Fast decoupled state-estimator with linear models

Fast decoupled state-estimator with linear models H P van M e e t e r e n De,lft University of Technolo~w, The Netherlands

Ioadflow is applied for contingency analysis, tile results are sometimes insufficiently accurate, especially in tile case o f long lines or snlall R/X-values. ttowever, this disadvantage of the DC model is overcome because of tile redundancy in tile measurement vector and tile iterative nature of tile algorithm. Tests have shown that the algorithnr will converge in a few iterations to an acceptable estimate of both tile active and tile reactive power flow. From statistical bad data analysis, tile reliability of tile nlethod appeared to be sufficient ;is compared with the so-called optimal estimator. I Tile properties (ff tile decmlpled state-estimator will be presented. and tile differences with an optimal estin/at(~r will be discussed.

,.1 static state-estimator aleorithm is proposed which is based ~ut linear models ./i>r b o t h real and reaetire power .~lows in [lie s l'sgeltl. A I t approximate mathematical model based mt redlteed ittedsltrellletlt vectors is presented. Because the reduetiott o.f the measttrement J,eetor is state-ctcT)emlent, the estimator al~,orithm is iteratil'e but linear. For better ('om'er.gettce, altentbm is giren to the ('r)rreetion vector. ,1 Ithottgh this deeoupled eslimator is not unbiased and trot optimal from a statistical point o f view, the aecttrac.v attd abililv m detect and Mentifv measurement errors is stt/fieient as compared with the optimal nonlinear esthnator. (~;,mparalivc results ~ff'some eompttter studies arc' presented.

II. D e c o u p l i n g Ill a p o w e r [ransllliSSiOll syslell/, tile illeasllrelllellt v e c t o r zm, tile stale vector x alld the ll/easllrel/lent error vector e are related to each other by tile nonlinear vector equation

I. I n t r o d u c t i o n Online state estimation is used to provide a realiable assessmeat of tile operating stale of tile power system u n d e r control. This estimate is used to update tile real-time database of the system. In order Io judge tile merits of a stateestimation algoritllm, tile following should be considered: • •

z,. = fix) + e

In mder Io apply file general least-squares slate-estunation apprtmch, tile network equations z = f(x) are linearized according Io ;1 l sl-order Taylor series appro×inration, which

whether the estimator is unbiased and optimal from a stalistical point of view

~lVeS

COlllplltatiollal aspecls SIIch as convergence. COI]lpllter

Az - / : ( x )

time trod lnenrory requirenlenls •

When tile network equations are decoupled 2 by Ihe sinlplifications

ability to detect and identify measurement errors.

F1~ml an opmationul point of view the algorithm should converge, while the con/purer til]le is at its lowest, on tile ccmdilion th:.il lhe shilislical properties, tile bad data handli~,g and memory requiren/cnls meet certain specifications. 1-¢} allain ~1minimal computer time. an algorithm based on li;te~ll equations is pretcrred. These equations are derived by al>proxilnations and transformations from tile well known DC load flow model, which describes the real power flow to a high degree o] accuracy. Altempls to describe tile reactive power with a linear n/odel are unacceptable for Ioadflow studies because el'high mmlinearilies in reactive power t]{~wequations. When the rcsuhs ~t the real power flow eslinlator based on I)(' Ioadflow equaiions are used to transform tile reactive pOWCl IIle;isHrelilCl/[S, [hell a f l e r SOUle approxilllaliOllS

which are usually made t,> derive lhc I)C equations, a linear m~,_k'] is obtained which ,.z'ivesthe csthnales {~i the voltage nla~znittldes. I~sl imate>; ~fi Ihc '~oltage a t g i l a / e l l i s and magniI/IdL's e l the ~,,de v<~ll:igesarc used to approximate the real ~ w c r h~n,~cs ~,.hich a~, ,di/>lcd 1~ the real power injections. i'~CC[II!SC lhC tC~tC[I\L' ptl\",C! I'h)w tr:.lllSlorlllaliol] iS nonlinear. Ihc c,limLlli~l :dg~)rJlhnl should be iterative. When the D r

V(~I 1 No 3 October 1979

Ax

bm(x) - ~ •

,

I

aP

- 0 I

and

t:'~m(x):

