Method, device and medium for analyzing static stability boundary of alternating current line based on phasor geometry

Abstract

The application discloses an alternating current line static stability boundary analysis method and device based on a phasor geometry, and a medium, and relates to the technical field of alternating current line static stability. The method is based on a line impedance model of a reference system of a first-end voltage, and a voltage phasor triangle composed of the first-end voltage, a last-end voltage and a line voltage drop is constructed. By analyzing the trajectory of the vertex of the triangle moving along the corresponding circumcircle arc when the last-end complex power changes, a core relationship that the apparent power is proportional to the area of the triangle is found. It is proved that when the triangle is an isosceles triangle, the apparent power reaches the maximum value and is a line static stability critical point. Therefore, the application proposes that the isosceles of the voltage phasor triangle is used as a geometric condition for judging the static stability boundary, and power, voltage, power angle and impedance boundaries of the line static stability are derived. The application converts the traditional complex nonlinear algebraic problem into clear geometric analysis, the physical concept is clear, and the solving process is significantly simplified.

Description

TECHNICAL FIELD

[0001] The application belongs to the technical field of alternating current line static stability, and particularly relates to an alternating current line static stability boundary analysis method, equipment and medium based on phasor geometry. BACKGROUND

[0002] Accurately, efficiently and intuitively determining the static stability boundary of each alternating current line has become a key problem to be solved urgently to ensure the safe and reliable operation of the power grid. The static stability boundary is not only the basis for evaluating the safety current-carrying capacity of the line, but also the theoretical premise for studying the dynamic stability and transient stability of the system. The static stability domain provides a recovery buffer zone for the dynamic oscillation of the system after suffering a small disturbance, and also provides a key deceleration area for the transient instability process under a large disturbance.

[0003] From the network structure, although the power grid structure is complex, from the perspective of the flow direction of the observed active power, the path presents a radial structure. In static stability analysis, the elements such as lines, transformers and generators can be equivalent to impedance branches, and the lower-level power grid and its load can be regarded as the equivalent end load of the upper-level transmission branch. Based on this, the upper-level branch must meet the static stability requirement. By analogy, the sufficient and necessary condition for the whole network to maintain static stability can be derived: that is, every impedance branch in the power grid must be in a static stable state. Since in actual operation, the line head voltage is usually maintained relatively stable through generator excitation control, reactive power compensation and other means, and is less affected by the end power change, establishing a "head reference system impedance branch model" with the head voltage as the reference phase to study the static stability boundary of the line itself is a reasonable and effective theoretical simplification approach.

[0004] At present, the traditional method for solving the static stability boundary is mainly the algebraic extreme value method. This method needs to establish a complex variable equation describing the power-voltage relationship of the line, and convert it into a high-order nonlinear real variable equation set for extreme value analysis. For example, the previous study (such as Dispersed Load Security Region of Power Grid and Its Application (II): Boundary-essence-morphology of Static Instability of Lines and Stability Margin of Load Security Region[J]. Proceedings of the Chinese Society of Electrical Engineering, 2024, 44(05): 1737-1750.) converted the problem into two three-dimensional independent real variable equations through complex transformation, and then analyzed to obtain the consistent power boundary of voltage instability and angle instability, and respectively solved the static voltage and angle boundaries. However, these real variable equations are all three-dimensional fourth-order nonlinear transcendental equations, and the process of analyzing the extreme value conditions, solving the extreme values and proving the partial derivatives (i.e. strict boundary proof) is extremely complex, and it is extremely difficult to analyze. More prominent is that the pure algebraic derivation process is difficult to reveal the physical image and internal logic of the state quantity change, resulting in that the analysis result is not intuitive, and it is difficult to provide clear physical perception for the operation personnel, and it is also difficult to form a simple and easy-to-engineer application stability boundary judgment tool.

[0005] Therefore, in the background of high proportion of new energy access leading to increasingly complex power grid operation characteristics, in order to overcome the disadvantages of traditional algebraic extremum method in terms of difficult analysis and obscure physical meaning, it is urgent to develop a new analysis method which can uniformly, concisely and intuitively depict the static stability boundary of power, voltage, power angle and impedance. SUMMARY

[0006] In view of the above-mentioned defects in the prior art, the purpose of the present application is to provide an AC line static stability boundary analysis method, device and medium based on phasor geometry, so as to solve the technical problems of traditional algebraic extremum method in terms of complex analysis and difficult calculation caused by high-order nonlinearity of equations, and poor physical intuition and difficulty in forming an intuitive judgment tool for engineering application. The present application aims to provide a unified, concise and clear physical concept analysis method by introducing phasor geometry analysis method, to intuitively reveal the evolution process and geometric nature of the line from stability to static instability, simplify the solution of the stability boundary, and strengthen the physical perception ability of the power grid operation state, thereby providing an efficient and intuitive analysis basis for the safe and stable operation of the power grid under the background of high proportion of new energy access.

