An iterated sample construction with path relinking method: Application to switch allocation in electrical distribution networks

Computers & Operations Research 40 (2013) 24–32

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An iterated sample construction with path relinking method: Application to switch allocation in electrical distribution networks Alexander J. Benavides a, Marcus Ritt a,n, Luciana S. Buriol a, Paulo M. Franc- a b a b

´tica, Universidade Federal do Rio Grande do Sul – UFRGS, Brazil Instituto de Informa Faculdade de Ciˆencias e Tecnologia, Universidade Estadual Paulista – UNESP, Brazil

a r t i c l e i n f o

a b s t r a c t

Available online 24 May 2012

We present a metaheuristic approach which combines constructive heuristics and local searches based on sampling with path relinking. Its effectiveness is demonstrated by an application to the problem of allocating switches in electrical distribution networks to improve their reliability. Our approach also treats the service restoration problem, which has to be solved as a subproblem, to evaluate the reliability benefit of a given switch allocation proposal. Comparisons with other metaheuristics and with a branch-and-bound procedure evaluate its performance. & 2012 Published by Elsevier Ltd.

Keywords: Metaheuristics Sample algorithms Path relinking Power systems reliability Switch allocation

1. Introduction Most of the faults of an electrical power system take place in the distribution network [1,2]. The installation of redundant lines and switches is a common method to improve the reliability of a distribution network. In this way, in case of failures, service providers can modify the network topology and reduce the areas without energy. The installation of switches in every line is impracticable due to high costs. For this reason, companies must carefully choose the lines to install switches. This combinatorial optimization problem is called the switch allocation problem. To assess the reliability of an allocation proposal for a set of switches, we have to determine the expected non-supplied energy for a given distribution of failures. Since the actual non-supplied energy depends on how much of the service area can be restored by reconfiguring the network, the service restoration with electrical constraints occurs as a subproblem of the switch allocation problem. In this paper, we propose a sample construction and a sample local search for the switch allocation problem, and, based on these algorithms, a new iterated sample construction with path relinking (ISCPR) to solve the switch allocation problem. Another contribution is that our treatment of the service restoration problem makes no assumption on the topology of the network and includes electrical restrictions.

In Section 2 we give a formal description of the problem. In Section 3 we describe the construction and local search algorithms based on sampling. In Section 4 we introduce the ISCPR algorithm which uses the sample construction to modify the current solution and the path relinking to perform a guided local search. Section 5 first compares the performance of sample based construction and local search methods to other such methods, and then compares several heuristics for the switch allocation problem. When compared to a heuristic based on Greedy Randomized Adaptive Search Procedure (GRASP) [3], ISCPR shows the best performance. Another comparison with a branch-and-bound procedure attests the quality of the solutions yielded by our approach.

2. Description of the problems Fig. 1a shows an example of an electric power distribution network in normal operation. The basic circuit of an operational distribution network has no cycles. It is composed of substations (square nodes), consumers (round nodes), and feeder lines (solid lines). Additionally, redundant lines (dotted lines) can restore the energy in areas affected by failures. Switches control the power flow. In normal operation, switches of redundant lines are open, and switches in the basic circuit are closed. 2.1. Graph model of distribution networks

n

Corresponding author. Tel.: þ55 5133086818. E-mail addresses: [email protected] (A.J. Benavides), [email protected] (M. Ritt), [email protected] (L.S. Buriol), [email protected] (P.M. Franc- a). 0305-0548/$ - see front matter & 2012 Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.cor.2012.05.006

We use an undirected graph G ¼(N,E) to model a distribution network. The set of nodes N represents substations and consumer load points, and the set of edges E represents the feeder lines. A Boolean value Be indicates the presence of a switch on a line

A.J. Benavides et al. / Computers & Operations Research 40 (2013) 24–32

1

3

2

1

25

3

2 a

4

8

13

8 10

4

14

b S(9)

9 5

6

11

12

7

16

5

15

6

d

S(f) 10

13 c

14

9

11

12

7

16

15

Fig. 1. Three-feeder example of a distribution network by Civanlar et al. [4]. (a) Normal operating conditions and (b) sectors affected by a failure.

