# LOAD-FLOW SOLUTIONS FOR ILL-CONDITIONED POWER SYSTEMS BY A NEWTON-LIKE METHOD by shaileshchvk

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3648                    IEEE Transactions on Power Apparatus and Systems, Vol. PAS-101, No. 10 October 1982

LOAD-FLOW SOLUTIONS FOR ILL-CONDITIONED POWER SYSTEMS BY A NEWTON-LIKE METHOD
S.C. Tripathy    G.     Durga Prasad                   O.P. Malik    G.S. Hope
Dept. of Electrical     Engineering                    Dept. of Electrical Engineering
Indian Institute of     Technology                     University of Calgary
New Delhi - 110016      India                          Calgary, Alta. T2N lN4, Canada

ABSTRACr                                                                        (v) choice of acceleration factors.
In this paper mathematician K.M. Brown's method is                    Networks which have the above features are des-
used to solve load-flow problems. The method is par-                  cribed by a system of ill-conditioned nonlinear alge-
ticularly effective for solving of ill-conditioned non-               braic equations.      In other words, a small change in
linear algebraic equations. It is a variation of New-                 parameter produces a large'change in the solution.
ton's method incorporating Gaussian elimination in such
a way that the most recent infonnation is always used                      In this paper a new algorithm using K.M. Brown's
at each step of the algorithm; similar to what is done                method [7,8] is presented. It is useful when formla-
in the Gauss-Seidel process. The iteration converges                  tion of the load-flow problem results in ill-condition-
locally and the convergence is quadratic in nature. A                 ed equation and the nodal admttance matrix. This is
general discussion of ill-conditioning of a system of                 a quadratically convergent Newton-like method based up-
algebraic equations is given, and 'it is also show by                 on Galussian elimination. In Brown's method, each equa-
the fixed-point formulation that the.-proposed method                 tion is expanded in the approximate Taylor's series;
falls in the general category of sucessive approxima-                 however, the most recent information available is im-
tion methods. Digital computer solutions by the pro-                  mediately used in the construction of the next fuction
posed method are given for cases for which the standard               argment      This procedure is similar to the procedure
load-flow methods failed to converge, namely 11-, 13-                 in the Gauss-Seidel process for the solution of nonlin-
and 43-bus ill-conditioned test systems. A comparison                 ear sets of equations. This contrasts sharply with
of this metlhod with the stanidard load-flow methods is               Newton's method in which all equations are treated sim-
also presented for the well-conditioned AEP 30-' and                  ultaneously     It has been proven that Brown's method
57-bus systemns.                                                       [7] is not equivalent to Newton's method. An optional
feature of Brown's method is that the partial deriva-
tives of the equations are replaced by their first dif-
ference quotient approximations similar to the discrete
INTRODUJCIION                                    Newton's [13] that It the proposed load-flow methodformu-
lation method.
is shown by the fixed-point falls
Load-flow calculations are performed in system                   into the general category of successive approximation
planning.  An early approach was the Gauss-Seidel iter-               tal comuter results of the and Newton's method. Digi-
methods such as Gauss-Seidel proposed Brown's load-fl
ative method [1] using the nodal-admittance-method and
this was further improved by using the nodal-impedance
algorithm are given for 11-, 13- and 43-bus ill-condi-
power 1,4] naey Gus-id,Nwo-
matrix method [2,9].- Newton's method [3] using a nodal               flw
cause of its quadratic convergence characteristics.                   Raphson and fast decoupled algorithms failed to con-
Although method [5] work Denmead method [2]
Fletcher-Powell the Brameller andfor ill-conditione'd
However, limitations in small-core- computer ap-
it has
plications where the weakly convergent Gauss-Seidel                   ~~~~~~~~~~~and shw
verge.
proble they         poor cn rg       A comaisoneo
methd isgenrall more suitable. Extensive meiry
method is generally resuitble Extnsie meory                           the proposed show poor convergence. A,co, arlson of
problems, they Brown's load-flow algorithm with the
raequiremt    inlare pow system       calculations
vatedthe xplotatio of parsty wih                 mor
ti-
orered              standard load-flow methods [1,3,4,5] test systems, for
11-, 13- and 43-bus ill-conditioned is presented -as
elimination and skilful progranuning in the Newt,on n~th              well as for1 AEP 14-, 30- and 57-bus well-conditioned
test
od f3]. Recently, advantage has been taken of loose
physical interaction between MW and MVAr flows by math-                       systems [9].
ematically decoupling the MW-e and MVAr-V-calculations                                                 LIST OF SYMBOLS
[4].
Despite the substantial progress there are still                         k = condition number
some difficulties with som'eof the above methods [5,6].                      [J] = Jacobian matrix
Features which cause instability and divergence in
(i) position .: the reference slack bus
' . of                                                           ~~~~~~~~~N
=total          number of ukown variables
(ii) existence of negative line reactance
(iii) certain types of radial systems                                        x° = initial assumption to the vector of knowns
(iv) high ratio of long-to-short line reactance for                         x* = solution vector
lines terminating on the same bus                                     f= ith fuction of system of equations f
gj = ith function of system of equations ,
F = iteration function of Brown's method
fi = af /ax. partial derivative of function fi
w.r.t. x.
82 WM 021-4 A paper recomended and approved by the3
IEEE Power System Engineering Committee of the IEEE                        gi         = ag./ax. partial       deriv5ative   of   fuction gi
Power Engineering Society for presentation at the IEEE                           3      w. r.t. x .
PES 1982 Winter Meeting, New York, Nlew York, January 31-                                          D
February 5, 1982. Man^uscript s-ubmitted January 21, 1981;                      T = transpose of matrix (superscript)
made available for printing Ootober 23, 1981.                                   n = iteration cout (superscript)
3649
P. = ith bus real power mismatch                      stable.   K.M. Brown's method is described in detail in
1. = ith bus reactive power mismatch
Q. = ith bus reactive power mismatch                 Appendix 1. The application of this method to a system
of nonlinear equations is also demonstrated in Appendix
Pi(cal) = ith bus calculated real power                      1. Brown's method can be formalized by writing terms
QQ(cal) = ith bus calculated reactive power                  of the iteration function F = (fl,         N) used, be-
ginning with a starting guess x°, to form successively:               f**.
