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An efficient heuristic algorithm for reconfiguration based on branch power flows direction
Electrical Power and Energy Systems 41 (2012) 71–75
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An efficient heuristic algorithm for reconfiguration based on branch power flows direction Antonio José Gil Mena ⇑, Juan Andrés Martín García 1 Department of Electrical Engineering, University of Cádiz, Escuela Politécnica Superior de Algeciras, Avda. Ramón Puyol, s/n, 11202 Algeciras (Cádiz), Spain
a r t i c l e
i n f o
Article history: Received 31 May 2011 Received in revised form 16 November 2011 Accepted 10 March 2012 Available online 21 April 2012 Keywords: Reconfiguration Heuristic Distribution systems Breakpoint node Loss minimization
a b s t r a c t This paper presents a meshed algorithm for optimal reconfiguration of distribution systems. In the reconfiguration problem, the final objective is to reach a radial network that optimizes some function like the network losses. Many algorithms start from a radial network where a switch closure is complemented by opening of another switch to ensure a radial network. These radial algorithms have an inherent inconvenient, that is, the final solution depends on the initial radial network selected. Other group of algorithms initially represent the distribution network as a meshed network and then open switches until a radial system is obtained. In this paper, to avoid the above aforementioned inconvenient of radial algorithms, a meshed algorithm is used. Furthermore, taking into account that breakpoint nodes are defined as the nodes where the branch power flows converge, the contribution of this paper is to provide a method for solving the problem when multiple loops are considered using an approach based on the breakpoint nodes, since it is complicated to associate each breakpoint node with its corresponding loop. On the other hand, one of the drawbacks of the reconfiguration problem is the need to solve a great number of power flow computations for calculating the losses in each stage of the algorithm. The algorithm proposed has the property that reduces the number of power flows. By this way, the execution time of the algorithm is improved. Besides, it is not necessary to check the network connectivity at each step of the procedure. To prove the effectiveness of the proposed algorithm several test systems have been used, achieving good results. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Losses in electric power systems are an inherent consequence of energy transmission from generation to consumers. The electric power systems consist of three subsystems: generation, transmission and distribution, being the last one the most numerous. Besides, due to the low voltage level, losses are greater in distribution systems than in transmission systems. This is why reducing losses in distribution systems remains an interesting line of research. In general, distribution systems are structurally meshed, but technical considerations make that they are radially operated. To achieve the final radial network switches are installed for changing the network topology. This process is denominated as the reconfiguration problem, which is widely studied. Nevertheless, new techniques are continuously appearing in order to achieve a global optimal solution and to reduce the computation time.
⇑ Corresponding author. Tel.: +34 956 028 015; fax: +34 956 028 001. E-mail addresses: [email protected] (A.J. Gil Mena), [email protected] (J.A. Martín García). 1 Tel.: +34 956 028 167; fax: +34 956 028 001. 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.03.009
There are many approaches to solve the reconfiguration problem. A first collection of the different techniques was published by Sarfi et al. [1], where a review and a classification of different techniques was presented, that is: blending heuristic and optimization, algorithms based solely on heuristic and artificial intelligent techniques. Regardless of the technique used, and considering the structure of the algorithm, these ones can be divided into two main groups, namely, algorithms that start from a network with all switches closed and algorithms that start from a radial network. Meshed algorithms are characterized by opening a branch in each step of the algorithm to achieve a radial network. What differentiates each meshed algorithm is the method or criterion used to decide the branch to open in each step of the algorithm. Moreover, radial algorithms start from a radial network with a number of branches initially opened. At each step of the algorithm, a branch is closed, so a mesh is created. Thereupon, similarly to meshed algorithms, different approaches are used to decide which branch is selected to open. This process is repeated until an acceptable solution is achieved. Merlin and Back [2] were the first to raise the issue of reconfiguration, using a meshed algorithm where a minimal-loss load flow was used to decide the branch to be opened. In contrast to Merlin
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A.J. Gil Mena, J.A. Martín García / Electrical Power and Energy Systems 41 (2012) 71–75
and Back, Civanlar et al. [3] were the first using a radial algorithm where a simple formula was used to evaluate the losses increase. Other papers using the same structures but with different approaches are continuously appearing. Among others, Refs. [4–7] make use of meshed algorithms and Refs. [8–10] of radial algorithms. In Ref. [4], Viswanadha and Bijwe made use of real power loss sensitivity with respect to the impedances of the candidate branches to select the branch to be opened. Another meshed algorithm was used by Singh et al. in [5], where proposed opening of a branch in a loop carrying minimum resistive power flow. de Oliveira et al. [6] made use of a mixed integer non-linear programming approach, in which a continuous function is used to handle the discrete variables and Chandramohan et al. [7] use a non-dominated sorting genetic algorithm (NSGA) for reconfiguring to minimize its operating costs considering the system real power transmission loss and the cost of reactive power purchased by the distribution system. With