[0007] The present application solves the above technical problems through the following technical solutions: an AC line static stability boundary analysis method based on phasor geometry, comprising:

[0008] Based on the reference system impedance model of the first end voltage of the AC line, the correlation between the line equivalent impedance, the line current phasor, the first end voltage phasor, the end voltage phasor and the end complex power is obtained;

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

[0010] When the power factor angle of the end complex power is a constant value and the apparent power changes, the moving track of the vertex of the voltage phasor triangle on the excircle arc corresponding to the power factor angle is analyzed;

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

[0012] According to the proportional relationship, when the voltage phasor triangle is an isosceles triangle, the apparent power reaches the maximum;

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

[0014] The application successfully converts the complex algebraic extremum problem of solving high-order nonlinear transcendental equations into a problem that can be reasoned according to intuitive geometric rules such as the central angle theorem and the sine theorem by constructing circuit physical quantities into a voltage phasor triangle and analyzing the trajectory of the vertex on the circumcircle arc. This conversion greatly reduces the computational complexity and analytical difficulty of the analysis. More importantly, the method discovers and establishes the unique and universal geometric determination condition that the voltage phasor triangle is an isosceles triangle, unifies the determination standards of the voltage, power angle, power and impedance boundaries that need to be handled separately in the traditional method, and solves the core defects of the traditional method that the analysis is complicated and the conclusions are scattered.

[0015] The method of the application dynamically reveals the complete physical process that the terminal complex power change drives the voltage phasor triangle vertex to move until the isosceles critical state is reached, making the evolution of static instability visible and perceptible. This not only completely overcomes the drawback of the traditional algebraic method that the physical meaning is obscure, but also strengthens the intuitive understanding of the running personnel. Moreover, the geometric proof clearly shows that voltage drop and power angle increase occur at the same time at the critical point, thereby deeply revealing the homogeneity of voltage stability and power angle stability at the static instability level and deepening the theoretical understanding of the nature of stability.

[0016] Based on the above simple geometric determination condition and clear physical image, the method of the application can be easily converted into a graphical and standardized engineering criterion and online monitoring algorithm. This enables front-line personnel to quickly assess the stability margin without complex calculations, greatly promoting the transformation of theoretical achievements into practical tools. In the face of the power fluctuation challenge brought by a high proportion of new energy access, the method provides strong technical support for the real-time and agile stability boundary perception and safety warning of the power grid, and helps to improve the resilience and safety level of power grid operation.

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

[0018] line equivalent impedance , line current phasor , first-end voltage phasor , and terminal voltage phasor satisfy the voltage equation: ;

[0019] and line current phasor , terminal voltage phasor , and terminal complex power The satisfied power equation is: wherein represents the conjugate complex of the line current phasor .

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

[0021] Further, the circular arc where the movement trajectory is located is a poor arc, a semicircle or an excellent arc, and the specific form is determined by the difference between the impedance angle of the line equivalent impedance and the power factor angle.

[0022] The present application fully reveals the geometric topology of the stable boundary under different operating conditions by clarifying the three forms (poor arc, semicircle, excellent arc) of the vertex trajectory circular arc and their determining factors, so that the analysis method can accurately depict all line operating states from inductive to capacitive loads, enhancing the completeness and adaptability of the method.

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

[0024] ;

[0025] wherein, represents the apparent power; represents the power factor angle; represents the modulus value of the line equivalent impedance; represents the area of the voltage phasor triangle; represents the impedance angle.

[0026] The present application provides a quantitative bridge for converting the abstract power transmission capacity problem into an intuitive geometric extreme value problem by establishing an accurate mathematical relationship between the apparent power and the area of the voltage phasor triangle, which is the key mathematical foundation for the method to finally derive the exact physical conclusion from geometric analysis.

[0027] Further, the condition that the voltage phasor triangle is an isosceles triangle is that the length of the side corresponding to the terminal voltage phasor is equal to the length of the side corresponding to the line voltage drop.

[0028] The present application accurately maps the abstract geometric condition "isosceles triangle" to a clear physical relationship, i.e., the terminal voltage amplitude is equal to the line voltage drop amplitude, so that the abstract geometric rule has clear, measurable and calculable engineering connotations, greatly facilitating the direct application and verification of the analysis method in engineering practice.

[0029] Further, the geometric condition is used to determine the static stability boundary of the alternating current line, and is specifically used to determine at least one of the following boundaries: a terminal voltage boundary, an angle of operation boundary, an apparent power boundary, and an equivalent load impedance boundary.

[0030] The present application clearly indicates that a single geometric determination condition can uniformly derive multi-dimensional stability boundaries (voltage, angle of operation, power, and impedance), fully demonstrates the systematic advantage of the method of "one principle, hundred principles", realizes comprehensive and integrated characterization of the stable state of the line, and greatly improves the engineering practical value and analysis efficiency of the method.

