observability theory monticelli wu 1985

7
IEEE Transactions on Power Apparatus and Systems, Vol. PAS-104, No. 5, May 1985 NETWORK OBSERVABILITY: THEORY Felix F. Wu Department of Electrical Engineering and Computer Sciences and the Electronics Research Laboratory University of California, Berkeley CA 94720 Abstract - A complete theory of network observability is presented. Starting from a fundamental notion of the observability of a network, a number of basic facts relating to network observability, unobservable states, unobservable branches, observable islands, relevancy of measurements, etc. are derived. Simple and efficient algo- rithms can be developed based on these basic facts to (i) test network observability, (ii) identify observable islands and (iii) place measure- ments for observability. I. INTRODUCTION State estimation processes a set of redundant measurements to estimate the state of the power system. The result of state estimation forms the basis for all real-time security analysis functions in a power control center [1]. There are three types of real-time measurements. (i) The analog measurements that include bus voltage magnitudes, real and reactive power injections, and real and reactive power flows. (ii) The logic measurements which consist of the status of switches and breakers. (iii) The pseudo-measurements that may include forecasted bus loads and generations and zero-injections in passive nodes. Analog and logic measurements are telemetered to the control center. Logic measurements are used in Topology Processor to determine the system configuration. The State Estimator uses a set of analog measurements, along with the system configuration supplied by the topology processor, network parameters such as line impedances, and perhaps some pseudo-measurements as its input. If the set of measurements is sufficient in number an well-distributed geographically, the state estima- tor will give an estimate of the system state. When there is enough redundancy in measurements, the state estimator will be able to process bad data [2]. In the design stage, the following questions concerning the meas- urement set arise naturally. (1) Are there sufficient measurements to make state estimation possi- ble? (2) If not, where additional meters should be placed so that state esti- mation is possible? If the set of measurements is sufficient to make state estimation possi- ble, we say the network is observable. Observability depends on the number of measurements available and their geographic distribution. The first question raised here is concerned with the test of observability. The second question is meter or measurement placement for observability. Usually a system is designed to be observable for most operating conditions. Temporary unobservability may still occur due to unantici- pated network topology changes or failures in the telecommunication systems. The following questions emerge naturally in conjunction with state estimation in system operation. On leave from Departamento de Engenharia Eletrica, UNICAMP, Campinas, S. P., Brazil. 84 SMI 581-5 A paper recommended and approved by the IEEE Power System Engineer-ing Committee of- the IEEE Power Engineering Society for presentation at the IEEE/PES 1984 Summer Meeting, Seattle, Washington, July 15 - 20, 1984. Manuscript submit- ted February 2, 1984; made available for printing June 6, 1984. (1) Are there enough real-time measurements to make state estima- tion possible? (2) If not, which part or parts of the network whose states can still be estimated with the available measurements? (3) How to estimate the states of these observable islands? (4) How to select additional pseudo-measurements to be included in the measurement set to make state estimation possible? (5) How to guarantee that the inclusion of the additional pseudo- measurements will not contaminate the result of the state estima- tion? The analysis which lead to the answers to these questions may be called observability analysis. The analysis includes observability test, identification of observability islands, and measurement plicenient. It should be performed prior to the state estimation. In practice most of the time the system is observable. The observability analysis becomes valuable only in those rare situations when the system becomes unob- servable. In this paper a complete theory of network observability is developed, which provides a theoretical foundation for the answers to all the questions raised above. A number of basic facts are derived. Based on these facts, algorithms can be developed for * testing observability * identification of observable islands * measurement placement for observability. The design and testing of the algorithm, including computational considerations, are presented separately [3]. The algorithms are charac- terized by * being extremely simple * using subroutines already in a state estimation program * incurring very little extra computation. Some algorithms for testing observability based on heuristic approaches were proposed earlier [4-71. Clerments, Krumpholz, Davis in a series of papers [8-11] proposed a graph-theoretic foundation for network observability. The algorithms that have resulted from their theory are combinatoric. These algorithms seem to be computationally complex. The organization of this paper is as follows. The state estimation models are reviewed in Section II. A fundamental definition of net- work observability is introduced in Section'III. Equivalent statements of network observability are then derived. Section IV discusses how to test observability. Section V deals with the questions of how to identify the observable islands and how to estimate the states of the observable islands. The measurement placement problem is studied in Section VI. A mathematical format (theorems, proofs) is adopted for the develop- ment of the theory in this paper for the purpose of maintaining rigor and precision. Physical interpretations of the results are provided throughout. II. STATE ESTIMATION MODELS In this section we review the models of standard WLS state esti- mation [12] and the model-decoupled state estimation [13]. Similar to Clements et al. [8-11], we use the linearized state estimation model for the network observability theory. 1. WLS State Estimator The non-linear equations relating the measurements and the state vector are (1) z = h(x) + w 0018-9510/85/0005-1042$01.00©1985 IEEE 1042 * A. Monticelli

