Power system state estimation Hessian matrix method based on optimal control

Abstract

The invention discloses a power system state estimation Hessian matrix method based on optimal control, and belongs to the technical field of power system state estimation. According to the method, a node power equation for state estimation of the alternating-current power distribution network is established, and a Jacobian matrix and a Hessian matrix of node power unbalance to node voltage and phase angle are deduced; a state estimation optimization model with the minimum sum of squares of node power measurement errors as a target is constructed, the power system state estimation Hessian matrix method based on optimal control is provided, the state estimation convergence is improved, meanwhile, the calculation speed is guaranteed, and compared with a traditional Newton-Raphson state estimation method, the method has the advantages that the calculation speed is improved, and the calculation efficiency is improved. The state estimation method is few in iteration times and good in convergence.

Description

Technical Field

[0001] This invention belongs to the field of power system state estimation technology, specifically relating to a Hessian matrix method for power system state estimation based on optimal control. Background Technology

[0002] State estimation has always been the core of power system energy management systems, serving as the foundation for power flow calculations, economic dispatch, and voltage security assessments. Traditional power system state estimation employs the weighted least squares (WLS) method, estimating node voltage magnitudes and phase angles by minimizing the weighted sum of squares of measurement residuals. However, the following scenarios can easily lead to ill-conditioned state estimation: measurement noise and poor data cause the residual function to exhibit high nonlinearity, deteriorating the condition number of the Jacobian matrix; parameter errors and topology errors result in inaccurate line parameters or false alarms of switch states, compromising the model's fit with reality; and insufficient observability due to low-redundancy measurements leads to underdetermined state equations and a rank deficiency in the Jacobian matrix.

[0003] Existing methods (such as Newton's method and Levenberg-Marquardt method) require multiple iterations and are computationally inefficient, making them difficult to meet real-time requirements. This invention significantly improves convergence speed and robustness under ill-conditioned conditions by introducing an optimal control method and Hessian matrix. Summary of the Invention

[0004] To address the aforementioned technical problems in existing technologies, this invention proposes a Hessian matrix method for power system state estimation based on optimal control. This method is rationally designed, overcomes the shortcomings of existing technologies, and has good performance.

[0005] To achieve the above objectives, the present invention adopts the following technical solution:

[0006] A Hessian matrix method for power system state estimation based on optimal control includes the following steps:

[0007] Step 1: Define the voltage amplitude V i and phase angle θ i Let x be the state variable of the power system;

[0008] Step 2: Connect the active and reactive power at the node with the state variable x through a nonlinear function, establish the measurement equation, and solve for the measurement errors ΔP and ΔQ of the load's active and reactive power.

[0009] Step 3: Calculate the power flow Jacobian matrix and power flow Hessian matrix for node voltage and phase angle using the active and reactive power measurement errors ΔP and ΔQ;

[0010] Step 4: Construct an objective function with the goal of minimizing the weighted sum of squared residuals as the objective of state estimation, and calculate the first and second partial derivatives of the objective function with respect to node voltage and phase angle;

[0011] Step 5: Calculate the voltage and phase angle correction values ​​according to the optimal control optimization method to obtain new voltage and phase angle iterative values;

[0012] Step 6: Determine whether the power flow calculation has converged. If it has not converged, return to step 2 to recalculate. If it has converged, the voltage and phase angle iteration values ​​in step 5 are the final output of the node voltages and phase angles.

[0013] Preferably, the measurement error equation in step 2 is expressed as shown in formula (1):

[0014]

[0015] In the formula: U i and U j Let P be the voltage of node i and node j to be determined, n be the number of nodes in the power grid, and P be the voltage of node i and node j to be determined. i,load Q i,load These represent the active and reactive power of the load measured at node i, respectively; U i,mea Let ΔP be the voltage measurement value at node i. i ΔQ i , ΔU i These represent the active power, reactive power, and voltage measurement error at node i, respectively, θ. ij G is the difference between the voltage phase angle at node i and the voltage phase angle at node j. ij Let B be the conductance matrix of the busbar connecting node i and node j. ij Let be the susceptance matrix connecting the busbars of nodes i and j.

[0016] Preferably, in step 3, based on the first-order partial derivatives of the node power measurement error with respect to the node voltage and phase angle in formula (3), the power flow Jacobian matrix J is expressed as shown in formula (2):

[0017]

[0018] According to formula (4), the second-order partial derivatives of the node power measurement error with respect to the node voltage and phase angle are obtained, and the power flow Hessian matrix is ​​expressed as:

[0019]

[0020]

[0021] Where k represents node k, k = 1, 2, ..., n.