[0031] Further, based on the geometric condition, the terminal voltage boundary, the angle of operation boundary, the apparent power boundary, and the equivalent load impedance boundary are respectively determined by the following formulas:

[0032] ;

[0033] ;

[0034] ;

[0035] ;

[0036] wherein, represents a terminal voltage stability boundary value; represents an amplitude of a first terminal voltage phasor; represents an impedance angle; represents a power factor angle; represents an angle of operation stability boundary value; represents a power stability boundary value; represents a modulus value of a line equivalent impedance; represents an equivalent load impedance stability boundary value; represents a load impedance.

[0037] The present application provides analytical formulas of accurate voltage boundaries, angle of operation boundaries, apparent power boundaries, and equivalent load impedance boundaries directly derived from geometric conditions, converts intuitive geometric determination into a mathematical tool that can be directly used for quantitative calculation and engineering design, so that the evaluation of stability limits becomes simple, clear, and efficient, and solves the complex problem of solving voltage boundaries, apparent power boundaries, and equivalent load impedance boundaries in traditional methods.

[0038] Based on the same concept, the present application also provides an electronic device comprising 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 geometric phasor-based alternating current line static stability boundary analysis method as described above.

[0039] Based on the same concept, the application also provides a computer readable storage medium, which stores a computer program or instructions, and the computer program or instructions are executed by a processor to realize the power geometry based alternating current line static stability boundary analysis method as described above.

[0040] Compared with the prior art, the application has the beneficial effects that:

[0041] The application converts the traditional stability boundary problem relying on complex algebraic solution into an intuitive geometric figure construction and trajectory analysis problem through the innovative "power geometry analysis method". The method unifies and simplifies the judgment process of voltage, power angle, power and other multi-dimensional stability boundaries by establishing the simple and universal geometric judgment condition that the voltage phasor triangle is an isosceles triangle, which not only greatly reduces the calculation complexity of the analysis, but also makes the physical process of static instability and the critical state clear and visible, provides a theoretical tool with clear physical concept and convenient rapid application for engineering personnel, and effectively overcomes the defects of the traditional method such as difficult analysis and poor physical intuition. BRIEF DESCRIPTION OF DRAWINGS

[0042] In order to more clearly illustrate the technical solutions of the application, the drawings needed in the embodiment description will be briefly introduced below. Obviously, the drawings in the following description are only one embodiment of the application, and other drawings can be obtained by those skilled in the art without creative labor on the basis of these drawings.

[0043] Figure 1 is the flow chart of the alternating current line static stability boundary analysis method in the embodiment of the application;

[0044] Figure 2 is the schematic diagram of the radial active path of the multi-level power grid in the embodiment of the application;

[0045] Figure 3 is the schematic diagram of the reference system impedance model of the first end voltage in the embodiment of the application;

[0046] Figure 4 is the schematic diagram of the power geometry relationship of the line in the embodiment of the application;

[0047] Figure 5 is the schematic diagram of the corresponding triangle and circumcircle of the line voltage phasor in the embodiment of the application;

[0048] Figure 6 is the schematic diagram of the power geometry relationship of the line in the embodiment of the application along different power factor angles direction mode change schematic diagram;

[0049] Figure 7 is the schematic diagram of the power geometry relationship of the line in the embodiment of the application Corresponding voltage phasor triangle circumscribed circle arc schematic diagram;

[0050] Figure 8 The relationship between the voltage and power factor angle of the line static stability voltage and power angle boundary in the embodiment of the application

[0051] Figure 9 The relationship between the power boundary and power factor angle of the line static stability in the embodiment of the application

[0052] Figure 10 The line external load impedance model schematic diagram in the embodiment of the application

[0053] Figure 11 The impedance boundary and stability domain schematic diagram of the line static stability in the embodiment of the application

[0054] Figure 12 The power flow distribution diagram of the IEEE9 node system load increase by 2.6 times in the embodiment of the application

[0055] Figure 13 The line L1 end complex power point and power boundary of the IEEE9 node system in the embodiment of the application when the power flow is 2.6 times

[0056] Figure 14 The line L2 end complex power point and power boundary of the IEEE9 node system in the embodiment of the application when the power flow is 2.6 times

[0057] Figure 15 The line L3 end complex power point and power boundary of the IEEE9 node system in the embodiment of the application when the power flow is 2.6 times

[0058] Figure 16 The line L4 end complex power point and power boundary of the IEEE9 node system in the embodiment of the application when the power flow is 2.6 times

[0059] Figure 17 The line L5 end complex power point and power boundary of the IEEE9 node system in the embodiment of the application when the power flow is 2.6 times

[0060] Figure 18 The line L6 end complex power point and power boundary of the IEEE9 node system in the embodiment of the application when the power flow is 2.6 times. DETAILED DESCRIPTION

[0061] ​​The technical solutions in the present application will be described clearly and completely below in connection with the drawings in the embodiments of the present application. Obviously, the described embodiments are only a part of the embodiments of the present application, rather than all the embodiments. Based on the embodiments in the present application, all other embodiments obtained by those skilled in the art without creative work fall within the scope of the present application.

[0062] The technical solutions of the present application will be described in detail below with specific embodiments. The following specific embodiments can be combined with each other, and the same or similar concepts or processes can not be described repeatedly in some embodiments.

