Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
Robust Three phase Newton Raphson Power Flows
P. R. Bijwe, Senior Member, IEEE, Abhijith B, G. K.Viswanadha Raju, Member, IEEE
Zhang et al. [4] developed a three-phase continuation power
flow method.
Optimal multiplier has been used for coupled, constant and
Fast Decoupled Newton power flows [12-17] for balanced
power systems, to overcome convergence difficulties due to
ill-conditioning, which are more likely to occur at higher
loading conditions in a weak network. This approach provides
an efficient alternative to a continuation power flow, for
determination of critical loading. This approach, however, has
not been extended to unbalanced system power flows. In view
of this, the motivation in this paper is to explore the same.
Abstract— In this paper, robust, three-phase, Newton Raphson
power flow (NRPF) is developed for transmission system, using
Optimal Multiplier. The non-divergent, three-phase power flow is
required for the analysis of ill-conditioned or highly stressed
systems. Such situations are more common in unbalanced
systems. Results for sample system, with different loading
conditions, demonstrate the potential of the proposed algorithm.
I. INTRODUCTION
Power flow solutions are generally obtained assuming
balanced network and operating conditions. Hence, most of the
research efforts have been directed towards this type of power
flow. These assumptions, however, are not always valid due to
the presence of unbalanced loading, lack of complete
transposition, unbalanced operation of controllers and some
abnormal operating conditions. Three-phase power flow is a
problem of higher dimension and complexity. The problems of
severe ill-conditioning are likely to be encountered in this
case.
Some of the three phase power flow algorithms have been
developed based on purely phase variables [1-4]. Many three
phase power flow algorithms have also been developed using
phase and sequence variables [5-10]. The former has the
advantage of ease of modeling line mutual coupling between
phases and phase shift introduced by some transformer
connections. The disadvantage is the very heavy computational
burden due to 3NX3N matrices involved. The latter approach
has the advantage of computational efficiency. This approach
faces difficulties in handling line mutual coupling and handling
of transformer introduced phase shifts. Efforts have been made
to overcome these difficulties. This approach also makes a
simplifying assumption that the generator positive sequence
power is 1/3rd of the total three-phase power.
The success which fast decoupled power flow approach
enjoyed for balanced network was not so easily extended to
unbalanced networks. Arrillaga and Watson [11] have
discussed these convergence concerns. These convergence
difficulties are expected to become serious while handling
highly stressed system conditions encountered while
performing voltage stability analysis. The effects of unbalance
may be quite important for such analysis. Keeping this in view,
II. MATHEMATICAL MODEL
A. Three-Phase Power Flow in Rectangular Co-ordinates
The complete set of specifications for three phase power
flow include, real and reactive bus power specifications for
each phase at the load and generator terminal buses, total three
phase real power specifications at internal generator buses,
positive sequence voltage specifications at the generator
terminal buses. In case of reactive limit violation at the PV
bus, the positive sequence voltage specification is replaced by
the total three-phase reactive power limit. The expression for
components of bus power specifications are given by [11]
ΔPip = (Pip )sp − Pip
n
= (Pip )sp − eip ∑
∑
k =1 m = a,b,c
n
(emk G ikpm − Bikpm f km ) −fip ∑
∑
(f km G ikpm + Bikpm e mk )
k =1 m = a ,b,c
(1)
ΔQip = (Qip )sp − Qip
n
= (Qip )sp + eip ∑
∑
k =1 m = a ,b,c
n
(f km G ikpm + Bikpm emk ) −fip ∑
∑
(e mk G ikpm − Bikpm f km )
k =1 m = a ,b,c
(2)
where,
i, k = 1, 2,...........No.of buses (n).
p, m ∈[a, b, c]
G, B real and imaginary parts of Y-bus, respectively
e, f
real and imaginary parts of load bus voltage vector.
p sp
(Pi ) , (Qip )sp real and reactive power load specifications of
ΔPip , ΔQip
P. R. Bijwe is with the Department of Electrical engineering, Indian
Institute of Technology Delhi, New Delhi, India. (e-mail: prbijwe@
ee.iitd.ac.in).
Abhijith B is with IBM India (p) Ltd., Bangalore, India. (e-mail:
abhijith.boompelly@gamil.com).
G. K. Viswanadha Raju is with Power Research and Development
Consultants (p) Ltd., Bangalore, India. (e-mail: gkvr_iitd@yahoo.co.in).
phase, p at bus i
real and reactive power mismatches of phase,
p at bus i
For every generator k having reactive generation within
limits, the bus voltage mismatch is given by
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Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
( ΔVreg )k2 = ((e1k )2 + (f k1 )2 )sp − ((e1k )2 + (f k1 )2 )cal
voltages;
Xe estimate of X;
ΔX bus voltage update vector;
J
Jacobian matrix containing partial derivatives of bus
specifications with respect to state variables.
