Optimal distributed generation location and size using a modified teaching–learning based optimization algorithm

Electrical Power and Energy Systems 50 (2013) 65–75

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Electrical Power and Energy Systems

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Optimal distributed generation location and size using a modified teaching–learningbased optimization algorithm

Juan Andrés Martín García 1, Antonio José Gil Mena ⇑Department of Electrical Engineering, University of Cádiz, Escuela Politécnica Superior de Algeciras, Avda. Ramón Puyol, s/n, 11202 Algeciras (Cádiz), Spain

a r t i c l e i n f o

Article history:Received 3 September 2012Received in revised form 7 February 2013Accepted 22 February 2013Available online 20 March 2013

Keywords:Distributed Generation (DG)Modified Teaching–Learning-BasedOptimization (MTLBO)Optimal placementDistribution systemsPower loss reduction

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⇑ Corresponding author. Tel.: +34 956 028 168; faxE-mail addresses: [email protected] (J.A. M

ca.es (A.J. Gil Mena).1 Tel.: +34 956 028 167; fax: +34 956 028 001.

a b s t r a c t

In this paper, a method which employs a Modified Teaching–Learning Based Optimization (MTLBO) algo-rithm is proposed to determine the optimal placement and size of Distributed Generation (DG) units indistribution systems. For the sake of clarity, and without loss of generality, the objective function consid-ered is to minimize total electrical power losses, although the problem can be easily configured as multi-objective (other objective functions can be considered at the same time), where the optimal location ofDG systems, along with their sizes, are simultaneously obtained. The optimal DG site and size problem ismodeled as a mixed integer nonlinear programming problem. Evolutionary methods are used byresearchers to solve this problem because of their independence from type of the objective functionand constraints. Recently, a new evolutionary method called Teaching–Learning Based Optimization(TLBO) algorithm has been presented, which is modified and used in this paper to find the best sites toconnect DG systems in a distribution network, choosing among a large number of potential combinations.A comparison between the proposed algorithm and a brute force method is performed. Besides this, it hasalso been carried out a comparison using several results available in other articles published by othersauthors. Numerical results for two test distribution systems have been presented in order to show theeffectiveness of the proposed approach.

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1. Introduction

Distributed Generation (DG) can be defined as electricity gener-ation by limited size generators connected to the distribution net-work or near the loads fed. During recent years, several factorshave been responsible for the appearance of DG in electric distribu-tions systems. Among them are environmental concerns to reduceemissions of greenhouse gases, depletion of fossil fuels, advancesin generation technologies, as well as the current global trend ofderegulation of the electricity market which implies the need formore flexible electric systems.

Research has shown that installation of DG sources in the powerdistribution system could lead to achieve many benefits, some ofwhich are voltage profile improvement, reduced lines losses,increased security for critical loads, grid reinforcement, reductionin the on-peak operation cost, etc. In order to optimize these ben-efits, it is essential to determine the optimal sizes of DG units andtheir best locations in distribution systems, otherwise, it could leadto adverse effects such as increased power losses. The reason for

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this is that DG affects the power flow in distribution network.However, the task of finding the optimal sizes and sites of DG unitsin distribution system is not easy.

Fortunately, many methods have been reported to solve thisproblem. A critical review of different methodologies used in solv-ing this optimization problem was published in [1], where authorspropose a classification according to the employed solution meth-odology. In summary, existing approaches could be grouped intothree different categories: classical optimization [2–4], analyticalapproaches [5–7] and the meta-heuristics [8–13]. Since then (year2010), a large number of articles have been published on this sub-ject and examples of this are Refs. [14–21]. This fact indicates thatthis topic remains an interesting line of research.

In the classical optimization approaches, among others, are in-cluded techniques such as the Optimal Power Flow (OPF), whichis able to optimize highly complex problems with many variables,although limited by the high dimensionality of power systems [2];linear programming, whose methodology is easy to implement, butis usually very difficult to reduce the models into a set of linearequations [3]; or the Lagrange multipliers, which also becomes lessefficient when the number of elements increases [4].

In relation to analytical approaches, Wang and Nehrir [5] wereprimarily concerned with finding the optimal locations of DG butfailed to optimize size. Acharya et al. [6] proposed an analytical

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66 J.A. Martín García, A.J. Gil Mena / Electrical Power and Energy Systems 50 (2013) 65–75

expression to calculate the optimal size, and an effective method-ology to identify the corresponding optimal location for DG place-ment based on an approximate loss formula. In addition, thismethodology is compared with the loss sensitivity factor method.The analytical procedure used by Gozel and Hocaoglu [7] is fasterand more accurate than previous analytical methods, since the for-mer does not make use of admittance, impedance or Jacobianmatrices; however, it is only suitable for radial systems.

