Novel method for solving radial distribution networks

D. Das H.S. Nagi D.P. Kothari

Indexing terms: Distribution loaddpow, Mathematical techniques, Radial networks

Abstract: The paper presents a novel method for solving radial distribution networks. The radial feature of the network has been fully exploited to develop an algorithm by a unique lateral, node and branch numbering scheme. The proposed method involves only the evaluation of simple algebraic voltage expressions without any trigono- metric functions. Thus, computationally, the pro- posed method is very efficient and requires less computer memory storage as all data is stored in vector form. The proposed method can easily handle different types of load characteristics. Several Indian rural distribution networks have been successfully solved using the proposed method.

Nomenclature

N B = total number of nodes (j) = branch number, j = 1,2, . . . , N B - 1 PL(i) = real power load of ith node QL(i) = reactive power load of ith node I V(i) 1 = voltage magnitude of ith node R ( j ) = resistance ofjth branch X ( j ) = reactance of jth branch I ( j ) = current flowing through branch j P(i + 1) = total real power load fed through node i + 1 Q(i + 1) = total reactive power load fed through node

6(i + 1) = voltage angle of node i + 1 LP(j) = real power loss of branch j LQ(j) = reactive power loss of branchj N L = total number of laterals [L] = lateral number, L = 1,2, ..., N L SN(L) = source node of lateral L EB(L) = end node of lateral L LB(L) = node, just ahead of source node of lateral L F(i) = integer variable

i + 1

1 Introduction

Not much work has been carried out on load flow analysis of distribution networks. Generally distribution networks are radial and the R / X ratio is very high. For

0 IEE, 1994 Paper 9966C (W), first received 17th February and in revised form 16th November 1993 D. Das and H.S. Nagi are with TERI, New Delhi, India D.P. Kothari is with IIT, Delhi, New Delhi, India

I E E Proc.-Gener. Transm. Distrib., Vol. 141, No. 4, July 1994

this reason distribution networks are ill-conditioned, and conventional Newton Raphson (NR) and fast decoupled load flow (FDLF) methods [l, 2, 15, 161 are inefficient in solving such networks.

Many researchers [3-51 have suggested modified ver- sions of the conventional load flow methods for solving ill-conditioned power networks. Recently some researchers have paid much attention to obtaining solu- tions for distribution networks.

Kersting and Mendive [6] and Kersting [7] have pre- sented a load flow technique based on the ladder network theory. Shirmohammadi et al. [SI have presented a com- pensation based power flow method for weakly meshed distribution and transmission systems. Baran and Wu [9] and Chiang [lo] have obtained the load flow solution in a distribution system by the iterative solution of three fundamental equations representing real power, reactive power and voltage magnitude. Renato [ l l ] has proposed a method for obtaining the load flow solution of radial distribution networks. His technique seems to be quite promising because it solves for bus voltage magnitudes only. Goswami and Basu [12] have presented a direct solution method for solving radial and meshed distribu- tion networks. However, the main limitation of their method is that no node in the network is the junction of more than three branches. Jasmon and Lee [13, 141 have proposed a new load flow method for obtaining the solu- tion of a radial distribution network using a single-line equivalent.

In India all the 11 kV rural distribution feeders are radial and too long. The voltages at the far end of many such feeders are very low with very high voltage regula- tion. Many of these practical rural distribution feeders have failed to converge while using NR and FDLF methods. Therefore, in this paper the main motivation of the authors has been to develop a new load flow tech- nique for radial distribution networks by using a unique lateral, node and branch numbering scheme. The pro- posed method solves a recursive relation of voltage mag- nitude and can be called a ‘forward sweeping method‘. Computationally the proposed method is very efficient. Another advantage of the proposed method is that all necessary data can be stored in vector form, thus saving a lot of computer memory. Convergence is always guar- anteed for any type of practical radial distribution network with realistic R / X ratios while using the pro- posed foward sweeping method. Loads have been rep- resented as constant power. However, composite load modelling can also be considered. The proposed load flow technique has been implemented on an IBM PC-AT. Several practical rural radial distribution feeders in India have been successfully solved using the proposed method.

