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Power Network Observability Guidelines for Measurement System Design
Copyright © IF AC Powe r Syste ms Modelling and Control Applicatio ns . Brussels. Belgium 1988
POWER NETWORK OBSERVABILITY: GUIDELINES FOR MEASUREMENT SYSTEM DESIGN E. D. Crainic, X. D. Do and D. F. Mukhedkar Departmmt of Electrica l E'l gineering, Ecole Polytechllique de M ontreal. p.a . B ox 6079 , Station " A ", M ontrea l, H3 C 3 A 7, Ca narla
Abstract. In order to increase the numerical value of its strength, general guidel ines for a measurement system design are proposed . This new concept, the absolute strength of a measurement system, is def ined as the total number of subsets of n dimensional I inearly independent measurements existing in a set of.A£measurements . with n ~..4£. For P-6 observability n = N-l, and for O-V observability n = N. where N represents the number of the network nodes. The procedure to evaluate the strength of a measurement system is based on the property of the measurement to branch incidence matrix H, and the node to branch incidence matrix 4 of being both totally unimodular matrices . For regular, simpler networks, general formulas for the strength of their corresponding measurement systems are derived . These formulas generate new or already known integer sequences . some of the latter ones related to Fibonacci series, or to coefficients of Chebyshev polynomials . Keywords. Power system planning ; state estimation ; observabi I ity; measurement design ; unimodular matrices; Fibonacci series; Chebyshev polynomials .
I.
INTRODUCTION
system
to it .
As a tool of power system secure operation, the static state estimator must provide a rei iable estimate of the state vector comprising the vol tage magnitudes and angles at each node of the network (Schweppe , 1970). The state vector thus obtained is subsequently used as an essential input for the security analysis and moni taring functions of any modern energy management system .
An optimal meter placement aiming the qual i ty of the state estimation is discussed by Kogl in (1975), whereas Phua and Di I Ion (1977) formulate it as a non- I inear programming problem. Albertos . Alvarez and Reig (1980) propose a performance index. weighting both accuracy and cost . to determine the optimal measurement equipment . Fetzer and Anderson (1975) answer the question of choosing the best measurement to add to an existing measurement set, as being the measurement increasing the smallest eigenvalue of an observability matri x. Monti celli and Wu (1984) propose an algori thm to introduce addi tional measurements such that they would not contaminate the state est imat ion resul ts .
of a state The rei iabi I ity and performance estimator depend on the qual ity and avai labi I ity of the measurements used as i ts input data . To provide a degree of security in the event of telemetry or transducer fai lures . or to detect and identi fy bad measurements, there are more measurements than the minimum number required to calculate the state vector.
A more general problem for meter placement in the p l anning stage , aiming the rei iabi I i ty of having an observable network under different contingenc ies, is addressed by Ar iatt i, Marzio and Ricci 11975) , and by elements , Krumpholz and Davis (1982 , 1983) .
The ratio between the existing measurements (i . e . voltage magnitudes, active and reactive power flows. active and reactive bus injections , etc . ) and the number of components in the state vector is cal led the redundancy level of a measurement system, its value being typically betweenl.1 and3 . As defined above , the notion of redundancy level is . of limited use in describing a measurement system. Its secur i ty depends not only on the number of existing measurements , but also on their type. their location and on the i r distribution on various remote telemetry uni ts (RTUs) .
In the present paper a new concept is introduced , i.e. the measurement system strength . It is def ined as the tota l number of subsets of n-dimensional linear Iy i ndependent measurements existing i n a set of..4£ measurements. For P-6 observabi I ity n = N-l, and for P-O observabi I ity n = N, with N the number of network nodes, and of cou r se.;ftt ~ n.
There are many papers deal ing with network observabi I ity analysis based on measurement systems already in place . but relatively few treating the problem of measurement system design .
To take into account the security of a measurement system, coefficients of its relative strength are also defined as being the ratio between the values of its absolute strength after and before different measurement or/and network contingenc ies have occurred .
Schweppe, Wi Ides and Rom (1970) treat the problem of meter placement with the aim of improving the accuracy of the estimate, their method being based on the error covariance matrix analysis . Handschin measuring al I and Bongers (1972) recommend quantit ies at every bus . i . e . voltage, power injection and al I I ine flows on branches incident
The procedure for evaluating the measurement system strength is purely algebraic in nature and it is based on the property of measurement to branch incidence matrix H. and node to branch incidence 481
482
E. D. Crainic. X. D. Do and D. F. Mukhedkar
matrix A of being both totally unimodular matrices.
rank (Ho)
N-l
(5)
The organizat ion of the paper is as follows. In Sec t i on I I i mpo r tan t concep t s and de fin i t ions 0 f network observabi I i ty are reviewed. An equivalent matrix for network algebraic observabi I ity analysis in then der ived. Sect ion III states and proves two theorems. It also introduces a new notion . i.e. measurement spanninu tree which. together wi th the notions of topological spanning tree and spannin tree of full rank . allows lor a clearer formulat ion of the .power network observabi I i ty problem. The concept of measurement system absolute strength and def ini t ion of its coeff icients of relat ive strength are introduced in Section IY . In Section Y for regular and simpler networks general formulas for the strength of thei r corresponding measurement systems are der ived . In Sect ion YI some guidel ines for a measurement system design are presented and discussed. Concluding remarks can be lound in the I as t sec t i on .
rank
N
(6)
The most widely used definit ions of power system observabi I ity are al I based on a I inearized. decoupled version 01 the standard WLS estimator. Belore giving precise defini tions of algebraic. ar i thmet ic . and topological observabi I i ty . it is necessary to briefly review this model. For a power network having N buses, a set of Jt( the state measurements (wi th Jt( = mp+mQ+my). estimator calculates an estimate of the network bus vol tage magnitudes and angles Irom the measurement vector: ~ = h(.!!.) + y (1) where ~ is theJt(dimensional measurements vector . y the gdimensional noise vector . .!!. the (2N-l )-dimensional state vector of N voltage magnitudes .!!.Y and (N-l) vol tage angles .!!.6 . and h(.!!.) a non-I inear measurement funct ion. The non-I inear model (1) may be approximated decoupled I inear system, i.e. :
r~
Ho
.!!.6
~Y
Hy
.!!.Y
by
a
yP
+
~
YQ
( 2)
Yy
where !P, ~, ~Y are. respectively . the measurement vector of active power . reactive power and voltage magnitude. and yP. YQ. Yy are the vector of measurement noise of . respect ively. act ive and reactive power and vol tage magnitude. From (2) the Jacobian matrix H is defined as being:
ah H - a'J1
Wi th the assumption that there are one bus vol tage magn i t ude and mo r eac t i ve powe r measu r emen t 5 • the condit ion (6) becomes :
'",' [
rank (Hy)
(3)
and evaluated wi th .!!. at flat start. In (3) the submatrix H6 is mp x(N-l)-dimensional and the submatrix Hy is (ma + my) x N-dimensional . According to the delini tion given by Krumpholz, Clements and Davis (1960), a power system is said to be algebraically observable, in the context of stat ic state est imat ion, i I (4) rank (H) = 2N-l that is the dimension of the state vector l!. . The matrix H being block diagonal, the network observabi I ity may be divided into separate P-6 and Q-Y observabi I ity problems, wi th the condition (4) becoming:
~1
(7)
wi th Hy representing the mo x (N-l)-dimensional reactive power submatrix of the Jacobian matrix H. Therefore the condition (6) may be replaced by: N-l
rank (Hy)
(8)
By introducing the mX(N-l)-dimensional submatrix Hm . with Hm; Ho for m = mp. and Hm; Hv for m = moo the network observabi I i ty analysis wi I I be generally done two times by verifying if: rank (Hm) = N-l
POWER SYSTEM OBSERYABILITY ANALYSIS: REYIEW OF KNOWN RESULTS
II
(Hy)
(9)
From now on we will deal only with the matrix Hm, keeping in mind that it may represent ei ther the matrix H6 . or the matrix Hy . The m rows of Hm in (9 ) wi II refer to the actual as well as the virtual ' me'a surements, if any, I t must be noted the distinction between algebraic observability and numerical observability, the latter concept implying that for a given power system its measurement model (1) can be iteratively solved for a state estimate &, from a flat start : l!.6 = Q, xY = l!.Y.O· As shown by Krumpholz , Clements and Davis (1980), the ma t r i x Hm i n (9) rep r esen t sap r oduc t 0 f t h r ee ma t r ices i . e. : 1
MYA t
( 10)
where M is the mxL-dimensional measurement to branch incidence matrix. Y the LxL-dimensional diagonal matrix of branch admittances. and A an (N-l)xL-dimensional node to branch incidence matrix (where L the number of network branches, and wi th t for " transposed") . Trying to evaluate the rank of matr ix Hm as defined in (10) by using some techniques borrowed from the network or graph theory is known as topological observabi I itv analysis (see section I I I) . For the purpose of measurement system design we are concerned only with the approximate model (2), and wi th the concept of algebraic observabi I i ty . Let us assume that in (10) al I branch admittances are nonzero . In that case . the following matr ix: I = MAt (11) defined as the mx(N-l)-dimensional measurement to node incidence matrix, has the same rank as matrix Hm, i.e .: rank(Hm) = rank(l) $ min[rank(M), rank(A)j (12) For a connected network, rank (A) = N-l, therelore the rank 01 matrix M must be thoroughlully i nves t i ga t ed, since even i I rank (M)
MATRICES M AND A VS . NETWORK ALGEBRAIC OBSERVABILITY
3.1. Prel jmjnary Remarks Krumpholz, formulated
Clements and Davis (1960) have the necessary and sulficient condition
483
Power :'>Jet work Obseryability: Guidelines for Measurement System Design LEGal) :
for the power system algebraic observabi I ity pr ob Iem , i . e. : !;!a_-:-'lnl1.e.!.tw!!.o!.!. !. .rk~-:-!.-'iSL-...Ja!.l!..lgd.!e
~ - BUS - I NJECT I ON hEASl.faENT
Ea - REOLN)ANT
In the context of the present paper the following concepts related to power system algebraic observabi I i ty analysis are def ined , i . e . : a) topological spanning tree (referred from now on as TST) ; b) measurement spanning tree (referred as MST) ; c) spanning tree of full rank (referred as FRST).
(OR NOT USED)
BUS - I NJECTI ON IoEASlJ'IaENT
o-
NO BUS -I NJECT ION IoEASlJ'IaENT LlI£-FLOW IEASlJ'IED BRAIOi BRANCH WITHOUT LlI£-FLOW IEASl.IeENT
The concept of TST is identical to the notion of spanning tree as it is used in network or graph theory, i . e. a tree containing all N nodes and having (N-l) branches . For a connected network the total number of TSTs wi I I be denoted by T .
1Qb. \
3
0--02
.~\,
An MST represents any subset of (N-l) linearly independent measurements (active or react ive). For an algebraically observable power system the total number of MSTs wi I I be denoted by ~.
