Alternating current line static stability boundary analysis method and device based on phasor geometry and medium

Abstract

The invention discloses an AC line static stability boundary analysis method and device based on phasor geometry and a medium, and relates to the technical field of AC line static stability. The method comprises the following steps: constructing a voltage phasor triangle formed by head end voltage, tail end voltage and line voltage drop based on a line impedance model of a head end voltage reference system; the core relation that the apparent power is in direct proportion to the area of the triangle is found by analyzing the moving track of the vertex of the triangle along the arc of the corresponding circumcircle when the complex power at the tail end changes. Therefore, when the triangle is an isosceles triangle, the apparent power reaches the maximum value and is a line static stability critical point. Therefore, according to the method, the voltage phasor triangular isosceles is used as a geometric condition for judging the static stability boundary, and the static stability power, voltage, power angle and impedance boundary of the line are deduced. According to the method, a traditional complex nonlinear algebraic problem is converted into clear geometric analysis, the physical concept is clear, and the solving process is remarkably simplified.

Description

Technical Field

[0001] This invention belongs to the field of AC line static stability technology, and particularly relates to a method, equipment and medium for AC line static stability boundary analysis based on phasor geometry. Background Technology

[0002] Determining the static stability boundary of each AC line accurately, efficiently, and intuitively has become a critical issue that urgently needs to be addressed to ensure the safe and reliable operation of the power grid. The static stability boundary is not only the basis for assessing the safe current-carrying capacity of the line, but also a theoretical premise for studying the dynamic and transient stability of the system: the static stability region provides a recovery buffer for the dynamic oscillations of the system after being subjected to small disturbances, and also provides a key deceleration region for the transient instability process under large disturbances.

[0003] From a network structure perspective, although the power grid structure is complex, its active power flow exhibits a radial pattern. In static stability analysis, components such as lines, transformers, and generators can be considered as equivalent impedance branches, and the lower-level grid and its loads as a whole can be regarded as the equivalent terminal load of the upper-level transmission branch. Based on this, the upper-level branch must meet the static stability requirements. By analogy, the necessary and sufficient condition for the entire network to maintain static stability can be derived: that is, every impedance branch in the power grid must be in a statically stable state. Since the voltage at the beginning of the line is usually maintained relatively stable through generator excitation control, reactive power compensation, etc., and is less affected by changes in the power at the end, establishing a "beginning-point reference system impedance branch model" with the beginning-point voltage as the reference phase to study the static stability boundary of the line itself is a reasonable and effective theoretical simplification approach.

[0004] Currently, the traditional method for solving static stability boundaries is mainly the algebraic extremum method. This method requires establishing complex variable equations describing the power-voltage relationship of a line and transforming them into a set of high-order nonlinear real variable equations for extremum analysis. For example, previous research (such as the distributed load safety domain of the power grid and its application (II): boundary-essence-morphology of line static instability and stability margin of load safety domain [J]. Proceedings of the CSEE, 2024, 44(05): 1737-1750.) transformed the problem into two three-dimensional independent real variable equations through complex transformations, and then analyzed and obtained the consistent power boundary of voltage instability and power angle instability, and obtained the static voltage and power angle boundaries respectively. However, these real variable equations are all ternary fourth-order nonlinear transcendental equations, and the process of performing extremum condition analysis, extremum solution and partial derivative proof (i.e., strict boundary proof) is extremely complicated and analytically very difficult. More importantly, the pure algebraic derivation process is difficult to reveal the physical picture and internal logic of the changes in state variables, resulting in unintuitive analysis results. It cannot provide operators with a clear physical perception, nor can it form a simple and easy-to-engineer stable boundary judgment tool.

[0005] Therefore, given the increasingly complex grid operation characteristics caused by the high proportion of renewable energy integration, there is an urgent need to develop a new analytical method that can uniformly, concisely, and intuitively characterize the static stability boundaries of power, voltage, power angle, and impedance in order to overcome the shortcomings of traditional algebraic extremum methods, such as analytical difficulties and obscure physical meaning. Summary of the Invention

[0006] To address the aforementioned deficiencies in existing technologies, the present invention aims to provide a method, device, and medium for static stability boundary analysis of AC lines based on phasor geometry. This solves the technical problems of traditional algebraic extremum methods, which suffer from analytical complexity and computational difficulties due to high-order nonlinearity of equations, as well as poor physical intuitiveness and difficulty in forming intuitive judgment tools suitable for engineering applications. The present invention aims to provide a unified, concise, and physically clear analysis method by introducing phasor geometry analysis. This method intuitively reveals the evolution process and geometric essence of a line from stability to static instability, simplifies the solution of stability boundaries, and enhances the physical perception of the power grid's operating state. Therefore, it provides an efficient and intuitive analytical basis for the safe and stable operation of the power grid under the background of high-proportion renewable energy integration.

[0007] This invention solves the above-mentioned technical problems through the following technical solution: a static stability boundary analysis method for AC lines based on phasor geometry, comprising:

[0008] Based on the impedance model of the reference system of the AC line's starting voltage, the correlation between the line's equivalent impedance, line current phasor, starting voltage phasor, ending voltage phasor, and ending complex power is obtained.

[0009] Based on the aforementioned correlation, a voltage phasor triangle is constructed with the first-end voltage phasor, the last-end voltage phasor, and the line voltage drop as its sides; wherein, the line voltage drop is equal to the product of the line current phasor and the line equivalent impedance.

