Non Gaussian State Estimation in Power Systems - CiteSeerX

International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 28-30, 2006

Non Gaussian State Estimation in Power Systems Roberto M´ınguez Department of Mathematics, University of Castilla-La Mancha.

Antonio J. Conejo Department of Electrical Engineering University of Castilla-La Mancha.

Ali S. Hadi Department of Mathematics American University in Cairo.

Abstract Most power system state estimators consider measurements as independent gaussian random variables and use a least square approach to estimate the most likely system state. In this paper we drop the questionable assumptions of independency and normality of measurements and develop techniques to accurately estimate the system state through appropriate transformations of correlated non-gaussian measurements. An example illustrates the proposed estimation technique. Conclusions are finally drawn.

Key Words: Least square estimation, Non gaussian random variables, Power system state estimation.

1

Introduction

Owing to the complexities of operating large, interconnected networks, more and more electric utilities replace their tradicional dispatch offices with modern EMS (Energy Management System). The purpose of an EMS is to monitor, control, and optimize the transmission and generation facilities with advanced computer technologies. The aim of the state estimation is to get the best estimate of the current system states processing a set of real-time redundant measures and network parameters available in the EMS database. The performance of state estimation, therefore, depends on the accuracy of the measured data as well as the parameters of the network model. The measured data are subject to noise or errors in the metering system and the communication process. Large errors in the analog measurements, so-called bad data, may happen in practice. Network ∗ Correspondence to: Roberto M´ınguez. Department of Mathematics. University of Castilla-La Mancha, Ciudad Real, Spain. email: [email protected]

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R. M´ınguez, A. J. Conejo and A. S. Hadi

parameters such as impedances of transmission lines may be incorrect as a result of inaccurate manufacturing provided data, error in calibration, etc. In addition, due to the lack of field information and possible errors in calculations, transformer tap positions may be erroneous. The purpose of a state estimator is to filter all these errors to achieve the best possible estimate of the state of the system. Background on state estimation can be found, for instance, in Larson et al. (1970), Schweppe and Wildes (1970), Schweppe and Rom (1970), Schweppe (1970), Garc´ıa et al. (1979), Holten et al. (1988), Monticelli and Garc´ıa (1990), Monticelli (2000), and Abur and Exp´osito (2004). This paper is related to and partly motivated by state estimation in distribution networks where the number of available real time measurements is generally low and load predictions are used as pseudomeasurements. Past work on statistical modeling of the states of distribution networks has shown that loads are not normally distributed, Borkowska (1974), Allan and Al-Shakarchi (1976), Dopazo et al. (1975), EPRI Report EA-3467 (1984), Anders (1990), and Herman and Kritzinger (1993), i.e., loads are not generally gaussian random variables. It is also recognize that certain measurements such as voltage and active and reactive power injections at a given bus, are correlated in both transmission and distribution networks. Due to these reasons variables in state estimation cannot be modeled as independent gaussian random variables. A general WLS (weighted-least-square) state estimator that does not require the measurements to be either Gaussian or statistically independent is proposed. It is based on the maximum likelihood estimation using appropriate statistical transformations. This paper is organized as follows. In Section 2 the maximum likelihood estimation method and associated statistical assumptions are presented. In Section 3 the likelihood estimation using different transformations (Transformed Likelihood Estimation, TLE) is explained. In Section 4 the TLE method is applied to the power system state estimation problem (General State Estimation, GSE). In Section 5 a method for bad data detection is provided. Section 6 gives results from an illustrative example to demonstrate the functioning of the method. And finally, Section 7 provides some conclusions.

Non Gaussian State Estimation in Power Systems

2

3

Maximum Likelihood Estimation

The objective of the state estimation is to determine the most likely state of a system based on the quantities that are measured, i.e., measurements. One way to accomplish this is by the maximum likelihood estimation (MLE), a method widely used in statistics. The measured errors are assumed to have a known probability distribution with unknown parameters. This function is referred to as the likelihood function and will attain its peak value if the unknown parameters are selected to be closest to their actual values. Hence, an optimization problem can be set up in order to maximize the likelihood function as a function of the unknown parameters. The solution gives the maximum likelihood estimates for the parameters of interest. Consider the joint probability density function which represents the probability of m measurements, fZ (z; θ) = fZ (z1 , z2 , . . . , zm ; θ)