e A E. We represent a solution for the switch allocation problem by a set EB ¼ feA E9Be ¼ 1g of lines that are selected to install new switches. The sector SðnÞ of a node n A N is the largest connected subgraph which contains n and is connected only with basic circuit feeder lines (normally closed) that have no switch installed (Be ¼0). Ellipses in Fig. 1b represent sectors. The sector of a line e A E is defined as the union of e with the sectors corresponding to the two incident nodes. The frontier F ðG0 Þ of a subgraph (or sector) G0 is the set of edges e A E which are incident to exactly one node in G0 . 2.2. Network reliability estimation In this paper, we use the expected energy non-supplied (EENS) [5] to measure the network reliability. It is defined as X X lf rf P n ðMW h=yearÞ, EENS ¼ f A Enc

n A Nf

where Enc is the set of lines that can fail (normally closed), Nf is the set of nodes affected by a failure f, lf is the average failure rate, rf is the average outage time, and Pn is the average energy consumption of node n. Our reliability estimation algorithm simulates a failure f in each sector S of the network, and opens the switches of the corresponding frontier F ðSðf ÞÞ to isolate the failure. Then, it determines the set Nf of consumers affected by the failure with a service restoration algorithm, and calculates their partial EENS. Evaluating sector by sector, the partial reliability of every feeder line of a sector is evaluated at once, reducing the number of service restoration problems that have to be solved. The same process is applied to the remaining lines that are not part of any sector. 2.3. The service restoration problem Network reconfiguration is the process of opening and closing some switches to change the network topology. When a power failure is detected, the network is reconfigured to isolate the failure and to restore energy by alternate lines. Consider a failure in line f of Fig. 1b. Without switches, the whole tree under substation 2 would be unattended. When the frontier switches F ðSðf ÞÞ ¼ fa,b,cg are opened, the failure is isolated in sector Sðf Þ. Consequently, sector Sð9Þ has neither failure nor power supply. The service on sector Sð9Þ is restored when the switch on line d is closed. The service restoration problem consists in choosing which switches must be opened or closed to minimize the unattended

area after the isolation of a failure. A solution has to satisfy electrical constraints. The electrical constraints considered in this paper are the radiality of the reconfigured network, the capacities of the lines and substations, and the maximum allowed voltage drop. Since we are interested in solving the service restoration problem as a subproblem of the switch allocation problem, we do not consider secondary objective functions, such as minimizing the number of switching operations. For the same reason, we do not consider additional constraints such as the priority of customers, switching times, or restoration cost. For a single substation, the service restoration problem can be modeled as finding a spanning tree minimizing the non-supplied energy subject to electrical constraints [6]. Different from the minimum weight spanning tree, this version is NP-hard [7,8]. Our service restoration algorithm calculates the affected consumers after a failure isolation. It expands the attended area starting from the substations sector by sector, when this is possible without violating the electrical constraints. The electrical simulation is computationally expensive, but important to obtain a realistic approximation of the attended area. A detailed description of the reliability estimation and the service restoration algorithm can be found in [9]. For a survey of approaches to service restoration we refer the reader to C´urcˇic´ et al. [10] and Sudhakar and Srinivas [11].

2.4. The switch allocation problem The number and position of the switches in the network influences the amount of non-supplied energy in case of contingencies. The switch allocation problem consists in selecting a set of feeder lines to install a given number of new switches in a distribution network. The objective is to maximize the reliability, and is subject to electrical and topological constraints. The switch allocation problem is NP-hard, since its solution requires to solve the NP-hard service restoration problem. For this reason the literature concentrates on heuristic solution methods. Exact methods for the switch allocation problem based on integer programming have been proposed by Zambon et al. [12], Bupasiri et al. [13], Soudi and Tomsovic [14,15], Sohn et al. [16]. These methods apply only to small-sized distribution networks. Among the heuristics proposed are simulated annealing [17], immune algorithms [18,19], genetic algorithms [20–22], divide-and-conquer [23], reactive tabu search [24], evolutionary algorithms [25], particle swarm optimization [26], and ant colony optimization [5]. Some of these methods address the more general problem of network reinforcement planning, allowing to

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A.J. Benavides et al. / Computers & Operations Research 40 (2013) 24–32

install besides the sectionalizing switches other devices, e.g. protective devices such as reclosers or fuses, or capacitor banks. Comparing the efficiency of these methods directly is not possible, since the authors use different test instances, which are often not completely specified or from private real networks. Further, reliability is measured differently, for example by SAIDI [20], SAIFI [21], EENS [23,5], or ECOST [17,26]. Some authors use combined costs for non-supplied energy and installation and maintenance [17–19,22–24,26,5], or multiple objectives, e.g., the number of installed switches versus the non-supplied energy [25], or the installation costs versus the EENS [5]. Many of these methods considerably simplify the underlying service restoration problem by assuming that any redundant line can restore energy supply and that the affected area or the overall cost are known or easy to compute.