~
An FRST is a tree containing al I N network nodes , with the property that to ea ch of its (N-l) branches a measurement is assigned in such a way that no two branches have the same measurement . The assignment of a measurement to a branch obeys to the following rules (see also Fig. 1) : (1) a l ine-flow measurment may only be assigned to the branch on which it is located ; (2) a bus-inject ion measurement may be assigned to any branch incident to this node. In other words an FRST represents an MST assigned to a TST . The total number of FRSTs wi 11 be denoted by p.
.•~,.l\,
As an example for the network shown in Fig . 1, whose measurement system consists of 2 bus-inject ion and 2 I ine-flow measurements , we have (see also Table 1 ) : T = 3 , ~ = 6 and p = 10. 3.2
Node to Branch Incidence Matrix A. Definition . Properties
Except for Theorem 1, all the informat ion concerning the node to branch incidence matrix given in this subsection represent known results.
.1\,
Fig . 1 Network with measurements .
,
I~ TST
th
It~
2
2 . 3 1.3 1.2 1
Ao
=2 3
Related
[~
-1
to
~ -~]
-1
0
A
=
2 [
3
2 .3
1. 3
1
o
-1
-1
1.2
-~
]
2 ("2 . 0113)
X
a ("2 ' ''3)
X
• ("'a'''1a)
X
5 (OIa."12)
network
algebraic
observabi I ity
3
the
g.n.r.t·~h by the I
MST I
X
2
X
X
3
x
X
2
I
I '
3
for
PI • •,,"'.
X
, I • FASTI 1 .. lg"~d la Ihe II TST
3
4
•
lu
10111 n _ . 01
FAST,
problem,the following properties of the matr i ces Aa and / or A are of interest (for their proofs see Ch en • 1971) : • Property 1: The number r of TST5 in a connected network is given by : r
= det (AAt)
• Property 2 : rank (Aa)
the
FRSTs
X
8 (.12,"'13)
The matrix Ao has N rows , one for each node, and L columns, one for each branch , an element aiK of it being defined as fol lows: 1 (or -1), if the ith Rode is the positive (negative) node of the kt branch ; 0 , if the kth branch is not incident on the i th node . As an example, for the netwo r k shown in Fig . 1, the matrix Ao, and the matrix A (row 1 removed) are expressed as fol lows :
branches and 4
3
I( I . 3 . 2) b(1,2,3) c(3.1 , 2)
MST
, ("2."'2)
The node to branch incidence matrix A is any submatrix obtained from the al I nodes to branch incidence matrix Ao , by removing from it a row corresponding to any node of the network.
nodes ,
Number of TSTs. MSTs and Network shown in Fig. 1.
TABLE
Let us f~sume an arbi trary assigned direction to each ~ brfRch of the network . For inf~ance , if the it and i nodes adiacYRt to the k branch are such that i < j , the i node is 1~signated as its pas i t i ve node (+) , and the j node is designated as its negative node (-) .
3
(13)
For a connected network we have :
= rank
(A) = N-1
( 14)
E. D. Crainic, X. D. Do and D. F. Mukhedkar
484
• Prooerty 3 : The matrices Aa and A are both totally unimodular matrices, i.e . the determinant of each of their square matrices is 1, -1, or 0 ; • Property 4 : If to Aa (or to A or to any totally unimodular matrix), one row (or more) is (are) added containing only one element non zero , and equal to 1 or -1 , the resulting matrix i s also a totally unimodular matrix ; For any set of (N-l) branches • Property 5 : forming a TST , the major of matrix A whose (N-l) columns correspond to those (N-l) branches , has the value ei ther +1 , or -1. A major determinant or briefly a!!!U.Q!. of a matrix is a determinant of maximum order in that matr ix. Let US consider the (LxL)-dimens ional product matrix A~Ao, and let US consider any of its square submatrix of order (N - l), whosy columns correspond to the (N-l) branches of the i h TST . and whofg rows correspond to the (N-l) branches of the j TST, with i, j 1, 2, T . We have the following: Theorem 1 : The determinant of any s qu a re submatr ix of order (N-l) of the product matrix A~Ao ' who s e rows and columns correspond to a TST (not necessary the same) is equal to N, the number of network nodes.
and type 2 rows mentioned I f M contains type above , the submatrix formed with the mb rows corresponding to the mb bus-injection measurements is totally unimodular (see Property 3). Adding to it, one by one, the remaining ma rows corresponding to the ma line-flow measurements , the resulting submatrices are always totally unimodular (Property 4 ).
The following submatrices of M may be defined : a) the submatrix of type Ma ' defined as any (N-l)xL dimensional submatrix M whose rows correspond to any (N- 1) I i near I y independent measurements (or to any MST). If there exists a t least one submatrix of type Ma the network is algebraically observable. True enough, i f we note with Mai the submatrix whose rows corres pond to the i h MST we wi I I have: (15 ) were Wi i s the i th square submatrix of order (N-l) of the measurement to node incident matrix • def ined by (11), whose determinant value wi is non zero , i . e . : W
The
proof
of
this theorem is given in subsection
3 . 3. 3.3
Measurement to Branch Incidence Matrix M. Definition . Properties
An element mik of the measurement to branch incidence matrix M is defined as fol lows (Krumpholz , Clements and Davis , 1980) : . 1 (or -1), if the ith measurement is either a located at, or a bus-injection I ine-flow measurement I~~ated on the positive (negative) node of the k branch ; • 0, otherwise . As an example, for the network shown in Fig . 1 . 0 , the matrix M is wri tten as fol lows: 1.3 1 . 2
i
= de t
( Wi)
-1
( 17)
The above formula impl ies the following : • Property 6 : For a chosen matri x A, the FRSTs generated by an MST have al I the same sign (or or i en tat ion) , i. e . e i the r + 1, 0 r - 1 . As an example, for the network shown in Fig . 1, if we take for instance,that matrix Ma whose rows correspond to the I ine-flow measurement m12 and to the bus-injection measurement m3' and that matrix A derived from Aa with row (or node) 1 removed , that is :
o
II
There are two types of rows in M: 1 . line-flow measurements : in this case therow of M contains only one element non zero , whose value i s e i the r + 1, 0 r - 1 ; 2. bus-injection measurements : in this case the row of M is ident ical to the corresponding row of AO , Concerning
matrix M we have the following :
The (mxL)-dimensional measurement to Theorem 2 : branch incidence matrix M is a totally unimodular matr ix .
-1
If M contains only type 2 rows , it is either Identical to the matrix Aa or equivalent to a submatr Ix of Aa, therefore I t is also totally unlmodular (see Property 3) .
~]
At=[~-1 :~] 0
one can easily verify that W = det (Ma At) P = IW I = 2 (see a I so Tab Iel , row 4) .
-2 and
The matrix Mai defined in (15) and sat isfy i ng the cond i t i an ( 16) is ca 11 ed a I so a non-deaenerated ~. There are as many matrices of this type as the total number ~ of MSTs. I t may exist subsets of (N - 1) measurements which are not I inearly independent . I f we note with M~k a matrix whose rows correspond to (N-1) linearly dependent measurements we wi I I have :
M~k At
(18)
de t (Wk)
~:
If matrix M contains only type 1 rows, the determinant of every square matrix of M of order 1 is +1, -1, or O. Adding one by one to a given row of M the remaining (m-l) rows, the resulting intermediate submatrices of M are totally unlmodular (see Property 4) .
( 16)
In (16) wi is a positive or neaative integer, its absolute value repr~senting the number Pi of FRSTs generated by the it MST, i . e . :
1
o o
(Ma i At) ~ 0
= de t
= de t
(M~k At)
=0
( 19)
and the matrix M~k wi I I be cal led a degenerated Let US note wi th 1" the number of all degenerated matrices . We have:
~.
I't
= I'
+ 1"
= C~-1
m! (N-l)! (m - N+1)!
(20)
b) the subma t r i x 0 f t vpe M,8, i s de fined as any m x (N-1) dimensional submatrix whose columns correspond to the branches of a TST. There are r matrices of this type, wi th r given
by
Power Network Observability: Guidelines for Measurement System Design (13) . II we note wi th P j the number 01 FRSTs having as topological support the jth TST . and with "f~ the submatrix whose columns correspond to that j TST. we have : ( 21) II C is a c x d matrix and D is a d x c matrix . with c S d. the Binet·Cauchy theorem states (Chen . 1971) that : det (CD) =
E(
product of the ) corresponding majors of C and 0 (22)
Let us consider a measurement system which has N bus· inject ion measurements. that is M = Ao. and let us consider two submatrices ~Pa . MPb whose (N·1) columns correspond to the at and . respectively. to the b th TST 01 the network. Both "Pa l "Pb are N x (N · 1)·dimensional matrices . each having N distinct square matrices of order (N·1) whose determinant value are either +1. or ·1 (see Property 5). By applying the Binet·Cauchy theorem (22) . we have: det (M~a MPa)
= det (M~b
M£lb) " N
("~a "Pb) " det ("~b "£la)
Fibonacci
.1 .1
.2
~1
.2
11
~1
•
~ =
. 2
3
=
PN " I"N
=0
wi th "0
4
N
•
N-1
4
Fig . 3 Loop free regular networks: I ine·llow measured at both ends.
branches
I"N_1' "N.2
(27)
= 1.
and 1"1
c. for the network shown in Fig . 4 (see also Table 2 . co I umn c):
~1
Single loop regular networks : one I ine·llow measurement on each branch .
4
(24)
Since lor a measurement system consisting of N bus· inject ion measurements M = Ao . the resul ts (23) and (24), represent the prool of theorem 1 (see subsection 3.2) . True enough the product matrix 01 order (N-1). (M~a MPa) or (M~a Mpb) represents any submatrix 01 order (N·1) of the matrix (AoA) whose (N-1) columns and rows correspond to the same TST or. respectively. to any two distinct TSTs . IV
as
b . lor the network shown in Fig. 3 (see also Table 2. co I umn b):
Fig . N
known
(23j
and also (see Property 6): det
with FO " 0 and F1 "1 being sequence (Hogatt. 1969).
485
PN
N • F2N
(28)
"N
N • F2N-1
(29)
For N large enough . it resu Its : PN (30) I"N " 'P where 'P 1. 61803 represents the posi tive root of the equation :
i .y
MEASUREMENT SYSTEM STRENGTH EVALUATION
4 . 1 Prel jmjnary Remarks The number, 01 TSTs given by lormula (13) is a measure of network connectivity. i.e. for a given network the larger the value of '. the stronger is the connectivity of the network . In trying to de I i ne a si mi I a r quan tit a t i ve cr iter ion lor a measurement system strength two candidates are I ikely to be considered. that is P the total number of FRSTs . and I" the total number of MSTs . Lat U8 dar Iv. tha numar leal values of PN .nd I"N for aome regular, Simpler networks (see also Sact Ion V): a. lor the network shown in Fie . 2 (see also Table 2. co Iumn a):
- 1 " 0
(31 )
and known from ant iquity as the "Golden the "Divine Proport ion " (Hogatt. 1969) .
