CN102185308B - Power system state estimating method for taking zero injection measurement equality constraint into consideration - Google Patents

Abstract

The invention relates to a power system state estimating method for taking zero injection measurement equality constraint into consideration, belonging to the technical field of power system dispatching automation and grid simulation. The method comprises the following step of: adding zero injection measurement as an equality constraint into an optimized model of state estimation to form an equality constraint-contained optimized model. As for the model, a voltage amplitude value and a phase angle of a zero injection node are used as basic variables and a voltage amplitude value and a phase angle of a nonzero injection node are used as nonbasic variables for participating in optimization; the basic variables and the nonbasic variables are solved by adopting a reduced gradient method; and the zero injection node voltage is confirmed according to nonzero injection node voltage. The method disclosed by the invention can ensure that a zero injection constraint equation is strictly met and solve the problem that an estimating result does not meet a flow equation because zero injection node power mismatching amount is larger in the traditional power system estate estimation.

Description

A kind of method considering zero injects the power system state estimation method that measures equality constraint

Technical field

The present invention relates to a kind of method considering zero and inject the power system state estimation method that measures equality constraint, belong to dispatching automation of electric power systems and grid simulation technical field.

Background technology

Power system state estimation is one of key foundation function of EMS (EMS), and the nearly all online and off-line of EMS is used and all carried out as the basis take the state estimation result.Exist in the practical power systems a large amount of injecting powers be 0 zero inject node.In least square method state estimation (WLS) program that the control centre commonly uses at present, common processing method is that the addition measured value is 0 pseudo-measurement of node injecting power, and these pseudo-measurements are arranged large weight.Arrange too smallly if zero injects the measurement weight, will there be larger power mismatch amount in zero injection node in the final estimated result, thereby cause estimated result not meet power flow equation; Otherwise excessive if zero injection measures weight, the information matrix conditional number in the least square method will worsen, and may cause state estimation to be dispersed.

In addition, the problem that affected by bad data in order to solve least square method state estimation the possibility of result, we have proposed the anti-poor estimation model of new exponential type, and concrete grammar has been applied for Chinese invention patent (application number: 200910082501.8).Also there is similarly zero injection measurement problem in the novel estimation model of this class.

The conventional algorithm of processing equality constraint on the mathematics is method of Lagrange multipliers, is about to RegionAlgorithm for Equality Constrained Optimization:

$\underset{x}{\min }J\left(x\right)$

s.t.c(x)＝0

Add Lagrange multiplier, be converted into unconstrained optimization problem:

$\underset{x,\mathrm{\λ}}{\min }J\left(x\right)+{\mathrm{\λ}}^{T}c\left(x\right)$

In the Power system state estimation problem of reality, nearly all 220kV and above node, and the three-circuit transformer Centroid all belongs to zero injection node.For provincial power network, zero injects the ratio of node often about 50%.Therefore, traditional method of Lagrange multipliers can increase the optimized variable number significantly, causes amount of calculation sharply to rise.This also is that method of Lagrange multipliers does not have the main cause in the practical application of Power system state estimation field for a long time.

Summary of the invention

The objective of the invention is to propose a kind of method considering zero and inject the power system state estimation method that measures equality constraint, at the power system dispatching center, be 0 to join optimization method as equality constraint with zero injecting power that injects node, and the Optimized model that adopts the reduced gradient method this to be contained equality constraint is found the solution.

May further comprise the steps:

(1) zero injecting power that injects node in the electric power system is set up a Power system state estimation model that contains equality constraint as equality constraint:

min J(x)

s.t c(x)＝0

Wherein J (x) is the target function of state estimation, and this target function is least square target function or can little non-least square target function, and c (x) injects the pseudo-measurement equation that measures, component c wherein for electric power system zero _i(x) be:

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

V wherein _iThe voltage magnitude of node i, V _jThe voltage magnitude of node j, θ _IjThe phase angle difference between node i and the j, G _IjAnd B _IjIt is respectively the element in the node admittance matrix;

(2) the Power system state estimation model that contains equality constraint to setting up in the above-mentioned steps (1) adopts the reduced gradient method to find the solution, and solution procedure is as follows:

(2-1) state variable x is divided into basic variable x _BWith nonbasic variable x _N, x wherein _BBe voltage and the phase angle of zero injection node, x _NInject voltage and the phase angle of node for non-zero in the electric power system;

(2-2) the initial value x of set condition variable x ⁽⁰⁾With convergence precision ε, iterations counter k is set, k=0;