[0022] Preferably, the objective function in step 4 is as shown in formula (12):

[0023]

[0024] According to the Jacobian matrix form, for any node j in the AC distribution network, the first-order partial derivatives of the objective function with respect to the voltage and phase angle of node j are expressed as shown in formula (13):

[0025]

[0026] The second-order partial derivatives of the objective function with respect to the node j voltage and phase angle, i.e. the power flow Hessian matrix, are expressed in equation form as shown in formula (14):

[0027]

[0028] Define the Hessian matrix H e Specifically, it is represented by formula (15):

[0029]

[0030] Preferably, the optimal control optimization method in step 5 is as follows:

[0031]

[0032] Where R is the control weight matrix, used to balance the trade-off between objective function optimization and control energy, and to adjust the convergence speed and stability of the algorithm; x m Let f'(x) be the state variable matrix for the m-th iteration; m ) represents the objective function f(x) in x m The first derivative at the point; f(x) m ) represents the objective function f(x) in x m The second derivative at that point;

[0033] The voltage and phase angle corrections are obtained as shown in formula (17):

[0034]

[0035] Wherein, ΔU S (m) , Δθ S (m) Let He be the node voltage correction matrix and phase angle correction matrix for the m-th iteration, respectively. Let J be the Jacobian matrix and He be the Hessian matrix.

[0036] Preferably, step 6 specifically includes:

[0037] The convergence threshold is used to determine whether the voltage and phase angle state variables of the iteration converge. If they converge, the voltage and phase angle state variables are output as the final voltage and phase angle. If they do not converge, the active and reactive power measurement errors are recalculated, and the iteration calculation is continued according to the steps until the desired voltage and phase angle converge.

[0038] Preferably, step 5 specifically comprises:

[0039] remember Then the voltage iteration value U (m+1) This is represented as shown in formula (21):

[0040] U (m+1) =U (m) +ΔU (m) (twenty one);

[0041] Among them, U (m) The node voltage value in the m-th iteration;

[0042] remember Then the voltage iteration value θ (m+1) This is represented as shown in formula (22):

[0043] θ (m+1) =θ (m) +Δθ (m) (twenty two);

[0044] Where, θ (m) Let be the phase angle in the m-th iteration.

[0045] Preferably, in step 6, if the maximum value of the voltage correction is less than the threshold, the maximum value of the phase angle correction is less than the threshold, the maximum value of the active power imbalance is less than the threshold, and the maximum value of the reactive power imbalance is less than the threshold, then convergence occurs; that is, the voltage and phase angle values ​​of each node in this iteration are output.

[0046] The beneficial technical effects of this invention are as follows:

[0047] This invention establishes the nodal power equations for state estimation of AC distribution networks, derives the Jacobian and Hessian matrices of nodal power imbalance with respect to nodal voltage and phase angle, constructs a state estimation optimization model with the objective of minimizing the sum of squared errors in nodal power measurements, and proposes a Hessian matrix method for power system state estimation based on optimal control. This method improves the convergence of state estimation while ensuring computational speed. Compared with the traditional Newton-Raphson state estimation method, the proposed state estimation method has fewer iterations and better convergence. Attached Figure Description

[0048] Figure 1 This is a flowchart of the method of the present invention. Detailed Implementation

[0049] The present invention will be further described in detail below with reference to the accompanying drawings and specific embodiments:

[0050] A Hessian matrix method for power system state estimation based on optimal control, the process of which is as follows: Figure 1 As shown, it includes the following steps:

[0051] Step 1: Define the voltage amplitude V i and phase angle θ i Let x be the state variable of the power system;

[0052] Step 2: Connect the active and reactive power at the node with the state variable x through a nonlinear function, establish the measurement equation, and solve for the measurement errors ΔP and ΔQ of the load's active and reactive power.

[0053] The measurement error equation is expressed as shown in formula (1):

[0054]

[0055] In the formula: U i and U j The voltages at nodes i and j are respectively, n is the number of nodes in the power grid, and P is the voltage at node i and node j. i,load Q i,load These represent the active and reactive power of the load at node i, respectively; ΔP i ΔQ i θ represents the measurement error of active and reactive power of the load at node i, respectively. ij G is the difference between the voltage phase angle at node i and the voltage phase angle at node j. ij Let B be the conductance matrix of the busbar connecting node i and node j. ij Let be the susceptance matrix connecting the busbars of nodes i and j.