(2.1)

where z T = (z1 , z2 , . . . , zm ) is the vector of measures, θ is the vector containing the probability distribution function parameters. Note that the joint probability distribution function of measurements is FZ (z). The objective of the maximum likelihood estimation is to maximize the likelihood function (2.1) by varying the assumed parameters θ. In determining the optimum parameter values, the function is commonly replaced by its logarithm in order to simplify the optimization procedure. The modified function is called log-likelihood function log(fZ (z; θ)). Thus the likelihood estimation problem for a given set of measurements z can be stated as: maximize log(fZ (z; θ)). (2.2) θ Note that the complexity of the log-likelihood function depends on the probability distribution selected. Its expression can be very simple, i.e., if the variables are assumed independent and normally distributed, or very complex. In this case, if the joint probability density function (2.1) is completely described, it is possible to obtain a set of independent normal ran-

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R. M´ınguez, A. J. Conejo and A. S. Hadi

dom variables which may then be used with regression methods, reducing the complexity of the problem. This paper analyzes this transformation. 2.1

Transformation of random variables

Consider the general random variable vector z with known joint probability density function (2.1) which is related to other vector of random variables y (Freeman (1963)) by a known function: y = y(z) having unique (i.e., one to one) inverse z = z(y). Then fY (y; θ) = fZ (z; θ)|J |

(2.3)

where J is the Jacobian of the transformation whose elements are Jij = ∂zj /∂yi . Transformation (2.3) is valid provided its uniqueness. In this context the next subsections present several methods for transforming arbitrary random variables into independent standard normal variables. 2.1.1

Rosenblatt Transformation

Consider a vector of uniformly distributed random variables denoted by U . Let these be the intermediaries between the random variables in the original space Z, and the standardized normal variables Y . Provided that the joint probability distribution function FZ (z) it is known, its conditional distributions Fi (zi |z1 , z2 , . . . , zi−1 ) are available. The Rosenblatt (1952) transformation in the m-dimensional space becomes: Φ(y1 ) = u1 = F1 (z1 ) Φ(y2 ) = u2 = F2 (z2 |z1 ) .. .. .. . . . Φ(yi ) = ui = Fi (zi |z1 , z2 , . . . , zi−1 ) .. .. .. . . . Φ(ym ) = um = Fm (zm |z1 , z2 , . . . , zm−1 )

(2.4)

Non Gaussian State Estimation in Power Systems

5

where Φ(·) is the standard normal cumulative distribution function and u1 , . . . , um are the uniformly distributed variables U (0, 1). The component of vector Y can be obtained by successive inversion: y1 = Φ−1 [F1 (z1 )] y2 = Φ−1 [F2 (z2 |z1 )] .. .. .. . . . yi = Φ−1 [Fi (zi |z1 , z2 , . . . , zi−1 )] .. .. .. . . .

(2.5)

ym = Φ−1 [Fm (zm |z1 , z2 , . . . , zm−1 )]. A previous step is to obtain the jacobian of the transformation. For simplicity, the inverse of the Jacobian is obtained first:  0 if i < j     fi (zi |z1 , . . . , zi−1 )  ∂yi if i = j Jij−1 = = (2.6) φ(yi )  ∂zj  1 ∂Fi (zi |z1 , . . . , zi−1 )   if i > j  φ(yi ) ∂zi where the Jacobian is given in terms of z. Note that J −1 and J are lower triangular matrices, where J can be obtained from J −1 by back substitution. 2.1.2

Nataf Transformation

If only marginal probability distributions and correlation data are available, even for non-normal random variables, the Nataf transformation may be applied to give a set of independent normal random variables. Note that as there is no information about the conditional distributions the Rosenblatt transformation cannot be applied. This transformation creates an approximation based on a joint normal distribution. Considering the marginal cumulative distribution functions FZi (zi ) the

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R. M´ınguez, A. J. Conejo and A. S. Hadi

transformation in (2.5) becomes: y1 = Φ−1 [F1 (z1 )] y2 = Φ−1 [F2 (z2 )] .. .. .. . . . yi = Φ−1 [Fi (zi )] .. .. .. . . .

