A power system transient stability analysis method based on spherical geometry

Abstract

The application relates to a power system transient stability analysis method based on spherical geometry, which comprises the following steps: step 1, dimension reduction processing is performed on a high-dimensional multi-machine power system, and a projection of a global manifold is obtained in a unit projection sphere of a three-dimensional phase space through a shrinkage projection transformation method; step 2, by constructing and observing a geometric distribution condition diagram of SAIs of the high-dimensional multi-machine power system on a sphere, two groups of SAIs about the center symmetry of the sphere can be obtained, and the spherical triangular area SAST of a spherical curved triangular area on the surface of the sphere is defined as an SAI evaluation parameter by using three SAIs on the same half sphere. The application solves the problem that the current power system transient stability analysis method cannot effectively distinguish the stability degree of different fault conditions when the same index appears in different fault conditions, simulation results show that the evaluation parameter has the advantage of higher accuracy, and the transient stability degree of the system can be effectively determined.

Description

Technical Field

[0001] This 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 Technology

[0002] With the continuous growth of electricity load and the integration of large-scale intermittent energy sources such as wind and photovoltaic power generation, modern power systems exhibit significant uncertainties and low inertia characteristics, leading to significant changes in system stability mechanisms. These 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] A comprehensive understanding of the dynamic characteristics of a power system is essential for studying its transient stability. Studying the trajectories of solutions to the equations yields the system's topological space. Analyzing the system's global topology requires examining the boundaries of its complete stability region. When studying the stability region of a specific system, it is necessary to determine the set of solutions to the system's operating equations. All trajectories originating from this set converge to a specified state space. Plotting the boundary of this specified state space allows for the assessment of the system's stability, i.e., determining the maximum deviation from the stable operating state.

[0004] Therefore, when studying the dynamic characteristics of a system, the first step is to accurately characterize the boundary of the stability region (BSR). Currently, methods for characterizing the BSR mainly include the energy function method, manifold estimation numerical methods, and topological and dynamic characteristic analysis methods, with the first two being the most commonly used. The core of the energy function method is to approximate the system's stability region using the convex boundary of the Lyapunov function. This method is simple and direct, avoiding complex numerical integration calculations of the trajectory. However, the construction methods of energy functions are diverse, with varying forms and a lack of universality. The manifold estimation method combines the advantages of the energy function, developing a computational method for stable manifolds, thus achieving a more efficient characterization of the BSR. In the early 19th century, Hilbert studied two-dimensional topological manifolds and first defined a plane through its neighborhood, using it as an ideal tool for describing manifolds, thereby laying the foundation for manifold geometry. In 1993, Guckenheimer et al. introduced the concept of geodesic level sets for stable manifolds that locally approximate saddle points, initially realizing the visualization of manifold geometry. Since then, a large number of research results have emerged on manifold calculation methods, such as the trajectory arc length method, the adaptive trajectory extension method, the simplex method, and the geometric perturbation method, which have provided a theoretical basis and research ideas for the BSR characterization of dynamical systems.

[0005] Subsequent studies have shown that the size of the stability region reflects the system's ability to maintain stability, enabling transient stability studies such as critical cut-off angle and stability margin analysis of power systems. This demonstrates that achieving comprehensive topological analysis of the manifold and providing dynamic characteristics analysis of the system at infinity would be of paramount importance for establishing and applying a 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 every manifold, there exists a neighborhood that is homeomorphic to some open set in Euclidean space, ensuring that the manifold has a smooth structure. By studying the topological structure, we can obtain the metric structure and natural coordinate system on the manifold. When describing the topological structure of a manifold, spherical geometry (SG) is used as a specific type of manifold. Although EG is very useful on a plane, SG is more suitable for dealing with curved surfaces, the Earth's surface, and other spaces with positive curvature. The key difference between EG and SG can be seen as the difference between a plane and a sphere. A sphere has positive curvature, while a plane has zero curvature. This means that a straight line on a sphere is a great circle arc, while on a plane it is a straight line, making geodesics on a sphere more natural and intuitive. At the same time, the area calculation of a fixed region on a sphere is closer to the actual geographical area and more intuitive. These characteristics have led to a focus of research in many fields on SG: in addition to its frequent use in astronomy, geophysics and satellite communications, its application in various fields such as VR panoramic images and surround sound and robot joints in spherical space has gradually attracted the attention of scholars.

[0007] Currently, the main methods for transient stability analysis of power systems include time-domain simulation, energy function method, and extended equal-area method. Among these, time-domain simulation is commonly used because it can visually demonstrate the electromechanical transient processes of the system, provide the time response of various variables in the system, and has broad model adaptability. The energy function method mainly includes the UEP method, PEBS method, and BCU method, the core of which is to analyze the stability of the system's power angle initial swing based on the energy function. The core of the extended equal-area method is to divide the multiple generators in the system into critical and non-critical units, thus equating it to a two-machine system, and using the "equal area" criterion to determine the transient stability of the system and perform quantitative analysis.

[0008] However, the current technical literature "Ma Meiling, Wang Jie, Li Penghan, et al. Global manifold analysis on the power angle stability domain and boundary of power system [J]. Proceedings of the CSEE, 2020, 40(18): 5865-5875" uses the SAIs evaluation parameter Q proposed by EG. 2The drawback is that different fault conditions can result in the same indicators, making it impossible to effectively distinguish the severity of their stability. Summary of the Invention

[0009] To address the issue that the evaluation parameter Q2 of SAIs proposed in the literature based on spherical geometry exhibits the same index under different fault conditions, resulting in a certain accuracy deficiency and an inability to effectively distinguish the severity of stability, a transient stability analysis method for power systems based on spherical geometry is proposed. This invention builds upon existing transient stability analysis methods by combining manifolds and their characteristics in SG space to construct a transient stability index for power systems. To expand the application scope of power system stability theory and understand the topological structure of the stability domain, this invention improves the BSR structure of power systems by studying the distribution characteristics of manifolds and singularities at infinity (SAIs) on the BSR. It then performs corresponding equivalent transformations on multi-machine power systems using a contraction projection transformation method and constructs a transient stability index for power systems based on the correlation characteristics between SAIs and the projection space.

