Power system output differential passivity-based observation method based on input-output trajectories

Abstract

The application discloses a power system output differential passivity measurement method based on input-output trajectory, which comprises the following steps: S1, input-output trajectory data is acquired; S2, data normalization processing is performed; S3, an extended state space model is constructed; S4, a linear matrix inequality constraint is constructed; and S5, an optimal output differential passivity index is solved. The power system output differential passivity measurement method based on input-output trajectory can check the output differential passivity of the power system, calculate the optimal output differential passivity index and realize the output differential passivity identification of a 'black box' device without relying on internal parameters of equipment, without observing internal states of the system and in the presence of measurement noise.

Description

Technical Field

[0001] This invention belongs to the field of power system stability analysis and control technology, specifically relating to a passive measurement method for the output differential of a power system based on input-output trajectories. Background Technology

[0002] With the rapid development of new energy power generation and DC transmission technologies, a large number of power electronic devices are being connected to the power grid, resulting in highly nonlinear and strongly coupled power systems. Against this backdrop, the output differential passivity theory provides a physically clear and model-independent system analysis framework by mapping complex internal dynamics to the input-output characteristics of port dynamic deviations. It ensures the convergence of the system state trajectory from the perspective of energy dissipation, thus becoming an important tool for ensuring the safe operation of power electronic power systems. Meanwhile, in actual operation scenarios, grid-connected equipment exhibits significant "black box" characteristics. Due to limitations imposed by commercial confidentiality, integrated packaging, and standard interfaces, grid dispatchers or third-party maintenance personnel find it difficult to obtain the internal topology, physical parameters, and real-time state variables of the equipment; they can only collect input-output data such as port voltage and current through external instrument transformers.

[0003] Current mainstream technical approaches fall into two categories: First, the Jacobian matrix analysis method based on physical models. Its core idea is to use the system's Jacobian matrix to determine the incremental energy characteristics of the system in state space through algebraic derivation. Second, the passivity analysis method based on input-output trajectories. This method uses port data to establish linear matrix inequalities and solves for whether the system satisfies conventional passivity through positive semidefinite optimization.

[0004] However, the two existing technologies mainly face the following technical challenges: First, accurate physical models are difficult to obtain. Existing methods rely on detailed internal parameters and topology of the device. Due to the high-order nonlinearity and time-varying parameter characteristics of heterogeneous devices, it is difficult to construct accurate mathematical models. Second, conventional passivity can only reflect the energy dissipation characteristics of the system about the origin and cannot characterize the incremental behavior of the system state between dynamic trajectories. Therefore, it is impossible to achieve effective measurement of more refined output differential passivity, and it is difficult to meet the needs of stability analysis for complex nonlinear systems. Summary of the Invention

[0005] The purpose of this invention is to provide a method for measuring the output differential passivity of a power system based on input-output trajectories. By measuring the input-output trajectories, the output differential passivity of the power system can be checked without relying on internal parameters of the equipment, without observing the internal state of the system, and even in the presence of measurement noise. The optimal output differential passivity index can be calculated, thereby enabling the identification of the output differential passivity of "black box" equipment.

[0006] The technical solution adopted in this invention is:

[0007] A passive measurement method for the output differential of a power system based on input-output trajectories is described below: S1. Acquire input-output trajectory data: Acquire and process input-output trajectory data under non-ideal measurement conditions, and construct input and output data matrices; S2. Data normalization processing: Mapping the input and output trajectory data to a dimensionless normalized space; S3. Constructing an extended state-space model: Constructing an implicit extended state-space model and generating an extended matrix; S4. Constructing linear matrix inequality constraints: Constructing linear matrix inequality constraints for the passive nature of the power system output differential; S5. Solve for the optimal output differential passivity index: Construct an optimization model with the goal of maximizing the output differential passivity index, and solve for the index using the positive semidefinite optimization method.

[0008] The invention is further characterized by: In S1, the non-ideal measurement environment specifically refers to the operating condition in which, due to the limitations of the accuracy of the instrument transformer or the influence of environmental interference, the input and output data collected at the ports of the power system are superimposed with random measurement noise during actual operation.