Due to the high dimension of the possible solutions and thenonlinear nature of this problem, meta-heuristic techniques havecome to be the most widely used way to solve it. Among thesetechniques there are many optimization algorithms inspired bynature. To mention but a few: In [8] a methodology based on TabuSearch (TS) is presented for finding the optimal location of DG unitsso as to minimize power losses. Ref. [9] adopts the Genetic Algo-rithm (GA) approach for optimal DG allocation and sizing in distri-bution systems. In [10] a method is proposed that uses a ParticleSwarm Optimization (PSO) to search a large range of combinations.Ref. [11] presents a procedure using Ant Colony Optimization(ACO) for distributed generation sources allocation and sizing indistribution systems, etc. On the other hand, many researchershave considered the combination of two optimization techniquestogether for obtaining a better solution. So, in [12] is implementeda technique based on a GA and an e-constrained method, and a newhybrid algorithm of GA and TS is proposed in [13] to avoid the ma-jor drawbacks of the classical simple GA.

Nowadays, these techniques are still used for trying to enhancethe solutions of this problem, and many proposals continue to ap-pear which are based on all the above approaches (classical optimi-zation [14], analytical [15] and meta-heuristic [16,17]). Researchcontinues in order to improve the existing algorithms, and thisenhancement is done by hybridizing the existing algorithms orby modifying the existing algorithms. New hybrid methods areintroduced in [18,19]. Ref. [18] employs discrete PSO and OPF toovercome the optimal DG placement and sizing in distribution sys-tems, while in [19] a combination of GA and PSO is used for thisproblem. Also, new GA-based modified approaches are still pre-sented in [20,21]. More recent examples related to this topic canbe found in [22–24].

The main limitation of these meta-heuristic techniques is thedifficulty in determining the optimal controlling parameters. Thus,they generally provide a near optimal solution for a problem with alarge number of variables, and change in the selection of the algo-rithm parameters changes the effectiveness of it. This difficulty forthe selection of parameters can increase if hybridization or modi-fications are carried out.

Recently, has been presented a new evolutionary method calledTeaching–Learning Based Optimization (TLBO) algorithm [25]. It isbased on the effect of the influence of a teacher on the output oflearners in a class. TLBO method has the major advantage of notrequiring any parameter of the algorithm for its operation withthe exception of the population size and maximum number of iter-ations. Furthermore the algorithm is easily implemented and re-quires less computational memory when compared with all theabove mentioned algorithms.

Later, a Modified Teaching–Learning Based Optimization(MTLBO) algorithm was presented [26]. This new MTLBO methodis obtained by modifying the TLBO algorithm to improve the per-formance for global search. Basically, TLBO algorithm can be di-vided into two phases: ‘Teacher Phase’ and ‘Learner Phase’. InMTLBO algorithm the proposed modification uses mutation opera-tions in a similar way to Differential Evolution (DE) algorithm, suchthat now the process of optimization is simulated with anotheradditional phase: ‘Mutation Phase’.

In Ref. [27] a new modification to TLBO procedure was pro-posed, in which the algorithm uses a crossover technique, forimproving the algorithm performance for global search too.

Finally, in an attempt to improve the modification process moreeffectively, a new modification to TLBO process is proposed in [28].This proposed modification gets use of two mutation operations aswell as two crossover operations to enhance the ability of the algo-rithm for both local and global search exploration adequately,although it may have the disadvantage of increasing the difficultyfor the selection of the parameters of the algorithm. Besides Refs.[26–28], other references which relate to the use of TLBO algorithmin power system problems are [29,30].

In this paper, a method which employs the MTLBO algorithm[26] is proposed to determine the optimal placement and size ofDG units in distribution systems, but it has been adapted in orderto solve optimization problems in discrete search space (TLBOalgorithm was introduced for the problems with continuous vari-ables). This evolutionary method is used to solve this problemdue to its independence from the type of the objective functionand constraints, and because of the magnitude of the problemand its combinatorial nature. Besides, the algorithm is easy toimplement and requires only adjustment of the following constantparameters to find the global solution: the population size, themaximum number of iterations and the crossover rate. For the sakeof clarity, and without loss of generality, the objective functionconsidered is to minimize total electrical power losses, althoughthe problem can be easily configured as multi-objective (otherobjective functions can be considered at the same time), wherethe optimal location of DG systems, along with their sizes, aresimultaneously obtained choosing among a large number of poten-tial combinations. The results obtained on two test distributionsystems suggest that the proposed approach can effectively ensurethe power loss minimization, and that the MTLBO algorithm is effi-cient in finding the optimal solution.

To conclude, the contents of this paper are briefly outlined be-low. In Section 2, the description and formulation of the problemare presented. In Section 3, the MTLBO algorithm is described.The procedure step by step for the application of the MTLBO algo-rithm to the problem is given in Section 4. Numerical results areexplained in Section 5 and finally, in Section 6 the final conclusionsare presented.

2. Problem formulation

The problem to solve is to determine the optimal location andsize of a given number of DG units, Nu, in a known distribution sys-tem. As mentioned above, for clarity and without loss of generality,this optimization problem is formulated in this work with the aimof minimizing of the real power losses. Other objectives such as thetotal cost of DG units or the production costs [18] are not consid-ered here, but they could be included within the same methodol-ogy used.