P i(sch) = ith bus scheduled real power
Q i(sch) = ith bus scheduled reactive power                              x
xn+1
= P(xn)
=F(
n = 0, 1,p2 .(
2,
)         n      0                                 3

P. = ith bus net real power
1
The iteration fuction, F, in this method is given by
Qi = ith bus net reactive power                                                                        i-l
V .,V = voltage magnitude at 1 bus i and j
i
F.(x1,
i
*        x1) = Xi
XN) Xi
- ,                     /gN-i+l,x.
(gN-i+l~~~~~~~~~~~~~~~,x. x1
e., O.= voltage phase angles at bus i and j                                                                 J                 '
Y..= G. +jB. = (i,j)th element of bus admittance                (F             x                                        i      1, 2, .,N (4)
matrix
n
[H] = transverse upper triangular matrix formulated    where I = 0 whenever m > n                                                                (5)
in forward part of Brown's method                           j=m
E = specified tolerance                              and         gi is defined as follows:
= machine tolerance
A= eigenvalue.                                                g= f1(x1, x2, ...,                      xN)
ILL-CONDITIONED SYSTEMS AND EIGENVAL[E ANALYSIS
gi =fi(Xl, X2                  ..
XN-i+l LN-i+2 *                    LN)     (6)
A computer formulation of a problem- is defined to
be ill-conditioned if computed values are very -sensi-
tive to small changes in input value. A matrix may                   N         fN(Xl1      L2 L3            N**,
LN)
have some eigenvalues which are very sensitive to small
changes in its elements while others are comparatively       The linear terms L                         1, **L** LN, i
N-l1
are themselves functlons oi x. and are obtained recur-
.

insensitive.      It is convenient to have some number
called a 'Condition Number' which,defines the condition      sively by successive substitution in the system:
of a matrix with respect to the computing problem.
Ideally, it should give. some overall assessment of the
rate of change of the solution with respect to changes                L = x. -             y        n                                ) (L -      Xi)
in coefficients.       It may be recalled that the size of             1   1                                                             J        )
the condition number k([J]) of a Jacobian matrix [J] is
defined to be IIJII IIJ-11 . It'gives a good indica-                          -n           /g        i = N, N-1, ..., 2                                (7)
tion of the sensitivity of [JI- to small perturbation                                      1. N4+1,X
in [JM. It is also called the spectral condition nun-
ber because of its dependence on the spectral norm           with' g = f1 and L1 = xl, so that gN is just a function
u 1*.
1                                                       of the single variable xl. Now expanding, linearizing
Thle condition number of a symmetric positive defi-   and solving for x we obtain
nite matrix [JTJ], whose eigenvalues are all real and                            x     n    n                    (8)
positive can be computed by,                                                     xl x         gN,x                  )
Ix I
k([JTJ])       max                 (1)   The   point xl thus obtained is used                            as   the   next    approxi-
min                       mation   xkhl li6nce, is   to the first
vec'tor x* H ve 1 i renam'ed as-.L, a the t sy.
and
component
ys
xl of the     solution
The condition nunber of [J] is the square root of -              o
k([JTJ]). For a well-conditioned system, the value of        tem of eqn (7) is back-solved to get improved approxi-
k is 1. A very high value of k indicates that the sys-       mations to the other compoents of x*. Hre            is
tem is ill-conditioned.        If k exceeds 0a, where a      taken as the value obtained. for L. when back-s6lving
equals decimal precision of the digital computer used,       eqn., (7). The "successive substitu%ion!' nature of the
I
it is not possible to obtain a solution [11]. Solutions      algorithm allows the mst recent information available
to eqn. (1) are. given in Table I for the AEP 30-bus         to. be used in the construction of next function argu-
well-conditioned and 11-,          , and 43-bus ill-condi-   ment, similar to the Gauss-Seidel process for linear
tioned test systems.                                         and nonlinear systems of equations.
K.M. BROWN'S METHOD                      M
M-tri             pe    a
-. -- -   -,               .~         M              reetatim                                                               .
Consider the following real continuously differen-
tiable system of N nonlinear equations in N unknowns,                For the sake of -definiteness, it may be stated
x. (i = 1, 2, ..., N). In vector notation                     that the variables are eliminated in the.order xN,
f. (x) = 0 ; i-= 1, 2, .., N              (2)    xN,         T,Using thechain rule for differentia-
x2.