[0056] Step 3: Calculate the power flow Jacobian matrix and power flow Hessian matrix for node voltage and phase angle using ΔP and ΔQ;

[0057] Based on the first-order partial derivatives of the node power measurement error with respect to the node voltage and phase angle in formula (3), the power flow Jacobian matrix J is expressed as shown in formula (2):

[0058]

[0059] According to formula (4), the second-order partial derivatives of the node power measurement error with respect to the node voltage and phase angle are obtained, and the power flow Hessian matrix is ​​expressed as:

[0060]

[0061]

[0062] Where k represents node k, k = 1, 2, ..., n.

[0063] Step 4: Construct an objective function with the goal of minimizing the weighted sum of squared residuals as the objective of state estimation, and calculate the first and second partial derivatives of the objective function with respect to node voltage and phase angle;

[0064] The objective function is shown in formula (12):

[0065]

[0066] According to the Jacobian matrix form, for any node j in the AC distribution network, the first-order partial derivatives of the objective function with respect to the voltage and phase angle of node j are expressed as shown in formula (13):

[0067]

[0068] The second-order partial derivatives of the objective function with respect to the node j voltage and phase angle, i.e. the power flow Hessian matrix, are expressed in equation form as shown in formula (14):

[0069]

[0070] Define the iteration matrix H e Specifically, it is represented by formula (15):

[0071]

[0072] Step 5: Calculate the voltage and phase angle correction values ​​according to the optimal control optimization method to obtain new voltage and phase angle iterative values;

[0073] The specific optimization method for optimal control is as follows:

[0074] x m+1 =x m -g^ m (x m )

[0075]

[0076] Where R is the control weight matrix, used to balance the trade-off between objective function optimization and control energy, and to adjust the convergence speed and stability of the algorithm; x m Let f'(x) be the state variable matrix for the m-th iteration; m ) represents the objective function f(x) in x m The first derivative at the point; f(x) m ) represents the objective function f(x) in x m The second derivative at that point;

[0077] The voltage and phase angle corrections are obtained as shown in formula (17):

[0078]

[0079] Wherein, ΔU S (m) , Δθ S (m)Let He be the node voltage correction matrix and phase angle correction matrix for the m-th iteration, respectively. Let J be the Jacobian matrix and He be the Hessian matrix.

[0080] remember Then the voltage iteration value U (m+1) This is represented as shown in formula (21):

[0081] U (m+1) =U (m) +ΔU (m) (twenty one);

[0082] Among them, U (m) The node voltage value in the m-th iteration;

[0083] remember Then the voltage iteration value θ (m+1) This is represented as shown in formula (22):

[0084] θ (m+1) =θ (m) +Δθ (m) (twenty two);

[0085] Where, θ (m) Let be the phase angle in the m-th iteration.

[0086] Step 6: Determine whether the power flow calculation has converged. If it has not converged, return to step 2 to recalculate. If it has converged, the voltage and phase angle iteration values ​​in step 5 are the final output of the node voltages and phase angles.

[0087] According to formula (6), for any node j in the AC distribution network, the first-order partial derivative of the objective function with respect to the voltage and phase angle of node j is expressed as shown in formula (18):

[0088]

[0089] The second-order partial derivatives of the objective function with respect to the node j voltage and phase angle, i.e. the power flow Hessian matrix, are expressed in equation form as shown in formula (19):

[0090]

[0091] Define the iteration matrix H e The specific representation is shown in formula (20):

[0092]

[0093] If the maximum value of the voltage correction is less than the threshold, the maximum value of the phase angle correction is less than the threshold, the maximum value of the active power imbalance is less than the threshold, and the maximum value of the reactive power imbalance is less than the threshold, then convergence occurs; that is, the voltage and phase angle values ​​of each node in this iteration are output.

[0094] Of course, the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A Hessian matrix method for state estimation of power systems based on optimal control, characterized in that: Includes the following steps: Step 1: Define the voltage amplitude V i and phase angle θ i Let x be the state variable of the power system; Step 2: Connect the active and reactive power at the node with the state variable x through a nonlinear function, establish the measurement equation, and solve for the measurement errors ΔP and ΔQ of the load's active and reactive power. Step 3: Calculate the power flow Jacobian matrix and power flow Hessian matrix for node voltage and phase angle using the active and reactive power measurement errors ΔP and ΔQ; Step 4: Construct an objective function with the goal of minimizing the weighted sum of squared residuals as the objective of state estimation, and calculate the first and second partial derivatives of the objective function with respect to node voltage and phase angle; Step 5: Calculate the voltage and phase angle correction values ​​according to the optimal control optimization method to obtain new voltage and phase angle iterative values; Step 6: Determine whether the power flow calculation has converged. If it has not converged, return to step 2 to recalculate. If it has converged, the voltage and phase angle iteration values ​​in step 5 are the final output of the node voltages and phase angles.