29 1

The relative speed and memory requirements of the pro- posed method have been compared with NR and FDLF methods.

2 Assumptions

A balanced three-phase radial distribution network is assumed and can be represented by its equivalent single- tine diagram. Line shunt capacitance is negligible at the distribution voltage levels.

3 Solution methodology

3.1 Case 1 : Radial main feeder case Consider a distribution system consisting of a radial main feeder only. The one line diagram of such a feeder comprising n nodes and R - 1 branches is shown in Fig. 1. Fig. 2 shows the electrical equivalent of Fig. 1. From Fig. 2, following equations can be written

3 2 ) - jQ(2) = V*(2)1(1)

From eqns. 1 and 2 we have

I V(2) I = C{(P(2)R(l) + Q(W(1) - 0.5 I U1) 12)2 - (R2(1) + X2(l))(P2(2) + Q2(2))}’”

-(P(2)R(1)+ Q(2)X(1)-0.51 F‘(1)12)11’2 (3)

Eqn. 3 can be written in generalised form

I V(i + 1)l = C{(P(i + W(i) + Q(i + 1)X(i)

- 0.5 I V(i) 1’)’ - (R2(i) + X2(i))(P2(i + 1) + Q2(i + 1)))’” - (P(i + 1)R(i)

+ Q(i + 1)X(i) - 0.5 I V(i) 12)]1’2 (4) Eqn. 4 is a recursive relation of voltage magnitude. Since the substation voltage magnitude I V(l)l is known, it is

Fig. 1 Radial main feeder

possible to 6nd out the voltage magnitudes of all other nodes. From Fig. 2, the total real and reactive power load fed through node 2 are given by

N B N E - 1

0 2 ) = Psi) + LP(i) i = 2 i = 2

N B

Q(2) = Qui) + ‘5 ‘LQ(i) i = 2 i = 2

From eqn. 5, it is clear that total load fed through node 2 is the load of node 2 itself plus the load of all other nodes plus the losses of all branches except branch 1. 292

The real and reactive power losses in branch 1 are

Eqn. 5 can be written in generalised form NE NE- 1

P(i+ I ) = PL(j)+ 1 LP(j) j = i + 1 j = i + l

f o r i = 1,2, ..., N B - 2 NII N B - 1 (7)

Q ( i + 1 ) = j = i + l QL(jJ+-Y-LQ(j) j = i + l

f o r i = 1,2, ..., N B - 2

and for the last node

P(NB) = PL(NB) Q(NB) = QYNB) Eqn. 6 can also be written in generalised form

R(i) * (P2(i + 1) + Q2(i + 1) LP(i) = I + 1)12

X(i) (P2(i + 1) + Q2(i + 1)) I V(i + 1)12

LQ(i) =

Initially, if LP(i + 1) and LQ(i + 1) are set to zero for all i, then the initial estimates of P(i + 1) and Q(i + 1) will be

N B

P(i + 1) =

Q(i + 1) =

1 PL(j)

QL(j) fori = 1, 2, ..., NB - 1

for i = 1,2, . . . , N B - 1 (10)

j = i + l

N B

j = i + l

Eqn. 10 is a very good initial estimate for obtaining the load flow solution of the proposed method.

The convergence criteria of the proposed method are that if the difference of real and reactive power losses in successive iterations in each branch is less than 1 Watt and 1 VAr, respectively, the solution has then converged. A flow chart of the proposed technique is given in Fig. 3.

