Novel non-iterative load flow for

voltage contingency analysis

S N Singh
UPSEB Study Cell. liT Kanpur. India

P K Kalra and S C Srivastava

Department of Electrical Engineering, I IT Kanpur India

as the number of contingencies in a real power system

This paper reports on the development of new finearized network is quite large. So, approximate load flow 2, DC
A C Ioadflow models based on a novel approach. The load load flow 3 and linearized AC load flowwa methods have
flow models have been developed based on the prhwiple of been used. These methods are much faster than the
linearizhTg nonlinear powerflow equations around complete standard AC load flow schemes. However they are, in
operathT9 range by mhdmizin9 the square and integral general, quite inaccurate. Hence, there exists a need for
square errors. The new methods involve the direct solution relatively more accurate and faster methods for con-
o/" linear simuhaneous power flow equations and thus are tingency analysis. In this paper, novel non-iterative
non-iterative in nature. The performance of the proposed methods are proposed for contingency analysis which
lhlearized models have been tested on IEEE 14-bus and have been found particularly suitable for voltage security
IEEE 30-bus systems and an 89-bus hTdian system for assessment.
analysing both a base case and contingemy cases. Their The proposed methods involve the linearization of the
results are compared with the exact Ioad flow and some power flow equations around their operating range
of the existing approxhnate load flow solutions. The utilizing two different approaches, one based on the
proposed linearized load flow models are found to be principle of least square error (LSE), and the other using
extremely fast and much more accurate, especially for integral square error (ISE) minimization.
voltage magnitude prediction, than other existh19 linear- Six different versions of linearized load flow models in
ized and approximate models. It is envisaged that the new polar as well as in rectangular coordinates have been
models have great potential for on-lhTe applications in developed and tested on IEEE 14-bus and IEEE 30-bus
power system studies and conthlgency simulation for systems and on an 89-bus Indian system for the base case
vohage securiO' evaluation. and contingency cases. The accuracies of these new
line~_rized load flow versions have been compared with
Keywords." IhTearized load flow, contingeno, analysis, a first iteration of the decoupled load flow (DLF) and
voltage security, reaLtime applications, computer algorithm the Newton-Raphson load flow (NRLF) methods. They
have also been tested against the full AC load flow results
based on a fast decoupled load flow (FDLF) model. It
has been found that the proposed methods provide more
I. I n t r o d u c t i o n accurate results than the other available approximate
Contingency analysis is usually carried out to check if a models. Being non-iterative in nature, they are also
system, which is currently in a normal state, will continue computationally efficient.
to be in a normal state when a contingency occurs. The
effect of contingencies (line outages and generator outages
etc.) on the line flows and bus voltages are determined. II. New linearized load flow models
The 'line or real power security' problem is concerned In an attempt to formulate a new load flow method which
with the line flows limits and the 'voltage security' is both accurate and fast, the linearization of the load
problem deals with the bus voltages remain within their flow equations within their operating range using least
operating limits, even when a contingency takes place. square error (LSE) minimization and integral square
One obvious approach to the contingency analysis error (ISE) minimization is proposed. The important
problem is to undertake a full AC load flow analysis for feature of this formulation is that as it is non-iterative,
each contingency ~.~2. This is extremely time consuming, it requires only a matrix inversion to calculate a new
voltage profile following an outage.
The load flow equations for real and reactive powers
Received 30 March 1993; revised 21 July 1993 at bus-j in polar coordinates for an n-bus system are

Volume 1 6 Number 1 1 994 0142-0615/94/01011-06 1994 Butterworth-Heinemann Ltd 11

Non-iterative load flow." S. N. Singh et al

given as or in compact form

~=A6
Pj = Z [ ViiVj(Gij c o s 6ij - Bit sin 31j)] (1)
i=1 The above m equations for each bus power expression
can be generated by considering different operating
Qj = Z [ ViVj(- G/~sin 6 u - Bij cos 6;fl] (2) conditions, where [6] ... 6~,~1,V( ... V,")] in equation (8)
i=l
represent the bus angles and voltage magnitudes for the
Equations (1) and (2) may be written in rectangular ith operational condition.
coordinates as follows The error vector can be defined as

Pj = ~ [Gii(eiei + f~) + Bi.i(elf~- ejf)] (3) g=A6-

i=1
Minimizing the norms of this error JlAf~-~lh equiva-
lent to minimizing PIA ? ; - ~ II2, means minimizing the sum
Qj = ~ [G,j(e,~- ejfi) -- Bo{eie j + f / fj)] (4) of the squared errors
i=1