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Page 1: Observability Theory Monticelli Wu 1985

IEEE Transactions on Power Apparatus and Systems, Vol. PAS-104, No. 5, May 1985

NETWORK OBSERVABILITY: THEORY

Felix F. Wu

Department of Electrical Engineering and Computer Sciencesand the Electronics Research Laboratory

University of California, Berkeley CA 94720

Abstract - A complete theory of network observability ispresented. Starting from a fundamental notion of the observability of a

network, a number of basic facts relating to network observability,unobservable states, unobservable branches, observable islands,relevancy of measurements, etc. are derived. Simple and efficient algo-rithms can be developed based on these basic facts to (i) test networkobservability, (ii) identify observable islands and (iii) place measure-

ments for observability.

I. INTRODUCTIONState estimation processes a set of redundant measurements to

estimate the state of the power system. The result of state estimationforms the basis for all real-time security analysis functions in a powercontrol center [1]. There are three types of real-time measurements.(i) The analog measurements that include bus voltage magnitudes, realand reactive power injections, and real and reactive power flows. (ii)The logic measurements which consist of the status of switches andbreakers. (iii) The pseudo-measurements that may include forecastedbus loads and generations and zero-injections in passive nodes. Analogand logic measurements are telemetered to the control center. Logicmeasurements are used in Topology Processor to determine the systemconfiguration. The State Estimator uses a set of analog measurements,along with the system configuration supplied by the topology processor,network parameters such as line impedances, and perhaps some

pseudo-measurements as its input. If the set of measurements issufficient in number an well-distributed geographically, the state estima-tor will give an estimate of the system state. When there is enoughredundancy in measurements, the state estimator will be able to process

bad data [2].In the design stage, the following questions concerning the meas-

urement set arise naturally.(1) Are there sufficient measurements to make state estimation possi-

ble?(2) If not, where additional meters should be placed so that state esti-

mation is possible?If the set of measurements is sufficient to make state estimation possi-ble, we say the network is observable. Observability depends on thenumber of measurements available and their geographic distribution.The first question raised here is concerned with the test ofobservability. The second question is meter or measurement placementfor observability.

Usually a system is designed to be observable for most operatingconditions. Temporary unobservability may still occur due to unantici-pated network topology changes or failures in the telecommunicationsystems. The following questions emerge naturally in conjunction withstate estimation in system operation.

On leave from Departamento de Engenharia Eletrica, UNICAMP, Campinas, S.P., Brazil.

84 SMI 581-5 A paper recommended and approvedby the IEEE Power System Engineer-ing Committee of-the IEEE Power Engineering Society for presentationat the IEEE/PES 1984 Summer Meeting, Seattle,Washington, July 15 - 20, 1984. Manuscript submit-ted February 2, 1984; made available for printingJune 6, 1984.

(1) Are there enough real-time measurements to make state estima-tion possible?

(2) If not, which part or parts of the network whose states can still beestimated with the available measurements?

(3) How to estimate the states of these observable islands?

(4) How to select additional pseudo-measurements to be included inthe measurement set to make state estimation possible?

(5) How to guarantee that the inclusion of the additional pseudo-measurements will not contaminate the result of the state estima-tion?

The analysis which lead to the answers to these questions may be calledobservability analysis. The analysis includes observability test,identification of observability islands, and measurement plicenient. Itshould be performed prior to the state estimation. In practice most ofthe time the system is observable. The observability analysis becomesvaluable only in those rare situations when the system becomes unob-servable.

In this paper a complete theory of network observability isdeveloped, which provides a theoretical foundation for the answers toall the questions raised above. A number of basic facts are derived.Based on these facts, algorithms can be developed for

* testing observability* identification of observable islands* measurement placement for observability.

The design and testing of the algorithm, including computationalconsiderations, are presented separately [3]. The algorithms are charac-terized by

* being extremely simple* using subroutines already in a state estimation program

* incurring very little extra computation.

Some algorithms for testing observability based on heuristicapproaches were proposed earlier [4-71. Clerments, Krumpholz, Davisin a series of papers [8-11] proposed a graph-theoretic foundation fornetwork observability. The algorithms that have resulted from theirtheory are combinatoric. These algorithms seem to be computationallycomplex.

