A modified honey bee mating optimization ... - Semantic Scholar

Applied Energy 88 (2011) 4817–4830

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Applied Energy journal homepage: www.elsevier.com/locate/apenergy

A modified honey bee mating optimization algorithm for multiobjective placement of renewable energy resources Taher Niknam a, Seyed Iman Taheri a, Jamshid Aghaei a, Sajad Tabatabaei b,⇑, Majid Nayeripour a a b

Department of Electrical and Electronics Engineering, Shiraz University of Technology, P.O. Box 71555–31, Shiraz, Iran Department of Electrical Engineering, Mahshahr Branch, Islamic Azad University, Mahshahr, Iran

a r t i c l e

i n f o

Article history: Received 3 November 2010 Received in revised form 19 May 2011 Accepted 15 June 2011 Available online 13 July 2011 Keywords: Renewable electricity generator (REG) placement Multiobjective Honey bee mating optimization (HBMO) Backward–forward load flow

a b s t r a c t Electrical generators of renewable electricity resources are quiet, clean and reliable. Optimal placement of renewable electricity generators (REGs) results in reduction of objective functions like losses, costs of electrical generation and voltage deviation. Because of recent technology developments of photovoltaic units, wind turbine and fuel cell units, only these generators are considered in this paper. This work presents a multiobjective optimization algorithm for the siting and sizing of renewable electricity generators. The objectives consist of minimization of costs, emission and losses of distributed system and optimization of voltage profile. This multiobjective optimization is solved by the Improved honey bee mating optimization (HBMO) algorithm. In the proposed algorithm, an external repository is considered to save non-dominated (Pareto) solutions found during the search process. Since the objective functions are not the same, a fuzzy clustering technique is used to control the size of the repository within the limits. This algorithm is executed on a typical 70-bus test system. Results of the case study show the proper siting and sizing of REGs are important to improve the voltage profile, reduce costs, emission and losses of distribution system. The main feature of the algorithm refers to its accuracy and calculation speed. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Using renewable electricity generators (REGs) impose a different set of operating factors on distribution network such as reverse power flow, voltage rise, decreasing fault level and reduction of power losses, harmonic distortion and stability problems [1,2]. The impacts of REG depend on the several factors such as REG location, technology and capacity. Optimal REG placement aimed to find the optimal REG location and size of the REG in order to maximize or minimize a specific objective function subject to the operating constraints [3–5]. It is noted that these impacts in some cases have conflicting manner which persuades system operators to trade-off among these operating factors. In this regard, using multiobjective optimization framework can provide flexible tool for system operators who are responsible for decision making. In Ref. [6], the idea of distribution systems reinforcement planning using REGs is clearly formulated. The authors have discussed the possibility to consider REG as a feasible alternative for traditional reinforcement planning. The authors of Ref. [7] have developed a method for generating combination of several construction plans of distribution systems, considering the yearly increase of network loads, but they do not consider the installation ⇑ Corresponding author. Tel.: +98 711 7264121; fax: +98 711 7353502. E-mail addresses: [email protected] (T. Niknam), [email protected] (S. Tabatabae). 0306-2619/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2011.06.023

of REG units. In many other papers, pros and cons of the installation of REG units are considered and all the technical aspects of this problem are examined [8,9]. The optimal siting and sizing of REG units on the distribution system has been continuously studied in order to achieve different aims. The optimization problem objective can be the minimization of the active losses of the feeder [10,11]; or the minimization of the total network supply costs, which includes generators operation and losses compensation [12–14]. In Refs. [15,16] the objective function is optimization of voltage profile. All the mentioned objective functions are singleobjective and in this paper all of them are considered with together as the multiobjective optimization algorithm. Various methods are used for solving the optimization problem. In Refs. [17,18] genetic algorithm (GA) and in Ref. [19] dynamic ant colony search algorithms are used to cope with the optimization problem. The authors of Ref. [20] have presented analytical methods to determine the optimal location to place a REG in radial networked systems with respect to the power losses. Refs. [21,22] present power flow algorithms to find the optimal size of REG at each load bus in a network assuming that every load bus can have a REG source. Placement and penetration of distributed generation with the objective of generation cost minimization is proposed in Ref. [23]. Ref. [24] applied Monte Carlo method and Refs. [25,26] used Tabu Search algorithm to find optimal REG siting. Many algorithms tend to drive toward local minima instead of a global minimum, i.e.

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early convergence. One of the recently proposed evolutionary algorithms that have shown great potential and good perspective for the solution of various optimization problems is honey bee mating optimization (HBMO). The HBMO algorithm has remarkable accuracy and calculation speed to deal with the optimization problem. Advantages of the HBMO algorithm are presented in Refs. [27–29]. Refs. [30,31] have used the HBMO algorithm for solving optimization problems on two separate applications. In this paper, a multiobjective optimization is used for the placement and sizing of REGs by the improved HBMO algorithm. Original HBMO often converges to local optima. In order to avoid this shortcoming, in this paper a new mating process is proposed for rising accuracy of the algorithm. The objectives consist of minimization of the total network supply costs, emission and real power losses of distributed system and optimization of voltage profile. In the proposed algorithm, several queens have been considered as a set of non-dominated solutions. An external memory has been used for the storage of nondominated solutions found along the search process. Since the objective functions have competing nature, a fuzzy clustering algorithm is utilized to control the size of the external memory. For the better illustration, the contributions of the paper can be summarized based on their importance order as follows: – Optimal siting and sizing of renewable electricity generators (REGs) in the distribution systems using multiobjective framework. In the proposed placement scheme, generation costs, emission and losses of distributed systems and optimization of voltage profile are treated as competing objective functions. – Improved HBMO algorithm equipped with a fuzzy decision making tool has been used to cope with the Pareto-based multiobjective optimization problem. – Original HBMO often converges to local optima. In order to avoid this shortcoming, the new mating process is proposed. The reminder of the paper is organized as follow: the problem formulation is presented in Section 2. Section 3 shows the proposed algorithm descriptions. Simulation results are presented in Section 4 and finally some relevant conclusions are drawn in Section 5. 2. Problem formulation The main goal of the proposed algorithm is to determine the best location and size of new distributed generation resources by minimizing different objective functions. This section proposes four types of objective functions and their practical constraints in distributed system.