Exumple I: Fig. 4 shows the single-line diagram of a physically existing rural distribution feeder. The system data are given in the Appendix. The solution of the load flow has been given in Table 1. It took three iterations to converge by the proposed method. The coupled NR method also takes three iterations to converge but the

Table 1 : Load flow solution of Examole 1 ~ ~~

Node no. Voltage

1 1 .oooo 2 0.99433 3 0.98903 4 0.98057 5 0.96982 6 0.96653 7 0.96374 8 0.95530 9 0.94727

10 0.94446 11 0.94356 12 0.94335

magnitude

Total real power loss = 20.71 kW Total reactive power loss = 8.04 kVAr Real power supplied from substation = 455.71 kW Reactive power supplied from substation = 41 3.04 kVAr

IEE proC.-Gener. T r a m . Distrib., Vol. 141, No. 4, July 1994

FDLF method fails to converge. However, the proposed method is 1.6 times faster than the coupled NR method and the memory requirement is only 27% of the coupled NR method.

from(D)

1.1.2.3, .NE1 by usingeqn 9

compute Dp(I)=LP(I)-PLoss(I)

f0ri:l. 2. ,NE1

no

solve eqn 4

write voltagemagnitude

feeder losses

1=1*1 9 to(D)

Fig. 3 Mainfeeder case

Flow chart for the algorithm of radial distribution network.

These two laterals are numbered as laterals 2 and 3, respectively. For lateral 2, it is seen that source node SN(2) = 2, node just ahead of source node LB(2) = 9, end node EB(2) = 10 and there are three nodes of lateral 2, including the source node (node 2). The remaining nodes

in 1

26

Fig. 5 Radial distribution feeder with laterals

are numbered as 9 and 10. For lateral 3, SN(3) = 2, LB(3) = 11, EB(3) = 14 and there are five nodes of lateral 3 including the source node (node 2). The remaining nodes are numbered from 1 1 to 14. The branch numbers of laterals 2 and 3 are also shown inside brackets (.) (Fig. 5). Next, we will examine node 3 of lateral 1. It is seen that only one lateral has come out from node 3 and this lateral is numbered as lateral 4. For lateral 4, source node SN(4) = 3, LB(4) = 15, EB(4) = 18 and nodes are num- bered from 15 to 18. Similarly, we have to examine each node of lateral 1 and lateral, source node, node just ahead of source node, end node, and branch numbering have to be completed by using the above mentioned tech- nique. Details are given in Table 2.

Table 2: Details of the numbering scheme of Fig. 5

Laterals number Source node Node just ahead End node SN(L) of source node €B(L)

LB(L)

Lateral 1 1 2 8

Lateral 2 2 9 10 Lateral 3 2 11 14 Lateral 4 3 15 18 Lateral 5 5 29 19 Lateral 6 5 20 22 Lateral 7 6 23 26

(main feeder)

Fig. 4 Single line diagram of a physically existing mainfeeder

3.2 Case 2: Main feeder with laterals

32.1 Technique of lateral, node and branching num- bering: Fig. 5 shows the single diagram of a radial dis- tribution feeder with laterals. The lateral number is shown inside square brackets [ .]. The branch number is also shown inside brackets (. ).

First, we will number the main feeder as lateral 1 (L = 1) and number the nodes and branches of lateral 1 (main feeder). For lateral 1, source node SN(l), node just ahead of source node LB(1) = 2 and end node EB(1) = 8. For lateral 1, there are eight nodes and seven branches.

Next, we will examine node 2 of lateral 1. It is seen that two laterals are directly connected with node 2.

IEE F'roc.-Genet. Transm. Disfrib., Vol. 141, No. 4, July 1994

After numbering each lateral and nodes we follow the steps described below. Let TP(L) = total real power load fed through the node LB(L) of lateral L. Let TQ(L) = total reactive power load fed through the node LB(L) of lateral L.

Generalised expressions for TP(L) and TQ(L) are given below :

for L = 1, 2, ..., N L (11) NNlLl NN(LI - 1

for L = 1, 2, ..., N L (12)

293

where

SPL(1) = real power load of node 2 which has just been left plus real power loss of branch 2 which has just been left.

