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Application of nonlinear programming in power system state estimation
Electric Power Systems Research, 12 (1987) 41 - 50
41
A p p l i c a t i o n o f N o n l i n e a r P r o g r a m m i n g in P o w e r S y s t e m S t a t e E s t i m a t i o n
N. H. ABBASY and S. M. SHAHIDEHPOUR
Department o f Electrical and Computer Engineering, Illinois Institute o f Technology, Chicago, IL 60616 (U.S.A.) (Received October 3, 1986)
SUMMARY
This paper presents a nonlinear programming approach to the power system state estimation problem. The proposed technique combines the estimation, detection and identification steps applied to the classical weighted least square method and rejects the corrupted data while estimating the state o f the system. The nonlinear programming approach is compared with least square and linear programming algorithms and the results are presented. This technique is very reliable, efficient, and does not require separate testing o f the system observability.
INTRODUCTION
System monitoring is an overriding factor in the operation o f power systems. This monitoring provides the system operators with pertinent up-to
power system state estimator employs a set of redundant and noisy data to estimate the actual state of the system. The state vector of the power system is defined as the vector of all the voltage magnitudes and bus angles. Accurate information a b o u t this state is sufficient to monitor the system operating conditions. Usually, p o w e r system static state estimators employ more measurements than the minimum number necessary to completely define the state of the system. Measurement redundancy is important for improving the estimated system state by detecting and identifying the bad data included in the measurements. Thus, a static state estimator employs a data processing algorithm which combines the telemetered redundant data with the information representing the s y s t e m model to provide the best estimate of the system state. This estimation process is usually modeled as an optimization problem and the system state is estimated by minimizing or maximizing a selected criterion. The o p t i m i z a t i o n techniques which are discussed in this paper are least square (LS), linear programming (LP), and nonlinear programming (NLP).
APPLICATION OF THE LEAST SQUARE METHOD IN P O W E R S Y S T E M S T A T E E S T I M A T I O N
In the LS method, the objective is to minimize the sum of the squares of the weighted deviations o f the estimated measurements z from the actual measurements. In order to estimate the actual value of a vector x using n m measurements, we express the objective function as
"m [Zi -- 5(X)] ~
min j(x) = ~
i=1
(1)
Oi2
© Elsevier Sequoia/Printed in The Netherlands
42
where fl(x) is the system equation representing the ith measurement, ot 2 the variance of the ith measurement, ](x) the weighted measurement residual, nm the number of independent measurements, and zi the ith measured quantity. If ft(x) is a linear function, eqn. (1) will have a closed form solution. Otherwise, an iterative technique must be applied to minimize .i(x). Since in electric power systems the real and reactive power flows are described by nonlinear equations, an iterative m e t h o d is adopted to determine the state of the system. A c o m m o n l y used technique is to calculate the gradient of i(x) and then force it to zero using Newton's method. The optimal estimate is then given by
One of the major drawbacks of the least square technique is that, if some of the measurements are erroneous, the least square estimator gives very poor results in the sense that the entire optimal estimate will be changed. A c o m m o n technique to remedy this problem is to filter the incoming data and determine its accuracy. The techniques such as zero flow detection or exponential smoothing can detect and identify some gross measurements, but will not solve the problem entirely.
(2)
The static state estimator usually deals with the following types of errors: (1} measurement errors (random metering and communication errors), (2) parameter uncertainty in the model parameters, (3) bad data (which is defined as large unexpected meter and communication errors). In the literature there are mainly three proposed techniques for the bad data processing: (1) the four-step procedure of detection, identification of bad data, removal of the identified bad data and reestimation; (2) the three-step procedure of detection and identification of bad data, replacement of the identified bad data and reestimation. (3) the combined process of state estimation and bad data suppression. The first technique bases its bad data detection on statistical hypothesis tests of certain estimates and its bad data identification on the search for the largest weighted or normalized residues [1 - 3]. This technique requires a repeated formation and factorization of the gain matrix (HTR - 1H}-1, a process which is expensive in terms of computer time. The second technique is more appropriate when used with estimators which have a constant gain matrix. This technique avoids reformation of the gain matrix by replacing the bad data with their estimated values. These two techniques feature a separate bad data processing procedure. In such a procedure, only one or two bad data can be removed or replaced at a time. The third m e t h o d , bad data suppression [4, 5], inte-
= (HTR-1H)-IHTR-I[z
--f(x)]
where x is the vector of system states, H the Jacobian matrix, R the covariance matrix, z the vector of measurements, and f(x) the vector of system equations corresponding to measurements. The covariance matrix R reflects the relative importance of each data point in the measurement set. If there is no interaction between the various measurements then R will be a diagonal matrix. The diagonal elements are the variances of the individual measurements. That is, ri = oi 2
(3)
where oi is the standard deviation of noise in the ith measurements. It is worth mentioning that the LS method works satisfactorily so long as the noise associated with the measurements is small. Even if the standard deviation of the measurement is computed accurately, the LS estimator does not account for some random events such as the failure in instruments or in the communication channels. These probable events result in an inaccurate estimation of the system state. Data obtained under these circumstances are called bad data. By definition, the term bad data is applied to measurements which deviate from their true values by at least two or three times the variance associated with the measurement. It has been found that the least square solution is only optimal when the measurement noise is Gaussian and has well-known statistics. This is n o t the case in most practical applications.
DATA PROCESSING IN POWER SYSTEM STATE ESTIMATION
43 grates the state estimation with the bad data processing in a single iterative process. The idea of bad data suppression is to assign less weight to 'unusually large residues'. This idea is realized b y using a certain combination of quadratic and nonquadratic criteria for the estimation problem. Intuitively, the weighted residual j(x) calculated b y the state estimator has its minim u m value if there are no bad data. By introducing a threshold tj, bad data are detected when the value of j(x) > tj. The threshold tj is obtained from the X2 distribution of the probability density function o f j ( x ) , as shown in ref. 3. A major drawback of this statistical test is that it may fail when multiple interacting bad data occur [ 6 - 10]. Since the least square residuals are a linear combination of errors in the raw data, it is possible that the bad data get small residual values, while good data acquire misleadingly large residuals. This interaction of multiple bad data may thus lead to nonidentification o f bad data or misidentification of good data. The other problem with the hypothesis test is that it requires a high degree of accuracy for the measurement error covariance matrix R. The known statistical tests for detection and identification of the bad data give satisfactory results, yet they require lengthy matrix calculations and seem impossible to use for large-size systems. Sparse inverse matrix techniques [11] have been combined with implementation of statistical tests so that detection can be performed more rapidly. Using this method, there is no need for direct inversion of the HTR - 1H matrix.
E F F E C T OF R E D U N D A N C Y IN POWER SYSTEM STATE ESTIMATION
Power system static state estimators usually employ more measurements than the minim u m number necessary to completely define the state of the system. System redundancy is defined as the ratio o f the number of measurements to the number o f estimated states. The measurement redundancy is important in order to delete bad data points without sacrificing the system observability. For example, current measurements in p o w e r systems cannot be used to provide system
observability. However, the addition of current measurements can increase the degree of measurement redundancy and is beneficial to the accuracy of the estimated states. Also, it is logically accepted that the o u t p u t of the state estimator is greatly influenced by the selected set o f measurements, and different sets of measurements result in different state estimates. Therefore, the judicious choice of a set of measurements not only influences the performance of the estimator but also the financial cost involved in purchase of data acquisition and transmission equipment. The criteria for selecting a set of quantities to be measured depend on the overall topology of the network, level of redundancy required, as well as the overall economics of the telemetry system. This problem has been recognized as a meter placement or measurement selection problem.