(2.7)

ym = Φ−1 [Fm (zm )]. It is now assumed that Y is jointly normal, with m-dimensional standard normal probability density function φm (y, ρ) having zero means, unit standard deviations and correlation matriz ρ = {ρij }. The Nataf (1962) approximation for the joint probability density function is given by: fZ (z; θ) = φm (y, ρ; θ)|J |

(2.8)

where the Jacobian determinant |J | is obtained, taking into account (2.6), by: φ(y1 )φ(y2 ) · · · φ(ym ) |J| = . (2.9) fZ1 (z1 )fZ2 (z2 ) · · · fZm (zm ) Methods for obtaining ρ are shown in Liu and Der Kiureghian (1986). Note that the resulting distribution may be transformed to an independent standarized distribution φm (t, I) through the orthogonal transformation shown below. 2.1.3

Orthogonal Transformation of Normal Random Variables

Let Z be a correlated vector of random variables, with mean µZ and covariance matrix C Z . This matrix is diagonal if the variables are uncorrelated. This is a sufficient measurement of dependence for normal distributions. An uncorrelated vector U , and a linear transformation matrix A, is now sought, such that U = AZ. (2.10) It is desirable that transformation (2.10) is also orthogonal, i.e., Euclidean distances remain unchanged. From matrix theory (Golub (1996)), this implies that AT = A(−1) .

Non Gaussian State Estimation in Power Systems

7

Under the linear transformation (2.10) the covariance matrix is also transformed, becoming: C U = AC Z AT . (2.11) To obtain an uncorrelated vector U , the matrix C U has to be strictly diagonal, with its off-diagonal elements (i.e. i 6= j) equal to zero. This can be achieved by finding the characteristic values (eigenvalues) of C Z , which can be transformed into: C Z = AZ C U ATZ ,

(2.12)

where matrix C U contains the characteristic values λii and the columns of matrix AZ are the characteristic vectors. Proceeding in this way matrix A in (2.11) is equal to ATZ . Note that under transformation (2.10) the inverse of the Jacobian becomes matrix A, known form matrix theory to be orthogonal. It follows readily that in this case |J −1 | = ±1. There is another alternative manner to get an orthogonal transforma1/2 tion. Note that the standard deviations of C U (λii ) must all be positive, since they have no physical meaning otherwise. This means that C Z must be a positive definite matrix, thus Cholesky decomposition can be applied C Z = LLT . Substituting this expression in (2.11) and considering that our aim is to get C U = I we have: CU

T = AC Z A ¡ T T¢ = (AL) L A = I.

(2.13)

For expression (2.13) to hold A must be equal to L−1 . This way matrix A will be lower-triangular. The last step is to transform the variables U ∼ N (µZ , I) into the standardized normal random variables Y ∼ N (0, I) by means of: (2.14) Y = A(Z − µZ ). Using transformation (2.14) equation (2.3) becomes: m Y

fYi (yi ; θ) = fZ (z; θ)|A|

i=1

where the Jacobian J corresponds to matrix A.

(2.15)

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R. M´ınguez, A. J. Conejo and A. S. Hadi

The Transformed Likelihood Estimation Problem

An alternative formulation of the Maximum Likelihood Estimation problem (2.2) consist in transforming the random variables Z into an independent standardized normal random set Y using the transformations stated in Section 2. Thus using expression (2.3) problem (2.2) becomes: maximize log(fY (y; θ)/|J |) θ

(3.1)

y = T (z; θ)

(3.2)

subject to

θ

min

≤ θ≤θ

max

(3.3)

where (3.2) is the transformation, i.e. Rosenblatt (2.5), Nataf (2.7) or the orthogonal transformation (2.14), J is the Jacobian of the corresponding transformation, and (3.3) is the constraint fixing limits for the parameters involved. As in the transformations above Y are standardized normal random variables, the objective function (3.1) is equivalent to: minimize θ

m X

yi2 .

(3.4)

i=1

Figure 1 shows a graphical interpretation of the transformed likelihood estimation problem where the optimal solution corresponds to the values of the parameters which make the sum of the square distances of the transformed points to the origin minimum.