SQL(1) = reactive power load of node 2 which has just been left plus reactive power loss of branch 2 which has just been left.

Generalised definitions of SPL(L) and SQL(L) are as follows:

SPL(L) = sum of real power loads of all the nodes of lateral L which have just been left plus the sum of real power losses of all the branches of lateral L which have just been left except the real power loss in branch {LB(L) - 1 ) of lateral L.

SQL(L) = sum of reactive power loads of all the nodes of lateral L which have just been left plus the sum of reactive power losses of all the branches of lateral L which have just been left except the reactive power loss in branch {LB(L) - 1 ) of lateral L.

Next, we have to obtain the value of K (Fig. 6). In this case K = 0 + F(2) = 2. K = 2 indicates that we have just left two laterals (laterals 2 and 3). After that we have to check whether F(2) is positive or not? But in this case F(2) > 0. Therefore it will compute PS(1) and QS(1) (Fig. 6)

K = 2 PS(1) = 0 + BP(1 + 1,) = BP(2) + BP(3)

I l = P , = 1

K = 2

I 3 = P * = 1 QS(1) = 0 + BQ(1 + I , ) = BQ(2) + BQ(3)

Generalised definitions of PS(L) and QS(L) are as follows: PS(L) = sum of the real power loads of all the nodes

(except source nodes) of all the laterals which have just been left plus the sum of the real power losses of all the branches of all the lat- erals which have just been left.

QS(L) = sum of the reactive power loads of all the nodes (except source nodes) of all the laterals which have just been left plus the sum of the reactive power losses of all the branches of all the laterals which have just been left.

BP(2), BP(NL) and BQ(2), BQ(NL) can easily be com- puted from eqn. 13 and P , = P , + F(2) = 1 + 2 = 3.

Therefore, real and reactive power loads fed through the node 3 are given as:

P(3) = T P ( 1 ) - PS(1) - SPL(1)

= T P ( 1 ) - BP(2) - BP(3) - PL(2) - "(2)

and

where

" ( 1 ) = N B

"(2) = EB(2) ..

N N ( L ) = EB(L)

We will also define two more variables BP(L) and BQ(L). BP(L) = sum of the real power loads of all the nodes

of lateral L (except source node) plus sum of all the branch real power losses of lateral L (L > 1).

BQ(L) = sum of the reactive power loads of all the nodes of lateral L (except source node) plus s u m of all the branch reactive power losses of lateral L ( L > 1) .

Generalised expressions for BP(L) and BQ(L) are given below :

fiB(L) fiB(L) - 1

i = LB(L) BP(L) = PL(i) + 1 LP(i)

i = LB(L)

(13) for L = 2, 3, . . . , N L

fiB(L) EB(L) - 1

BQW) = QUO + 1 LQ(O i = LB(L) i = LB(L)

for L = 1 , 2, ..., N L

Now we will define one integer variable F(i), i = 1 , 2, . . . , N B - 1 , the meaning of which is as follows:

From Fig. 5, it can be seen that six laterals are con- nected with different nodes of lateral 1 (main feeder). Lat- erals 2 and 3 are connected with node 2, i.e. two laterals are connected with node 2, therefore F(2) = 2. Only one lateral is conected with node 3, i.e. F(3) = 1 . Similarly other values of F(i) can easily be obtained. From Table 3

Table 3: Nonzero integer values of F( i )

Source node F( i ) S N U )

2 F ( 2 ) = 2 3 F ( 3 ) = 1 5 F ( 5 ) = 2 6 Ff6) = 1

it is clear that F(i) is positive only at the source nodes { i = SN(L), L > 1). Other values of F(i) are zeros.

The significance of TP(L), TQ(L), BP(L), BQ(L) and F(i) will be described in Section 3.2.2.