where a and b are:

a¼

Capital cost ð$=kWÞ Capacity ðkWÞ Gr Life time ðYearÞ 365 24 LF

b ¼ Fuel cost ð$=kW hÞ þ O & M cost ð$=kW hÞ

f1 ðXÞ ¼

N fc X

CðpÞ ¼ a þ b p

ð1Þ

C fc ðPfc Þ þ

i¼1

N wind X

C wind ðPwind Þ þ

i¼1

N pv X

C pv ðPpv Þ þ Cost sub

i¼1

Costsub ¼ Psub þ Q sub F 1 ðXÞ ¼ min½f1 ðXÞ

ð4Þ

where Pfc, Pwind and Ppv are power generated by the fuel cell units, wind units and the photovoltaic units, respectively. Psub is substation power rating. Nfc, Nwind and Npv are the numbers of the fuel cell units, wind units and the photovoltaic units, respectively. Qsub is the cost of installation of substation. X is the vector that will be defined in Section 3.4 and shows the location and the power of fuel cell units, wind units and the photovoltaic units. f1(X) is the first objective function that should be minimized. 2.1.2. Minimizing the deviation of the bus voltage Bus voltage is one the most significant security and power quality indices, which can be described as follows:

f2 ðXÞ ¼

Nbus X jV Rating V i j V Rating i¼1

F 2 ðXÞ ¼ minðf2 ðXÞÞ

ð5Þ

where Nbus is the total number of the buses. Vi is the real voltage of the ith bus and VRating is the nominal voltage [28]. 2.1.3. Minimization of power losses Minimizing the real power loss is selected as the third objective function for the placement of REGs. Reducing the real power loss of the distribution feeders is an important purpose of implementing REGs. The minimization of total real power losses of feeders over 10 years study period can be calculated as follows: Nd X N br X t¼1

2.1.1. Minimization of cost Generally, cost per kW h electrical energy produced by REGs is a function of the capital cost, fuel cost, and the operation and maintenance cost [23]. Table 1 shows these economic specifications. According to this table the function of cost is modeled as:

ð3Þ

LF is the load factor, Gr is the annual interest rate and the O & M cost is the operation and maintenance cost. Minimizing cost function can be modeled as follows:

f3 ðXÞ ¼

2.1. Objective functions

ð2Þ

ðRi jIi j2 DtÞ

i¼1

F 3 ðXÞ ¼ minðf3 ðXÞÞ

ð6Þ

where Ri and Ii are the resistance and the actual current of the ith branch, respectively. Nbr is the number of the branches, Dt is time step that in this study equals to one year and Nd is the number of years that in this study equals to 10 [32].

Table 1 Economic specification of REG. REG type

Rated capacity (kW)

Capital cost ($/kW)

Fuel cost ($/kW h)

Operation & maintenance cost ($/kW h)

Life time (year)

Fuel cell with CHP Photovoltaic Small wind turbine Big wind turbine

200 100 10 1000

3674 6675 3866 1500

0.029 0 0 0

0.01 0.005 0.005 0.005

10 20 20 20


2.1.4. Minimization of emission The total ton/h emission Et of atmospheric pollutants such as SOx and NOx caused by fossil-fueled thermal units can be calculated as follows:

Et ¼

m X

Ei ðPi Þ ¼

i¼1

m X ðai P2i þ bi Pi þ ci þ ni expð9ki Pi ÞÞ i¼1

9j 2 f1; 2; . . . ; Nobj g : f j ðX 1 Þ < fj ðX 2 Þ ð7Þ

where ai, bi, ci, ni and ki are coefficients of the ith generator emission characteristics, Pi is the electrical output of ith generator, m is the number of generators committed to the operation system. f4(X) is the fourth objective function that should be minimized.

The constraints can be listed as follows: – Voltage limits The voltage of the network should always be kept within the permissible limits:

ð8Þ

where Vmin and Vmax are the lower and upper voltage limits, respectively. Also, |Vk| is the voltage magnitude at bus k. – Number of REGs In fact, losses in the system could be totally eliminated if all loads would be supplied by their local generators, as in the ‘ideal’ case. However, this is very unrealistic as the cost of capital investment is too much. Therefore, we would like to suggest the implementation of the objective function to reduce the power losses with a given number of REGs.

nDG 6 NDG

ð9Þ

where nDG and NDG are the number of REGs determined and the number of REGs specified, respectively. – REG size Depending on the maximum allowable investment of REG, the total REG size should be governed by nREG X

KW KREG 6 gPLoad

ð10Þ

K¼1

where KW KREG is the capacity of the Kth REG and Pload is the total load power. Also, g is a constant such that 0 < g < 1. 3. Multiobjective approach using Pareto dominance criterion In the multiobjective optimization problems the concept of optimality is replaced with that of efficiency or Pareto optimality. The efficient (or Pareto optimal, non-dominated, non-inferior) solution is the solution that cannot be improved in one objective function without deteriorating its performance in at least one of the rest [33,34]. It can be expressed as: If point X⁄ is Pareto-optimal solution and v is the search space:

8k 2 f1; 2; 3; . . . ; Kg : 8X 2 v fX g; f k ðX Þ 6 fk ðXÞ & 9m 2 f1; 2; 3; . . . ; Kg : f m ðX Þ < fm ðXÞ

ð13Þ

If the above two mentioned conditions are validated, X1 dominates X2. The solutions that are non-dominated within the entire feasible search space are denoted as Pareto-optimal and known the Paretooptimal set or Pareto-optimal front. 3.1. Original and multiobjective HBMO (MHBMO)

