harmony search algorithm for transmission network expansion planning

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Published in IET Generation, Transmission & DistributionReceived on 12th June 2009Revised on 18th January 2010doi: 10.1049/iet-gtd.2009.0611

ISSN 1751-8687

Harmony search algorithm for transmissionnetwork expansion planningA. Verma B.K. Panigrahi P.R. BijweDepartment of Electrical Engineering, Indian Institute of Technology, Delhi, Hauz Khas, New Delhi 110016, IndiaE-mail: [email protected]

Abstract: Transmission network expansion planning (TNEP) is a very important problem in power systems. It is amixed integer, non-linear, non-convex optimisation problem, which is very complex and computationallydemanding. Various meta-heuristic optimisation techniques have been tried out for this problem. However,scope for even better algorithms still remains. In view of this, a new technique known as harmony search ispresented here for TNEP with security constraints. This technique has been reported to be robust andcomputationally efficient compared to other meta-heuristic algorithms. Results for three sample test systemsare obtained and compared with those obtained with genetic algorithm and bacteria-foraging differentialevolution algorithm to verify the potential of the proposed algorithm.

1 IntroductionTransmission network expansion planning (TNEP) problem[1] deals with the least cost expansion of new lines such thatno overloads are produced during the planning horizon. Thisis a large scale, non-linear, mixed integer, non-convexoptimisation problem. The problem is very complex andcomputationally demanding because of large number ofoptions to be investigated and the discrete nature of theoptimisation variables. Further, the number of options to beanalysed increases exponentially with the size of thesystem, hence the problem is non-polynomial time-hard.Conventional optimisation techniques provide very successfulstrategies to obtain global optimum in simple and idealmodels. However, real-world engineering optimisationproblems like TNEP are very complex and difficult to solvewith these methods. In view of this, many new meta-heuristictechniques have been proposed for TNEP in last few yearsbecause of their ability to find global optimal solutions forsuch combinatorial problem. Some of them are discussed below.

A simulated annealing (SA) approach for TNEP is proposedin [2] for long-term TNEP. The SA approach is ageneralisation of Monte-Carlo method for examining theequations of states and frozen states of n-body system. Theconcept is based on the manner in which liquids freeze ormetals recrystallise in an annealing process. A parallel Tabusearch algorithm for TNEP is discussed in [3]. The proposed

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method is a third generation Tabu search procedure, whichincludes features of a variety of other approaches such asheuristic search, SA and genetic algorithms (GAs). A newvariant of Tabu search is presented in [4] for static TNEP(STNEP). The intensification and diversification phases aredesigned using medium and long-term memory concepts.Applications of GA have been proposed by many researchersin [5–7]. An improved GA is proposed for TNEP in [5].Some special features have been added to the basic GA toimprove its performance. The GA works on the set ofcandidate solutions known as population, and performs anumber of operations. These operators recombine theinformation contained in the individuals to create newsolutions (populations). A procedure based on SA approach isimplemented to improve the mutation mechanism. A GA-based approach for multistage and coordinated planning oftransmission expansions is presented in [6]. An efficient formof generation of initial population is used in the proposedapproach. A specialised GA is proposed in [7] for static andmultistage TNEP. The proposed GA has the followingspecial characteristic: (i) it uses fitness and unfitness functionsto identify the value of objective function and unfeasibility ofthe tested solution; (ii) applies efficient strategy of localimprovement for each individual tested and (iii) it substitutesonly one individual in the population for each iteration. Agreedy randomised adaptive search procedure (GRASP) forsolving TNEP is presented in [8]. GRASP is an expertiterative sampling technique that has two phases for each

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iteration. The first phase is construction phase that finds out thefeasible solution. The second phase is a local search procedurethat seeks for improvements on the construction phasesolution by local search. The application of a new discretemethod in particle swarm optimisation for TNEP has beendiscussed in [9].

A new technique known as harmony search (HS) used forsolving engineering optimisation problems was first presentedin [10]. The HS algorithm is based on the musical process ofsearching for a perfect state of harmony. The harmony inmusic is analogous to the optimisation solution vector, and themusicians improvisations are analogous to local and globalsearch schemes in optimisation techniques. Instead of agradient search, the HS algorithm uses a stochastic randomsearch [11] based on the harmony memory considering rate(HMCR) and the pitch adjustment rate (PAR), so thatderivative information is unnecessary. Compared to earliermeta-heuristic optimisation algorithms, the HS algorithmimposes fewer mathematical requirements and can be easilyadapted for various types of engineering optimisation problems.The HS algorithm has been very successful in wide variety ofoptimisation problems [12–17]. An improved version ofharmony search (IHS) is presented in [18] which employ anovel method for generating new solution vectors thatenhances the accuracy and convergence rate of the classical HS.