respect to deal with the problem of reconfiguration through radial algorithms, in Ref. [8], Zhang et al. proposed a joint optimization algorithm of combining network reconfiguration and capacitor control. Bernardon et al. [9] proposed a new fuzzy multicriteria decision making algorithm for the proper processing of the information sources available at the utilities in the context of distribution network reconfiguration. And recently, Ababei and Kavasseri [10] have proposed an efficient heuristic algorithm to solve the distribution network reconfiguration problem for loss reduction. Apart from these two large groups, a third group is formed by the union of a meshed and a radial algorithm, that is, a meshed algorithm is executed to give an initial solution for the radial algorithm. The paper proposed by Gomes et al. [11] belongs to this latter type. Finally, there are other algorithms that do not belong to the aforementioned groups like the heuristic constructive algorithm proposed in [12] by McDermott et al. Finally, Zhang et al. [13], have proposed a reliability-oriented reconfiguration (ROR) method that deals with uncertainties in loads, generations, electrical and economic parameters and ambiguous reliability parameters for distribution network reconfiguration. This work is an improvement of that published in [14] where a radial algorithm based on branch power flow directions was used. However, that approach did not consider the possibility that the algorithm started from a meshed network, where multiple loops were taken into account. The contribution of this paper is to provide a method for solving the problem when multiple loops are considered using the proposed approach. Overall, radial algorithms have the inconvenient that the final solution depends on the initial radial network selected. So, to avoid the above aforementioned inconvenient of radial algorithms, a meshed heuristic algorithm for optimal reconfiguration of distribution systems is now presented in this paper based on the direction of branch active and reactive power flows. Taking into account the above, a set of candidate branches to be opened are selected in each step of the algorithm in order to reach the radial network. To conclude, the paper is organized as follows: Section 2 covers the proposed heuristic technique and the study of the candidate branches to be selected. In Section 3, the algorithm is described. Test systems and numerical results are explained in Section 4 and finally, in Section 5 the final conclusions are presented.
2. Heuristic technique An extended description of the heuristic is described in Ref. [14] when only one loop is considered. That heuristic is based on the direction of power flows through the branches of the network.
Since the complex power in alternating current networks has no direction, active and reactive powers are used instead of complex power, and thus, two different power flows are represented separately in the network. In this way, two directed graphs are obtained. Once the network is represented by two directed graphs, particular nodes (buses) in each graph have the characteristic that the flows entering in the node are more or equal than two. These characteristic nodes are called active breakpoint node (P-breakpoint) and reactive breakpoint node (Q-breakpoint), and the branches entering in the breakpoint nodes are chosen as candidates to be opened. Thus, the number of candidate branches to be opened is generally reduced to two, instead of all the branches forming the loop. When the network has only one loop, the set of the candidate branches depends on the situation of the breakpoints in the loop. Two different cases can appear in a single loop if capacitor banks are not considered: P-breakpoint and Q-breakpoint are the same node and P-breakpoint and Q-breakpoint are different nodes. The candidate branches in these two cases are respectively: only the two branches entering in the breakpoint node and the collection of the branches entering in the P and Q nodes plus the branches of the path between the two nodes. The cases commented in paragraph above are valid for a single loop. Considering a network with l loops, in general, there will be so many breakpoint nodes as loops. This is true when all breakpoint nodes have in degree equal to two (the number of power flows entering in the node is two). It could happen that the number of power flows entering in a node was more than two, for example, three. In this case, the number of breakpoint nodes decreases in a unit. Therefore, if the in degree of a node is two, the number of candidate branches is two, if it is three, the set of candidate branches is three taking two by two, and so on. For a network with n nodes and b branches, the relation between the number of P-breakpoint or Q-breakpoint nodes (nBN) and their respective in degree is expressed by:
nBN ¼ b ðn 1Þ
X ðgðiÞ 2Þ;
8i 2 BN=gðiÞ > 2
ð1Þ
where BN is the set of breakpoint nodes; g(i) in degree of the breakpoint; b the number of branches; and n is the number of nodes. On the other hand, the number of candidate branches for each breakpoint node will be equal to the in degree of the node. Generally, the active and reactive breakpoint nodes in a mesh network do not have to coincide and moreover, the in degree of the breakpoint nodes could be greater than two. This is an inconvenient because the goal is to get so many breakpoint nodes as loops has the network. So, it is necessary to pair off each P-breakpoint to its corresponding Q-breakpoint. To solve this drawback, an adequate set of rules have been developed. These rules are: – There will be as many pairings as loops in the network. – The number of times that a breakpoint node will be paired off will be equal to its in degree less one. – When P-breakpoint and Q-breakpoint were the same node, they will be paired off. These nodes will be paired off as many times as they coincide according to their in degrees. – For the rest of breakpoint nodes, a path is looked for trough the branches that have opposite active and reactive power flow directions starting at a P-breakpoint node until a Q-breakpoint node is reached. To illustrate how the pairings are done, a hypothetical network has been considered.
