An analytical method for frequency full-response model of equivalent three-area interconnected power system

Abstract

The application discloses an analytical method for a full frequency response model of an equivalent three-area interconnected power system, and belongs to the field of power system frequency stability analysis and control. First, a single-machine frequency full response model is established, and dynamic equations and parameter expressions of the single-machine frequency full response model are obtained. Second, considering the frequency space-time distribution characteristics and the influence of power exchange of a tie line, the network structure of an interconnected system is simplified based on DC power flow, and equivalent aggregation and order reduction are performed based on three-area division of a disturbance area and a non-disturbance area, so as to establish an equivalent frequency response model of a three-area system. Finally, a three-area system frequency full response model is constructed, and analytical expressions of frequency differences of each area are obtained. The application can accurately depict the full frequency response process of the system, and simultaneously reflect the frequency space-time distribution characteristics and the influence of power exchange of the tie line between areas, and is suitable for large-disturbance frequency safety analysis of a three-area interconnected power system, and has the advantages of fast calculation speed, clear physical meaning and convenience for online fast evaluation.

Description

Technical Field

[0001] This invention belongs to the field of frequency stability analysis and control of power systems, and in particular relates to a frequency response analysis method for an equivalent three-zone interconnected power system (hereinafter referred to as a three-zone system). Background Technology

[0002] Solving for the frequency response under large disturbances is fundamental to frequency stability analysis and control of power systems. Traditional power systems rely primarily on conventional power sources, with relatively sufficient system inertia and primary frequency regulation capabilities, making frequency stability issues less prominent. At this stage, due to the relatively small grid size and close internal electrical connections, the spatial distribution of frequencies is not significant, and the dynamic changes in frequency across nodes within the system under large disturbances exhibit a high degree of consistency. Therefore, traditional frequency response analysis typically simplifies the analysis by equivalently aggregating all generator units within the system into a single-unit model with lumped parameters (i.e., the system center of inertia, COI).

[0003] However, with the continuous expansion of regional power grid interconnection and the integration of a high proportion of renewable energy into the grid, the overall frequency support capacity of the system shows a significant non-uniform downward trend, and the frequency stability situation is becoming increasingly severe. Simultaneously, due to the geographical distribution of long-distance transmission lines and the differences in dynamic parameters of generating units between regions, the evolution of frequency at nodes in each region under large disturbances exhibits strong spatiotemporal distribution characteristics. In the frequency stability analysis and control of modern power grids, if a single frequency model is still used while ignoring spatial characteristics, serious engineering risks will arise: 1) It is difficult to effectively observe the frequency exceedance risk of local regional nodes, easily leading to misjudgments of the overall frequency security situation; 2) Using a single equivalent frequency as the control basis can easily cause overshoot or undershoot in some regional generating units, resulting in maloperation or failure to operate emergency frequency control strategies (such as low-frequency load shedding). Therefore, accurately considering the spatiotemporal distribution characteristics of frequency has become a prerequisite for the safe and stable operation of new interconnected power grids.

[0004] Current methods for solving frequency response mainly include time-domain simulation, data-driven methods, and mathematical analytical methods. Time-domain simulation offers high accuracy but incurs high computational costs, making it difficult to meet the demands of online real-time applications. Data-driven methods have made breakthroughs in solution efficiency, but their accuracy is highly dependent on sample quality, and their generalization ability is limited under extreme conditions. In contrast, mathematical analytical methods demonstrate unique advantages in rapid online security assessments due to their fast computation speed, clear physical mechanisms, and strong generalization ability. However, in pursuit of ease of solution, existing analytical methods often significantly simplify the network topology and dynamic processes, making it difficult to balance speed and accuracy.

[0005] Specifically, existing analytical models face two major bottlenecks when dealing with complex spatial and temporal dynamic characteristics. First, in the spatial dimension, existing models are mostly limited to lumped parameters of a single machine, making it difficult to reflect the spatiotemporal distribution characteristics of frequency. While existing two-zone equivalent models have laid a theoretical foundation for revealing the basic dynamic coupling mechanism between regions, the multimodal inter-machine oscillations and nonlinear interactions involved in three-zone and above interconnected systems, which are increasingly complex in modern power grid topologies, are completely beyond the representation scope of two-zone models. Second, existing analytical solutions often fail to strictly consider the initial state of dynamic components (such as generator rotors and governors) at the moment of disturbance, resulting in mathematical discontinuities in the model at the point of operating condition switching, making it difficult to accurately capture the trajectory of the entire response process. Summary of the Invention

[0006] To address the problems existing in the prior art, this invention provides an analytical method for the full frequency response of a three-zone system, specifically an analytical method for the full frequency response model of an equivalent three-zone interconnected power system. Based on the existing two-zone equivalent analysis approach, this method constructs a higher-dimensional spatial analytical framework and rigorously considers the physical constraints of the initial state on the full response process, thereby achieving a rapid and accurate characterization of the spatiotemporal evolution trajectory of the frequency of a large-scale interconnected power grid.

[0007] To achieve the above objectives, the technical solution adopted by the present invention is as follows: An analytical method for a frequency full response model of an equivalent three-zone interconnected power system, the analytical method comprising the following steps: S1: Establish a single-unit frequency full response model, propose a full response analysis theory considering the initial state of dynamic components before disturbance, and obtain the dynamic equations and parameter expressions of the single-unit frequency full response model, providing a foundation for the subsequent construction and analysis of the frequency full response model of the three-zone system. Specifically: S1.1: To accurately depict the continuous evolution of frequency response at different time periods, this paper adopts a single-machine frequency full response model for analysis. Based on the single zero-state response of the classical system frequency response model, this model explicitly incorporates the zero-input response caused by the initial state, thus making it suitable for multi-boundary-point disturbance scenarios.

[0008] The relevant symbols for the single-machine frequency total response model are defined here as follows: This is the complex frequency domain expression for the active power disturbance; Let be the system's inertial time constant; The system damping coefficient; This refers to the droop coefficient of the speed controller; The time constant of the speed controller; This refers to the power coefficient of the high-pressure steam turbine. For mechanical power gain; This is the initial value for the frequency change; This represents the initial power value of the low-pressure cylinder in the steam turbine. The damping ratio; It is the natural frequency; This is the actual natural frequency.

[0009] S1.2: The zero-state response in the single-machine frequency full response model can be expressed as: (1) The expressions for the relevant parameters are as follows: (2) S1.3: In the single-machine frequency full response model, the zero-input response is the sum of the initial input of the frequency change and the initial input of the medium and low pressure cylinder, which can be expressed as: (3) In the formula, As the first parameter, This is the second parameter. The expressions for the relevant parameters are as follows: (4) Summing equations (1) and (3), we obtain the complex frequency domain expression for the single-machine frequency full response model: (5) Since the equivalent frequency response model of the subsequent three-zone system needs to simultaneously characterize the regional frequency coupling process and the influence of the initial state of the dynamic components on the entire response process, it is necessary to first establish a single-machine frequency full response model that takes into account the initial state as the basis for subsequent multi-zone model coupling analysis.