[0010] The analysis shows the trajectory of the vertex of the voltage phasor triangle on the circumcircle arc corresponding to the power factor angle when the power factor angle of the terminal complex power is a constant value and the apparent power changes.

[0011] Based on the movement trajectory, a proportional relationship is established between the area of ​​the voltage phasor triangle and the apparent power;

[0012] Based on the aforementioned proportional relationship, it is determined that the apparent power reaches its maximum when the voltage phasor triangle is an isosceles triangle;

[0013] Based on the apparent power reaching its maximum, which corresponds to the static stability critical state of the AC line, whether the voltage phasor triangle is an isosceles triangle is determined as the geometric condition for determining whether the AC line is at the static stability boundary; based on the geometric condition, the static stability boundary of the AC line is determined.

[0014] This invention transforms the complex algebraic extremum problem of solving high-order nonlinear transcendental equations into a problem that can be reasoned based on intuitive geometric rules such as the inscribed angle theorem and the sine theorem by constructing the physical quantities of the circuit as a voltage phasor triangle and analyzing the trajectory of its vertices on the circumcircle arc. This transformation greatly reduces the computational complexity and analytical difficulty of the analysis. More importantly, this method discovers and establishes the unique and universal geometric criterion that "the voltage phasor triangle is an isosceles triangle," unifying the criteria for determining the boundaries of voltage, power angle, power, and impedance that traditionally required separate processing, thus solving the core defects of traditional methods, such as cumbersome analysis and scattered conclusions.

[0015] This invention dynamically reveals the complete physical process by which changes in terminal complex power drive the movement of the vertices of the voltage phasor triangle until an isosceles critical state is reached, making the evolution of static instability visible and perceptible. This not only completely overcomes the drawbacks of the obscure physical meaning of traditional algebraic methods and enhances the intuitive understanding of operators, but also clarifies through geometric proof that voltage drops and power angle increases occur simultaneously at the critical point. This profoundly reveals the homogeneity of voltage stability and power angle stability at the level of static instability, deepening the theoretical understanding of the essence of stability.

[0016] Based on the aforementioned concise geometric criteria and clear physical picture, the method of this invention can be easily transformed into graphical and standardized engineering criteria and online monitoring algorithms. This allows frontline personnel to quickly assess stability margins without complex calculations, greatly promoting the transformation of theoretical results into practical engineering tools. Facing the power fluctuation challenges brought about by a high proportion of renewable energy integration, this method provides strong technical support for the power grid to achieve real-time, agile stability boundary perception and safety early warning, helping to improve the resilience and safety level of power grid operation.

[0017] Furthermore, in the impedance model of the first-end voltage reference system, the correlation includes:

[0018] Line equivalent impedance Line current phasors First-terminal voltage phasor and terminal voltage phasor Satisfied voltage equation: ;

[0019] and line current phasors Terminal voltage phasor and terminal complex power The power equation that is satisfied is: ,in Represents line current phasor The conjugate of complex numbers.

[0020] This invention establishes a solid and unambiguous physical and mathematical foundation for core geometric analysis by simultaneously clarifying the voltage equation and the power equation, ensuring the rigor, completeness, and engineering feasibility of subsequent geometric analysis methods.

[0021] Furthermore, the arc on which the movement trajectory lies is a minor arc, a semicircle, or a major arc, and its specific shape is determined by the difference between the impedance angle of the equivalent impedance of the line and the power factor angle.

[0022] This invention elucidates the three forms of the vertex trajectory arc (minor arc, semicircle, major arc) and their determining factors, fully revealing the geometric topology of the stable boundary under different operating conditions. This enables the analysis method to universally and accurately characterize the operating states of all lines from inductive to capacitive loads, enhancing the completeness and adaptability of the method.

[0023] Furthermore, the proportional relationship is determined by the following formula:

[0024] ;

[0025] in, Indicates apparent power; Indicates the power factor angle; This represents the magnitude of the equivalent impedance of the line. Represents the area of ​​the voltage phasor triangle; This indicates the impedance angle.

[0026] This invention provides a quantitative bridge for transforming the abstract problem of power transmission capacity into an intuitive problem of geometric extrema by establishing a precise mathematical relationship between the apparent power and the area of ​​the voltage phasor triangle. This is the key mathematical foundation for the method to ultimately derive accurate physical conclusions from geometric analysis.

[0027] Furthermore, the condition that the voltage phasor triangle is an isosceles triangle specifically means that the side corresponding to the terminal voltage phasor has the same length as the side corresponding to the line voltage drop.

[0028] This invention precisely maps the abstract geometric criterion "isosceles triangle" into a clear physical relationship, namely, the voltage amplitude at the end is equal to the voltage drop amplitude of the line. This gives the abstract geometric criterion a clear, measurable, and calculable engineering connotation, greatly facilitating the direct application and verification of this analysis method in engineering practice.

[0029] Furthermore, when the geometric conditions are used to determine the static stability boundary of the AC line, they are specifically used to determine at least one of the following boundaries: end voltage boundary, power angle boundary, apparent power boundary, and equivalent load impedance boundary.

[0030] This invention clearly points out that a single geometric criterion can uniformly derive multi-dimensional stability boundaries (voltage, power angle, power, impedance), fully demonstrating the systematic advantage of this method of "understanding one principle and clarifying all principles", realizing a comprehensive and integrated characterization of the line stability state, and greatly improving the engineering practical value and analysis efficiency of the method.