A.J. Gil Mena, J.A. Martín García / Electrical Power and Energy Systems 41 (2012) 71–75
In Fig. 1, when the directions of the active and reactive power flows through the branches coincide, these ones are drawn on the branches by an arrow. In contrast, when the flows are opposite, the active power flow is drawn on the branch while the reactive power flow is printed beside the branch. So, the two directed graphs and the P-breakpoint and Q-breakpoint nodes have been represented for this hypothetical network in Fig. 1. A summary of the breakpoint nodes, their in degrees and their entering branches are shown in Table 1. Once the breakpoint nodes are obtained and their in degree is known, and after applying the aforementioned rules, the resultant pairings are presented in Table 2. With each pairing, a set of candidate branches are selected. The choice of the candidate branches is the union of the branches entering in the P and Q-breakpoint nodes and the branches of the path between the two breakpoint nodes which have opposite power flow directions. One of the best advantages that derives from the way to make the pairings and the selection of candidate branches is that it is not necessary to check the network connectivity at each step of the procedure.
3. Algorithm description The proposed algorithm starts from a meshed network. Firstly, a power flow is calculated in order to obtain the P and Q breakpoint nodes. Then, the breakpoint node pairings obtained are sorted by a criterion, like its voltage, and the one with the maxima voltage is chosen. Different ordering criteria will produce different algorithms. In case that P and Q-breakpoint nodes were different nodes, its voltage is calculated as the mean of the P and Q-breakpoint node voltages. Once the pairing (P,Q) has been selected, and for each one of the candidate branches, a power flow is calculated opening the respective branch. To finish, the branch that has resulted in the smallest losses is opened. Now, the network has a loop less. By this way, the
Fig. 1. Directed graphs and breakpoint nodes.
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algorithm continues with the rest of pairings until the network has no loops and therefore the radial configuration is reached. It is noticed that each time a loop is opened the branch power flows vary and therefore the breakpoint nodes can change. The meshed algorithm can be enhanced if each time a loop is opened a power flow is calculated and so the P and Q-breakpoint nodes are recalculated. The flow diagram of the proposed algorithm is shown in Fig. 2. 4. Test results and discussion The proposed algorithm has been validated through three test networks used in the technical literature. These networks have been proposed by Das [15], Zhang et al. [16] and Romero-Ramos et al. [17]. Table 3 shows the main characteristics of these test systems: the number of nodes and branches, the published best solutions found in the literature and the losses for each solution. The commercial software MATLABÒ has been used for implementing the proposed algorithms. For solving the power flow by the Newton–Raphson method, a package so called MATPOWERÓ and supplied by PSERC has been used. The solutions reached by the proposed algorithm are presented in Table 4, where the number of load flow calculations, the switches to be opened and the losses are presented. Similarly, the results obtained by the algorithm proposed in [14] are pre-
Fig. 2. Flow diagram of the proposed algorithm.
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A.J. Gil Mena, J.A. Martín García / Electrical Power and Energy Systems 41 (2012) 71–75
Table 1 Breakpoint nodes, in degree and entering branches. P-nodes
In degree
Entering branches
Q-nodes
In degree
Entering branches
P1 P2 P3 P4 P5 P6 P7
4 2 2 2 2 2 2
s1, s2, s3, s4 s7, s8 s21, s22 s18, s19 s14, s15 s11, s12 s24, s25
Q1 Q2 Q3 Q4 Q5 Q6 Q7
3 2 2 2 2 2 3
s1, s2, s3 s6, s7 s14, s15 s9, s10 s16, s17 s20, s23 s13, s27, s28
Table 2 Pairings of P and Q-breakpoint nodes and their respective candidate branches. Pairings
Candidate branches
(P1,Q1) (P1,Q1) (P2,Q2) (P5,Q3) (P1,Q4) (P3,Q6) (P4,Q5) (P6,Q7) (P7,Q7)
s1, s2, s3, s4 s1, s2, s3, s4 s6, s7, s8 s14, s15 s1, s2, s3, s4, s5, s6, s8, s9, s10 s20, s21, s22, s23 s16, s17, s18, s19 s11, s12, s13, s27, s28 s13, s24, s25, s26, s27, s28
sented in Table 5, where the above test systems have been used to carry out the comparison with the proposed method in this paper. It can be seen that the proposed method have obtained better results in terms of solution quality and computational effort in the three distribution systems aforementioned. Respect to the computational effort, Table 4 also shows that the number of load flow calculations is slightly greater than three times the number of loops. This number will be exactly three times the number of loops in the case that the number of candidate branches for each pairing were two (P and Q breakpoint nodes coincide).
