Treatment of inequality constraints in power system state ... - IEEE Xplore

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IEEE Transactions on Power System, Vol. 10,No. 2, May 1995

Treatment of Inequality Constraints in Power System State Estimation Kevin A. Clements

Paul W. Davis Worcester Polytechnic Institute Worcester, MA 01609

Abstract A new formulation of the power system state estimation problem and a new solptioa technique are pro sented. The formulation allows for inequality constraints such aa Var limits on generators and transformer tap ratio limits. In addition, unmeasured loads can be modeled as unknown but bounded quantities. The solution technique is an interior point method that uses logarithmic barrier functions to treat the inequality constraints. We describe computational issues arising in the implementation of the algorithm. Numerical results are given for systems ranging in size from six to 118 buses. Keywords: Power system monitoring, State estimation, interior point methods.

1

Introduction

The static state of a power system is estimated from measurements made at various points in the network. The network is assumed to be in steady state, and the state vector consists of the bus voltage magnitudes and phase angles throughout the network. The state vector z is of dimension n and €he measurement vector z is of dimension rn. Measurements typically consist of bus voltage magnitudes, real and reactive line flows, real and reactive bus injections and line current magnitudes. It is assumed that the number and topological distribution of the measurements are sufficient for observability [SI. The measurement vector is related to the state vector through the measurement equation 2

= h(z) + €,

where h ( z ) is the vector function relating the state vector to the error-free measurements and e ia the measurement error with zero mean and diagonal covariance matrix R. At buses with neither load nor generation, injected power is identically zero. Such perfect measurements are modeled as equality constraintti,

where the dimension of the vector function g(z) is c. The power system static state estimation problem was formulated over twenty years ago. Over the intervening two decades, researchers have developed efficient solution algorithms, studied the question of network observability, and developed methods for the detection and identification of bad data. A summary of the history of developments in the area of power system state estimation may be found in [2]. The major papers written on the topic are also referenced in 121* T h e state estimation problem is customarily formulated as an equality constrained nonlinear leastsquares problem:

'\

I

i t

Minimize

!jrTR-' r

(1)

92 WM 111-5 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the IEEE/PES T992 Winter Meeting, New York, New York, January 26 30, 1992. Manuscript submitted September 3, 1991; made available for printing January 13. 1992.

-

Karen D. Rey

The necessary condition for optimality is that p, a,2 , r be a stationary point of the Lagrangian funetion 1 2

L = -rTR-'r

- p T g ( z ) - rT[r- z + h(z)],

(4)

where p and I are vectors of Lagrange multipliers. This condition requires that 2 , p, and I satisfy

0885-8950/95/$04.00@ 1992 IEEE

568

The Logarithmic Barrier Function Method

2

where G and H are the Jacobian matrices of g and h respectively. These nonlinear equations are typically solved by an iterative technique such as the GaussNewton method. This formulation, which is representative of the problem formulations that have appeared in the literature to date, does not address inequality constraints that are present in the system. These constraints arise because of Var limits on generators and transformer tap ratio limits. A problem arises when the computed state estimate results in violation of one or more inequality constraints. These constraints are usually treated in a heuristic manner by setting any violated limits to equality constraints during the final iterations of the solution [lo]. By directly incorporating inequality constraints into the problem, not only can such constraints be treated in a rigorous manner, an unknown but bounded model for unmeasured loads in the network can be used as well. Such a model is often preferable for unmeasured loads. Modeling the inequality constraints by f(z) 5 0 gives rise to the following minimization problem: Minimize

Interior point methods for nonlinear programming problems have been in the mathematical programming literature since the early 1950's and have been studied in detail by Fiacco and McCormick [5]. Interest in these methods has been rekindled recently by the projective method of Karmarkar [7]. Gill e2 al. showed that under certain assumptions, Karmarkar's method was equivalent to an interior point method known as the logarithmic barrier function method. In this method, the inequality constraints on the slack variable s are treated by appending a logarithmic barrier function to the objective function of (6), D

p is the number of inequality constraints and sk is

the L'th element of the slack variable vector s. The bam.er parameier p > 0 is forced t o decrease towards zero as the iterations progress. As u , -, 0 the solution of the subproblems approaches that of (6). The Karush-Kuhn-Tucker (KKT) first order necessary conditions for an optimal solution of the subproblems are expressed in terms of a stationary point of the Lagrangian function