(2-3) calculate at current some x ^(k)The target function J of place (x) is to nonbasic variable x _NFull gradient as follows:

$g_N^{\left(k\right)}=\frac{\mathrm{dJ}\left(x\right)}{{\mathrm{dx}}_N^{\left(k\right)}}=\frac{\∂J\left(x\right)}{{\∂x}_N^{\left(k\right)}}+\frac{\∂J\left(x\right)}{{\∂x}_B^{\left(k\right)}}\frac{{\∂x}_B^{\left(k\right)}}{{\∂x}_N^{\left(k\right)}}$

$=\frac{\∂J\left(x\right)}{{\∂x}_N^{\left(k\right)}}-{\left[{\left(\frac{\∂c\left(x\right)}{{\∂x}_B^{\left(k\right)}}\right)}^{-1}\frac{\∂c\left(x\right)}{{\∂x}_N^{\left(k\right)}}\right]}^T\frac{\∂J\left(x\right)}{{\∂x}_B^{\left(k\right)}}$

If (2-4) set Calculating is withdrawed from then state estimation convergence, if

It is as follows then to replace gradient direction with conjugated gradient direction:

$d^{\left(k\right)}=-g_N^{\left(k\right)}+{\mathrm{\β}}_k{d}^{(k-1)},k\≥1$

$d^{\left(k\right)}={-g}_N^{\left(0\right)},k=0$

Factor beta wherein _kAdopt PRP (Polak, Ribiere, Polyak) conjugate gradient method to ask for:

(2-5) carry out linear search according to the definite search direction of above-mentioned steps (2-4), obtain optimal step size

Then

K=k+1, repeating step (2-3)-(2-5).

The method considering zero that the present invention proposes injects the power system state estimation method that measures equality constraint, its advantage is: inject measurement joins state estimation as equality constraint Optimized model with zero, and adopt the reduced gradient method to find the solution, only non-zero is injected the node voltage x of node _NAs the optimizing variable, zero injects node node voltage x _BInject the node voltage x of node according to non-zero _NDetermine.The target function of method and state estimation model is irrelevant.Therefore the present invention has the following advantages:

1, in the methods of the invention, zero injecting power that injects node is processed as equality constraint, and its estimated value strictly is 0, and state estimation result strictly satisfies power flow equation.

2, in the inventive method, the optimizing variable only injects node voltage x for non-zero _N, the optimizing Spatial Dimension is little, and computational efficiency is high.

When 3, calculating full gradient in the inventive method, only relate to equality constraints functions c (x) to basic variable x _BThe Jacobian matrix of differentiate is inverted, and this Jacobian matrix is sparse, adopts factorization method to calculate, and calculates fast.

4, the inventive method highly versatile is applicable to the non-least-squares estimation that least-squares estimation and other target functions can be little.

5, the inventive method adopts gradient class algorithm, and convergence is reliable, and iteration can not dispersed.Under extreme case, can when estimating not restrain, estimate to calculate by abort state, the estimated result that obtains still strictly satisfies power flow equation.

Embodiment

The method considering zero that the present invention proposes injects the power system state estimation method that measures equality constraint, may further comprise the steps:

1, be 0 to regard equality constraint as with the zero node injecting power that injects node, and set up a state estimation model that contains equality constraint such as (1):

min J(x) (1)

s.t c(x)＝0

(1) J (x) is the target function of state estimation in, and c (x) is that the gaining merit of node that does not articulate load and generator injected the measurement equation of pseudo-measurement with idle zero.Its component C _i(x) can be expressed as:

$\left\{\begin{array}{c}{V}_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}{\sin \mathrm{\θ}}_{\mathrm{ij}})=0\\ V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}{\sin \mathrm{\θ}}_{\mathrm{ij}}-B_{\mathrm{ij}}{\cos \mathrm{\θ}}_{\mathrm{ij}})=0\end{array}\right.---\left(2\right)$

V wherein _iThe voltage magnitude of node i, V _jThe voltage magnitude of node j, θ _IjThe phase angle difference between node i and node j, G _IjAnd B _IjIt is respectively the element in the node admittance matrix.

