them, and respectively represent the derivative of u with respect to time t; Let d Solving the equation, three sets of solutions are obtained on the phase plane S1(1, 0, 0), that is, three sets of singular points A V (1, 0, 0), B V (1, 0, k A ) and C V (1, -k S , 0); Through the mapping relationship between any SAIs in the high-dimensional multi-machine power system in Equation (11) and the unit sphere surface through the phase plane, two sets of spherical coordinates symmetric about the center of the sphere can be obtained; the specific coordinates of the 6 SAIs mapped on the spherical surface are: Among them After the A V (1, 0, 0) coordinate transformation, two fixed points projected on the unit sphere are obtained And After the B V (1, 0, k A ) coordinate transformation, the coordinates projected on the unit sphere are obtained And Are symmetric about the center of the sphere and are distributed on the circumferential line of the sphere surface; after the C V (1, -k S , 0) coordinate transformation, the coordinates projected on the unit sphere are obtained And Are symmetric about the center of the sphere and are distributed on the circumferential line of the sphere surface.

5. The transient stability analysis method of a power system based on spherical geometry according to claim 1, characterized in that The calculation steps of the spherical triangle area in Step 3 are as follows: First, determine the radius of the sphere as R and obtain the three vertices H, J, and L of the spherical triangle; next, calculate the side lengths of the spherical triangle, that is, the spherical distances between the three vertices, and use the spherical distance formula to obtain them as a = R arccos(HJ), b = R arccos(JL), and c = R arccos(LH) respectively, where "·" represents the dot product calculation; after the spherical distances are determined, the relationship between the side lengths and the corresponding interior angles of the spherical triangle can be described by the spherical cosine theorem, and the magnitudes of the three spherical interior angles α, β, and γ can be further determined: cos(α) = (cos a - cos b cos c) / (sin b sin c), cos(β) = (cos b - cos a cos c) / (sin a sin c), cos(γ) = (cos c - cos a cos b) / (sin a sin b); finally, determine the surface area S of the spherical triangle by calculating the spherical excess E = α + β + γ - π 球Δ The calculation formula is as follows: S 球Δ = R 2 (α + β + γ - π) (16) For a three-dimensional sphere, any closed region distributed on the sphere is based on spherical triangles, and the area S of a spherical triangle 球Δ is an important measure for describing the physical properties of the sphere.

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