Nov 5, 1998 - Method for load-flow solution of radial distribution networks. S.Ghosh and D.Das. Abstract: A simple and efficient method for solving radial ...
Method for load-flow solution of radial distribution networks S.Ghosh and D.Das Abstract: A simple and efficient method for solving radial distribution networks is presented. The proposed method involves only the evaluation of a simple algebraic expression of receiving-end voltages. Computationally, the proposed method is very efficient. The effectiveness of the proposed method is demonstrated through three examples.
List of symbols NB LN1 j PL(9 QL(z) V(z) R(ii) Xh]
= = = = =
Zh) Zu)
=
IL(9 LPh) LQ(ii)
=
IScii)
=
IRW yo(z)
= =
ZC(9
=
= =
= = = =
DVMAX = 1
total number of nodes total number of branches branch number,j = 1,2, 3, ..., LN1 real power load at ith node reactive power load at ith node voltage of ith node resistance o f a h branch reactance o f a h branch impedance of branchjj current that flows through branchj load current of node i real-power loss of branchjj reactive-power loss of branchj sending-end node of branchj receiving-end node of branchj charging admittance at node i charging current at node i maximum voltage dfference
Introduction
Little attention has been given to load-flow analysis of distribution systems, unlike load-flow analysis of transmission systems. However, some work has been carried out on load-flow analysis of distribution networks, but the choice of a solution method for a practical system is often dsicult. Generally, distribution networks are radial and the R/ Xratio is very high. For t h s reason, conventional NewtonRaphson (NR) [l] and fast decoupled load-flow [2] methods do not converge. Many researchers have suggested modfied versions of the conventional load-flow methods for solving power networks with hgh R/X ratio [3-51. Kersting and Mendive [q and Kersting [I have developed a load-flow techque for solving radial distribution networks using ladder-network theory. They have OIEE, 1999 ZEE Proceedhgs online no. 19990464 DO1 l O . l 0 4 9 / i p g t d : l ~ Paper fmt received 5th November 1998 and in revised form 26th March 1999 The authors are with the ElectricalEngineering Department, Indian Institute of T E ~ o I o ~Kharagpur Y, - 721302, WB, India IEE Proc.-Gener. Transm. Distrib.. Vol. 146, No. 6, November 1999
developed the ladder technique from basic ladder-network theory into a working algorithm, applicable to the solution of radial load-flow problems. Stevens et al. [8] have shown that the ladder technique is found to be fastest but did not converge in five out of 12 cases studied. Shirmohammadi er al. [9] have proposed a method for solving radial distribution networks based on the direct application of Kirchhoff's voltage and current laws. They have developed a branch-numbering scheme to enhance the numerical performance of the solution method. They have also extended their method for solution of weakly meshed networks. Baran and Wu [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. They have computed the system Jacobian matrix using a chain rule. In their method, the mismatches and the Jacobian matrix involve only the evaluation of simple algebraic expressions and no trigonometric functions. They have also proposed decoupled and fast decoupled distribution load-flow algorithms. Chiang [l 11 has also proposed three different algorithms for solving radial distribution networks based on the method proposed by Baran and Wu [lo]. He has proposed decoupled, fast decoupled and very fast decoupled distribution load-flow algorithms. In fact decoupled and fast decoupled dlstribution load-flow algorithms proposed by Chiang [I 13 are similar to that of Baran and Wu [lo]. However, the very fast decoupled distribution load flow proposed by Chiang [111 is very attractive because it does not require any Jacobian matrix construction and factorisation. Renato [12] has proposed one method for obtaining a load-flow solution of radial distribution networks. He has calculated the electrical equivalent for each node summing all the loads of the network fed through the node including losses and then, starting from the source node, the receiving-end voltages of all the nodes are calculated. Goswami and Basu [13] have presented a direct method for solving radial and meshed distribution networks. However, the main h t a t i o n of their method is that no node in the network is the junction of more than three branches, i.e. one incoming and two outgoing branches. Jasmon and Lee [14, 151 have proposed a new load-flow method for obtaining the solution of radial distribution networks. They have used the three fundamental equations representing real power, reactive power and voltage magnitude derived in [lo]. They have solved the radial distribution network using these three equations by reducing the whole network into a single h e equivalent. 641