Despite the promise shown by meta-heuristic methods forTNEP, better techniques are still required. Hence, this paperpresents an application of IHS to TNEP with securityconstraints. The transmission expansion problem is a verycomplex problem. Hence, it is not possible to consider allaspects of TNEP in this paper. The objective in this paperis to investigate the potential of IHS algorithm for TNEP.Hence, a simple STNEP with security constraints based onDC model, is considered. The N 2 1 contingency analysisis used to ensure system security. However, the algorithm isgeneral enough to consider all other aspects of TNEP. Theplanners can use this technique for TNEP incorporatingthe issues relevant to the individual systems.

2 Transmission networkexpansion planningThe TNEP can be formulated as a mixed integer non-linearoptimisation problem. System security is an important aspectand must be considered in a TNEP. The N 2 1 contingencyanalysis looks at the system state after a single line outage. Acomprehensive model for TNEP with security constraints ispresented in [19], which is used as a base for formulatingTNEP with security constraints in this paper. The TNEPwith security constraints can be stated as follows

min v =∑l[V

cl nl (1)

s.t.

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Sf k + g = d (2)

f kl − gl (n

0l + nl )(Du

kl ) = 0,

for l [ 1, 2 . . . , nl and l = k (3)

f kl − gl (n

0l + nl − 1)(Duk

l ) = 0, for l = k (4)

|f kl | ≤ (n0

l + nl )f l , for l [ 1, 2 . . . , nl and l = k (5)

|f kl | ≤ (n0

l + nl − 1)f l , for l = k (6)

0 ≤ nl ≤ nl (7)

f k

land uk

l are unbounded, nl ≥ 0 and integer, forl [ 1, 2 . . . , nl and l = k, (nl + n0

l − 1) ≥ 0 and integerfor l = k, l [ V and k = 0, 1, . . ., NC, where k = 0represents the base case without any line outage

S is the branch-node incidence transposed matrix of thepower system, f k is the vector with elements fl

k, gl is thesusceptance of the circuit that can be added to lth right-of-way, nl is the number of circuits added in lth right-of-way,nl

0 is the number of circuits in the base case, Dulk is the

phase angle difference in lth right-of-way when kth line isout, fl

k is the total real power flow by the circuit in lthright-of-way when kth-line is out, f l is the maximumallowed real power flow in the circuit in lth right-of-way, nl

is the maximum number of circuits that can be added inlth right-of-way, V is the set of all right-of-ways, nl is thetotal number of lines in the circuit and NC is the numberof credible contingencies (taken as equal to nl in thepresent case).

The objective is to minimise the total investment cost ofthe new transmission lines to be constructed, satisfying theconstraint on real power flow in the lines of the network,for base case and N 2 1 contingency cases. Constraint (2)represents the power balance at each node. Constraints (3)and (4) are the real power flow equations in DC network.Constraints (5) and (6) represent the line real power flowconstraint. Constraint (7) represents the restriction on theconstruction of lines per corridor.

3 HS algorithmThe HS algorithm has recently been used for variety ofengineering optimisation problems. The HS algorithm isdeveloped based on an analogy with music improvisationprocess, where music players improvise the pitches of theirinstruments to obtain better harmony [10]. Musicalperformers seek to find pleasing harmony (a perfect state)as determined by aesthetic standard, just as theoptimisation process seeks to find a global optimumsolution. The pitch of each musical instrument determinesthe aesthetic quality, just as the objective function value isdetermined by the set of values assigned to each decisionvariable. The IHS algorithm proposed in [18] is used forsolving the STNEP problem. Brief outline of the methodis as follows.

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The steps involved in the procedure of HS are:

1. Initialise the problem and algorithm parameters.

2. Initialise the harmony memory.

3. Improvise a new harmony.

4. Update the harmony memory.

5. Check for the stopping criteria.

For better understanding, these steps are briefly describedin the following five subsections:

1. Initialise the problem and algorithm parameters: Specifythe optimisation problem as follows

minimise f (x) (8)

subject to xi [ Xi, i = 1, 2, . . . , N (9)

where f (x) is an objective function, x is set of decisionvariables xi, N is the number of decision variables and Xi

represents the possible range of values for each decisionvariables.

