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An exact method for loss allocation in radial distribution systems
Electrical Power and Energy Systems 36 (2012) 100–106
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An exact method for loss allocation in radial distribution systems J.S. Savier, Debapriya Das ⇑ Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, 721 302 West Bengal, India
a r t i c l e
i n f o
Article history: Received 14 August 2007 Received in revised form 21 October 2011 Accepted 29 October 2011 Available online 6 December 2011 Keywords: Distribution system Deregulation Loss allocation
a b s t r a c t This paper presents an exact method for real power loss allocation to consumers connected to radial distribution networks in a deregulated environment. The proposed method has the advantage that no assumptions are made in the allocation of real power losses as opposed to the algorithms available in the literature. A detailed comparison of the real loss allocation obtained with the proposed ‘Exact method’ with two alternative algorithms, namely, pro rata (PR), and quadratic loss allocation schemes are presented. Pro rata procedure is based on the load demand of each consumer, quadratic allocations are based on identifying the real and reactive parts of current in each branch and the losses are allocated to each consumer, and the proposed ‘exact method’ is based on the actual contribution of real power loss by each consumer. A case study based on 30 node distribution system is provided. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Deregulation of the power industry was intended to break up the industry’s traditional, vertically integrated structure of generators, transmission lines, and distribution facilities into separate entities with generators competing for sales across common transmission lines to local distribution outlets. Under such a system, generators would be in competition with each other to serve more than one utility distributor. Thus, the generators selling electricity at the lowest cost would have the most utility customers. Unlike generation and sale of electrical energy, activities of transmission and distribution are generally considered as a natural monopoly. The cost of transmission and distribution activities needs to be allocated to the users of these networks. Allocation can be done through network use tariffs, with a focus on the true impact they have on these costs. Among others, distribution power losses are one of the costs to be allocated. The main difficulty faced in allocating losses is the nonlinearity between the losses and delivered power which complicates the impact of each user on network losses [1]. Different techniques have been published in the literature for allocation of losses, most of them dedicated to transmission networks and can be classified into three broad categories – pro rata procedures, marginal procedures and proportional sharing procedures. Pro rata procedure [2,3] is the simplest one, in which, the total losses are allocated to loads or generators based on the load active power demand or bus generation. In marginal procedures [4–7], losses are assigned to generators and demands through
⇑ Corresponding author. E-mail address: [email protected] (D. Das). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.10.030
the so-called incremental transmission loss (ITL) coefficients. In proportional sharing procedures [8–13], losses are allocated to the generators and consumers by using the results of a converged power flow plus a linear proportional sharing principle. Conejo et al. [14] have presented a new procedure for allocating transmission losses to generation and loads based on the networks Z-bus matrix. Conejo et al. [15] have also presented a comparison of four different practical algorithms for transmission loss allocation. Fang and Ngan [16] proposed a succinct method for allocation of network losses which takes into consideration the influence of both active and reactive power injected into grids. Costa and Matos [17] have addressed the allocation of losses in distribution networks with embedded generation by considering quadratic loss allocation technique. Daniel et al. [18] presented an approach for transmission loss allocation using a modified Ybus. Abhyankar et al. [19,20] presented methods for transmission fixed cost allocation based on min–max fairness criteria and optimization approach. Carpaneto et al. [21] proposed a branch current decomposition method for loss allocation in radial distribution systems with distributed generation. Carpaneto et al. [22] also presented a detailed characterization of different loss allocation techniques for radial distribution systems with distributed generation. Belati and da Costa [23] presented a transmission loss application based on perturbation of optimum theorem. In their approach, from a given optimal operating point obtained by the optimal power flow the loads are perturbed and a new operating point that satisfies the constraints are obtained by sensitivity analysis, which is used for obtaining the coefficients of the losses. Lim et al. [24] introduced a new method for allocating losses in a power system using a loop-based representation of system behavior. Li et al. [25] presented a means of allocating transmission losses and costs taking into account pool and bilateral contract hybrid
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106
Nomenclature P kVAp Ploss NB IL(p) PLp QLp Vp Jj I(jj) N(jj) ie(jj, i)
a{ie(jj, i)}, b{ie(jj, i)} loss allocation factors for the consumer at
node number kVA load demand of consumer at node p total real power loss in the system total number of nodes in the system load current at pth node real power load at pth node reactive power load at pth node voltage at pth node branch number, jj = 1, 2, . . . , NB1 current of branch-jj total number of nodes beyond branch-jj nodes beyond branch-jj, for i = 1, 2, . . . , N(jj)
node ie(jj, i) IL{ie(jj, i)} load current of consumer at node ie(jj, i) ILD{ie(jj, i)} real component of load current at node ie(jj, i) ILQ{ie(jj, i)} reactive component of load current at node ie(jj, i) PLOSS(jj) real power loss of branch-jj R(jj) resistance of branch-jj Rfxg Real part of complex quantity, x ploss(jj, ‘) real power loss of branch-jj allocated to consumer connected at node ‘ Tploss (‘) real power loss supported by consumer ‘
deregulated power market. Bhakar et al. [26] presented probabilistic game approaches for network cost allocation. Alturki and Lo [27] presented a loss allocation method using current adjustment factors to allocate real and reactive losses simultaneously without any additional calculation except the substitution of line reactance instead of resistance. Ferreira et al. [28] presented a method for transmission cost allocation based on optimal re-dispatch. Savier and Das [29] presented allocation of power losses to consumers connected to radial distribution network before and after network reconfiguration in a deregulated environment. Parastar et al. [30] presented an approach of transmission loss allocation in which, a relationship between the bus current injections and the generator or load currents is first determined using a power invariant matrix and then Z-bus matrix is modified, which allows the real power loss of the network to be expressed in terms of generator or load currents. Choudhury and Goswami [31] solved the problem of transmission loss allocation of deregulated power system through the application of artificial neural network. As there is no unique or existing ideal procedure is available, any loss allocation algorithm should have most of the desirable properties such as consistent with the results of power flow, depend upon the amount of energy either produced or consumed, depend on the relative location within the network, easy to understand, and easy to implement [15]. In the light of the above developments, this work proposes a simple loss allocation method, called ‘Exact Method’ for real power loss allocation to consumers in radial distribution systems. It is assumed that consumers have to pay for losses. Real power loss allocation to each consumer obtained with the proposed method is compared with two different practical approaches for loss allocation, namely, pro rata (PR) and quadratic loss allocation schemes.
If the total number of nodes beyond branch-jj is N(jj), the current through that branch can be written as:
IðjjÞ ¼
NðjjÞ X ðILDfieðjj; kÞg jILQfieðjj; kÞgÞ
Real power loss of branch-jj with sending end and receiving end voltages Vi and Vj is given by:
PLOSSðjjÞ ¼ RfðV i V j Þ IðjjÞg ( i:e:; PLOSSðjjÞ ¼ R ðV i V j Þ
NðjjÞ X ðILDfieðjj; kÞg jILQfieðjj; kÞgÞ
In a radial distribution system, load current of a consumer connected to node ie(jj, k) beyond branch-jj can be written in the form:
ð1Þ
)
ð4Þ ⁄
Let (Vi Vj) = a{ie(jj, k)} + jb{ie(jj, k)} )PLOSSðjjÞ ¼ R
( NðjjÞ ) X ðafieðjj;kÞg þ jbfieðjj; kÞgÞ:ðILDfieðjj; kÞg jILQ fieðjj; kÞgÞ k¼1
ð5Þ Hence,
PLOSSðjjÞ ¼
NðjjÞ X
ðafieðjj; kÞgILDfieðjj; kÞg þ bfieðjj; kÞgILQ fieðjj; kÞgÞ
k¼1
ð6Þ Using Eq. (6), real power loss in branch-jj can be allocated to consumers beyond branch-jj. Real power loss of branch-jj allocated to consumer connected to node ie(jj, k) is given by:
plossfjj; ieðjj; kÞg ¼ afieðjj; kÞgILDfieðjj; kÞg þ bfieðjj; kÞgILQ fieðjj; kÞg for jj ¼ 1; 2; . . . ; NB 1 and k ¼ 1; 2; . . . ; NðjjÞ
2.1. Proposed exact method
ILfieðjj; kÞg ¼ ILDfieðjj; kÞg jILQ fieðjj; kÞg
ð3Þ
k¼1
2. Proposed method for distribution loss allocation In radial distribution systems, power is fed at substations and the power flows from substation to downstream. In this section, proposed exact method for loss allocation is first presented. For the purpose of comparison, pro rata (PR) and quadratic loss allocation schemes are used. Hence, loss allocation procedure with these two methods is also briefly presented.