Finally, the proposed algorithm has produced the same or better configuration than those obtained from other published articles in the technical literature. Particularly, the attained results for the 205 buses network [17] have produced a better configuration.
5. Conclusion In this paper an efficient meshed algorithm has been developed in order to solve the reconfiguration problem in distribution systems minimizing the network losses. A strategy has been designed to pair off active and reactive breakpoint nodes in a meshed network. Due to that strategy, it is not necessary to check the network connectivity at each step of the procedure, reducing therefore the computation time. Summarizing, the number of power flow calculations is approximately three times, especially for real networks where the P and Q-breakpoints tend to coincide. With relation to algorithm validation, three test systems from the technical literature have been used. The obtained results for these test systems have been equal or better than those published in the literature. Finally, using breakpoint nodes in a meshed algorithm has demonstrated its usefulness for finding the solution of the reconfiguration problem.
Table 3 Test systems characteristics. Test system
Buses
Branches
Published best solution in the technical literature
Losses (kW)
Das [15] Zhang et al. [16] Romero-Ramos et al. [17]
69 118 205
79 132 224
s13, s30, s45, s51, s66, s70, s75, s76, s77, s78, s79 s23, s26, s34, s39, s42, s51, s58, s71, s74, s95, s97, s109, s122, s129, s130 s13, s58, s61, s69, s70, s83, s85, s91, s97, s131, s134, s147, s148, s161, s168, s169, s206, s209, s217, s222
201.412 869.726 87.099
Table 4 Proposed algorithm results. Test system
No load flows
Open switches
Losses (kW)
Das [15] Zhang et al. [16] Romero-Ramos et al. [17]
36 53 71
s13, s30, s45, s51, s66, s70, s75, s76, s77, s78, s79 s23, s26, s34, s39, s42, s51, s58, s71, s74, s95, s97, s109, s122, s129, s130 s12, s58, s61, s69, s70, s85, s98, s131, s133, s142, s143, s147, s148, s161, s168, s169, s206, s209, s217, s222
201.412 869.726 85.986
Table 5 Method of Martín and Gil [14] results. Test system
No load flows
Open switches
Losses (kW)
Das [15] Zhang et al. [16] Romero-Ramos et al. [17]
83 177 213
s13, s28, s45, s51, s67, s70, s73, s75, s76, s78, s79 s23, s34, s39, s42, s48, s50, s58, s71, s74, s95, s97, s109, s119, s129, s130 s14, s58, s61, s69, s70, s85, s131, s134, s137, s142, s143, s147, s148, s162, s206, s209, s217, s218, s222, s224
203.861 874.858 87.718
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References [1] Sarfi RJ, Salama MMA, Chikhani AY. A survey of the state of the art in distribution system reconfiguration for system loss reduction. Electr Power Syst Res 1994;31:61–70. [2] Merlin A, Back H. Search for a minimal loss operating spanning tree configuration in an urban power distribution system. In: Proceedings of 5th power system computation conference, Cambridge; 1975. p. 1–18. [3] Civanlar S, Grainger JJ, Yin H, Le SSH. Distribution feeder reconfiguration for loss reduction. IEEE Trans Power Deliv 1998;3:1217–23. [4] Viswanadha Raju GK, Bijwe PR. An efficient algorithm for minimum loss reconfiguration of distribution system based on sensitivity and heuristics. IEEE Trans Power Syst 2008;23(3):1280–7. [5] Singh SP, Raju GS, Rao GK, Afsari M. A heuristic method for feeder reconfiguration and service restoration in distribution networks. Int J Electr Power Energy Syst 2009;31:309–14. [6] de Oliveira LW, Carneiro Jr S, de Oliveira EJ, Pereira JLR, Silva Jr IC, Costa JS. Optimal reconfiguration and capacitor allocation in radial distribution systems for energy losses minimization. Int J Electr Power Energy Syst 2010;32:840–8. [7] Chandramohan S, Naresh Atturulu, Kumudini Devi RP, Venkatesh B. Operating cost minimization of a radial distribution system in a deregulated electricity market through reconfiguration using NSGA method. Int J Electr Power Energy Syst 2010;32:126–32. [8] Zhang D, Fu Z, Zhang L. Joint optimization for power loss reduction in distribution systems. IEEE Trans Power Syst 2008;23(1):161–9.