,-1 -              * *                  ~~~~~~tion o expand each derivative g. , gives the follow-
t
K.M. Brown proposed a. local method [7,8] which         ing matrix representation for tlieXforward part of the
handles the functions of eqn. (2) one at a time so that       method:
information obtained from working with f1 can be. incor-                                   -n+    n
porated. when working with f2, etc. A successive sub-                               [H] (x     - x ) = - [g]         (9)
stitution scheme is used rather thanl the. simTultaneous
treatment of the f. which is the characteristic of New-      where the muatrix [H] = (h..) is given by
ton's method.      It is a quadratically convergent local
technique (in the vicinity of a root). It is fast and
3650

TABLE I
Maximum and Minimum Eigenvalues and Condition Number

Type of                 No. of             Maximum            Minimum                  Condition                        Re ]rs
System                  Buses             Eigenvalue         Eigenvalue               Number (k) *                        emar

syteml-conditione
WVell-conditioned |
system
30           .1087xl13           .2322x10 0                     X13
.4681x10                 Moeael
deratelywll-
el
~~~~~~~~~~~~~~~conditio'ned (fair   k)
11           .1222xlO3           .1126x100                .1086xl04                Ill-conditioned
2                     14
Ill-conditioned                     13           .2905x10            .1442xlO-1               .2014x10                 Ill-conditioned
43           .2426xl04           .9476xl0_1               . 2560x105               Ill-conditioned
* Ideal value of condition number: k = 1
f
flj               f                    f
~~~~~~j=1,..,N                        lInitialize          all the        variables-
lj     1,N-i+2 1N
f 2j     ff 2N
2,N-i+2                                                                    -Statieainc-_t>
S
h.. =      i+l   Ifij       f
'
f
|   i = 2,     i,      N             |        Create an array which permits
(-l)                                                                                ~~~~~~~~~~~~~~a
partial pivoting                e f'ect    witi
1,N-i                                J      1           N                      out   having to physically in-
f1N-+2
1-l1,N-i+2           fi i-1,N                                              Lerchange rows
D                     and      columns
, . ..(10)                                   |        Jwhere K - runction numberl
where the argunents f.j are progressive arguments gen-                                                                                            + ve
erated successively anA [g] is a vector-valued function                                                             K-
given by eqn. (10). It is observed that hi; = 0 for
j > N-i+l, that is, the matrix [H] is transverse upper                                                                   =                 KMIN . K-I
triangular. The matrix [H] can be obtained by trans-
forming the Jacobian matrix [J] into transverse upper                                                                        Solue the first KMIN rows of
triangular form using Gaussian elimination with partial                                                                      a triangularized linear syst-
pivoting.                                                                                                                    em for improved values ofe x
in terms of' previous one*s
I
When the condition number of the matrix [H] << the
condition number of the Jacobian [J] convergence oc-
curs. Thus Brown's method gives convergence in cases
where Newton's method fails.                                                                   vet     up approximate partial derviati-
es of the kth function and find the
Computational Efficiency of Brown's Method                                                     largest absolute alue or thp a
A count of the number of function values of the fi i
needed per iteration of Brown's method, is given in Ap-
e
l~~argest
absolut
apro.artial der
YS
YES
pendix 1. The first step requires N+l evaluations of                                                                 deO
vprox-partial
fl, the second step N evaluations of f2, the third N-l                                                                                       Try a different i
in
evaluations of f3, etc., so that the total is                                                                            NO                  tial .approximation
N+1        1      2
12~ N+3)(1                                                Set up coefricients for kth ro
i2   =                                                             or' triangular linear system us-
i=2                                                                  ed to back solve for the f'irst
Newton's. method requires N partial derivative
evaluations and N function component evaluations per                                               B   xck
substitute to obtain
iterative step. Thus there is a corresponding savings
in storage locations required: from (N2+N) for Newton's
method to   -I(N
+3N) locations for Brown's method. How-                                                                                            YES__j
ever, it must be stressed that here the savings in
function values applies only to saving function values
of the fi of the original system (2), because Brown's
method adds a number of other fuctions to be evaluated                                        Fig. 1.          Simnplified flow-chart for Brow's method
namely the linear functions, L}{. The evaluation of the
Lkr o included in the cout above, 1(N2+N                                            show that Brow's method also falls into the general
L ar no                                 '-N+3)                                      category of successive approximation methods like the
The equnceof omptatonsis how ina smpl                                    Gass-Seidel method and Newton-Raphson method, differ-
flow-chart for Brown's method in Fig. 1                                              ing only slightly in their iteration fuction.
-
FIXED-POINT FORMUATION
~~~~~~~solution of a vector set of equations f. (x) = 0 The
sought by means of a suitable recursive Tormla [14]
iS
put into the form:
The fixed-point formulation of Ref. 13 is used to
3651

xn+l       =       p(xn)                         (12)        same. The stopping criterion is when all the
function values are less than the prespecified
where the mapping 1p(xn) is given by                                                         tolerance.
xn
n(x ) = -                     f(xn)
[Rn] -                        (13)        Calculate all bus powers and line flows, and
5. print.
By a successive approximation procedure, solution to
the fixed-point formulation is given by
xn+l = xn [R'] h(xn, xn+l)
-
n = 0 1 2,... (14)                                        Nodes are numbered so that at each step of the
-      -         - '           '        ' ' '                                           Gaussian elimination the next node to be eliminated is
where [Rn] is a square matrix and h(.,.) is a continu-                                  the one having the least number of nonzero elements.
ous vector-valued fumction.                                                             This is generally preferred for Newton-Raphson method
[3,16]. The same strategy is used,for Brown's method
The Gauss-Seidel method uses a simple diagonal ma-                                 also. Henice, the above procedure can be easily incor-
trix [R] having scalar acceleration factors for the                                     porated into the existing Newton-Raphson load-flow pro-
real and imaginary parts of bus voltages. For Newton's                                  grams.
method matrix [R] corresponds to the inverse of the
Jacobian matrix, whereas in Brown's method it corres-                                   c) Results and CIarison: Line diagrams of 11-, 13-,
ponds to the inverse of Brown's Jacobian matrix [H]                                     and 3-us ill-conditioned test systems are shown in
given in eqn. (9).                                                                      Figs. 2, 3 and 4 respectively. The in
ill-conditioned systems are given
system data of these
LOAD-FLOW BY BROWN'S METHOD                                       and generation data are shown only for 13-bus system,
whereas, for 11-, and 43-bus systems only injection in-
a) Problem formulation: The load-flow problem involves                                  formation is available. The system in Fig. 3 is diffi-
the solution of a system of nonlinear algebraic equa-                                   cult to solve [6] because of the two series capacitors
tions, that is                                                                          and the position of the slack-generator. The systems
in Figs. 2 and 4 are also difficult to solve [4,12] be-
f. (x) = 0 * i = 1, 2, ..., N          (2)                                   cause of low X/R ratios and some negative line reac-
tances. This is also clear from Table I which gives the
For the solution to be unique, onee equation must be                                    eigenvalue analysis and condition number of the Jacob-
Forcifiedsolution unknwni                      oloig
specified for each unknown variable, h following                   eqaThe               ian matrix I [J]show the load-flow equations. The results
in Table
of
that the condition number of the ill-
equations define more clearly the load-flow problem on                                     i oned          s t a         conditi much highe in
the basis of the bus admittance matrix. Corresponding                                   conditioned to the well-conditioned systems.