[0010] S2: Based on the single-machine frequency full response model described in S1, considering the spatiotemporal distribution characteristics of frequency and the influence of tie-line power exchange, the network structure of the interconnected system is simplified based on DC power flow. Furthermore, based on the division into three zones—disturbed and undisturbed—regional equivalent aggregation and order reduction are performed to establish an equivalent frequency response model for the three-zone system. Specifically: S2.1: Let the baseline capacity be... The network equation expression for the extended model of the DC power flow network is as follows: (6) In the formula, the subscript g represents the virtual node of the generator internal potential, the subscript G represents the generator terminal bus node, and the subscript LD represents the load node; The generator rotor angle, The phase angle of the generator terminal bus voltage. The phase angle of the load node voltage; The generator output power Power injected into the generator terminal bus Power injected into load nodes The column vector formed; This is a submatrix of the DC power flow network admittance matrix. This is the admittance submatrix between generator terminal bus nodes. This is the admittance submatrix between the generator terminal bus node and the load node. This is the admittance submatrix between the load node and the generator terminal bus node. This is the admittance submatrix between load nodes; To extend the admittance matrix between the node and the original generator node, Let be the admittance submatrix between virtual nodes of generator internal potential. This is the admittance submatrix between the virtual node of the generator's internal potential and the bus node at the generator terminal. This is the admittance submatrix between the generator terminal bus node and the virtual node of the generator internal potential; , ,in N The number of generator nodes. L The number of load nodes. For generator node index, For load node indexing.

[0011] When the virtual node of the generator's internal potential When the corresponding generator is out of service, its corresponding generator terminal bus node No longer participating in power frequency regulation, it is incorporated into the extended non-generator node set in the incremental model. Unlike ordinary load nodes, the generator terminal bus node... After being incorporated into the extended set of non-generator nodes, it lacks power frequency regulation capability. Equation (6) can be rewritten and expressed as an incremental equation: (7) In the formula, This represents the increase in the electromagnetic power of the generator. To inject power increments into the extended non-generator nodes; , , , This is a submatrix of the DC power flow network admittance matrix after the system is modified. Let be the admittance submatrix between virtual nodes of generator internal potential. To expand the admittance submatrix between non-generator nodes and generator internal potential virtual nodes, This is the admittance submatrix between the virtual node of the generator's internal potential and the extended non-generator node. To provide the admittance submatrix between extended non-generator nodes; This represents the generator rotor angle increment. for Change in node phase angle; For generator end bus node Non-generator node Form an extended set of non-generator nodes, containing the number of nodes. N+L indivual.

[0012] When power loss occurs The expression for the change in nodal phase angle is: (8) Substituting equation (8) into equation (7) will eliminate the problem. The node yields the increase in the generator's electromagnetic power. The expression is: (9) In the formula, For inter-machine oscillation matrix, The disturbance power allocation matrix has the following expressions: (10) S2.2: Based on the three-zone division of the disturbed and undisturbed regions, the mathematical equations of the three-zone equivalent frequency response model are as follows: (11) In the formula, The rotor angle column vector, The rated angular frequency, It is a frequency difference column vector. The diagonal matrix is ​​the inertial time constant. This is the diagonal matrix of damping coefficients. To simplify the admittance matrix, , To simplify the susceptance matrix, This is the mechanical power increment matrix. Let be the perturbation matrix.

[0013] S2.3: Since the order of the three-zone equivalent frequency response model described in S2.2 is still very high, it needs to be further simplified to reduce the order.

[0014] The mechanical power increment is expressed as the sum of the proportional feedback power and the feedback power of the first-order inertial element, which, combined with equation (11), yields: (12) In the formula, For proportional feedback power, This represents the feedback power of a first-order inertial element.

[0015] Because first-order inertial elements have low-pass filtering characteristics, they can be used as low-pass filters. Let be the frequency difference column vector after equalization of the three regions, where Indicates the first Frequency difference Then the first... Zone frequency difference Decomposed into inertial center frequency and the Zone oscillation frequency Furthermore, since the inertial center frequency of the three-zone system reflects the overall low-frequency dominant response of the system, it is possible to adopt... Approximate substitution Simultaneously, the loop is unblocked at the frequency feedback point of the first-order inertial element, that is... and Decouple, and then use As an approximate input, where, ,in For the first Feedback power of the first-order inertial element in the region.

[0016] The above simplified steps utilize the characteristics of low-pass filtering to filter out... Therefore, adopt approximate Equations (11) and (12) are rewritten as follows: (13) in This represents the offset of the approximate feedback power value of the first-order inertial element. , This represents the power gain coefficient of the steam turbine. This is the adjustment coefficient. It is a unit column vector. This is the proportional coefficient of the steam turbine. This is a 0-1 vector used for region aggregation; The equivalent damping matrix of the system; This is the rotor angle column vector.

[0017] S3: Based on the single-machine frequency full response model established in S1 and the equivalent frequency response model of the three-zone system established in S2, a three-zone system frequency full response model is constructed. The closed-form solution of this model is then solved using the modal superposition method to obtain the analytical expression for the frequency difference in each zone. Specifically: S3.1: Based on the analysis method of S1, the transfer function of the equivalent frequency response model of the three-zone system is obtained according to equation (11): (14) In the formula, Let be the transfer function matrix of the generator rotor in its initial state. Let be the transfer function matrix of the initial state of the first-order inertial feedback power.

[0018] S3.2: Formula (14) in S3.1 represents a typical second-order two-degree-of-freedom forced vibration system with viscous damping. To solve formula (14), the modal superposition method in vibration theory is used.

[0019] Solve the eigenvalue problem based on the undamped free vibration equation corresponding to equation (14). To obtain its eigenvalues ​​and eigenvectors, a regular transformation is performed to... Represented as a mode matrix With modal coordinates The product is shown below: (15) in, Mode 1; Mode 2; It is mode 3.

[0020] Substituting equation (15) into equation (14), multiply both sides of the equation. The dynamic equations describing the modal coordinates are obtained through inverse Laplace transform, as shown below: (16) In the formula, M The main mass matrix, K The principal stiffness matrix, C Here is the modal damping matrix. F ( t () represents the excitation in modal coordinates; Modal coordinates; The first derivative of the modal coordinates; It is the second derivative of the modal coordinates.

