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A new power flow method for radial networksManuel A. Matos, Member, IEEE
Abstract —The need of fast algorithms for radial distribution
networks that take advantage of their particular structure has
been increasing, namely due to the use of genetic algorithms and
meta-heuristics for optimization in planning and operation.
In this paper, a new method for power flow calculation in
radial networks is presented. It uses an iterative process along
the branches, in a way similar to other methods, but the main
idea is very different from previous approaches, since it is based
on the exact power flow solution for a single branch and also
because it provides a complete solution (not only voltage
magnitudes). The method is fast and robust for different types of
networks and loads, including heavy loads.
The paper includes the theoretical derivation of the method,
an illustration example and tests with benchmarking networks.
Index Terms —Power Distribution, Load flow analysis,
Iterative methods, Planning, Operation.
I. I NTRODUCTION
The special structure of radial networks has lead, in the past, to a number of specialized algorithms that tried totake advantage of the absence of meshes to simplify the
calculations and save memory [1]-[6]. In some cases, the
methods are extended to weekly meshed networks with some
success.
Calculation of the power flow in radial networks was not a
priority in the past, since approximate methods were sufficientto have a general picture of the power flow and, if necessary,
a general-purpose method (like Newton-Raphson) could
always be used. Use of genetic algorithms and meta-heuristics
in the optimization of distribution networks, however, lead to
the need of fast calculation methods with some degree of
accuracy [7]. On the other hand, more and more DMS
(Distribution Management Systems)[7] are being developed
and installed, and fast methods for radial networks are again
welcomed. Finally, dispersed generation connected to
distribution networks is growing, and adequate algorithms are
needed to deal with it.
II. FOUNDATION OF THE METHOD
The main idea of the method is to use the exact power flow
solution for one branch, when we know the voltage in the
sending end (V0) and the injected power in the receiving end
(S1), z1 being the impedance of branch 0-1 (see Fig. 1).
M. A. Matos ([email protected]) is with INESC Porto – Instituto de
Engenharia de Sistemas e Computadores do Porto, Campus da FEUP, Rua Dr.
Roberto Frias, nº 378, 4200-465 Porto, Portugal. Phone: +351.22.2094230
Fax: +351.22.2094050, and also with FEUP – Faculdade de Engenharia da
Universidade do Porto, Portugal
V0 V1z1
-S1
Fig. 1 - Network with a single branch
Of course, we have:*
1
1110
V
S.zVV
or
0S.zVV.V 1*1
2
11*0 = (1)
The analytical solution of this complex quadratic equation
may be found in a number of ways. For instance, changing to
rectangular coordinates and assuming that θ0=0 (so V0=e0),V1=e1+jf 1, z1=r 1+jx1 and S1=P1+jQ1, we get:
0 jQP jxr f e jf ee 11112
121110 =
and, after separating real and imaginary parts:
0Q.xP.r f eee 11112
12110 = (2a)
0P.xQ.r f e 111110 = (2b)
Now, from (2b) we take immediately the value of f 1, since
all the other quantities are known, and then we find e1 by
substituting f 1 in (2a) and solving the (real) quadratic
equation, where e1 will be the biggest solution. This analytical
solution of the power flow problem is well known, but
generally not used, since it is only applicable to the trivial case
of two buses.
V0 V1 V2 Vn-1 Vn
-S1 -S2 -Sn-1 -Sn
z1 z2 zn
Fig. 2 – Simple radial network model
If we have now a radial network with successive branches
1, 2,..n, where node 0 is the root, with a specified voltage
V0=e0 (see Fig. 2).
0-7803-7967-5/03/$17.00 ©2003 IEEE
Paper accepted for presentation at 2003 IEEE Bologna Power Tech Conference, June 23th-26th, Bologna, Italy
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In the first branch, (1) transforms to:
*n
k k
k
V
S . z V V
∑
=1
110
or, multiplying by V1 and conjugating:
∑
=
=
n
k k k
*
V
V .S . z V V .e1
11
2
110 0 (3)
Similar expressions may be established for the remaining
nodes, in each case using the predecessor node voltage as a
constant. The general expression (i=1..n) is therefore:
02
=
∑
∈ )i( succk k
ik i
*iii
* )i( pred
V
V .S S . z V V .V (4)
The idea is then to apply (4), beginning in the first node
after the root - which corresponds to (3) – in order tosuccessively calculate the voltages, until getting the leaves of
the tree. This corresponds to an iteration that can be repeated
with the updated values of the voltages, until some
convergence criterion is met.