These power flow equations are nonlinear in nature e.i"~ +e2 + ... +e,,2
and can only be solved using iterative numerical solution where, e{ are the square of the distances or errors (e3.
techniques. Several full AC load flow models 1'2'x2 have The least square solution 9 for the vector ? is given by
been developed based on this principle and on certain
properties of power system networks, such as decoupling F= lAVA] -'AVb- (9)
of real and reactive power equations. It was felt worth Once the constants are determined, it is necessary to
exploring more accurate linearized models based on LSE compute the bus voltages and angles for the given
and ISE minimization principles. injections as
In order to develop these models let us consider a
nonlinear function 'f' of variables x, y, z and assume that
[ S i - Ki] = [ B][ ~] (10)
in linearized form it may be written as Ax + By + Cz + D.
The error in linearization can be defined as
or
e = f(x, y, z) - Ax - B y - Cz - D (5)
This error function e can be minimized by using either
an LSE method or an ISE minimization principle,
resulting in two different models oflinearized load flows. The matrix [B] in equations (10) and (11)is the constant
coefficient matrix defined as
I1.1 Linearized model A
This model is based on minimization of a squared error

I
F11 ... FI. M~I ... M~. 1
(e2). Assume that the bus power St (either Pi or Qi) at [B]= i " " "
bus-/can be expressed as a nonlinear function of voltage
magnitudes (V) and angles (~-) by t-F2,, I "'" F2..l M2,,,I "'" M ....
S, = f(g, P') (6)
11.2 Linearized model B
In linearized form, it may be written as
The problem can also be formulated so as to minimize
the integral of the squared errors, written as
j=l j=l

where K~, F~j and M;~ are coefficients. If there were no E= (f(x, y, z)-- Ax - B y - Cz - D) 2 dz dy dx
I '1 I
errors in this approximation, one could have obtained
unknowns K~, F;j, M o by directly equating equation (7)
(12)
at known operating points. For minimization of E, one needs to equate its derivatives
However, this approximation is bound to have some with respect to A, B, C, D to zero and obtain four linear
error because of the nonlinear nature of the function. The equations to solve for A, B, C and D. Once the A, B, C
least squares method can be used to compute these and D coefficients are known, the nonlinear function
coefficients requiring a greater number of equations than f(x,y, z) can be replaced by the linearized relationship.
the number of unknowns. For every power equation (P Depending on the approximations considered, it is
or (2) the unknowns are 2n + 1. To compute the required possible to derive various versions of these linearized
coefficients, m ( m > 2 n + 1) linear equations are required model load flow equations. Five different versions of the
to be generated, which can be written as linearized load flows based on ISE minimization have
been explored and are presented below
-Ki
st 71 61 -- a,. VII I1",l - F,
11.2.1 Linearized version B1
a~, .. V.'- In this version the trigonometric functions are linearized
Fin as
S?- l] 1 6'/' -1 .- 67-l V~-t V,"- ' m, cos (6~j)= A6~j + B (13)
s? d 1 aT .. 67 v7 v." _ sin (61fl= Cflj + D (14)
Min In equations (13) and (14) the coefficients A, B, C, D can
(8) be obtained by minimizing ISE (equation (12)). The