2.2. Constraints

V min 6 jV k j 6 V max

ð12Þ

where fi(X) is ith objective function and X is a feasible solution. For any two solutions X1 and X2 can have one of two possibilities: one dominates the other or none dominates the other. In a minimization problem, a solution X1 dominates X2 if the following two conditions are satisfied:

8i 2 f1; 2; . . . ; Nobj g : f i ðX 1 Þ 6 fi ðX 2 Þ

f4 ðXÞ ¼ Etfc þ Etwind þ Etpv F 4 ðXÞ ¼ minðf4 ðXÞÞ

min FðXÞ ¼ ½f1 ðXÞf2 ðXÞ . . . fNobj ðXÞ

4819

ð11Þ

where K is the number of objective functions. In other words for a multiobjective minimization problem given by:

In order to deal with multiobjective problems, some modifications in the HBMO algorithm should be made. After the generation of the initial population and respective evaluation of objective functions, the selection of the ‘‘best solutions’’ (queens) should be made, but no longer based only on the comparison of single objective function values. Under a multiobjective approach, a new concept, such as the Pareto dominance concept, is needed for dealing with different solutions, i.e. classifying them as dominated or non-dominated solutions. The ‘‘best solutions’’ (queens) selected from the initial population are the non-dominated solutions. Once identified the non-dominated solutions (queens), the iterative process is initiated in the same way as in the single objective case (mating flights, generation of new queens, improvement of the queens and of the new generation and selection of new queens). Each non-dominated solution will generate a certain number of solutions after each iteration. The criteria for generation and improvement of the solutions specially best solution (queen) are the same as employed in the uni-objective version. With the new generated solutions and the non-dominated solutions from the previous iteration, the new set of non-dominated solutions is identified, which forms the Pareto front. These new solutions are saved in the repository and will generate the new solutions in the next iteration. The process is repeated until the stop criterion is satisfied. Frequently, the number of solutions that belong to the Pareto front increases as the algorithm evolves, thus each non-dominated solution is a potential generator (queen) of new solutions in the next iteration of the algorithm. It is noted that these non-dominated solutions saved in the repository are not the final non-dominated solutions because the repository will be updated after generation of broods in the next iterations. Besides, our experiences in the implementation of the proposed algorithm shows that it is safe to say the repository of the non-dominated solutions will be significantly updated in each new iteration with respect to the previous ones in the initial iterations. However, after some iteration the results of the repository may be saturated. That is the non-dominated solutions may be remained unchanged. Indeed the new solutions in the higher number of the iterations may be equal to the same of the repository solutions or they will be dominated by the repository ones. Therefore, it can be concluded from this manner that the non-dominated solutions of the repository after some iteration are trustworthy. This would make the algorithm slower and more inefficient as the number of iteration increases, since each solution would generate a number of new solutions, escaping the user control. In order to increase the algorithm efficiency, a clustering method is used to select the non-dominated solutions among the front solutions [35]. The clustering technique used here not only promotes a better

4820


distribution of the solutions along the front, but also improves the performance of the algorithm. The HBMO algorithm combines a number of different procedures. Each of them corresponds to a different phase of the mating process of the Honey-Bee that is presented in Fig. 1 [29]. A drone mates with queen can be expressed using a probabilistic-based annealing function as follows [30]:

ProbðDÞ ¼ exp½Dðf Þ=SðtÞ

ð14Þ

where Prob(D) is the probability of adding the sperm of drone D to the spermatheca of the queen, D(f) is the absolute difference between the fitness of D and the fitness of the queen, and S(t) is the speed of the queen at time t. The probability of mating is high when the queen is with the high speed level, or when the fitness of the drone is as well as the queen’s. After each transition in space, the queen’s speed decreases according to the following equation:

Sðt þ 1Þ ¼ a SðtÞ

ð15Þ

where a e (0, 1) is the amount of speed and energy reduction after each transition and each step. Workers may represent a set of different heuristics to improve the brood’s genotype. The rate of improvement in the brood’s genotype, as result of a heuristic application to that brood, defines the heuristic fitness value [31]. In the proposed MHBMO algorithm, several queens are utilized. They are considered as a set of non-dominated solutions and stored in an external memory (repository). The repository is initialized by the non-dominated individuals presented in the initial population. Since the objective functions are not the same, the best solution is extracted using a fuzzy-based mechanism. During the search process, if the best solution dominates any individual in the repository, the corresponding individual is immediately removed from it. A great size of the repository is better. However, it is necessary to limit repository size since there are memory constraints. Also, a large repository increases computational time. In the MHBMO algorithm, the probability of a drone mates with a queen can be alculated as:

ProbðDÞ ¼ expðjF queen F drone j=SðtÞÞ

ð16Þ

jF queen F drone j qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 ¼ ðf1queen f1drone Þ þ ðf2queen f2drone Þ þ ðf3queen f3drone Þ þ ðf4queen f4drone Þ

where fiqueen and fidrone (i = 1, 2, 3, 4) are the values of the ith objective function for the queen and the drone, respectively. Consequently the above-mentioned procedure can be summarized as: firstly, one drone is randomly selected from initial population and then the probability of mating can be calculated from Eqs. (14) and (16) for single and multiobjective problems, respectively. Then, if the probability of mating is greater than the another random number between [0, 1], the selected drone is stored in the spermatheca of the queen. In order to avoid the non-dominated solutions towards densely populated regions and thereby obtaining a uniformly-distributed Pareto front, a form of fitness sharing has been incorporated to select the globally best solution from the non-dominated solutions stored in the repository. The fitness sharing is based on the fundamental idea. Thus, the higher the number of neighboring individuals the lesser will be the fitness. The performance of the fitness sharing is sensitive to the population size N. The goal of fitness sharing is to distribute the population over a number of different peaks in the search space, with each peak receiving a fraction of the population in proportion to the height of that peak. Furthermore, if we have one non-dominated solution the algorithm selects it as the queen and if we have more than one non-dominated solution, e.g. 10 non-dominated solutions, using sharing procedure, the best of them is chosen as the queen of the algorithm and the algorithm will be continued. Consequently, in the next iteration the selected queen can produce more non-dominated solutions. 3.2. Best compromise solution Upon having Pareto-optimal set of non-dominated solution, the decision maker needs to select one solution as the best compromise solution among the Pareto set. Because of imprecise nature of the decision maker’s judgment, the ith objective function is represented by a membership function lfi(X) defined as [36]:

Fig. 1. The HBMO algorithm.