III. CALCULATION PROCESS DETAILS
A. Equations
In order to conduct the iterative process, (4) is conveniently
transformed to:
0V
V.SS.zVV.V)i(succk
)1 p(k
)1 p(i
k i*i
2ii
) p(*)i( pred =
++− ∑
∈−
−
(5)
where (p) denotes the iteration. Once it’s clear that the
voltage of the predecessor node is always known when we
calculate the updated value of Vi, we may write, simply (the
meaning of S’i is obvious):
0'S.zVV.V i*i
2
ii*
)i( pred = (6)
corresponding to the model of Fig. 3. As mentioned before,
a formulation in rectangular coordinates is the best way tosolve the equation in order to get V i, avoiding the need for
trigonometric calculations with small angles.
Vpred(i) Vizi
-S’i
Fig. 3 – General branch model
Note that other modified forms of (4) could be used, but
our results shown that they are less efficient than (5).
With the proposed formulation, it is not necessary to
estimate initial values for the voltages, but only to consider
that:
all k ∈succ(i) (7))0(i)0(
k VV =
This corresponds to using, in the first iteration,
02
=
∑
∈ )i( succk
k i*iii
* )i( pred S S . z V V .V
instead of (6). Of course, if good initial values are known,
they may always be used with the normal version of the
equation.
B. Iterative process
As mentioned before, the method progresses, in each
iteration, from the root to the leaves, with successive use of
(6). The updated values of Vi are immediately used in their
successors’ equations.
The sequence of calculations is very straightforward and
similar to other forward sweep methods, so we’ll only sketch
it with the help of Fig. 4.
0 1 2 3 4
5 6
Fig. 4 - Example distribution network
Since node 0 is the root (V0 is known), calculation of V1 is
the first step of an iteration. In the second step, V2 and V5 may
be calculated independently. V3 and V6 will follow, and V4
will be updated in the final step of the iteration. This shows
the possibility of partial parallel calculation in typical
distribution networks, with important time savings.
The updated values of the voltages are then used in a new
top-down iteration, and the process is repeated until a
specified tolerance on the voltages' successive values is met.
A final convergence test on the specified injected power is
recommended.
C. Branch model
Up to this point, a simple model for the branches has been
used, considering only the branch impedance, which is
common in distribution networks. However, a more detailed
model can be used if necessary, with minor changes in the
equations. This feature may be important in underground
networks, where the capacitance of the cables is not
negligible.
Fig. 5 shows the typical π model for a branch, where ysi isthe semi-admittance of the branch.
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Vpred(i) Vizi
-S’iysi ysi
Fig. 5 - Detailed branch model
The other variables are the same of Fig. 3. Because there ismore than one branch connected to each node, it is now
convenient to define a branch admittance for each node i:
(8)∑
∈
)i( sconk
sk si si y yY
where scon(i) is the set of successors connected to node i.
In this case, the total current through branch i includes terms
related to all the admittances in i and in its succesors:
∑
∈ )i( succk
k sk i si
*
i
iii )i( pred V .Y V .Y V
' S
. z V V
and (6) transforms into
012
=
∑
∈
i*ii
* si
*ii
)i( succk
*k
* sk
*i
* )i( pred ' S . z V Y . z V .V .Y . z V
In order to maintain the structure of the equation, and since
is a constant, we may now write:*
si*i Y . z 1
011
2=
∑
∈
i* si
*i
*i
ii* si
*i
)i( succk
*k * sk *i* )i( pred
' S .Y . z
z V V .
Y . z
V .Y . z V
(9)Q
With this formulation, the resolution process in rectangular
coordinates is not altered. Obviously, if all the Ys are
negligible, (9) reduces to (6).
D. Node admittances
It is also easy to include constant node admittances Yi, i.e.,
capacitors or reactors connected to node i, or loads
represented by a constant impedance. In fact, it is sufficient to
change (8) in order to include Yi, and then use (9) as thegeneral expression of the algorithm:
(8’)∑
∈
)i( sconk
i sk si si Y y yY
IV. SOME ENHANCEMENTS
A. PV nodes
Although the method is primary intended to deal with PQ
(or impedance) nodes, it possible to consider PV nodes as
well. The existence of a PV node affects the calculations of
itself and of all its predecessors. Regarding the first issue, we
start by rearranging (4) in order to isolate the constant terms
(note that, to deal with the more general equation (9), a similar
process is possible):
02
=
∑
∈
)i( succk k
ik
*iii
*ii
* )i( pred
V
V .S . z V S . z V .V
02
=
∑
∈ )i( succk k
ik
*iii
*ii
*ii
* )i( pred
V
V .S . z V P . z jQ. z V .V
(10)0C jQ. z V .V i*ii
* )i( pred
where C is a complex constant with obvious meaning.