3.2.2 Explanation of the proposed algorithm: From Fig. 5 it is seen that for L = 1, total real and reactive power loads fed through node 2 are TP(1) and TQ(1) (eqn. 1 1 ) . At any iteration voltage magnitude of node 2 can easily be obtained by using eqn. 4 {P(2) = TP(1) and Q(2) = T Q ( l ) } . After solving the voltage magnitude of node 2 one has to obtain the voltage magnitude of node 3 and so on. Before proceeding to node 3, we will define here four more variables which are extremely important for obtaining exact load feeding through nodes 3, 4, ..., EB(1) of lateral 1 or in general obtaining exact load feeding through LB(L) + 1 , LB(L) + 2, ... ., EB(L) of lateral L. It is seen from the flow chart (Fig. 6) that

SPL(1) = 0 + PL(2) + LP(2) = PL(2) + LP(2)

SQL(1) = 0 + Q Y 2 ) + LQ(2) = QW) + 294

After computing P(3) and Q(3), eqn. 4 has to be solved to obtain the voltage magnitude at node 3. Before obtaining the voltage magnitude of node 4, computer logic will perform the following computations (Fig 6):

SPL(1) = PL(2) + LP(2) + PL(3) + LP(3)

S Q W ) = QW) + LQU + Q U 3 ) + LQ(3) and K = 2 + F(3) = 2 + 1 = 3.

IEE Proc.-Gener. Transin. Distrib., Vol. 141, No. 4, July 1994

read initialise

M i ) I,linepammeters LQ(i)= 0 55vdtagemagNtude LP(i)= 0

~

gnd load data fori4.2. NB-1

SPL(L).O.O sQL(U.0 0

QS(L)=O.O

- compute TP(L)and TQ( L) byusingeqn 11

+ IT:1 -

4 to (A)

to(B) t IV(Kl)I=IV(K2)1

- set set

from(8)

K=O 1.1 PLoss(I)=LP(I) Pl=l L.1 - QLOSS(I):LQ(O

P2.1 fwt-1.2. NB-1

Fig. 6 Flow chart for the algm’thm of radial distribution networks having laterals and sublaterals

compute TP(L)and TQ(L) byusingeqn. 13

Next it will check whether F(3) is positive or not? But Total real and reactive loads fed through the node 4 are:

P(4) = TP(1) - PS(1) - SPL(1) F(3) = 1, therefore K = 3

Is=Pi = 3 PS(1) = BP(2) + BP(3) + 1 BP(1 + I , ) = TP(1) - BP(2) - BP(3) - BP(4)

- PL(2) - LP(2) - PL(3) - LP(3) = BP(2) + BP(3) + BP(4)

Q(4) = TQ(1) - Qs(1) - SQL(1)

= TQ(1) - BQ(2) - BQ(3) - BQ(4) QS(1) = BQ(2) + BQ(3) + Ki3 BQ(1 + 13) 1 3 =Pi = 3

= BQQ) + BQ(3) + BQ(4) - QL(3 - LQ(2) - QL(3) - LQW

295 I E E Proc.-Gew. Transm. Distrib., Vol. 141, No. 4, July 1994

--

and solve eqn. 4 for obtaining the voltage magnitude of node 4.

For lateral 1 (L = 1, main feeder) similar computa- tions have to be repeated for all the nodes. At any iter- ation, after solving the voltage magnitudes of all the nodes of lateral 1 one has to obtain the voltage magni- tudes of all the nodes of laterals 2, and so on. Before solving voltage magnitudes of all the nodes of lateral 2 the voltage magnitude of all the nodes of lateral 1 is stored in the name of another variable, say V1, i.e. I V l ( J ) I = 1 V(J) I for J = P , to EB(1) (Fig. 6). For lateral 1 (main feeder) P , = 1 and EB(1) = 8. For lateral 2,

K 2 = SN(L) = SN(2) = 2, or I V(8)I = I V(2)I and solve the voltage magnitudes of all the nodes of lateral 2 using eqn. 4. The proposed com- puter logic will follow the same procedure for all the lat- erals. This will complete one iteration. After that it will compute total real and reactive power losses and update the loads. This iterative process continues until the solu- tion converges.