State estimation model (1) does not have special requirement for the form of target function, if adopt conventional least square method state estimation, then state estimation model can be write as:

$\min J\left(x\right)=\underset{i}{\mathrm{\Σ}}\frac{1}{R_{\mathrm{ii}}}{(z_i-h_i\left(x\right))}^2---\left(3\right)$

s.t c(x)＝0

Or a maximum negative exponent square anti-poor estimation model can be write as: (concrete grammar has been applied for Chinese patent (application number is 200910082501.8))

$\max J\left(x\right)=\underset{i}{\mathrm{\Σ}}\exp (-\frac{4}{R_{\mathrm{ii}}}{(z_i-h_i\left(x\right))}^2)---\left(4\right)$

s.t c(x)＝0

In above-mentioned (3) and (4) formula, z _iBe measuring value, comprise the active-power P of circuit or transformer _IjAnd reactive power Q _Ij, node voltage magnitude V _i, generator and load active-power P _iAnd reactive power Q _iDeng, R _IiBe the variance of real-time measurement, x is the state variable of electric power system, comprises voltage magnitude V and the phase angle theta of all nodes, h _i(x) be the system measurements equation, for common measurement type, it is defined as:

The real-time measurement equation of circuit is:

$\left\{\begin{array}{c}{P}_{\mathrm{ij}}^s=V_i^2{g}_{\mathrm{ij}}-V_i{V}_j(g_{\mathrm{ij}}{\cos \mathrm{\θ}}_{\mathrm{ij}}+b_{\mathrm{ij}}{\sin \mathrm{\θ}}_{\mathrm{ij}})\\ Q_{\mathrm{ij}}^s=-V_i^2(b_{\mathrm{ij}}+y_c)-V_i{V}_j(g_{\mathrm{ij}}{\sin \mathrm{\θ}}_{\mathrm{ij}}-b_{\mathrm{ij}}{\cos \mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.---\left(5\right)$

The real-time measurement equation of transformer is:

$\left\{\begin{array}{c}{P}_{\mathrm{ij}}^s=-\frac{1}{k}{V}_i{V}_j{b}_{\mathrm{ij}}{\sin \mathrm{\θ}}_{\mathrm{ij}}\\ Q_{\mathrm{ij}}^s={-V}_i^2{b}_{\mathrm{ij}}+\frac{1}{k}{V}_i{V}_j{b_{\mathrm{ij}}\cos \mathrm{\θ}}_{\mathrm{ij}}\end{array}\right.---\left(6\right)$

In the following formula, P _Ij ^sAnd Q _Ij ^sRespectively active power and the reactive power estimated value of circuit or transformer branch road, V _iAnd V _jRespectively the voltage magnitude of node i and node j, θ _IjIt is the phase angle difference between node i and the node j; To the circuit of (5) formula, g _Ij, b _IjAnd y _cRespectively line conductance, susceptance and charging susceptance; To the transformer branch road of (6) formula, b _IjWith k be respectively susceptance and the no-load voltage ratio of transformer branch road, regulation j side is non-standard no-load voltage ratio side.

The voltage real-time measurement equation of node i:

$V_i^s=V_i---\left(7\right)$

V _i ^sVoltage estimated value for node i.

The injection real-time measurement equation of node i:

$\left\{\begin{array}{c}{P}_i^s=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\cos {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}{\sin \mathrm{\θ}}_{\mathrm{ij}})\\ Q_i^s=V_i\underset{j\∈i}{\mathrm{\Σ}}{V}_j(G_{\mathrm{ij}}\sin {\mathrm{\θ}}_{\mathrm{ij}}+B_{\mathrm{ij}}{\cos \mathrm{\θ}}_{\mathrm{ij}})\end{array}\right.---\left(8\right)$

In the following formula, P _i, Q _iMeritorious injecting power and the idle injecting power estimated value of node i.

2, the state estimation model that contains equality constraint to setting up in the step 1 adopts the reduced gradient method to find the solution.According to equality constraint, system variable x is divided into basic variable x _BWith nonbasic variable x _NBasic variable x _BBy equality constraint by nonbasic variable x _NDetermine that namely formula (1) can be write as following unconstrained optimize model:

min J(x _N，x _B(x _N)) (9)

X wherein _B(x _N) expression basic variable x _BValue be nonbasic variable x _NFunction, in the Optimized model of (8) formula, only with nonbasic variable x _NAs optimized variable, basic variable x _BValue by nonbasic variable x _NDetermine.Guaranteed that so not only optimum results satisfies equality constraint c (x)=0, and significantly reduced the optimizing space dimensionality, improved computational efficiency.

In Power system state estimation model (1), x _BBe taken as node voltage and the phase angle of zero injection node, x _NBe taken as node voltage and phase angle that non-zero injects node.Then can use gradient method, to finding the solution without optimizing restricted problem of (8) formula.Concrete steps are as follows:

(2-1) establish initial value for state variable x

Given convergence precision ε arranges iterations counter k, k=0.