[0009] The specific method for S1 is as follows: First, consider the following linear time-invariant system: (1) in, ;where the matrix A, B Unknown, matrix C Known; For a continuously acquired input-output sequence under a non-ideal measurement environment, the following input data matrix and output data matrix are constructed: Assume the time window length for data collection is... The matrix is ​​constructed as follows: Input data matrix : ; Output data matrix : ; Defined based on input data matrix Constructed Hankel matrix for: (2) For a finite length input sequence Its structure Hankel matrix The row must satisfy the full rank condition, i.e. Then the input sequence is called satisfy Continuous incentives; Set the data length to be collected N The input sequence must be satisfied. for Continuous incentives; Considering the non-ideal measurement environment under real-world conditions, namely measurement noise and system nonlinear characteristics in the actual power grid environment, a multi-cycle averaging method is used to extract input and output trajectory data, which are then combined into an input data matrix. Output data matrix .

[0010] Continuous incentive refers to constructing a depth of The Hankel matrix must be full-rank, and the spectral richness of the input signal itself must satisfy the aforementioned higher-order excitation conditions to ensure that it contains complete dynamic modal information of the system; if the input sequence does not meet these conditions... A continuous stimulus indicates that the dynamic information contained in the data is incomplete and cannot uniquely determine the system behavior; if this condition is met, it indicates that the data meets the stimulus requirements. The multi-period averaging method refers to the method of superimposing and averaging multiple periodic data points by taking advantage of the periodicity of the system response under the premise of applying a periodic excitation signal, thereby canceling random noise and extracting the true steady-state trajectory of the system.

[0011] The specific method for S2 is as follows: For the output data matrix after multi-period averaging in S1, calculate the Euclidean norm of each row of state variables and construct a diagonal scaling matrix. , : (3) (4) in Representing the Euclidean norm of the time series of each output component; Define the normalized output trajectory and input trajectory The relationship between them is as follows: (5) in, These are the normalized dimensionless state matrix, output matrix, and input matrix; After matrix transformation, normalized input and output trajectory data were obtained. .

[0012] The specific method for S3 is as follows: S3.1 First, construct the normalized extended state vector: using the normalized input and output trajectory data obtained in S2. Define the moment Normalized extended state vector This vector is from the past It is composed of the normalized inputs and outputs stacked at each moment: (6) in The input data matrix defined in S1 U Data stack length; due to All have been normalized, therefore It is also a dimensionless normalized state vector; At this point, an extended state-space dynamic equation is established, and the extended state vector satisfies the following discrete linear time-invariant state-space equation: (7) in The system matrix in the normalized space; S3.2 Arrange the extended state vector, input vector, and output vector in the time series according to the time edge, and assemble them into the following block matrix to represent the full trajectory information of the system's input-state-output: Current extended state matrix: ; Next time step extended state matrix: ; Current control input matrix: ; Current system output matrix: ; The system output matrix at the next moment: ; The aforementioned data matrix is ​​used for subsequent data-driven control solutions.

[0013] The specific method for S4 is as follows: Using the corresponding input matrix in the extended state space of S3 Output matrix and the output matrix at the next time step Define the following intermediate variable matrix : (8) in, Let be the output differential passive performance index matrix in the normalized space to be solved; Based on the theory of output differential passivity, a sufficient condition for a system to satisfy output differential passivity is the existence of a positive definite symmetric matrix. This makes the following linear matrix inequality hold: (9) in, and These are the current extended state matrix and the next extended state matrix constructed in S3, respectively.

[0014] The specific method for S5 is as follows: Based on the linear matrix inequalities in S4, a method is constructed to maximize the output differential passivity index. The trace is the target, and it satisfies the positive definite symmetric matrix. The semidefinite programming convex optimization model is as follows:

[0015] (10)

[0016] Solving the above optimization model yields the optimal solution, which is the optimal output differential passive performance index in the normalized space. To ensure the convergence stability of the algorithm under noise interference, a numerical tolerance relaxation configuration is introduced: The solver's original feasibility tolerance, dual feasibility tolerance, and relative gap criterion are all relaxed to... Construct a relaxed constraint system; The optimal solution in the normalized space is obtained based on the above relaxed constraint system. Then, it needs to be mapped back to physical space to evaluate the actual output differential passivity of the system; according to the normalization transformation relation constructed in S2, the output differential passivity matrix in physical space is... It can be obtained from normalization Solution: (11) Finally, extract the matrix. The minimum value of the diagonal elements is taken as the final measurement index characterizing the physical dissipation properties of the system. : (12) at this time, This refers to the passive measurement index of the output differential of the power system under test under the current operating conditions.

