A Backtracking Search Algorithm for Distribution Network Reconfiguration Problem Thuan Thanh Nguyen, Hiep Ngoc Pham, Anh Viet Truong, Tuan Anh Phung and Thang Trung Nguyen
Abstract This paper proposes a distribution network reconfiguration (DNR) methodology based on a backtracking search algorithm (BSA) for minimizing active power loss and minimizing voltage deviation. The BSA is a new evolutionary algorithm for solving of numerical optimization problems. It uses a single control parameter and two crossovers and mutation strategies for powerful exploration of the problem’s search space. The effectiveness of the proposed BSA has been tested on 69-node distribution network system and the obtained test results have been compared to those from other methods in the literature. In addition to BSA, two other algorithms—particle swarm optimization (PSO) and cuckoo search algorithm (CSA)—are implemented for comparisons. The simulation results show that the proposed BSA can be an efficient and promising method for distribution network reconfiguration problems.
Keywords Distribution network reconfiguration Cuckoo search algorithm Particle swarm optimization Backtracking search algorithm Power loss reduction
T.T. Nguyen H.N. Pham A.V. Truong Faculty of Electrical and Electronics Engineering, HCMC University of Technology and Education, 1 Vo Van Ngan Street, Thu Duc District, Ho Chi Minh City, Vietnam T.T. Nguyen Dong an Polytechnic, 30/4 Street, Di an Dist, Binh Duong Province, Vietnam T.A. Phung Ha Noi University of Technology, 1 Dai Co Viet Street, Ha Noi, Vietnam T.T. Nguyen (&) Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, No. 19 Nguyen Huu Tho Street, District 7, Ho Chi Minh City, Vietnam e-mail:
[email protected] © Springer International Publishing Switzerland 2016 V.H. Duy et al. (eds.), AETA 2015: Recent Advances in Electrical Engineering and Related Sciences, Lecture Notes in Electrical Engineering 371, DOI 10.1007/978-3-319-27247-4_20
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1 Introduction An electric power power distribution network (DN) system carries electricity from the transmission system to individual consumers. Recently, DN systems are becoming large and complex leading to higher system losses and poor voltage regulation. Therefore, loss reduction and voltage profile enhancement in DN systems have constituted one of the most important objectives for researchers. There are many methods available for reducing power loss and improvement of voltage profile at distribution level: capacitor installation, load balancing, reconfiguration and distributed generator installation. Reconfiguration is one of the most economic method among them. In the last two decades, many researches have been carried out to solve DNR problems using different methods. Merlin and Back [1] were the first to report a method for distribution network reconfiguration to minimize feeder loss. Later on, several intelligent algorithms have been developed for loss minimization and/or voltage profile improvement. The most important algorithms in this category are genetic algorithm (GA) [2, 3] tabu search [4, 5] particle swam optimization (PSO) [6, 7] ant colony optimization (ACO) [8, 9] Recently, several novel methods based on artificial intelligence techniques have been implemented for DNR problems such as shuffled frog leaping algorithm (SFLA) [10], fireworks algorithm (FWA) [11], hybrid big bang-big crunch algorithm (HBB-BC) [12]. In general, most of the above intelligence techniques all inevitably involve a large number of computation requirements and have a lot of control parameters. The backtracking search optimization algorithm (BSA) developed by Pinar Civicioglu is a new evolutionary algorithm for solving optimization problems [13]. The BSA has a unique mechanism for generating a trial individual which enables it to solve numerical optimization problems successfully and quickly. The BSA uses three basic genetic operators: selection, mutation and crossover to generate trial individuals. In this paper, the BSA is proposed for solving DNR problem considering power losses in transmission systems and voltage profile improvement. The effectiveness of the proposed BSA has been tested on 69-node distribution network system and the obtained results have been compared to those from other methods available in the literature. This paper is organized as follows: The problem formulation is given in Section ‘Problem formulation’. The implementation of BSA for the problem is presented in Section ‘Backtraking search optimization algorithm for DNR’. The numerical results are provided in Section ‘Numerical results’. Finally, the conclusion is given.
