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Electric Power Systems Research 81 (2011) 13881402

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Electric Power Systems Research

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e p s r

An integer-arithmetic algorithm for observability analysis of systems with

SCADA and PMU measurements

G.N. Korres

School of Electrical and Computer Engineering, National Technical University of Athens, 9, Iroon Polytechneiou Street, Zografou 15780, Athens, Greece

a r t i c l e i n f o

Article history:

Received 3 December 2010

Accepted 8 February 2011Available online 10 March 2011

Keywords:

Observable islands

Measurement placement

Standard Gaussian elimination

Fraction-free Gaussian elimination

Gain matrix

Gram matrix

a b s t r a c t

This paper presents an efficient numerical method for observability analysis of systems including both

conventional (SCADA) measurements and synchronized phasor (PMU) measurements, using integer pre-

serving Gaussian elimination of integer coefficient matrices. The observable islands are identified in a

noniterative manner, by performing backward substitutions on the integer triangular factors of the inte-

gergainmatrix. Multiple placementof conventional andphasor measurementsfor a systemthat is found

to be unobservable is done by a direct method, using the integer triangular factors of a Gram matrix

associated with a reduced size Jacobian matrix. Since all computations performed are exact, no round-

off error, numerical instability, or zero identification problems occur. The IEEE 14-bus system is used to

illustrate the steps of the proposed method. Test results for the IEEE 300-bus and the FRCC 3949-bus

systems are provided to demonstrate the features of the proposed algorithms.

2011 Elsevier B.V. All rights reserved.

1. Introduction

The power system observability analysis determines if a state estimation solution for the entire system can be obtained using the

available set of measurements. Therefore it is a very important component in the EMS and it is usually carried out before the execution ofstate estimation. Observability is dependent on the location and types of available measurements as well as the network topology. When

a network is found to be unobservable, it is divided into observable islands and a minimal set of additional measurements has to be placed

to render the entire network observable.

Observability analysis algorithms can be classified as topological,numerical,hybrid, andsymbolic. The topological algorithms[16]use

graph theory concepts and they are combinatorial and rather complex. The numerical algorithm[7,8]is computationally simpler and is

based on the triangular factorization of the gain matrix of the normal equations. It uses an iterative scheme to determine the observable

islands. For measurement placement, one candidate boundary injection is placed at a time, each time updating the observable islands

until enough candidates are placed to merge all observable islands together. The numerical approach is extended to state estimation

using orthogonal transformations[9], state estimation with equality constraints [10,11], and the Hachtels based methods [12,13]. An

alternative numerical approach is presented in [14], based on the echelon reduction of decoupled Jacobian matrices. A direct (noniterative)

numerical procedure based on gain matrix triangular factorization is presented in[15]to determine the observable islands and in[16]for

multiple measurement placement. In [17] all the information neededfor observability analysis is extracted from the Jacobian measurement

matrix using Gaussian elimination techniques. The numerical observability algorithm of[18],[19]relies on the Gram matrix related with

the Jacobian measurement matrix. In[20,21] a pivoting algorithm which calculates the null space of the Jacobian matrix is used for

observability analysis. A technique for observability checking based on Gaussian elimination and binary arithmetic is provided in [22].An

algebraic approach to observability checking and restoration is proposed in[23],based on transferring rows to columns and vice versa

in the Jacobian measurement matrix. In [24]a hybrid (numerical-topological) method is proposed for network observability. The flow

measured branches form islands that determine a reduced network. Numerical techniques are then applied to this reduced network for

observability purposes. In[25,26], a hybrid procedure based on graph properties of the entire and a reduced network respectively and

echelonformof a corresponding test matrixis used forobservabilityanalysis. Thealgorithms of[27,28] combine factorization pathconcepts

and triangularization of the gain and Jacobian matrix respectively. The observability procedures of[29,30] are based on a symbolic iterative

reduction of the Jacobian matrix, which allows the identification of redundant measurements and observable islands.

Tel.: +30 772 3621; fax: +30 772 3659.

E-mail address: [email protected]

0378-7796/$ see front matter 2011 Elsevier B.V. All rights reserved.

doi:10.1016/j.epsr.2011.02.005
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G.N. Korres / Electric Power Systems Research 81 (2011) 13881402 1389

The above observability analysis methods were developed for traditional SCADAmeasurement systems but can be enhanced for systems

including PMUmeasurements. Unlike traditional SCADA systems, the phasor measurement unit (PMU) is a measuring device synchronized

via signals from global positioning system (GPS) satellite transmission[31]and can measure the voltage phasor of the installed bus and

the current phasors of all the lines connected with that bus. Several algorithms have been published in the literature for PMU placement,

with or without the presence of traditional measurements. These methods are mainly used for measurement system planning purposes.

