A network-constructing inverter state-parameter joint estimation and node inertia mapping method based on PINN gray box extended Kalman filter

Abstract

The application discloses a network-constructed inverter state-parameter joint estimation and node inertia mapping method based on PINN gray-box extended Kalman filtering, which fuses a network-constructed inverter steady-state measurement physical model and a PINN residual compensation model, constructs a gray-box measurement equation capable of representing transient response, and can simultaneously identify dynamic state, virtual inertia and damping parameters of the inverter. The method provided by the application can improve the state-parameter joint estimation accuracy without complex transient measurement modeling, and can map the source side inertia to the distribution network node, thereby providing technical support for node frequency support capability evaluation.

Description

Technical Field

[0001] This invention proposes a method for joint estimation of state-parameters and node inertia mapping of grid-type inverters based on PINN gray-box extended Kalman filtering, which belongs to the field of distribution networks. Background Technology

[0002] With the high proportion of new energy sources and energy storage devices being connected to the distribution network, the power system is gradually shifting from being dominated by synchronous machines to being dominated by power electronic devices, and its dynamic characteristics and stability mechanisms are also changing accordingly. Grid-connected inverters can actively support voltage phasors and frequency, playing a frequency support role in high-penetration distribution networks. Their phase angle, angular frequency, virtual inertia, and damping coefficient directly affect the dynamic response after disturbances, but these variables are usually difficult to obtain directly from existing measurement devices, so online identification using dynamic state estimation methods is necessary.

[0003] Existing dynamic state estimation is typically based on explicit state-space models and measurement models. However, for grid-connected inverters, the measurement equations for active and reactive power measurements often employ steady-state physical relationships, which are insufficient to reflect transient changes during disturbances. If a complete transient model is used, it requires knowledge of the inverter's internal structure, control parameters, and dynamic components, which is difficult to implement in engineering applications. If simplified steady-state physical relationships are used, deviations will occur between the measurement equations and the actual output, causing a decrease in the estimation accuracy of the extended Kalman filter during disturbances or operating condition switching.

[0004] Existing research has explored dynamic estimation of grid-connected inverters through methods such as order reduction modeling, robust filtering, and data-driven modeling. However, relying solely on physical models is insufficient to cover unknown dynamics, while relying solely on data models suffers from strong sample dependence, insufficient extrapolation capabilities, and weak physical interpretability. Therefore, it is necessary to introduce a residual compensation model for surface-to-surface measurement equation errors, while preserving the physical mechanisms of grid-connected inverters, to form a gray-box estimation framework with the physical model as the main body and the data model as a correction.

[0005] Furthermore, the virtual inertia of a grid-connected inverter is primarily reflected in the parameters of the source-side equipment, while the node frequency response is also affected by network topology, line reactance, and power flow distribution. Obtaining only the inertia of a single device or the total system inertia is insufficient to reflect the differences in frequency support capabilities among distribution network nodes. Therefore, it is necessary to further map the source-side state-parameter estimation results into equivalent node inertia distributions to provide a basis for online monitoring and frequency security assessment of the distribution network. Summary of the Invention

[0006] Purpose of the invention: To address the above problems, this invention provides a method for joint estimation of state-parameters and mapping of node inertia of grid-connected inverters based on PINN gray-box extended Kalman filtering. This method can estimate the state and parameters of grid-connected inverters even when there are model biases in the measurement equations, and further maps the source-side inertia to the equivalent inertia distribution of distribution network nodes, providing a basis for operation monitoring, frequency support capability assessment and stability control of high-proportion power electronic distribution networks.

[0007] Technical Solution: To achieve the aforementioned objectives, this invention proposes a method for joint state-parameter estimation and node inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering. This method includes the following steps:

[0008] Step 1: Based on the virtual synchronous machine control mechanism of the grid-type inverter, establish the state-parameter joint state space equation, and construct the extended state vector by combining the phase angle, angular frequency, voltage integral intermediate state, virtual inertia and damping coefficient, and incorporate the virtual inertia and damping coefficient as parameters to be identified into the state estimation process.

[0009] Step 2: Obtain real-time measurements at the port side of the grid-connected inverter, including active power, reactive power, and phase angle measurements, and construct an extended Kalman filter measurement vector;

[0010] Step 3: Establish basic measurement equations based on the steady-state physical relationship of the grid-type inverter, and use the basic measurement equations as the main measurement model; In order to address the problem that the basic measurement equations cannot fully reflect the actual measurement output, the PINN residual model is introduced to compensate for the error of the measurement equations, thus forming the PINN gray box measurement equations.

[0011] Step 4: Train the PINN residual model based on historical operating measurement samples, so that it learns the residual relationship between the output of the basic measurement equation and the actual measurement, and through residual amplitude constraints and physical consistency constraints, make the gray box measurement equation take the steady-state basic measurement equation of the grid inverter as the main body.

[0012] Step 5: Perform extended Kalman filter prediction based on the state space equation described in Step 1 to obtain the prior estimate of the extended state vector and the prior error covariance matrix;

[0013] Step 6: Calculate the measurement prediction value and the Jacobian matrix of the measurement equation according to the PINN gray box measurement equation described in Step 3, and calculate the Kalman gain in combination with the real-time measurement vector described in Step 2. Update the prior estimate value to obtain the posterior estimate value of the extended state vector.

[0014] Step 7: Perform steps 2 to 6 on multiple grid-type inverters respectively to obtain the source-side inertia estimate of each grid-type inverter; construct a frequency allocation matrix based on the distribution network topology and line reactance, construct a proportional power flow tracking matrix based on the source-side output, and calculate the equivalent inertia of each node in the distribution network accordingly.