The losses in a system are dependent on the system operatingconditions and they are given by:

PL ¼XNb

i¼1

XNb

j¼1

½aijðPiPj þ Q iQ jÞ þ bijðQiPj � PiQ jÞ� ð1Þ

In Eq. (1) which is popularly known as ‘exact loss’ formula [6],subscripts i and j vary from 1 to the total number of buses, Nb; Pand Q are net real power and reactive power injection on eachbus respectively; and aij and bij can be calculated by Eqs. (2).

aij ¼rij

V iVjcosðdi � djÞ ð2aÞ

J.A. Martín García, A.J. Gil Mena / Electrical Power and Energy Systems 50 (2013) 65–75 67

bij ¼rij

ViVjsinðdi � djÞ ð2bÞ

where rij is the real part of the element at row i and column j of theZbus matrix, and V and d are the voltage and load angle at corre-sponding buses.

However, in this paper a simplified loss formula has been used.The objective function (the real power losses) has been expressedas:

PL ¼XNb

i¼1

XNb

j¼1

rijðPiPj þ Q iQjÞ ð3Þ

Eq. (3) is obtained from Eq. (1) matching V = 1 and d = 0 for allbuses (flat voltage profile), and they are equivalent when loss coef-ficients, a and b, are calculated in the case of a DG penetration rateof 100% in the distribution system, that is, when there is a DG unitat each bus of sizing equal to the value of the load at that bus.Numerical results show that the accuracy gained by updating aand b is small and negligible [6], so these coefficients have beencalculated in the aforementioned case (flat voltage profile) insteadof in the base case.

The above problem is solved using a MTLBO algorithm. Notethat in order to evaluate the objective function given by (3) it isnecessary to know the location and size of DG units. The first setof variables (location) is coded for each candidate, in the MTLBOalgorithm. However, the size of DG units is calculated by an analyt-ical expression proposed, as described below.

For a given set of buses where the new DG units will be placed,the optimal capacity for these can be found using a generalizationof previous work of Achayra et al. [6], who determined the optimalsize of the DG to place in a single bus, so that losses are minimized.Following the same methodology, if it is known the Nu buses whereto place the Nu DG units then the sizes of these units are deter-mined knowing that at the point of minimum losses, the rate ofchange of losses with respect to injected powers is zero. Thus, bydifferentiating Eq. (3) with respect to each of the powers injectedon the Nu buses, Pi with i = 1, 2, . . . ,Nu, and equating to zero it ispossible to obtain the following matrix equation (linear systemof equations):

r11 r12 � � � r1Nu

r21 r22 � � � r2Nu

..

. ... . .

. ...

rNu1 rNu2 � � � rNuNu

0BBBB@

1CCCCA

P1

P2

..

.

PNu

0BBBB@

1CCCCA ¼ �

XNb

i¼Nuþ1

r1iPi

XNb

i¼Nuþ1

r2iPi

..

.

XNb

i¼Nuþ1

r3iPi

0BBBBBBBBBBBBBB@

1CCCCCCCCCCCCCCA

ð4Þ

In Eq. (4) the system matrix is a sub-matrix of the Rbus matrixwhich is defined as the real part of the Zbus matrix, and it is as-sumed that the buses where the DG units have to be placed arenumbered from 1 to Nu.

It can be seen that it is necessary to solve a linear system ofequations to determine the optimal sizes of the DG units, sincethe unknowns of the system are the net real power injected atthe Nu nodes given, Pi with i = 1, 2, . . . ,Nu which are the differencesbetween the real power generated, PGi, and the real power con-sumed, PCi, at each bus, so that it follows that:

PGi ¼ Pi þ PCi i ¼ 1; 2; :::;Nu ð5Þ

where PGi is the real power injection from DG unit placed at bus i,and PCi is the load demand at node i.

In this way, given a set candidate nodes to place the Nu DG units(which represents an individual in the MTLBO algorithm), it can be

calculated optimal sizes of these DG units using Eqs. (4) and (5),and then it is possible to evaluate the objective function given byEq. (3) which indicates the goodness of that individual.

The proposed mathematical modeling, a priori, does not takeinto account operational constraints such as the current limits onthe branches and the voltage modules on system buses. Therefore,it might be thought the found solutions could violate such con-strains. However, an improvement in the radial system losses leadsto improve the voltage profile and, hence, to decrease the currentsflowing through the lines. Nevertheless, the feasibility of the solu-tion found (constraints) is checked, but only once at the end of thealgorithm when performing a load flow to calculate the exactlosses.

3. Modified Teaching–Learning-Based Optimization (MTLBO)algorithm

In this section, the basic philosophy of the Modified Teaching–Learning Based Optimization (MTLBO) method is explained.

