CN103400303A - Static equivalence method for external network based on internal network routine Ward equivalent circuit - Google Patents

Abstract

The invention relates to a static equivalence method for an external network based on an internal network routine Ward equivalent circuit. The static equivalence method comprises the steps of first inputting a topological structure, parameters thereof and status information through programs by using a computer; then calculating to obtain a routine Ward equivalent network of the internal network and parameter values of the routine Ward equivalent network; thereafter establishing virtual measurement equation of equivalent circuit parameters of the external network based on the restraint relationship between the equivalent circuit parameters of the external network and the routine Ward equivalent circuit parameters of the internal network; and finally solving the virtual measurement equation by adopting a least square method to acquire parameter values of the equivalent network of the external network. The static equivalence method can be widely applied to interconnected networks to solve an equivalence problem of completely unknown adjacent subnets, provides an effective solution for carrying out equivalence on completely unknown adjacent subsystems in the interconnected networks, providing a profound theoretical basis for improving the precision of safe and stable correction for independent subsystems, and being conducive to safe and stable running of the interconnected networks.

Description

Outer net Static Equivalent method based on the conventional Ward equivalent circuit of Intranet

Technical field

The present invention relates to the method for a kind of electric system based on Intranet information estimator outer net Static Equivalent model and parameter thereof, be specifically related to a kind of outer net Static Equivalent method based on the conventional Ward equivalent circuit of Intranet.

Background technology

Along with the continuous propelling of China's electric Power Reform, Power Market In China will progressively form.Under Power Market, interconnection subsystem represents different interest groups usually, and the electric network data of its grasp is thought to be rich in the secret of commercial value more and more, between each grid company, just according to rules, shares some essential informations.Cause the real-time exchange that can't carry out data in system between subsystems, the safety and stability of independent particle system is checked and be there will be larger deviation, thereby affects the safe and stable operation of electric system.Therefore, in the situation that, only based on the real measured data of Intranet, estimate that the Equivalent Model parameter of outer net has important theory significance and practical value.

Traditional external network equivalent method has Ward equivalent method and REI equivalent method, but these class methods all require known complete outer net topological structure and status information, can not solve the equivalent problem under outer net information unknown situation.Only based on aspect Intranet information estimator external network equivalent model parameter, the Dai Weinan equivalent parameters method of estimation comparative maturity of one port network.The most basic method is the analytical expression of deriving voltage source and impedance in the Dai Weinan equivalent parameters, utilizes the measurement voltage and current at boundary node place to solve the Dai Weinan equivalent parameters.For multiport network, very rarely seen based on the research of Intranet information estimator external network equivalent parameter.

There is the scholar to propose the outer net method for parameter estimation based on a plurality of period flow datas of Intranet abroad, but require to represent with the flow state that the Intranet branch road repeatedly cut-offs the measurement equation of different periods.Its design conditions harshness, be difficult to be applied to reality; Before and after the method hypothesis Intranet line disconnection, the boundary node injecting power is constant simultaneously, and this hypothesis is difficult to set up, and may bring larger model systematic error.The domestic scholar of having has proposed the two-port outer net Static Equivalent method for parameter estimation based on the Intranet real measured data, the method is based on the power equation at boundary node place, set up the virtual measurement equation of external network equivalent parameter and solve, but the method has only been utilized the measurement information at boundary node place.

Therefore, there is following major defect in existing method based on Intranet information estimator external network equivalent model parameter: 1. required condition harshness is difficult to meet in real system; 2. only depend on the measurement information at boundary node place, do not take full advantage of more Intranet measurement information.

Summary of the invention

Technical matters to be solved by this invention, just be to provide a kind of outer net Static Equivalent method based on the conventional Ward equivalent circuit of Intranet, the condition that this method needs is simple, in real system, easily meets, and taken full advantage of more Intranet measurement information, result is more reasonable.

Solve the problems of the technologies described above, the technical solution used in the present invention is:

A kind of outer net Static Equivalent method based on the conventional Ward equivalent circuit of Intranet, at first utilize computing machine, by program, at first inputs topological structure and parameter and the status information of Intranet; Next calculates conventional Ward equivalent network and the parameter value thereof of Intranet; Then based on the restriction relation between external network equivalent circuit parameter and the conventional Ward Equivalent Circuit Parameter of Intranet, set up the former virtual measurement equation; Finally adopt least square method to solve and obtain the parameter value of external network equivalent network.

Described method specifically comprises the following steps:

S1 inputs basic data

At first input the basic data of Intranet, comprise the topological structure of Intranet and the measurement information of a plurality of periods; Topological structure refers to contain the basic data of node, namely node serial number, node type (balance node, PV node, PQ node), node zone of living in (Intranet, outer net, boundary node), electricity is led and susceptance over the ground; The basic data of generator, comprise node serial number, meritorious and idle exerting oneself; The basic data of circuit, i.e. circuit first and last node serial number, resistance, reactance, susceptance; The basic data of transformer, namely place, transformer both sides node serial number, resistance, reactance, electricity lead, susceptance, no-load voltage ratio;

The Intranet measurement information of a plurality of periods comprises under the condition of a plurality of different conditions, the voltage magnitude of node and phase angle, injecting power, the power that transmit at circuit and transformer two ends;

S2 calculates conventional Ward equivalent circuit and the parameter thereof of Intranet

After completing the S1-step, according to conventional Ward equivalence method, can ask for corresponding under a plurality of different conditions, a plurality of conventional Ward equivalent circuit and the parameter value thereof of Intranet, detailed process comprises following sub-step:

Asking for of the equivalent branch road parameter of S1-1

Based on the parameter of the topological structure of Intranet and circuit, transformer, according to the definition of self-admittance and transadmittance, can ask for the bus admittance matrix of Intranet; Utilize Gaussian elimination method, the contact node that keeps Intranet and outer net is boundary node, and other nodes of cancellation Intranet obtain only containing the Equivalent admittance matrix of boundary node, suc as formula (28), represent:

Y′ _BB=Y _BB-Y _BIY _II ^-1Y _IB (28)

In formula: Y is bus admittance matrix, and subscript B represents boundary node, and subscript I represents other Intranet nodes except the node of border;

Definition by admittance matrix is as can be known, Y ' _BBTo be obtained by the parameter of branch road over the ground of boundary and the route parameter calculation that is connected with boundary node; Therefore, when known admittance matrix, can obtain conventional Ward Equivalent Circuit Parameter value by Extrapolation;

With in conventional Ward equivalent circuit in dotted line frame in Fig. 1 between boundary node the interconnection impedance parameter be calculated as example: through type (28) calculates the Equivalent admittance matrix Y ' at boundary node place _BBAfter, can obtain the equivalent transadmittance between boundary node, and should the equivalence transadmittance by the parameter of original line parameter circuit value between boundary node and conventional Ward equivalent circuit medium value interconnection, be formed, therefore the part that equivalent transadmittance is deducted to the former just can obtain the latter, i.e. Z in conventional Ward equivalent circuit ₁₂Parameter;

In like manner, can calculate equivalence branch road Y over the ground ₁And Y ₂Parameter value;

Asking for of the equivalent injecting power of S1-2

Based on the status information of a plurality of period Intranets, the injecting power of cancellation other nodes of Intranet except the node of border can obtain the equivalent injecting power of a plurality of periods at boundary node place suc as formula shown in (29):

${\tilde{S}}_{\mathrm{eq}}=\left(\mathrm{diag}{\left({\stackrel{.}{V}}_B\right)}^{*}\right)Y_{\mathrm{BI}}{Y}_{\mathrm{II}}^{-1}{\left(\frac{{\tilde{S}}_I}{{\stackrel{.}{V}}_I}\right)}^{*}$ (29)

In formula: Expression node injecting power,

The voltage of expression node;

Be the injecting power of Intranet through conversion assignments to the equivalent injecting power on boundary node, by it, can calculate in Fig. 1 in the dotted line frame the equivalent injecting power at boundary node place in conventional Ward equivalent circuit

With

By two above steps, namely can calculate the parameter value of the whole elements of conventional Ward equivalent circuit;

S3 sets up virtual measurement equation and the least-squares estimation model of external network equivalent circuit parameter

After completing the S2 step, can based on the outer net shown in Fig. 1, simplify the restriction relation between Ward Equivalent Circuit Parameter and outer net expansion voltage source Ward Equivalent Circuit Parameter and the conventional Ward Equivalent Circuit Parameter of Intranet respectively, set up outer net and simplify the virtual measurement equation of Ward Equivalent Circuit Parameter and outer net expansion voltage source Ward Equivalent Circuit Parameter, and further obtain the least square model of two external network equivalent circuit parameters, finally can solve the parameter value that obtains two external network equivalent circuit based on least square method, and using the latter as final external network equivalent circuit;

The S3-1 outer net is simplified virtual measurement equation and the least-squares estimation model of Ward Equivalent Circuit Parameter

Based on the KCL theorem in circuit theory, the Injection Current sum at boundary node place is zero; As shown in Fig. 1 (a), take boundary node 1 as example, the electric current of Inside and outside network injection boundary node is opposite number each other, that is:

${\stackrel{.}{I}}_1^{\mathrm{eq}}-\frac{{\stackrel{.}{V}}_1^t-{\stackrel{.}{V}}_2^{\mathrm{<figure-callout\; id="12"\; label="t\; Z"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">t</figure-callout>}}}{Z_{}^{}}=\frac{{\stackrel{.}{V}}_1^t-{\stackrel{.}{V}}_2^{\mathrm{<figure-callout\; id="12"\; label="t\; Z"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">t</figure-callout>}}}{Z_{}}+Y_1{\stackrel{.}{V}}_1^t-{\stackrel{.}{I}}_{L,1}^t$ (30)