ð2Þ
k¼1
ð7Þ
2.2. Pro rata (PR) allocation method The PR method proportionally allocates losses to the consumers based on the kW load demand [15]. In order to take into account reactive load of each consumer, instead of considering real power demand of the consumer, kVA load demand is considered. The total kVA demand of the system can be written as:
TkVAD ¼
NB X
kVAp
ð8Þ
p¼2
Note that substation is marked as node 1, i.e., p = 1. Hence, power loss allocated to consumer at node p is given by:
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kVAp TkVAD
plossp ¼ Ploss
i:e:; plossp ¼ K D
9
ð9Þ
kVAp ;
KD ¼
Ploss TkVAD
(8)
6
ð10Þ 8
It may be noted that the demand loss allocation factors KD are identical for all buses. Hence, pro rata procedure is simple to understand and implement but they ignore the relative location of the consumers within the network. That is, two identical demands located respectively near substation and far away from the substation are equally treated, and this is unfair to the load located near the substation.
(5)
1
(7) 3
(1)
2
(2)
(3)
½ILDfieðjj; iÞg
¼
afieðjj; kÞg ½ILDfieðjj; kÞg
ð11Þ
2
and
bfieðjj; iÞg ½ILQfieðjj; iÞg2
¼
bfieðjj; kÞg
ð12Þ
½ILQfieðjj; kÞg2
Based on this principle, the power loss of the branch-jj of the network allocated to consumers beyond branch-jj, for i = 1, 2, . . . , N(jj) are (Derivation given in Appendix A):
plossfjj; ieðjj; iÞg ¼ RðjjÞ
þ
NðjjÞ X
8 > < > :
½ILDfieðjj; iÞg2 þ ½ILQ fieðjj; iÞg2
ILDfieðjj; iÞg ILDfieðjj; kÞgafieðjj; iÞg
k¼1 k–i
9 > = ILQfieðjj; iÞg ILQfieðjj; kÞgbfieðjj; iÞg ð13Þ þ > ; k¼1 NðjjÞ X
k–i
The global value of losses to be supported by consumer ‘ results from the sum of the losses allocated to it in each branch-jj of the network, i.e.,
Tplossð‘Þ ¼
NB1 X
plossðjj; ‘Þ for
(4)
(6)
In quadratic loss allocation scheme, since the power losses grow quadratically with power flows, the following constraint is imposed [1]. 2
5
S/S
2.3. Quadratic allocation method
afieðjj; iÞg
4
‘ ¼ 2; 3; . . . ; NB
ð14Þ
jj¼1
Note that node 1 is substation. Eq. (14) indicates that each consumer has allocated losses only at branches to which power flow contributes. 3. Case study In order to validate the proposed method, two radial distribution systems are considered. In the first example, a nine node radial distribution system is used to demonstrate the proposed method and in the second example, a detailed comparison of loss allocation results obtained with the proposed method with two methods available in the literature is presented. 3.1. Example – 1 In this example, a distribution system having nine nodes, as shown in Fig. 1 is considered. The line and load data for this system are given in Appendix B. In Case-1, the real and reactive power loads as shown in Appendix B is taken. For the purpose of explanation of
7 Fig. 1. Sample distribution system with nine nodes.
the proposed technique when there is current injection, Case-2 is considered in which the line and load data is same as those given in Appendix B except that the reactive power load at node no. 8 is taken as negative (injection). In Case-1, total real power loss of the radial distribution system after a converged load flow was 24.0406 kW. Allocation of real power losses to different consumers, voltages and load currents at different nodes are given in Table 1. It can be seen from the table that the real power losses allocated to consumer at node nos. 8 and 9 are 7.4571 kW and 7.9504 kW. It can be seen that the real power loss allocated to consumer at node no. 9 is higher than that at node no. 8, even though the real and reactive power losses at node no. 9 is lower than that at node no. 8. This is due to the fact that the proposed method allocates losses based on actual contribution of the losses. In Case-2, total real power loss of the distribution system after a converged load flow is 16.863 kW. It can be seen from Table 1 that the real power losses allocated to all the nodes are reduced in this case due to the fact that the total power loss has reduced in this case. Real power allocated to consumers at node nos. 8 and 9 in Case-2 are 6.5651 kW and 4.8895 kW respectively as compared to 7.4571 kW and 7.9504 kW in Case-1.
3.2. Example – 2 A practical 11 kV distribution system having 30 nodes, as shown in Fig. 2 is considered to compare the four distribution loss allocation methods. The line and load data for this system are given in Appendix C. The loads considered are either industrial or commercial type. The total real power loss of the system, obtained from the results of the load flow is 146.09 kW. Table 2 shows the real power loss allocated to each consumer with the three different approaches. Power flow data and results are used as input data for the four loss allocation algorithms. The loss allocation based on pro rata procedure allocates losses irrespective of the geographical location of consumers. Hence, consumers having same load demands are allocated same losses, even though the power loss contribution of the consumer electrically closer to the substation is less as compared to those consumers electrically away from the substation. For example, consumers at nodes 10 and 28 are having the same load demand. Hence, loss allocation based on pro rata procedure allocates same losses to both the consumers, i.e., 4.3624 kW. Whereas, loss allocation to consumers at nodes 10 and 28 based on quadratic and exact method are different. Quadratic loss allocation scheme allocated 3.3636 kW and 3.6851 kW to the consumers at nodes 10 and 28, respectively, and proposed exact method allocated 4.2312 kW
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106 Table 1 Loss allocation to consumers for a nine node system. Node no.
Case-1
1 2 3 4 5 6 7 8 9
Case-2
Voltage (pu)
Current (pu)
Real power loss allocated (kW)
Voltage (pu)
Current (pu)
Real power loss allocated (kW)
1 0.9844 0.9767 0.974 0.9739 0.9841 0.9765 0.9643 0.9575 Total
0 0.0016 0.0010i 0.0015 0.0014i 0.0001 0.0001i 0.0003 0.0002i 0.0005 0.0003i 0.0001 0.0001i 0.0019 0.0014i 0.0016 0.0012i
0 2.6413 3.7695 0.3311 0.8307 0.7634 0.2971 7.4571 7.9504 24.0406
1 0.9871 0.9811 0.9794 0.9793 0.9868 0.981 0.9716 0.9649 Total
0 0.0016 0.0010i 0.0015 0.0014i 0.0001 0.0001i 0.0003 0.0002i 0.0005 0.0003i 0.0001 0.0001i 0.0019 + 0.0014i 0.0016 0.0013i
0 1.8183 2.1586 0.2061 0.5182 0.5060 0.2012 6.5651 4.8895 16.8630
13 12 11 1
S /S
28
29
30
5
6
7
10 2
3
23
4
8
20
14
19
21
15
18
22
9
4. Conclusions
24 16
17
25 27
26
Fig. 2. Distribution system with 30 nodes.