Das et al. [16] have proposed a load-flow technique for solving radial distribution networks by calculating the total real and reactive power fed through any node. They have proposed a unique node, branch and lateral numbering scheme which helps to evaluate exact real- and reactivepower loads fed through any node and receiving-end voltages. In this paper, the main aim of the authors has been to develop a new load-flow technique for solving radial distribution networks. The proposed method involves only the evaluation of a simple algebraic expression of receiving-end voltages. The proposed method is very efficient. It is also observed that the proposed method has good and fast convergence characteristics. Loads in the present formulation have been presented as constant power. However, the proposed method can easily include composite load modelling, if the composition of the loads is known. Several radial distribution 'feeders have been solved successively by using the proposed method. The speed requirement of the proposed method has also been compared with other existing methods. 2
Assumption
It is assumed that the three-phase radial distribution networks are balanced and can be represented by their equivalent single-line diagrams. 1
2
3
6
5
4
similarly for branch 2,
V ( 3 )= V ( 2 )- 1 ( 2 ) 2 ( 2 )
(2)
As the substation voltage V(1) is known, so if Z(l) is known, i.e. current of branch 1, it is easy to calculate V(2) from eqn. 1. Once V(2) is known, it is easy to calculate V(3) from eqn. 2, if the current through branch 2 is known. Similarly, voltages of nodes 4, 5, ..., NB can easily be calculated if all the branch currents are known. Therefore, a generalised equation of receiving-end voltage, sending-end voltage, branch current and branch impedance is
V(m2)= V(rn1) - I ( j j ) Z ( j j ) where j j is the branch number. m2 = I R ( j j )
(3)
(4)
ml = IS(jj)
(5) Eqn. 3 can be evaluated forjj = 1, 2, ..., LN1 (LN1 = NB - 1 = number of branches). Current through branch 1 is equal to the sum of the load currents of all the nodes beyond branch 1 plus the sum of the charging currents of all the nodes beyond branch 1, i.e. LN1
LN1
IL(2) +
I(1)=
IC(2)
(6)
i=2
i=2
The current through branch 2 is equal to the sum of the load currents of all the nodes beyond branch 2 plus the sum of the charging currents of all the nodes beyond branch 2, i.e.
I
12
+
1
+ IL(6)
+ I C ( 3 ) + IC(4) + I C ( 5 ) + I C ( 6 ) +
Fig. 1 Single-line diagram of radial distribution network
3
+
I(2)= IL(3) IL(4) IL(5) + IL(10) + IL(11)
11
Solution methodology
Fig. 1 shows single-line diagram of a distribution feeder. The branch number sending-end and receiving-end node of this feeder are given in Table 1. Consider branch 1. The receiving-end node voltage can be written as
V ( 2 )= V ( l ) - 1(1)2(1)
+
IC(10) IC(11) (7) Therefore, if it is possible to identify the nodes beyond all the branches, it is possible to compute all the branch currents. Identification of the nodes beyond all the branches is realised through an algorithm as explained in Section 4. The load current of node i is
(1)
Table 1: Branch number (jj, sending-end (ml /S(jj))node, receiving-end node (m2 = (/R(jll) and nodes beyond branches 1,2,3. ..., 11 of Fig. 1 Branch number ( j j
642
Sending end
rnl = /S(hl
1
1
2 3 4 5 6 7 8 9 10 11
2 3 4 5 2 7
a 4 10 8
Receiving end m 2 = /R(jj
2 3 4 5 6 7 8 9 10 11 12
Nodes beyond branch j j
2,3,7,8,5,10,9,12,6,11 3, 4,5,IO, 6,1 1 4,5, IO, 6,1 1 5,6 6 7,8,9,12 8,9,12 9 IO, 1 1 11 12
Total number of nodes Mjl) beyond branch j
11
6 5 2 1 4 3 1 2 1
1
IEE Proc.-Gener. Transm. Disfrib., Vol. 146, No. 6,November 1999
