Electr Eng (2014) 96:335–345DOI 10.1007/s00202-014-0302-5

ORIGINAL PAPER

Optimal power flow for distribution networks using gravitationalsearch algorithm

Jordan Radosavljevic · Miroljub Jevtic ·Nebojša Arsic · Dardan Klimenta

Received: 13 July 2013 / Accepted: 20 June 2014 / Published online: 12 July 2014© Springer-Verlag Berlin Heidelberg 2014

Abstract This paper presents a gravitational search algo-rithm (GSA)-based approach to solve the optimal power flow(OPF) problem in a distribution network with distributed gen-eration (DG) units. The OPF problem is formulated as a non-linear optimization problem with equality and inequality con-straints, where optimal control settings in case of fuel costminimization of DG units, power loss minimization in thedistribution network, and finally simultaneous minimizationof the fuel cost and power loss are obtained. The proposedapproach is tested on an 11-node test system and on a mod-ified IEEE 34-node test system. Simulation results obtainedfrom the proposed GSA approach are compared with thatobtained using a genetic algorithm approach. The resultsshow the effectiveness and robustness of the proposed GSAapproach.

Keywords Optimal power flow · Distributed generation ·Distribution network · Gravitational search algorithm

1 Introduction

Recently, due to the growth of distributed generation (DG)penetration, the nature of distribution network is altered froma passive network to an active one and by a distributioncompany market which is operated by distribution systemoperator (DSO) [1]. DSO is responsible for operation of thedistribution network in an effective manner, maintaining therequired reliability of electricity supply and the quality ofsupply as well as coordinating the operation of the coordi-nated HV grid in cooperation with the transmission system

J. Radosavljevic (B) · M. Jevtic · N. Arsic · D. KlimentaFaculty of Technical Sciences, University of Priština in KosovskaMitrovica, Kneza Miloša 7, 38220 Kosovska Mitrovica, Serbiae-mail: jordan.radosavljevic@pr.ac.rs

operator. If properly planned and controlled, the DG unitsmay offer the improved voltage profile and power lossesreduction of the distribution network, better economics anda reduced dependence on the local utility [2]. Safe, reliableand economic operation of the active distribution network canbe achieved only by efficient coordination of the DG unitsoperation, voltage regulation (voltage regulators, ULTC) andreactive power compensation (VAR compensators). Accord-ingly, the optimal power flow (OPF) calculation is one of theimportant tools for DSO to perform this duty.

Main objective of the OPF problem for a distribution net-work is to minimize the fuel cost of DG units [3,4], voltageprofile improvement [5,6], var/volt coordination [7] and min-imization of power losses [8–10], though optimal settings ofthe control variables while at the same time satisfying variousdistribution system operating constraints.

In its most general formulation, the OPF is a nonlin-ear, nonconvex, large-scale, static optimization problem withboth continuous and discrete control variables. In recent year,many population-based optimization techniques have beenused to solve complex constrained optimization problems.Some of the population-based methods such as genetic algo-rithm, differential evolution algorithm, gravitational searchalgorithm (GSA) have been successfully applied to find theoptimal solution for OPF problem in transmission networks.Lai et al. [11] develop an improved genetic algorithm (GA)to solve the OPF problem and show that the method is betterin searching for global optimal point than the gradient-basedconventional method. An enhanced GA for solving OPF ispresented in [12]. The main advantage of the GA solutionof the OPF problem is its modeling flexibility: nonconvexunit cost functions, discrete control variables, and complex,nonlinear constraints can be easily modeled. The main dis-advantage of GAs is that they are stochastic algorithms andthe solution they provide to the OPF problem is not guaran-

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teed to be optimum. In [13], differential evolution algorithmis applied to solve OPF problems with different objectivefunctions that reflect fuel cost minimization, voltage profileimprovement and voltage stability enhancement. The resultspresented in [14,15] proved the robustness and superiority ofthe GSA approach to solve the OPF problem in transmissionnetworks.

In this paper, a novel GSA-based approach is proposed tosolve the OPF problem in distribution networks. The OPF isformulated as a nonlinear optimization problem with equalityand inequality constraints. The following objective functionsare considered: (Case 1) fuel cost minimization for DG units,(Case 2) power loss minimization and (Case 3) simultaneousminimization of the fuel costs and power losses. The pro-posed approach is tested on an 11-node test system and ona modified IEEE 34-node test system. The results are com-pared to those obtained using a reference method that is theGA approach. The effectiveness and robustness of the pro-posed GSA approach are demonstrated.