[0017] The beneficial effects of this invention are: (1) The present invention is based on the measurement method of power system output differential passivity based on input-output trajectory. It constructs a convex optimization solution model containing two linear matrix inequalities. By checking whether the noisy data satisfies the inequality constraints, it realizes the judgment of the system output differential passivity and calculates the optimal output differential passivity index. The method has stronger anti-interference ability and nonlinear dynamic capture ability. It can realize the accurate evaluation of the system output differential passivity without knowing the model parameters, without observing the internal state of the system and with measurement noise. (2) The present invention is a passive measurement method for the output differential of power systems based on input-output trajectories. It abandons the dependence on the internal physical parameters of the equipment and directly uses the sampled input sequence and output sequence to construct a data matrix containing the dynamic information of the system. The calculation method of the present invention is simple and efficient, and transforms the complex passive analysis of the output differential of the system into a linear matrix inequality constraint problem. It can be applied to various types of dynamic equipment, has a good ability to suppress measurement noise, has a wider range of applications, and is more practical in engineering. Attached Figure Description

[0018] Figure 1 This is a schematic diagram illustrating the principle of extracting the input-output trajectory using the multi-cycle averaging method under non-ideal measurement conditions. Detailed Implementation

[0019] The present invention will now be described in detail with reference to the accompanying drawings and specific embodiments.

[0020] First, define the energy storage function of the discrete-time system as a quadratic form:

[0021] in, for The system state vector at time t. Let be the positive definite symmetric matrix to be solved, where ; According to the theory of output differential passivity, if the system is output differential passivity, then the change in its incremental storage function should be less than or equal to the incremental supply rate, using the following form of differential supply rate:

[0022] in, For system input, For system output, The output is a differential passivity index, the magnitude of which reflects the system's ability to suppress port disturbances; Therefore, the dissipation inequality that satisfies the output differential passivity condition can be expressed as:

[0023] That is, the increase in system energy should not exceed the incremental energy of the port input.

[0024] This invention relates to a passive measurement method for the output differential of a power system based on input-output trajectories. The specific method is as follows: S1. Acquire input-output trajectory data: Acquire and process input-output trajectory data under non-ideal measurement conditions, and construct input and output data matrices; Specifically, a non-ideal measurement environment refers to the operating condition in which, due to the limitations of instrument transformer accuracy or environmental interference, random measurement noise is superimposed on the input and output data collected at the ports of a power system during actual operation.

[0025] The specific method for S1 is as follows: First, consider the following linear time-invariant system: (1) in, ;where the matrix A, B Unknown, matrix C Known; For a continuously acquired input-output sequence under a non-ideal measurement environment, this invention constructs the following input data matrix and output data matrix: Assume the time window length for data collection is... The matrix is ​​constructed as follows: Input data matrix : ; Output data matrix : ; To capture the dynamic characteristics of the system, a method based on the input data matrix is ​​defined. Constructed Hankel matrix for: (2) For a finite length input sequence Its structure Hankel matrix The row must satisfy the full rank condition, i.e. Then the input sequence is called satisfy Continuous incentives; To ensure that the subsequent extended state trajectory spans a complete state space, this invention sets the data length for collection. N The input sequence must be satisfied. for Continuous incentives; Furthermore, Continuous incentive refers to constructing a depth of The Hankel matrix must be full-rank, and the spectral richness of the input signal itself must satisfy the aforementioned higher-order excitation conditions to ensure that it contains complete dynamic modal information of the system; if the input sequence does not meet these conditions... A continuous stimulus indicates that the dynamic information contained in the data is incomplete and cannot uniquely determine the system behavior; if this condition is met, it indicates that the data meets the stimulus requirements. Considering the non-ideal measurement environment under real-world conditions, namely measurement noise and system nonlinear characteristics in the actual power grid environment, this invention employs a multi-cycle averaging method to extract input and output trajectory data and combines them into an input data matrix. Output data matrix .

[0026] Furthermore, the multi-period averaging method refers to the method of superimposing and averaging multiple periodic data points by utilizing the periodicity of the system response under the premise of applying a periodic excitation signal, thereby canceling random noise and extracting the true steady-state trajectory of the system.

[0027] To visually illustrate the denoising effect of this method, we will use... Figure 1 The following example uses a standard signal model for illustration. Figure 1 As shown, a unit sine function disturbed by 5% random noise is set as the signal to be processed. The dashed line represents the trajectory of each period, and the solid line represents the trajectory after multi-period averaging. It can be seen that through multi-period averaging, random noise is effectively filtered out, restoring the true dynamics of the system.