[0063] Embodiment one

[0064] The core of the AC line static stability boundary analysis method based on phasor geometry provided by the embodiments of the present application is to intuitively reveal and determine the static stability boundary of the line through geometric figures. The specific implementation steps of the method of the present application will be described in detail below in connection with Figure 1 .

[0065] Step S1: Based on the reference system impedance model of the first end voltage of the AC line, the correlation between the equivalent impedance of the line, the line current phasor, the first end voltage phasor, the end voltage phasor and the end complex power is obtained.

[0066] Based on the actual structure and operating characteristics of the power grid, the analysis is carried out. The power grid presents a complex topology in physics, but from the transmission nature of active power, the path has a radial characteristic: the active power flows from a few power generation nodes to a large number of load nodes, and the path does not form a closed loop, as shown in Figure 2 . Figure 2 In the figure, 1, 2, 3, 4, 5 and 6 represent bus numbers, T1 represents a transformer connected between bus 1 and bus 2, T2 represents a transformer connected between bus 3 and bus 4, L 2-3 represents a line connecting bus 2 and bus 3, L 4-5 represents a line connecting bus 4 and bus 5, L 4-6 represents a line connecting bus 4 and bus 6, F2, F3, F4, F5 and F6 represent loads connected to bus 2, bus 3, bus 4, bus 5 and bus 6 respectively, Q C4 represents a parallel capacitor compensation device at bus 4.

[0067] In static stability analysis, the generators, transformers and transmission lines and other elements in the system can be equivalent to corresponding impedance branches. Among them, the static stability boundary of the AC transmission line (especially the line with voltage grade of 110 kV and above, which undertakes the function of point-to-point power transmission) is the key link to limit the power transmission capacity of the power grid, and is also the main analysis object of the method of the present application.

[0068] Based on the above understanding, the whole network static stability problem can be reasonably decoupled: since the static instability of any transmission branch (manifested as the imbalance of its terminal power-voltage relationship) will lead to the instability of its downstream power supply area, therefore, the sufficient and necessary condition to ensure the static stability of the whole network is to ensure that each branch in the network meets the static stability. Therefore, it is allowed to focus on the single line model.

[0069] For a single AC transmission line, a head-end voltage reference system impedance model (as shown in Figure 3 ) is established for analysis. The core assumptions and parameter definitions of this model are as follows:

[0070] The line is equivalent to a lumped parameter impedance , whose modulus and impedance angle are determined by the line parameters and are considered as known constants in the analysis.

[0071] Since in actual operation, the line head-end voltage is usually maintained relatively stable through generator excitation control, transformer voltage regulation and reactive power compensation, etc., and is less affected by the change of the terminal load, it can be set as a known reference phasor, denoted as .

[0072] The line terminal voltage phasor is , and the phase difference between the head-end voltage phasor and the terminal voltage phasor is defined as the power angle . , are the phase of the head-end voltage phasor and the terminal voltage phasor, respectively.

[0073] The line current phasor is .

[0074] The line terminal load is characterized by the terminal complex power . Wherein, and are the active power and the reactive power, respectively, is the apparent power, is the power factor angle (i.e. the angle lags behind ).

[0075] In the head-end voltage reference system impedance model, the following two basic relationships between the above physical quantities determined by the basic laws of circuit are satisfied: voltage equation and power equation.

[0076] According to Kirchhoff's voltage law, the line equivalent impedance , the line current phasor , the head-end voltage phasor and the terminal voltage phasor​ The voltage equation satisfied by is: ;

[0077] From the definition of complex power, the line current phasor , the terminal voltage phasor and the terminal complex power satisfy the power equation: where denotes the conjugate complex of the line current phasor .

[0078] The above voltage reference frame impedance model and the two basic equations form the strict physical and mathematical basis for all subsequent geometric analysis and derivation.

[0079] Step S2: Based on the association relationship, construct a voltage phasor triangle with the first end voltage phasor, the terminal voltage phasor and the line voltage drop as the sides.

[0080] Based on the first end voltage reference frame impedance model and the basic equations in step S1, the first end voltage phasor , the terminal voltage phasor and (line voltage drop) form a closed voltage phasor triangle, the geometric relationship of which is shown in Figure 4 .

[0081] According to phasor geometry, the angle relationship of each side of the voltage phasor triangle can be derived from the circuit phasor. Specifically, the line voltage drop phasor is in the same direction as the line current phasor , and the included angle is the line impedance angle . At the same time, the included angle between the line voltage drop phasor and the terminal voltage phasor is . This relationship can be expressed as:

[0082] (1)

[0083] Therefore, the three internal angles of the voltage phasor triangle shown in Figure 4 can be determined as:

[0084] (2)

[0085] Where, when the line type is determined (i.e. the impedance angle is constant), the angle only changes with the terminal power factor angle . This characteristic is the key to the subsequent analysis of the vertex geometric locus.

