Design of an advanced electric power distribution systems using seeker optimization algorithm

Electrical Power and Energy Systems 63 (2014) 196–217

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Design of an advanced electric power distribution systems using seeker optimization algorithm Deepak Kumar, S.R. Samantaray ⇑ School of Electrical Sciences, Indian Institute of Technology, Bhubaneswar, India

a r t i c l e

i n f o

Article history: Received 14 February 2014 Received in revised form 17 May 2014 Accepted 31 May 2014

Keywords: Advanced power distribution system (APDS) Seeker optimization algorithm (SOA) Automatic reclosers (RAs) Contingency-load-loss index (CLLI) Distribution system planning

a b s t r a c t The power distribution network design problem has a growing impact on secure and economical operation of distribution power system. This issue is well known as a non-linear, multi-modal and multi-objective optimization problem where global optimization techniques are required in order to avoid local minima. In this study, a new approach using seeker optimization algorithm (SOA) is proposed for distribution system planning problem with simultaneous placement of automatic reclosers (RAs), considering total system economic cost, overall system reliability, system power losses and voltage deviations as an objective functions. Normally, conventional power distribution systems (CPDS) are radial in nature and momentary fault in the system causes large area of the grid to be blacked out, which leads to huge load interruptions. Reliable distribution system minimize this effect by allowing faults to clear themselves by protection device operations such as automatic reclosers (RAs) and quickly restores the power through system reconfiguration by minimizing the loads affected. Thus, in order to prevent the system from momentary faults, simultaneous placement of RAs has been also done which leads to a design of an advanced power distribution system (APDS). Thus, the reliability issues are of major importance for effective planning of an APDS. For evaluation of reliability measure, the contingency-load-loss index (CLLI) is assessed in this paper, which is independent of the failure rate and fault repair duration of the feeder branches. The performance of the proposed algorithm is extensively assessed and comparisons are made with Particle Swarm Optimization (PSOs), Genetic Algorithm (GA) and graph theoretical approach (GTA) applied on the 54 and 100-bus primary power distribution systems. The simulation results show that the proposed approach performed better than the other listed algorithms and can be efficiently used for the optimal design of an advanced power distribution system. Ó 2014 Elsevier Ltd. All rights reserved.

Introduction Planning of power distribution network is highly complex as it has to consider various important issues including technical and environmental constraints while meeting the customer needs. The deregulation in power industry has opened a competitive market for power industries to hold market while keeping customers satisfied. This affects more to distribution sector due to their direct link with the customers. Thus an efficient planning of distribution network is essential for all the utilities. Mostly, the distribution networks are designed to be radial for operational convenience and lower protection cost [1–15]. The distribution system planning is basically an optimization process that considers simultaneous optimization of overall system cost and network reliability.

⇑ Corresponding author. Tel.: +91 9437305131; fax: +91 6742301983. E-mail address: [email protected] (S.R. Samantaray). http://dx.doi.org/10.1016/j.ijepes.2014.05.073 0142-0615/Ó 2014 Elsevier Ltd. All rights reserved.

Overall system cost (total installation and operational cost) is minimized by optimizing the number of feeders, their routes, and the number and locations of the automatic reclosers. Past researchers have considered network reliability evaluation using nondelivered energy due to faults [10–13,16–18]. Computation of non-delivered energy is very difficult as the estimation of it depends upon the actual failure rates and repair duration of the feeder branches. Also, most of these heuristic based approaches have considered average failure rates and the repair durations of all the feeder branches for evaluation of reliability based on expected energy not served (EENS). In some works [19,20], the outage cost due to faults, as seen by utilities, is also used to optimize network reliability. These reliability objectives are basically functions of failure rate and fault repair duration of each feeder branch. Hence, they are generally optimized by choosing the branch conductor sizes with lower failure rates. Further, the fault in the feeder branches is unpredictable and it occurs due to various non-technical reasons, such as short circuit due to contact of small tree branches, animals etc. [19]. Moreover,

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Section 1 LP6

DS

CB

Feeder 1

LP1

LP2

LP7

LP8

LP3

LP4

RA

LP5

LP9 LP10 LP11 LP12 Section 2 CB or RA DS Distribution substation LP Load Points

Fig. 1. Single feeder radial network with/without RA.