First define the variables: j j = 1, 2, 3, ..., LN1 cij indicates branch of Fig. 1, see also Table 1); ip is the node count (identifies the number of nodes beyond a particular branch); ZK(ip) is the node identifier (helping to identify nodes beyond all the branches); Mjj] is the total number of nodes beyond branchjj; and IEGJ, ip + 1) is the receiving-end node. ZE(J, ip + 1) will now be explained. Consider the fxst branch in Fig. 1, i.e. j j = 1; the receiving-end node of branch 1 is 2, i.e. ZRb] = ZR(1) = 2. Therefore ZEb, ip + 1) = ZE(1,ip + 1) will help to identify all the nodes beyond branch 1. This will help to find the exact current flowing through branch 1. S i a r l y , consider branch 2, i.e. jj = 2; the receiving-end node of branch 2 is 3, i.e. IR(jj] = ZR(2) = 3. Therefore, E ( J , ip + 1) = ZE(2, ip + 1) will identify all
The charging current at node i is
i = 2 , 3 , . . . ,N B
I C ( i )= yo(i)V(i) (9) Load currents and charging currents are computed iteratively. Initially, a flat voltage of all the nodes is assumed and load currents and charging currents of all the loads are computed using eqns. 8 and 9. A detailed load-flow-calculation procedure is described in Section 5. The real and reactive power loss of branchjj are given by: Wjj)=
IW)12Wj)
L Q ( j j ) = Il(jj)12X(jj) 4
(10)
(11)
Identificationof nodes beyond all the branches
Before the detailed algorithm is given, the details of the methodology of identifying the nodes beyond all branches will be discussed. This wdl help in finding the exact current flowing through all the branches.
(7) from A
read sending-end and receiving-end nodes and total number of nodes and branches
in 5 ip to F no
ni iil nc=l
1
k=jj+l
I
no
I II
from B
.( IE(jj, ip+l)=lR(jj)
from E
iP =i P+1 IK(ip)=i
from C
Li nc=O
IE(jj, ip+l)=iR(jj) N(jj) =ip+ 1
to E
L, d
r -
i=i+l
in=i
f IE(jj, ip+l)=IR(jj) N(jj)=ip+1
IE(LN,I)=IR(LNI) N(LNl) = 1
1
stop
Fig.2 IEE Proc.
643
the nodes beyond branch 2. This will help to compute the exact current flowing through branch 2. For each node identitication beyond a particular branch, '@' will be incremented by 1. Note here that before identification of nodes beyond a particular branch, '@' has to be reset to zero. For jj = 1 (first branch of Fig. 1 , Table I), ZRG) = ZR(1 ) = 2; check whether IR(1)= ZS(i) or not for i = 2, 3, 4, ..., LN1. It is seen that ZR(1)= ZS(2) = 2, IR(1)= IS(6) = 2; the corresponding receiving-end nodes are ZR(2) = 3 and ZR(6) = 7. Therefore, IE(1,1) = 2, IE(1, 2) = 3 and ZE(1, 3) = 7 . Note that there should not be any repetition of any node while identifying nodes beyond a particular branch, and this logic has been incorporated in the proposed algorithm and further explained in the flowchart given in Fig. 2. From the above discussion, it is seen that node 2 is connected to nodes 3 and 7. Smilarly, the proposed logic will identify the nodes which are connected to nodes 3 and 7. First it w d check whether node 3 appears in the left-hand column of Table 1 . It is seen that node 3 is connected to node 4. Therefore, ZE(1, 4) = 4. Then it will check whether node 7 appears in the left-hand column of Table 1 . It is
seen that the node 7 is connected to node 8. Therefore, ZE(1, 5 ) = 8. From the above dlscussion, it is again seen that node 3 is connected to node 4 and node 7 is connected to node 8. Similarly, the proposed logic will check whether nodes 4 and 8 are connected to any other nodes. Thls process will continue unless all nodes are identified beyond branch 1. The nodes beyond branch 1 are also given in Table 1. The total current flowing through branch 1 is equal to the sum of the load currents of all nodes beyond branch 1 plus the sum of the charging currents of all the nodes beyond branch 1 . For jj = 2 (second branch in Fig. 1; Table l), ZRG] = ZR(2)= 3, check whether ZR(2) = ZS(z] or not for i = 3, 4, ..., LN1. It is seen that IR(2) = IS(3) = 3. The corresponding receiving-end node is IR(3) = 4. Therefore, IE(2, 1) = 3 and ZE(2, 2) = 4. From the above discussion, it is seen that node 3 is connected to node 4. The proposed logic will identify the nodes which are connected to node 4. It will check whether node 4 appears in the left-hand column of Table 1. It is seen that node 5 and node 10 are connected to node 4. Therefore
-+ from A
start
-1 read substation voltage V(l), line data and load data.