3. Construction and local search This section describes four construction algorithms (random, sample, greedy and semi-greedy) and three local search strategies (sample search, first and best improvement) for the switch allocation problem.

3.1. Construction algorithms Random construction selects k switches randomly and evaluates the resulting solution. Greedy, semi-greedy and sample constructions build a feasible solution by adding one element at each iteration. Greedy construction evaluates all the elements to select the best each time. Semi-greedy construction (Fig. 2a) evaluates every possible element to build a restricted candidate list which contains a fraction of a elements with the highest reliability. Finally, one element is selected randomly from the restricted candidate list. (Setting a ¼ 0 selects the best element, while a ¼ 1 selects randomly between all the elements.) Sample construction (Fig. 2b) first selects randomly a sample of b elements. Then, it evaluates the sample candidate list to choose the best. (A value of b ¼ 0% corresponds to a random construction, while b ¼ 100% corresponds to a greedy construction). The most expensive operation is the reliability estimation. When installing k switches, the number of reliability estimations is Oðk9E9Þ for the greedy and semi-greedy constructions, Oðk9E9bÞ for sample construction, and O(1) for random construction.

3.2. Local search algorithms A local search algorithm iteratively replaces the current solution with a better neighbour. The entire neighbourhood for our local search strategies is defined by the relocation of one switch at a time from its current position e A EB to a feeder line e0 A E\EB . First improvement local search explores the neighbourhood until a better neighbour is found. This neighbour becomes the current solution and the search continues with the next iteration. The number of evaluated neighbours varies with the iterations. Best improvement local search explores all the neighbourhood to select the best neighbour for the next iteration. Sample local search (Fig. 3) reduces the size of the explored neighbourhood and the number of reliability estimations. It samples b percent of the lines with switches ða A EB Þ and b percent of lines without switches ðbA E\EB Þ to relocate one switch. If the algorithm finds a better solution in the sample, it becomes the current solution for the next iteration. Sample and best improvement evaluate a constant number of neighbours in each iteration, but sample search reduces the neighbourhood size. First and best improvement stop when there are no better solutions in the neighbourhood. Sample neighbourhood exploration is not exhaustive and does not guarantee to find a local minimum. Thus, the stop criterion may be a maximum number of iterations or a number of iterations without improvement. To guarantee that a local minimum is reached, a best improvement local search can be performed after the sample search. When installing k switches, the number of reliability estima2 tions of each iteration is Oðk9E9b Þ for the sample local search and Oðk9E9Þ for the other two strategies.

4. Iterated sample construction with path relinking (ISCPR) ISCPR is a variant of iterated local search (ILS) [27]. At each iteration, ISCPR perturbs the last solution with a partial random destruction followed by a sample construction, and uses path relinking as a guided local search. Path relinking [28] explores the neighbourhood between two solutions. It repeatedly relocates one switch of the initial solution, to a position defined by the guiding solution. It can be seen as a local search where the neighbourhood is limited to the differences between initial and guiding solutions. By sampling and searching only in a limited neighbourhood, ISCPR reduces the number of reliability estimations during construction and local search phases.

Fig. 2. Construction algorithms. (a) Semi-greedy and (b) sample.

A.J. Benavides et al. / Computers & Operations Research 40 (2013) 24–32

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Table 1 Instances. Inst.

SS

Cns.

Lns

Pw.D.

Op.V.

BR BU R3 B4 R4 AR AU R5 R6 R7

1 1 1 3 11 5 5 8 3 7

32 32 32 38 83 80 80 135 201 873

39 39 37 72 96 109 109 156 216 900

3.70 3.70 4.55 24.58 35.20 28.60 28.60 19.96 32.44 148.99

12.66 12.66 12.66 11.00 12.66 20.00 20.00 12.66 33.60 126.60

SS, Cns, Lns: Number of substations, consumers, and lines, respectively. Op.V.: Operation voltage (kV). Pw.D.: Total power demand (MW).