[0021] S3.3: Based on the modal coordinate description obtained in S3.2, the dynamic equations are used to ignore off-diagonal elements using the forced decoupling method. Equation (16) is decoupled into equations with respect to the modal coordinates of S3.2. y 1. y Differential equations for y2 and y3: (17) (18) (19) in, The moment of inertia of region 1, The moment of inertia of region 2. The moment of inertia of zone 3; The damping coefficient for zone 1. The damping coefficient for zone 2. The damping coefficient for zone 3; The time constant of the speed controller in Zone 1. The time constant of the speed controller in Zone 2. The time constant of the speed controller in Zone 3; This is the power gain coefficient related to the generator power factor and reserve factor in Zone 1; The power gain coefficient related to the generator power factor and reserve factor in Zone 2; The power gain coefficient related to the generator power factor and reserve factor in Zone 3; The initial value of the frequency difference instant before the disturbance in zone 1 occurs. The initial value of the frequency difference instant before the disturbance in zone 2 occurs. The initial value of the frequency difference instant before the disturbance in Zone 3 occurs; The Dirac impulse function; The equivalent power change of the inertial element of the turbine in Zone 1 is input. The equivalent power change of the inertial element of the turbine in zone 2 is input. Input the equivalent power change of the inertial element of the turbine in Zone 3; This represents the disturbance power in Zone 1. This represents the disturbance power in zone 2. This represents the disturbance power in zone 3. , , The elements are the equivalent matrix obtained by multiplying the inter-machine oscillation matrix by the rated angular frequency, where the subscript indicates the row and column of the element in the matrix; This refers to the output power of the low- and intermediate-pressure cylinders of the turbine in zone 1 at the moment of system state transition. This refers to the output power of the low- and intermediate-pressure cylinders of the turbine in zone 2 at the moment of system state transition. This refers to the output power of the low-pressure cylinder of the steam turbine in zone 3 at the moment of system state switching; The element in the 2nd row and 1st column of the modality matrix. The element in the second row and second column of the modality matrix. The element in the 2nd row and 3rd column of the modality matrix. The element in the 3rd row and 1st column of the modality matrix. The element in the 3rd row and 2nd column of the modality matrix. This is the element in the 3rd row and 3rd column of the modality matrix; The rated angular frequency; For the second-order differential of mode one, For the first-order differential of mode one, For the second-order differential of mode two, For the first-order differential of mode two, For the second-order differential of mode three, It is the first-order differential of mode three.

[0022] S3.4: Based on equation (17) in S3.3, its essence is a low-order average frequency response model containing a single rotor and three feedback loops. The actual meaning is that the system's angular frequency has a named value. This equation can be further transformed into a low-order system frequency response model. Therefore, The closed-form solution is as follows: (20) In the formula, For the first-order differential of mode one, The disturbance power experienced by the system; This is the system droop coefficient; The system damping coefficient; For the system's mechanical power gain; The damping ratio; It is the natural frequency; The time constant of the speed controller; This refers to the actual natural frequency; This is the initial frequency difference; The first coefficient; The second coefficient; The initial state of the first-order inertial feedback power; specifically: (twenty one) in, The rated angular frequency; It is the natural frequency; This refers to the actual natural frequency; The damping ratio; Let the system's rotational inertia be denoted by . For system load damping; It is the reciprocal of the system droop coefficient; The power ratio of the high-pressure cylinder in the system; The system time constant; The disturbance received by the system; These are the coefficients in the system's frequency response expression; These are the coefficients in the system's frequency response expression.

[0023] S3.6: Based on equation (18) in S3.5, due to the offset of the approximate value of the feedback power of the first-order inertial element. The resulting response is much smaller Therefore, it can be ignored. right Due to the influence of accuracy, solving equation (18) yields: (twenty two) In the formula, For the first-order differential of mode two, a 2 The first coefficient; 2 is the second coefficient; 2 is the third coefficient; It is the fourth coefficient; It is the fifth coefficient; It is the sixth coefficient; 2 represents the damping ratio corresponding to the second mode; It is the natural frequency corresponding to the second mode. It is the actual natural frequency corresponding to the second mode; The time constant of the speed controller in Zone 1. The time constant of the speed controller in Zone 2. The time constant of the speed controller in Zone 3; specifically: (twenty three) in, The element in the 2nd row and 1st column of the modality matrix. The element in the second row and second column of the modality matrix. This is the element in the 2nd row and 3rd column of the modality matrix.

[0024] S3.7: Based on equation (19) in S3.5, due to the offset of the approximate value of the feedback power of the first-order inertial element. The resulting response is much smaller Therefore, it can be ignored. right Due to the influence of accuracy, solving equation (19) yields: The closed-form solution.

[0025] (twenty four) In the formula, For the first-order differential of mode three, a 3 The first coefficient; b 3 The second coefficient; c 3 It is the third coefficient; It is the fourth coefficient; It is the fifth coefficient; It is the sixth coefficient; The damping ratio corresponding to the third mode; The natural frequency corresponding to the third mode; This refers to the actual natural frequency corresponding to the third mode; specifically: (25) In the formula, The element in the 3rd row and 1st column of the modality matrix. The element in the 3rd row and 2nd column of the modality matrix. This is the element in the 3rd row and 3rd column of the modality matrix.

[0026] S3.8: Based on the closed-form solutions of modal coordinates in S3.4, S3.5, S3.6, and S3.7, and equation (15), the closed-form solution expression of the three-zone frequency full response model is obtained as follows: (26) in, Here is the modal matrix; The rated angular frequency; The regional frequency difference of region 1. The regional frequency difference of region 2. This represents the regional frequency difference of region 3.

[0027] The beneficial effects of this invention are as follows: (1) By taking into account the initial state of the generator rotor and the first-order inertial feedback loop of the governor before the disturbance in the single-machine frequency full response model, the present invention ensures the continuity of the model description before and after the disturbance, thereby enabling a more accurate characterization of the system frequency full response process and avoiding the problem of mathematical description discontinuity at the working condition switching point in the existing analytical method.

[0028] (2) This invention simplifies the network structure of the interconnected system based on DC power flow, and performs regional equivalent aggregation and order reduction processing on the basis of the three-zone division of the disturbance zone and the non-disturbance zone, and establishes an equivalent frequency response model of the three-zone system. This model can simultaneously reflect the spatiotemporal distribution characteristics of frequency and the influence of power exchange between regional tie lines, and is more suitable for the large disturbance frequency security analysis of the three-zone interconnected power system.