(3)
where,
e1k , f k1 real and imaginary parts of positive sequence
voltage of kth generator terminal bus
An important feature of (8) is that it does not contain terms
beyond third because of the quadratic nature of the defining
functions in (6). Another significant feature of (8) is that the
third term represents a second order power term evaluated
simply by substituting ΔX for X in quadratic expressions for
the defining functions.
Iwamoto and Tamura [12] proposed following modification
of (8) using optimal multiplier:
For every generator i, with the exception of slack machine,
the total three phase power generation mismatch is given by
sp
(ΔPgen )i = (Pgen
)i − (Pgen )i
∑
sp
= (Pgen
)i −
eip
p = a ,b,c
+
∑
p = a,b,c
eip
∑
∑
(eim Gg ipm − Bg ipm f im ) −
m = a,b,c
(E i Gg ipm − Bg ipm Fi ) −
m = a,b,c
∑
f ip
p = a,b,c
∑
p = a,b,c
fip
∑
∑
(fim Gg ipm + Bg ipm eim )
m = a,b,c
(Fi Gg ipm + Bg ipm E i )
m = a,b,c
(4)
Ys − Y(X e ) − μJΔX − μ 2 Y( ΔX) = 0
where,
Ggi, Bgi real and imaginary parts of admittance matrix
(Ygi), determined by positive, negative and zero
sequence impedances of ith generator.
real and imaginary parts of internal voltage of ith
Ei, Fi
generator
sp
(Pgen )i
total real power generation specification of ith
(9)
The value of μ is obtained by minimizing following
objective function
Minimize
T
F = 0.5 ⎡⎣a + μb + μ 2 c ⎤⎦ ⎡⎣a + μb + μ 2 c ⎤⎦
(10)
generator
where,
a = Ys − Y(X e ), b = − JΔX = −a, c = −Y(ΔX)
The state variables are the real and imaginary parts of bus
voltages (e, f) for each phase, at all the buses. It may be noted
that since the internal generator voltages are balanced, only
one phase voltage components (E, F) need to be determined.
The optimality condition for the minimization is given by
∂F
=0
∂μ
The linear set of equations for three phase NRPF is as
follows.
∂P / ∂E
∂P / ∂f
⎡ ΔP ⎤ ⎡ ∂P / ∂e
⎢ ΔP ⎥ ⎢ ∂P / ∂e ∂P / ∂E ∂P / ∂f
gen
gen
⎢ gen ⎥ = ⎢ gen
⎢ ΔQ ⎥ ⎢ ∂Q / ∂e
∂Q / ∂E
∂Q / ∂f
⎢ 2 ⎥ ⎢ 2
2
2
⎣⎢ ΔVreg ⎦⎥ ⎣⎢∂Vreg / ∂e ∂Vreg / ∂E ∂Vreg / ∂f
∂P / ∂F ⎤ ⎡ Δe ⎤
∂Pgen / ∂F ⎥⎥ ⎢ ΔE ⎥
⎢ ⎥
∂Q / ∂F ⎥ ⎢ Δf ⎥
⎥⎢ ⎥
2
∂Vreg
/ ∂F ⎦⎥ ⎣ ΔF ⎦
(12)
which results in following cubic equation
(5)
g 0 + g1μ + g 2 μ 2 + g 3μ3 = 0
(13)
where,
g 0 = a T b, g1 = ⎡⎣ bT b + 2a T c ⎤⎦ , g 2 = 3b T c, g 3 = 2cT c
B. Optimal Multiplier for Newton Raphson Method in
Rectangular Coordinates
As explained in [12], the nonlinear power flow equations in
rectangular coordinates are as follows
Ys = Y(X)
(11)
(14)
If more than one real solution for μ exist, then the
lowermost solution is used for updating the state variables, that
is
(6)
X = X e + μ ΔΧ
(15)
The Taylor series expansion of (6) can be written as:
Ys = Y(X e ) + JΔX + Y( ΔX)
This procedure is adopted for updating the state variables in
every iteration until convergence is obtained.
(7)
C. Optimal Multiplier for Newton Raphson Method in Polar
Coordinates [13]:
Optimal multiplier evaluation using rectangular coordinates
is quite different as compared to that in polar coordinates
because
1) Taylor series in former contains only three terms as
Ys − Y(X e ) − JΔX − Y(ΔX) = 0
(8)
where,
Ys vector of bus power and voltage specifications;
X vector of real and imaginary parts of complex bus
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Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
Pi = Pio (1 + λ ), Q i = Q io (1 + λ ) i=1,2,....... no. of buses
compared to infinite terms in the latter.