LP6 Section 2

Section 1 CB

RA

Feeder 1

DS LP1 CB

LP2

LP3

LP8

TS

LP5 Loop Configuration

RA

Feeder 2

LP7

LP4

LP9

LP10

LP11

LP12

Section 4

Section 3

TS (Normally opened Tie Switch) Fig. 2. Multi feeder radial network with two RA and a TS.

I(k)

5

RE

SE

6

7

1

8 Branch k

V (m1) < δ (m1)

2 3

9

4

Fig. 3. Single line diagram of a sample radial distribution network.

the fault repair duration varies with the location and severity of the fault. Thus, this reliability evaluation may suffer from considerable inaccuracy. Thus, a new reliability index is assessed in this proposed work called contingencyload-loss index (CLLI) [19]. The distribution system planning problem is a non-linear, nonconvex, non-differentiable, constrained optimization problem with integer and continuous decision variables. Research works reported using classical optimization techniques such as simplex programming [2], Branch and bound algorithm [3,4], Lagrange method [5], and quadratic programming [6] etc. for designing power distribution system. However, these classical optimization techniques have limited scope in practical applications as almost

V (m2) < δ (m2)

P(RE)+jQ(RE)

Fig. 4. Equivalent diagram of one branch of a radial distribution network.

all the practical problems involves objective function that are non-linear, non-convex, and non-differentiable in nature. In this regard, the heuristics-based algorithms have distinct advantages, i.e., they can handle non-linear, non-convex problems, and do not require any gradient information. Some of the heuristics-based algorithms proposed for this planning problem are: genetic algorithm (GA) [11,13–16], network flow programming [12], and Tabu search [18] etc. Another powerful heuristics based algorithm, successfully used in many complex problems, is the particle swarm optimization (PSO) [20]. The advantages of the PSO over the other evolutionary algorithms are easy implementation, effective

( +1) pj

(0) pj

0

(0) pj

rj

( −1) pj (0) ( +1) pj + pj

Fig. 5. The proportional selection rule of search direction.

1

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2

3

4

7

6

5

1

Fig. 8. SOA procedure.

Fig. 6. The action part of the fuzzy reasoning.

memory use, less number of function evaluations, and an efficient maintenance of diversity. Although GA and PSO have received increasing interest from the evolutionary computation community, several studies have shown that the GA is prone to not only premature convergence but also stagnation [21], and that a successful location of the global optimum depends on choosing the correct control parameters. Meanwhile, the performance of PSO also depends on its parameters [22] and may be influenced by premature convergence and stagnation problem [23]. In general radial distribution systems are inherently insecurefailure of a major components leads to huge load interruptions. Reliable distribution system minimize this effect by allowing faults to clear themselves by protection device operations such as automatic reclosers and quickly restores the power through system reconfiguration by minimizing the loads affected. Thus, in order to prevent the system from momentary faults, simultaneous placement of automatic reclosers (RAs) has been also done, to enhance the overall system reliability by automatically restoring the power to the line after a momentary fault. In electric power distribution system, automatic recloser (RA) is a circuit breaker equipped with a mechanism that can automatically close the breaker after it has been opened due to a fault. Reclosers are used on overhead distribution systems to detect and interrupt momentary faults. Since many short-circuit on overhead lines clear themselves, a recloser improves service continuity by automatically restoring power to the line after a temporary fault. Thus, the power distribution system (PDS) planning including automatic reclosers (RAs) using a new meta-heuristic evolutionary optimization technique called Seeker optimization algorithm (SOA) is proposed [24]. SOA is a new population-based heuristic search algorithm, which attempts to simulate the act of human searching for real-parameter optimization. In comparison with [24], four main improvements in SOA have been done in this paper: first, the algorithm in this directly uses search direction and step length to update the position. Hence, the implementation of the algorithm is relatively easier. Second, the calculation of search direction applies selection rule, which can improve the population diversity to improve global search ability and decrease the

Seekers classified into K sub-population

Xt

numbers of control parameters so as to make it simpler to implement. In [25], the cloud reasoning was used which can be viewed as an extension of fuzzy theory, for the evaluation of step length but its implementation is relatively complicated and time-consuming. Thus, fuzzy reasoning is used to generate the step length in this paper. Finally, mixed variable handling method is used in this paper and thus the proposed SOA can handle both continuous and discrete variable. The performance of the proposed planning algorithm is illustrated on typical 54-bus and 100-bus distribution systems. The performance of the proposed approach is compared with four versions of PSOs, GA and GTA by statistical tests. The Paper is organized as follows: Description of objective functions (Section 2), Multi-objective problem formulation (Section 3), Implementation of SOA for design of APDS (Section 4), Formation of radial configurations using Seeker optimization algorithm (Section 5), Presentation of results and discussion (Section 6) and Conclusions (Section 7).