8 1 for f i ðXÞ 6 fimin > > < for f i ðXÞ P fimax lfi ðXÞ ¼ 0 max > > fi fi ðXÞ : for fimin 6 fi ðXÞ 6 fimax f max f min i

ð17Þ

i

The membership function consists of lower and upper boundaries, together with a strictly monotonically decreasing and continuous function [30], shown in Fig. 2. The lower and upper bounds (fimin and fimax ) of each objective function subject to the given constraints are established to elicit a membership function lfi(X) for each objective function, fi(X). For each non-dominated solution k, the normalized membership function lk can be calculated as [37,38]:

PNobj

k i¼1 li

lk ¼ PM PNobj k¼1

i¼1

ð18Þ

lki

where M is the number of non-dominated solutions. The best compromise solution is the one having the maximum value of lk. 3.3. Modification of HBMO In the original HBMO the jth brood is generated using the following process: A population of broods is generated based on the mating between the queen and the drones stored in the queen’s spermatheca. The jth brood is generated using the following process:

X best ¼ ½x1best ; x2best ; . . . ; xnbest Spi ¼ ½s1i ; s2i ; . . . ; sni Broodj ¼ X best þ b ðX best Spi Þ;

ð19Þ j ¼ 1; 2; . . . ; NBrood

where b is a random number between 0 and 1 and Broodj is the jth brood. Xbest is the queen place. Spi is the ith drone in the queen’s spermatheca. The original mating process combines the features of two parent structures to form a similar offspring. Its purpose is the maintenance and exchange of queen’s place. However, this cannot guarantee the convergence to the optimal point and sometimes causes premature convergence to local minima. In order to improve the broods produced by the mating flight of the queen, in this paper a new approach is proposed to generate the broods as follows: Initially, three drones ðSpz1 ; Spz2 ; Spz3 Þ are randomly selected from the queen’s spermatheca so that z1 – z2 – z3.Then, an improved drone position vector is calculated as follows:

X imp;1 ¼ Spz1 þ randðÞ ðSpz2 Spz3 Þ X imp;1 ¼ x1im1 ; x2im1 ; . . . ; xnim1 1 X Brood;1 ¼ xbr1 ; x2br1 ; . . . ; xnbr1 ( xiim1 ; if /1 6 /2 xibr1 ¼ sim1 ; otherwise X imp;2 ¼ X best þ randðÞ ðSpz2 Spz3 Þ X imp;2 ¼ x1im2 ; x2im2 ; . . . ; xnim2 X Brood;2 ¼ x1br2 ; x2br2 ; . . . ; xnbr2 ( xiim2 ; if /3 6 /2 xibr2 ¼ xibest ; otherwise

4821

ð20Þ

where /1, /2, /3, /4 are random numbers between 0 and 1. The objective functions are evaluated for XBrood,1 and XBrood,2. The best solution between them is considered as the new brood. 3.4. Finding the non-dominated set of solution The HBMO has widespread applications in different fields because of its speed and accuracy to achieve objectives. However, it often converges to local optima. Generally, a probabilistic method has a large possibility of exploring the search space freely in the beginning and slowly few valleys while the run progresses. In order to avoid this shortcoming, an improved HBMO method is proposed in the paper. Consider a set of N population members, each having Nobj objective function values. The following procedure can be used to find the non-dominated set of solution: Step 1: Begin with i = 1. Step 2: Pick randomly two candidates for selection X1 and X2. Step 3: Pick randomly a comparison set of individuals from the population. Step 4: Compare each candidate, X1 and X2, against each individual in the comparison set for domination using the conditions for domination given in Eqs. (12) and (13). Step 5: If one candidate is dominated by the comparison set while the other is not, then select the later for reproduction and go to Step 7, else proceed to step 6. Step 6: If neither or both candidates are dominated by the comparison set, then use sharing to choose winner. Step 7: If the criteria i = N is reached, stop selection procedure, else set i = i + 1 and go to Step 2. 3.4.1. Sharing procedure To prevent the HBMO drift problem, a form of sharing should be carried out when there is no preference between two candidates. This form of sharing maintains the HBMO diversity along the population fronts and allows the HBMO to develop reasonable representation of the Pareto-optimal front [35]. The sharing procedure can be described as: Given a set P whose size exceeds the maximum allowable size N, it is required to form a subset P⁄ with the size N. The algorithm is illustrated in the following steps. Step 1: Initialize cluster set C; each individual i e p constitutes a distinct cluster. Step 2: If number of clusters 6N, then go to Step 5, else go to Step 3. Step 3: Calculate the distance of all possible pairs of clusters. The distance dc of two clusters c1 and c2 e C is given as the average distance between pairs of individuals across the two clusters:

dc ¼ Fig. 2. A membership functions for objective function.

1 n1 n2

X i1 2c1 ;i2 2c2

dði1 i2 Þ

ð21Þ

4822


where n1 and n2 are the number of individuals in cluster c1 and c2, respectively. The function d reflects the distance in the objective space between individuals i1 and i2. Step 4: Determine two clusters with minimal distance dc. Combine these clusters into a larger one. Go to Step 2.

Step 5: Find the centroid of each cluster. Select the nearest individual in this cluster to the centroid as a representative individual and remove all other individuals from the cluster. Step 6: Compute the reduced non-dominated set P⁄ by unitizing the representatives of the clusters.

