1
Fast Decoupled Load Flow with Optimal Axes
Rotation
M. Firmino de Medeiros Jr., Dep. de Eng. de Comp. e Automação, and Josélia dos Anjos Lucas, PPgEE
UFRN - Natal - Brazil
Abstract-- Load flow calculation by the fast decoupled
method [2] presents advantages on economy of core and CPUtime requirements. This last characteristic takes greater
importance by supervisory system applications. Nevertheless,
previous experience have shown the decoupling hypothesis by
this method are valid for over-head transmission lines, but
neither for distribution networks, nor for underground cable
systems. By these cases, the original decoupled method often
doesn’t converge. Looking for exploring the advantages of
decoupling and otherwise trying to avoid inconveniences of not
converging calculations, this paper presents a method based on
decoupling hypothesis independent of average values X/R of the
network. It has been shown the new method is much faster than
the Newton’s method [1], even reaching quick convergence by
those cases where fast decoupled method doesn’t converge.
I. INTRODUCTION
C
ONSIDERING the need to implement fast programs for
load flow calculation, decoupled methods were
developed, starting from Newton-Raphson algorithm.
Those methods are based on decoupling hypotheses between
real power and voltage angle, as well reactive power and
voltage magnitude, in the iteration’s equations of Newton's
method. The first method undoubtedly effective in reducing
computational time was the Fast Decoupled Load Flow
(FDLF) presented in [2]. The great advantage due to the
decoupling consists in computer time economy of achieved by
solving two linear equation systems of order (n-1), instead of
just one system of order (2n-2). Previous researches have
demonstrated that convergence of the traditional decoupled
method is dependent on the average value of the ratio between
branches’ equivalent series reactance and resistance of the
whole system. X/R ratio values of numerical order 3 imply an
extremely favorable condition to the convergence. As those
values decrease the convergence becomes more and more
difficult. X/R ratio values of order 1 and lower certainly lead
the iterative process to divergence. The first methods
developed looking for avoiding divergence problems of FDLF
were based on techniques of either series or parallel
compensation ([5], [6]). The inconveniences of those methods
were demonstrated by calculations in networks with many
branches having low ratio X/R. A great improvement on
convergence of the FDLF has been reached by the method
proposed in [7] and [9]. It has introduced a complex operator
in the load flow equations in order to produce a rotation in the
injected complex powers, as well as in the corresponding
elements of the admittance matrix. In that way, the ratio X/R
can be adjusted, in principle, to any real value. However, this
method presents the disadvantage of not allowing parameters
adjustment of specific branches, separately.
Reference [8] introduces, through algebraic manipulation
on the load flow equations, a new function that works as a base
for approximations used to obtain a low effort calculation of
voltage magnitude corrections, V. However, those
approximations are partly identical to the hypotheses that have
originated FDLF (small ik). In addition, adjustment factors are
introduced, heuristically, to correct matrix B’.
The General Purpose Version of FDLF (GPFD), presented
in [10], conserves all the decoupling hypotheses of original
FDLF, except for the fact of choosing matrix B'' to ignore the
resistances, instead of matrix B'. Furthermore, the iterations P and Q-V are strictly successive. The results obtained by this
method are, on average, comparable to the results produced by
the method presented in [8]. The situation in which one or the
other method presents better performance depends on the ratio
X/R, as shown in the comparative analysis reported in [10].
Reference [12] presents a method for decoupling the load
flow equations, grounded in the application of an average
rotation angle for the complex powers at nodes connecting
branches with unfavorable ratios X/R. The same rotation is
also applied to the admittances of these branches. Considering
a limit rotation angle of -36o, obtained through experience with
some networks, increases the convergence performance of the
method. In addition, convergence is also improved through
adjustments in admittance and power residues, achieved by
introducing two other X/R-dependent factors. Those fittings
are also accomplished through experimental calculations in
test-networks. It is demonstrated in [12], through several
examples, that the proposed method is more effective than
GPFD method of [10].
The method here proposed allows, on one side, to explore
the advantages of decoupling, because it introduces corrections
in the iteration equations system, whose result is a formal
reduction of the system in two smaller sub-systems. On the
other hand, the inconveniences of an usual decoupling are
avoided, since the interdependence degree among the network
variables is not explored. All empiric parameters fittings as
well generalized rotation angles in powers and admittances are
avoided. An optimal rotation angle is calculated for each bus
of the network. In that way, it is possible to avoid low
convergence performance, even in networks containing an
only branch with unfavorable X/R-ratio.
2
II. FAST DECOUPLED LOAD FLOW WITH AXES ROTATION
In spite of the popularity acquired by the original fast
decoupled method ([2]) for calculating electric power systems,
several researches accomplished on it demonstrate that its
convergence becomes more and more difficult by increasing
values of the ratio R/X.
Among other techniques adopted with the purpose of
avoiding convergence problems the Axes Rotation’s algorithm
become more attractive because the rotation affects directly
the impedance angles of the network. That technique consists
of changing, temporarily, the complex coordinates system by
rotating both axes so that the impedances represented in the
new reference system can have new R/X ratios, favorable to
decoupling.
Applying a rotation angle ψ to a branch impedance Z, it
becomes:
 R'  R cos   X sen 
Z '  Ze j  
 X '  R sen   X cos 
(1)
(2)
Where the underline means complex values. In that way, the
ratio R'/X' can be expressed by:
R' R cos   X sen 