systems Jacobian matrix is much higher in
s

to fi in eqn. (2), the load-flow mismatch equations                                     comparison
equal to the number of unknown voltage phase-angles and                                                       Slack-node
magnitudes are given by                                                                                         1         2
Real Power
Mismatch:             APi. =P. S
I      i(Sch)
i(Cal)=0
icl
(15)                             (
Reactive Power                                                                                                              7
Mismatch:    AQi =             Qi(Sch)              Qi(Cal)                      (16)
The equations for the calculated real and reactive bus
powers at bus i are given by
n                                                                                                                 (
P. =     I V.V.(G.. cos(6. - 0.)
1
+ B..   sin(e. O.))
-
1j=           1 3   13           1                       3       1    J                                               1
* * (17)                                        11
n
Qi   =       l V.V.(G.. sin(0.                -   0.) - Bi; cos(0. - 0.))                  Fig. 2. 11 Bus ill-conditioned power system
P a 500                   Slack                Pa O
(18)
0a ?                     generotor             0.?
The total number of nonlinear                             algebraic load-flow
equations (15)      and (16) that                              are to be solved by               Paso                                            650
Brown's method is equal to: (n - 1)                            real power mismatch
equations and (n - 1 - number of                                PV buses) reactive                                    Pao
power mismatch equations.
b) Solution algorithm: Equations (15) and (16) are to
be solved alternately as in [3], for each bus by K.M.
Brown's method as follows:
and form the Y-bus matrix elements according to                                            Piaso _2
the optimal ordering described below.                                                        .2
2. Formulate the system of equations for real and re-                                                               P'OPa 500
active bus power mismatch equations (16 and 17) by                                                                sa?
using optimal ordering.
3. Initialize the unknown vector xn with alternate                                             Q a30'
voltage phase-angles and voltage miagnitudes, i.e.,
voltage magnitudes to 1.0 p.u. and angles to 0°.                                       Fig. 3.    13 Bus ill-conditioned power system
4. Solve the bus power mismatch equations for the new
estimates xl+± by Brown's method and update the
3652

Slack-.node                                                                                     A comparison of the proposed Brown's load-flow
method with the standard methods [1,3,4,5] in terms of
convergence characteristics is given in Table II. The
proposed Brown's method converges in fewer iterations
for both ill- and well-conditioned systems.
A comparison for the well-conditioned systems be-
tween Brown's load-flow method and the standard New-
ton's method in terms of ratio of computing time per
12         23               16D1
(!) (9                 (           iteration is given in Table III. Since the Newton-
Raphson method does not work for any of the 11-, 13-,
and 43-bus ill-conditioned test systems, comparison re-
sults are not given for this method. Digital computer
22                      28              37                     31       @results were obtained by using non-optimized Fortran
compilation on ICL 2960 computer with George 2+ operat-
ing systems. Even though Brown's method takes 15% more
21)   30              38                     g
33) (       time per iteration than Newton's method, for large sys-
tems the total number of iterations is less. This
makes Brown's load-flow method comparable to Newton's
36       - \9method in total computing time.
-The storage requirement of Brown's method is
slightly more (not exceeding 10%) than that of Newton's
method. However, this disadvantage is offset by the
superiority of Brown's method for ill-conditioned sys-
tems.

CONCLUSIONS
The main objective of this paper is to show that
the K.M. Brown technique is suitable for solution of
3                        13              18                                 ill-conditioned power systems equations. The technique
has quadratic convergence characteristics. The pro-
Fig. 4.            43 Bus ill-conditioned power system                                   cedure adopted in this method is relatively simple and
can be easily incorporated in the existing Newton-Raph-
The proposed Brown's load-flow method has been ap-                                           son algorithm. The optimal ordering strategy and
plied to the ill-conditioned test systems and AEP 14-,                                            sparsity programming technique with compact storage
30-, and 57-bus well-conditioned test systems. The re-                                            scheme can be easily implemented. Also transformers
sults are shown in Fig. 5 for the convergence rate of                                             with no-load tap changers and phase shifters can be
the largest absolute mismatch for ill-conditioned as                                              taken into account while developing production type
well as for well-conditioned test systems. It can be                                              load-flow programs. It is not necessary to include
observed that the convergence rate of the absolute max-                                           transformer resistance in the algorithm to help the
imum mismatch is quite rapid. The proposed method con-                                            load-flow to converge to a solution. Brown's load-flow
verges close to the solution on the first iteration and                                           algorithm converges to the solution of ill-conditioned
requires few iterations to obtain the final solution.                                             system in a few iterations, whereas the standard load-
flow methods either show poor convergence or diverge.
11 bus system                                                                 ACKNOWLEDGEMNTS
13 bus system   I ll - cond; t ioned
-3 bus system                                                The authors would like to acknowledge Dr. S.
1-                bs-                                                 Iwamoto of the Department of Electrical Engineering,
\1I. bus system                                          Tokai University, Japan for providing data for 11- and
30 bus syslem Well-conditioned                     43-bus ill-conditioned systems.
- 101
-      \        *-- 57 bus system
t                                                                                                                 REFERENCES
X        X0
o°                                                                           [1] A.F. Glimn, and G.W. Stagg, "Automatic calculation
of load-flows", AIEE Trans., 1957, PAS-76, pp.
817-825.
[2] A. Brameller, and J.K. Demnead, "Some improved
methods for digital network analysis", Proc. IEE,
l0.2                                                                                1962, 109, A, pp. 109-116.
w

iJ3L
A.
'<>--                           \          \     \[3] W.F. Tinney, and C.E. Hart, "Power flow solution
by   Newton's      method"v,
I        EEE Trans., 1967,   PAS-86,
: \\ X                                           ~~~~~~pp.
