TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, VOL. PAS-89 NO. 1, JANUARY 1970

APPENDIX

DERIVATION OF ESTIMATE

The best estimate x of xtr,e made from z = BXtrue + n isdefined as the value of x which minimizes J(x),

J(x) = [z - Bx]'4V'[z - Bx]

which can be rewritten as

J(x) = z'[01 - 0-OBIB'0-']z

+ [x - B'6-lz]'6-l[x - 2B'-lz]

I [B'O-1B]-l.

It is obvious that

x = B'O-lz.

It also follows that

E{x} XXtrue

x - Xtrue][X-Xtrue]'}

This derivation uses the same basic argument used in [3], i.e., aweighted least squares criterion followed by matrix manipulationto perform the desired minimization. It is possible to show thatthat resulting x actually yields the minimum error covariancematrix of any linear function of z yielding an unbiased estimate.

ACKNOWLEDGMENT

The authors wish to thank American Electric Power ServiceCorporation engineers for their help in explaining power systemproblems. These engineers may not agree with the results, butthe paper would have been impossible without their help.

REFERENCES[11 T. Anderson, An Introduction to Mutltivariate Statistical Analysis.

New York: Wiley, 1958.[21 D. B. Rom, "Real power redistribution after system outages

error analysis," Power System Engrg. Group, Dept. of Elec.Engrg., Massachusetts Institute of Technology, Cambridge,Rept. 7, August 19, 1968.

[31 F. C. Schweppe and J. Wildes, "Power system static-state esti-mation, pt. I: exact model," this issue, pp.120-125.

[4] F. C. Schweppe, "Power system static-state estimation, pt.III: implementation," this issue, pp. 130-135.

[5] G. W. Stagg and A. H. El-Abiad, Computer Methods in PowerSystem Analysis. New York: McGraw-Hill, 1968.

Power System Static-State Estimation,Part III: Implementation

FRED C. SCHWEPPE, MEMBER, IEEE

Abstract-The static state of an electric power system is definedas the vector of the voltage magnitudes and angles at all networkbuses. The static-state estimator is a data processing algorithmfor converting redundant meter readings and other available in-formation into an estimate of the static-state vector. Discussionscenter on implementation problems associated with computationtime requirements, dimensionality resulting from a large number ofbuses, and the actual time-varying (nonstatic) character of powersystems. Various potentially useful approaches are discussed andcompared.

Paper 69 C 2-PWR, recommended and approved by the PowerSystem Engineering Committee of the IEEE Power Group forpresentation at the IEEE PICA Conference, Denver, Colo., May18-21, 1969. Manuscript submitted January 20, 1969; made avail-able for printing August 1, 1969. This research was supported by theAmerican Electric Power Service Corporation and NSF Fellow-ships.The author is with the Massachusetts Institute of Technology,

Cambridge, Mass. 02139.

INTRODUCTIONTHIS PAPER is the third of a three-part sequence on static-

state estimation and related detection and identificationproblems. The static-state estimator is a data-processingalgorithm for use by a digital computer to reduce meter measure-ments and other information on an electric power system into anestimate of the system's steady state or static state. Thestatic state is a vector composed of the voltage magnitudes andangles at all the buses.

In the first paper [2], discussion covers the overall problem,mathematical modeling, and general algorithms for state estima-tion, detection, and identification. In the second paper [11an approximate mathematical model and the resulting simplifica-tions in estimation, detection, and identification are discussed.In the present paper various implementation problems associatedwith dimensionality, computer speed and storage, and the time-varying nature of actual power systems are discussed.

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One important point underlies many of the discussions tofollow. It is desired to implement digital-computer programswhich generate satisfactory results with a minimum and/oracceptable amount of computation expense. It is rarely desiredto implement some equation of [1]; [2] with high accuracy.Precise, highly accurate answers are usually not needed. Further-more, inherent measurement and modeling errors always causethe results to be in error no matter how much calculation is done.Thus in actual implementation many engineering liberties canand usually should be taken with the equations. This attitude isreflected throughout the discussion to follow.The following notational conventions are used. A bold face

letter denotes a vector or matrix. All vectors are column vectors.A prime denotes matrix transpose, and -1 denotes matrixinverse. The letter E denotes the expectation or averagingoperation on a random variable or vector. Complex numbers ormatrices are denoted by a tilde; i.e., Y is a real-number while Yis a complex number.