In formula:

For outer net, simplify the equivalent current source of Ward equivalent circuit, Z ' ₁₂For outer net is simplified the impedance of interconnection between boundary node in the Ward equivalent circuit,

Be the electric current representation of the equivalent injecting power of boundary node, subscript " t " (t=1,2,3, m are the time hop count of required Intranet metric data) represents period t; The numbering of subscript " 1,2 " expression boundary node;

Z ' is multiply by on the following formula both sides simultaneously ₁₂Z ₁₂, and launch according to real part and imaginary part, can obtain formula (31), (32), similar above thinking is analyzed and can be obtained formula (33), (34) boundary node 2:

$({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}{R}_{12}-{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;}{X}_{12})(I_{1,\mathrm{Re}}^{\mathrm{eq}}+I_{L,1,\mathrm{Re}}^t-G_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t+B_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+R_{12}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(I_{1,\mathrm{Im}}^{\mathrm{eq}}+I_{L,1,\mathrm{Im}}^t-$ (31)

$G_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t-B_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t)-(R_{12}+{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;})(V_{1,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t)+({\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(V_{1,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t)=0$

$({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}{R}_{12}-{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;}{X}_{12})(I_{1,\mathrm{Im}}^{\mathrm{eq}}+I_{L,1,\mathrm{Im}}^t-G_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t+B_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t)+(R_{12}^{\&prime;}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+R_{12}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(I_{1,\mathrm{Re}}^{\mathrm{eq}}+I_{L,1,\mathrm{Re}}^t-$ (32)

$G_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t+B_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t)-(R_{12}+{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;})(V_{1,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(V_{1,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t)=0$

$({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}{R}_{12}-{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;}{X}_{12})({\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{Re}}^{\mathrm{eq}}+I_{L,2,\mathrm{Re}}^t-G_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t+B_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+R_{12}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(I_{2,\mathrm{Im}}^{\mathrm{eq}}+I_{L,2,\mathrm{Im}}^t-$ (33)

$G_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t-B_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t)-(R_{12}+{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;})(V_{2,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t)+({\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(V_{2,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t)=0$

$({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}{R}_{12}-{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;}{X}_{12})({\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{Im}}^{\mathrm{eq}}+I_{L,2,\mathrm{Im}}^t-G_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t+B_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t)+({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+R_{12}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(I_{2,\mathrm{Re}}^{\mathrm{eq}}+I_{L,2,\mathrm{Re}}^t-$ (34)

$G_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t+B_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t)-(R_{12}+{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;})(V_{2,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(V_{2,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t)=0$

In formula: subscript " Re " and " Im " are real part and the imaginary parts of variable; Formula (31)-(34) can represent with the form of concentrating:

$f_m^t\left(x_s\right)=0$ m=1～4 (35)

In formula: $x_s=[I_{1,\mathrm{Re}}^{\mathrm{eq}},{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">I</figure-callout>}}_{1,\mathrm{Im}}^{\mathrm{eq}},{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{Re}}^{\mathrm{eq}},{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{Im}}^{\mathrm{eq}},{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;},{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;}],$

The expression formula of segmentation during for difference (31)-(34);

Therefore the least-squares estimation model of first stage outer net simplification Ward equivalent parameters is:

${\mathrm{<figure-callout\; id="1"\; label="J"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">J</figure-callout>}}_1=\min \underset{t=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{4}{\mathrm{\&Sigma;}}}{\left(f_m^t\left(x_s\right)\right)}^2$ (36)

Wherein N is the time hop count that institute's taken amount is surveyed data, unknown quantity x to be solved _sBe 6, measurement equation has 4N, consider the solvability of model, redundancy and the equivalent precision of measurement equation, and after a large amount of simulation analysis, N is taken as 3; Virtual measurement equation and the least-squares estimation model of S3-2 outer net expansion voltage source Ward Equivalent Circuit Parameter

Simplify the Ward equivalent circuit and adopt constant Injection Current to carry out the impact of equivalent outer net on the Intranet trend, can not truly reflect the voltage support effect of outer net to boundary node; Therefore, the equivalent Injection Current of simplifying in the Ward equivalent circuit is next equivalent with voltage source series impedance branch road, so just obtained expansion voltage source Ward equivalent circuit;

Consider that simplifying the Ward equivalent circuit and expanding voltage source Ward equivalent circuit is two kinds of equivalent circuits of same external electrical network, should have approximately equalised relation between its equivalent parameters; And based on the conventional Ward equivalent circuit of Intranet, can effectively estimate the former equivalent parameters, therefore, utilize two approximation relations between Equivalent Circuit Parameter, and using the constraint condition that the estimates of parameters of simplifying the Ward equivalent circuit estimates as expansion voltage source branch road Ward equivalent parameters, can improve the latter's ornamental;

Based on above-mentioned thought, can obtain restriction relation equation (37)-(42) of two Equivalent Circuit Parameters:

R′ ₁₂-R′′ ₁₂=0 (37)

X′ ₁₂-X′′ ₁₂=0 (38)

${\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">I</figure-callout>}}_{1,\mathrm{Re}}^t-I_{2,\mathrm{Re}}^{\mathrm{eq}}=0$ (39)

$I_{1,\mathrm{Im}}^t-I_{1,\mathrm{Im}}^{\mathrm{eq}}=0$ (40)

${\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{Re}}^t-I_{2,\mathrm{Re}}^{\mathrm{eq}}=0$ (41)

$I_{2,\mathrm{Im}}^t-I_{2,\mathrm{Im}}^{\mathrm{eq}}=0$ (42)

Electric current on the expansion branch road can be expressed as

Both sides are with taking advantage of Z _i, it is launched according to real part and imaginary part:

${\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t-R_1{I}_{1,\mathrm{Re}}^t+X_1{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">I</figure-callout>}}_{1,\mathrm{Im}}^t=0$ (43)

${\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t-R_1{I}_{1,\mathrm{Im}}^t-X_1{I}_{1,\mathrm{Re}}^t=0$ (44)

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,Re}^t-R_2{I}_{2,\mathrm{Re}}^t+X_2{I}_{2,\mathrm{Im}}^t=0$ (45)

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t-R_2{I}_{2,\mathrm{Im}}^t-X_2{I}_{2,\mathrm{Re}}^t=0$ (46)

In addition, in expansion voltage source Ward equivalent circuit, voltage source is considered as to the PV node, so its voltage magnitude is constant:

${\left({\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^t\right)}^2+{\left({\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^t\right)}^2-{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_1^2=0$ (47)

${\left({\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^t\right)}^2+{\left({\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^t\right)}^2-{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_2^2=0$ (48)

Similar with S3-1, same based on the KCL theorem in circuit theory, the electric current sum at boundary node place is 0; Can obtain following equation:

$({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}{R}_{12}-{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;}{X}_{12})(I_{1,\mathrm{Re}}^t+I_{L,1,\mathrm{Re}}^t-G_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t+B_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+R_{12}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})(I_{1,\mathrm{Im}}^t+I_{L,1,\mathrm{Im}}^t-$ (49)

$G_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t-B_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t)-({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;})({\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t)+({\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})({\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t)=0$

$({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}{R}_{12}-{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;}{X}_{12})(I_{1,\mathrm{Im}}^t+I_{L,1,\mathrm{Im}}^t-G_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t+B_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t)+({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+R_{12}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})(I_{1,\mathrm{Re}}^t+I_{L,1,\mathrm{Re}}^t-$ (50)

$G_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t+B_1{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;})({\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})({\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t)=0$

$({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}{R}_{12}-{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;}{X}_{12})(I_{2,\mathrm{Re}}^t+I_{L,2,\mathrm{Re}}^t-G_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t+B_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+R_{12}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})(I_{2,\mathrm{Im}}^t+I_{L,2,\mathrm{Im}}^t-$ (51)

$G_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t-B_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t)-({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;})({\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t)+({\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})({\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t)=0$

$({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}{R}_{12}-{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;}{X}_{12})(I_{2,\mathrm{Im}}^t+I_{L,2,\mathrm{Im}}^t-G_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t+B_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t)+({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+R_{12}{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})(I_{2,\mathrm{Re}}^t+I_{L,2,\mathrm{Re}}^t-$ (52)

$G_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t+B_2{\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;})({\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Im}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Im}}^t)-({\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})({\mathrm{<figure-callout\; id="2"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">V</figure-callout>}}_{2,\mathrm{Re}}^t-{\mathrm{<figure-callout\; id="1"\; label="V"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">V</figure-callout>}}_{1,\mathrm{Re}}^t)=0$

Formula (37)-(52) can represent with the form of concentrating:

$g_m^t\left(x_e\right)=0$ m=1～16 (S3)

In formula: $x_e=[I_{1,\mathrm{Re}}^t,I_{1,\mathrm{Im}}^t,I_{2,\mathrm{Re}}^t,I_{2,\mathrm{Im}}^t,{\mathrm{<figure-callout\; id="1"\; label="R"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_1,{\mathrm{<figure-callout\; id="1"\; label="X"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="R"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">R</figure-callout>}}_2,{\mathrm{<figure-callout\; id="2"\; label="X"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">X</figure-callout>}}_2,{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;},{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;},{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^{\&prime;\&prime;},{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{<figure-callout\; id="2"\; label="Im\; t\; E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Im</figure-callout>}}^t{E}_{}^{}{}_{2,\mathrm{Re}}^t,{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{<figure-callout\; id="1"\; label="Im\; t\; E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^t{E}_{}{}_1,E_2],$