The HS algorithm parameters to be initialised are asfollows:

† harmony memory size (HMS): this indicates the numberof solution vectors in the harmony memory;

† HMCR;

† PAR;

† number of improvisations (NI) or stopping criteria.

2. Initialise the harmony memory: The harmony memory isinitialised with as many randomly generated vectors as theHMS

HM =

x11 x1

2 . . . x1N−1 x1

N

x21 x2

2. . . x2

N−1 x2N

..

. ... ..

. ... ..

.

xHMS−11 xHMS−1

2 . . . xHMS−1N−1 xHMS−1

N

xHMS1 xHMS

2 . . . xHMSN−1 xHMS

N

⎡⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎦

(10)

3. Improvise a new harmony: A new harmony vectorxt = (xt

1, xt2, . . . , xt

i ) is generated based on three rules: (i)memory consideration, (ii) pitch adjustment and (iii)


random selection

xti �

xti [ {x1

i , x2i . . . xHMS

i } with probability HMCR

xti [ Xi with probability(1−HMCR)

⎧⎨⎩

(11)

A HMCR of 0.90 indicates that the HS algorithm willchoose the decision variable from the stored values in theHM with 90% probability and from the entire range with(100–90%) probability. Every component chosen byharmony consideration is examined for pitch adjustmentbased on the following rule:

Pitch adjusting decision for xit is given as

xti �

Yes with probability PARNo with probability (1− PAR)

{(12)

The value of (1 2 PAR) sets the rate of doing nothing. Ifpitch adjustment decision for xi

t is yes, xit is modified as

follows

xti = xt

i + rand( )∗bw (13)

where bw is an arbitrary distance bandwidth and rand() is arandom number between 0 and 1.

The PAR and bw are adjusted as follows

PAR(gn) = PARmin +(PARmax − PARmin)

NI× gn (14)

where gn ¼ 1, 2, . . ., NI, PAR(gn) is the pitch adjusting ratefor generation or improvisation of gn, PARmin is theminimum pitch adjusting rate and PARmax is themaximum pitch adjusting rate.

To explore the search space, the control parameterbandwidth ‘bw’ is adjusted depending upon the variance ofthe population in each improvisation [20], and is given by

bw(gn) = var(X )

√(15)

4. Update the harmony memory: The new memory isjudged in terms of the objective function (fitness function)value and if the new memory is better than the previousmemory in the HM, then new harmony memory isincluded in the HM and the existing worst harmony isexcluded from the HM.

5. Check for stopping criteria: If maximum number ofimprovisations is reached, then stop, otherwise steps 3 and4 are repeated.

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3.1 HS for TNEP with security constraints

The f (x) represents the objective function represented by (1)for STNEP. x defines the set of candidate lines presenting a

solution to STNEP. Each element in x represents the right-of-way in which a candidate line is constructed. The range ofeach variable defined by Xi indicates the list of available right-of-ways. If two lines are added in a particular right-of-way,

Figure 1 Flowchart IHS for TNEP

6 IET Gener. Transm. Distrib., 2010, Vol. 4, Iss. 6, pp. 663–673The Institution of Engineering and Technology 2010 doi: 10.1049/iet-gtd.2009.0611

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then two elements with same number (indicating number ofthe right-of-way) will come in the vector x.

3.1.1 Fitness function evaluation: To check theworth of a vector (solution) in the harmony memory,fitness function is evaluated using the following equation

f =∑

l

cl nl +W1

∑NC

k=0

∑ol

(abs(f kl )− f l )+W2(nl − �n

l)

(16)

Here ol represents the set of overloaded lines.

The objective of STNEP is to find the set of transmissionlines to be constructed such that the cost of expansion plan isminimum and no overloads are produced during the planninghorizon. Hence, the first term in (16) indicates the totalinvestment cost of a transmission expansion plan. Thesecond term is added to the objective function for the realpower flow constraint violations in the base case, andN 2 1 contingency cases. The third term is added to theobjective function if maximum number of circuits that canbe added in lth right-of-way exceeds the maximum limit.W1 and W2 are constants. The second and third terms areadded to the fitness function only in case of violations.