2. The Hessian matrix method for power system state estimation based on optimal control according to claim 1, characterized in that: The measurement error equation for step 2 is expressed as shown in formula (1): In the formula: U i and U j Let P be the voltage of node i and node j to be determined, n be the number of nodes in the power grid, and P be the voltage of node i and node j to be determined. i,load Q i,load These represent the active and reactive power of the load measured at node i, respectively; U i,mea Let ΔP be the voltage measurement value at node i. i ΔQ i , ΔU i These represent the active power, reactive power, and voltage measurement error at node i, respectively, θ. ij G is the difference between the voltage phase angle at node i and the voltage phase angle at node j. ij Let B be the conductance matrix of the busbar connecting node i and node j. ij Let be the susceptance matrix connecting the busbars of nodes i and j.

3. The Hessian matrix method for power system state estimation based on optimal control according to claim 2, characterized in that: In step 3, based on the first-order partial derivatives of the node power measurement error with respect to the node voltage and phase angle in formula (3), the power flow Jacobian matrix J is expressed as shown in formula (2): According to formula (4), the second-order partial derivatives of the node power measurement error with respect to the node voltage and phase angle are obtained, and the power flow Hessian matrix is ​​expressed as: Where k represents node k, k = 1, 2, ..., n.

4. The Hessian matrix method for power system state estimation based on optimal control according to claim 3, characterized in that: The objective function in step 4 is shown in formula (12): According to the Jacobian matrix form, for any node j in the AC distribution network, the first-order partial derivatives of the objective function with respect to the voltage and phase angle of node j are expressed as shown in formula (13): The second-order partial derivatives of the objective function with respect to the node j voltage and phase angle, i.e. the power flow Hessian matrix, are expressed in equation form as shown in formula (14): Define the Hessian matrix H e Specifically, it is represented by formula (15):

5. The Hessian matrix method for power system state estimation based on optimal control according to claim 4, characterized in that: The optimal control optimization method in step 5 is as follows: Where R is the control weight matrix, used to balance the trade-off between objective function optimization and control energy, and to adjust the convergence speed and stability of the algorithm; x m Let f'(x) be the state variable matrix for the m-th iteration; m ) represents the objective function f(x) in x m The first derivative at the point; f(x) m ) represents the objective function f(x) in x m The second derivative at that point; The voltage and phase angle corrections are obtained as shown in formula (17): Wherein, ΔU S (m) , Δθ S (m) Let He be the node voltage correction matrix and phase angle correction matrix for the m-th iteration, respectively. Let J be the Jacobian matrix and He be the Hessian matrix.

6. The Hessian matrix method for power system state estimation based on optimal control according to claim 5, characterized in that: Step 6 specifically involves: The convergence threshold is used to determine whether the voltage and phase angle state variables of the iteration converge. If they converge, the voltage and phase angle state variables are output as the final voltage and phase angle. If they do not converge, the active and reactive power measurement errors are recalculated, and the iteration calculation is continued according to the steps until the desired voltage and phase angle converge.

7. The Hessian matrix method for power system state estimation based on optimal control according to claim 6, characterized in that: Step 5 specifically involves: remember Then the voltage iteration value U (m+1) This is represented as shown in formula (21): U (m+1) =U (m) +ΔU (m) (21); Among them, U (m) The node voltage value in the m-th iteration; remember Then the voltage iteration value θ (m+1) This is represented as shown in formula (22): i (m+1) =θ (m) +Δθ (m) (22); Where, θ (m) Let be the phase angle in the m-th iteration.

8. The Hessian matrix method for power system state estimation based on optimal control according to claim 7, characterized in that: In step 6, if the maximum value of the voltage correction is less than the threshold, the maximum value of the phase angle correction is less than the threshold, the maximum value of the active power imbalance is less than the threshold, and the maximum value of the reactive power imbalance is less than the threshold, then convergence occurs; that is, the voltage and phase angle values ​​of each node in this iteration are output.

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