[0031] Furthermore, based on the aforementioned geometric conditions, the terminal voltage boundary, power angle boundary, apparent power boundary, and equivalent load impedance boundary are determined by the following formulas:

[0032] ;

[0033] ;

[0034] ;

[0035] ;

[0036] in, Indicates the terminal voltage stability boundary value; Indicates the magnitude of the voltage phasor at the beginning of the circuit; Indicates the impedance angle; Indicates the power factor angle; Indicates the stability boundary value of the work angle; Indicates the power stability boundary value; This represents the magnitude of the equivalent impedance of the line. This represents the stability boundary value of the equivalent load impedance; This represents the load impedance.

[0037] This invention provides analytical formulas for accurate voltage boundaries, power angle boundaries, apparent power boundaries, and equivalent load impedance boundaries directly derived from geometric conditions. It transforms intuitive geometric determinations into mathematical tools that can be directly used for quantitative calculations and engineering design, making the evaluation of stability limits simple, clear, and efficient to calculate. It solves the problem of complex solutions for voltage boundaries, apparent power boundaries, and equivalent load impedance boundaries in traditional methods.

[0038] Based on the same concept, the present invention also provides an electronic device, including a memory, a processor, and a computer program or instructions stored in the memory, wherein the processor executes the computer program or instructions to implement the phasor geometry-based AC line static stability boundary analysis method as described above.

[0039] Based on the same concept, the present invention also provides a computer-readable storage medium having a computer program or instructions stored thereon, which, when executed by a processor, implements the phasor geometry-based AC line static stability boundary analysis method as described above.

[0040] Compared with the prior art, the beneficial effects of the present invention are as follows:

[0041] This invention, through an innovative "phasor geometric analysis method," transforms the traditional stability boundary problem, which relies on complex algebraic solutions, into an intuitive problem of geometric graph construction and trajectory analysis. By establishing the simple and universal geometric criterion that "the voltage phasor triangle is an isosceles triangle," this method unifies and simplifies the determination process of multi-dimensional stability boundaries such as voltage, power angle, and power. This not only greatly reduces the computational complexity of the analysis but also makes the physical process and critical state of static instability clearly visible. It provides engineers with a theoretical tool that has clear physical concepts and is easy to apply quickly, effectively overcoming the shortcomings of traditional methods, such as difficult analysis and poor physical intuitiveness. Attached Figure Description

[0042] To more clearly illustrate the technical solution of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. Obviously, the drawings described below are only one embodiment of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a flowchart of the AC line static stability boundary analysis method in an embodiment of the present invention;

[0044] Figure 2 This is a schematic diagram of the radial active power path of a multi-level power grid in an embodiment of the present invention;

[0045] Figure 3 This is a schematic diagram of the impedance model of the first-end voltage reference system in an embodiment of the present invention;

[0046] Figure 4 This is a schematic diagram of the phasor geometry of the circuit in an embodiment of the present invention;

[0047] Figure 5 This is a schematic diagram of the corresponding triangle and circumcircle of the line voltage phasor in an embodiment of the present invention;

[0048] Figure 6 This is an embodiment of the present invention. Along different power factor angles Schematic diagram of directional pattern changes;

[0049] Figure 7 These are different power factor angles in the embodiments of the present invention. A schematic diagram of the circumcircle arc of the corresponding voltage phasor triangle;

[0050] Figure 8 The static stability voltage and power angle boundaries and power factor angle of the line in the embodiments of the present invention are... Relationship diagram;

[0051] Figure 9 The power boundary and power factor angle of the line in the embodiment of the present invention are the static stability of the line. Relationship diagram;

[0052] Figure 10 This is a schematic diagram of the external load impedance model of the line in an embodiment of the present invention;

[0053] Figure 11 This is a schematic diagram of the impedance boundary and stability region of the line in the embodiment of the present invention; Figure 12 This is a power flow distribution diagram of the IEEE 9-node system when the load is increased by 2.6 times in an embodiment of the present invention;

[0054] Figure 13 This refers to the complex power point and power boundary at the end of line L1 of the IEEE 9-node system during 2.6 times the power flow in this embodiment of the invention.

[0055] Figure 14 This refers to the line L2 end complex power point and power boundary of the IEEE 9-node system under 2.6 times power flow in this embodiment of the invention.

[0056] Figure 15 This refers to the line L3 end complex power point and power boundary of the IEEE 9-node system under 2.6 times power flow in this embodiment of the invention.

[0057] Figure 16 This refers to the complex power point and power boundary at the L4 end of the line in the IEEE 9-node system during 2.6 times the power flow in this embodiment of the invention.

[0058] Figure 17 This refers to the complex power point and power boundary at the L5 end of the line in the IEEE 9-node system during 2.6 times the power flow in this embodiment of the invention.

[0059] Figure 18 This refers to the complex power point and power boundary at the L6 end of the line in the IEEE 9-node system during 2.6 times the power flow in this embodiment of the invention. Detailed Implementation

[0060] The technical solutions of the present invention will be clearly and completely described below with reference to the accompanying drawings of the embodiments. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0061] The technical solution of the present invention will be described in detail below with reference to specific embodiments. These specific embodiments can be combined with each other, and the same or similar concepts or processes may not be described again in some embodiments.

[0062] Example 1

[0063] The static stability boundary analysis method for AC lines based on phasor geometry provided in this invention lies in its ability to intuitively reveal and determine the static stability boundary of the line through geometric figures. The following is a combination of... Figure 1 The specific implementation steps of the method of the present invention will be described in detail.