•

Set N Sperm

the size of queen’s spermatheca

• • • • •

Set α Set Smax Set Smin Set NDrone Set N Worker

the speed reduction schema the speed of queen at the start of a mating flight the speed of queen at the end of mating flight the number of drone the number of workers

•

Set N Brood

the number of broods

• •

Set Iter_max Number of iteration Set X(equation 22) Begin Generate NDrone drone with X length randomly and set them as D Calculate their objective functions Calculate non-dominate solutions and save in repository Select the best drone and set it as queen (Q) For Iter=1 to Iter_max For i1=1 to NDrone Select a Di from D randomly S=Smax Calculate Δ( f ) from equation 16 Generate r randomly If exp(- Δ( f ) /S) > r Add D i to spermatheca Else S= α *S Until capacity of spermatheca completed(equal N Sperm ) or S=Smin End i1 For i2=1 to NDrone Select a crossover function from list NWor ker , according to its probability Select a Si randomly and generate new solution by crossover Si and Q and set it as Bi Calculate crossover fitness Update probability matrices of crossover function selection Until Number of broods equal to N Brood Select a mutation function from list NWor ker , according to its probability Mutation Bi and set it as Ei If f(Ei)>f(Q) swap Ei and Q Else keep the previous solution Calculate mutation fitness Update probability matrices of mutation function selection Calculate objective functions for all broods Calculate non-dominate solutions and save in repository2 End i2 Generate new drones list randomly Until the termination criteria satisfied For i3=1 to size of repository Select Q as Spz1 Select Spz2 randomly from repository Select Spz3 randomly from repository2 Calculate equation 20 Calculate objective functions Using sharing procedure and Calculate best non-dominate solutions and save in repository3 End i3 End Iter Print repository3 End.

Fig. 3. Pseudo codes of proposed algorithm.

4823

2.92 2.87 2.52 2.40 2.32 2.30 2.30 2.29 2.27 2.25 2.876 10 1.8851 106 1.2021 106 1.0797 106 1.0742 106 1.0617 106 1.0541 106 1.0512 106 1.0484 106 1.0471 106

7

3.02 2.91 2.61 2.45 2.40 2.38 2.37 2.37 2.36 2.35

6

Best solution

f1 f4 f3

Pn

135.64 132.81 126.13 123.12 122.89 121.66 121.31 121.21 121.13 120.75

P2

7.13 10 6.921 107 6.631 107 6.497 107 6.483 107 6.476 107 6.432 107 6.430 107 6.481 107 6.627 107

P1

3.12 3.02 2.76 2.49 2.48 2.47 2.46 2.45 2.44 2.43 137.182 135.876 128.371 125.837 124.123 123.352 122.612 122.098 121.971 121.754 7.23 10 7.01 107 6.73 107 6.545 107 6.540 107 6.538 107 6.535 107 6.534 107 6.531 107 6.530 107

These bold face values show the optimum values of objective functions optimized separately.

0.9 0.9 1 1 1.1 1.1 1.2 1.2 1.2 1.2 1 2 3 4 5 6 7 8 9 10

0.1 0.1 0.2 0.2 0.2 0.3 0.3 0.3 0.4 0.4

0.90 0.91 0.92 0.92 0.93 0.94 0.95 0.96 0.96 1.0

85 90 95 100 105 110 120 130 140 150

85 90 95 100 100 110 120 130 140 150

100 250 350 500 600 700 800 900 900 1000

80 85 90 100 110 120 130 140 150 150

59.35 87.20 100.32 118.245 201.21 256.72 298.22 333.19 400.22 521.98

7

f2

f3

f4

6

2.1231 10 1.9753 106 1.2362 106 1.0853 106 1.0773 106 1.0623 106 1.0512 106 1.0445 106 1.0412 106 1.0401 106

Average

f1

Worst solution

where N is the number of drones and F1N, F2N, F3N and F4N are the values of four objective functions for Nth drone. n is the number of REGs, locationn is location of nth REGs, Pn is size of nth REG. locationn change between 1 and maximum number of the buses and Pn change between 0 and maximum capacity of nth REG. It is noted that there are two kinds of variables in the problem: continuous and discrete variables. The former refers to the active power values of renewable energy resources and the later is related to location of renewable energy resources. For the discrete variables a Round function is used. In REG placement problem X1, X2, . . . , XN are locations and sizes of REGs in each bus of the distribution system. Step 3: i = 1. Step 4: select the ith drone. The values of the objective functions for the ith drone are evaluated using results of the distribution load flow. Step 5: If the drone is a non-dominated solution, then the drone is stored into the repository and the fuzzy clustering is used to control the repository’s size. Step 6: if all drones are selected, go to Step 8, otherwise i = i + 1 and return to Step 5. Step 7: Select a queen (Xbest) from the repository randomly. Step 8: Generate the queen speed. Step 9: Select the population of the drones. Step 10: Generate the queen’s spermatheca matrix. Step 11: Feed the selected broods and queen with the royal jelly by workers. Step 12: The values of the objective functions for each brood are evaluated by using the results of the distribution load flow. If each brood is a non-dominated, then, store the brood into repository and use the fuzzy clustering to control its size, else the termination criteria should be checked. If the termination criteria is satisfied, then finish the algorithm, else discard all

Average CPU time (s)

F 4N

NBrood

F 3N

NSperm

F 2N

ð22Þ

NDrone

...

NWorker

...

7 F 42 7 7 7 ... 5

a

F 32

Smin

F 1N

F 22

Smax

N

F 31

Case

XN

F 41

3

F 21

Table 2 Simulation results for impact of parameters of proposed algorithm.