4

General State Estimation (GSE) Formulation

Most state estimation models in practical use are formulated as overdetermined systems of nonlinear equations. Consider the nonlinear measurement model z = h(xtrue ) + e, (4.1)

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Non Gaussian State Estimation in Power Systems y2

z2

µz2

Joint probability density function contours

y2 Joint probability density function contours

fy(y) y1

fz(z) µz1

y1

z1

(a)

(b)

Figure 1: Graphical illustration of the the maximum likelihood estimation problem for two measurements: (a) Measurements in the original space, (b) measurements in the standard normal random space.

where z is the vector of measurements, xtrue is the true state vector, i.e. the parameters in the above statistical model (θ), h is a nonlinear function vector relating measurements to states, and e is the measurement error. There are m measurements and n state variables, n < m. In general, the state estimation problem can be formulated mathematically as an optimization problem including equality and inequality constraints as: m X minimize yi2 (4.2) x i=1 subject to y = T (z; h(x))

(4.3)

c(x) = 0

(4.4)

g(x) ≤ 0

(4.5)

where x is the vector of state variables (parameters to be obtained), h(x) are the nonlinear functions relating measurements to states, i.e., power flow quantities (dependent variables), c(x) are the equality constraints representing very accurate measurements (zero injections), and g(x) are inequality constraints normally used to represent physical operating limits. Note that constraint (3.3) is included in (4.5).

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R. M´ınguez, A. J. Conejo and A. S. Hadi

Note that in this paper we consider as state variables nodal voltages and angles. Power flows in branches that follow Kirchoff’s laws are dependent variables and can be determined from the state variables. However, in branches where the application of the Kirchoff’s law is not fruitful, such as branches with unknown impedances, flows can be introduced as additional state variables. 4.1

Independent Gaussian probability density function

Traditionally, measurement errors in state estimation are assumed to be independent and to have a Gaussian (Normal) distribution. Under these assumptions, and considering (2.14), problem (4.2)-(4.5) becomes: m X

minimize x

yi2

(4.6)

i=1

subject to zi − hi (x) σi c(x) = 0 yi =

g(x) ≤ 0,

(4.7) (4.8) (4.9)

where (4.7) corresponds to the Rosenblatt transformation for this particular statistical assumption. Considering the residual as r = z − h(x) and the weight Wi = σi−1 the above problem can be expressed as: m X

Wi ri2

(4.10)

ri = zi − hi (x)

(4.11)

minimize x

i=1

subject to

c(x) = 0

(4.12)

g(x) ≤ 0,

(4.13)

which is the classical weighted least squares formulation.

Non Gaussian State Estimation in Power Systems

4.2

11

Dependent Gaussian probability density function

If measurement errors are assumed to be dependent and to have a Gaussian (Normal) distribution with correlation matrix ρ and standard deviation vectors σ known. Problem (4.2)-(4.5) becomes: minimize x

m X

yi2

(4.14)

i=1

subject to y = A(z − h(x))

(4.15)

c(x) = 0

(4.16)

g(x) ≤ 0,

(4.17)

where matrix A is the inverse of the Cholesky transformation matrix of the covariance matrix whose elements are σi ρij σj . 4.3

Dependent Non-Gaussian probability density function

The most general case corresponds to the case where measurement errors are assumed to be dependent and to have a Non-Gaussian distribution. This case corresponds to problem (4.2)-(4.5).

5

Bad Data Detection

One advantage of the proposed method is that the Chi-squares test can be applied in a simple manner. Note that the transformation (3.2) allows us to obtain a set of independent standard normal random variables Yi ∼ N (0, 1), and considering that the objective function (3.4) of the transformed likelihood estimation problem is the sum of squares of the Yi variables, therefore, the objective function has a χ2 distribution with at most (m − n) degrees of freedom, since in power systems, at least n measurements will have to satisfy the power balance equations. Thus, χ2 test for detecting bad data can be used as follows: 1. Solve the estimation problem and obtain the objective function optiP 2. mal value m y i=1 i

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R. M´ınguez, A. J. Conejo and A. S. Hadi

2. Get the value of the Chi-squares distribution function corresponding to a detection confidence with probability β, χ2m−n,β . P 2 2 3. Check if m i=1 yi ≥ χm−n,β ; if yes, bad data is suspected; else, data are assumed to be free of error.