Ratio " or
d , for the network shown in Fig , 5 (see also Table 2, co I umn d ): 5 ~1
O:3A,:O:'30 . :.'.
Fig , 5 Single loop regular networks: branches I ine-flow measured at both ends . TABLE 2 Number 01 FRSTs and MSTs lor the Regylar Networks shown In fig. 2 to Fig. 5. (a)
(b)
PN:I'N
PN:I'N
(d)
(c)
N
.....1 _..... 1fIo2_ _ ~
•••
Elg . 2 Loop frae regular networks: measurement on each branch.
-.--. N-1
one
N
line-flow (25)
where F2N represents the (2N)th Fibonacci number . the integer sequence having the recurrence lormula: EN
= EN_1
• FN·2
(26)
1
1
PH 1
PN
I'N 1
I'N
1
1
1
2
3
4
6
4
8
8
3
8
15
24
15
45
33
4
21
56
84
52
224
184
5
5~
209
275
170
1 045
785
6
144
780
884
534
4 eao
3 425
7
377
2 911
2 839
1 e31
20 377
14 917
8
967
10 864
7 898
4 880
9 2 584
40 545 23 258 14 373
se 912
e3 824
90~
2e7 129
384
10 8 785 151 31e 87 850 41 810 1 513 leo 1 107 710
486
E. D. Crainic . X. D. Do and D . F. Mukhedkar PN-2
4 PH-1
(--,;r-r-
N
PH
o and
wi th PO
I'N = N
P1
= 1. and
I'N-1
4
I'N-2
(33)
(--,;r-r- - R72)
defined for N > 2 and wi Ih 1'1 N large enough, i I resul Is:
PNil'N
=" = 1.36602
where v the equat ion
y2 _ Y _ 0.5 4.2 Measurement
Strength .
measurement system strength after a second worst contingency has occured. If we note with I1n_k the coeff icient of the measurement system relat ive strength after a k-elements cont ingency, the following inequal ities hold true:
(32)
R72)
1
o
For
6.
and 1'2
(34)
represenls Ihe posi live rool
=0
SYstem Oefini tion
01
(35) Absolute
and
The concept of FRST is very useful to answer the question of a power network as being. or not. algebraically observable. However . larger values for p merely reflect a greater number of bus-injection existing in a measurement system than describing ils strength. As an example. let us conSider two di Iferent measurement systems M, and M2 for the network shown in Fig . 1 . 0 . The measurement system M, consists of three I ine-flow measured branches {m'2' m23. m,3} . I'(M,) = 3. The measurement which gives p(M,) system M2 consists of three bus-injection measurements {m1. m2 ' m3}' therefore I'(M2) = 3 and p(M2) = 9 . We have I'(M,) I'(M2). but p(M2) > p(M,) . Let us add a bus-injection measurement to M,. the new measurement system becoming M'{m,2 ' m23, m,3 . m,} . for which I'(Ml> = 6 and p(Ml> = 7 . We slill have p(M2) = 9 > p(M\) = 7. but the measurement system M\ is, objeclively . stronger than M2' True enough if any two measurements are removed from M\ the network remains observable . whereas by removing from M2 any Iwo measurements the network is no longer observable. Therefore. it is more appropriate to relale the strength of a measurement system to the concepl of MST . We define the total number of MSTs associated to a power network ~a~s~t~h~e-2a~b~s~o~l~u~te~s~t~re~n~gLtuh~oLf~~i~t~s measurement system. In assessing the Isecur i Iy · .. of a measurement system. very useful are the cQefficients ~n-k 21 i Is relat ive strength def ined as follows (when k I) :
(36)
1'0
1'0 - I'n-2 I1n-2
with
=
~n.2
(37)
1'0
representing
system
the
new
value
of
the
< ~n-k
.. .
(38)
In designing a measurement system, or when enforcing an existing one. the meter placement must be done to obtain the smal lest values for its coefficients of relat ive strength . The smaller the value of a coefficient "the more secure is the measurement system for the given con I i ngency. of
Ihe Measurement System Absolute
St rength In order to obtain Ihe lolal number P of FRSTs the general formula may be used, i.e.: P=
L
I'
det (MbjMpj)
L
j=1
Ide t
( Mc> i At)
I
139)
i=1
To evaluate 1'0. the measurement system absolute strength, two approaches are possible: one based on the non-degenerated (N-I)xL-dimensional submalrix Mc> , and the second based on the mx(N-l)-dimensional submatrix Mp , both derived from ma t r i x M (see Sec t i on I I I ) . A. Enumerat ion submatrices
of
non-degenerated
*
A submatrix Mc> has the property det (M"A t ) 0, and the total number of MSTs equals the number of al I submatrices Mc> of M having this property . As may be seen from (20) the matrix W has Pt distinct submatrices of order (N-I), therefore the procedure has, at least teoretica"y. Pt steps each step evaluating the determinant of a very sparse matrix whose elements are integers . B. Use of
the Hp type submatrix
The jth m x (N-I)-dimensional submatrix Mpi has the property (see also Fig. 1 and Table I): P j = P j = de t (Hb jMp j )
(40)
that is the num~~r of FRSTs having as topological suppor t the j TST equal s the number of MSTs generating them. Let us consider th~ following example : hi f i, k represent the it , respectively, the kt TST, the resulting number of MSTs is: I'j+k
where: the absolute value of the measurement system 1'0. strength. when al I its elements are in normal operation state (no disconnected I ine. no telemetry system malfunction. etc .. ); I'n-l. the new value of the measurement system strength. corresponding to the worst case of the following possible contingencies : a) losing a measurement, b) loosing an RTU, c) losing a branch I ine-flow measured or having at least one bus-injection measurement as its adjacent nodes. The coefficient ~n-2 of the measurement relative strength is defined as fol lows:
~n-I < ~n-2 <
4 . 3 Evaluation
Re I at i ve
1'0 - I'n-I
$
I'i
+
I'k - I'jk = det (HbiMpj +
+ det (MbkMpk) - det
(H~jMpk)
(41)
Note that Pl k = Pkj ' and if I'jk = I'k (or I'jk = 1'1') it results rom (41) that I'i+k = Pi (or I'++k = I'k ' or if Pjk equals zero I'j+k = 1'+' + Pk ' . her~f~re, only a limited number of TS s, that IS a limited number of Mp type submatrices, should be investigated to obtain the value of ~. As compared to the previous approach, thi s one is a I ittle more complex to COdify, but is recommended for radial and topological'y weak connected networks. If expressions on closed form exist to evaluate al I network T5Ts (see (16)) and al I network FR5Ts (see (401 it is not possible, at least for the moment, to provide general formulas for enumerating al I network MSTs . except for the special cases of
487
Power Network Observability: Guidelines for Measurement System Design
regular network section. V
as
we
wi I I
see
in
the
next
REGULAR AN:) QUAS I - REGULAR t£TWORKS .
IEASlJ9IENT SYSTEM
The term ·regular· as used in this section refers to the .. asurement system of a power network, Implying al I Its N nodes having bus-injection .. asurements, and/or al I branches having line-flow ..asurements located on them at one or both 01 their ends. ·Ouasi-regular" term means that one (or IIOre) _asurement(s) is(are) removed , as specified, from a regular network . Any two nodes are I inked by one branch only, and the one node network having no physical meaning, is considered lor mathematical reasons only as the starting point of an integer sequence . Most 01 the general formulas given in this section were establ ished by induction and, as some admit several forms, the simplest 01 them is presented . Networks wjth only Bus-Injection Measurements When all N network lleasurements PN
nodes
have
Ouasi-Regular Networks having N For the network shown in Fig. 7 , bus-injection and N line-flow .. asur ...nts (for
.1 Q1 A 4U14'f!t1-:1 JC
23
Fig . 7
N ~ 2), and an additional number of N(N-3)/2 branches (for N ~ 4) not I ine-flow measured, the strength of its measurement system is given by: N-1 PN = E (~~-I-i) = C~1 i=O
Number 01 MSTs :
(42)
•
Networks wjth only Ljne-Flow Measurements When al I branches have a I ine-Ilow measurement placed on them, doesn't matter at which end, there is a direct correspondence between an MST and a TST, i.e .:
(47)
of a Chebyshev given in Table 3
and when al I branches have I ine-Ilow measurements at both 01 their ends we have : N 1 2 - r
(44)
Ljne-Flow
For the networks shown in Fig. 8 having N bus-injection measurements and one, respectively two, line-flow measurement(s) placed on each branch, the lormulas expressing their corresponding measurement systems strength are (see Table 3, co I umn a , co Iumn b), N-1 PN = E (i+1) ~-1 = (N+1) 2N-2 i=O
(45)
N
•
N-1
'280
1 4 15 54 189 141 2187 7290 24 OS7
281.
78732
20
5
41 112 251 578
I
e
d
,
1 4
15 56
3 10 33 10S 324
210 7;2 3003
177
" 440 43 751 187 lao
2195 1412 24415
f 0 3 10 32
19 299
"7 2585 7501 21 528
9 1 2 I 19 59 179
S33 1 514
4538 13 031
0 1
3
•
28 74 209 587 '141 45118
I f the network shown in Fig. 4 • is taken, whose in it i a I measurement sys tem conta i ns N bus-injection, and N I ine-Ilow measurements, by successively modifying it, its strength wi I I also be modi lied, i .e. : • one I ine-flow measurement missing ( Table 3 and column d) : N-2 /AN =NF2N_' E (_1)i (N-1-i)F2(N-1-i) (48) i=O • one bus- inject i on measurement mi ss i ng (Tab le 3 .. column e) : N-2 /AN=.,F2N - 1+ 1:: ( _,)i (N-'-i)F2(N-1-i) i=O
(49)
• two I ine-flow measurements missing ( Table 3- t): (for N '>1)· [(N-3)/2]
= F2N-1
+
1::
i=1
(N-2-2i) F2(N-2-2i)
(50)
where [(N-3)/2] represents the integer part of the division . • one bus-Injection and its two adjacent II~e-llow ..nurements missing (Table 3. col~g):
2
Loop free network •. of a
c
b
4
10
PN 3
a 1 3
7
A Comparison .
IEASUeEHf SYSTD STllENGTli
1 2 3
• ••
(43 )
which gen.rat •• coefficient. polynomial (Sloane . 1973), and:
(2NI ! :N+1)! (N-1)!
which generates coeffiCients po I ynomi a I (SI oane, 1973) as col umn c"
a known result, where ~-1 represents combinations of (N-1) measurements Irom a set of N measurements . (see Theor811 1 and Section Ill) .