Teaching–Learning Based Optimization (TLBO) algorithm is anefficient optimization method which was first introduced by Raoet al. [25,31–33]. Like other evolutionary optimization techniques,TLBO is an algorithm inspired from the nature and works on the ef-fect of the influence of a teacher on the results achieved by learnersin a class. So, TLBO also uses a population of solutions to proceed tothe global solution. The population is considered as a group oflearners or a class of students. In the optimization algorithms, anindividual of the population consists of different variables thatmust be obtained. In TLBO, the different variables are analogousto the different subjects (i.e. courses) offered to learners and thestudent score is analogous to the ‘fitness’, as in other optimizationtechniques based on the population.

A teacher tries to disseminate knowledge among the learners,aims increase the level of knowledge of the class and helps stu-dents to get good marks or grades, according to his capability. Infact, a good teacher is one who brings his students up to his levelin term of knowledge. From this stage on, the learners are in needwith a new teacher, of superior quality than themselves. As theteacher is considered as the most knowledgeable person in thesociety, who shares his knowledge with the learners, so, the bestsolution (the best individual of the population) acts as a teacherat that iteration. But it is necessary to take into account that stu-dents will gain knowledge according to the quality of teachingdelivered by the teacher and the quality of the students presentin class. Besides, students also learn from interaction betweenthemselves (group of discussions, presentations, formal communi-cations, etc.), which also helps in their outcomes.

Then, based on the above explanation, the process of TLBO issimulated into two phases. The first phase or ‘Teacher Phase’ con-sists of learning from the teacher, and the second phase or ‘LearnerPhase’ means learning through the interaction among learners.After generation of the initial population and evaluation of theobjective function for each individual separately, the Teacher Phaseand the Learner Phase are expressed as follows:

Teacher Phase: The teacher will try to change the mean score ofthe class (population) toward his position, but in practice this isnot possible and a teacher can only move the mean of a class upto some extent depending on many factors. So, this phase followsa random process in which for each individual or position a newposition is generated given by:

Xnew;D ¼ Xold;D þ rðXteacher;D � TFMDÞ ð6Þ

In above equation subscript D indicates the number of subjectsor courses, Xold,D is the old individual, when this still had to learnfrom his teacher for increasing his level of knowledge, and consists

Yes

No

BEGINOptimization problem has been defined

Initialize the optimization parameters:Pn, Gn, Nu,ULi and LLi with i = 1, 2, …, Nu

Generate randomly the initial population

Ensure outcomes are within the feasibleregion and calculate the objectivefunction value of all individuals

Teacher phase:Modify the population based on Eq. (6)

and discretize it

Accept each new individual if it givesa better function value than the original


Learner phase:Modify the population based on Eq. (7)

and discretize it



Mutation phase:Modify the population based onEqs. (8) and (9), and discretize it



Present the final values of the solutionEND

Is the terminationcriterion satisfied?

Fig. 1. Flow diagram of the MTLBO algorithm.


in a vector (1 x D) in dimension which contains his outcomes foreach particular subject or course, r is a random number in therange [0,1], Xteacher,D is the best individual of the population in thisiteration who will try to change the mean of the class (population)toward his position, TF is a teaching factor, and MD is a vector (1 xD) in dimension which contains the mean or average values of theclass outcomes for each particular subject or course. In Eq. (6) thevalue of TF can be either 1 or 2 which is again a heuristic step and itis decided randomly with equal probability. The new individualXnew,D is accepted if he is better than the old individual.

Learner Phase: In order to simulate the increase of knowledge ofeach learner because of the interaction randomly with other stu-dents, the below equation is applied to all learners, so that, a stu-dent learns something new if the other student has moreknowledge than him.

Xnew;i ¼ Xold;i þ riðXj � XkÞ ð7Þ

In Eq. (7) subscript i varies from 1 to the total number of indi-viduals, Xold,i is the old individual, when he still had not learnedfrom his interaction with other students, ri is a random numberin the range [0,1], and Xj and Xk are two students randomly se-lected with j – k and where Xj gives a better objective function va-lue than Xk. The individual Xnew,i is accepted if he is better than theold individual Xold,i.

On the other hand, a Modified Teaching–Learning Based Opti-mization (MTLBO) algorithm was presented later by Hosseinpouret al. [26]. The proposed modification uses mutation operationssimilar to Differential Evolution (DE) algorithm. Mutation is a co-gent strategy to diversify the population in order to improve theperformance or ability of the algorithm for global search ade-quately, and to avoid premature convergence to local optima ofthe objective function. Therefore, now the process of optimizationis simulated with another additional phase. This new phase or‘Mutation Phase’ is described below:

Mutation Phase: This phase can also be understood as a modifi-cation of the previous phase or Learner Phase. The process which isshown in Eqs. (8) and (9) should be repeated for all students suchthat the population would be updated completely in this phase. So,in each iteration (from i = 1 to the total number of individuals) anew mutant vector or modified student is generated as:

Xmut ¼ Xrand1 þ rðXrand2 � Xrand3Þ ð8Þ

In Eq. (8) Xrand1, Xrand2 and Xrand3 are three students randomly se-lected for the ith learner, Xi, from the population, such thatrand1 – rand2 – rand3 – i, and r (the mutation scaling factor) isa random number in the range [0,1]. As r is randomly generatedin each iteration of the algorithm, then this parameter is notadjustable in the proposed algorithm.