Table 2 Real power loss allocation to consumers. Node nos.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Total
and 4.4907 kW to the consumers at nodes 10 and 28 respectively. It can be seen that all the methods except pro rata method allocated less power loss to consumer at node 10, which is electrically closer to the substation, as compared to losses allocated to consumer at node 28, which is electrically away from the substation. Quadratic loss allocation technique makes use of assumption to obtain power loss of each branch allocated to consumers beyond that branch. The proposed ‘exact method’ allocates branch losses to different consumers based on actual contribution of the branch power losses by each consumer beyond that branch. In the present work, a load flow algorithm developed in [32] for solving the radial distribution network has been used.
Real power loss allocated Proposed method
Pro rata method
Quadratic method
0.000 6.079 9.414 0.876 2.326 3.801 1.036 1.731 16.709 4.231 5.832 3.239 1.080 2.674 4.755 5.179 13.384 7.761 4.104 0.995 4.722 9.820 3.328 10.346 6.134 3.456 2.688 4.491 3.395 2.507 146.09
0.000 13.691 14.819 1.018 2.544 4.118 0.976 1.676 14.310 4.362 5.390 2.926 0.976 2.793 4.362 4.362 11.738 5.853 3.702 0.976 4.936 8.725 2.932 9.275 5.196 2.793 1.951 4.362 3.146 2.186 146.09
0.000 8.227 15.381 0.093 0.894 2.577 0.102 0.403 24.118 3.364 5.194 1.623 0.123 1.235 3.420 3.700 19.831 6.531 2.755 0.098 4.206 12.025 1.609 13.788 5.612 1.730 0.814 3.685 1.998 0.956 146.09
In the present work, a simple loss allocation method for consumers connected to radial distribution networks has been proposed. A detailed comparison of real power loss obtained with the proposed ‘exact’ method with two different methods namely, pro rata, and quadratic loss allocation methods has been presented. From the case study, it can be seen that even though pro rata procedure is simple and easy to implement, power loss allocated to consumers having same load demands are the same, which is injustice to the consumer electrically nearer to the substation. Quadratic loss allocation scheme is based on branch current flow and it allocates branch power loss to only those consumers beyond that branch. Quadratic loss allocation scheme makes the assumption that the loss allocation factor of a particular consumer is proportional to the square of real/reactive load current of that consumer. In the proposed ‘exact method’, losses are allocated to consumers without making any assumptions and can be implemented easily. Appendix A For the purpose of explanation, consider a sample distribution network as shown in Fig. A1. In Fig. A1, branch numbers are shown in (). The branch number, sending end and receiving end nodes are given in Table A1. The load current at any node p, is given as:
ILðpÞ ¼
PLp jQLp ; V p
p ¼ 2; 3; . . . ; NB
ðA1Þ
Consider branch-8 of Fig. A1. Number of node beyond branch-8 is one and this is node 9. Therefore, current through branch-8 is:
Ið8Þ ¼ ILð9Þ
ðA2Þ
Similarly consider branch-7. Total number of nodes beyond branch-7 is three and these nodes are 8, 9 and 12. Therefore, current through branch-7 is:
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106
2
11
NðjjÞ X
jIðjjÞj ¼ 4
(1 0 )
!2 ILDfieðjj; iÞg
þ
i¼1
NðjjÞ X
!2 312 ILQ fieðjj; iÞg 5
ðA7Þ
i¼1
10 (9 )
1 2
3
4
5
)jIðjjÞj2 ¼
6
NðjjÞ X 2 ½ILDfieðjj; iÞg þ ½ILQfieðjj; iÞg2 i¼1
S/S (1 )
(2 )
(3 )
(4 )
(5 )
þ2
(6 )
NðjjÞ1 NðjjÞ X X i¼1
7
ðILDfieðjj; iÞg:ILDfieðjj; kÞg
k¼iþ1
þ ILQ fieðjj; iÞg:ILQfieðjj; kÞgÞ (7 )
Power loss of branch-jj can be written as: 12
8
PLOSSðjjÞ ¼ RðjjÞ:jIðjjÞj2
(1 1 ) (8 )
ðA9Þ
From Eqs. (A8) and (A9), we get,
9
( NðjjÞ X 2 2 PLOSSðjjÞ ¼ RðjjÞ ½ILDfieðjj; iÞg þ ½ILQ fieðjj; iÞg
Fig. A1. Sample distribution network with 12 nodes.
i¼1
Ið7Þ ¼ ILð8Þ þ ILð9Þ þ ILð12Þ
ðA3Þ
From Eqs. (A2) and (A3), it is clear that, if we identify the nodes beyond all the branches and if the load currents are known, then it is extremely easy to compute the branch currents. In [32], an algorithm for identifying the nodes beyond all the branches has been proposed. A load flow algorithm based on the identification of these nodes beyond all the branches has also been proposed in [32]. In the present work, this load flow algorithm is used to compute the load currents and branch currents. General expression of branch current through branch-jj is given by [32]:
IðjjÞ ¼
NðjjÞ X
ILfieðjj; iÞg
ðA4Þ
In Table A1, total number of nodes (consumers) beyond branchjj (N(jj)) and nodes (consumers) beyond branch-jj (ie(jj, i); i = 1, 2, . . . , N(jj)) are given for Fig. A1 for the purpose of explanation. Let us define:
ILfieðjj; iÞg ¼ ILDfieðjj; iÞg jILQfieðjj; iÞg
ðA5Þ
From Eqs. (A4) and (A5), we obtain: NðjjÞ X
½ILDfieðjj; iÞg jILQ fieðjj; iÞg
ðA6Þ
i¼1
Magnitude of current in branch-jj is expressed as:
Table A1 Branch number, sending end node, receiving end node and nodes beyond different branches. Branch number (jj) 1 2 3 4 5 6 7 8 9 10 11
i¼1
Sending end node IS(jj)
Receiving end node IR(jj)
1 2 3 4 5 2 7 8 4 10 8
2 3 4 5 6 7 8 9 10 11 12
Total number of nodes beyond branch-jj N(jj)
Nodes beyond branchjj ie(jj,i) i = 1, 2, . . . , N(jj)
ðILDfieðjj; iÞg ILDfieðjj; kÞg
k¼iþ1
)
þILQfieðjj; iÞg ILQfieðjj; kÞgÞ
ðA10Þ
The development of expression given by Eq. (A10), the consumer ie(jj, i) (consumers beyond branch-jj for i = 1, 2, . . . , N(jj)) has the impact on the terms [ILD{ie(jj, i)}]2 and [ILQ{ie(jj, i)}]2 of the power loss of branch-jj, which are exclusively due to consumers ie(jj, i) for i = 1, 2, . . . , N(jj). The terms,
2
NðjjÞ1 NðjjÞ X X i¼1
i¼1
IðjjÞ ¼
þ2
NðjjÞ1 NðjjÞ X X
ðILDfieðjj; iÞg ILDfieðjj; kÞg þ ILQ fieðjj; iÞg ILQfieðjj; kÞgÞ
k¼iþ1
are the results of the simultaneous influence of consumers ie(jj, i) and the remaining consumers ie(jj, k) in the components of the current in the branch-jj. These crossed terms must also be allocated. The allocation of crossed terms has been done in a quadratic way, as mentioned in [1]. The crossed terms 2 ILD{ie(jj, i)} ILD{ie(jj, k)} and 2 ILQ{ie(jj, i)} ILQ{ie(jj, k)} are split into two components as described here. Let us define,
afieðjj; iÞg ½ILDfieðjj; iÞg ILDfieðjj; kÞg þ afieðjj; kÞg ½ILDfieðjj; iÞg ILDfieðjj; kÞg ¼ 2 ½ILDfieðjj; iÞg ILDfieðjj; kÞg for i ¼ 1; 2; . . . ; NðjjÞ 1 for k ¼ i þ 1; i þ 2; . . . ; NðjjÞ
ðA11Þ
and
bfieðjj; iÞg ½ILQ fieðjj; iÞg ILQ fieðjj; kÞg þ bfieðjj; kÞg ½ILQ fieðjj; iÞg ILQ fieðjj; kÞg ¼ 2 ½ILQfieðjj; iÞg ILQfieðjj; kÞg
11 6 5 2 1 4 3 1 2 1 1
2,3,4,5,6,7,8,9,10,11,12 3,4,5,6,10,11 4,5,6,10,11 5,6 6 7,8,9,12 8,9,12 9 10,11 11 12
for i ¼ 1; 2; . . . ; NðjjÞ 1 for k ¼ i þ 1; i þ 2; . . . ; NðjjÞ
ðA12Þ
Eqs. (A11) and (A12) can be simply written as:
afieðjj; iÞg þ afieðjj; kÞg ¼ 2:0
ðA13Þ
bfieðjj; iÞg þ bfieðjj; kÞg ¼ 2:0
ðA14Þ
where a{ie(jj, i)} and b{ie(jj, i)} are the variable loss allocation factors for the consumer at node ie(jj, i).