6

Illustrative Example

The 6-bus electric energy system depicted in Fig. 2 is considered in this example (Shebl´e (1999)). The data of a power flow for this system is provided in Conejo et al. (2005). Note that a power flow provides all variables to define a state of the system. Bus 2 (GENCO 2)

Bus 3 (GENCO 3)

Bus 6 (ESCO 3) Bus 1 (GENCO 1)

Bus 5 (ESCO 2) Bus 4 (ESCO 1)

Figure 2: Six-bus system.

The measurements considered are the following: 1. Voltage magnitude for every node. 2. Active and Reactive power injection at every node. 3. Active and Reactive power flow at both ends of every line or transformer.

Non Gaussian State Estimation in Power Systems

13

Hence the total number of measurements and the number of degrees of freedom for this 6-bus system are: 6 + 2 × 6 + 2 × 2 × 11 = 62, and 62 − 2 × 6 + 1 = 51, respectively. 6.1

Statistical assumptions

To show the importance of the dependency between measurements in the estimation process the following statistical model has been used and the provided results are compared with those obtained using a model that uses the traditional independent normal random variable assumption. Note that the proposed model has been selected for illustration purposes, as additional research is needed on the right estimation of the best measurements probability distribution. The formula used by the American Electric Company for determining variance values for measurements in its state estimator is (Allemong et al. (1982)):  q 2 2    qPFkm + QFkm for flow k − m σi = 0.0067Si +0.0016FSi where Si = Pk2 + Q2k for bus k    vk for voltage k (6.1) and PFkm , QFkm are the active and reactive power flow magnitudes, respectively, Pk , Qk are the active and reactive power injection magnitudes, respectively, and vk is the voltage magnitude in bus k. FSi is the scale factor which in this paper will be taken as max(PFkm , QFkm ) for flow measurements, max(Pk , Qk ) for power injection measurements, and max(vk ) for voltage measurements, respectively. Additionally, measurement systems generally establish a correlation between voltage measurements and the remainder measurements (active and reactive power injections, active and reactive power flows) in the same bus. Therefore, dependency between voltage measurements and the remainder measurements is considered. To model this complex mechanism, a correlation coefficient α is used, so that, α = ±1 implies the maximum correlation possible (positive or negative) whereas α = 0 implies independency. Figure 3 shows the pattern of the covariance matrix using this model. Additionally, as the voltage measurements are positive variables they

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R. M´ınguez, A. J. Conejo and A. S. Hadi

v

0

P

Q

PF

QF v P

10

Q 20

PF

30

40

50

QF

60 0

10

20

30 nz = 174

40

50

60

Figure 3: Covariance matrix pattern for the six-bus system example.

are modeled using random variables with a log-normal distribution ln(Vi ) ∼ 2 N (µln Vi , σln Vi ) whose parameters can be obtained form those corresponding to the normal variable as follows: ¢2 ¡ V vi 2 µln Vi = ln q and σln (6.2) Vi = ln(1 + σi /vi ). ¡ V ¢2 (1 + σi /vi ) where σiV is the standard deviation for bus i obtained using equation (6.1). The General State Estimation problem (GSE) (4.2)-(4.5) for this particular example is stated below: Minimize vi , δi , Pi , Qi ; i = 1, . . . , 6; PFij , QFij ; (i, j) ∈ ΩPF

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Non Gaussian State Estimation in Power Systems

J(x) =

X

(yiV )2 +

i∈ΩV

X

X

(yiP )2 +

i∈ΩP

(yiQ )2 +

i∈ΩQ

X

PF 2 (yij ) +

(i,j)∈ΩPF

X

QF 2 (yij )

(i,j)∈ΩQF

(6.3) subject to 



yiV

=

yiP

=

yiQ = PF yij

QF yij

Pi

=

vi  ln(vim ) − ln  q 1 + (σiV /vj ) q ; i ∈ ΩV V ln 1 + (σi /vi )

(6.4)

Pim − Pi − ασiP Q yiV ; i ∈ ΩP P Q√ 2 σi 1−α

(6.5)

PQ V Qm i − Qi − ασi yi ; i ∈ ΩQ √ PQ 2 σi 1−α

(6.6)

P QF V PFmij − PFij − ασij yi ; (i, j) ∈ ΩPF √ P QF 2 σij 1−α

(6.7)