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2 Problem Formulation The reconfiguration is defined as the process of changing the topology of system for a certain objective. The DNR is accomplished by changing open/close state of switches. In this study, the objective is to minimize total system active power loss and voltage deviation. The objective function can be described as follows [11, 14]: minimize F ¼ DPRloss þ DVD
ð1Þ
The net power loss reduced (DPRloss ) is taken as the ratio of total power loss before and after the reconfiguration of the system: DPRloss ¼
Prec: loss P0loss
ð2Þ
The total power loss of the system is determined by the summation of losses in all line sections: Ploss ¼
Nbr X i¼1
2 Pi þ Q2i Ri Vi2
ð3Þ
The voltage deviation index (DVD ) can be defined as follows: DVD ¼ max
V1 Vi V1
8i ¼ 1; 2; . . .; Nbus
ð4Þ
The reconfigured process will try to minimize the DVD closer to zero and thereby improves voltage stability and network performance. The constraints of objective function are as follows: (1) For the proposed configuration, the computed voltages and currents should be in their premising range. Vmin Vi Vmax ; 0 Ii Imax;i ;
i ¼ 1; 2; . . .; Nbus i ¼ 1; 2; . . .; Nbr
ð5Þ ð6Þ
(2) The radial nature of DN must be maintained and all loads must be served.
3 Backtracking Search Optimization Algorithm for DNR BSA comprises five processes: initialization, selection-I, mutation, crossover, and selection-II. The BSA method is implemented for DNR problem as follows.
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Initialization
To maintain the radial topology of the network in DNR process, the number of open branches should always be equal to the number of tie-switches (Nts) and could be obtained through Eq. 7: Nts ¼ Nbr ðNbus Nss Þ
ð7Þ
Therefore, the number of switches which must be open after reconfiguration is specific and must be used as a variable in algorithm. These switches are called tie-switches (SW). Therefore, every member of the initial population is a radial structure of the network. In DNR process using BSA, each radial structure of the network is considered as an individual. A population is represented by Pi ¼ i P1 ; . . .; PiNts1 ; PiNts with i = 1, 2,…, N. Where N and Nts are the population size, problem dimension, respectively. In which each Pi represents a solution vector of variables given by: Pi ¼ SWdi ; with d ¼ 1; 2; . . .Nts ð8Þ where SWdi (d = 1, 2,…, Nts) are the tie-switches of corresponding to individuals i to maintain the radial topology of the network. In order to reduce searching space of each tie-switch, the codification presented in [15] was used for the method proposed in this work. According to [15], the number of tie-switches is also equal the number of fundamental loops. Thus, an individual i is a vector with Nts positions in which each position corresponds to a fundamental loop. In the BSA, each individual can be regarded as a solution which is randomly generated in the initialization. Therefore, each individual i of the population is randomly initialized as follows: h i i i i Pi ¼ round SWmin;d þ rand SWmax;d SWmin;d ð9Þ i i and SWmax;d are minimum tie-switch and maximum tie-switch which where SWmin;d are encoded in fundamental loop d. Based on the initialized population, the load flow using Newton-Raphson load flow method is run then the fitness of each individual is calculated by the objective function Eq. 1. In addition, BSA is a dual-population algorithm that uses both current and historical populations. It remembers the population of a randomly selected generation for use in calculating the search direction matrix. The historical population is also initialized by Eq. 10 h i i i i oldPi ¼ round SWmin;d þ rand ðSWmax;d SWmin;d Þ ð10Þ
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Selection-I
In this stage, BSA generates the historical population oldP, used as the search direction. At the beginning of each iteration, BSA redefines oldP by comparing between two randomly generated numbers according to Eq. 11: If rand ð0; 1Þ \rand ð0; 1Þ then oldP ¼ P
ð11Þ
BSA memory remembers oldP as the historical population in each iteration until it is changed. This ensures that BSA designates a population belonging to a randomly selected previous generation. After determining the historical population based on the aforementioned equation, BSA changes the order of the individuals in oldP randomly through Eq. 12. oldP ¼ permutingðoldPÞ
ð12Þ
where permuting function is a random shuffling function.
