Adaptive and Dynamic Ant Colony Search ... - Semantic Scholar

Applied Intelligence 24, 31–42, 2006 c 2006 Springer Science + Business Media, Inc. Manufactured in The Netherlands.

Adaptive and Dynamic Ant Colony Search Algorithm for Optimal Distribution Systems Reinforcement Strategy S. FAVUZZA Dipartimento di Ingegneria Elettrica, Elettronica e delle Telecomunicazioni, Universit`a di Palermo, Italia G. GRADITI Centro ricerche ENEA, Ente per le Nuove Tecnologie, l’Energia e l’Ambiente, Portici, Napoli, Italia E. RIVA SANSEVERINO Dipartimento di Ingegneria Elettrica, Elettronica e delle Telecomunicazioni, Universit`a di Palermo, Italia

Abstract. The metaheuristic technique of Ant Colony Search has been revised here in order to deal with dynamic search optimization problems having a large search space and mixed integer variables. The problem to which it has been applied is an electrical distribution systems management problem. This kind of issues is indeed getting increasingly complicated due to the introduction of new energy trading strategies, new environmental constraints and new technologies. In particular, in this paper, the problem of finding the optimal reinforcement strategy to provide reliable and economic service to customers in a given time frame is investigated. Utilities indeed need efficient software tools to take decisions in this new complex scenario. In past times, utilities project the load growth for several years and then estimate when the capacity limit will be exceeded. Designers then consider some feasible alternatives and select the optimal one in terms of performance and costs. In this paper, the Distributed Generation, DG, technology considered in compound solutions with the installation of feeder and substations is viewed as a new option for solving distribution systems capacity problems, along several years. The objective to be minimized is therefore the overall cost of distribution systems reinforcement strategy in a given timeframe. An application on a medium size network is carried out using the proposed technique that allows the identification of optimal paths in extremely large or non-finite spaces. The proposed algorithm uses an adaptive parameter in order to push exploration or exploitation as the search procedure stops in a local minimum. The algorithm allows the easy investigation of these kinds of complex problems, and allows to make useful comparisons as the intervention strategy and type of DG sources vary. Keywords: 1.

ant colony search, dynamic optimization problems, electrical distribution systems

Introduction

Metaheuristics and heuristic methods have been extensively used in the field of power optimization problems and in general in electrical engineering. Indeed, these methods allow a more accurate formulation of the problem and the attainment of results in reasonable computation times. On the other hand, special attention must be paid to the choice of the appropriate method, depending on the particular features of the problem at hand. In power systems, heuristic methods offer a viable alternative to classical optimization techniques to face complicated problems and to attain general guidelines for design and management of electrical systems. In particular the Ant Colony Optimization paradigm, which is a relatively new algorithm [1, 2] for the solution of combinatorial optimization problems, has been used in the field of power systems, to solve different problems.

In [3, 4], the Authors use the ACO, Ant Colony Optimization, to solve the generation unit commitment problem. They have created a discrete search space, by considering a multi-stage scheduling and a suitable transition cost from one stage to the other. In this case, the problem was to identify the output level of a given set of units in a time frame of 24 h. Test runs have been carried out in both papers to make a comparison between the Ant Colony approach and other methods (a dynamic programming approach, a Lagrangian relaxation method and a Genetic algorithm). In all cases the Ant Colony approach has given better results. In both papers a ‘constructive’ approach is used and the solution is actually created by each ant. Another power systems problem dealt with by means of the same paradigm is that of the switch relocation problem. In [5] the Authors minimize the interruption costs for distribution feeders by rearranging the existent switches. The