And then the following implementation is used:

Xnew ¼Xmut if r1 P r2Xi otherwise

�ð9Þ

In Eq. (9) r1 and r2 are two random numbers in the range [0,1],Xi is the ith individual or student from the population and Xmut is amutated vector which has been generated for Xi. With regard to thecrossover rate used in the algorithm, according to Eq. (9), theexpression Xnew = Xmut if r1 P r2 is equivalent to use a crossoverrate equal to 0.5, that is, Xnew = Xmut if r1 P 0.5.

Finally, the modified individual Xnew is accepted if he is betterthan the original individual Xi.

Based on the above Eqs. (6)–(9), a mathematical model can beprepared for the optimization of a non-linear continuous function,but if all individuals are discretized by rounding to the nearestinteger after each phase, such that they are positive integer

numbers, then it can be applied to the optimization of a discretefunction.

S

1 92 3 4 5 6 7 8 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

28 29 30 31 32 33 34 35

36 37 38 39 40 41 42 43 44 45 46

53 54 55 56 57 58 59 60 61 62 63 64 65

47 48 49 50

51 52

66 67

68 69

Fig. 2. 69-Bus test distribution system considered as a case study.


4. Application of the MTLBO algorithm to the problem

This section describes the procedural steps for the implementa-tion of the algorithm to the problem described above. The flow-chart of the MTLBO algorithm is given in Fig. 1.

The encoding of the solutions is a very important aspect to takeinto account in the implementation of the algorithm. In this paper,an individual (a learner), or possible solution (a set of candidatebuses), is represented by a vector (matrix-row) in which each ele-ment indicates the result obtained by the student in a subject, orthe bus where a DG unit is located. Then, the dimension of this vec-tor is the number of subjects that the learner must study, or thenumber of DG units that is necessary to place in the distributionsystem, Nu.

Step 1: Initialize the following optimization parameters: pop-ulation size or number of learners, Pn, number of generations,Gn, number of design variables or subjects (courses) offeredwhich coincides with the number of units to place in the dis-tribution system, Nu, and limits of design variables (upperand lower of each case), ULi and LLi with i = 1, 2, . . . ,Nu.Step 2: Generate the initial population. A random populationis created according to Pn and Nu. This population isexpressed as a matrix (Pn x Nu) in dimension which containsthe class outcomes for each particular subject or course, orfor this problem, the buses where the DG units are located,so they must be positive integer numbers. Each row repre-sents an individual of the population, and each column cor-responds to the bus where each DG unit is placed (themark for each subject), so the number of rows is equal tothe size of the population, while the number of columnsmust be equal to DG units. Note that the location of DG unitscan be restricted to a given set of buses using any selectioncriterion (e.g. those with greater demand, those further awayfrom the substation, etc.). In any case, in this step it must beensured that values of the design variables are within thefeasible region, that is, they not exceed the limit values (asit is not possible to place a GD unit in a bus that does not

belongs to the distribution system), and they are notrepeated for a given individual (as it is not possible to placeseveral GD units in a same node). In addition, to reduce thesearch space, the rows representing the individuals aresorted from the smallest value to the largest (placing twoDG units at the buses i and j is the same as locate them atthe buses j and i). Finally, the corresponding value of theobjective function is calculated for each individual. For eachindividual, first the optimal sizes of the DG units are calcu-lated using Eqs. (4) and (5), and then the objective functiongiven by Eq. (3) is evaluated.Step 3: Teacher phase. In this step the population is modifiedusing Eq. (6) to decide next positions using continuous vari-ables. Subsequently, all individuals are discretized by round-ing to the nearest integer, such that they are integers, in orderto solve this optimization problem in the discrete searchspace. Now, as in the previous step, it must be ensured thatthe population is within the feasible region (bounded bymaximum and minimum values, and without repetitions ofthe positions for the same individual). Similarly, the searchspace is reduced by ordering the rows of individuals fromthe smallest value to the largest, and the corresponding val-ues of the objective function are calculated as in Step 2.Finally, each new individual is accepted if it gives a betterfunction objective value than the old individual.Step 4: Learner or Student phase. Learner modification isexpressed as Eq. (7). As in Step 3, now the population is dis-cretized to integer numbers. Then, it must be ensured that allindividuals belong to the feasible region, their rows aresorted, and the corresponding values of the objective func-tion are evaluated. And finally, the individuals generated inthis phase are accepted only if they give a better functionvalue.Step 5: Mutation phase. The population is changed again inthis step using Eqs. (8) and (9). With the exception of this dif-ference, this phase is identical to the previous. Each new gen-erated individual is also accepted only if it gives a betterfunction value than the original individual.