Algorithm [31] finds the minimal set of PMUs for placement, by using graph theory and simulated annealing method. Simulated annealing

methodis used in [32] to solvethe problem of communication-constrained PMU placement, basedon incomplete observability.A minimum

number of strategically located PMUs is selected in[33]to improve the bad data processing capability of state estimation. A generalized

ILP formulation under full and incomplete observability, considering situations with and without zero injections, is presented in[34].A

method to determine the minimum number of PMUs that make a system topologically observable for normal operating conditions, as well

as for single branch outages, is presented in[35].Besides the placement of mere PMUs, algorithm[36]also considers the placement of

PMUs when conventional measurements are present in the system.

Since floating point arithmetic is used, numerical observability methods are prone to round-off errors. Detection of zero pivots encoun-

tered during triangular factorization is based on a suitable zero threshold. The choice of threshold may lead to observable or unobservable

situations or even different number of observable islands for the same network. As the network size and the ratio of the number of injec-

tions to the number of flows grow, the gain matrix may become ill-conditioned[37]and the standard approach may give incorrect results

if the zero threshold is not suitably selected.

In this paper an efficient integer numerical procedure for observability analysis of systems including SCADA as well as PMU measure-

ments is proposed. The method is based on the fraction-free Gaussian elimination of the gain matrix, that avoids floating point operations.

Since the proposed algorithm uses exact arithmetic, is not prone to rounding errors or numerical instabilities and requires no criterion for

zero identification. The observable islands of a non-observable system are determined in a noniterative manner, by using the triangular

factors of the singular gain matrix. A minimal set of pseudo measurements that make the network observable is chosen, by using the

triangular factors of the Gram matrix associated with a reduced size Jacobian matrix.

The paper is organized as follows. Section2presents the standard and the proposed fraction free Gaussian elimination of a matrix andthe solution of a linear system by a forward and a backward substitution. Section3presents the observability checking algorithm. Section

4presents a noniterative algorithm for the identification of the observable islands. Section5presents a multiple measurement placement

algorithm based on the Gram matrix of a reduced network Jacobian. In Section6 the IEEE 14-bus system is used to illustrate the details

of the proposed method. Section 7presents numerical results for the IEEE 300-bus and the FRCC 3949-bus systems, to demonstrate the

computational behavior of the proposed algorithms. Finally, Section8shows the conclusions.

2. Solution of an integer linear system by standard and fraction free Gaussian elimination

We consider the following linear system

Ax = b (1)

whereA = [ai,j] is annsymmetric and positive definite integer matrix,bis an1 integer vector, andx is the unknownn1 vector. The

standard Gaussian elimination admits the following factorization ofA, as shown inAppendix A.

A = LDU (2)

whereL is a unit lower triangular matrix, D is a diagonal matrix, and Uis a unit upper triangular matrix, all having real entries. Since

the matrixA is positive definite, we have U= LT. The linear system(1)can be solved forx by performing a standard forward substitution

(LDy = b) and a backward substitution (U x = y), as shown inAppendix B.Note that all the operations to compute the LDUfactors ofA and

the unknown vectorxare done in floating-point arithmetic.

An alternative procedure to solve(1) is the integer-preserving or fraction free Gaussian elimination ofA [3842],which admits the

following factorization.

A = LD1U (3)

whereL,D, andUhave integer entries (seeAppendix A).Then we solve the linear system

As = b det(A) (4)

by performing a forward substitution (LD1y = b) and a backward substitution (Us =ydet(A)) and we compute the vector x as

x =s

det(A) (5)

All entries of vector s are integer[42].Nevertheless, the vector x has not integer entries, but they are expressed as fractions of integer

entries ofsscaled by the determinantdet(A) of matrixA.

It is to be noted that in both standard and fraction free Gaussian elimination approaches, the factors L,D, andUhave identical sparsity

pattern and they require the same number of numerical operations for their computation.

3. Observability checking

For observability purposes, we consider the linearized measurement vector equation

z= Hx (6)

wherez(m1) is the measurement vector composed of SCADA and PMU measurements, (n1) is the vector of all bus phase angles,

H(mn) is the measurement Jacobian matrix, e(m1) is the measurement error vector, m is the number of measurements, and n is


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1390 G.N. Korres / Electric Power Systems Research 81 (2011) 13881402

the number of buses. If no PMUs exist in the measurement system, then an artificial zero phase angle measurement is introduced at an

arbitrarily selected reference bus,without excludingthat bus fromthe problem formulation. It should be notedthat the system observability

is independent of the branch parameters as well as the operating state of the system. Therefore, all system branches can be assumed to

have an impedance ofj1.0p.u.and all bus voltages can be set equal to 1.0p.u.for the purpose of observability analysis.