Fig. 3. Sample local search algorithm.

Consumer demand for instances BR and AR are random values, and for BU and AU are constant values. Instance B4 is the RBTS Bus 4 from Billinton and Jonnavithula [17]. Instances R3, R4, R5, R6, and R7 are from the REpository of Distribution Networks (REDS) [29]. To complete the necessary information, we assume an outage time of r ¼ 2 h, a resistance R ¼ 0:257 O=km, a reactance X ¼ 0:087 O=km, a failure rate l ¼ 0:065 f=km=year, and a maximum current IMAX ¼ 500 A for every line. The problem instances can be found at [30]. 5.2. Choice of parameters

Fig. 4. ISCPR algorithm.

The ISCPR algorithm (Fig. 4) starts generating a solution with the sample construction algorithm. After initialization, it iterates over three steps. First, some elements are removed from the previous solution. Then, a solution is reconstructed using the sample construction algorithm. Once a new solution is complete, path relinking guides a local search from the current solution to the best solution. Finally, the best solution found is returned.

5. Computational experiments All tests have been executed on a PC with an Intel Core i7 processor (2.8 GHz, 12 GB RAM). The algorithms have been implemented in Cþþ, and compiled with GNU Cþþ with optimization level 2 (-O2).

The proposed algorithms depend on five parameters, as shown in Table 2. To find a good setting we conducted experiments with the construction and local search procedures on instances B4 and R6. In some preliminary tests we found that the sample local search rarely improved after 10 iterations without improvement, so we fixed the stop criterion at I¼10. We tested values of a A f0:1,0:2, . . . ,0:9g and b A f10%,20%, . . . ,90%g and compared the averages of 100 replications with different random seeds of the results after construction and after local search. We observe that the goal of tuning the construction procedures is not only to provide solutions of good quality, but also to maintain a good variation of the constructed solutions that can be exploited in the subsequent optimization by a local search or another heuristic. We found that for both instances the solution quality of the semi-greedy construction increased with a and the solution quality of the sample construction increased with b. After the local search, the variation of the solution quality did not depend significantly on a for semi-greedily constructed solutions, so we chose to fix a ¼ 0:5 at the midpoint of its domain. The solutions constructed by sampling produced the largest variation for b ¼ 10%. Since the main function of the sample construction is to provide diverse solutions we fixed b at this value. Another advantage of a smaller sample size is that the construction is faster. Finally, for the parameter d in ISCPR we found that d ¼ bs=2c, where s is the number of switches to install to be a good compromise between perturbing the solution sufficiently to escape a local minimum, but not too much to produce a random solution.

5.1. Problem instances

5.3. Construction and local search methods

Table 1 presents details of the problem instances that were used in our tests. Instances AU and AR are from Augugliaro et al. [7], and BU and BR are from Baran and Wu [6]. The total power demand was divided between the consumers load points.

To compare construction and local search methods, we tested all combinations C–L of construction algorithms and local search strategies. The construction algorithm C can be greedy (Gr), semigreedy (SGr), sample (Spl), or random (Rnd). The local search L

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Table 2 Parameters and values used in the experiments. Par. Description

Value

a

0.5

I d

10% 10 bs=2c

19

18

60

Local search BI FI Spl

17 EENS (MWh/year)

70

Construction Rnd SGr Spl Gr

50

16

40

15 30 14 20 13 10 12

Relative deviation from best known solution (11.262 MWh/year)

b

Greediness in semi-greedy construction Sample size in sample construction Maximum iterations without improvement in SLS and SBI Number of elements destroyed and reconstructed in ISCPR of a total of s

0 0

2

4

6

8

10

Time (seconds)

Construction Rnd SGr Spl Gr

5.5

EENS (MWh/year)

5

Local search BI FI Spl

4.5

4

200

160

120

3.5 80 3

40

2.5

Relative deviation from best known solution (1.793 MWh/year)

240

6

2 0

20

40

60 Time (seconds)

80

0 100

Fig. 5. Average performance for construction and local search combinations. (a) Instance B4 and (b) instance R6.