[0028] Based on the above principles, in the actual measurement of the power system in this example, the above multi-cycle averaging method is also used for preprocessing of the collected input-output trajectory data, and finally the sequence is used for data normalization processing in step 2.

[0029] S2, Data Normalization Processing: Map the input-output trajectory data obtained in S1 to a dimensionless normalized space; This step constructs a diagonal scaling matrix. , This is used to map input and output trajectory data from physical space to a dimensionless normalized space. The specific method is as follows: For the output data matrix after multi-period averaging in S1, calculate the Euclidean norm of each row of state variables and construct a diagonal scaling matrix. , : (3) (4) in Representing the Euclidean norm of the time series of each output component; Define the normalized output trajectory and input trajectory The relationship between them is as follows: (5) in, These are the normalized dimensionless state matrix, output matrix, and input matrix; After matrix transformation, normalized input and output trajectory data were obtained. .

[0030] S3. Constructing an extended state-space model: Based on the normalized input and output trajectory data obtained in S2, construct an implicit extended state-space model and generate an extended matrix; The specific method is as follows: S3.1 First, construct the normalized extended state vector: using the normalized input and output trajectory data obtained in S2. Define the moment Normalized extended state vector This vector is from the past It is composed of the normalized inputs and outputs stacked at each moment: (6) in The input data matrix defined in S1 U Data stack length; due to All have been normalized, therefore It is also a dimensionless normalized state vector; At this point, an extended state-space dynamic equation is established, and the extended state vector satisfies the following discrete linear time-invariant state-space equation: (7) in The system matrix in the normalized space; It should be noted that this invention does not require explicit identification of the system parameter matrix, but rather implicitly incorporates these dynamic characteristics using the subsequently constructed data matrix, thereby achieving fully data-driven control.

[0031] S3.2 Arrange the extended state vector, input vector, and output vector in the time series according to the time edge, and assemble them into the following block matrix to represent the full trajectory information of the system's input-state-output: Current extended state matrix: ; Next time step extended state matrix: ; Current control input matrix: ; Current system output matrix: ; The system output matrix at the next moment: ; The aforementioned data matrix is ​​used for subsequent data-driven control solutions.

[0032] S4. Constructing linear matrix inequality constraints: Based on the extended state-space model obtained in S3, construct linear matrix inequality constraints for the passive nature of the power system output differential. The specific method is as follows: To characterize the energy supply relationship between the system's input and output rates of change, the corresponding input matrix in the extended state space of S3 is used. Output matrix and the output matrix at the next time step Define the following intermediate variable matrix : (8) in, Let be the output differential passive performance index matrix in the normalized space to be solved; Based on the theory of output differential passivity, a sufficient condition for a system to satisfy output differential passivity is the existence of a positive definite symmetric matrix. This makes the following linear matrix inequality hold: (9) in, and These are the current extended state matrix and the next extended state matrix constructed in S3, respectively.

[0033] It should be noted that the process of constructing the above linear matrix inequality constraints depends solely on the normalized data matrix output by S3. , , , , It does not involve any internal mechanism parameters of the equipment.

[0034] The solution obtained by satisfying the above linear matrix inequality constraints This is the optimal passive performance index in the normalized space, which can then be restored to the true performance index of the physical system through the inverse mapping of S2 normalization.

[0035] S5. Solving for the optimal output differential passivity index: Construct an optimization model with the objective of maximizing the output differential passivity index, and solve for the index using the positive semidefinite optimization method. The specific method is as follows: Based on the linear matrix inequality in S4, this invention transforms the identification of the system's output differential passivity into a semi-positive definite programming convex optimization problem, constructing a method to maximize the output differential passivity index. The trace is the target, and it satisfies the positive definite symmetric matrix. The semidefinite programming convex optimization model is as follows:

[0036] (10)

[0037] Solving the above optimization model yields the optimal solution, which is the optimal output differential passive performance index in the normalized space. To ensure the convergence stability of the algorithm under noise interference, a numerical tolerance relaxation configuration is introduced: The solver's original feasibility tolerance, dual feasibility tolerance, and relative gap criterion are all relaxed to... We construct a relaxed constraint system; this allows the objective function to float within a small range, thereby avoiding iteration interruptions caused by data noise and ensuring the smooth calculation of the optimal solution. The optimal solution in the normalized space is obtained based on the above relaxed constraint system. Then, it needs to be mapped back to physical space to evaluate the actual output differential passivity of the system; according to the normalization transformation relation constructed in S2, the output differential passivity matrix in physical space is... It can be obtained from normalization Solution: (11) Finally, extract the matrix. The minimum value of the diagonal elements is taken as the final measurement index characterizing the physical dissipation properties of the system. : (12) at this time, This refers to the passive measurement index of the output differential of the power system under test under the current operating conditions.