1
jj 5 LN1
assume a flat voltage starI,i.e. V(i) =V(l)=I~O'for i=2,3 ,........, NB set W(i)=V(i) for i=2,3 ,......., NB set lSS(ij)=lS(jj) and IRR(jj)= IR(jj) for jj=1,2,3 ,........, LNl set iteration count k=O
calculate IL(i) and IC(i) using eqns. 8 and 9 for i=2,3, .... NB
I
calculate branch currents by using eqn. 12
1
1
no
,-,
L7J converged
W(m2)=V(m2) for m2=2,3,.... NB
4
1
calculate line losses, line flows etc. and print required dyta
1
to B
set m l =ISS(jj) and rnS=IRR(jj) compute receiving-end voltage V(m2) by using eqn. 3 calculate absolute change in voltage at node m2, DV(m2)=ABS(/V(m2)/-/W(m2)/)
I Fig.3 644
jj=jj+l
-1
to A Fibwehartfor load-jhw calculation of r& dimibution network IEE Proc-Gener. Transm. Distrib., Vol. 146, No. 6,November 1999
ZE(2, 3) = 5 and IE(2, 4) = 10. The proposed logic will check whether nodes 5 and 10 are connected to any other nodes. This process will continue unless all nodes are identified beyond branch 2. The nodes beyond branch 2 are given in Table 1. Similarly it is necessary to consider the receiving-end node of branch 3, branch 4, ..., branch LN1 in Fig. 1 and, in a similar way to that discussed above, the nodes have to be identified beyond these branches. The nodes beyond all the branches are also given in Table 1. Note that, if the receiving-end node of any branch in Fig. 1 is an end node of a particular lateral, the total current of this branch is equal to the load current of this node plus the charging current of this node itself. For example, consider node 6 in Fig. 1 (branch 5, Table 1); this is an end node. Therefore, the branch current 1(5)is equal to the load current of node 6 plus the charging current of node 6 (i.e. 4 5 ) = ZL(6) + ZC(6)). Sirmlarly, 9, 11 and 12 are end nodes of Fig. 1. The proposed computer logic will identify all the end nodes automatically. The concept of identifying the nodes beyond all the branches, whch helps in computing the exact current flowing through all the branches, has been realised using an algorithm (Fig. 2) and applied in the load-flow technique as shown in the flowchart in Fig. 3. 5
Load-flow calculation
Once all nodes beyond each branch are identified, it is very easy to calculate the current flowing through each branch as described in Section 3. For this purpose, the load current and charging current of each node are calculated by using eqns. 8 and 9. Once the nodes are idenflied beyond each branch, the expression of branch current is given as N(33)
I ( j j )=
I L { I E ( j j ,i)} 2=1
+
Nb3)
I C { I E ( j j ,i)} 2=1
(12) Initially, a constant voltage of all the nodes is assumed and load currents and charging currents are computed using eqns. 8 and 9. After load currents and charging currents have been calculated, branch currents are computed using eqn. 12. The voltage of each node is then calculated by using eqn. 3 with eqn. 4. Real and reactive power loss of each branch is calculated by using eqns. 10 and 11, respectively. Once the new values of the voltages of all the nodes are computed, convergence of the solution is checked. If it does not converge, then the load and charging currents are computed using the most recent values of the voltages and the whole process is repeated. The convergence criterion of the proposed method is that if, in successive iterations the maximum difference in voltage magnitude (DVMAX) is less than O.OOOlp.u., the solution has then converged. The convergence analysis of the proposed method is given in Appendjx 1 (Section 9.1). The proposed distribution load-flow algorithm for solving radial distribution networks is given in the form of a flowchart in Fig. 3. 6
Example
To demonstrate the effectiveness of the proposed method, three examples are selected. The first example is a 29-node 11kV rural distribution feeder of India. Data for this 29node system are given in Appendix 2 (Section 9.2). Table 2 gives the load-flow results of a 29-node radial distribution network. The real- and reactive-power loss of this system are 303.78kW and 124.74kVAr, respectively. IEE Proc -Gener Transm Distrcb , Vol 146, No. 6, November 1999
Table 2: Load-flow solution of example 1: 29-node radial distribution network ~
~
Node number
Voltage magnitudes (P.u.)