E F F E C T OF SYSTEM OBSERVABILITY POWER SYSTEM STATE ESTIMATION
ON
A system is considered to be observable if the entire state vector of the bus voltages and angles can be estimated based on the available set of measurements. For economic reasons and also due to transducer failures, a sufficient number of measurements may not be available for system identification [ 12, 13 ]. Theoretically, a system is observable if the gain matrix is nonsingular. This requires the rank of the Jacobian matrix to be equal to 2 n b - 1, where n b is equal to the number of buses in the system. Nonsingularity of the gain matrix is a necessary but insufficient condition to maintain system observability. System unobservability indicates that the connectivity of the network buses through the given set of measurements is unrealizable [14, 15]. Since most measurement schemes restrict themselves to P, Q, and V quantities, different procedures have been developed to test the system observability. These algorithms will flag the system unobservability if there is no path traceable between t w o buses in the network, in terms of P-Q and P - V measurements. So, the measurements must be distributed properly throughout the network to satisfy the observability conditions. The optimal choice o f measurements has been discussed in ref. 16. The optimization problem
44 was modeled as a nonlinear programming problem and the solution was achieved by minimizing an objective function representing the estimation error and the cost of telemetry system. A combined observability-reliabflity algorithm was introduced in ref. 17. This algorithm determines the conditions for system observability, in addition to identification of the critical measurements. According to this criterion, if a critical measurement is unavailable, the system becomes unobservable. The system observability has also been investigated by a symbolic reduction technique applied to the measurement Jacobian matrix [18]. Using this technique, it is possible to determine all the observable islands of a power network b y eliminating the corresponding variables from the linearized measurement equations. It is also possible to apply the available set o f measurements to the observable portion of the network, and model the unobservable part as an external network. Several methods have been investigated to determine the largest observable subnetwork [ 19, 20]. An external network is defined as an interconnected network for which very little or no real-time data are available. One of the algorithms used for the determination of the boundaries of the external system is to figure o u t real-time data for the external network in such a way that the mismatch values at the boundary are minimized. An alternative to the least square method in power systems is the weighted least absolute value technique [21]. This method detects and rejects bad data simultaneously with the estimation process of the system state. The whole idea is that the system state is estimated by interpolating the points in the measurement set, that is, it rejects bad data during the interpolation procedure. This m e t h o d works well providing that (1) the number of redundant measurements is greater than the number of bad data points, and (2) the bad data points are extremely erroneous. It has also been shown that the computation time for the weighted least absolute value estimators is comparable with that of the linear programming estimator.
One of the other applications of the state estimators in power systems is the possibility of estimation of the quantities which are n o t measured. This application is extremely useful if the data from one of the substations are unavailable. However, with a limited number of measurements, the estimated state may be less accurate than the case with a high degree of redundancy. This problem can be solved by adding pseudomeasurements to the set of available data [22]. Pseudomeasurements consist of the generated data based on the available historical information in the system data file. These measurements are used to maintain system observability when some of the real-time measurements are lost. An apparent disadvantage of this technique is that, if the accuracy of the pseudomeasurement is n o t compatible with that of the real-time data, the interaction of these two sets of information may cause degradation of the state estimation, unless a proper weighting factor is assigned to the pseudomeasurements. POWER SYSTEM STATE ESTIMATION USING LINEAR PROGRAMMING Power system state estimation can be formulated as a linear programming (LP) problem. This formulation leads to a state estimator that combines the advantages of noise filtering and bad data elimination. Via various examples, the advantages of the LP method are confirmed in refs. 23 and 24, especially where the data are corrupted by a number of gross errors. For real-time power system monitoring and control where process variables have u n k n o w n statistics, the linear programming m e t h o d has shown better results. Linear approximation of the equations of a large-scale power system is accomplished b y replacing the nonlinear equations with their first-order Taylor series approximation expanded at the operating point. By linearizing nonlinear functions at each intermediate solution, a sequence of system states is generated which under suitable circumstances converges to an optimal solution of the original nonlinear programming problem. The major drawback of this method is the assumption of small variations of the system states. To illustrate the idea, we modeled the original equations of the system in the linearized form, using a F O R T R A N program.
45 According to this m e t h o d , the nonlinear equations of the power systems are linearized in the neighborhood of the operating point and the objective is to minimize the effect of measured noise in the evaluation of the system states. This m e t h o d is quite fast and efficient as long as the variations of the system states are small. If the initial vector of the system state is estimated incorrectly, the linearization process applied to the power system would result in inaccurate results. On the other hand, if the quadratic approximation is used for modeling the system equation and formulating the objective function, the estimated state will be closer to the actual values. Generally, quadratic approximation is considered as a better approximation of a nonlinear equation and, in the neighborhood of the optimal solution, the system represented by the quadratic approximation converges much faster.
POWER SYSTEM MODELING
The nonlinear equations of a power system which represent the injected power and line flows are given by
its measured value. So, it can be regarded as either a positive or a negative value. In order to model the behavior of a large-scale power system, using the nonlinear programming technique, v is replaced by two positive slack variables. These two variables will account for the uncertainty in determination of the exact value of the noise magnitude. Therefore, the state estimation problem can be modeled as follows: n m
minimize
~ ai + bi
(9)
i=l
subject to zi - - f t ( x ) = a~ - - bi a i >/ O,
b i >/ 0
where zi is the ith measurement, fi the system equation corresponding to the ith measurement, and ai and bi are slack variables representing the ith measurement error. In order to reduce the number of unknowns, the given model can be restated as follows: n m
minimize
~ 2b~ - - f i ( x ) + zt
(10)
i=1
nb
Pi = ~, E i E I Y i i cos(5i -- 5j -- 0~i)
(4)
j=l
subject to .[t,(X) - - bi ~< z t
nb
Q~ = ~, E i E j Y i i sin(St -- 5i -- 0ii)
(5)
bi>~0
i=I
P~i = EiEIYiJ c o s ( 5 ~ - - 51
--0ii)
- - E~2Yii c o s 0~1
(6) Qii = E t E I Y i i sin(Si -- 5i --0il) + Ei2Yii sin 0ii
i = 1, ..., n b
(7)
j= l, ...,nb
where n b is the number of buses in the system. If the measurement is denoted by z, then z = f(x)
+ v
(8)
where f ( x ) is the vector representing the system equations, and v is the vector of measurem e n t errors. Generally, v represents the difference between the actual value of the function and
NONLINEAR PROGRAMMING METHOD POWER SYSTEM STATE ESTIMATION
FOR
In power system state estimation, if the measurement noise is Gaussian with a given covariance, the least square m e t h o d gives the maximum-likelihood minimum-variance estimate. However, in practice there is a possibility that the noise statistics are not determined very accurately and, if some of the measurements are in error, the least square estimator will give deceptive results. This is due to the fact that the erroneous measurem e n t is weighted according to the square of its residual and has a significant effect on the estimated state. Geometrically, the nonlinear programming (NLP) m e t h o d can be interpreted as minimiz-