[0038] Example 1 This embodiment uses a droop-controlled power electronic power supply example to illustrate the effects of the present invention.

[0039] It should be noted that the following parameters are only used for data generation and theoretical verification of subsequent results in the simulation environment of this embodiment. The linear matrix inequality identification method proposed in this invention does not rely on or call these internal physical parameters during the execution process. Instead, it only performs calculations based on the collected input and output time series data.

[0040] The dynamic model commonly used for droop-controlled power electronic power supplies is described by the following second-order equations: (13) in To control the time constant; These represent the phase angle and magnitude of the voltage in the DQ coordinate system, respectively. These are the set reference output active power, reactive power, voltage phase angle, and voltage amplitude, respectively. and These are the adjustment coefficients for active and reactive power. The active and reactive power generated by the equipment.

[0041] The physical parameters are designed as follows: , , , , , , , .

[0042] In this embodiment, the node voltage phase angle and amplitude are selected as the system state, i.e. Select the active and reactive power output of the equipment as the control input, that is... In this case, the output vector is consistent with the state vector, that is... .

[0043] This embodiment presents a passive measurement method for the output differential of a power system based on input-output trajectories. The specific method is as follows: Step 1: Obtain input and output trajectory data: Obtain input-output trajectory data under non-ideal measurement conditions and process it to construct input and output data matrices.

[0044] Define the parameters related to the multi-period averaging method: , , ;in The period of the input signal, For the total measurement time, This is the sampling step size.

[0045] In the range of 0~440 s The system acquires raw input-output response data in a noisy environment over a period of time, and the acquired 440 s The original data is according to The data is divided into 22 periodic segments; an average is calculated along the periodic dimension of these 22 periods to obtain... Single-period average trajectory data input data matrix and output data matrix .

[0046] Set data window length Based on the processed single-period data, a depth of [missing information] is constructed. Hankel matrix Calculations show that the matrix satisfies the rank condition: (14) in For the input dimension, The system order; This result indicates that the extracted input signal satisfies The continuous excitation conditions contain complete system dynamic modal information and meet the data quality requirements of this invention.

[0047] Step 2: Data normalization: Map the input and output trajectory data to a dimensionless normalized space.

[0048] Based on the single-period truncated data matrix obtained in step 1, the row vector norms of the input and output are calculated respectively, and a diagonal scaling matrix is ​​constructed.

[0049] Output scaling matrix Calculate the state matrix For each row's Euclidean norm, take the reciprocal of its row norm to construct... diagonal matrix .

[0050] Input scaling matrix Construct using the row norm of the output matrix Input scaling matrix .

[0051] Using the scaling matrix described above, the original physical data is normalized and dimensionless mapped to obtain the normalized current-time input. and the output at the current moment .

[0052] Step 3: Construct the extended state space model: Construct an implicit extended state space model and generate the extended matrix.

[0053] The extended state vector, input vector, and output vector in the time series are arranged along the time edge and assembled into the following block matrix to represent the full trajectory information of the system's input-state-output: Current extended state matrix: ; Next time step extended state matrix: ; Current control input matrix: ; Current system output matrix: ; The system output matrix at the next moment: .

[0054] The matrix constructed above Substitute directly into the linear matrix inequality in step 4.

[0055] Step 4: Construct linear matrix inequality constraints: Construct linear matrix inequality constraints for the passive nature of the power system output differential.

[0056] Using the numerical matrix determined in step 3 Build intermediate items : (15) It should be noted that at this time All values ​​are known. Only matrices A linear function.

[0057] Based on this, the following specific linear matrix inequality constraints are established: Positive definiteness constraints: (16) Output differential passivity constraint: (17) At this point, the system of linear matrix inequalities based on the sampled data is complete.

[0058] Step 5: Solve for the optimal output differential passivity index: Construct an optimization model with the goal of maximizing the output differential passivity index, and solve for the index using the positive semidefinite optimization method.

