CN105512502B - One kind is based on the normalized weight function the least square estimation method of residual error - Google Patents

Abstract

The invention discloses a weight function least square state estimation method based on residual normalization, which belongs to the field of electric power system scheduling automation. The method of the present invention uses a computer to first input and initialize the SCADA data, network structure and parameter information of any time section collected by the data terminal through a program, and then calculate the node admittance matrix of the network to form a zero-injection equality constraint equation, and then Comprehensively consider the voltage amplitude measurement equation, injection power measurement equation, and branch power measurement equation, and use the node voltage amplitude and phase angle as state variables to calculate the corresponding residual, Jacobian matrix, and weight function, and finally update the state Variables, to judge the convergence, to realize the state estimation of the power grid. The invention can effectively suppress bad lever measurement, has strong error resistance and good convergence, and has high calculation efficiency, and has good engineering application prospect.

Description

A Weighted Function Least Squares State Estimation Method Based on Residual Normalization

technical field

The invention belongs to the field of power system scheduling automation, and in particular relates to a weight function least square state estimation method based on residual normalization.

Background technique

Power system state estimation is an important part of the energy management system and the basis for running other application software in the power system. The results directly affect the intelligent analysis and decision-making of power grid dispatching. Since Schweppe introduced the state estimation into the power system for the first time in 1970, scholars and engineers at home and abroad have conducted a large number of in-depth research and practice on state estimation, and various state estimation methods have emerged during this period.

At present, Weighted least squares (WLS) is one of the most widely used methods in state estimation. Its model is simple and easy to solve, but it is not anti-interference, so robust state estimation appears. The existing robust state estimation methods include the exponential objective function robust state estimation method (Maximum exponential square, MES), the weighted function least squares state estimation method based on the reciprocal of the standardized residual, and the exponentially weighted least squares robust state estimation method (Exponential function weighted least squares, EFWLS) and so on. However, the calculation of these methods consumes a lot of time for each iteration because of the calculation of the standardized residual, which affects their application in the actual system.

Based on this, there have been proposed methods that can improve computational efficiency to a certain extent, but the solution process is complex, and the amount of calculation for some large power systems is too large, so it is not easy to be widely used. Some people have also proposed a state estimation algorithm (Exponential least absolute value, E-LAV) of the exponential weight function, which improves the calculation efficiency, but the ability to resist bad leverage measurement is low. Therefore, it is necessary to provide an estimation method for power system state estimation that can not only meet the requirements of computational efficiency, but also have good tolerance performance, so as to meet the needs of practical engineering applications.

Contents of the invention

The purpose of the present invention is to address the shortcomings of existing state estimation research methods, and propose a weight function least squares state estimation method based on residual normalization that has high computational efficiency, good convergence, and can effectively suppress bad leverage measurements. (Residuals normalized weighted least squares, RNWLS).

The technical scheme for realizing the present invention is: a kind of weight function least square state estimation method based on residual normalization, utilize computer, through program, at first input the SCADA data of any time section collected by data terminal, network structure and The parameter information is initialized, and then the node admittance matrix of the network is calculated to form a zero-injection equality constraint equation, and then the voltage amplitude measurement equation, injection power measurement equation, and branch power measurement equation are comprehensively considered, and the node voltage amplitude The value and phase angle are state variables, and the corresponding residuals, Jacobian matrix, and weight function are calculated, and finally the state variables are updated to judge the convergence, so as to realize the state estimation of the power grid. The concrete steps of described method are as follows:

(1) Input basic data and initialization

1) Enter basic data

First input the data acquisition and monitoring control system of the power grid under any time section, namely SCADA data, network structure and parameter information; the SCADA data of the power grid includes node voltage amplitude, node injection power and branch power; the network structure and The parameter information includes resistance, reactance, susceptance, rated voltage and power reference of network elements; said network elements include lines and transformers.

2) Parameter initialization

Set the m-order identity matrix R ^-1 , the diagonal elements of the identity matrix R ^-1 are all 1, and the off-diagonal elements are all 0, m is the actual number of measured variables in the state estimation; when setting the state estimation The initial value of the node voltage amplitude is 1 p.u., and the phase angle is 0. The impedance and susceptance of the network elements are all reduced to the p.u. value; the node injection power and branch power are also reduced to the p.u. value; initialize the maximum number of iterations Tmax to 40~60; the convergence precision ε is 10 ^-3 to 10 ^-5 , and the detection threshold r _min =0.002 for small residual error in weight calculation, and set the number of iterations time=1.

(2) Calculate node admittance matrix

After step (1) is completed, use the formula (1) to calculate the node admittance matrix Y:

In the formula, the diagonal element of the node admittance matrix is expressed as Y _ii , and the off-diagonal element of the node admittance matrix is expressed as Y _ij , both of which are calculated by the above formula; y _ij is the distance between node i and node j Reciprocal of branch impedance z _ij ; symbol j∈i means that node j is directly connected to node i, and when node i has a grounding branch, it should also include the case of j=0.

(3) Form the zero-injection equality constraint equation

After step (2) is completed, select a node that is neither a generator node nor a load node in the quantity measurement, and use the injected power of the node as the zero injection equality constraint equation, expressed as:

c(x)=0 (2)

In the formula, c(x) is the measurement equation of the zero-injected node, and the injected active power and reactive power value of the node represented by it can be calculated by formula (3); x is the n-dimensional state quantity, and n is the actual The number of state variables of . The calculation formula of c(x) is:

In the formula, P _i ′ is the active power injected by node i; Q _i ′ is the reactive power injected by node i; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node _j ; The voltage phase angle of i, θ _j is the voltage phase angle of node j, and θ _ij = θ _i -θ _j , which means the voltage phase angle difference between node i and node j; G _ij and B _ij are respectively Corresponding to the real part and imaginary part of the elements between nodes i and j; N is the number of nodes in the grid.

