CN101599643B - Robust state estimation method in electric power system based on exponential type objective function - Google Patents

Abstract

The invention relates to a robust state estimation method in an electric power system based on exponential type objective function, belonging to the field of electric power system dispatching automation. The method comprises the following steps: building a new state estimation mathematical model of the exponential type objective function according to the electric network model of the actual electric network and real-time measuring data; estimating the busbar voltage and phase angle of the electric network, thus calculating the active power and reactive power of the circuit, a transformer, a generator, loads and other devices. The invention can automatically realize that the impact of measurement with big errors on the objective function tends to zero, while the impact of correct measurement on the objective function tends to one. Therefore, the state estimation algorithm proposed in the invention can process in the absence of bad data identification, thus the invention has convenient realization and high computing efficiency; furthermore, if the system is absent of bad data, the estimated performance in the invention is similar to the classical weighted least squares method.

Description

A kind of anti-difference of electric power system method for estimating state based on the exponential type target function

Technical field

The present invention relates to a kind of anti-difference of electric power system method for estimating state, belong to dispatching automation of electric power systems and grid simulation technical field based on the exponential type target function.

Background technology

State estimation is the basic function of EMS, and it utilizes the metric data of actual measurement and the running status that the electric network model parameter Estimation goes out electrical network.The method for estimating state of extensive use is a weighted least-squares method the most.When measuring error profile when being the normal distribution of no rough error, can prove that the weighted least-squares method state estimation is the maximum likelihood estimation when the measurement weight of weighted least-squares method state estimation algorithm is got measurement variance reciprocal.But least square method does not have anti-difference ability, when promptly in measuring, bad data (rough error) occurring, and the estimated result rapid deterioration.So practical least square method must embedded bad data identification program.Bad data discrimination method commonly used at present is mostly based on the regularization residual error, and this method can't pick out the bad data of a plurality of associations, therefore uses least square method might calculate failure.

In order to address this problem, the scholar has proposed anti-poor method for estimating state.So-called anti-difference estimates that actual is under the inevitable situation of rough error (bad data), by selecting suitable method of estimation, makes the least possible influence that is subjected to rough error of unknown quantity estimated value, draws the best estimate under the normal mode.

Handschin E., Schweppe F.C, Kohlas J., Fiechter A., " Bad Data Analysis for Power SystemState Estimation " .IEEE Transactions on Power Apparatus and Systems, Vol.PAS-94, No.2, pp.329-337, March/April, 1975. propose and can realize detecting automatically and compressing the influence of bad data to estimated result by revising algorithm for estimating.Paper has proposed the target function of the non-quadratic forms of three classes:

(1) secondry constants Quadratic-Constant (QC)

r _iBe i residual error that measures, σ _iThe standard deviation of i error that measures, a is a threshold value

(2) the linear Quadratic-Linear (QL) of secondary

(3) square root Square Root (SR)

The greatest problem of above method is that all target functions are not continuously differentiable, needs to adjust the measurement target function in iterative computation, calculation of complex, seldom practical application.

Summary of the invention

The objective of the invention is to propose a kind of anti-difference of electric power system method for estimating state based on the exponential type target function, in grid dispatching center, electric network model and real-time measuring data according to actual electric network are set up new state estimation Mathematical Modeling, estimate electrical network busbar voltage and phase angle, thereby calculate the meritorious and idle of equipment such as circuit, transformer, generator, load, whole estimation procedure need not carry out the bad data identification, with the raising computational efficiency, and makes realization convenient.