Description of objective functions The multi-objective functions of the APDS include the technical and economic goals. The economic goal is mainly to minimize the active power loss and overall system cost. The technical goals are to minimize the contingency-based-load-loss index (CLLI) and load bus voltage deviation from the ideal voltage. Hence, the objectives of the APDS model in this paper are overall system cost, overall system reliability (CLLI), active power loss (PLoss), and voltage deviation (DVL). System economic cost The overall annual cost representing the yearly investment costs plus the yearly operation (variable) costs of the network, consisting of the cost of the energy losses and interruption costs in the circuit branches, subject to equality as well as inequality constraints, represents one of the main concerns in planning the APDS. The details of the various cost functions are defined as follows [7]:

Selection based on sorting of overall fitness function value

Inter sub-population learning mechanism

Sub-Pop k X1K

Sub-Pop 1

Sub-Pop 2

Sub-Pop k-1

X11

X12

X1(K-1)

X2K

X21

X22

X2(K-1)

X3K

X31

X32

X3(K-1)

XSK

XS1

XS2

XS(K-1)

Mutated Pop.

New population based on “movement operation”

Fig. 7. Method for maintaining radiality.

X t+1

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199

Start Load Network Data Initialize initial radial configurations of size NND, each configuration represents one seeker. Divide the population into NSP subpopulations randomly. Evaluate the objective function values of each seeker using (15).

Obtain the personal best position, neighborhood best position and global best position.

Compute step length of each seeker

Compute the search direction of each seeker

Update the position of each seeker using (19). Apply the subpopulation learning strategy.

Calculate the objective function value of each seeker using (15) Update the position of personal best, neighborhood best and global best. Perform seeker movement operation

Now, select the best seekers of size NND depending on their fitness values(15). No

Stop criterion Satisfied? Yes Outputs the results Fig. 9. Flow chart of the proposed seeker optimization algorithm.

Investment costs The yearly fixed cost recovery is the annual equivalent of the capital or total investment cost of obtaining the asset plus the salvage. In most of the engineering problems, the total capital cost has two components. First component is the initial investment required, and the second component is the salvage value of assets received at the end of their useful life. Thus, the fixed cost recovery can be defined as [7]:

C f ¼ g

X

Cb

ð1Þ

b2Xb

where Cb is the cost of branch b of the main feeder and g is the yearly recovery rate of the fixed cost. Costs of branches emanating

from the source substation include the line, the corresponding substation costs, and the associated circuit breaker costs. Xb denotes the set of all possible branches in a particular radial path in the network configuration under consideration. Operational costs The second component is interruption cost and as there are no alternative supply routes available to energize the load nodes in a radial network, thus an outage of a branch interrupts the delivery to all consumers supplied through this branch. Thus, the cost of supply interruption can be expressed as

C i ¼ ci ad

X pffiffiffi kb 3RefIb gU r

b2Xb

ð2Þ

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L5

5

L6

2

L7

L1

Case 1: Single feeder radial configuration without RA and with RA in the branch between lode points (LPs), LP 6 and LP 12.

3

Case 2: Two feeder radial configuration with RA’s are at position between the nodes (4, 5) and (10, 11) and one Tie switch (TS, a normally open switch) between LP 6 and LP 12.