[0029] (3) This invention constructs a three-zone system frequency full response model by coupling the single-machine frequency full response model with the three-zone system equivalent frequency response model, and further obtains the closed-form solution expression of the frequency difference of each zone. Compared with the time-domain simulation method that relies on numerical integration, it has the advantages of fast calculation speed, clear physical meaning and convenient online rapid evaluation. Attached Figure Description

[0030] Figure 1 The single-machine frequency full response model provided by this invention; Figure 2 The equivalent frequency response model of the three-zone system provided by this invention; Figure 3 The analytical and simulation results of the three-zone system frequency full response model provided by this invention;Figure 3 (a) in the figure is a comparison diagram of the frequency difference of the disturbance region; Figure 4 (b) and (c) in the diagram are frequency difference comparison diagrams of the non-disturbance region; Figure 4 This is a flowchart of the method of the present invention. Detailed Implementation

[0031] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, but this is not intended to limit the invention.

[0032] This invention introduces an analytical method for the frequency full response model of a three-zone system. The method comprises three parts: analytical analysis of the single-machine frequency full response model, analytical analysis of the equivalent frequency response model of the three-zone system, and analytical analysis of the frequency full response model of the three-zone system. It includes the following steps: S1: Establish a single-unit frequency full response model, propose a full response analysis theory considering the initial state of dynamic components before disturbance, and obtain the dynamic equations and parameter expressions of the single-unit frequency full response model, providing a foundation for the subsequent construction and analysis of the frequency full response model of the three-zone system. Specifically: S1.1: To accurately depict the continuous evolution of frequency response at different time periods, this paper adopts a single-machine frequency full response model for analysis. Based on the single zero-state response of the classical system frequency response model, this model explicitly incorporates the zero-input response caused by the initial state, thus making it suitable for multi-boundary-point disturbance scenarios.

[0033] The relevant symbols for the single-machine frequency total response model are defined here as follows: This is the complex frequency domain expression for the active power disturbance; Let be the system's inertial time constant; The system damping coefficient; This refers to the droop coefficient of the speed controller; The time constant of the speed controller; This refers to the power coefficient of the high-pressure steam turbine. For mechanical power gain; This is the initial value for the frequency change; This represents the initial power value of the low-pressure cylinder in the steam turbine. The damping ratio; It is the natural frequency; This is the actual natural frequency.

[0034] S1.2: The zero-state response in the single-machine frequency full response model can be expressed as: (1) The expressions for the relevant parameters are as follows: (2) S1.3: In the single-machine frequency full response model, the zero-input response is the sum of the initial input of the frequency change and the initial input of the medium and low pressure cylinder, which can be expressed as: (3) In the formula, As the first parameter, This is the second parameter. The expressions for the relevant parameters are as follows: (4) Summing equations (1) and (3), we obtain the complex frequency domain expression for the single-machine frequency full response model: (5) Since the equivalent frequency response model of the subsequent three-zone system needs to simultaneously characterize the regional frequency coupling process and the influence of the initial state of the dynamic components on the entire response process, it is necessary to first establish a single-machine frequency full response model that takes into account the initial state as the basis for subsequent multi-zone model coupling analysis.

[0035] S2: Based on the single-machine frequency full response model described in S1, considering the spatiotemporal distribution characteristics of frequency and the influence of tie-line power exchange, the network structure of the interconnected system is simplified based on DC power flow. Furthermore, based on the division into three zones—disturbed and undisturbed—regional equivalent aggregation and order reduction are performed to establish an equivalent frequency response model for the three-zone system. Specifically: S2.1: Let the baseline capacity be... The network equation expression for the extended model of the DC power flow network is as follows: (6) In the formula, the subscript g represents the virtual node of the generator internal potential, the subscript G represents the generator terminal bus node, and the subscript LD represents the load node; The generator rotor angle, The phase angle of the generator terminal bus voltage. The phase angle of the load node voltage; The generator output power Power injected into the generator terminal bus Power injected into load nodes The column vector formed; This is a submatrix of the DC power flow network admittance matrix. This is the admittance submatrix between generator terminal bus nodes. This is the admittance submatrix between the generator terminal bus node and the load node. This is the admittance submatrix between the load node and the generator terminal bus node. This is the admittance submatrix between load nodes; To extend the admittance matrix between the node and the original generator node, Let be the admittance submatrix between virtual nodes of generator internal potential. This is the admittance submatrix between the virtual node of the generator's internal potential and the bus node at the generator terminal. This is the admittance submatrix between the generator terminal bus node and the virtual node of the generator internal potential; , ,in N The number of generator nodes. L The number of load nodes. For generator node index, For load node indexing.

[0036] When the virtual node of the generator's internal potential When the corresponding generator is out of service, its corresponding generator terminal bus node No longer participating in power frequency regulation, it is incorporated into the extended non-generator node set in the incremental model. Unlike ordinary load nodes, the generator terminal bus node... After being incorporated into the extended set of non-generator nodes, it lacks power frequency regulation capability. Equation (6) can be rewritten and expressed as an incremental equation: (7) In the formula, This represents the increase in the electromagnetic power of the generator. To inject power increments into the extended non-generator nodes; , , , This is a submatrix of the DC power flow network admittance matrix after the system is modified. Let be the admittance submatrix between virtual nodes of generator internal potential. To expand the admittance submatrix between non-generator nodes and generator internal potential virtual nodes, This is the admittance submatrix between the virtual node of the generator's internal potential and the extended non-generator node. To provide the admittance submatrix between extended non-generator nodes; This represents the generator rotor angle increment. for Change in node phase angle; For generator end bus node Non-generator node Form an extended set of non-generator nodes, containing the number of nodes. N+L indivual.

[0037] When power loss occurs The expression for the change in nodal phase angle is: (8) Substituting equation (8) into equation (7) will eliminate the problem. The node yields the increase in the generator's electromagnetic power. The expression is: (9) In the formula, For inter-machine oscillation matrix, The disturbance power allocation matrix has the following expressions: (10) S2.2: Based on the three-zone division of the disturbed and undisturbed regions, the mathematical equations of the three-zone equivalent frequency response model are as follows: (11) In the formula, The rotor angle column vector, The rated angular frequency, It is a frequency difference column vector. The diagonal matrix is ​​the inertial time constant. This is the diagonal matrix of damping coefficients. To simplify the admittance matrix, , To simplify the susceptance matrix, This is the mechanical power increment matrix. Let be the perturbation matrix.

[0038] S2.3: Since the order of the three-zone equivalent frequency response model described in S2.2 is still very high, it needs to be further simplified to reduce the order.

[0039] The mechanical power increment is expressed as the sum of the proportional feedback power and the feedback power of the first-order inertial element, which, combined with equation (11), yields: (12) In the formula, For proportional feedback power, This represents the feedback power of a first-order inertial element.