32

Favuzza, Graditi and Riva Sanseverino

customer interruption cost indeed depends on the amount of load disconnected during an outage, if there is no possibility to supply it from another source point. In this way, the search space is again arranged in stages and, at each stage, a new switch is newly positioned. Therefore, the search space is constituted by a number of nodes equivalent to the number of switches multiplied by the number of candidate locations of the same switches. Each ant will rearrange one switch at each stage. The approach essentially tries to ‘improve’ the starting solution. Finally in [6] the economic dispatch problem is considered. The Authors try to minimize the operating cost of the power system based on the power output of the generation units and subjected to some technical and economical constraints. The constraints are included into the main objective function as penalty terms and the optimization is carried out trying to improve a randomly selected but feasible solution. A parameter allowing to control the scale of the exploration called ‘visibility’ is introduced. The algorithm also uses elements of neighborhood search and of the Genetic algorithms search and the solution is actually searched by each ant which in this case acts like a search agent. So again the approach tries to ‘improve’ the solution. Other papers, such as [7], use the ACO paradigm to carry out the optimization of functions by interesting ‘constructive’ approaches. None of the above cited papers uses a ‘constructive’ approach in a non-finite or extremely large space at each stage. In the present paper, the authors try to solve a dynamic optimization problem such as the reinforcement of electrical distribution systems, by dividing the search space in stages. Each stage, this time, represents a possible intervention on the system at a given year. The intervention consists of the installation of one or more elements in order to increase the electrical system’s capacity, namely the ability of the system to supply a larger amount of electric load. Thus in this problem it is required to solve an optimization sub-problem at each stage, even though the overall optimization requires the search for the minimum cost strategy in the considered time-frame. Each ant tour represents a strategy that can be followed. At each stage, the ant itself creates a set of candidate nodes related to the subsequent stage. The nodes are possible scenarios of new installations, in which the minimum condition of supplying the entire load is satisfied. The ant can choose the next node among the newly created ones and a set of existing nodes, created by ants that had previously gone through the same path. Of course, the generation of this set is guided by some criteria, in order to guarantee a good exploration of the search space at that considered stage. During its tour, the ant leaves a trace of pheromone on the path it has gone through. This path is kept in memory using a dynamic search tree that keeps track of all the visited routes. The search tree dimension is limited by the branching level of the tree and the number of stages/interventions considered. Each time an ant has to decide the following node, it creates a set of new candidate scenarios and the choice is made

among the set comprising the existing paths and this new set. Another feature that distinguishes the present approach from the standard ACS is the adoption of an adaptive, instead of fixed, parameter, q0 , which rules the elitism of the algorithm. Each time the ant has to choose the following node, this can be done either on a deterministic base or on a probabilistic base and the choice of the criterion is guided by the parameter q0 .

2.

The Problem of Distribution Systems Reinforcement Planning

The identification of a good reinforcement strategy in order to supply reliable and economical service to customers is a huge problem for distribution utilities. Traditionally planners start thinking about new capacity only when the load reaches a certain level; then, only a few options is considered and these are restricted to substations or feeders reinforcement. This methodology works when the economic environment is stable and the number of technically available options is limited. The new deregulated energy market and the new important issues concerning sustainable development make the environment change rapidly and technology provides new viable options for the task of expanding the systems capacity. In this paper, the problem of identifying the best network reinforcement strategy along a given timeframe is dealt using a new formulation. The installation of Dispersed Generation units, and in particular renewables (Photovoltaic units), is considered as a viable alternative to traditional network reinforcement methods, such as cables and substation transformers installation. This is valid under the hypothesis that the loading peaks occur during the maximum insulation hours. In this formulation, the reinforcement consists in introducing parallel Medium Voltage, MV, cables and Medium Voltage/Low Voltage, MV/LV, transformers as well as PV, Photovoltaic, units at LV level. Moreover, since cables are installed underground, it has been hypothesized that they cannot be overloaded. Figure 1 shows a generic MV/LV load node of the considered system. In the figure, OT is the Old Transformer, whereas RT is the Reinforcement Transformer. Therefore, the problem here dealt with results in a combinatorial optimization problem, where the main objective is that of minimizing the overall cost while meeting some technical constraints such as the voltage drop limitation and the elements capacity consideration. The overall cost can be expressed as the summation of different terms. Some are installation costs, others are operational costs such as losses and maintenance costs. In addition, also external costs such as incentives for the production of energy from renewables (‘clean energy’) have been considered. In what follows, all the cost terms are described in detail. In the present formulation of the posed problem, for a given load factor increase, L, each node at the LV level can be equipped with distributed Photovoltaic, PV, systems and the capacity of each branch can be multiplied by installing some