[0015] Furthermore, in step 1, the establishment of the state-parameter joint state-space equation based on the virtual synchronous machine control mechanism of the grid-connected inverter is as follows:

[0016] The phase angle, per-unit angular frequency, and voltage integral intermediate state of the grid-connected inverter are used as dynamic state variables, and the virtual inertia and damping coefficient are used as parameters to be identified, which together form the extended state vector. :

[0017] ;

[0018] In the formula, For the phase angle of the grid-connected inverter, For grid-connected inverters, the per-unit frequency is specified. This represents the intermediate state of voltage integration. The damping coefficient is... For virtual inertia, Let the time be numbered, the state-parameter joint state-space equation is:

[0019] ;

[0020] ;

[0021] ;

[0022] ;

[0023] ;

[0024] in, This is the first derivative of the phase angle with respect to time for a grid-connected inverter. The first derivative of the per-unit angular frequency with respect to time for a grid-connected inverter. This is the first derivative of the voltage integral with respect to time at the intermediate state. The first derivative of the damping coefficient with respect to time, The first derivative of the virtual inertia with respect to time. The rated angular frequency, This is the baseline value for the capacity of grid-connected inverters. This is the active power setpoint. For grid-connected inverters to output active power, The coefficient of the voltage integral element. This is the voltage reference value. This represents the port voltage amplitude.

[0025] Furthermore, in step 2, the measurements include active power, reactive power, and phase angle measurements of the grid-connected inverter. Real-time measurements are acquired by a synchronous sampling measurement device to reflect the current port operating status of the grid-connected inverter. An extended Kalman filter measurement vector is constructed based on the port active power, reactive power, and phase angle measurements. :

[0026] ;

[0027] In the formula, This is to output reactive power for grid-type inverters.

[0028] Furthermore, in step 3, the process of establishing the PINN gray box measurement equation is as follows:

[0029] Establish fundamental physical measurement equations based on the steady-state physical relationships of grid-connected inverters :

[0030] ;

[0031] in,

[0032] ;

[0033] In the formula, For input parameters of grid-connected inverters, The measurement equation is derived from steady-state physical relations. This is the predicted active power at the port, obtained from steady-state physical relationships. This is the predicted reactive power at the port, obtained from steady-state physical relationships. To measure noise, The equivalent internal potential amplitude of the grid-connected inverter. This is the equivalent grid connection point voltage value. This is the equivalent reactance between the inverter port and the equivalent network node. The equivalent network node voltage phase angle;

[0034] In conjunction with the voltage control loop, the equivalent internal potential amplitude is expressed as:

[0035] ;

[0036] in, and These are the proportional coefficient and integral coefficient for voltage control, respectively.

[0037] To address the discrepancy between the basic measurement equation and the actual measurement output, the PINN residual model is introduced:

[0038] ;

[0039] In the formula, This is the residual compensation term in the active power measurement equation. This is the residual compensation term in the reactive power measurement equation. For the PINN residual model;

[0040] Therefore, the PINN gray box measurement equation is obtained:

[0041] ;

[0042] The extended Kalman filter is updated using:

[0043] ;

[0044] Among them, the basic measurement equation Based on the steady-state physical relationships established by the grid-connected inverter, this model describes the mapping relationship between port measurements and state variables. Since the steady-state measurement equations struggle to characterize the transient changes in measurement outputs during disturbances, and the complete transient measurement model is complex to derive and has limited applicability, a PINN residual model is introduced to compensate for the system deviation between the steady-state measurement equations and the actual measurement outputs. Furthermore, the PINN residual model can also equivalently characterize measurement mapping deviations caused by factors such as filtering, coordinate transformation, power calculation, limiting, sampling delay, and unknown internal controller states, thereby improving the adaptability of the gray-box measurement equations to different inverter structures and actual operating conditions.

[0045] Furthermore, the specific method in step 4 is as follows:

[0046] Based on historical operational samples, the actual measurement vectors are compared with the output of the basic measurement equations to obtain measurement residual samples. :

[0047] ;

[0048] The PINN residual model takes phase angle, angular frequency, voltage integral intermediate state, time parameters, and inverter numbering characteristics as inputs, and measures residual samples. The training objective is to use this as supervised information for training.

[0049] ;

[0050] In the formula, Let be the training loss function for the PINN residual model. The number of historical running samples, This is the set of parameters to be trained in the PINN residual model. Represents the square of the L2 norm;

[0051] Furthermore, step 5 involves performing extended Kalman filter prediction based on the state-space equations, specifically including:

[0052] Prediction is made based on the state-space equations of the virtual synchronous machine in a grid-type inverter:

[0053] ;

[0054] In the formula, for The predicted value of the extended state vector at each time step. for The posterior estimate of the state vector at time step [time]. It is a nonlinear state transition function;

[0055] The state prediction model is as follows:

[0056] ;

[0057] ;

[0058] ;

[0059] ;

[0060] ;

[0061] In the formula, The sampling interval; for Predicted phase angle of grid-connected inverters at any given time. for Predicted per-unit frequency of the grid-connected inverter. for The predicted value of the voltage integral at the intermediate state at time step. for Predicted value of the damping coefficient at time step. for Predicted value of virtual inertia at any given time;

[0062] By linearizing the state prediction function to the first order, we obtain the state transition Jacobian matrix:

[0063] ;

[0064] In the formula, for The first-order linearized Jacobian matrix of the time-state prediction equation This represents the partial derivative of the state transition function with respect to the extended state vector;

[0065] Calculate the prior error covariance matrix:

[0066] ;

[0067] In the formula, Let be the posterior error covariance matrix of the previous time step. Let the prior error covariance matrix be at the current time. Let be the process noise covariance matrix.