(4) Calculation of residuals, Jacobian matrix and weight function

After the step (3) is completed, the node voltage amplitude in the grid, the node injection power after removing the zero injection node and the branch power are measured, and the residual error, Jacobian matrix and weight function of the grid measurement are calculated. The specific steps are as follows:

I) Calculation of residuals

Calculate the parameters measured by each quantity in state estimation based on formula (4):

r=z-h(x) (4)

In the formula, z is the m-dimensional quantity measurement, m is the actual number of measured variables in the state estimation; x is the n-dimensional state quantity, and n is the actual number of state variables in the state estimation. h(x) is the measurement equation, including the measurement equation corresponding to the injected power, the measurement equation corresponding to the line branch power, the measurement equation corresponding to the transformer branch power and the measurement equation corresponding to the node voltage; h(x) It can be calculated by Formula (5)-Formula (8); r is the measurement residual.

Based on formula (5), the measurement equation corresponding to the node injection power after removing the zero injection node is obtained:

In the formula, P _i is the active power injected by node i; Q _i is the reactive power injected by node i; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node j; θ _i is the voltage amplitude of node i Voltage phase angle, θ _j is the voltage phase angle of node j, and θ _ij = θ _i -θ _j , which means the voltage phase angle difference between node i and node j; G _ij and B _ij are the corresponding nodes in the node admittance matrix The real and imaginary parts of the elements between i and j.

Based on the formula (6), the measurement equation corresponding to the line branch power is obtained:

In the formula, P _ij and Q _ij are the active power and reactive power of the line branch node i side respectively, and P _ji and Q _ji are the active power and reactive power of the branch node j side. u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node j; θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij = θ _i -θ _j , Indicates the voltage phase angle difference between node i and node j. g is the line conductance, b is the line susceptance, and y _c is the line-to-ground susceptance.

Based on formula (7), the measurement equation corresponding to the transformer branch power is obtained:

In the formula, P _ijk , Q _ijk are the active power and reactive power at the node i side of the transformer branch respectively, and P _jik and Q _jik are the active power and reactive power at the node j side of the transformer branch respectively. u _i is the voltage amplitude of node i, and u _j is the voltage amplitude of node j. θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij = θ _i - θ _j represents the voltage phase angle difference between node i and node j. K is the non-standard transformation ratio of the transformer: j is the standard side, the transformation ratio is 1, i is the non-standard side, and the transformation ratio is K; b _T is the susceptance of the standard side of the transformer.

Based on the formula (8), the measurement equation corresponding to the node voltage is obtained:

U _i =u _i (8)

In the formula, U _i and u _i both represent the voltage amplitude of node i.

II) Calculate the Jacobian matrix

The Jacobian matrices H and C are formed based on formula (9) - formula (19). Among them, H is the Jacobian matrix of quantity measurement, including the Jacobian matrix elements formed by injection power, line branch power, transformer branch power and node voltage amplitude. C is the Jacobian matrix constrained by the zero-injection equality, which is composed of elements of the Jacobian matrix formed by the injection power of the zero-injection node.

For the injected power, its Jacobian matrix elements are formed based on equations (9) and (10):

In the formula, P _i is the active power injected by node i; Q _i is the reactive power injected by node i. u _i is the voltage amplitude of node i, and u _j is the voltage amplitude of node j. θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij =θ _i -θ _j represents the voltage phase angle difference between node i and node j. G _ij , B _ij are the real part and imaginary part of the elements between corresponding nodes i and j in the node admittance matrix, respectively. G _ii and B _ii are the real part and imaginary part of the elements on the main diagonal corresponding to node i in the nodal admittance matrix, respectively.

For the power of the branch on the line i side, its Jacobian matrix elements are formed based on formulas (11) and (12):

In the formula, P _ij and Q _ij are the active power and reactive power of the line branch node i side. u _i is the voltage amplitude of node i, and u _j is the voltage amplitude of node j. θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij =θ _i -θ _j represents the voltage phase angle difference between node i and node j. g is the line conductance, b is the line susceptance, and y _c is the line-to-ground susceptance.

For the branch power on the side of line j, its Jacobian matrix elements are formed based on formulas (13) and (14):

In the formula, P _ji and Q _ji are the active power and reactive power at node j side of the line branch. u _i is the voltage amplitude of node i, and u _j is the voltage amplitude of node j. θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij =θ _i -θ _j represents the voltage phase angle difference between node i and node j. g is the line conductance, b is the line susceptance, and y _c is the line-to-ground susceptance.

For the branch power on the i side of the transformer, its Jacobian matrix elements are formed based on formulas (15) and (16):

In the formula, P _ijk and Q _ijk are the active power and reactive power of the transformer branch node i side. u _i is the voltage amplitude of node i, and u _j is the voltage amplitude of node j. θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij = θ _i - θ _j represents the voltage phase angle difference between node i and node j. K is the non-standard transformation ratio of the transformer: j is the standard side, the transformation ratio is 1, i is the non-standard side, the transformation ratio is K; b _T is the susceptance of the standard side of the transformer.