ROLE OF APPROXIMATE MODEL

The approximate model of [1] is based on four assumptions;1) transmission lines with high X/R, 2) uncorrelated observationerrors, 3) small voltage angle differences across lines, and 4)voltage magnitudes close to nominal (i.e., one). Systems withhigh X/R and uncorrelated errors are often of interest. The lasttwo assumptions of the approximate model are also valid formost normal and some abnormal operating conditions but mayfail when the transmission system (or some portion thereof) isbeing stretched to its limits. Since the state estimator is mostneeded in times of stress, implementation of only the approximatemodel is not recommended. However, the approximate modelcan be used to save computer time in the routine daily processingwhen the system is not being stretched. It provides a practicalapproach to the problem of handling interconnected systems.It yields insight into the behavior of the more complete model.Finally, the approximate model provides equations which cansometimes be used when implementing the exact solution. Allthese points are discussed further in the following sections. Therecommendation not to rely soley on the approximate model ismade without consideration of computer and metering costs.In particular applications, a real power-voltage angle approxi-mate solution may be all that can be economically justified.

CHOICE OF MODEL PARAMETERS

Implementation requires the choice of numerical values for thenetwork structure (admittance matrix Y) and the error structure(error covariance matrix 6).The choice of a good network model Y is crucial to satisfactory

operation. Unfortunately, the only advice is to make use ofengineering judgment. A precise error model 0 would vary withload conditions, time of day, weather conditions, and the moodof system operators and would include off-diagonal elements.However, a precise model is usually not needed. The matrix 0plays the role of a weighting matrix which causes the estimateto fit the good measurements closer than the bad measurements.This weighting is important when wattmeter readings of 2 per-cent accuracy are to be combined with pseudoload powers of50 percent accuracy and when watt, var, and voltage measure-

ments are to be combined together. However, relatively largeerrors in 6 usually have only small effects on the actual estimate.Therefore, in most cases 0 can be assumed to diagonal, andrough engineering guesses for the main diagonal elements suffice.

ITERATIVE STATE ESTIMATION

In [2] a state estimate x is defined as the value of x whichminimizes J(x), where

J(x) = [z -f(x) ''-I [z -f(x)] (1)where z is the vector of measurements, f(x) is a nonlinear func-tion of x determined by Yj and Kirchoff's laws which give themeasurements of ideal meters, and 6 is the measurement errorcovariance matrix. The algorithm for obtaining x is the iteration

Xn+l = Xn + 1(x.)F'(x.)6-[z - f(x.) ]

X (x,) = [F'(xn)O-'F(xn)]-l

(2)

(3)

where F(x.) is the Jacobian matrix of f(x). Hopefully x,n convergesto x. Methods to simplify this iteration and decrease computa-tional requirements are now discussed.

Consider (2). It can be viewed as a feedback loop whichattempts to drive x,, to a steady-state value (i.e., x,, = xn+±)by feeding back a correction term through a gain. Two possibleinterpretations are the 1-gain and W-gain cases defined as

I gain

gain [F (xn)correction term F'(x.)O-fz - f(xn)]

W gain

W(xn) = Z(x,)F'(x.)O-'[z - f(x.)] J

Consider the definition of x. Since x minimizes J(x), it followsthat x is a value of x which satisfies the equation

dJ(x)/dx = F'(x)0-1[z - f(x)] = 0. (4)

It is clear from (2) that if x,n converges, it converges to a solutionof (4) independent of the value of the I-gain matrix. This ob-servation leads immediately to the concept of simplifying I(x,)to reduce the computer load. One obvious simplification is to usevalues which do not vary during the iteration; i.e., I(xn) =

I0, where Io is 1(x) for one nominal value of x. Other approxi-mations are discussed later when dimensionality problems areconsidered. Simplification of the I-gain matrix reduces computertime per iteration without affecting the final answer (if conver-gence occurs).