%J

irTR-'r

Hence, 1 Le K K T conditions for this problem are

where s is a vector of slack variables used to convert the inequality constraint to an equality constraint. Interior point optimization methods compute a s e quence of solution points that are interior to the feasible region of the problem. These solution points lie on a path that progresses towards the optimal solution. An interior point method for solving the nonlinear least-squares problem with inequality constraints is described in the following section. Section 3 discusses computational issues and Section 4 presents some numerical results. Conclusions are presented in Section 5.

-pS-'e-X V,t = -f(z)-s VAL = -g(z) V,t = V,t = -r+z- h(z) vrc = R-lr-n V,L = - F ~ X - G + H ~ T

= 0 = 0

=

0

= 0

(9)

= 0

= o

s L 0, where S = diag(s1,. ..,sm) and e = [l,. .. , 1IT. These nonlinear equations can be solved iteratively using the Gauss-Newton method. In this method, the following linearizing approximations are made at each iteration:

569

Initialization

3.1 f(zk)+FAz g ( z ) w g(z')+GAz h ( z ) w h(zk)+HAz

f(z)

fi:

F w F(P) G w G(zk) H w H(2') S-'e

It is assumed that an initial state vector, 8,is available such that f(zo)< 0. The initial state vector can (10)

-

(Sk)-'e (Sk)-2As

After these approximations are made and As and r are eliminated, the following system of equations results:

where

1 D = -(Sk)2

The coefficient matrix in the in Equation (11) is symmetric with non-negative diagonal elements. Consequently Equation (1 1) is a sparse positive semidefinite symmetric system whose solution is discussed in the following section of this paper. The solution of (11) is used to compute the direction of changes in t , A, p and a. It may not be possible to take a full Newton step computed by wlving (1 1) without violating the inequality constraints. Consequently, new values of these variables are computed from

Pk+'

=

d+' =

+

Pk 4 P - P E ) rk+a(a-ak)

(12)

The scalar step size a is chosen so khat xk+' r e mains interior to the feasible region, i.e., so that f(z"+') < 0. It should be noted that, while the sign restrictfon on s does not appear explicitly Equation (ll), it is satisfied due to the procedure used to choose the scalar step size.

3

Solving for the Newton Direction

3.2

c1

z*+' = z k + a A z Ak+1 = Ak + a(A - Ak)

be chosen as the previously computed estimate or as a flat start. Either one of these initial state vectors will usually satisfy the inequality constraints. If this is not the case then a modified problem similar to a Phase I problem in linear programming can be formulated. Iterations of the modified problem are solved until an interior feasible point is found. The initial slack vector is so = -f(zo). The only other variable that needs to be initialized is the barrier parameter p. In the computational results reported in the following section, p was initialized at 1.0. The optimal choice for p may be problem dependent although, in our experience, the performance of the barrier function method does not appear to be very sensitive to this choice.

The principal computational task at each iteration is solving the semi-definite system of equation (11). Equations of this type appear in many applications and much effort has been spent on developing efficient solution methods for them. Duff and Reid [4] developed a general purpose method based on Cholesky factorization that has been widely used. Other more specialized approaches take advahtage of the particular sparsity structure arising in power systems have also been developed [1,3,10]. We have used the method in [3]. This method reduces to the normal equations method when no equality constraints are present. In order to apply the block matrix approaches of [l]or [lo], it is necessary to rearrange the equations;

[: 2

O G @][;]I=[-s(zk) tF

P

ET

o

]

(13)

Az

where

Computational Issues

In this section we discuss computational issues and give details of the algorithm as we have implemented it. We discuss initialization in Section 3.1, solution for the Newton direction in Section 3.2 and adjustment of the barrier parameter, calculation of step length and the stopping criterion in Section 3.3.

The vector AF can be partitioned as AZ = [As,A%lT. AYA contains measurements and inequality constraints whose Jacobian structure is that of a bus injection measurement, A& contains the remaining measurements and inequality constraints.