2) Evaluation of second order power term in the former is
very straightforward and efficient as compared to that in the
latter.
where, λ is a scalar load multiplier.
Table I shows the comparison of the convergence behavior
(in terms of iterations) of both the methods with and without
optimal multiplier, without Q-limits. For both the methods
optimal multiplier could arrest divergence for loading beyond
the critical one (infeasible region).
In the polar co-ordinate version of NRPF, the bus voltage
angle and magnitude update vectors in an iteration k are
evaluated from following equations
⎡ ΔP ⎤ ⎡ ∂P / ∂θ
⎢ ΔP ⎥ ⎢ ∂P / ∂θ
⎢ gen ⎥ = ⎢ gen
⎢ ΔQ ⎥ ⎢ ∂Q / ∂θ
⎢
⎥ ⎢
⎢⎣ ΔVreg ⎥⎦ ⎢⎣ ∂Vreg / ∂θ
∂P / ∂θ int
∂P / ∂V
∂Pgen / ∂θ int
∂Pgen / ∂V
∂Q / ∂θint
∂Vreg / ∂θint
∂Q / ∂V
∂Vreg / ∂V
∂P / ∂Vint ⎤ ⎡ Δθ ⎤
∂Pgen / ∂Vint ⎥⎥ ⎢ Δθ int ⎥
⎢
⎥
∂Q / ∂Vint ⎥ ⎢ ΔV ⎥
⎥⎢
⎥
∂Vreg / ∂Vint ⎥⎦ ⎣ ΔVint ⎦
(16)
TABLE I
ITERATIONS REQUIRED BY NRPF METHODS TO CONVERGE WITH AND WITHOUT
USING OM WITHOUT Q-LIMITS
Without Q-limits
S.
Power
No
Flow
Lambda=0
lambda=0.47
Lambda>0.47
w/o
with
w/o
with
w/o
with OM
OM
OM
OM
OM
OM
1
NRPF
6
6
8
8
Diverge
Arrest
(polar)
divergence
2
NRPF
6
5
7
7
Diverge
Arrest
(rect)
divergence
The evaluation of optimal multiplier for polar co-ordinate
NRPF is as follows
−1 k
1. Evaluate the state vector updates, ΔX kp = J pk
a
where
a k is the mismatch vector and J −pk1 is the polar co-
ordinate Jacobian inverse in kth iteration.
2. Transform the updates, ΔX kp , into rectangular co-ordinates,
Similarly, Table II shows the comparison of the
convergence behavior (in terms of iterations) of both the
methods with and without optimal multiplier, with Q-limits. In
this case also in both the methods, optimal multiplier could
arrest divergence for loading beyond the critical one
(infeasible region). In these studies, the checking of Q-limits
has been suppressed for three iterations.
ΔX kr
3. Compute ak, bk and ck vectors.
Evaluation of vector bk is, however, quite different in the
polar coordinates as compared to that in rectangular
coordinates. Vector bk is evaluated in the proposed method
from
b k = − J ΔX
k
r
k
r
TABLE II
ITERATIONS REQUIRED BY NRPF METHODS TO CONVERGE WITH AND WITHOUT
USING OM WITH Q-LIMITS
With Q-limits
S.
Power
No
Flow
Lambda=0
lambda=0.279
Lambda>0.279
w/o
with
w/o
with
w/o
with OM
OM
OM
OM
OM
OM
1
NRPF
6
6
10
8
Diverge
Arrest
(polar)
divergence
2
NRPF
6
5
9
8
Diverge
Arrest
(rect)
divergence
(17)
where J kr is the exact variable Jacobian in rectangular
coordinates. This Jacobian, however, need not be stored. It is
easy to see that in the present case b k ≠ a k . This is because
a k = J pk ΔX kp
(18)
c k = Y(ΔX kr )
(19)
From Tables I and II it can be seen that, the rectangular
version of NRPF has better convergence than the polar version
of NRPF.
It can be seen from the tables that the optimal multiplier
does not improve convergence much at light/medium loading
conditions. Similar trend was reported for balanced system
Newton power flow using optimal multiplier. Hence, the use of
optimal multiplier is primarily for difficult convergence cases.
However, non-divergent feature is a vital attribute required in
practical power flow algorithms.
4. Compute g 0 , g1 , g 2 , g 3 and coefficients of the cubic
5. Evaluate μ by solving cubic equation (13)
6. Modify voltages using X kp +1 = X kp + μΔX kp .
III.
(20)
RESULTS
In order to compare the convergence behavior of
rectangular and polar versions of NRPF with and without
Optimal Multiplier (OM), results for 24-bus [9], test systems
have been obtained. The convergence criteria used in these
studies is 10-4. Constant power loads are uniformly scaled as
follows. This scenario has been assumed in the absence of any
practical data. However, any other load increase scenario can
be used, if data is available.