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

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

[0012] Step 1: Dimensionality reduction is performed on the high-dimensional multi-machine power system. The global manifold projection is obtained in the unit projection sphere of the three-dimensional phase space by using the shrinking projection transformation method. After obtaining the geometric distribution of the infinite singularities SAIs of the high-dimensional multi-machine power system on the surface of the unit projection sphere, observation and analysis are then performed.

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

[0014] Step 3: Construct SAIs evaluation parameters based on spherical geometry SG to measure the transient stability of the power system: The area of ​​the spherical surface triangle SAST formed by three SAIs on the same hemisphere is defined as the SAIs evaluation parameter.

[0015] Furthermore, SAST is represented as follows:

[0016] SAST=R 2 (α+β+γ-π)×100% (17)

[0017]

[0018] Where 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 the three points.

[0019] Furthermore, the specific implementation of the shrinking projection transformation method in step 1 is as follows:

[0020] In a high-dimensional multi-machine power system, any point (x1, x2, x3) can be mapped to a corresponding point (y1, y2, y3) in a finite space. The corresponding coordinate mapping is as follows:

[0021]

[0022] in, After the transformation, when a point in a high-dimensional multi-machine power system approaches infinity, the trajectory will be mapped onto a new sphere:

[0023]

[0024] When x i As we approach infinity, h also tends towards 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) in the finite space S, respectively.

[0026] The finite space S is a unit sphere with coordinates (y1, y2, y3) and satisfies the equation of the sphere. 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. With these six points as the center, six corresponding phase planes can be constructed.

[0027] Let the coordinate system 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 a point in the system is a SAI, it will be contracted and projected onto the surface of the unit sphere; other points will be contracted and projected into the interior of the sphere. Taking the phase plane where S1(1, 0, 0) is located as an example, any point N(x1, x2, x3) in a high-dimensional multi-machine power system, assuming that this point is a SAI, will be contracted and projected onto the surface of the unit sphere. Through this point, a mapping relationship can be formed with point N on the phase plane. * The collinear relationship between (1, u1, u2) and point N′(y1, y2, y3) on the surface of the sphere can determine the specific coordinate relationship of the contraction projection transformation; based on the collinear relationship between N(x1, x2, x3) and N * The collinear relationship of (1, u1, u2) can be obtained as follows:

[0028]

[0029] Let u3 = 1 / x1, and take the partial derivative with respect to u, we can obtain the definition of the contraction projection transformation:

[0030]

[0031] Let the equation There is a solution as Sure and After obtaining the value, the specific coordinates of the point (1, u1, u2) on the phase plane can be determined. That is, the SAIs coordinates mapped onto the phase plane;

[0032] Based on N′(y1,y2,y3) and N * The collinear relationship of (1, u1, u2) must exist for any scalar This makes the following relation hold:

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

[0034] Combining the equation of the sphere The value of d can be determined:

[0035]

[0036] Therefore, the coordinates of a point on the phase plane mapped onto the surface of the sphere can be determined as follows:

[0037]

[0038] The coordinates of SAIs in the high-dimensional multi-machine power system obtained by combining equation (7) mapped to the phase plane The coordinates of the SAIs coordinates in a high-dimensional multi-machine power system mapped onto the surface of a sphere can be obtained as follows:

[0039]

[0040] Due to the symmetry of the sphere, the remaining SAIs can be determined 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 a high-dimensional multi-machine power system is disturbed, the first step is to identify the corresponding coherent generator group based on the generator swing curve characteristics. When a fault occurs at a certain point in the system, there are two groups: one group is the leading generator group S, which is severely disturbed and whose power angle curve changes rapidly; the other group is the remaining generator group A, whose power angle curves are relatively stable. The rotor motion equations of the two groups are then added together to obtain...

[0043]

[0044] For the leading fleet S and the remaining fleet A, where k S and k A δ represents the ratio of the damping coefficient D to the inertial time constant M for the two generator groups; m and n represent the number of generators in the two generator groups, respectively; S and δ A These are the equivalent rotor angle centers of the two machine groups, respectively; ω S and ω A The equivalent electrical angular velocities of the two groups of machines are respectively; P ms and P ma The equivalent mechanical power of the two groups of machines are respectively; P es and P ea These are the equivalent electromagnetic power of the two groups of generators, respectively. and Let Δδ = δt, where δ is the derivative of δt with respect to time t. To visually represent the positional distribution of SAIs in three-dimensional phase space, it is assumed that no oscillations occur between the two groups of machines, and let Δδ = δt. S -δ A The third-order equation of equation (12) is shown below:

[0045]

[0046] According to the definition of contraction projection transformation mentioned in equation (7), the state variables (Δδ, ω) of the above multi-machine power system in equation (13) S ,ω A Point N on phase plane S1(1,0,0) *There is always a collinear relationship between (1, u1, u2): 1 / Δδ=u1 / ω S =u2 / ω A Let u3 = 1 / Δδ, then taking the derivative with respect to u, we get:

[0047]

[0048] in, and Let these be the derivatives of u with respect to time t;

[0049] make Solving the equation yields three sets of solutions on the phase plane S1(1,0,0), which are the three sets of singularities A. V (1,0,0), B V (1,0,k A ) and C V (1,-k S ,0);

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

[0051]

[0052] in via A V After the coordinate transformation (1,0,0), two fixed points are projected onto the unit sphere. and via B V (1,0,k A Coordinate transformation yields the coordinates projected onto the unit sphere. and Symmetric about the center of the sphere and distributed along the circumference of the sphere's surface; via C V (1,-k S After coordinate transformation, the coordinates projected onto the unit sphere are obtained. and Symmetric about the center of the sphere and distributed on the circumference of the sphere's surface.