Fig. 8
•••
2
AI I nodes connected, quasi-regular networks .
TABLE 3
Networks wjth Bus-Injectjon and Measurements (see also Section IV)
2 3
bus-injection
= ~-1 = N
PN =
,.,
23
Chebyshev (46)
PN
= F2N-2
N-4
+ t
I
(-1)
(N-3-i) F2(N-3-i)
(51)
1=0
In formulas (48) to (51) if k number Is zero .
<
0 the kth Fibonaccl
488
E. D. Crainic, X. D. Do and D. F. Mukhedkar
VI Ge£RAL ClUIDELltES FOR hEASlJIDENT SYSTEM DESIGN
As may be seen from the previous section,the formulas expressing the strength of the measurement system corresponding to simple but regular networks (i . e. combined bus-injection and line-flow measurements) generate integer sequences which start increasing rapidly with the network nodes N.
1'0 = 6 Fig. 9
On the other hand, as soon as the measurement system regulari ty is lost, its strength diminishes dramatically (see Table 2 column c compared to columns d to g of Table 3) . The measurement system for real power network is far from being regular . For networks wi th the number of branches L = 1.4 N to 2 . 5 N, their measurement system may have , theoretically , 4 . B N to 7 N measurements (N bus-injection , N 5 voltage magnitudes, 2 . B N to 5 N line-flow measurements), as compared to a redundancy level of 1. 1 N to 3 N for real life networks .
The following guidel ines are intended to make a bet ter, if not opt imal, use for the number and type of the avai lable measurements for a new measurement design . Measurement System Absolute Strength
I'b = 104
I'c
= 111
Line-flow measurements placement: Equidistancy procedure .
Fig . 9 . 0 already having 6 bus-injection measurements . The best plscement on this network of 3 additional I ine-flow measurements is shown in Fig . 9.c , whose messurement system strength equals 111 .
00_ 0 6
6
240.-24
3
1'0 = 6
6
8
15
4
a I t is not unusual that for networks wi th relatively high redundancy level, even in normal operation states , to have one or more critical measurements which indicate a poorly designed measurement system . Therefore, for real-I i fe operational measurement systems , their strength may take very low numerical values compared with to regular ones .
I'a = 93
Hi
3
15
24
3
I'a = 74
I'b = 7B
1
2
3
I'c = Bl
Fig. 10 Bus-injection measurements placement: Equidistancy procedure . Let us also consider the network shown in Fig. 10 . 0 already having 6 I ine-flow measurements . The best placement on this network of 3 additional bus-injection measurements is shown in Fig . 10 . c, whose Neasurement system strength equals Bl. if an additional c) loop-closing procedure : I ine-flow measurement is avai lable to close a loop of line-flow measured branches, it is better to p I ace i I I n such a way Ihe resu I1 i ng loop to contain the largesl number of branches .
For a given network and for a given number and type of measurements to improve the initial value ~O of the resulting measurement system absolute strength , the following recommendations may be made: a) regularity procedure, stated as follows : if some addi t ional I ine-f low measurements are avai lable, it is better to place them on branches not yet I ine-flow measured, if any. 1
o.
(1)---
a, b,
c,
3 4 [email protected]'O 2
Fig . 11 4
~
~
.--~I'a
18
~
• a"-~I'b
17
~I'c
19
.---. • •
"I'd = 21 Fig. 8 Line-flow measurements placement: Regularity procedure . As an example, for the network shown in Fig. 8 . 0, bus-Injection measurements, al I which has 4 distinct hypothesis of placing 3 addi tional I ine-flow measurements were conSidered (a double I ine means a branch line-flow measures at both of i III ends) . The i r bes t p Iacemen t is the configuration shown In Fig . B. d for which the mealurement system strength is equal to 21 . d,
i)
"a=175
I)
b) eguj-djstancy procedure, stated 85 follows: if a limited number of line-flow (bus-injection) measurement are to be placed on an already regular network zone, in terms of bus-Injection (1Ine-f low) lIB8Surementl, It II better to place them at equal intervall . As an example, let us consider the network shown in
Pb::182
Line-flOW measurements placement : Loop-closing procedure.
An example of this procedure is given in Fig. 11 for which an add it i ona I I i ne- flow measurement may be placed on anyone ( 1 ,2) , ( 1 , 3) , or. (1: 4) The best placement is shown In Fig. branches . " . c, whose measurement system strength equals 190 . d) bus-iniectjon to maximum-degree node p~0gedu~e, consists in placing an additional bUS-InJection measurement at the node with the largest number of branches incident on it .
4« :,14, : '149 : "14« c~>t'
3~23~23~23~2 1'0 ,. 3
~a
:: 6
I'b :: 6
I'c :: 9
Fig . 12
Bus-injection measurements placement : Bus-injection to maximum-degree node procedure. An example of this procedure 15 given In Fig. 12 for which an additional bus-Injection measur ... nt may be placed at any node 2. 3 or 4 . The best placement Is shown in Fig. 12,c, whose measur ...nt system strength equals 9. Even IIOre iMportant than the absolute value of a measurement syatem strength is itl rei lablllty, caplblllty to preserve the network I .e . Its
Powe r Ne two rk Obse rvabilit y: Guid elines fo r Meas ure me nt S ~ s t e lll Desig n observabi I ity following measurement and/or topology contingencies .
network
Measurement System Relative Strength Let us define the following notions : proper measurements of a node : the set • consisting of its bus - injection measurement. and al I I ine-flow measurements located on ItS sIde ; the set • adjacent measurements of a node : conSisting of al I bus-injection measurements existing at its adjacent nodes . and all Ilne:flow measurements located on its adjacent nodes SIde ; • branch priority I ist: a I ist contai~ing all the branches in descending order as functlo~h of the number 'k of TSTs to which the k branch belongs . A branch with 'k : ' . where is the total number of the network TSTs is called a crjtical branch; • proper measurements of a branch : the set consisting of the line-flow measurement(s) located on it. and of bus-injection measurements. i f any. ex i s t i ng at its ad j acen t nodes . The rei iabi I i ty level of a measurement system may also be characterised by its coefficients of relative strength "n-l ' " n-2 . .. . . "n-k which must be strictly less than 1 (see Section IV). To assure that . (concerning only may be made :
the the
following recommendations coefficients "n-l and "n-2)
a) place the avai lable number 01 line-flow measurements in the order given by the pr ior i ty I ist of branches. as delined above ;
~o
:
1
~o
=3
~O
: 4
"n-l = 1 "n-l = 1 "n-l = 0 . 75 F i g . 13 Pr i 0 r i t Y lis tal branches I i ne - fI ow measurement placement . As an example (Fig . 13) the best placement 01 an additional I ine-flow measurement is on branch 1-2 . ~ince for it "n-l = 0 . 75 < 1. b) i l two additional line-flow measurements are avai lable to be placed on two or more parallel branches not yet I ine-flow measured . they must be on di Iferent branches and not both located at the same end; C) the proper measurements of a node should not al I be put on the same RTU (this requ i rement is the most difficult . if not impossible . to be met in practice) ; d) the proper and adjacent measurements of a node should not al I be distributed on any comb ination of two RTUs ; e) the proper measurements of a critical branch ahould not al I be distributed on the same RTU. or on any combination of two RTUs; f) the l ine-flow measurements forming a loop should not al I be distributed on any combination of two RTUs. VI I
CONCLUSION
In this paper it is shown that the me~surement system design for a power network may .entlrel y ~e done on the measurement to branch inclden~e matrIx M and on the node to branch inc idence matrIx A. The property of measurement to branch in~idence matrix M. and node to branch Incidence matrIx A of
489
both being totally unimodular matrices allows for the definition and evaluation o f the absolute strength and various coefficients of relative strength of a measurement system . The absolute strength of a measurement system is defined as the total number of subsets of n-dimensional I inearly independent measurements existing in a set of.$measurements (n N- l for P-6 observabi I i ty . and n=N for Q-V observabi I ity. with N representing the number of network nodes . Also important is the and ..1ft ~ n). measurement system secur i ty. i . e . its capabil i ty to preserve the network observabi I i ty following measurement and / or network topology cont ingencies . For this purpose . coefficients "n-l . "n -2 ' "n-k of relative strength are introduced . They g i ve an indicat i on of changes in the strength of a measurement system following s i ngle or mult iple measurement and / or network contingencies . The assessement of a measurement absolute and relative strength may constitute a valuable objective tool in comparing di Iferent measurement plans for a given network . The use of . the der i ved general guidel ines for meter placement to analyse the ex isting measurement sys t ems may also provide necessary information for their improvement . REFERENCES Albertos. P .. C. Alvarez and A. Reig ( 1980) . Quantitative analysis of an on-I ine measurement equ i pmen t 10 r s tat i c s tat e est imation purpose . IFAC Conference . Rabat . ( 1975) . Ariatti. F .. L. Marz i o and P . Ricc i in v iew of Designing state estimation Cambr idge . rei iabi I ity . Proc. 5th PSCC . Paper 2 . 3/8 . Chen . \fI . K. ( 197 1 ) . ~A!.Ilp!.llp~I..!.i.s:e:Qd_-llgLr.ll.aI20!lh_-l..th!..!:elZ'OlLr!..y X. North-Hol land . Amsterdam. Clements . K. A. . G.R. Krumpholz and P .W. Davis (1982). State est imator measurement system rei iabi I ity evaluation : an efficient algorithm based on topolog i cal observabi I i ty theory . IEEE Trans PAS . ~. 997-1004. Clements . K. A. . G.R. Krumpholz and P . W. Davis (1983). Power system state estimation with measurement placement algorithm . IEEE Trans ~ . ~ . 2012-2020. Fetzer . E.E. and P .M. Anderson (1975) . Observabi I ity in the state est imation of power systems . IEEE Trans . PAS . ~. 1981-1988 . In E. Handschin . E. and C. Bongers (1972) . Handsch i n (Ed . ) . Real Time Control 01 Electric Power Systems . Elsevier. Amsterdam . Hogatt. V.E . Jr . (1969). Fibonacci and Lucas ~. Houghton Miffl in . Boston . Kogl in. M. J . (1975) . Optimal measuring system for state estimation . Proc . 5th PSOC . Cambridge . Paper 2 . 3/12 . Krumpholz. G. R. . K.A. Clements and P .W. Davis (1980) . Powe r system observability : a practical algorithm using network topology . IEEE Irans . PAS. ~. 1534-1542 . Monticel I i. A. and F . F. \flu (1984). Network observability : identification of observable islands and measurements placement. IEEE PES Summer Meeting . Seattle . Paper 580-7 . Phua. K. and T. S . Di I Ion (1977) . Optimal choice of measurements for state estimation . PICA 77 . Ioronto . Schweppe . F.C .• J. \fIi Ides and D.P . Rom (1970) . ~ Power system stat i c state estimation . Trans. PAS . ~ . 120-135 . Sloane. N. J.A. (1973). A Handbggk of Inteqer Sequences . Academic Press. New York .