Adaptive and Dynamic Ant Colony Search Algorithm

33

Figure 1. A generic MV/LV node of the distribution system, at time t, on the left, and at time t + t, on the right, with a load increase and with all the possible reinforcement means.

other cables in parallel with the existing ones. As a consequence, in the latter case, the capacity of each substation can be multiplied by installing other transformers in parallel with the existing ones. It must be considered that the required number of interventions depends on the load course. In this paper, the load evolves with the following exponential law: P(t) = P0 (1 + r )t

(2)

where, C P V [ /kW] is the cost per kW and P P V [kW] is the rated real power of the PV system. (b) The cables cost can be expressed as:

(5)

= cman

TR

× n tr

[ ]

where n tr is the number of transformers; cman here 600 .

(6) TR

is

(d) The losses cost can be expressed as: [ ]

(7)

where C P [ /kWh] is the cost of losses per kWh and Er[kWh] are the total energy losses evaluated in one year. C P is here set to 0.05 /kWh. (e) Savings due to PV systems installations. Two terms must be considered: (e1) Savings from not buying energy from the transmission level; per year:

(b1) Installation cost: C2 = CCa (I ) × L

[ ]

(3)

where CCa [ /km] is the cost of the cable for a given capacity I [A] and length L [km]. (b2) Maintenance costs per year:

cman is here 70

TR

C4 = C P × Er [ ]

[ ]

where C Sk [ ] is the cost of the required transformer having rated power Ank [kVA]. (c2) Maintenance cost per year; for all transformers is evaluated as: Cman

(a) The PV systems installation cost:

Cman

C3 = C Sk (An k )

(1)

where P0 is the rated load required by the customers at year 0, P(t) is the expected load at year t and r is the rate of increase of the loads. In what follows, the different cost terms are described.

C1 = C P V × PP V

(c1) Installation cost

cables

= cman × L

[ ]

R1 = Renergy PPV

(8)

where Renergy is equal to 6.5 /kW and has been obtained considering that the cost of 1 kg of fuel is 0.02 . (e2) Incentives for those who produce from renewables:

(4)

/km.

(c) The MV/LV substation transformer cost can be expressed as:

R2 = Rc × PPV

(9)

where Rc can range between 43.8 /kW and 820 /kW. The variation of this parameter allows the

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Favuzza, Graditi and Riva Sanseverino identification of a threshold value above which the massive installation of PV units is economically convenient.

3.

The Objective Function

The objective function is the overall cost. The overall yearly cost can be therefore obtained as the summation of the different terms above described: CTOT = (C1 + C2 + C3 ) × (e) + C4 + Cman + Cman

TR + C man

cables

PV − (R1 + R2 )

(10)

where the different cost or saving terms have been defined in Section 2 above; e is the rate (7%) for the calculation of the yearly cost. The constraints concern voltage drop within limits and current below branches capacity. Another constraint requires that all the loads have to be supplied. A single solution is coded into a string having two times the number of branches elements (br), so that each couple of elements is related to one branch and to its ending bus. The solution string is therefore a vector composed as follows: [S1 , P1 , S2 , P2 , . . . Sbr , Pbr ]

(11)

where Si [mm2 ] is the section of the reinforcement cable at the i-th branch, Pi [kW] is the sizing power of the PV plants below the i-th node. Each of these elements can vary in a discrete fashion among a set of possible available sizes. The optimisation problem here dealt with is then a combinatorial optimization problem; therefore the fitness can be expressed as the inverse of the cost function terms (10), calculated at each intervention in terms of the new installations, actualized at year 0 and summed up. In [9] the Authors have focused their attention on the definition of the minimum cost reinforcement plan with a simplified objective function and with an algorithm, the Evolutionary Parallel Tabu Search, which was suitable for static combinatorial optimization. In this paper, the Authors propose a dynamic design strategy derived from the ACO paradigm. It is aimed at the optimisation of the expansion strategy of a distribution system through a discrete number of interventions within a time frame of 24 years with reference to a given course of load increase in the served area. The solution of this problem requires the identification of the optimal expansion strategy for the system. The problem is combinatorial and non-linear. 4.