It is seen from (2) that convergence of x,, for an arbitrary W-gain matrix does not necessarily lead to a solution to (4). How-ever, this may not be important as it is only desired to obtain anestimate which is close to the value of x which minimizes J(x).Hence, it may often be satisfactory to use W(xn) = Wo for somenominal Wo. Equation (2) can be written in the form

[F'(xn)f-WF(xn)][Xn+l - xn ] = F'(xn)0-'[z - f(x.)] (5)

which can be viewed as a set of linear equations which are to besolved to find the value of x,n+± - xn. Matrix inversion (to get 1)solves this equation, but there are much more efficient ways tosolve a system of linear equations. However, such alternateapproaches lose their appeal when a simplified I- or W-gainmatrix can be used.No matter what iteration equation is used, the stopping rule

should not be chosen to keep iterating until the minimum ofJ(x) has been located exactly. It is only desired to find some xwhich yields an approximate minimum.

DIMENSIONALITY PROBLEMS

Many transmission networks of interest involve hundreds andeven thousands of buses. Several aspects of this dimensionalityproblem are now discussed.

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, JANUARY 1970Y

Calculation and Storage of s-Gain MatrixApproximations for and calculations of the s-gain matrix

are discussed. Most of the ideas also apply to the W-gain matrix.The approximate model of [1] yields

X=approx [ (6)

where 16 and 1, are the voltage angle and voltage magnitudesgains (error covariance matrices), respectively. Equation (6)can be used as a s-gain matrix.Approximations of a given X, a full matrix (or a given 2a, 1 u)

with a sparce matrix saves both computer storage and time. Oneapproach is to simply set all the off-diagonal terms in I equalto zero before inversion. To ensure stability, multiplication byconstant, 0 < a < 1, may be required.Three approaches to the actual matrix inversion problem are

now discussed. Since F is a sparce matrix and I is diagonal,-1 = F'O-1F is also a sparce matrix (although not as sparce as

F), and sparce matrix inversion techniques can be used. A secondapproach can be based on the following formulas for inverting apartitioned positive definite matrix:

Bil B12 -

B12' B22_-a-1 ~ ~~~~A-lB12B22-1

(7)- B22-FB,'-1 B22-1 +B22-1B,2"A '1B,2B2 '1

A = Bil -B12B22-IB12f. (8)

Equation (7) can be used to convert the problem of invertingone large-dimensional matrix into a problem of inverting manysmaller dimensional matrices. The smaller submatrices whicharise can be approximated at each step by sparce matrices toreduce storage and computer time. A third approach is really aspecial case of the matrix partitioning technique wherein a zcorresponding to many measurements can be builtup one mea-surement at a time. The basic matrix identity to be used is

[B + bO-1b']- = B-1 -B-lb[b'Bb + 0-1]-lb'B-' (9)

where B is a positive definite matrix and b is a column vector.To use this identity, B-1 may correspond to I for some minimaldata set, and b may correspond to the column vector Jacobianof the first redundant measurement. The buildup can then berepeated until all redundant measurements have been added.The buildup can also be started with the first measurement if theinitial B-1 is a diagonal matrix with values which are muchlarger than the main-diagonal elements of the final 1. At eachstep in this buildup, the full s matrix can be approximated by asparce matrix.

Spatial QuantizationTo simplify the discussion, the overall network (own system

(OS) with or without interconnected systems (IS) as desired)is assumed to be spatially quantized into three geographicregions or sets of buses which are chained together. Some region 1buses are connected to region 2 buses by quasi-tie lines, and someof the region 2 and 3 buses are similarly connected. However,no region 1 and region 3 buses are directly connected. To furthersimplify tbe discussion, the need for a reference node is ignored.