2 Problem formulation

The goal of the OPF is to minimize a selected objective func-tion via optimal settings of the control variables, subjectedto various equality and inequality constraints. Generally, theOPF problem can be formulated as follows [11–16]:

min F (x, u) (1)

Subject to : g (x, u) = 0 (2)

h (x, u) ≤ 0 (3)

u ∈ U (4)

where F is objective function to be minimized, x and u arevectors of dependent and control variables, respectively.

For distribution network with DG units, the vector ofdependent variables (x) consisting of:

• The active power bought/sold from/to the utility grid Pgr.This is active power that is delivered through the root nodeof the distribution network.

• Load node voltage, including DG units which are modeledas PQ nodes, VL.

• The reactive power outputs of the DG units which are mod-eled as PV nodes QDG.

• Branch flow Sl.

Therefore, x can be expressed as:

x = [Pgr, VL1 . . . VLNL, QDG1 . . . QDGNPV, Sl1 . . . SlN

]T

(5)

where NL, NPV and N are the number of load nodes, numberof PV nodes, and number of nodes (namely branches), in thedistribution network, respectively.

The vector of control variables (u) consisting of:

• The active power output of the DG units with nonrenewableenergy sources PDG.

• Root node voltage V0.• The terminal voltage at PV nodes VPV.• Transformer (voltage regulation) tap setting t .• The output of shunt VAR compensators QC.

Therefore, the vector of control variables can be expressedas:

u = [PDG1 . . . PDGNNR, V0, VPV1 . . . VPVNPV,

t1 . . . tNT, QC1 . . . QCNC]T (6)

where NNR, NPV, NT and NC are the number of the non-renewable DG units, number of PV nodes (that is DG units,modeled as PV nodes), number of regulating transformers,and number of VAR compensators, respectively.

2.1 Objective function

The objective function can take different forms. Several caseshas been considered in this paper.

Case 1: Fuel cost minimization

F = fgr(Pgr

) +NNR∑

i=1

fi (PDGi ) (7)

where fgr(Pgr) is the cost characteristics for theactive power bought/sold from/to the utility grid andfi (PDGi ) is the fuel cost characteristics of the i th DGunit.In this paper, the cost characteristics are defined asquadratic cost function of active power, as follow:

fi (Pi ) = ai + bi Pi + ci P2i

($/h

)(8)

where Pi stands for Pgr or PDGi . ai , bi and ci arethe appropriate cost coefficients.

Case 2: Power losses minimization

F =N∑

i=1

Plossi (9)

where Plossi is the power loss in branch i .

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Case 3: Simultaneous minimization of the fuel cost andpower losses:

F = fgr(Pgr

)+NNR∑

i=1

fi (PDGi )+wPloss

N∑

i=1

Plossi (10)

where wPloss is the weighting factor for powerlosses.

2.2 Constraints

The equality constraints (2) represent typical power balanceand power flow equations. The power balance equation indistribution network with DG units with renewable and non-renewable energy sources can be expressed as follows:

NNR∑

i=1

PDGi + Pgr =NL∑

i=1

PLi +N∑

i=1

Plossi −NR∑

i=1

PDGi (11)

where NR are the number of DG units with renewable energysources.

The backward/forward sweep power flow equations aregiven in [17,18].

The inequality constraints (3) are the functional operatingconstraints that contain load bus voltage magnitude limits,DG units reactive power capabilities and branch flow limits:

V mini ≤ Vi ≤ V max

i i = 1, . . . , NL (12)

QminDGi ≤ QDGi ≤ Qmax

DGi i = 1, . . . , NPV (13)

Si ≤ Smaxi , i = 1, . . . , N (14)

Constraints (4) define the feasibility region of the problemcontrol variables such as DG unit active power output limits,root node voltage limits, PV bus voltage limits, transformertap setting limits and shunt VAR compensation limits:

PminDGi ≤ PDGi ≤ Pmax

DGi i = 1, . . . , NNR (15)

V min0 ≤ V0 ≤ V max

0 (16)

V minDGi ≤ VDGi ≤ V max

DGi i = 1, . . . , NPV (17)

tmini ≤ ti ≤ tmax

i i = 1, . . . , NT (18)

QminCi ≤ QCi ≤ Qmax

Ci i = 1, . . . , NC (19)

The inequality constraints of dependent variables containload bus voltage magnitude, DG units reactive power outputand branch loading that are added to the objective functionas quadratic penalty terms [16].