The Dynamic Ant Colony Search Algorithm

The ACS algorithm, proposed by Dorigo and Gambardella [2], is an algorithm simulating the behaviour of natural ant colonies. The algorithm uses a set of agents which cooperate for the research of new solutions acting simultaneously.

This algorithm has been applied to different problems in engineering, in particular to those applications where a length measure must be optimized such as in the Traveling Salesman Problem, (TSP).1 This algorithm has rarely been applied for optimization strategy problems. However there are some papers regarding this aspect for different engineering fields [4, 10]. The natural metaphor for ant algorithms is ant colonies. Real ants find the shortest path from a food source to their nest, without using visual cues, by exploiting pheromone information. While the ants go towards the food, each ant deposits on the ground a certain quantity of pheromone, which can be recognized by the other ants, and continues its tour. At the beginning all the ants move randomly. When the pheromone evaporates, the traces that can still be recognized are those that have been left on the shortest paths since they can be followed more rapidly. In this way, the number of ants that choose to go through the shortest paths grows and the pheromone trace gets stronger as more ants follow it [2]. The ACS algorithm has been presented and first implemented for the Traveling Salesman Problem, TSP, since there is an explicit similarity between the ‘tour length’ in the problem and the ants path length. The key to the application of the ACS to a new problem is to identify an appropriate representation for the problem, namely an appropriate spatialization. The latter can be attained by means of a graph representation (when it is possible) of the considered engineering problem. Any solution must also be represented by means of a tour through the edges of the graph. Besides a suitable expression of the distance between any two nodes of the graph must also be determined. Then the probabilistic interaction among the artificial ants mediated by the pheromone trial deposited on the graph edges will generate good, and often optimal, problem solutions. Some problem may arise when the ‘spatialization’ is not straightforward, namely, when other physical measures have to be turned into ‘distances’. In the application here proposed, the different reinforced configurations of the electrical system at different years (with the relevant load factor) represent the nodes of the graph, the distances between them are transition costs, suitably actualized in order to make them comparable at year zero. The transition costs are the installation and operating costs to expand the system from the current configuration to another to be reached in the following time. The problem of identifying the best reinforcement strategy of distribution systems has been here dealt with by looking for an analogue with the TSP. It is essentially a minimum cost tour problem but it is different from the TSP problem, since there is not a unique set of points to be visited for a single solution, but there may be infinite possible paths through different points. One of the key differences with traditional ACS is that the algorithm dynamically creates new candidate solutions and eliminates unpromising search directions.

Adaptive and Dynamic Ant Colony Search Algorithm In this way, at each year a number of candidate solutions is identified. It is realistic to assume that utilities do not operate each year but they carry out interventions only every n years years. A solution strategy is therefore a ‘tour’ comprising all the system’s configurations between year zero and year 24 with T intervals of n years years; the cost of a strategy is the summation of the transition costs in the considered time horizon. These costs have all been actualized in order to make different strategies comparable. The distance between two solutions, namely between configuration r, related to year i (j–n years) and configuration s related to year j, is given by d(r,s) defined as follows: d(r, s) =

j k= j−n

j

CTOT ni (r, s) (1 + a)k years

(12)

j

where CTOT ni (r, s) is the new installations, external, maintenance and losses costs for configuration s at year j, when it evolves from configuration r at year j–n years and a is the actualization rate. The following quantities are used in the algorithm: • τ (r, s) is the pheromone amount between configurations r and s; • Mk is the set of configurations that have been identified for the year configuration r belongs to; • β is the parameter weighting the importance of the transition cost from configuration r to configuration s; • α is the pheromone updating parameter, ruling its decay and its reinforcement; • τ0 is the pheromone initialization value which is given to any possible tuple such as (r, s); • n ants is the number of ants constituting the artificial colony. Configuration s that follows the starting configuration r is identified by means of the following law, if q ≤ q0 : s = arg max (τ (r, s) · δ(r, s)−β ) u ∈M / k