[0040] Because first-order inertial elements have low-pass filtering characteristics, they can be used as low-pass filters. Let be the frequency difference column vector after equalization of the three regions, where Indicates the first Frequency difference Then the first... Zone frequency difference Decomposed into inertial center frequency and the Zone oscillation frequency Furthermore, since the inertial center frequency of the three-zone system reflects the overall low-frequency dominant response of the system, it is possible to adopt... Approximate substitution Simultaneously, the loop is unblocked at the frequency feedback point of the first-order inertial element, that is... and Decouple, and then use As an approximate input, where, ,in For the first Feedback power of the first-order inertial element in the region.

[0041] The above simplified steps utilize the characteristics of low-pass filtering to filter out... Therefore, adopt approximate Equations (11) and (12) are rewritten as follows: (13) in This represents the offset of the approximate feedback power value of the first-order inertial element. , This represents the power gain coefficient of the steam turbine. This is the adjustment coefficient. It is a unit column vector. This is the proportional coefficient of the steam turbine. This is a 0-1 vector used for region aggregation; The equivalent damping matrix of the system; This is the rotor angle column vector.

[0042] S3: Based on the single-machine frequency full response model established in S1 and the equivalent frequency response model of the three-zone system established in S2, a three-zone system frequency full response model is constructed. The closed-form solution of this model is then solved using the modal superposition method to obtain the analytical expression for the frequency difference in each zone. Specifically: S3.1: Based on the analysis method of S1, the transfer function of the equivalent frequency response model of the three-zone system is obtained according to equation (11): (14) In the formula, Let be the transfer function matrix of the generator rotor in its initial state. Let be the transfer function matrix of the initial state of the first-order inertial feedback power.

[0043] S3.2: Formula (14) in S3.1 represents a typical second-order two-degree-of-freedom forced vibration system with viscous damping. To solve formula (14), the modal superposition method in vibration theory is used.

[0044] Solve the eigenvalue problem based on the undamped free vibration equation corresponding to equation (14). To obtain its eigenvalues ​​and eigenvectors, a regular transformation is performed to... Represented as a mode matrix With modal coordinates The product is shown below: (15) in, Mode 1; Mode 2; It is mode 3.

[0045] Substituting equation (15) into equation (14), multiply both sides of the equation. The dynamic equations describing the modal coordinates are obtained through inverse Laplace transform, as shown below: (16) In the formula, M The main mass matrix, K The principal stiffness matrix, C Here is the modal damping matrix. F ( t () represents the excitation in modal coordinates; Modal coordinates; The first derivative of the modal coordinates; It is the second derivative of the modal coordinates.

[0046] S3.3: Based on the modal coordinate description obtained in S3.2, the dynamic equations are used to ignore off-diagonal elements using the forced decoupling method. Equation (16) is decoupled into equations with respect to the modal coordinates of S3.2. y 1. y Differential equations for y2 and y3: (17) (18) (19) in, The moment of inertia of region 1, The moment of inertia of region 2. The moment of inertia of zone 3; The damping coefficient for zone 1. The damping coefficient for zone 2. The damping coefficient for zone 3; The time constant of the speed controller in Zone 1. The time constant of the speed controller in Zone 2. The time constant of the speed controller in Zone 3; This is the power gain coefficient related to the generator power factor and reserve factor in Zone 1; The power gain coefficient related to the generator power factor and reserve factor in Zone 2; The power gain coefficient related to the generator power factor and reserve factor in Zone 3; The initial value of the frequency difference instant before the disturbance in zone 1 occurs. The initial value of the frequency difference instant before the disturbance in zone 2 occurs. The initial value of the frequency difference instant before the disturbance in Zone 3 occurs; The Dirac impulse function; The equivalent power change of the inertial element of the turbine in Zone 1 is input. The equivalent power change of the inertial element of the turbine in zone 2 is input. Input the equivalent power change of the inertial element of the turbine in Zone 3; This represents the disturbance power in Zone 1. This represents the disturbance power in zone 2. This represents the disturbance power in zone 3. , , The elements are the equivalent matrix obtained by multiplying the inter-machine oscillation matrix by the rated angular frequency, where the subscript indicates the row and column of the element in the matrix; This refers to the output power of the low- and intermediate-pressure cylinders of the turbine in zone 1 at the moment of system state transition. This refers to the output power of the low- and intermediate-pressure cylinders of the turbine in zone 2 at the moment of system state transition. This refers to the output power of the low-pressure cylinder of the steam turbine in zone 3 at the moment of system state switching; The element in the 2nd row and 1st column of the modality matrix. The element in the second row and second column of the modality matrix. The element in the 2nd row and 3rd column of the modality matrix. The element in the 3rd row and 1st column of the modality matrix. The element in the 3rd row and 2nd column of the modality matrix. This is the element in the 3rd row and 3rd column of the modality matrix; The rated angular frequency; For the second-order differential of mode one, For the first-order differential of mode one, For the second-order differential of mode two, For the first-order differential of mode two, For the second-order differential of mode three, It is the first-order differential of mode three.

[0047] S3.4: Based on equation (17) in S3.3, its essence is a low-order average frequency response model containing a single rotor and three feedback loops. The actual meaning is that the system's angular frequency has a named value. This equation can be further transformed into a low-order system frequency response model. Therefore, The closed-form solution is as follows: (20) In the formula, For the first-order differential of mode one, The disturbance power experienced by the system; This is the system droop coefficient; The system damping coefficient; For the system's mechanical power gain; The damping ratio; It is the natural frequency; The time constant of the speed controller; This refers to the actual natural frequency; This is the initial frequency difference; The first coefficient; The second coefficient; The initial state of the first-order inertial feedback power; specifically: (twenty one) in, The rated angular frequency; It is the natural frequency; This refers to the actual natural frequency; The damping ratio; Let the system's rotational inertia be denoted by . For system load damping; It is the reciprocal of the system droop coefficient; The power ratio of the high-pressure cylinder in the system; The system time constant; The disturbance received by the system; These are the coefficients in the system's frequency response expression; These are the coefficients in the system's frequency response expression.

[0048] S3.6: Based on equation (18) in S3.5, due to the offset of the approximate value of the feedback power of the first-order inertial element. The resulting response is much smaller Therefore, it can be ignored. right Due to the influence of accuracy, solving equation (18) yields: (twenty two) In the formula, For the first-order differential of mode two, a 2 The first coefficient; 2 is the second coefficient; 2 is the third coefficient; It is the fourth coefficient; It is the fifth coefficient; It is the sixth coefficient; 2 represents the damping ratio corresponding to the second mode; It is the natural frequency corresponding to the second mode. It is the actual natural frequency corresponding to the second mode; The time constant of the speed controller in Zone 1. The time constant of the speed controller in Zone 2. The time constant of the speed controller in Zone 3; specifically: (twenty three) in, The element in the 2nd row and 1st column of the modality matrix. The element in the second row and second column of the modality matrix. This is the element in the 2nd row and 3rd column of the modality matrix.