[0068] Furthermore, in step 6, an extended Kalman filter measurement update is performed based on the PINN gray-box measurement equation, specifically including:

[0069] The result obtained in step 5 Predicted value of the extended state vector at time step Substituting into the PINN gray box measurement equation, calculate the predicted measurement value at the current time:

[0070] ;

[0071] In the formula, for Time measurement calculation value, for The input vector at time step;

[0072] The PINN gray-box measurement equation is linearized to the first order, yielding the Jacobian matrix of the measurement equation:

[0073] ;

[0074] In the formula, for Time measurement equation Jacobian matrix This represents the partial derivative of the PINN gray box measurement equation with respect to the extended state vector;

[0075] Calculate the Kalman gain based on the prior error covariance matrix, the measurement equation Jacobian matrix, and the measurement noise covariance matrix:

[0076] ;

[0077] In the formula, for Moment Kalman gain, for The prior error covariance matrix at time step, To observe the noise covariance matrix;

[0078] The prior estimates are updated using Kalman gain to obtain the extended posterior estimates of the state vector:

[0079] ;

[0080] In the formula, for Time measurement value, for The posterior estimate of the extended state vector at time step [time], which contains [the following information]. Posterior estimate of phase angle of grid-connected inverter , Posterior estimate of the per-unit angle frequency of a grid-connected inverter , Posterior estimate of the intermediate state of voltage integral at time step , Posterior estimate of the damping coefficient at time step , Posterior estimate of virtual inertia at time step ,Right now:

[0081]

[0082] Update the posterior error covariance matrix:

[0083] ;

[0084] In the formula, for The posterior error covariance matrix at time step [time]. It is an identity matrix.

[0085] Furthermore, in step 7, steps 2 to 6 are performed on each of the multiple grid-connected inverters to obtain the source-side inertia estimate for each inverter. The equivalent node inertia is then calculated based on the frequency allocation matrix and the upstream power distribution matrix, specifically including:

[0086] Each grid-type inverter is considered as a source-side device with inertia support capability. The source-side inertia constant is calculated based on the virtual inertia estimate obtained in step 6:

[0087] ;

[0088] In the formula, For the first Inertia constant of a source-side grid-type inverter For the first Virtual inertia estimate of a source-side grid-connected inverter For rated capacity, At the rated angular frequency, the estimated inertia values ​​from each source side are arranged into a diagonal matrix:

[0089] ;

[0090] In the formula, The number of source-side devices with inertia support capability. Numbering for source-side grid-type inverters; For the first Taiwan-based grid-connected inverters in The estimated source-side inertia at time t. This is the source-side inertia diagonal matrix;

[0091] Establish the network admittance matrix based on the distribution network topology and line reactance. And construct the source-side node-to-ground equivalent susceptance matrix. and the susceptance correlation matrix between source-side nodes and network nodes Based on this, a frequency allocation matrix is ​​constructed:

[0092] ;

[0093] In the formula, Assign a matrix to the frequency, with dimension . , The number of network nodes;

[0094] Based on the active power output of the source side and the power flow direction of the distribution network, an upstream distribution matrix is ​​constructed using the upstream power tracking method, and the upstream distribution matrix that corresponds to the power supply relationship between the source-side equipment and each network node is selected. The matrix satisfies:

[0095] ;

[0096] In the formula, This represents the active power output vector of the source-side equipment. Inject power vectors into network nodes, matrices The intermediate element is determined by the branch power flow direction and the proportion of inflow power at the node:

[0097] ;

[0098] In the formula, i and j are the network node numbers, respectively. For the node Flow to Node The active power of the branch circuit, For nodes Total inflow power;

[0099] Construct the weight matrix:

[0100] ;

[0101] In the formula, This is the weighting matrix for the impact of node power perturbation on node frequency variation;

[0102] According to the weight matrix Calculate the first Equivalent inertia of each node:

[0103] ;

[0104] In the formula, For the first Estimated equivalent inertia of each node. Show node Power imbalance on nodes The influence weight of frequency changes is calculated by repeating the above calculation for all nodes to obtain the equivalent inertia distribution results of the distribution network nodes.

[0105] By repeating the above calculations for all nodes, the equivalent inertia distribution of the distribution network nodes is obtained.

[0106] Beneficial effects: Compared with the prior art, the technical solution of the present invention has the following beneficial technical effects:

[0107] This invention proposes a joint state-parameter estimation and node inertia mapping method for grid-connected inverters based on PINN gray-box extended Kalman filtering to address the challenges of accurately representing the measurement equations and directly obtaining the internal state and control parameters of grid-connected inverters. To address the limitations of traditional physical models in accommodating transient processes and unknown control elements, and the insufficient physical constraints of purely data-driven models, this invention uses the grid-connected inverter's state-space equations as the master prediction model and steady-state physical measurement relationships as the foundation measurement model. It also utilizes the PINN residual model to compensate for the deviation between the measurement equations and actual measurement outputs, thereby improving the estimation accuracy of extended Kalman filtering under disturbance processes and operating condition switching scenarios. This gray-box estimation framework can identify key dynamic quantities of the grid-connected inverter online, such as phase angle, angular frequency, virtual inertia, and damping coefficient. Furthermore, by combining network topology, line reactance, and power flow distribution, it maps the source-side inertia estimation results to the equivalent inertia distribution of distribution network nodes. This invention provides a new solution for assessing the dynamic sensing and frequency support capabilities of distribution networks under conditions of high-proportion grid-connected inverters, taking into account physical interpretability, estimation accuracy, and engineering applicability. Attached Figure Description

[0108] Figure 1 This is a flowchart of the method for joint estimation of state-parameters and node inertia mapping of grid-type inverters based on PINN gray-box extended Kalman filtering as described in this invention.

[0109] Figure 2 This is a diagram of the 33-node distribution network topology;

[0110] Figure 3 These are the estimation results of the virtual inertia and damping coefficient of a 1-node grid inverter;

[0111] Figure 4 These are the phase angle and angular frequency estimation results for a 1-node grid-connected inverter;

[0112] Figure 5 This is the intermediate state estimation result of the voltage integral of a 1-node grid-connected inverter;

[0113] Figure 6 It is a diagram of the equivalent inertia distribution of nodes;

[0114] Figure 7 This is a comparison chart of the actual reference values ​​and estimated values ​​of the nodal inertia of nodes 3 and 8.