~O

: 0

30

the storage requircnlenls are halved. The resulling decoupled estinlator algorithm will converge t~ the same sohHion as the optimal estimator, because lhe full AC Ioadl]ow equations are used to compute file residual vector A z - z m fix). Because tile J a c o b i a n / : ( x ) is slate dependent, il is necessary to repeat tile Irianglflar factorization of tile inverse of the state-covari:mce malrix when the systenl state has changed signilicantly. To reduce the computation lime. a decoupled stale-estimator algorillnu based on a linear state-independent model is proposed. If we confine mlrselves to the real power Ioadflow. it is possible, after some simplifications, to denote a linear relationship between tile real power flows in tile transmission elenlenls and tile injected power flows at tile nodes of die network. AI first sight, it seems to be :ldv;mlageous to choose the injected powers as slalc vector. I lowever, llle matrices in tile cstima!or equ:llion will he c~mqqetdy Iilled,

0142-0615/79/030187-06

S02.00 @ 1979 IPC Business Press

187

which should he avoided. When lhe vector o f the wfllage [ll-gUll]elltS 0 is lakell as lhe stute vector in tile P-illode] ~tlld the voltage magnitudes :is tile state vector in tile trans/'orllled ()-nlodel, the estimator equations will ]lave tile sallle structure :is in the optimal state-estimator.

III.

Mathematical

L'[C//1CIH ill t h e I I O t w < ! t k , L'(IIiH[ iOl;

p;

(L+)iL'~,ti]{ ",lii l i b

cqtJ',lth)ll

+6,

,

If we define lhc reduced rc',d power fimv ~,ecl.~ Ih ,-,.d eclu~lt[Ol] (0) yields

Ih,,-,,a- (},0

model

The transmission element between nodes i :rod j of tile network will he presented by a ~-section, :is given in Figure I.

The iniected real power al node i is £iveu M t ' .~m.i

When this representation is used, the power flow llon~ node i h) lit)de / iS given hy p ..

,2

(-)t, ij =

~i?l)l)t.i.i UiU/( ])t,i/c°sOij + ,. .... t.,/

~,n = (': '.'.~'~ ii I/i(i/(,~.~.ii cos Oii+ ht.iisin Oii) SiR

O/j)

(1 ) (2)

with

k'i.('[ "~ + ~

p . r,~/

t,~

~ i i h .k'i, a fcsJs|;.ll/Ce 1o earth HI/d QEi [he se{ o i [r'dusmissi~ut elements, which arc incidem to node i. ,"~fler 'a ledliccd ;c:i] power iRicciion Pini, i, red is defined, we can derive lI{mi ectualions ( 4 ) a n d (~)

Pi,,i.i. ,cd - Pi,,.i.i +.k'i.U[

<~<~ft.#! ,~t, tl

:J;; I: +

<~t,n

(IPl.i/ + t'p.i: )

which gives

ht)t ii = /st.ii + Dt.ii If Wre neglect the real t)ower losses and apl~roxhllalo Ihe voltage mugnitudes by their mmfinal values ( I p.u.), equalion ( 1 ) yields

p., = t,t/

p .. -. t,/t

/)t, ii Sill Oij

~ ½('hr. ii+bt./i J <~i
t~z, ijOij

(3)

Tile losses ',ire ;idiusied Io nt)de in.ieclions ;it hoih sides t)l the line according it) Figure 2.

I

~(ht, ii + ht.ii)(O i

(
'7

--

=

~')hDm/+

&)ht,,-i

' 1 ("/Oii(20iibt, ij + L't, ii)

w i l i l the ',lppr{}xin~.ation cosOii=

1

0/) + l'p,i/

(4)

wilh

:.7,.,/:

-',.,)

(s)

lJec:.ltlse eqtlatiolqs (4) ~lIld (5) apply it) each trzlllSlllission

~'~hl~t.ii+ &}ht,:i L"iO/,(½Oiibmi ,~':::v

After d e f i n i l i o n
1"i/.,'i=

~')Oi/(lOi/bt,il + :"t,,'i)

t:~c ii =

, 10 i/bt, ii ~'iOi/(2

(I I )

,~t, ii)

we c:.ln express bb analogy It) equations ( G l a n d (7)

i
iqt, red

gt "J +)bt. U

:

-t

I-

(1())

½0~ ~:Illd Sill 0ii - 0i/.