Let Nb,k denote the number of buses in region k, k = 1,2,3.Let Xk denote the state vector (voltage magnitude and anglesof all buses) for region k, k = 1,2,3. Then the complete state-vector is x' = [xl,x2,x3] which has 2(Nb,l + Nb,2 + Nb,3) dimen-sions (since the reference node problem is being ignored). LetNTL,12 and NTL,23 denote the number of quasi-tie lines connectingregion 1 to 2 and 2 to 3, respectively. Introduce a fake bus in the-middle of each tie line. Let x12 and X23 denote the vectors of the-voltage magnitudes and angles at these NTL,12 + NTL,23 buses.Let Zk denote the vector of measurements available on the busesand lines of region k, k = 1,2,3 (including any measurements madeon the kth end of the quasi-tie lines). The complete measure--ment vector is z' = [Zl,' Z2,' Z3']. Assume that the correspondingerror covariance matrix 0 is diagonal. Assume that the informa-tion in Zk, k = 1,2,3 is sufficient so that z, can be used to estimatexi and X12; Z2 to estimate X12, X2, and x13; and Z3 to estimate X13 andX3. Then instead of one estimation problem where x has 2(Nb,l +Nb,2 + Nb,3) dimensions, there are three separate estimationproblems of 2(Nb,l + NTL,12), ete. dimensions. The three separateestimates of course result in a mismatch at the fake tie linebuses; i.e., x12 estimated from z, will not equal x12 estimated fromZ2. However, each estimate also has an associated error covariancematrix, and the estimates plus or minus their errors will overlap.

Spatial quantization extends in an obvious fashion to anynumber of regions. The assumed chain structure is not essential,i.e., loops (region 3 connected to region 1) are possible.

Spatial SweepSpatial quantization is a simple useful concept but it has

several disadvantages. One disadvantage is the mismatch prob-lem which may be unacceptable in some applications. Anotherdisadvantage is the fact that sufficient metering is required ineach region so that z, is sufficient to estimate xi, X12, ete. A thirddisadvantage is a loss in accuracy and detection ability as Z2and Z3 actually contain useful information about xi, etc. Thespatial sweep technique removes some of these problems.The basic principle underlying spatial sweep is best seen in

terms of two connected regions with states xi and X2, a fake tieline bus state X12, and measurements z1 and Z2. Let ixl(zi) andx,2(z1) denote region 1 estimates obtained by processing z, as inspatial quantization. Let x'2(zl,z2) and x2(zl,z2) denote estimatesmade using z, and Z2 SO X2(Zl,Z2) is the standard estimate of x2obtained by processing z, and Z2. Now view &x2(z1) as a measure-ment on x12 with measurement error covariance matrix 112(l),where 112(1) is actually the estimate error covariance matrixof x12(z,) as obtained from a formula like (3). Let x12[x12(z1),z2]and x2[x,2(z1),z2] denote estimates of X12 and x2 obtained byprocessing x12(z,) and Z2. The key to spatial sweep lies in assuming

(10)

Equation (10) can be interpreted as saying that all the informa-tion contained in z, about x2 is also contained in x2(zl). Equation(10) can be shown to be exact when f(x) is a linear function of xas in the approximate model. In the general case, (10) is only anapproximation.Now consider the simple chain of three regions. The spatial

sweep concept can be divided into three main steps:

1) a forward sweep starting with region 1 and going to region 32) a backward sweep starting with region 3 and going back to

region 13) combination of the results of the forward and backward

sweeps.

132

X2[XI2(Zl),Z2] = X2(Zl,Z2)-

,SCHWEPPE: SYSTEM STATIC-STATE ESTIMATION, III

The forward sweep is done in three steps:

1) calculate xii(z1), x12(zl) from zi.2) calculate x12(zl,z2), x2(zl,z2), and X23(z1,z2) from x12(z,) and Z2.3) calculate x23(zl,z2,z3) and x3(zi,z2,z3) from x23(z1,z2) and Z3.

The backward sweep yields in a similar fashion X23(Z3), X3(Z3))X12(z2,zO), X2(z2,z3), x23(z2,z3), x1(z1,z2,z3), and xi2(z1,z2,z3). The for-ward sweep yields final estimates for X23 and X3 while the back-ward sweep yields final estimates for xi and X12. The third mainstep gets a final estimate for x2 by, say, combining X2(zi,z2), x12-(Zl,Z2), and x23(z,z2) with x23(z3).For the nonlinear case, the spatial sweep technique is not

precise, and repeated sweeps through the data (and regions) arerequired, i.e., an iteration solution as expected. When the spatialsweep is used, only the first and last regions have to have com-plete metering. These end regions could be a single bus if de-sired. If each region is completely metered so that xl(zD), x12(z,),i2(z2), etc. are calculated first, the spatial sweep can be modifiedinto a technique to combine these estimates into one consistentestimate with no mismatch. The spatial sweep discussion wasbased on a simple chain configuration of three regions. Extensionto longer chains is obvious. Extension to more complex figurescontaining loops introduces complexities which require someextra covariance matrices to be carried along. However, it shouldbe remembered that the various error covariance matrices areeffectively the Z-gain matrices, and hence various approxima-tions can be used.