[0049] S3.7: Based on equation (19) in S3.5, due to the offset of the approximate value of the feedback power of the first-order inertial element. The resulting response is much smaller Therefore, it can be ignored. right Due to the influence of accuracy, solving equation (19) yields: The closed-form solution.

[0050] (twenty four) In the formula, For the first-order differential of mode three, a 3 The first coefficient; b 3 The second coefficient; c 3 It is the third coefficient; It is the fourth coefficient; It is the fifth coefficient; It is the sixth coefficient; The damping ratio corresponding to the third mode; The natural frequency corresponding to the third mode; This refers to the actual natural frequency corresponding to the third mode; specifically: (25) In the formula, The element in the 3rd row and 1st column of the modality matrix. The element in the 3rd row and 2nd column of the modality matrix. This is the element in the 3rd row and 3rd column of the modality matrix.

[0051] S3.8: Based on the closed-form solutions of modal coordinates in S3.4, S3.5, S3.6, and S3.7, and equation (15), the closed-form solution expression of the three-zone frequency full response model is obtained as follows: (26) in, The modal matrix, For the first-order differential of mode one, For the first-order differential of mode two, The first-order differential of mode three; The rated angular frequency; The regional frequency difference of region 1. The regional frequency difference of region 2. This represents the regional frequency difference of region 3.

[0052] The above model can be combined with specific scenario information, and MATLAB can be used to solve the frequency dynamic curve of the system under large disturbances.

[0053] The frequency full response model of the above three-zone system can be solved analytically using MATLAB in conjunction with specific examples, and compared with the time-domain simulation results of Matlab / Simulink to verify the accuracy and effectiveness of the analytical method proposed in this invention.

[0054] This embodiment constructs a 9-node, three-zone interconnected system as a case study. The system includes 3 generator nodes and 6 non-generator nodes, with a rated system frequency of 60Hz and a unified system base capacity of 100MVA. In the case study, node parameters, line parameters, and generator parameters together constitute the parameter basis of the equivalent frequency response model of the three-zone system. Branch parameters are determined by the branch's starting and ending nodes, resistance, reactance, and charging susceptance. Load frequency characteristics are represented using a load frequency sensitivity coefficient matrix. Based on these parameters, the inter-generator oscillation matrix, disturbance power distribution matrix, and relevant equivalent parameters required for solving the three-zone system's full frequency response model are further calculated.

[0055] This embodiment divides the entire frequency response process into two stages for analysis. The total simulation time is set to 30s, and the simulation step size is set to 0.01s. The first stage is from 0 to 10s, during which an initial disturbance power with an amplitude of -0.2pu is applied at the disturbance node; the second stage is from 10 to 30s, during which an additional disturbance power with an amplitude of 0.05pu is superimposed on the state at the end of the first stage to analyze the subsequent full frequency response process of the system under state switching conditions.

[0056] In the specific solution process, firstly, an equivalent frequency response model of the three-zone system is established based on the parameters of the example system, and the inertia matrix, equivalent damping matrix, equivalent stiffness matrix, and parameters required for mode decomposition are calculated. Then, the frequency response term of the system's inertial center is obtained based on the single-machine frequency full response model. At the same time, the frequency response components corresponding to the two oscillation modes are solved by combining the inter-machine oscillation modes, thereby obtaining the analytical curves of the frequency difference in the three zones within the first stage.

[0057] At the end of the first stage, the state variables at the boundary moments of the three regions are extracted. These state variables include the initial values ​​of the frequency difference of each region and the initial values ​​of the power of the low-pressure cylinder in the turbine. These are used as the initial input conditions for the analytical solution in the second stage. Based on these boundary point state variables, the analytical curves of the full frequency response of the three-region system under the action of the additional power regulation are solved in the second stage. Finally, the analytical results of the first and second stages are concatenated in chronological order to obtain the analytical results of the full frequency response of the three-region system within a 30-second time domain.

[0058] To verify the correctness of the analytical method proposed in this invention, a Matlab / Simulink time-domain simulation model was further constructed under the same system parameters and perturbation conditions, and the time-domain simulation results of the frequency response in three regions were extracted. To ensure that the analytical results and the time-domain simulation results are compared on the same time scale, the time-domain simulation results were interpolated to the time points corresponding to the analytical calculations. Then, the root mean square error (RMSE) and mean absolute percentage error (MAPE) were used as goodness-of-fit evaluation indicators. RMSE reflects the absolute error level between the analytical results and the time-domain simulation results, while MAPE reflects the relative error level between the two.

[0059] Table 1: Summary Table of Multi-Region Fit Indicators

[0060] The above results show that the analytical results of the proposed three-zone system frequency full response model are highly consistent with the Matlab / Simulink time-domain simulation results. The RMSE for all three zones is less than 0.005Hz, and the MAPE is much less than 0.01%, indicating that the method of this invention can accurately characterize the dynamic frequency evolution of the three-zone system under large disturbances and reflect the frequency coupling between zones and the propagation characteristics of inter-machine oscillations. Especially in the case of phased state transitions, this invention, by explicitly considering the frequency difference at the boundary point and the initial power state of the low-pressure cylinders of the turbine, ensures that the analytical results maintain good continuity and high computational accuracy at the stage transition points.

[0061] Therefore, this embodiment verifies the effectiveness of the analytical method for the equivalent three-zone interconnected power system frequency full response model proposed in this invention. Compared with time-domain simulation methods that rely solely on numerical integration, the method of this invention maintains high computational accuracy while possessing advantages such as clear physical meaning, fast solution speed, and ease of rapid analysis across multiple scenarios. It can provide effective support for frequency security analysis and control strategy research of three-zone interconnected systems.

[0062] The embodiments described above are merely illustrative of the implementation methods of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the protection scope of the present invention.