For the power of the side branch of the transformer j, its Jacobian matrix elements are formed based on formulas (17) and (18):

In the formula, P _jik and Q _jik are the active power and reactive power at the node j side of the transformer branch. u _i is the voltage amplitude of node i, and u _j is the voltage amplitude of node j. θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij = θ _i - θ _j represents the voltage phase angle difference between node i and node j. K is the non-standard transformation ratio of the transformer: j is the standard side, the transformation ratio is 1, i is the non-standard side, and the transformation ratio is K; b _T is the susceptance of the standard side of the transformer.

For the node voltages, the elements of its Jacobian matrix are formed based on formula (19):

In the formula, U _i and u _i are the voltage amplitude of node i, u _j is the voltage amplitude of node j, θ _i is the voltage phase angle of node i, and θ _j is the voltage phase angle of node j.

The Jacobian matrix of the zero-injection equality constraint is Among them, c(x) is the measurement equation of the zero injection node, which represents the injected active power and reactive power value of the node; x is the n-dimensional state quantity, and n is the actual number of state variables in the state estimation.

III) Calculate the weight function

Based on the formula (20) to calculate the diagonal matrix elements of the weight function W, the calculation formula is:

In the formula, r _min is the detection threshold of small residual error, take 0.002, w _i ^* is the weight function of measurement i, is the fixed weight of measurement i; r _i is the residual error of measurement i.

(5) State variable update and convergence judgment

i) State variable update

After step (4) is completed, calculate the correction amount Δx ^(time) of the state variable according to the formula (21), and then update the state variable to obtain the new value of the state variable x ^(time+1) = x ^(time) + Δx ^(time) , time=time+1.

In the formula, time is the number of calculation iterations; x ^(time) is the state quantity at the time-th iteration; W is the diagonal matrix of the weight function, and its diagonal elements are equal to the weight function, that is, W _ii =w _i ^* . is the Jacobian matrix of quantity measurement, and H ^T is its transpose. c(x ^(time) ) is the zero injection equality constraint when the iteration value is x ^(time) , Inject the equality-constrained Jacobian matrix for zero, and C ^T its transpose. zh(x ^(time) ) indicates the residual error when the iteration value is x ^(time) ; λ ^(time) is the Lagrangian multiplier vector at the time-th iteration.

ii) Judgment of convergence

When the correction amount Δx ^(time) of the state variable satisfies max(|Δx ^(time) |)<ε, the iterative calculation is ended and the result is output; when max(|Δx ^(time) |)≥ε and the number of iterations time≥Tmax, Then stop the iteration and output "does not converge!".

When max(|Δx ^(time) |)≥ε and the number of iterations time<Tmax, increase the number of iterations time by 1, and return to step (3) for re-iteration calculation.

After the present invention adopts above-mentioned technical scheme, mainly have following effect:

Compared with the weight coefficient assignment method in the prior art, the present invention adopts a weight function based on residual normalization, and only needs to change the weight during the iterative process to automatically eliminate the influence of bad data. The robustness performance is good, and it can effectively suppress the bad influence of the bad leverage measurement on the estimation result.

The present invention introduces a threshold criterion in the calculation of the weight function, that is, the weights of the measurements whose residuals are smaller than the threshold are taken as 1, and the weights of the measurements whose residuals are greater than the threshold are all less than 1. Therefore, the effective information in the quantity measurement is fully utilized, and the state estimation does not converge due to the large fluctuation of the quantity measurement weight, so the convergence and numerical stability when bad data and measurement errors coexist are greatly improved.

The present invention directly uses the residual error of the quantity measurement as a variable when calculating the weight function, avoids the calculation of the standardized residual error, thus greatly improves the calculation speed, saves calculation time, and is very suitable for engineering practical application.

Additional aspects and advantages of the invention will be set forth in the description which follows, and in part will be obvious from the description, or may be learned by practice of the invention.

Description of drawings

Fig. 1 is a schematic flow chart of the weight function least squares state estimation method based on residual normalization of the present invention;

Figure 2 shows the network parameters and measurement distribution of the 3-node system;

Figure 3 is a wiring diagram of the IEEE-9 node system.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and embodiments, but it should not be understood that the scope of the subject matter of the present invention is limited to the following embodiments. Without departing from the above-mentioned technical ideas of the present invention, various replacements and changes made according to common technical knowledge and conventional means in this field shall be included in the protection scope of the present invention.

As shown in 1, the specific steps of a weight function least squares state estimation method based on residual normalization are as follows:

(1) Input basic data and initialization

1) Enter basic data

First, input the SCADA data, network structure and parameter information of the power grid under any time section. That is, the node voltage amplitude, node injection power and branch power of the input grid at any time section, the network structure and parameter information are the resistance, reactance and susceptance parameters of network components (including lines and transformers), as well as rated voltage and power benchmark.

2) Parameter initialization

According to the standard data of IEEE-9 node system in the article "Power system control and stability" written by Anderson PM and Fouad AA, input the network structure, load data and related parameter information. The node voltage amplitude, node injection power, and branch power obtained from the power flow calculation are measured. Assuming that there is no bad data in the measurement, and the normal measurement has a certain measurement error, the error conforms to the normal distribution. Set the m-order identity matrix R ^-1 (that is, the diagonal elements of the m-order identity matrix R ^-1 are all 1, and the off-diagonal elements are all 0), m is the actual number of measured variables in the state estimation, which is 57; The initial value of the node voltage amplitude when setting the state estimation is 1 per unit, and the phase angle is 0; the impedance and susceptance of the network elements are all reduced to the per unit value; the node injection power and branch power are also normalized to Calculated to the per unit value, the reference power is set to 100MVA; and the detection threshold value r _min =0.002 of the small residual error during weight calculation, and the iteration number time=1 is set. For the selection and adoption of the maximum number of initialization iterations and convergence accuracy, Tmax=40 and ε=10 ⁻⁴ are preferred in this embodiment.