Equation (39)-(52) of corresponding equation (37), (38) and different periods;

The least square model of outer net expansion voltage source Ward equivalent parameters is:

${\mathrm{<figure-callout\; id="2"\; label="J"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">J</figure-callout>}}_2=\min (\underset{\min }{\overset{2}{\mathrm{\&Sigma;}}}{\left(g_m\left(x_e\right)\right)}^2+\underset{t=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=3}{\overset{16}{\mathrm{\&Sigma;}}}{\left(g_m^t\left(x_e\right)\right)}^2)$ (54)

Unknown quantity x to be solved _eIndividual for (8+8N), measurement equation has 2+14N, is similar to the 1.2.1 joint, and N is taken as 3;

S4 calculates the external network equivalent circuit parameter

After completing the S3 step, can be based on the conventional Ward equivalent circuit of the Intranet that calculates in the S2 step and parameter thereof, priority is simplified the least square model of Ward equivalent circuit and outer net expansion voltage source Ward Equivalent Circuit Parameter according to formula (36) and the definite outer net of formula (54), utilize least square method to solve and obtain the equivalent parameters value that outer net is simplified Ward equivalent circuit and outer net expansion voltage source Ward equivalent circuit, and using the latter as final external network equivalent circuit; Specifically comprise the following steps:

The S4-1 outer net is simplified the calculating of Ward Equivalent Circuit Parameter

Based on conventional Ward equivalent circuit and the parameter thereof of 3 period Intranets, the outer net that can set up the formula of being similar to (36) is simplified the least square model of Ward Equivalent Circuit Parameter, and the initial value of getting outer net simplification Ward Equivalent Model parameter is x _s=[1,1,1,1,0.03,0.3], utilize least square method to try to achieve the parameter of equivalent circuit, comprises real part and the imaginary part of the equivalent Injection Current at boundary node place

And between boundary node interconnection resistance R ' ₁₂With reactance X ' ₁₂

The calculating of S4-2 outer net expansion voltage source Ward Equivalent Circuit Parameter

Be similar to S4-1, equally based on conventional Ward equivalent circuit and the parameter thereof of 3 period Intranets, can set up the least square model of the outer net expansion voltage source Ward Equivalent Circuit Parameter of the formula of being similar to (54), and estimation in S4-1 is obtained

And R ' ₁₂, X ' ₁₂In outer net expansion voltage source Ward equivalent circuit And R ' ' ₁₂, X ' ' ₁₂Initial value, the initial value of outer net expansion voltage source Ward Equivalent Circuit Parameter is like this $x_e=[I_{1,\mathrm{Re}}^{\mathrm{eq}},I_{1,\mathrm{Im}}^{\mathrm{eq}},I_{2,\mathrm{Re}}^{\mathrm{eq}},I_{2,\mathrm{Im}}^{\mathrm{eq}}\mathrm{0.03,0.3,0.03,0.3},{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;},{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;},\mathrm{1,0,1,0,1,0}];$ Then utilize least square method to try to achieve the parameter of outer net expansion voltage source Ward equivalent circuit, comprise the voltage magnitude E of virtual PV node ₁, E ₂, the resistance R of expansion branch road ₁, R ₂With reactance X ₁, X ₂And between boundary node interconnection resistance R ' ' ₁₂With reactance X ' ' ₁₂

Then based on the relation between power and voltage, electric current, calculate the meritorious P of exerting oneself of virtual PV node in outer net expansion voltage source Ward equivalent circuit ₁, P ₂So just, calculated the value of 10 parameters in outer net expansion voltage source Ward equivalent circuit; And will expand the final equivalent circuit of voltage source Ward equivalent circuit as outer net.

After the present invention adopts technique scheme, mainly contain following characteristics:

1,, without any need for topological structure and the status information of external electrical network, be applicable to solve the Static Equivalent problem of black box outer net.

2, the topological structure and the parameter thereof that only need Intranet, and the status data of 3 periods, condition is loosely easily obtained, and is suitable for extending to practical power systems.

3, the introduction of the conventional Ward equivalent circuit of Intranet, utilized more Intranet information, makes the estimated value of external network equivalent parameter more reasonable.

The accompanying drawing explanation

Fig. 1 is that outer net is simplified the Ward equivalent circuit;

Fig. 2 is outer net expansion voltage source Ward equivalent circuit;

Fig. 3 is the algorithm flow chart of outer net Static Equivalent method;

Fig. 4 is IEEE-39 node simulation system.

In Fig. 1 and Fig. 2, in the dotted line frame, be the conventional Ward equivalent circuit of Intranet, wherein

The equivalent injecting power at boundary node place in the conventional Ward equivalent circuit of Intranet, Z ₁₂=R ₁₂+ jX ₁₂The impedance of interconnection between boundary node, Y ₁=G ₁+ jB ₁And Y ₂=G ₂+ jB ₂It is respectively the admittance over the ground at boundary node place.

Fig. 1 dotted line frame is simplified the Ward equivalent circuit for outer net outward, wherein

For outer net, simplify the equivalent electric current at boundary node place in the Ward equivalent circuit, Z ' ₁₂=R ' ₁₂+ jX ' ₁₂Impedance for interconnection between boundary node.

Fig. 2 dotted line frame is outer net expansion voltage source Ward equivalent circuit, wherein E outward ₁, E ₂For the voltage magnitude of virtual PV node in outer net expansion voltage source Ward equivalent circuit, P ₁, P ₂That the meritorious of virtual PV node exerted oneself; Z ' ' ₁₂=R ' ' ₁₂+ jX ' ' ₁₂For the impedance of interconnection between boundary node, Z ₁=R ₁+ jX ₁And Z ₂=R ₂+ jX ₂For the impedance of expansion branch road, For the electric current on the expansion branch road.

Embodiment

Below in conjunction with embodiment, further illustrate the present invention.

Embodiment

Below using IEEE-39 node simulation system as research object, to proved of the present invention:

(1) input basic data

The basic data of input IEEE-39 node simulation system, comprise its topological structure and parameter value thereof, then simulating the burden with power of the whole network node and load or burden without work increases in 0.3% ratio, meritorious the exerting oneself of the whole network PV node increases in proportion by 0.3%, imbalance power is shared naturally by balance node and PV node, with this real measured data that obtains 3 periods, comprise voltage magnitude and the phase angle of node, and the performance number of branch road two ends transmission.And get node 3 and 17 as boundary node, node 1,2,25-30 and 37-39 is as the outer net node, and all the other are the Intranet node.

(2) calculate conventional Ward equivalent circuit and the parameter thereof of Intranet

1. equivalent branch road parameter asks for

After completing (1) step, can be calculated by formula (28) the Equivalent admittance matrix at boundary node place:

$Y_{\mathrm{BB}}^{\&prime;}=\left[\begin{array}{cc}10.593-j124.74& -4.918+j60.869\\ -4.918+j60.869& 9.2308-j115.65\end{array}\right]$ (55)

By the transadmittance between formula (55) boundary node 3 as can be known and 17, be Y _3,17=-4.918+j60.869, observe between IEEE-39 node simulation system boundary node 3 as can be known and 17 and to be originally not having circuit, therefore, this transadmittance value is the negative inverse of interconnection impedance between boundary node 3 and 17 in the conventional Ward equivalent circuit of Intranet, can obtain thus interconnection impedance in the conventional Ward equivalent circuit of Intranet and be:

Z ₁₂=-1/Y _3,17=0.001319+j0.01632 (56)

Self-admittance by formula (55) boundary node as can be known 3 places is Y _3,3=10.593-j124.74, the self-admittance of observing IEEE-39 node simulation system boundary node 3 as can be known by the equivalence at the interconnection admittance between boundary node 3,17 and boundary node 3 places in the admittance of former power network line 3-2 and ground capacitance part and the conventional Ward equivalent circuit of Intranet over the ground admittance form, therefore, remove Y _3,3Interconnection admittance part between the admittance of middle circuit 3-2,3-18 and ground capacitance part and boundary node 3,17 can obtain the equivalence admittance over the ground at boundary node 3 places in the conventional Ward equivalent circuit of Intranet:

Y ₁=Y _3,3-y _3,2-y ₃₂₀-y _3,17=(10.593-j124.74)-(5.6596-j65.7379) (57)

-j0.1286-(4.918-j60.869)=0.0154-j1.7383

In like manner can obtain the equivalence admittance over the ground at boundary node 17 places:

Y ₂=Y _17,17-y _17,27-y _17,27,0-y _3,17=(9.2308-j115.65)-(4.3192-j57.4789) (58)

-j0.1608-(4.918-j60.869)=-0.0064-j2.5371

2. equivalent injecting power asks for

By formula (29), can be in the hope of boundary node 3 and the equivalent injecting power of 3 periods of 17 places:

${\tilde{S}}_{L,1}^1=-494.980-j311.571$

${\tilde{S}}_{L,1}^2=-496.408-j311.971$ (59)

${\tilde{S}}_{L,1}^3=-497.836-j312.372$

${\tilde{S}}_{L,2}^1=132.169-j295.259$

${\tilde{S}}_{L,2}^2=132.565-j295.178$ (60)

${\tilde{S}}_{L,2}^3=132.961-j295.096$

In formula: subscript " 1,2,3 " represents respectively the 1st, 2,3 periods.

(3) set up virtual measurement equation and the least-squares estimation model of external network equivalent circuit parameter

After completing (2) step, can set up virtual measurement equation and the least-squares estimation model of external network equivalent circuit parameter.