[0115] Figure 8 This is a comparison chart of the actual reference values ​​and estimated values ​​of the nodal inertia of nodes 18 and 22;

[0116] Figure 9 This is a comparison chart of the actual reference values ​​and estimated values ​​of the nodal inertia for nodes 27 and 32.

[0117] Figure 10 This is a comparison chart of the actual and estimated values ​​of the rate of change of frequency at the inverter nodes connected to the grid after the disturbance. Detailed Implementation

[0118] The technical solution of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0119] like Figure 1 As shown, this invention proposes a method for joint estimation of state-parameters and node inertia mapping of grid-type inverters based on PINN gray-box extended Kalman filtering. The method includes the following steps:

[0120] Step 1: Based on the virtual synchronous machine control mechanism of the grid-type inverter, establish the state-parameter joint state space equation, and construct the extended state vector by combining the phase angle, angular frequency, voltage integral intermediate state, virtual inertia and damping coefficient, and incorporate the virtual inertia and damping coefficient as parameters to be identified into the state estimation process.

[0121] Step 2: Obtain real-time measurements at the port side of the grid-connected inverter, including active power, reactive power, and phase angle measurements, and construct an extended Kalman filter measurement vector;

[0122] Step 3: Establish basic measurement equations based on the steady-state physical relationship of the grid-type inverter, and use the basic measurement equations as the main measurement model; In order to address the problem that the basic measurement equations cannot fully reflect the actual measurement output, the PINN residual model is introduced to compensate for the error of the measurement equations, thus forming the PINN gray box measurement equations.

[0123] Step 4: Train the PINN residual model based on historical operating measurement samples, so that it learns the residual relationship between the output of the basic measurement equation and the actual measurement, and through residual amplitude constraints and physical consistency constraints, make the gray box measurement equation take the steady-state basic measurement equation of the grid inverter as the main body.

[0124] Step 5: Perform extended Kalman filter prediction based on the state space equation described in Step 1 to obtain the prior estimate of the extended state vector and the prior error covariance matrix;

[0125] Step 6: Calculate the measurement prediction value and the Jacobian matrix of the measurement equation according to the PINN gray box measurement equation described in Step 3, and calculate the Kalman gain in combination with the real-time measurement vector described in Step 2. Update the prior estimate value to obtain the posterior estimate value of the extended state vector.

[0126] Step 7: Perform steps 2 to 6 on multiple grid-type inverters respectively to obtain the source-side inertia estimate of each grid-type inverter; construct a frequency allocation matrix based on the distribution network topology and line reactance, construct a proportional power flow tracking matrix based on the source-side output, and calculate the equivalent inertia of each node in the distribution network accordingly.

[0127] Furthermore, in step 1, the establishment of the state-parameter joint state-space equation based on the virtual synchronous machine control mechanism of the grid-connected inverter is as follows:

[0128] The phase angle, per-unit angular frequency, and voltage integral intermediate state of the grid-connected inverter are used as dynamic state variables, and the virtual inertia and damping coefficient are used as parameters to be identified, which together form the extended state vector. :

[0129] ;

[0130] In the formula, For the phase angle of the grid-connected inverter, The per-unit frequency for grid-connected inverters, This represents the intermediate state of voltage integration. The damping coefficient is... For virtual inertia, Let the time be numbered, the state-parameter joint state-space equation is:

[0131] ;

[0132] ;

[0133] ;

[0134] ;

[0135] ;

[0136] in, This is the first derivative of the phase angle with respect to time for a grid-connected inverter. The first derivative of the per-unit angular frequency with respect to time for a grid-connected inverter. This is the first derivative of the voltage integral with respect to time at the intermediate state. The first derivative of the damping coefficient with respect to time. The first derivative of the virtual inertia with respect to time. The rated angular frequency, This is the baseline value for the capacity of grid-connected inverters. This is the active power setpoint. For grid-connected inverters to output active power, The coefficient of the voltage integral element. This is the voltage reference value. This represents the port voltage amplitude.

[0137] Furthermore, in step 2, the measurements include active power, reactive power, and phase angle measurements of the grid-connected inverter. Real-time measurements are acquired by a synchronous sampling measurement device to reflect the current port operating status of the grid-connected inverter. An extended Kalman filter measurement vector is constructed based on the port active power, reactive power, and phase angle measurements. :

[0138] ;

[0139] In the formula, This is to output reactive power for grid-type inverters.

[0140] Furthermore, in step 3, the process of establishing the PINN gray box measurement equation is as follows:

[0141] Establish fundamental physical measurement equations based on the steady-state physical relationships of grid-connected inverters :

[0142] ;

[0143] in,

[0144] ;

[0145] In the formula, For input parameters of grid-connected inverters, The measurement equation is derived from steady-state physical relations. This is the predicted active power at the port, obtained from steady-state physical relationships. This is the predicted reactive power at the port, obtained from steady-state physical relationships. To measure noise, The equivalent internal potential amplitude of the grid-connected inverter. This is the equivalent grid connection point voltage value. This is the equivalent reactance between the inverter port and the equivalent network node. The equivalent network node voltage phase angle;

[0146] In conjunction with the voltage control loop, the equivalent internal potential amplitude is expressed as:

[0147] ;

[0148] in, and These are the proportional coefficient and integral coefficient for voltage control, respectively.