[0086] According to the voltage equation in step S1 and power equation (For three-phase system, it is often simplified as in per-unit or single-phase equivalent analysis), the expression of terminal apparent power is:

[0087] (3)

[0088] where, is the amplitude of terminal voltage phasor (i.e. terminal voltage), is the amplitude of line voltage drop. If is a constant, then:

[0089] (4)

[0090] Equation (4) shows that the terminal apparent power is proportional to the product of the lengths of two sides (corresponding to and ) in the voltage phasor triangle. This constant relationship is the mathematical basis for directly linking the power physical quantity and the geometric characteristics of the triangle.

[0091] Step S3: Analyze the moving track 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.

[0092] To facilitate purely geometric analysis, construct a corresponding triangle Figure 4 similar to the voltage phasor triangle in , as shown in Figure 5 . Figure 5 In , the triangle side and the voltage phasor amplitude have a clear corresponding relationship:

[0093] (5)

[0094] where, is the amplitude of the source voltage phasor (i.e. source voltage). Considering a specific variation mode of the terminal complex power : its power factor angle remains a fixed value unchanged, only its modulus changes. In this mode, according to equation (2) in step S2, the angle opposite to side AB in triangle becomes a constant value. At the same time, the length of side AB is known and unchanged, only the lengths of sides b and c and the position of point C change with .

[0095] Since the angle Invariable, its side AB (chord) is fixed, according to the circumference angle theorem, the locus of vertex C (corresponding to the end voltage phasor endpoint) is constrained on a specific circle - the circle with AB as chord, and the circumference angle opposite to AB chord is constant . Therefore, when changes, vertex C will move on a circular arc of the circle. The diameter 2R of the circumscribed circle can be determined by the sine theorem:

[0096] (6)

[0097] Generalized to the general case: if the end complex power changes in different power factor angles direction (i.e. takes different values), for each fixed , there is a corresponding angle , thus determining a circumscribed circle with AB as the common chord but different diameters. As shown in Figure 6 and Figure 7 , when (i = 1, 2, 3,...) changes, vertex C will move on the circular arc corresponding to the current value. The specific form of the circular arc (minor arc, semicircle, or major arc) is determined by the angle :

[0098] (7)

[0099] Therefore, the voltage phasor triangle of the line is a "common chord inscribed triangle" family with the amplitude of the primary voltage phasor as the fixed chord length. When the end complex power changes in a certain power factor angle direction, the vertex C of the triangle moves on the circular arc determined by the circumscribed circle corresponding to the . This geometric constraint relationship is the bridge connecting power change and voltage phasor form evolution.

[0100] Step S4: Based on the moving locus, establish the proportional relationship between the area of the voltage phasor triangle and the apparent power.

[0101] For the triangle corresponding to a specific power factor angle analyzed in step S3, according to formula (3), its apparent power can be expressed as:

[0102] (8)

[0103] Let The height of the triangle corresponding to the base AB is h, and the area of the triangle is may be expressed as

[0104] (9)

[0105] According to the sine theorem, for we have

[0106] (10)

[0107] where , and R is the radius of the circumscribed circle. Using the relationship of the sine theorem , formula (9) can be further transformed into

[0108] (11)

[0109] By combining formula (8) and formula (11), the apparent power and the direct relationship between the area of the triangle can be obtained:

[0110] (12)

[0111] Formula (12) shows that for a fixed , the apparent power is proportional to the area of the triangle , and the proportional coefficient is . At the same time, since the height h is proportional to the area (fixed base ), there is also .

[0112] Extending the above relationship to any power factor angle , for the corresponding triangle , the apparent power is

[0113] (13)

[0114] The derivation in step S4 proves an important geometric and physical law: for a given line (fixed , and power factor angle , the end apparent power is proportional to the area of the corresponding voltage phasor triangle . This relationship (13) is the key mathematical basis for transforming the physical limit problem of power transmission into the area extremum problem of geometric figures.

[0115] Step S5: According to the proportional relationship, it is determined that the apparent power reaches the maximum when the voltage phasor triangle is an isosceles triangle.

[0116] Based on the analysis of step S3, for any given power factor angle , the corresponding voltage phasor triangle vertex C can only move on a specific circular arc with AB as the chord . Combined with the key relationship obtained in step S4 that the apparent power is proportional to the triangle area , the problem of finding the maximum apparent power is equivalent to finding the position of the C point on this circular arc that maximizes the triangle area.

[0117] According to the principles of plane geometry, for a triangle with fixed base AB and moving vertex C on a given circular arc, when the vertex C moves to the intersection point of the median of the base AB and the circular arc (such as the Cs point in Figure 5 , Figure 6 ), the vertical distance (height h) of the vertex C to the base AB reaches the maximum, and thus the area of the triangle also reaches the maximum on this circular arc. Accordingly, the apparent power at this time also reaches its maximum .

[0118] Observe the shape of the triangle at this time: since the vertex C is located on the median of the base AB, it satisfies AC = BC, that is , an isosceles triangle. Mapping back to the original physical quantities, this geometric condition corresponds to:

[0119] (14)

[0120] The first equation indicates that the amplitude of the terminal voltage phasor is equal to the amplitude of the line voltage drop, and the second equation gives the relationship between the specific power angle and the internal angle of the triangle when the maximum apparent power is reached.