A.J. Benavides et al. / Computers & Operations Research 40 (2013) 24–32

29

local minimum increases the runtime, but it remains about half of the runtime of BI. Rnd-FI is the most expensive combination. The cheapest combination is Spl–Spl. The fastest combination that finds the best known solution in our experiments is Spl–SBI.

can be first improvement (FI), best improvement (BI), sample (SLS), or sample followed by best improvement (SBI). For example, Spl–SBI is the combination of a sample construction and a sample local search followed by a best improvement local search. We repeated each experiment 1000 times for B4, and 100 times for R6 (except Gr–BI which is deterministic) allocating 20 switches. Fig. 5 and Table 3 report average results. Table 3 shows the average relative deviation from the best known solution after the construction (c) and after the local search (l), the average runtime after construction (tc) and local search (tl), and the percentage of tests that reach the best known solution (lBK), or the relative deviation of the best found solution in parentheses when the best known solution is not found. The best solution found in our tests for 20 switches has an EENS of 11.262 MW h/year for B4 and 1.793 MW h/year for R6. Fig. 5 shows the average EENS and the relative deviation (in percent) from the best known solution over time. Four points show the average result of the construction algorithms and curves depict the average performance of the local search methods. In the following we refer to values for instance B4 and give the values for R6 in parentheses. Initial solutions created by the semigreedy algorithm are better than random solutions by 2.2 (1.2) MW h/year, and require 0.5 (4.7) s while random solutions require less than 4 ms. Greedy construction generates always the best initial solution at almost the same computational effort than semi-greedy construction, but a following local search does not find the best known solution. Solutions created by the sample algorithm are better than random solutions by 5.0 (3.4) MW h/year. Sample construction creates better solutions than the semi-greedy algorithm in about 10% of the corresponding time, having the best cost/benefit among all construction algorithms. The average final solutions after local search shown in Table 3 are less than 3.5 (4.6)% from the best known solution, except after sample local search with 6.9 (12.8)%. All search strategies find the best known solution at least once, except sample search, which yields solutions 0.4 (0.9)% above the best known value. The search strategies differ in their performance over time. The curves in Fig. 5 show a quick initial progress for FI, but BI becomes better later. FI finds better neighbours in early iterations, but spends most of its runtime in later iterations with small improvements. The runtime of sample search is very small, due to the restricted neighbourhood. Performing BI after sample search (SBI) to reach a

5.4. Comparison of metaheuristics In this section, we compare ISCPR with a Greedy Randomized Adaptive Search Procedure (GRASP) [3]. GRASP is a metaheuristic which repeatedly constructs a solution and applies a local search. Both the semi-greedy and the sample construction algorithms present the greedy, randomized and adaptive characteristics of GRASP as described below. The semi-greedy construction restricts greedily the candidate list and selects randomly one of them, while the sample construction restricts randomly the candidate list and selects greedily one of them. Both constructions are adaptive when selecting a new element because they take into account the benefits of the previously selected elements. Four variants of GRASP were implemented combining the construction and local search algorithms tested before (SGr–BI, SGr–SBI, Spl–BI, Spl–SBI). We also tested variants of GRASP with path relinking to the current best solution after the local search (GRASP-PR), since often this can improve GRASP [31]. An iterated semi-greedy construction with path relinking, ICPR (SGr-PR) is also compared to ISCPR (Spl-PR). Fig. 6 and Table 4 present results for the installation of 20 switches. Experiments were repeated 100 times, except for instance R7 with 10 repetitions). The stop criterion is a fixed time tmax given in Table 4, and 10 min for results in Fig. 6. Fig. 6 presents a time-to-target plot [32,33] which shows the percentage of tests that reach a given target value within a given execution time. It uses instance B4. The target is the best known solution for 20 switches. Comparing the average runtime of each heuristic to standard GRASP (SGr-BI), the sample search GRASP (SGr-SBI) is faster by a factor of 0.75, the sample construction GRASP (Spl–BI) is slower by a factor of 2.4, GRASP with both sample algorithms (Spl–SBI) is slower by a factor of 1.8, and GRASP-PR (SGR-BI-PR) is slower by a factor of 1.4. The other GRASP-PR variants that we tested took longer than their GRASP counterparts to obtain the same percentage of target

Table 3 Comparison of construction and local search algorithms. Algorithm

Instance B4

Instance R6

Constr.