A stepwise flowchart of IHS algorithm for TNEP (Fig. 1)is given in Section 3.2 as follows.

3.2 Flowchart of HS algorithm

3.3 Handling of unconnected buses in thesystem

3.3.1 Handling of unconnected generators/loads:In cases where the unconnected generators are present in thenetwork, the total available candidate lines are divided into‘total number of unconnected generators+1’ parts (sayNGU+ 1). The initial 1, 2, . . ., NGU parts define theavailable candidate lines that can be directly connected togenerator 1, 2, . . ., NGU, respectively. The last part or(NGU+ 1)th part defines all other available candidate linesfor the given network (this includes indirect connections tothe generators also).

Before calculating the fitness function given by (16) usingDC load flow, the harmony vector is also divided intoNGU+ 1 parts

x = [x1:x2: · · · :xNGU:xNGU+1]

where x1 = [x11, x2

1, . . . , xgen1 ] indicates the set of decision

variables that define the candidate lines directly connectedto generator 1 for a given topology; xNGU ¼ [x1

NGU, x2NGU,

. . ., xgenNGUNGU ] indicates the set of decision variables that

define the candidate lines directly connected to generatorNGU for a given topology and xNGU+1 ¼ [x1

NGU+1,


x2NGU+1, . . ., xgenNGU+1

NGU+1 ] indicates the set of decisionvariables that define all other candidate lines connected tothe network for a given topology (this includes indirectconnections to the generators also).

It must be noted that this division is carried out onlybefore evaluating the fitness function using DC load flowto ensure the connectivity of unconnected generators. Forall the other meta-heuristic operations like improvisation,and so on the harmony vector is a single vector only, likein the case where unconnected generators/loads are notpresent.

Note: Unconnected loads are also treated as unconnectedgenerators.

3.3.2 Handling of unconnected interconnectingbuses: In case of the candidate lines, which are connectedthrough an unconnected interconnecting bus, for example,lines ni2j and nj2k are connected through an unconnectedinterconnecting bus j (Fig. 2), a single line from bus jcannot be connected until and unless at least one line existsbetween bus j and other two buses, i and k.

These types of cases are handled by imposing an additionalconstraint on such candidate paths considered for TNEP.The constraint can be defined as if ni2j . 0, thennecessarily nj2k . 0; that is a line cannot be added in i andj without adding a line between j and k. This constraint is

Figure 2 Candidate lines connected through anunconnected interconnecting bus

Figure 3 IEEE 24 bus system [22]

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imposed till at least one such connection exists between busesi 2 j and k. After that possibility of addition of only one line,ni2j or nj2k can be considered.

The connectivity of the generators/loads and unconnectedinterconnecting buses are checked before the evaluation

of fitness function, and if the network is found unconnectedfor certain topology, fitness function is not actuallyevaluated and the ‘f ’ in (16) is assigned a very high value(penalty). So, with subsequent improvisations such typeof topologies are omitted and we approach towards thesolution.

Table 1 Results for TNEP with security constraints for generation plan G1 –G4

Plan TNEP with IHS TNEP with BF-DEA TNEP with GA

G1 n7 – 8 ¼ 3, n1 – 5 ¼ 2, n1 – 2 ¼ 1,n3 – 24 ¼ 2, n4 – 9 ¼ 1, n6 – 10 ¼ 2,

n9 – 12 ¼ 1, n10 – 11 ¼ 1,n11 – 14 ¼ 1, n12 – 13 ¼ 1,n14 – 16 ¼ 2, n15 – 24 ¼ 1,n15 – 21 ¼ 2, n15 – 16 ¼ 1,

n19 – 22 ¼ 1

n7 – 8 ¼ 3, n6 – 10 ¼ 2, n1 – 5 ¼ 2,n15 – 24 ¼ 2, n15 – 21 ¼ 2,

n15 – 16 ¼ 1, n4 – 9 ¼ 1, n2 – 6 ¼ 1,n3 – 24 ¼ 2, n14 – 16 ¼ 2,n10 – 11 ¼ 1, n11 – 13 ¼ 1,

n19 – 22 ¼ 1

n7 – 8 ¼ 3, n6 – 10 ¼ 2, n1 – 5 ¼ 2,n1 – 2 ¼ 1, n15 – 24 ¼ 2, n15 – 21 ¼ 2,n15 – 16 ¼ 1, n4 – 9 ¼ 1, n2 – 6 ¼ 1,

n3 – 24 ¼ 2, n14 – 16 ¼ 2,n10 – 11 ¼ 1, n11 – 13 ¼ 1,

n19 – 22 ¼ 1

total numberof lines

22 21 22

investmentcost [106 US$]