The SCADA measurements are modelled as follows:

- Active power flow measurement at busiof branchij:

Pmeasij = i j (7)

- Active power injection measurement or zero injection at bus i:

Pmeasi =

ja(i)

Pij =

ja(i)

(i j) (8)

wherea(i) is the set of buses connected with busi. Note thatPi = 0 for a zero injection at busi.

A PMU located at busi can measure the phase angleibus i:

measi = i (9)

and the real partIij,rof current phasorIijof all incident branches ij, assuming thatIijis represented in rectangular coordinates. Then, Iij,rcan be approximated as[44]:

Imeasij,r

= i j (10)

The Jacobian matrix will have the following form:

(11)

whereniis the number of busesj,k, . . .connected with busi.

The gain matrix can be formed as:

G = HTH (12)

where the measurement error covariance matrix is assumed to be the identity matrix without loss of generality.

The system is said to be observable if and only if the Jacobian matrix Hor the gain matrixGhas full rank:

rank(H) = rank(G) = n (13)

nullity(H) = nullity(G) = 0 (14)

Note that(13) provides a necessary but not sufficient condition for observability. For most power systems under normal operating

conditions, Eq.(13)will guarantee a reliable state estimate. The symmetric gain matrix G is positive definite for an observable system and

positive semi-definite for an unobservable system[16].The matrixG can be decomposed into itsLDUfactors. If the diagonal factor D has

no zeros on its diagonal, thennullity(G) = 0 and the system is observable. If the diagonal factorDhas at least one zero on its diagonal, then

nullity(G) =p > 0 and the system is unobservable. Note that if theith diagonal entryDiiis zero, then all entriesUij,j > iofUare zero[43].

Note that under the above assumptions the entries ofG will be integer. Although the matrix G is integer, the standard Gaussianelimination uses floating point arithmetic and the factors L, D, and Uhave real entries. Hence, it is prone to rounding errors and needs

a suitable threshold to detect zero pivots (seeAppendix B).In the proposed fraction free algorithm exact integer arithmetic is used, the

factorsL,D, andUhave integer entries and no criterion for zero pivot identification is required (Appendix B).

4. Identification of observable islands

We assumethatthe network branches aretreatedas directed (theirfromandto ends aredefined). Theelementsof thebranch-to-node

incidence matrix are defined as follows:

bij =

1 if branch i is directed away from node j1 if branch i is directed towards node j

0 if branch i is not incident to node j

An unobservable system with nullity(G) =p > 0 is divided in rp observable islands, which canbe determined by thefollowing algorithm

[16], that is based on the LDUfactorization described inAppendix B:


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Step 1. Form the gain matrixG = HTHand compute its LDUfactors by Gaussian elimination.

Step 2. Compute the vector ciby a backward substitution Uci = uiieifor those is for which dii = 0, where eiis a vector with all entries zero

except for a 1 as itsith entry. Note that the number of such zero diagonals in Dwill be exactly equal to p. When the fraction free approach

is used the elements of vectors ciwill be integer.

Step 3. Form the matrix Ccomposed of column vectors ciand compute the product BC, where B is the branch-to-node incidence matrix.

If at least one entry in a row ofBCis not zero, then the corresponding branch is unobservable.

Step 4. Remove all the unobservable branches. The resulting connected subnetworks define the observable islands.

Step 5. Stop.

It should be noted that factor Ucan be obtained either by the standard or the fraction-free Gaussian elimination of the gain matrix,

providing equivalent results.

5. Measurement placement

Let an unobservable system, withnullity(H) =p > 0,be separatedin robservable islands. In orderto makethe network globallyobservable,

we needp non-redundant measurements with regard to the existing measurements. Candidates for placement can be boundary injections

or PMUs. If a candidate PMU is installed at an internal bus of an observable island, its associated voltage phasor is sufficient to merge

it to another island, but its incident branch current phasors cannot be exploited, since they are redundant with the existing internal

measurements. A candidate PMU located at an unmeasured boundary bus can be used to merge more observable islands together, through

the current phasors of its incident unobservable branches, thus reducing the number of PMUs used for observability restoration. The

candidate boundary buses to place PMUs are ordered in descending order of number of connected neighbour islands. The candidate

boundary buses to place injections may be ordered in descending order of accuracy.