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106
In quadratic loss allocation scheme, since the power losses grow quadratically with power flows, the following constraint is imposed [1].
afieðjj; iÞg 2
½ILDfieðjj; iÞg
¼
afieðjj; kÞg ½ILDfieðjj; kÞg
ðA15Þ
2
½ILQfieðjj; iÞg2
¼
Tplossð‘Þ ¼
NB1 X
plossðjj; ‘Þ for ‘ ¼ 2; 3; . . . ; NB
ðA20Þ
jj¼1
and
bfieðjj; iÞg
The global value of losses to be supported by consumer ‘ results from the sum of the losses allocated to it in each branch-jj of the network, i.e.,
bfieðjj; kÞg
ðA16Þ
½ILQ fieðjj; kÞg2
Note that node 1 is substation. Eq. (A20) indicates that each consumer has allocated losses only at branches to which power flow contributes.
From Eqs. (A13) and (A15), we get,
afieðjj; iÞg ¼
2:½ILDfieðjj; iÞg
Appendix B
2
2
½ILDfieðjj; iÞg þ ½ILDfieðjj; kÞg
ðA17Þ
2
Similarly, from Eqs. (A14) and (A16), we get,
bfieðjj; iÞg ¼
2 ½ILQ fieðjj; iÞg2
ðA18Þ
½ILQ fieðjj; iÞg2 þ ½ILQ fieðjj; kÞg2
Based on this principle, the power loss of the branch-jj of the network allocated to consumers beyond branch-jj, for i = 1, 2, . . . , N(jj) are:
(
plossfjj; ieðjj; iÞg ¼ RðjjÞ ½ILDfieðjj; iÞg2 þ ½ILQ fieðjj; iÞg2 þ
NðjjÞ X
ILDfieðjj; iÞg ILDfieðjj; kÞg afieðjj; iÞg
Branch Sending Receiving R (X) end node number end IR(jj) jj node IS(jj)
X (X) PL(IR(jj)) QL(IR(jj)) (kW) (kVar)
1 2 3 4 5 6 7 8
1.1019 0.7346 0.3673 0.1836 0.3673 0.3896 0.7792 1.1688
1 2 3 4 2 3 4 8
2 3 4 5 6 7 8 9
1.632 1.088 0.5440 0.2720 0.5440 1.3760 2.7520 4.1280
162 150 12 30 45.6 12 180 156
96 138 7.2 18 33.6 6 140.4 120
k¼1 k–i
9 > = ILQ fieðjj; iÞg ILQ fieðjj; kÞg:bfieðjj; iÞg ðA19Þ þ > ; k¼1 NðjjÞ X
Appendix C
k–i
Table C1.
Table C1 Line and load data of 30 node radial distribution system. Branch number jj
Sending end node IS(jj)
Receiving end node IR(jj)
R (X)
X (X)
PL(IR(jj)) (kW)
QL(IR(jj)) (kVar)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
1 2 3 4 5 6 7 8 4 10 11 12 11 28 29 5 20 21 22 21 24 25 26 6 14 15 16 16 18
2 3 4 5 6 7 8 9 10 11 12 13 28 29 30 20 21 22 23 24 25 26 27 14 15 16 17 18 19
1.6320 1.0880 0.5440 0.2720 0.5440 1.3760 2.7520 4.1280 3.6432 0.9108 0.4554 0.4554 1.3760 1.3760 4.1280 0.9108 1.8216 2.7324 0.9108 2.7520 3.0272 2.7520 2.7520 0.9108 1.8216 1.8216 0.9108 1.3760 1.3760
1.1019 0.7346 0.3673 0.1836 0.3673 0.3896 0.7792 1.1688 1.5188 0.3797 0.1898 0.1898 0.3896 0.3896 1.1688 0.3797 0.7594 1.1391 0.3797 0.7792 0.8571 0.7792 0.7792 0.3797 0.7594 0.7594 0.3797 0.3896 0.3896
162 150 12 30 45.6 12 18 156 48 64.8 36 12 48 36 26.4 12 48 96 32.4 96 54 30 24 30 48 48 120 72 36
96 138 7.2 18 33.6 6 14.4 120 36 36 18 6 36 24 14.4 6 48 72 24 84 46.8 24 12 24 36 36 108 36 36
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[17] Costa PM, Matos MA. Loss allocation in distribution networks with embedded generation. IEEE Trans Power Syst 2004;19:384–9. [18] Daniel JS, Salgado RS, Irving MR. Transmission loss allocation through a modified Ybus. Proc Inst Elect Eng Gen Transm Distrib 2005;152(2):208–14. [19] Abhyankar AR, Soman SA, Khaparde SA. Optimization approach to real power tracing: an application to transmission fixed cost allocation. IEEE Trans Power Syst 2006;21(3):1350–61. [20] Abhyankar AR, Soman SA, Khaparde SA. Min–max fairness criteria for transmission fixed cost allocation. IEEE Trans Power Syst 2007;22(4):2094–104. [21] Carpaneto E, Chicco G, Akilimali JS. Branch current decomposition method for loss allocation in radial distribution systems with distributed generation. IEEE Trans Power Syst 2006;21(3):1170–9. [22] Carpaneto E, Chicco G, Akilimali JS. Characterization of the loss allocation techniques for radial systems with distributed generation. Int J Electr Power Syst Res 2008;78:1396–406. [23] Belati EA, da Costa GRM. Transmission loss allocation based on optimal power flow and sensitivity analysis. Int J Electr Power Energy Syst 2008;30(4):291–5. [24] Lim VSC, McDonald JDF, Saha TK. Development of a new loss allocation method for a hybrid electricity market using graph theory. Int J Electr Power Syst Res 2009;79:301–10. [25] Li X, Yamashiro S, Wu L, Liu Z, Ouyang M. Generation scheduling in deregulated power market taking into account transmission loss allocation. IET Gen Transm Dist 2010;4(7):883–92. [26] Bhakar R, Sriram VS, Padhy NP, Gupta HO. Probabilistic game approaches for network cost allocation. IEEE Trans Power Syst 2010;25(1). [27] Alturki YA, Lo KL. Real and reactive power loss allocation in pool-based electricity markets. Int J Electr Power Energy Syst 2010;32:262–70. [28] Ferreira J, Vale Z, Sousa T, Canizes B, Puga J. Transmission costs allocation based on optimal re-dispatch. In: 8th International conference on european energy market (EEM). Croatia; May 2011. p. 473–8. [29] Savier JS, Das Debapriya. Loss allocation to consumers before and after reconfiguration of radial distribution networks. Int J Electr Power Energy Syst 2011;33(3):540–9. [30] Parastar A, Mozafari B, Pirayesh A, Omidi H. Transmission loss allocation through modified Z-bus. Int J Energy Convers Manage 2011;52: 752–6. [31] Choudhury NBD, Goswami SK. Artificial intelligence solution to transmission loss allocation problem. Int J Expert Syst Appl 2011;38:3757–64. [32] Ghosh S, Das D. Method of load flow solution of radial distribution networks. Proc IEE, Gen Transm Dist 1999;146(6):641–8.