L8

L3 4

L4 6

L2 Node 1 : Substation Nodes 2,3,4,5,6 : Load nodes Fig. 10. Diagram of a 6 bus meshed network.

where ci is the Cost per unit of energy not delivered; a Load factor; d the Repair duration; kb the Branch failure rate; Ur the Network rated voltage; Ib is the Branch current at peak load. The third component is cost of energy losses (Cl):

C l ¼ 8760cl b

X

r b RefIb g2

ð3Þ

b2Xb

and; LossfactorðbÞ ¼ 0:15a þ 0:85a2

ð4Þ

where cl is the cost per unit of energy lost; b the loss factor; rb the branch resistance; Ib is the Branch current at peak load; a is the load factor. The total annual cost that should be minimized is,

C ¼ Cf þ Ci þ Cl

ð5Þ

Contingency-load-loss-index (CLLI) [19] An original method has been developed in order to evaluate the function CLLI related to the distribution network reliability to carry out the optimal multi-objective design. In this paper, a contingency based reliability assessment named as contingency-load-loss index (CLLI) is proposed as a new reliability index. This index can be defined as the ratio of the average of non-delivered load due to failure of all branches, taken one at a time, to the total load, where the non-delivered load (NDL) can be defined as a set of load demands disconnected from the main feeder because of sudden interruption or faults in the feeder branches. The advantage of CLLI is that the information regarding the failure rate and fault repair duration of the feeder branches are not required to compute CLLI, whereas most of the system performance indices (SAIFI, SAIDI, CAIDI, ASAI, ASUI, MAIFI, and CAIFI, etc.) [20] are calculated as the weighted averages of the basic load point indices (failure rate, and fault repair duration). CLLI is calculated considering all single contingency events (all branch failures, taken one at a time). The occurrences of simultaneous contingencies, such as simultaneous faults in two or more branches are not considered in this planning work as the frequency of occurrence of such faults are very less for any practical distribution system. Thus the objective function CLLI can be mathematically expressed as, NF X

CLLI ¼

NDLAvg: ¼ LTotal

A small distribution network (DN) is considered as shown in Fig. 1, with one substation, two RA and twelve load nodes with each node have a fixed load of 100 kW. From Fig. 1 it is seen that, if RA is not connected in the branch between node (6, 12), for a fault at any branch in the DN the circuit breaker (CB) will isolate all the load nodes from the supply. Thus, the total NDL for a branch failure will be 1200 KW. Since any branch fault causes the same total NDL, the average NDL for all the branch fault, considered one at a time is 1200 KW. Thus, the CLLI of this network is one. This is the maximum possible value of CLLI; hence this is the least reliable network from the non-delivered point of view. Now, if RA is placed in between node 6 and 12, and for a fault occurs in any branch of Section 2, the RA will be opened to maintain the supply to Section 1 and it causes the total NDL of only six load points with a total load capacity of 600 kW. But if the fault occurs in Section 1, it causes the complete blackout of the system and the total NDL becomes 1200 kW. As, each section have six LPs, thus the total NDL is 900 kW and the CLLI of this network is 0.75. Now from Fig. 2, it can be observed that for two feeder radial configuration with two RAs, the CLLI becomes 0.375 when the tie switch is open. Thus this network is more reliable than the single feeder radial network. Now if TS is closed, thus it forms a loop configuration and it causes the most reliable network with CLLI index of 0.23. Thus the loop network is most reliable amongst all the configurations. Some power utilities like Hong Kong Electric Company, Singapore power, etc. have adopted normally closed loop configurations to serve their customers with high reliability [26] but it is more costlier than the radial network due to more number of branches or more costly breakers or switchgears for higher shortcircuit level. Thus, from several decades the configuration of PDS has been typically designed in a radial form for easy access and control, lower protection cost and operational convenience. The active power loss (PLoss) Fig. 3 shows the single line diagram of a sample radial distribution network, and the electrical equivalent of one branch of a radial distribution network is shown in Fig. 4. The mathematical model of radial distribution networks can easily be derived from [27], from Fig. 4, the current flowing through branch k can be defined as,

IðkÞ ¼

jVðSEÞj\dðSEÞ  jVðREÞj\dðREÞ ZðkÞ

5

L1 NDLi =NF

i¼1

LTotal

L5

ð7Þ

L6

2

3

L7 L8

L3 4

ð6Þ

where NDLi is the non-delivered load due to fault in branch i. NDLAvg the average non-delivered load. LTotal is the total load in KVA. Two cases have been considered for the computation of CLLI.

L4 6

L2 Fig. 11. Representation of a single radial network.

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Start Node 1

1

1

2

3

End Node

2

6

3

4

5

Step length Search Direction New End Node 2

4

6

4

5

New seeker (One new radial configuration) Start Node 1

1

1

2

3

End Node

4

6

5

4

2

Fig. 12. Formation of a new seeker using SOA.