The organization of this paper is as follows. The state estimationmodels are reviewed in Section II. A fundamental definition of net-work observability is introduced in Section'III. Equivalent statementsof network observability are then derived. Section IV discusses how totest observability. Section V deals with the questions of how to identifythe observable islands and how to estimate the states of the observableislands. The measurement placement problem is studied in Section VI.A mathematical format (theorems, proofs) is adopted for the develop-ment of the theory in this paper for the purpose of maintaining rigorand precision. Physical interpretations of the results are providedthroughout.

II. STATE ESTIMATION MODELSIn this section we review the models of standard WLS state esti-

mation [12] and the model-decoupled state estimation [13]. Similar toClements et al. [8-11], we use the linearized state estimation model forthe network observability theory.1. WLS State Estimator

The non-linear equations relating the measurements and the statevector are

(1)z = h(x) + w

0018-9510/85/0005-1042$01.00©1985 IEEE

1042

*

A. Monticelli

Page 2: Observability Theory Monticelli Wu 1985

1043

where z is the (mxl) measurement vector, h(-) is the (mxl) vector ofnon-linear functions, x is the (2nxl) true state vector, w is the (mxl)measurement error vector, m is the number of measurements, and n isthe number of buses.

The estimate of the unknown state vector x is designated byx andis obtained by minimizing the weighted least squares function

J(x) = [z-h(x)] TW[z-h(x)] (2)

where W is a diagonal (mxm) matrix whose elements are the measure-ment weighting factors.

The condition for optimality is that the gradient of J vanishes atthe optimal solution x i.e.,

H T (x)W[z-h(x)] = 0 (3)

where the Jacobian matrix -HPQ (x) is

HpQ (x) = h (4)

OxAs for computing the estimate x the method that has received

wide acceptance is the iterative method which computes the correctionsOxk at each iteration by solving eq. (5).

GPQ(Xk)8Xk = HJQ(Xk)W(Z-h(Xk)) (5)

Xk+± = Xk + 8Xk (6)

for k = 0,1,2,... until appropriate convergence is attained. Here thegain matrix GpQ is

GpQ (Xk) = HJQ (Xk)WHpQ (Xk) (7)

{i= 0,k; 0 k = voltage angle at bus k.

Ui = Qkim/ Vk; Qkcm = reactive power flow from bus k to m.

Ki = Qk/ Vk; Qk = reactive power injection into bus k.

E, = Vk; Vk = voltage magnitude at bus k.

The state vector x is given by

x= (O,V) (12)

where V is is a (nxl) vector whose elements are the bus voltage mag-nitudes, and 0 is the (nxl) vector whose elements are the bus voltageangles, including the angular reference (or angular references).

The Jacobian matrix is

HpQ - ah lax HQr QV

where

(13)

Hpg = ahp/aO,Hpv = ahp/8V,HQO = ahQ/IO,

and HQV = OhQ/QV.By applying the decoupling principle to the matrix HpQ, we

obtain the decoupled gain matrix

Ge

GPQ = OGII

GV

(14)

In general, sparsity-oriented LDU decomposition is used in solving eq.(5).

The gain matrix GpQ in (7) can be written as

T -m hik Oh1iGpQ = HPQWHPQ= [x W [ i

where is the ith row of the Jacobian matrix HpQ, and w, is the

weighting factor associated with measured zi. The above equation sug-gests that in forming the gain matrix one can process the measurementone at a time.

2. DecouplingEquation (1) can be rewritten as

Zp = hp(x) + wp (8)

zQ = hQ(x) + WQ (9)

where zp is the (mpxl) vector of the "real" measurements (real powerflows, injections, and voltage angles), and zQ is the (mQxl) vector ofthe "reactive" measurements (reactive power flows, injections, and vol-tage magnitudes):

Zp= (T,I,O) (11)

zQ = (U,K,E)

Here the components of vectors T, I,1, U, K, and E are respectively:T,= Pkin' Vk;Pk,, = real power flow from bus k to m.

I= Pk/ Vk;Pk = real power injection into bus k.

where

Go= HTWPHpH (15)

Gv= HQvWQHQV (16)

Here the matrices Wp and WQ contain the weighting factorscorresponding to the measurements zp and zQ, respectively.

Similar to (7) the matrices Go and G v may be formed by process-ing the measurement one at a time.