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[9] Bernardon DP, Garcia VJ, Ferreira ASQ, Canha LN. Electric distribution network reconfiguration based on a fuzzy multi-criteria decision making algorithm. Electr Power Syst Res 2009;79:1400–7. [10] Ababei C, Kavasseri R. Efficient network reconfiguration using minimum cost maximum flow-based branch exchanges and random walks-based loss estimations. IEEE Trans Power Syst 2011;26(1):30–7. [11] Gomes FV, Carneiro Jr S, Pereira JLR, Vinagre MP, Garcia PAN, Araujo LR. A New heuristic reconfiguration algorithm for large distribution systems. IEEE Trans Power Syst 2005;20(3):1373–8. [12] McDermott TE, Drezga I, Broadwater RP. A heuristic nonlinear constructive method for distribution system reconfiguration. IEEE Trans Power Syst 1999;14(2):478–83. [13] Zhang P, Li W, Wang S. Reliability-oriented distribution network reconfiguration considering uncertainties of data by interval analysis. Int J Electr Power Energy Syst 2012;34:138–44. [14] Martín JA, Gil AJ. A new heuristic approach for distribution system loss reduction. Electr Power Syst Res 2008;78:1953–8. [15] Das D. A fuzzy multi objective approach for network reconfiguration of distribution systems. IEEE Trans Power Deliv 2006;21(1):202–9. [16] Zhang D, Fu Z, Zhang L. An improved TS algorithm for loss-minimum reconfiguration in large-scale distribution systems. Electr Power Syst Res 2007;77(5–6):685–94. [17] Romero-Ramos E, Gomez Expósito A, Riquelme-Santos J, Llorens Iborra F. Pathbased distribution network modelling: application to reconfiguration for loss reduction. IEEE Trans Power Syst 2005;20(2):556–64.
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
An efficient heuristic algorithm for reconfiguration based on branch power flows direction Antonio José Gil Mena ⇑, Juan Andrés Martín García 1 Department of Electrical Engineering, University of Cádiz, Escuela Politécnica Superior de Algeciras, Avda. Ramón Puyol, s/n, 11202 Algeciras (Cádiz), Spain
a r t i c l e
i n f o
Article history: Received 31 May 2011 Received in revised form 16 November 2011 Accepted 10 March 2012 Available online 21 April 2012 Keywords: Reconfiguration Heuristic Distribution systems Breakpoint node Loss minimization
a b s t r a c t This paper presents a meshed algorithm for optimal reconfiguration of distribution systems. In the reconfiguration problem, the final objective is to reach a radial network that optimizes some function like the network losses. Many algorithms start from a radial network where a switch closure is complemented by opening of another switch to ensure a radial network. These radial algorithms have an inherent inconvenient, that is, the final solution depends on the initial radial network selected. Other group of algorithms initially represent the distribution network as a meshed network and then open switches until a radial system is obtained. In this paper, to avoid the above aforementioned inconvenient of radial algorithms, a meshed algorithm is used. Furthermore, taking into account that breakpoint nodes are defined as the nodes where the branch power flows converge, the contribution of this paper is to provide a method for solving the problem when multiple loops are considered using an approach based on the breakpoint nodes, since it is complicated to associate each breakpoint node with its corresponding loop. On the other hand, one of the drawbacks of the reconfiguration problem is the need to solve a great number of power flow computations for calculating the losses in each stage of the algorithm. The algorithm proposed has the property that reduces the number of power flows. By this way, the execution time of the algorithm is improved. Besides, it is not necessary to check the network connectivity at each step of the procedure. To prove the effectiveness of the proposed algorithm several test systems have been used, achieving good results. Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction Losses in electric power systems are an inherent consequence of energy transmission from generation to consumers. The electric power systems consist of three subsystems: generation, transmission and distribution, being the last one the most numerous. Besides, due to the low voltage level, losses are greater in distribution systems than in transmission systems. This is why reducing losses in distribution systems remains an interesting line of research. In general, distribution systems are structurally meshed, but technical considerations make that they are radially operated. To achieve the final radial network switches are installed for changing the network topology. This process is denominated as the reconfiguration problem, which is widely studied. Nevertheless, new techniques are continuously appearing in order to achieve a global optimal solution and to reduce the computation time.