1. outer net is simplified virtual measurement equation and the least-squares estimation model of Ward Equivalent Circuit Parameter

According to formula (31)-(34), based on branch road parameter and the equivalent injecting power of the 1st period of the conventional Ward equivalent circuit of the Intranet of trying to achieve in (2) step, the outer net simplification Ward Equivalent Circuit Parameter virtual measurement equation that can set up the 1st period is as follows: $(0.00132{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}-0.0163{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(I_{1,\mathrm{Re}}^{\mathrm{eq}}-3.9435)-(0.0163{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}+0.00132X_{12}^{\&prime;})(I_{1,\mathrm{Im}}^{\mathrm{eq}}+5.7675)$ (61)

+0.0075(0.00132+R′ ₁₂)-0.0181(0.0163+X′ ₁₂)=0

$(0.00132{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}-0.01632{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(I_{1,\mathrm{Im}}^{\mathrm{eq}}+2.0511)+(0.01632{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}+0.00132X_{12}^{\&prime;})(I_{1,\mathrm{Re}}^{\mathrm{eq}}-3.9436)$ (62)

-0.0006(0.00132+R′ ₁₂)-0.0004(0.01632+X′ ₁₂)=0

$(0.00132{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}-0.01632{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(I_{2,\mathrm{Re}}^{\mathrm{eq}}+1.3090)-(0.01632{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}+0.00132X_{12}^{\&prime;})(I_{2,\mathrm{Im}}^{\mathrm{eq}}+0.04142)$ (63)

-0.0866(0.00132+R′ ₁₂)+0.0181(0.01632+X′ ₁₂)=0

$({0.00132\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}-0.01632{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;})(I_{2,\mathrm{Im}}^{\mathrm{eq}}+5.2244)+(0.01632{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;}+0.00132X_{12}^{\&prime;})(I_{2,\mathrm{Re}}^{\mathrm{eq}}+1.3090)$ (64)

-0.0181(0.00132+R′ ₁₂)+0.0037(0.01632+X′ ₁₂)=0

In like manner, according to branch road parameter and the equivalent injecting power of the 2nd, 3 periods of the conventional Ward equivalent circuit of Intranet, can set up the outer net of the 2nd, 3 periods and simplify Ward Equivalent Circuit Parameter virtual measurement equation, because process is similar, not repeat them here.

According to the outer net of 3 periods of setting up, simplify Ward Equivalent Circuit Parameter virtual measurement equation, the least-squares estimation model that can set up the formula of being similar to (36) is as follows:

$(0.0118+0.0008{\left(I_{1,\mathrm{Im}}^{\mathrm{eq}}\right)}^2)+0.0027I_{1,\mathrm{Im}}^{\mathrm{eq}}-0.0055{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">I</figure-callout>}}_{1,\mathrm{Re}}^{\mathrm{eq}}+0.0008{\left(I_{1,\mathrm{Re}}^{\mathrm{eq}}\right)}^2+$

$0.0007I_{2,\mathrm{Im}}^{\mathrm{eq}}+0.0008{\left(I_{2,\mathrm{Im}}^{\mathrm{eq}}\right)}^2+0.0003I_{2,\mathrm{Re}}^{\mathrm{eq}}+0.0008{\left(I_{2,\mathrm{Re}}^{\mathrm{eq}}\right)}^2{\left(R_{12}\right)}^2-0.0001X_{12}$ (65)

$+(0.0119+0.0027I_{1,\mathrm{Im}}^{\mathrm{eq}}+0.0008{\left(I_{1,\mathrm{Im}}^{\mathrm{eq}}\right)}^2-0.0055{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">I</figure-callout>}}_{1,\mathrm{Re}}^{\mathrm{eq}}+0.0008{\left(I_{1,\mathrm{Re}}^{\mathrm{eq}}\right)}^2+$

${0.0007I}_{2,\mathrm{Im}}^{\mathrm{eq}}+0.0008{\left(I_{2,\mathrm{Im}}^{\mathrm{eq}}\right)}^2+0.0003I_{2,\mathrm{Re}}^{\mathrm{eq}}+0.0008{\left(I_{2,\mathrm{Re}}^{\mathrm{eq}}\right)}^2){\left(X_{12}\right)}^2$

2. outer net is expanded virtual measurement equation and the least-squares estimation model of voltage source Ward Equivalent Circuit Parameter

Be similar to 1., according to formula (37)-(52), based on branch road parameter and the equivalent injecting power of the 1st period of the conventional Ward equivalent circuit of the Intranet of trying to achieve in (2) step, the outer net expansion voltage source Ward Equivalent Circuit Parameter virtual measurement equation that can set up the 1st period is as follows:

R′′ ₁₂-0.7699=0 (66)

X′′ ₁₂-2.0007=0 (67)

$I_{1,\mathrm{Re}}^1-3.4116=0$ (68)

$I_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+1.6928=0$ (69)

$I_{2,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+0.1608=0$ (70)

$I_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+0.4085=0$ (71)

${\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^1-1.0124-R_1{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+X_1{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1=0$ (72)

${\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+0.1748-R_1{I}_{1,\mathrm{Im}}^1-X_1{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1=0$ (73)

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^1-1.0199-R_2{I}_{2,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+X_2{I}_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1=0$ (74)

${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+0.1567-R_2{I}_{2,\mathrm{Im}}^1-X_2{I}_{2,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1=0$ (75)

${\left({\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^1\right)}^2+{\left({\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^1\right)}^2-{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_1^2=0$ (76)

${\left({\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^1\right)}^2+{\left({\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^1\right)}^2-{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_2^2=0$ (77)

$({0.0013\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}-0.0163{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})(I_{1,\mathrm{Re}}^1-4.5514)-({0.0163\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}+{0.0013X}_{12}^{\&prime;\&prime;})(I_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+2.1500)$ (78)

-0.0075(0.0013+R′′ ₁₂)-0.0181(0.0163+X′′ ₁₂)=0

$({0.0013\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}-0.0163{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})(I_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+4.3975)+({0.0163\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}+0.0013X_{12}^{\&prime;\&prime;})(I_{1,\mathrm{Re}}^1-4.5514)$ (79)

+0.0181(0.0013+R′′ ₁₂)+0.0075(0.0163+X′′ ₁₂)=0

$({0.0013\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}-0.0163{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})(I_{2,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+1.3090)-({0.0163\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}+0.0013X_{12}^{\&prime;\&prime;})(I_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+0.0413)$ (80)

-0.0075(0.0013+R′′ ₁₂)+0.0181(0.0163+X′′ ₁₂)=0

$({0.0013\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}-0.0163{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;\&prime;})(I_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+4.4069)+(0.0163{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;\&prime;}+0.0013X_{12}^{\&prime;\&prime;})(I_{2,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+1.4345)$ (81)

-0.0181(0.0013+R′′ ₁₂)-0.0075(0.0163+X′′ ₁₂)=0

In like manner, branch road parameter and the equivalent injecting power of the 2nd, 3 periods according to the conventional Ward equivalent circuit of Intranet, can set up the outer net expansion voltage source Ward Equivalent Circuit Parameter virtual measurement equation of the 2nd, 3 periods, because process is similar, not repeat them here.

According to the outer net of 3 periods of setting up, simplify Ward Equivalent Circuit Parameter virtual measurement equation, the least-squares estimation model that can set up the formula of being similar to (54) is as follows:

$50.4488+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^1(0.3496+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^1)+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^2(0.3508+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^2)+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^3(0.3520+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^3)+$

${\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^1(-2.0247+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^1)+(-2.0239+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^2){\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{<figure-callout\; id="2"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Re</figure-callout>}}^2+(-2.032+{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^3)+3{\left(E_2\right)}^2+$

${0.3133\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+{0.3144\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{<figure-callout\; id="2"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Im</figure-callout>}}^2+{0.3155\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^3-{2.0397\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^1-2.0380{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^2-2.0382{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{<figure-callout\; id="3"\; label="Re"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^3$

$+{\left(I_{1,\mathrm{Im}}^1\right)}^2+{\left(I_{1,\mathrm{Im}}^2\right)}^2+{\left(I_{1,\mathrm{Im}}^3\right)}^2+3.3856(I_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+I_{1,\mathrm{<figure-callout\; id="2"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Im</figure-callout>}}^2+I_{1,\mathrm{Im}}^3)+{\left(I_{1,\mathrm{Re}}^1\right)}^2{+\left(I_{1,\mathrm{Re}}^2\right)}^2+{\left(I_{1,\mathrm{Re}}^3\right)}^2$

$-6.8232(I_{1,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+I_{1,\mathrm{<figure-callout\; id="2"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Re</figure-callout>}}^2+I_{1,\mathrm{Re}}^3)+{\left(I_{2,\mathrm{Im}}^1\right)}^2+{\left(I_{2,\mathrm{Im}}^2\right)}^2+{\left(I_{2,\mathrm{Im}}^3\right)}^2+0.817(I_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1{+I}_{2,\mathrm{<figure-callout\; id="2"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Im</figure-callout>}}^2+I_{2,\mathrm{Im}}^3)$

${+\left(I_{2,\mathrm{Re}}^1\right)}^2+{\left(I_{2,\mathrm{Re}}^2\right)}^2+{\left(I_{2,\mathrm{Re}}^3\right)}^2+0.3216(I_{2,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+I_{2,\mathrm{<figure-callout\; id="2"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Re</figure-callout>}}^2+I_{2,\mathrm{Re}}^3)-0.3496R_1{I}_{1,\mathrm{Im}}^1-{2R}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^1{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1$

