CN104242304A - Power system state estimation method based on phasor measurement - Google Patents

Abstract

The invention relates to a power system state estimation method based on phasor measurement, and belongs to the technical field of power system operation and control. The method comprises the steps that a complex number measurement equation based on full PMU measurement is set up in a complex number field, a measurement quantity and a state quantity in the measurement equation are in a linear relation, meanwhile, an optimized objective function is directly set up for an overdetermination measurement equation in the complex number field, the measurement equation is solved through a complex number relative least-square method, and therefore state estimation and computation based on the full PMU measurement are directly completed in the complex number field. According to the method, power system state estimation is expanded to the complex number field from a real number field, and the state estimation result can be directly obtained through one-time computation. Due to the fact that the complex number method is adopted, high robustness is achieved on phase angle errors, algorithm efficiency is high, meanwhile, the self-weighting effect of the least square method is considered, and therefore the accuracy of power system state estimation is improved.

Description

A kind of power system state estimation method based on Phasor Measurements

Technical field

The present invention relates to a kind of power system state estimation method based on Phasor Measurements, belong to power system operation and control technology field.

Background technology

In recent years, along with phasor measurement unit (Phasor Measurement Unit, hereinafter referred to as PMU) the quick layout in electrical network, a large amount of Phasor Measurements is that Electrical power system analysis and computing provides new data.But active power measures relatively, not enough, and due to situations such as network communication interruption, measuring equipment faults, in PMU measurement, bad data may be there is in the precision of the phase angle measurements of PMU, present stage is arranged on the PMU in electrical network simultaneously, and its certainty of measurement is very not satisfactory.Therefore PMU metric data can not directly use, and the state estimation measured based on full PMU is necessary.

Weighted least-squares method (the Weighted Least Squares of the method for estimating state measured based on full PMU at present mainly real number field, hereinafter referred to as WLS), the phasor that PMU measures by the method represents by its real part and imaginary part respectively, thus the linear measurement equation of real number under setting up rectangular coordinate, and utilize WLS method to solve this state estimation problem.But the method introduces transformed error when phasor being converted to real part, imaginary part; And the dependency relation that have ignored between the real part of same phasor and imaginary part; After phasor converts real part, imaginary part in addition, the scale of coefficient matrix increases, and adds amount of calculation; The method does not have Robustness least squares to phase angle error simultaneously.

Summary of the invention

The object of the invention is to propose a kind of power system state estimation method based on Phasor Measurements, measure as phasor i.e. this brass tacks of plural number for PMU in prior art, directly set up in complex field the plural measurement equation measured based on full PMU, Power system state estimation problem is extended to complex field from real number field, and propose with plural number least square method (Complex Number Relative Least Squares relatively, hereinafter referred to as CRLS) solve this plural estimation problem, thus in complex field, directly complete the state estimation calculating measured based on full PMU.

The power system state estimation method based on Phasor Measurements that the present invention proposes, comprises the following steps:

(1) set up the plural measurement equation for Power system state estimation based on Phasor Measurements, detailed process is as follows:

(1-1) voltage measurement equation and the current measurement equation of electric power system is set up:

In each Phasor Measurements, voltage phasor measurement equation is:

${\stackrel{\·}{U}}_{m,i}={\stackrel{\·}{U}}_{T,i}+{\stackrel{\·}{\mathrm{\ϵ}}}_{U,i}$

Wherein with be respectively the measuring value of i-th voltage phasor, true value and error in measurement;

In each Phasor Measurements, electric current phasor measurement equation is:

${\stackrel{\·}{I}}_{m,j}={\stackrel{\·}{I}}_{T,j}+{\stackrel{\·}{\mathrm{\ϵ}}}_{I,j}$

Wherein with be respectively the measuring value of a jth electric current phasor, true value and error in measurement;

By above-mentioned voltage phasor measurement equation and electric current phasor measurement equation composition equation group, and be rewritten into following matrix form:

$Z_m=\left[\begin{array}{c}{U}_m\\ I_m\end{array}\right]=\left[\begin{array}{c}{U}_{\mathrm{mT}}\\ I_{\mathrm{mT}}\end{array}\right]+\left[\begin{array}{c}{\mathrm{\ϵ}}_U\\ {\mathrm{\ϵ}}_I\end{array}\right]$

Wherein U _m, U _mTand ε _ube respectively voltage phasor in Phasor Measurements and measure the measurement vector, the true value vector error in measurement vector that form, I _m, I _mTand ε _ithe electric current phasor be respectively in Phasor Measurements measures the measurement vector, the true value vector error in measurement vector that form;

(1-2) relation set up between electric current true value vector in above-mentioned Phasor Measurements and voltage true value vector is as follows:

$\left\{\begin{array}{c}{\stackrel{\·}{I}}_{\mathrm{pq}}=Y_{\mathrm{pp}}{\stackrel{\·}{U}}_p+Y_{\mathrm{pq}}{\stackrel{\·}{U}}_q\\ {\stackrel{\·}{I}}_{\mathrm{qp}}=Y_{\mathrm{qp}}{\stackrel{\·}{U}}_p+Y_{\mathrm{qq}}{\stackrel{\·}{U}}_q\end{array}\right.$

Wherein, for in the measurement branch road pq of electric power system, from node q to the electric current phasor of node p, with be respectively in the measurement branch road pq of electric power system, the voltage phasor of node p and node q, Y _pp, Y _pq, Y _qp, Y _qqfor each branch admittance, expression is as follows:

$\left\{\begin{array}{c}{Y}_{\mathrm{qq}}=\frac{1}{r+\mathrm{jx}}+j\frac{b}{2}\\ Y_{\mathrm{qp}}=-\frac{1}{(r+\mathrm{jx})K{e}^{\mathrm{j\θ}}}\\ Y_{\mathrm{pp}}=\frac{Y_{\mathrm{qq}}}{{\left|K\right|}^2}\\ Y_{\mathrm{qp}}=-\frac{1}{(r+\mathrm{jx}){\left(K{e}^{\mathrm{j\θ}}\right)}^{*}}\end{array}\right.$

Wherein, K is the no-load voltage ratio of transformer in electric power system, and θ is the phase shifting angle of phase shifter in electric power system, and r, x and b are respectively the resistance of branch road in electric power system, reactance and susceptance, and j is imaginary number unit;

The electric current phasor of all branches in electric power system is measured composition equation group, and is rewritten as following matrix form:

I _mT＝Y _b·U _T

Wherein $U_T=\left[\begin{array}{c}{U}_{\mathrm{mT}}\\ U_{\mathrm{umT}}\end{array}\right],$ U _umTfor in electric power system the vector that forms with or without the voltage phasor true value of Phasor Measurements node, Y _bfor electric power system measures the admittance matrix of branch road, the line number of matrix is the electric current phasor measurement number that electric power system measures branch road, matrix column number is the nodes of electric power system, when the Phasor Measurements configuration of the topological structure of electric power system, the network parameter of electric power system and electric power system is constant, and Y _bit is a sparse complex constant matrix;

(1-3) relational expression set up between the measurement amount of above-mentioned Phasor Measurements and quantity of state is as follows:

$Z_m=\left[\begin{array}{cc}E& 0\\ Y_{b1}& Y_{b2}\end{array}\right]U_T+\left[\begin{array}{c}{\mathrm{\ϵ}}_U\\ {\mathrm{\ϵ}}_I\end{array}\right]=A\·U_I+\mathrm{\ϵ}$

Wherein, E is unit matrix, and 0 is null matrix, Y _b=[Y _b1y _b2];

(2) target function based on the Power system state estimation of Phasor Measurements is set up:

$J\left(U_T\right)=\mathrm{\Σ}{\left(\frac{\left|{\mathrm{\ϵ}}_k\right|}{\left|Z_{\mathrm{mk}}\right|}\right)}^2={\mathrm{\ϵ}}^T\mathrm{P\ϵ}={(Z_m-A\·U_T)}^{T}P(Z_m-A\·U_T)$