LetHb(mb n) be the submatrix ofHcorresponding to the existing boundary injections and the voltage phasors of existing PMUs and

Hc(mcn) be the Jacobian matrix corresponding to candidates for placement (boundary injections, voltage phasors and current phasors

in unobservable branches of boundary PMUs). We define by Wb(mb r) andWc(mb r) the reduced order Jacobian matrices:

Wb = (wb1 wbi wbr) (15)

Wc= (wc1 wci wcr) (16)

where columnwbi(wci) is the sum of columns ofHb(Hc) which correspond to the buses of the ith island. It can be proven that[24]:

nullity(H) = nullity(Wb) (17)

nullity

HHc

= nullity

WbWc

(18)

LetMbe the Gram matrix[43]associated with

WbWc

:

M=

WbWc

WbWc

T(19)

From linear algebra we have the following properties:

rank(M) = rank

WbWc

= r nullity

WbWc

(20)

The condition for making the network globally observable is achieved if and only if:

nullity

HHc

= nullity

WbWc

= 0 rank(M) = r (21)

The Gram matrixMis decomposed into itsLDUfactors. If thekth diagonal entry ofDis zero (nonzero), then thekth row ofMis linearly

dependent(independent) from theset of rows 1, . . ., k1 ofM[43] andthe kth measurement is redundant (non-redundant)with respect tothe previousk1 measurements. Hence, candidate measurements associated with nonzero diagonals ofDare chosen for placement until

the number of selected measurements reachesp. If atleast the voltage phasor measurement ata bus i or a current phasor measurement at a

branch ij is selected for placement, then all the available measurements of the PMU at bus i will be added for restoration, thus increasing

the system redundancy and reliability.

Comparing the proposed method with those of[16,17,19],we see that:

- In [16,17] the mp product E= HcCis used, where matrix C(np) is computed byp back substitutions with the factors of the gain matrix

G(nn) or the Jacobian matrix H(mn), respectively. Following the columns with nonzero pivots in the Echelon form ofEwill yield the

corresponding candidate measurements as the minimal set for observability restoration.

- In [19], the (m + mc) (m + mc) Gram matrix F=

HHc

HHc

Tis used. Following the rows with nonzero pivots in the triangular factors

ofFwill yield the corresponding candidate measurements as the minimal set for observability restoration.

Note that the number of such candidate measurements for restoration will be exactly equal to the rank deficiency pof the gain matrix.

Since mb + mcn and mb + mcm + mc, the Gram matrix Min (19)has much smaller dimensions than matrices Hc, Cand F. Note also


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: power flow measurement

: power injection measurement

: current phasor measurement

: voltage phasor measurement

112

16

4131

11

Island 1 Island 4

Island 2

10

Island 3

11 14

9

5

12 1

8 7

Island 5

3

Island 6

Fig. 1. IEEE 14-bus system and measurement configuration.

that[16,17]require additional computations to form the product E= HcC. Hence, fewer arithmetic operations are needed for the proposed

algorithmand that leads to significant savings in computational effort. It should be emphasized that theproposedmethod does notdepend

on the type of algorithm (topological, numerical, hybrid, or symbolic) used for observability analysis and needs no intermediate results,

like matrixCused in[16,17].

6. Illustrative example

The IEEE 14-bus system, with the mixed conventionaland phasor measurement configuration ofFig. 1, is used to illustrate the proposed

algorithms. The measurement system consists of conventional power flows in branches 1-2, 1-5, 3-4, 4-9, and 6-13, conventional power

injections at buses 1, 6, and 9, and PMUs at buses 7 and 12. The PMU at bus 7 measures the voltage phasor of bus 7 and the current phasors

of branches 7-4, 7-8, and 7-9. The PMU at bus 12 measures the voltage phasor of bus 12 and the current phasors of branches 12-6 and

12-13.

6.1. Observability checking

The measurement Jacobian matrixH, associated to the measurement set ofFig. 1, is as follows:


6/15


The corresponding gain matrixG = HTHis:

G =

6 3 3

3 2 1

1 1

1 4 5 1 1

3 1 3 4 1 1 1

4 18 4 5 55 1 5 1 1

1 1

5 5 18 4 4

1 4 1 1 1

1 1 4 1 1

1 5 1 4

1 5 1 3

1 1 4 1 1

The triangular factors ofG = HTHwith the standard Gaussian elimination are:

U=

1.0 0.5 0.51.0 1.0

1.0 1.0

1.0 1.6667 0.3333 0.33331.0 4.0 1.0 1.0 1.0

1.0 0.5 0.51.0 0.2 1.0 0.2 0.2

1.0 1.25 0.25 0.251.0 0.3171 0.3171

1.01.0 1.0

1.0 0.61.0

1.0

The triangular factors ofG = HTHwith the fraction free Gaussian elimination are:

U=

6 3 3

3 3

3 3

9 15 3 3

9 36 9 9 9

18 9 9

90 18 90 18 1872 90 18 18

246 78 78

246

18 18

45 27

27

27

Since the factor D has two zero diagonal entries at positions 10 and 14, the rank deficiency of the measurement system is equal

to two (nullity(H) = nullity(G) = 2) and the network is unobservable. Notice thatD1,1 = U1,1, Di,i = Ui1,i1Ui,i, 2 i9,D10,10 = 0,D11,11 =18,

Di,i = Ui1,i1Ui,i, 12 i13, D14,14 =0(see (A16)(A18) in Appendix A), which indicates that matrix D is completely predicable (its diagonals

are computed from already computed diagonal elements ofU). Hence, only the positions of the zero diagonals (pivots) in D need to be

stored.