P QF V Qm yi Fij − QFij − ασij = ; (i, j) ∈ ΩQF P QF √ 2 σij 1−α X = vi vj (Gij cos(θi − θj ) + Bij sin(θi − θj )) ;

(6.8)

j

Qi

i = j; (i, j) ∈ Ωi ; i = 1, . . . , 6 X = vi vj (Gij sin(θi − θj ) − Bij cos(θi − θj )) ;

(6.9)

j

i = j; (i, j) ∈ Ωi ; i = 1, . . . , 6 PFij

= vi vj (Gij cos(θi − θj ) + Bij sin(θi − θj )) −Gij vi2 ; (i, j) ∈ ΩPF

QFij

(6.10) (6.11)

= vi vj (Gij sin(θi − θj ) − Bij cos(θi − θj )) +vi2 (Bij − bsij /2); (i, j) ∈ ΩQF

(6.12)

Pimin ≤ Pi ≤ Pimax ;

i = 1, · · · , 3

(6.13)

Qmin i

i = 1, · · · , 3

(6.14)

≤ Qi ≤

Qmax i

−π ≤ δi ≤ π ; δ1 = 0,

;

i = 2, · · · , 6

(6.15) (6.16)

where constraint (6.3) is the objective function including the sums of the

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R. M´ınguez, A. J. Conejo and A. S. Hadi x 10 -4 16

50

14

GSE

45

12 10

40

etrue

J(x) 35

WLS

8 6

WLS

30

4 GSE

2 25

0 -1

-0.5

0

0.5

1 α

(a)

-1

-0.5

0 (b)

0.5

1 α

Figure 4: Graphical illustration of the the estimation state performance considering WLS and GSE: (a) Objective function, (b) true quadratic error.

squares of the normalized errors corresponding to voltage, active and reactive power injection, and active and reactive power flow measurements, respectively, (6.4)-(6.8) are the equations corresponding to the Rosenblatt P QF transformation, being σiV , σiP Q , and σij the standard deviation values associated with voltage, active and reactive power injection, and active and reactive power flow measurements, respectively, which are obtained using equation (6.1). Constraints (6.9)-(6.12) are the active and reactive power injections and flows equations, and the last constraints are physical limits, such us minimum and maximum reactive power generation and angle extreme values (−π,π). Note also that Gij , Bij , and bSij are the element ij of the real part of the admittance matrix, the element ij of the imaginary part of the admittance matrix, and the charging susceptance of line ij, respectively. Measurements vm , Pm , Qm , PF m and QF m are synthetically generated by adding randomly generated errors to the true values vtrue , Ptrue , Qtrue , PF true and QF true . Note that these process is done generating independent standard normal random numbers and using the inverse of the Rosenblatt transformation to obtain the measurements. Several synthetically generated measurements using different values of the correlation factor α have been performed. Next, the general state estimation problem (GSE) (6.3)-(6.16) and the usual weighted least square

Non Gaussian State Estimation in Power Systems

17

estimation problem (WLS), considering independent normal random variables, are solved. The corresponding objective functions J(x) and the regression curves for both approaches are shown inPFigure 4 (a). In Figure est true )2 , i.e., 4 (b) the true quadratic errors defined as etrue = m i=1 (xi − xi the square sum over the total number of measurements of the difference between the true value and the estimation. The following observations are pertinent: 1. For α = 0 the objective function and the true quadratic error are almost the same, the small difference being due to the use of the lognormal distribution for the voltage measurements in the GSE model. 2. The WLS objective function decreases if the correlation factor absolute value increases, whereas the GSE objective function increases slightly. Note that the maximum differences in the objective functions for the WLS and GSE models are 49.9% and 3.6%, respectively. These result could suggest that the WLS is a better approach because the chi-square value (objective function) is smaller, but the true errors (etrue ) are clearly smaller for the GSE estimation, as shown in Figure 4 (b). 3. True errors decrease if the correlation absolute value increases, and the maximum difference between errors using both approaches occurs for α ± 1. Additionally, to check the behavior of both models if outliers (bad measurements) exist, a synthetically bias over the measurement in the voltage for bus 1 (a typical bus) equal to 6 standard deviations (a clear outlier) is included. The corresponding objective functions J(x) and the regression curves for both models (WLS and GSE) are shown in Figure 5 (a), whereas in Figure 5 (b) the true quadratic errors are shown. The following observations are pertinent: 1. The behavior of the objective function for both models is analogous to the previous case (see Figure 4 (a)) but with a larger objective function due to the outlier. 2. Figure 5 (a) also shows horizontal lines corresponding to the χ2(m−n),β distribution function corresponding to a detection confidence with