3.3
Mutation
The mutation process of BSA generates Mutant as the initial form of the trial population through Eq. 13. Mutant ¼ P þ round ½F ðoldP PÞ
ð13Þ
where F is a function that controls the amplitude of the search direction matrix, which is the difference between the population and the historical population. In this paper, the standard Brownian movement with F = 3 × rand (0,1) is used.
3.4
Crossover
This operation generates the final form of the trial population. In this process, the initial trial population Mutant is changed to the final trial population T through a crossover operator. Trial individuals with better fitness values for the optimization problem are used to evolve the target population individuals. The crossover process has two steps. The first step set the value of T to Mutant, then a binary integer-valued matrix (map) with N rows and d columns is generated to select the individuals that have to be manipulated. If map (i,j) = 1, where i = {1, 2,…, N} and j = {1, 2, 3,…, d}, the individual T(i,j) is updated with T(i,j) = P(i,j). The second step applies a procedure related to a mix-rate parameter (mixrate) which is the only
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control parameter during optimization that controls the number of elements of individuals that will modify in a trial. Some individuals of T may violate the boundary condition of the optimization problem, so they are redefined at the end of the crossover process, as follows: If T(i,j) < SWminj or T(i,j) > SWmaxj then: h i i i i Ti ¼ round SWmin;d þ rand ðSWmax;d SWmin;d Þ
3.5
ð14Þ
Selection-II
In Selection-II stage, the individuals of population T with better fitness values than the corresponding particles of population P are used to update P based on a greedy selection. The global minimum among all the individuals is also updated according to the fitness values for T and P.
3.6
Stopping Criteria
In the proposed BSA method, the stopping criterion for the algorithm is based on the maximum number of iterations (Itermax). The algorithm is terminated when the number of iterations (Iter) reaches the maximum number of iterations (Fig. 1). The flowchart of the proposed BSA for DNR problem is given in Fig. 2.
4 Numerical Results To demonstrate the performance and effectiveness of the proposed method using BSA, it is applied to 69-node test system and compared the results with those of PSO which has been applied for many optimal problems related to power system [16] as well as cuckoo search algorithm (CSA), which is a recently developed optimization algorithm [17] and has been applied successfully applied for DNR problem [18]. The BSA based methodology was developed by Matlab R2014a in 2 GHz, i3, personal computer.
4.1
Selection of Parameters
In the proposed BSA method, there are three control parameters to be handled including population size, maximum number of iterations (Itermax) and mix rate
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Begin Randomize a number of population P = [tie-sw1, tie-sw2, …, tie-swd] Randomize a number of historical population oldP = [tie-sw1, tie-sw2, …, tie-swd] Check the radial topology, run power flow and evaluate the fitness function Selection-I Redefine historical population oldP based on population P via Eq. 10 Permute randomly the order of the individuals in oldP Mutation Generate Mutant poulation: Mutant = P + round[F x (oldP - P)], with F = 3 x rand (0,1) Crossover Generate the manipulated matrix map = zeros(N,Nts) Generate a vector U containing a random permutation of the integers 1:Nts map(i, U(1: mix-rate x Rand x Nts)) = 1, i=1,..,N Generate the offspring population: T = Mutant .x not(map) + map .x P Check the boundary condition of the population T via Eq. 14 Selection-II Check the radial topology, run power flow and evaluate the fitness function If fitness(Ti) < fitness(Pi) then fitness(Pi) = fitness(Ti) and Pi=Ti with i = 1,…, N
Yes Iter = Iter + 1
Iter