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An exact method for loss allocation in radial distribution systems J.S. Savier, Debapriya Das ⇑ Department of Electrical Engineering, Indian Institute of Technology, Kharagpur, 721 302 West Bengal, India
a r t i c l e
i n f o
Article history: Received 14 August 2007 Received in revised form 21 October 2011 Accepted 29 October 2011 Available online 6 December 2011 Keywords: Distribution system Deregulation Loss allocation
a b s t r a c t This paper presents an exact method for real power loss allocation to consumers connected to radial distribution networks in a deregulated environment. The proposed method has the advantage that no assumptions are made in the allocation of real power losses as opposed to the algorithms available in the literature. A detailed comparison of the real loss allocation obtained with the proposed ‘Exact method’ with two alternative algorithms, namely, pro rata (PR), and quadratic loss allocation schemes are presented. Pro rata procedure is based on the load demand of each consumer, quadratic allocations are based on identifying the real and reactive parts of current in each branch and the losses are allocated to each consumer, and the proposed ‘exact method’ is based on the actual contribution of real power loss by each consumer. A case study based on 30 node distribution system is provided. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Deregulation of the power industry was intended to break up the industry’s traditional, vertically integrated structure of generators, transmission lines, and distribution facilities into separate entities with generators competing for sales across common transmission lines to local distribution outlets. Under such a system, generators would be in competition with each other to serve more than one utility distributor. Thus, the generators selling electricity at the lowest cost would have the most utility customers. Unlike generation and sale of electrical energy, activities of transmission and distribution are generally considered as a natural monopoly. The cost of transmission and distribution activities needs to be allocated to the users of these networks. Allocation can be done through network use tariffs, with a focus on the true impact they have on these costs. Among others, distribution power losses are one of the costs to be allocated. The main difficulty faced in allocating losses is the nonlinearity between the losses and delivered power which complicates the impact of each user on network losses [1]. Different techniques have been published in the literature for allocation of losses, most of them dedicated to transmission networks and can be classified into three broad categories – pro rata procedures, marginal procedures and proportional sharing procedures. Pro rata procedure [2,3] is the simplest one, in which, the total losses are allocated to loads or generators based on the load active power demand or bus generation. In marginal procedures [4–7], losses are assigned to generators and demands through
⇑ Corresponding author. E-mail address: [email protected] (D. Das). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.10.030
the so-called incremental transmission loss (ITL) coefficients. In proportional sharing procedures [8–13], losses are allocated to the generators and consumers by using the results of a converged power flow plus a linear proportional sharing principle. Conejo et al. [14] have presented a new procedure for allocating transmission losses to generation and loads based on the networks Z-bus matrix. Conejo et al. [15] have also presented a comparison of four different practical algorithms for transmission loss allocation. Fang and Ngan [16] proposed a succinct method for allocation of network losses which takes into consideration the influence of both active and reactive power injected into grids. Costa and Matos [17] have addressed the allocation of losses in distribution networks with embedded generation by considering quadratic loss allocation technique. Daniel et al. [18] presented an approach for transmission loss allocation using a modified Ybus. Abhyankar et al. [19,20] presented methods for transmission fixed cost allocation based on min–max fairness criteria and optimization approach. Carpaneto et al. [21] proposed a branch current decomposition method for loss allocation in radial distribution systems with distributed generation. Carpaneto et al. [22] also presented a detailed characterization of different loss allocation techniques for radial distribution systems with distributed generation. Belati and da Costa [23] presented a transmission loss application based on perturbation of optimum theorem. In their approach, from a given optimal operating point obtained by the optimal power flow the loads are perturbed and a new operating point that satisfies the constraints are obtained by sensitivity analysis, which is used for obtaining the coefficients of the losses. Lim et al. [24] introduced a new method for allocating losses in a power system using a loop-based representation of system behavior. Li et al. [25] presented a means of allocating transmission losses and costs taking into account pool and bilateral contract hybrid
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106
Nomenclature P kVAp Ploss NB IL(p) PLp QLp Vp Jj I(jj) N(jj) ie(jj, i)
a{ie(jj, i)}, b{ie(jj, i)} loss allocation factors for the consumer at
node number kVA load demand of consumer at node p total real power loss in the system total number of nodes in the system load current at pth node real power load at pth node reactive power load at pth node voltage at pth node branch number, jj = 1, 2, . . . , NB1 current of branch-jj total number of nodes beyond branch-jj nodes beyond branch-jj, for i = 1, 2, . . . , N(jj)
node ie(jj, i) IL{ie(jj, i)} load current of consumer at node ie(jj, i) ILD{ie(jj, i)} real component of load current at node ie(jj, i) ILQ{ie(jj, i)} reactive component of load current at node ie(jj, i) PLOSS(jj) real power loss of branch-jj R(jj) resistance of branch-jj Rfxg Real part of complex quantity, x ploss(jj, ‘) real power loss of branch-jj allocated to consumer connected at node ‘ Tploss (‘) real power loss supported by consumer ‘
deregulated power market. Bhakar et al. [26] presented probabilistic game approaches for network cost allocation. Alturki and Lo [27] presented a loss allocation method using current adjustment factors to allocate real and reactive losses simultaneously without any additional calculation except the substitution of line reactance instead of resistance. Ferreira et al. [28] presented a method for transmission cost allocation based on optimal re-dispatch. Savier and Das [29] presented allocation of power losses to consumers connected to radial distribution network before and after network reconfiguration in a deregulated environment. Parastar et al. [30] presented an approach of transmission loss allocation in which, a relationship between the bus current injections and the generator or load currents is first determined using a power invariant matrix and then Z-bus matrix is modified, which allows the real power loss of the network to be expressed in terms of generator or load currents. Choudhury and Goswami [31] solved the problem of transmission loss allocation of deregulated power system through the application of artificial neural network. As there is no unique or existing ideal procedure is available, any loss allocation algorithm should have most of the desirable properties such as consistent with the results of power flow, depend upon the amount of energy either produced or consumed, depend on the relative location within the network, easy to understand, and easy to implement [15]. In the light of the above developments, this work proposes a simple loss allocation method, called ‘Exact Method’ for real power loss allocation to consumers in radial distribution systems. It is assumed that consumers have to pay for losses. Real power loss allocation to each consumer obtained with the proposed method is compared with two different practical approaches for loss allocation, namely, pro rata (PR) and quadratic loss allocation schemes.