12 Electrical Power & Energy Systems

 Non-iterative load flow: S. N. Singh et al
following values of coefficients are found. f/fj = A 6f/--I- B 6 f j "[" C 6 (24)
(a) For 6;t varying between 0 to 10, we get eifj = A 7ei + B 7 f j + C7 (25)
A-- -0.0870934; B = 1.0025311 Coefficients of e and f in equations (23)-(25) can be
C=0.9954362; D =0.0001770 calculated as mentioned earlier. If one assumes the
voltage magnitudes to vary from 0.9 to I.I p.u. and angles
(b) For 6ij varying between 0 to 20 , we get
from - 2 0 to 0 , then one can assume the variation of
A = -0.1731205; B = 1.0100307 e to be between 0.85 to 1.1 a n d f t o be between - 0 . 6 to
C = 0.9818125; D = 0.0014092 0.1. Once equations (4) and (5) are linearized, a single
Normally voltages are close to unity. Hence, one can matrix inversion gives the solution of voltages.
assume that 11.2.5 Linearized version B 5
V=I+AV (15) The e component of voltage is known to be normally
Substituting equations (13), (14) and (15) in equations (1) close to unity and hence one can model the e component
as given below
and (2) and ignoring (A V)2 terms and A Vf~jterms, we get
e = 1 + Ae (26)
Pj ----~ [Kijt~ij "{-Lo(A Vi + A I/i) + Lij ] (16) The remaining terms can be linearized as in version B4, i.e.
i=1
f J i = Asf~ + BsfJ + C8 (27)
Qt = ~ [Mijbij+ Ni:(AV,.+ AVt)+ Nij ] (17)
i=1 Aei.fj = A9Ae i + B 9 f j + C 9 (28)
where The result of this version can be improved further by
incorporating a linearized version of the product term of
Kij = A G I j - CBIj AeiAe j. This version also results in a linearized relation-
L o = B G i j - DB~j ship for P and Q and hence one matrix inversion will
yield the voltage.
M~j = - C G i j - ABo
N 0 = - D G q - BBq
The equations (16) and (17) form the set of linearized III. System studies
equations of the following form All five versions of model B have been tried out for the
IEEE 14-bus and IEEE 30-bus systems t and 89-bus
e ]L j (18)
Indian system ~ to obtain voltages at the load buses for
both the base case and for outages.
The range of &ij for linearization has been taken to be
The solution of equation (18) directly provides the voltage
0 to 10 for IEEE 14-bus and IEEE 30-bus systems and
magnitude and angle values. This requires one inversion
0 to 20 has been used for 89-bus Indian system. These
of matrix [A].
ranges were established from the base case studies. The
11.2.2 Linearized version B 2 voltage range for all three systems was taken as 0.85 to
1.1 p.u.
In version B1 the product terms 6;tAV~ or 6~jAVj have
The three systems mentioned above have been used to
been ignored, assuming that their contribution is small.
carry out the investigations on HP-9000 computer. The
However, to improve the accuracy of equations (16) and
investigations presented in Tables 1 to 4 include com-
(17), one can also linearize these product terms as
parison of results of the proposed five linearized load
6qA Vi ~- A i6it + B i A Vi + C l (19) flow ~,ersions with the first iteration of N R L F and D L F
methods for the base case and contingengies.
6~tA V~j~- A 26q + B2A Vj+ C 2 (20)
Table 1 gives a summary of errors in voltages and
With the above terms, the elements of matrix I-A] become CPU time, excluding time required for calculation of
modified accordingly. constants, for IEEE 14-bus, IEEE 30-bus and 89-bus
Indian systems for the base case. The r.m.s. (p.u.) and
11.2.3 Linearized version B3 maximum (percentage) error values are with respect to
This linearized version considers the linearization of the results obtained from the full AC load flows. In the
V~V~cos &it and Vi V~sin 6 u terms as follows present study the fast decoupled load flow method aa has
been used to obtain full AC load flow results.
ViVjcos flj= A3 Vi + B3 Vj + C36ij + D3 (21) The following observations can be made from the
V/V~sin 6ij = A 4 Vi + B4 l/')q'-C 4it 3r"D4 (22) results presented in Table 1.

Use of equations (21) and (22) will lead to a set of linear (a) Of the proposed five linearized versions, version B2
equations for Pt and Qt in terms of variables V~, Vjand 6~j. predicts the bus voltages most accurately. However,
the performance of versions B1, B2 and B3 are
11.2.4. Linearized version B4 comparable.
This version deals with linearization of P, Q equations (b) All the proposed versions outperform the results of
in rectangular coordinates (equations (3) and (4)). The first iteration of N R L F and D L F for IEEE 14-bus,
products of e (real part of voltage) and f(imaginary part IEEE 30-bus and 89-bus Indian systems.
of voltage) can easily be linearized as (c) R.m.s. error in voltage is quite small for all the
versions.
eiet = A sei + Bsej + C5 (23) (d) The performance of version B4 and version B5 for

Volume 16 Number 1 1994 13

Non-iterative load flow: S. N. Singh et al

Table 1. Comparison of load flow results for base case

Error System Version Version Version Version Version 1st 1st

B1 B2 B3 B4 B5 iteration iteration
NRLF DLF

R.m.s. error V(p.u.) 14 bus 0.0019 0.0016 0.0022 0.0060 0.0060 2.2400 0.0140
30 bus 0.0021 0.0022 0.0022 0.0091 0.0091 0.0420 0.0230
89 bus 0.0070 0.0065 0.0085 0.0065 0.0065 0.270 0.0970