46
ing the sum of the moduli of distances of the solution point to the measurement hyperplanes. The values estimated by the NLP method lie on the intersection o f p hyperplanes in p-dimensional space and the estimator will select a set of p hyperplanes from nm available values to minimize the objective function. So, the most accurate p measurements are chosen to define a solution point [24]. The proposed method will combine the automatic bad data rejection with a reliable degree of noise filtering. Unlike the application of LP in solving the linear optimization problem, no one method or class of nonlinear programming methods can be expected to solve all problems accurately and efficiently. Each method has its own particular strengths and weaknesses. Recent investigations have proved that quadratic approximation is especially efficient in terms of the number of required functions and gradient evaluation [25]. The NLP method used in this paper is based on Powell's algorithm which solves a sequence of positive definite quadratic programming subproblems. Other algorithms include the augmented Lagrangian or multiplier method, the penalty function method, and the generalized reduced gradient method. The general nonlinear constraint optimization problem can be stated as minimize
(11)
c(x)
subject to
hi(x )
=
0
i = 1, ..., m
and gf(x) >1 0
i = m + 1, ..., n
where the objective function c and the n constraint functions are functions of p system variables. The linear and the quadratic programming problems are considered as two special cases of the general nonlinear programming problem. When the objective function c and the constraints are all linear, a linear programming case is obtained. If the objective function is quadratic and the constraints are linear, the optimization technique is one of quadratic programming. In Powell's technique, the objective function is derived based on the quadratic approximation of a Lagrangian, and the linear approximation of
g(x) and h(x). The Lagrangian function L(x) is given by m
L(x, u, w) = c ( x ) + ~ w f h i ( x ) +
i=l
uigi(x) i=rn+l
(12) where w and u are the Lagrange multipliers. This function is approximated by considering the first t w o terms in the Taylor expansion of the nonlinear equations of the power system. So, the nonlinear optimization problem can be represented as follows: minimize
L(xh-1, Uk-1, Wk-1)
+ v T L ( x k _ I , U k - 1,
Wk-1)(Xk--Xk-1)
1 (xk - - x k - 1 ) T V2L(xk-I, uk-1, wk-1) + --£
× (xk - - x k - 1 )
(13)
subject to hi(Xk_l) +
vThl(xk-1)(Xk--Xk-1)
= 0
i = 1, ..., m and gi(xk-1) + vWgt(xh-l)(Xk --Xk--1) /> 0 i=m+
l,...,n
where V2L(x) is the Hessian of L(x), Vh(x) the gradient of h(x), and VT the transpose of V. The solution of (13) will determine the value of x during the kth iteration. The minimized function achieves both requirements of decreasing the objective function as well as reducing the amount by which the constraints fail to be satisfied. This iterative algorithm is designed to converge to a point that satisfies the necessary K u h n - T u c k e r conditions. During each iteration, t w o major steps are performed: solving a positive definite quadratic programming problem, and minimizing a one
47 TABLE l ( a ) Line impedances and line charging for the sample system
-Z
<
yes
no
Fig. 1. Flow chart of the iterative process (NLP): 1, initialization; 2, k = 1; 3, determine a search direction and Lagrange multiplier estimates by solving a quadratic programming problem; 4, convergence criterion satisfied? 5, stop; 6, determine a new solution estimate by approximately minimizing a function of one variable which depends on both the objective function and those constraints which are not satisfied; 7, k = k + 1.
Bus code p -q
Impedance Zpq (p.u.)
ypq/ ' 2 (p.u.)
1 1 2 2 2 3 4
0.02 0.08 0.06 0.06 0.04 0.01 0.08
0.0 0.0 0.0 0.0 0.0 0.0 0.0
Bus code
-
Iwikhi(xk-1)[
i=1 n
~
lu~gi(xk-1)l
+ j0.030 + j0.025 + j0.020 + j0.020 + j0.015 + j0.010 + j0.025
Load flow output for the sample system
m
+
+ j0.06 + j0.24 + j0.18 + j0.18 + j0.12 + j0.03 + j0.24
TABLE l ( b )
A c o n v e r g e n c e t e s t is m a d e o n e a c h i t e r a tion, after the quadratic programming p r o b l e m is s o l v e d . T h e a l g o r i t h m t e r m i n a t e s i f t h e f o l l o w i n g c o n d i t i o n is s a t i s f i e d :
I~Tc(xk-1)(Xk--Xk--1)I + Z
-2 -3 -3 -4 -5 -4 -5
Line charging
(14)
Line flows
2 3 1 3 4 5 1 2 4 2 3 5 2 4
(MW)
(MVAR)
88.8 40.7 --87.4 24.7 27.9 54.8 --39.5 --24.3 18.9 --27.5 --18.9 6.3 --53.7 --6.3
--8.6 1.1 6.2 3.5 3.0 7.4 --3.0 --6.8 --5.1 --5.9 3.2 --2.3 --7.2 --2.8
i= rn+l
w h e r e e is a u s e r - s u p p l i e d e r r o r t o l e r a n c e .
EXAMPLES In order to compare applications of the least square, linear programming, and nonlinear programming techniques in p o w e r system state estimation, a five-bus system was modeled on a VAX 11/780. The results of the s t u d y r e p r e s e n t e d t h e e f f e c t o f G a u s s i a n distributed noise, bad data, and an invalid operating point (starting point) on the estimation of power system state. Figure 2 represents the five-bus system. T h e s y s t e m d a t a a r e g i v e n in T a b l e l ( a ) , a n d t h e l o a d f l o w o u t p u t d a t a in T a b l e l ( b ) . These values represent the true estimates of t h e s y s t e m s t a t e a n d w i l l b e u s e d as t h e b a s e case.
2
I
1
5
4
Fig. 2. Five-bus system. In the case of low-level noise (with G a u s s i a n d i s t r i b u t i o n ) , all t h r e e t e c h n i q u e s produce fairly accurate results. Generally, the L S m e t h o d r e q u i r e s less c o m p u t a t i o n t i m e compared with the other two techniques if the noise has a Gaussian distribution. The r e s u l t s o f t h i s s t u d y a r e g i v e n in T a b l e 2 ( a ) .
48 TABLE 2(a)
TABLE 3(a)
Estimation with 0.01% Gaussian noise level
Estimation in the case of reverse flow measurement (bad data point)
NLP
LP
LS
V 6 (p.u.)
V 8 (p.u.)
V 8 (p.u.)
1.060 1.047 1.024 1.024 1.018
1.060 1.047 1.024 1.024 1.018
1.060 1.047 1.024 1.024 1.018
0.000 --0.049 --0.087 --0.093 --0.107
0.000 --0.049 --0.087 --0.093 --0.107
0.000 --0.049 --0.087 --0.093 --0.107
NLP
LP
LS
V 5 (p.u.)
V 6 (p.u.)
V (p.u.)
6
0.9832 0.9698 0.9486 0.9482 0.9407
0.0000 --0.0494 --0.0862 --0.0924 --0.1166
1.059 0.0000 1.065 0.0000 1.045 0.0500 1.045 --0.0456 1.023 --0.0870 1.026 --0.0849 1.023 --0.0929 1.025 --0.0908 1.017 --0.1070 1.026 --0.1070
TABLE 2(b) TABLE 3(b) Percentage error in the estimation of the bus voltages
Estimation with 0.1% Gaussian noise level NLP
LP
LS
V 8 (p.u.)
V (p.u.)
V (p.u.)
1.060 1.047 1.024 1.024 1.018
1.0565 1.0460 1.0260 1.0250 1.0260
0.0000 --0.0496 --0.0873 --0.0933 --0.1070
0.0000 --0.0456 --0.0849 --0.0909 --0.1070
1.071 1.058 1.036 1.036 1.032
0.0000 --0.0490 --0.0868 --0.0925 --0.1067
It is i m p o r t a n t to emphasize t h a t the LP and N L P m e t h o d s are b o t h required t o determ i n e the same n u m b e r o f u n k n o w n s f o r the same r e d u n d a n c y level. H o w e v e r , the results o f t h e s t u d y based o n t h e N L P algorithm are closer t o the base case quantities. When the level o f t h e Gaussian noise increases, the results given in Table 2(b) give a m o r e accurate e s t i m a t i o n o f the s y s t e m state based o n the N L P application. T h e s u p e r i o r i t y o f the N L P algorithm in this application is d u e to the fact t h a t LS results are e x t r e m e l y sensitive to the level o f noise. T h e N L P results are also m o r e accurate t h a n the LP results because the system e q u a t i o n s are m o d e l e d m o r e a c c u r a t e l y using t h e N L P a l g o r i t h m . Bad d a t a suppression is also simulated b y these t h r e e algorithms. This s i m u l a t i o n represents the application o f the p o w e r s y s t e m state e s t i m a t o r in bad d a t a suppression. T h e e f f e c t o f bad d a t a is s h o w n b y reversing t h e sign o f t h e real p o w e r flow in line 23. This s t u d y simulates the case o f the reverse conn e c t i o n o f t h e meters and the results are given in Table 3(a). These results indicate t h a t the LS t e c h n i q u e is very m u c h a f f e c t e d b y the presence o f bad data. In this case the LP and
Bus voltage No.