[0064] Step S1: Based on the impedance model of the AC line's head-end voltage reference system, obtain the correlation between the line's equivalent impedance, line current phasor, head-end voltage phasor, tail-end voltage phasor, and tail-end complex power.

[0065] This analysis is based on the actual structure and operational characteristics of the power grid. While the power grid physically exhibits a complex topology, from the perspective of active power transmission, its path has a radial characteristic: active power flows from a few generating nodes to most load nodes, and this path does not form a closed loop. Figure 2 As shown. Figure 2 In the diagram, 1, 2, 3, 4, 5, 6 represent busbar numbers, T1 represents the transformer connected between busbar 1 and busbar 2, T2 represents the transformer connected between busbar 3 and busbar 4, and L... 2-3 L represents the line connecting busbar 2 and busbar 3. 4-5 L represents the line connecting busbar 4 and busbar 5. 4-6 This indicates the line connecting busbar 4 and busbar 6. F2, F3, F4, F5, and F6 represent the loads connected to busbar 2, busbar 3, busbar 4, busbar 5, and busbar 6, respectively. Q C4 This indicates the parallel capacitor compensation device at busbar 4.

[0066] In static stability analysis, components such as generators, transformers, and transmission lines in the system can be equated to corresponding impedance branches. Among them, the static stability boundary of AC transmission lines (especially those with voltage levels of 110kV and above, and those undertaking point-to-point power transmission) is a key factor limiting the power grid's transmission capacity and is also the main object of analysis in the method of this invention.

[0067] Based on the above understanding, the static stability problem of the entire network can be reasonably decoupled: since the static instability of any transmission branch (manifested as an imbalance in its power-voltage relationship at the end) will lead to instability in its downstream power supply area, the necessary and sufficient condition for ensuring the static stability of the entire network is to ensure that every branch in the power grid satisfies static stability. Therefore, it is permissible to focus the research on a single-line model.

[0068] For a single AC transmission line, an impedance model with a reference voltage at the head-end (e.g., ...) is established for analysis. Figure 3 (As shown). The core assumptions and parameter definitions of this model are as follows:

[0069] The line is equivalent to a lumped parameter impedance. Its modulus and impedance angle Determined by the line parameters, it is considered a known constant in the analysis.

[0070] In actual operation, the voltage at the beginning of the line is usually maintained relatively stable through generator excitation control, transformer voltage regulation, and reactive power compensation, and is less affected by changes in the load at the end. Therefore, it can be set as a known reference phasor, denoted as . .

[0071] The voltage phasor at the end of the line is Its phasor to the first-end voltage The phase difference is defined as the power angle. , , These are the phases of the first-end voltage phasor and the last-end voltage phasor, respectively.

[0072] The line current phasor is .

[0073] The load at the end of the line is composed of end-complex power. Characterization, .in, and These are active power and reactive power, respectively. Apparent power, The power factor angle (i.e.) Lagging behind (angle).

[0074] In the impedance model of the first-end voltage reference system, the above physical quantities satisfy the following two fundamental relationships determined by the basic laws of circuits: the voltage equation and the power equation.

[0075] According to Kirchhoff's voltage law, the equivalent impedance of the line is... Line current phasors First-terminal voltage phasor and terminal voltage phasor The voltage equation that is satisfied is: ;

[0076] From the definition of complex power, the line current phasor can be obtained. Terminal voltage phasor and terminal complex power The power equation that is satisfied is: ,in Represents line current phasor The conjugate of complex numbers.

[0077] The aforementioned impedance model of the first-end voltage reference system and the two fundamental equations constitute the rigorous physical and mathematical foundation for all subsequent geometric analyses and derivations.

[0078] Step S2: Based on the correlation, construct a voltage phasor triangle with the first-end voltage phasor, the last-end voltage phasor, and the line voltage drop as sides.

[0079] Based on the impedance model and fundamental equations of the first-end voltage reference system in step S1, the first-end voltage phasor Terminal voltage phasor and The (line voltage drop) forms a closed voltage phasor triangle, and its geometric relationship is as follows: Figure 4 As shown.

[0080] Based on phasor geometry, the angle relationships between the sides of the voltage phasor triangle can be derived from the circuit phasors. Specifically, the line voltage drop phasor... With line current phasor In the same direction, the included angle is the line impedance angle. At the same time, the line voltage drop phasor With terminal voltage phasor The included angle between them is This relationship can be expressed as:

[0081] (1)

[0082] Therefore, it can be determined Figure 4 The three interior angles of the voltage phasor triangle shown are:

[0083] (2)

[0084] Among them, when the line type is determined (i.e., impedance angle) When the angle is a constant, Only with the terminal power factor angle Change. This characteristic is key to subsequent analysis of vertex geometric trajectories.

[0085] According to the voltage equation in step S1 and power equation (For three-phase systems, in per-unit or single-phase equivalent analysis, it is often simplified to) The apparent power at the end can be obtained. The expression is:

[0086] (3)

[0087] in, This is the magnitude of the terminal voltage phasor (i.e., the terminal voltage). This represents the magnitude of the line voltage drop. If... If is a known constant, then:

[0088] (4)

[0089] Equation (4) shows that two sides of the apparent power and voltage phasor triangle at the end (corresponding to) and The product of the lengths of the triangles is directly proportional. This quantitative relationship forms the mathematical basis for directly linking the physical quantity of power with the geometric characteristics of a triangle.