964 975 978

G2 n7 – 8 ¼ 2, n1 – 5 ¼ 1, n2 – 4 ¼ 1,n3 – 24 ¼ 2, n3 – 9 ¼ 1, n6 – 10 ¼ 2,

n10 – 12 ¼ 2, n12 – 13 ¼ 1,n14 – 16 ¼ 2, n15 – 24 ¼ 2,n15 – 21 ¼ 1, n16 – 17 ¼ 2,n17 – 18 ¼ 2, n21 – 22 ¼ 1

n7 – 8 ¼ 2, n6 – 10 ¼ 2, n1 – 5 ¼ 1,n15 – 24 ¼ 2, n15 – 21 ¼ 2,n15 – 16 ¼ 1, n3 – 24 ¼ 2,n14 – 16 ¼ 2, n10 – 12 ¼ 2,

n19 – 22 ¼ 1, n3 – 9 ¼ 1, n2 – 4 ¼ 1,n12 – 13 ¼ 1

n7 – 8 ¼ 2, n6 – 10 ¼ 2, n1 – 5 ¼ 1,n1 – 2 ¼ 1, n15 – 24 ¼ 2, n15 – 21 ¼ 2,

n15 – 16 ¼ 1, n3 – 24 ¼ 2,n14 – 16 ¼ 2, n10 – 11 ¼ 1,

n10 – 12 ¼ 1, n19 – 22 ¼ 1, n3 – 9 ¼ 1,n2 – 4 ¼ 1, n11 – 13 ¼ 1


22 20 21


942 974 977

G3 n1 – 5 ¼ 2, n3 – 9 ¼ 2, n3 – 24 ¼ 1,n4 – 9 ¼ 1, n6 – 10 ¼ 2, n7 – 8 ¼ 3,

n9 – 12 ¼ 1, n10 – 12 ¼ 1,n12 – 23 ¼ 1, n13 – 14 ¼ 1,n14 – 23 ¼ 1, n15 – 21 ¼ 1,n20 – 23 ¼ 1, n21 – 22 ¼ 1

n7 – 8 ¼ 3, n6 – 10 ¼ 2, n1 – 5 ¼ 1,n1 – 2 ¼ 1, n15 – 21 ¼ 1, n20 – 23 ¼ 1,n1 – 3 ¼ 1, n3 – 9 ¼ 1, n14 – 23 ¼ 1,

n13 – 14 ¼ 1, n9 – 12 ¼ 1,n12 – 23 ¼ 1, n3 – 24 ¼ 1,

n10 – 12 ¼ 1, n21 – 22 ¼ 1, n2 – 4 ¼ 1,n10 – 11 ¼ 1

n7 – 8 ¼ 3, n6 – 10 ¼ 3, n1 – 5 ¼ 2,n15 – 21 ¼ 1, n20 – 23 ¼ 1,n3 – 24 ¼ 1, n16 – 19 ¼ 1,n14 – 23 ¼ 2, n15 – 24 ¼ 1,n9 – 12 ¼ 1, n10 – 12 ¼ 1,

n12 – 13 ¼ 1, n19 – 22 ¼ 1, n4 – 9 ¼ 1


19 20 20


837 898 903

G4 n7 – 8 ¼ 3, n6 – 10 ¼ 2, n15 – 24 ¼ 2,n3 – 24 ¼ 2, n12 – 13 ¼ 1, n2 – 4 ¼ 1,n1 – 5 ¼ 1, n1 – 2 ¼ 1, n14 – 23 ¼ 2,

n10 – 12 ¼ 1, n15 – 21 ¼ 1,n21 – 22 ¼ 1, n10 – 11 ¼ 1

n7 – 8 ¼ 3, n6 – 10 ¼ 2, n15 – 24 ¼ 2,n3 – 24 ¼ 2, n12 – 13 ¼ 1, n2 – 4 ¼ 1,n1 – 5 ¼ 1, n1 – 2 ¼ 1, n14 – 23 ¼ 2,