Now, we must solve (10) to calculate V i and Qi. The best way
is to eliminate Qi from the two real equations that result from
(10) and then use the fact that we know |Vi| to obtain the real
and imaginary parts of V i. Then, using (10) again, we’ll obtain
Qi.
For control purposes, it is convenient to calculate also the
generated reactive power, using:
load ii
Gi QQQ
If is outside its limits, the adequate limit must be used
instead, while the bus is temporarily classified as a PQ bus.
GiQ
The values of Qi obtained in the process will be used in the
next iteration, in the calculation of the predecessors of node i.
However, this is not sufficient for the algorithm to work, since
in the first calculation of a node, we need the values of Qk forall its successors. We may get ahead of this problem by using
00=k or γ. P k k =
0Q (with some typical value for γ) as
initial values for all PV nodes or, if non-trivial initial values
are available for the V k , by using (10) from the leaves to the
root (that is, in the opposite direction of the normal algorithm)
before beginning the iteration process.
We summarize now the inclusion of PV nodes in the
general algorithm:
a) Estimate initial values for Qk of all the PV buses;
b) When reaching a PV node in the iterative process, use
(10) to obtain V i and Qi);
c) Save the value of Qi to be used in the next iteration.An important point is that this process may suffer from
some instability, if the initial values for the voltages in the
buses are too far from the correct ones. Because of this, it is
advisable to wait two or three iterations before using b) and
c).
B. Dispersed generation
Dispersed generation is now frequent in distribution
networks, namely by means of asynchronous machines.
Modeling these nodes as traditional PQ or PV nodes has been
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tried, but that approach doesn’t capture correctly the behavior
of asynchronous generators [8].
We’ll implement here the idea of “PX node”, developed in
[8] and [9]. Briefly, the generated reactive power of one such
unit can be approximated by iiGi X V
2 Q (negative, since
the machine actually gets reactive power from the network),
with .ii V f X =
In order to integrate this kind of node in our method, wemay use tabulated values for ii V f X = , or a simple model
like 02 iii X V X , where is the value of the
magnetizing reactance at nominal voltage [9]. In any case,
we’ll use |V
0i X
i| to get Xi and then include it in the expression of
Ysi:
i )i( sconk
i sk si si X
jY y yY 1
∑
∈
(8”)
In this case, (8”) must be updated for all the PX nodes in
each iteration (for the remaining nodes, it is a constant). On
the other hand, if a battery is installed in the node, as it usually
happens, it may be included as Yi in (8”), as explained before.
No other changes in the algorithm are necessary.
V. ILLUSTRATION EXAMPLE
The performance of the method was tested with case
studies found in the literature.
First, a 12-bus network (data and results in [1]) was
calculated as a base case (12 PQ buses). Comparison with [1]
and other power flow calculation tools showed that the
method gives the correct results, as displayed in TABLE I.
Note that [1] (like many similar methods) only gives the
voltage magnitudes and total losses, while the new method
also calculates the voltage angles (displayed in degrees) and
therefore all the usual power flow results.
TABLE I
R ESULTS (12- NODE NETWORK )
Node
Voltage
(method)
Argument
(method)
Voltage
([1])
1 1.00000 0 1.00000
2 0.99433 0.116 0.99433
3 0.98903 0.223 0.98903
4 0.98058 0.402 0.98057
5 0.96982 0.629 0.96982
6 0.96654 0.698 0.96653
7 0.96375 0.758 0.96374
8 0.95531 1.011 0.95530
9 0.94728 1.242 0.94727
10 0.94446 1.318 0.94446
11 0.94356 1.342 0.94356
12 0.94335 1.349 0.94335
The new method was also tested against more demanding
systems, like a 69-bus network [10] [4] and an 85-bus rural
network [2]. In both cases, detailed data and results can be
found in the references and will not be repeated here.
TABLE II shows the number of iterations needed in these
cases, and also in the tests with PV and PX nodes that will be
mentioned later. All the results correspond to a convergence
error of 10-6 p.u. in the real and imaginary parts of the
voltages. Execution times are only indicative of the relativecomputational effort, since no optimized code was developed.
Informal tests that need further confirmation shown the new
method is faster than other methods for radial networks and
also faster than Newton-Raphson based methods, for the same
tolerance.