[0090] Step S3: Analyze the trajectory of the vertex of the voltage phasor triangle on the circumcircle arc corresponding to the power factor angle when the power factor angle of the terminal complex power is a constant value and the apparent power changes.

[0091] To facilitate purely geometric analysis, a model is constructed that... Figure 4 Corresponding triangles similar to the voltage phasor triangles (The angular relationship is exactly the same as that of the voltage phasor triangle), such as Figure 5 As shown. Figure 5 In the diagram, there is a clear correspondence between the triangular sidecar and the voltage phasor amplitude:

[0092] (5)

[0093] in, This represents the magnitude of the first-terminal voltage phasor (i.e., the first-terminal voltage). Consider the complex power at the end. A specific variation pattern: its power factor angle Keep it at a fixed value Unchanged, only its modulus remains unchanged. Changes. In this mode, as can be seen from equation (2) in step S2, the triangle The angle opposite to side AB in the middle It becomes a constant value. At the same time, the length of edge AB... Given and unchanged, only the side lengths b and c, and the position of point C change. change.

[0094] Due to the angle With the opposite side AB (chord) fixed, according to the inscribed angle theorem, the trajectory of vertex C (corresponding to the endpoint of the terminal voltage phasor) is constrained to a specific circle—a circle with AB as a chord, and the inscribed angle subtended by AB is always 0. Therefore, when When the change occurs, vertex C will be on a segment of the arc of the circle. Move upwards. The diameter 2R of the circumcircle can be determined by the sine theorem:

[0095] (6)

[0096] Generalization: If the end-of-line complex power Along different power factor angles Directional change (i.e.) (taking different values), then for each fixed value... Each corresponds to a corner. This allows us to determine a circumcircle with AB as its common chord but different diameters. For example... Figure 6 and Figure 7 As shown, when When (i = 1, 2, 3, ...) changes, vertex C will be in relation to the current... The value corresponds to the circumcircle arc. Move upwards. The specific shape of this arc (minor arc, semicircle, or major arc) is determined by the angle. Decide:

[0097] (7)

[0098] Therefore, the voltage phasor triangle of the line is a triangle with the amplitude of the voltage phasor at the beginning of the line as the phasor. This is a family of "common-chord inscribed triangles" with a fixed chord length. When the complex power at the end is along a specific power factor angle... When the direction changes, vertex C of the triangle is in contact with the... The corresponding circumcircle determines the movement along the arc. This geometric constraint serves as a bridge connecting power changes with the evolution of voltage phasor forms.

[0099] Step S4: Based on the movement trajectory, establish a proportional relationship between the area of ​​the voltage phasor triangle and the apparent power.

[0100] For the specific power factor angle analyzed in step S3 triangle From formula (3), it can be seen that its apparent power can be expressed as:

[0101] (8)

[0102] set up In the triangle, the height corresponding to the base AB is h. What is the area of ​​this triangle? It can be represented as:

[0103] (9)

[0104] According to the law of sines, for have:

[0105] (10)

[0106] in, , R is the radius of the circumcircle. Using the relationship of the sine theorem... Formula (9) can be further transformed into:

[0107] (11)

[0108] Combining equations (8) and (11), the apparent power can be obtained. Direct relationship with the area of ​​a triangle:

[0109] (12)

[0110] Formula (12) shows that for a fixed Apparent power and area of ​​triangle Proportional, with a proportionality constant of At the same time, due to the relationship between height h and area... Proportional (due to the base) (fixed), therefore also .

[0111] Extending the above relationship to any power factor angle For the corresponding triangle Its apparent power is:

[0112] (13)

[0113] The derivation in step S4 proves an important geometrical physical law: for a given path ( , (Fixed) and power factor angle Apparent power at the end Its corresponding voltage phasor triangle area It is directly proportional. This relationship (13) is the key mathematical basis for transforming the physical limit problem of power transmission into a geometric surface value problem.

[0114] Step S5: Based on the proportional relationship, determine that the apparent power reaches its maximum when the voltage phasor triangle is an isosceles triangle.

[0115] Based on the analysis in step S3, for any given power factor angle The corresponding voltage phasor triangle vertex C can only lie on a specific arc with AB as the chord. Move upwards. Combine this with the key relationship derived in step S4—apparent power. Area of ​​a triangle Proportional to the maximum apparent power The problem is equivalent to finding the point C on this arc that maximizes the area of ​​the triangle.

[0116] According to the principles of plane geometry, for a triangle with a fixed base AB and a vertex C moving along a given arc, when vertex C moves to the intersection of the perpendicular bisector of side AB and the arc (e.g., ... Figure 5 , Figure 6 In triangle ABC, the perpendicular distance (height h) from vertex C to the base AB is maximized, thus increasing the area of ​​the triangle. It also reaches its maximum value on that arc. Correspondingly, the apparent power also reaches its maximum value at this point. .

[0117] Observe the shape of the triangle at this point: Since vertex C lies on the perpendicular bisector of side AB, AC = BC, that is... It is an isosceles triangle. Mapping back to the original physical quantity, this geometric condition corresponds to:

[0118] (14)

[0119] The first equation shows that the magnitude of the terminal voltage phasor is equal to the magnitude of the line voltage drop, and the second equation gives the specific power angle at which maximum apparent power is achieved. With the interior angle of the triangle The relationship.