1449-1460.
10.4l              2           3      4
5                            [4] B. Stott, and 0. Alsac,          "'Fast
ITERATIONS                                                         IEEE Trans., 1974, PAS-93, pp. 859-869.
[5] A.M. Sasson, "tNonlinear programming solutions for
Fig. 5.         Gonvergence rate of Brown' s load-flow method                                        load-flow, minimum loss and economic dispatch
for well-conditioned and ill-conditioned                                             problems", IEEE Trans., 1969, PAS-88, pp. 399-409.
systems
f.(x)=0*i1,. N(9)
3653

TABLE II
Convergence Characteristics of Different Methods
(Number of iterations)

System          No. of              Gauss-         Newton-             Fast                Fletcher             Brown's
Type           Buses               Seidel         Raphson           Decoupled             (Steps)              Method

Ill-                   11         divergent         divergent          divergent            divergent                     5
conditioned            13         divergent         divergent          divergent                  47                      4
systems                43         divergent         divergent             22                divergent                     5
Wlell-                 14               24              3                  4                    32                        2
conditioned            30               33              3                  4                    68                        3
systems                57               59              4                  41                  135                        3

TABLE III                                                  tems of equations",                     (Academic Press, New York,
1966).
Comparison of Computing Time per Iteration
[15] R. Fletcher, and M.J.D. Powell, "A rapidly conver-
gent descent method for minimization", Computer
One iteration time in terms                         Journal, Vol. 6, pp. 163-168, 1963.
System
System   No. of
obuss                       ~~~~~of
ratios
(Newton's method as 1 unit)                    [16]   W.F. Tinney and J.W. Walker, "Direct solutions of
Type        buses                                                               sparse network equations by optimally ordered tri-
Newton's method            Brown's method                angular factorization", Proc. IEE, Vol. 55, pp.
1801-1809, Nov. 1967.
Well-           14             1.0                         1.17
conditioned     30             1.0                         1.12            [17] S.     Iwamoto, Personal communication with the au-
systems         57             1.0                         1.16
APPENDIX I
K.M. Brown's Method
[6] K. Zollenkopf, "Load-flow calculation using loss
minimization techniques", Proc. IEE, 1968, 115(1),                         A step by step synthesis of Brown's method [7] of
pp. 121-127.                                                          solving the system of nonlinear algebraic equations is
as follows:
[7] K.M. Brown, "A quadratically convergent Newton-
like method based upon Gaussian elimination", SIAM                            Let the system be in vector notation as
J. Numer. Anal., 1969, pp. 560-569.
[8] K.M. Brown, "Computer oriented algorithms for                                              f(x)         0      ;i -1            2,
solving systems of simultaneous nonlinear alge-                       Step 1
braic equations", in G.D. BRYNE and C.A. HALL,
(eds.), "Nunerical solutions of systems of nonlin-                         Let x' denote an approximation to the solution x*
ear algebraic equations", (Academic Press, 1973),                     of (9). Eipand the first function in the Taylor series
pp. 281-348.                                                          expansion about point _,. Alternatively f, can be ex-
panded in an approximate Taylor series with the actual
[9] L.L. Freris, and A.M. Sasson, "Investigation of                       partial derivatives replaced by first difference quo-
the load-flow problem", Proc. IEE, 1968, 115, (10)                    tient approximations. In the later case the. user does
pp. 1459-1470.                                                        not have to provide the partial derivative expressions
in his program. Retaining only first order terms gives
[10] G.W. Stagg, and A.E. El-Abiad, "Computer methods                     a first order approximation
in power system a;nalysis", (McGraw-Hill, 1968).
[11] J.H. Wlilkinson, "The algebraic eigenvalue prob-                 1
(x) e    f1 (x )     +   f     (xn)     (x   -   xn)   +       f 2(   n)   rx
2-   x)

led'l, (Clarendon P-ress, Oxford, 1965).
[12] S. Iwamoto, and Y. Tamura, "A load-flow calcula-                                        +fx (x%(x -xE)
l                                                             (20)
tion method for ill-conditioned power systems"',                                                  N-
presented at the IEEE PES Summer Meeting, Vani-
couver, Canada, 15-20th July, 1979, paper A79 441-                                   af.          f.(x'n + -n
)                   _          i = ,     ,..,
~~~~~~~~f
[13] J. Meisel, and R.D. Barnard,. "Application of                            )               )                      h~                                 . .. (21)
fixed-point techniques to load-flow studies", IEEE
Trans., 1970, PAS-89, pp. 136-140.                                   where ej denotes the j th unit vector and the scalar hn
is normally chosen such that hn = 0(1 If(xn) |). With
[14] A.M. Ostrowski,    "Solutions of equations and sys-                  this choice it can be proven 1181 that the discrete
3654

Newton's method has second order convergence. This is                           the Ws are obtained by back-substitution in the N-1
the same rate of convergence as the ordinary Newton's                           rowed Itriangular linear system which now has the form
method.
If xn is close enough to x*, fl(x ) Owe can
equate (70) to zero and solve for the ffrst variable.                                    L=            - I
J=1                  /gf   Nl             i   )(Lg   -   xj,
This is the variable, say XN, whose corresponding par-                                          - g
tial derivative, f (xD, is largest in absolute val-                                                           /g-NN+l,x.2           5
ue. This gives    lxN                                                                                                                            J,

N-1     n                                                  with g- f1 and L1 = xl so that gn is just a function
XN         .
j=1   (fXNflx     N )(Xj
-X.   X      f/f           (22)   of the single variable xl. Now expanding, linearizing
j                                   N          and solving for xl, we obtain
The constants f'1 /f'n1          are saved for later use. Brown                                                        =        -'n/a                                        (26)
lxi lxN
[7] has shown that under the usual hypotheses for New-
X, i1        glg,        1
2)
ton's method there is always at least one non-zero par-                         Thepoint x thus obtained is used as the next approxi-
tial derivative, and, of course, the corresponding ap-                          T
proximate partial (21) is non-vanishing. Thus the sol-                          mation, x'1+ to the first coponent, xl, of the solu-
ution procedure (22) is well defined. By choosing the                           tion vector x*. Renane xl as L1 and back solve the Li
approximate partial derivative of largest absolute val-                         system (25) to get an improved approximation of' comon-
ue as the division, a partial pivoting effect is a-                             ents of x*. Here x4+1 is taken as the value obtained
chieved similar to what is often done when using the                            for L. wheii back solving (25).
