Multi-objective optimization for optimal DG allocation and ...journal.esrgroups.org/jes/iceedt/icraesCD/ICRAES... · The DG placement and sizing problem is formulated as a multi-objective

Multi-objective optimization for optimal DGallocation and sizing in radial distribution systems

M. Mosbah, R. D. MohammediLaboratoire d’Analyse et de Commande des Systèmes d’Energie et Réseaux Electriques (LACoSERE),

University of Laghouat, Laghouat, [email protected]

Abstract— The current paper describes a multi-objectiveoptimization approach to determine the optimal placement andsizing of Distributed Generation (DG) units in radial distributionsystems. The main objectives are to minimize the active power lossand to improve the voltage stability. Non-dominated SortingGenetic Algorithms II (NSGA-II) is applied to determine the set ofPareto optimal solutions, which form a numerous set of non-dominated solutions and will be employed by the Decision Makerto decide on the best compromise solution. A detailed performanceanalysis is applied on 12, 33, 69 and 85 bus systems to illustrate theeffectiveness of the proposed methodology.

Keywords— Distributed generation, Radial distribution systems,NSGA-II, Multi-objective, Active power losses, Voltage stability.

I. INTRODUCTION

Distributed generation (DG) generally describes small-scale(1 kW to 50 MW) power generation that produce electricity ata site closer to customers than central generation plants [1].Recently, DG has gained many momentum in the power sectorbecause of its ability in power loss reduction, improvedreliability, low investment price, and most significantly, to takeadvantage of renewable energy resources. The installation ofDG units at non-optimal locations may lead to an increase insystem losses, meaning an increase in costs, and thus, havingan opposite effect to what is wanted [2].

It is seen from the literature review that many attempts weremade to resolve the optimization problem related to size andplacement of the DG. Most of the research related to size andplacement were aimed at minimizing the active power losses ofthe distribution system, energy losses and cost of generation.Less work has been done related to the voltage support andstability which is also a major concern of the power systems.

In reference [3], authors have used analytical approach foroptimum DG placement with an objective function ofminimization of system losses. Reference [4] has proposed anew analytical approach for DG placement and sizing indistribution systems based on a new power stability index (PSI)to improve voltage profile and minimize the burden of systemlosses. Reference [5] has developed a genetic-based algorithmto find the optimum size and placement of multiple DGs toreduce the active power losses and the power supplied by the

distribution network. A Cuckoo Search Algorithm (CSA) wasalso applied in references [6] and [7] for DG placement andsizing problem in order to reduce the active power losses andto enhance the network voltage profile. The Particle SwarmOptimization (PSO) technique was adopted in reference [8] tosearch for the optimal locations and sizes of DGs producing thebest loss reduction of distribution systems. In reference [9], anovel method based on PSO was proposed to evaluate theoptimal DG sizing and placement in order to minimize thepower losses in the distribution system considering voltagestability index. A combined genetic algorithm (GA)/(PSO) waspresented in [10] for optimal location and sizing of DG ondistribution systems. A new approach based on ModifiedTeaching-Learning Based Optimization (MTLBO) algorithmwas used in reference [11], which consists of two phases, i.e.teacher phase and learn phase. The objective was to find theoptimal placement and size of DG units in distribution systemsto minimize active power losses. In the same direction, thereference [12] has proposed the use of new optimizationtechnique called Artificial Bee Colony (ABC) algorithm to findout the optimum size of DGs, power factor, and location so asto reduce the system active losses. A number of papers havefocused on the use of EAs (evolutionary algorithms) tooptimize the DG placement and its sizing [13, 14]. The differentmethods with their merits and demerits can be seen in reference[15].

Within the above-mentioned literature, the DG placementand sizing problem was presented as a multi-objectiveoptimization problem. Nevertheless, the multi-objectiveproblem was converted to a single objective problem bysumming all the objectives (Weighted Sum Method) [7, 9, 10].This method requires multiple runs as many times as thenumber of desired optimal solutions. Furthermore, the valuesof weighting factors have a significant impact on the finalsolution.