3. Linearized (DC) State EstimatorIn this subsection we derive a linearized (DC) state estimator that

has the same features as the DC load flow (BO = P)The same approximations used in obtaining the matrix B' of the

fast-decoupled load flow (or the matrix B of the DC load flow) can beused to simplify the Jacobian matrix Hpq as well as the gain matrix GO:a) Flat voltage profile, i.e., V = 1 p.u. and 0 = 0.b) Line susceptances approximated by l/x, where x is the line reac-

tance.Let us call G° the resulting gain matrix.

The DC state estimator computes the bus voltage angles by solv-ing the equation:

Go0=HPh Wpzp (17)

which basically corresponds to performing the first 0-iteration of thefast model-decoupled state estimator [131.

The linearized state estimator (17) is equivalent to a linear leastsquare problem [14, pp. 208-249]. Let us write H = WJ/2Hp0 andC = Wj/2zp. Then (17) is equivalent to the solution of the followingleast square problem: determining a vector 0 that minimizes the sumof squares of the residual vector.

r= C-HO (18)

Page 3: Observability Theory Monticelli Wu 1985

1044

Equation (17) becomes

HTHO = HTC (19)

The network observability theory we are going to present is basedon the linearized state estimator (17). We may make the same approxi-mations to HQV as well as the gain matrix Gv in the reactive powermiodel to obtained similarly a linearized Q-V state estimator model.

III. NETWORK OBSERVABILITYWe shall use the linearized state estimator model to develop a

theory of network observability. For ease of presentation, the realpower model is used. The development is very similar for the reactivepower model, however, whenever the difference becomes important, itwill be pointed out. It should also be pointed out here that in the fol-lowing development, the vector 0 represents a true state vector.

For network observability, we are concerned with the power flowsin the network and the measurements made on the network. Let usfirst elaborate on the flows and the measurements. Given is a statevector 0, the power flow through the branch connecting buses k and mis equal to i (ok - m). For observability, we will only be concerned

Xiwith the fact whether the flow is zero or not, not the actual numericalvalue of the flow when it is nonzero. Therefore for simplicity let us setxi= 1 and call the "flow" to be the same as the angle difference8i = Ck-Om. Using the (unreduced) network incidence matrix A theset of "flows," 8, can be written as

8 = ATE (20)

Thus,

k81=

m

1 0. (21)

On the other hand, given the state vector 0, the set of measurements iswritten as

Intuitively we call a network observable if any flow in the networkcan be observed by some sort of indication in the set of measurements.In other words, whenever there is any nonzero flow in the network, atleast one of the measurements should read nonzero. This is equivalentto saying that a network is observable if, whenever all measurementsare equal to zero implies that all flows are zero. When a network is notobservable, it means that it is possible to have all measurements zero,yet there still are nonzero flows in the network. In such a case, thosebranches having nonzero flows will be called unobservable branches.These definitions will be made formally below.Definitions. A network is said to be observable if for all 0 such thatH0= 0, AT0= 0. Any state 0 for which HO = 0,AT .X 0, iscalled an unobservable state. For an unobservable state 0, let8 = ATo@, if 87; 0, then we call the corresponding branch anunobservable branch.

Theorem 1 below follows immediately from the definitions. Thetheorem supplies us with two other equivalent statements of observabil-ity.

Theorem 1. Assume that there is no voltage measurement, then thefollowing statements are equivalent.(i) The network is observable.(ii) Let H be obtained from H by deleting any column, then H is of

full rank.(iii) The triangular factorization reduces the gain matrix G = HTH

into the following form:

(26)

0= HO (22)

For the real power model there are two types of measurements(i) line flow. If measurement i is the line flow from bus k to bus m,

then

k m

h. -h. 0

(ii) injection. If measurement i is the injection at bus k, where thereare branches connecting bus k to buses m, n, l, then

_ rk m n

(i= 1-hz -hm -h 0

where S = hm + hn + hi.

For the reactive power model there is an additional type of meas-urement, which is the voltage-magnitude measurement. Thecorresponding one in the real power model would be the voltage-anglemeasurement. Even though the voltage angle measurement is notavailable in real life, it is still helpful to include this in the considera-tion. Later in the paper we will show that a key result in our theory ofnetwork observability may be interpreted as adding 0-pseudo measure-ments.(iii) voltage. If measurement i is the voltage-angle at bus i, then

(25)

(23)

where the shaded area corresponds to possibly nonzero elements.

Proof. (i) <=# (ii). Because ATo = 0 if and only if 0 = a 1 where1 = (1,...1) and a is any real number, we have (i) <=> (a)HO = 0iff 0 = a 1. Now we show (a) <= (ii).