⇑ Corresponding author. Tel.: +34 956 028 015; fax: +34 956 028 001. E-mail addresses: [email protected] (A.J. Gil Mena), [email protected] (J.A. Martín García). 1 Tel.: +34 956 028 167; fax: +34 956 028 001. 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.03.009
There are many approaches to solve the reconfiguration problem. A first collection of the different techniques was published by Sarfi et al. [1], where a review and a classification of different techniques was presented, that is: blending heuristic and optimization, algorithms based solely on heuristic and artificial intelligent techniques. Regardless of the technique used, and considering the structure of the algorithm, these ones can be divided into two main groups, namely, algorithms that start from a network with all switches closed and algorithms that start from a radial network. Meshed algorithms are characterized by opening a branch in each step of the algorithm to achieve a radial network. What differentiates each meshed algorithm is the method or criterion used to decide the branch to open in each step of the algorithm. Moreover, radial algorithms start from a radial network with a number of branches initially opened. At each step of the algorithm, a branch is closed, so a mesh is created. Thereupon, similarly to meshed algorithms, different approaches are used to decide which branch is selected to open. This process is repeated until an acceptable solution is achieved. Merlin and Back [2] were the first to raise the issue of reconfiguration, using a meshed algorithm where a minimal-loss load flow was used to decide the branch to be opened. In contrast to Merlin
72
A.J. Gil Mena, J.A. Martín García / Electrical Power and Energy Systems 41 (2012) 71–75
and Back, Civanlar et al. [3] were the first using a radial algorithm where a simple formula was used to evaluate the losses increase. Other papers using the same structures but with different approaches are continuously appearing. Among others, Refs. [4–7] make use of meshed algorithms and Refs. [8–10] of radial algorithms. In Ref. [4], Viswanadha and Bijwe made use of real power loss sensitivity with respect to the impedances of the candidate branches to select the branch to be opened. Another meshed algorithm was used by Singh et al. in [5], where proposed opening of a branch in a loop carrying minimum resistive power flow. de Oliveira et al. [6] made use of a mixed integer non-linear programming approach, in which a continuous function is used to handle the discrete variables and Chandramohan et al. [7] use a non-dominated sorting genetic algorithm (NSGA) for reconfiguring to minimize its operating costs considering the system real power transmission loss and the cost of reactive power purchased by the distribution system. With respect to deal with the problem of reconfiguration through radial algorithms, in Ref. [8], Zhang et al. proposed a joint optimization algorithm of combining network reconfiguration and capacitor control. Bernardon et al. [9] proposed a new fuzzy multicriteria decision making algorithm for the proper processing of the information sources available at the utilities in the context of distribution network reconfiguration. And recently, Ababei and Kavasseri [10] have proposed an efficient heuristic algorithm to solve the distribution network reconfiguration problem for loss reduction. Apart from these two large groups, a third group is formed by the union of a meshed and a radial algorithm, that is, a meshed algorithm is executed to give an initial solution for the radial algorithm. The paper proposed by Gomes et al. [11] belongs to this latter type. Finally, there are other algorithms that do not belong to the aforementioned groups like the heuristic constructive algorithm proposed in [12] by McDermott et al. Finally, Zhang et al. [13], have proposed a reliability-oriented reconfiguration (ROR) method that deals with uncertainties in loads, generations, electrical and economic parameters and ambiguous reliability parameters for distribution network reconfiguration. This work is an improvement of that published in [14] where a radial algorithm based on branch power flow directions was used. However, that approach did not consider the possibility that the algorithm started from a meshed network, where multiple loops were taken into account. The contribution of this paper is to provide a method for solving the problem when multiple loops are considered using the proposed approach. Overall, radial algorithms have the inconvenient that the final solution depends on the initial radial network selected. So, to avoid the above aforementioned inconvenient of radial algorithms, a meshed heuristic algorithm for optimal reconfiguration of distribution systems is now presented in this paper based on the direction of branch active and reactive power flows. Taking into account the above, a set of candidate branches to be opened are selected in each step of the algorithm in order to reach the radial network. To conclude, the paper is organized as follows: Section 2 covers the proposed heuristic technique and the study of the candidate branches to be selected. In Section 3, the algorithm is described. Test systems and numerical results are explained in Section 4 and finally, in Section 5 the final conclusions are presented.
2. Heuristic technique An extended description of the heuristic is described in Ref. [14] when only one loop is considered. That heuristic is based on the direction of power flows through the branches of the network.