Wherein, ε ^tfor the conjugate transpose of ε, ε _kfor a kth element of ε, Z _mkfor a kth measurement amount of electric power system, P represents diagonal matrix, and a kth diagonal element is | Z _mk| ^-2;

(3) adopt plural number least square method relatively, the plural measurement equation of simultaneous solution step (1) and the electric power system Phasor Measurements state estimation target function of step (2), comprise the following steps:

(3-1) setting C is a mn rank real/complex matrix, the complex vector of a to be length be n, the partial derivative computing complex field is defined as follows:

$\begin{array}{cc}\frac{\∂\left(\mathrm{Ca}\right)}{\∂a}=C& \frac{\∂{\left(\mathrm{Ca}\right)}^T}{\∂a}=C^T;\end{array}$

(3-2) target function of above-mentioned steps (2) is asked partial derivative to quantity of state:

$\frac{\∂J\left(U_T\right)}{\∂U}=-Z_m^T\mathrm{PA}-{\left(A^{T}P{Z}_m\right)}^T+U_T^T{A}^T\mathrm{PA}+{\left(A^T\mathrm{PA}{U}_T\right)}^T,$

By partial derivative zero setting, obtain the minimizing following expression of target function of Power system state estimation:

$-Z_m^T\mathrm{PA}=U_T^T{A}^T\mathrm{PA},$

According to this minimizing expression formula, the estimation formulas obtaining electric system state quantity is:

U _e＝(A ^TPA) ^-1A ^TPZ _m＝GZ _m。

The power system state estimation method based on Phasor Measurements that the present invention proposes, its advantage is, the inventive method is solving state estimation problem in complex field directly, only need once calculate the estimated value that can obtain quantity of state, and measurement transformed error can not be introduced, the dimension of coefficient matrix can not be increased, there is very strong Robustness least squares to phase angle error simultaneously, the inventive method considers the weighting effect of least square method simultaneously, therefore compare existing least square method, have better Power system state estimation accuracy.

Embodiment

The power system state estimation method based on Phasor Measurements that the present invention proposes, comprises the following steps:

(1) set up the plural measurement equation for Power system state estimation based on Phasor Measurements, detailed process is as follows:

(1-1) voltage measurement equation and the current measurement equation of electric power system is set up:

In each Phasor Measurements, voltage phasor measurement equation is:

${\stackrel{\·}{U}}_{m,i}={\stackrel{\·}{U}}_{T,i}+{\stackrel{\·}{\mathrm{\ϵ}}}_{U,i}$

Wherein with be respectively the measuring value of i-th voltage phasor, true value and error in measurement, measuring value, true value and error in measurement are plural form;

In each Phasor Measurements, electric current phasor measurement equation is:

${\stackrel{\·}{I}}_{m,j}={\stackrel{\·}{I}}_{T,j}+{\stackrel{\·}{\mathrm{\ϵ}}}_{I,j}$

Wherein with be respectively the measuring value of a jth electric current phasor, true value and error in measurement, measuring value, true value and error in measurement are plural number;

By above-mentioned voltage phasor measurement equation and electric current phasor measurement equation composition equation group, and be rewritten into following matrix form:

$Z_m=\left[\begin{array}{c}{U}_m\\ I_m\end{array}\right]=\left[\begin{array}{c}{U}_{\mathrm{mT}}\\ I_{\mathrm{mT}}\end{array}\right]+\left[\begin{array}{c}{\mathrm{\ϵ}}_U\\ {\mathrm{\ϵ}}_I\end{array}\right]$

Wherein U _m, U _mTand ε _ube respectively voltage phasor in Phasor Measurements and measure the measurement vector, the true value vector error in measurement vector that form, I _m, I _mTand ε _ithe electric current phasor be respectively in Phasor Measurements measures the measurement vector, the true value vector error in measurement vector that form;

(1-2) relation set up between electric current true value vector in above-mentioned Phasor Measurements and voltage true value vector is as follows:

For the current measurement in electric power system , what it was measured is the electric current phasor of branch road from p to q, if represented by branch road π type equivalent circuit, then has following relational expression to set up:

$\left\{\begin{array}{c}{\stackrel{\·}{I}}_{\mathrm{pq}}=Y_{\mathrm{pp}}{\stackrel{\·}{U}}_p+Y_{\mathrm{pq}}{\stackrel{\·}{U}}_q\\ {\stackrel{\·}{I}}_{\mathrm{qp}}=Y_{\mathrm{qp}}{\stackrel{\·}{U}}_p+Y_{\mathrm{qq}}{\stackrel{\·}{U}}_q\end{array}\right.$

Wherein, for in the measurement branch road pq of electric power system, from node q to the electric current phasor of node p, with be respectively in the measurement branch road pq of electric power system, the voltage phasor of node p and node q, Y _pp, Y _pq, Y _qp, Y _qqfor each branch admittance, expression is as follows:

$\left\{\begin{array}{c}{Y}_{\mathrm{qq}}=\frac{1}{r+\mathrm{jx}}+j\frac{b}{2}\\ Y_{\mathrm{qp}}=-\frac{1}{(r+\mathrm{jx})K{e}^{\mathrm{j\θ}}}\\ Y_{\mathrm{pp}}=\frac{Y_{\mathrm{qq}}}{{\left|K\right|}^2}\\ Y_{\mathrm{qp}}=-\frac{1}{(r+\mathrm{jx}){\left(K{e}^{\mathrm{j\θ}}\right)}^{*}}\end{array}\right.$

Wherein, K is the no-load voltage ratio of transformer in electric power system, and K is plural number; If measure branch road pq not containing transformer, then K=1, θ are the phase shifting angle of phase shifter in electric power system, unit is radian, if measure branch road pq not containing phase shifter, then θ=0, r, x and b are respectively the resistance of branch road in electric power system, reactance and susceptance, and j is imaginary number unit;

The electric current phasor of all branches in electric power system is measured composition equation group, and is rewritten as following matrix form:

I _mT＝Y _b·U _T

Wherein $U_T=\left[\begin{array}{c}{U}_{\mathrm{mT}}\\ U_{\mathrm{umT}}\end{array}\right],$ U _umTfor in electric power system the vector that forms with or without the voltage phasor true value of Phasor Measurements node, Y _bfor electric power system measures the admittance matrix of branch road, the line number of matrix is the electric current phasor measurement number that electric power system measures branch road, and matrix column number is the nodes of electric power system, Y _brelevant with the configuration of the topological structure of electric power system, the network parameter of electric power system and Phasor Measurements, when the Phasor Measurements configuration of the topological structure of electric power system, the network parameter of electric power system and electric power system is constant, Y _bit is a sparse complex constant matrix;

(1-3) relational expression set up between the measurement amount of above-mentioned Phasor Measurements and quantity of state (voltage phasor) is as follows:

$Z_m=\left[\begin{array}{cc}E& 0\\ Y_{b1}& Y_{b2}\end{array}\right]U_T+\left[\begin{array}{c}{\mathrm{\ϵ}}_U\\ {\mathrm{\ϵ}}_I\end{array}\right]=A\·U_I+\mathrm{\ϵ}$

Wherein, E is unit matrix, and 0 is null matrix, Y _b=[Y _b1y _b2], the vector in above-mentioned relation formula or the element in matrix are plural number;

(2) target function based on the Power system state estimation of Phasor Measurements is set up:

$J\left(U_T\right)=\mathrm{\Σ}{\left(\frac{\left|{\mathrm{\ϵ}}_k\right|}{\left|Z_{\mathrm{mk}}\right|}\right)}^2={\mathrm{\ϵ}}^T\mathrm{P\ϵ}={(Z_m-A\·U_T)}^{T}P(Z_m-A\·U_T)$

Wherein, ε ^tfor the conjugate transpose of ε, ε _kfor a kth element of ε, Z _mkfor a kth measurement amount of electric power system, P represents diagonal matrix, and a kth diagonal element is | Z _mk| ^-2;