[0053] Furthermore, the steps for calculating the area of ​​the spherical triangle 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. Next, calculate the side lengths of the spherical triangle, i.e., the spherical distances between the three vertices. Using the spherical distance formula, we obtain a = Rarccos(H·J), b = Rarccos(J·L), and c = Rarccos(L·H), where "·" indicates the dot product. After determining the spherical distances, we can further determine the sizes of the three spherical interior angles α, β, and γ by describing the relationship between the side lengths of the spherical triangle and their corresponding interior angles using the spherical cosine theorem: cos(α) = (cosα - cosbcosc) / sinβsinc, cos(β) = (cosb - cosαcosc) / sinαsinc, and cos(γ) = (cosc - cosαcosb) / sinαsinb. Finally, by calculating the spherical angle E = α + β + γ - π, we can determine the surface area S of the spherical triangle. 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 region distributed on the surface is based on a spherical triangle, and the area S of the spherical triangle is... 球Δ It is an important measure for describing the physical properties of a sphere.

[0057] The beneficial effects of this invention are as follows:

[0058] With the rapid development of my country's interconnected power grid and new energy sources, the stability mechanism of new power systems has undergone tremendous changes, posing severe challenges to the safe and stable operation of the power grid. Manifold and spherical geometry theories provide new insights into the dynamic characteristics of nonlinear dynamic systems. This invention, from the perspective of stability domain structure, elaborates on the role of the infinite singularity in characterizing the manifold topology and the boundary of the complete stability domain, aiming to provide more comprehensive theoretical support for the stable operation of the power grid. Based on Poincaré compactness theory, the coordinates of the infinite singularity in a multi-machine power system are derived. Combining the concept of spherical geometry, the geometric distribution of the infinite singularity located on the (hyper)sphere is analyzed. Using the mapping relationship between the infinite singularity and the power system stability domain, an infinite singularity evaluation parameter based on spherical geometry principles is proposed to determine the transient stability of the power system. Simulation results show that the evaluation parameter proposed in this invention has the advantage of higher accuracy and can effectively determine the transient stability of the system. Attached Figure Description

[0059] Figure 1 This is a diagram showing the location distribution of stable manifolds on the boundary of this invention.

[0060] Figure 2This is a schematic diagram of the six coordinate ranges of the present invention;

[0061] Figure 3 This is a geometric distribution diagram of the multi-machine power system SAIs of the present invention;

[0062] Figure 4 This is a diagram of the 16-machine, 68-node test system of the present invention;

[0063] Figure 5 The SAIs evaluation parameter Q proposed in the references 2 picture;;

[0064] Figure 6 The SAST diagram is the evaluation parameter for SAIs proposed in this invention.

[0065] Figure 7 This is a power angle curve of the test system under fault 1 of the present invention;

[0066] Figure 8 This is a power angle curve of the test system under fault condition 4 of the present invention;

[0067] Figure 9 This is a power angle curve of the test system under fault condition 7 of the present invention. Detailed Implementation

[0068] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments. These embodiments are based on the technical solution of the present invention and provide detailed implementation methods and specific operating procedures. However, the scope of protection of the present invention is not limited to the following embodiments.

[0069] A transient stability analysis method for power systems based on spherical geometry includes the following steps:

[0070] (Fundamentals 1.2, 1.3) This section covers the calculation method of SAIs in three-dimensional space and how to apply this method to multi-machine power systems. These two foundational steps effectively map out SAIs, facilitating the subsequent construction of relevant indicators based on the geometric distribution of SAIs. This includes indicators obtained in previous articles (where the indicator was previously named SVI, but has been renamed Q). 2 Both the standard SAIs and the metric SAST used in this paper require these two basic calculation methods. (Section 1.1 only briefly explains the necessity of SAIs research.)

[0071] (Improved implementation of Section 2.2) Construct the metric SAST. (Section 2.1 compares and illustrates the superiority of SG over EG.)

[0072] (Improved Implementation and Comparative Verification) (Section 3.1) Through actual simulation verification, the two indicators Q are compared and verified. 2Compared with SAST, the previous indicator Q was found. 2 The drawback is that different fault conditions can result in identical indicators, making it impossible to effectively distinguish the severity of their stability. However, the SAST index established in this invention effectively avoids this drawback. Therefore, this invention concludes that SAST has superior accuracy.

[0073] (Section 3.2) The effectiveness of the SAST index was verified by comparing it with the "critical settling time tcr under fault conditions".

[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 nonlinear systems, the core of studying their stability lies in analyzing the stability of the system's equilibrium points. When calculating the equilibrium points of a nonlinear system, consider a system of the following form:

[0078]

[0079] Let any equation dx / dt = f(x) with respect to the unknown x be a nonlinear system equation;

[0080] Here, the equilibrium point of the system refers to the point that satisfies the equation f(x) s The solution with t=0 has a state transition process φ(t,x(0)) representing the trajectory of the solution starting from time t=0. In phase space, these solutions converge to the equilibrium point x. s The set of all initial points is called the stability region of the equilibrium point:

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

[0082] When x = x s When x = 0, we can find f(x) = 0, and at this time we consider x to be... s For the equilibrium point of system f(x), formula (2) is to let x start from the initial point and change with the increase of time t, so that x eventually approaches the equilibrium point x we ​​want. s This formula allows us to find an expression that eventually approaches x over time. s These x's constitute the equilibrium point x'. s Stability region; Ω(x) s This means that x eventually approaches x over time. sStored therein are all x values ​​that meet the requirements, which are also the equilibrium points x. s The surrounding stable region can also be called the stable boundary;

[0083] Based on the above definition, it is not difficult to find Ω(x) s The boundary of ) is dotted with critically stable equilibrium points, and such equilibrium points (denoted as x) are denoted as x. u The Jacobian matrix of has eigenvalues ​​containing both positive and negative real parts. This means that the equilibrium point x... u The neighborhood of contains both unstable manifolds W u (x u )={x|φ(t,x)→x u ,t→-∞} and stable manifold W s (x u )={x|φ(t,x)→x u ,t→+∞}. Where, W s (x u Starting from a distance, they eventually converge at x. u W u (x u Starting from that point and gradually moving away from x u The union of the stable manifolds at these specific equilibrium points on the boundary constitutes the stable equilibrium point x. s BSR, that is

[0084]

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

[0086] It is evident that in order to determine an equilibrium point x s For the BSR, it is not necessary to calculate the set of all trajectories that can converge to the point; it is only necessary to calculate the union of the stable manifolds on the boundary.

