CN102185308A - 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 power system state estimation method of considering zero injection measurement equality constraint

Technical field

The present invention relates to a kind of power system state estimation method of considering zero injection measurement 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 it is that carry out on the basis with the state estimation result all that the nearly all online and off-line of EMS is used.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 uses always at present, common processing method is that the addition measured value is 0 pseudo-measurement of node injecting power, and these pseudo-measurements are provided with big weight.Be provided with too smallly if zero injects the measurement weight, will there be bigger power mismatch amount in zero injection node in the final estimated result, thereby cause estimated result not meet power flow equation; Otherwise if zero injection measurement weight is excessive, the information matrix conditional number in the least square method will worsen, and may cause state estimation to be dispersed.

In addition, in order to solve the problem that least square method state estimation the possibility of result is influenced by bad data, we have proposed the anti-difference of new exponential type estimation model, 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 handling equality constraint on the mathematics is a method of Lagrange multipliers, is about to the equality constraint optimization problem:

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

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 electric pressure 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 optimization 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 power system state estimation method of considering zero injection measurement equality constraint, at the power system dispatching center, with zero injecting power that injects node is 0 to join as equality constraint and to optimize equation, and the Optimization Model that adopted both about 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 a 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 _iBe the voltage magnitude of node i, V _jBe the voltage magnitude of node j, θ _IjBe the 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) adopted both about 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 the voltage and the phase angle of zero injection node, x _NInject the 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:

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

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

(2-4) set if

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 _kEmploying PRP (Polyak) conjugate gradient method is asked for for Polak, Ribiere:

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

Then

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

The power system state estimation method that measures equality constraint is injected in the consideration that the present invention proposes zero, its advantage is: inject measurement joins state estimation as equality constraint Optimization Model with zero, and adopted both about 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 handled as equality constraint, and its estimated value strictness is 0, and state estimation strictness as a result 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, the computational efficiency height.

When 3, calculating full gradient in the inventive method, only relate to equality constraint function 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 strictness satisfies power flow equation.

Embodiment

The power system state estimation method that measures equality constraint is injected in the consideration that the present invention proposes zero, 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 as (1):

min?J(x) (1)

s.t?c(x)＝0

(1) middle J (x) is the target function of state estimation, 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.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 _iBe the voltage magnitude of node i, V _jBe the voltage magnitude of node j, θ _IjBe the 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 the 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-difference 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 power system state variable, comprises the 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 ^sBe respectively the active power and the reactive power estimated value of circuit or transformer branch road, V _iAnd V _jBe respectively 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 _cBe respectively line conductance, susceptance and charging susceptance; To the transformer branch road of (6) formula, b _IjWith k be respectively the 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 _iBe the meritorious 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 adopted both about 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 promptly formula (1) can be write as following unconstrained optimization 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 Optimization Model of (8) formula, only with nonbasic variable x _NAs optimizing variable, basic variable x _BValue by nonbasic variable x _NDetermine.Guaranteed that so not only the optimization result 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 the 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, the nothing of (8) formula be optimized restricted problem find the solution.Concrete steps are as follows:

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

Given convergence precision ε is provided with 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 constitute 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 influence that target function is applied.For the Power system state estimation problem, target function J (x) is to x _B, x _NThe part gradient by the decision of the form of target function, and equality constraint is to x _B, x _NDerivative

Corresponding submatrix for the measurement equation Jacobian matrix.

(2-3) if

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 near convergence the time, can produce shortcomings such as reforming phenomena, available conjugated gradient direction replaces gradient direction.Be following definite optimizing 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 _k(Polyak) conjugate gradient method is asked for for Polak, Ribiere, factor beta to adopt PRP _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, obtain optimal step size according to (2-4) definite optimizing direction

The algorithm of linear search does not have specific (special) requirements, and rule of thumb, more effective linear search algorithm is the cubic curve fitting process.Linear search may need nonbasic variable x _NAsk for along optimizing 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 is that linearizing network equation was as follows after zero injection node and non-zero injected node:

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

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

Can be by current optimizing 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 optimizing 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:

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

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

Promptly

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

k＝k+1。Repeating step (2-2)-(2-6).

Claims (1)

1. consider that zero injects the power system state estimation method that measures equality constraint for one kind, it is characterized in that this 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 a 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 _iBe the voltage magnitude of node i, V _jBe the voltage magnitude of node j, θ _IjBe the 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) adopted both about 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 the voltage and the phase angle of zero injection node, x _NInject the 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:

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

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

(2-4) set if

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, obtain optimal step size according to the definite optimizing direction of above-mentioned steps (2-4) Then

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