NLP
LP
LS
1 2 3
0.094 0.190 0.097
0.470 0.190 0.190
7.260 7.440 7.420
TABLE 3(c) Estimated bus voltages and angles in the case of meter failure NLP
LS
V (p.u.)
~
V (p.u.)
1.059 1.045 1.023 1.022 1.015
0.0000 --0.0500 --0.0852 --0.0930 --1.068
1.024 1.011 0.990 0.989 0.983
0.0000 --0.0510 --0.0890 --0.0951 --0.1100
N L P m e t h o d s give b e t t e r estimations o f the s y s t e m state. Table 3(b) gives the p e r c e n t a g e error in t h e e s t i m a t i o n o f t h e bus voltages. T h e presence o f bad d a t a m a y also be due to c o m m u n i c a t i o n and m e t e r failures. These effects can be simulated b y replacing t h e real and reactive p o w e r flows o n line 24 with zeros. This s t u d y leads t o the results given in Table 3(c) w h i c h again proves t h a t the existence o f bad d a t a in t h e p o w e r s y s t e m m a y cause i n a c c u r a c y in the c o n v e n t i o n a l LS solution. The p r o b l e m s with the e x a c t d e t e r m i n a t i o n o f t h e s y s t e m o p e r a t i n g p o i n t {initial estim a t e ) m a y c o n t r i b u t e t o an i n a c c u r a t e estimat i o n o f states if the LP t e c h n i q u e is used. The o p e r a t i n g p o i n t o f the system consists o f
49 cate t hat the NLP m e t h o d yields bet t er results for unmeasured quantities.
TABLE4 Estimation withanimproperinitialpoint NLP
LP
LS
V 6 (p.u.)
V 6 (p.u.)
V 6 (p.u.)
CONCLUSION
1.060 1.047 1.024 1.024 1.017
1.057 1.046 1.025 1.025 1.023
1.071 1.058 1.036 1.036 1.032
The application of nonlinear programming in power system state estimation has been considered in this study. This algorithm has been tested on a five-bus system and the results are presented in the paper. However, the proposed m e t h o d can be applied to an n-bus system. The NLP technique has also been compared with linear programming and weighted least square m e t h o d s and it appears t hat NLP is m ore accurate than the o t h e r two m e t h o d s in several applications such as bad data detection, suppression of Gaussian distributed noise with large magnitude, bad data rejection, inaccurate estimation of the initial operating point, estimation of the quantities n o t being measured, etc. One of the difficulties with application of the least square m e t h o d in pow er system state estimation is t hat it is ext rem el y noise sensitive, so bad data need to be detected before t h e y can be rejected from the set o f measurements. In addition, it assumes that the noise has a Gaussian distribution. In m any applications, the determination of the exact value for the standard deviation o f the noise measurements is a difficult task. The NLP m e t h o d is relatively fast and does not restrict the system studies to Gaussian distributed noise. Neither does it require the bad data det ect i on and identification techniques applied t o the LS m e t h o d . For this application of nonlinear programming, a quadratic approxi m at i on has been considered in modeling the system equations, which is generally considered as a m ore precise representation of the nonlinear equation. In particular, if the operating point o f t he system has been estimated improperly, the NLP m e t h o d is able to determine the new state o f the system to a higher degree of accuracy than that of the LP m e t h o d . This is due to the fact that, using the LP technique, the linear approxi m at i on is considered in modeling the system equations and the accuracy o f this approximation is limited t o small deviations o f the system states.
0.0000 --0.0496 --0.0879 --0.0930 --0.1060
0.0000 --0.0239 --0.064 --0.0688 --0.0848
0.0000 --0.0494 --0.0868 --0.0925 --0.106
TABLE 5 Estimation of quantities not being measured (0.1% noise level) Quantity
Load flow value (p.u.)
Estimated value (p.u.) NLP LP LS
Pl2 P13 P2s
0.888 0.407 0.548
0.8809 0.4040 0.5440
0.815 0.388 0.551
0.916 0.412 0.516
some estimated values and some guesses. Since the LP equations are linearized in the n e i g hb o r h o o d o f the system operating point, any significant deviation o f the linearized system equations from the actual operating point may result in erroneous estimation o f the system state. In contrast, the nonlinear m e t h o d will lead to b e t t e r estimated values owing to the quadratic nature o f the approximation. Table 4 represents the results o f system state estimation while the system operating p o i n t (initial value) is estimated improperly. These results show that LP fails to give an accurate estimation o f the system states. In this case, b o th NLP and LS invoke bet t er solutions for the estimation o f the system state. Power system state estimation m ay also be used to determine the quantities which have n o t been measured in the pow er n e t w o r k and the unmeasured values estimated by different estimators. Table 5 represents the unmeasured values which are c o m p u t e d according to the estimated states o f the system. The estimated states are the same as those given in Table 3 and the c o m p u t e d quantities are c om par e d with the base case values. These results indi-
50 REFERENCES
1 F. Zhung and R. Balasubramanian, Bad data processing in power system state estimation by direct bad data deletion and hypothesis tests,
IEEE PES Winter Meeting, New York, 1986, Paper No. 86 WM 108-5. 2 L. Milli, T. V. Cutsem and M. R. Pavella, Bad data identification methods in power system state estimation -- a comparative study, IEEE Trans., PAS-104 (11) (1985) 3037 - 3049. 3 A. J. Wood and B. F. Wollengberg, Power Generation, Operation and Control, Wiley, New York, 1984. 4 F. Zhuang and R. Balasubramanian, Bad data suppression in power system state estimation with a variable quadratic-constant criterion, IEEE Trans., PAS-104 (4) (1985) 857 - 862. 5 K. L. Lo, P. S. Ong, R. D. McColl, A. M. Moffatt and J. L. Sulley, Development of a static state estimator, part I: estimation and bad data suppression, IEEE Trans., PAS-102 (8) (1983) 2486 2491. 6 M. Ayrer and P. Haley, Bad data groups in power system state estimation, IEEE PES Winter Meet: ing, New York, 1986, Paper No. 86 WM 040-0. 7 A. Monticelli and A. Garcia, Reliable bad data processing for real time state estimation, IEEE Trans., PAS-102 (5) (1983) 1126 - 1139. 8 A. Monticelli, F. Wu and M. Yen, Multiple bad data identification for state estimation by combinatorial optimization, PICA Conf., 1985, IEEE, New York, 1985, pp. 452 - 460. 9 K. A. Clements and P. W. Davis, Multiple bad data detectability and identifiability: a geometric approach, PICA Conf., 1985, IEEE, New York, 1985, pp. 461 - 466. 10 Xiang Nian, Wang Shi-Ying and Yu Er-Keng, A new approach for detection and identification of multiple bad data in power system state estimation, IEEE Trans., PAS-101 (2) (1982) 4 5 4 461. 11 F. Broussole, State estimation in power systems: deleting bad data through the sparse inverse matrix methods, IEEE Trans., PAS-97 (3) (1977) 678 - 682. 12 K. A. Clements, G. R. Krumpholz and P. W. Davis, Power system state estimation with measurement deficiency. An observability]measurement placement algorithm, IEEE Trans.. PAS-102 (7) (1983) 3044 - 3052.