[0087] For an equilibrium point, the manifold that approaches that point is called a stable manifold, and the manifold that moves away from that point is called an unstable manifold. Figure 1 A schematic diagram of the location of stable manifolds on the boundary is given. In this case, there exists a class of situations where the number of equilibrium points on the boundary is small, and the local manifolds within the neighborhood of these equilibrium points are insufficient to form a closed BSR. For example... Figure 1 The diagram shows that stable equilibrium points are surrounded by unstable equilibrium points; this region is not closed, and there are no closed curves at either end. To obtain a complete closed BSR, SAIs are needed. SAIs start or end at infinity and end or start at the unstable equilibrium points in the diagram, respectively, to form a closed BSR.

[0088] From a topological perspective, local information from the BSR alone is insufficient to represent the complete dynamic characteristics of a nonlinear system. Whether the manifolds on the stability domain boundary can curl inward to form a closed BSR requires further analysis. To obtain the global structure of the BSR, a contraction projection transformation method is typically used to project the system onto a finite spatial range, observing the topological structure of the system's stability domain and its complete BSR within the projected space.

[0089] 1.2 Calculation of SAIs in 3D Space

[0090] The following uses a 3D nonlinear system as an example to illustrate the contraction projection transformation method and the morphology and distribution of SAIs in 3D space. Any point (x1, x2, x3) in the original system can be mapped to a corresponding point (y1, y2, y3) in a finite space, with the corresponding coordinate mappings as follows:

[0091]

[0092] in, After the transformation, when a point in the original system approaches infinity, the trajectory will be mapped onto a new sphere:

[0093]

[0094] When x i As we approach infinity, h also tends towards infinity, and correspondingly... Specifically, all trajectories in the original system will be contracted and projected into the interior of a closed finite space S, while trajectories at infinity will be contracted and projected onto the surface of the finite space S. If SAIs exist in the original system at infinity, they must be distributed on the surface of the finite space S.

[0095] 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) in the finite space S, respectively.

[0096] like Figure 2 As shown, the finite space S is a unit sphere with coordinates (y1, y2, y3) and satisfies the equation of the sphere. S1, S′1, S2, S′2, S3, and S′3 are the points on point S that intersect the coordinate system (y1, y2, y3) of the unit sphere. Using these six points as centers, six phase planes can be constructed. These six phase planes form a cube circumscribed around the unit sphere, allowing for effective observation of the global structure of the unit sphere. The process of establishing the phase planes is illustrated using vertices S1, S2, and S3 as examples: 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 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 a point in the system is a SAI, it will be contracted and projected onto the surface of the unit sphere; other points will be contracted and projected onto the interior of the sphere. Taking the phase plane containing S1(1, 0, 0) as an example, any point N(x1, x2, x3) in the original system, assuming this point is a SAI, will be contracted and projected onto the surface of the unit sphere. Through this point, a mapping relationship can be formed with point N on the phase plane. * The collinearity between (1, u1, u2) and point N′(y1, y2, y3) on the surface of the sphere can determine the specific coordinate relationship of the contraction projection transformation. Based on the collinearity between N(x1, x2, x3) and N... * The collinear relationship of (1, u1, u2) can be obtained as follows:

[0098]

[0099] Let u3 = 1 / x1, and take the partial derivative with respect to u, we can obtain the definition of the contraction projection transformation:

[0100]

[0101] Let the equation There is a solution as Sure and After obtaining the value, the specific coordinates of the point (1, u1, u2) on the phase plane can be determined. That is, the SAIs coordinates mapped onto the phase plane.

[0102] Based on N′(y1,y2,y3) and N * The collinear relationship of (1, u1, u2) must exist for any scalar This makes the following relation hold:

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

[0104] Combining the equation of the sphere The value of d can be determined:

[0105]

[0106] Therefore, the coordinates of a point on the phase plane mapped onto the surface of the sphere can be determined as follows:

[0107]

[0108] The coordinates of SAIs in the original system obtained by combining equation (7) mapped to the phase plane The coordinates of the SAIs in the original system mapped onto the surface of the sphere can be obtained as follows:

[0109]

[0110] The above describes the calculation process of projecting the SAIs at a certain point on the sphere within the phase plane S1(1,0,0) of the original system. Due to the symmetry of the sphere, the remaining SAIs can be calculated in other phase planes.

[0111] 1.3 SAIs Calculation for Multi-Machine Power Systems

[0112] The equations of motion for multi-machine power systems have high dimensions, making direct calculation of high-dimensional manifolds challenging. To apply the coordinate transformation method mentioned earlier to multi-machine power systems, it is necessary to reduce the dimensionality of the high-dimensional multi-machine system, then obtain the projection of the global manifold in three-dimensional phase space through a shrinking projection transformation, and finally perform observation and analysis.

[0113] When a multi-machine power system is disturbed, the first step is to identify the corresponding coordinating generator group based on the generator swing curve characteristics. When a fault occurs at a point in the system, there are two groups: one group is the leading generator group S, which is severely disturbed and whose power angle curve changes rapidly; the other group is the remaining generator group A, whose power angle curves are relatively stable. Adding the rotor motion equations of the two groups yields...

[0114]

[0115] For the leading fleet S and the remaining fleet A, where k S and k A The damping coefficients of the two groups of aircraft are respectively D The ratio to the inertial time constant M. m and n are the number of generators in the two generator groups, respectively. δ S and δ A These are the equivalent rotor angle centers of the two machine groups, respectively. ω S and ωA P represents the equivalent electric angular velocity of the two machine groups. ms and P ma These represent the equivalent mechanical power of the two machine groups, respectively. es and P ea These are the equivalent electromagnetic power of the two groups of generators, and Let Δδ = δt, where δ is the derivative of δt with respect to time t. To visually represent the positional distribution of SAIs in three-dimensional phase space, it is assumed that no oscillations occur between the two groups of machines, and let Δδ = δt. S -δ A The third-order equation of equation (12) is shown below:

[0116]

[0117] According to the definition of contraction projection transformation mentioned in equation (7), the state variables (Δδ, ω) of the above multi-machine power system in equation (13) S ,ω A Point N on phase plane S1(1,0,0) * There is always a collinear relationship between (1, u1, u2): 1 / Δδ=u1 / ω S =u2 / ω A Let u3 = 1 / Δδ, then differentiating with respect to u, we get:

[0118]

[0119] make Solving the equation yields three sets of solutions on the phase plane S1(1,0,0), which are the three sets of singularities A. V (1,0,0), B V (1,0,k A ) and C V (1,-k S ,0).