Claims

1. An analytical method for a frequency full response model of an equivalent three-zone interconnected power system, characterized in that, The analytical method for the frequency full response model of the equivalent three-zone interconnected power system includes the following steps: S1: Establish a single-machine frequency full response model, propose a full response analysis theory that considers the initial state of the dynamic components before disturbance, and obtain the dynamic equations and parameter expressions of the single-machine frequency full response model; specifically: S1.1: The single-machine frequency full response model is used for analysis to characterize the continuous evolution of the frequency response in each time period; based on the single zero-state response of the classical system frequency response model, the zero-input response caused by the initial state is explicitly included, which is suitable for multi-boundary-point disturbance scenarios. S1.2: Zero-state response in the single-machine frequency total response model Represented as: (1) in, This is the complex frequency domain expression for the active power disturbance; This refers to the droop coefficient of the speed controller; It is the natural frequency; The system damping coefficient; For mechanical power gain; The time constant of the speed controller; For the Laplace operator in the complex frequency domain; The damping ratio; This refers to the actual natural frequency; S1.3: Zero-input response in a single-machine frequency full response model It is the sum of the initial input of frequency change and the initial input of the medium and low pressure cylinder; S1.4: Obtain the single-machine frequency full response model Complex frequency domain expression: (2) First, a single-machine frequency full response model considering the initial state is established as the basis for subsequent multi-region model coupling analysis; S2: Based on the single-machine frequency full response model obtained in S1, considering the spatiotemporal distribution characteristics of frequency and the influence of tie line power exchange, the network structure of the interconnected system is simplified based on DC power flow. Based on the division of the three regions into disturbed and non-disturbed regions, regional equivalent aggregation and order reduction are performed to establish the equivalent frequency response model of the three-region system. S3: Based on the single-machine frequency full response model established in S1 and the three-zone system equivalent frequency response model established in S2, construct the three-zone system frequency full response model, and use the modal superposition method to solve the closed-form solution of the model to obtain the analytical expression of the frequency difference of each zone.

2. The analytical method for the frequency full response model of an equivalent three-zone interconnected power system according to claim 1, characterized in that, In S1.2, the expressions for the relevant parameters are as follows: (3) in, Let be the system's inertial time constant; This represents the power coefficient of the high-pressure steam turbine.

3. The analytical method for the frequency full response model of an equivalent three-zone interconnected power system according to claim 1, characterized in that, In S1.3, the zero-input response is: (4) In the formula, This is the initial value for the frequency change; This represents the initial power value of the low-pressure cylinder in the steam turbine. As the first parameter, This is the second parameter; the expressions for the relevant parameters are as follows: (5)。 4. The analytical method for the frequency full response model of an equivalent three-zone interconnected power system according to claim 3, characterized in that, Specifically, S2 is: S2.1: Let the baseline capacity be... The network equation expression for the extended model of the DC power flow network is as follows: (6) In the formula, the subscript g represents the virtual node of the generator internal potential, the subscript G represents the generator terminal bus node, and the subscript LD represents the load node; The generator rotor angle, The phase angle of the generator terminal bus voltage. The phase angle of the load node voltage; The generator output power Power injected into the generator terminal bus Power injected into load nodes The column vector formed; This is a submatrix of the DC power flow network admittance matrix. This is the admittance submatrix between generator terminal bus nodes. This is the admittance submatrix between the generator terminal bus node and the load node. This is the admittance submatrix between the load node and the generator terminal bus node. This is the admittance submatrix between load nodes; To extend the admittance matrix between the node and the original generator node, Let be the admittance submatrix between virtual nodes of generator internal potential. This is the admittance submatrix between the virtual node of the generator's internal potential and the bus node at the generator terminal. This is the admittance submatrix between the generator terminal bus node and the virtual node of the generator internal potential; , ,in N The number of generator nodes. L The number of load nodes. For generator node index, Index for load nodes; When the virtual node of the generator's internal potential When the corresponding generator is out of service, its corresponding generator terminal bus node No longer participating in power frequency regulation, it is incorporated into the extended non-generator node set in the incremental model; and the generator terminal bus node. After being incorporated into the extended set of non-generator nodes, it lacks power frequency regulation capability; Equation (6) is rewritten and expressed as an incremental equation: (7) In the formula, This represents the increase in the electromagnetic power of the generator. To inject power increments into the extended non-generator nodes; , , , This is a submatrix of the DC power flow network admittance matrix after the system is modified. Let be the admittance submatrix between virtual nodes of generator internal potential. To expand the admittance submatrix between non-generator nodes and generator internal potential virtual nodes, This is the admittance submatrix between the virtual node of the generator's internal potential and the extended non-generator node. To provide the admittance submatrix between extended non-generator nodes; This represents the generator rotor angle increment. for Nodal phase angle change; subscript For generator end bus node Non-generator node Form an extended set of non-generator nodes, containing the number of nodes. N+L indivual; When power loss occurs The expression for the change in nodal phase angle is: (8) In the formula, Admittance submatrix The inverse matrix; Substituting equation (8) into equation (7), we can eliminate... The node yields the increase in the generator's electromagnetic power. The expression is: (9) In the formula, For inter-machine oscillation matrix, The disturbance power allocation matrix has the following expressions: (10) S2.2: Based on the three-zone division of the disturbed and undisturbed regions, the mathematical equations of the three-zone equivalent frequency response model are as follows: (11) In the formula, This is the rotor angle column vector; The rated angular frequency; This is the frequency difference column vector after equalization of the three zones; This is a diagonal matrix representing the inertial time constant. This is a diagonal matrix of damping coefficients; To simplify the admittance matrix, , To simplify the susceptance matrix; This is the mechanical power increment matrix; Here is the perturbation matrix; S2.3: Simplify and reduce the order of the three-zone equivalent frequency response model obtained in S2.2; The mechanical power increment is expressed as the sum of the proportional feedback power and the feedback power of the first-order inertial element, which, combined with equation (11), yields: (12) In the formula, This is the proportional feedback power; The feedback power of the first-order inertial element; The disturbance power; Let be the frequency difference column vector after equalization of the three regions, where Indicates the first Frequency difference ; The first Zone frequency difference Decomposed into inertial center frequency and the Zone oscillation frequency ,use Approximate substitution Simultaneously, the loop is unblocked at the frequency feedback point of the first-order inertial element, that is... and Decouple, and then use As an approximate input; where, , For the first Feedback power of the first-order inertial element in the region; The above simplified steps utilize the characteristics of low-pass filtering to filter out... ,use approximate Equations (11) and (12) are rewritten as follows: (13) in, This represents the offset of the approximate feedback power value of the first-order inertial element. , This represents the power gain coefficient of the steam turbine. This is the adjustment coefficient. It is a unit column vector. This is the proportional coefficient of the steam turbine. This is a 0-1 vector used for region aggregation; The equivalent damping matrix of the system; This is the rotor angle column vector; This represents the disturbance power.