Node number
Voltage magnitudes (P.u.)
1 2 3 4
1 .ooooo 0.95884 0.91877 0.89598 0.83736 0.85068 0.82855 0.82189 0.81233 0.80821
0.88945
12
0.94989 0.94655
15 16 17 18 19 20 21 22 23 24 25 26
13 14
0.93980 0.93684
27 28
5 6 7 8 9 10 11
0.88560 0.88296 0.87556 0.86914 0.86378 0.86240 0.86120 0.81710 0.81079 0.80725 0.80467 0.80884 0.80774
Table 3: Load-flow solution of example 2:69-node radial distribution network Node number
Voltage magnitudes (P.u.)
Node number
Voltage magnitudes (P.u.)
1 2 3 4 5 6 7 8 9
1 .ooooo 0.99997 0.99993 0.99984 0.99902 0.99009 0.98079 0.97858 0.97745
36 37 38 39 40 41 42 43 44
10
0.97245 0.97135 0.96819 0.96526 0.96237 0.95950 0.95897 0.95809 0.95808 0.95761 0.95731 0.95683 0.95683 0.95675 0.95660 0.95643 0.95636 0.95634 0.99993 0.99985 0.99973 0.99971 0.99961 0.99935 0.99901 0.99895
45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69
0.99992 0.99975 0.99959 0.99954 0.99954 0.99884 0.99855 0.99851 0.99850 0.99841 0.99840 0.99979 0.99854 0.99470 0.99415 0.97854 0.97853 0.97466 0.97142 0.96694 0.96257 0.94010 0.92904 0.92476 0.91974 0.91234 0.91205 0.91166 0.90976 0.90919 0.97129 0.97129 0.96786 0.96786
11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
645
Table 4 Load-flow results of example 3: 33-node radial distribution network Status of the network
Real-power loss (kW)
Reactive-power loss (kVAr)
Minimum voltage (P.u.)
Without charging admittance
202.67
135.14
Vi8 = 0.91309
With charging admittance
196.54
131.03
V78 = 0.91513
Table 5: Comparison of speed of proposed method with other existing methods
Proposed method and other four existing methods
Example 1: 29-node distribution network
Example 2: 69-node distribution network
Example 3: 33-node distribution network
CPU time
Iteration number
CPU time
(s)
Iteration number
CPU time
(S)
(S)
Iteration number
Proposed method
0.07
3
0.16
3
0.09
3
Chiang [ I l l very fast decoupled distribution load flow
0.09
3
0.24
3
0.11
3
Load flow using single-line equivalent [14, 151
0.1 1
3
0.29
3
0.13
3
Renato [I21 load flow using forward-sweeping method
0.12
4
0.33
4
0.14
4
Kersting [71 load flow using ladder 0.14 technique
4
0.37
4
0.16
4
The second example is a 69-node radial distribution network. Data for this system are available in [lo]. Load-flow results of this system are given in Table 3. Real- and reactive-power losses of this system are 224.96kW and 114.15kVAr, respectively. The third example is a 33-node radial distribution network. Data for this system are available in [17, 151. T h ~ s 33-node radial network is solved with and without considering charging admittance. Table 4 gives real- and reactivepower losses and the minimum voltage with and without charging admittance. The proposed method is also compared with four other existing methods. Table 5 shows the CPU time and number of iterations of all three examples. All these three examples were simulated on a Meteor 400VT with a 66MHz clock. From Table 5, it is seen that the proposed method is better than the other four existing methods. It is worth mentioning here that the authors have tried to solve the above three examples using the Newton-Raphson (NR) and Gauss-Seidel (GS) methods. However, for all these three examples, the NR and GS methods did not converge. 7
Conclusions
A simple and efficient load-flow technique has been proposed for solving radial distribution networks. It has been found from the cases with which the method was tested that the method has good and fast convergence characteristics compared with some other existing methods. The proposed method has been implemented on a Meteor 400VT with a 66MHz clock. Several radial distribution networks have been solved successfully using the proposed method. 8