(3) adopt plural number least square method relatively, the plural measurement equation of simultaneous solution step (1) and the electric power system Phasor Measurements state estimation target function of step (2), comprise the following steps:

(3-1) setting C is a mn rank real/complex matrix, the complex vector of a to be length be n, the partial derivative computing complex field is defined as follows:

$\begin{array}{cc}\frac{\∂\left(\mathrm{Ca}\right)}{\∂a}=C& \frac{\∂{\left(\mathrm{Ca}\right)}^T}{\∂a}=C^T;\end{array}$

(3-2) target function of above-mentioned steps (2) is the Convex quadratic function of quantity of state, and the target function of above-mentioned steps (2) is asked partial derivative to quantity of state:

$\frac{\∂J\left(U_T\right)}{\∂U}=-Z_m^T\mathrm{PA}-{\left(A^{T}P{Z}_m\right)}^T+U_T^T{A}^T\mathrm{PA}+{\left(A^T\mathrm{PA}{U}_T\right)}^T,$

By partial derivative zero setting, obtain the minimizing following expression of target function of Power system state estimation:

$-Z_m^T\mathrm{PA}=U_T^T{A}^T\mathrm{PA},$

According to this minimizing expression formula, the estimation formulas obtaining electric system state quantity is:

U _e＝(A ^TPA) ^-1A ^TPZ _m＝GZ _m。

Claims (1)

1., based on a power system state estimation method for Phasor Measurements, it is characterized in that the method comprises the following steps:

(1) set up the plural measurement equation for Power system state estimation based on Phasor Measurements, detailed process is as follows:

(1-1) voltage measurement equation and the current measurement equation of electric power system is set up:

In each Phasor Measurements, voltage phasor measures and is:

${\stackrel{\·}{U}}_{m,i}={\stackrel{\·}{U}}_{T,i}+{\stackrel{\·}{\mathrm{\ϵ}}}_{U,i}$

Wherein with be respectively the measuring value of i-th voltage phasor, true value and error in measurement;

In each Phasor Measurements, electric current phasor measures and is:

${\stackrel{\·}{I}}_{m,j}={\stackrel{\·}{I}}_{T,j}+{\stackrel{\·}{\mathrm{\ϵ}}}_{I,j}$

Wherein with be respectively the measuring value of a jth electric current phasor, true value and error in measurement;

By above-mentioned voltage phasor measurement equation and electric current phasor measurement equation composition equation group, and be rewritten into following matrix form:

$Z_m=\left[\begin{array}{c}{U}_m\\ I_m\end{array}\right]=\left[\begin{array}{c}{U}_{\mathrm{mT}}\\ I_{\mathrm{mT}}\end{array}\right]+\left[\begin{array}{c}{\mathrm{\ϵ}}_U\\ {\mathrm{\ϵ}}_I\end{array}\right]$

Wherein U _m, U _mTand ε _ube respectively voltage phasor in Phasor Measurements and measure the measurement vector, the true value vector error in measurement vector that form, I _m, I _mTand ε _ithe electric current phasor be respectively in Phasor Measurements measures the measurement vector, the true value vector error in measurement vector that form;

(1-2) relation set up between electric current true value vector in above-mentioned Phasor Measurements and voltage true value vector is as follows:

$\left\{\begin{array}{c}{\stackrel{\·}{I}}_{\mathrm{pq}}=Y_{\mathrm{pp}}{\stackrel{\·}{U}}_p+Y_{\mathrm{pq}}{\stackrel{\·}{U}}_q\\ {\stackrel{\·}{I}}_{\mathrm{qp}}=Y_{\mathrm{qp}}{\stackrel{\·}{U}}_p+Y_{\mathrm{qq}}{\stackrel{\·}{U}}_q\end{array}\right.$

Wherein, for in the measurement branch road pq of electric power system, from node q to the electric current phasor of node p, with be respectively in the measurement branch road pq of electric power system, the voltage phasor of node p and node q, Y _pp, Y _pq, Y _qp, Y _qqfor each branch admittance, expression is as follows:

$\left\{\begin{array}{c}{Y}_{\mathrm{qq}}=\frac{1}{r+\mathrm{jx}}+j\frac{b}{2}\\ Y_{\mathrm{qp}}=-\frac{1}{(r+\mathrm{jx})K{e}^{\mathrm{j\θ}}}\\ Y_{\mathrm{pp}}=\frac{Y_{\mathrm{qq}}}{{\left|K\right|}^2}\\ Y_{\mathrm{qp}}=-\frac{1}{(r+\mathrm{jx}){\left(K{e}^{\mathrm{j\θ}}\right)}^{*}}\end{array}\right.$

Wherein, K is the no-load voltage ratio of transformer in electric power system, and θ is the phase shifting angle of phase shifter in electric power system, and r, x and b are respectively the resistance of branch road in electric power system, reactance and susceptance, and j is imaginary number unit;

The electric current phasor of all branches in electric power system is measured composition equation group, and is rewritten as following matrix form:

I _mT＝Y _b·U _T

Wherein $U_T=\left[\begin{array}{c}{U}_{\mathrm{mT}}\\ U_{\mathrm{umT}}\end{array}\right],$ U _umTfor in electric power system the vector that forms with or without the voltage phasor true value of Phasor Measurements node, Y _bfor electric power system measures the admittance matrix of branch road, the line number of matrix is the electric current phasor measurement number that electric power system measures branch road, matrix column number is the nodes of electric power system, when the Phasor Measurements configuration of the topological structure of electric power system, the network parameter of electric power system and electric power system is constant, and Y _bit is a sparse complex constant matrix;

(1-3) relational expression set up between the measurement amount of above-mentioned Phasor Measurements and quantity of state is as follows:

$Z_m=\left[\begin{array}{cc}E& 0\\ Y_{b1}& Y_{b2}\end{array}\right]U_T+\left[\begin{array}{c}{\mathrm{\ϵ}}_U\\ {\mathrm{\ϵ}}_I\end{array}\right]=A\·U_I+\mathrm{\ϵ}$

Wherein, E is unit matrix, and 0 is null matrix, Y _b=[Y _b1y _b2];

(2) target function based on the Power system state estimation of Phasor Measurements is set up:

$J\left(U_T\right)=\mathrm{\Σ}{\left(\frac{\left|{\mathrm{\ϵ}}_k\right|}{\left|Z_{\mathrm{mk}}\right|}\right)}^2={\mathrm{\ϵ}}^T\mathrm{P\ϵ}={(Z_m-A\·U_T)}^{T}P(Z_m-A\·U_T)$

Wherein, ε ^tfor the conjugate transpose of ε, ε _kfor a kth element of ε, Z _mkfor a kth measurement amount of electric power system, P represents diagonal matrix, and a kth diagonal element is | Z _mk| ^-2;

(3) adopt plural number least square method relatively, the plural measurement equation of simultaneous solution step (1) and the electric power system Phasor Measurements state estimation target function of step (2), comprise the following steps:

(3-1) setting C is a mn rank real/complex matrix, the complex vector of a to be length be n, the partial derivative computing complex field is defined as follows:

$\begin{array}{cc}\frac{\∂\left(\mathrm{Ca}\right)}{\∂a}=C& \frac{\∂{\left(\mathrm{Ca}\right)}^T}{\∂a}=C^T;\end{array}$

(3-2) target function of above-mentioned steps (2) is asked partial derivative to quantity of state:

$\frac{\∂J\left(U_T\right)}{\∂U}=-Z_m^T\mathrm{PA}-{\left(A^{T}P{Z}_m\right)}^T+U_T^T{A}^T\mathrm{PA}+{\left(A^T\mathrm{PA}{U}_T\right)}^T,$

By partial derivative zero setting, obtain the minimizing following expression of target function of Power system state estimation:

$-Z_m^T\mathrm{PA}=U_T^T{A}^T\mathrm{PA},$

According to this minimizing expression formula, the estimation formulas obtaining electric system state quantity is:

U _e＝(A ^TPA) ^-1A ^TPZ _m＝GZ _m。