[0120] when When the triangle is isosceles, its maximum area can be calculated directly using geometric relationships. Let the length of the leg of the isosceles triangle be l (i.e., AC = BC = l), and the base be... The vertex (located at vertex C) is Its maximum area can be derived using trigonometric relationships:

[0121] (15)

[0122] Substituting formula (15) into formula (13) and simplifying using trigonometric identities, we can obtain the line at a given power factor angle. The expression for the maximum apparent power that can be transmitted, i.e., its static steady power boundary:

[0123] (16)

[0124] Therefore, the apparent power transmitted through the line reaches its maximum value. When the voltage phasor triangle is at this point, it is an isosceles triangle. This simple geometric shape is the key geometric condition for determining whether the line has reached the static stable power boundary. Formula (17) gives the specific calculation method for this boundary value.

[0125] Step S6: Based on the static stability critical state of the AC line corresponding to the maximum apparent power, determine whether the voltage phasor triangle is an isosceles triangle as the geometric condition for determining whether the AC line is at the static stability boundary; based on the geometric condition, determine the static stability boundary of the AC line.

[0126] In actual power system operation, it is generally required that the end of the line operate in a state of lagging power factor, that is, there is usually... This makes the corresponding triangle The circumcircle is Figure 7 The minor arc is shown. Under this premise, combined with formula (12), a rigorous geometric division and proof can be made for the static stable state.

[0127] Observe vertex C (representing) The position of the endpoint on the arc:

[0128] Stable region: When vertex C moves along the right arc of the perpendicular bisector of side AB (the magnitude of the first-end voltage phasor) (e.g.) Figure 7 Within this region, with apparent power... As the triangle increases, the height h of the triangle increases, but the magnitude of the voltage phasor at the end increases. Reduce, work angle Increase. Its geometric characteristics correspond to the classical criteria for static stability:

[0129] (17)

[0130] At this point, the side length of the triangle satisfies That is, AC > BC, the AC line is in a statically stable load state in terms of voltage and power angle.

[0131] Instability region: When vertex C crosses the perpendicular bisector and lies on the arc to its left. Within this region, as the apparent power... Decrease the magnitude of the terminal voltage phasor Instead, the work angle decreases. It also decreases, exhibiting negative damping characteristics:

[0132] (18)

[0133] At this point, the side length of the triangle satisfies That is, AC < BC, the AC line is in a state of static instability in both voltage and power angle.

[0134] Critical boundary: When vertex C lies exactly on the perpendicular bisector of side AB ( When the apparent power is on the boundary between static stability and instability, Reaching the maximum value And satisfy:

[0135] (19)

[0136] This is equivalent to active power. and reactive power The classical static stability critical condition where the partial derivatives with respect to each state variable are zero:

[0137] (20)

[0138] At this point, the triangle is an isosceles triangle, which satisfies the geometric conditions shown in formula (14).

[0139] Therefore, the perpendicular bisector of side AB is a voltage phasor. The geometric boundary between the statically stable region and the statically unstable region. And the "voltage phasor triangle" The geometric condition of "isosceles triangle" is the only and universal criterion for determining whether a circuit has reached a statically stable critical state.

[0140] Based on the above isosceles triangle criterion, the static stability boundaries of the line can be uniformly derived (e.g., Figure 8 , Figure 9 (as shown)

[0141] Voltage boundary and angle of attack boundary: can be directly obtained from the geometric relationship of isosceles triangles, the terminal voltage boundary (i.e. the terminal voltage at the static stability critical point or the terminal voltage stability boundary value). Sum of work and boundary They are respectively:

[0142] (twenty one)

[0143] The two satisfy the relationship This reveals the essence that voltage instability and power angle instability occur simultaneously and are interconnected in a static sense.

[0144] Power boundary: The apparent power corresponding to the critical state is the maximum transmission power. Therefore, the power stability boundary (i.e., the apparent power at the static stability critical point) is the power boundary. That is equal to the maximum apparent power :

[0145] (twenty two)

[0146] Formula (22) can also be expressed in parabolic form through trigonometric identities, that is:

[0147] (twenty three)

[0148] Based on the above boundaries, the voltage phasor at the end of the line and complex power The statically stable operating domain can be described as:

[0149] (twenty four)

[0150] Despite each boundary value , , Both are power factor angles The functions are denoted by , but the geometric conditions upon which they depend are simple, unique, and constant, i.e., " "It is an isosceles triangle." This fully demonstrates the core advantages of phasor geometry: clear physical concepts and intuitive judgments.

[0151] Furthermore, although in practice, when the apparent power at the end... When there are large changes, the system regulation may cause the amplitude of the first-terminal voltage phasor to change. Small changes may occur, thus affecting the specific magnitude of the boundary value. However, for the established "first-end voltage reference system", the geometric analysis logic and judgment criteria of the method of the present invention remain unchanged, and the boundary relationship forms revealed by formulas (21) and (22) still hold, which ensures the robustness and universality of the method.

[0152] The geometric criteria of the method of this invention also apply to the analysis of the stability boundary from the perspective of impedance. If the load connected to the end of the line is equivalent to a load impedance... ,like Figure 10 As shown, its impedance angle is the terminal power factor angle. ,Right now .

[0153] Based on the circuit relationship, the line current... Simultaneously flowing through the line impedance and load impedance ,satisfy:

[0154] (25)

[0155] Based on the static stability critical geometric condition determined in steps S5 and S6, namely, the voltage phasor triangle is an isosceles triangle ( Substituting into equation (25), we can deduce the equivalent load impedance at the end of the line when it reaches the static stability critical point. The following boundary conditions must be met:

[0156] (26)

[0157] in, This represents the critical load impedance that ensures the static stability of the line.