n10 – 12 ¼ 1, n15 – 21 ¼ 1,n21 – 22 ¼ 1, n10 – 11 ¼ 1

n7 – 8 ¼ 3, n6 – 10 ¼ 2, n1 – 2 ¼ 2,n1 – 5 ¼ 1, n15 – 21 ¼ 1, n10 – 11 ¼ 1,n3 – 9 ¼ 2, n19 – 22 ¼ 1, n14 – 16 ¼ 2,n12 – 23 ¼ 1, n4 – 9 ¼ 1, n20 – 23 ¼ 1,

n10 – 12 ¼ 1, n9 – 12 ¼ 1,n11 – 13 ¼ 1


19 19 21


882 882 899


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4 ResultsThe proposed algorithm has been tested for three standard testsystems, IEEE 24 bus system, south Brazilian 46 bus systemand 93 bus Colombian system. The comparison of results ispresented for IEEE 24 bus system and 46 bus southBrazilian system, with the one obtained with basic binaryGA and bacteria foraging-differential evaluation algorithm(BF-DEA) to confirm the potential of the proposedapproach. The GA and BF-DEA were implemented tocompare the results. The detailed results are shown only forthe case of TNEP with security constraints; however, theresults for TNEP without security constraints are also available.

4.1 IEEE 24 bus system

This system consists of 24 buses, 41 candidate circuits and8550 MW of total demand. A maximum of three lines percorridor can be added. The initial network can be found in[21] and also shown in Fig. 3. The electrical data andgeneration/load data have been taken for plans G1–G4 of [22].

4.1.1 TNEP without security constraints: The finalsolution obtained with IHS for TNEP without securityconstraints results in an investment cost of US$ 390 × 106

for plan G1, US$ 336 × 106 for plan G2, US$ 214 × 106

for plan G3 and US$ 292 × 106 for plan G4. The resultsobtained with BF-DEA and GA matches exactly with theones obtained with IHS. The cost of expansion planobtained for generation plan G1 is 11% and for generationplan G4 1.8% lesser, with three methods applied, than theone given in [22].

4.1.2 TNEP with security constraints: The finalsolutions obtained with IHS, BF-DEA and GA for planG1–G4, for TNEP with security constraints are presentedin Table 1 and the number of fitness function evaluationsrequired are presented in Table 2.

The percentage reductions in cost and number of fitnessfunction evaluations required by IHS as compared to BF-DEA and GA for plans G1–G4 are given in Table 3.

It can be observed from Table 3 that the IHS providesbetter results (lower cost), in all cases except one in whichthe solutions are same. It can also be seen that IHSrequires much lesser number of fitness function evaluationsfor the solutions of all cases as compared to other twomethods. The range of reduction is being 82–97%.

A rigorous study of algorithm parameters done by varyingthem for the above system between the permissible ranges isgiven in Tables 4 and 5. The parametric study presented inTables 4 and 5 are done for plan G3. Initially, the effect ofvariations of HMCR is observed by keeping HMSconstant. Then with the best value of HMCR obtained,the effect of variations of HMS is observed.

It can be observed from Tables 4 and 5 that HMS ¼ 50and HMCR ¼ 0.98 provided the best results in terms ofnumber of fitness function evaluations and standarddeviations, whereas according to the parametric studycarried out for TNEP without security constraints for thesame plan (G3), (not included in this paper), the

Table 3 Percentage reduction in cost and number of fitness function evaluation

Plan % Reduction in cost of expansion plan obtained with IHS % Reduction in number of fitness functionevaluation required with IHS

Compared to BF-DEA Compared to GA Compared to BF-DEA Compared to GA

G1 1.13 1.43 89.78 93.92

G2 3.29 3.58 97.23 93.47

G3 6.79 7.31 89.45 97.88

G4 no change 1.89 82.95 91.81

Table 2 Comparison of number of fitness function required by three algorithms, for plans G1 –G4

Plan Cost [106 US$] Number of fitness function evaluation requiredto obtain the final solution

IHS BF-DEA GA IHS BF-DEA GA

G1 964 975 978 118 280 1 157 900 1 945 090

G2 942 974 977 20 450 737 300 313 167

G3 837 898 903 58 400 553 500 2 753 166

G4 882 882 899 220 500 1 293 100 2 690 833

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Table 4 Effect of HMCR variation on performance of IHS algorithm for TNEP (with HMS ¼ 50)