[0149] To address the discrepancy between the basic measurement equation and the actual measurement output, the PINN residual model is introduced:

[0150] ;

[0151] In the formula, This is the residual compensation term in the active power measurement equation. This is the residual compensation term in the reactive power measurement equation. For the PINN residual model;

[0152] This leads to the PINN gray box measurement equation:

[0153] ;

[0154] The extended Kalman filter is updated using:

[0155] ;

[0156] Among them, the basic measurement equation Based on the steady-state physical relationships established by the grid-connected inverter, this model describes the mapping relationship between port measurements and state variables. Since the steady-state measurement equations struggle to characterize the transient changes in measurement outputs during disturbances, and the complete transient measurement model is complex to derive and has limited applicability, a PINN residual model is introduced to compensate for the system deviation between the steady-state measurement equations and the actual measurement outputs. Furthermore, the PINN residual model can also equivalently characterize measurement mapping deviations caused by factors such as filtering, coordinate transformation, power calculation, limiting, sampling delay, and unknown internal controller states, thereby improving the adaptability of the gray-box measurement equations to different inverter structures and actual operating conditions.

[0157] Furthermore, the specific method in step 4 is as follows:

[0158] Based on historical operational samples, the actual measurement vectors are compared with the output of the basic measurement equations to obtain measurement residual samples. :

[0159] ;

[0160] The PINN residual model takes phase angle, angular frequency, voltage integral intermediate state, time parameters, and inverter numbering characteristics as inputs, and measures residual samples. The training objective is to use this as supervised information for training.

[0161] ;

[0162] In the formula, Let be the training loss function for the PINN residual model. The number of historical running samples, This is the set of parameters to be trained in the PINN residual model. Represents the square of the L2 norm;

[0163] Furthermore, step 5 involves performing extended Kalman filter prediction based on the state-space equations, specifically including:

[0164] Prediction is made based on the state-space equations of the virtual synchronous machine in a grid-type inverter:

[0165] ;

[0166] In the formula, for The predicted value of the extended state vector at each time step. for The posterior estimate of the state vector at time step [time]. It is a nonlinear state transition function;

[0167] The state prediction model is as follows:

[0168] ;

[0169] ;

[0170] ;

[0171] ;

[0172] ;

[0173] In the formula, The sampling interval; for Predicted phase angle of grid-connected inverters at any given time. for Predicted per-unit frequency of the grid-connected inverter. for The predicted value of the voltage integral at the intermediate state at time step. for Predicted value of the damping coefficient at time step. for Predicted value of virtual inertia at any given time;

[0174] By linearizing the state prediction function to the first order, we obtain the state transition Jacobian matrix:

[0175] ;

[0176] In the formula, for The first-order linearized Jacobian matrix of the time-state prediction equation This represents the partial derivative of the state transition function with respect to the extended state vector;

[0177] Calculate the prior error covariance matrix:

[0178] ;

[0179] In the formula, Let be the posterior error covariance matrix of the previous time step. Let the prior error covariance matrix be at the current time. Let be the process noise covariance matrix.

[0180] Furthermore, in step 6, an extended Kalman filter measurement update is performed based on the PINN gray-box measurement equation, specifically including:

[0181] The result obtained in step 5 Predicted value of the extended state vector at time step Substituting into the PINN gray box measurement equation, calculate the predicted measurement value at the current time:

[0182] ;

[0183] In the formula, for Time measurement calculation value, for The input vector at time step;

[0184] The PINN gray-box measurement equation is linearized to the first order, yielding the Jacobian matrix of the measurement equation:

[0185] ;

[0186] In the formula, for Time measurement equation Jacobian matrix This represents the partial derivative of the PINN gray box measurement equation with respect to the extended state vector;

[0187] Calculate the Kalman gain based on the prior error covariance matrix, the measurement equation Jacobian matrix, and the measurement noise covariance matrix:

[0188] ;

[0189] In the formula, for Moment Kalman gain, for The prior error covariance matrix at time step, To observe the noise covariance matrix;

[0190] The prior estimates are updated using Kalman gain to obtain the extended posterior estimates of the state vector:

[0191] ;

[0192] In the formula, for Time measurement value, for The posterior estimate of the extended state vector at time step [time], which contains [the following information]. Posterior estimate of phase angle of grid-connected inverter , Posterior estimate of the per-unit angle frequency of a grid-connected inverter , Posterior estimate of the intermediate state of voltage integral at time step , Posterior estimate of the damping coefficient at time step , Posterior estimate of virtual inertia at time step ,Right now:

[0193]

[0194] Update the posterior error covariance matrix:

[0195] ;

[0196] In the formula, for The posterior error covariance matrix at time step [time]. It is an identity matrix.

[0197] Furthermore, in step 7, steps 2 to 6 are performed on each of the multiple grid-connected inverters to obtain the source-side inertia estimate for each inverter. The equivalent node inertia is then calculated based on the frequency allocation matrix and the upstream power distribution matrix, specifically including:

[0198] Each grid-type inverter is considered as a source-side device with inertia support capability. The source-side inertia constant is calculated based on the virtual inertia estimate obtained in step 6.

[0199] ;

[0200] In the formula, For the first Inertia constant of a source-side grid-type inverter For the first Virtual inertia estimate of a source-side grid-type inverter For rated capacity, At the rated angular frequency, the estimated inertia values ​​from each source side are arranged into a diagonal matrix:

[0201] ;

[0202] In the formula, The number of source-side devices with inertia support capability. Numbering for source-side grid-type inverters; For the first Taiwan-based grid-connected inverters in The estimated source-side inertia at time t. This is the source-side inertia diagonal matrix;

[0203] Establish the network admittance matrix based on the distribution network topology and line reactance. And construct the source-side node-to-ground equivalent susceptance matrix. and the susceptance correlation matrix between source-side nodes and network nodes Based on this, a frequency allocation matrix is ​​constructed:

[0204] ;

[0205] In the formula, Assign a matrix to the frequency, with dimension . , The number of network nodes;

[0206] Based on the active power output of the source side and the power flow direction of the distribution network, an upstream distribution matrix is ​​constructed using the upstream power tracking method, and the upstream distribution matrix that corresponds to the power supply relationship between the source-side equipment and each network node is selected. The matrix satisfies:

[0207] ;

[0208] In the formula, This represents the active power output vector of the source-side equipment. Inject power vectors into network nodes, matrices The intermediate element is determined by the branch power flow direction and the proportion of inflow power at the node:

[0209] ;

[0210] In the formula, i and j are the network node numbers, respectively. For the node Flow to Node The active power of the branch circuit, For nodes Total inflow power;

[0211] Construct the weight matrix:

[0212] ;

[0213] In the formula, This is the weighting matrix for the impact of node power perturbation on node frequency variation;

[0214] According to the weight matrix Calculate the first Equivalent inertia of each node:

[0215] ;

[0216] In the formula, For the first Estimated equivalent inertia of each node. Show node Power imbalance on nodes The influence weight of frequency changes is calculated by repeating the above calculation for all nodes to obtain the equivalent inertia distribution results of the distribution network nodes.