BV al/alog_v lil t.'quuli()ll ( l O ) We c:.iil express

derive

l
Pt.ii =

/TpO

C) t.i/

I,,,.,,

(ht,ii + l~t,ji)Oii

Call

0 i)

[ rr.illsft,rn/al ion o f Otlliatioll ( 2 ) glV0S

The leal power losses can he a p p r o x i n u l t e d by ]) ,3 ,~ . . . . + ( ' ,l,'~,',t,] ..... ' ' ,- ii + <~ , ~.,1~ .. ) .il =it,-i<~<.t,t/ l'i[:/(.~z,

[;lonl eCltlatiolls ( I ) and (3) we

(Oi

Pi,,i,i,,-~,d =

--

iqt

( 12 ) f,/

The reaciive power injection at node i is given by

T

Q)ilii, i --

/}i<,U/ + E

(-)t, ii

i c~i

which :.lfler lruIlsftlrll/u|ioI1 yields Figure 1 Flepresentation of a transmission e l e m e n t

Pt

°

lqi,,.i.i =

[qini,i /'

', i I

I

i ?.

, :j

(13)

Substitution o f equations (IO) and (1 l ) in equation (13) gives

P 4

~>,

hi,,/_,~ + ~- l q t d

z""J i

]~io[-'i + 2 (bi~t.iJ~'i 1; c~i

ht.ijUj)

l ) e f i n i l i o n o f ;i reduced q u a n t i t y [qini. L red tlccordillg ltt 4/iili, i. red - /qini, i + Di<,U i

~

t';t.i i

gives Figure2 the line

188

Line losses adjusted to injections at both sides of

iqini, r,, d = BqU

t i.4-1

Electrical Power & Energy Systems

IV.

Estimator algorithm

The general least-squares estimator, applied to a linear model zm = A x +c, results in an optimal estimate'~ of z given by the solution of

(A/'WA) x = A'l'Wzm Equations (7), (q), (12) and ( 1 4 ) m a y be combined to zp. ,-ed= Ap0 Zq. red =

(15)

Aq()

Whel/ r e d u c e d nleastlrelilenl v e c t o r s

Z.,p,

red a n d

Zmq' red

are

used, the state-estimator equations can be expressed as (A/(W,, r~.dA,) 0 = Apr ~/p, redZmp, red T

(Aq

Wq, redAq) ['

r-

Aq Wq, redZmq, retl

where Wp, r~d and Wq,~,,d are the variance matrices of the reduced measurement vectors. As the statistical properties {}t the measurement errors of the reduced measurements are unknown, they are approximated by the statistical properties o f the original measurements. From tests it appeared that this assumption has a minor influence on the results. The justification for the approximation can be derived from the orthogonality principle applied to the unreduced quantities. From equations (15) and (16) it appears that the model and the estimator equations are decoupled. However, the coupling between the real power model and the reactive power model is brought on by the reduction vectors fp and fq, which are nonlinear functions of the state vectors 0 and U. As Cp and Cq have the structure of the branch node i~tcidence matrix, and Bp and Bq of the node admittance matrix, the matrices A/~ WpAp and ATWqAq both have the same structure as the matrix FTWF which is used in the optimal estimator.

V. Bad data analysis Bad data processing can be based on either a suppression principle or a statistical analysis of the residuals and the cost functiol]. For online operation of a state estimator, a simple suppression of large residuals seems to be more advantageous than a statistical analysis, ttowever, as the generalized least-squares method can be regarded as a sl:atistical analysis of the mfommtion available, it is

6J

5J

4J

3J

2J

1J

Jp(O)

= (Z,np, red -

J q ( U ) = (Zmq ' red

(J6)

T

obviously best to use the statistical properties of the results to analyse the performance of both the optimal estimator and the DC estimator when subjected to faulty measurements, A description o f the bad data decision and identification analysis is given in references 3 and 4. Bad data detection is based on a chi-square test applied either to the usual costfunction .I(0, U) in the case of the optimal state-estimator or to the costfunction Yp(O)and Jq(U) in the case of the estimator described in this paper. These costfunctions which are based on the reduced measurements, are given by

ApO)PWp(Z,np,red

ApO)

AqU) Wq(Zmq ' red

Aq U)

The test to identify the faulty measurements is applied to both the residuals and the normalized residuals.

Vl.