REAL-TIME STATE ESTIMATION

Thus far discussions have centered on using one set of measure-ments made on a power system in steady state. However, thealgorithms actually needed are for real time, 24-hour a dayoperation using measurements made on a power system withcontinually changing conditions. Various aspects of this problemare now discussed.The timing and sequence used in sending the metered data to

the computer is important, and two basic types of operation canbe defined. "Snapshot" operation denotes the case where allthe meters are read at the same instant of time and fed into thecomputer as one vector. These snapshots are repeated periodi-cally. (The steady-state model can be viewed as one snapshot.)"Sequential time scan" denotes the case where the meters areread and sent to the computer sequentially so that the computerinput is a time series of scalar meter readings. After one scanis completed, another is started. Actual telemetry and computerinput systems may be some combination of the pure snapshotand pure sequential time scan, but discussion of the two extremecases conveys the basic ideas.

Let r denote the "system time constant" which is looselydefined as the time it takes the state vector to change an apprecia-ble amount due to normal load changes and the resulting genera-tor and other regulator responses. If tl-t2 < r, it is assumedthat x(ti) = x(t2). The actual value of r depends on many factors(including time of day), but for many applications r is probablybetween 10 seconds and 10 minutes. Effects of network changesor major generation pattern changes are not considered here.

Snapshot OperationLet tC, n = 1, * denote the times the snapshots are taken.

Let z, (ta), n = 1, - denote the corresponding measurementvectors (of both actual and pseudomeasurements). If It. -t.+d< r, a real time state estimation can be obtained by simply turn-ing the iteration of (2) into a recursion by rewriting (2) as

X(tn+l) = k(tn) + W(tn) [z(tn+l) - f[k(tn) I (11)

W(tn) = (tn) If [X (tn) I0 (12)

where x(t.) is the estimate based on z(tn). It is often useful tointerpret (11) as a feedback loop (a digital servomechanism)with input, z(4), which attempts to track a time-varying quan-tity, the true state x(t,,). In practice, time invariant sparceI-gain or W-gain matrices would probably be used. Equation(11) could be called a "tracking" estimator.

If Jtn - t,+il > r, a reasonable approach is to combine re-cursions and iterations. Thus given z(tn+i), a recursion like (11)is done first, and then iterations like (2) are repeated until asatisfactory result obtained.

Sequential Time Scan OperationLet tk, k = 1, * * * denote the times the individual meters are

read and sent to the computer. Let the scalars Z(tk) denote thecorresponding measurements. Assume that there are M meters.Then z(tk) and Z(tk+M) are readings made by the same meter, andtM+k - t is the time needed to complete one complete scan of allthe meters. To simplify the discussions, assume that no pseudo-measurements are present.

Consider first the case of a fast scan, where tM+k - tk < r.This can be treated as a snapshot by simply storing the Z(tk) . . .

z(tk+M), forming them into one vector, and doing a recursionlike (11) at tM, t2M, t3M, etc. However, instead of storing andwaiting, the measurements can be processed recursively as theyarrive. Assume for simplicity that a time invariant W-gainmatrix would be used for the corresponding snapshot operation.Then a recursive method, tracking estimator, is

X(tk+l) = i(tk) + Wtk+l[Z(tkil) -ftkl['(tk)] (13)

where ft,,(x) is the ideal meter measurement corresponding tothe meter read at time tk+i and Wtk+l is the corresponding columnvector of the W-gain matrix.Equation (13) can of course be interpreted as a feedback loop

just as (11). Equation (11) is a vector input-vector feedbacksystem while (13) is a scalar input-scalar feedback system.Equation (11) can be operated with a time-invariant gain matrix.Equation (13) must be operated with a gain vector that varieswith the meter reading being processed.Now consider the case of a slow scan, where tM+k - tk > r.