[0120] By using the mapping relationship between any SAIs in the original system in equation (11) and 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. Their positional distribution is as follows: Figure 3 As shown, the specific coordinates of the 6 SAIs mapped onto the surface of the sphere are:

[0121]

[0122] in

[0123] via A V After the coordinate transformation (1,0,0), two fixed points are projected onto the unit sphere. and via B V (1,0,k A Coordinate transformation yields the coordinates projected onto the unit sphere. and Symmetric about the center of the sphere and distributed along the green circumference of the sphere's surface, the collinear relationship between two points is represented by a green line segment with an arrow; through C V (1,-k S After coordinate transformation, the coordinates projected onto the unit sphere are obtained. and Symmetric about the center of the sphere and distributed on the yellow circumference of the sphere's surface, the collinear relationship between two points is represented by a yellow line segment with an arrow.

[0124] 2SG and Transient Stability Index Construction Method

[0125] In practical transient analysis of multi-machine power systems, the grid structure of the system after a fault is determined the instant the fault is cleared, and the corresponding equivalent system parameters are also determined. That is, it is not necessary to wait for the actual power angle to reach infinity; the location distribution of SAIs and their evaluation parameters can be obtained the instant the fault is cleared, thus effectively measuring the transient stability of the system. When constructing evaluation parameters for the changes in the location distribution of SAIs, SAIs projected onto a sphere can be observed and analyzed from different geometric domains. Different geometric domains have different representation methods and advantages. Next, this section will compare the characteristics of EG space and SG space in detail, and construct SAIs evaluation parameters based on SG space characteristics; its superiority will be demonstrated below.

[0126] By observing the geometric distribution of SAIs in the aforementioned multi-machine system on the sphere, two sets of SAIs can be obtained, and these two sets of SAIs... and They are symmetric about the center of the unit sphere, respectively. Based on their positional distribution on the sphere, two inscribed planar triangles symmetric about the center of the sphere can be constructed based on EG. Reference (Ma Meiling, Wang Jie, Li Penghan, et al. Global manifold analysis on the power angle stability domain and boundary of power system [J]. Proceedings of the CSEE, 2020, 40(18): 5865-5875.) Based on a certain set of SAIs (with Taking the distribution of SAIs (for example), the area of ​​the inscribed triangle formed by them is used as the evaluation parameter Q for the change in the location distribution of SAIs. 2 Based on these evaluation parameters, a stability analysis of the multi-machine system was conducted, such as... Figure 5 As shown.

[0127] Corresponding to the planar case of EG, if we analyze the coordinate positions of these two sets of SAIs from the perspective of SG, we can theoretically obtain more accurate singularity change evaluation parameters. Next, this invention will demonstrate this from the perspective of SG, constructing SG-based SAIs evaluation parameters to measure the transient stability of the power system.

[0128] 2.1SG and spherical triangle

[0129] SG is a field of geometry that deals with the properties of points, curves, and angles on a three-dimensional sphere. SG differs from the more commonly used plane EG in many ways. The main differences between the two regarding the transformation from a plane to a sphere are shown in Table 1.

[0130] Table 1 Comparison of EG and SG

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

[0132]

[0133] For SG (Spherical Geometry), the characteristics of a region on a sphere can be effectively observed from any direction, and the geometric shape represented on the sphere changes with the numerical value. In SG, the simplest geometric shape is a spherical triangle, consisting of three vertices on the sphere and the great circle segment between them. When analyzing the characteristics of a spherical triangle, its area is an important indicator. The steps to calculate the area of ​​a spherical triangle are as follows:

[0134] 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, i.e., the spherical distances between the three vertices. Using the spherical distance formula, we obtain a = Rarccos(H·J), b = Rarccos(J·L), and c = Rarccos(L·H), where "·" indicates the dot product. After determining the spherical distances, we can further determine the sizes of the three spherical interior angles α, β, and γ by describing the relationship between the side lengths of the spherical triangle and their corresponding interior angles using the spherical cosine theorem: cos(α) = (cosa - cosbcosc) / sinbsinc, cos(β) = (cosb - cosacosc) / sinasinc, and cos(γ) = (cosc - cosacosb) / sinasinb. Finally, by calculating the spherical angle excess E = α + β + γ - π, we determine the surface area S of the spherical triangle. 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 surface is based on a spherical triangle, and the area S of the spherical triangle is... 球Δ It is an important measure for describing the physical properties of a sphere. For example... Figure 3 As shown, after a contraction projection transformation, the surface areas (SAIs) of a multi-machine power system are typically distributed on the surface of a unit sphere in three-dimensional projection space. As the base plate (BSR) changes, the SAIs will move along the sphere. As endpoints of the BSR, the stability and location of the SAIs depend on the structural characteristics of the system after a fault. The addition of SAIs helps to outline the complete structure of the BSR; furthermore, the distribution characteristics of the SAIs can reflect the changes in the BSR, thus reflecting the stability of the system.

[0137] Therefore, it can be seen that the changes in SAIs are related to the stability of the system. The following text uses the SAIs distribution parameters as a benchmark to construct an index of the transient stability of the power system.

[0138] 2.2 Constructing a transient stability index for power systems

[0139] Projecting the distribution of SAIs positions on the sphere onto an inscribed plane triangle inside the sphere and representing the spherical image with a planar image, the SAIs evaluation parameters constructed based on this are not accurate. Although this EG-based representation can achieve the measurement objective to some extent, it inevitably leads to discrepancies between data correlation and reality, resulting in some errors. Specifically, the SAIs evaluation parameter Q proposed based on EG... 2 It can be determined that when the location distribution of SAIs in a multi-machine power system changes, different areas of the inscribed planar triangles will be generated. The three vertices of these planar triangles are the three SAIs mapped onto the surface of the same hemisphere. In this case, it is possible for two different inscribed planar triangles to have the same area. The evaluation parameter Q in this situation... 2 It will be impossible to measure the difference between the two scenarios.