POWER NETWORK OBSERVABILITY: GUIDELINES FOR MEASUREMENT SYSTEM DESIGN E. D. Crainic, X. D. Do and D. F. Mukhedkar Departmmt of Electrica l E'l gineering, Ecole Polytechllique de M ontreal. p.a . B ox 6079 , Station " A ", M ontrea l, H3 C 3 A 7, Ca narla
Abstract. In order to increase the numerical value of its strength, general guidel ines for a measurement system design are proposed . This new concept, the absolute strength of a measurement system, is def ined as the total number of subsets of n dimensional I inearly independent measurements existing in a set of.A£measurements . with n ~..4£. For P-6 observability n = N-l, and for O-V observability n = N. where N represents the number of the network nodes. The procedure to evaluate the strength of a measurement system is based on the property of the measurement to branch incidence matrix H, and the node to branch incidence matrix 4 of being both totally unimodular matrices . For regular, simpler networks, general formulas for the strength of their corresponding measurement systems are derived . These formulas generate new or already known integer sequences . some of the latter ones related to Fibonacci series, or to coefficients of Chebyshev polynomials . Keywords. Power system planning ; state estimation ; observabi I ity; measurement design ; unimodular matrices; Fibonacci series; Chebyshev polynomials .
I.
INTRODUCTION
system
to it .
As a tool of power system secure operation, the static state estimator must provide a rei iable estimate of the state vector comprising the vol tage magnitudes and angles at each node of the network (Schweppe , 1970). The state vector thus obtained is subsequently used as an essential input for the security analysis and moni taring functions of any modern energy management system .
An optimal meter placement aiming the qual i ty of the state estimation is discussed by Kogl in (1975), whereas Phua and Di I Ion (1977) formulate it as a non- I inear programming problem. Albertos . Alvarez and Reig (1980) propose a performance index. weighting both accuracy and cost . to determine the optimal measurement equipment . Fetzer and Anderson (1975) answer the question of choosing the best measurement to add to an existing measurement set, as being the measurement increasing the smallest eigenvalue of an observability matri x. Monti celli and Wu (1984) propose an algori thm to introduce addi tional measurements such that they would not contaminate the state est imat ion resul ts .
of a state The rei iabi I ity and performance estimator depend on the qual ity and avai labi I ity of the measurements used as i ts input data . To provide a degree of security in the event of telemetry or transducer fai lures . or to detect and identi fy bad measurements, there are more measurements than the minimum number required to calculate the state vector.
A more general problem for meter placement in the p l anning stage , aiming the rei iabi I i ty of having an observable network under different contingenc ies, is addressed by Ar iatt i, Marzio and Ricci 11975) , and by elements , Krumpholz and Davis (1982 , 1983) .
The ratio between the existing measurements (i . e . voltage magnitudes, active and reactive power flows. active and reactive bus injections , etc . ) and the number of components in the state vector is cal led the redundancy level of a measurement system, its value being typically betweenl.1 and3 . As defined above , the notion of redundancy level is . of limited use in describing a measurement system. Its secur i ty depends not only on the number of existing measurements , but also on their type. their location and on the i r distribution on various remote telemetry uni ts (RTUs) .
In the present paper a new concept is introduced , i.e. the measurement system strength . It is def ined as the tota l number of subsets of n-dimensional linear Iy i ndependent measurements existing i n a set of..4£ measurements. For P-6 observabi I ity n = N-l, and for P-O observabi I ity n = N, with N the number of network nodes, and of cou r se.;ftt ~ n.
There are many papers deal ing with network observabi I ity analysis based on measurement systems already in place . but relatively few treating the problem of measurement system design .
To take into account the security of a measurement system, coefficients of its relative strength are also defined as being the ratio between the values of its absolute strength after and before different measurement or/and network contingenc ies have occurred .
Schweppe, Wi Ides and Rom (1970) treat the problem of meter placement with the aim of improving the accuracy of the estimate, their method being based on the error covariance matrix analysis . Handschin measuring al I and Bongers (1972) recommend quantit ies at every bus . i . e . voltage, power injection and al I I ine flows on branches incident
The procedure for evaluating the measurement system strength is purely algebraic in nature and it is based on the property of measurement to branch incidence matrix H. and node to branch incidence 481
482
E. D. Crainic. X. D. Do and D. F. Mukhedkar
matrix A of being both totally unimodular matrices.
rank (Ho)
N-l
(5)
The organizat ion of the paper is as follows. In Sec t i on I I i mpo r tan t concep t s and de fin i t ions 0 f network observabi I i ty are reviewed. An equivalent matrix for network algebraic observabi I ity analysis in then der ived. Sect ion III states and proves two theorems. It also introduces a new notion . i.e. measurement spanninu tree which. together wi th the notions of topological spanning tree and spannin tree of full rank . allows lor a clearer formulat ion of the .power network observabi I i ty problem. The concept of measurement system absolute strength and def ini t ion of its coeff icients of relat ive strength are introduced in Section IY . In Section Y for regular and simpler networks general formulas for the strength of thei r corresponding measurement systems are der ived . In Sect ion YI some guidel ines for a measurement system design are presented and discussed. Concluding remarks can be lound in the I as t sec t i on .
rank
N
(6)
The most widely used definit ions of power system observabi I ity are al I based on a I inearized. decoupled version 01 the standard WLS estimator. Belore giving precise defini tions of algebraic. ar i thmet ic . and topological observabi I i ty . it is necessary to briefly review this model. For a power network having N buses, a set of Jt( the state measurements (wi th Jt( = mp+mQ+my). estimator calculates an estimate of the network bus vol tage magnitudes and angles Irom the measurement vector: ~ = h(.!!.) + y (1) where ~ is theJt(dimensional measurements vector . y the gdimensional noise vector . .!!. the (2N-l )-dimensional state vector of N voltage magnitudes .!!.Y and (N-l) vol tage angles .!!.6 . and h(.!!.) a non-I inear measurement funct ion. The non-I inear model (1) may be approximated decoupled I inear system, i.e. :
r~
Ho
.!!.6
~Y
Hy
.!!.Y
by
a
yP
+
~
YQ
( 2)
Yy
where !P, ~, ~Y are. respectively . the measurement vector of active power . reactive power and voltage magnitude. and yP. YQ. Yy are the vector of measurement noise of . respect ively. act ive and reactive power and vol tage magnitude. From (2) the Jacobian matrix H is defined as being:
ah H - a'J1
Wi th the assumption that there are one bus vol tage magn i t ude and mo r eac t i ve powe r measu r emen t 5 • the condit ion (6) becomes :
'",' [
rank (Hy)
(3)
and evaluated wi th .!!. at flat start. In (3) the submatrix H6 is mp x(N-l)-dimensional and the submatrix Hy is (ma + my) x N-dimensional . According to the delini tion given by Krumpholz, Clements and Davis (1960), a power system is said to be algebraically observable, in the context of stat ic state est imat ion, i I (4) rank (H) = 2N-l that is the dimension of the state vector l!. . The matrix H being block diagonal, the network observabi I ity may be divided into separate P-6 and Q-Y observabi I ity problems, wi th the condition (4) becoming:
~1
(7)
wi th Hy representing the mo x (N-l)-dimensional reactive power submatrix of the Jacobian matrix H. Therefore the condition (6) may be replaced by: N-l
rank (Hy)
(8)
By introducing the mX(N-l)-dimensional submatrix Hm . with Hm; Ho for m = mp. and Hm; Hv for m = moo the network observabi I i ty analysis wi I I be generally done two times by verifying if: rank (Hm) = N-l
POWER SYSTEM OBSERYABILITY ANALYSIS: REYIEW OF KNOWN RESULTS
II
(Hy)
(9)
From now on we will deal only with the matrix Hm, keeping in mind that it may represent ei ther the matrix H6 . or the matrix Hy . The m rows of Hm in (9 ) wi II refer to the actual as well as the virtual ' me'a surements, if any, I t must be noted the distinction between algebraic observability and numerical observability, the latter concept implying that for a given power system its measurement model (1) can be iteratively solved for a state estimate &, from a flat start : l!.6 = Q, xY = l!.Y.O· As shown by Krumpholz , Clements and Davis (1980), the ma t r i x Hm i n (9) rep r esen t sap r oduc t 0 f t h r ee ma t r ices i . e. : 1
MYA t
( 10)
where M is the mxL-dimensional measurement to branch incidence matrix. Y the LxL-dimensional diagonal matrix of branch admittances. and A an (N-l)xL-dimensional node to branch incidence matrix (where L the number of network branches, and wi th t for " transposed") . Trying to evaluate the rank of matr ix Hm as defined in (10) by using some techniques borrowed from the network or graph theory is known as topological observabi I itv analysis (see section I I I) . For the purpose of measurement system design we are concerned only with the approximate model (2), and wi th the concept of algebraic observabi I i ty . Let us assume that in (10) al I branch admittances are nonzero . In that case . the following matr ix: I = MAt (11) defined as the mx(N-l)-dimensional measurement to node incidence matrix, has the same rank as matrix Hm, i.e .: rank(Hm) = rank(l) $ min[rank(M), rank(A)j (12) For a connected network, rank (A) = N-l, therelore the rank 01 matrix M must be thoroughlully i nves t i ga t ed, since even i I rank (M)
MATRICES M AND A VS . NETWORK ALGEBRAIC OBSERVABILITY
3.1. Prel jmjnary Remarks Krumpholz, formulated
Clements and Davis (1960) have the necessary and sulficient condition
483
Power :'>Jet work Obseryability: Guidelines for Measurement System Design LEGal) :
for the power system algebraic observabi I ity pr ob Iem , i . e. : !;!a_-:-'lnl1.e.!.tw!!.o!.!. !. .rk~-:-!.-'iSL-...Ja!.l!..lgd.!e
~ - BUS - I NJECT I ON hEASl.faENT
Ea - REOLN)ANT
In the context of the present paper the following concepts related to power system algebraic observabi I i ty analysis are def ined , i . e . : a) topological spanning tree (referred from now on as TST) ; b) measurement spanning tree (referred as MST) ; c) spanning tree of full rank (referred as FRST).
(OR NOT USED)
BUS - I NJECTI ON IoEASlJ'IaENT
o-
NO BUS -I NJECT ION IoEASlJ'IaENT LlI£-FLOW IEASlJ'IED BRAIOi BRANCH WITHOUT LlI£-FLOW IEASl.IeENT
The concept of TST is identical to the notion of spanning tree as it is used in network or graph theory, i . e. a tree containing all N nodes and having (N-l) branches . For a connected network the total number of TSTs wi I I be denoted by T .
1Qb. \
3
0--02
.~\,
An MST represents any subset of (N-l) linearly independent measurements (active or react ive). For an algebraically observable power system the total number of MSTs wi I I be denoted by ~.