X ' R sen   X cos 
(3)
Choosing an appropriate value for the angle , a new ratio
R'/X' could be adjusted, looking for an improvement at the
decoupling conditions. After defining the angle , common to
the whole network, all impedances are modified, resulting in a
new system ([12]). In order to maintain unaffected the state
(voltages and angles) obtained by application of the Fast
Decoupled Method, the real (P) and reactive (Q) power
injections must be also modified, ensuring validity of the
relationships among complex power, complex voltage and
complex impedance as described below:
voltages (, i.e., the network state) won't be changed. After
expanding Si for real and imaginary parts it results in:
P'  P cos   Q sen
Q'  P sen  Q cos 
(4)
III. OPTIMAL AXES ROTATION
By the method described above the angle  is the same for
the whole network. The most adequate value of this angle for a
network is found repeating load flow calculations so many
times as necessary, by a trial and error procedure.
The modification proposed here consists of obtaining a
specific rotation angle i for the complex variables of each
network bus i. It is achieved by applying an optimization
technique that considers this angle as an adjustment parameter,
and the decoupling hypotheses as objective. Thus, the need of
repeated load flow calculations is eliminated and moreover, it
is guaranteed that each nodal equation will have its respective
optimum angle. Another modification in comparison to the
basic algorithm of the Fast Decoupled Method is that no line
parameter is ignored.
A generic load flow equation for the node i can be written
as:
n
Piesp  jQiesp 
n
*
*
*
ViVkYik cos ik  j senik 
k 1
Rotating the axes of the specified complex power by an
angle i will result in:
P
esp
i

 jQiesp e j i 

n
ViVk e j
k 1
ik
Gik  jBik e j i

esp
esp

 Pi cos  i  Qi sen i  

 esp

esp

 j Qi cos  i  Pi sen i 




n



j Gik cos  i  Bik sen i    
 ViVk e ik 


 j Gik sen i  Bik cos  i   
k 1


k 1
S i  E i  I i  E i   Y ik E k
(9)
It must be attempted that values of impedances and powers
at the new axes system should only be used for iterations
purpose. Calculation of power flows in the branches, e.g., must
be carried out either using original impedance values or after
producing a reverse rotation in the parameters.
n
I i   Y ik E k
(8)
(5)
k 1
Thus, one can define:
Substituting
Y ik  Y ' ik e j in equation (5), it will result
in:
S 'i  E i I '
*
i
(6)
Or:
n
S ' i  E i   Y '*ik E k
*
(7)
Pi esp  Pi esp cos  i  Qiesp sen  i
(10)
Q iesp  Q iesp cos i  Pi esp sen i
(11)
and similarly:
k 1
Where
S 'i  S i e
 j
It is obvious from equation (7) that, if a symmetrical rotation
angle -ψ was applied to the complex powers, the complex
Gik  Gik cos  i  Bik sen  i
(12)
3
Bik  Bik cos i  Gik sen i
(13)
Consequently, the calculated complex power will be
formulated as:
Pi  jQi 
n
ViVk Gik  jBik cos ik  j senik 
k 1
that, after separating real and imaginary parts, results in:
Pical 
n
ViVk Gik cos ik  Bik senik 
(14)
k 1
Qical
any influence of the branch ik on the decoupling. Obviously,
another branch also connected to same node, e.g. ij, will
require a different value for i. The optimum rotation angle for
the node i, therefore, must be found, looking for minimizing
the global influence of parameters of all branches connected to
that node. This task can be solved by using the least square
method for optimization, as following:
N i  tg i 
n