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6.2. Identification of observable islands

To identify the observable islands, we perform the following two backward substitutions:

standard procedure:

Uc10 = u10,10e10 = 1 e10 = ( 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 0.0 0.0 0.0 0.0 )T

Uc14 = u14,14e14 = 1 e14 = ( 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 1.0 )T

fraction free procedure:

Uc10 = u10,10e10 = 246 e10 = ( 0 0 0 0 0 0 0 0 0 246 0 0 0 0 )T

Uc14 = u14,14e14 = 27 e14 = ( 0 0 0 0 0 0 0 0 0 0 0 0 0 27 )T

and form the matrixC= (c10c14) given below, for the standard and the fraction free approach, respectively:

C=

27 0

27 0

0 0

0 0

27 0

0 0

0 0

0 0

0 0

27 0

0 27

0 0

0 0

0 27

Bus1

Bus2

Bus3

Bus4

Bus5

Bus6

Bus7

Bus8

Bus9

Bus10

Bus11

Bus12

Bus13

Bus14

C=

1.0 0.01.0 0.00.0 0.00.0 0.01.0 0.0

0.0 0.00.0 0.00.0 0.00.0 0.0

1.0 0.00.0 1.00.0 00.0 0

0.0 1.0

Bus1

Bus2

Bus3

Bus4

Bus5

Bus6

Bus7

Bus8

Bus9

Bus10

Bus11

Bus12

Bus13

Bus14

The branch-to-node incidence matrixBis given by:

21 876543 109 11 12 13 14

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

Br 1-2-1

Br 1-5-1

Br 2 - 3-1

Br 2 - 4-1

Br 2 - 5-1

Br 3-4-1

Br 4 - 5-1

Br 4 - 7-1

Br 4 - 9-1

B Br 5 -6-1

Br-1

-1

-1

-1

-1

-1

-1

-1

-1

-1

6-10

Br 6-12

Br 6-13

Br 7 - 8

Br 7 - 9

Br 9-11

Br 9-14

Br 10-11

Br 12-13

Br 13-14

The corresponding matrixBCfor the standard and the fraction free approach is as follows:


8/15


00

00

27 0

27 0

00

00

27 0

00

00

27 0

27 0

00

00

00

00

270

270

27 27

00

270

Br 1-2

Br 1-5

Br 2-3

Br 2- 4

Br 2 - 5

Br 3-4

Br 4-5

Br 4 - 7

Br 4 - 9

Br 5-6B.C

Br 6-10

Br 6-12

Br 6-13

Br 7 - 8

Br 7 - 9

Br 9-11

Br 9-14

Br 10-11

Br 12-13

Br 13-14

0.0 0.0

0.0 0.0

1.0 0.0

1.0 0.0

0.0 0.0

0.0 0.0

1.0 0.0

0.0 0.0

0.0 0.0

1.0 0.0

1.0 0.0

0.0 0.0

0.0 0.0

0.0 0.0

0.0 0.0

0.0 1.0

0.0 1.0

1.0 1.0

0.0 0.0

0.0 1.0

Br 1-2

Br 1-5

Br 2-3

B.C

Br 2 - 4

Br 2 - 5

Br 3-4

Br 4 - 5

Br 4 - 7

Br 4 - 9

Br 5 -6

Br 6-10

Br 6-12

Br 6-13

Br 7 - 8

Br 7 - 9

Br 9-11

Br 9-14

Br 10-11

Br 12-13

Br 13-14Nonzero rows 3, 4, 7, 10, 11, 16, 17, 18, and 20 ofBCcorrespond to unobservable branches 2-3, 2-4, 4-5, 5-6, 6-10, 9-11, 9-14, 10-11,

and 13-14 respectively. Removing all unobservable branches, the resulting connected subnetworks define the observable islands: island 1

{6, 12, 13}, island 2{10}, island 3{11}, island 4{14}, island 5{1, 2, 5}, and island 6{3, 4, 7, 8, 9}, as shown inFig. 1.

6.3. Measurement placement

Since the rank deficiency of the measurement Jacobian H is equal to two, we need two non-redundant (critical) measure-

ments to make the whole system barely observable. Based on the identified observable islands, we form the reduced Jacobian

matrix Wb associated with the existing boundary injection measurements P6 and P9 and the phase angle measurements 7 and12.

whereI1,I2,I3,I4,I5and I6denote the observable islands.