The new expanded objective function to be minimizedbecomes [11,14].

Fp = F +λV

NL∑

i=1

(Vi −V lim

i

)2+λQDG

NPV∑

i=1

(QDGi −Qlim

DGi

)2

+λS

N∑

i=1

(Si − Slim

i

)2(20)

Where λV , λQDG and λS are defined as penalty factors. xlim

is the limit value of the dependent variable x and given as:

x lim = xmax if x > xmax, and x lim = xmin if x < xmin (21)

3 Gravitational search algorithm

The gravitational search algorithm (GSA) is a newly sto-chastic search algorithm developed by Rashedi et al. [19].This algorithm has a great potential to be a break-thoughoptimization method [14]. GSA has been verified as havinghigh-quality performance in solving different optimizationproblems in literature [14,20–23].

In GSA, the search agents are a collection of masses whichinteract with each other based on the Newtonian gravity andthe laws of motion. In this algorithm, agents are considered asobjects and their performances are measured by their masses.All these objects attract each other by the gravity force, andthis force causes a global movement of all objects towards theobjects with heavier masses [19]. The position of the masscorresponds to the solution of the problem, and its gravita-tional and inertial masses are determined using a fitness func-tion. In other words, each mass presents a solution. The algo-rithm is navigated by properly adjusting the gravitational andinertial masses. By lapse of time, the masses will be attractedby the heaviest mass which it represents an optimum solutionin the search space.

The GSA could be considered as an isolated system ofmasses. It is like a small artificial world of masses obeyingthe Newtonian laws of gravitation and motion.

In a system with N agents (masses), the position of thei th agent is defined by:

Xi =(

x1i , . . . , xd

i , . . . , xni

)for i = 1, 2, . . . , N (22)

where n is the search space dimension of the problem and xdi

defines the position of the i th agent in the dth dimension.After evaluating the current population fitness, the mass

of each agent is calculated as follows [19,23]:

Mi (t) = mi (t)N∑

j=1m j (t)

(23)

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Table 1 Steps of GSA

Step 1 Search space identification. Initialize GSA parameters like N , T, G0 and α

Step 2 Initialization: generate random population of N agents and form the agent matrix X. The initial positions of each agent are randomlyselected between minimum and maximum values of the control variables (i.e. DG active power outputs, voltage magnitudes of PVnodes, tap settings of regulating transformers, reactive power outputs of shunt compensators and root node voltage). Each set in theagent matrix X represents a potential solution of the OPF problem

Step 3 Run the backward/forward sweep power flow program using each individual set of the agent matrix to determine the dependentvariables of (5) to check whether they satisfy the inequality constraints of (12)–(14). Calculate the fitness value for each member ofthe agent matrix X. In GSA the fitness values of agents are represented by their masses. In this paper, different objective functionsare considered. Their fitness values are calculated with the help of (7)–(10) and (20)

Step 4 Update the G(t) (32), best (t) (25), worst (t) (26) and Mi (t) (23) for i = 1, 2, . . . , N

Step 5 Calculation of the total force in different directions using (27)

Step 6 Calculation of acceleration and velocity of each agent using (29) and (30)

Step 7 Update each agent’s position using (31)

Step 8 Repeat Steps 3–7 until the stop criteria is reached. That is a predefined number of iteration, T

Step 9 Return best solution. Stop

Table 2 GSA control parameters

Population size N 50Maximum iteration number T 100

Initial gravitational constant G0 100

Constant α 20

where:

mi (t) = fiti (t) − worst (t)

best (t) − worst (t)(24)

where fiti (t) represents the fitness value of the agent i at timet . best(t) and worst(t) are the best and worst fitness of allagents, respectively and defined as follows (for a minimiza-tion problem):

best(t) = minj∈{1,...,N } fit j (t) (25)

worst(t) = maxj∈{1,...,N } fit j (t) (26)

According to Newton gravitation theory, the total forcethat acts on the i th agent in the dth dimension at t time isspecified as follows:

Fdi (t)=

∑

j∈K best, j �=i

rand j G(t)M j (t)×Mi (t)

Ri, j (t)+ε

(xd

j (t)−xdi (t)

)

(27)

where rand j is a random number in the interval [0, 1]. G(t) isgravitational constant at time t, Mi (t) and M j (t) are massesof agents i and j, ε is a small constant and Ri j (t) is theEuclidian distance between the two agents i and j given bythe following equation:

Fig. 1 Single line diagram of the 11-node test system

Ri j (t) = ∥∥Xi (t) , X j (t)∥∥

2 (28)

Kbest is the set of first K agents with the best fitness value andbiggest mass, which is a function of time, initialized to K0

at the beginning and decreased with time. In such a way, atthe beginning, all agents apply the force, and as time passes,Kbest is decreased linearly and at the end there will be justone agent applying force to the others.

By the law of motion, the acceleration of the i th agent, att time in the dth dimension, is given by following equation:

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Table 3 Line data and Loaddata for 11-node test system Send. node Rec. node Line parameters Load at Rec. node

R(�) X (�) B (S) PL (kW) QL (kVAr)

0 1 0.20900 0.11700 0.000116 1,262.4 256.2

0 2 0.20900 0.11700 0.000116 2,416.6 490.5

0 3 0.20900 0.11700 0.000116 612.7 124.4

1 4 0.41800 0.23400 0.000232 435.1 88.3

2 5 0.52250 0.29250 0.000290 1,148.0 233.0

3 6 0.41800 0.23400 0.000232 0 0

3 7 0.41800 0.23400 0.000232 575.4 116.8

4 8 0.83600 0.46800 0.000464 0 0

5 9 0.41800 0.23400 0.000232 1,188.2 241.2

7 10 0.41800 0.23400 0.000232 0 0

Table 4 DG units data for 11-node test system

DG unit Type Mode PDGnom (kW) QDG (kVAr) Cost coefficients

a ($/h) b ($/kWh) c ($/kW2h)

DG8 MT PQ 1,000 cos ϕ = 0.9 1 15 · 10−3 14 · 10−6

DG9 FC PV 1,800 −1,350 ÷ 1, 350 3 11 · 10−3 7.9 · 10−6

DG10 FC PQ 600 cos ϕ = 0.9 3 1.2 · 10−3 35 · 10−6

Market HV network Slack node – – 0 25 · 10−3 0

adi (t) = Fd

i (t)

Mi (t)(29)

The searching strategy on this notion can be defined tofind the next velocity and next position of an agent. Nextvelocity of an agent is defined as a function of its currentvelocity added to its current acceleration. Hence, the nextposition and next velocity of an agent can be computed asfollows:

vdi (t + 1) = randi × vd

i (t) + adi (t) (30)

xdi (t + 1) = xd

i (t) + vdi (t + 1) (31)

where randi is a uniform random variable in the interval [0,1]. This random number is utilized to give a randomized char-acteristic to the search. xd

i represents the position of agent iin dimension d, vd

i is the velocity and adi is the acceleration.

It must be pointed out that the gravitational constant G(t)is important in determining the performance of GSA. It isinitialized at the beginning and will be reduced with time tocontrol the search accuracy. In other words, the gravitationalconstant is a function of the initial value G0 and time t :

G(t) = G0e−α tT (32)

where α is a user specified constant, t the current iterationand T is the maximum iteration number. The parameters

of maximum iteration T , population size N , initial gravita-tional constant G0 and constant α control the performance ofGSA.

3.1 GSA implementation

Proposed GSA has been applied to solve the OPF problemfor distribution network. The control variables of the OPFproblem constitute the individual position of several massesthat represent a complete solution set. In a system with Nagents (masses), the position of the i th agent is definedby:

Xi =(

x1i , . . . , xd

i , . . . , xni

)for i = 1, 2, . . . , N

Several agents together form the agent matrix X

X =

⎡

⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

x11 , x2

1 , . . . , xd1 , . . . , xn

1

x12 , x2

2 , . . . , xd2 , . . . , xn

2...

x1i , x2

i , . . . , xdi , . . . , xn

i...

x1N , x2

N , . . . , xdN , . . . , xn

N

⎤

⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

Different steps to solve the OPF problem using GSA arelisted in Table 1.