[0217] By repeating the above calculations for all nodes, the equivalent inertia distribution of the distribution network nodes is obtained.

[0218] Case Analysis

[0219] 1) Example Explanation

[0220] This method uses the IEEE 33-bus distribution system as the test object, and the system topology is as follows: Figure 2 As shown, to simulate a high-proportion renewable energy integration scenario, grid-connected inverters are connected at nodes 1, 13, and 33 to provide voltage and frequency support and participate in state-parameter joint estimation and node inertia mapping; grid-connected inverters and energy storage devices are connected at nodes 18 and 22 to characterize renewable energy power injection and energy storage regulation. This example can reflect the coupling relationship between the distribution network operation state, source-side inertia, and node equivalent inertia under multi-type inverter integration.

[0221] 2) Estimation accuracy

[0222] The verification method for the estimation accuracy is as follows: The measurement configuration includes node-injected active power, reactive power, and voltage phase angle. Considering the fast time scale of the dynamic process of the grid-connected inverter, the measurement data is obtained using a synchronous sampling measurement device, where the uncertainty level of active power, reactive power, and phase angle measurements is set to 1%. Figure 3 , Figure 4 , Figure 5 Estimates for the virtual inertia and damping coefficient, phase angle and angular frequency, and intermediate state of the voltage integral are presented for the node 1 grid-type inverter. Figure 3 It is evident that the proposed method can quickly track changes in virtual inertia. Although the damping coefficient exhibits short-term transient adjustments, it converges overall to near the true value. Figure 4 As can be seen, the estimated values ​​of phase angle and angular frequency are in high agreement with the true values, and they maintain good tracking performance, especially when the frequency undergoes a step change; Figure 5 As can be seen, the voltage integral state estimation result almost completely coincides with the true value, with minimal steady-state error. The results indicate that this method can effectively achieve multi-state tracking and online identification of key parameters in grid-connected inverters under synchronous sampling measurement conditions.

[0223] Table 1. Estimation results of state parameters for grid-connected inverters

[0224] GFM1 0.0012659 0.00017828 50 49.9998 2735.672 2735.542 GFM2 0.0016320 0.00030607 22 22.0023 1215.854 1214.480 GFM3 0.0012198 0.00015062 39 39.00004 2127.745 2127.330

[0225] Table 1 presents the state tracking error and key parameter estimation results for the three grid-connected inverters. It can be seen that the grid-connected inverters at different connection locations all achieve good state estimation results, with phase angle and angular frequency errors generally at a low level, indicating that the proposed method has good adaptability to multi-inverter scenarios. From the parameter estimation results, the estimated virtual inertia values ​​of the three inverters converge to near the true values, and the damping coefficient estimation results also maintain good consistency. These results demonstrate that the proposed method can not only effectively track the dynamic state of grid-connected inverters but also identify key control parameters such as virtual inertia and damping coefficient using operational measurement information, providing a reliable source-side parameter basis for subsequent node inertia mapping.

[0226] 3) Node inertia mapping

[0227] Figure 6 This displays the distribution of equivalent inertia at the nodes under a reference load condition. The upper part shows the inertia at the source node, and the lower part shows the ranking of the equivalent inertia of the remaining nodes. Figure 6 It is evident that nodes 1, 13, and 33, due to their connection to the grid-connected inverters, possess strong virtual inertia support capabilities, resulting in significantly higher equivalent inertia compared to ordinary load nodes. For the remaining non-source nodes, the equivalent inertia is primarily generated by the grid-connected inverters through network coupling, and generally decreases gradually with increasing electrical distance from the inertia source. Furthermore, due to variations in distribution network topology and power flow distribution, the degree of inertia attenuation among the non-source nodes is not entirely consistent. This result indicates that the node inertia level depends not only on the inertia parameters of the source-side grid-connected inverters but also on the combined influence of line reactance, network topology, and power distribution relationships.

[0228] Figure 7 , Figure 8 , Figure 9 Time-domain comparisons of the equivalent inertia reference values ​​and estimated values ​​for nodes 3 and 8, 18 and 22, and 27 and 32 are presented. The estimated equivalent inertia of each node is calculated in real-time from the virtual inertia estimation results of the three network-type inverters to obtain the source-side inertia, and then mapped to each node based on network topology, line reactance, and power flow tracing relationships. Figure 7 It can be seen that the equivalent inertia estimation curves of nodes 3 and 8 undergo a brief adjustment in the initial stage, and then quickly converge to the reference value, with the estimated value being basically consistent with the reference value throughout the entire time domain; Figure 8 As can be seen, the estimated values ​​of nodes 18 and 22 can also quickly track changes in the reference value, with relatively small steady-state tracking errors; Figure 9As can be seen, the estimated curves for nodes 27 and 32 stably approach the reference values ​​after initial adjustment, further validating the effectiveness of the method. All three sets of results show that the source-side virtual inertia estimation error is not significantly amplified after network mapping, indicating that the proposed method can effectively transfer the state-parameter estimation results of the grid-connected inverter to the node inertia assessment stage. The differences in steady-state inertia levels among different nodes further reflect the characteristics of node inertia being affected by electrical distance and network coupling. Overall, these results validate the effectiveness of the proposed method in online estimation of node equivalent inertia.