(13)

where q ∈ [0, 1] is a random number and q0 ∈ [0, 1] is a parameter allowing to regulate the elitism of the algorithm, namely to establish a compromise between exploration and exploitation of the search space. Indeed, if q0 is very close to the unity it is highly possible that the random parameter q is lower than q0 and therefore that configuration s is the maximum of the function (τ (r, s) · δ(r, s)−β ). If q > q0 configuration s is chosen following the probabilistic law:  −β  τ (r, s) · δ(r, s) if s ∈ / Mk −β pk (r, s) = / k τ (r, u) · δ(r, u)  u ∈M 0 otherwise (14)

35

The probability that the k-th ant moves towards a configuration of the same year must be zero, whereas a biased law has been used to select the solution towards which the ant shall move. In this paper, parameter q0 varies adaptively with the condition of flattening on the currently found best solution. The law with which it varies is reported below: q0 = qbase + (flat + y)−1

(15)

where qbase is the lowest value that the parameter q0 can reach, flat is the number of iterations without improvement and y sets the maximum value for q0 . The parameter flat is limited and when it reaches a certain maximum value the algorithm can be stopped. When this parameter is high, the parameter q0 gets the lowest possible value, whereas when flat is zero it takes the maximum value. The maximum possible value for q0 also depends on y. If this is set to 1, and qbase is necessarily set to zero, the maximum value for q0 is 1. As it can be noted, as the number of iterations without improvement, flat, grows, the parameter q0 controlling the elitism of the algorithm decays, thus allowing a larger exploration of the search space. The local updating of the pheromone is applied to the traveled paths. This mechanism is used to prevent premature convergence and simulate the natural phenomenon of evaporation; it is executed by means of the function: τ (r, s) = (1 − α)τ (r, s) + aτ0

(16)

where the symbols have been above defined. Global updating is executed when all ants have completed an entire tour for exploration and it is aimed at the reinforcement of the pheromone of those transitions (r, s) belonging to the best tour, namely to the minimum cost strategy. It is performed using the function:  −1  (1 − α)τ (r, s) + αL gb if (r, s) ∈ global best tour τ (r, s) =  (1 − α)τ (r, s) otherwise (17) where Lgb is the ‘length’ or the cost of the best strategy which has been identified at the end of one iteration. L gb =

min

k=1,...n ants

×

Ctot(l) k

n edges l=1

− Ctot(l−1) k

(1 + a)αl−1

(18)

where: • n edges is the total number of edges of the k-th path (ant); • αl−1 is the year in which new installations related to the configuration l have been executed;

36


• n ants is the number of ants constituting the artificial colony. Note that, when l equals the total number of edges of the considered path, the ant has reached the target configuration and the strategy is complete, therefore expression (18) gives the cost of the entire strategy actualized at the starting year. In this way, the pheromone of the transitions belonging to the best tour is increased whereas the pheromone of other transitions is decreased. The local updating encourages the exploration of the search space because it prevents premature convergence, whereas global updating encourages the exploitation of the most promising solutions, namely the overall less costly solutions. In traditional ACS algorithm a number of ants start exploring the search space which is constituted by a finite number of points. Then they gradually converge towards the best path. When the search space of the faced problem is intrinsically of finite type, as in the TSP, the algorithm works quite well. When instead a ‘spatialization’ (discretization or the reduction of the search space dimension) of the problem is required, the above cited approach may be no more valuable. Indeed, it is well known that in these cases, the attempts of reducing the search space dimensions may bring to misleading results. Real ants indeed move randomly before finding the optimal way from their nest to the food. They do not have discrete points in the search space through which they move. This happens especially at the beginning. After some time, the ants go one after the other towards the food, because of the mechanism of the pheromone release. This is the idea behind the proposed algorithm, called Dynamic Ant Colony Search, DACS, algorithm. From the programming point of view, it has been implemented using dynamic data structures, such as a dynamic tree, representing the ideal paths over which the ants have moved. The pheromone release mechanism allows the identification of good traces and in the end an increasing number of ants will tend to go through these paths. In this representation of the problem, the depth of the tree is the timing at which the interventions over the distribution system are carried out, see Fig. 2. So the maximum depth of the tree is fixed by the user. Since the overall time frame in which the strategy is designed is also fixed, the number of years between one intervention and the other is also known. The basic hypothesis is that components are installed in order to face the load increase relevant to the following time interval and at the beginning of the same time interval. Moreover expenses are amortized in 24 years. Each branch of the tree represents the move from one year to n years later; it is weighted with the operational costs and with the cost of possible installations of new components or with the cost of the yearly amortization expense of components already installed. The algorithm starts with the generation of a given number of candidate solutions, N, each of which is created accordingly with a given percentage of penetration of the Pho-