[0121] When is an isosceles triangle, its maximum area can be directly calculated through geometric relationships. Let the length of the isosceles triangle's leg be l (i.e., AC = BC = l), the base be , and the vertex angle (located at vertex C) be . Through the relationship of trigonometric functions, it can be deduced that its maximum area is:

[0122] (15)

[0123] Substituting formula (15) into formula (13) and simplifying using trigonometric identities, the line at a given power factor angle The maximum apparent power that the line can transmit, i.e. the expression of its static stability power boundary:

[0124] (16)

[0125] Therefore, when the line transmission apparent power reaches the maximum value , its corresponding voltage phasor triangle presents as an isosceles triangle. This simple geometric form is the key geometric condition for determining whether the line reaches the static stability power boundary. Formula (17) gives the specific calculation method of the boundary value.

[0126] Step S6: Based on the static stability critical state corresponding to the AC line when the apparent power reaches the maximum, whether the voltage phasor triangle is an isosceles triangle is determined as the geometric condition for determining whether the AC line is in the static stability boundary; based on the geometric condition, the static stability boundary of the AC line is determined.

[0127] In actual operation of power systems, it is usually required that the line end operates in a lagging power factor state, i.e. generally , which makes the circumscribed arc of the corresponding triangle as shown in the inferior arc Figure 7 . Under this premise, combined with formula (12), the static stability state can be strictly geometrically divided and proved.

[0128] Observe the position of the vertex C (representing the end point) on the circular arc:

[0129] Stable region: when the vertex C moves on the right circular arc of the perpendicular line of the edge AB (the amplitude of the first end voltage phasor) (such as Figure 7 ). In this region, as the apparent power increases, the height h of the triangle increases, but the amplitude of the end voltage phasor decreases, and the power angle increases. Its geometric characteristics correspond to the classical judgment of static stability:

[0130] (17)

[0131] At this time, the triangle side length satisfies , i.e. AC>BC, and the AC line is in a voltage and power angle static stability carrying state.

[0132] Unstable region: when the vertex C crosses the perpendicular line and is located on the left circular arc. In this region, as the apparent power decreases, the amplitude of the end voltage phasor decreases instead, and the power angle also decreases, showing a negative damping characteristic:

[0133] (18)

[0134] At this time, the triangle side length satisfies , i.e. AC < BC, the AC line is in a state of voltage and power angle static instability.

[0135] Critical boundary: when the vertex C is located on the perpendicular bisector of the AB side ( the static stability and instability demarcation line), the apparent power reaches the maximum value , and satisfies:

[0136] (19)

[0137] This is equivalent to the classical static stability critical condition that the partial derivative of active power and reactive power to each state variable is zero:

[0138] (20)

[0139] At this time, the triangle is an isosceles triangle, i.e. it satisfies the geometric condition shown in formula (14).

[0140] Therefore, the perpendicular bisector of the AB side is the voltage phasor geometric demarcation line from the static stability region to the static instability region. And the geometric condition that the "voltage phasor triangle is an isosceles triangle" is the only and universal criterion for the line to reach the static stability critical state.

[0141] Based on the above isosceles triangle criterion, the various static stability boundaries of the line (such as shown in Figure 8 , Figure 9 ) can be uniformly derived:

[0142] Voltage boundary and attack angle boundary: directly derived from the geometric relationship of the isosceles triangle, the terminal voltage boundary (i.e. the terminal voltage at the static stability critical state or the terminal voltage stability boundary value) and the power angle boundary are respectively:

[0143] (21)

[0144] Both satisfy the relationship , which reveals the nature that voltage instability and power angle instability occur simultaneously and are related to each other in the static sense.

[0145] Power boundary: the apparent power corresponding to the critical state is the maximum transmission power, so the power stability boundary (i.e. the apparent power at the static stability critical state) That is equal to the maximum apparent power :

[0146] (twenty two)

[0147] Formula (22) can also be expressed in parabolic form through trigonometric identities, that is:

[0148] (twenty three)

[0149] 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:

[0150] (twenty four)

[0151] 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.

[0152] 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.

[0153] 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 .

[0154] Based on the circuit relationship, the line current... Simultaneously flowing through the line impedance and load impedance ,satisfy:

[0155] (25)

[0156] According to the static stability critical geometric condition determined in steps S5 and S6, that is, the voltage phasor triangle is an isosceles triangle , by substituting the relationship of formula (25), it can be deduced that the boundary condition that must be met by the equivalent load impedance of the line when it reaches the static stability criticality is

[0157] (26)

[0158] wherein, represents the critical load impedance that ensures the static stability of the line.