(2-2) calculate at current some x ^(k)The target function J of place (x) is to nonbasic variable x _NFull gradient:

$g_N^{\left(k\right)}=\frac{\mathrm{dJ}\left(x\right)}{{\mathrm{dx}}_N^{\left(k\right)}}=\frac{\∂J\left(x\right)}{{\∂x}_N^{\left(k\right)}}+\frac{\∂J\left(x\right)}{{\∂x}_B^{\left(k\right)}}\frac{{\∂x}_B^{\left(k\right)}}{{\∂x}_N^{\left(k\right)}}---\left(10\right)$

$=\frac{\∂J\left(x\right)}{{\∂x}_N^{\left(k\right)}}-{\left[{\left(\frac{\∂c\left(x\right)}{{\∂x}_B^{\left(k\right)}}\right)}^{-1}\frac{\∂c\left(x\right)}{{\∂x}_N^{\left(k\right)}}\right]}^T\frac{\∂J\left(x\right)}{{\∂x}_B^{\left(k\right)}}$

J (x) is to nonbasic variable x _NGradient consisted of by two parts, be respectively that J (x) itself is to x _NGradient, and because x _NVariation, cause basic variable x by under equality constraint c (x)=0 effect _BChange, thus the impact that target function is applied.For the Power system state estimation problem, target function J (x) is to x _B, x _NThe part gradient determined that by the form of target function and equality constraint is to x _B, x _NDerivative Corresponding submatrix for the measurement equation Jacobian matrix.

If (2-3)

Calculating is withdrawed from then state estimation convergence, if

Then carry out step (2-4)

(2-4) for improving convergence of algorithm speed, overcome gradient method and when approaching convergence, can produce the shortcomings such as reforming phenomena, available conjugated gradient direction replaces gradient direction.Be following definite search direction:

$d^{\left(k\right)}=-g_N^{\left(k\right)}+{\mathrm{\β}}_k{d}^{(k-1)},k\≥1---\left(11\right)$

$d^{\left(0\right)}={-g}_N^{\left(0\right)},k=0$

Factor beta wherein _kAdopt PRP (Polak, Ribiere, Polyak) conjugate gradient method to ask for factor beta _kBe taken as:

${\mathrm{\β}}_k=\frac{{g_N^{\left(k\right)}}^T(g_N^{\left(k\right)}-g_N^{(k-1)})}{{g_N^{(k-1)}}^T{g}_N^{(k-1)}}---\left(12\right)$

(2-5) carry out linear search according to (2-4) definite search direction, obtain optimal step size

The algorithm of linear search does not have specific (special) requirements, and rule of thumb, more effective linear search algorithm is By Cubic Curve Fitting.Linear search may need nonbasic variable x _NAsk for along search direction d ^(k)Target function value J (x behind the step-length of the advancing α _N+ α d ^(k)), and at the derivative of this some place target function value to step-length

With the node division of electric power system be zero inject node and non-zero and inject node after, linearizing network equation is as follows:

$\left[\begin{array}{cc}{Y}_{\mathrm{BB}}& Y_{\mathrm{BN}}\\ Y_{\mathrm{NB}}& Y_{\mathrm{NN}}\end{array}\right]\left[\begin{array}{c}{\stackrel{\·}{V}}_B\\ {\stackrel{\·}{V}}_N\end{array}\right]=\left[\begin{array}{c}{\stackrel{\·}{I}}_B\\ {\stackrel{\·}{I}}_N\end{array}\right]---\left(13\right)$

Wherein subscript B is zero injection node (basic variable), and subscript N is nonbasic variable.Wherein

Can be by current search direction d ^(k)And step-length α is definite, is known quantity.For zero injection node, its node Injection Current is always 0.So we can obtain:

$Y_{\mathrm{BB}}{\stackrel{\·}{V}}_B+Y_{\mathrm{BN}}{\stackrel{\·}{V}}_N={\stackrel{\·}{I}}_B=0---\left(14\right)$

Therefore have:

${\stackrel{\·}{V}}_B=-Y_{\mathrm{BB}}^{-1}\left(Y_{\mathrm{BN}}{\stackrel{\·}{V}}_N\right)---\left(15\right)$

Basic variable x _BValue can be directly determine by finding the solution system of linear equations, need not any iterative computation.