IV. CONCLUSION
Non-divergent, three phase, Newton Raphson power flow
algorithms in rectangular and polar coordinates have been
developed in this paper. Such a feature is extremely desirable
for analyzing ill-conditioned and highly stressed systems.
These situations are more likely in unbalanced systems.
Results obtained for a sample 24-bus power system, have
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Fifteenth National Power Systems Conference (NPSC), IIT Bombay, December 2008
demonstrated the potential of the proposed algorithms.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
K. A. Birt, J. J. Graf, J. D. McDonald, and A. H. El-Abiad, “Three phase
power flow program,” IEEE Trans. Power App. Syst., vol. PAS-95, no.
1, pp. 59–65, 1976.
R. G. Wasley and M. A. Shlash, “Newton-Raphson algorithm for three
phase load flow,” Proc. Inst. Elect. Eng., vol. 121, no. 7, pp. 631–638,
1974.
J. Arrillaga and B. J. Harker, “Fast-decoupled three-phase load flow,”
Proc. Inst. Elect. Eng., vol. 125, no. 8, pp. 734–740, 1978.
X. P. Zhang, P. Ju and E. Handschin, “Continuation three-phase power
flow: A tool for voltage stability analysis of unbalanced three-phase
power systems,” IEEE Trans. Power Syst., vol. 20, no. 3, pp. 13201329, August 2005
X. P. Zhang and H. Chen, “Asymmetrical three phase load flow based
on symmetrical component theory,” Proc. Inst. Elect. Eng., Gen.,
Transm., Distrib., vol. 137, pp. 248–252, May 1994.
X. P. Zhang, “Fast three phase load flow methods,” IEEE Trans. Power
Syst., vol. 11, no. 3, pp. 1547–1553, August 1996.
B. C. Smith, J. Arrillaga, “Improved three-phase load flow using phase
and sequence components,” Proc. Inst. Elect. Eng., vol. 145, no. 3, pp.
245–250, 1998.
Mamdouh Abdel-Akher, Khalid Mohamed Nor and Abdul Halim Abdul
Rashid, “Improved Three-Phase Power-Flow Methods Using Sequence
Components,” IEEE Trans. Power App. Syst., vol. 20, no. 3, pp. 1389–
1397, August 2005.
B. K. Chen, M. S. Chen, R. R. Shoults, and C. C. Liang, “Hybrid three
phase load flow,” Proc. Inst. Elect. Eng., Gen., Transm., Distrib., vol.
137, pp. 177–185, May 1990.
K. L. Lo and C. Zhang, “Decomposed three-phase power flow solution
using the sequence component frame,” Proc. Inst. Elect. Eng., Gen.,
Transm., Distrib., vol. 140, no. 3, pp. 181-188, May 1993.
J. Arrillaga and N. R. Watson, “Computer modeling of electrical power
systems,” 2nd edition. New York: Wiley, 2001.
S. Iwamoto and Y. Tamura, “A load flow method for ill-conditioned
power systems,” IEEE Trans. Power Apparat. Syst., vol. PAS-100,
pp.1736–1743, April 1981.
P. R. Bijwe and S. M. Kelapure, ”Non-divergent fast power flow
methods,” IEEE Trans. Power Syst., vol. 18, no. 2, pp.633-638, May
2003.
L. M. C. Braz, C. A. Castro, and A. F. Murari, “A critical evaluation of
step size based optimization methods,” IEEE Trans. Power Syst.,
vol.15, no. 1, pp. 202–207, February 2000.
J. E. Tate and T. J. Overbye, “A comparison of the optimal multiplier in
polar and rectangular coordinates,” IEEE Trans. Power Syst., vol. 20,
no. 4, pp. 1667-1674, November 2005.
M. D. Schaffer and D. J. Tylavsky, “A non-diverging polar form
Newton based power flow,” IEEE Trans. Industry Appl., vol. 24, no. 5,
pp. 870-877, 1988.
T. J. Overbye, “A power flow measure for unsolvable cases,” IEEE
Trans. Power Syst., vol. 9, no. 3, pp. 1359-1365, August 1994.
P. R. Bijwe (SM’99) is a Professor of electrical engineering at Indian
Institute of Technology, Delhi, India. His research interests include power
system analysis and optimization, Voltage stability, deregulation, fuzzy
systems and distribution system analysis and optimization.
Abhijith. B is a Software engineer at IBM India (p) Ltd, Bangalore, India.
His field of interest includes computer applications to power systems.
G. K. Viswanadha Raju (S’05 – M’08) is a senior R&D engineer at Power
Research and Development Consultants (p) Ltd, Bangalore, India. His
research interests include power system analysis and optimization and
distribution system analysis and optimization.
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