L.S.

c

tc

l

lBK

tl

c

tc

l

lBK

tl

Gr

BI

3.95

0.826

3.95

(3.95)

2.51

7.37

6.450

0.06

(0.06)

113.54

SGr

FI BI SLS SBI

49.53

0.444

2.18 2.24 6.18 2.64

17.5 21.3 0.5 15.1

29.56 18.19 1.12 9.01

164.41

4.699

2.99 3.51 11.34 2.78

10.0 5.0 (1.06) 1.0

808.58 309.60 15.33 152.72

Rnd

FI BI SLS SBI

69.20

0.001

2.40 2.36 6.22 2.76

8.6 14.5 0.2 13.5

44.42 22.11 0.74 8.53

229.47

0.003

4.56 3.79 12.76 3.66

1.0 4.0 (2.14) 4.0

1182.59 328.93 11.04 137.41

Spl

FI BI SLS SBI

25.04

0.056

3.37 3.33 6.87 3.45

5.3 7.8 (0.41) 5.8

18.98 13.12 0.59 8.38

43.01

0.584

2.76 2.57 11.08 2.57

15.0 7.0 (0.91) 2.0

351.71 223.36 8.65 133.42

c, l: average percentage over best known solution of construction and local search, respectively. lBK: percentage of tests that reach the best known solution (or relative deviation in parentheses if not reached). tc, tl: average runtime in seconds of construction and local search, respectively.

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A.J. Benavides et al. / Computers & Operations Research 40 (2013) 24–32

1 0.9 0.8 0.7

Probability

0.6 0.5 0.4 0.3 ISCPR (Spl-PR) ICPR (SGr-PR) GRASP (SGr-SBI) GRASP (SGr-BI) GRASP-PR (SGr-BI-PR) GRASP (Spl-SBI) GRASP (Spl-BI)

0.2 0.1 0 0

100

200 300 400 Time (in seconds) to target (11.262 MWh/year)

500

600

Fig. 6. Metaheuristics comparison. Table 4 Comparison of ISCPR with GRASP. Inst

BR BU R3 B4 R4 AR AU R5 R6 R7

BK

758.7 782.3 3277.8 11 262.1 3055.9 21 186.6 21 487.8 7552.6 1793.0 351 958.7

tmax

40 s 40 s 40 s 5m 5m 30 m 30 m 30 m 30 m 6h

ISCPR

GRASP

Itmax

S

SBK

t

It

Itmax

S

SBK

t

It

303 320 162 452 314 1875 1827 977 772 683

0.044 0.000 0.000 0.000 0.152 0.233 1.006 0.129 0.054 1.344

87 100 100 100 28 84 43 57 64 20

7 2 5 25 81 303 337 484 292 7782

56 15 20 39 86 315 340 269 126 244

30 32 15 15 7 34 36 10 6 2

0.003 0.000 0.000 0.000 0.235 2.494 3.635 0.620 0.197 6.606

99 100 100 100 1 5 3 8 28 (3.4)

11 7 2 77 109 936 879 903 890 13 701

9 5 1 4 2 18 17 5 3 1

BK: Best known solution (in kW/year). tmax, Itmax: Time limit, average iterations executed within time limit. S: average percentage over best known solution. SBK: percentage of tests that reach the best known solution (or relative deviation if not reached). t, It: average runtime (in seconds) and iterations to reach best solution.

solutions; they are not shown in Fig. 6 for this reason. This indicates that the effectiveness of path relinking is reduced when applied to local minima after a local search for this problem. ICPR (SGr-PR) performs slightly faster than the best GRASP combination. This results from the faster semi-greedy reconstruction that rebuilds only half of the switches, and the reduced neighbourhood exploration of path relinking. ISCPR always reaches the best known solution in less than 2 min, about 25 s in average. ISCPR is more than two times faster than all other tested combinations. This is a consequence of the fast execution of the sample reconstruction and of the guided search of path relinking over the reduced neighbourhood. Sample local search reduces the search time of GRASP. Sample construction also reduces the runtime, but it generates biased initial solutions and GRASP needs more iterations to compensate this. On the other hand, sample construction combined with path relinking (ISCPR) reduces the runtime without loosing quality, and performs faster than semi-greedy construction combined with path relinking and than GRASP variants.