The anti-difference of the electric power system method for estimating state based on the exponential type target function that the present invention proposes may further comprise the steps:

(1) sets up a state estimation model based on the exponential type target function;

$\max J\left(X\right)=\underset{i}{\mathrm{\Σ}}\exp (-\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(X\right))}^2)$

s.t c(X)＝0

Wherein, Z _iBe real-time measurement values, comprise the active power P of power network line or transformer _IjAnd reactive power Q _Ij, bus voltage magnitude V _i, generator active power P _iAnd reactive power Q _i, power system load active power P _iAnd reactive power Q _i, R _IiBe the variance of real-time measurement values, h _i(X) be to measure equation in real time, X is the state variable of electrical network, comprises the voltage magnitude V and the phase angle theta of all nodes, and c (X) is the pseudo-measurement equation that measures of meritorious and idle zero injection that does not articulate the node of load and generator, measures equation h in real time _i(X) be defined as:

The real-time measurement equation of circuit or transformer is:

$\left\{\begin{array}{c}{P}_{\mathrm{ij}}=V_i^2{g}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+b_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_{\mathrm{ij}}=-V_i^2(b_{\mathrm{ij}}+y_c)-V_i{V}_j(g_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-b_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.$

In the following formula, P _IjBe the wattful power messurement value of circuit or transformer, Q _IjBe the wattless power measurement value of circuit or transformer, g _Ij, b _Ij, y _cThe electricity that is circuit or transformer is respectively led, V is held in susceptance and charging _iBe the voltage magnitude of node i, V _jBe the voltage magnitude of node j, θ _IjIt is the phase angle difference between node i and node j;

The voltage of bus i is measured equation: V in real time _i=V _i

Equation is measured in the injection of bus i in real time:

$\left\{\begin{array}{c}{P}_i=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_i=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.$

In the following formula, P _i, Q _iBe the arbitrarily meritorious injecting power real-time measurement values and the idle injecting power real-time measurement values of generator or load in the overall electrical network, G _IjAnd B _IjIt is respectively the element in the node admittance matrix;

C (X)=the 0th, bus zero injects the pseudo-real-time measurement equation that measures:

$\left\{\begin{array}{c}0=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ 0=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.$

(2) adopt method of Lagrange multipliers, above-mentioned state estimation model found the solution, specifically may further comprise the steps:

(2-1) initial value of variable X and λ is set, wherein X is the voltage and the phase angle of state variable node, and node voltage is made as 1, and phase angle is made as 0, and λ is that Lagrange multiplier is made as 1;

(2-2) iterations counter k, k=0 are set;

(2-3) to iteration variable X ^K+1And λ ^K+1Revise:

$\left[\begin{array}{c}{X}^{k+1}\\ {\mathrm{\λ}}^{k+1}\end{array}\right]=\left[\begin{array}{c}{X}^k\\ {\mathrm{\λ}}^k\end{array}\right]+{\left[\begin{array}{cc}Q& C^T\\ C& 0\end{array}\right]}^{-1}\left[\begin{array}{c}H{\left(X^k\right)}^{T}W\left(X^k\right)(Z-h\left(X^k\right))\\ H{\left(X^k\right)}_0^T\mathrm{\λ}\end{array}\right]$ Wherein, H is the measurement Jacobian matrix of m * n, and m measures number in real time, and n is a POWER SYSTEM STATE variable number,

W (X) is the diagonal matrix of m * m, and wherein diagonal element is $W_{\mathrm{ii}}\left(X\right)=2\frac{{\mathrm{\ω}}_i\left(X\right)}{R_{\mathrm{ii}}},$

${\mathrm{\ω}}_i\left(X\right)=\exp (-\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(X\right))}^2);$

$Q=-H{\left(X^k\right)}^{T}W\left(X^k\right)[I-\mathrm{diag}\left(\frac{2}{R_{\mathrm{ii}}}\right)(Z-h\left(X^k\right))]H\left(X^k\right)$

$C=\frac{\∂c\left(X\right)}{\∂X}$ Be that zero of p * n dimension is injected the measurement Jacobian matrix that measures, p is the zero pseudo-number that measures of injecting;