$-0.3508R_1{I}_{1,\mathrm{Im}}^2-{2R}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^2{I}_{1,\mathrm{Im}}^2-0.3520{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">I</figure-callout>}}_{1,\mathrm{Im}}^3-2R_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^3{I}_{1,\mathrm{<figure-callout\; id="3"\; label="Im"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^3+{2R}_1{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+2.0240R_1{I}_{1,\mathrm{<figure-callout\; id="2"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Re</figure-callout>}}^2$

${-2R}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^2{I}_{1,\mathrm{<figure-callout\; id="2"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Re</figure-callout>}}^2+2.0232R_1{I}_{1,\mathrm{Re}}^3-{2R}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^3{I}_{1,\mathrm{<figure-callout\; id="3"\; label="Re"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^3+{\left(R_1\right)}^2({\left(I_{1,\mathrm{Re}}^1\right)}^2+{\left(I_{1,\mathrm{Re}}^2\right)}^2+{\left(I_{1,\mathrm{Re}}^3\right)}^2+$

${\left(I_{1,\mathrm{Im}}^1\right)}^2+{\left(I_{1,\mathrm{Im}}^2\right)}^1+{\left(I_{1,\mathrm{Im}}^3\right)}^2)+0.0119{\left(R_{12}\right)}^2+0.0009{\left(R_{12}\right)}^2+0.0012{\left(R_{12}\right)}^2{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+$

$0.0003{\left(R_{12}\right)}^2{\left(I_{1,\mathrm{Im}}^2\right)}^2+0.0009{\left(R_{12}\right)}^2{I}_{1,\mathrm{<figure-callout\; id="2"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Im</figure-callout>}}^2+0.0003{\left(R_{12}\right)}^2{\left(I_{1,\mathrm{Im}}^2\right)}^2+0.0009{\left(R_{12}\right)}^2{I}_{1,\mathrm{<figure-callout\; id="3"\; label="Im"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^3$

$+0.0003{\left(R_{12}\right)}^2{\left(I_{1,\mathrm{Im}}^3\right)}^2-0.0018{\left(R_{12}\right)}^2{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+0.0003{\left(R_{12}\right)}^2{\left(I_{1,\mathrm{Re}}^1\right)}^2-0.0018{\left(R_{12}\right)}^2{I}_{1,\mathrm{<figure-callout\; id="2"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Re</figure-callout>}}^2$

$+0.0003{\left(R_{12}\right)}^2{\left(I_{1,\mathrm{Re}}^2\right)}^2-0.0018{\left(R_{12}\right)}^2{I}_{1,\mathrm{<figure-callout\; id="3"\; label="Re"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^3+0.0003{\left(R_{12}\right)}^2{\left(I_{1,\mathrm{Re}}^3\right)}^2+0.0003{\left(R_{12}\right)}^2{I}_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1$

$+0.0003{\left(R_{12}\right)}^2{\left(I_{2,\mathrm{Im}}^1\right)}^2+0.0002{\left(R_{12}\right)}^2{I}_{2,\mathrm{<figure-callout\; id="2"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Re</figure-callout>}}^2+0.0003{\left(R_{12}\right)}^2{\left(I_{2,\mathrm{Im}}^2\right)}^2+0.0002{\left(R_{12}\right)}^2{I}_{2,\mathrm{<figure-callout\; id="3"\; label="Im"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^3$

$+0.0003{\left(R_{12}\right)}^2{\left(I_{2,\mathrm{Im}}^3\right)}^2+0.0003{\left(R_{12}\right)}^2{\left(I_{2,\mathrm{Re}}^1\right)}^2+0.0003{\left(R_{12}\right)}^2{\left(I_{2,\mathrm{Re}}^2\right)}^2+$

$0.0003{\left(R_{12}\right)}^2{\left(I_{2,\mathrm{Im}}^3\right)}^2+0.3133R_{12}{I}_{2,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+0.0003{\left(R_{12}\right)}^2{\left(I_{2,\mathrm{Re}}^3\right)}^2-0.3133R_2{I}_{2,\mathrm{Im}}^1-$

${2R}_2{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^1{I}_{2,\mathrm{Im}}^1-0.3144R_2{I}_{2,\mathrm{Im}}^2-{2R}_2{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^2{I}_{2,\mathrm{Im}}^2-0.3155R_2{I}_{2,\mathrm{Im}}^3-{2R}_2{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^3{I}_{2,\mathrm{<figure-callout\; id="3"\; label="Im"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^3+$

${2.0397R}_2{I}_{2,\mathrm{Re}}^1-{2R}_2{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^1{I}_{2,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1+{2.0390R}_2{I}_{2,\mathrm{Re}}^2-{2R}_2{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^2{I}_{2,\mathrm{<figure-callout\; id="2"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Re</figure-callout>}}^2+{2.0382R}_2{I}_{2,\mathrm{<figure-callout\; id="3"\; label="Re"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^3$

$-{2R}_2{\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^3{I}_{2,\mathrm{<figure-callout\; id="3"\; label="Re"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^3+{\left(R_2\right)}^2({\left(I_{2,\mathrm{Re}}^1\right)}^2+{\left(I_{2,\mathrm{Re}}^2\right)}^2+{\left(I_{2,\mathrm{Re}}^3\right)}^3+{\left(I_{2,\mathrm{Im}}^1\right)}^2+{\left(I_{2,\mathrm{Im}}^2\right)}^2+{\left(I_{2,\mathrm{Im}}^3\right)}^2)$

${-2.0247X}_1{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+{2X}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^1{I}_{1,\mathrm{Im}}^1-2.0240X_1{I}_{1,\mathrm{<figure-callout\; id="2"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">Im</figure-callout>}}^2+{2X}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^2{I}_{1,\mathrm{Im}}^2-2.0232X_1{I}_{1,\mathrm{<figure-callout\; id="3"\; label="Im"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^3+$

${2X}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Re}}^3{I}_{1,\mathrm{Im}}^3-0.3496X_1{I}_{1,\mathrm{Re}}^1-0.3508X_1{I}_{1,\mathrm{Re}}^2-{2X}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^2{I}_{1,\mathrm{Re}}^2-0.3520X_1{I}_{1,\mathrm{Re}}^3-$

${2X}_1{\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^3{I}_{1,\mathrm{<figure-callout\; id="3"\; label="Im"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^3+{\left(X_1\right)}^2({\left(I_{1,\mathrm{Re}}^1\right)}^2+{\left(I_{1,\mathrm{Re}}^2\right)}^2+{\left(I_{1,\mathrm{Re}}^3\right)}^2+{\left(I_{1,\mathrm{Im}}^1\right)}^2+{\left(I_{1,\mathrm{Im}}^2\right)}^2{\left(I_{1,\mathrm{Im}}^3\right)}^2)$

${-2\mathrm{<figure-callout\; id="1"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">E</figure-callout>}}_{1,\mathrm{Im}}^1(R_1{I}_{1,\mathrm{<figure-callout\; id="1"\; label="Im"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^1+X_1{I}_{1,\mathrm{Re}}^1)-0.0001{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}+0.0119{\left(X_{12}\right)}^2+0.0009I_{1,\mathrm{Im}}^1{\left(X_{12}\right)}^2+$

$0.0003{\left(I_{1,\mathrm{Im}}^1\right)}^2{\left(X_{12}\right)}^2+0.0009I_{1,\mathrm{Im}}^2{\left(X_{12}\right)}^2+0.0003{\left(I_{1,\mathrm{Im}}^2\right)}^2{\left(X_{12}\right)}^2+$

$0.0009I_{1,\mathrm{Im}}^3{\left(X_{12}\right)}^2+0.0003{\left(I_{1,\mathrm{Im}}^3\right)}^2{\left(X_{12}\right)}^2-0.0018{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">I</figure-callout>}}_{1,\mathrm{Re}}^1{\left(X_{12}\right)}^2+$

$0.0003{\left(I_{1,\mathrm{Re}}^1\right)}^2{\left(X_{12}\right)}^2-0.0018{\mathrm{<figure-callout\; id="1"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">I</figure-callout>}}_{1,\mathrm{Re}}^2{\left(X_{12}\right)}^2+0.0003{\left(I_{1,\mathrm{Re}}^2\right)}^2{\left(X_{12}\right)}^2-$

${0.0018I}_{1,\mathrm{Re}}^2{\left(X_{12}\right)}^2+0.0003{\left(I_{1,\mathrm{Re}}^3\right)}^2{\left(X_{12}\right)}^2+0.0002I_{2,\mathrm{Im}}^1{\left(X_{12}\right)}^2+$

$0.0003{\left(I_{2,\mathrm{Im}}^1\right)}^2{\left(X_{12}\right)}^2+0.0002I_{2,\mathrm{Im}}^2{\left(X_{12}\right)}^2+0.0003{\left(I_{2,\mathrm{Im}}^2\right)}^2{\left(X_{12}\right)}^2+$

$0.0002I_{2,\mathrm{Im}}^3{\left(X_{12}\right)}^2+0.0003{\left(I_{2,\mathrm{Im}}^3\right)}^2{\left(X_{12}\right)}^2+0.0003{\left(I_{2,\mathrm{Re}}^1\right)}^2{\left(X_{12}\right)}^2+$

$0.0003{\left(I_{2,\mathrm{Re}}^2\right)}^2{\left(X_{12}\right)}^2+0.0003{\left(I_{2,\mathrm{Re}}^3\right)}^2{\left(X_{12}\right)}^2+I_{2,\mathrm{Im}}^1({2\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^1-2.0397)$ (82)

$+X_2(I_{2,\mathrm{Im}}^2({2\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^2-2.0390)-2.0382{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{<figure-callout\; id="3"\; label="Im"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">Im</figure-callout>}}^3+{2\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Re}}^3{I}_{2,\mathrm{Im}}^3-0.3133{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{<figure-callout\; id="1"\; label="Re"\; filenames="HDA00003457700900011.png,HDA00003457700900021.png"\; state="\{\{state\}\}">Re</figure-callout>}}^1$