Table 4 compares ISCPR and standard GRASP (SGr-BI) for other instances. It shows the best known solution value (BK), the time limit (tmax), the number of iterations (Itmax) reached within the time limit, the average relative deviation from the best known solution (S), the percentage of tests that reach the best known solution (SBK), or the relative deviation of the best found solution in parentheses when the best known solution is not found, the average runtime (t), and the average iterations (It) needed to reach a best solution. ISCPR presents better average results than GRASP and reaches the best known solutions more often. Within the time limit, ISCPR iterates at least 10 times more than GRASP. Consequently, ISCPR finds its best solution faster than GRASP. The better performance of ISCPR is more evident in large instances such as R7. It finds the best solution in 130 min and 244 iterations in average for R7, while GRASP completes only one iteration in an average of 172 min. With the exception of instances BR and R3, ISCPR finds better solutions faster and with higher probability. In particular, for large instances with more than 80 lines, it finds the best

A.J. Benavides et al. / Computers & Operations Research 40 (2013) 24–32

known value in about half the execution time with a 13 times higher average probability.

5.5. Quality of the solutions In this section, we compare the solutions of GRASP and ISCPR with optimal solutions to validate their quality. The optimal solutions were obtained with a branch-and-bound procedure. Our implementation of branch-and-bound starts from a network with no switches installed. The lines are processed in non-increasing order of the EENS reduction that produces the installation of a unique switch in that line of the network. For each partial solution, it generates two new solutions by allowing or forbidding the allocation of a switch in the processed line. To lower-bound the EENS value of a partial solution, the network is evaluated supposing that every unprocessed line holds a switch. Observe that the solutions of the branch-and-bound algorithm are only optimal relative to the heuristic service restoration algorithm, since the branch-and-bound searches for the best switch allocation. Due to the complexity of the problem, the branch-and-bound algorithm is able to solve only small instances. Our experiments with branch-and-bound install five or 10 switches. Table 5 shows the optimum value and the runtime for instances that were solved by branch-and-bound within a time limit of 16 h. We tested GRASP and ISCPR 100 times for each of those instances with a time limit of 30 s. GRASP and ISCPR always reach the optimal solutions within the time limit. Table 5 shows the average time, the maximum time and the average number of iterations needed to reach the optimal solution. Runtimes of GRASP and ISCPR are considerably smaller than branch-andbound runtime. Since we can only compare with branch-andbound in small instances, the difference between ISCPR and GRASP is less than in Table 4, but ISCPR still finds the optimal solution faster in seven of the 12 instances, confirming the overall observations of Section 5.4.

6. Concluding remarks Our results show that sample construction is efficient and can create high quality solutions for the switch allocation problem. A restriction of the neighbourhood by random samples also can speed up the local search, leading to good solutions in short times. Table 5 Comparison of branch and bound with ISCPR and GRASP. Sw.

Inst.

B&B

ISCPR

Opt

t

t

GRASP tmax

It

t

tmax

It

5

BR BU R3 B4 R4 AR AU R5

1957.3 2220.0 6469.3 17 057.9 4353.9 37 268.5 38 288.3 11 388.6

1.5 1.8 7.7 39.7 4553.0 7484.0 7968.0 55 625.0

0.02 0.02 0.17 0.26 0.28 1.68 1.35 1.72

0.05 0.08 0.89 1.32 0.70 6.90 6.22 7.30

8.2 8.8 70.3 17.7 7.0 60.5 49.7 27.6

0.04 0.06 0.15 1.17 1.43 1.46 5.64 3.18

0.11 0.22 0.79 6.36 2.78 3.44 17.05 4.61

1.2 1.7 3.1 2.2 1.1 1.1 4.7 1.0

10

BR BU R3 B4

1174.1 1285.1 4026.4 14 235.0

342.3 273.3 12 089.0 50 362.0

3.83 3.65 2.02 3.52

12.48 13.27 5.68 15.11

131.4 129.5 32.5 22.3

1.00 0.91 0.70 4.93

3.88 4.09 1.96 21.58

3.9 3.9 1.3 1.7

Opt: Optimal solution value (in kW/year). t, tmax, It: Average runtime, largest runtime (in seconds) and average iterations to reach the optimal solution value.

31

The use of path relinking as guided local search can improve the solutions efficiently. These components together form an efficient heuristic, ISCPR, that is capable of reducing the number of reliability estimations, and is able to perform better than GRASP for this problem.

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