X (Km)

(0,0) (2)

X (54)

(39) (40)

(4) (51)

(9)

(41) (49)

(17) (8)

(25)

(16)

(26) (15)

(48)

(27)

Y (Km)

(43)

(18)

(13)

(24) (7)

(38) (6)

(1) (12)

(32) Substation Load Node

(34) (14)

(30)

(21) (44)

(53)

(11) (31)

(20) (33)

(28)

(37)

(23)

(45)

(10)

(22)

Y

(42)

(46)

(5) (3)

(47)

(19)

(29)

(50) (35)

(36)

Fig. 13. A 54-bus primary distribution system load points (LPs) and a substation at node 1.

and

IðkÞ ¼

Voltage deviation

PðREÞ  jQðREÞ V  ðREÞ

ð8Þ

where Z(k) = R(k) + jX(k), SE and RE, the sending and receiving end nodes, respectively; P(RE), sum of the real power loads of all the nodes beyond node RE plus the real power load of the node RE itself plus sum of the real power losses of all the branches beyond node RE; Q(RE), sum of the reactive power loads of all the nodes beyond node RE plus the reactive power load of the node RE itself plus sum of the reactive power losses of all the branches beyond node RE. From Eqs. (7) and (8):

RðkÞ ðP2 ðREÞ þ Q 2 ðREÞÞ jVðREÞj

2

 NL  X V N  V i    min DV L ¼  N  i¼1

ð10Þ

L

where DVL is the average voltage deviation in per unit, NL is the total number of the system load buses, VN is the nominal voltage magnitude and Vi is the actual voltage magnitude at bus i. Problem formulation



PLossðkÞ ¼

It can be defined as the difference between the nominal voltage and the actual voltage. The smaller the deviation of bus voltage from the nominal one, the better the voltage condition of the system. The average voltage deviation in per unit can be defined as [28]:

ð9Þ

The real power loss in all branches can be evaluated in the same manner. The system real power loss is taken to be the sum of the real power losses in all branches.

Considering different sub-objective functions have different ranges of function values, every sub-objective uses a transform to keep itself within [0, 1]. The four sub-objective functions, overall system cost, system reliability index (CLLI), active power loss and voltage deviation, are normalized:

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2 40

54 41

4

51 49

17 26

9

16

52

25

48

15 8

27

47

18 19 38

24 43

7 13

6

5

42

1

46 10

23

45

20

21

33

44

3 28

37 12 32

22

34

11

53

14

50

29

Substation

35 31

Load node

30 36 Fig. 14. Possible paths between source and LPs for a 54-bus system.

f1 ¼

8 > <0

Cost < Costmin

CostCostmin

Costmax Costmin > : 1

Costmin 6 Cost 6 Costmax Cost > Costmax

9 > =

ð11Þ

> ;

f2 ¼

8 > <0 > :

CLLI < CLLImin

CLLICLLImin CLLImax CLLImin

CLLImin 6 CLLI 6 CLLImax

1

CLLI > CLLImax

9 > =

ð12Þ

> ;

Table 1 Power demand requirements, in KVA and X–Y coordinate, for the distribution network nodes. X

Y

Load (kVA)

X

Y

Load (kVA)

X

Y

Load (kVA)

0.2(1) 1(2) 2(3) 3 4 5 6 7 1.5 11.5 7.5 8.5 12.5 11 8 11 5.5 3.5 From node (1) to

0.11 2 15 4 12 11.5 10 7 5.5 13.5 17.5 15.5 10.5 17.5 7.5 6 5.5 8.5 node(54)

0 25 25 25 50 63 63 50 25 16 16 25 50 63 63 25 16 16

13 14 16.5 5.5 20.5 8 5 8 10.5 10.5 9 7.5 5.5 3 13 14 12.5 11(36)

8 13 14 17 12 9 7 5.5 8 15 19 19.5 19.5 17.5 15.5 16.5 19 20

16 63 25 25 50 100 100 100 50 50 25 63 63 25 50 50 25 25

5 2 3 6 9 4 15 15 15.5 12 14.5 13.5 13 13.5 4 9.5 9.5(53) 12(54)