In this case, the meter readings are not made on the same statevectors, i.e., x(t,) # x(tM). Slow sequential time scans cannot beconverted to snapshots, and it is necessary to use an implementa-tion like (13). The choice of gain vectors can be made by judg-ing and by evaluating behavior of similation trials or by thedynamic state-estimation theory mentioned in the next section.The sequential time scan concept can be easily modified to

handle a time sequence of small snapshots taken on differentportions of the system. This concept then blends nicely with thespatial sweep concept. The two techniques can be combined bydropping the backward sweep and just continually scanning andsweeping around the system in space and time.

Dynamic State EstimationIt is not philosophically satisfying to employ a static logic on a

time-varying system. There are also some practical disadvan-tages. One was already encountered in the case of a slow sequen-tial time scan. Another possible disadvantage is the loss in in-formation. For example, consider two closely time spaced snap-shots z(t,) and z(t2). z(t,) contains useful information about

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IEEE TRANSACTIONS ON POWER APPARATUS AND SYSTEMS, JANUARY 1970

x(t2) which is ignored by the static-state estimator. Dynamic-state estimation concepts overcome these problems and havestill other advantages (especially in detection). Unfortunatelydynamic-state estimation for power systems is a large topic,and only a few basic ideas are outlined here. Complete discussionshave to be deferred to a future paper.The crucial step in obtaining a dynamic-state estimator is the

development of a mathematical model for the time evolution ofthe state vector. One particularly useful model is of the form

X*(t+l) = M[x*(tn),tn] + w(t) (14)

x(tn) = y[x*(tn)] (15)

where x*(tn) is the state vector of the dynamic model and w(tn)is a stochastic process with known statistics. In the simplestcase, x*(tn) could be just x(tn), but in general it contains addi-tional variables such as frequency, amount of steam stored inboilers, rate of change of load, etc. Such dynamic models can becombined with the static measurement models to yield dynamicstate estimation algorithms similar to (11) in the sense that theycan be viewed as feedback tracking systems. The I-gain matrixbecomes a function of 4) and the statistics of w(t) as well as fand 0. Such a dynamic state estimator analysis could automat-ically yield the desired gain vectors Wtk to be used in (13)for the case of slow sequential scan.

Dynamic-state estimation will probably be used eventually,but the static-state estimation concept has been emphasizedhere since an understanding of and experience with the static-state estimator is needed as a first step. Furthermore, the static-state estimator should provide satisfactory performance in manypractical situations so that the additional difficulties associatedwith the dynamic-state estimator need not be faced immediately.

REAL-TIME DETECTION AND IDENTIFICATION

In the static case, detection techniques for spotting modelerrors (lost transmission lines, bad data points, etc.) can bebased on evaluating 3 = J(x) for all possible models or for justthe assumed model by using probability distributions. Real-time operation on a time sequence of measurements reveals athird method of detection.

Consider snapshot operation with measurements z(tn) andcorresponding x(tn) and J(tn) = J[x(tn)], n = 1, . e -. If the modelis valid, the J(tn), n = 1,- remain relatively constant withthe only variations being caused by the changing measurementerrors. If the model suddenly becomes invalid (say, when atransmission line is lost), J(tn) jumps in magnitude. If themodel gradually becomes invalid (say, when pseudomeasure-ments slowly become bad), J(tn), n = 1,..- has an averagepositive slope. Thus, detection can be accomplished by merelymonitoring J(t.) and looking for jumps or upward trends. Suchmonitoring can be done by the computer or the system operator.

If spatial quantization or sweep processing techniques areused for state estimation, the detection logics can be similarlyquantized. This can effect sensitivity and has the definite advan-tage of localizing the source of trouble once it has been detected.The preceding real-time detection logic can also be con-

veniently applied to the bad data point detection and identifica-tion scheme of [1], based on the normalized residuals.

LOST DATA POINTS

In actual operation, some of the real-time meters or the corre-sponding communications links will fail at various times. If thisfailure is known to the computer, these are called lost data points.(If the failure is not known, it is a bad data point.)

Consider first the case where the lost data point is not criticalin the sense that an estimate is still possible without the lostdata. If Okk denotes the error variance associated with the lostdata point, 0 is simply modified by setting Okk- = 0. The neededmodification to I is made using formulas like (9). Thus, a com-plete matrix inversion is not needed.