TABLE II
TESTS WITH DIFFERENT SYSTEMS
# nodes # iterations time (s)
12 PQ 5 < 0.005
69 PQ 6 0.005
85 PQ 6 0.016
11 + 1 PV 4 < 0.005
11 + 1 PX 7 < 0.005
In order to test the PV model, the last node of the 12-node
network was first reversed (still as a PQ node) to a “negative
load”. Then, the voltage magnitude was used as the specified
voltage for the PV test (that’s why V = 0.95258) is not a
“round” value). Results, shown in TABLE III, of the PV test
were validated against the preliminary PQ run and also with a
Newton-Raphson based tool. As expected, voltages increase
in all the nodes, due to the injection of reactive power in node
12.
sp12
TABLE III
R ESULTS ( NODE 12 AS PV)
Node Voltage Argument
1 1.00000 0.000
2 0.99477 0.107
3 0.98993 0.204
4 0.98231 0.365
5 0.97281 0.563
6 0.96995 0.623
7 0.96756 0.674
8 0.96066 0.876
9 0.95458 1.041
10 0.95276 1.083
11 0.95238 1.089
12 0.95258 1.082
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Finally, the PX model was also tested in a similar way,
showing again the ability of the method to deal with that type
of nodes, although a greater number of iterations was
necessary. Validation of these results is more difficult, since
usual methods do not include this feature, so we checked all
the power flow equations and also the relation between Q, V
and X in the asynchronous generators.
VI. ACKNOWLEDGMENT
The author would like to thank students Agostinho Sousa,
and Nuno Ribeiro, both from the Faculty of Engineering of
the University of Porto. Agostinho implemented the first
version of the code of the new method, and Nuno developed it
and helped in the test studies.
VII. R EFERENCES
[1] D. Das, H. S. Nagi and D. P. Kothari, "Novel method for solving radial
distribution networks", IEE Proc.Gen. Transm. Distrib., Vol. 141, pp.
291-298, July 1994.
[2] D. Das, D. P. Kothari and A. Kalam, "Simple and efficient method for
load flow solution of radial distribution networks", Electric Power & Energy Systems, Vol. 17, pp. 335-346, 1995.
[3] A. G. Expósito and E. R. Ramos, “Reliable Load Flow Technique for
Radial Distribution Networks”, IEEE Trans. Power Systems, Vol. 14,
No. 3, pp. 1063-1069, August 1999.
[4] S. Ghosh, D. Das, "Method for load flow solution of radial distribution
networks", IEE Proc.-Gener. Transm. Distrib., Vol. 146, No. 6,
November 1999
[5] S. F. Mekhamer, S. A. Soliman, M. A. Mostafa, M. E. El-Hawary, “Load
Flow Solution of Radial Distribution Feeders: A new Approach”, in
Proc. 2001 IEEE Porto Power Tech., Vol.3 , Porto, September 2001.
[6] A. Augugliaro, L. Dusonchet, M. G. Ippolito, E. R. Sanseverino, “An
efficient iterative method for load-flow solution in radial distribution
networks”, in Proc. 2001 IEEE Porto Power Tech., Vol.3 , Porto,
September 2001.
[7] V. Miranda, M. Matos, J. P. Lopes J. T. Saraiva, J. N. Fidalgo, M. T.
Ponce de Leão, “Intelligent tools in a real-world DMS environment”,
Proc. IEEE Power Engineering Society Summer Meeting , Vol: 1, pp:
163-168, 2000.
[8] J.P. Lopes, F. M. Barbosa, J. C. Pidré, “Operation simulation of MV
distribution networks with asynchronous local generation sources”,
Proc. MELECON’91, May 1991.
[9] J. C. Pidré, J. M. Velasco, J.P. Lopes, F. M. Barbosa, “Modeling of non-
linear nodal admittances in load-flow analysis”, Proc. IFAC Symposium
on Power Plants and Systems, Munich, March 1992.
[10] M. Chakravorty, D. Das, "Voltage stability analysis of radial distribution
networks", Electrical Power & Energy Systems 23 (2001) 129-135
VIII. BIOGRAPHY
Manuel A. Matos (El. Eng., Ph.D., Aggregation)
was born in 1955 in Porto (Portugal). He is presently
Full Professor at the Faculty of Engineering of theUniversity of Porto, Portugal, and the Manager of the
Power Systems Unit of INESC Porto. He also
collaborates with the Management School of the
University of Porto. His research interests include
fuzzy modeling of power systems, optimization and
decision-aid methods. He is a member of IEEE.