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Pivoting can be performed on those rows of the coefficient matrix corresponding to A& to yield equations of the following form:

6 is a positive quantity if the primal and dual variables are feasible and it is zero at the optimum point. For linear programming problems, McShane et al. [9] suggest making the barrier parameter proportional to the duality gap. We have adopted their approach for our problem choosing

-

(14) where F = H z E i l g ~ . Schemes for pairing elements of x with those of either p or %A are described in [ l ]and [lo].By using either of these schemes it is possible to rearrange the equations into a block structure of the form

Ky=b

(15)

where the elements of K are 4x 4 blocks of the general form

The step length is chosen such that the primal solution remains feasible and interior to the feasible region. This is done by taking

a = 0.9995h

(19)

where

and As = -FAX. This step size is based on linearization of the inequality constraints. The nonlinear constraints are also tested to insure that f(zk a A t ) < 0. If the nonlinear constraints are not satisfied, the step size is further reduced until they are satisfied. The iterations are terminated when the relative duality gap is sufficientlysmall, i.e. when for some specified v

+

where Jij-is a 2 x 2 Jacobian block which is either a block of H A or G or zero for buses where no measurement or inequality constraint is present. &j is zero if i # j . R,i is either a diagonal 2 x 2 block of RA or zero for rows corresponding to G. (Note that if a decoupled solution algorithm is used, then the blocks of K will be 2 x 2 rather than 4 x 4 blocks.) The block structure of K is the same as that of a network admittance matrix and well known ordering techniques can be used for optimal ordering and block Cholesky factorization.

3.3

Adjusting the Barrier Parameter and Step Length; Stopping Criterion

The barrier parameter must be adjusted so that it approaches zero as the iterations progress. Our barrier adjustment is calculated using the difference between the primal and dual objective functions computed at each iteration. The problem dual to (6) is Maximize A, p, a Subject to:

- ATf(x) g ( x ) - aT [r - z

4r;R-lr -p

+ h(x)]

- F ~ A - G ~ ~ - =H o~ ~ R"r-a = 0 A S 0

(16) The duality gap, the difference between the primal and objective functions, is 6 =XTf(x)

+ p T g ( x ) + aT[r- z + h ( x ) ] ;

(17)

b

rTR-lr

4

< U.

Computational Results

This interior point method was applied to three systems ranging in size from 6 to 118 buses. Table 1 lists the number of buses, branches, and measurements in these networks. Test cases in which generator Var limits are modeled as inequality constraints are shown in Tables 2 and 3. The number of inequality constraints reflects the fact that every generator bus in the network has both an upper and lower Var limit. In the case when no nonlinear inequality constraints become active, the interior point method converges to the least-squares estimate. When a leastsquares estimate was calculated for the systems listed in Table 1, all three case8 took one iteration less than those required by the interior point method. In the case where nonlinear inequality constraints were not satisfied and the step size was modified, the three networks did not need any additional iterations to reach convergence. The 6, 30, and 118 bus networks were tested with up t o one, two, and three, respectively, inequality constraints that were not satisfied. The iterations in the interior point method are terminated when the relative duality gap is 0.000001.

57 1

Table 3 shows how the duality gap in equation (17) decreases per iteration in the 6, 30, and 118 bus networks. Tables 4 and 5 list the results for unmeasured loads test cases. In the 6 bus model, three unmeasured bads are modeled as unknown but bounded; in the 30 bus model, 12 loads are unmeasured; and in the 118 bus model 15 losds are unmeasured. Modeling each such load requires four inequality constraints; thus the number of inequality constraints is 12, 48, and 60 respectively. In the 6 bus model, one of the inequality constraints was reached; in the 30 bus model, two Constraints were reached; and in the 118 bus model, three constraints were reached. The number of iterations required for convergence was again comparable to the number required for unconstrained least squares estimation. Table 5 shows the duality gap versus iter* tion count for each of these cases. Table 6 lists the CPU time requirements for the unconstrained and constrained estimators. The unconstrained estimator uses the normal equations formulation with block Cholesky factorhation followed by forward and backward solutions to wlve tor Az at each iteration. In our implementation, for eases with no equality constraints, we computed Az by solving the equation