13 G. R. Krumpholz, K. A. Clements and P. W. Davis, Power system observability: a practical algorithm using network topology, IEEE Trans., PAS-99 (4) (1980) 1534 - 1542. 14 A. Monticelli and F. Wu, Observability analysis for orthogonal transformation based state estimation, PICA Conf., 1985, IEEE, New York, 1985, pp. 148 - 152. 15 J. W. Gu, K. A. Clements, G. R. Krumpholz and P. W. Davis, Topologically partitioned power system state estimation, IEEE PES S u m m e r Meeting, 1982, Paper No. 82 SM 423-2. 16 K. Phua and T. S. Dillon, Optimal choice of measurements for state estimation, PICA Conf., 1977, IEEE, New York, 1977, pp. 431 - 441. 17 I. Slutsker and J. Scudded, Network observability analysis through measurement, IEEE PES Winter Meeting, New York, 1986, Paper No. 86 WM 095-4. 18 K. A. Clements, G. R. Krumpholz and P. W. Davis, State estimator measurement system reliability evaluation -- an efficient algorithm based on topological observability theory, IEEE Trans., PAS-101 (4) (1982) 997 - 1004. 19 K. Geisler and A. Bose, State estimation based external network solution for on line security analysis, IEEE Trans., PAS-102 (8) (1983) 2447 2454. 20 K. A. Clements, G. R. Krumpholz and P. W. Davis, Power system state estimation with measurement deficiency: an algorithm that determines the maximal observable subnetwork, IEEE Trans., PAS-101 (9) (1982) 2012 - 2020. 21 W. Kotinga and M. Vidyasagar, Bad data rejection properties and weighted least absolute value techniques applied to static state estimation, IEEE Trans., PAS-101 (4) (1982) 844 - 851. 22 K. L. Lo, P. S. Ong, R. D. McColl, A. M. Moffatt and J. L. Sulley, Development of a static state estimator part 2: bad data placement and of pseudomeasurements, IEEE Trans., PAS-102 (8) (1983) 2 4 9 2 - 2500. 23 K. L. Lo and Y. M. Mahmoud, Decoupled linear programming technique for power system state estimation, IEEE Trans., PAS-105 (1986) 1 5 4 160. 24 M. R. Irving, R. C. Owen and M. J. Sterling, Linear programming, Proc. Inst. Electr. Eng., 125 (9) (1978) 879 - 885. 25 R. L. Crane, K. E. Hillstrom and M. Minkoff, Solution of the general nonlinear programming problem with subroutine VMCON, Rep. No. ANL-80-64 (Argonne National Laboratory).
41
A p p l i c a t i o n o f N o n l i n e a r P r o g r a m m i n g in P o w e r S y s t e m S t a t e E s t i m a t i o n
N. H. ABBASY and S. M. SHAHIDEHPOUR
Department o f Electrical and Computer Engineering, Illinois Institute o f Technology, Chicago, IL 60616 (U.S.A.) (Received October 3, 1986)
SUMMARY
This paper presents a nonlinear programming approach to the power system state estimation problem. The proposed technique combines the estimation, detection and identification steps applied to the classical weighted least square method and rejects the corrupted data while estimating the state o f the system. The nonlinear programming approach is compared with least square and linear programming algorithms and the results are presented. This technique is very reliable, efficient, and does not require separate testing o f the system observability.
INTRODUCTION
System monitoring is an overriding factor in the operation o f power systems. This monitoring provides the system operators with pertinent up-to
power system state estimator employs a set of redundant and noisy data to estimate the actual state of the system. The state vector of the power system is defined as the vector of all the voltage magnitudes and bus angles. Accurate information a b o u t this state is sufficient to monitor the system operating conditions. Usually, p o w e r system static state estimators employ more measurements than the minimum number necessary to completely define the state of the system. Measurement redundancy is important for improving the estimated system state by detecting and identifying the bad data included in the measurements. Thus, a static state estimator employs a data processing algorithm which combines the telemetered redundant data with the information representing the s y s t e m model to provide the best estimate of the system state. This estimation process is usually modeled as an optimization problem and the system state is estimated by minimizing or maximizing a selected criterion. The o p t i m i z a t i o n techniques which are discussed in this paper are least square (LS), linear programming (LP), and nonlinear programming (NLP).
APPLICATION OF THE LEAST SQUARE METHOD IN P O W E R S Y S T E M S T A T E E S T I M A T I O N
In the LS method, the objective is to minimize the sum of the squares of the weighted deviations o f the estimated measurements z from the actual measurements. In order to estimate the actual value of a vector x using n m measurements, we express the objective function as
"m [Zi -- 5(X)] ~
min j(x) = ~
i=1
(1)
Oi2
© Elsevier Sequoia/Printed in The Netherlands
42
where fl(x) is the system equation representing the ith measurement, ot 2 the variance of the ith measurement, ](x) the weighted measurement residual, nm the number of independent measurements, and zi the ith measured quantity. If ft(x) is a linear function, eqn. (1) will have a closed form solution. Otherwise, an iterative technique must be applied to minimize .i(x). Since in electric power systems the real and reactive power flows are described by nonlinear equations, an iterative m e t h o d is adopted to determine the state of the system. A c o m m o n l y used technique is to calculate the gradient of i(x) and then force it to zero using Newton's method. The optimal estimate is then given by
One of the major drawbacks of the least square technique is that, if some of the measurements are erroneous, the least square estimator gives very poor results in the sense that the entire optimal estimate will be changed. A c o m m o n technique to remedy this problem is to filter the incoming data and determine its accuracy. The techniques such as zero flow detection or exponential smoothing can detect and identify some gross measurements, but will not solve the problem entirely.
(2)
The static state estimator usually deals with the following types of errors: (1} measurement errors (random metering and communication errors), (2) parameter uncertainty in the model parameters, (3) bad data (which is defined as large unexpected meter and communication errors). In the literature there are mainly three proposed techniques for the bad data processing: (1) the four-step procedure of detection, identification of bad data, removal of the identified bad data and reestimation; (2) the three-step procedure of detection and identification of bad data, replacement of the identified bad data and reestimation. (3) the combined process of state estimation and bad data suppression. The first technique bases its bad data detection on statistical hypothesis tests of certain estimates and its bad data identification on the search for the largest weighted or normalized residues [1 - 3]. This technique requires a repeated formation and factorization of the gain matrix (HTR - 1H}-1, a process which is expensive in terms of computer time. The second technique is more appropriate when used with estimators which have a constant gain matrix. This technique avoids reformation of the gain matrix by replacing the bad data with their estimated values. These two techniques feature a separate bad data processing procedure. In such a procedure, only one or two bad data can be removed or replaced at a time. The third m e t h o d , bad data suppression [4, 5], inte-
= (HTR-1H)-IHTR-I[z
--f(x)]
where x is the vector of system states, H the Jacobian matrix, R the covariance matrix, z the vector of measurements, and f(x) the vector of system equations corresponding to measurements. The covariance matrix R reflects the relative importance of each data point in the measurement set. If there is no interaction between the various measurements then R will be a diagonal matrix. The diagonal elements are the variances of the individual measurements. That is, ri = oi 2
(3)
where oi is the standard deviation of noise in the ith measurements. It is worth mentioning that the LS method works satisfactorily so long as the noise associated with the measurements is small. Even if the standard deviation of the measurement is computed accurately, the LS estimator does not account for some random events such as the failure in instruments or in the communication channels. These probable events result in an inaccurate estimation of the system state. Data obtained under these circumstances are called bad data. By definition, the term bad data is applied to measurements which deviate from their true values by at least two or three times the variance associated with the measurement. It has been found that the least square solution is only optimal when the measurement noise is Gaussian and has well-known statistics. This is n o t the case in most practical applications.