Maximum error V(%) 14 bus 0.43 0.34 0.52 1.15 1.15 3.26 2.59
30 bus 0.52 0.53 0.62 1.52 1.52 5.66 7.73
89 bus 0.56 1.40 1.90 14.71 14.71 11.50 34.83

CPU time (s) 14 bus 0.12 0.12 0.13 0.13 0.14 0.13 0.12
30 bus 0.45 0.46 0.48 0.48 0.49 0.43 0.39
89 bus 2.92 2.92 2.95 5.20 5.23 2.90 2.75

Table 2. Contingency analysis for IEEE 14-bus system

Outage of
Voltage
Method error Gen-2 Line 1-2 Line 3-11 Line 7-14

Vet B 1 rms 0.0019 0.0087 0.0023 0.0018

max. 0.42 2.65 0.41 0.46

Ver B2 rms 0.017 0.0088 0.0019 0.0014

max. 0.35 2.67 0.31 0.27

Ver B3 rms 0.0022 0.0086 0.0023 0.0023

max. 0.52 2.61 0.52 0.46

Ver B4 rms 0.0052 0.0052 0.0065 0.0061

max. 1.23 6.21 1.55 1.56

Ver B5 rms 0.0052 0.0052 0.0065 0.0061

max. 1.2 6.21 1.55 1.56

NRLF rms 0.0270 0.1320 0.0270 0.0250

1st iteration max. 3.62 15.29 3.99 4.18

DLF rms 0.150 0.0200 0.0280 0.0160

1st iteration max. 2.60 3.48 2.86 4.04

all three systems is the same and the errors produced (7-14) and (3-11) in sequence. Hence, the outage of these
by these versions are relatively much higher than three,lines has been considered for the study. In addition,
versions B1 to B3. Gen-2 outage has also been simulated. In the case of the
(e) CPU time taken by the proposed B1, B2 and B3 IEEE-30 bus system, outages of lines (1-2), (2-5) and
versions are comparable with the first iterations of (13-28) and Gen-2 have been simulated. In case of the
N R L F and D L F methods. However, versions B4 and 89-bus system, outages of lines (74-75), 48-30), and
B5 take comparatively more CPU time. (48-50) and Gen-10 have been simulated. The errors in
In order to establish the potential of the linearized load the various methods for different contingencies are
flow to contingency cases, outages were considered in the summarized in Tables 2 to 4. The following observations
14-bus, 30-bus and 89-bus systems. The contingencies are made from the results presented in these tables.
considered include single line/transformer outages and (a) In the majority of cases considered the overall
single generator outages. Line outage cases in all the performance of version B2 has been found to be
systems have been considered for some of those lines superior to the first iteration of N R L F and DLF.
which were carrying maximum power in the base case. (b) It was found that a full F D L F diverges for two
For example, in the 14-bus system, the line between buses contingencies in the 89-bus system. These include
1 and 2 was carrying maximum power followed by lines transformer outages between 48-30 buses and 74-75

14 Electrical Power & Energy Systems

 Non-iterative load flow: S. N. Singh et al

Table 3. Contingency analysis for IEEE 30-bus system

Outage of
Voltage
Method error Gen-2 Line 1-2 Line 2-5 Line 13-28

Ver B1 rms 0.0023 0.0057 0.0020 0.0054

max. 0.57 5.37 1.52 1.35

Ver B2 rms 0.0023 0.0151 0.0059 0.0021

max. 0.591 5.37 1.15 0.54

Ver B3 rms 0.0023 0.0147 0.0020 0.0062

max. 0.63 5.35 1.12 1.39

Ver B4 rms 0.0096 0.0940 " 0.0100 0.0297

max. 1.65 11.62 2.43 1.84

Ver B5 rms 0.0096 0.0940 0.0100 0.0297

max. 1.65 11.62 2.43 1.84

NRLF rms 0.0460 0.1950 0.0610 0.0440

1st iteration max. 6.09 21.99 7.74 6.43

DLF rms 0.0250 0.0820 0.0270 0.0230

1st iteration max. 9.80 29.33 9.32 6.86

Table 4. Contingency analysis for 89-bus Indian s~stem

Outage of
Voltage
Method error Gen-10 48-30(TR) Line 48-50 74-75(TR)