Gaussian elimination process for solving linear sys-
tems. This enhances the numerical stability of the                              Choice of h in the Computer Implementation
n

method. We observe from (22) that xN is a linear func-
tion of the N-1 variables xl, x2, ..., XN . For pur-
X                                            In order to guarantee quadratic convergence, the
poses of clarity the right-hand side of
as L.(x1, x2, *.., xNl)
is renamed (221                    following strategy is used for choosing hM, by which to
increment x' when working with the i finction fi in
eqn. (21):
Step 2
hn7    =    max Cn.         5 x 10 -6+2)
Define a new function g2 of the N-1 variables xl,                                                                     1)
.., xN_1 which is related to the second function f2 Of                         where
(19) as follows
{ fln                   2^***^|gn| )
n
;   *001   x     xn |}
2(X     **...,   XN-1) f2(X1i,       ...,   XN-1   Ln(xi,
1'      '                               n fN-1m
..(27)
Now expand g2 in an approximate Taylor series
**(23)
expansion
where S is the machine tolerance, i.e. the nunber of
significant digits carried by the machine.
about the point (x, ..., N_). Linearize (ignore
higher order terms) and solve for that variable, say                            EXAMPLE: Application of K.M. Brown's method
xN_, whose corresponding partial derivative, x    is
largest in magnitude:                         N-1                                     Let us consider a nonlinear system with two var-
N-2                                                              iables:
xN- j1~N-11
=
22X/2xj - ) 92/2x
-     )
f (X)=X -2x + 1 = °
j=1

(24)
~~~~~~~~~~~~
...
f (x) = x + 2x - 3 = O
X_l is a linear function of the remaining N-2 vari-                                              2(      1     2
ables. Remember the right-hand side of (24) is denoted                         which has the solution at [1,1] T. For x° = [0,0]T as
by Li    (xl,  ...`, xN2). Again this forms the ratio,                          the initial approximation, the application procedure is
N2NI/gteration
92x 2N-1          1, ..., N-2, and               gn/gn
922N-1Ieato
is stored                         demonstrated here using exact partial derivatives.
for future use in any camputer inplementation of this
algorithm. Each step of the algorithm adds one more
linear expression to a linear system. During the                                 0    Expand f1(x) i the Taylor series about the poit
(k+l)st step of the algorithm, it is necessary to eval-                         x;   retain only first-order terms and thus obtain the
uate gk+l, i.e fk1 for various arguments. The values                            Iinear approximation:+
of the last k components of the argument of fk+1 are                                             0        x        + x
obtained by back-substitution in the linear system LN,
LN-1, .. ., 'N-k+1 which has been built up. These argu-
f1(X)         f1(X         +   f1X (Xi X0            +        f1X (X2
2
XP (2)
...
ments are required to determine the quantities gktl and                         where
,n+l , j = 1, ..., N-k, needed for the elimination of
the (k+l)st variable, say xN..k, by the basic process of
expansion, linearization and solution of the resulting                                     teRSo
1-' lx
q.(9          n ov o h    ozr
x2
expression. The process results, for each k, in the                             Equ.ate           te          oeq. eoad ov o h
.2)t
(k+l)st variable, say ._,i expressed as a linear                                variable x2, whose corresponding partial derivative is
combination, L-k of th~ekremaining N-k+l variables,                             largest in absolute value:
Step N                                                                                                  xl = 0. 5
At this stage gN EN
f(x1 L2, L3, ... LN) where                             Substituting this value into function f2 (x), we have
3655

Substitute this value in function                         f2(x) in (28) and      re-
define as g2(x):
Thus2-
xl 2.0
=                                                                     g2(x)      =   2.5x2 -        1.95
Expanding the function g2(x) into the Taylor series
2                                                          9 2(x) = g2( + g2x (x2 - x2)
g                                    (31)
2                =j
Iteration 211                                                                      where                    g(x ) = -1.0 = 2.5          gl
By repeating the samne procedure with new estimates                                                           2
Solving for the variable x2 from (31) gives the new
f1(x(X 4.0 ; f = 4.0 ; f1
f1
1                                    2.0
) 4.0              4.0     f      -2.0lx2                         estimate after the second iteration
2
Substituting in                                                                                                        x1 =1.20
f (x) = f(x1) + f
f1x f1(x
1                     (x1--
f x2 - X
fx o2 en(30) x(
xl     +
2
22
2x20)9
1;nd equatingh                                          =
and equating the RHS of eqn. (30) to ze-ro gives      By repeating this same procedure the problem converged
x= 0.5x2 + 0.75                   in five iterations to xi = 1.0 and x2 = 1.0.

APPENDIX II                                        A2.2 Description of the 13 bus ill-conditioned system
System Data for Ill-conditioned Systems                                                     The line and bus data has been taken from [6].