${-2\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^1{I}_{2,\mathrm{Re}}^1-0.3144{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{Re}}^2-{2\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^2{I}_{2,\mathrm{Re}}^2-0.3155{\mathrm{<figure-callout\; id="2"\; label="I"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">I</figure-callout>}}_{2,\mathrm{Re}}^3-{2\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="HDA00003457700900011.png,HDA00003457700900012.png"\; state="\{\{state\}\}">E</figure-callout>}}_{2,\mathrm{Im}}^3{I}_{2,\mathrm{Re}}^3)+$

${\left(X_2\right)}^2({\left(I_{2,\mathrm{Re}}^1\right)}^2+{\left(I_{2,\mathrm{Re}}^2\right)}^2+{\left(I_{2,\mathrm{Re}}^3\right)}^2+{\left(I_{2,\mathrm{Im}}^1\right)}^2+{\left(I_{2,\mathrm{Im}}^2\right)}^2+{\left(I_{2,\mathrm{Im}}^3\right)}^2)$

(4) calculate the external network equivalent circuit parameter

After completing (3) step, can be based on the conventional Ward equivalent circuit and the parameter thereof that in (2) step, calculate Intranet, the outer net of successively determining according to formula (65) and formula (82) is simplified the least square model of Ward equivalent circuit and outer net expansion voltage source Ward Equivalent Circuit Parameter, utilizes least square method to solve to obtain outer net to simplify the equivalent parameters value that Ward equivalent circuit and outer net are expanded voltage source Ward equivalent circuit.Concrete steps are as follows:

1. outer net is simplified the calculating of Ward Equivalent Circuit Parameter

The real part of obtaining current and the initial value of imaginary part are 1, and the initial value of resistance and reactance is respectively 0.03 and 0.3, and the initial value of outer net simplification Ward Equivalent Circuit Parameter is x like this _s=[1,1,1,1,0.03,0.3], utilize least square method to calculate and can obtain the parameter value that outer net is simplified the Ward equivalent circuit.Wherein, the resistance R of interconnection between boundary node ' ₁₂=0.7700 and reactance X ' ₁₂=2.0007, real part and the imaginary part of the equivalent Injection Current at boundary node place are respectively $I_{1,\mathrm{Re}}^{\mathrm{eq}}=3.4116,I_{1,\mathrm{Im}}^{\mathrm{eq}}=-1.6928,I_{2,\mathrm{Re}}^{\mathrm{eq}}=-0.1608,I_{2,\mathrm{Im}}^{\mathrm{eq}}=-0.4085.$

2. the calculating of outer net expansion voltage source Ward Equivalent Circuit Parameter

By what estimate in 1. to obtain

And R ' ₁₂, X ' ₁₂In outer net expansion voltage source Ward equivalent circuit

And R ' ' ₁₂, X ' ' ₁₂Initial value, the initial value of power taking resistance and reactance is respectively 0.03 and 0.3, the real part of voltage and the initial value of imaginary part are respectively 1 and 0, the initial value of voltage magnitude is made as 1, outer net is expanded the initial value of voltage source Ward Equivalent Circuit Parameter and is like this $x_e=[I_{1,\mathrm{Re}}^{\mathrm{eq}},I_{1,\mathrm{Im}}^{\mathrm{eq}},I_{2,\mathrm{Re}}^{\mathrm{eq}},I_{2,\mathrm{Im}}^{\mathrm{eq}}\mathrm{0.03,0.3,0.03,0.3},{\mathrm{<figure-callout\; id="12"\; label="R"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">R</figure-callout>}}_{12}^{\&prime;},{\mathrm{<figure-callout\; id="12"\; label="X"\; filenames="HDA00003457700900021.png"\; state="\{\{state\}\}">X</figure-callout>}}_{12}^{\&prime;},\mathrm{1,0,1,0,1,0}].$ Utilize least square method to try to achieve the parameter of outer net expansion voltage source Ward equivalent circuit, comprise the voltage magnitude E of virtual PV node ₁=1.1659, E ₂=1.0617, the resistance R of expansion branch road ₁=0.02211, R ₂=0.0609 and reactance X ₁=0.0455, X ₂=0.0829 and boundary node between interconnection resistance R ' ' ₁₂=0.7698 and reactance X ' ' ₁₂=2.0002.

Then based on the relation between power and voltage, electric current, calculate the meritorious P of exerting oneself of virtual PV node in outer net expansion voltage source Ward equivalent circuit ₁=3.7497, P ₂=-0.1000.So just, calculate the value of outer net expansion voltage source Ward equivalent circuit medium value parameter, and will expand the final equivalent circuit of voltage source Ward equivalent circuit as outer net.

Experiment effect

After adopting the external network equivalent method for parameter estimation of the present embodiment, Intranet is carried out to the static security stability analysis, by it and existingly only based on the measurement information at boundary node place, estimate that the method for external network equivalent parameter contrasts, thus the validity of checking the technology of the present invention.

The evaluation criteria of systematic error: when carrying out the static security evaluation, whether forecast accident can threaten to the safe and stable operation of system, not only relevant with the absolute error of trend calculating, and relevant with the electric pressure of circuit, in general, the power swing amplitude that the circuit that electric pressure is higher can bear is larger.Similar state estimation qualification rate index, adopt the average of circuit to gain merit and idle error on the safe side (ε _{Ave_P}And ε _{Ave_Q}) and maximum meritorious and idle error on the safe side (ε _{Max_P}And ε _{Max_Q}) two kinds of indexs pass judgment on the precision that this paper puies forward outer net Static Equivalent method.

Average gaining merit is defined as with idle error on the safe side:

(83)

In formula: P _l(Q _l) and

Be respectively true value and the estimated value of Intranet circuit meritorious (idle),

For all line set of Intranet, l is the Intranet circuit number, n _lNumber for the Intranet circuit.S _BaseFor the power reference value, for the 220KV circuit, get S _Base=304MVA, for 500KV, get S _Base=1082MVA.

Maximum gaining merit is defined as with idle error on the safe side:

(84)

In formula: max represents the expression formula maximizing, and the implication of other variable symbol is similar to formula (83).

Outer net node in deletion IEEE-39 node system, and the external network equivalent circuit that calculates more than inciting somebody to action splices to Intranet, changing the Intranet load and cut-offfing in two kinds of situations of Intranet circuit, relatively the precision of this paper invention technology and existing outer net Static Equivalent method for parameter estimation.

Table 1 has provided under 3 kinds of load levels, the systematic error of two kinds of outer net static estimation methods.Wherein, the corresponding ground state trend of 1 load level, the node load that load level 1.02(1.05) is corresponding is the 1.02(1.05 of ground state trend load) doubly.As seen from Table 1, the systematic error of two kinds of methods is all little, but in general, the systematic error of the inventive method is less than existing method.

The systematic error of two kinds of methods under table 1 different load level

Table 2 has provided the systematic error result of three kinds of branch breaking schemes.Therefrom visible, under the branch breaking condition, the systematic error of two kinds of methods all increases to some extent, and while especially cut-offfing branch road 14-15, existing methodical maximum active power security error has reached 5.9%, and the inventive method has also reached 1.8%.In general, the equivalent precision of the inventive method is equally higher than existing.

The systematic error of two kinds of methods during table 2 line disconnection

Claims (1)

1. outer net Static Equivalent method based on the conventional Ward equivalent circuit of Intranet is characterized in that comprising the following steps:

S1 inputs basic data

At first input the basic data of Intranet, comprise topological structure and the measurement information of 3 periods of Intranet;

Topological structure basic data: comprise node serial number, node type, node zone of living in, electricity is led and susceptance over the ground;

The basic data of generator: comprise node serial number, meritorious and idle exerting oneself;

The basic data of circuit: comprise circuit first and last node serial number, resistance, reactance, susceptance;

The basic data of transformer: comprise that place, transformer both sides node serial number, resistance, reactance, electricity are led, susceptance, no-load voltage ratio;

The Intranet measurement information of 3 periods: comprise under the condition of a plurality of different conditions the voltage magnitude of node and phase angle, injecting power, the power that transmit at circuit and transformer two ends;

S2 calculates conventional Ward equivalent circuit and the parameter thereof of Intranet

Asking for of the equivalent branch road parameter of S1-1

Based on the parameter of the topological structure of Intranet and circuit, transformer, according to the definition of self-admittance and transadmittance, ask for the bus admittance matrix of Intranet;

Utilize Gaussian elimination method, the contact node that keeps Intranet and outer net is boundary node, and other nodes of cancellation Intranet obtain only containing the Equivalent admittance matrix of boundary node, suc as formula (28), represent:

Y′ _BB=Y _BB-Y _BIY _II ^-1Y _IB （1）

In formula: Y is bus admittance matrix, and subscript B represents boundary node, and subscript I represents other Intranet nodes except the node of border;

Definition by admittance matrix is as can be known, Y ' _BBTo be obtained by the parameter of branch road over the ground of boundary and the route parameter calculation that is connected with boundary node; Therefore, when known admittance matrix, can obtain conventional Ward Equivalent Circuit Parameter value by Extrapolation;

In like manner, calculate equivalence branch road Y over the ground ₁And Y ₂Parameter value;

Asking for of the equivalent injecting power of S1-2

Based on the status information of a plurality of period Intranets, the injecting power of cancellation other nodes of Intranet except the node of border, obtain the equivalent injecting powers of a plurality of periods at boundary node place, shown in (29):