(a) => (ii). Let H be obtained from H by deleting the k-th column h.Suppose HO = 0. Let 0 = (-,0). We thus must have 0 = 0, whichsays H is of full rank.

fii) => (a). Since the column sum of H is always zero, i.e.,HI = -h. Thus (HjTjj)-1fjTi= -1. Now suppose HO = 0 orHO+hOk = 0. Then 0 =-(HHT)-l HT0k = lOk.

(ii) <=# (iii). Let H = (H,h), we have

(24) G= HTH= |THH jThi[hTR hThJ

-T-Note that HT is nonsingular if and only if the triangular factorizationreduces it to a triangular matrix.

U

Comments1) Statement (ii) of Theorem 1 relates the intuitive concept of

observability to the solvability (existence of a unique solution) ofthe state estimation problem. As a matter of fact, the solvabilityof state estimation has been used as the definition of observabilityin the literature. To see the relation, note that H is of full rank ifand only if (HTf) is nonsingular, this is exactly the conditionthat is needed for the state estimator (19) to have a unique solu-tion. Thus Theorem 1 implies that a network is observable if andonly if the state estimation problem can be solved with a uniquesolution.

i

II

Page 4: Observability Theory Monticelli Wu 1985

2) Statement (iii) of Theorem 1 provides a numerically stablemethod for testing observability based on the triangular factoriza-tion of the gain matrix. As a matter of fact the triangularfactorization of the gain matrix will also provide the informationabout the observable islands as will be shown in the subsequentsections.

3) When there are voltage measurements, statement (ii) will bereplaced by: H is of full rank.

IV. DETERMINATION OF UNOBSERVABLE STATESWhen the network is not observable, we proceed to find an unob-

servable state, which is a solution of HO = 0. The solution of HO = 0is sensitive to the numerical values of the elements of H, as well as theerrors introduced during the solution process [14, pp. 317-318]. This ishighly undesirable. Lemma 1 below provides an alternative to solvingHO = 0.Lemma 1. HO = 0 if and only if (HTH)O = 0Proof. (=-) Premultipl, HO = 0 by HT.(<== ) Premultiply (H H)O = 0 by 0T results in|11011= 0, whichimplies HO = 0.

When the given network is not observable, the triangular factori-zation with complete pivoting reduces G = HTH to the following form[14, pp. 126-127]:

0.

Proof. Let H = [Hlh2H3] where h2 is a column.

ITHH H/Th2 HITH3G = HTH = h2TH1 h2Th2 h2/H3

H/TH1 H3Th2 H3TH3

The triangular factorization reduces G to

(27)

where

For any arbitrary O,, for example, Ob = (0,1,2,...) T, solving the upperhalf of the above equation (27) yields a Oa) then (0a,O,,) is an unob-servable state.

An alternative way to obtain the same unobservable state (Oa,Ob)is (i) replacing the diagonal elements of the lower right matrix by 1's,and (ii) replacing the corresponding right-hand side of (27) by(0,1,...) T, (iii) solving the resulting equation

p = h2Th2-h2TH1(H/THI)-1H/Th2

q = h2TH3 - h2TH1 (HlTH)-'HTH3

The situation here is that we have p = 0.proved.

HIH1l nonsingular and

(31)

(32)

The following claim is first

p = h2Th2-h2THI(H/TH1)-lHITh2= 0

if and only if

(28) columns of H1 are linearly independent and

h2 = H1a for some vector a.

By substituting h2 = H1a into the expression for p we obtain 0.

Note that (28) is identical to the state estimation equation (19) withthe pseudo-measurements of the voltage angles at buses correspondingto Ob present and all other measurements set to zero. This importantobservation will be used again.

Triangular factorization with complete pivoting involves permuta-tion of rows and columns of G and the corresponding reordering of thevector 0. Since for the solution of large systems by triangular factoriza-tion, the ordering of the matrix is done based on sparsity considera-tions, it is desirable not to have two different orderings. Theorem 2

below shows that the reordering for complete pivoting above is actuallynot necessary for observability test for obtaining an unobservable state.