(2) Calculate node admittance matrix

After step (1) is completed, the node admittance matrix of the network is calculated, and the calculation formula is the formula (1) in the technical scheme.

According to the network structure and parameter information of the IEEE-9 node system, according to the formula (1) in the technical solution, the node admittance matrix Y of the network is calculated as:

(3) Form the zero-injection equality constraint equation

After step (2) is completed, the nodes that are neither generator nodes nor load nodes in the quantity measurement are selected, and their injected power is used as the zero injection equality constraint equation. According to the formula (2) in the technical plan, the first iteration As an example of the calculation results, the calculated zero injection power c(x) is:

(4) Calculation of residuals, Jacobian matrix and weight function

After step (3) is completed, take the node voltage amplitude in the power grid, the node injection power after removing the zero injection node, and the branch power as the measurement, and calculate the residual error, Jacobian matrix and weight function of the power grid measurement, The calculation formula is formula (4)-formula (20) in the technical scheme.

I) Calculation of residuals

According to the definition of residual error and measurement equation, according to the formula (4)-formula (8) in the technical solution, taking the result of the first iterative calculation as an example, the calculated residual r of the quantity measurement is:

II) Calculate the Jacobian matrix

According to the definition of the elements of the Jacobian matrix, calculate the Jacobian matrix H and C according to the formula (9)-formula (19) in the technical solution, and take the result of the first iterative calculation as an example, the calculated Jacobian matrix H and C are:

III) Calculate the weight function

According to the definition of the weight function, according to the formula (20) in the technical solution, taking the result of the first iteration calculation as an example, the calculated weight function W is:

(5) State variable update and convergence judgment

i) State variable update

After the 4th step is finished, calculate the correction amount Δx ^(time) of state variable according to formula (21) in the technical scheme, then update state variable, obtain the new value of state variable, namely: x ^(time+1) =x ^{( time)} +Δx ^(time) , time=time+1.

Taking the result of the first iterative calculation as an example, according to the formula (21) in the technical solution, the correction amount Δx ⁽¹⁾ of the state variable is calculated as:

ii) Judgment of convergence

When the correction amount Δx ^(time) of the state variable satisfies max(|Δx ^(time) |)<ε, the iterative calculation is ended and the result is output; when max(|Δx ^(time) |)≥ε and the number of iterations time≥Tmax, Then stop the iteration and output "does not converge!". When max(|Δx ^(time) |)≥ε and the number of iterations time<Tmax, increase the number of iterations time by 1, and return to step (3) for re-iteration calculation.

Taking the result of the first iterative calculation as an example, at this time, ε=10 ⁻⁴ , Tmax=40, max(|Δx ⁽¹⁾ |)=0.16419>ε, time=1<Tmax. According to the convergence judgment, the following steps are performed: time=time+1=2, return to step (3), and perform iterative calculation again.

According to the previous steps, the convergence condition is satisfied after 3 iterations, at this time max(|Δx ⁽³⁾ |)=7.4934E ^-05 <ε, the estimated results of the state variables are shown in the following table:

Table 1 Estimation results of IEEE-9 node system state variables

Comparative Analysis of Experimental Effects

In order to enable those skilled in the art to better understand the present invention and understand the advantages of the present invention over the prior art, the applicant further explains in conjunction with specific embodiments.

(1) Comparison of tolerance performance

The inventors compared the RNWLS of the present invention with other state estimators to test the robustness of the RNWLS.

1) 3-node system

Taking the 3-node system shown in 2 as an example, the measurement No. 1 is lever measurement and is set as bad data, and the other measurement values take the true value (calculated value of power flow), where P and Q represent active and reactive power, respectively. Using four state estimators (RNWLS, EFWLS, E-LAV, WLS) to simulate and analyze the above example, the results are as follows:

Table 2 Comparison of estimation results of various methods in 3-node system

Among them, the data marked in bold is bad measurement data. It can be seen from the above calculation results that in terms of the ability to deal with bad leverage measurement, the results estimated by the method RNWLS and EFWLS of the present invention are closest to the true value of power flow, indicating that it can accurately estimate the real state of the system, and the amount of bad leverage of No. 1 However, the active power estimated by E-LAV is closer to the true value of the power flow, while the reactive power has a large difference; the difference between the estimated result of WLS and the true value of the power flow is the largest, indicating that its ability to resist bad leverage measurement is limited , the true state of the system cannot be estimated.

2) IEEE-118 node system

In order to analyze the tolerance performance of the RNWLS method of the present invention on a larger scale system, taking the IEEE-118 node system as an example, 11 cases in which the proportion of bad data is 0% to 10% are tested respectively. Among them, the normal measurement has a certain measurement error, and the error conforms to the normal distribution; the bad data is the opposite number of the corresponding proportional voltage amplitude, active power, and reactive power value in the true value of the power flow (among them, active power, The reactive powers are all set at the maximum (bad data). Using four state estimators (RNWLS, EFWLS, E-LAV, WLS) to simulate and analyze the above example, the results are as follows:

Table 3 Voltage amplitude error comparison of different estimation methods for IEEE-118 node systems

Among them, ΔU _max and ΔU _mean represent the maximum estimation error and average estimation error of the node voltage amplitude respectively. Considering that the voltage phase angle of the node in the power flow information obtained by SCADA is 0, only the node voltage amplitude Perform simulation analysis. From the above calculation results, it can be seen that in the process of bad data changing from 0% to 10%, the errors of the results obtained by E-LAV and WLS are relatively large; while the errors of the estimated results of RNWLS and EFWLS are close to 0, indicating that in the process of large-scale In a large system, these two estimation methods can eliminate bad data, obtain accurate estimation results, and have good tolerance performance.