15.5 10.5 3.5 4 4.5 11.5 10 14.5 12.5 12 7.5 6 4.5 18 5 6.5 17 2.5

50 50 63 25 25 50 50 25 25 63 63 25 16 16 25 16 25 50

203

D. Kumar, S.R. Samantaray / Electrical Power and Energy Systems 63 (2014) 196–217 Table 2 Parameters for the different techniques. Parameters

SOA

SGA

CPSO

SPSO

PSO-w

PSO-cf

Population size (lmax, lmin) (wmax, wmin) Crossover probability Mutation probability Max Gen Independent run

100 (0.95,0.0111) (0.1,0.9) – – 100 20

100 – – 0.8 0.02 100 20

100 – (0.1,0.9) – – 100 20

100 – – – – 100 20

100 – (0.1,0.9) – – 100 20

100 – (0.1,0.9) – – 100 20

feeder branches on which power flow are more than the maximum allowable value. DVL, DSG, and DIF can be defined as:

Table 3 Specification of circuit breaker. CB type

Short circuit current rating (kA)

Installation cost($)

1 2 3

25 35 50

20,000 40,000 60,000

(

DV L ¼ (

DSG ¼

f3 ¼

f4 ¼

8 > <0 > :

PLossPLossmin PLossmax PLossmin

PLoss < PLossmin PLossmin 6 PLoss 6 PLossmax

1

PLoss > PLossmax

8 > <0

DV L DV Lmin

DV Lmax DV Lmin > : 1

9 > = > ;

9 DV L < DV Lmin > = DV Lmin 6 DV L 6 DV Lmax > ; DV L > DV Lmax

ð13Þ

ð14Þ

V L  V max L

if VL > V max L ) min if SG < SG

Smin  SG G SG  Smax G

 DIF ¼ IF  Imax F

) ð16Þ

ð17Þ

if SG > Smax G  max if IF > IF

ð18Þ

The capacity constraint limits of the feeders

jIf ;i j 6 Imax f ;i

X 2 X DV L þ kS DS2G N lim V

X 2 þ kF DI F

if VL < V min L

The simultaneous minimization of objective functions are subjected to some technical constraints, which are:

where the subscript ‘‘min’’ and ‘‘max’’ denote the corresponding expectant minimum and possible maximum value, respectively. As all the sub-objective function is a minimization problem; thus, the overall fitness function can be defined as minimization optimization problem using the weighted aggregation of all the four subobjective functions. Control variables are self-constrained, and the dependent variables are constrained using penalty terms. The infeasible solutions are penalized, by applying a constant penalty to the solutions. Therefore, penalty functions correspond to voltage violations at all load buses, capacity limit violations at all substation buses, and power flow violations at all feeder branches have considered. Thus, the overall fitness function can be normalized as follows:

min F ¼ w1 f1 þ w2 f2 þ w3 f3 þ w4 f4 þ kV

V min  VL L

i ¼ 1; 2; . . . ; Nf

where |If,i| and Imax are the current amplitude and its maximum f ;i allowable value of the ith feeder and Nf is the number of feeders. Substation capacity limit constraint

Simin 6 Scapi 6 Simax where Simax is the maximum capacity of the ith substation. Simin the minimum capacity of the ith substation. Scapi is the capacity of the ith substation. The voltage level in load buses The voltage level at every node in the radial distribution network must be within the acceptable voltage limit.

N lim S

ð15Þ

V min 6 V i 6 V max i i

N lim F

Conservation of power flow where wi(i = 1, 2, 3, 4) is the user defined constants which are used to weight the contributions from different sub-objective functions. kV , kS , and kF are the penalty factors. N lim is the set of total number V of load buses on which voltages are outside the permissible limits, N lim is the set of total number of substation buses on which power S are outside the permissible limits, N lim is the set of total number of F

The vector of power flow through each of the nL feeder is equal to the vector of power demand at each of the nL nodes, where nL is the total number of load nodes.

AS ¼ D () AðS1 ; S2 : . . . SnL Þ ¼ ðD1 ; D2 . . . DnL Þ

Table 4 Specification of conductor sizes. Sl. No.