This result greatly simplifies computation.Theorem 2. In the triangular factorization of the gain matrix G, if azero pivot is encountered, then the remaining row and column are all

zeros, i.e., G is reduced to the form:

Suppose the columns of (HI h2) are linearly independent, thenthe matrix

[H/TH H h2T|TH h/h2

(33)

is nonsingular. The determinant of matrix (33) is equal to theproduct det(H/THI)det (p). Since (H17) is nonsingular, p #: 0.We reach a contradiction. Therefore the claim is proved. Usingthe above result that p = 0 implies h2 = Hla, we substituteh = Hla into (32) and obtain q = 0.2

1045

(29)

(30)

ea 0

(9a

Page 5: Observability Theory Monticelli Wu 1985

1046

Comment. The implication of Theorem 2 is that for any given orderingof the gain matrix G whenever a zero pivot is encountered in theprocess of triangular factorization, the corresponding 0 belongs to 0band it can be assigned an arbitrary value in obtaining an unobservablestate. Or equivalently, one adds a 0-pseduo-measurement for thatnode. In other words, the zero pivot in G is replaced by a 1, and thecorresponding right-hand side 0 is replaced by the value assigned to the0-pseudo-measurement. The triangular factorization may then con-tinue. A network is observable if and only if there is only one zeropivot, which necessarily happens at the end, whereas when the networkis not observable one encounters more than one zero pivots in the tri-angular factorization of G.

V. IDENTIFICATION OF OBSERVABLE ISLANDSWhen the network is not observable, we would like to know

which part or parts of the network whose states can be estimated byprocessing the available measurements. These subnetworks will becalled observable islands. Two questions are answered in this section:(a) How to identify the observable islands.(b) How to estimate the states of the observable islands.

We start from a given unobservable state 0 obtained from thesolution of GO = 0. Let us arrange 0 so that the components havingidentical values are grouped together. For example, suppose there arethree groups of values Oa, HP,0. in 0, we have 0 = (0a,"p,0y) whereO= (Oa,O,a,..Ga)5 etc. Subnetwork a consists of nodes in Gatogether with the branches connecting them. Similarly for subnetworks,8 and y.

Let us further group together (i) line flow measurements for linesin the same subnetwork, and (ii) injection measurements for which allthe nodes connected to the injection node belong to the same subnet-work. It can easily be shown that it is impossible to have line flowmeasurements for lines connecting different subnetworks, for if thiswere true the 0 values for these two components would have to be thesame. Therefore, the remaining measurements are (iii) those injectionsfor which the nodes connected to it belong to different subnetworks.According to this grouping the matrix H becomes (rearrange rows): .

Theorem 3 below asserts that as far as the subnetworks a, /3, andy are concerned, the given unobservable state 0 is observable, hencethe subnetworks are candidates for observable islands. The reaon thatthey are merely candidates is because the given 0 is only one unobserv-able state, there may be other states that make these subnetworksunobservable. An observability test on each subnetwork is stillrequired.

Theorem 3. 0a is not an unobservable state for the subnetwork ax withmeasurements Ha, Similarly for 0 and 0,.

Proof. Equations (34) and (35) give us: *a is a solution ofHaOa = 0 and AaT0a = 0.

Subnetworks a,43, and y are candidates for observable islands.The measurement sets that are relevant at this time in the determina-tion of observable islands are Ha,Hg, and H.. Subsets of them willeventually be used in the state estimation of the observable islands.Therefore the measurements in Ha,y are irrelevant as far as state esti-mation of the observable islands is concerned. We shall call thesemeasurements in Hasy irrelevant measurements. They are discarded forfurther analysis of observability.

Theorem 3 suggests an iterative scheme to identify observableislands. Since each subnetwork a identified in (35) is a candidate forobservable island, we may test its observability with respect to themeasurement set H,. If the answer is yes, we find an observableisland. If not, we proceed to find an unobservable state in subnetworka and further decompose it by discarding unobservable branches.

After observable islands are identified, we use the relevant meas-urements to estimate the states of the observable islands. Note that inthe process of identifying observable islands, we have introduced a setof 0-pseudo-measurements in (28). A question arises as to whetherthese 0-speudo-measurements would affect the final state estimationresults. Theorem 4 below guarantees that they would not.

Theorem 4. Consider the state estimation model

C =HO +r (36)

r-ytt1.

0

(34)

The rows of Ha, Hp, and H. correspond to the measurements (i) and(ii) of subnetworks a, /3, and y respectively, and the rows of Hap,Pcorrespond to the measurements (iii). Also if we group branches in thesame network together, AT becomes:

Oat

[S 0 0=

(35)

Suppose that the measurement set consists of the 1-pseudo-measurements O' introduced in (28), and all other measurements equalto zero. Then the residuals r = 0.

Proof. We can renumber the network nodes and the measurementssuch that

0

Os

as

1

m-Q

m-Q+l.

m

I'2

I

n-Q.

n-)+l

n

regul armeasurements

(37)0-pseudomeasurements

voltage angleat other nodes

(38)voltage angle atnodes with 0-pseudomeasurements

Accordingly, the gain matrix and the Jacobian matrix can be partitionedas follows:

where A,, Ap, and A., correspond to the branches in the subnetworksa, /, and y, respectively, and AaB correspond to the branches con-necting different components. In other words, the rows of Aa#v,,correspond to the set of unobservable branches.