(2) Comparison of computational efficiency

In order to compare the efficiency, the inventor took the IEEE-118 node system as an example, assuming that there is no bad data in the measurement, and the normal measurement has a certain measurement error, and the error conforms to the normal distribution. Using four state estimators (RNWLS, EFWLS, E-LAV, WLS) to simulate and analyze the above example, the average time of each iteration and the total calculation time are shown in the following table:

Table 4 Computational efficiency comparison of each method in IEEE-118 node system

It can be seen from the above calculation results that the RNWLS and EFWLS estimation methods of the present invention have better convergence, but for the average time of each iteration, the WLS estimation is the shortest, but it does not have tolerance; the EFWLS estimation is the longest; the RNWLS estimation and E- The LAV estimate is intermediate.

To sum up, the RNWLS estimation proposed by the present invention is similar to the EFWLS estimation in terms of robustness, has good robustness, and can effectively suppress the adverse effects of bad leverage measurements on the results; and is similar in computational efficiency to E- LAV estimation has high computational efficiency and good convergence. Therefore, the advantages of existing state estimation methods in robustness and calculation efficiency are effectively concentrated, and it is very suitable for practical engineering applications.

Claims (1)

1. A weight function least squares state estimation method based on residual normalization, utilize computer, by program, realize the state estimation of power grid, it is characterized in that: the concrete steps of described method comprise the following contents; (1) Input basic data and initialization 1) Enter basic data First input the data acquisition and monitoring control system of the power grid under any time section, namely SCADA data, network structure and parameter information; the SCADA data of the power grid includes node voltage amplitude, node injection power and branch power; the network structure and The parameter information includes resistance, reactance, susceptance, rated voltage and power reference of network elements; the network elements include lines and transformers; 2) Parameter initialization Set the m-order identity matrix R ^-1 , the diagonal elements of the identity matrix R ^-1 are all 1, and the off-diagonal elements are all 0, m is the actual number of measured variables in the state estimation; when setting the state estimation The initial value of the node voltage amplitude is 1 p.u., and the phase angle is 0. The impedance and susceptance of the network elements are all reduced to the p.u. value; the node injection power and branch power are also reduced to the p.u. value; initialize the maximum number of iterations Tmax to 40~60; the convergence precision ε is 10 ^-3 to 10 ^-5 , and the detection threshold of small residual error r _min =0.002 during weight calculation, and set the number of iterations time=1; (2) Calculate node admittance matrix After step (1) is completed, use the formula (1) to calculate the node admittance matrix Y; <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><msub><mi>Y</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub><mo>=</mo><munder><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>&amp;Element;</mo><mi>i</mi></mrow></munder><msub><mi>y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mi>y</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow> In the formula, the diagonal element of the node admittance matrix is expressed as Y _ii , and the off-diagonal element of the node admittance matrix is expressed as Y _ij , both of which are calculated by the above formula; y _ij is the distance between node i and node j The reciprocal of the branch impedance z _ij ; the symbol j∈i means that the node j is directly connected to the node i, and when the node i has a grounding branch, it should also include the case of j=0; (3) Form the zero-injection equality constraint equation After the (2) step is completed, select a node that is neither a generator node nor a load node in the quantity measurement, and use the injected power of the node as a zero injection equation constraint equation, expressed as; c(x)=0 (2) In the formula, c(x) is the measurement equation of the zero-injected node, and the injected active power and reactive power value of the node represented by it can be calculated by formula (3); x is the n-dimensional state quantity, and n is the actual The number of state variables; the calculation formula of c(x) is; <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><msubsup><mi>P</mi><mi>i</mi><mo>&amp;prime;</mo></msubsup><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><munderover><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>&amp;lsqb;</mo><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>&amp;rsqb;</mo></mrow></mtd></mtr><mtr><mtd><mrow><msubsup><mi>Q</mi><mi>i</mi><mo>&amp;prime;</mo></msubsup><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><munderover><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>&amp;lsqb;</mo><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>&amp;rsqb;</mo></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></mrow> In the formula, P′ _i is the active power injected into node i; Q′ _i is the reactive power injected into node i; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node _j ; The voltage phase angle of i, θ _j is the voltage phase angle of node j, and θ _ij = θ _i -θ _j , which means the voltage phase angle difference between node i and node j; G _ij and B _ij are respectively Corresponding to the real part and imaginary part of the elements between nodes i and j; N is the number of nodes in the grid; (4) Calculation of residuals, Jacobian matrix and weight function After step (3) is completed, take the node voltage amplitude in the power grid, the node injection power after removing the zero injection node, and the branch power as the measurement, and calculate the residual error, Jacobian matrix and weight function of the power grid measurement; specifically The steps are as follows; I) Calculation of residuals