Figure 2. Dynamic tree representing the different strategies, p ranges from 0 to T.

tovoltaic plants. In this way, in the first solution, a value randomly chosen between 0 and the 100/N % of the load increase will be covered with PV plants, in the second solution a value randomly chosen between 100/N % and 200/N%, and so on. Then the ant performs a choice among the candidate solutions using expressions (13) and (14). The local pheromone updating is performed using expression (16), and the solutions that have not been chosen are abandoned. A new set of solutions hypothesizing a different penetration of PV plants is again considered and so on, till the last year. Then the second ant starts again from year zero and finds a pheromone trace of the ant that has just passed. So it generates again N new solutions, different from the previous one, and the choice is made among N + 1 candidate solutions, because of the existing path. In the following years the second ant will find N candidate solutions, if the node from which it comes is newly created, otherwise, it finds N + 1 candidate solutions, N newly generated and 1 belonging to the path chosen by the first ant. At the end of the iteration, all the ants have gone through the 24 years and probably have generated a number of new nodes. The maximum possible number of nodes of the dynamic tree containing the traces is of order NT , where T is the number of intervals of n years. Of course N is variable and can be set by the user. It defines the precision with which the algorithm can work out a solution. But on the other hand, the opportunity to generate a new set of solutions at each step gives some confidence about finding the solution with appropriate diversification. Also T is variable and can be set by the user, but it is a design variable since the timing with which interventions can be carried out on the system is also a design choice. When the ant finds N existing paths it can still generate N new solutions with different PV penetrations and it will again activate the selection using expressions (13) and (14). If an existing path is selected, the pheromone trace is modified according to equation (16), if instead a new trace is selected; the worst, in terms of pheromone and cost, among

Adaptive and Dynamic Ant Colony Search Algorithm the old paths is selected and eliminated, together with the branches and nodes downstream, and replaced with the new chosen trace. The pheromone trace is then modified according to Eq. (16). In Fig. 3 the flowchart of the DACS is reported.

5.

Applications

The studied system is a radial power distribution network with 20 kV of rated voltage. It has 23 branches and 23 load nodes as depicted in Fig. 4. Each load node is represented in Fig. 1. The distribution system loads are supplied by one High Voltage/Medium Voltage, HV/MV, substation and two main feeders. The reinforcement strategy is carried out in a time frame of 24 years and with interventions that can be executed every n years years. The yearly rate of increase of the loads, r, is 0.03. The used load model is with constant power and the algorithm to solve the network is a ‘backward-forward’ algorithm. The runs have been carried out using the parameters reported in Table 1. In Table 1 α is the pheromone updating parameter, ruling its decay and its reinforcement, β is the parameter weighting the importance of the transition cost from one configuration to another, q0 is the parameter which varies according to Equation (15), τ0 is the pheromone initialization value and the parameter flat counts the number of iterations for which the algorithm does not find better solution. The parameter q0 controlling the elitism property has an adaptive behaviour since its course depends on the value of flat. It is important to underline that for each setting of the electrical and strategy parameters (Rc , T, n years) the Authors have carried out a tuning of the above reported parameters, which, nevertheless, remain confined in a small neighbourhood. Parameter y, which rules the course of q0 using Eq. (15), in all applications ranges between 2 and 3 according to the desired course of q0 . As an example, it has been observed that, as the number of required interventions decreases, it is more important to have higher values of q0 , namely higher values of qbase and smaller values of y. In Fig. 5, the course of the best-so-far solution strategy is reported, together with the course of q0 reported for a generic run with T = 8 and n years = 3 (8 × 3 strategy) and a value of Rc of 760 /kW per year. Parameter y has been set to 2 and parameter qbase to 0.5. Table 1. Branching (N ) 10