3.3 Case 3: Main feeder with laterals and sublaterals The proposed technique described above can easily be applied to any radial distribution network consisting of any numer of laterals and sublaterals.

Fig. 7 shows another radial distribution network having many laterals and sublaterals. However, for ease of explanation we will also call a sublateral a lateral.

We will now explain the laterals and sublaterals num- bering technique and the integer variable F(i). Here main feeder is numbered as lateral 1. For lateral 1 (L = l), source node SN(1) = 1, node, just ahead of source node LB(1) = 2 and end node EB(1) = 8. Next, we examine

P , = E B ( l ) + 1 = 8 + 1 = 9 . L = L + 1 = 1 + 1 = 2 , 1 V(EB(1)) I = I V ( K 2 ) I

12

t

25

Fig. 7

296

Radial distribution network with laterals and sublaterals

node 2. It is seen that two laterals have come out from node 2 and many laterals (sublaterals) are connected with these two laterals. We number one of these two laterals as lateral 2. For the lateral 2, SN(2) = 2, LB(2) = 9 and EB(2) = 12. Next we must check all nodes of lateral 2 to see whether any lateral (sublateral) is connected. It is seen that at node 1 1 of lateral 2 one lateral (sublateral) is con- nected. This lateral is numbered as lateral 3. For lateral 3, S N ( 3 ) = 11, LB(3) = 13 and EB(3) = 13. Another lateral also has come out from node 2 of lateral 2 (main feeder) and is numbered as lateral 4. For lateral 4, SN(4) = 2, LB(4) = 14, and EB(4) = 18.

Now we must examine each node of lateral 4 and it is seen that from node 15 one lateral (sublateral) has come out and this lateral is numbered as lateral 5. For lateral 5, SN(5) = 15, LB(5) = 19 and EB(5) = 19. Next, it is also seen that from node 16 of lateral 4 another lateral (sublateral) has come out. This lateral is numbered as lateral 6. For lateral 6, SN(6) = 16, LB(6) = 20 and EB(6) = 22. Again it is seen that from node 20 of lateral 6, one more lateral (sublateral) has come out and this lateral is numbered as lateral 7 and for lateral 7, SN(7) = 20, LB(7) = 23 and EB(7) = 25. After numbering all the laterals directly or indirectly associated with node 2, we have to examine node 3 and this lateral, node and branch numbering technique will be the same as described above. The details of numbering schemes of Fig. 7 are shown in Table 4.

Table 4: Details of the numberina scheme of Fia. 7

Laterals Source Node, just ahead End node Number node of source node €B(L) ( L ) SN(L) LB(L)

Lateral 1 1 2 8

Lateral 2 2 9 12 Lateral 3 11 13 13 Lateral 4 2 14 18 Lateral 5 15 19 19 Lateral 6 16 10 22 Lateral 7 20 23 25 Lateral 8 3 26 27 Lateral 9 4 28 30 Lateral 10 6 31 31

32 6 32 Lateral 11 Lateral 12 6 33 33

(main feeder)

Now we have to obtain nonzero integer values of F(i). First we examine node 2 in Fig. 7. It is seen that two laterals have come out from node 2 of lateral 1, one lateral has come out from node 11 of lateral 2, two later- als have come out from nodes 15 and 16 of lateral 4 and one more lateral has come out from node 20 of lateral 6. Therefore, two laterals are directly associated with node 2 and four laterals (sublaterals) are indirectly associated with node 2. So F(2) = 2 + 4 = 6. Similarly, F(3) = 1, F(4) = 1, F(6) = 3, F(11) = 1, F(15) = 1, F(16) = 2 and F(20) = t . Other values of F(i) are zeros.