Page 6: Observability Theory Monticelli Wu 1985

1047

H Hr *a

I.

I

HTH HTrtrr r t

.HTHIir

m-Q

m-Q+l

m

where II is the unit matrix of order 1. Performing Gaussian elimina-tion in (40) we get

n-Qn-k+l

n

where

A = HITHa - HaTHr (HrTHrY)71HrJRa = 0

The estimate is given byGo = HTC

Introducing (37)-(42) into (43), and inverting G-1, we get:

O= (HjTHr-HHaOS

#4 =es

and from (36) we get:rt = [ - Hr(HrTHr)-IHrTHa + HJIs

rS =0

Now considerlkrll = (ir) T(rr)

By substituting (42), (46) into (47), we obtainlkrll = 0. Hence r =

VI. MEASUREMENT PLACEMENT

Given an unobservable network, we would like to know

* what is a minimal set of additional measurements whose inclusionwill make the network observable?

(44)

(45)

(46)

* would the additional measurements contaminate the state estima-tion of the observable islands?Theorem 5 below can be proved mathematically. Due to space

limitation the proof is omitted here.

Theorem 5. If a minimal set of additional non-redundant (pseudo)-measurements is so selected that they make the network barely observ-able, then the estimated states of the already observable islands will notbe affected by these pseudo-measurements.

It is clear that the candidates for these additional measurementsare (i) injection pseudo-measurements for which the nodes it is con-nected to belong to different observable islands, and (ii) line flowpseudo-measurements for lines connecting different observable islands.The second type of pseudo-measurements are seldom available. Weshall concentrate on the first type.

This set of pseudo-measurements may be obtained sequentially byadding a pseudo-measurement from the candidate list one at a time.The additional pseudo-measurement will make some unobservablebranches observable, thus coalescing observable islands into largerones. Computationally this selection process can be accomplished by a

0) scheme which adds a pseudo-measurement from the candidate list andthen recalculates the observable islands at each iteration until the wholenetwork becomes one observable island.

VII. CONCLUSIONA complete theory of network observability is presented. The

major points are summarized below:* A network is said to be observable if, whenever there is any

nonzero flow in the network, at least one of the measurementshuld read nonzero. In other words, a network is observablewhen, all measurements are zero implies that all flows are zero.

* A network is observable if and only if the state estimation can besolved with a unique solution.

* An observability test can be performed based on the triangularfactorization of the gain matrix. It goes as follows. In carryingout triangular factorization of the gain matrix G. whenever a zeropivot is encountered, we add a 0-pseudo-measurement for thecorresponding node, the effect of that is that the 0 is replaced by aI in the triangular factor. The network is observable if only oneiero pivot is encountered in the process and not observable ifmore than one zero pivots are encountered.

* By setting the 0-pseudo-measurements of the nodes of zero pivotsto different values and all the other measurements to zero, thesolution of the state estimation equations yields an unobservablestate of the system.

* Consider the subnetwork formed by the set of nodes havingidentical value in the unobservable state, together with thebranches connecting them. For the measurement set, considerthe line flow measurements for which all the nodes connected tothe injection node belong to the same subnetwork. This subnet-work along with the measurement set is a candidate for an observ-able island.

* Branches between two candidates of observable islands are unob-servable branches.

* Those injection measurements for which there is an unobservablebranch connected to the injection, are irrelevant as far as observa-bility of the network is concerned and may be discarded for thepurpose of identifying observable islands.

* Using the relevant measures and the 0-pseudo-measurements, thestates of the observable islands may be estimated. The 0-pseudo-measurements do not affect the state estimation results on lineflows.

* If a minimal set of additional non-redundant (pseudo) -

(47) measurements is so selected that they make the network barelyobservable, then the esitmated states of the already observable

0 islands will not be affected by these pseudo-measurements.

The actual design and testing of efficient algorithms based on

these results are presented in a companion paper [3].

H =

.

Page 7: Observability Theory Monticelli Wu 1985

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ACKNOWLEDGEMENTSResearch sponsored by the Electric Power Research Institute,

Power System Planning and Operations Program, under contract RP1999-6.