If the total number of nodes beyond branch-jj is N(jj), the current through that branch can be written as:
IðjjÞ ¼
NðjjÞ X ðILDfieðjj; kÞg jILQfieðjj; kÞgÞ
Real power loss of branch-jj with sending end and receiving end voltages Vi and Vj is given by:
PLOSSðjjÞ ¼ RfðV i V j Þ IðjjÞg ( i:e:; PLOSSðjjÞ ¼ R ðV i V j Þ
NðjjÞ X ðILDfieðjj; kÞg jILQfieðjj; kÞgÞ
In a radial distribution system, load current of a consumer connected to node ie(jj, k) beyond branch-jj can be written in the form:
ð1Þ
)
ð4Þ ⁄
Let (Vi Vj) = a{ie(jj, k)} + jb{ie(jj, k)} )PLOSSðjjÞ ¼ R
( NðjjÞ ) X ðafieðjj;kÞg þ jbfieðjj; kÞgÞ:ðILDfieðjj; kÞg jILQ fieðjj; kÞgÞ k¼1
ð5Þ Hence,
PLOSSðjjÞ ¼
NðjjÞ X
ðafieðjj; kÞgILDfieðjj; kÞg þ bfieðjj; kÞgILQ fieðjj; kÞgÞ
k¼1
ð6Þ Using Eq. (6), real power loss in branch-jj can be allocated to consumers beyond branch-jj. Real power loss of branch-jj allocated to consumer connected to node ie(jj, k) is given by:
plossfjj; ieðjj; kÞg ¼ afieðjj; kÞgILDfieðjj; kÞg þ bfieðjj; kÞgILQ fieðjj; kÞg for jj ¼ 1; 2; . . . ; NB 1 and k ¼ 1; 2; . . . ; NðjjÞ
2.1. Proposed exact method
ILfieðjj; kÞg ¼ ILDfieðjj; kÞg jILQ fieðjj; kÞg
ð3Þ
k¼1
2. Proposed method for distribution loss allocation In radial distribution systems, power is fed at substations and the power flows from substation to downstream. In this section, proposed exact method for loss allocation is first presented. For the purpose of comparison, pro rata (PR) and quadratic loss allocation schemes are used. Hence, loss allocation procedure with these two methods is also briefly presented.
ð2Þ
k¼1
ð7Þ
2.2. Pro rata (PR) allocation method The PR method proportionally allocates losses to the consumers based on the kW load demand [15]. In order to take into account reactive load of each consumer, instead of considering real power demand of the consumer, kVA load demand is considered. The total kVA demand of the system can be written as:
TkVAD ¼
NB X
kVAp
ð8Þ
p¼2
Note that substation is marked as node 1, i.e., p = 1. Hence, power loss allocated to consumer at node p is given by:
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kVAp TkVAD
plossp ¼ Ploss
i:e:; plossp ¼ K D
9
ð9Þ
kVAp ;
KD ¼
Ploss TkVAD
(8)
6
ð10Þ 8
It may be noted that the demand loss allocation factors KD are identical for all buses. Hence, pro rata procedure is simple to understand and implement but they ignore the relative location of the consumers within the network. That is, two identical demands located respectively near substation and far away from the substation are equally treated, and this is unfair to the load located near the substation.
(5)
1
(7) 3
(1)
2
(2)
(3)
½ILDfieðjj; iÞg
¼
afieðjj; kÞg ½ILDfieðjj; kÞg
ð11Þ
2
and
bfieðjj; iÞg ½ILQfieðjj; iÞg2
¼
bfieðjj; kÞg
ð12Þ
½ILQfieðjj; kÞg2
Based on this principle, the power loss of the branch-jj of the network allocated to consumers beyond branch-jj, for i = 1, 2, . . . , N(jj) are (Derivation given in Appendix A):
plossfjj; ieðjj; iÞg ¼ RðjjÞ
þ
NðjjÞ X
8 > < > :
½ILDfieðjj; iÞg2 þ ½ILQ fieðjj; iÞg2
ILDfieðjj; iÞg ILDfieðjj; kÞgafieðjj; iÞg
k¼1 k–i
9 > = ILQfieðjj; iÞg ILQfieðjj; kÞgbfieðjj; iÞg ð13Þ þ > ; k¼1 NðjjÞ X
k–i
The global value of losses to be supported by consumer ‘ results from the sum of the losses allocated to it in each branch-jj of the network, i.e.,
Tplossð‘Þ ¼
NB1 X
plossðjj; ‘Þ for
(4)
(6)
In quadratic loss allocation scheme, since the power losses grow quadratically with power flows, the following constraint is imposed [1]. 2
5
S/S
2.3. Quadratic allocation method
afieðjj; iÞg
4
‘ ¼ 2; 3; . . . ; NB
ð14Þ
jj¼1
Note that node 1 is substation. Eq. (14) indicates that each consumer has allocated losses only at branches to which power flow contributes. 3. Case study In order to validate the proposed method, two radial distribution systems are considered. In the first example, a nine node radial distribution system is used to demonstrate the proposed method and in the second example, a detailed comparison of loss allocation results obtained with the proposed method with two methods available in the literature is presented. 3.1. Example – 1 In this example, a distribution system having nine nodes, as shown in Fig. 1 is considered. The line and load data for this system are given in Appendix B. In Case-1, the real and reactive power loads as shown in Appendix B is taken. For the purpose of explanation of
7 Fig. 1. Sample distribution system with nine nodes.
the proposed technique when there is current injection, Case-2 is considered in which the line and load data is same as those given in Appendix B except that the reactive power load at node no. 8 is taken as negative (injection). In Case-1, total real power loss of the radial distribution system after a converged load flow was 24.0406 kW. Allocation of real power losses to different consumers, voltages and load currents at different nodes are given in Table 1. It can be seen from the table that the real power losses allocated to consumer at node nos. 8 and 9 are 7.4571 kW and 7.9504 kW. It can be seen that the real power loss allocated to consumer at node no. 9 is higher than that at node no. 8, even though the real and reactive power losses at node no. 9 is lower than that at node no. 8. This is due to the fact that the proposed method allocates losses based on actual contribution of the losses. In Case-2, total real power loss of the distribution system after a converged load flow is 16.863 kW. It can be seen from Table 1 that the real power losses allocated to all the nodes are reduced in this case due to the fact that the total power loss has reduced in this case. Real power allocated to consumers at node nos. 8 and 9 in Case-2 are 6.5651 kW and 4.8895 kW respectively as compared to 7.4571 kW and 7.9504 kW in Case-1.
3.2. Example – 2 A practical 11 kV distribution system having 30 nodes, as shown in Fig. 2 is considered to compare the four distribution loss allocation methods. The line and load data for this system are given in Appendix C. The loads considered are either industrial or commercial type. The total real power loss of the system, obtained from the results of the load flow is 146.09 kW. Table 2 shows the real power loss allocated to each consumer with the three different approaches. Power flow data and results are used as input data for the four loss allocation algorithms. The loss allocation based on pro rata procedure allocates losses irrespective of the geographical location of consumers. Hence, consumers having same load demands are allocated same losses, even though the power loss contribution of the consumer electrically closer to the substation is less as compared to those consumers electrically away from the substation. For example, consumers at nodes 10 and 28 are having the same load demand. Hence, loss allocation based on pro rata procedure allocates same losses to both the consumers, i.e., 4.3624 kW. Whereas, loss allocation to consumers at nodes 10 and 28 based on quadratic and exact method are different. Quadratic loss allocation scheme allocated 3.3636 kW and 3.6851 kW to the consumers at nodes 10 and 28, respectively, and proposed exact method allocated 4.2312 kW
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106 Table 1 Loss allocation to consumers for a nine node system. Node no.