[0158] Considering the line resistance in the actual system and load resistance Generally positive ( >0, >0), on the complex impedance plane, the impedance boundary satisfies equation (26). The trajectory is a circle centered at the origin with a radius of... semicircular arc (such as) Figure 11 (As shown). This is because, according to the conclusion of step S6, stable operation requires meeting the following conditions. Combining equation (25), it can be equivalently derived that Therefore, line impedance The static stability region (i.e., the load impedance that ensures line stability) The range of values ​​for ( ) can be described as:

[0159] (27)

[0160] This area corresponds to Figure 11 The shaded area to the right of the semicircular arc. Within this stability region, if the magnitude of the load impedance... The larger the voltage (i.e., the lighter the load), the higher the terminal voltage. The higher the value, the better the static stability of the line; conversely, The smaller (the heavier the load), the better. The lower the value, the worse the stability.

[0161] The static stability boundary equations (21) and (22) for voltage, power angle, and power obtained from geometric analysis are completely consistent with the results obtained by algebraic extremum method in previous studies (such as the distributed load safety domain of power grid and its application (II): boundary-essence-morphology of line static instability and stability margin of load safety domain [J]. Proceedings of the CSEE, 2024, 44(05): 1737-1750.). The boundary curves are as follows: Figure 8 , Figure 9 As shown.

[0162] As can be seen, the phasor geometry method intuitively demonstrates the process by which changes in the complex power at the terminal lead to changes in the voltage phasor until instability. It intuitively obtains the stability boundaries of power, voltage, and power angle, and clearly explains the homogeneity of voltage and power angle instability. Compared with the algebraic extremum method, this method is simple, intuitive, and has clear physical concepts.

[0163] Example 2

[0164] To verify the relationship between the static stability of the power grid and its branches, and to demonstrate the relationship between the power of each line and the static stability boundary when the power grid approaches static instability, a simulation of the IEEE 9-bus system is conducted.

[0165] Simulation settings: Taking the output bus of generator G1 as the balancing node, starting from the initial power flow state, the active and reactive power of all loads in the system, as well as the active power output of generators G2 and G3 (with the output bus set as the PU node), are gradually increased in the same proportion. Figure 12 As shown. Figure 12 In this table, G1, G2, and G3 represent generators, T1, T2, and T3 represent transformers, and L1 to L6 represent lines. Node (bus) voltage information is expressed in the format of "amplitude ∠phase angle," with units of per-unit value (pu) and degree (°). For example, the voltage information at G2 is 1.02∠30.80°. Branch (line / transformer) power flow information is expressed in the format of "active power + j reactive power," with units of per-unit value (pu). For example, the power flow at the end of L1 is 1.35 + j1.32.

[0166] Critical state: When the growth rate reaches 2.7 times, the PSASP power flow calculation program fails to converge, indicating that the system has lost static stability. We will analyze the power flow state at a rate close to instability of 2.6 times.

[0167] Power boundary verification: The terminal complex power operating points of the six key lines in the system under 2.6 times power flow are plotted on the complex power plane and compared with the power boundary curves of each line calculated by formula (22) of the present invention. Figures 13 to 18 . Figures 13 to 18 In the diagram, pu on the coordinate axis represents the corresponding power value (active power P or reactive power Q), which has been divided by a selected reference power to obtain a dimensionless per-unit value. The results show that when the system as a whole approaches instability, the operating point of line L2 is closest to its power boundary, while other lines still have a significant margin. This verifies that: 1) the static stability of the entire network depends on the stability of each branch; 2) the power boundary derived by the method of this invention can accurately identify the stability limit of the actual system.

[0168] The phasor geometry analysis method for the static stability boundary of AC lines proposed in this invention establishes a direct proportional relationship between apparent power and the area of ​​the triangle by constructing and analyzing the voltage phasor triangle and its circumcircle trajectory. This leads to the discovery of a unified and intuitive static stability geometric criterion centered on the condition that "the voltage phasor triangle is an isosceles triangle." Based on this condition, analytical expressions for the static stability boundary of power, voltage, power angle, and impedance are uniformly derived. Theoretical derivation and simulation of the IEEE 9-bus system both verify the correctness of this method.

[0169] Compared to the traditional algebraic extremum method that relies on solving complex nonlinear equations, the method of this invention transforms the abstract stability limit problem into a clear geometric analysis. The physical concepts are clear, and the analysis process is intuitive and concise. It effectively overcomes the shortcomings of traditional methods, such as difficult analysis and obscure physical meaning. It provides a powerful theoretical tool and intuitive engineering criteria for the rapid assessment of stability boundaries in power grid planning, operation and monitoring, and has both important theoretical value and broad engineering application prospects.

[0170] Example 3

[0171] This invention also provides an electronic device, which includes a memory, a processor, and a computer program or instructions stored in the memory. The processor executes the computer program or instructions to implement the AC line static stability boundary analysis method based on phasor geometry in this invention.

[0172] Although not shown, the electronic device includes a processor that can perform various appropriate operations and processes based on programs and / or data stored in read-only memory (ROM) or loaded from a storage portion into random access memory (RAM). The processor can be a multi-core processor or may contain multiple processors. In some embodiments, the processor may include a general-purpose main processor and one or more specialized coprocessors, such as a central processing unit, graphics processing unit (GPU), neural network processor (NPU), digital signal processor (DSP), etc. Various programs and data required for device operation are also stored in RAM. The processor, ROM, and RAM are interconnected via a bus. Input / output (I / O) interfaces are also connected to the bus.