54 6 7 8

10 11 15 16 17

9

29 47 53 55 56 58 59

3

2

12 13 14

18 19 20 21 22 23 24 25 26 27

28 60 61 6250 51 52 54 57

48 49

30 31 32 33 34 35 36 37

39 40 42 51 44 45 4638 41

108 110 111 113 114115 117 118100 106 107 109 112

101 102 116103 104 105

9263 64 76 88 89 90 91 93 94 95 96 97 98 99

65 66 70 71 72 73 74 75

67 68 69

79 80 8177 78 82 83

84 85 86 87

S

1

Fig. 3. 119-Bus test distribution system considered as a case study.


Step 6: Termination criterion. The process stops if the maxi-mum number of generations has been achieved, otherwisethe process must be repeated from Step 3. However, others

different termination criteria could be adopted (e.g. stop ifthe best solution found does not change after a predefinednumber of generations).

Table 1Optimal locations and sizes of DG units for 69-bus test system using brute force algorithm.

Simulation case Connection buses Power to connect (MW) Total capacity added (MW) Approximate losses (MW) Accurate losses (MW)

1 DG unit 61 1.819691 1.819691 0.079316 0.0833232 DG units 17 0.519705 2.251709 0.068898 0.071776

61 1.7320043 DG units 11 0.493830 2.544743 0.066977 0.069539

18 0.37838561 1.672528

5 DG units 11 0.492986 3.261656 0.064991 0.06748118 0.37838550 0.71794561 1.38306964 0.289271

7 DG units 9 0.336903 3.380480 0.064584 0.06697112 0.33427917 0.16767321 0.17623350 0.71730061 1.35882164 0.289271

Table 2Optimal locations and sizes of DG units for 119-bus test system using brute force algorithm.

Simulation case Connection buses Power to connect (MW) Total capacity added (MW) Approximate losses (MW) Accurate losses (MW)

1 DG unit 93 2.871431 2.871431 0.896151 1.0170572 DG units 93 2.871431 5.640866 0.731308 0.805934

115 2.7694353 DG units 58 2.835841 8.476707 0.618455 0.668013

93 2.871431115 2.769435

5 DG units 58 2.835841 11.620797 0.536289 0.57565570 2.07500484 1.66371096 2.276807

115 2.7694357 DG units – – – – –

Volta

ge (p

.u.)

Fig. 4.bus tes


It is seen from the above steps that no provision is made to han-dle the constraints in the problem. That is not necessary because itis always ensured that the candidate solutions (population) arewithin the feasible region. If any of the variables (integers) repre-senting an individual in the population exceeds a limit value, thenthat variable is simply equated to the limit value which has been

10 20 30 40 50 601 69690.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

0.99

1

Buses

Without DG

1 DG

7 DGs

Comparison of voltage profile without and with connected DGs on the 69-t system.

exceeded, and only it should be verified that all these variablesare different from each other.

5. Test results and discussion

In this section, two different test cases are discussed, to showthat the proposed methodology can be implemented in

10 20 30 40 50 60 70 80 90 100 1101 1180.86

0.88

0.9

0.92

0.94

0.96

0.98

1

Buses

Volta

ge (p

.u.)

Without DG

1 DG

7 DGs

Fig. 5. Comparison of voltage profile without and with connected DGs on the 119-bus test system.


distribution systems of various configuration and size. The firstsystem is the widely used 69-bus test system presented in [34].This test system is shown in Fig. 2, where the numbering ofbranches and nodes coincide with the numbering used in Ref.[6]. It is a 68-branches radial system with the total load demandof 3.8022 MW and 2.6946 MVAR. The second test system contains119 buses and 117 branches. It is a radial system with the totalload of 22.7097 MW and 17.0409 MVAR [35]. The nodes of this testsystem have been renumbered as depicted in Fig. 3, and tie-switches (switches normally open) have been removed in orderto preserve its radiality.

The commercial software MATLAB� has been used for imple-menting the proposed algorithm, which has been performed on a3.4-GHz PC with 4 GB RAM. In order to show the effectiveness ofthe proposed approach, a brute force method has been developedto carry out a comparison of the numerical results obtained. Thisbrute force algorithm has also been implemented in MATLAB,and it is used to define optimal capacities of DG units and optimalconnection nodes in the test systems considering several cases.Each case corresponds to a given number of DG units to be con-nected to a test distribution network. The cases considered repre-sent groups of 1, 2, 3, 5, and 7 DG units. Considering the 69-bus testsystem used, these cases represent search spaces of 68, 2278,50116, 10424128, and 969443904 possible combinations, respec-tively, where each possible combination represents a set of buseswhere the DG can be connected. On the other hand, the numberof possible combinations is 117, 6786, 260130, 167549733, and49594720968, in each case, respectively, for the 119-bus testsystem. Furthermore, the raised problem is very complex sincethe search space has many local minima. The brute force algorithmgoes over all possible combinations in each case and, for each com-bination; the optimal sizes of DG units are calculated using Eqs. (4)and (5) and the objective function is evaluated using Eq. (3). So, theoptimal location corresponds to the combination that is associatedwith the lowest value of the objective function. The optimal place-ments and capacities for the five simulated cases are shown in Ta-bles 1 and 2. It is seen in Table 2, where 119-bus test system isconsidered, that this brute force algorithm has not been evaluatedfor the case of 7 DG units, since the computational time for thatcase is extremely excessive (more than 5 months).