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R. M´ınguez, A. J. Conejo and A. S. Hadi

90

β

80

0.995 0.98 0.96 0.90

GSE

70 J(x) 60

χ2

51, β

50 40

α ~ 0.525 WLS

α ~ -0.5

30 -1

-0.5

0 (a)

0.5

α

1

x 10 -3 10 9 8 7 6 etrue 5 4 3 2 1 0 -1

WLS

GSE

-0.5

0 (b)

0.5

α

1

Figure 5: Graphical illustration of the the estimation state performance considering WLS and GSE and including an outlier: (a) Objective function, (b) true quadratic error.

probability β. Note that the WLS estimation fails to detect suspicious data if the correlation coefficient absolute value increases. For instance, if the confidence probability is 96%, WLS estimation fails to detect bad data if α ≤ −0.5 and α ≥ 0.525. The range where the WLS estimation fails increases with the confidence probability. On the other hand GSE model detects bad data for all values of α and β considered. 3. The WLS true error is always greater than the GSE true error. The difference increases as the correlation factor absolute value increases, attaining the maximum value (−0.0091) when α = −1. 4. The GSE is more robust because the maximum difference between the true error with and without outlier is ≈ 13.1% whereas in the WLS model the difference is ≈ 83.6%. Unlike common practice in state estimation, the proposed method is directly based on solving an optimization problem. We advocate this approach due to the versatility, efficiency and robustness of currently available software. We emphasize that the available optimization codes efficiently account for sparsity and possible numerical ill-conditioning.

Non Gaussian State Estimation in Power Systems

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19

Conclusion

This paper provides a procedure for solving the state estimation problem considering that the measurements are non gaussian and can be statistically correlated. The method is useful to increase the confidence level of the model and its predictions. It has the following advantages with respect the standard WLS model: 1. It is more robust with respect to outliers. 2. It increases accuracy in identifying bad data. 3. It is flexible allowing the use of complex statistical models. An example is used to illustrate the performance of the method.

Acknowledgment Authors are partly supported by the Ministry of Science and Education of Spain through CICYT Projects DPI2003-01362 and BIA2005-07802-C0201; by the Junta de Comunidades de Castilla-La Mancha, through project PBI-05- 053.

Notation The main notation used throughout the paper is stated below for quick reference. Other symbols are defined as required in the text. State variables vi Voltage magnitude at bus i. θi Voltage angle at bus i. x Vector of state variables.

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Dependent variables Pi Active power injection at bus i. Qi Reactive power injection at bus i. PFij Active power flow from bus i to bus j. QFij Reactive power flow from bus i to bus j. Functions J(·) Quadratic measurement error function. Measurements vim Voltage magnitude measurement at bus i. Pim Active power injection measurement at bus i. Qm i Reactive power injection measurement at bus i. PFmij Active power flow measurement from bus i to bus j. Qm Fij Reactive power flow measurement from bus i to bus j. Physical limits Pimax Maximum active power generation at bus i. Qmax Maximum reactive power generation at bus i. i Pimin Minimum active power generation at bus i. Qmin Minimum reactive power generation at bus i. i

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Constants π Number π. xtrue Vector of state variable true values. Gij Real part of Yij . Bij Imaginary part of Yij . bsij Shunt susceptance of the π equivalent model of the line connecting nodes i and j. Sets Ωi Set of buses adjacent to bus i. Ω0 Set of transit nodes associated with zero injections. ΩV Set of available voltage magnitude measurements. ΩPi Sets of available active power injection measurements, where subindex i = g, d, b refers to only generation, only demand and both. ΩQi Sets of available reactive power injection measurements, where subindex i = g, d, b refers to only generation, only demand and both. ΩPF Set of available active power flow measurements. ΩQF Set of available reactive power flow measurements. Numbers m Number of measurements. n Number of state variables. It should be noted that a variable, function or parameter written in bold without index is a vector form representing the corresponding quantities. For example, the symbol θ represents the vector of bus voltage angles.

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