Test results

Both the optimal estimator and tile decoupled linear estimator, as described in this paper, have been implemented on a 16-bit process computer. Where necessqry, a double precision format (34-bit fraction, 7-bit exponent) is used. The programs can work in two modes: online, using realtime measurements from an AC dynamic network analyser: or offline, using recorded measurements or computed measurements corrupted by simulated measurement errors from a noise generator. For reasons o f comparison, only results which refer to c o m p u t e d measurements are presented. The sample system consisted of a 13-node 150kV transmission network (Figure 3). The transmission lines are overhead, double-circuit lines with a capacity of 200 MVA/ circuit. The total loading of the system is 786 MW. The results presented are obtained by applying both algorithms from a fiat start. In the optimal method, the Jacobian was computed at each iteration step, while the process was stopped at e = 10 s or after four iteration steps. The decoupled linear algorithm was stopped after three iterations from a flat start. Measurement noise was added to the computed measurements by using a standard deviation of 0.02 p.u. for MW and MVAR measurements and 0.005 p.u. for voltage measurements. Tests were performed with different noise vectors as a result of different start values for the random generator. Only a test which shows relatively large differences between the results of the two methods is presented. To express the effect of the approximations, an extended network is used in which the distances between the stations are twice as long using the same type of transmission line. This is indicated as networks I and 11. Different measurement redundancies are used, given by voltage measurements at nodes 1 J , 6 J , 7J and 3K in addition to: (a) all MW and MVAR measurements at one side o f the line and injection measurements at 2J, 5J and 6K (redundancy = 1.6)

(b) As (a) with additional injection measurements at 3K, 6J, 4J, 7K and 1K (redundancy = 2.0) (c) As (a) with all injection measurements (redundancy = 2.4)

47 ~

F 30km 4

(d) As (a) with all line measurements at both sides of the lines (redundancy = 2.8) (e) As (d) with all injection measurements (redundancy =

Figure3 150kVtest network

Vol 1 No 3 October 1979

3.6).

189

Table 1 Magnitude of normalized fault vector vj. correction vector v c and angular displacement 0 of vfand ~, for both the optimal estimator (oe) and the approximated estimator (ae) applied to the original network (Network i ) and the extended network (Network ll) ! Network 1 r

1.6 2.0 "~ 4 218 3.6

oe

92.2 / 93.7 ] 9"~ "~ /881; / 92.0 !

1

Network II

ivj, i

0

Iv,, I

ae

oe

ae

oe

tie

9 3._~ "

0.82 0.75 0.69 0.56 0.56

0.83 0.74 0.70 0.57 0.56

0.60 0.70 0.76 0.81 0.85

0.61 0.72 0.76 0.81 0.85

93.9 93.6 88.7 02.4


i

Iv:.l

[ oe 0.92 0.94 0.92 0.89 0.92

i~

lie

OC

ac

t )L'

LIe

0.o4 0.96 0.95 0.92 03)3

0.82 0.76 0.67 0.56 0.56

0.82 0.75 0.68 0.50 0.57

0.61 0.71 0.76 0.82 0.85

(J.O ; (). 75 ().7 ~) 0.82 ().F,~,

Table 2 True ( 0 , measured (m) and estimated (e) injections, their differences and normalized residuals for both the optimal (oe) and approximated (ae) estimator

ZI

Znz

Active power injection 7] 0.817 0.816 3K 0.856 0.88 I 1] 0.946 0.951 6] 0.765 0.766 4K 0.400 0.387 6K 0.443 0.458 5K 0.42(-.) 0.412 1K 0.397 0.407 5J 0.000 0.000 4J 0.400 0.418 2J 0.439 0.424 3J 0.407 0.381 7K 0.431 0.432 Reactive 73 3K lJ 6J 4K 6K 5K 1K 5J 4J 2J 3.I 7K

power injection 0.779 0.771 0.943 0.906 0.158 0.187 0.102 0.091 0.212 0.200 0.235 0.281 0.217 0.221 0.221 0.263 0.000 0.000 0.208 0.227 0.210 0.207 0.226 0.241 0.203 O. 176