A2.1 Description          of the 11 bus            ill-conditioned system                                            TABLE A2.3
The line and bus data has been taken from [17] and                                      Line   data     for 13 bus ill-conditioned system
is available only in the form of Y-bus matrix elements
and net bus powers.                                                                                                                                           Suscep-
Branch       From          To       Resistance              Reactance          tance
TABLE A2.1                                              number      node          node             (p.u.)            (p.u.)        (pu)
Y-bus matrix elements of 11 bus system                                   1          0             1              0.0040            0.0850        0
2          0             2              0.0040            0.0947        0
Gi km
k -m                          Bkm         k - m              Gkm         B           3          4             3
2              0.0040
0.0074            0.0947
0.1430        0
0.436
km      4          3
1       1     0.0     -14.939             5   2.5815                    -5.889                     5
5           1             0.0481            0.4590        0.246
1       2     0.0      14.148             6   0.0  6                   -55.556         6           5           6             0.,0090           0.1080        0.016
2       2    12.051 -33.089               7   3.2267                    -4.304                     7           2             0.0121            0.2330        0.712
2       3     0.0       6.494             7  -2.213T
8                        2.959870                                                                0
0.620
2       4 -12.051      13.197             8   2.8938                    -5.468       9     8      9                          0.0105            0.2020
3       3     2.581 -10.282               8  -0.1389                     1.379      10     9     10                          0                -0.1500        0
3       5 4 7-2.581     3.789
-0.592 0.786              8
11  -0.851
10
0.283
11                     1.163
-2.785      11    10 MVA 11
Base 7-01000
6                                 0.,0086           0.1665        0.508
048
4
4
4
5
12.642 -74.081
0.0       2.177
9
10
0.104
1.346
9
10
-1.042
-6.110
12    11     12                          0075               .16
10       0
4       6     0.0      56.689            10  -0.374
11                     3.742
-2.785
4       7    -0.592     0.786
Operating~
~            ~ ~ ~ 0.283~
11
~  11
~ ~~       ~1codiio                    12      1107  systm4-8
0bus6
TABLE A2.4
TABLE A2.2                                                           Transformer data for 13 bus system
Operating condition of               11 bus system__________________

Branch              From           TO                 Tap
Bus          Voltage          Phase-          Net real.
- 0. 090            Net reactive
- 0 . 068~~Bas           number              node          node              setting
No.         magnitude6        angle            power                  powe-r                 100 W
+ 5%90
V (p.u.)      e    (deg.)         P    (p.u.)            Q (p.u.)                          1             0                 1
2             0                 2            +10%
1        1.024*            0.0*                003                                                               4                              +10%

4                                              0.0                0.0
3                                             -0.128             -0.080
3                                             -0.1028            -0 .062

6                                         -0.090             -0.068

*Slack-bus       input data
3656
TABLE A2. 5                                                                  TABLE A2. 7
Bus data for 13 bus system                                              Operating condition of 43 b-us system
Assumed
Busbus             phasue-
Lea         Bus         Voltage     Phase-    Net real       Net    reactive
Bus     voltage       agle        pG          QG             PL          QL   No.        magnitude    angle      power               power
V (p.u.)      e (deg.)    4)        (4VAr))          (M)     (WAr)          *    V (p.u.)    e (deg.)   P     (p.u.)         (P.U.)
0       1. 0*          0. 0*-       -     1650 560                             1          1. 136*     0. 0*           -              -
1       1.0            0.0  0       0        0     0                           2                                    0.0          0.0
2      '1.0            0.0  0       0        0     0                           3                                   -0.160       -0.120
3       1.0            0.0  0       0         0    0                           4                                    0.0          0.0
4       1.0*           0.0  0       -         0    0                           5                                   -0.530       -0.400
5     1.037*           0.0 500      -       50    30                           6                                    0.0          0.0
6     1.063            0.0  0       0        0     0                           7                                   -1.600       -1.200
7     1.100*           0.0  0       -         0    0                           8                                    0.0          0.0
8     0.943*           0.0 500      -         0    0                           9                                    0.0          0.0
9     1.100*           0.0  0       -         0    0                         10                                      0.0            0.0
10     1.0              0.0  0       0       50    30                         11                                     0.0          0.0
11     1.0              0.0  0       0       50    32                         12                                    -0.800       -0.600
12     1.0              0.0  0       0         0    0                         13                                     0.0          0.0
*Input data                                                                   14                                    -0.800       -0.600
15                                     0.0          0.0O
A2.3 Description of the 43 bus ill-conditioned system                         16                                    -0.640       -0 .480
The line and bus data has been taken from [17] and                       17                                     0.0          0.0
is available only in the form of Y-bus matrix elements                        18                                    -0.240       -0.180
and net bus powers.                                                           19                                     0.0          0.0
20                                    -0.880       -0.660
TABLE A2.6                                              21                                     0.0          0.0
Y-bus matrix elements of 43 bus system                                22                                     0.0          0.0
:_____________________________________________ _ .                           -23                                    0.0          0.0
k -m       Gk       Bk      k-        G         B
.km~           -ki
24
25
-0.640
0.0
-0.480
0.0
1      1      0.0     -30.609 1        2    0.0     30.609                    26                                   -0.800       -0.600
2      2    481.288 -1545.194 2        5 -277.195 873.583                     27                                   -0.320       -0.240
2      6    -34.368 108.124 2         15 -169.726 534.322                     28                                    0.0          0.0
3      3      0.0      -5.714 3        4    0.0-     6.015                    29                                    0.0          0.0
4      4     61.331 -69.160 4         13 -61.331    62.874                    30                                    0.0.         0.0
5      5    277.195 -916.892 5         7    0.0     21.277                    31                                    1.160        0.520
5      8      0.0      20.513 6        6   34.368 -118.699                    32                                    2.900        2.570
6     12      0.0      10.638 7        7    0.0    -20.000                    33                                    0.285        0.300
8      8    452.840 -482.861 8         9 -288.938 295.777                     34                                    0.0          0.0
8     23   -163.902 167.191 9          9 300.983 -317.044                     35                                    0.580        0.560
9     10    -12.045    12.342 9       16    0.0      8.796                    36                                   -0.005        0.030
10     10     12.045 -20.855 10        11    0.0      2.857                    37                                    0.0          0.0
10     17      0.0       5.714 11      11    0.0     -2.857                    38                                   -1.440       -1.020
12     12      0.0     -10.000 13      13   92.381 -100.709                    39                                    0.0          0.0
13     18      0.0       6.015 13      25 -31.050    31.640                    40                                    0.0          0.0
14     14      0.0     -15.015 14      43    0.0     15.400                    41                                   -0.800       -0.300
15     15    340.398 -916.783 15       19    0.0      8.649                    42                                   -2.240       -1.680
15     20      0.0      15.791 15      28 -170.673 357.003                     43                                    0.0          0.0
16     16      0.0      -8.576 17      17    0.0     -5.714
18     18      0.0a     -5.714 19      19 164.292 -280.783                     *Slack-bus input data
19     22   -164.292 272.805 20        20          0.0  -15.002
21     21    104.312 -143.609 21       24        0.0      9.267
21     29   -104.312 133.623 22        22      164.292 -282.281                    Dr. S.C. Tripathy, Professor of Electrical Engin-
22     26      0.0       9.023 23      23      321.579 -328.810               eering at I.I.T. Delhi, India, is presently a visiting
23     29   -157.677 161.760 24        24        0.0     -8.572               professor at The University of Calgary. His research
25     25     87.150 -106.814 25       27    0.0      9.023                   interests are in power system analysis and control.