HMCR 0.99 0.98 0.95 0.9

cost of expansion plan [US$ 106] 837 837 837 1026

standard deviation 38.31 25.82 76.77 79.69

number of fitness function evaluations 61 250 58 400 113 950 114 300

Table 5 Effect of HMS variation on performance of IHS algorithm for TNEP (with HMCR ¼ 0.98)

HMS 25 50 75 100

cost of expansion plan [US$ 106] 837 837 837 925

standard deviation 82.88 25.82 19.50 85.0815


parameters HMS ¼ 35 and HMCR ¼ 0.98 provided thebest results. Hence, these settings are system dependent.For experiments with large number of systems and withdifferent levels of complexity, it has been observed that forTNEP problem, HMCR value of 0.98 provides betterresults. However, HMS needs to be adjusted dependingupon the size of the system and level of complexity ofTNEP problem.

The final parameters used for IHS, BF-DEA and GA forthe results shown in Table 1 are as follows.

For IHS: HMS ¼ 50; HMCR ¼ 0.98; maximum PAR(PARmax) ¼ 0.99; minimum PAR (PARmin) ¼ 0.1;number of improvisations (NI) or stopping criteria are 2500.

For BF-DEA: swimming length (Ns) ¼ 1, cross-overrate (CR) ¼ 0.75, scale factor for DE-type mutationF ¼ 1.0, number of bacteria S ¼ 100 and number ofiterations ¼ 3000.

For GA: population size ¼ 750, CR ¼ 0.8, mutationrate ¼ 0.01, number of generations ¼ 4000.

4.2 South Brazilian 46 bus system

This system has 46 buses, 79 right-of-ways on which circuitscan be constructed, and a total demand of 6880 MW.The maximum number of lines which can be added toeach corridor is six. This is a realistic system representing agood test case for the proposed algorithm. The relevantdata can be found in [23] and the network configuration isshown in Fig. 4.

4.2.1 TNEP without security constraints: Thefinal solution obtained by HS for TNEP without securityconstraints results in an investment cost ofUS$154 420 000, which matches exactly with the onereported in [24].

0The Institution of Engineering and Technology 2010

4.2.2 TNEP with security constraints: The finalsolution obtained by HS for TNEP with securityconstraints results in an investment cost of US$337 809 000 with the addition of following 34 lines

n31−32 = 1, n28−30 = 1, n26−29 = 3, n29−30 = 2,

n17−19 = 1, n27−38 = 1, n46−11 = 3, n11−5 = 6,

n12−14 = 1, n42−43 = 3, n23−24 = 1, n20−21 = 1,

n24−25 = 3, n25−21 = 1, n25−32 = 1, n26−27 = 1,

n32−43 = 1,n2−5 = 1, n19−21 = 1, n20−21 = 1

Figure 4 46 Bus south Brazilian system [23]

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Table 7 Effect of HMS variation on performance of IHS algorithm for TNEP (with HMCR ¼ 0.98)

HMS 25 50 75 100

cost of expansion plan [US$ 106] 340.679 337.809 337.809 337.809

standard deviation 42 18 19.50 17.809


Table 6 Effect of HMCR variation on performance of IHS algorithm for TNEP (with HMS ¼ 50)

HMCR 0.99 0.98 0.95 0.9

cost of expansion plan [US$ 106] 337.809 337.809 337.809 340.679

standard deviation 21.39 18 15.55 48


However, the TNEP with security constraints for abovesystem is also obtained with BF-DEA and basic binary GA.

The final solution obtained by BF-DEA results in aninvestment cost of US $361 863 000 with the addition offollowing 36 lines

n31−32 = 1, n28−30 = 1, n26−29 = 3, n29−30 = 2, n46−6 = 2,

n6−5 = 4, n24−25 = 5, n25−32 = 1, n25−21 = 1, n32−43 = 1,

n42−43 = 3, n12−14 = 1, n37−39 = 2, n2−5 = 3, n20−21 = 2,

n19−21 = 1, n17−19 = 1, n27−38 = 1

The final solution obtained by GA results in an investmentcost of US$ 432 350 000 with the addition of following 43lines

n26−29 = 3, n29−30 = 2, n28−30 = 1, n31−32 = 1,

n46−11 = 1, n11−5 = 3, n24−25 = 4, n25−32 = 1,

n25−21 = 1, n46−6 = 2 , n6−5 = 2, n12−14 = 3, n2−5 = 2,

n16−28 = 1, n26−27 = 1, n1−7 = 2, n19−21 = 1, n9−14 = 1,

n27−36 = 1, n28−43 = 1, n42−43 = 3, n37−39 = 2, n34−35 = 1,

n20−21 = 2, n2−4 = 1

The rigorous algorithm parametric study for the above systemis given in Tables 6 and 7.