If only power injections are used for placement, candidates will be the injections at the non-measured boundary buses 2, 3, 4, 5, 10, 11,

13, and 14. The corresponding reduced Jacobian matrix Wcwill be:


9/15


We form the reduced Jacobian matrixW=

WbWc

and its Gram matrixM= WWT as follows:

M=

6 0 0 2 2 1 2 4 4 1 2 2

0 6 2 0 4 2 4 2 1 4 1 4

0 2 1 0 2 1 2 1 0 1 0 1

2 0 0 1 0 0 0 1 1 0 1 12 4 2 0 8 4 8 6 0 2 0 2

1 2 1 0 4 2 4 3 0 1 0 1

2 4 2 0 8 4 8 6 0 2 0 2

4 2 1 1 6 3 6 6 1 1 1 2

4 1 0 1 0 0 0 1 6 4 1 1

1 4 1 0 2 1 2 1 4 6 0 1

2 1 0 1 0 0 0 1 1 0 2 3

2 4 1 1 2 1 2 2 1 1 3 6

P6P97

12P2P3P4P5

P10P11P13P14

The factorsDandUofMusing the fraction free Gaussian elimination are:

U=

6 2 2 1 2 4 4 1 2 2

36 12 24 12 24 12 6 24 6 24

12 24 12 24 12 12 12 12 12

4 8 4 8 4 4 4 4 4

8 4 8 8 8 4

8

8

8

4 8 4 8

44

4

P6P97

12P2P3P4P5

P10

P11P13P14

The injections at buses 2 and 10 are nonredundant with respect to injections at buses 6 and 9 and phase angles at buses 7 and 12, since

they correspond to nonzero diagonal entries ofD, and hence they are chosen as the minimal measurement set to restore observability.

If only PMUs are used for placement, the candidates will be PMUs located at the non-measured boundary buses, in descending order

of connections to other islands. Buses 5, 10, 11, 14, which are connected with two neighbour islands, are ordered first and buses 2, 4, 3,

13, which are connected with one neighbour island, are ordered last. It is to be noted that only current measurements in interconnec-

tion (unobservable) branches will be considered for observability restoration, since current measured internal (observable) branches are

redundant with regard to existing measurements.

The corresponding reduced Jacobian matrix Wcwill be:


10/15


We form the reduced Jacobian matrix W=

WbWc

and its Gram matrixM= WWT as follows:

M=

6 2 1 1 3 1 3 1 1 2 1 1 1 1 1 1 2 26 2 2 1 1 3 1 1 3 1 2 2 2 2 2 2 2 12 1 1 1 1 1 1 1 1 1 1

2 1 1 1 1 1 11 1 1 1 1 1 1 1 1 11 2 1 1 2 1 1 1 1 2 2 1 2 2 1 23 1 1 1 2 1 1 1 1 1 1 1 1 1 11 1 1 1 13 1 1 1 2 1 1 1 1 11 1 1 1 2 1 1 2

1 1 1 1 13 1 1 1 1 2 1 1 1 1 1 1 1 1 1

1 1 1 1 2 1 1 21 1 1 1 13 1 1 1 1 2 1 1 1 1 1 1 1 1 1

2 1 1 1 1 1 1 2 1 21 1 1 1 1 1 1 1 1 11 2 1 1 2 1 1 1 1 2 2 1 2 2 1 21 2 1 1 2 1 1 1 1 2 2 1 2 2 1 2

2 1 1 1 1 1 1 1 1 1 1 11 2 1 1 2 1 1 1 1 2 2 1 2 2 1 21 2 1 1 2 1 1 1 1 2 2 1 2 2 1 2

2 1 1 1 1 1 1 1 1 1 1 11 2 1 1 2 1 1 1 1 2 2 1 2 2 1 22 1 1 1 1 1 12 1 1 1 1 1 1 2 1 2

P6P97125

I54I5610

I106I1011

11I119

I111014

I149I1413

2I23I244

I42I453

I3213

I1314


11/15


Table 1

Test results for the IEEE 300-bus system.

Case No. of power

flows and

current phasors

No. of voltage

phasors

No. of power

injections

Standard

Gaussian

elimination

Fraction free

Gaussian

elimination

= 104 = 105 = 106 No. of

pivots

No. of

islands

No. of pivots No. of

islands

No. of

pivots

No. of

islands

No. of

pivots

No. of

islands

1 0 0 279 21 243 21 243 21 224 21 243

2 95 25 279 3 6 3 6 3 6 3 6

3 146 30 279 4 6 4 6 4 6 4 6

The factorsDandUofMusing the fraction free Gaussian elimination are:

U=

6 2 1 1 3 1 3 1 1 2 1 1 1 1 1 1 2 236 12 12 6 6 18 6 6 18 6 12 12 12 12 12 12 12 12 6

12 12 12 12 12 12 12 12 12 12 12 12 12 12 12 124 4 4 4 4 4 4 4 4 4 4 4 4 4 4

2 2 2 2 2 2 2 2 2 2 2 2 222

22

1 1 1 1 1 1 11

11

11

11

11

11

11

11

1

P6P9712

5I54I5610

I106I1011

11I119I1110

14I149I1413

2I23I244

I42I453

I32

13I1314

The diagonal entries 1, 2, 3, 4, 5 and 10 ofD are nonzero, which means that rows 5 and 10 ofW, corresponding to 5 andI10-11, arelinearly independent from rows 1, 2, 3, and 4 ofW. Hence, the candidate phase angle measurement 5and branch current measurementI10-11 are nonredundant with respect to existing measurements P6,P97andI10-11, and PMUs at buses 7 and 10 are chosen as the minimal

set of PMUs to make the network globally observable.