Evolutionary techniques have been widely applied to multi-objective problems because of their global searching capabilityand the ability to obtain multiple Pareto-optimal solutions in asingle run. One of the most popular multi-objectiveevolutionary programming strategies, the NSGA-II is used in

this paper to solve the DG placement and sizing problem. Usingthe fundamentals of evolution, non-domination and solutiondiversity, the NSGA-II has the capability to obtain multiplePareto-optimal solutions in one run of the algorithm that arewell spread out over the Pareto frontier. This is clearly shownthrough a bi-objective optimization problem presented in thispaper.

II. PROBLEM FORMULATION

A. Objective FunctionsThe DG placement and sizing problem is formulated as a

multi-objective problem by considering minimization of activepower losses and enhancement of voltage stability as objectiveswhile satisfying system and unit constraints.

1) First objective function (Active power loss)In radial distribution networks, each receiving bus is fed by

only one sending bus. From Fig. 1, The line losses between thereceiving and sending end buses ( ),lossP i can be calculatedusing eq. 1.

Fig. 1. One line diagram of a two-bus system

2 2i i

loss i 2i

P QP ( i ) r

V

(1)

Therefore, the first objective function is calculated asfollows:

busN

1 loss lossi 2

f P P ( i )

(2)

where i iV is complex voltage at the ith bus; i ir jxis the impedance of the line connecting buses i 1 and i ; iP ,

iQ are the active (resp. reactive) power injections at the ithbus; busN is the number of buses.

2) Second objective function (Voltage stability)Voltage stability has been defined by the IEEE Power

System Engineering Committee in the following way [16]:“Voltage stability is the ability of a system to maintain voltageso that when load admittance is increased, load power willincrease and so that both power and voltage are controllable”.

The fast indicator of voltage stability (SI Index) proposed byChakravorty and Das [17] is selected as the objective for thevoltage stability enhancement.

From Fig. 1 i 1 i 1 i i i iV V I r jx (3)

i i i iV I P jQ (4)where I is the current amplitude and ‘*’ symbolizes the

complex conjugate operator.From Eqs. (3) and (4), we get

2 2 2 2 2i i i 1 i i i iV V V P Q r x 0 (5)

Roots of Equation (5) are real if

2 2 2 2 2i 1 i i i iV 4 P Q r x 0 (6)

From this, the voltage stability index for bus i iSI is derivedas

24i i 1 i i i i

2 2i i i i i 1

SI V 4 P x Q r

4 P r Q x V 0

(7)

Under the stable operation, the value of SI should be greaterthan zero for all buses, i.e. iSI busi 2 ,3 , , N 0 .When the value of SI becomes closer to one, all buses becomemore stable. The bus having the SI minimum value is the mostsensitive to voltage collapse. In the proposed algorithm, SIvalue is calculated for each bus in the network. For the bushaving the minimum value of SI, will be considered in thesecond fitness function.

2min

1f1 SI

(8)

where minSI is the minimum SI value of all the buses.

B. Constraints

1) Equality ConstraintsThe equality constraints are active/reactive power flow

equations as:

bus

bus

N

Gi Di i j ij ij j ij 1

N

Gi Di i j ij ij j ij 1

P P V V Y cos 0

,

Q Q V V Y sin 0

busi 1 , 2 , , N (9)where GiP and GiQ are the active (resp. reactive) power

generated at the ith bus; DiP and DiQ are the active (resp.reactive) load demand at the same bus; ijY and ij are theadmittance magnitude (resp. angle) of branch connecting bus iand j.

2) Inequality Constraints

a. Generation constraint

busN

DG Dii 1

0 size of DG( P ) P

(10)

b. Voltage constraintsmaxmin

i i i busV V V , i 2 ,3 , , N (11)

Sending End Receiving End Load

where miniV and max

iV are the minimum (resp.maximum) voltages of the ith bus.

c. Line thermal limitmax

li li busS S , i 2 ,3 , , N (12)

where liS is a line loading and maxliS is a maximum

permissible loading limit of branch incident to the ith node.