If the lost data point is critical, it is necessary to replace thelost meter reading by a pseudomeasurement. The obviouspseudomeasurement to use is the last available meter reading.The corresponding 0 value is increased with time to account forthe error buildup as the last available reading is extrapolatedfurther and further in time.

SAMPLE IMPLEMENTATION

There is no best implementation. However, one possible com-plete system is briefly described as an example.The sample implementation is a sophisticated multimode

system which requires extensive software development. It usessimple, fast algorithms for the day to day routine processing withmore advanced, slower algorithms used when needed. Theextra computer capacity could normally be used for other pur-poses such as billing, record keeping, unit commitment, main-tenance scheduling, etc. This extra capacity might also be ob-tained from a second backup computer.The sample implementation can be divided into four operating

modes; 1) tracking-state estimation, 2) iterative-state estima-tion, 3) detection, and 4) identification. The tracking-stateestimator is a simple feedback loop (with a constant approximatel- or W-gain matrix) which operates almost all of the time, track-ing the normal system time evolution. It is based heavily on theapproximate model. The iterative-state estimator is essentiallythe complete iteration of (2) (with some simplifications in the 1-gain matrix) but is used only after major system changes whichoccurred rapidly or as an intermittent check on the tracking-state estimator. The detectors are based on evaluating J(tn), n =1, ... and are in operation all the time. They are tied directlyinto the tracking-state estimator. There are two levels of detec-tion, simple tests based on the approximate model for continualuse and more advanced versions used to check alarms called bythe simple tests. The identification algorithms rely on operatorassistance and guidance furnished through a sophisticated man-machine interface. Lost data points are automatically accountedfor, but the operator is warned when critical data is lost and isbeing replaced by pseudomeasurements.The tracking-state estimator and detectors operate automat-

ically although a display of the history of 1(t,) is made availableto the system operator. The iterative-state estimator is called onautomatically by the detection logics but can also be used at thediscretion of the operator. Identification is only done underoperator command and control.

CONCLUSIONS

Some of the problems of actually implementing state estima-tion, detection, and identification algorithms were discussed.The best way to proceed in any actual implementation dependson the type of power system, the type of metering and communi-cations, the available resources (hardware, software, and man-power), and the uses to be made of the static estimate. Engineer-ing judgment is an essential ingredient in any successful imple-mentation.The main conclusion of this paper is the author's personal

opinion that there is no major obstacle to successful implementa-tion.

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ACKNOWLEDGMENT

The author wishes to thank American Electric Power ServiceCorporation engineers for their help in explaining power systemproblems. These engineers may not agree with the results, butthe paper would have been impossible without their help.

REFERENCESt[] F. C. Schweppe and D. Rom, "Power system static-state

estimation, pt. II: approximate model," this issue, pp. 125-130.t2] F. C. Schweppe and J. Wildes, "Power system static-state

estimation, pt. I: exact model," this issue, pp. 120-125.

I am impressed by the complexity of operations and controlproblems in power systems. At the same time, I believe that moderncontrol techniques can make some contributions to these problems.The papers under discussion make such a contribution.

REFERENCES[3] S. Pines, H. Wolf, D. Woolston, and R. Squires, "Goddard

minimum variance orbit determination program," NASA-GSFCRept. X-640-62-191, October 1962.

[4] S. Pines, H. Wolf, A. Bailie, and J. Mohan, "Modifications of theGoddard minimum variance program for the processing of realdata," Analytical Mechanics Associates, Inc., Contract NAS5-2535, Rept., October 1964.

[5] S. F. Schmidt et al., "Case study of Kalman filtering in the C-5aircraft navigation system," Reprints, 1968 IEEE G-AC CaseStudies in System Control Workshop, University of Michigan,Ann Arbor, June 1968.

[6] A. H. Jazwinski, Stochastic Processes and Filtering Theory.New York: Academic Press, 1969.

Discussion

Graham Allan Jones (Pacific Gas and Electric Company, San Fran-cisco, Calif. 94106): In power systems planning and operation, it iscommon to obtain near-simultaneous data from the system duringsome critical time of operation. Of course, much of the data is redun-dant. It is necessary to reduce the system data into a useable formto develop a real system load pattern, a real life version of the load-flow solution. A pattern so obtained is most useful in determiningtypical system operating characteristics and in checking with simula-tions.