[0229] 4) Rate of change of frequency

[0230] Figure 10 Given The actual and estimated values ​​of the rate of change of frequency (ROCOF) at typical nodes are compared after an active load step disturbance. This disturbance is used to simulate the power imbalance caused by a sudden load increase in the distribution network, with a disturbance power of 400kW. ROCOF reflects the severity of the node frequency change in the initial stage of the disturbance and is an important dynamic indicator characterizing the equivalent inertia support capacity of a node.

[0231] Depend on Figure 10 As can be seen, after the disturbance, the ROCOF of each node showed a significant transient shift, which gradually recovered to near zero under the support of the virtual inertia of the grid-connected inverter and the damping adjustment, indicating that the system transitioned from the disturbance transient process to a new stable operating state. The estimated ROCOF is consistent with the actual ROCOF in terms of the direction of change, peak position, and recovery trend, indicating that the node equivalent inertia obtained by the proposed method can effectively reflect the frequency dynamic response after the disturbance.

Claims

1. A method for joint state-parameter estimation and nodal inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering, characterized in that, The method includes the following steps: Step 1: Based on the virtual synchronous machine control mechanism of the grid-type inverter, establish the state-parameter joint state space equation, and construct the extended state vector by combining the phase angle, angular frequency, voltage integral intermediate state, virtual inertia and damping coefficient, and incorporate the virtual inertia and damping coefficient as parameters to be identified into the state estimation process. Step 2: Obtain real-time measurements at the port side of the grid-connected inverter, including active power, reactive power, and phase angle measurements, and construct an extended Kalman filter measurement vector; Step 3: Establish basic measurement equations based on the steady-state physical relationship of the grid-type inverter, and use the basic measurement equations as the main measurement model; In order to address the problem that the basic measurement equations cannot fully reflect the actual measurement output, the PINN residual model is introduced to compensate for the error of the measurement equations, thus forming the PINN gray box measurement equations. Step 4: Train the PINN residual model based on historical operating measurement samples, so that it learns the residual relationship between the output of the basic measurement equation and the actual measurement, and through residual amplitude constraints and physical consistency constraints, make the gray box measurement equation take the steady-state basic measurement equation of the grid inverter as the main body. Step 5: Perform extended Kalman filter prediction based on the state space equation described in Step 1 to obtain the prior estimate of the extended state vector and the prior error covariance matrix; Step 6: Calculate the measurement prediction value and the Jacobian matrix of the measurement equation according to the PINN gray box measurement equation described in Step 3, and calculate the Kalman gain in combination with the real-time measurement vector described in Step 2. Update the prior estimate value to obtain the posterior estimate value of the extended state vector. Step 7: Perform steps 2 to 6 on multiple grid-type inverters respectively to obtain the source-side inertia estimate of each grid-type inverter; construct a frequency allocation matrix based on the distribution network topology and line reactance, construct a proportional power flow tracking matrix based on the source-side output, and calculate the equivalent inertia of each node in the distribution network accordingly.

2. The method for joint state-parameter estimation and nodal inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering according to claim 1, characterized in that, In step 1, the establishment of the state-parameter joint state-space equations based on the virtual synchronous machine control mechanism of the grid-connected inverter is as follows: The phase angle, per-unit angular frequency, and voltage integral intermediate state of the grid-connected inverter are used as dynamic state variables, and the virtual inertia and damping coefficient are used as parameters to be identified, which together form the extended state vector. : ； In the formula, For the phase angle of the grid-connected inverter, For grid-connected inverters, the per-unit frequency is specified. This represents the intermediate state of voltage integration. The damping coefficient is... For virtual inertia, Let the time be numbered, the state-parameter joint state-space equation is: ； ； ； ； ； in, This is the first derivative of the phase angle with respect to time for a grid-connected inverter. The first derivative of the per-unit angular frequency with respect to time for a grid-connected inverter. This is the first derivative of the voltage integral with respect to time at the intermediate state. The first derivative of the damping coefficient with respect to time, The first derivative of the virtual inertia with respect to time. The rated angular frequency, This is the baseline value for the capacity of grid-connected inverters. This is the active power setpoint. For grid-connected inverters to output active power, The coefficient of the voltage integral element. This is the voltage reference value. This represents the port voltage amplitude.

3. The method for joint state-parameter estimation and nodal inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering according to claim 2, characterized in that, In step 2, the measurements include active power, reactive power, and phase angle measurements of the grid-connected inverter. Real-time measurements are acquired by a synchronous sampling measurement device and are used to reflect the current port operating status of the grid-connected inverter. An extended Kalman filter measurement vector is constructed based on the port active power, reactive power, and phase angle measurements. : ； In the formula, This is to output reactive power for grid-type inverters.

4. The method for joint state-parameter estimation and nodal inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering according to claim 3, characterized in that, In step 3, the process of establishing the PINN gray box measurement equation is as follows: The basic physical measurement equation is established based on the steady-state physical relationship of the grid-connected inverter. : ； in, ； In the formula, For input parameters of grid-connected inverters, The measurement equation is derived from steady-state physical relations. This is the predicted active power at the port, obtained from steady-state physical relationships. This is the predicted reactive power at the port, obtained from steady-state physical relationships. To measure noise, The equivalent internal potential amplitude of the grid-connected inverter. This is the equivalent grid connection point voltage value. This is the equivalent reactance between the inverter port and the equivalent network node. The equivalent network node voltage phase angle; In conjunction with the voltage control loop, the equivalent internal potential amplitude is expressed as: ； in, and These are the proportional coefficient and integral coefficient for voltage control, respectively. To address the discrepancy between the basic measurement equation and the actual measurement output, the PINN residual model is introduced: ； In the formula, This is the residual compensation term in the active power measurement equation. This is the residual compensation term in the reactive power measurement equation. The PINN residual model is used; from this, the PINN gray box measurement equation is obtained: ； The extended Kalman filter is updated using: 。 5. The method for joint state-parameter estimation and nodal inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering according to claim 4, characterized in that, The specific method in step 4 is as follows: Based on historical operational samples, the actual measurement vectors are compared with the output of the basic measurement equations to obtain measurement residual samples. : ； The PINN residual model takes phase angle, angular frequency, voltage integral intermediate state, time parameters, and inverter numbering characteristics as inputs, and measures residual samples. The training objective is to use this as supervised information for training. ； In the formula, Let be the training loss function for the PINN residual model. The number of historical running samples, This is the set of parameters to be trained in the PINN residual model. This represents the square of the L2 norm.