5. The analytical method for the frequency full response model of an equivalent three-zone interconnected power system according to claim 4, characterized in that, Specifically, S3 is: S3.1: Based on the analysis method of S1, the transfer function of the equivalent frequency response model of the three-zone system is obtained according to equation (11): (14) In the formula, Let be the transfer function matrix of the generator rotor in its initial state. Let be the transfer function matrix of the initial state of the first-order inertial feedback power; S3.2: Formula (14) in S3.1 represents a typical second-order two-degree-of-freedom forced vibration system with viscous damping. To solve formula (14), the modal superposition method in vibration theory is used. Solve the eigenvalue problem based on the undamped free vibration equation corresponding to equation (14). We obtain its eigenvalues ​​and eigenvectors, and then perform regularization transformations to... Represented as a mode matrix With modal coordinates The product is shown below: (15) in, This is the first mode; This is the second mode; It is the third mode; Substituting equation (15) into equation (14), multiply both sides of the equation. The dynamic equations describing the modal coordinates are obtained through inverse Laplace transform, as shown below: (16) In the formula, M The main mass matrix, K The principal stiffness matrix, C Here is the modal damping matrix. F ( t () represents the excitation in modal coordinates; Modal coordinates; The first derivative of the modal coordinates; The second derivative of the modal coordinates; S3.3: Based on the modal coordinate description obtained in S3.2, the dynamic equations are used to ignore off-diagonal elements using the forced decoupling method. Equation (16) is decoupled into equations with respect to the modal coordinates of S3.

2. y 1. y Differential equations for y2 and y3: (17) (18) (19) in, The moment of inertia of region 1, The moment of inertia of region 2. The moment of inertia of zone 3; The damping coefficient for zone 1. The damping coefficient for zone 2. The damping coefficient for zone 3; The time constant of the speed controller in Zone 1. The time constant of the speed controller in Zone 2. The time constant of the speed controller in Zone 3; This is the power gain coefficient related to the generator power factor and reserve factor in Zone 1; The power gain coefficient related to the generator power factor and reserve factor in Zone 2; The power gain coefficient related to the generator power factor and reserve factor in Zone 3; The initial value of the frequency difference instant before the disturbance in zone 1 occurs. The initial value of the frequency difference instant before the disturbance in zone 2 occurs. The initial value of the frequency difference instant before the disturbance in Zone 3 occurs; The Dirac impulse function; The equivalent power change of the inertial element of the turbine in Zone 1 is input. The equivalent power change of the inertial element of the turbine in zone 2 is input. Input the equivalent power change of the inertial element of the turbine in Zone 3; This represents the disturbance power in Zone 1. This represents the disturbance power in zone 2. This represents the disturbance power in zone 3. , , The elements are the equivalent matrix obtained by multiplying the inter-machine oscillation matrix by the rated angular frequency, where the subscript indicates the row and column of the element in the matrix; This refers to the output power of the low- and intermediate-pressure cylinders of the turbine in zone 1 at the moment of system state transition. This refers to the output power of the low- and intermediate-pressure cylinders of the turbine in zone 2 at the moment of system state transition. This refers to the output power of the low-pressure cylinder of the steam turbine in zone 3 at the moment of system state switching; The element in the 2nd row and 1st column of the modality matrix. The element in the second row and second column of the modality matrix. The element in the 2nd row and 3rd column of the modality matrix. The element in the 3rd row and 1st column of the modality matrix. The element in the 3rd row and 2nd column of the modality matrix. This is the element in the 3rd row and 3rd column of the modality matrix; The rated angular frequency; For the second-order differential of mode one, For the first-order differential of mode one, For the second-order differential of mode two, For the first-order differential of mode two, For the second-order differential of mode three, The first-order differential of mode three; S3.4: Equation (17) in S3.3 is a low-order average frequency response model containing a single rotor and three feedback loops. The actual meaning is that the system's angular frequency has a named value; this is further converted into a low-order system frequency response model; therefore, The closed-form solution is as follows: (20) In the formula, The disturbance power experienced by the system; This is the system droop coefficient; The system damping coefficient; For the system's mechanical power gain; The damping ratio; It is the natural frequency; The time constant of the speed controller; This refers to the actual natural frequency; This is the initial frequency difference; The first coefficient; The second coefficient; This represents the initial state of the first-order inertial feedback power. S3.6: Based on equation (18) in S3.5, due to the offset of the approximate value of the feedback power of the first-order inertial element. The resulting response is much smaller Therefore, ignore right Due to the influence of accuracy, solving equation (18) yields: (22) In the formula, a 2 The first coefficient; 2 is the second coefficient; 2 is the third coefficient; It is the fourth coefficient; It is the fifth coefficient; It is the sixth coefficient; It is the first-order differential of mode two; 2 represents the damping ratio corresponding to the second mode; It is the natural frequency corresponding to the second mode. It is the actual natural frequency corresponding to the second mode; S3.7: Based on equation (19) in S3.5, due to the offset of the approximate value of the feedback power of the first-order inertial element. The resulting response is much smaller Therefore, ignore right Due to the influence of accuracy, solving equation (19) yields: The closed-form solution; (24) In the formula, a 3 The first coefficient; b 3 The second coefficient; c 3 It is the third coefficient; It is the fourth coefficient; It is the fifth coefficient; It is the sixth coefficient; The first-order differential of mode three; The damping ratio corresponding to the third mode; The natural frequency corresponding to the third mode; This represents the actual natural frequency corresponding to the second mode; S3.8: Based on the closed-form solutions of modal coordinates in S3.4, S3.5, S3.6, and S3.7, and equation (15), the closed-form solution expression of the three-zone frequency full response model is obtained as follows: (26) in, The modal matrix, For the first-order differential of mode one, For the first-order differential of mode two, The first-order differential of mode three; The rated angular frequency; The regional frequency difference of region 1. The regional frequency difference of region 2. This represents the regional frequency difference of region 3.

6. The analytical method for the frequency full response model of an equivalent three-zone interconnected power system according to claim 5, characterized in that, In S3.4: (21) in, The rated angular frequency; It is the natural frequency; This refers to the actual natural frequency; The damping ratio; Let the system's rotational inertia be denoted by . For system load damping; It is the reciprocal of the system droop coefficient; The power ratio of the high-pressure cylinder in the system; The system time constant; The disturbance received by the system; These are the coefficients in the system's frequency response expression; These are the coefficients in the system's frequency response expression.

7. The analytical method for a frequency full response model of an equivalent three-zone interconnected power system according to claim 6, characterized in that, In S3.6: (23) in, The element in the 2nd row and 1st column of the modality matrix. The element in the second row and second column of the modality matrix. This is the element in the 2nd row and 3rd column of the modality matrix.

8. The analytical method for the frequency full response model of an equivalent three-zone interconnected power system according to claim 7, characterized in that, In S3.7: (25) In the formula, The element in the 3rd row and 1st column of the modality matrix. The element in the 3rd row and 2nd column of the modality matrix. This is the element in the 3rd row and 3rd column of the modality matrix.

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