~
An FRST is a tree containing al I N network nodes , with the property that to ea ch of its (N-l) branches a measurement is assigned in such a way that no two branches have the same measurement . The assignment of a measurement to a branch obeys to the following rules (see also Fig. 1) : (1) a l ine-flow measurment may only be assigned to the branch on which it is located ; (2) a bus-inject ion measurement may be assigned to any branch incident to this node. In other words an FRST represents an MST assigned to a TST . The total number of FRSTs wi 11 be denoted by p.
.•~,.l\,
As an example for the network shown in Fig . 1, whose measurement system consists of 2 bus-inject ion and 2 I ine-flow measurements , we have (see also Table 1 ) : T = 3 , ~ = 6 and p = 10. 3.2
Node to Branch Incidence Matrix A. Definition . Properties
Except for Theorem 1, all the informat ion concerning the node to branch incidence matrix given in this subsection represent known results.
.1\,
Fig . 1 Network with measurements .
,
I~ TST
th
It~
2
2 . 3 1.3 1.2 1
Ao
=2 3
Related
[~
-1
to
~ -~]
-1
0
A
=
2 [
3
2 .3
1. 3
1
o
-1
-1
1.2
-~
]
2 ("2 . 0113)
X
a ("2 ' ''3)
X
• ("'a'''1a)
X
5 (OIa."12)
network
algebraic
observabi I ity
3
the
g.n.r.t·~h by the I
MST I
X
2
X
X
3
x
X
2
I
I '
3
for
PI • •,,"'.
X
, I • FASTI 1 .. lg"~d la Ihe II TST
3
4
•
lu
10111 n _ . 01
FAST,
problem,the following properties of the matr i ces Aa and / or A are of interest (for their proofs see Ch en • 1971) : • Property 1: The number r of TST5 in a connected network is given by : r
= det (AAt)
• Property 2 : rank (Aa)
the
FRSTs
X
8 (.12,"'13)
The matrix Ao has N rows , one for each node, and L columns, one for each branch , an element aiK of it being defined as fol lows: 1 (or -1), if the ith Rode is the positive (negative) node of the kt branch ; 0 , if the kth branch is not incident on the i th node . As an example, for the netwo r k shown in Fig . 1, the matrix Ao, and the matrix A (row 1 removed) are expressed as fol lows :
branches and 4
3
I( I . 3 . 2) b(1,2,3) c(3.1 , 2)
MST
, ("2."'2)
The node to branch incidence matrix A is any submatrix obtained from the al I nodes to branch incidence matrix Ao , by removing from it a row corresponding to any node of the network.
nodes ,
Number of TSTs. MSTs and Network shown in Fig. 1.
TABLE
Let us f~sume an arbi trary assigned direction to each ~ brfRch of the network . For inf~ance , if the it and i nodes adiacYRt to the k branch are such that i < j , the i node is 1~signated as its pas i t i ve node (+) , and the j node is designated as its negative node (-) .
3
(13)
For a connected network we have :
= rank
(A) = N-1
( 14)
E. D. Crainic, X. D. Do and D. F. Mukhedkar
484
• Prooerty 3 : The matrices Aa and A are both totally unimodular matrices, i.e . the determinant of each of their square matrices is 1, -1, or 0 ; • Property 4 : If to Aa (or to A or to any totally unimodular matrix), one row (or more) is (are) added containing only one element non zero , and equal to 1 or -1 , the resulting matrix i s also a totally unimodular matrix ; For any set of (N-l) branches • Property 5 : forming a TST , the major of matrix A whose (N-l) columns correspond to those (N-l) branches , has the value ei ther +1 , or -1. A major determinant or briefly a!!!U.Q!. of a matrix is a determinant of maximum order in that matr ix. Let US consider the (LxL)-dimens ional product matrix A~Ao, and let US consider any of its square submatrix of order (N - l), whosy columns correspond to the (N-l) branches of the i h TST . and whofg rows correspond to the (N-l) branches of the j TST, with i, j 1, 2, T . We have the following: Theorem 1 : The determinant of any s qu a re submatr ix of order (N-l) of the product matrix A~Ao ' who s e rows and columns correspond to a TST (not necessary the same) is equal to N, the number of network nodes.
and type 2 rows mentioned I f M contains type above , the submatrix formed with the mb rows corresponding to the mb bus-injection measurements is totally unimodular (see Property 3). Adding to it, one by one, the remaining ma rows corresponding to the ma line-flow measurements , the resulting submatrices are always totally unimodular (Property 4 ).
The following submatrices of M may be defined : a) the submatrix of type Ma ' defined as any (N-l)xL dimensional submatrix M whose rows correspond to any (N- 1) I i near I y independent measurements (or to any MST). If there exists a t least one submatrix of type Ma the network is algebraically observable. True enough, i f we note with Mai the submatrix whose rows corres pond to the i h MST we wi I I have: (15 ) were Wi i s the i th square submatrix of order (N-l) of the measurement to node incident matrix • def ined by (11), whose determinant value wi is non zero , i . e . : W
The
proof
of
this theorem is given in subsection
3 . 3. 3.3
Measurement to Branch Incidence Matrix M. Definition . Properties
An element mik of the measurement to branch incidence matrix M is defined as fol lows (Krumpholz , Clements and Davis , 1980) : . 1 (or -1), if the ith measurement is either a located at, or a bus-injection I ine-flow measurement I~~ated on the positive (negative) node of the k branch ; • 0, otherwise . As an example, for the network shown in Fig . 1 . 0 , the matrix M is wri tten as fol lows: 1.3 1 . 2
i
= de t
( Wi)
-1
( 17)
The above formula impl ies the following : • Property 6 : For a chosen matri x A, the FRSTs generated by an MST have al I the same sign (or or i en tat ion) , i. e . e i the r + 1, 0 r - 1 . As an example, for the network shown in Fig . 1, if we take for instance,that matrix Ma whose rows correspond to the I ine-flow measurement m12 and to the bus-injection measurement m3' and that matrix A derived from Aa with row (or node) 1 removed , that is :
o
II
There are two types of rows in M: 1 . line-flow measurements : in this case therow of M contains only one element non zero , whose value i s e i the r + 1, 0 r - 1 ; 2. bus-injection measurements : in this case the row of M is ident ical to the corresponding row of AO , Concerning
matrix M we have the following :
The (mxL)-dimensional measurement to Theorem 2 : branch incidence matrix M is a totally unimodular matr ix .
-1
If M contains only type 2 rows , it is either Identical to the matrix Aa or equivalent to a submatr Ix of Aa, therefore I t is also totally unlmodular (see Property 3) .
~]
At=[~-1 :~] 0
one can easily verify that W = det (Ma At) P = IW I = 2 (see a I so Tab Iel , row 4) .
-2 and
The matrix Mai defined in (15) and sat isfy i ng the cond i t i an ( 16) is ca 11 ed a I so a non-deaenerated ~. There are as many matrices of this type as the total number ~ of MSTs. I t may exist subsets of (N - 1) measurements which are not I inearly independent . I f we note with M~k a matrix whose rows correspond to (N-1) linearly dependent measurements we wi I I have :
M~k At
(18)
de t (Wk)
~:
If matrix M contains only type 1 rows, the determinant of every square matrix of M of order 1 is +1, -1, or O. Adding one by one to a given row of M the remaining (m-l) rows, the resulting intermediate submatrices of M are totally unlmodular (see Property 4) .
( 16)
In (16) wi is a positive or neaative integer, its absolute value repr~senting the number Pi of FRSTs generated by the it MST, i . e . :
1
o o
(Ma i At) ~ 0
= de t
= de t
(M~k At)
=0
( 19)
and the matrix M~k wi I I be cal led a degenerated Let US note wi th 1" the number of all degenerated matrices . We have:
~.
I't
= I'
+ 1"
= C~-1
m! (N-l)! (m - N+1)!
(20)
b) the subma t r i x 0 f t vpe M,8, i s de fined as any m x (N-1) dimensional submatrix whose columns correspond to the branches of a TST. There are r matrices of this type, wi th r given
by
Power Network Observability: Guidelines for Measurement System Design (13) . II we note wi th P j the number 01 FRSTs having as topological support the jth TST . and with "f~ the submatrix whose columns correspond to that j TST. we have : ( 21) II C is a c x d matrix and D is a d x c matrix . with c S d. the Binet·Cauchy theorem states (Chen . 1971) that : det (CD) =
E(
product of the ) corresponding majors of C and 0 (22)
Let us consider a measurement system which has N bus· inject ion measurements. that is M = Ao. and let us consider two submatrices ~Pa . MPb whose (N·1) columns correspond to the at and . respectively. to the b th TST 01 the network. Both "Pa l "Pb are N x (N · 1)·dimensional matrices . each having N distinct square matrices of order (N·1) whose determinant value are either +1. or ·1 (see Property 5). By applying the Binet·Cauchy theorem (22) . we have: det (M~a MPa)
= det (M~b
M£lb) " N
("~a "Pb) " det ("~b "£la)
Fibonacci
.1 .1
.2
~1
.2
11
~1
•
~ =
. 2
3
=
PN " I"N
=0
wi th "0
4
N
•
N-1
4
Fig . 3 Loop free regular networks: I ine·llow measured at both ends.
branches
I"N_1' "N.2
(27)
= 1.
and 1"1
c. for the network shown in Fig . 4 (see also Table 2 . co I umn c):
~1
Single loop regular networks : one I ine·llow measurement on each branch .
4
(24)
Since lor a measurement system consisting of N bus· inject ion measurements M = Ao . the resul ts (23) and (24), represent the prool of theorem 1 (see subsection 3.2) . True enough the product matrix 01 order (N-1). (M~a MPa) or (M~a Mpb) represents any submatrix 01 order (N·1) of the matrix (AoA) whose (N-1) columns and rows correspond to the same TST or. respectively. to any two distinct TSTs . IV
as
b . lor the network shown in Fig. 3 (see also Table 2. co I umn b):
Fig . N
known
(23j
and also (see Property 6): det
with FO " 0 and F1 "1 being sequence (Hogatt. 1969).
485
PN
N • F2N
(28)
"N
N • F2N-1
(29)
For N large enough . it resu Its : PN (30) I"N " 'P where 'P 1. 61803 represents the posi tive root of the equation :
i .y
MEASUREMENT SYSTEM STRENGTH EVALUATION
4 . 1 Prel jmjnary Remarks The number, 01 TSTs given by lormula (13) is a measure of network connectivity. i.e. for a given network the larger the value of '. the stronger is the connectivity of the network . In trying to de I i ne a si mi I a r quan tit a t i ve cr iter ion lor a measurement system strength two candidates are I ikely to be considered. that is P the total number of FRSTs . and I" the total number of MSTs . Lat U8 dar Iv. tha numar leal values of PN .nd I"N for aome regular, Simpler networks (see also Sact Ion V): a. lor the network shown in Fie . 2 (see also Table 2. co Iumn a):
- 1 " 0
(31 )
and known from ant iquity as the "Golden the "Divine Proport ion " (Hogatt. 1969) .
