CN105119276B - A kind of electric system WLAV Robust filter methods based on ADMM - Google Patents

Abstract

The electric system WLAV Robust filter methods based on ADMM that the invention discloses a kind of, noise is measured by standardization and measurement equation is linearized based on Newton method, it is the optimization problem containing equality constraint by WLAV Robust filter model conversations, it is solved using ADMM, the decoupling substep for realizing state variable and auxiliary variable solves, the time is calculated to significantly reduce WLAV, while remaining its estimated accuracy.Ieee standard system test result shows that the estimated efficiency of WLAV greatly improved compared to PDIPM, ADMM, maintains the basic robustness of WLAV.The method provided by the present invention is the premise that the practical application of WLAV Robust filters provides technical support and ensures.

Description

A kind of electric system WLAV Robust filter methods based on ADMM

Technical field

The electric system WLAV Robust filter methods based on ADMM that the present invention relates to a kind of belonging to power system monitoring, divide Analysis and control technology field.

Background technology

State estimation estimates the real-time operating status of electric system, the knot based on state estimation according to the life data of telemetering Fruit, Energy Management System (energy management system, EMS) carry out a series of follow-up advanced applied software analysis meters It calculates (including Real-Time Scheduling, optimum control, safety analysis etc.), thus state estimation is the vital parts of EMS.Traditional Weighted least-squares (weighted least estimation, WLS) estimation obeys stringent Gaussian Profile in measurement noise When, it can efficiently estimate the best state of system.However since the aging of measuring instrument, the remote of data are transmitted, very To the bad data for being artificial malice injection, inevitably make the estimated result of WLS by the shadow of bad data (or measuring rough error) It rings, to deviate actual true value.

Due to that can inhibit to measure influence of the rough error to estimated accuracy, Robust filter causes grinding extensively for domestic and foreign scholars Study carefully.In numerous Robust filter devices, weighting least absolute value (weighted least absolute values, WLAV) is estimated Robustness is good, can resist a plurality of types of errors in measurement, has become the research emphasis in Robust filter field in recent years.It is existing WLAV estimations are generally basede on nonlinear primal-dual interior-point algorithm (primary dual interior point method, PDIPM) It solves, a large amount of auxiliary variable need to be quoted, considerably increase the solution scale of state estimation.WLAV is to increase computation complexity Cost improves the precision of state estimation, however compared to WLS estimators, lower computational efficiency also limits to a certain extent Its application in engineering practice.

For this purpose, the present invention proposes that one kind being based on alternating direction Multiplier Algorithm (alternating direction method Of multipliers, ADMM) electric system WLAV Robust filter methods.Noise is measured by standardization and is based on Newton method Measurement equation is linearized, is the optimization problem containing equality constraint by WLAV Robust filter model conversations, is solved using ADMM, it is real The decoupling substep for having showed state variable and auxiliary variable solves, and calculates the time to significantly reduce WLAV, while remaining it Estimated accuracy.Institute's extracting method computation complexity of the present invention is close with WLS, but estimated accuracy higher, is expected to as WLAV Robust filters Practical application provide it is technical support and ensure.

Invention content：

Goal of the invention：The present invention proposes a kind of electric system WLAV Robust filter methods based on ADMM, to improve electric power The computational efficiency of system WLAV Robust filters provides safeguard for the functionization of WLAV Robust filters.

Technical solution：The present invention provides following technical scheme：A kind of Robust filter sides electric system WLAV based on ADMM Method includes the following steps：

1) construction compressed format ADMM alternating direction Multiplier Algorithms iteratively solve step；

2) construction standardization measures noise；

3) it introduces and measurement equation is linearized based on Newton method；

4) it is formed and disregards the WLAV estimation equations for measuring weight；

5) ADMM iterative solution WLAV estimations are based on；

Construction compressed format ADMM iterative solutions step includes in the step 1)：

It is defined as follows the optimization problem containing equality constraint：

min f(x)+g(z)

S.t.Ax+Bz=c；

In formula：X, z are optimized variable, and A, B are constant matrix, and c is constant vector；

The Augmented Lagrangian Functions of above formula are：

In formula：y^TFor Lagrange multiplier, ρ is penalty factor；

R=Ax+Bz-c is defined, above-mentioned Augmented Lagrangian Functions can be converted into：

In formula：U=(1/ ρ) y is the dual variable of compression；

It is solved based on ADMM, the step of above formula is：

u^k+1=u^k+Ax^k+1+Bz^k+1-c；

Construction standardization measures noise and includes the following steps in the step 2)：

For measurement equation

Z=h (x)+v；

(1/ σ of L=diag defined in it₁,…,1/σ_m), L is multiplied by simultaneously in above formula both sides, can be obtained：

z^*=h^*(x)+e；

In formula：z^*=Lz, h^*(x)=Lh (x), e_i~N (0,1).