It appears that the author has developed a useful technique thatmight be applied to reduction of off-line load-pattern data to obtainconsistent results. However, in practice, much data is obtained bydirect operator reading. Thus, it may be necessary to account for thenonsimultaneous nature of the data and the meter resolution andreading errors as well as the actual meter accuracy. Perhaps theauthor would comment on the application of his state-estimationmethod to the real load-pattern problem.

Manuscript received June 3, 1969.

A. H. Jazwinski (Analytical Mechanics Associates, Inc., Seabrook,Md. 20801): The authors have presented an important contributionto the problems of bulk power system security, reliability, and on-

line control. The state estimator is the heart of a system monitor.Because of its ability to simultaneously process redundant datafrom diverse sources, automatically take into account the relativevalue of each piece of data, detect, identify, and eliminate bad data,the state estimator is ideally suited for this function. Redundancyimplies- reliability. It permits the detection of bad data, which thestate estimator can do automatically, based on sound statisticalprinciples. Estimation, of one sort or another, is a prerequisite forcontrol. To change the system state, one must obviously know whatthe state is.The principle of least squares estimation is of course quite old.

Practical realizations of this principle for complex systems are rela-tively new (e.g., [3]-[5]). Hundreds of man years of effort have beendevoted by the aerospace community over the past 10 years todevelop reliable computer algorithms implementing the stateestimator. This body of experience is partly science and partly art.Much of it is presented in [6]. This experience is today availableto the power systems community in developing state estimators forpower systems.

Manuscript received June 9, 1969.

F. C. Schweppe: Mr. Jones raises an important point which shouldhave been made in the papers. The static-state estimator is directlyapplicable to off-line studies. In fact, such an application should be afirst step in the development of an on-line system. Off-line studieswill help shake down the logic and models and provide operatingexperience which is needed before attempting on-line implementa-tion. As Mr. Jones indicates, such off-line calculations are also ofdirect value in their own right. If recording meters with clocksare available, the data tapes can be brought back every month to anoff-line computer and processed by a static-state estimator in a rou-tine fashion. In addition to load patterns, such processing can indicatesystem weaknesses such as unmetered but overloaded lines andtransformers and unsuspected voltage deviations. Such processingcan also provide a reliable and convenient data base for other studiessuch as developing short term (0-36 hours) load forecasts by areawith time and weather dependences. If some of the data for off-linestate estimation is obtained by direct operator readings, the non-simultaneous nature of the readings can be handled in various ways(the problem is closely related to the slow sequential scan problemdiscussed in Part III). If one wanted to be sophisticated, dynamicmodels of various types could be used. However, the simple conceptof modifying the measurement error covariance matrix 0 shouldsuffice in most cases. To illustrate the concept, assume that a realpower reading at a load bus is known to have been made sometimeduring a 5-minute interval. Assume that the particular load isknown to vary with a standard deviation of 10MW during 5 minutes.Then the corresponding value of 0 is set to (10)2 = 100 (assumingthe meter itself is much more accurate than 10MW). If manyoperator obtained readings are used, bad data points (operatorrecording errors) become very likely so that bad data point detectionand identification schemes should be implemented as part of thestate estimator. If men have to be sent to unattended switchyardsto take readings, studies based on X, the estimate error covariancematrix, should be made beforehand to determine which readings areactually needed and how simultaneous the readings have to be (i.e.,how large a 0 can be tolerated).Mr. Jazwinski's general comments are valid, but further elabora-

tion seems to be appropriate. Many modern control techniquesoriginally developed for aerospace can be definitely applied to powersystems, but much research and development is needed to adaptand extend them to meet power system needs. I worked on aerospaceproblems for many years before converting to power systems, and,in my opinion at least, power problems are tougher in many respects.In the state-estimation area for on-line control systems, it is neces-sary to develop a reliable metering-communication-data processingsystem for continual 24 hour a day operation under adverse fieldconditions. The number of variables is huge, and many types of un-certainties are present. The interconnected system is at least partiallyunobservable. Cost is a very important factor. Few if any aerospacestate-estimation problems yield such a challenging set of conditions.

Manuscript received June 30, 1969.

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