Ratio " or
d , for the network shown in Fig , 5 (see also Table 2, co I umn d ): 5 ~1
O:3A,:O:'30 . :.'.
Fig , 5 Single loop regular networks: branches I ine-flow measured at both ends . TABLE 2 Number 01 FRSTs and MSTs lor the Regylar Networks shown In fig. 2 to Fig. 5. (a)
(b)
PN:I'N
PN:I'N
(d)
(c)
N
.....1 _..... 1fIo2_ _ ~
•••
Elg . 2 Loop frae regular networks: measurement on each branch.
-.--. N-1
one
N
line-flow (25)
where F2N represents the (2N)th Fibonacci number . the integer sequence having the recurrence lormula: EN
= EN_1
• FN·2
(26)
1
1
PH 1
PN
I'N 1
I'N
1
1
1
2
3
4
6
4
8
8
3
8
15
24
15
45
33
4
21
56
84
52
224
184
5
5~
209
275
170
1 045
785
6
144
780
884
534
4 eao
3 425
7
377
2 911
2 839
1 e31
20 377
14 917
8
967
10 864
7 898
4 880
9 2 584
40 545 23 258 14 373
se 912
e3 824
90~
2e7 129
384
10 8 785 151 31e 87 850 41 810 1 513 leo 1 107 710
486
E. D. Crainic . X. D. Do and D . F. Mukhedkar PN-2
4 PH-1
(--,;r-r-
N
PH
o and
wi th PO
I'N = N
P1
= 1. and
I'N-1
4
I'N-2
(33)
(--,;r-r- - R72)
defined for N > 2 and wi Ih 1'1 N large enough, i I resul Is:
PNil'N
=" = 1.36602
where v the equat ion
y2 _ Y _ 0.5 4.2 Measurement
Strength .
measurement system strength after a second worst contingency has occured. If we note with I1n_k the coeff icient of the measurement system relat ive strength after a k-elements cont ingency, the following inequal ities hold true:
(32)
R72)
1
o
For
6.
and 1'2
(34)
represenls Ihe posi live rool
=0
SYstem Oefini tion
01
(35) Absolute
and
The concept of FRST is very useful to answer the question of a power network as being. or not. algebraically observable. However . larger values for p merely reflect a greater number of bus-injection existing in a measurement system than describing ils strength. As an example. let us conSider two di Iferent measurement systems M, and M2 for the network shown in Fig . 1 . 0 . The measurement system M, consists of three I ine-flow measured branches {m'2' m23. m,3} . I'(M,) = 3. The measurement which gives p(M,) system M2 consists of three bus-injection measurements {m1. m2 ' m3}' therefore I'(M2) = 3 and p(M2) = 9 . We have I'(M,) I'(M2). but p(M2) > p(M,) . Let us add a bus-injection measurement to M,. the new measurement system becoming M'{m,2 ' m23, m,3 . m,} . for which I'(Ml> = 6 and p(Ml> = 7 . We slill have p(M2) = 9 > p(M\) = 7. but the measurement system M\ is, objeclively . stronger than M2' True enough if any two measurements are removed from M\ the network remains observable . whereas by removing from M2 any Iwo measurements the network is no longer observable. Therefore. it is more appropriate to relale the strength of a measurement system to the concepl of MST . We define the total number of MSTs associated to a power network ~a~s~t~h~e-2a~b~s~o~l~u~te~s~t~re~n~gLtuh~oLf~~i~t~s measurement system. In assessing the Isecur i Iy · .. of a measurement system. very useful are the cQefficients ~n-k 21 i Is relat ive strength def ined as follows (when k I) :
(36)
1'0
1'0 - I'n-2 I1n-2
with
=
~n.2
(37)
1'0
representing
system
the
new
value
of
the
< ~n-k
.. .
(38)
In designing a measurement system, or when enforcing an existing one. the meter placement must be done to obtain the smal lest values for its coefficients of relat ive strength . The smaller the value of a coefficient "the more secure is the measurement system for the given con I i ngency. of
Ihe Measurement System Absolute
St rength In order to obtain Ihe lolal number P of FRSTs the general formula may be used, i.e.: P=
L
I'
det (MbjMpj)
L
j=1
Ide t
( Mc> i At)
I
139)
i=1
To evaluate 1'0. the measurement system absolute strength, two approaches are possible: one based on the non-degenerated (N-I)xL-dimensional submalrix Mc> , and the second based on the mx(N-l)-dimensional submatrix Mp , both derived from ma t r i x M (see Sec t i on I I I ) . A. Enumerat ion submatrices
of
non-degenerated
*
A submatrix Mc> has the property det (M"A t ) 0, and the total number of MSTs equals the number of al I submatrices Mc> of M having this property . As may be seen from (20) the matrix W has Pt distinct submatrices of order (N-I), therefore the procedure has, at least teoretica"y. Pt steps each step evaluating the determinant of a very sparse matrix whose elements are integers . B. Use of
the Hp type submatrix
The jth m x (N-I)-dimensional submatrix Mpi has the property (see also Fig. 1 and Table I): P j = P j = de t (Hb jMp j )
(40)
that is the num~~r of FRSTs having as topological suppor t the j TST equal s the number of MSTs generating them. Let us consider th~ following example : hi f i, k represent the it , respectively, the kt TST, the resulting number of MSTs is: I'j+k
where: the absolute value of the measurement system 1'0. strength. when al I its elements are in normal operation state (no disconnected I ine. no telemetry system malfunction. etc .. ); I'n-l. the new value of the measurement system strength. corresponding to the worst case of the following possible contingencies : a) losing a measurement, b) loosing an RTU, c) losing a branch I ine-flow measured or having at least one bus-injection measurement as its adjacent nodes. The coefficient ~n-2 of the measurement relative strength is defined as fol lows:
~n-I < ~n-2 <
4 . 3 Evaluation
Re I at i ve
1'0 - I'n-I
$
I'i
+
I'k - I'jk = det (HbiMpj +
+ det (MbkMpk) - det
(H~jMpk)
(41)
Note that Pl k = Pkj ' and if I'jk = I'k (or I'jk = 1'1') it results rom (41) that I'i+k = Pi (or I'++k = I'k ' or if Pjk equals zero I'j+k = 1'+' + Pk ' . her~f~re, only a limited number of TS s, that IS a limited number of Mp type submatrices, should be investigated to obtain the value of ~. As compared to the previous approach, thi s one is a I ittle more complex to COdify, but is recommended for radial and topological'y weak connected networks. If expressions on closed form exist to evaluate al I network T5Ts (see (16)) and al I network FR5Ts (see (401 it is not possible, at least for the moment, to provide general formulas for enumerating al I network MSTs . except for the special cases of
487
Power Network Observability: Guidelines for Measurement System Design
regular network section. V
as
we
wi I I
see
in
the
next
REGULAR AN:) QUAS I - REGULAR t£TWORKS .
IEASlJ9IENT SYSTEM
The term ·regular· as used in this section refers to the .. asurement system of a power network, Implying al I Its N nodes having bus-injection .. asurements, and/or al I branches having line-flow ..asurements located on them at one or both 01 their ends. ·Ouasi-regular" term means that one (or IIOre) _asurement(s) is(are) removed , as specified, from a regular network . Any two nodes are I inked by one branch only, and the one node network having no physical meaning, is considered lor mathematical reasons only as the starting point of an integer sequence . Most 01 the general formulas given in this section were establ ished by induction and, as some admit several forms, the simplest 01 them is presented . Networks wjth only Bus-Injection Measurements When all N network lleasurements PN
nodes
have
Ouasi-Regular Networks having N For the network shown in Fig. 7 , bus-injection and N line-flow .. asur ...nts (for
.1 Q1 A 4U14'f!t1-:1 JC
23
Fig . 7
N ~ 2), and an additional number of N(N-3)/2 branches (for N ~ 4) not I ine-flow measured, the strength of its measurement system is given by: N-1 PN = E (~~-I-i) = C~1 i=O
Number 01 MSTs :
(42)
•
Networks wjth only Ljne-Flow Measurements When al I branches have a I ine-Ilow measurement placed on them, doesn't matter at which end, there is a direct correspondence between an MST and a TST, i.e .:
(47)
of a Chebyshev given in Table 3
and when al I branches have I ine-Ilow measurements at both 01 their ends we have : N 1 2 - r
(44)
Ljne-Flow
For the networks shown in Fig. 8 having N bus-injection measurements and one, respectively two, line-flow measurement(s) placed on each branch, the lormulas expressing their corresponding measurement systems strength are (see Table 3, co I umn a , co Iumn b), N-1 PN = E (i+1) ~-1 = (N+1) 2N-2 i=O
(45)
N
•
N-1
'280
1 4 15 54 189 141 2187 7290 24 OS7
281.
78732
20
5
41 112 251 578
I
e
d
,
1 4
15 56
3 10 33 10S 324
210 7;2 3003
177
" 440 43 751 187 lao
2195 1412 24415
f 0 3 10 32
19 299
"7 2585 7501 21 528
9 1 2 I 19 59 179
S33 1 514
4538 13 031
0 1
3
•
28 74 209 587 '141 45118
I f the network shown in Fig. 4 • is taken, whose in it i a I measurement sys tem conta i ns N bus-injection, and N I ine-Ilow measurements, by successively modifying it, its strength wi I I also be modi lied, i .e. : • one I ine-flow measurement missing ( Table 3 and column d) : N-2 /AN =NF2N_' E (_1)i (N-1-i)F2(N-1-i) (48) i=O • one bus- inject i on measurement mi ss i ng (Tab le 3 .. column e) : N-2 /AN=.,F2N - 1+ 1:: ( _,)i (N-'-i)F2(N-1-i) i=O
(49)
• two I ine-flow measurements missing ( Table 3- t): (for N '>1)· [(N-3)/2]
= F2N-1
+
1::
i=1
(N-2-2i) F2(N-2-2i)
(50)
where [(N-3)/2] represents the integer part of the division . • one bus-Injection and its two adjacent II~e-llow ..nurements missing (Table 3. col~g):
2
Loop free network •. of a
c
b
4
10
PN 3
a 1 3
7
A Comparison .
IEASUeEHf SYSTD STllENGTli
1 2 3
• ••
(43 )
which gen.rat •• coefficient. polynomial (Sloane . 1973), and:
(2NI ! :N+1)! (N-1)!
which generates coeffiCients po I ynomi a I (SI oane, 1973) as col umn c"
a known result, where ~-1 represents combinations of (N-1) measurements Irom a set of N measurements . (see Theor811 1 and Section Ill) .
Fig. 8
•••
2
AI I nodes connected, quasi-regular networks .