The WLAV estimation equations that measurement weight is disregarded in the middle formation of the step 3) include the following steps：

Measurement equation can be obtained with Newton method approximate linearization：

r^*=Δ z^*-H^*Δx；

In formula：Δz^*=z^*-h^*(x), H^*=LH；

It is represented by without the WLAV weighting least absolute value estimations for measuring weight：

Included the following steps based on ADMM iterative solution WLAV estimations in the step 4)：

Variable Δ x, r are solved with ADMM substeps^*,u：

The first step：

Obtain Δ x^k+1Afterwards, x is updated^k+1, H^*(x^k+1), Δ z^k+1：

1.a)x^k+1=x^k+Δx^k+1；

1.b)H^*(x^k+1)=LH (x^k+1)；

1.c)Δz^k+1=z^*-h^*(x^k+1)；

Second step：

In formula：

Third walks：

u^k+1=u^k+r^k+1-Δz^k+1+H^*(x^k+1)Δx^k+1。

Technique effect：The present invention is compared with prior art：It is solved using ADMM, realizes state variable and auxiliary variable Decoupling substep solves, and calculates the time to significantly reduce WLAV, while remaining its estimated accuracy.Ieee standard system testing The result shows that compared to PDIPM, the estimated efficiency of WLAV greatly improved in ADMM, maintains the basic Robustness least squares of WLAV Energy.The method provided by the present invention is the premise that the practical application of WLAV Robust filters provides technical support and ensures.

Description of the drawings：

Fig. 1：The method of the present invention flow chart.

Specific implementation mode：

The techniqueflow of invention is described in detail below in conjunction with the accompanying drawings：

ADMM brief introductions

ADMM algorithms can formally be divided into general type and compressed format, and the form and solution procedure of the two are slightly not Together, but essence is identical.The optimization aim and solution procedure of ADMM will be mainly introduced below.

ADMM is substantially to solve for the optimization problem containing equality constraint, for solving the optimization problem of following form

min f(x)+g(z)

S.t.Ax+Bz=c；

In formula：X, z are optimized variable, and A, B are constant matrix, and c is constant vector.

The Augmented Lagrangian Functions of above formula are：

In formula：y^TFor Lagrange multiplier, ρ is penalty factor.

It is solved based on ADMM, the solution procedure of above formula is：

y^k+1=y^k+ρ(Ax^k+1+Bz^k+1-c)；

ADMM compressed formats

ADMM can usually be expressed as another more succinct form, define r=Ax+Bz-c, then have：

In formula：U=(1/ ρ) y is the dual variable of compression.

It is solved based on ADMM, the step of above formula is：

u^k+1=u^k+Ax^k+1+Bz^k+1-c；

General solution form compared to ADMM, the ADMM of compressed format is substantially identical, but form is more succinct. The present invention will use the compressed format of ADMM.

WLAV estimations are solved based on ADMM

Measure the standardization of noise

For measurement equation

Z=h (x)+v；

Define L=diag (1/ σ₁,…,1/σ_m).L is multiplied by simultaneously in above formula both sides, can be obtained：

z^*=h^*(x)+e；

In formula：z^*=Lz, h^*(x)=Lh (x), e_i~N (0,1).

After measuring noise normalization, the noise for measuring z* obeys standardized normal distribution, thus the target letter of state estimation Number is without meter and weight matrix.It is worth noting that, traditional state estimator, which is not necessarily to standardization, measures noise, optimization Object function can directly count and weight matrix, however solution of the standardisation process of above-mentioned measurement noise for hereafter model It is indispensable.

Newton method approximate linearization

Measurement equation can be obtained with Newton method approximate linearization：

r^*=Δ z^*-H^*Δx；

In formula：Δz^*=z^*-h^*(x), H^*=LH.

According to above-mentioned residual equation, it is represented by without the WLAV estimations for measuring weight：

ADMM is solved

The Augmented Lagrangian Functions form of WLAV estimations is represented by：

In formula：u∈R^m, it is dual variable.

Variable Δ x, r are solved with ADMM substeps^*,u：

Step1：

Obtain Δ x^k+1Afterwards, x is updated^k+1, H^*(x^k+1), Δ z^k+1：

1.a)x^k+1=x^k+Δx^k+1；

1.b)H^*(x^k+1)=LH (x^k+1)；

1.c)Δz^k+1=z^*-h^*(x^k+1)；

Step2：

In formula：

Step3：

u^k+1=u^k+r^k+1-Δz^k+1+H^*(x^k+1)Δx^k+1；

From the above equation, we can see that ADMM algorithms realize the decoupling decoupled method of quantity of state x, residual error r, dual variable u, and The form of calculation of step1 is similar to WLS, and difference is only that the right side Δ z according to u, r update equation^k-(r^*)^k-u^k, thus be based on The WLAV that ADMM is solved have with computational efficiency similar in WLS, but robust effect is more preferable.And compared to being solved based on PDIPM WLAV, ADMM estimated accuracy are close, but computational efficiency is more preferably.

Sample calculation analysis

The sample calculation analysis of this paper using Intel Core i3 3.3GHz, 4GB RAM PC machine as test platform, be based respectively on ADMM and PDIPM solves WLAV, tests the estimated accuracy under different service conditions, measure configuration, bad data ratio.This selected works Multiple standard examples in MATPOWER are taken, metric data is in different service conditions, different measure configurations, different measurement noises Under randomly generate：

1) not different degrees of load (between 50%~150%, not considering the idle constraint of generator).