(2-4) judge inequality H (X respectively ^K+1) ^TW (X ^K+1) (Z-h (X ^K+1))+C (X ^K+1) ^Tλ ^K+1≤ ξ ₁And c (X ^K+1)≤ξ ₂Whether set up simultaneously,, then make k=k+1, and forward step (2-3) to, if establishment, then output state variable, wherein ξ if be false ₁And ξ ₂Span be 10 ^-5-10 ^-6

The anti-difference of the electric power system method for estimating state based on the exponential type target function that the present invention proposes is characterized in: the target function of an exponential type has been proposed, and continuously differentiable.Finding the solution of this model is simple, can compress the influence of bad data to estimated result automatically in the iterative process, has very strong bad data identification and compressed capability.Therefore the inventive method has the following advantages:

1, all measurements do not have under the situation of bad data, and the estimated performance of the inventive method is consistent with traditional weighted least-squares method;

2, when there is bad data in measurement, the inventive method has very strong anti-poor ability, can estimate right value;

3, calculate simply, realize that easily iteration form and least square method are similar, do not need to adjust any parameter in the iterative process.

Description of drawings

Fig. 1 is the target function figure that measures in the inventive method.

Fig. 2 is two independent target function figure that measure in the inventive method.

Fig. 3 is the IEEE-9 node system that utilizes the inventive method.

Embodiment

The anti-poor method for estimating state that the present invention proposes may further comprise the steps:

(1) set up a state estimation model based on the exponential type target function, the target function of this model is continuously differentiable.One and two measure target functions and distribute respectively as depicted in figs. 1 and 2, and when measuring residual error greater than 2 the time, its target function is near 0, and therefore measuring bad data not have to influence to target function substantially.And when measuring residual error hour, then its target function is near 1.

$\max J\left(X\right)=\underset{i}{\mathrm{\Σ}}\exp (-\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(X\right))}^2)---\left(1\right)$

s.t c(X)＝0

Wherein,

Wherein, Z _iBe real-time measurement values, comprise the active power P of power network line or transformer _IjAnd reactive power Q _Ij, bus voltage magnitude V _i, generator active power P _iAnd reactive power Q _i, power system load active power P _iAnd reactive power Q _i, R _IiBe the variance of real-time measurement values, h _i(X) be to measure equation in real time, X is the state variable of electrical network, comprises the voltage magnitude V and the phase angle theta of all nodes, and c (X) is the pseudo-measurement equation that measures of meritorious and idle zero injection that does not articulate the node of load and generator, measures equation h in real time _i(X) be defined as:

The real-time measurement equation of circuit or transformer is:

$\left\{\begin{array}{c}{P}_{\mathrm{ij}}=V_i^2{g}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+b_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_{\mathrm{ij}}=-V_i^2(b_{\mathrm{ij}}+y_c)-V_i{V}_j(g_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-b_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.---\left(2\right)$

In the following formula, P _IjBe the wattful power messurement value of circuit or transformer, Q _IjBe the wattless power measurement value of circuit or transformer, g _Ij, b _Ij, y _cThe electricity that is circuit or transformer is respectively led, V is held in susceptance and charging _iBe the voltage magnitude of node i, V _jBe the voltage magnitude of node j, θ _IjIt is the phase angle difference between node i and node j;

The voltage measurement equation of bus i:

V _i＝V _i (3)

The injection measurement equation of bus i:

$\left\{\begin{array}{c}{P}_i=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_i=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.---\left(4\right)$

In the following formula, P _i, Q _iBe the arbitrarily meritorious injecting power real-time measurement values and the idle injecting power real-time measurement values of generator or load in the overall electrical network, G _IjAnd B _IjIt is respectively the element in the node admittance matrix;

Bus i is the node that does not articulate load and generating, then its zero injection measurement equation:

$\left\{\begin{array}{c}0=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ 0=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.---\left(5\right)$

(2) adopt method of Lagrange multipliers to find the solution above-mentioned state estimation model;

For state estimation model (1), adopt method of Lagrange multipliers to find the solution, obtain following Lagrange's equation:

$L=\underset{i}{\mathrm{\Σ}}\exp (-\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(X\right))}^2)+{\mathrm{\λ}}^{T}c\left(X\right)---\left(6\right)$

If ${\mathrm{\ω}}_i\left(X\right)=\exp (-\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(X\right))}^2),$ Then Lagrange's equation (6) can be write as:

$L=\underset{i}{\mathrm{\Σ}}{\mathrm{\ω}}_i\left(X\right)+{\mathrm{\λ}}^{T}c\left(X\right)---\left(7\right)$

Formula (7) is asked the single order optimal conditions:

$\frac{\∂L}{\∂X}=\underset{i}{\mathrm{\Σ}}\frac{\∂{\mathrm{\ω}}_i}{\∂X}+\underset{j}{\mathrm{\Σ}}{\mathrm{\λ}}_j\frac{\∂c_j\left(X\right)}{\∂X}---\left(8\right)$

$=H^{T}W\left(X\right)(Z-h\left(X\right))+C^T\mathrm{\λ}=0$

$\frac{\∂L}{\∂X}=c\left(X\right)=0$

Wherein,

H is the measurement Jacobian matrix of m * n, and is just the same with the measurement Jacobian matrix in the traditional least square method state estimation; M measures number, and n is the state variable number, generally is two demultiplications 1 of grid nodes number;

W (X) is the diagonal matrix of m * m, and wherein diagonal element is $W_{\mathrm{ii}}\left(X\right)=2\frac{{\mathrm{\ω}}_i\left(X\right)}{R_{\mathrm{ii}}};$

$C=\frac{\∂c\left(X\right)}{\∂X}$ Be that zero of p * n dimension is injected the measurement Jacobian matrix that measures, p is the zero number that measures of injecting, and does not promptly articulate the twice of the bus number of generator and load.

(2) specifically may further comprise the steps:

(2-1) initial value of variable X and λ is set, wherein X is the voltage and the phase angle of state variable node, and node voltage is made as 1, and phase angle is made as 0, and λ is that Lagrange multiplier is made as 1;

(2-2) iterations counter k, k=0 are set;

(2-3) to iteration variable X ^K+1And λ ^K+1Revise:

$\left[\begin{array}{c}{X}^{k+1}\\ {\mathrm{\λ}}^{k+1}\end{array}\right]=\left[\begin{array}{c}{X}^k\\ {\mathrm{\λ}}^k\end{array}\right]+{\left[\begin{array}{cc}Q& C^T\\ C& 0\end{array}\right]}^{-1}\left[\begin{array}{c}H{\left(X^k\right)}^{T}W\left(X^k\right)(Z-h\left(X^k\right))\\ H{\left(X^k\right)}_0^T\mathrm{\λ}\end{array}\right]$ Wherein, H is the measurement Jacobian matrix of m * n, and is just the same with the measurement Jacobian matrix in the traditional least square method state estimation; M measures number in real time, and n is a POWER SYSTEM STATE variable number,

W (X) is the diagonal matrix of m * m, and wherein diagonal element is $W_{\mathrm{ii}}\left(X\right)=2\frac{{\mathrm{\ω}}_i\left(X\right)}{R_{\mathrm{ii}}},$

${\mathrm{\ω}}_i\left(X\right)=\exp (-\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(X\right))}^2);$

$Q=-H{\left(X^k\right)}^{T}W\left(X^k\right)[I-\mathrm{diag}\left(\frac{2}{R_{\mathrm{ii}}}\right)(Z-h\left(X^k\right))]H\left(X^k\right)$

$C=\frac{\∂c\left(X\right)}{\∂X}$ Be that zero of p * n dimension is injected the measurement Jacobian matrix that measures, p is the zero pseudo-number that measures of injecting;