[0140] When the above problems occur, the SAIs evaluation parameters constructed using the spherical triangle area based on SG exhibit better performance and also have certain advantages in numerical calculation. This invention uses the area of ​​the spherical triangle formed by three SAIs on the same hemisphere to define the surface area of ​​a spherical triangle (SAST) as the SAIs evaluation parameter. Based on the previous analysis of SG and spherical triangles, the SAST representation can be obtained as follows:

[0141] SAST=R 2 (α+β+γ-π)×100% (17)

[0142]

[0143] Where 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 the three points. This evaluation parameter reflects the changing trend of the equilibrium point at infinity, embodies the expansion and contraction of the stability region, and can also measure the change in the transient stability of the system.

[0144] 3. Simulation Analysis and Verification

[0145] Transient stability assessment of power systems needs to consider both the system's structural stability and its state changes after disturbances. To verify the effectiveness of these assessment parameters in large-scale power grids, this section presents simulations on the IEEE 16-machine 68-bus system model, such as... Figure 4 As shown, the transient power angle stability of the system after different faulty lines are disconnected is analyzed. The distribution characteristics of SAIs (Short-term Aspects of Instability) of the multi-machine system after shrinkage projection transformation are investigated, and the evaluation parameter Q of the two SAIs is compared. 2 The difference between SAST and QAST is verified. 2 The advantages of SAST are demonstrated. Simultaneously, the effectiveness of SAST in measuring the transient power angle stability of power systems is verified. The specific steps are as follows:

[0146] 1) Disconnect different faulty lines, simplify the multi-machine system model into an equivalent model according to the grouping of generator sets, and calculate the corresponding equivalent parameters;

[0147] 2) Obtain the distribution of SAIs using the shrinking projection transformation method given in Section 1.2;

[0148] 3) Based on the content of Section 2.2, calculate the SAIs evaluation parameters as an indicator of the change in stability under different fault line conditions.

[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). Figure 4 The fault duration is set to 300ms. The system grouping results are completely different under different fault conditions. After the fault occurs, when the leading group S is identified, all other groups are assigned to the remaining group A. After equivalent processing, the equivalent power angle δ of the two groups can be obtained. S and δ A and the corresponding damping coefficient k Sand k A According to the damping coefficient k S and k A The evaluation parameters SAST and Q of SAIs can be further calculated. 2 The transient stability under different fault conditions was analyzed.

[0150] 3.1 Comparison of different SAI evaluation parameters

[0151] As shown in Table 2 (the fault locations are randomly selected here to facilitate the analysis of the superiority of the coordinate SAST proposed in this invention), Fault 1 refers to setting a three-phase short-circuit fault at node 31 and clearing the faulty lines 28-26 after 300ms; Fault 2 refers to setting a three-phase short-circuit fault at node 26 and clearing the faulty lines 24-68 after 300ms; Fault 3 refers to setting a three-phase short-circuit fault at node 81 and clearing the faulty lines 67-66 after 300ms; Fault 4 refers to setting a three-phase short-circuit fault at node 15 and clearing the faulty lines 15-42 after 300ms. Fault 5 refers to setting a three-phase short-circuit fault at node 20 and clearing the faulty line 19-68 after 300ms; Fault 6 refers to setting a three-phase short-circuit fault at node 10 and clearing the faulty line 10-31 after 300ms; Fault 7 refers to setting a three-phase short-circuit fault at node 13 and clearing the faulty line 13-17 after 300ms. It is not difficult to see that two sets of faults (faults 1 and 4, and faults 2 and 3) have appeared in the system. 2 Even though the indexes are equal in magnitude, the SAST evaluation parameter for the two faults is not equal in magnitude. This indicates that the evaluation parameter SAST is related to Q. 2 It has higher accuracy.

[0152] Among them, G i' {i'∈(1,16)} represents the 16 generators in the system, which will generate different leading generator groups under different fault conditions; t cr This represents the critical settling time of the system under fault conditions, and its duration is positively correlated with the transient stability of the power system. (t) cr The explanation in Section 3.2 regarding the use of SAST to determine the transient power angle stability of a multi-machine system exists; this is achieved by defining t as a node with a power angle difference greater than 180°. cr The effectiveness of SAST was verified by comparing the critical resection time with the SAST index.

[0153] Table 2 SAI evaluation parameters (Q) under different fault conditions 2 (with SAST)

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

[0155]

[0156] 1) SAIs evaluation parameter Q for fault 1 and fault 4 2 Both are 0.2817, but the SAST of fault 1 is 37.0086 and the SAST of fault 4 is 32.8762, which are not the same.

[0157] 2) SAIs evaluation parameter Q for faults 2 and 3 2 Both are the same at 0.2614, but the SAST of fault 2 is 34.118 and the SAST of fault 3 is 33.9754.

[0158] Both of the above situations indicate that the evaluation parameter Q 2 There are certain accuracy limitations in measuring the transient stability of power systems. To clearly illustrate the geometric distribution of the evaluation parameters for the two types of SAIs, a comparative analysis is conducted using fault 1 and fault 4 as examples. Table 3 shows the system equivalent parameters and their corresponding coordinate distributions for the two sets of SAIs under the two fault conditions.

[0159] The system SAIs evaluation parameter Q under two fault conditions 2 The geometric distribution of parameters corresponding to SAST is as follows: Figure 5 and Figure 6 As shown. Among them, This is a set of SAIs under fault condition 1. The blue shaded area represents the geometric distribution of the evaluation parameters of the SAIs under fault condition 1. This is a set of SAIs under fault condition 4. The red shaded area represents the geometric distribution of the evaluation parameters for the SAIs under fault condition 4. The same two fault types... Figure 5 Evaluation parameter Q in 2 The statement indicated that the two inscribed plane triangles had equal areas, but no correct conclusion was given. Figure 6 The areas of the curved triangles on the sphere surface represented by the evaluation parameter SAST under the two fault conditions are not equal, which can effectively distinguish the transient stability of the power system. SAST is superior to Q... 2 It can more accurately determine the transient stability of the system under different fault conditions.