${\tilde{S}}_{\mathrm{eq}}=\left(\mathrm{diag}{\left({\stackrel{.}{V}}_B\right)}^{*}\right)Y_{\mathrm{BI}}{Y}_{\mathrm{II}}^{-1}{\left(\frac{{\tilde{S}}_I}{{\stackrel{.}{V}}_I}\right)}^{*}$ (2)

In formula:

Expression node injecting power,

The voltage of expression node;

Be the injecting power of Intranet through conversion assignments to the equivalent injecting power on boundary node, by it, can calculate in Fig. 1 in the dotted line frame the equivalent injecting power at boundary node place in conventional Ward equivalent circuit

With

By two above steps, namely can calculate the parameter value of the whole elements of conventional Ward equivalent circuit;

S3 sets up virtual measurement equation and the least-squares estimation model of external network equivalent circuit parameter

After completing the S2 step, based on outer net, simplify the restriction relation between Ward Equivalent Circuit Parameter and outer net expansion voltage source Ward Equivalent Circuit Parameter and the conventional Ward Equivalent Circuit Parameter of Intranet respectively, set up outer net and simplify the virtual measurement equation of Ward Equivalent Circuit Parameter and outer net expansion voltage source Ward Equivalent Circuit Parameter, and further obtain the least square model of two external network equivalent circuit parameters, finally can solve the parameter value that obtains two external network equivalent circuit based on least square method, and using the latter as final external network equivalent circuit;

The S3-1 outer net is simplified virtual measurement equation and the least-squares estimation model of Ward Equivalent Circuit Parameter

Based on the KCL theorem in circuit theory, the Injection Current sum at boundary node place is zero; Take boundary node 1 as example, the electric current of Inside and outside network injection boundary node is opposite number each other, that is:

${\stackrel{.}{I}}_1^{\mathrm{eq}}-\frac{{\stackrel{.}{V}}_1^t-{\stackrel{.}{V}}_2^t}{Z_{12}^{\&prime;}}=\frac{{\stackrel{.}{V}}_1^t-{\stackrel{.}{V}}_2^t}{Z_{12}}+Y_1{\stackrel{.}{V}}_1^t-{\stackrel{.}{I}}_{L,1}^t$ (3)

In formula: For outer net, simplify the equivalent current source of Ward equivalent circuit, Z ' ₁₂For outer net is simplified the impedance of interconnection between boundary node in the Ward equivalent circuit,

Be the electric current representation of the equivalent injecting power of boundary node, subscript " t " (t=1,2,3, m are the time hop count of required Intranet metric data) represents period t; The numbering of subscript " 1,2 " expression boundary node;

Z ' is multiply by on the following formula both sides simultaneously ₁₂Z ₁₂And launch according to real part and imaginary part, can obtain formula (31), (32), similar above thinking is analyzed and can be obtained formula (33), (34) boundary node 2:

$(R_{12}^{\&prime;}{R}_{12}-X_{12}^{\&prime;}{X}_{12})(I_{1,\mathrm{Re}}^{\mathrm{eq}}+I_{L,1,\mathrm{Re}}^t-G_1{V}_{1,\mathrm{Re}}^t+B_1{V}_{1,\mathrm{Im}}^t)-(R_{12}^{\&prime;}{X}_{12}+R_{12}{X}_{12}^{\&prime;})(I_{1,\mathrm{Im}}^{\mathrm{eq}}+I_{L,1,\mathrm{Im}}^t-$ (4)

$G_1{V}_{1,\mathrm{Im}}^t-B_1{V}_{1,\mathrm{Re}}^t)-(R_{12}+R_{12}^{\&prime;})(V_{1,\mathrm{Re}}^t-V_{2,\mathrm{Re}}^t)+(X_{12}+X_{12}^{\&prime;})(V_{1,\mathrm{Im}}^t-V_{2,\mathrm{Im}}^t)=0$

$(R_{12}^{\&prime;}{R}_{12}-X_{12}^{\&prime;}{X}_{12})(I_{1,\mathrm{Im}}^{\mathrm{eq}}+I_{L,1,\mathrm{Im}}^t-G_1{V}_{1,\mathrm{Im}}^t+B_1{V}_{1,\mathrm{Re}}^t)+(R_{12}^{\&prime;}{X}_{12}+R_{12}{X}_{12}^{\&prime;})(I_{1,\mathrm{Re}}^{\mathrm{eq}}+I_{L,1,\mathrm{Re}}^t-$

(5)

$G_1{V}_{1,\mathrm{Re}}^t+B_1{V}_{1,\mathrm{Im}}^t)-(R_{12}+R_{12}^{\&prime;})(V_{1,\mathrm{Im}}^t-V_{2,\mathrm{Im}}^t)-(X_{12}+X_{12}^{\&prime;})(V_{1,\mathrm{Re}}^t-V_{2,\mathrm{Re}}^t)=0$

$(R_{12}^{\&prime;}{R}_{12}-X_{12}^{\&prime;}{X}_{12})(I_{2,\mathrm{Re}}^{\mathrm{eq}}+I_{L,2,\mathrm{Re}}^t-G_2{V}_{2,\mathrm{Re}}^t+B_2{V}_{2,\mathrm{Im}}^t)-(R_{12}^{\&prime;}{X}_{12}+R_{12}{X}_{12}^{\&prime;})(I_{2,\mathrm{Im}}^{\mathrm{eq}}+I_{L,2,\mathrm{Im}}^t-$ (6)

$G_2{V}_{2,\mathrm{Im}}^t-B_2{V}_{2,\mathrm{Re}}^t)-(R_{12}+R_{12}^{\&prime;})(V_{2,\mathrm{Re}}^t-V_{1,\mathrm{Re}}^t)+(X_{12}+X_{12}^{\&prime;})(V_{2,\mathrm{Im}}^t-V_{1,\mathrm{Im}}^t)=0$

$(R_{12}^{\&prime;}{R}_{12}-X_{12}^{\&prime;}{X}_{12})(I_{2,\mathrm{Im}}^{\mathrm{eq}}+I_{L,2,\mathrm{Im}}^t-G_2{V}_{2,\mathrm{Im}}^t+B_2{V}_{2,\mathrm{Re}}^t)+(R_{12}^{\&prime;}{X}_{12}+R_{12}{X}_{12}^{\&prime;})(I_{2,\mathrm{Re}}^{\mathrm{eq}}+I_{L,2,\mathrm{Re}}^t-$ (7)

$G_2{V}_{2,\mathrm{Re}}^t+B_2{V}_{2,\mathrm{Im}}^t)-(R_{12}+R_{12}^{\&prime;})(V_{2,\mathrm{Im}}^t-V_{1,\mathrm{Im}}^t)-(X_{12}+X_{12}^{\&prime;})(V_{2,\mathrm{Re}}^t-V_{1,\mathrm{Re}}^t)=0$

In formula: subscript " Re " and " Im " are real part and the imaginary parts of variable; Formula (31)-(34) can represent with the form of concentrating:

$f_m^t\left(x_s\right)=0$ m=1～4 (8)

In formula: $x_s=[I_{1,\mathrm{Re}}^{\mathrm{eq}},I_{1,\mathrm{Im}}^{\mathrm{eq}},I_{2,\mathrm{Re}}^{\mathrm{eq}},I_{2,\mathrm{Im}}^{\mathrm{eq}},R_{12}^{\&prime;},X_{12}^{\&prime;}],$

The expression formula of segmentation during for difference (31)-(34);

Therefore the least-squares estimation model of first stage outer net simplification Ward equivalent parameters is:

$J_1=\min \underset{t=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=1}{\overset{4}{\mathrm{\&Sigma;}}}{\left(f_m^t\left(x_s\right)\right)}^2$ (9)

Wherein N is the time hop count that institute's taken amount is surveyed data, unknown quantity x to be solved _sBe 6, measurement equation has 4N, consider the solvability of model, redundancy and the equivalent precision of measurement equation, and after a large amount of simulation analysis, N is taken as 3; Virtual measurement equation and the least-squares estimation model of S3-2 outer net expansion voltage source Ward Equivalent Circuit Parameter

Simplify the Ward equivalent circuit and adopt constant Injection Current to carry out the impact of equivalent outer net on the Intranet trend, can not truly reflect the voltage support effect of outer net to boundary node; Therefore, the equivalent Injection Current of simplifying in the Ward equivalent circuit is next equivalent with voltage source series impedance branch road, so just obtained expansion voltage source Ward equivalent circuit;

Consider that simplifying the Ward equivalent circuit and expanding voltage source Ward equivalent circuit is two kinds of equivalent circuits of same external electrical network, should have approximately equalised relation between its equivalent parameters; And based on the conventional Ward equivalent circuit of Intranet, can effectively estimate the former equivalent parameters, therefore, utilize two approximation relations between Equivalent Circuit Parameter, and using the constraint condition that the estimates of parameters of simplifying the Ward equivalent circuit estimates as expansion voltage source branch road Ward equivalent parameters, improve the latter's ornamental;

Based on above-mentioned thought, can obtain restriction relation equation (37)-(42) of two Equivalent Circuit Parameters:

R′ ₁₂-R′′ ₁₂=0 (10)

X′ ₁₂-X′′ ₁₂=0 (11)

$I_{1,\mathrm{Re}}^t-I_{1,\mathrm{Re}}^{\mathrm{eq}}=0$ (12)

$I_{1,\mathrm{Im}}^t-I_{1,\mathrm{Im}}^{\mathrm{eq}}=0$ (13)

$I_{2,\mathrm{Re}}^t-I_{2,\mathrm{Re}}^{\mathrm{eq}}=0$ (14)