[0173] The processor and memory described above are used together to execute programs / instructions stored in the memory. When the program / instructions are executed by the computer, they can implement the methods, steps, or functions described in the above embodiments.

[0174] Although not shown, embodiments of the present invention also provide a computer-readable storage medium having a computer program or instructions stored thereon, which, when executed by a processor, implements the AC line static stability boundary analysis method based on phasor geometry in embodiments of the present invention.

[0175] Readable storage media include both permanent and non-permanent, removable and non-removable media that can store information by any method or technology. Information can be computer-readable instructions, data structures, program modules, or other data. Examples of computer storage media include, but are not limited to, phase-change memory (PRAM), static random access memory (SRAM), dynamic random access memory (DRAM), other types of random access memory (RAM), read-only memory (ROM), electrically erasable programmable read-only memory (EEPROM), flash memory or other memory technologies, CD-ROM, digital versatile optical disc (DVD) or other optical storage, magnetic tape, disk storage or other magnetic storage devices, or any other non-transferable medium that can be used to store information accessible by a computing device. As defined herein, computer-readable media does not include transient computer-readable media, such as modulated data signals and carrier waves.

[0176] The above description only discloses specific embodiments of the present invention, but the scope of protection of the present invention is not limited thereto. Any changes or modifications that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for static stability boundary analysis of AC lines based on phasor geometry, characterized in that, The analytical method includes: Based on the impedance model of the reference system of the AC line's starting voltage, the correlation between the line's equivalent impedance, line current phasor, starting voltage phasor, ending voltage phasor, and ending complex power is obtained. Based on the aforementioned relationship, a voltage phasor triangle is constructed with the first-end voltage phasor, the last-end voltage phasor, and the line voltage drop as its sides; wherein, the line voltage drop is equal to the product of the line current phasor and the line equivalent impedance. The analysis shows the trajectory of the vertex of the voltage phasor triangle on the circumcircle arc corresponding to the power factor angle when the power factor angle of the terminal complex power is a constant value and the apparent power changes. Based on the movement trajectory, a proportional relationship is established between the area of ​​the voltage phasor triangle and the apparent power; Based on the aforementioned proportional relationship, it is determined that the apparent power reaches its maximum when the voltage phasor triangle is an isosceles triangle; Based on the apparent power reaching its maximum, which corresponds to the static stability critical state of the AC line, whether the voltage phasor triangle is an isosceles triangle is determined as the geometric condition for determining whether the AC line is at the static stability boundary; based on the geometric condition, the static stability boundary of the AC line is determined.

2. The AC line static stability boundary analysis method based on phasor geometry according to claim 1, characterized in that, In the impedance model of the first-end voltage reference system, the correlation includes: Line equivalent impedance Line current phasors First-terminal voltage phasor and terminal voltage phasor Satisfied voltage equation: ; and line current phasors Terminal voltage phasor and terminal complex power The power equation that is satisfied is: ,in Represents line current phasor The conjugate of complex numbers.

3. The AC line static stability boundary analysis method based on phasor geometry according to claim 1, characterized in that, The arc on which the movement trajectory lies is a minor arc, a semicircle, or a major arc, and its specific shape is determined by the difference between the impedance angle of the equivalent impedance of the line and the power factor angle.

4. The AC line static stability boundary analysis method based on phasor geometry according to claim 1, characterized in that, The proportional relationship is determined by the following formula: ； in, Indicates apparent power; Indicates the power factor angle; This represents the magnitude of the equivalent impedance of the line. Represents the area of ​​the voltage phasor triangle; This indicates the impedance angle.

5. The AC line static stability boundary analysis method based on phasor geometry according to claim 1, characterized in that, The condition that the voltage phasor triangle is an isosceles triangle specifically means that the side corresponding to the terminal voltage phasor has the same length as the side corresponding to the line voltage drop.

6. The static stability boundary analysis method for AC lines based on phasor geometry according to any one of claims 1 to 5, characterized in that, When the geometric conditions are used to determine the static stability boundary of the AC line, they are specifically used to determine at least one of the following boundaries: end voltage boundary, power angle boundary, apparent power boundary, and equivalent load impedance boundary.

7. The AC line static stability boundary analysis method based on phasor geometry according to claim 6, characterized in that, Based on the aforementioned geometric conditions, the terminal voltage boundary, power angle boundary, apparent power boundary, and equivalent load impedance boundary are determined by the following formulas: ； ； ； ； in, Indicates the terminal voltage stability boundary value; Indicates the magnitude of the voltage phasor at the beginning of the circuit; Indicates the impedance angle; Indicates the power factor angle; Indicates the stability boundary value of the work angle; Indicates the power stability boundary value; This represents the magnitude of the equivalent impedance of the line. This represents the stability boundary value of the equivalent load impedance; This represents the load impedance.

8. An electronic device comprising a memory, a processor, and a computer program or instructions stored in the memory, characterized in that, The processor executes the computer program or instructions to implement the AC line static stability boundary analysis method based on phasor geometry as described in any one of claims 1 to 7.

9. A computer-readable storage medium having a computer program or instructions stored thereon, characterized in that, When the computer program or instructions are executed by the processor, they implement the AC line static stability boundary analysis method based on phasor geometry as described in any one of claims 1 to 7.

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