According to current x _B, x _N, nonbasic variable x _NAlong search direction d ^(k)Target function value J (x behind the step-length of the advancing α _N+ α d ^(k)) just can determine.

And this some place target function value is to the derivative of step-length For:

$\frac{\mathrm{dJ}(x_N+{\mathrm{\αd}}^{\left(k\right)})}{\mathrm{d\α}}=\frac{\∂J(x_N+{\mathrm{\αd}}^{\left(k\right)})}{\∂(x_N+{\mathrm{\αd}}^{\left(k\right)})}\frac{d(x_N+{\mathrm{\αd}}^{\left(k\right)})}{\mathrm{d\α}}+\frac{\∂J(x_N+{\mathrm{\αd}}^{\left(k\right)})}{{\∂x}_B}\frac{{\∂x}_B}{\∂\mathrm{\α}}---\left(16\right)$

$=\frac{\∂J\left(x\right)}{{\∂x}_B}{d}^{\left(k\right)}-\left[{\left(\frac{\∂h}{{\∂x}_B}\right)}^{-1}\frac{\∂h}{{\∂x}_N}\right]\frac{\∂J\left(x\right)}{{\∂x}_B}{d}^{\left(k\right)}=g(x_N+{\mathrm{\αd}}^{\left(k\right)})d^k$

Namely

Numerical value be nonbasic variable x _NAlong this search direction d ^(k)The reduced gradient of the new point that obtains behind the displacement α and this search direction d ^(k)Dot product.

k＝k+1。Repeating step (2-2)-(2-6).

Claims (1)

1. a method considering zero injects the power system state estimation method that measures equality constraint, it is characterized in that the method may further comprise the steps:

(1) zero injecting power that injects node in the electric power system is set up a Power system state estimation model that contains equality constraint as equality constraint:

min J(x)

s.t c(x)＝0

Wherein J (x) is the target function of state estimation, and this target function is least square target function or can little non-least square target function, and c (x) injects the pseudo-measurement equation that measures, component c wherein for electric power system zero _i(x) be:

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

V wherein _iThe voltage magnitude of node i, V _jThe voltage magnitude of node j, θ _IjThe phase angle difference between node i and the j, G _IjAnd B _IjIt is respectively the element in the node admittance matrix;

(2) the Power system state estimation model that contains equality constraint to setting up in the above-mentioned steps (1) adopts the reduced gradient method to find the solution, and solution procedure is as follows:

(2-1) state variable x is divided into basic variable x _BWith nonbasic variable x _N, x wherein _BBe voltage and the phase angle of zero injection node, x _NInject voltage and the phase angle of node for non-zero in the electric power system;

(2-2) the initial value x of set condition variable x ⁽⁰⁾With convergence precision ε, iterations counter k is set, k=0;

(2-3) calculate at current some x ^(k)The target function J of place (x) is to nonbasic variable x _NFull gradient as follows:

$g_N^{\left(k\right)}=\frac{\mathrm{dJ}\left(x\right)}{{\mathrm{dx}}_N^{\left(k\right)}}=\frac{\∂J\left(x\right)}{\∂x_N^{\left(k\right)}}+\frac{\∂J\left(x\right)}{\∂x_B^{\left(k\right)}}\frac{\∂x_B^{\left(k\right)}}{\∂x_N^{\left(k\right)}}$

$=\frac{\∂J\left(x\right)}{{\∂x}_N^{\left(k\right)}}-{\left[{\left(\frac{\∂c\left(x\right)}{\∂x_B^{\left(k\right)}}\right)}^{-1}\frac{\∂c\left(x\right)}{{\∂x}_N^{\left(k\right)}}\right]}^T\frac{\text{\∂J}\left(x\right)}{\∂x_B^{\left(k\right)}}$

If (2-4) set

Calculating is withdrawed from then state estimation convergence, if

It is as follows then to replace gradient direction with conjugated gradient direction:

$d^{\left(k\right)}=-g_N^{\left(k\right)}+{\mathrm{\β}}_k{d}^{(k-1)},k\≥1$

$d^{\left(0\right)}=-g_N^{\left(0\right)},k=0$

Factor beta wherein _kAdopt the PRP conjugate gradient method to ask for:

(2-5) carry out linear search according to the definite search direction of above-mentioned steps (2-4), obtain optimal step size

Then $x_N^{(k+1)}=x_N^{\left(k\right)}+\widehat{\mathrm{\α}}{d}^{\left(k\right)},k=k+1;$

(2-6) repeating step (2-3)-(2-5).