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Table 5 The limits of the control variables for 11-node test system

Control variables Min Max

PDG8 (kW) 0 1,000

PDG9 (kW) 0 1,800

PDG10 (kW) 0 600

QC1 (kVAr) 0 500

QC2 (kVAr) 0 500

QC3 (kVAr) 0 500

VDG9 (p.u.) 0.98 1.03

V0 (p.u.) 0.97 1.03

The computer program was developed in MATLAB 2011bcomputing environment and run on a 2.20 Ghz, PC with3.0 GB RAM. The control parameters of GSA are shownin Table 2.

4 Simulation results

The proposed GSA solution of the OPF is evaluated usingan 11-node test system and a modified IEEE 34-node testsystem. To verify the simulation results and compare the per-formance of the proposed approach, the same OPF problemis solved using a GA, which is the most popular and mostwidely used heuristic optimization method. The MATLABrealization of the GA is applied, where population size is setto 50, maximum number of generation is set to 100, and otherparameters as the default parameters are used. Ten consecu-tive test runs have been performed for each case examined.The results shown are the best values obtained over these 10runs.

4.1 11-node test system

The 11-node test system is a MV distribution network con-sisting of three feeders shown in Fig. 1 [24]. Line data andload data of the distribution network are listed in Table 3. Itis supposed that three DG units are integrated in the distri-bution network. The DG units at node 8 consist of severalfuel cell stacks and DG units at nodes 9 and 10 consist ofmicroturbine units, whose characteristics are given in [3].We assume that the DG8 and DG10 operate in PQ mode at0.9 power factors (produce reactive power) in all operatingconditions, and DG9 is capable to control active power andvoltage magnitude independently, and therefore operate inPV mode. The DG units data, including the cost coefficients,are given in Table 4. The limits for the control variables aregiven in Table 5. The maximum and minimum values for volt-ages at load buses, including DG units which are modeled asPQ nodes, are 1.05 and 0.95 p.u., respectively.

The proposed GSA approach has been applied to solve theOPF problem for three different objective functions. Table 6shows the optimal settings of control variables and objectivefunction values.

Case 1: Fuel cost minimization As can be seen from theTable 6, the total fuel cost obtained by the proposed GSA isreduced to be 186.28 $/h compared to the base case (the con-trol variables are set to nominal values) fuel cost of 201.02$/h. It is clear that the control variables’ settings correspond-ing to fuel cost-based OPF result in a reduction of 7.33 % infuel cost, but insignificant reduction in power losses.

The comparison results are shown in Table 7. From theTable 7, it can be seen that the results obtained from theGSA are better than those obtained from the reference GA-based approach. Figure 2 shows the comparative convergenceprofiles of GSA and GA for Case 1. It may be observed from

Table 6 Optimal settings ofcontrol variables for 11-nodetest system

Control variableand obj. fun.values

Base case Case 1: Fuel costminimization

Case 2: Lossminimization

Case 3: Fuel cost andloss minimization

PDG8 (kW) 1,000 342.7 381.9 346.5

PDG9 (kW) 1,800 892.2 1,800.0 1,795.6

PDG10 (kW) 600 317.9 476.3 372.2

QC1 (kVAr) 500 287.8 72.1 209.9

QC2 (kVAr) 500 250.0 2.8 10.0

QC3 (kVAr) 500 369.0 42.5 35.7

VDG9 (p.u.) 1 0.9991 1.0005 1.0009

V0 (p.u.) 1 1.0039 1.0016 1.0015

Fuel cost ($/h) 201.0167 186.2800 193.3815 192.7196

Ploss (kW) 13.63 13.36 7.94 8.84

Pgr (kW) 4,252 6,099 4,988.1 5,132.9

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Fig. 2 Comparative convergence profiles of GSA and GA for Case 1 in the 11-node test system

Table 7 Comparison of the simulation results for Case 1 for 11-nodetest system

Methods Fuel cost ($/h) The averagevalue ofcomputationaltime (s)Min Average Max

GSA 186.2800 186.2901 186.3489 13.5958

GA 186.2823 186.3289 186.3601 17.8755

this figure that the GSA tends to find the optimal solutionfaster than the GA.