[0159] Considering that the line resistance and the load resistance in the actual system are generally positive >0, >0), in the complex impedance plane, the trajectory of the impedance boundary satisfying formula (26) is a semicircular arc with the center at the origin and the radius of (as shown in Figure 11 ). This is because, according to the conclusion of step S6, it needs to satisfy during stable operation, and in combination with formula (25), it can be equivalently deduced that . Therefore, the static stability domain of the line impedance (that is, the value range of the load impedance that ensures the stability of the line) can be described as:

[0160] (27)

[0161] This area corresponds to the shaded area on the right side of the semicircular arc in Figure 11 . Within this stability domain, if the modulus of the load impedance is larger (that is, the load is lighter), then the terminal voltage is higher, and the static stability of the line is better; on the contrary, is smaller (the load is heavier), then is lower, and the stability is worse.

[0162] The static stability boundary formulas (21) and (22) of voltage, power angle, and power obtained through geometric analysis are completely consistent with the results obtained by the algebraic extreme value method in the previous research (such as "Dispersed Load Safety Domain of Power Grid and Its Application (II): Boundary-essence-form of Static Instability of Line and Stability Margin of Load Safety Domain" [J]. Proceedings of the Chinese Society of Electrical Engineering, 2024, 44(05): 1737-1750.). The boundary curves are shown in Figure 8 , Figure 9 .

[0163] ​It can be seen that the phasor geometry method intuitively shows the process of the terminal complex power change leading to the voltage phasor change until instability, and intuitively obtains the stability boundary of power, voltage and power angle, and clearly explains the homogeneity of voltage and power angle instability. Compared with the algebraic extreme value method, this method is simple and intuitive, and the physical concept is clear.

[0164] Embodiment Two

[0165] To verify the relationship between the static stability of the power grid and the branch, and show the relationship between the power of each line and the static stability boundary when the power grid approaches static instability, the IEEE 9-node system simulation example is simulated and verified.

[0166] Simulation setting: Taking the outlet bus of generator G1 as the balance node, starting from the initial power flow state, gradually increasing the active and reactive power of all loads in the system and the active power output of generators G2 and G3 (the outlet bus is set as a PU node) by the same proportion, as shown in Figure 12 . Figure 12 Among them, G1, G2 and G3 respectively represent generators; T1, T2 and T3 respectively represent transformers, and L1-L6 represent lines; the node (bus) voltage information is represented in the format of "amplitude ∠ phase angle", and the unit is per unit (pu) and degree (°), for example, the voltage information at G2 is 1.02 ∠ 30.80°; the branch (line / transformer) power flow information is represented in the format of "active+jreactive", and the unit is per unit (pu), for example, the terminal power flow of L1 is 1.35+j1.32.

[0167] Critical state: when the growth ratio reaches 2.7 times, the PSASP power flow calculation program cannot converge, indicating that the system has lost static stability. Take the 2.6 times power flow state close to instability for analysis.

[0168] Power boundary verification: the terminal complex power operating points of the 6 key lines in the system under the 2.6 times power flow are plotted on the complex power plane, and compared with the power boundary curve of each line calculated by the formula (22) of the method, as shown in Figures 13 to 18 . Figures 13 to 18 In the figure, pu on the coordinate axis represents that the corresponding power value (active power P or reactive power Q) 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 the other lines still have a large margin. This verifies that: 1) the static stability of the whole network depends on the stability of each branch; 2) the power boundary derived by the method can accurately identify the stability limit of the actual system.

[0169] The phasor geometry analysis method for alternating current line static stability boundary provided by the application establishes a direct proportional relationship between the apparent power and the triangle area by constructing and analyzing the voltage phasor triangle and its circumscribed circle trajectory, and finds a unified and intuitive static stability geometry determination condition with the core of "voltage phasor triangle as an isosceles triangle". Based on the condition, the static stability boundary analytical expression of power, voltage, power angle and impedance is derived. The correctness of the method is verified by theoretical derivation and IEEE 9-node system simulation.

[0170] Compared with the traditional algebraic extremum method which depends on the solution of complex nonlinear equations, the method of the application converts the abstract stability limit problem into clear geometric figure analysis, has clear physical concept, and the analysis process is intuitive and simple, effectively overcomes the shortcomings of the traditional method that the analysis is difficult and the physical meaning is obscure, provides a powerful theoretical tool and intuitive engineering criterion for fast evaluation of the stability boundary in power grid planning, operation and monitoring, and has important theoretical value and wide engineering application prospect.

[0171] Embodiment three

[0172] The embodiment of the application also provides an electronic device, which comprises a memory, a processor and a computer program or instructions stored in the memory, and the processor executes the computer program or instructions to realize the phasor geometry based alternating current line static stability boundary analysis method in the embodiment of the application.

[0173] Although not shown, the electronic device comprises a processor which can perform various appropriate operations and processes according to programs and / or data stored in a read-only memory (ROM) or programs and / or data loaded from a storage section into a random access memory (RAM). The processor can be a multi-core processor or can include a plurality of processors. In some embodiments, the processor can include a general-purpose main processor and one or more special-purpose coprocessors, such as a central processing unit, a graphics processing unit (GPU), a neural network processing unit (NPU), a digital signal processor (DSP), etc. In the RAM, various programs and data required for device operation are also stored. The processor, the ROM and the RAM are connected to each other through a bus. An input / output (I / O) interface is also connected to the bus.