[HTR-'H

HTR-'

[%

+ FTD-lF] Az = - h ( Z L ) ] + FTD-'f(zL)

Table 1: Description of test cases; m is the total number of voltage, injection, and flow measurements. rBuaee I Linea 1

I

I

I

6 30 118

I

I

I

I

'Lbble 2: Var limit test cases: number of Newton iteratiom required for convergence with v = 0.000001.

hequality Buees Constraints 6 6 10 30 108 118

Active * Constraints 1 2 3

It&&ions 5

I

4 7

Table 3: Var limit test cases: the duality gap per iteration

Iterations I BBu6 I 3 0 B w 1 1 1 8 B ~ I 5.90005 19.018392 I 106.29086641 1 0.44271 0.020839 64.25252297 0.03323 0.000750 0.26679723 0.00250 0.000007 0.00111018 0.00004619 0.00019 o.moo19 0.00000001

(22)

The difference between the CPU times of the unconstrained and the constrained estimators is due to the additional iterations required by the constrained eatimator (typically one additional iteration) BB well as by the additional effort required required to form FTD-lF and FTD"f(zk). In the 6 bus model, the additional CPU time is rather modest, increasing by less than 14% for the Var constraints and by less than 20% for the unmeasured loads. The additional time required for the 30 bus model is more substantial. This is due to the fact that there are more inequality constraints. In the unmeasured loads case, twelve of the thirty bus loads are modeled as unknown but bounded. In terma of computational effort per iteration, this is equivalent to adding twenty four injection measurements to the measurement set. The percent increase in CPU time for the 118 bus model is more modest for the unmeasured loads case. This is due to the fact that the number of inequality constraints is lower relative to the number of buses in the model than for the 30 bus model.

M'eaeutements Volt Inj Flow 11 1 1 ' 2 2 7 41 I 52 1 10 41 178 I254 51 25 178 m

n

Table 4: Unmeasured loads test caaes: number of Newton iterations required for convergence with Y = 0.000001.

I Ineauality I

r -

I Buses I Conitrainits I Iterations 30 ~

118

5 7

48 60

Tabk 6: Unmebsured 1

I

Active Contraints

i7-i

1

2 3

4 test cases: the duality

gap per iteratioa

Iterations I 6 Bus I 9.63729 1

I

30Bus

I

118 Bus

1 94.007394 142.509101347

0.000000021 0.000000001

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I.S. Duff and J.K. Reid, MA27

Table 6: CPU times (seconds) for the unconstrained and constrained estimators.

I

Buses 6

30 118

5

I

Unconstrained Estimator

I

0.82 1.10 8.35

VAR I Unmeasured Constraints Loads

0.93 1.81 14.56

0.98 4.12 12.42

Conclusions

We have presented a new formulation of the power system state estimation problem. This formulation incorporates inequality constraints in the BYStem model. Inequality constraints can model reactive power limits at generators as well as limits on transformer tap settings. The use of inequality constraints also enables unmeasured loads to be modeled as unknown but bounded quantities. W e applied an interior point optimization technique known as the logarithmic barrier function technique to solve the reformulated problem. The computational experience reported here indicates that the number of iterations needed t o converge to a solution is comparable to conventional least-squares. The principal computational step, solving the symmetric positive semi-definite system of (11), is identical to that of the conventional least-squares method. Consequently, this method can be conveniently implemented using procedures from existing estimation software.

Acknowledgement The research of the authors was sponsored by the National Science Foundation under grant ECS-88-14046.

I

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a set of Fortran subroutines for solving sparse symmetric linear equations, Report AERE R-10533,Computer Science and Systems Division, AERE Harwell, Harwell England, (1982).