Calculate the parameters measured by each quantity in the state estimation based on formula (4); r=z-h(x) (4) In the formula, z is the m-dimensional quantity measurement, m is the actual number of measurement variables in the state estimation; x is the n-dimensional state quantity, n is the actual number of state variables in the state estimation; h(x) is the quantity The measurement equations include the measurement equations corresponding to the injected power, the measurement equations corresponding to the line branch power, the measurement equations corresponding to the transformer branch power and the measurement equations corresponding to the node voltage; h(x) can be expressed by the formula (5)- Calculated by formula (8); r is the measurement residual; Based on formula (5), the measurement equation corresponding to the node injection power after removing the zero injection node is obtained; <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><msub><mi>P</mi><mi>i</mi></msub><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><munderover><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>&amp;lsqb;</mo><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>&amp;rsqb;</mo></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Q</mi><mi>i</mi></msub><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><munderover><mo>&amp;Sigma;</mo><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mi>N</mi></munderover><mo>&amp;lsqb;</mo><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow><mo>&amp;rsqb;</mo></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>5</mn><mo>)</mo></mrow></mrow> In the formula, P _i is the active power injected by node i; Q _i is the reactive power injected by node i; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node j; θ _i is the voltage amplitude of node i Voltage phase angle, θ _j is the voltage phase angle of node j, and θ _ij = θ _i - θ _j , which means the voltage phase angle difference between node i and node j; G _ij and B _ij are the corresponding nodes in the node admittance matrix the real and imaginary parts of the elements between i and j; Obtain the measurement equation corresponding to the line branch power based on formula (6); <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><msup><msub><mi>u</mi><mi>i</mi></msub><mn>2</mn></msup><mi>g</mi><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mo>-</mo><msup><msub><mi>u</mi><mi>i</mi></msub><mn>2</mn></msup><mrow><mo>(</mo><mi>b</mi><mo>+</mo><msub><mi>y</mi><mi>c</mi></msub><mo>)</mo></mrow><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub><mo>=</mo><msup><msub><mi>u</mi><mi>j</mi></msub><mn>2</mn></msup><mi>g</mi><mo>+</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mo>-</mo><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub><mo>=</mo><mo>-</mo><msup><msub><mi>u</mi><mi>j</mi></msub><mn>2</mn></msup><mrow><mo>(</mo><mi>b</mi><mo>-</mo><msub><mi>y</mi><mi>c</mi></msub><mo>)</mo></mrow><mo>+</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>6</mn><mo>)</mo></mrow></mrow> In the formula, P _ij , Q _ij are the active power and reactive power of the line branch node i side respectively, P _ji and Q _ji are the active power and reactive power of the branch node j side respectively; u _i is the power of node i Voltage amplitude, u _j is the voltage amplitude of node j; θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij = θ _i - θ _j means node i and node j The voltage phase angle difference; g is the conductance of the line, b is the susceptance of the line, and y _c is the susceptance of the line to the ground; Based on the formula (7), the corresponding measurement equation of the transformer branch power is obtained; <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><msub><mi>P</mi><mrow><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mi>mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><msup><mi>K</mi><mn>2</mn></msup></mfrac><msup><msub><mi>u</mi><mi>i</mi></msub><mn>2</mn></msup><msub><mi>b</mi><mi>T</mi></msub><mo>+</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub><mo>=</mo><mo>-</mo><msub><mi>b</mi><mi>T</mi></msub><msup><msub><mi>u</mi><mi>j</mi></msub><mn>2</mn></msup><mo>+</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>7</mn><mo>)</mo>mo></mrow></mrow> In the formula, P _ijk , Q _ijk are the active power and reactive power of the transformer branch node i side respectively, P _jik , Q _jik are the active power and reactive power of the transformer branch node j side respectively; u _i is the node i , u _j is the voltage amplitude of node j; θ _i is the voltage phase angle of node i, θ _j is the voltage phase angle of node j, and θ _ij = θ _i -θ _j , which means node i and node The voltage phase angle difference of j; K is the non-standard transformation ratio of the transformer: j is the standard side, the transformation ratio is 1, i is the non-standard side, the transformation ratio is K; b _T is the susceptance of the standard side of the transformer; Obtain the measurement equation corresponding to the node voltage based on formula (8); U _i =u _i (8) In the formula, U _i and u _i both represent the voltage amplitude of node i; II) Calculate the Jacobian matrix Based on formula (9)-formula (19), the Jacobian matrices H and C are formed; where H is the Jacobian matrix of quantity measurement, including the Jacobian matrix formed by injected power, line branch power, transformer branch power and node voltage amplitude Ratio matrix elements; C is the Jacobian matrix constrained by the zero injection equality, which is composed of the Jacobian matrix elements formed by the injection power of the zero injection node; For the injected power, its Jacobian matrix elements are formed based on equations (9) and (10); <mrow><mo>{</mo><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><msub><mi>u</mi><mi>i</mi></msub></mfrac><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub><msup><msub><mi>u</mi><mi>i</mi></msub><mn>2</mn></msup><mo>+</mo><msub><mi>P</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub><msup><msub><mi>u</mi><mi>i</mi></msub><mn>2</mn></msup><mo>-</mo><msub><mi>Q</mi><mi>i</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow></mi>msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>9</mn><mo>)</mo></mrow></mrow> <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><msub><mi>u</mi><mi>i</mi></msub></mfrac><mrow><mo>(</mo><mo>-</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub><msup><msub><mi>u</mi><mi>i</mi></msub><mn>2</mn></msup><mo>+</mo><msub><mi>Q</mi><mi>i</mi></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>i</mi></mrow></msub><msup><msub><mi>u</mi><mi>i</mi></msub><mn>2</mn></msup><mo>+</mo><msub><mi>P</mi><mi>i</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><msub><mi>G</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><msub><mi>B</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mi>mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>10</mn><mo>)</mo></mrow></mrow> In the formula, P _i is the active power injected by node i; Q _i is the reactive power injected by node i; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node j; θ _i is the voltage amplitude of node i Voltage phase angle, θ _j is the voltage phase angle of node j, and θ _ij = θ _i -θ _j , which means the voltage phase angle difference between node i and node j; G _ij and B _ij are the corresponding nodes in the node admittance matrix The real part and imaginary part of the element between i and j; G _ii and B _ii are respectively the real part and imaginary part of the element on the main diagonal at the corresponding node i in the node admittance matrix; For the branch power of the line i side, the Jacobian matrix elements are formed based on formulas (11) and (12); <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mn>2</mn><msub><mi>u</mi><mi>i</mi></msub><mi>g</mi><mo>-</mo><msub><mi>u</mi><mi>j</mi></msub><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><msub><mi>u</mi><mi>j</mi></msub><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></mi>msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>11</mn><mo>)</mo></mrow></mrow> <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><msub><mi>u</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>b</mi><mo>+</mo><msub><mi>y</mi><mi>c</mi></msub><mo>)</mo></mrow><mo>-</mo><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></mi>msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mi>mo><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>12</mn><mo>)</mo></mrow></mrow> In the formula, P _ij and Q _ij are the active power and reactive power of node i side of the line branch respectively; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node j; θ _i is the voltage amplitude of node i Voltage phase angle, θ _j is the voltage phase angle of node j, and θ _ij = θ _i -θ _j , which means the voltage phase angle difference between node i and node j; g is the line conductance, b is the line susceptance, y _c is line-to-ground susceptance; For the branch power on the side of line j, its Jacobian matrix elements are formed based on formulas (13) and (14); <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mo>-</mo><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mn>2</mn><msub><mi>u</mi><mi>j</mi></msub><mi>g</mi><mo>+</mo><msub><mi>u</mi><mi>i</mi></msub><mrow><mo>(</mo><mo>-</mo><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi>mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>13</mn><mo>)</mo></mrow></mrow> <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>b</mi><mo>+</mo><msub><mi>y</mi><mi>c</mi></msub><mo>)</mo></mrow><mo>+</mo><msub><mi>u</mi><mi>i</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>+</mo><mi>b</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mrow><mo>(</mo><mi>g</mi><mi></mi><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>-</mo><mi>b</mi><mi></mi><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>)</mo></mrow></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>14</mn><mo>)</mo></mrow></mrow> In the formula, P _ji and Q _ji are the active power and reactive power of node j side of the line branch respectively; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node j; θ _i is the voltage amplitude of node i Voltage phase angle, θ _j is the voltage phase angle of node j, and θ _ij = θ _i -θ _j , which means the voltage phase angle difference between node i and node j; g is the line conductance, b is the line susceptance, y _c is line-to-ground susceptance; For the branch power of the transformer i side, the Jacobian matrix elements are formed based on formulas (15) and (16); <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mi>mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>15</mn><mo>)</mo></mrow></mrow> <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>2</mn><msup><mi>K</mi><mn>2</mn></msup></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>b</mi><mi>T</mi></msub><mo>+</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mo>mfrac><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><mi>&amp;theta;</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>16</mn><mo>)</mo></mrow></mrow> In the formula, P _ijk and Q _ijk are the active power and reactive power of node i side of the transformer branch respectively; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node j; θ _i is the voltage amplitude of node i Voltage phase angle, θ _j is the voltage phase angle of node j, θ _ij = θ _i - θ _j , indicating the voltage phase angle difference between node i and node j; K is the non-standard transformation ratio of the transformer: j is the standard side, and the transformation ratio is 1, i is the non-standard side, and the transformation ratio is K; b _T is the susceptance of the standard side of the transformer; For the power of the side branch of the transformer j, its Jacobian matrix elements are formed based on formulas (17) and (18); <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>P</mi><mrow><mi>i</mi><mi>j</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>17</mn><mo>)</mo>mo></mrow></mrow> <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mo>-</mo><mn>2</mn><msub><mi>b</mi><mi>T</mi></msub><msub><mi>u</mi><mi>j</mi></msub><mo>+</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>cos&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></mi>msub></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>Q</mi><mrow><mi>j</mi><mi>i</mi><mi>k</mi></mrow></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>K</mi></mfrac><msub><mi>u</mi><mi>i</mi></msub><msub><mi>u</mi><mi>j</mi></msub><msub><mi>b</mi><mi>T</mi></msub><msub><mi>sin&amp;theta;</mi><mrow><mi>i</mi><mi>j</mi></mrow></msub></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>18</mn><mo>)</mo></mrow></mrow> In