25     29    -56.100    65.824 26      26    0.0     -8.572
27 27          0.0      -8.572 28      28  373.447 -612.837                             r. G. Dlurga Prasad is a research scholar at I.I.T.
28 39       -202.775 256.136 29        29  318.089 -372.311                   DelTli, with interests in     power system analysis.
29 30          0.0       3.766 29      37    0.0      7.895
30 30        125.789 -524.464 30       32    0.0     30.769                       'Dr. O.P. Malik, SMIEEE, Professor of Electrical
30 38          0.0       4.131 30      40 -125.789 485.547                    Engineering afid Associate Dean - Academic, The Univer-
31 31         0.0       -13.038 31 37              0.0              13.038    sity of Calgary is interested in the real-time digital
32 32         0.0       -30.769 33 33              0.0              -3.320    control of electrical machines and power systems.
33 38         0.0         3.320 34- 34             0.0              -7.365
34 38         0.0         6.852 35 35              0.0              -6.180         Dr. G.S. Hope, SIEBE, Professor of Electrical En-
35 38         0.0         6.180 36 36              0.0              -2.703    gineering, Thie University of Calgary, has research in-
36 38         0.0         2.703 37 37              0.0             -21.348    terests in the area of digital systems, anld real-time
38     38     0.0    -22.398 39        39 512.581 -663.260                    control and protection of power systems.
39     41     0.0     15.015 39        43 -309. 806 392.255
40     40   125.789 -508.837 40        42    0.0     21.622
41     41     0.0    -15.015 42        42    0.0    -20.000
43     43   309.806 -408. 029
3657

Discussions                                      As regards 11 bus system results. The discussers have the following
observations to make:
J. Nanda, D. P. Kothari and D. L. Shenoy (Indian Institute of                 1) 11 bus test system as reported by the authors pertains to an
Technology, New Delhi,. India): The authors are to be commended for           Australian Distribution System originally reported by Mr. B. Stott.
establishing their stand that K. M. Brown's method is reliable for solv-      2) For the same 11 bus system the discussers made several tests which
ing ill-conditioned power systems.                                            revealed that this system has converged by Newton-Raphson (rec-
Authors' definition and consequently the condition they impose for         tangular co-ordinate version) with a flat voltage start (1 +jo) for all
"condition number" for ill-conditioning of computer formulation of a          load buses in 6 iterations within an accuracy of 0.01 MW/MVAR, while
problem are quite interesting indeed. It is stated in the text of the paper   B. Stott's FDLF failed (1]. The results-for the 11 bus system (obtained
that for a well conditioned system, the value of the condition number K       by Newton-Raphson method) are given in Table-1. Could the authors
is unity. However, in Table I which shows the maximum and minimum             please clarify why they could not get convergence?
eigen values and the condition number K for well-conditioned 30 bus
and ill-conditioned 11.13 and 43 bus systems. The condition number for
the 30 bus test system is given as 0.468 x 103, which is quite large.
Authors have indicated in the remarks column this number K as fair.
Would the authors indicate the ranges in which the condition number
could be said to be good (well-conditioned), fair (moderately well-                                        REFERENCE
[1] P. G. Murthy, D. L. Shenoy, J. Nanda and D. P. Kothari, "Perfor-
Table I                                          mance of Typical Power Flow Algorithms with reference to Indian
11 Bus System Solution as obtained by Newton-Raphson (rectangular                 Power Systems", presented at Second Symposium on Power Piant
co-ordinate version) method.                                                      Dynamics and Control, Bharat Heavy Electricals Ltd., Hyderabad,
India, February, 1979.
MW          MVAR         Mag            Angle in
in p.u.        degree
i         -          -           1.024           0.000(slack bus)
2        0.0        0.0          1.0552           1.215                     S. C. Tripathy, G. D. Prasad, O. P. Malik, and G. S. Hope: The
3       12.8        6.2          1.0444           4.608                     authors are thankful for a stimulating discussion on the present paper.
4        0.0        0.0          1.0306           2.830                     The condition number k is a pointer to the extent of separation of
5       16.5        8.0          1.0329           4.830                     system eigenvalues and ideally its value is one if the minimum and max-
6        9.0        6.8          1.0501           2.916                     imum values are -equal. It gives a qualitative assessment of the ill-
7        0.0        0.0          0.8310          12.066                     conditioning effect, so it is improper to define the range of its values
8        0.0        0.0          0.9131          14.857                     quantitatively. Regarding the second question on the solution of the
9        2.6        0.9          1.1264          15.911                     1 1-bus system, the results of the angle in degrees given in Table 1 by the
10       0.0        0.0          0.8266          21.033                     discussers are in doubt because of their positive signs.
11      15.8        5.7          1.0186          23.996