Case-1
1 2 3 4 5 6 7 8 9
Case-2
Voltage (pu)
Current (pu)
Real power loss allocated (kW)
Voltage (pu)
Current (pu)
Real power loss allocated (kW)
1 0.9844 0.9767 0.974 0.9739 0.9841 0.9765 0.9643 0.9575 Total
0 0.0016 0.0010i 0.0015 0.0014i 0.0001 0.0001i 0.0003 0.0002i 0.0005 0.0003i 0.0001 0.0001i 0.0019 0.0014i 0.0016 0.0012i
0 2.6413 3.7695 0.3311 0.8307 0.7634 0.2971 7.4571 7.9504 24.0406
1 0.9871 0.9811 0.9794 0.9793 0.9868 0.981 0.9716 0.9649 Total
0 0.0016 0.0010i 0.0015 0.0014i 0.0001 0.0001i 0.0003 0.0002i 0.0005 0.0003i 0.0001 0.0001i 0.0019 + 0.0014i 0.0016 0.0013i
0 1.8183 2.1586 0.2061 0.5182 0.5060 0.2012 6.5651 4.8895 16.8630
13 12 11 1
S /S
28
29
30
5
6
7
10 2
3
23
4
8
20
14
19
21
15
18
22
9
4. Conclusions
24 16
17
25 27
26
Fig. 2. Distribution system with 30 nodes.
Table 2 Real power loss allocation to consumers. Node nos.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Total
and 4.4907 kW to the consumers at nodes 10 and 28 respectively. It can be seen that all the methods except pro rata method allocated less power loss to consumer at node 10, which is electrically closer to the substation, as compared to losses allocated to consumer at node 28, which is electrically away from the substation. Quadratic loss allocation technique makes use of assumption to obtain power loss of each branch allocated to consumers beyond that branch. The proposed ‘exact method’ allocates branch losses to different consumers based on actual contribution of the branch power losses by each consumer beyond that branch. In the present work, a load flow algorithm developed in [32] for solving the radial distribution network has been used.
Real power loss allocated Proposed method
Pro rata method
Quadratic method
0.000 6.079 9.414 0.876 2.326 3.801 1.036 1.731 16.709 4.231 5.832 3.239 1.080 2.674 4.755 5.179 13.384 7.761 4.104 0.995 4.722 9.820 3.328 10.346 6.134 3.456 2.688 4.491 3.395 2.507 146.09
0.000 13.691 14.819 1.018 2.544 4.118 0.976 1.676 14.310 4.362 5.390 2.926 0.976 2.793 4.362 4.362 11.738 5.853 3.702 0.976 4.936 8.725 2.932 9.275 5.196 2.793 1.951 4.362 3.146 2.186 146.09
0.000 8.227 15.381 0.093 0.894 2.577 0.102 0.403 24.118 3.364 5.194 1.623 0.123 1.235 3.420 3.700 19.831 6.531 2.755 0.098 4.206 12.025 1.609 13.788 5.612 1.730 0.814 3.685 1.998 0.956 146.09
In the present work, a simple loss allocation method for consumers connected to radial distribution networks has been proposed. A detailed comparison of real power loss obtained with the proposed ‘exact’ method with two different methods namely, pro rata, and quadratic loss allocation methods has been presented. From the case study, it can be seen that even though pro rata procedure is simple and easy to implement, power loss allocated to consumers having same load demands are the same, which is injustice to the consumer electrically nearer to the substation. Quadratic loss allocation scheme is based on branch current flow and it allocates branch power loss to only those consumers beyond that branch. Quadratic loss allocation scheme makes the assumption that the loss allocation factor of a particular consumer is proportional to the square of real/reactive load current of that consumer. In the proposed ‘exact method’, losses are allocated to consumers without making any assumptions and can be implemented easily. Appendix A For the purpose of explanation, consider a sample distribution network as shown in Fig. A1. In Fig. A1, branch numbers are shown in (). The branch number, sending end and receiving end nodes are given in Table A1. The load current at any node p, is given as:
ILðpÞ ¼
PLp jQLp ; V p
p ¼ 2; 3; . . . ; NB
ðA1Þ
Consider branch-8 of Fig. A1. Number of node beyond branch-8 is one and this is node 9. Therefore, current through branch-8 is:
Ið8Þ ¼ ILð9Þ
ðA2Þ
Similarly consider branch-7. Total number of nodes beyond branch-7 is three and these nodes are 8, 9 and 12. Therefore, current through branch-7 is:
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106
2
11
NðjjÞ X
jIðjjÞj ¼ 4
(1 0 )
!2 ILDfieðjj; iÞg
þ
i¼1
NðjjÞ X
!2 312 ILQ fieðjj; iÞg 5
ðA7Þ
i¼1
10 (9 )
1 2
3
4
5
)jIðjjÞj2 ¼
6
NðjjÞ X 2 ½ILDfieðjj; iÞg þ ½ILQfieðjj; iÞg2 i¼1
S/S (1 )
(2 )
(3 )
(4 )
(5 )
þ2
(6 )
NðjjÞ1 NðjjÞ X X i¼1
7
ðILDfieðjj; iÞg:ILDfieðjj; kÞg
k¼iþ1
þ ILQ fieðjj; iÞg:ILQfieðjj; kÞgÞ (7 )
Power loss of branch-jj can be written as: 12
8
PLOSSðjjÞ ¼ RðjjÞ:jIðjjÞj2
(1 1 ) (8 )
ðA9Þ
From Eqs. (A8) and (A9), we get,
9
( NðjjÞ X 2 2 PLOSSðjjÞ ¼ RðjjÞ ½ILDfieðjj; iÞg þ ½ILQ fieðjj; iÞg
Fig. A1. Sample distribution network with 12 nodes.