7. Test cases

This section presents and discusses results from realistic case studies based on the IEEE 300-bus system[45]and the FRCC (Florida

Reliability Coordinating Council) system [46] having3949 busesand 6038branches. Different simulations are performedinvolving different

proportions of SCADA measurements (branch power flows and bus power injections) and PMU measurements (voltage phasors at the

installed buses and current phasors on all lines incident to those buses). For observability purposes, power flows and current phasors areequivalent. The number of voltage phasors is equal to thenumber of PMUs. For simplicity, only active power measurements areconsidered.

The proposed algorithm has been implemented with minor modifications to the factorization and substitution routines of an existing state

estimation program written in Compaq Visual Fortran for Windows. All simulations have been carried out using an Intel Core i3 processor,

clocking at 2.13 GHz and 2 GB of RAM, with the 64-bit Windows 7 operating system.

In all cases the networks are found to be unobservable. Tables 1 and 2show comparative results for the two test systems by applying

both the standard and the fraction free algorithms. Observing Table 1(IEEE 300-bus system) it can be concluded that for an injection only

measurement set, the number of zero pivots and observable islands are different for different values of. For the FRCC system (Table 2),we obtain different number of zero pivots and observable islands for different values of, when only injections are present or when thenumber of injections is higher than the number of flows. In all cases, the observability was successfully restored for any combination of

SCADA and PMU candidate measurements.

While no numerical problem is observed in detecting zero pivots and observable islands with the proposed approach, incorrect results

are observed with the standard approach because of floating-point errors and inappropriate choice of zero threshold value . The reasonis that, as the network size and the ratio of number of injections to number of flows grow, the gain matrix becomes ill-conditioned [37]

and the standard approach gives incorrect results for any zero threshold choice.


12/15


Table 2

Test results for the FRCC system.

Case No. of power

flows and

current phasors

No. of voltage

phasors

No. of power

injections

Standard

Gaussian

elimination

Fraction free

Gaussian

elimination

= 104 = 105 = 106 No. of

pivots

No. of

islands

No. of

pivots

No. of

islands

No. of

pivots

No. of

islands

No. of

pivots

No. of

islands

1 1553 45 0 2793 2793 2793 2793 2793 2793 2793 2793

2 1553 65 3015 576 1675 573 1558 565 1546 577 1558

3 4842 150 3015 106 172 106 172 106 172 106 172

4 0 0 3015 930 2897 929 2886 927 2815 932 2899

8. Conclusions

This paper presents an efficient observability analysis method, for systems composed of traditional as well as synchronized phasor

measurements, using the integer factors of integer coefficient matrices. Observable islands are identified in a noniterative manner and

additional measurements for placement are provided by a direct method, for a network found to be unobservable. Since the proposed

method does not use floating-point computations is not prone to rounding errors. The proposed algorithms are computationally very

attractive, yet simple to implement in existing state estimators by minor modifications to the existing sparse factorization andsubstitution

operations. Numerical examples are provided to illustrate the steps and performance of the proposed algorithms.

Appendix A. Standard and fraction free LDU factorization

LetA = [ai,j] be annsymmetric positive definite matrix. The standard Gaussian elimination gives the following LDUfactorization ofA.

A =

1

l2,1 1...

..

.. . .

ln1,1 ln1,2 1ln,1 ln,2 ln,n1 1

d1,1d2,2

. . .

dn1,n1dn,n

1 u1,2 u1,n1 u1,n1 u2,n1 u2,n

. . ....

..

.

1 un1,n1

= ldu (A1)

where

a(0)i,1 = ai,1,

a(j

1)i,j =

a(j2)i,j

a(j2)

j1,j1 a

j2i,j1

a(j2)

j1,j

a(j2)

j1,j1

(2 j i n) (A2)

li,j = uj,i =a(j1)

i,j

aj1j,j

(1 j i n) (A3)

di,j = a(i1)i,i

(1 i n) (A4)

The determinant ofAis given as[40,42]:

det(A) =

ni=1

di,i = a(0)1,1

a(1)2,2

a(2)3,3

. . . a(n2)n1,n1

a(n1)n,n (A5)

We introduce byA(k)i,j

the following determinant:

(A6)

From[42]we have the relations:

A(1)

0,0 = 1, A(0)

i,j = ai,j (1 i, j n)

A(k)i,j =

A(k1)k,k

A(k1)i,j

A(k1)i,k

A(k1)k,j

A(k2)k1,k1

(k < i,j n)(A7)


13/15


Combining(A2) and (A7)we have:

a(i1)i,j

=

A(i1)i,j

A(i2)i1,i1

(1 i, j n) (A8)

Combining(A3), (A4),and(A8)we have:

li,j = uj,i =A

(j1)i,j

A

(j1)

j,j

, di,i =A

(i1)i,i

A

(i2)

i1,i1

(1 j < i n) (A9)

Based on(A9),we have the integer-preserving or fraction free Gaussian elimination, which admits the following LD1Ufactorization

[42]ofA.

A =

L1,1L2,1 L2,2

......

. . .

Ln1,1 Ln1,2 Ln1,n1Ln,1 Ln,2 Ln,n1 1

D1,1D2,2

. . .

Dn1,n1Dn,n

U1,1 U1,2 U1,n1 U1,nU2,2 U2,n1 U2,n

. . ....

...

Un1,n1 Un1,nUn,n

= LD1U(A10)

where

Li,j = Uj,i =A(j1)i,j

(1 j i 1) (A11)

Li,i = Ui,i =A

(i1)

i,i (1 i n 1) (A12)

Un,n =A(n1)n,n = det(A) (A13)

Di,i =A(i2)i1,i1

A(i1)i,i

(1 i n 1) (A14)

Dn,n = A(n2)n1,n1

(A15)

From(A7), (A12), (A14)and(A15)we obtain:

D1,1 = U1,1 (A16)

Di,i = Ui1,i1Ui,i (2 i n 1) (A17)

Dn,n = Un1,n1 (A18)

Notice that since the matrix D is completely predicable, it does not need to be stored. If the matrixA has integer entries, in the standard

Gaussian elimination all divisions are floating-point operationsand L, D,and Uhave real entries, butin thefractionfree Gaussian eliminationall divisions are integer operations with exact integer results and L,D, andUhave integer entries[40,42].

Appendix B. Standard and fraction free forward and backward substitution

LetA be a nn symmetricand positive definite or semi-definite matrix. If no zero pivotis encounteredduring its triangularfactorization,

then the matrixAis positive definite else is positive semi-definite.

For the standard Gaussian elimination (Appendix A)we have the followingLDUfactorization algorithm. Positive thresholdis used fordetecting zero pivots during standard Gaussian elimination. Diagonal entries whose absolute value is less than are considered as zero.

(B1)

The linear systemAx = bcan be solved by performing a forward and a backward substitution as follows.


14/15


forward substitution:

LDy = b (B2)

backward substitution:

Ux =y (B3)

For the fraction free Gaussian elimination (Appendix A)we have the followingLD1Ufactorization algorithm:

(B4)

Note that if the matrix A has integer entries, there is no need for any zero threshold in fraction free Gaussian elimination, since all

computations are exact (integer). The solution of the linear system Ax = b can be obtained by solving the linear system As = bunn and

computingx = s/unn. All entries of vector s will be also integer[42].However the vectorx will not have integer entries, but they will be

expressed as fractions of integer entries ofsscaled byunn.

The vectorscan be computed by performing a forward and a backward substitution as follows.

forward substitution:

LD1y = b (B5)

backward substitution:

Us =yunn (B6)

Taking into account that lii = 1 ( 1 in) for the standard Gaussian elimination and d11/l11 = 1, dnn/ln-1,n-1 = 1 , (wii/lii)/li1,i1 =

1, lii/dii = 1/li1,i1 (1< i n 1) for the fraction free Gaussian elimination, we obtain a common forward substitution algorithmfor both cases, as follows.

(B7)

wherewii = 1/diifor the standard Gaussian elimination andwii = diifor the fraction free Gaussian elimination.

The backward substitution algorithm for both cases will be as follows:

(B8)

wherew =x,t=yfor the standard Gaussian elimination and w = s,y =yunnfor the fraction free Gaussian elimination.


15/15


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GeorgeN. Korreswasborn in Athens,Greece,in 1960. He receivedthe Dipl.Eng. andthe Ph.D. degreesin electricalengineeringfromthe NationalTechnical University of Athens,Greece, in 1984 and 1988, respectively. Currently, he is an Associate Professor in the School of Electrical and Computer Engineering at the National Technical Universityof Athens, Greece. His fields of interest include estimation and identification techniques in power systems. He is a senior member of IEEE and member of CIGRE and theTechnical Chamber of Greece.
http://www.ee.washington.edu/research/pstca/http://www.powerworld.com/http://www.powerworld.com/http://www.powerworld.com/http://www.ee.washington.edu/research/pstca/

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