Table 3Results for proposed MTLBO algorithm after 100 runs on the 69-bus test system.

Simulationcase

Best solution reached (optimal) Optimal sorate (%)

Connectionbuses

Power to connect(MW)

Approx. losses(MW)

1 DG unit 61 1.819691 0.079316 97

2 DG units 17 0.519705 0.068898 9561 1.732004

3 DG units 11 0.49383018 0.378385 0.066977 9161 1.672528

5 DG units 11 0.49298618 0.37838550 0.717945 0.064991 9061 1.38306964 0.289271

7 DG units 9 0.33690312 0.33427917 0.16767321 0.176233 0.064584 9750 0.71730061 1.35882164 0.289271

In this work, DG units are modeled as PQ nodes. These units canbe classified into four types based on real and reactive power deliv-ering capability as follows: Type 1 (Active power supply only),Type 2 (Reactive power supply only), Type 3 (Active and reactivepower supply) and Type 4 (Active power supply and reactivepower consumption). DG units of Type 1 could be photovoltaic, mi-cro-turbines and fuel cells, which are interface to the grid by elec-tronic converters. Typical synchronous compensators are units ofType 2. All the units that use a synchronous machine as a generatorfall in Type 3, and those units with asynchronous generators (windfarms and mini hydraulics) belong to Type 4.

The power factor of DG units depends on operating conditionsand type of DG. When the power factor of DG is given, the optimalsize of DG units, located at each set of buses considered, can befound using Eq. (4). With the proposed methodology is possibleto handle the four different types of DG, however, in this work,the DG units are modeled as Type 1. So, all DG units consideredin this study were assumed to have a pre-specified unity powerfactor.

It can be seen in Tables 1 and 2 that the accurate and approxi-mate losses have been calculated. A standard Newton–Raphsonalgorithm based load flow routine is used to compute the accuratepower losses in each case. However, the approximate losses areevaluated using Eq. (3). Thus, accurate total power loss (objectivefunction) is equal to 0.2250 MW for 69-bus test system and1.2981 MW for 119-bus test system, without connected DG units,whereas approximate total power loss is equal to 0.1915 MWand 1.1026 MW, for both test systems, respectively. It can also beappreciated in Tables 1 and 2 that the total losses of the systemare reduced with the incorporation of DG units. Nevertheless, thedifference in the total losses decreases with increasing the numberof groups to be connected. Regarding the connection nodes of theDG sources in 69-bus test system, bus 61 appears in all cases. Notethat it is the bus with greater demand. Buses 11 and 18 appear intwo cases, for search of 3 and 5 DG units. Similarly, buses 64 and 50appear in two cases, for search of 5 and 7 DG systems. Finally, bus17 also appears in two cases, for search of 2 and 7 DG sources.Buses 9, 12, and 21 only appear for search of 7 DG units. In 119-bus test distribution network, bus 93 appears in three cases, forsearch of 1, 2, and 3 DG systems. Likewise, bus 115 also appears

lution Cpu time(s)

Worst solution reached

Connectionbuses


Approx. losses(MW)

0.3 9 2.665136 0.150900

0.7 61 1.686944 0.07171166 0.786768

18 0.396365 0.0672022.5 61 1.680111

66 0.430905

9 0.336903 0.06520712 0.366607

54.5 21 0.31157750 0.71730061 1.648092

11 0.22372112 0.24722321 0.311577 0.064617

392.2 50 0.71735451 0.21796461 1.36326464 0.289271

Table 4Results for proposed MTLBO algorithm after 100 runs on the 119-bus test system.

Simulationcase

Best solution reached (optimal) Optimal solutionrate (%)

Cpu time(s)

Worst solution reached

Connectionbuses


Approx. losses(MW)

Connectionbuses


Approx. losses(MW)

1 DG unit 93 2.871431 0.896151 95 0.5 58 2.835841 0.989766

2 DG units 93 2.871431 58 2.835841 0.783298115 2.769435 0.731308 90 1.2 93 2.871431

3 DG units 58 2.835841 58 2.835841 0.61845593 2.871431 0.618455 100 6.5 93 2.871431

115 2.769435 115 2.769435

5 DG units 58 2.835841 58 2.835841 0.53699070 2.075004 66 2.25714584 1.663710 0.536289 94 106.9 84 1.63585596 2.276807 95 2.335922

115 2.769435 115 2.769435

7 DG units 20 1.793079 20 1.793079 0.48277842 1.263298 42 1.26329858 2.717474 58 2.71747470 2.075004 0.482778 100 441.2 70 2.07500484 1.663710 84 1.66371096 2.276807 96 2.276807

115 2.769435 115 2.769435

0 5 10 15 20 25 30 35 40 45 500.06

0.07

0.08

0.09

0.1

0.11

0.12

0.13

0.14

0.15

0.16

0.17

Iterations

Obj

ectiv

e Fu

nctio

n (M

W)

1 DG

2 DGs

3 DGs5 DGs

7 DGs

Fig. 6. Convergence curves of the used algorithm on the 69-bus test system.