6. The method for joint state-parameter estimation and nodal inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering according to claim 5, characterized in that, Step 5 involves performing extended Kalman filter prediction based on the state-space equations, specifically including: Prediction is made based on the state-space equations of the virtual synchronous machine in a grid-type inverter: ； In the formula, for The predicted value of the extended state vector at each time step. for The posterior estimate of the state vector at time step [time]. It is a nonlinear state transition function; The state prediction model is as follows: ； ； ； ； ； In the formula, The sampling interval; for Predicted phase angle of grid-connected inverters at any given time. for Predicted per-unit frequency of the grid-connected inverter. for The predicted value of the voltage integral at the intermediate state at time step. for Predicted value of the damping coefficient at time step. for Predicted value of virtual inertia at any given time; By linearizing the state prediction function to the first order, we obtain the state transition Jacobian matrix: ； In the formula, for The first-order linearized Jacobian matrix of the time-state prediction equation This represents the partial derivative of the state transition function with respect to the extended state vector; Calculate the prior error covariance matrix: ； In the formula, Let be the posterior error covariance matrix of the previous time step. Let the prior error covariance matrix be at the current time. Let be the process noise covariance matrix.

7. The method for joint state-parameter estimation and nodal inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering according to claim 6, characterized in that, In step 6, the extended Kalman filter measurement update is performed according to the PINN gray-box measurement equation, specifically including: The result obtained in step 5 Predicted value of the extended state vector at time step Substituting into the PINN gray box measurement equation, calculate the predicted measurement value at the current time: ； In the formula, for Time measurement calculation value, for The input vector at time step; The PINN gray-box measurement equation is linearized to the first order, yielding the Jacobian matrix of the measurement equation: ； In the formula, for Time measurement equation Jacobian matrix This represents the partial derivative of the PINN gray box measurement equation with respect to the extended state vector; Calculate the Kalman gain based on the prior error covariance matrix, the measurement equation Jacobian matrix, and the measurement noise covariance matrix: ； In the formula, for Moment Kalman gain, for The prior error covariance matrix at time step, To observe the noise covariance matrix; The prior estimates are updated using Kalman gain to obtain the extended posterior estimates of the state vector: ； In the formula, for Time measurement value, for The posterior estimate of the extended state vector at time step [time], which contains [the following information]. Posterior estimate of phase angle of grid-connected inverter , Posterior estimate of the per-unit angle frequency of a grid-connected inverter , Posterior estimate of the intermediate state of voltage integral at time step , Posterior estimate of the damping coefficient at time step , Posterior estimate of virtual inertia at time step ,Right now: ； Update the posterior error covariance matrix: ； In the formula, for The posterior error covariance matrix at time step [time]. It is an identity matrix.

8. The method for joint state-parameter estimation and nodal inertia mapping of a grid-type inverter based on PINN gray-box extended Kalman filtering according to claim 7, characterized in that, In step 7, steps 2 to 6 are executed for each of the multiple grid-connected inverters to obtain the source-side inertia estimate for each inverter. The equivalent node inertia is then calculated based on the frequency allocation matrix and the upstream power distribution matrix, specifically including: Each grid-type inverter is considered as a source-side device with inertia support capability. The source-side inertia constant is calculated based on the virtual inertia estimate obtained in step 6: ； In the formula, For the first Inertia constant of a source-side grid-type inverter For the first Virtual inertia estimate of a source-side grid-connected inverter For rated capacity, At the rated angular frequency, the estimated inertia values ​​from each source side are arranged into a diagonal matrix: ； In the formula, The number of source-side devices with inertia support capability. Numbering for source-side grid-type inverters; For the first Taiwan-based grid-connected inverters in The estimated source-side inertia at time t. This is the source-side inertia diagonal matrix; Establish the network admittance matrix based on the distribution network topology and line reactance. And construct the source-side node-to-ground equivalent susceptance matrix. and the susceptance correlation matrix between source-side nodes and network nodes Based on this, a frequency allocation matrix is ​​constructed: ； In the formula, Assign a matrix to the frequency, with dimension . , The number of network nodes; Based on the active power output of the source side and the power flow direction of the distribution network, an upstream distribution matrix is ​​constructed using the upstream power tracking method, and the upstream distribution matrix that corresponds to the power supply relationship between the source-side equipment and each network node is selected. The matrix satisfies: ； In the formula, This represents the active power output vector of the source-side equipment. Inject power vectors into network nodes, matrices The intermediate element is determined by the branch power flow direction and the proportion of inflow power at the node: ； In the formula, i and j are the network node numbers, respectively. For the node Flow to Node The active power of the branch circuit, For nodes Total inflow power; Construct the weight matrix: ； In the formula, This is the weighting matrix for the impact of node power perturbation on node frequency variation; According to the weight matrix Calculate the first Equivalent inertia of each node: ； In the formula, For the first Estimated equivalent inertia of each node. Show node Power imbalance on nodes The influence weight of frequency changes is calculated by repeating the above calculation for all nodes to obtain the equivalent inertia distribution results of the distribution network nodes.

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