Zn.l

Zt

Ze

Zm

I~

zt I zzl

and

Normalized residuals

Zl

ae

oe

tie

t)c

0.001 0.024 0.005 0.001 0.013 0.015 0.017 0.010

0.830 0.875 0.934 0.762 0.411 0.456 0.422 0.413

0.831 0.876 0.934 0.762 0.411 0.457 0.422 0.415

0.014 0.006 0.017 0.004 0.025 0.002 0.010 0.006

0.015 0.005 0.017 0.004 0.024 0.001 0.010 0.008

0.013 0.018 0.012 0.003 0.011 0.013 0.008 0.017

0.000

0.000

0.000

0.000

0.000

0.000

0.000

0.018 0.014 0.026 0.002

0.410 0.416 0.390 0.434

0.419 0.416 0.390 0.433

0.001 0.008 0.009 0.002

0.001 0.008 0.009 0.001

0.019 0.022 0.017 0.003

0.019 0.022 0.017 0.003

0.008 0.037 0.029 0.010 0.012 0.046 0.004 0.042 0.000 0.019 0.003 0.015 0.027

0.750 0.931 0.170 0.112 0.195 0.243 0.213 0.251 0.000 0.222 0.205 0.227 0.157

0.750 0.935 0.170 0.112 0.106 0.245 0.214 0.250 0.000 0.222 0.206 0.228 0.t 56

0.020 0.025 0.017 0.021 0.0(/5 0.038 0.008 0.012 0.000 0.006 0.002 0.014 0.01 ¢)

0.020 0.029 0.017 0.021 0.004 0.037 0.007 0.012 0.000 0.005 0.001 0.014 0.020

0.028 0.012 0.012 0.01 I 0.016 0.008 0.004 0.030 0.000

0.028 0.008 0.012

0.014

0.015

0.004 0.000 0.046

0.003 0.002 0.048

Iz,,,

~i

iv, l = Iz,,,

zrI

w i l h z t equal to the true veclor, z m equal to ihe i/leastlrcment veclor, and ~ equal Io the estimated vech)l.

1o check For the optimality criterion, v.t.is tested lo,r orlhogonalily 1o vc. which is represented by the angular disphJce-

190

Zc

()e

To present lhe results in a condensed fl)rm, the length o1 the normalized fauh vector v£ = ~ z r and the normalized correction vector vc = z,,, k are computed, given b3, try. l= Iz,,

Ze

ae

o{2

HC

0.014 0.020 0.012 0.003 0.011 0.014 0.008 0.0t8

0.~) I 7 1.4q I ().S~)()

0.010

1.00q

0.015 0.009 0.003 0.029 0.000

0.275 2.672 0..470 ().8~G (1.(10() (1.3 I *) (),o3s i).~72

}.4~a ,

merit d#. 1he 0esuhs are prcsenlcd in condensed iorn~ h~ Table 1, and in extended form in Table 2 for measuremcul system (c) ( l e d u n d a n c y - 2.41. On[5 injection qutm[iticx :lie plesented m Table 2. The picltu-e of lhe line l]o~ qtl;il;lili,.'> is the same as that o1 lhe ilqecthm quantifies, arid lhercl,,Ec [}lO~,' :Ire 1/o| i q e S C l l t c d

]lele.

All qmmti[ic>, arc ~ivcn in p.u. v~du0s wil[i 1 p.!L = .;()(! \1\~ The hlenlifh::lIion procedure for bad dala is 51illted i!

,/(~):~'.\2(m

it).,6'

with

~--O.l

Eleetrical Power & Enerqy Sysl,m>

Table 3 s 2, s~ and s~ for both tim optimal estimator (oe) and the approximated estimator (ae) applied to the original network (Network 1) and the extended network (Network ll) Network ! r

1.6 2.0 2.4 2.8 3.6

Network i l

s2

sp 2

sq 2

s2

S2

ae sp 2

sq 2

1.480 1.340 1.185 1.220 1.094

1.639 0.901 0.739 1.070 0.891

1.338 1.620* 1.550" 1.353 1.278

1.347 1.212 1.168 1.227 1.075

1.485 1.275 1.279 1.262 1.077

1.857* 0.936 0.913 1.128 0.886

1.238 1.541 * 1.587" 1.367 1.251

(/o

ae

oe

1.400 1.260

1.160 1.220 1.074

* Bad data i d e n t i f i e d

Table 4 Largest normalized residuals (lrl), second largest normalized residuals (lr2), quotient of Ir2 and/rl (q) for measurement errors of 0.08 p.u. in consecutively the active and reactive line flow measurements from node 1K to 4K, obtained with the optimal estimator (oe) and approximated estimator (ae) applied to both the original network (Network I) network (Network i) and the extended network (Network 11) Error i n P 1 K ae