DATA PROCESSING IN POWER SYSTEM STATE ESTIMATION
43 grates the state estimation with the bad data processing in a single iterative process. The idea of bad data suppression is to assign less weight to 'unusually large residues'. This idea is realized b y using a certain combination of quadratic and nonquadratic criteria for the estimation problem. Intuitively, the weighted residual j(x) calculated b y the state estimator has its minim u m value if there are no bad data. By introducing a threshold tj, bad data are detected when the value of j(x) > tj. The threshold tj is obtained from the X2 distribution of the probability density function o f j ( x ) , as shown in ref. 3. A major drawback of this statistical test is that it may fail when multiple interacting bad data occur [ 6 - 10]. Since the least square residuals are a linear combination of errors in the raw data, it is possible that the bad data get small residual values, while good data acquire misleadingly large residuals. This interaction of multiple bad data may thus lead to nonidentification o f bad data or misidentification of good data. The other problem with the hypothesis test is that it requires a high degree of accuracy for the measurement error covariance matrix R. The known statistical tests for detection and identification of the bad data give satisfactory results, yet they require lengthy matrix calculations and seem impossible to use for large-size systems. Sparse inverse matrix techniques [11] have been combined with implementation of statistical tests so that detection can be performed more rapidly. Using this method, there is no need for direct inversion of the HTR - 1H matrix.
E F F E C T OF R E D U N D A N C Y IN POWER SYSTEM STATE ESTIMATION
Power system static state estimators usually employ more measurements than the minim u m number necessary to completely define the state of the system. System redundancy is defined as the ratio o f the number of measurements to the number o f estimated states. The measurement redundancy is important in order to delete bad data points without sacrificing the system observability. For example, current measurements in p o w e r systems cannot be used to provide system
observability. However, the addition of current measurements can increase the degree of measurement redundancy and is beneficial to the accuracy of the estimated states. Also, it is logically accepted that the o u t p u t of the state estimator is greatly influenced by the selected set o f measurements, and different sets of measurements result in different state estimates. Therefore, the judicious choice of a set of measurements not only influences the performance of the estimator but also the financial cost involved in purchase of data acquisition and transmission equipment. The criteria for selecting a set of quantities to be measured depend on the overall topology of the network, level of redundancy required, as well as the overall economics of the telemetry system. This problem has been recognized as a meter placement or measurement selection problem.
E F F E C T OF SYSTEM OBSERVABILITY POWER SYSTEM STATE ESTIMATION
ON
A system is considered to be observable if the entire state vector of the bus voltages and angles can be estimated based on the available set of measurements. For economic reasons and also due to transducer failures, a sufficient number of measurements may not be available for system identification [ 12, 13 ]. Theoretically, a system is observable if the gain matrix is nonsingular. This requires the rank of the Jacobian matrix to be equal to 2 n b - 1, where n b is equal to the number of buses in the system. Nonsingularity of the gain matrix is a necessary but insufficient condition to maintain system observability. System unobservability indicates that the connectivity of the network buses through the given set of measurements is unrealizable [14, 15]. Since most measurement schemes restrict themselves to P, Q, and V quantities, different procedures have been developed to test the system observability. These algorithms will flag the system unobservability if there is no path traceable between t w o buses in the network, in terms of P-Q and P - V measurements. So, the measurements must be distributed properly throughout the network to satisfy the observability conditions. The optimal choice o f measurements has been discussed in ref. 16. The optimization problem
44 was modeled as a nonlinear programming problem and the solution was achieved by minimizing an objective function representing the estimation error and the cost of telemetry system. A combined observability-reliabflity algorithm was introduced in ref. 17. This algorithm determines the conditions for system observability, in addition to identification of the critical measurements. According to this criterion, if a critical measurement is unavailable, the system becomes unobservable. The system observability has also been investigated by a symbolic reduction technique applied to the measurement Jacobian matrix [18]. Using this technique, it is possible to determine all the observable islands of a power network b y eliminating the corresponding variables from the linearized measurement equations. It is also possible to apply the available set o f measurements to the observable portion of the network, and model the unobservable part as an external network. Several methods have been investigated to determine the largest observable subnetwork [ 19, 20]. An external network is defined as an interconnected network for which very little or no real-time data are available. One of the algorithms used for the determination of the boundaries of the external system is to figure o u t real-time data for the external network in such a way that the mismatch values at the boundary are minimized. An alternative to the least square method in power systems is the weighted least absolute value technique [21]. This method detects and rejects bad data simultaneously with the estimation process of the system state. The whole idea is that the system state is estimated by interpolating the points in the measurement set, that is, it rejects bad data during the interpolation procedure. This m e t h o d works well providing that (1) the number of redundant measurements is greater than the number of bad data points, and (2) the bad data points are extremely erroneous. It has also been shown that the computation time for the weighted least absolute value estimators is comparable with that of the linear programming estimator.
One of the other applications of the state estimators in power systems is the possibility of estimation of the quantities which are n o t measured. This application is extremely useful if the data from one of the substations are unavailable. However, with a limited number of measurements, the estimated state may be less accurate than the case with a high degree of redundancy. This problem can be solved by adding pseudomeasurements to the set of available data [22]. Pseudomeasurements consist of the generated data based on the available historical information in the system data file. These measurements are used to maintain system observability when some of the real-time measurements are lost. An apparent disadvantage of this technique is that, if the accuracy of the pseudomeasurement is n o t compatible with that of the real-time data, the interaction of these two sets of information may cause degradation of the state estimation, unless a proper weighting factor is assigned to the pseudomeasurements. POWER SYSTEM STATE ESTIMATION USING LINEAR PROGRAMMING Power system state estimation can be formulated as a linear programming (LP) problem. This formulation leads to a state estimator that combines the advantages of noise filtering and bad data elimination. Via various examples, the advantages of the LP method are confirmed in refs. 23 and 24, especially where the data are corrupted by a number of gross errors. For real-time power system monitoring and control where process variables have u n k n o w n statistics, the linear programming m e t h o d has shown better results. Linear approximation of the equations of a large-scale power system is accomplished b y replacing the nonlinear equations with their first-order Taylor series approximation expanded at the operating point. By linearizing nonlinear functions at each intermediate solution, a sequence of system states is generated which under suitable circumstances converges to an optimal solution of the original nonlinear programming problem. The major drawback of this method is the assumption of small variations of the system states. To illustrate the idea, we modeled the original equations of the system in the linearized form, using a F O R T R A N program.
45 According to this m e t h o d , the nonlinear equations of the power systems are linearized in the neighborhood of the operating point and the objective is to minimize the effect of measured noise in the evaluation of the system states. This m e t h o d is quite fast and efficient as long as the variations of the system states are small. If the initial vector of the system state is estimated incorrectly, the linearization process applied to the power system would result in inaccurate results. On the other hand, if the quadratic approximation is used for modeling the system equation and formulating the objective function, the estimated state will be closer to the actual values. Generally, quadratic approximation is considered as a better approximation of a nonlinear equation and, in the neighborhood of the optimal solution, the system represented by the quadratic approximation converges much faster.