Case 2: Power loss minimization Minimization of powerloss as objective in OPF results in reduction of 41.75 % inlosses, and 3.8 % in fuel cost over the base case. In relationto Case 1, the power loss is reduced by 40.56 % but the fuelcost increased by 3.8 %.

The comparison results are shown in Table 8. From theresults in Table 8, it is clear that in GSA approach, Ploss is7.94 kW which is reduced to 12.46 % than that of GA algo-rithm. Also, the comparative convergence profiles of GSAand GA for Case 2 are shown in Fig. 3. According to thisfigure, GSA tends to find the optimal solution faster than GAand hence has a higher convergence rate.

Case 3: Fuel cost and power loss minimization Simul-taneous minimization of the fuel cost and power loss is amulti-objective OPF problem. The best compromise solutionobtained by the proposed GSA is 192.7196 $/h and 8.84 kW,which shows 4.13 % reduction in fuel cost and 35.14 % reduc-tion in power losses of base case values. The results obtainedfrom the proposed GSA technique are compared to those ofthe reference GA method. Table 9 reports the comparisonresults. Based on these results it is clear that the GSA hasbetter performance in comparison to the GA approach.

Table 8 Comparison of the simulation results for Case 2 for 11-nodetest system

Methods Power loss (kW) The averagevalue of com-putationaltime (s)Min Average Max

GSA 7.940 8.915 9.620 11.3548

GA 9.070 10.00 12.17 17.5018

4.2 IEEE 34-node test system

To evaluate the effectiveness and efficiency of the proposedGSA approach in solving larger distribution network, a mod-ified IEEE 34-node test system shown in Fig. 4 is considered.The original IEEE 34-node test system [25] is modified usingassumptions given in [26]. Three different DG units wereintroduced in the modified test system. Line data and loaddata of the test system are listed in Table 10. The DG unitsdata, including the cost coefficients, are given in Table 11.It has a total of eight control variables as follows: three DGunit active power outputs, two shunt VAR compensator reac-tive power outputs, two voltage regulation tap settings andone root node voltage magnitude. The limits for the controlvariables are given in Table 12. The maximum and minimumvalues for voltages at load buses, including DG units whichare modeled as PQ nodes, are 1.05 and 0.95 p.u., respectively.

Two optimization cases were considered. Case 1: fuel costminimization; Case 3: simultaneous minimization of the fuelcost and power losses. The test results obtained using GSAalong with GA for the two objective functions are presentedin Table 13.

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Fig. 3 Comparative convergence profiles of GSA and GA for Case 2 in the 11-node test system

Table 9 Comparison of the simulation results for Case 3 for 11-nodetest system

Methods Fuel cost ($/h) Power loss (kW) The averagevalue of com-putationaltime (s)

Average Average

GSA 192.8601 10.0370 11.6222

GA 192.8844 10.1910 18.2849

Case 1: Fuel cost minimization The total fuel cost obtainedby the proposed GSA is reduced to 41.20098 $/h comparedto the base case (the control variables are set to nominal val-ues) fuel cost of 50.4585 $/h. The control variables’ settingscorresponding to fuel cost-based OPF result in a reduction of

18.34 % in fuel cost and significant loss reduction of 31.82 %.It can be seen that the results obtained from the GSA are betterthan those obtained from the reference GA-based approach.The GSA tends to find the optimal solution faster thanthe GA.

Case 3: Fuel cost and power loss minimization Simul-taneous minimization of the fuel cost and power loss is amulti-objective OPF problem. The best compromise solutionobtained by the proposed GSA is 43.21132 $/h and 55.05 kW,which shows 14.36 % reduction in fuel cost and 45.65 %reduction in power losses of base case values. It can be seenthat the minimum fuel cost and power loss obtained by GSAare better than those obtained by reference GA. Addition-ally, the proposed GSA approach is more computationallyefficient than the GA approach.