3 2 X1 F 11 7 6 6 6 X2 7 6 F 12 7 )F¼6 Pop ¼ 6 7 6 6 45 4 ... 2

f1

7

location2 . . . locationn

f2

X i ¼ ½location1 i ¼ 1; 2; . . . ; N

f2

This section presents the proposed hybrid algorithm based on the combination of HBMO and Pareto sets for multiobjective REGs placement. To apply the proposed algorithm in the REGs placement problem, the following steps have to be taken: Step 1: Define input data: The input data including the network configuration, line impedance, loads, the number of drone (NDrone), the number of workers (NWorker), the number of broods (NBrood), fimin , fimax , the speed of queen at the start of a mating flight (Smax), the speed of queen at the end of mating flight (Smin), the speed reduction schema (a), the number of iteration and the size of queen’s spermatheca (NSperm) should be defined in this step. Step 2: Generate initial population: An initial population is generated as follows:

133.2 130.9 124.4 121.9 121.2 120.6 120.4 120.3 120.2 120.1

f3

3.5. Solution of MHBMO for REGs placement

7.02 10 6.822 107 6.523 107 6.437 107 6.434 107 6.432 107 6.429 107 6.427 107 6.426 107 6.425 107

f4

The utilization of improvements in the breeding process can be useful to escape more easily from local minima than with the traditional mating.

2.0132 106 1.8213 106 1.1986 106 1.0726 106 1.0712 106 1.0611 106 1.0567 106 1.0560 106 1.0558 106 1.0550 106


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Fig. 4. A single line diagram of 11 KV distribution test system.

Table 3 Results obtained by optimizing the total cost. Method

Cost ($)

CPU time (s)

REGs placement (Bus Number)

Genetic algorithm [17] The proposed algorithm Particle swarm optimization [23] HBMO [28]

6.512 107 6.437 107 6.488 107 6.491 107

572.63 102.3 354.2 120

1, 2, 3, 13, 21, 29, 31, 33, 34, 44, 52, 53 1, 2, 3, 14, 21, 29, 31, 33, 34, 44, 52, 53 1, 2, 3, 14, 21, 29, 31, 33, 34, 44, 52, 53 1, 2, 3, 14, 21, 29, 31, 33, 34, 44, 52, 53

Table 4 Results obtained by optimizing the voltage deviation of buses. Method

Voltage deviation (Pu)

CPU time (s)



2.493844 2.4063 2.485545 2.485545

445.12 124.01 256.18 155.02

10, 12, 13, 18, 28, 34, 35, 44, 51, 53, 66, 67 11, 12, 13, 18, 28, 34, 35, 44, 51, 53, 66, 67 11, 12, 13, 18, 28, 34, 35, 44, 51, 53, 66, 67 11, 12, 13, 18, 28, 34, 35, 44, 51, 53, 66, 67

previous trial solutions and return to the Step 3 until convergence criteria is met. Pseudo codes of the proposed algorithm are given in Fig. 3.

4. Simulation results In this section, the proposed algorithm is employed to solve REGs placement problem for a distribution test system. The

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parameters required for implementation of the proposed algorithm are the speed of queen at the start of a mating flight (Smax), the speed of queen at the end of mating flight (Smin), the speed reduction schema (a), the number of drone (NDrone), the number of workers (NWorker), the number of broods (NBrood), the number of iteration and the size of queen’s spermatheca (NSperm). Table 2 shows the

impacts of different parameters of the proposed optimization algorithm. Note that the accuracy and calculation speed have conflicting manner and operators should trade-off between these goals. For instance, if the value of Nsperm is increased, the accuracy of algorithm will enhance with the cost of lower calculation speed of the algorithm. Therefore, the experience of the expert operator can

Table 5 Results obtained by optimizing the total power losses. Method

Power losses (kW h)

CPU time (s)



129.5982 121.9012 128.9817 127.5179

147.21 123.89 306.81 153.81

6, 7, 8, 23, 30, 31, 41, 44, 58, 59, 66, 67 6, 8, 9, 23, 30, 31, 41, 44, 58, 59, 66, 67 7, 8, 9, 23, 30, 31, 41, 44, 58, 59, 66, 69 6, 8, 9, 23, 30, 31, 41, 44, 58, 59, 66, 67

Table 6 Results obtained by optimizing the emission. Method Genetic algorithm [17] The proposed algorithm Particle swarm optimization [32] HBMO [28]

Emission (kg/h) 6

1.2100 10 1.0726 106 1.0700 106 1.0780 106

CPU time (s)

REGs locations (Bus Number)

444.46 122.78 252.49 151.28

6, 7, 8, 23, 30, 31, 41, 44, 58, 59, 66, 67 7, 9, 10, 23, 30, 31, 41, 44, 58, 59, 66, 67 7, 9, 10, 23, 30, 31, 41, 44, 58, 59, 66, 67 7, 9, 10, 23, 30, 31, 41, 44, 58, 59, 66, 67

Fig. 5. Convergence characteristics of the objective function.

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help to find good parameters by try-and-error method. Based on the results of Table 2, the best values for the mentioned parameters are: Smax = 1, Smin = 0.2, a = 0.92, NDrone = 100, NWorker = 100, NBrood = 100, NSperm = 500, determined by 100 runs of algorithm. The number of the initial population is 400. The values that given for mentioned parameters result in the best accuracy and speed calculation of HBMO algorithm.

Implemented test system to show the efficiency of the proposed framework is a 70-Bus 11 KV Radial Distribution system [37]. A single line diagram of the system with two substations, four feeders, 70 nodes and 78 branches (including tie branches) is shown in Fig. 4. The system data is given in Ref. [39].

Fig. 6. Pareto front obtained using MHBMO.