Number of ants

Termination criterion

50

Flattening on one solution for 20 iterations (flat = 20)

α

β

q0

τ0

0.00001

2

variable

0.2

37

The usefulness of the proposed approach has been tested by running the algorithm with fixed q0 and with variable q0 . In the latter case, the algorithm has proved in all the performed executions, an improved capacity of exploring and of exploiting the search space, by finding lower values of the objective function. The differences in the objective function value, for Rc = 43.8 /kW, have been evaluated and range between 0.74% and 8.7%, the latter value being referred to the hardest possible problem, namely T = 12 and n years = 2 (12 × 2 strategy) strategy. With the fixed q0 , the algorithm showed all times premature convergence in a few iterations, whereas with variable q0 it was able to improve the optimal solution till the end of the run. Moreover, the effect of a lower value of q0 on the convergence property of the algorithm consists in the ability to find better solutions and is justified by the possibility to explore in a wider sense. Another interesting issue concerns the ability of the algorithm to overcome local minima and find out a low cost strategy. This characteristic must be attained by an accurate calibration of the algorithm’s parameters. Indeed, a too strong reinforcement of the pheromone trace at local level, may give out misleading results in overall terms. The local reinforcement of the pheronome trace is guided by the alpha parameter, as well as by the number of ants. Indeed, a larger number of ants in the same iteration, making choices about the paths to be followed and implementing Eq. (16) over the trace, means a stronger pheromone information at local level. A comparison between an optimization algorithm that treats the entire strategy as optimization variable and the proposed DACS algorithm which is a constructive method has been carried out. On purpose a Genetic Algorithm [11], suitably adapted in order to handle the considered problem, has been implemented. In the GA implementation, the optimization variable is composed of T substrings each indicating the state of the installations at the n years × jth year ( j ∈ [0, T ]). The Genetic Algorithm uses a Roulette Wheel Selection and a two points crossover. Both crossover and mutation operators are performed in order to prevent the generation of unfeasible solutions. It is indeed possible that the variation of one bit randomly results in the removal of one or more PV modules, thing which is not allowed in the considered formulation. Considering one solution as the entire strategy string allows to take fully into account the mutual influence between substrings. In this way, a wide exploration of the search space has been carried out even though the GA proves to be unable to identify better solutions than the proposed search technique. In Fig. 6 the results obtained with the GA and with the proposed DACS technique (Rc = 43.8 /kW) are reported. The figure clearly indicates that as much as the problem gets similar to a static optimization problem, the GA performs better, this is also motivated by the fact that in this case the search space is smaller. On the other hand, the savings that can be attained using as optimization technique the DACS algorithm instead of a GA range between

38


Figure 3.

Flowchart of DACS.

the 5% and 30%, stopping both algorithms after 20 iterations without improvement. These differences range between 5% and 10% if the GA is let run for a much greater number of iterations, but with an extremely high computational effort. The following figure reports results about this issue.

As far as the computing cost is concerned, five runs per algorithm have been carried out in order to detect the statistical behaviour of the two algorithms. Figure 7 reports a comparison of the two algorithms in terms of the number of objective function evaluations necessary to get the best solution they could attain. Again Rc has been set to 43.8


Figure 4.

Test system.

Figure 5.

Course of q0 and of the overall cost of the strategy, the effect of the adaptive q0 is that to allow a larger exploration of the search space.