[0059] This step, based on the linear matrix inequality constraints established in step 4, uses a positive semidefinite optimization algorithm to solve for the optimal output differential passivity index. To obtain the most compact output differential passivity index for the system within the current window, the output differential passivity analysis is transformed into the following convex optimization problem:

[0060] (18)

[0061] In this example, the MOSEK solver is used to solve the above problem. The solution accuracy tolerance is set to [value missing]. At this point, we obtain... .

[0062] The result calculated after normalization can be used to calculate the original result: (198) Finally, extract the matrix. The minimum value of the diagonal elements is taken as the final measurement index characterizing the physical dissipation properties of the system. : (20) at this time, This refers to the passive measurement index of the output differential of droop control under the current operating conditions.

[0063] To verify the accuracy of the data-driven method proposed in this invention, the above measurement results are compared with the theoretical true values ​​derived based on the physical model parameters. The system physical parameters are pre-set as inherent properties of the object under test and are used only to generate simulation data and provide a theoretical verification benchmark. The theoretical maximum under this operating condition... The value is obtained by taking the intersection of the lower bounds of the following two inequalities: (twenty one) Based on the physical parameters set above, the theoretical output differential passivity index of the system is... The calculation is as follows: (twenty two) The comparison shows that the measurement results obtained by the present invention under the condition of unknown model parameters are basically consistent with the theoretical true value, with a relative error of only 1.26%, which verifies the effectiveness of the method of the present invention.

[0064] Example 2 This embodiment presents a passive measurement method for the output differential of a power system based on input-output trajectories. The specific method is as follows: S1. Acquire input-output trajectory data: Acquire and process input-output trajectory data under non-ideal measurement conditions, and construct input and output data matrices; Specifically, a non-ideal measurement environment refers to the operating condition in which, due to the limitations of instrument transformer accuracy or environmental interference, random measurement noise is superimposed on the input and output data collected at the ports of a power system during actual operation.

[0065] S2. Data normalization processing: Mapping the input and output trajectory data to a dimensionless normalized space; S3. Constructing an extended state-space model: Constructing an implicit extended state-space model and generating an extended matrix; S4. Constructing linear matrix inequality constraints: Constructing linear matrix inequality constraints for the passive nature of the power system output differential; S5. Solve for the optimal output differential passivity index: Construct an optimization model with the goal of maximizing the output differential passivity index, and solve for the index using the positive semidefinite optimization method.

[0066] Example 3 The specific method for S1 is as follows: First, consider the following linear time-invariant system: (1) in, ;where the matrix A, B Unknown, matrix C Known; For a continuously acquired input-output sequence under a non-ideal measurement environment, the following input data matrix and output data matrix are constructed: Assume the time window length for data collection is... The matrix is ​​constructed as follows: Input data matrix : ; Output data matrix : ; Defined based on input data matrix Constructed Hankel matrix for: (2) For a finite length input sequence Its structure Hankel matrix The row must satisfy the full rank condition, i.e. Then the input sequence is called satisfy Continuous incentives; Set the data length to be collected N The input sequence must be satisfied. for Continuous incentives; Considering the non-ideal measurement environment under real-world conditions, namely measurement noise and system nonlinear characteristics in the actual power grid environment, a multi-cycle averaging method is used to extract input and output trajectory data, which are then combined into an input data matrix. Output data matrix .

[0067] in, Continuous incentive refers to constructing a depth of The Hankel matrix must be full-rank, and the spectral richness of the input signal itself must satisfy the aforementioned higher-order excitation conditions to ensure that it contains complete dynamic modal information of the system; if the input sequence does not meet these conditions... A continuous stimulus indicates that the dynamic information contained in the data is incomplete and cannot uniquely determine the system behavior; if this condition is met, it indicates that the data meets the stimulus requirements. The multi-period averaging method refers to the method of superimposing and averaging multiple periodic data points by taking advantage of the periodicity of the system response under the premise of applying a periodic excitation signal, thereby canceling random noise and extracting the true steady-state trajectory of the system.

[0068] Example 4 The specific method for S2 is as follows: For the output data matrix after multi-period averaging in S1, calculate the Euclidean norm of each row of state variables and construct a diagonal scaling matrix. , : (3) (4) in Representing the Euclidean norm of the time series of each output component; Define the normalized output trajectory and input trajectory The relationship between them is as follows: (5) in, These are the normalized dimensionless state matrix, output matrix, and input matrix; After matrix transformation, normalized input and output trajectory data were obtained. .