In this case there is no need to change the computer logic as given in the flow chart (Fig. 6). Here we have to change only eqn. 12. Eqn. 12 as described in the preced- ing Section can be written as

"(1) = NB

"(2) = EB(3)

"(3) = EB(3)

"(4) = EB(7)

"(5) = EB(5) IEE Proc.-Gener. Transm. Distrib., Vol. 141, No. 4, July I994

“(6) = EB(7)

“(7) = EB(7)

“(8) = EB(8)

“(9) = EB(9)

“(10) = EB(10)

“(11) = EB(11)

“(12) = EB(12)

From the above discussions it is clear that any type of radial distribution network can be described by the above mentioned technique. We have solved several rural distribution feeders by using proposed algorithm.

Example 2: The system data for a physically existing 28-node rural distribution feeder is given in the Appen- dix. Table 5 gives the load flow solution. It has taken four iterations to converge by the proposed method. The coupled NR method takes three iterations to converge, whereas FDLF method takes seven iterations to con- verge. However, the proposed method is 2.2 times faster than the coupled NR and 4.3 times faster than the FDLF methods. Memory requirement is 25% of the coupled NR and 50% of the FDLF methods.

Table 5: Load flow solution of Example 2

Node no. Voltage Node no. Voltage magnitude magnitude

1 (substation) 1.OOOOOO 15 0.94279 2 0.98621 16 0.93704 3 0.96644 17 0.92585 4 0.95233 18 0.92487 5 0.93817 19 0.92320 6 0.92763 20 0.92234 7 0.91846 21 0.921 70 8 0.91600 22 0.91 557 9 0,91572 23 0,91403

10 0.91547 24 0.91 286 11 0.94614 25 0.91 260 12 0.94437 26 0.91 243 13 0.94331 27 0.91 550 14 0.94303 28 0.91 538

Total real power loss = 68.8458 kW Total reactive power 1065 = 46.0695 kVAr Total real power supplied from substation = 829.8858 kW Total reactive power supplied from substation = 822.0695 kVAr

4

All loads, including shunt capacitors for reactive power compensation were represented by their active (Po) and reactive (eo) components at 1.0 per unit. The effect of voltage variation is represented as follows:

Additional application of the proposed method (load modelling)

P = POI VIk (14)

Q = Qol VIk

where I VI is the voltage magnitude and k = 0, 1, and 2 for constant power, constant current and constant impedance loads, respectively.

The value of k may be different according to the load characteristics. The load flow solution depends on the type of real and reactive loads. It is extremely easy to

IEE Proc.-Gmer. T r a m . Dism’b., Vol. 141, No. 4, July 1994

include real and reactive power loads in the proposed algorithm. For constant current and constant impedance loads, real and reactive power have to be computed after every iteration.

5 Conclusions

A novel load flow technique, named ‘forward sweeping method’, has been proposed for solving radial distribu- tion networks. It completely exploits the radial feature of the distribution network. A unique lateral, node and branch numbering scheme has been suggested which helps to obtain the load flow solution of the radial dis- tribution network. The forward sweeping method always guarantees convergence of any type of practical radial distribution network with a realisic R / X ratio. Computa- tionally, the proposed method is extremely efficient, as compared to the coupled NR and FDLF methods, as it solves simple algebraic recursive expressions of voltage magnitude only. Another advantage of the proposed method is that all data can be stored in vector form, thus saving an enormous amount of computer memory. The method can easily handle the composite loads if the break up of the loads is known. The proposed method has been implemented on an IBM PC-AT. Several Indian rural distribution networks have been successfully solved using the proposed forward sweeping method.