References
1 TINNY, W.F., and HART, C.E.: ‘Power flow solution by Newton’s
method’, IEEE Tram., 1967, PAS86, pp. 1449-1456 2 SCOlT, B., and ALSAC, 0.: ‘Fast decoupled load flow’, ZEEE Trans., 1974, PAS-93, pp. 859-869 3 RAJICIC, D., and TAMURA, Y.: ‘A modification to fast decoupled power flow for networks with hgh R/X ratios’, IEEE Tram., 1988, PWRS-3, pp. 743-746 4 IWAMOTO, S., and TAMURA, Y.: ‘A load flow calculation method for illconditioned power systems’, IEEE Trans., 1981, PAS-100, pp. 17361713 646
TRIPATHY, S.C., DURGAPRASAD, G., MALIK, O.P., and HOPE, G.S.: ‘Load flow solutions for ill-conditioned power system by a Newton like method’, ZEEE Tram., 1982, PAS-101, pp. 36843657 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 KERSTING, W.H.: ‘A method to the design and operation of a distribution system’, ZEEE Trans., 1984, PAS-103, pp. 1945-1952 STEVENS, R.A., RIZY, D.T., and PURUCKER, S.L.: ‘Performance of conventional power flow routines for real-time distribution automation application’. Proceedings of 18th southeastern symposium on System theory, 1986, (IEEE Computer Society), pp. 196200 SHIRMOHAMMADI. D.. HONG. H.W.. SEMLYEN. A.. and LUO, G.X.: ‘A compensa6on-based power ’flow method for weakly meshed distribution and transmission networks’, IEEE Trans., 1988, PWRS-3, pp. 753-743 10 BARAN, M.E., and WU, F.F.: ‘Optimal sling of capacitors placed on a radial distribution system’, IEEE Trans., 1989, P-2, pp. 735743 11 CHIANG, H.D.: ‘A decoupled load flow method for distribution power network algorithms, analysis and convergence study’, Electr. Power Energy Syst., 1991, 13, (3), pp. 13C138 12 RENATO, C.G.: ‘New method for the analysis of distribution networks’, IEEE Trans., 1990, PWRD-5, (l), pp. 9-13 13 GOSWAMI, S.K., and BASU, S.K.: ‘Direct solution of distribution
systems’, IEE Proc. C. , 1991, 188, (I), pp. 78-88 14 JASMON, G.B., and LEE, L.H.C.C.: ‘Distributionnetwork reduction for voltage stability analysis and load flow calculations’, Electr. Power Energy Syst., 1991, 13, ( I ) , pp. 9-13 15 JASMON. G.B.. and LEE. L.H.C.C.: ‘Stabilitv of load flow techniques for’distribution system voltage stability ahysis’, IEE Proc. C, 1991, 138, (6), pp. 479484 16 DAS, D., NAGI, H.S., and KOTHARI, D.P.: ‘Novel method for solving radial distribution networks’, IEE Proc. C, 1994, 141, (4), pp. 391-3911 , I
I , -
17 BARAN, M.E., and WU, F.F.: ‘Network reconfiguration in distribution systems for loss reduction and load balancing’, ZEEE Tram. Power Deliv., 1989, PWRD-4, pp. 1401-1407
9
Appendices
9.1 Convergence analysis It has been explained in Section 3 that the proposed loadflow algorithm starts with an initial set of node voltages and load currents and charging currents are then computed using eqns. 8 and 9. Now, for the convergence analysis of the proposed method, consider the electrical equivalent of Fig. 1. Fig. 4 shows the electrical equivalent of Fig. 1. In Fig. 4, j j = branch number m l = ISfij) = sending-end node m2 = IRfij) = receiving-end node IEE Proc-Gener. Transm. Distrib., Vol. 146, No. 6, November 1999
P(m2) = total real power load fed through node m2 Q(m2) = total reactive power load fed through node m2
From Fig. 4, the following equations can be written:
) { W j )+ j X ( j j ) }
(13)
P(m2)- jQ(m2)= V * ( m 2 ) l ( j j ) In fact, eqn. 13 is similar to eqn. 3:
(14)
V(m2)= V(m1) - W
V(m1)
sending end
-
Mi)
V(r”2)
receiving end
I 1P(mZ)+jQ(mZ)
RW+iXUi)
mi =IS(ij)
Fig. 4
fore, if e(m2) for m2 = 2, 3, ..., NB converges then, in eqn. 3, V(m2),for m2 = 2, 3, ..., NB, will also converge. Also note that, when one is solving the receiving-end voltage, the sending-end voltage is known.