TABLE 3
Networks wjth Bus-Injectjon and Measurements (see also Section IV)
2 3
bus-injection
= ~-1 = N
PN =
,.,
23
Chebyshev (46)
PN
= F2N-2
N-4
+ t
I
(-1)
(N-3-i) F2(N-3-i)
(51)
1=0
In formulas (48) to (51) if k number Is zero .
<
0 the kth Fibonaccl
488
E. D. Crainic, X. D. Do and D. F. Mukhedkar
VI Ge£RAL ClUIDELltES FOR hEASlJIDENT SYSTEM DESIGN
As may be seen from the previous section,the formulas expressing the strength of the measurement system corresponding to simple but regular networks (i . e. combined bus-injection and line-flow measurements) generate integer sequences which start increasing rapidly with the network nodes N.
1'0 = 6 Fig. 9
On the other hand, as soon as the measurement system regulari ty is lost, its strength diminishes dramatically (see Table 2 column c compared to columns d to g of Table 3) . The measurement system for real power network is far from being regular . For networks wi th the number of branches L = 1.4 N to 2 . 5 N, their measurement system may have , theoretically , 4 . B N to 7 N measurements (N bus-injection , N 5 voltage magnitudes, 2 . B N to 5 N line-flow measurements), as compared to a redundancy level of 1. 1 N to 3 N for real life networks .
The following guidel ines are intended to make a bet ter, if not opt imal, use for the number and type of the avai lable measurements for a new measurement design . Measurement System Absolute Strength
I'b = 104
I'c
= 111
Line-flow measurements placement: Equidistancy procedure .
Fig . 9 . 0 already having 6 bus-injection measurements . The best plscement on this network of 3 additional I ine-flow measurements is shown in Fig . 9.c , whose messurement system strength equals 111 .
00_ 0 6
6
240.-24
3
1'0 = 6
6
8
15
4
a I t is not unusual that for networks wi th relatively high redundancy level, even in normal operation states , to have one or more critical measurements which indicate a poorly designed measurement system . Therefore, for real-I i fe operational measurement systems , their strength may take very low numerical values compared with to regular ones .
I'a = 93
Hi
3
15
24
3
I'a = 74
I'b = 7B
1
2
3
I'c = Bl
Fig. 10 Bus-injection measurements placement: Equidistancy procedure . Let us also consider the network shown in Fig. 10 . 0 already having 6 I ine-flow measurements . The best placement on this network of 3 additional bus-injection measurements is shown in Fig . 10 . c, whose Neasurement system strength equals Bl. if an additional c) loop-closing procedure : I ine-flow measurement is avai lable to close a loop of line-flow measured branches, it is better to p I ace i I I n such a way Ihe resu I1 i ng loop to contain the largesl number of branches .
For a given network and for a given number and type of measurements to improve the initial value ~O of the resulting measurement system absolute strength , the following recommendations may be made: a) regularity procedure, stated as follows : if some addi t ional I ine-f low measurements are avai lable, it is better to place them on branches not yet I ine-flow measured, if any. 1
o.
(1)---
a, b,
c,
3 4 [email protected]'O 2
Fig . 11 4
~
~
.--~I'a
18
~
• a"-~I'b
17
~I'c
19
.---. • •
"I'd = 21 Fig. 8 Line-flow measurements placement: Regularity procedure . As an example, for the network shown in Fig. 8 . 0, bus-Injection measurements, al I which has 4 distinct hypothesis of placing 3 addi tional I ine-flow measurements were conSidered (a double I ine means a branch line-flow measures at both of i III ends) . The i r bes t p Iacemen t is the configuration shown In Fig . B. d for which the mealurement system strength is equal to 21 . d,
i)
"a=175
I)
b) eguj-djstancy procedure, stated 85 follows: if a limited number of line-flow (bus-injection) measurement are to be placed on an already regular network zone, in terms of bus-Injection (1Ine-f low) lIB8Surementl, It II better to place them at equal intervall . As an example, let us consider the network shown in
Pb::182
Line-flOW measurements placement : Loop-closing procedure.
An example of this procedure is given in Fig. 11 for which an add it i ona I I i ne- flow measurement may be placed on anyone ( 1 ,2) , ( 1 , 3) , or. (1: 4) The best placement is shown In Fig. branches . " . c, whose measurement system strength equals 190 . d) bus-iniectjon to maximum-degree node p~0gedu~e, consists in placing an additional bUS-InJection measurement at the node with the largest number of branches incident on it .
4« :,14, : '149 : "14« c~>t'
3~23~23~23~2 1'0 ,. 3
~a
:: 6
I'b :: 6
I'c :: 9
Fig . 12
Bus-injection measurements placement : Bus-injection to maximum-degree node procedure. An example of this procedure 15 given In Fig. 12 for which an additional bus-Injection measur ... nt may be placed at any node 2. 3 or 4 . The best placement Is shown in Fig. 12,c, whose measur ...nt system strength equals 9. Even IIOre iMportant than the absolute value of a measurement syatem strength is itl rei lablllty, caplblllty to preserve the network I .e . Its
Powe r Ne two rk Obse rvabilit y: Guid elines fo r Meas ure me nt S ~ s t e lll Desig n observabi I ity following measurement and/or topology contingencies .
network
Measurement System Relative Strength Let us define the following notions : proper measurements of a node : the set • consisting of its bus - injection measurement. and al I I ine-flow measurements located on ItS sIde ; the set • adjacent measurements of a node : conSisting of al I bus-injection measurements existing at its adjacent nodes . and all Ilne:flow measurements located on its adjacent nodes SIde ; • branch priority I ist: a I ist contai~ing all the branches in descending order as functlo~h of the number 'k of TSTs to which the k branch belongs . A branch with 'k : ' . where is the total number of the network TSTs is called a crjtical branch; • proper measurements of a branch : the set consisting of the line-flow measurement(s) located on it. and of bus-injection measurements. i f any. ex i s t i ng at its ad j acen t nodes . The rei iabi I i ty level of a measurement system may also be characterised by its coefficients of relative strength "n-l ' " n-2 . .. . . "n-k which must be strictly less than 1 (see Section IV). To assure that . (concerning only may be made :
the the
following recommendations coefficients "n-l and "n-2)
a) place the avai lable number 01 line-flow measurements in the order given by the pr ior i ty I ist of branches. as delined above ;
~o
:
1
~o
=3
~O
: 4
"n-l = 1 "n-l = 1 "n-l = 0 . 75 F i g . 13 Pr i 0 r i t Y lis tal branches I i ne - fI ow measurement placement . As an example (Fig . 13) the best placement 01 an additional I ine-flow measurement is on branch 1-2 . ~ince for it "n-l = 0 . 75 < 1. b) i l two additional line-flow measurements are avai lable to be placed on two or more parallel branches not yet I ine-flow measured . they must be on di Iferent branches and not both located at the same end; C) the proper measurements of a node should not al I be put on the same RTU (this requ i rement is the most difficult . if not impossible . to be met in practice) ; d) the proper and adjacent measurements of a node should not al I be distributed on any comb ination of two RTUs ; e) the proper measurements of a critical branch ahould not al I be distributed on the same RTU. or on any combination of two RTUs; f) the l ine-flow measurements forming a loop should not al I be distributed on any combination of two RTUs. VI I
CONCLUSION
In this paper it is shown that the me~surement system design for a power network may .entlrel y ~e done on the measurement to branch inclden~e matrIx M and on the node to branch inc idence matrIx A. The property of measurement to branch in~idence matrix M. and node to branch Incidence matrIx A of
489
both being totally unimodular matrices allows for the definition and evaluation o f the absolute strength and various coefficients of relative strength of a measurement system . The absolute strength of a measurement system is defined as the total number of subsets of n-dimensional I inearly independent measurements existing in a set of.$measurements (n N- l for P-6 observabi I i ty . and n=N for Q-V observabi I ity. with N representing the number of network nodes . Also important is the and ..1ft ~ n). measurement system secur i ty. i . e . its capabil i ty to preserve the network observabi I i ty following measurement and / or network topology cont ingencies . For this purpose . coefficients "n-l . "n -2 ' "n-k of relative strength are introduced . They g i ve an indicat i on of changes in the strength of a measurement system following s i ngle or mult iple measurement and / or network contingencies . The assessement of a measurement absolute and relative strength may constitute a valuable objective tool in comparing di Iferent measurement plans for a given network . The use of . the der i ved general guidel ines for meter placement to analyse the ex isting measurement sys t ems may also provide necessary information for their improvement . REFERENCES Albertos. P .. C. Alvarez and A. Reig ( 1980) . Quantitative analysis of an on-I ine measurement equ i pmen t 10 r s tat i c s tat e est imation purpose . IFAC Conference . Rabat . ( 1975) . Ariatti. F .. L. Marz i o and P . Ricc i in v iew of Designing state estimation Cambr idge . rei iabi I ity . Proc. 5th PSCC . Paper 2 . 3/8 . Chen . \fI . K. ( 197 1 ) . ~A!.Ilp!.llp~I..!.i.s:e:Qd_-llgLr.ll.aI20!lh_-l..th!..!:elZ'OlLr!..y X. North-Hol land . Amsterdam. Clements . K. A. . G.R. Krumpholz and P .W. Davis (1982). State est imator measurement system rei iabi I ity evaluation : an efficient algorithm based on topolog i cal observabi I i ty theory . IEEE Trans PAS . ~. 997-1004. Clements . K. A. . G.R. Krumpholz and P . W. Davis (1983). Power system state estimation with measurement placement algorithm . IEEE Trans ~ . ~ . 2012-2020. Fetzer . E.E. and P .M. Anderson (1975) . Observabi I ity in the state est imation of power systems . IEEE Trans . PAS . ~. 1981-1988 . In E. Handschin . E. and C. Bongers (1972) . Handsch i n (Ed . ) . Real Time Control 01 Electric Power Systems . Elsevier. Amsterdam . Hogatt. V.E . Jr . (1969). Fibonacci and Lucas ~. Houghton Miffl in . Boston . Kogl in. M. J . (1975) . Optimal measuring system for state estimation . Proc . 5th PSOC . Cambridge . Paper 2 . 3/12 . Krumpholz. G. R. . K.A. Clements and P .W. Davis (1980) . Powe r system observability : a practical algorithm using network topology . IEEE Irans . PAS. ~. 1534-1542 . Monticel I i. A. and F . F. \flu (1984). Network observability : identification of observable islands and measurements placement. IEEE PES Summer Meeting . Seattle . Paper 580-7 . Phua. K. and T. S . Di I Ion (1977) . Optimal choice of measurements for state estimation . PICA 77 . Ioronto . Schweppe . F.C .• J. \fIi Ides and D.P . Rom (1970) . ~ Power system stat i c state estimation . Trans. PAS . ~ . 120-135 . Sloane. N. J.A. (1973). A Handbggk of Inteqer Sequences . Academic Press. New York .