3) Constraints handling mechanismThe constraints are handled as follows:

a. Equality constraints

Generally, distribution feeders have a high R/X ratio inaddition to their radial configuration. These reasons make thatdistribution systems are ill-conditioned and thus traditionalmethod as Gauss-Seidel and Newton-Raphson are notappropriate for solving the load flow problem in most cases andmay often fail to converge. Based on this matter, for solving theload flow of distribution networks, the known backward-forward technique is used. The backward-forward sweeptechnique has been widely used to solve load flow in radialdistribution networks because it converges very fast andconsumes less computational memory [18].

b. Inequality ConstraintsThe voltage limits (11) and the line thermal limits (12) are

handled by introducing penalty functions, because they aresimple and easy to implement [19].

bus

line

N2Penalized lim

n n v i ii 1

N2lim

s Li Lii 1

F F K V V

K S S

(13)

where nF is nth objective function value, limiV and lim

LiS aredescribed as

max maxi i i

lim min mini i i i

maxmini i i i

V if V VV V if V V

V if V V V

(14)

max maxLi Li Li

lim min minLi Li Li Li

maxminLi Li Li Li

S if S SS S if S S

S if S S S

(15)

where vK and sK are the penalty factors. In this paper thevalues of penalty factors have been considered 10000.

III. ALGORITHM BASED ON NSGA-IIThe main components of the algorithm are summarized

below.

A. Initial populationAn initial population, 1P of size pop vN n , where popN

is the number of individuals (chromosomes) and vn is thenumber of variables (discrete and continuous), is generatedrandomly within the lower and upper boundaries.

B. Non-dominated SortAfter the initialization stage, the population is sorted on the

basis of on non-domination [20] into each front as shown in Fig.2.

Fig. 2. The Pareto-optimal front in a bi-objective minimization problem

C. Crowding distanceTo provide the diversity in population, the crowding distance

is calculated [20]. The crowding distance d of point s is ameasure of the objective space around s which is not occupiedby any other solution in the population. In Fig. 2, for a bi-objective problem, the crowding distance of the solution s isthe average side-length of the cuboid (shown using a dashedbox).

D. The main loopLet us consider the t-th generation of NSGA-II algorithm.

Suppose the parent population at this generation is tP and itssize is popN , while the offspring population created from tP

is tQ having popN chromosomes. The first step is to select thebest popN chromosomes from the combined parent and

offspring population t t tR P Q (of size pop2 N ), thusallowing to preserve elite chromosomes of the parentpopulation. To achieve this, first, the combined population tRis sorted according to different non-domination levels ( 1 2F , Fand so on). Then, each non-domination level is selected one ata time to construct a new population t 1P . This processcontinues until it reaches a front that can not be completelyaccommodated by t 1P ’s size popN restriction. When it

reaches this front, call it nF , each chromosome must be rankedaccording to its crowding distance, where larger crowdingdistance means better rank. The best-ranked chromosomesfrom nF are then placed into t 1P one at a time until the

Objective function 1

Obj

ectiv

efu

nctio

n2

Pareto-optimalfrontRank 1 SolutionsRank 2 SolutionsRank n SolutionsInfeasible Solutions

nF

overall size of t 1P contains popN chromosomes. With theelite parent population formed, the evolutionary operators(tournament selection [20], BLX-α crossover [21] andpolynomial mutation [20]) are invoked to create the nextgeneration child population t 1Q . This ends one NSGA-II'smain evolution loop iteration. The NSGA-II procedure issummarized in Fig. 3.

E. Stopping criteriaDifferent stopping criteria could be specified like: the

number of generations, an acceptable solution has been found,and no improvement is observed over a number of generations.