REFERENCES[1] T. E. Dy Liacco, "State Estimation in Control Centers," Electric

Power and Energy Systems, vol. 5, no. 4, pp 218-221.[21 A. Garcia, A. Monticelli, and P. Abreu, "Fast Decoupled State

Esimtation and Bad Data Processing," IEEE Trans. PowerApparatus and Systems, vol. PAS-98, no. 5, Sept./Oct. 1979, pp.1645-1652.

[3] A. Monticelli and F. F. Wu, "Network Observability: Identificationof Observable Islands and Measurement Placement," submittedfor presentation at IEEE PES Summer Meeting, Seattle, WA,1984.

[4] K. A. Clements and B. F. Wollenberg, "An Algorithm for Obser-vability Determination in Power System State Estimation," PaperA75 447-3, IEEE PES Summer Power Meeting, San Francisco,CA, 1975.

[5] J. S. Horton and R. D. Masiello, "On-Line Decoupled Observabil-ity Processing," PICA Proc., Toronto, Ontario, pp. 420-426.

[6] J. J. Allemong, G. D. Irisarri, and A. M. Sasson, "An Examina-tion of Solvability for State Estimation Algorithms," Paper A80008-3, IEEE PES Winter Meeting, New York, NY, 1980.

[7] E. Handschin and C. Bongers, "Theoretical and Practical Con-siderations in the Design of State Estimators for Electric PowerSystems," in Computerized Operation of Power Systems, S. C.Savulescu, Editor, Elsevier, pp. 104-136, 1972.

[8] G. R. Krumpholz, K. A. Clements, and P. W. Davis, "Power Sys-tem Observability: A Practical Algorithm Using Network Topol-ogy," IEEE Trans. Power Apparatus and Systems, vol. 99, pp.1534-1542, July/Aug. 1980.

[9] K. A. Clements, G. R. Krumpholz and P. W. Davis, "Power Sys-tem State Estimation Residual Analysis: An Algorithm Using Net-work Topology," IEEE Trans. Power Apparatus and Systems, vol.100, no. 4, pp. 1779-1787, April 1981.

[10] K. A. Clements, G. R. Krumpholz and P. W. Davis, "Power Sys-tem State Estimation with Measurement Deficiency: An Algo-rithm that Determines the Maximal Observable Subnetwork,"IEEE Trans. Power Apparatus and Systems, vol. 101, no. 9, pp.3044-3052, September 1982.

[11] K. A. Clements, G. R. Krumpholz and P. W. Davis, "Power Sys-tem State Estimation with Measurement Deficiency: AnObservability/Measurement Placement Algorithm," IEEE Trans.Power Apparatus and Systems, vol. PAS-102, pp. 2012-2020, July1983.

[12] F. C. Schweppe, J. Wildes, and D. P. Rom, "Power System StaticState Estimation," Parts I, II, and III, IEEE Trans. PowerApparatus and Systems, vol. 89, pp. 120-135, January 1970.

[13] A. Monticelli and A. Garcia, "Reliable Bad Data Processing forReal-Time State Estimation," IEEE Trans. PAS, voL. 102, no. 5, pp.1126-1139, May 1983.

[14] G. W. Stewart, Introduction to Matrix Computations, AcademicPress, 1973.

DiscussionF. L. Alvarado (University of Wisconsin, Madison, WI): This work isimportant, comprehensive and well presented and is bound to becomean important reference. A few years ago, Fetzer and Anderson [Al]presented a paper on dynamic observability. The static case of [Al]resulted in not only topological but quantitative measures of degree ofobservability of a system with a given set of measurements, and was basedon a HHT matrix much like the gain matrix G. Can the authors com-ment on the relationship (if any) between the two papers?

REFERENCE[Al] E. E. Fetzer and P. M. Anderson, "Observability in the StateEstimation of Power Systems," IEEE Trans. on Power Apparatus andSystems, vol. PAS-94, no. 6, Nov./Dec. 1975.

Manuscript received July 26, 1984.

A. Monticelli and F. F. Wu: We thank Prof. Alvarado for his comments.We do not see much relation between our paper and Ref. [Al]. Ref. [Al]has two parts in it. The first part is an attempt to derive network obser-vability condition (H full rank) from the observability condition in LinearSystem Theory for linear dynamical systems. The derivation in [Al] isnot correct. The steady-state of a dynamical system x = Ax + fis reach-ed when x = 0, which does not imply A = 0, at the starting point in[Al]. The second part suggests the use of "well-conditioning" of thegain matrix as the interior for selecting the "best" set of measurements.The relation between this criterion and the objective of state estimationis questionable. There are other proposed criteria for measurement place-ment cited in Refs. [10-15] in the companion paper [3].

Manuscript received December 26, 1984.