4Kof0.08p.u. lr2 oe ae

oe

ae

oe

ae

4Kof0.08p.u. lr2 tie ae oe

lr 1 r

()e

Error i n O l K q

lr 1

q ae

Network I

1.6

3.52

2.0 2.4 2.8 3.6

4.53 5.26 5.83 6.83

2.38 3.17 3.77 4.33 5.04

1.35 1.44 2.32 1.53 1.70

0.95 1.09 1.78 1.08 1.26

0.38 0.32 0.44 0.26 0.25

0.40 0.34 0.47 0.25 0.25

3.19 4.00 4.59 5.14 5.91

2.59 3.28 3.67 4.07 4.87

2.29 2.35 2.36 1.57 2.02

1.90 1.95 1.99 1.30 1.70

0.72 0.59 0.51 0.31 0.34

0.73 0.59 0.54 0.32 0.35

Network 1I 1.6 3.51 2.0 4.53 2.4 5.28 2.8 5.82 3.6 6.83

2.36 3.18 3.75 4.31 5.02

1.37 1.46 2.35 1.55 1.70

1.02 1.15 1.84 1.11 1.26

0.39 0.32 0.44 0.26 0.25

0.43 0.36 0.49 0.26 0.25

3.28 4.13 4.72 5.23 6.06

2.63 3.32 3.72 4.06 4.88

2.20 2.44 2.18 1.59 1.81

1.75 2.00 i.94 1.25 1.68

0.67 0.59 0.46 0.30 0.30

0.67 0.60 0.52 0.31 0.34

Tests are performed on J(O, U) for the optimal estimator and 0,3 d(0", I~), Jp(O), Jq(U) as a function of the reduced measurements for the decoupled linear estimator.

2 corresponds result of bad data detection by the value Of Sq, to the largest measurement error. Although the detection may be considered to be incorrect, the identification process performs satisfactorily.

In Table 3

s2--a(O,O)lOn

==J(O)/(,,,,,

Sq

,)

½,, ½)

~ire given for both the original and the extended network using the same noise vector as in Table 1. From Tables 1 and 2 the conclusion can be drawn that the differences between the results of both the optimal and approximated estimator are not significant. When both estimators are applied to the extended network, the largest differences between the estimated values are 0.002 and 0.007 for the real and reactive power, respectively. From Table 3, it appears that separate detection with Sp2 and sq2 differs significantly from detection with s. Part of the discrepancy can be accounted for by the difference between the standard deviation of the measurement errors of both the active and reactive measurements. As can be seen from Table 2, the largest normalized residuals, calculated as a

V o l 1 No 3 O c t o b e r 1979

Measurement errors in line f l o w measuren~ents larger than six times the standard deviation are detected and identified by using the results of both estimators. Because of the decoupling, the detection applied to the resuhs of the approximated method is more distinct. To illustrate the performance of the bad data analysis, small measurement errors of 0.08 p.u. for both active and reactive power measurements are added to the true values. The results are presented in Table 4. Ill Table 4, it appears from tim value of the quotient q,equal to the second largest mmnalized residual and the largest residual, that the identification process performs better when the optimal estimator is used.

V I I . Conclusion From numerous tests, the algorithm appears to perforn~ satisfactorily as compared with tile optimal estimator. Tests have been performed on networks of different size and

191

X/R-values. For

networks with mixed X/R-values, as in those composed o f both cables and overhead lines, it appeared that the convergence of the reduction fq was slower and oscillatory (7 to 10 iterations from a fiat start). Without had data in the measurement vector, the differences between the results obtained with both estimators are not significant. Bad data analysis applied to the results o f tile approximated estimator performs well for a redundancy larger than 2.0. Care should be taken that the local redundancies for both the active and reactive power measurements are sufficient. When the estimator is applied to a specific system, it would be worthwhile to investigate if filrther approximations in the algoritlun are permitted.

192

V I I I . References 1 Aschmoneit, F 'Optimal Power System Static-State Estimator' Proc. 5th Power Syst. Comp. Conf. Canqbrldge 1975 2 Stott, B 'Decoupled Newton Load Flow' Trans. IEEE Vol PAS-91 (1972) pp 1955-1959 Dopazo, J F 'State Estimation for Power Systems: Detection and Identification of Gross Measurement errors' Proc. 8th IEEE PICA Conf. (1973) 4 Guttman, I and Wilks, S S Introductory Engineering Statistics Wiley (1965)

Electrical Power & Energy Systems