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

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

[0162]

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

[0164] Taking faults 1, 4, and 7 from Table 2 as examples, the effectiveness of the evaluation parameter SAST for transient power angle stability analysis is verified. Under different fault conditions, the 16 generators of the system are divided into a leading group S and a remaining group A according to the homogeneity of their power angle curves. The generator power angle curves of the simulated system after faults 1, 4, and 7 occur individually are shown below. Figure 7 , Figure 8 and Figure 9 As shown.

[0165] The curve trend indicates that the system has been split into two operating modes, reflecting the changing trends of the power angles of the two groups of generators. For example... Figure 7 As shown, after fault 1, the system is divided into: a leading group S = {2,3,4,5,6,7,8,9} and a remaining group A = {1,10,11,12,13,14,15,16}. The equivalent power angle δ of the leading group is... S The equivalent power angle δ of the remaining units is represented by a thick black dashed line in the diagram. A This is indicated by a thick red dashed line in the diagram. Then, the following are listed in order: Figure 8 The system following fault 4 is divided into: the leading group S = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,16} and the remaining group A = {15}, δ S Represented by a thick black dashed line, δ A Represented by a thick red dashed line. For example... Figure 9 The system following fault 7 is divided into: the leading group S = {1,2,3,4,5,6,7,8,9,10,11,12} and the remaining group A = {13,14,15,16}, δ S Represented by a thick black dashed line, δ A Represented by a thick red dashed line.

[0166] After determining the equivalent power angles of the two generator groups, a threshold of 180° is set between their equivalent power angles. When the difference between the equivalent power angles of the two generator groups exceeds 180°, the multi-machine system loses stability, and the corresponding time t is the critical stabilization time t under this fault condition. cr The duration of this time is positively correlated with the transient stability of the power system. When tcr Over a longer period, the system has more time to adapt to the fault, thus slowing down the steady-state process. This means the system has better transient stability and can better maintain the stability of frequency, voltage, and power, thereby reducing the risk of system instability. If t cr If the fault is very short, the system may not be able to adapt to it, resulting in drastic fluctuations in frequency and voltage, which could lead to system failure. This is detrimental to the system's transient stability.

[0167] By comparison Figure 7 , Figure 8 and Figure 9 The system power angle curve can be used to visually obtain:

[0168] 1) The system's t after fault 1 cr The time interval is 1038ms, and the system's time interval after fault 4 is t. cr The time was 812ms. Comparing fault 1 and fault 4, it was found that the system had a longer time to adapt to the fault after fault 1, and the transient stability of the system after fault 1 was stronger than that after fault 4.

[0169] 2) The system's t after fault 7 cr The critical clearance time of the system after fault 7 is 605ms. A comparison between fault 7 and fault 4 shows that the critical clearance time of the system after fault 7 is shorter than that after fault 4, and the transient stability of the system after fault 7 is weaker than that after fault 4.

[0170] Therefore, it can be determined 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 in this invention will be compared with the critical resection time t. cr Comparative analysis was conducted to verify the effectiveness of SAST.

[0171] After performing a shrinking projection transformation on the equivalent model, the SAST (Short-Short Angle Allocation) evaluation parameter based on SG (Short-Short Angle) can be further obtained. Different faults correspond to different SAST and critical resection times t. cr By comparing the degree of change 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 for faults 1 to 7 gradually decreases from 37.0086 for fault 1 to 24.6918 for fault 7, and its critical fault clearing time t cr The time also decreased sequentially from 1038ms for fault 1 to 605ms for fault 7. The sequential decrease between the two verifies the effectiveness of the evaluation parameter SAST in measuring the transient stability of the power system.

[0172] 4. Summary

[0173] This invention explores the importance of stability regions and manifolds in nonlinear dynamic systems, emphasizing the role of SAI (Stable Area Transformation) for the boundary of a complete stability region. By introducing the contraction projection transformation method and the SAI calculation method in three-dimensional space, and through the derivation of mathematical models and equations, the coordinate mapping relationship between them is elucidated. Equivalent calculations were performed on a multi-machine power system to obtain the SAI-based projective spatial topology. Combining the concepts of spherical geometry and spherical triangles, and taking the system's stability and post-disturbance state changes as references, the SAI evaluation parameters based on EG (Geometric Evolution) were optimized, and a SAI evaluation parameter based on SG (Spherical Stability) was proposed. The superiority of this parameter in terms of accuracy was verified, as well as its effectiveness in measuring the transient stability of power systems. Overall, this invention provides a new perspective and theoretical support for the transient stability analysis of power systems, and offers beneficial ideas and methods for expanding the application scope of power system stability theory.

[0174] The above-described embodiments are merely one implementation of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these all fall within the protection scope of the present invention. Therefore, the protection scope of this invention should be determined by the appended claims.

Claims

1. A transient stability analysis method for power systems based on spherical geometry, characterized in that, Includes the following steps: Step 1: Dimensionality reduction is performed on the high-dimensional multi-machine power system. The global manifold projection is obtained in the unit projection sphere of the three-dimensional phase space by using the shrinking projection transformation method. After obtaining the geometric distribution of the infinite singularities SAIs of the high-dimensional multi-machine power system on the surface of the unit projection sphere, observation and analysis are then performed. Step 2: By constructing and observing the geometric distribution of SAIs (Self-Alignment Aspects) of a high-dimensional multi-machine power system on a sphere, two sets of SAIs can be obtained. and in, and For any SAIs in a high-dimensional multi-machine power system, two sets of spherical coordinates symmetric about the center of the sphere are obtained through the mapping relationship between the phase plane and the surface of a unit sphere. Step 3: Construct SAIs evaluation parameters based on spherical geometry SG to measure the transient stability of the power system: The area of ​​the spherical surface triangle SAST formed by three SAIs on the same hemisphere is defined as the SAIs evaluation parameter.

2. The transient stability analysis method for power systems based on spherical geometry according to claim 1, characterized in that, SAST is represented as follows: SAST=R 2 (a+b+c-p)×100% (17) Where 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 the three points.