the formula, P _jik and Q _jik are the active power and reactive power of node j side of the transformer branch respectively; u _i is the voltage amplitude of node i, u _j is the voltage amplitude of node j; θ _i is the voltage amplitude of node i Voltage phase angle, θ _j is the voltage phase angle of node j, θ _ij = θ _i - θ _j , indicating the voltage phase angle difference between node i and node j; K is the non-standard transformation ratio of the transformer, j is the standard side, and the transformation ratio is 1, i is the non-standard side, and the transformation ratio is K; b _T is the susceptance of the standard side of the transformer; For node voltages, form its Jacobian matrix elements based on formula (19); <mrow><mfenced open = "{" close = ""><mtable><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>U</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mn>1</mn></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>U</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>i</mi></msub></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>U</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>u</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></mtd></mtr><mtr><mtd><mrow><mfrac><mrow><mo>&amp;part;</mo><msub><mi>U</mi><mi>i</mi></msub></mrow><mrow><mo>&amp;part;</mo><msub><mi>&amp;theta;</mi><mi>j</mi></msub></mrow></mfrac><mo>=</mo><mn>0</mn></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>19</mn><mo>)</mo></mrow></mrow> In the formula, U _i and u _i are the voltage amplitude of node i, u _j is the voltage amplitude of node j, θ _i is the voltage phase angle of node i, and θ _j is the voltage phase angle of node j; The Jacobian matrix of the zero-injection equality constraint is Where c(x) is the measurement equation of the zero injection node, which represents the injected active power and reactive power value of the node; x is the n-dimensional state quantity, and n is the actual number of state variables in the state estimation; III) Calculate the weight function Based on the formula (20) to calculate the diagonal matrix elements of the weight function W, the calculation formula is: <mrow><msup><msub><mi>w</mi><mi>i</mi></msub><mo>*</mo></msup><mo>=</mo>< mfenced open = "{" close = ""><mtable><mtr><mtd><mn>1</mn></mtd><mtd><mrow><mo>|</mo><msub><mi>r</mi><mi>i</mi></msub><mo>|</mo><mo>&amp;le;</mo><msub><mi>r</mi><mi>min</mi></msub></mrow></mtd></mtr><mtr><mtd><mrow><msup><msub><mi>R</mi><mi>i</mi></msub><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><mrow><mo>|</mo><mn>1</mn><mo>/</mo><msub><mi>r</mi><mi>i</mi></msub><mo>|</mo></mrow><mrow><mo>|</mo><mn>1</mn><mo>/</mo><msub><mi>r</mi><mi>min</mi></msub><mo>|</mo></mrow></mfrac><mo>=</mo><msup><msub><mi>R</mi><mi>i</mi></msub><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfrac><msub><mi>r</mi><mi>min</mi></msub><mrow><mo>|</mo><msub><mi>r</mi><mi>i</mi></msub><mo>|</mo></mrow></mfrac></mrow></mtd><mtd><mrow><mo>|</mo><msub><mi>r</mi><mi>i</mi></msub><mo>|</mo><mo>&gt;</mo><msub><mi>r</mi><mi>min</mi></msub></mrow></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>20</mn><mo>)</mo></mrow></mrow> In the formula, r _min is the detection threshold of small residual error, which is 0.002; w _i ^* is the weight function of measurement i; is the fixed weight of measurement i; r _i is the residual of measurement i; (5) State variable update and convergence judgment i) State variable update After step (4) is completed, calculate the correction amount Δx ^(time) of the state variable according to the formula (21), and then update the state variable to obtain the new value of the state variable x ^(time+1) = x ^(time) + Δx ^(time) , time=time+1; <mrow><mfenced open = "[" close = "]"><mtable><mtr><mtd><mrow><msup><mi>&amp;Delta;x</mi><mrow><mo>(</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>)</mo></mrow></mo>msup></mrow></mtd></mtr><mtr><mtd><msup><mi>&amp;lambda;</mi><mrow><mo>(</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>)</mo></mrow></msup></mtd></mtr></mtable></mfenced><mo>=</mo><msup><mfenced open = "[" close = "]"><mtable><mtr><mtd><mrow><msup><mi>H</mi><mi>T</mi></msup><mi>W</mi><mi>H</mi></mrow></mtd><mtd><msup><mi>C</mi><mi>T</mi></msup></mtd></mtr><mtr><mtd><mi>C</mi></mtd><mtd><mn>0</mn></mtd></mtr></mtable></mfenced><mrow><mo>-</mo><mn>1</mn></mrow></msup><mfenced open = "[" close = "]"><mtable><mtr><mtd><msup><mi>H</mi><mi>T</mi></msup><mi>W</mi><mo>(</mo><mi>z</mi><mo>-</mo><mi>h</mi><mrow><mo>(</mo><msup><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>)</mo></mrow></msup><mo>)</mo></mrow><mo>)</mo></mtd></mtr><mtr><mtd><mo>-</mo><mi>c</mi><mo>(</mo><msup><mi>x</mi><mrow><mo>(</mo><mi>t</mi><mi>i</mi><mi>m</mi><mi>e</mi><mo>)</mo></mrow></msup><mo>)</mo></mtd></mtr></mtable></mfenced><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>21</mn><mo>)</mo></mrow></mrow> In the formula, time is the number of calculation iterations; x ^(time) is the state quantity during the time iteration; W is the weight function diagonal matrix, and its diagonal elements are equal to the weight function, that is, W _ii =w _i ^* ; is the measured Jacobian matrix, H ^T is its transpose; c(x ^(time) ) is the zero-injection equality constraint when the iteration value is x ^(time) , is the Jacobian matrix of equality constraints injected into zero, C ^T is its transpose; zh(x ^{(time) )} indicates the residual error when the iteration value is x ^(time) ; Grangian multiplier vector; ii) Judgment of convergence When the correction amount Δx ^(time) of the state variable satisfies max(|Δx ^(time) |)<ε, the iterative calculation is ended and the result is output; when max(|Δx ^(time) |)≥ε and the number of iterations time≥Tmax, Then stop the iteration and output "no convergence!"; When max(|Δx ^(time) |)≥ε and the number of iterations time<Tmax, increase the number of iterations time by 1, and return to step (3) for re-iteration calculation.