ViVk Gik sen ik  Bik cos  ik 
(15)
k 1
tg i 
Starting from the nodal equations above for the powers P'
and Q' as functions of the modified values of conductance and
susceptance, one obtains as decoupling hypotheses:
Pi
 cos  ik  Bik
 sen ik   0
Vk  ViVk Gik
Vk
Qi
 
 cos  ik  Bik
 sen ik   0
J ik
 ViVk  Gik
 k
 
Nik
(16)
(17)
Thus, the following relationships can be obtained, from both
equation (16) or from equation (17):
Gik   Bik tg ik
Gik cos  i  Bik sen i 
 Bik cos  i  Gik sen i  tg ik
Gik cos  i cos  ik  Bik sen i cos  ik 
 Bik cos  i sen  ik  Gik sen i sen  ik 
Gik cos  ik  Bik sen  ik cos  i 
Gik sen  ik  Bik cos  ik  sen i
tg i 
2
 tg  Rik




i
X
ik


k i
df i
R
  dtg i  0
  2 tg i  ik

X
ik 
d i ki 
d i
fi 
Gik cos  ik  Bik sen  ik
Gik sen  ik  Bik cos  ik
1
N i
R
ki
R
k i
ik
ik
X ik
X ik
(19)
Where N i is the number of buses connected to bus i.
After defining a rotation angle for each node according to
equation (19) the admittances of all network branches must be
corrected according to equations (12) and (13). Following,
decoupling hypotheses must be considered, and both matrices
B' and B' ' of the original FDLF algorithm must be assigned,
starting from the new admittances.
IV. RESULTS
In order to give an idea about the effectiveness of the
proposed method, three other methods were selected for
comparison with it: Newton-Raphson (NR), fast decoupled
(FDLF) and fast decoupled with axes rotation (FDAR).
Twelve networks were used as test systems in this
investigation: seven distribution feeders with rated voltage of
13.8 kV and different R/X ratios; three transmission networks,
where two of them were obtained from technical literature and
the other is part of an European real network and, finally, two
real mixed sub-transmission/distribution 69/13.8 kV networks.
Table 1 shows the load flow results for the most critical mixed
network. It was selected for presentation because of its greatest
difference between maximal and minimal R/X ratio. Table 2
shows a performance measure of proposed method, by
comparing its iterations' number with those corresponding to
the other methods, for the most representative test systems.
The same tolerance in power residuals of 10-2 was used for all
methods, and a rotation angle of 45o was admitted for use by
FDAR method.
Considering cos  ik  1 , sen ik  0 , then:
tg i  
Gik
R sér
 iksér
Bik X ik
(18)
Table 1 – Load flow results for most critical test system
(Values of R/X ratios between 0.41 and 3.25)
V. LOAD FLOW RESULTS
Equation (18) defines the angle that should be used to rotate
powers and admittances associated to node i, in order to avoid
Node
number
Voltage
magnitudes
Voltage
angles
Rotation
angles
4
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
(p.u.)
1.0600
0.9741
0.9229
0.9056
0.9015
0.8856
0.8786
0.8679
0.8532
0.8442
0.8410
0.8402
0.8395
0.8335
0.8304
0.8936
0.8929
0.8909
0.8781
0.8778
0.8530
0.8437
0.8395
0.8330
0.8786
0.8303
0.8929
0.8909
0.8781
0.8530
0.8395
(degrees) (degrees)
0
22.7
-5.0
22.7
-14.5
26.1
-15.9
26.1
-16.2
54.7
-17.7
26.1
-18.4
49.7
-19.5
31.5
-20.7
57.6
-21.4
57.6
-21.7
31.5
-21.8
26.1
-21.8
47.5
-22.1
54.3
-22.3
54.3
-16.2
66.7
-16.2
63.6
-16.2
73.5
-18.4
73.3
-18.4
72.9
-20.7
73.5
-21.4
72.9
-21.8
54.3
-22.2
54.3
-18.4
26.1
-22.3
54.3
-16.2
54.3
-16.2
74.0
-18.4
74.0
-20.7
74.0
-21.8
54.3
Here must be pointed out that, although the value of R/X
ratio was found to be the most important parameter to control
the convergence, it isn't the only one. Certainly, network
loading is also important, but not so decisive as the former.
Note that low voltages were obtained by the load flow
calculation for the network, whose results are presented in
table 1. It has been happened because of a high system
loading. Nevertheless, the method of optimal axes rotation
has converged after a relatively slow number of iterations,
although other decoupled methods haven’t been converged.
4
3
93P /
82Q
diverges
6P /
4Q
diverges
0.21 - 0.48
0.41 - 3.25
4
diverges
diverges
0.39 - 2.47
4
diverges
diverges
5P /
4Q
5P /
4Q
8P /
8Q
5P /
5Q
V. CONCLUSION
As should be expected, Newton-Raphson's method (NR) has
presented no convergence difficulties for the tested networks.
Even in mixed sub-transmission/distribution networks a small