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4.1. Case study 1 At first, the total power losses, emission, voltage deviation and the cost are separately optimized to find the extreme points of the trade-off front. In this case 12 fuel cells are used that their maximum generation is 100 kW and coefficients of emission are:

a ¼ 4:285

kg 2

hMW kg c ¼ 4:586 ; h

;

kg ; hMW kg n ¼ 1:0 106 ; k ¼ 8:000 MW1 h b ¼ 5:094

The best results obtained by optimizing all the four objectives separately are shown in Tables 3–6, respectively. In these tables results of the proposed algorithm have been compared with the other research works and the best buses for installation of REGs are obtained. As shown in the tables, the algorithm is capable of finding the better solutions for each objective function with respect to the other methods. The generated tradeoffs by the MHBMO algorithm are illustrated in these tables. It can be seen that the proposed algorithm can achieve the proper Pareto front. The simulation results in these tables show that depending on the system priorities, the distribution system operator can choose different configurations. The convergence characteristics of all four objective functions have been shown in Fig. 5. The Pareto front obtained by the MHBMO is shown in Fig. 6. It is evident that the Pareto solutions are well-distributed over the trade-off curve primarily due to several diversity-preserving mechanisms used throughout the entire optimization process in the proposed algorithm. The Pareto fronts obtained from Multiobjective Genetic Algorithm (MOGA) [35] and Multiobjective Particle Swarm Optimization (MOPSO) [32] are shown in Fig. 7. By comparison, it can be seen that the solutions obtained by these two approaches are dominated by those derived through MHBMO. Also the diversity of the solutions obtained from MHBMO is the highest among all the three approaches due to the diversity preservation mechanisms used throughout the optimization process. In Fig. 7 only 3 positions of 6 positions are shown for comparison. Fig. 8 shows three objective functions with together in three dimensional diagrams. Fig. 8 helps the distribution system operator to choose better, different configurations. Because of the fact that the MATLAB software cannot show four objective functions together in four dimensional diagrams, Table 7 is presented. Table 7 shows only 10 statuses of 100. Indeed, repository is limited to 100. In this table, distribution system operator can see four objective functions together and can select the best status based on his experiences. To better illustrate the multiobjective DG placement problem, the results achieved from the proposed algorithm are classified into the following cases (Cases I–IV are reserved for single objective optimization results) in Table 8: Case Case Case Case Case

V: Considering functions f1, f2 and f3. VI: Considering functions f1, f3 and f4. VII: Considering functions f2, f3 and f4. VIII: Considering functions f1, f2 and f4. IX: Considering functions f1, f2, f3 and f4.

The three-dimensional Pareto optimal set achieved by optimizing the 3-objective problem are shown in Fig. 8 for Cases V–VIII. It is note that each Pareto optimal solution is an alternative choice for a decision-maker. System operator can adopt one of them for the respective system based on its preferences over the objective functions. In fact, after obtaining the Pareto optimal

Fig. 7. Comparison of different Pareto fronts obtained.

solutions, the decision-maker needs to choose one best compromised solution according to the specific preference for different conditions. After applying the MHBMO to generate Pareto sets, the best compromised solution among them is selected according to the following equation:

Pn k¼1 wk ljk Nl ðjÞ ¼ Pm Pn j¼1 k¼1 wk ljk

ð23Þ

where wk is the weight factor for the kth objective functions, n is the number of objective functions and m is the number of non-dominated solutions. It is noted that in the above equation, the importance of the objective functions should be determined by decision P maker such that 3i¼1 wi ¼ 1. The results of selecting the best compromised over Pareto optimal solutions in Cases I–IX are proposed in Table 8. For comparison between single-objective and multiobjective optimization, Cases I– IV are devoted to the single objective optimization results. By analyzing the results, the following observations are made: – In Case V, when the weight factors (w1, w2) are decreased or increased, the objective functions did not decrease or increase, it means that these two objective functions have not conflicted to each other. – The objectives f1 and f2 have the same behavior. The reasons for this claim can be inferred from the results of Cases I, II, III, IV and V. In Cases I and IV when either f1 or f2 is minimized individuality, the other is also minimized.

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Fig. 8. Three objective functions together in three dimensional diagrams.

Table 7 Four objective functions for placement of 12 fuel cell units in proposed system. f1 ($) 6.437 107 6.497 107 6.505 107 6.511 107 6.521 107 6.529 107 6.533 107 6.537 107 6.542 107 6.549 107

f2 (Pu)

f3 (kW h)

f4 (ton/h)

Fuel cell locations

2.712455 2.556823 2.690091 2.725545 2.454002 2.718884 2.566400 2.649225 2.4063 2.486489

138.9573 135.6973 129.3975 121.9012 126.3971 127.3614 128.3975 129.3358 135.6587 129.1937

1.08 106 1.78 106 1.74 106 1.1 106 1.79 106 1.75 106 1.79 106 1.69 106 1.073 106 1.0726 106

1, 2, 3, 14, 21, 29, 31, 33, 34, 44, 52, 53 3, 12, 23, 34, 35, 43, 47, 49, 55, 59, 62, 67 6, 8, 18, 23, 31, 33, 39, 48, 54, 59, 66, 67 6, 8, 9, 23, 30, 31, 41, 44, 58, 59, 66, 67 1, 9, 11, 18, 40, 47, 50, 59, 62, 64, 67, 69 7, 9, 11, 19, 40, 45, 54, 57, 60, 64, 65, 69 8, 8, 11, 17, 43, 44, 52, 57, 62, 64, 65, 68 8, 9, 11, 13, 40, 45, 50, 59, 60, 64, 65, 69 4, 8, 11, 22, 30, 31, 41, 46, 58, 59, 66, 69 7, 12, 14, 19, 25, 33, 44, 52, 54, 66, 67, 69


4.2. Case study 2 In second case study, types of REGs are changed firstly to photovoltaic, secondly to small wind turbine and thirdly big wind turbine. Distributed test system is the same as the previous system. Table 9 shows the results of using different types of REGs. It is obvious that the wind turbine and the photovoltaic are considered when they can generate power. The results of this table show the cost function has its minimum value when all 12 REGs are fuel cell units. This is because of the fact that the capital cost of fuel cell

units is lower than the capital cost of photovoltaic and big wind turbine units. On the other hand, fuel cell units have emission while the other two types of generators have no emission. Renewable electricity production from sources such as wind power and solar power is sometimes criticized for being variable or intermittent. For this reason, they should be used in combination with other types of generators in the network. The reason of using all REGs of the case study in photovoltaic or wind types is the fact that the simulation results will help the system operator in selecting desired combination of REGs types.