2) measurement of different redundancies (between 2.5~4.5, and system Observable).

3) standard deviation of the random noise of Gaussian distributed, power measurement noise is that voltage measures the standard deviation of noise.

4) bad data of different proportion (between 1% to 10%).

Metric data is generated according to above-mentioned rule, stochastic simulation 200 times compares estimated accuracy and the calculating of two methods Efficiency.

Estimated accuracy compares

It is based respectively on PDIPM and ADMM and solves WLAV, stochastic simulation 200 times counts S^VWith S^θMean value and the coefficient of variation. Mean value can compare S on the whole^VWith S^θSize, i.e. the estimated accuracy of algorithm entirety；And the coefficient of variation then weighs S^VWith S^θFrom The degree of dissipating, the i.e. stability of algorithm.Table 1 is respectively PDIPM, ADMM S under different examples from table 2^VWith S^θMean value and discrete journey Degree compares.

By table 1 and table 2 it is found that being based respectively on ADMM, PDIPM solves WLAV, difference tests S under example^VWith S^θMean value with Coefficient of dispersion is substantially similar, illustrates that the two has more similar overall estimation precision and estimation stability.

S under the different examples of table 1^VWith S^θMean value compare

S under the different examples of table 2^VWith S^θThe coefficient of variation compare

The computation complexity of ADMM and PDIPM under the different examples of table 3

ADMM is compared with the calculating time of PDIPM under the different test examples of table 4

Note：Speed-up ratio is the ratio that PDIPM calculates time and ADMM calculating times.

Computation complexity compares

Compared to PDIPM, the feature of ADMM maximums is that the decoupling for realizing quantity of state, residual error, dual variable is counted step by step It calculates, thus significantly reduces the dimension of correction matrix, table 3 is the update equation dimension of different test examples and non-zero element Number.

As shown in Table 3, ADMM correction matrixs dimension, non-zero element number are significantly less than PDIPM, thus computation complexity is more It is low.It is based respectively on ADMM, PDIPM and solves WLAV, the calculating time under different examples is as shown in table 4.

As shown in Table 4, under different system scale, ADMM iterations are held essentially constant, and PDIPM iterations are with being System scale increases and increases.Compared to PDIPM, ADMM computational efficiencies improve 1.5~4 times, and system scale is bigger, computational efficiency On advantage it is more apparent.

Claims (1)

1. a kind of electric system WLAV based on ADMM alternating direction Multiplier Algorithms weights least absolute value Robust filter method, It is characterized in that, includes the following steps：

1) construction compressed format ADMM alternating direction Multiplier Algorithms iteratively solve step；

2) construction standardization measures noise；

3) it introduces and measurement equation is linearized based on Newton method；

4) it is formed and disregards the WLAV estimation equations for measuring weight；

5) ADMM iterative solution WLAV estimations are based on；

Construction compressed format ADMM iterative solutions step includes in the step 1)：

It is defined as follows the optimization problem containing equality constraint：

min f(x)+g(z)

S.t. Ax+Bz=c；

In formula：X, z are optimized variable, and A, B are constant matrix, and c is constant vector；

The Augmented Lagrangian Functions of above formula are：

In formula：y^TFor Lagrange multiplier, ρ is penalty factor；

R=Ax+Bz-c is defined, above-mentioned Augmented Lagrangian Functions can be converted into：

In formula：U=(1/ ρ) y is the dual variable of compression；

It is solved based on ADMM, the step of above formula is：

u^k+1=u^k+Ax^k+1+Bz^k+1-c；

Construction standardization measures noise and includes the following steps in the step 2)：

For measurement equation

Z=h (x)+v；

(1/ σ of L=diag defined in it₁,…,1/σ_m), L is multiplied by simultaneously in above formula both sides, can be obtained：

z^*=h^*(x)+e；

In formula：z^*=Lz, h^*(x)=Lh (x)；

The WLAV estimation equations that measurement weight is disregarded in the middle formation of the step 3) include the following steps：

Measurement equation can be obtained with Newton method approximate linearization：

r^*=Δ z^*-H^*Δx；

In formula：Δz^*=z^*-h^*(x), H^*=LH；

It is represented by without the WLAV weighting least absolute value estimations for measuring weight：

Included the following steps based on ADMM iterative solution WLAV estimations in the step 4)：

Variable Δ x, r are solved with ADMM substeps^*,u：

The first step：

Obtain Δ x^k+1Afterwards, x is updated^k+1, H^*(x^k+1), Δ z^k+1：

1.a)x^k+1=x^k+Δx^k+1；

1.b)H^*(x^k+1)=LH (x^k+1)；

1.c)Δz^k+1=z^*-h^*(x^k+1)；

Second step：

In formula：

Third walks：

u^k+1=u^k+r^k+1-Δz^k+1+H^*(x^k+1)Δx^k+1。