6 References

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2 S T O n , B., and ALSAC, 0.: ’Fast decoupled load flow’, IEEE Trans., 1974, PAS-93, pp. 859-869

3 RAJICIC, D., and BOSE, A.: ‘A modification to the fast decoupled power flow for networks with high R/X ratios’, IEEE Tram., 1988, PWRS-3, pp. 743-746

4 IWAMOTO, S., and TAMURA, Y.: ‘A load flow calculation method for ill-conditioned power systems’, IEEE Trans., 1981, PAS-100, 1736-1713

S TRIPATHY, S.C., DURGAPARASAD, G., MALIK, O.P., and HOPE, G.S.: ‘Load flow solutions for ill-conditioned power system by a Newton like method‘, IEEE Trans., 1982, PAS-101. pp. 3648- 3657

6 KERSTING, W.H., and MENDIVE, D.L.: ‘An application of ladder network theory to the solution of three phase radial load flow problem’. IEEE PES winter meeting, New York, January 1976, paper A 76 044-8

7 KERSTING, W.H.: ‘A method to teach the design and operation of a distributon system’, IEEE Trans., 1984, PAS-103, pp. 1945-1952

8 SHIRMOHAMMADI, D., HONG, H.W., SEMLYEN, A., and LUO, G.X.: ‘A compensation-based power flow method for weakly meshed distribution and transmission networks’, IEEE Trans., 1988, PWRS-3, pp. 753-762

9 BARAN, M.E., and WU, F.F.: ‘Optimal sizing of capacitors placed on a radial distribution system’, IEEE Trans., 1989, PWRD-2, pp. 735-743

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297

Elecn. Power & E M g y Sj’St., 1991,13, (I), pp. 9-13

7 Appendix

Table 6: Line data of Example 1

Branch Sending Receiving R (ohms) X (ohms) no end end

1 1 2 2 2 3 3 3 4 4 4 5 5 5 6 6 6 7 7 7 8 8 8 9 9 9 10

10 10 11 11 11 12

1.093 1.184 2.095 3.1 88 1.093 1.002 4.403 5.642 2.89 1.514 1.238

0.455 0.494 0.873 1.329 0.455 0.41 7 1.21 5 1.597 0.81 8 0.428 0.351

Table 7: Load data of Example 1

Node no. PL (kW) OL (kVAr)

1 0 0 2 60 60 3 40 30 4 55 55 5 30 30 6 20 15 7 55 55 8 45 45 9 40 40

10 35 30 11 40 30 12 15 15

Table 8: Line data of Example 2

Branch Sending Receiving R (ohms) X (ohms) no. end end

1 1 2 1.197 0.82 2 2 3 1.796 1.231 3 3 4 1.306 0.895 4 4 5 1.851 1.268 5 5 6 1.524 1.044 6 6 7 1.905 1.305 7 7 8 1.197 0.82 8 8 9 0.653 0.447 9 9 10 1.143 0.783

10 4 11 2.823 1.172 11 11 12 1.184 0.491 12 12 13 1.002 0.416 13 13 14 0.455 0.189 14 14 15 0.546 0.227 15 5 16 2.55 1.058 16 6 17 1.366 0.567 17 17 18 0.819 0.34 18 18 19 1.548 0.642 19 19 20 1.366 0.567 20 20 21 3.552 1.474 21 7 22 1.548 0.642 22 22 23 1.092 0.453 23 23 24 0.91 0.378 24 24 25 0.455 0.189 25 25 26 0.364 0.151 26 8 27 0.546 0.226 27 27 28 0.273 0.113

Table 9: Load data of ExamDle 2

Node no.

1 2 3 4 5 6 7 8 9

10 11 12 13 14

PL (kW)

0 35.28 14 35.28 14 35.28 35.28 35.28 14 14 56 35.28 35.28 14

Node no.

15 16 17 18 19 20 21 22 23 24 25 26 27 28

PL (kW)

35.28 35.28 8.96 8.96

35.28 35.28 14 35.28 8.96

56 8.96

35.28 35.28 35.28

Power factor of the load is taken as cos cp = 0.7 and reactive power load OL = PL tan cp.

298 I E E Proc.-Gew. Transm. Distrib., Vol. 141, No. 4, July 1994