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T. Niknam et al. / Applied Energy 88 (2011) 4817–4830 Table 8 Objective function values in all cases. Cases

Importance W1

W2

W3

W4

Case I

–

–

–

– –

Case II Case III Case IV

–

f1 ($)

–

–

–

–

–

–

–

–

– –

f2 (kW h)

f3 (Pu)

f4 (kg/h)

6.437 107

123.5494

2.4152

1.0727 106

7.9284 10

7

157.1672

2.8519

1.0726 106

6.4580 10

7

121.9012

2.4122

1.0732 106

6.4666 10

7

123.3046

2.4063

1.0747 106

7

Case V

0.33 0.2 0.4 0.4

0.33 0.4 0.2 0.4

0.33 0.4 0.4 0.2

– – – –

6.4668 10 6.4451 107 6.4451 107 6.4451 107

129.8938 121.0178 121.0178 121.0178

2.4000 0.1979 0.3958 0.3958

– – – –

Case VI

0.33 0.2 0.4 0.4

– – – –

0.33 0.4 0.2 0.4

0.33 0.4 0.4 0.2

6.4828 107 6.4733 107 6.4733 107 6.4733 107

– – – –

2.5298 0.3891 0.3891 0.1946

1.5170 106 1.0842 106 1.0842 106 1.0842 106

Case VII

– – – –

0.33 0.2 0.4 0.4

0.33 0.4 0.2 0.4

0.33 0.4 0.4 0.2

–– – – –

141.9335 141.9335 141.9335 139.5985

0.3224 0.3908 0.3908 0.1761

1.0755 106 1.0755 106 1.0755 106 1.0878 106

Case VIII

0.33 0.2 0.4 0.4

0.33 0.4 0.2 0.4

– – – –

0.33 0.4 0.4 0.2

7.6070 107 6.8066 107 6.8066 107 6.8066 107

161.4555 0.3527 0.3527 0.1764

– – – –

1.0727 106 1.0877 106 1.0877 106 1.0877 106

Case IX

0.25 0.1 0.3 0.3 0.3

0.25 0.3 0.1 0.3 0.3

0.25 0.3 0.3 0.1 0.3

0.25 0.3 0.3 0.3 0.1

7.6821 107 7.7180 107 7.6071 107 7.5784 107 7.6278 107

147.9335 145.2145 148.7865 144.8321 140.3472

1.4231 1.3251 1.2947 1.7214 0.4109

1.2727 106 1.1729 106 1.1624 106 1.2513 106 1.2953 106

Table 9 Comparison of usage kind of REGs. REG type

f1 ($)

f2 (Pu)

f3 (kW h)

f4 (ton/h)

Fuel cell with CHP Photovoltaic Small wind turbine Big wind turbine

6.437 107 12.927 107 6.892 107 300.369 107

2.4063 2.578253 2.925442 2.166975

121.9012 125.3987 95.6284 148.2258

1.6 106 0 0 0

4.3. Case study 3 In third case study, combination of three types of REGs (fuel cell, photovoltaic and small wind turbine) is applied. In this case, 8 fuel cell units, two photovoltaic units and two small wind turbines are used. The maximum generation of fuel cell, photovoltaic and small wind turbine are 200 kW, 100 kW and 10 kW, respectively. It is assumed that small wind turbines can generate power all the time and photovoltaic units can generate power only from 6:00 am to 6:00 pm in each day (24 h). Lack of photovoltaic units should com-

pensate by another REGs. Table 10 shows simulation results of this case study. In this table only 10 statuses of 100 statuses are shown. Results show that in this case, emission is decreased, cost is littledecreased and deviations of voltage and losses have very little changes.

5. Conclusions A Pareto-based multiobjective optimization equipped with a fuzzy decision making tool is presented for sitting and sizing of REGs by the improved HBMO algorithm. In the proposed placement scheme, generation costs, emission and losses of distributed system and optimization of voltage profile are treated as competing objective functions. One of the most important advantages of the proposed multiobjective formulation is that it obtains several non-dominated solutions allowing the system operator (decision maker) to exercise his personal preference in selecting each of those solutions based on the operating conditions of the system. Moreover, in the proposed optimization algorithm, the mating pro-

Table 10 Four objective functions and placement of REGs in propose system. f1 ($) 6.472 107 6.499 107 6.502 107 6.509 107 6.519 107 6.514 107 6.527 107 6.532 107 6.539 107 6.546 107

f2 (Pu)

f3 (kW h)

f4 (ton/h)

Location of fuel cell units

Location of photovoltaic units

Location of wind units

2.712451 2.556821 2.690097 2.725544 2.454007 2.718882 2.566495 2.649221 2.385541 2.486493

138.9577 135.6978 129.3979 125.4165 126.3976 127.3619 128.3978 129.3361 135.6591 129.1940

1.075 106 1.72 106 1.65 106 1.1 106 1.69 106 1.66 106 1.64 106 1.61 106 1.51 106 1.24 106

4, 7, 12, 14, 21, 29, 31, 33 5, 11, 22, 34, 35, 44, 47, 49 6, 8, 18, 23, 31, 33, 39, 48 9, 23, 30, 31, 41, 44, 58, 59 11, 18, 40, 47, 50, 59, 62, 64 11, 19, 40, 44, 56, 57, 60, 64 11, 17, 43, 44, 52, 57, 62, 64 8, 9, 11, 13, 50, 59, 65, 69 4, 8, 11, 22, 41, 46, 58, 66 7, 12, 25, 33, 44, 52, 67, 69

22, 69 15, 66 66, 69 22, 45 45, 66 15, 45 15, 22 22, 45 45, 69 22, 66

56, 57 56, 60 5, 26 57, 60 32, 35 54, 60 5, 60 60, 64 32, 57 54, 56


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