A.V. Fiacco and G.P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley & Sons, New York, (1968). P.E. Gill, W. Murray, M.A. Saunders, and M.H. Wright, On projected Newton barn-er methods f o r linear programming and equivalence to Karmarkar’s method, Mathematical Programming, 36, (1986),183-209. N.K. Karmarkar, A new polynomial-time algorithm f o r linear programming, Combinatorica 4,

(1984),373-295. G.R. Krumpholz, K.A. Clements and P.W. Davis, Power sysiem observability: a practical algorithm using network topology, IEEE Trans. PAS-99 (1980),15341542. K.A. McShane, C.L. Monma and D.Shanno, An implementation of a primal-dual interior point method f o r linear programming, ORSA Journal on Computing, 1 (1989),70-83. R. Nucera and M. Gilles, A blocked sparse matriz formuhation f o r the solution of equalityconstrained state estimation, IEEE PES Winter Meeting, Paper 90 WM 234-5 PWRS, Atlanta, February 1990.

Biographies

References [l] F.L. Alvarado and W.F. Tinney, State estimaiion using augmented block matrices, IEEE PES Summer Meeting, Paper 90 SM 273-3 PWRS, Minneapolis, July 1990.

A. Bose and K.A. Clements, Real-iime modeling of power networks, Proceedings of the IEEE, Vol. 75,pp. 1607-1622,Dec. 1987.

K.A. Clements, G.W. Woodzell, and R.C. Burchett, A new method for solving equalityconstrained power system estimation, IEEE PES Summer Meeting, Paper 89 SM 687-5 PWRS, Long Beach, July 1990.

Kevin A. Clements (Fellow, IEEE) received the B.S. degree in electrical engineering from Manhattan College, Bronx, NY, in 1963,and the M.S.and Ph.D. degrees from the Polytechnic Institute of Brooklyn, Brooklyn, NY,in 1967 and 1970,respectively. From 1963 t o 1968,he was employed by the General Electric Company, Pittsfield, MA. He was at the General Precision division of Singer, Little Falls, NJ, from 1968 t o 1970. In 1970 he joined the Electrical Engineering Department of Worcester Polytechnic Institute, Worcester, MA, where he is currently a Professor. His research interests include monitoring and control of electric power systems, power system stability, and methods for solution of large-scale systems.

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Paul W. Davis (Member, IEEE) received the B.S., M.S.,and Ph.D. in applied mathematics from Rensselaer Polytechhic Institute. He is a Profeasor of Mathematical Sciences at Worcester Polytechnic Institute, and he servea as Managing Editor of SIAM Review. His research interests include estimation, detection, and i d e n t h t t i o n for electric power uystenm as well as error analysis and computational methods in optical intedemxnetry.

Karen D. Frey (Student Meniber, IEEE)received the B.S., M.S. atld Ph.D. degrees in electrical engineering from Worcester Pdytechnic Institute in 1982, 1985, and 1991, respectively. Her industrial experience includes work at Honeywell Electro-Optics Division, Lexington, MA, and General Electric Company, Pittsfield, MA,where she was involved in the design and analysis of adaptive control systems. She had previouely been engaged in digital design applications at Texas Instrumats, Equipment Group, Dallas, Texas.

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Discussion

mented Block Matrices,” IEEE Transactions on Power Systems, vo1.5, No.3, August 1990.

Manuscript received February 19, 1992.

We like to ask the authors some questions regardin the practical use of the method. The first question is how t o a a n d l e the bad data roblem. Will an accurate measurement become a bad data when the aygorithm mana e8 to satisf all the constraints that might be wrongly posed an$ if the bounss for the unknown quantities unmeasured loids. is incorrect. is there a way of knowins that’f:: the solution o f -the method. This.leads to the second cuestion of how to take care of the infeasibility problem. Is it always true that. the flat starting point will be i,n the ipterior of the feasible region defined by the constraints, and if the flat starting point is not a feasible T i n t how the modified problem can be setup. The author mentioned t at the solution always stays in the interior.of the region defined by the cpnstraints, then can the solution satisfy all the equality constraints exactly. We found the paper very interesting and we enjoyed it very much.

