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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

mailto:[email protected]:[email protected]


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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.

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