number of iterations has been resulted as necessary.
The General Purpose (GPFD) method didn't converge for
the studied systems.
The fast decoupled load flow (FDLF) has presented
convergence for the tested distribution systems, although with
a very large number of iterations: about 100. This fact
discourages Engineers to use the method for calculating such
systems. Moreover, no convergence was obtained for the
simulated mixed networks.
The excessive iterations number of FDLF for distribution
systems could be avoided by introducing a fixed rotation angle
of 45o to correct impedances and powers, according to FDAR
method. Even so, there were cases for which convergence
couldn't be obtained. A further adjustment process in rotation
angle should be necessary.
Although adopting an adjustment process of trial-and-error
to obtain the most adequate rotation angle, FDAR method has
not been converged for the investigated mixed test systems.
This led to the need of defining an ideal rotation angle for each
node of the system. This procedure, however, duplicates the
number of admittances, producing asymmetry in network
matrices. This fact shouldn't be considered a disadvantage, if
compared with the benefits of decoupling.
By the investigated test systems, rotation angles lie between
130 and 720, and the iterations' number was never larger than 8,
for both voltage magnitudes and angles calculations.
VI. REFERENCES
[1]
[2]
[3]
[4]
Table 2: Performance of the methods for most critical test
systems
(Number of iterations)
System
Method
(R/X range) NR
FDLF
FDAR
FDOAR
0.11 - 0.48
4
117P /
diverges
11P /
96Q
11Q
1.38 - 3.25
4
142P /
5P /
5P /
137Q
4Q
4Q
1.29 - 2.47
[5]
[6]
[7]
[8]
Tinney, W.F. & Hart, C.E., (1967): "Power Flow Solution by Newton’s
Method"; IEEE Trans. on PAS, vol. 86: 1449-1456
Stott, B. & Alsaç, O., (1974): "Fast Decoupled Load Flow"; IEEE
Trans. on PAS, Vol. 93, pp 859-867.
Monticelli, A.J., (1983): Fluxo de Carga em Redes de Energia Elétrica;
Cepel-Eletrobrás; Editora Edgard Blücher
Elgerd, Olle I., (1973). Electric Energy Systems Theory: An
Introduction; TMH Edition, McGraw-Hill.
Dy Liacco, T.E.; Ramarao, K.A.: "Discussion on “Wu, F.F.: Theoretical
Study of ...”. IEEE Trans on PAS, Vol. PAS 96, Jan/Feb 1977, pp 268275.
Deckmann, S.; Pizzolante, A.; Moticelli, A.; Stott, B.; Alsaç, O.:
"Numerical Testing of Power System Load Flow Equivalents". IEEE
Trans on PAS, Vol. PAS-99, Nov/Dec 1980, pp 2292-2300.
Haley, P.H.;Ayres, M.: "Super Decoupled Loadflow with Distributed
Slack Bus". IEEE Trans on PAS, Vol. PAS 104, pp 104-113, 1985
Rajicic, D.; Bose, A.: "A modification to the fast decoupled power flow
for networks with high R/X ratios". IEEE Trans on PS, Vol. 3, No. 2,
May 1988.
5
[9]
Monticelli, A.; Garcia, A. Saavedra, O.R.: "Fast decoupled loadflow:
hypotesis, derivations and testing".
IEEE/PES Winter Meeting, NY,
29 Jan - 3 Feb 1989, pp No. 89 WM 172-8 PWRS.
[10] Van Amerongen, R.A.M.: "A general-purpose version of the fast
decoupled loadflow". IEEE Trans on PS, Vol. 4, No. 2, May 1989.
[11] Chang, S.-K.; Brandwajn: "Solving the adjustment interactions in fast
decoupled loadflow". IEEE Trans on PS, Vol. 6, No. 2, May 1991.
[12] Patel, S.B.: "Fast super decoupled loadflow". IEE Proceedings-C, Vol.
139, No. 1, Jan 1992.
VII. BIOGRAPHIES
M. Firmino de Medeiros Jr. was born in Macaíba, Brazil, on July 11, 1954.
He graduated from the UFRN, Natal, obtained his MSc. Degree at the UFPb,
Campina Grande, both cities in Brazil, and obtained his Dr.-Ing. Degree at the
Techinical University of Darmstadt in Germany.
His employment experience was as Engineering Director at Companhia
Energética do Rio Grande do Norte - COSERN in Natal, from 1987 to 1990.
His special fields of interest include Numerical Computation and
Optimization in Power Systems. Since 1990 he is Professor at UFRN, he is
head of Dep. of Computer Egineering and Automation.
Josélia dos Anjos Lucas. was born in Natal, Brazil, on November 11, 1976.
She graduated from the UFRN, Natal and obtained her MSc. Degree at the
same University, in Brazil.
Her employment experience is as Engineer, working for Companhia
Energética do Rio Grande do Norte - COSERN in Natal. Her special fields of
interest include Numerical Computation and Optimization in Power Systems.