POWER SYSTEM MODELING
The nonlinear equations of a power system which represent the injected power and line flows are given by
its measured value. So, it can be regarded as either a positive or a negative value. In order to model the behavior of a large-scale power system, using the nonlinear programming technique, v is replaced by two positive slack variables. These two variables will account for the uncertainty in determination of the exact value of the noise magnitude. Therefore, the state estimation problem can be modeled as follows: n m
minimize
~ ai + bi
(9)
i=l
subject to zi - - f t ( x ) = a~ - - bi a i >/ O,
b i >/ 0
where zi is the ith measurement, fi the system equation corresponding to the ith measurement, and ai and bi are slack variables representing the ith measurement error. In order to reduce the number of unknowns, the given model can be restated as follows: n m
minimize
~ 2b~ - - f i ( x ) + zt
(10)
i=1
nb
Pi = ~, E i E I Y i i cos(5i -- 5j -- 0~i)
(4)
j=l
subject to .[t,(X) - - bi ~< z t
nb
Q~ = ~, E i E j Y i i sin(St -- 5i -- 0ii)
(5)
bi>~0
i=I
P~i = EiEIYiJ c o s ( 5 ~ - - 51
--0ii)
- - E~2Yii c o s 0~1
(6) Qii = E t E I Y i i sin(Si -- 5i --0il) + Ei2Yii sin 0ii
i = 1, ..., n b
(7)
j= l, ...,nb
where n b is the number of buses in the system. If the measurement is denoted by z, then z = f(x)
+ v
(8)
where f ( x ) is the vector representing the system equations, and v is the vector of measurem e n t errors. Generally, v represents the difference between the actual value of the function and
NONLINEAR PROGRAMMING METHOD POWER SYSTEM STATE ESTIMATION
FOR
In power system state estimation, if the measurement noise is Gaussian with a given covariance, the least square m e t h o d gives the maximum-likelihood minimum-variance estimate. However, in practice there is a possibility that the noise statistics are not determined very accurately and, if some of the measurements are in error, the least square estimator will give deceptive results. This is due to the fact that the erroneous measurem e n t is weighted according to the square of its residual and has a significant effect on the estimated state. Geometrically, the nonlinear programming (NLP) m e t h o d can be interpreted as minimiz-
46
ing the sum of the moduli of distances of the solution point to the measurement hyperplanes. The values estimated by the NLP method lie on the intersection o f p hyperplanes in p-dimensional space and the estimator will select a set of p hyperplanes from nm available values to minimize the objective function. So, the most accurate p measurements are chosen to define a solution point [24]. The proposed method will combine the automatic bad data rejection with a reliable degree of noise filtering. Unlike the application of LP in solving the linear optimization problem, no one method or class of nonlinear programming methods can be expected to solve all problems accurately and efficiently. Each method has its own particular strengths and weaknesses. Recent investigations have proved that quadratic approximation is especially efficient in terms of the number of required functions and gradient evaluation [25]. The NLP method used in this paper is based on Powell's algorithm which solves a sequence of positive definite quadratic programming subproblems. Other algorithms include the augmented Lagrangian or multiplier method, the penalty function method, and the generalized reduced gradient method. The general nonlinear constraint optimization problem can be stated as minimize
(11)
c(x)
subject to
hi(x )
=
0
i = 1, ..., m
and gf(x) >1 0
i = m + 1, ..., n
where the objective function c and the n constraint functions are functions of p system variables. The linear and the quadratic programming problems are considered as two special cases of the general nonlinear programming problem. When the objective function c and the constraints are all linear, a linear programming case is obtained. If the objective function is quadratic and the constraints are linear, the optimization technique is one of quadratic programming. In Powell's technique, the objective function is derived based on the quadratic approximation of a Lagrangian, and the linear approximation of
g(x) and h(x). The Lagrangian function L(x) is given by m
L(x, u, w) = c ( x ) + ~ w f h i ( x ) +
i=l
uigi(x) i=rn+l
(12) where w and u are the Lagrange multipliers. This function is approximated by considering the first t w o terms in the Taylor expansion of the nonlinear equations of the power system. So, the nonlinear optimization problem can be represented as follows: minimize
L(xh-1, Uk-1, Wk-1)
+ v T L ( x k _ I , U k - 1,
Wk-1)(Xk--Xk-1)
1 (xk - - x k - 1 ) T V2L(xk-I, uk-1, wk-1) + --£
× (xk - - x k - 1 )
(13)
subject to hi(Xk_l) +
vThl(xk-1)(Xk--Xk-1)
= 0
i = 1, ..., m and gi(xk-1) + vWgt(xh-l)(Xk --Xk--1) /> 0 i=m+
l,...,n
where V2L(x) is the Hessian of L(x), Vh(x) the gradient of h(x), and VT the transpose of V. The solution of (13) will determine the value of x during the kth iteration. The minimized function achieves both requirements of decreasing the objective function as well as reducing the amount by which the constraints fail to be satisfied. This iterative algorithm is designed to converge to a point that satisfies the necessary K u h n - T u c k e r conditions. During each iteration, t w o major steps are performed: solving a positive definite quadratic programming problem, and minimizing a one
47 TABLE l ( a ) Line impedances and line charging for the sample system
-Z
<
yes
no
Fig. 1. Flow chart of the iterative process (NLP): 1, initialization; 2, k = 1; 3, determine a search direction and Lagrange multiplier estimates by solving a quadratic programming problem; 4, convergence criterion satisfied? 5, stop; 6, determine a new solution estimate by approximately minimizing a function of one variable which depends on both the objective function and those constraints which are not satisfied; 7, k = k + 1.
Bus code p -q
Impedance Zpq (p.u.)
ypq/ ' 2 (p.u.)
1 1 2 2 2 3 4
0.02 0.08 0.06 0.06 0.04 0.01 0.08
0.0 0.0 0.0 0.0 0.0 0.0 0.0
Bus code
-
Iwikhi(xk-1)[
i=1 n
~
lu~gi(xk-1)l
+ j0.030 + j0.025 + j0.020 + j0.020 + j0.015 + j0.010 + j0.025
Load flow output for the sample system
m
+
+ j0.06 + j0.24 + j0.18 + j0.18 + j0.12 + j0.03 + j0.24
TABLE l ( b )
A c o n v e r g e n c e t e s t is m a d e o n e a c h i t e r a tion, after the quadratic programming p r o b l e m is s o l v e d . T h e a l g o r i t h m t e r m i n a t e s i f t h e f o l l o w i n g c o n d i t i o n is s a t i s f i e d :
I~Tc(xk-1)(Xk--Xk--1)I + Z
-2 -3 -3 -4 -5 -4 -5
Line charging
(14)
Line flows
2 3 1 3 4 5 1 2 4 2 3 5 2 4
(MW)
(MVAR)
88.8 40.7 --87.4 24.7 27.9 54.8 --39.5 --24.3 18.9 --27.5 --18.9 6.3 --53.7 --6.3
--8.6 1.1 6.2 3.5 3.0 7.4 --3.0 --6.8 --5.1 --5.9 3.2 --2.3 --7.2 --2.8
i= rn+l
w h e r e e is a u s e r - s u p p l i e d e r r o r t o l e r a n c e .
EXAMPLES In order to compare applications of the least square, linear programming, and nonlinear programming techniques in p o w e r system state estimation, a five-bus system was modeled on a VAX 11/780. The results of the s t u d y r e p r e s e n t e d t h e e f f e c t o f G a u s s i a n distributed noise, bad data, and an invalid operating point (starting point) on the estimation of power system state. Figure 2 represents the five-bus system. T h e s y s t e m d a t a a r e g i v e n in T a b l e l ( a ) , a n d t h e l o a d f l o w o u t p u t d a t a in T a b l e l ( b ) . These values represent the true estimates of t h e s y s t e m s t a t e a n d w i l l b e u s e d as t h e b a s e case.
2
I
1
5
4
Fig. 2. Five-bus system. In the case of low-level noise (with G a u s s i a n d i s t r i b u t i o n ) , all t h r e e t e c h n i q u e s produce fairly accurate results. Generally, the L S m e t h o d r e q u i r e s less c o m p u t a t i o n t i m e compared with the other two techniques if the noise has a Gaussian distribution. The r e s u l t s o f t h i s s t u d y a r e g i v e n in T a b l e 2 ( a ) .
48 TABLE 2(a)
TABLE 3(a)
Estimation with 0.01% Gaussian noise level
Estimation in the case of reverse flow measurement (bad data point)
NLP
LP
LS
V 6 (p.u.)