F. Selecting the best compromise solutionA fuzzy-based technique is employed in this paper to select

the best compromise solution from the obtained Pareto-optimalset. For each objective if , a fuzzy membership function iscalculated as follows [22]:

i

maxi i

f max mini i

f ff f

(16)

Where minif , max

if are the minimum (resp. maximum)value of ith objective function of all Pareto optimal solutions.For each Pareto solution k, the normalized membershipfunction is found as follows:

i

i

mkf

i 1kD m

kf

k 1 i 1

(17)

where D is the total number of Pareto solutions and m isthe total number of objective functions. The best compromisesolution is that having maximum value of k .

IV. APPLICATION OF THE NSGA-II TO THE PROPOSED PROBLEM

The DG placement and sizing problem can be expressed as:

obj 1 2minimize F f , f (18)Subject to the constraints in Eqs. (9) – (12).

The solution process of the proposed method is given asfollows:

Step 1: Input data (which includes network topology, networkdata and algorithm parameters).Step 2: Create the initial population 0P taking intoconsideration the defined information in the previous step. Setthe generation count t =0.

0 1 2 NpopP x x x (19)

i pl sizex DG DG (20)

pl busDG round 2 rand ( N 2 ) ; (21)

min max minsize DG DG DGDG P rand P P . (22)

where plDG and sizeDG are , respectively, the placement andsizing of DG, round(•) is a function which returns the nearestnatural number, and rand denotes a uniformly distributedrandom number within [0, 1].Step 3: For each individual in tP , run the load flow algorithmusing backward/forward sweep approach and evaluate theobjective functions (f 1 and f2) by Eqs. (2) and (8).Step 4: Check the constraints considering the results of loadflow analysis. If all constraints are satisfied, go to the next step,otherwise add penalty term to the objective functions.Step 5: Generate offspring population tQ from tP byperforming tournament selection, BLX-α crossover andpolynomial mutation operators.Step 6: Combine the parent population tP and offspringpopulation tQ to get the intermediate population

t t tR P Q of size pop2 N .Step 7: Carry out the non-dominated sorting of intermediatepopulation tR . (i.e., sorting the population based on value ofobjective function value in ascending order).

Non-dominatedsorting

Rejected SolutionsBased on crowding

Distances

Crowdingdistance sorting

Rejected SolutionsBased on Domination

Evol

utio

nary

oper

ator

s*

EvolutionaryOperators*

(*): - tournament selection- BLX-α crossover- polynomial mutation

Fig. 3. NSGA-II Main Loop Strategy

Step 8: The new population t 1P of size popN is filled byadding the highest ranked front sets until the size of thepopulation popN . The last front solutions are chosen based on

crowded comparison operator to fill exactly popN slots.Step 9: Increment the generation number and repeat the stepsfrom 3 to 8 until the maximum number of generation is reached.Step 10: Choose the best compromise solution using fuzzy settheory.

The flowchart of the proposed algorithm is represented in Fig.4.

V. SIMULATIONS AND RESULTS

In this section, the proposed algorithm is tested on 12-bus[23], 33-bus [24], 69-bus [25] and 85-bus [26] radialdistribution networks. Table 1 shows the effectiveness of theproposed algorithm on system performance in comparison withbase case (without DG). To find out the maximum systemloadability (λmax) of the system, active and reactive power loadis increased on all buses, till the voltage collapse is observed(see Fig.7).

The proposed algorithm is also compared with theanalytical method [3], Cuckoo search algorithm [7], andParticle swarm optimization [9]. The optimum DG position andsizes are tabulated in Table 2. The proposed NSGA-IImethodology is programmed in MATLAB 7.1 running on IntelCore i3 (3.1 GHz, 4 Go of main memory) with MS Windows 7installed.

VI. DISCUSSION

As can be seen from the comparison of columns (2) and (7)in Table I, the total active power losses of all test systems arereduced significantly after installation of DG units. Also, it canbe observed that the voltage stability is improved, and thesystem loadability is increased after proper allocation of DGunits. More loads can be added without experiencing theproblem of voltage collapse.