3. The transient stability analysis method for power systems 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: In a high-dimensional multi-machine power system, any point (x1, x2, x3) can be mapped to a corresponding point (y1, y2, y3) in a finite space. The corresponding coordinate mapping is as follows: in, After the transformation, when a point in a high-dimensional multi-machine power system approaches infinity, the trajectory will be mapped onto a new sphere: When x i As we approach infinity, h also tends towards infinity, and correspondingly... 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) in the finite space S, respectively. The finite space S is a unit sphere with coordinates (y1, y2, y3) and satisfies the equation of the sphere. 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. With these six points as the center, six corresponding phase planes can be constructed. Let the coordinate system 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 a point in the system is a SAI, it will be contracted and projected onto the surface of the unit sphere; other points will be contracted and projected into the interior of the sphere. Taking the phase plane where S1(1, 0, 0) is located as an example, any point N(x1, x2, x3) in a high-dimensional multi-machine power system, assuming that this point is a SAI, will be contracted and projected onto the surface of the unit sphere. Through this point, a mapping relationship can be formed with point N on the phase plane. * The collinear relationship between (1, u1, u2) and point N′(y1, y2, y3) on the surface of the sphere can determine the specific coordinate relationship of the contraction projection transformation; based on the collinear relationship between N(x1, x2, x3) and N * The collinear relationship of (1, u1, u2) can be obtained as follows: Let u3 = 1 / x1, and take the partial derivative with respect to u, we can obtain the definition of the contraction projection transformation: Let the equation There is a solution as Sure and After obtaining the value, the specific coordinates of the point (1, u1, u2) on the phase plane can be determined. That is, the SAIs coordinates mapped onto the phase plane; Based on N′(y1,y2,y3) and N * The collinear relationship of (1, u1, u2) must exist for any scalar This makes the following relation hold: (y1,y2,y3)=d(1,u1,u2) (8) Combining the equation of the sphere The value of d can be determined: Therefore, the coordinates of a point on the phase plane mapped onto the surface of the sphere can be determined as follows: The coordinates of SAIs in the high-dimensional multi-machine power system obtained by combining equation (7) mapped to the phase plane The coordinates of the SAIs coordinates in a high-dimensional multi-machine power system mapped onto the surface of a sphere can be obtained as follows: Due to the symmetry of the sphere, the remaining SAIs can be determined in other phase planes.

4. The transient stability analysis method for power systems based on spherical geometry according to claim 1, characterized in that, The process of obtaining the two sets of spherical coordinates symmetric about the center of the sphere in step 2 is as follows: When a high-dimensional multi-machine power system is disturbed, the first step is to identify the corresponding coherent generator group based on the generator swing curve characteristics. When a fault occurs at a certain point in the system, there are two groups: one group is the leading generator group S, which is severely disturbed and whose power angle curve changes rapidly; the other group is the remaining generator group A, whose power angle curves are relatively stable. The rotor motion equations of the two groups are then added together to obtain... For the leading fleet S and the remaining fleet A, where k S and k A δ represents the ratio of the damping coefficient D to the inertial time constant M for the two generator groups; m and n represent the number of generators in the two generator groups, respectively; S and δ A These are the equivalent rotor angle centers of the two machine groups, respectively; ω S and ω A The equivalent electrical angular velocities of the two groups of machines are respectively; P ms and P ma The equivalent mechanical power of the two groups of machines are respectively; P es and P ea These are the equivalent electromagnetic power of the two groups of generators, respectively. and Let Δδ = δt, where δ is the derivative of δt with respect to time t. To visually represent the positional distribution of SAIs in three-dimensional phase space, it is assumed that no oscillations occur between the two groups of machines, and let Δδ = δt. S -δ A The third-order equation of equation (12) is shown below: According to the definition of contraction projection transformation mentioned in equation (7), the state variables (Δδ, ω) of the above multi-machine power system in equation (13) S ,ω A Point N on phase plane S1(1,0,0) * There is always a collinear relationship between (1, u1, u2): 1 / Δδ=u1 / ω S =u2 / ω A Let u3 = 1 / Δδ, then taking the derivative with respect to u, we get: in, and Let these be the derivatives of u with respect to time t; Let d Solving the equation yields three sets of solutions on the phase plane S1(1,0,0), which are the three sets of singularities A. V (1,0,0), B V (1,0,k A ) and C V (1,-k S ,0); By using the mapping relationship between any SAIs in the high-dimensional multi-machine power system and the surface of the unit sphere in equation (11), two sets of spherical coordinates symmetric about the center of the sphere can be obtained; the specific coordinates of the 6 SAIs mapped onto the surface of the sphere are: in via A V After the coordinate transformation (1,0,0), two fixed points are projected onto the unit sphere. and via B V (1,0,k A Coordinate transformation yields the coordinates projected onto the unit sphere. and Symmetric about the center of the sphere and distributed along the circumference of the sphere's surface; via C V (1,-k S After coordinate transformation, the coordinates projected onto the unit sphere are obtained. and Symmetric about the center of the sphere and distributed on the circumference of the sphere's surface.

5. The transient stability analysis method for power systems based on spherical geometry according to claim 1, characterized in that, The steps for calculating the area of ​​the spherical triangle 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, i.e., the spherical distances between the three vertices. Using the spherical distance formula, we obtain a = Rarccos(HJ), b = Rarccos(JL), and c = Rarccos(LH), where "·" indicates the dot product calculation. After determining the spherical distances, we can further determine the sizes of the three spherical interior angles α, β, and γ by describing the relationship between the side lengths of the spherical triangle and their corresponding interior angles using the spherical cosine theorem: cos(α) = (cosα - cosbcosc) / sinβsinc, cos(β) = (cosb - cosαcosc) / sinαsinc, and cos(γ) = (cosc - cosαcosb) / sinαsinb. Finally, by calculating the spherical angle E = α + β + γ - π, we can determine the surface area S of the spherical triangle. 球Δ The calculation formula is as follows: S 球Δ =R 2 (a+b+c-p) (16) For a three-dimensional sphere, any closed region distributed on the surface is based on a spherical triangle, and the area S of the spherical triangle is... 球Δ It is an important measure for describing the physical properties of a sphere.

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