[0069] Example 5 The specific method for S3 is as follows: S3.1 First, construct the normalized extended state vector: using the normalized input and output trajectory data obtained in S2. Define the moment Normalized extended state vector This vector is from the past It is composed of the normalized inputs and outputs stacked at each moment: (6) in The input data matrix defined in S1 U Data stack length; due to All have been normalized, therefore It is also a dimensionless normalized state vector; At this point, an extended state-space dynamic equation is established, and the extended state vector satisfies the following discrete linear time-invariant state-space equation: (7) in The system matrix in the normalized space; S3.2 Arrange the extended state vector, input vector, and output vector in the time series according to the time edge, and assemble them into the following block matrix to represent the full trajectory information of the system's input-state-output: Current extended state matrix: ; Next time step extended state matrix: ; Current control input matrix: ; Current system output matrix: ; The system output matrix at the next moment: ; The aforementioned data matrix is ​​used for subsequent data-driven control solutions.

[0070] Example 6 The specific method for S4 is as follows: Using the corresponding input matrix in the extended state space of S3 Output matrix and the output matrix at the next time step Define the following intermediate variable matrix : (8) in, Let be the output differential passive performance index matrix in the normalized space to be solved; Based on the theory of output differential passivity, a sufficient condition for a system to satisfy output differential passivity is the existence of a positive definite symmetric matrix. This makes the following linear matrix inequality hold: (9) in, and These are the current extended state matrix and the next extended state matrix constructed in S3, respectively.

[0071] Example 7 The specific method for S5 is as follows: Based on the linear matrix inequalities in S4, a method is constructed to maximize the output differential passivity index. The trace is the target, and it satisfies the positive definite symmetric matrix. The semidefinite programming convex optimization model is as follows:

[0072] (10)

[0073] Solving the above optimization model yields the optimal solution, which is the optimal output differential passive performance index in the normalized space. To ensure the convergence stability of the algorithm under noise interference, a numerical tolerance relaxation configuration is introduced: The solver's original feasibility tolerance, dual feasibility tolerance, and relative gap criterion are all relaxed to... Construct a relaxed constraint system; The optimal solution in the normalized space is obtained based on the above relaxed constraint system. Then, it needs to be mapped back to physical space to evaluate the actual output differential passivity of the system; according to the normalization transformation relation constructed in S2, the output differential passivity matrix in physical space is... It can be obtained from normalization Solution: (11) Finally, extract the matrix. The minimum value of the diagonal elements is taken as the final measurement index characterizing the physical dissipation properties of the system. : (12) at this time, This refers to the passive measurement index of the output differential of the power system under test under the current operating conditions.

Claims

1. A passive measurement method for the output differential of a power system based on input-output trajectories, characterized in that, The specific method is as follows: S1. Acquire input-output trajectory data: Acquire and process input-output trajectory data under non-ideal measurement conditions, and construct input and output data matrices; S2. Data normalization processing: Mapping the input and output trajectory data to a dimensionless normalized space; S3. Constructing an extended state-space model: Constructing an implicit extended state-space model and generating an extended matrix; S4. Constructing linear matrix inequality constraints: Constructing linear matrix inequality constraints for the passive nature of the power system output differential; S5. Solve for the optimal output differential passivity index: Construct an optimization model with the goal of maximizing the output differential passivity index, and solve for the index using the positive semidefinite optimization method.

2. The passive measurement method for the output differential of a power system based on input-output trajectories according to claim 1, characterized in that, The non-ideal measurement environment in S1 specifically refers to the operating condition in which, due to the limitations of the accuracy of the instrument transformer or the influence of environmental interference, the input and output data collected at the port of the power system are superimposed with random measurement noise during actual operation.

3. The passive measurement method for the output differential of a power system based on input-output trajectories according to claim 1, characterized in that, The specific method of S1 is as follows: First, consider the following linear time-invariant system: （1） in, ;where the matrix A, B Unknown, matrix C Known; For a continuously acquired input-output sequence under a non-ideal measurement environment, the following input data matrix and output data matrix are constructed: Assume the time window length for data collection is... The matrix is ​​constructed as follows: Input data matrix : ; Output data matrix : ; Defined based on input data matrix Constructed Hankel matrix for: （2） For a finite length input sequence Its structure Hankel matrix The row must satisfy the full rank condition, i.e. Then the input sequence is called satisfy Continuous incentives; Set the data length to be collected N The input sequence must be satisfied. for Continuous incentives; Considering the non-ideal measurement environment under real-world conditions, namely measurement noise and system nonlinear characteristics in the actual power grid environment, a multi-cycle averaging method is used to extract input and output trajectory data, which are then combined into an input data matrix. Output data matrix .