(2-4) judge inequality H (X respectively ^K+1) ^TW (X ^K+1) (Z-h (X ^K+1))+C (X ^K+1) ^Tλ ^K+1≤ ξ ₁And c (X ^K+1)≤ξ ₂Whether set up simultaneously,, then make k=k+1, and forward step (2-3) to, if establishment, then output state variable, wherein ξ if be false ₁And ξ ₂Span be 10 ^-5-10 ^-6

Formula (1) is in Z-h (X)=0 Taylor expansion,

s.t c(X)＝0

Following formula is equivalent to following formula:

s.t c(X)＝0

If do not have bad data in measuring, then measure residual error z _i-h _i(x) very little, so high-order term can ignore, then following formula can be rewritten as:

s.t c(X)＝0

And following formula is the estimation model of classical least square state estimation method.Therefore, when not having bad data in measuring, then the inventive method and classical least square state estimation method have similar estimated performance.

Below introduce an embodiment of the inventive method:

IEEE 9 node systems as shown in Figure 3 design the estimated result comparison of three examples to the inventive method and weighted least-squares method.

Example 1: true value is all got in all measurements, and 8 bad datas are set

As table 1,8 measurements in front are bad datas, and its measuring value is the polarity negate of true value.

The estimated result that two kinds of methods of table 1. measure and are closely related and measure bad data in the example 1

The brief summary as a result of table 2. pair example 1 state estimation

As can be seen from Table 1, no matter the inventive method is the also correct measurement of bad data, the deviation of its estimated value is all very little, and the estimated bias of least square method is then generally bigger.This is because the bad data identification program in the least square method program can't pick out the 7th and the 8th bad data, and has assigned the 9th and the 10th measurement as bad number.The inventive method is all having great advantage aspect estimated accuracy and the convergence than least square method as can be seen from Table 2.

Example 2: the error of all measurements becomes normal distribution, does not have bad data

In this example, all measurements produce the normal distribution error in measurement by following formula near true value:

$S_m=S_t+\frac{1}{3}(a_m{S}_t+b_m{S}_f)a_t---\left(6\right)$

Wherein,

S _mThe expression measuring value.

S _tIt is the true value that measures

a _mBe and the relevant coefficient of measurement true value, measure a for voltage _m=0.003, for meritorious and idle measurement a _m=0.02

b _mBe error coefficient, measure b for voltage _m=0.003, for meritorious and idle measurement b _m=0.035

S _fBe the full scale value of measuring equipment, S herein _fGet the twice that measures true value

a _t～N(0，1)

The brief summary as a result of table 3. pair example 2 state estimation

When the measurement error became normal distribution, the inventive method was all having great advantage aspect estimated accuracy and the convergence than least square method as can be seen from Table 3.

Example 3: the error of all measurements becomes normal distribution, and wherein 8 measure the polarity negate, and the measurement production method all as bad data is the same with example 2, but the negate of the same wherein 8 measurement polarity with example, as bad data.

Table 4: the brief summary as a result of the state estimation under the example 3

When the measurement error became normal distribution and contains 8 bad datas, the inventive method was all having great advantage aspect estimated accuracy and the convergence than least square method as can be seen from Table 4.This is because in the methods of the invention, and wherein therefore the target function of 8 bad datas influences very little to the state estimation result all near 0.

Claims (1)

1. the electric power system based on the exponential type target function resists the difference method for estimating state, it is characterized in that this method may further comprise the steps:

(1) sets up a state estimation model based on the exponential type target function;

$\max J\left(X\right)=\underset{i}{\mathrm{\Σ}}\exp (-\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(X\right))}^2)$

s.t c(X)＝0

Wherein, Z _iBe real-time measurement values, comprise the active power P of power network line or transformer _IjAnd reactive power Q _Ij, bus voltage magnitude V _i, generator active power P _iAnd reactive power Q _i, power system load active power P _iAnd reactive power Q _i, R _IiBe the variance of real-time measurement values, h _i(X) be to measure equation in real time, X is the state variable of electrical network, comprises the voltage magnitude V and the phase angle theta of all nodes, and c (X) is the pseudo-measurement equation that measures of meritorious and idle zero injection that does not articulate the node of load and generator, measures equation h in real time _i(X) be defined as:

The real-time measurement equation of circuit or transformer is:

$\left\{\begin{array}{c}{P}_{\mathrm{ij}}=V_i^2{g}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+b_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_{\mathrm{ij}}=-V_i^2(b_{\mathrm{ij}}+y_c)-V_i{V}_j(g_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-b_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.$

In the following formula, P _IjBe the wattful power messurement value of circuit or transformer, Q _IjBe the wattless power measurement value of circuit or transformer, g _Ij, b _Ij, y _cThe electricity that is circuit or transformer is respectively led, V is held in susceptance and charging _iBe the voltage magnitude of node i, V _jBe the voltage magnitude of node j, θ _IjIt is the phase angle difference between node i and node j;

The voltage of bus i is measured equation: V in real time _i=V _i

Equation is measured in the injection of bus i in real time:

$\left\{\begin{array}{c}{P}_i=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ Q_i=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.$

In the following formula, P _i, Q _iBe the arbitrarily meritorious injecting power real-time measurement values and the idle injecting power real-time measurement values of generator or load in the overall electrical network, G _IjAnd B _IjIt is respectively the element in the node admittance matrix;

C (X)=the 0th, bus zero injects the pseudo-real-time measurement equation that measures:

$\left\{\begin{array}{c}0=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}})\\ 0=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}-B_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.$

(2) adopt method of Lagrange multipliers, above-mentioned state estimation model found the solution, specifically may further comprise the steps:

(2-1) initial value of variable X and λ is set, wherein X is the voltage and the phase angle of state variable node, and node voltage is made as 1, and phase angle is made as 0, and λ is that Lagrange multiplier is made as 1;

(2-2) iterations counter k, k=0 are set;

(2-3) to iteration variable X ^K+1And λ ^K+1Revise:

$\left[\begin{array}{c}{X}^{k+1}\\ {\mathrm{\λ}}^{k+1}\end{array}\right]=\left[\begin{array}{c}{X}^k\\ {\mathrm{\λ}}^k\end{array}\right]+{\left[\begin{array}{cc}Q& C^T\\ C& 0\end{array}\right]}^{-1}\left[\begin{array}{c}H{\left(X^k\right)}^{T}W\left(X^k\right)(Z-h\left(X^k\right))\\ H{\left(X^k\right)}_0^T\mathrm{\λ}\end{array}\right]$ Wherein, H is the measurement Jacobian matrix of m * n, and m measures number in real time, and n is a POWER SYSTEM STATE variable number,

W (X) is the diagonal matrix of m * m, and wherein diagonal element is $W_{\mathrm{ii}}\left(X\right)=2\frac{{\mathrm{\ω}}_i\left(X\right)}{R_{\mathrm{ii}}},$ ${\mathrm{\ω}}_i\left(X\right)=\exp (-\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(X\right))}^2);$

$Q=-H{\left(X^k\right)}^{T}W\left(X^k\right)[I-\mathrm{diag}\left(\frac{2}{R_{\mathrm{ii}}}\right)(Z-h\left(X^k\right))]H\left(X^k\right)$

$C=\frac{\∂c\left(X\right)}{\∂X}$ Be that zero of p * n dimension is injected the measurement Jacobian matrix that measures, p is the zero pseudo-number that measures of injecting;

(2-4) judge inequality H (X respectively ^K+1) ^TW (X ^K+1) (Z-h (X ^K+1))+C (X ^K+1) ^Tλ ^K+1≤ ξ ₁And c (X ^K+1)≤ξ ₂Whether set up simultaneously,, then make k=k+1, and forward step (2-3) to, if establishment, then output state variable, wherein ξ if be false ₁And ξ ₂Span be 10 ^-5-10 ^-6

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