[0174] The above processor and memory are used together to execute programs / instructions stored in the memory, and the programs / instructions can realize the methods, steps or functions described in the above embodiments when executed by a computer.

[0175] Although not shown, the embodiments of the present application also provide a computer readable storage medium having stored thereon a computer program or instructions, which, when executed by a processor, implement the method for analyzing the static stability boundary of an AC power system based on the phase geometry in the embodiments of the present application.

[0176] The computer readable storage medium includes permanent and non-permanent, removable and non-removable media, which can realize information storage by any method or technology. The information can be computer readable instructions, data structures, program modules or other data. Examples of the storage medium of the computer 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 technology, compact disc read-only memory (CD-ROM), digital versatile disc (DVD) or other optical storage, magnetic cassette, disk storage or other magnetic storage device, or any other non-transmission medium that can be used to store information accessible by a computing device. According to the definition herein, the computer readable medium does not include transitory media such as modulated data signals and carriers.

[0177] The above only discloses specific embodiments of the present application, but the protection scope of the present application is not limited thereto. Any person skilled in the art can easily think of changes or modifications within the technical scope disclosed by the present application, which should be covered within the protection scope of the present application.

Claims

1. A method for power system static stability margin analysis based on phasor geometry, characterized in that, The analysis method comprises: obtaining a correlation between line equivalent impedance, line current phasor, head-end voltage phasor, tail-end voltage phasor and tail-end complex power based on a head-end voltage reference system impedance model of the AC line; constructing a voltage phasor triangle with the head-end voltage phasor, the tail-end voltage phasor and line voltage drop as sides based on the correlation; wherein the line voltage drop is equal to the product of the line current phasor and the line equivalent impedance; analyzing the moving track of the vertex of the voltage phasor triangle on the excircle arc corresponding to the power factor angle when the power factor angle of the tail-end complex power is a constant value and the apparent power varies; establishing a proportional relationship between the area of the voltage phasor triangle and the apparent power based on the moving track; determining that the apparent power reaches the maximum when the voltage phasor triangle is an isosceles triangle according to the proportional relationship; determining the geometric condition for judging whether the AC line is in the static stability boundary based on whether the voltage phasor triangle is an isosceles triangle, which corresponds to the static stability critical state of the AC line when the apparent power reaches the maximum; determining the static stability boundary of the AC line based on the geometric condition.

2. The method of claim 1, wherein the method is based on a phasor geometry. In the head-end voltage reference system impedance model, the correlation comprises: Line equivalent impedance Line current phasor Source voltage phasor and end voltage phasor Voltage equation satisfied by ; and line current phasor terminal voltage phasor and terminal complex power satisfied power equation: where denotes the conjugate complex of the line current phasor .

3. The method of claim 1, wherein the method is based on a phasor geometry of the AC power system. The arc where the moving track is located is a poor arc, a semicircle or an excellent arc, and the specific form is determined by the difference between the impedance angle of the line equivalent impedance and the power factor angle.

4. The method of claim 1, wherein the method is based on a phasor geometry of the AC power system. The proportional relationship is determined by the following formula: ； wherein, denotes the apparent power; denotes the power factor angle; denotes the modulus of the line equivalent impedance; denotes the area of the voltage phasor triangle; denotes the impedance angle.

5. The method of claim 1, wherein the method is based on a phasor geometry of the AC power system. The condition that the voltage phasor triangle is an isosceles triangle specifically refers to that the length of the side corresponding to the tail-end voltage phasor is equal to the length of the side corresponding to the line voltage drop.

6. The phasor geometry based alternating current line static stability boundary analysis method according to any one of claims 1-5, wherein, The geometric condition is used for determining at least one of the following boundaries when it is used for judging the static stability boundary of the AC line: tail-end voltage boundary, power angle boundary, apparent power boundary and equivalent load impedance boundary.

7. The method of phasor geometry based ac line static stability margin analysis as claimed in claim 6, wherein, Based on the geometric condition, the tail-end voltage boundary, the power angle boundary, the apparent power boundary and the equivalent load impedance boundary are respectively determined by the following formulas: ； ； ； ； wherein, denotes the end voltage stability boundary value; denotes the amplitude of the head voltage phasor; denotes the impedance angle; denotes the power factor angle; denotes the power angle stability boundary value; denotes the power stability boundary value; denotes the modulus of the line equivalent impedance; denotes the equivalent load impedance stability boundary value; denotes the load impedance.

8. An electronic device comprising a memory, a processor, and a computer program or instructions stored on the memory, wherein the computer program or instructions, when executed by the processor, cause the electronic device to perform the method of any one of claims 1-7. The processor executes the computer program or instruction to realize the AC line static stability boundary analysis method based on phasor geometry according to any one of claims 1-7.

9. A computer readable storage medium having stored thereon a computer program or instructions, characterized in that, The computer program or instruction is executed by the processor to realize the AC line static stability boundary analysis method based on phasor geometry according to any one of claims 1-7.

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