The voltage profile of all test systems are depicted in Figs.5. As it can be seen, the voltage levels at all nodes for the radialdistribution systems are improved and placed in an acceptablemargin.

Fig. 6 shows the voltage stability indices of all test systems.The weakness of voltage stability indices for all buses in thedistribution system before installing DGs is vivid. But afterinstalling DGs, the stability buses indexes for all distributionsystems are considerably improved.

It can be seen in Table II and Fig. 6 that the proposedalgorithm reaches the same or better results than other methodspublished by other authors.

Fig. 7 demonstrates that the optimum size and the bestlocation of DG have improved the system loadability.

VII. CONCLUSION

This paper has presented a multi-objective approach foroptimum DG placement, considering minimization of powerlosses and enhancement of voltage stability. NSGA-II wasemployed to find the Pareto optimal solutions. Finally, a fuzzy-based method has been utilized to choose the best compromisesolution. The proposed algorithm was tested on 12-bus, 33-bus,69-bus and 85-bus radial distribution test systems. The resultshave proved the efficiency of this method for enhancement ofvoltage profile, reduction of active power losses and also anincrease in voltage stability margin.

Fig. 4. Flowchart of the DG placement and sizing algorithm.

Start A

NSGA-II

B

Non-dominatedSorting of Rt

(BLX-α) Crossover

Polynomial mutationMutation

Check the StopCriterion

Y

Stop

Run load flow analysisand evaluate fitness

values for each individual

Read data:1. load and line data2. network topology3. algorithm parameters

Select NSGA-II parameters

Creates a newoffspring

population Qt

Combine Rt=Pt Qt

Crowding distancesorting

Select the best compromisesolution using fuzzy settheory

AN

BGenerate initial population P0

Tournament selection

TABLE I

APPLICATION OF PROPOSED APPROACH ON DIVERSE TEST SYSTEMS

Without DG Proposed method

Test System Real

Pow

erLo

ss (k

W)

min

(SI)

Syste

mlo

adab

ility

Opt

imal

Bus

Opt

imal

Size

(MV

A)

Real

Pow

erLo

ss (k

W)

Loss

redu

ctio

n%

min

(SI)

Syste

mlo

adab

ility

12-bus 20.69 0.7920 5.30 8 0.43500 13.66 33.97 0.9827 6.0833-bus 202.66 0.6952 3.62 7 3.22991 114.16 35.92 0.8618 4.8669-bus 225.00 0.6833 3.20 61 2.66380 103.96 53.80 0.8956 4.2485-bus 316.12 0.5764 2.54 25 2.48451 180.98 42.75 0.7825 2.96

TABLE II

APPLICATION OF ANALYTICAL METHOD, PSO AND CS FOR DG PLACEMENT AND SIZING

TestSystem

Analytical method [3] PSO [9] CS [7]

Opt

imal

Bus

Opt

imal

Size

(MV

A)

Loss

redu

ctio

n%

min

(SI)

Syste

mlo

adab

ility

Opt

imal

Bus

Opt

imal

Size

(MV

A)

Loss

redu

ctio

n%

min

(SI)

Syste

mlo

adab

ility

Opt

imal

Bus

Opt

imal

Size

(MV

A)

Loss

redu

ctio

n%

min

(SI)

Syste

mlo

adab

ility

12-bus 9 0.22715 47.95 0.9311 5.92 9 0.2539 47.70 0.9413 6.0333-bus 6 2.49078 47.31 0.7839 3.70 7 2.8951 45.55 0.8149 3.7869-bus 61 1.80782 62.95 0.8778 3.92 61 2.0264 62.65 0.8824 4.03 61 2.2000 62.80 0.8840 4.0685-bus 8 2.20886 44.28 0.7306 2.88

(a) 12-bus System (b) 33-bus System

(c) 69-bus System (d) 85-bus System

Fig. 5. Voltage profile of radial distribution systems with and without DG

Fig. 6. Comparison of Bus Voltage Stability Index (SI) using different DG placement algorithms.

Fig. 7. Loadability curve with and without DG

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