i¼1
Ið7Þ ¼ ILð8Þ þ ILð9Þ þ ILð12Þ
ðA3Þ
From Eqs. (A2) and (A3), it is clear that, if we identify the nodes beyond all the branches and if the load currents are known, then it is extremely easy to compute the branch currents. In [32], an algorithm for identifying the nodes beyond all the branches has been proposed. A load flow algorithm based on the identification of these nodes beyond all the branches has also been proposed in [32]. In the present work, this load flow algorithm is used to compute the load currents and branch currents. General expression of branch current through branch-jj is given by [32]:
IðjjÞ ¼
NðjjÞ X
ILfieðjj; iÞg
ðA4Þ
In Table A1, total number of nodes (consumers) beyond branchjj (N(jj)) and nodes (consumers) beyond branch-jj (ie(jj, i); i = 1, 2, . . . , N(jj)) are given for Fig. A1 for the purpose of explanation. Let us define:
ILfieðjj; iÞg ¼ ILDfieðjj; iÞg jILQfieðjj; iÞg
ðA5Þ
From Eqs. (A4) and (A5), we obtain: NðjjÞ X
½ILDfieðjj; iÞg jILQ fieðjj; iÞg
ðA6Þ
i¼1
Magnitude of current in branch-jj is expressed as:
Table A1 Branch number, sending end node, receiving end node and nodes beyond different branches. Branch number (jj) 1 2 3 4 5 6 7 8 9 10 11
i¼1
Sending end node IS(jj)
Receiving end node IR(jj)
1 2 3 4 5 2 7 8 4 10 8
2 3 4 5 6 7 8 9 10 11 12
Total number of nodes beyond branch-jj N(jj)
Nodes beyond branchjj ie(jj,i) i = 1, 2, . . . , N(jj)
ðILDfieðjj; iÞg ILDfieðjj; kÞg
k¼iþ1
)
þILQfieðjj; iÞg ILQfieðjj; kÞgÞ
ðA10Þ
The development of expression given by Eq. (A10), the consumer ie(jj, i) (consumers beyond branch-jj for i = 1, 2, . . . , N(jj)) has the impact on the terms [ILD{ie(jj, i)}]2 and [ILQ{ie(jj, i)}]2 of the power loss of branch-jj, which are exclusively due to consumers ie(jj, i) for i = 1, 2, . . . , N(jj). The terms,
2
NðjjÞ1 NðjjÞ X X i¼1
i¼1
IðjjÞ ¼
þ2
NðjjÞ1 NðjjÞ X X
ðILDfieðjj; iÞg ILDfieðjj; kÞg þ ILQ fieðjj; iÞg ILQfieðjj; kÞgÞ
k¼iþ1
are the results of the simultaneous influence of consumers ie(jj, i) and the remaining consumers ie(jj, k) in the components of the current in the branch-jj. These crossed terms must also be allocated. The allocation of crossed terms has been done in a quadratic way, as mentioned in [1]. The crossed terms 2 ILD{ie(jj, i)} ILD{ie(jj, k)} and 2 ILQ{ie(jj, i)} ILQ{ie(jj, k)} are split into two components as described here. Let us define,
afieðjj; iÞg ½ILDfieðjj; iÞg ILDfieðjj; kÞg þ afieðjj; kÞg ½ILDfieðjj; iÞg ILDfieðjj; kÞg ¼ 2 ½ILDfieðjj; iÞg ILDfieðjj; kÞg for i ¼ 1; 2; . . . ; NðjjÞ 1 for k ¼ i þ 1; i þ 2; . . . ; NðjjÞ
ðA11Þ
and
bfieðjj; iÞg ½ILQ fieðjj; iÞg ILQ fieðjj; kÞg þ bfieðjj; kÞg ½ILQ fieðjj; iÞg ILQ fieðjj; kÞg ¼ 2 ½ILQfieðjj; iÞg ILQfieðjj; kÞg
11 6 5 2 1 4 3 1 2 1 1
2,3,4,5,6,7,8,9,10,11,12 3,4,5,6,10,11 4,5,6,10,11 5,6 6 7,8,9,12 8,9,12 9 10,11 11 12
for i ¼ 1; 2; . . . ; NðjjÞ 1 for k ¼ i þ 1; i þ 2; . . . ; NðjjÞ
ðA12Þ
Eqs. (A11) and (A12) can be simply written as:
afieðjj; iÞg þ afieðjj; kÞg ¼ 2:0
ðA13Þ
bfieðjj; iÞg þ bfieðjj; kÞg ¼ 2:0
ðA14Þ
where a{ie(jj, i)} and b{ie(jj, i)} are the variable loss allocation factors for the consumer at node ie(jj, i).
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106
In quadratic loss allocation scheme, since the power losses grow quadratically with power flows, the following constraint is imposed [1].
afieðjj; iÞg 2
½ILDfieðjj; iÞg
¼
afieðjj; kÞg ½ILDfieðjj; kÞg
ðA15Þ
2
½ILQfieðjj; iÞg2
¼
Tplossð‘Þ ¼
NB1 X
plossðjj; ‘Þ for ‘ ¼ 2; 3; . . . ; NB
ðA20Þ
jj¼1
and
bfieðjj; iÞg
The global value of losses to be supported by consumer ‘ results from the sum of the losses allocated to it in each branch-jj of the network, i.e.,
bfieðjj; kÞg
ðA16Þ
½ILQ fieðjj; kÞg2
Note that node 1 is substation. Eq. (A20) indicates that each consumer has allocated losses only at branches to which power flow contributes.
From Eqs. (A13) and (A15), we get,
afieðjj; iÞg ¼
2:½ILDfieðjj; iÞg
Appendix B
2
2
½ILDfieðjj; iÞg þ ½ILDfieðjj; kÞg
ðA17Þ
2
Similarly, from Eqs. (A14) and (A16), we get,
bfieðjj; iÞg ¼
2 ½ILQ fieðjj; iÞg2
ðA18Þ
½ILQ fieðjj; iÞg2 þ ½ILQ fieðjj; kÞg2
Based on this principle, the power loss of the branch-jj of the network allocated to consumers beyond branch-jj, for i = 1, 2, . . . , N(jj) are:
(
plossfjj; ieðjj; iÞg ¼ RðjjÞ ½ILDfieðjj; iÞg2 þ ½ILQ fieðjj; iÞg2 þ
NðjjÞ X
ILDfieðjj; iÞg ILDfieðjj; kÞg afieðjj; iÞg
Branch Sending Receiving R (X) end node number end IR(jj) jj node IS(jj)
X (X) PL(IR(jj)) QL(IR(jj)) (kW) (kVar)
1 2 3 4 5 6 7 8
1.1019 0.7346 0.3673 0.1836 0.3673 0.3896 0.7792 1.1688
1 2 3 4 2 3 4 8
2 3 4 5 6 7 8 9
1.632 1.088 0.5440 0.2720 0.5440 1.3760 2.7520 4.1280
162 150 12 30 45.6 12 180 156
96 138 7.2 18 33.6 6 140.4 120
k¼1 k–i
9 > = ILQ fieðjj; iÞg ILQ fieðjj; kÞg:bfieðjj; iÞg ðA19Þ þ > ; k¼1 NðjjÞ X
Appendix C
k–i
Table C1.
Table C1 Line and load data of 30 node radial distribution system. Branch number jj
Sending end node IS(jj)
Receiving end node IR(jj)
R (X)
X (X)
PL(IR(jj)) (kW)
QL(IR(jj)) (kVar)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29
1 2 3 4 5 6 7 8 4 10 11 12 11 28 29 5 20 21 22 21 24 25 26 6 14 15 16 16 18
2 3 4 5 6 7 8 9 10 11 12 13 28 29 30 20 21 22 23 24 25 26 27 14 15 16 17 18 19
1.6320 1.0880 0.5440 0.2720 0.5440 1.3760 2.7520 4.1280 3.6432 0.9108 0.4554 0.4554 1.3760 1.3760 4.1280 0.9108 1.8216 2.7324 0.9108 2.7520 3.0272 2.7520 2.7520 0.9108 1.8216 1.8216 0.9108 1.3760 1.3760
1.1019 0.7346 0.3673 0.1836 0.3673 0.3896 0.7792 1.1688 1.5188 0.3797 0.1898 0.1898 0.3896 0.3896 1.1688 0.3797 0.7594 1.1391 0.3797 0.7792 0.8571 0.7792 0.7792 0.3797 0.7594 0.7594 0.3797 0.3896 0.3896
162 150 12 30 45.6 12 18 156 48 64.8 36 12 48 36 26.4 12 48 96 32.4 96 54 30 24 30 48 48 120 72 36
96 138 7.2 18 33.6 6 14.4 120 36 36 18 6 36 24 14.4 6 48 72 24 84 46.8 24 12 24 36 36 108 36 36
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J.S. Savier, D. Das / Electrical Power and Energy Systems 36 (2012) 100–106
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