It can be observed from Tables 6 and 7 that HMCR ¼ 0.98and HMS ¼ 50 provided best results in terms of numberof fitness function evaluations and standard deviation forthe above system. Hence, the algorithm parameters used forthis system are as follows: HMS ¼ 50; HMCR ¼ 0.98;PARmax¼ 0.99;PARmin¼ 0.1;NIor stoppingcriteria ¼ 2500.

The parameters used for BF-DEA and GA forcomparison are as follows.


For BF-DEA: swimming length (Ns) ¼ 1, CR ¼ 0.75, scalefactor for DE-type mutation F ¼ 1.0, number of bacteriaS ¼ 100 and number of iterations ¼ 4000.

For GA: number of population ¼ 1000, CR ¼ 0.8, mutationrate ¼ 0.01 and number of generations ¼ 4000.

The number of fitness function evaluations required by allthe methods for the results shown above is given in Table 8.It can be observed from the above results that the HSalgorithm provides much better (low cost) results withlesser number of fitness function evaluations as comparedto basic binary GA and BF-DEA. The cost of expansionplan obtained with IHS is 6.65 and 21.86% lesser than theone obtained with BF-DEA and GA, respectively. Thenumber of fitness function evaluations required by IHS is19.46% and 91.101% lesser as compared to that requiredby BF-DEA and GA, respectively.

4.3 93 Bus Colombian system

This system has 93 buses and 155 circuits, and a maximumof five circuits can be added to each corridor. The initialconfiguration for the above system can be seen in [6] andalso presented in Fig. 5. After taking results for twosystems, IHS has been proved to be the best for all the

Table 8 Cost and number of fitness functions required bydifferent methods for 46 bus south Brazilian system

Method Number of fitnessfunction

evaluations

Cost of expansionplan [US$ 106]

HS 2.40 × 105 337.809

BF-DEA 2.98 × 105 361.863

basic binary GA 2.67 × 106 432.350

671


67

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case studies (TNEP with/without security constraints).Hence, for this system the results are obtained withIHS only.

4.3.1 TNEP without security constraints: Thesolution to TNEP without security constraints results in aninvestment cost of US$ 562 417 000 with the addition offollowing 19 lines

n43−88 = 2, n19−82 = 2, n30−65 = 1, n68−86 = 1,

n50−54 = 1, n62−73 = 1, n55−62 = 1, n30−72 = 1,

n55−57 = 1, n82−85 = 1, n72−73 = 1, n55−84 = 1,

n27−64 = 1, n56−57 = 1, n54−56 = 1, n15−18 = 1, n27−29 = 1

4.3.2 TNEP with security constraints: The solution toTNEP with security constraints results in an investment costof US$ 1 194 448 000 with the addition of following 46 lines.

n43−88= 3, n1−59= 1, n2−83= 1, n3−71= 1, n7−90= 1,

n9−83= 1, n13−14= 1, n15−18= 1, n18−21= 1, n18−58= 1

n19−82= 3, n27−80= 1, n27−64= 2, n28−29= 1,

n29−31= 1, n30−65= 1, n30−72= 1, n45−81= 2,

n45−50= 1, n48−63= 1, n55−62= 2, n55−57= 1,

n55−84= 2, n57−81= 3, n57−84= 1, n59−67= 1,

n62−73= 2, n67−68= 1, n68−86= 2, n72−73= 2

n82−85= 2, n83−85= 1

Figure 5 93 Bus Colombian system [6]


IETdo

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5 ConclusionsIn this paper, application of IHS algorithm has beeninvestigated for TNEP with security constraints. Thecomparison of results is made with the ones obtained withBF-DEA and basic binary GA. Results for three samplesystems confirms the potential of the proposed approach.The proposed IHS algorithm provides better (low cost)solution in all the cases with lesser number of fitnessfunction evaluations.

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