/kW. It must be noted that using a PC with a pentium 4 processor, 1.5 GHz, the GA took, in the best case, almost one hour to get a solution 5% worst than that attained by the DACS in 29 seconds. The proposed DACS algorithm is quite efficient also in this dimension, since it reduces the calculation time up to 98%. In order to test robustness 50 runs of the DACS algorithm have been carried out, with the same settings, and a small dispersion of the results has been observed. Figure 8 indeed shows the frequency distribution of results for the lowest cost strategy outputted by the algorithm in the 50 runs (3×8 strategy, Rc = 43.8 /kW). In Fig. 9, a portion of the search tree created by the ants during the DACS algorithm is reported. In the figure, the lines widths indicate the pheromone intensity, therefore paths with low pheronome (about 0.2) are represented by light lines, whereas higher values of pheromone are represented by wider lines. As it can be noted, the paths with higher value of the pheronomone have a higher branching as compared to those having a low pheromone value. This means that the algorithm actually searches ‘in the neighbourhood’ of the best solutions in a more intensive way as the iterations proceed.

39

From the electrical engineering point of view, other results can be observed. Several runs have been carried out with different values of Rc . In particular, ranging from 43.8 /kW up to 820 /kW. More robust installations of PV modules can be observed only after a value of Rc of 700 /kW, even though solutions with PV modules can be observed starting from the minimum proposed value of 43.8 /kW, which is a realistic value. Other runs have been carried out varying the number of years between one intervention and the other (parameter n years), keeping the same timeframe (24 years). In particular, the Authors have considered the parameter T ranging from 12 to 2 and accordingly the parameter n years ranging from 2 to 12 years. In this way, in what follows, for example ‘strategy 6 × 4’ indicates a strategy with n years equal to 4 and T equal to 6. In this case, the tests have been carried out varying the frequency of interventions (type of strategy) and varying the parameter Rc . The costs of the strategy have been considered reporting them at the year zero. In Fig. 10, the course of the strategy costs as the environmental benefit, Rc, varies and the different modes of intervention in the considered time frame. As it can be observed,

40


Figure 6.

Comparison of performance GA vs. DACS, minimization of the objective function.

Figure 7.

Comparison of performance GA vs. DACS, number of objective function evaluations.

the 2 × 12 and 3 × 8 strategies are the most interesting in economic terms. This means that for the considered load growth, a small number of interventions on the electrical system is advisable.

6.

Future Developments and Conclusions

In this paper, a new heuristic cooperative algorithm based on the ACO paradigm, suitable for dynamic optimization


Figure 8.

Frequency distribution of results of the DACS algorithm for a 3 × 8 strategy and Rc = 43.8

Figure 9. Portion of the search tree outputted after one run with a 3 × 8 strategy and Rc = 43.8 /kW.

problems, is proposed. The problem dealt with is an electrical engineering problem, namely that of the reinforcement of distribution systems using renewable sources (PhotoVoltaic systems) together with traditional means, such as cables and transformers.

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/kW.

The minimum cost strategy has been therefore attained for different values of some parameter taking into account environmental issues. The problem, of course, results in a non-linear, dynamic optimization problem with mixedinteger variables. The Authors have set up an optimization algorithm which derives from the Ant Colony Search algorithm suitably adapted to treat large search spaces and mixed integer variables. Traditional ACS algorithm was indeed devoted to the solution of minimum path length problems and therefore able to treat problems with a discrete search space made of a finite number of points. In this paper, the discretization of the search space is made step-by-step by the ants themselves. They locally choose the best way to follow among a given number of candidate sites that are updated at each iteration. The authors have also introduced an adaptive parameter q0 which suitably controls the elitism during the search. The algorithm has proved to be robust and to work quite well with the proposed problem of finding the optimal reinforcement strategy for a distribution system. The proposed algorithm is also a valuable tool to carry out further studies with different renewable sources such as biomass, wind power and fuel cells, or also a combination of these. It is also easily possible to change the times of interventions or even make this parameter an optimization variable. The algorithm can easily be modified to push the research towards exploration or towards exploitation by tuning the parameters and in a possible future development may be the possibility to change the criterion with which it chooses the candidate solutions. Moreover, the algorithm can be modified in order to better keep track of the best path

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Figure 10.

Comparison of different strategies.

found and to let the ants explore the neighbourhood of the best path. In this way, it would be possible to improve the precision of the best solution found. Note

5.

6.

1. The TSP is the problem of finding, given a finite number of ”cities” along with the cost of travel between each pair of them, the cheapest way of visiting all the cities and returning to the starting point. 7.

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