0 5 10 15 20 25 30 35 40 45 500.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

1.05

Iterations

Obj

ectiv

e Fu

nctio

n (M

W)

1 DG

2 DGs

3 DGs5 DGs

7 DGs

Fig. 7. Convergence curves of the used algorithm on the 119-bus test system.


in three cases, but this time for search of 2, 3, and 5 DG sources. Be-sides, bus 58 appears in two cases, for search of 5 and 7 DG units.Finally, buses 70, 84, and 96 only appear in one case, for search of 5DG systems.

As regards the impact of the connection of the DG units on thedistribution network, it should be noted that, as shown in Figs. 4and 5, DG connections cause a voltage rise in, at least, the neigh-boring nodes. This effect can be used to improve the voltage profileon long sections of distribution network.

Several simulations have been run to find the optimal locationbuses and the optimal sizes of DG sources for sets of 1, 2, 3, 5,and 7 DG units using the proposed MTLBO algorithm. Only thenodes with demand have been selected as possible sites for theconnection of DG systems in these simulations. In this way, only48 possible sites appear when 69-bus test network is considered.So, search spaces have been reduced to 48, 1128, 17296,1712304, and 73629072 combinations, respectively. However,search spaces have not been reduced when considering 119-bustest system. The population size and the number of generations

have been selected to guarantee the convergence of the algorithmto a satisfactory solution. So, the population sizes used to obtainthe connection buses and capacities for sets of 1, 2, 3, 5, and 7DG sources have been of 10, 20, 60, 600, and 2000, respectively.Moreover, the values of the number of generations used have beenof 10, 15, 25, 45, and 50, respectively. The values of these parame-ters have been the same for both test systems.

In Tables 3 and 4 are presented the results obtained after 100runs for each case on the two test networks, among these, the com-putational time of the proposed algorithm (cpu time) and the num-ber of times the optimal solution has been reached in% (optimalsolution rate). The optimal value of objective function is reachedby the proposed method with high optimal solution rate (MTLBOalgorithm converges to solutions of great quality) and with lowvalues of the parameters (number of particles and iterations).Increasing the values of these parameters increases the computa-tional cost. Figs. 6 and 7 show the convergence curves of MTLBOalgorithm where the mean value of the objective function versusiterations is plotted for different cases.

Table 5Comparison of the results between the proposed method and other existing methods.

Simulation case Test system Method Connection buses Power to connect (MW) Accurate losses (MW)

1 DG unit 69 bus Proposed approach 61 1.8197 0.083323Acharya et al. [6] 61 1.8078 0.083372Gozel et al. [7] 61 1.8078 0.083372Alrashidi et al. [17] 61 1.9042 0.083259

2 DG units 69 bus Proposed approach 17, 61 0.5197, 1.7320 0.071776Alrashidi et al. [17] 21, 61 0.3220, 1.5820 0.075325

3 DG units 69 bus Proposed approach 11,18,61 0.4938, 0.3784, 1.6725 0.069539Alrashidi et al. [17] 21, 61, 64 0.3240, 1.2780, 0.3010 0.074777


The computational cost of MTLBO algorithm is much lower thanthat required by the brute force algorithm which has been devel-oped in this paper and also uses Eq. (3). Specifically, to find theoptimal connection buses and sizes for groups of 7 DG units on69-bus test system, the runtime for MTLBO algorithm is 950 timeslower than that for the exact technique with exhaustive search.Moreover, in the same case, if 119-bus test system is considered,then the runtime for MTLBO algorithm is more than 30000 timeslower than that for brute force algorithm. Therefore, the use ofMTLBO algorithm in a planning program can be interesting.

Besides this, it has also been carried out a comparison usingseveral results available in other articles published by othersauthors. The obtained solutions by proposed method and theseothers methods are shown in Table 5.

It can be seen in Table 5 that the proposed MTLBO algorithmreaches the same or better results than other methods published.

6. Conclusion

A new approach has been presented in this paper in order todetermine the optimal location and size of a given number of DGunits for a known distribution system. This algorithm allows find-ing the best places to connect a number of DG units, among a largenumber of combinations, by optimizing an objective function. Theproposed approach is a discrete version of the MTLBO algorithm. Ithas been assessed using two different test systems, a 69-bus testsystem and another 119-bus test system. The simulation resultshave shown the good performance and effectiveness of theproposed method. Acceptable solutions have been reached withfew evaluations, therefore, computational costs are enough lowerthan that required for exhaustive search. These solutions have pro-duced equal or better results when compared with a number of ap-proaches available in the technical literature.

Acknowledgements

This work was supported by the Spanish Government (Ministryof Science and Innovation) under Grant ENE2010-19744/ALT.

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Optimal distributed generation location and size using a modified teaching–learning based optimization algorithm