rnP=IR(jj)
Electrical equivalent of Fig. I
P(m2) = sum of the real-power loads of all the nodes beyond node m2 plus the real-power load of the node m2 itself plus real-power losses of all the branches beyond node m2. Q(m2) = sum of the reactive-power loads of all the nodes beyond node m2 plus the reactive-power load of the node m2 itself plus reactive-power losses of all the branches beyond node m2. From eqns. 13 and 14 one obtains
To study the convergence analysis of the proposed method, assume that
From eqns. 23 and 22 one obtains
V(m2)= V(m1)
(15) Now the voltages at nodes m2 and ml are expressed as
+ V(m1)= e(m1)+ j f ( m 1 )
V(m2)= e(m2) j f ( m 2 )
+
(16)
(17)
Substituting eqns. 16 and 17 in eqn. 15 and separating real and imamary parts gives
+
e(m2)= g{e(m2)} (24) Now consider the graph (Fig. 5) of g(e(m2)). In Fig. 5, e0(m2)is the initial approximation of e(m2), 5(m2) is the final solution of e(m2), and ~,(rn2) and .en+,(m2)are the errors in the nth and (n+l)th iterations. From Fig. 5,
+
en(m2)= t(m2) ~ , ( m 2 )
+
en+l (m2)= ((ma) ~ ~ + l ( m 2 ) (26) The solution of e(m2) at various iterative stages can be written using eqn. 24:
e2(m2) f’(m2) = {e(ml)e(m2) f ( m l ) f ( m 2 ) )
el (m2)= g(eo(m2)) e2 (ma)= d e l (4)
- { W ’ j ) P ( m 2+ ) X(jj)Q(m2)1 (18)
First let us consider eqn. 18. In fact in a distribution system the voltage angle is extremely small. Hence, the imaginary parts of the voltages are extremely small. Therefore, the termsf2(m2) andjfmlMm2) in eqn. 18 can be dropped. Therefore, eqn. 18 can be written as
(25)
en+l(m2)= g{en(m2)) From eqns. 27 and 25 one obtains
(27)
en+l(m2)= g(r(m2)+ En(m2)) From eqns. 28 and 26 one obtains
(28)
+
+
[(ma) Enfl(m2)= g(E(m2) En(m2)) (29) Expanding the right-hand term of eqn. 29 by Taylor series and considering the first three terms, one obtains t(m2)+ En+l(m2)= t(m2)+ &n(m2)g’{t(m2)}
+ *g”{t(m2)} 2 Assuming that
en+1(m2)= E n ( m 2 ) g ’ { t ( m 2 ) } + ~ g ’ r { F ( m 2 ) } 2
(30) From eqns. 20 and 21 one obtains
Now the first and second derivatives of eqn. 23 are:
4 j j ) g’(e(m2))= ez(m2) Now consider eqn. 19. In eqn. 19, the terms on the righthand side are independent of jfm2), and jfm2) can be obtained directly after computing e(m2) iteratively. ThereIEE Proc.-Gener. Transm. Distrib., Vol. 146, No. 6, November 1999
g”(e(m2))=
-2Au(jj) e3 (m2) 641
negligible and can be dropped from eqn. 36.
when e(m2) = gm2)
(33) and (34)
From eqns. 30, 33 and 34, one obtains
From eqns. 35 and 33, one obtains
During the iterative process, it has been observed that, for all the examples, (~,(m2)/5(m2)}for m2 = 2, 3, ..., NB is
.‘. E n S l ( m 2 ) = &n(m2)g’{l(m2))
(37)
From eqn. 37 it is seen that the error at the (n + 1)th iteration is proportional to the error at the nth iteration. Hence convergence of the proposed method is linear. When r, x, P and Q are expressed in P.u., the term Aubj) (eqn. 21) is 2)} extremely small and hence g’(Krn2)) = { A u ( ~ j ] / ~ ( mis extremely small, i.e. g’(gm2))