V 8 (p.u.)
V 8 (p.u.)
1.060 1.047 1.024 1.024 1.018
1.060 1.047 1.024 1.024 1.018
1.060 1.047 1.024 1.024 1.018
0.000 --0.049 --0.087 --0.093 --0.107
0.000 --0.049 --0.087 --0.093 --0.107
0.000 --0.049 --0.087 --0.093 --0.107
NLP
LP
LS
V 5 (p.u.)
V 6 (p.u.)
V (p.u.)
6
0.9832 0.9698 0.9486 0.9482 0.9407
0.0000 --0.0494 --0.0862 --0.0924 --0.1166
1.059 0.0000 1.065 0.0000 1.045 0.0500 1.045 --0.0456 1.023 --0.0870 1.026 --0.0849 1.023 --0.0929 1.025 --0.0908 1.017 --0.1070 1.026 --0.1070
TABLE 2(b) TABLE 3(b) Percentage error in the estimation of the bus voltages
Estimation with 0.1% Gaussian noise level NLP
LP
LS
V 8 (p.u.)
V (p.u.)
V (p.u.)
1.060 1.047 1.024 1.024 1.018
1.0565 1.0460 1.0260 1.0250 1.0260
0.0000 --0.0496 --0.0873 --0.0933 --0.1070
0.0000 --0.0456 --0.0849 --0.0909 --0.1070
1.071 1.058 1.036 1.036 1.032
0.0000 --0.0490 --0.0868 --0.0925 --0.1067
It is i m p o r t a n t to emphasize t h a t the LP and N L P m e t h o d s are b o t h required t o determ i n e the same n u m b e r o f u n k n o w n s f o r the same r e d u n d a n c y level. H o w e v e r , the results o f t h e s t u d y based o n t h e N L P algorithm are closer t o the base case quantities. When the level o f t h e Gaussian noise increases, the results given in Table 2(b) give a m o r e accurate e s t i m a t i o n o f the s y s t e m state based o n the N L P application. T h e s u p e r i o r i t y o f the N L P algorithm in this application is d u e to the fact t h a t LS results are e x t r e m e l y sensitive to the level o f noise. T h e N L P results are also m o r e accurate t h a n the LP results because the system e q u a t i o n s are m o d e l e d m o r e a c c u r a t e l y using t h e N L P a l g o r i t h m . Bad d a t a suppression is also simulated b y these t h r e e algorithms. This s i m u l a t i o n represents the application o f the p o w e r s y s t e m state e s t i m a t o r in bad d a t a suppression. T h e e f f e c t o f bad d a t a is s h o w n b y reversing t h e sign o f t h e real p o w e r flow in line 23. This s t u d y simulates the case o f the reverse conn e c t i o n o f t h e meters and the results are given in Table 3(a). These results indicate t h a t the LS t e c h n i q u e is very m u c h a f f e c t e d b y the presence o f bad data. In this case the LP and
Bus voltage No.
NLP
LP
LS
1 2 3
0.094 0.190 0.097
0.470 0.190 0.190
7.260 7.440 7.420
TABLE 3(c) Estimated bus voltages and angles in the case of meter failure NLP
LS
V (p.u.)
~
V (p.u.)
1.059 1.045 1.023 1.022 1.015
0.0000 --0.0500 --0.0852 --0.0930 --1.068
1.024 1.011 0.990 0.989 0.983
0.0000 --0.0510 --0.0890 --0.0951 --0.1100
N L P m e t h o d s give b e t t e r estimations o f the s y s t e m state. Table 3(b) gives the p e r c e n t a g e error in t h e e s t i m a t i o n o f t h e bus voltages. T h e presence o f bad d a t a m a y also be due to c o m m u n i c a t i o n and m e t e r failures. These effects can be simulated b y replacing t h e real and reactive p o w e r flows o n line 24 with zeros. This s t u d y leads t o the results given in Table 3(c) w h i c h again proves t h a t the existence o f bad d a t a in t h e p o w e r s y s t e m m a y cause i n a c c u r a c y in the c o n v e n t i o n a l LS solution. The p r o b l e m s with the e x a c t d e t e r m i n a t i o n o f t h e s y s t e m o p e r a t i n g p o i n t {initial estim a t e ) m a y c o n t r i b u t e t o an i n a c c u r a t e estimat i o n o f states if the LP t e c h n i q u e is used. The o p e r a t i n g p o i n t o f the system consists o f
49 cate t hat the NLP m e t h o d yields bet t er results for unmeasured quantities.
TABLE4 Estimation withanimproperinitialpoint NLP
LP
LS
V 6 (p.u.)
V 6 (p.u.)
V 6 (p.u.)
CONCLUSION
1.060 1.047 1.024 1.024 1.017
1.057 1.046 1.025 1.025 1.023
1.071 1.058 1.036 1.036 1.032
The application of nonlinear programming in power system state estimation has been considered in this study. This algorithm has been tested on a five-bus system and the results are presented in the paper. However, the proposed m e t h o d can be applied to an n-bus system. The NLP technique has also been compared with linear programming and weighted least square m e t h o d s and it appears t hat NLP is m ore accurate than the o t h e r two m e t h o d s in several applications such as bad data detection, suppression of Gaussian distributed noise with large magnitude, bad data rejection, inaccurate estimation of the initial operating point, estimation of the quantities n o t being measured, etc. One of the difficulties with application of the least square m e t h o d in pow er system state estimation is t hat it is ext rem el y noise sensitive, so bad data need to be detected before t h e y can be rejected from the set o f measurements. In addition, it assumes that the noise has a Gaussian distribution. In m any applications, the determination of the exact value for the standard deviation o f the noise measurements is a difficult task. The NLP m e t h o d is relatively fast and does not restrict the system studies to Gaussian distributed noise. Neither does it require the bad data det ect i on and identification techniques applied t o the LS m e t h o d . For this application of nonlinear programming, a quadratic approxi m at i on has been considered in modeling the system equations, which is generally considered as a m ore precise representation of the nonlinear equation. In particular, if the operating point o f t he system has been estimated improperly, the NLP m e t h o d is able to determine the new state o f the system to a higher degree of accuracy than that of the LP m e t h o d . This is due to the fact that, using the LP technique, the linear approxi m at i on is considered in modeling the system equations and the accuracy o f this approximation is limited t o small deviations o f the system states.
0.0000 --0.0496 --0.0879 --0.0930 --0.1060
0.0000 --0.0239 --0.064 --0.0688 --0.0848
0.0000 --0.0494 --0.0868 --0.0925 --0.106
TABLE 5 Estimation of quantities not being measured (0.1% noise level) Quantity
Load flow value (p.u.)
Estimated value (p.u.) NLP LP LS
Pl2 P13 P2s
0.888 0.407 0.548
0.8809 0.4040 0.5440
0.815 0.388 0.551
0.916 0.412 0.516
some estimated values and some guesses. Since the LP equations are linearized in the n e i g hb o r h o o d o f the system operating point, any significant deviation o f the linearized system equations from the actual operating point may result in erroneous estimation o f the system state. In contrast, the nonlinear m e t h o d will lead to b e t t e r estimated values owing to the quadratic nature o f the approximation. Table 4 represents the results o f system state estimation while the system operating p o i n t (initial value) is estimated improperly. These results show that LP fails to give an accurate estimation o f the system states. In this case, b o th NLP and LS invoke bet t er solutions for the estimation o f the system state. Power system state estimation m ay also be used to determine the quantities which have n o t been measured in the pow er n e t w o r k and the unmeasured values estimated by different estimators. Table 5 represents the unmeasured values which are c o m p u t e d according to the estimated states o f the system. The estimated states are the same as those given in Table 3 and the c o m p u t e d quantities are c om par e d with the base case values. These results indi-
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