Fig. 4 Single line diagram of the modified IEEE 34-node test system

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Table 10 Line data and Load data for modified IEEE 34-node test system

Send. node Rec. node Section parameters Load at Rec. node

R (�) X (�) PL (kW) QL (kVAr) Load model

800 802 0.5473 0.4072 27.5 14.5 PQ

802 806 0.367 0.273 27.5 14.5 PQ

806 808 6.8373 5.0864 16 8 I

808 812 7.9553 5.9181 0 0 PQ

812 814 6.3069 4.6919 0 0 PQ

814 850 0.0032 0.0016 0 0 PQ

850 816 0.0992 0.0494 171.5 88 PQ

816 824 3.2681 1.6265 44.5 22 I

824 828 0.2689 0.1338 5.5 2.5 PQ

828 830 6.5426 3.2562 48.5 21.5 Z

830 854 0.1664 0.0828 4 2 PQ

854 852 11.7889 5.8672 0 0 PQ

852 832 0.0032 0.0016 7.5 3.5 Z

832 858 1.5684 0.7806 25.5 13 PQ

858 834 1.8661 0.9288 89 45 Z

834 860 0.6466 0.3218 174 106 PQ

860 836 0.8578 0.4269 61 31.5 PQ

836 840 0.2753 0.137 47 31 I

836 862 0.0896 0.0446 28 14 PQ

834 842 0.0896 0.0446 4.5 2.5 PQ

842 844 0.4321 0.2151 432 329 Z

844 846 1.1651 0.5799 34 17 PQ

846 848 0.1696 0.0844 71.5 53.5 PQ

832 888 11.78 25.2864 0 0 PQ

888 890 80.26 59.7059 450 225 I

Table 11 DG units data for modified IEEE 34-node test system

Location Type Mode PDGnom (kW) QDG (kVAr) Cost coefficientsa ($/h) b ($/kWh) c ($/kW2h)

816 MT PQ 1,000 cos ϕ = 0.9 1 15 · 10−3 14 · 10−6

854 FC PV 900 cos ϕ = 0.9 3 11 · 10−3 7.9 · 10−6

888 FC PQ 600 cos ϕ = 0.9 3 1.2 · 10−3 35 · 10−6

800 Electric grid Slack node − – 0 25 · 10−3 0

Table 12 The limits of the control variables for modified IEEE 34 nodetest system

Control variables Min Max

PDG1 (kW) 0 1,000

PDG2 (kW) 0 900

PDG3 (kW) 0 600

QC1 (kVAr) 0 300

QC2 (kVAr) 0 450

tVR1 (p.u.) 0.90 1.10

tVR2 (p.u.) 0.90 1.10

V800 (p.u.) 0.97 1.05

5 Conclusion

The GSA-based approach has been proposed to solve theoptimal power flow problem in distribution networks withDG units. The OPF problem is formulated as a nonlinear opti-mization problem with the main operating constraints andcharacteristics of distribution network taking into accountalso the different DG units and load models. The proposedalgorithm has been tested by two test systems. This approachwas successfully applied to find the optimal settings of thecontrol variables in three optimization cases: fuel cost min-imization for DG units, power loss minimization in the dis-

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Table 13 Optimal settings ofcontrol variables forIEEE34-node test system

Control variable settings Base case Case 1: Fuel costminimization

Case 3: Fuel cost and lossminimization

GA GSA GA GSA

PDG1 (kW) 1,000 356.6 313.1 395.1 290.0

PDG2 (kW) 900 867.0 837.7 763.1 778.1

PDG3 (kW) 600 371.1 401.8 600.0 600.0

QC1 (kVAr) 300 147.6 8.3 86.4 237.5

QC2 (kVAr) 450 306.5 360.2 250.2 114.2

tVR1 (p.u.) 1 0.9504 0.9764 0.9638 0.9384

tVR2 (p.u.) 1 0.9643 0.9263 1.0029 1.0017

V800 (p.u.) 1 0.9765 0.9701 1.0082 0.9833

Fuel cost ($/h) 50.4585 41.20233 41.20098 43.22020 43.21132

Ploss (kW) 101.28 67.66 69.05 55.20 55.05

Computational time (s) 56.2087 47.9132 70.0320 52.3600

tribution network, and finally simultaneous minimization ofthe fuel cost and power loss. The simulation results demon-strate the effectiveness and robustness of the proposed GSAapproach to solve the OPF problem. Additionally, the resultsof the proposed GSA approach have been compared to thoseof a reference method that is the GA approach. Based on thiscomparison, it is clear that the GSA has better performanceover the reference GA approach, especially in the computa-tional efficiency. The proposed algorithm can act as a decisionsupporting tool for distribution system operators.

Acknowledgments This paper presents the part of results of theProject TR 33046 funded by the Ministry of Education, Science andTechnological Development of the Republic of Serbia.

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