Ver B 1 rms 0.0054 A 0.0085 A

max. 1.35 C 1.30 C

Ver B2 rms 0.0051 L 0.0052 L

max. 1.40 O 1.31 O
A A-
Ver B3 rms 0.0062 D 0.0069 D
max. 1.39 1.49
F F--
Ver B4 rms 0.0297 L 0.0690 L
max. 13.88 O 16.03 O
W W--
Ver B5 rms 0.0297 0.0690
max. 13.88 D 16.03 D
I I -
NRLF rms 0.0730 V 0.0300 V
1st iteration max. 26.12 E 10.85 E
R R---
DLF rms 0.1000 G 0.0990 G
1st iteration max. 35.13 E 34.86 E
D D
TR = t r a n s f o r m e r

versions (B4 and B5) and first iteration of N R L F and

buses. It has also been observed that the full N R L F
D L F methods.
also diverges for transformer outage between 74-75
buses, whereas all linearized load flow versions Model A was tried for only the I E E E 14-bus system 9
provide results in both these cases. for base case and contingencies. For the LSE minim-
(c) Maximum and r.m.s, errors in voltages are very small ization, the number of cases have been obtained by
for models B1, B2 and B3 as compared with other running off-line load flows for different loadings. The

Volume 16 Number 1 1994 15

Non-iterative load flow: S. N. Singh et al

number of coefficients to be computed were 22 for which provided by D.S.T. New Delhi (India) under project No.
50 operating points were generated. For base case study, DST/EE/9266.
the maximum voltage error with Model A is 0.86% but
in the case of line and generator outages the error is large
and is found to be as much as 11.34% for one of the Vl. References
contingencies. Since Model A was found to be quite 1 Tinney, W F and Hart, C E 'Power flow solution by
inaccurate, especially for contingency cases, it was not Newton's Method', IEEE Trans. Power Appar. Syst. Vol
tried for other systems. PAS-86 (1967) pp 583-588
From these observations, the conclusion was made that 2 Stott, B'Decoupled Newton load flow' IEEE Trans. Power
version B2 was superior to most of the other methods Appar. Syst. Vol PAS-91 (1972) pp 1955-1959
for voltage contingency analysis in the networks studies.
3 Lauby, M G 'Evaluation of local DC load flow screening
method for branch contingency selection of overloads'
Trans. Power Syst. Vol PWRS-3 (1988) pp 923-928
IV. Conclusions
(1) In this paper the concepts of minimization of least 4 Patterson, N M, Tinney, W F and Bree, O W 'Iterative
square error and integral square error have been linear AC power flow solution for fast approximate outage
studies' IEEE Trans. Power Appar. Syst. Vol PAS-91 (1972)
explored, for the first time, in linearizing the power pp 2048-2056
flow equations over the possible operating range.
(2) Amongst the six proposed linearized versions based 5 Wells, D W 'Methods for secure loading of a power system'
on ISE and LSE minimization, version B2 in polar Proc. lEE Vol 15 No 8 (1968) pp 1190--1194
coordinates provides more accurate results compared 6 Leonidopoloalos, G 'Linear power system equations and
with the other versions. These are far superior to the security assessment" Int. d. Electr. Power Energy Syst. Vol
results obtained with other approximate load flow 13 No 2 (1991) pp 100-102
models such as the first iteration of N R L F and D L F 7 Stott, B 'Review of load flow calculation methods' Proc.
methods. IEEE Vol 62 No 7 (1974) pp916-929
(3) The linearized load flow version B2 predicts bus
8 Mikolinnas, T A and Wollenberg, B F 'An advance
voltages with about 1% accuracy in most cases.
contingency selection algorithms' IEEE Trans. Power
(4) The method is fast as it is non-iterative in nature. Appar. Syst. Vol PAS-100 No 2 (1981) pp608-617
(5) For some of the contingency cases, where AC load
flow methods diverged, the proposed method pro- 9 Barnett, S Matrices, methods and applications Clarendon
vides a possible load flow solution. Press, Oxford (1990)
l0 Freris, L L and Sasson, A M 'Investigation of the load flow
In view of the above, the proposed linearized load flow problem' Proc. IEEE Vol 105 No 8 (1968) pp 1459-1469
models, especially version B2, can be used for on-line
voltage contingency analysis. Il Srivastava, S C 'On some aspects of load flow and optimal
load flow of interconnected power system' PhD thesis in
Electrical Engg. Department, IIT Delhi (May 1987)
12 Stott, B and AIsac, O 'Fast decoupled load flow' IEEE
V. A c k n o w l e d g m e n t Trans. Power Appar. Syst. Vol PAS-93 No 3 (1974)
This work has been carried out under financial support pp 859-869

16 Electrical Power & Energy Systems