Manuscript received February 18, 1992

H. Singh and F.L. Alvarado (University of Wisconsin-Madison) The authors are t o be commended for an interesting and well written paper which illustrates the use of an interior point method for solving the power system state estimation problem. Interestingly, the authors had used a n interior point method for a Weighted Least Absolute Value (WLAV) formulation of the problem previously [All. A major contribution of the paper as indicated by the title is the handling of inequality constraints in the Weighted Least Squares (WLS) formulation of the state estimation problem. These inequality constraints usually refer t o generator var limits and transformer t a p ratios whose violation after convergence of the state estimator is a problem that occurs in the solution for the external system model. As far as the observable internal portion of the network is concerned, the violation of these limits is not a problem. Did the authors test their method for solution of the external system? The representation of umeasured loads as intervals rather than as k e d numbers by means of inequality constraints is an interesting idea. Once again, this would be of greater benefit for the external system state estimation rather than the internal system estimation where sufficient redundancy of measurements has been assumed. Would the authors give a sense of what the typical intervals were. If these are made too small, the solution would move closer t o t h a t obtained by using pseudo-measurement s. As far as the solution method is concerned, the interior point methods can often be influenced by the choice of the initial feasible point. Did the authors ever have t o use anything other than a flat start. If so, was there any effect on the convergence? Finally, reference [I] in the paper has been incorrectly uted, as the paper was presented at the winter meeting in Atlanta in February 1990. For the benefit of readers, the correct reference is given below [A2]. [All K.A. Clements, P.D. Davis and K.D. Frey, “ An Interior Point Algorithm for Weighted Least Absolute Value Power System State Estimation Paper 91 WM 235-2 PWRS, IEEE Winter

I.W.SLUTSKER, N.VEMPATI, Empros Systems International, Plymouth, Minnesota: The authors are congratulated on the development of a new method of solving the power system state estimation providing an enforcement of inequality constraints . Their method is the first in which the inequality constraints are explicitly included in the formulation of the state estimation problem. We would like to solicit authors’ responses to the following questions. In the authors‘ approach all inequality constraints to be enforced must be explicitly modelled in the solution, regardless of whether they actually will end up being violated. Every constraint to be processed by the method must be represented by a separate equation. A measured quantity that is also bounded will, therefore, end up contributing three equations to the solution process rather than one as in classical formulation of the state estimation problem. The constraints that are usually enforced by state estimator may include generator MW and MVAR limits, voltage ranges at regulated buses, LTC tap ranges and unmeasured load limits. The total number of conhtrainta for a real life problem may approach or even exceed the number of measurements. As a result, it is not unrealistic to envision the size of the solution matrix doubling. The impact of the state estimator solution time, which can, to some extent, be seen from the test results in section 4, may be much more pronounced and can be dramatic. The increased execution time penalty is present even if no constraints are violated. Have authors attempted to reformulate the problem limiting the number of constraints that need to be modelled? It is frequently observed in practice that constraints to be enforced are in conflict and can not be all enforced. For example, under certain loading conditions a generator may not have sufficient W A R range to force the regulated bus voltage to be within the specified range. Only one of the constraints, the generator MVAR range or the regulated bus voltage range, can be enforced, but not both. In estimators that perform the limit enforcement in a heuristic manner (by increasing weights of the measurements in violation thereby forcing their estimates back into the range), the constraint conflicts can be handled. With such an approach it is possible to recognize a conflict and resolve it in favor of one of the constraints (based on the user-specified rules) and thus obtain an estimator solution acceptable to the user. It appears that the presented method in the presence of constraint conflicts will simply diverge. Can authors share their views on how constraint conflicts can be detected with their approach?

As the iterations progress, the step length is chosen such that the succeeding point remains feasible. This bears a striking resemblance t o the Generalized Reduced Gradient (GRG) method utilized for the OPF in yesteryears. The GRG was notorious for getting bogged down in attempting to reach an optimal solution. Did the authors experience o r foresee similar problems with their algorithm? The traditional normal equations method terminated the iterations when the increment to the state variables fell below a certain tolerance. However, t h e interior point algorithm is terminated when the duality gap is smaller than a specified threshold. What is the threshold and how it is chosen? Is this threshold problem dependent? Can the authors elaborate on this and give us a feel for the relationship between the two stopping criteria? Once again, the authors are commended on an analytically elegant method of solving a state estimation problem including the enforcement of inequality constraints. Their answers to our questions will be awaited with great interest. Manuscript received February 21, 1992.

Power Meeting, New York, 1991. [A21 F.L. Alvarado and W.F. Tinney, “State Estimation using Aug-

Closure was not provided by the author.