Since the complex power in alternating current networks has no direction, active and reactive powers are used instead of complex power, and thus, two different power flows are represented separately in the network. In this way, two directed graphs are obtained. Once the network is represented by two directed graphs, particular nodes (buses) in each graph have the characteristic that the flows entering in the node are more or equal than two. These characteristic nodes are called active breakpoint node (P-breakpoint) and reactive breakpoint node (Q-breakpoint), and the branches entering in the breakpoint nodes are chosen as candidates to be opened. Thus, the number of candidate branches to be opened is generally reduced to two, instead of all the branches forming the loop. When the network has only one loop, the set of the candidate branches depends on the situation of the breakpoints in the loop. Two different cases can appear in a single loop if capacitor banks are not considered: P-breakpoint and Q-breakpoint are the same node and P-breakpoint and Q-breakpoint are different nodes. The candidate branches in these two cases are respectively: only the two branches entering in the breakpoint node and the collection of the branches entering in the P and Q nodes plus the branches of the path between the two nodes. The cases commented in paragraph above are valid for a single loop. Considering a network with l loops, in general, there will be so many breakpoint nodes as loops. This is true when all breakpoint nodes have in degree equal to two (the number of power flows entering in the node is two). It could happen that the number of power flows entering in a node was more than two, for example, three. In this case, the number of breakpoint nodes decreases in a unit. Therefore, if the in degree of a node is two, the number of candidate branches is two, if it is three, the set of candidate branches is three taking two by two, and so on. For a network with n nodes and b branches, the relation between the number of P-breakpoint or Q-breakpoint nodes (nBN) and their respective in degree is expressed by:
nBN ¼ b ðn 1Þ
X ðgðiÞ 2Þ;
8i 2 BN=gðiÞ > 2
ð1Þ
where BN is the set of breakpoint nodes; g(i) in degree of the breakpoint; b the number of branches; and n is the number of nodes. On the other hand, the number of candidate branches for each breakpoint node will be equal to the in degree of the node. Generally, the active and reactive breakpoint nodes in a mesh network do not have to coincide and moreover, the in degree of the breakpoint nodes could be greater than two. This is an inconvenient because the goal is to get so many breakpoint nodes as loops has the network. So, it is necessary to pair off each P-breakpoint to its corresponding Q-breakpoint. To solve this drawback, an adequate set of rules have been developed. These rules are: – There will be as many pairings as loops in the network. – The number of times that a breakpoint node will be paired off will be equal to its in degree less one. – When P-breakpoint and Q-breakpoint were the same node, they will be paired off. These nodes will be paired off as many times as they coincide according to their in degrees. – For the rest of breakpoint nodes, a path is looked for trough the branches that have opposite active and reactive power flow directions starting at a P-breakpoint node until a Q-breakpoint node is reached. To illustrate how the pairings are done, a hypothetical network has been considered.
A.J. Gil Mena, J.A. Martín García / Electrical Power and Energy Systems 41 (2012) 71–75
In Fig. 1, when the directions of the active and reactive power flows through the branches coincide, these ones are drawn on the branches by an arrow. In contrast, when the flows are opposite, the active power flow is drawn on the branch while the reactive power flow is printed beside the branch. So, the two directed graphs and the P-breakpoint and Q-breakpoint nodes have been represented for this hypothetical network in Fig. 1. A summary of the breakpoint nodes, their in degrees and their entering branches are shown in Table 1. Once the breakpoint nodes are obtained and their in degree is known, and after applying the aforementioned rules, the resultant pairings are presented in Table 2. With each pairing, a set of candidate branches are selected. The choice of the candidate branches is the union of the branches entering in the P and Q-breakpoint nodes and the branches of the path between the two breakpoint nodes which have opposite power flow directions. One of the best advantages that derives from the way to make the pairings and the selection of candidate branches is that it is not necessary to check the network connectivity at each step of the procedure.
3. Algorithm description The proposed algorithm starts from a meshed network. Firstly, a power flow is calculated in order to obtain the P and Q breakpoint nodes. Then, the breakpoint node pairings obtained are sorted by a criterion, like its voltage, and the one with the maxima voltage is chosen. Different ordering criteria will produce different algorithms. In case that P and Q-breakpoint nodes were different nodes, its voltage is calculated as the mean of the P and Q-breakpoint node voltages. Once the pairing (P,Q) has been selected, and for each one of the candidate branches, a power flow is calculated opening the respective branch. To finish, the branch that has resulted in the smallest losses is opened. Now, the network has a loop less. By this way, the
Fig. 1. Directed graphs and breakpoint nodes.
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algorithm continues with the rest of pairings until the network has no loops and therefore the radial configuration is reached. It is noticed that each time a loop is opened the branch power flows vary and therefore the breakpoint nodes can change. The meshed algorithm can be enhanced if each time a loop is opened a power flow is calculated and so the P and Q-breakpoint nodes are recalculated. The flow diagram of the proposed algorithm is shown in Fig. 2. 4. Test results and discussion The proposed algorithm has been validated through three test networks used in the technical literature. These networks have been proposed by Das [15], Zhang et al. [16] and Romero-Ramos et al. [17]. Table 3 shows the main characteristics of these test systems: the number of nodes and branches, the published best solutions found in the literature and the losses for each solution. The commercial software MATLABÒ has been used for implementing the proposed algorithms. For solving the power flow by the Newton–Raphson method, a package so called MATPOWERÓ and supplied by PSERC has been used. The solutions reached by the proposed algorithm are presented in Table 4, where the number of load flow calculations, the switches to be opened and the losses are presented. Similarly, the results obtained by the algorithm proposed in [14] are pre-
Fig. 2. Flow diagram of the proposed algorithm.