4. The passive measurement method for the output differential of a power system based on input-output trajectories according to claim 3, characterized in that, The Continuous incentive refers to constructing a depth of The Hankel matrix must be full-rank, and the spectral richness of the input signal itself must satisfy the aforementioned higher-order excitation conditions to ensure that it contains complete dynamic modal information of the system; if the input sequence does not meet these conditions... The continuous stimulation indicates that the dynamic information contained in the data is incomplete and cannot uniquely determine the system behavior; If this condition is met, it indicates that the data meets the incentive requirements; The multi-period averaging method refers to the method of superimposing and averaging multiple periodic data points by utilizing the periodicity of the system response under the premise of applying a periodic excitation signal, thereby canceling random noise and extracting the true steady-state trajectory of the system.

5. The passive measurement method for the output differential of a power system based on input-output trajectories according to claim 4, characterized in that, The specific method of S2 is as follows: For the output data matrix after multi-period averaging in S1, calculate the Euclidean norm of each row of state variables and construct a diagonal scaling matrix. , : （3） （4） in Representing the Euclidean norm of the time series of each output component; Define the normalized output trajectory and input trajectory The relationship between them is as follows: （5） in, These are the normalized dimensionless state matrix, output matrix, and input matrix; After matrix transformation, normalized input and output trajectory data were obtained. .

6. The passive measurement method for the output differential of a power system based on input-output trajectories according to claim 5, characterized in that, The specific method of S3 is as follows: S3.1 First, construct the normalized extended state vector: using the normalized input and output trajectory data obtained in S2. Define the moment Normalized extended state vector This vector is from the past It is composed of the normalized inputs and outputs stacked at each moment: （6） in The input data matrix defined in S1 U Data stack length; due to All have been normalized, therefore It is also a dimensionless normalized state vector; At this point, an extended state-space dynamic equation is established, and the extended state vector satisfies the following discrete linear time-invariant state-space equation: （7） in The system matrix in the normalized space; S3.2 Arrange the extended state vector, input vector, and output vector in the time series according to the time edge, and assemble them into the following block matrix to represent the full trajectory information of the system's input-state-output: Current extended state matrix: ; Next time step extended state matrix: ; Current control input matrix: ; Current system output matrix: ; The system output matrix at the next moment: ; The aforementioned data matrix is ​​used for subsequent data-driven control solutions.

7. The passive measurement method for the output differential of a power system based on input-output trajectories according to claim 6, characterized in that, The specific method of S4 is as follows: Using the corresponding input matrix in the extended state space of S3 Output matrix and the output matrix at the next time step Define the following intermediate variable matrix : （8） in, Let be the output differential passive performance index matrix in the normalized space to be solved; Based on the theory of output differential passivity, a sufficient condition for a system to satisfy output differential passivity is the existence of a positive definite symmetric matrix. This makes the following linear matrix inequality hold: （9） in, and These are the current extended state matrix and the next extended state matrix constructed in S3, respectively.

8. The passive measurement method for the output differential of a power system based on input-output trajectories according to claim 7, characterized in that, The specific method of S5 is as follows: Based on the linear matrix inequalities in S4, a method is constructed to maximize the output differential passivity index. The trace is the target, and it satisfies the positive definite symmetric matrix. The semidefinite programming convex optimization model is as follows: （10） Solving the above optimization model yields the optimal solution, which is the optimal output differential passive performance index in the normalized space. To ensure the convergence stability of the algorithm under noise interference, a numerical tolerance relaxation configuration is introduced: The solver's original feasibility tolerance, dual feasibility tolerance, and relative gap criterion are all relaxed to... Construct a relaxed constraint system; The optimal solution in the normalized space is obtained based on the above relaxed constraint system. Then, it needs to be mapped back to physical space to evaluate the actual output differential passivity of the system; according to the normalization transformation relation constructed in S2, the output differential passivity matrix in physical space is... It can be obtained from normalization Solution: （11） Finally, extract the matrix. The minimum value of the diagonal elements is taken as the final measurement index characterizing the physical dissipation properties of the system. : （12） at this time, This refers to the passive measurement index of the output differential of the power system under test under the current operating conditions.

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