$I_{2,\mathrm{Im}}^t-I_{2,\mathrm{Im}}^{\mathrm{eq}}=0$ (15)

Electric current on the expansion branch road can be expressed as

Both sides are with taking advantage of Z _i, it is launched according to real part and imaginary part:

$E_{1,\mathrm{Re}}^t-V_{1,\mathrm{Re}}^t-R_1{I}_{1,\mathrm{Re}}^t+X_1{I}_{1,\mathrm{Im}}^t=0$ (16)

$E_{1,\mathrm{Im}}^t-V_{1,\mathrm{Im}}^t-R_1{I}_{1,\mathrm{Im}}^t-X_1{I}_{1,\mathrm{Re}}^t=0$ (17)

$E_{2,\mathrm{Rer}}^t-V_{2,\mathrm{Re}}^t-R_2{I}_{2,\mathrm{Re}}^t+X_2{I}_{2,\mathrm{Im}}^t=0$ (18)

$E_{2,\mathrm{Im}}^t-V_{2,\mathrm{Im}}^t-R_2{I}_{2,\mathrm{Im}}^t-X_2{I}_{2,\mathrm{Re}}^t=0$ (19)

In addition, in expansion voltage source Ward equivalent circuit, voltage source is considered as to the PV node, so its voltage magnitude is constant:

${\left(E_{1,\mathrm{Re}}^t\right)}^2+{\left(E_{1,\mathrm{Im}}^t\right)}^2-E_1^2=0$ (20)

${\left(E_{2,\mathrm{Re}}^t\right)}^2+{\left(E_{2,\mathrm{Im}}^t\right)}^2-E_2^2=0$ (21)

Similar with S3-1, same based on the KCL theorem in circuit theory, the electric current sum at boundary node place is 0; Can obtain following equation:

$(R_{12}^{\&prime;\&prime;}{R}_{12}-X_{12}^{\&prime;\&prime;}{X}_{12})(I_{1,\mathrm{Re}}^t+I_{L,1,\mathrm{Re}}^t-G_1{V}_{1,\mathrm{Re}}^t+B_1{V}_{1,\mathrm{Im}}^t)-(R_{12}^{\&prime;\&prime;}{X}_{12}+R_{12}{X}_{12}^{\&prime;\&prime;})(I_{1,\mathrm{Im}}^t+I_{L,1,\mathrm{Im}}^t-$ (22)

$G_1{V}_{1,\mathrm{Im}}^t-B_1{V}_{1,\mathrm{Re}}^t)-(R_{12}+R_{12}^{\&prime;\&prime;})(V_{1,\mathrm{Re}}^t-V_{2,\mathrm{Re}}^t)+(X_{12}+X_{12}^{\&prime;\&prime;})(V_{1,\mathrm{Im}}^t-V_{2,\mathrm{Im}}^t)=0$

$(R_{12}^{\&prime;\&prime;}{R}_{12}-X_{12}^{\&prime;\&prime;}{X}_{12})(I_{1,\mathrm{Im}}^t+I_{L,1,\mathrm{Im}}^t-G_1{V}_{1,\mathrm{Im}}^t+B_1{V}_{1,\mathrm{Re}}^t)+(R_{12}^{\&prime;\&prime;}{X}_{12}+R_{12}{X}_{12}^{\&prime;\&prime;})(I_{1,\mathrm{Re}}^t+I_{L,1,\mathrm{Re}}^t-$ (23)

$G_1{V}_{1,\mathrm{Re}}^t+B_1{V}_{1,\mathrm{Im}}^t)-(R_{12}+R_{12}^{\&prime;\&prime;})(V_{1,\mathrm{Im}}^t-V_{2,\mathrm{Im}}^t)-(X_{12}+X_{12}^{\&prime;\&prime;})(V_{1,\mathrm{Re}}^t-V_{2,\mathrm{Re}}^t)=0$

$(R_{12}^{\&prime;\&prime;}{R}_{12}-X_{12}^{\&prime;\&prime;}{X}_{12})(I_{2,\mathrm{Re}}^t+I_{L,2,\mathrm{Re}}^t-G_2{V}_{2,\mathrm{Re}}^t+B_2{V}_{2,\mathrm{Im}}^t)-(R_{12}^{\&prime;\&prime;}{X}_{12}+R_{12}{X}_{12}^{\&prime;\&prime;})(I_{2,\mathrm{Im}}^t+I_{L,2,\mathrm{Im}}^t-$ (24)

$G_2{V}_{2,\mathrm{Im}}^t-B_2{V}_{2,\mathrm{Re}}^t)-(R_{12}+R_{12}^{\&prime;\&prime;})(V_{2,\mathrm{Re}}^t-V_{1,\mathrm{Re}}^t)+(X_{12}+X_{12}^{\&prime;\&prime;})(V_{2,\mathrm{Im}}^t-V_{1,\mathrm{Im}}^t)=0$

$(R_{12}^{\&prime;\&prime;}{R}_{12}-X_{12}^{\&prime;\&prime;}{X}_{12})(I_{2,\mathrm{Im}}^t+I_{L,2,\mathrm{Im}}^t-G_2{V}_{2,\mathrm{Im}}^t+B_2{V}_{2,\mathrm{Re}}^t)+(R_{12}^{\&prime;\&prime;}{X}_{12}+R_{12}{X}_{12}^{\&prime;\&prime;})(I_{2,\mathrm{Re}}^t+I_{L,2,\mathrm{Re}}^t-$ (25)

$G_2{V}_{2,\mathrm{Re}}^t+B_2{V}_{2,\mathrm{Im}}^t)-(R_{12}+R_{12}^{\&prime;\&prime;})(V_{2,\mathrm{Im}}^t-V_{1,\mathrm{Im}}^t)-(X_{12}+X_{12}^{\&prime;\&prime;})(V_{2,\mathrm{Re}}^t-V_{1,\mathrm{Re}}^t)=0$

Formula (37)-(52) can represent with the form of concentrating:

$g_m^t\left(x_e\right)=0$ m=1～16 (26)

In formula: $x_e=[I_{1,\mathrm{Re}}^t,I_{1,\mathrm{Im}}^t,I_{2,\mathrm{Re}}^t,I_{2,\mathrm{Im}}^t,R_1,X_1,R_2,X_2,R_{12}^{\&prime;\&prime;},X_{12}^{\&prime;\&prime;},E_{1,\mathrm{Re}}^{\&prime;\&prime;},E_{1,\mathrm{Im}}^t{E}_{2,\mathrm{Re}}^t,E_{2,\mathrm{Im}}^t{E}_1,E_2],$

Equation (39)-(52) of corresponding equation (37), (38) and different periods;

The least square model of outer net expansion voltage source Ward equivalent parameters is:

$J_2=\min (\underset{\min }{\overset{2}{\mathrm{\&Sigma;}}}{\left(g_m\left(x_e\right)\right)}^2+\underset{t=1}{\overset{N}{\mathrm{\&Sigma;}}}\underset{m=3}{\overset{16}{\mathrm{\&Sigma;}}}{\left(g_m^t\left(x_e\right)\right)}^2)$ (27)

Unknown quantity x to be solved _eIndividual for (8+8N), measurement equation has 2+14N, and N is taken as 3;

S4 calculates the external network equivalent circuit parameter

The S4-1 outer net is simplified the calculating of Ward Equivalent Circuit Parameter

Based on conventional Ward equivalent circuit and the parameter thereof of 3 period Intranets, set up the least square model that the outer net that is similar to formula (36) is simplified the Ward Equivalent Circuit Parameter, the initial value of getting outer net simplification Ward Equivalent Model parameter is x _s=[1,1,1,1,0.03,0.3], utilize least square method to try to achieve the parameter of equivalent circuit, comprises real part and the imaginary part of the equivalent Injection Current at boundary node place

And between boundary node interconnection resistance R ' ₁₂With reactance X ' ₁₂

The calculating of S4-2 outer net expansion voltage source Ward Equivalent Circuit Parameter

Be similar to S4-1, equally based on conventional Ward equivalent circuit and the parameter thereof of 3 period Intranets, can set up the least square model of the outer net expansion voltage source Ward Equivalent Circuit Parameter of the formula of being similar to (54), and estimation in S4-1 is obtained

And R ' ₁₂, X ' ₁₂In outer net expansion voltage source Ward equivalent circuit

And R ' ' ₁₂, X ' ' ₁₂Initial value, the initial value of outer net expansion voltage source Ward Equivalent Circuit Parameter is like this $x_e=[I_{1,\mathrm{Re}}^{\mathrm{eq}},I_{1,\mathrm{Im}}^{\mathrm{eq}},I_{2,\mathrm{Re}}^{\mathrm{eq}},I_{2,\mathrm{Im}}^{\mathrm{eq}}\mathrm{0.03,0.3,0.03,0.3},R_{12}^{\&prime;},X_{12}^{\&prime;},\mathrm{1,0,1,0,1,0}];$ Then utilize least square method to try to achieve the parameter of outer net expansion voltage source Ward equivalent circuit, comprise the voltage magnitude E of virtual PV node ₁, E ₂, the resistance R of expansion branch road ₁, R ₂With reactance X ₁, X ₂And between boundary node interconnection resistance R ' ' ₁₂With reactance X ' ' ₁₂

Then based on the relation between power and voltage, electric current, calculate the meritorious P of exerting oneself of virtual PV node in outer net expansion voltage source Ward equivalent circuit ₁, P ₂So just, calculated the value of 10 parameters in outer net expansion voltage source Ward equivalent circuit; And will expand the final equivalent circuit of voltage source Ward equivalent circuit as outer net.

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