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Table 1 Breakpoint nodes, in degree and entering branches. P-nodes
In degree
Entering branches
Q-nodes
In degree
Entering branches
P1 P2 P3 P4 P5 P6 P7
4 2 2 2 2 2 2
s1, s2, s3, s4 s7, s8 s21, s22 s18, s19 s14, s15 s11, s12 s24, s25
Q1 Q2 Q3 Q4 Q5 Q6 Q7
3 2 2 2 2 2 3
s1, s2, s3 s6, s7 s14, s15 s9, s10 s16, s17 s20, s23 s13, s27, s28
Table 2 Pairings of P and Q-breakpoint nodes and their respective candidate branches. Pairings
Candidate branches
(P1,Q1) (P1,Q1) (P2,Q2) (P5,Q3) (P1,Q4) (P3,Q6) (P4,Q5) (P6,Q7) (P7,Q7)
s1, s2, s3, s4 s1, s2, s3, s4 s6, s7, s8 s14, s15 s1, s2, s3, s4, s5, s6, s8, s9, s10 s20, s21, s22, s23 s16, s17, s18, s19 s11, s12, s13, s27, s28 s13, s24, s25, s26, s27, s28
sented in Table 5, where the above test systems have been used to carry out the comparison with the proposed method in this paper. It can be seen that the proposed method have obtained better results in terms of solution quality and computational effort in the three distribution systems aforementioned. Respect to the computational effort, Table 4 also shows that the number of load flow calculations is slightly greater than three times the number of loops. This number will be exactly three times the number of loops in the case that the number of candidate branches for each pairing were two (P and Q breakpoint nodes coincide).
Finally, the proposed algorithm has produced the same or better configuration than those obtained from other published articles in the technical literature. Particularly, the attained results for the 205 buses network [17] have produced a better configuration.
5. Conclusion In this paper an efficient meshed algorithm has been developed in order to solve the reconfiguration problem in distribution systems minimizing the network losses. A strategy has been designed to pair off active and reactive breakpoint nodes in a meshed network. Due to that strategy, it is not necessary to check the network connectivity at each step of the procedure, reducing therefore the computation time. Summarizing, the number of power flow calculations is approximately three times, especially for real networks where the P and Q-breakpoints tend to coincide. With relation to algorithm validation, three test systems from the technical literature have been used. The obtained results for these test systems have been equal or better than those published in the literature. Finally, using breakpoint nodes in a meshed algorithm has demonstrated its usefulness for finding the solution of the reconfiguration problem.
Table 3 Test systems characteristics. Test system
Buses
Branches
Published best solution in the technical literature
Losses (kW)
Das [15] Zhang et al. [16] Romero-Ramos et al. [17]
69 118 205
79 132 224
s13, s30, s45, s51, s66, s70, s75, s76, s77, s78, s79 s23, s26, s34, s39, s42, s51, s58, s71, s74, s95, s97, s109, s122, s129, s130 s13, s58, s61, s69, s70, s83, s85, s91, s97, s131, s134, s147, s148, s161, s168, s169, s206, s209, s217, s222
201.412 869.726 87.099
Table 4 Proposed algorithm results. Test system
No load flows
Open switches
Losses (kW)
Das [15] Zhang et al. [16] Romero-Ramos et al. [17]
36 53 71
s13, s30, s45, s51, s66, s70, s75, s76, s77, s78, s79 s23, s26, s34, s39, s42, s51, s58, s71, s74, s95, s97, s109, s122, s129, s130 s12, s58, s61, s69, s70, s85, s98, s131, s133, s142, s143, s147, s148, s161, s168, s169, s206, s209, s217, s222
201.412 869.726 85.986
Table 5 Method of Martín and Gil [14] results. Test system
No load flows
Open switches
Losses (kW)
Das [15] Zhang et al. [16] Romero-Ramos et al. [17]
83 177 213
s13, s28, s45, s51, s67, s70, s73, s75, s76, s78, s79 s23, s34, s39, s42, s48, s50, s58, s71, s74, s95, s97, s109, s119, s129, s130 s14, s58, s61, s69, s70, s85, s131, s134, s137, s142, s143, s147, s148, s162, s206, s209, s217, s218, s222, s224
203.861 874.858 87.718
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