Conductor type

Size (kcmil)

R (X/mi)

X (X/mi)

Branch installation cost ($/km)

Preventive maintenance cost ($/km/year)

Corrective maintenance cost ($/km/year)

Current rating (A)

1 2 3 4 5 6

Turkey Swan Sparrow Raven Pigeon Penguin

6 4 2 1/0 3/0 4/0

4.257 2.719 1.753 1.145 0.762 0.628

0.760 0.723 0.674 0.614 0.578 0.556

10,000 10,000 10,000 10,000 10,000 10,000

53.5 53.5 53.5 53.5 53.5 53.5

6.5 6.5 6.5 6.5 6.5 6.5

105 140 184 242 315 357

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Table 5 Results of overall objective function values for various algorithms on 54-bus system over 20 runs (p.u.).

Table 8 Results of power loss values for various algorithms on 54-bus system over 20 runs (p.u.).

Algorithms

Best

Worst

Mean

Std. Dev.

h

Algorithms

Best

Worst

Mean

Std. Dev.

h

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.19334 0.10413 0.11412 0.10286 0.09849 0.09671 0.09671

0.507300 0.262718 0.313773 0.265421 0.313910 0.187458 0.158598

0.284700 0.195963 0.220670 0.130727 0.182511 0.127549 0.099747

11.964  102 8.1713  102 8.3200  102 7.9800  102 9.2696  102 3.9639  102 3.8939  102

1 1 1 1 1 1 –

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.0789000 0.0010955 0.0003302 0.0004092 0.0002669 0.0001131 0.0001131

0.926000 0.023360 0.048617 0.020638 0.025680 0.066325 0.025390

0.290400 0.010984 0.013893 0.006457 0.013952 0.008734 0.005706

0.322600 0.007334 0.013557 0.019510 0.009985 0.007862 0.007296

1 1 1 1 1 1 –

Table 6 Results of cost function values for various algorithms on 54-bus system over 20 runs (p.u.). Algorithms

Best

Worst

Mean

Std. Dev.

h

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.195400 0.148748 0.142566 0.141163 0.154094 0.142465 0.142465

0.638000 0.608727 0.661089 0.614754 0.651995 0.596602 0.568700

0.255300 0.378460 0.431900 0.261250 0.349677 0.242175 0.201492

0.15180 0.37846 0.43190 0.20877 0.22223 0.18522 0.16229

1 1 1 1 1 1 –

where S the vector of power flow through each of the nL feeder.D the vector of power demand at each of the nL nodes. A the node-element incidence matrix. Radiality constraint A radial network containing nL load nodes must have exactly nL number of conducting branches regardless of the number of supply nodes. The PDS topology can be considered as a graph consisting of m arcs ðm ¼ jXb Þ and n: (nS + nL) number of nodes ðn ¼ jXn Þ where nL the total number of load nodes and nS is the supply nodes. From reference [29] it is known that a tree is a connected graph without any loop and thus, it is possible to compare the radial topology with a tree. The tree of a graph is a sub-graph connected with (nL) number of arcs. Hence, one can state that topology of an electrical PDS satisfies radiality constraint with nL load nodes if and only if it satisfies the two following conditions:  Condition 1: The solution must have (nL) number of arcs.  Condition 2: The solution must be connected, i.e. solution must have n: (nS + nL) (sum of supply nodes plus the load nodes) number of nodes and nL number of branches [29]. Also the obtained solution satisfies conservation of power flow constraint, i.e. it must supply the power demand at every load bus, so that a path between the substation and each other bus exists. Therefore, every bus is linked with the substation bus, forming a connected graph.

Table 7 Results of reliability function values for various algorithms on 54-bus system over 20 runs (p.u.). Algorithms

Best

Worst

Mean

Std. Dev.

h

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.192558 0.151961 0.158427 0.159028 0.142169 0.131500 0.131500

0.628000 0.404359 0.462694 0.443701 0.420603 0.389623 0.340714

0.398000 0.243632 0.211038 0.262310 0.229691 0.194632 0.166107

0.277200 0.021820 0.023193 0.061220 0.107802 0.039463 0.037039

1 1 1 1 1 1 –

Seeker optimization algorithm This section introduces the basic concepts of Seeker optimization algorithm [28]; afterward the implementations of this algorithm for multi-objective (MO) optimal planning of PDS are described in detail. Implementation of seeker optimization algorithm Step 1: Initialization process: Initialize the good acceptable initial set of seekers of size N IS using fundamental loop configurations that preserve the radiality operation [30] where, each seeker represents one radial configuration of dimension P, where first N (N < P) dimension represents N load nodes that will constitute a radial configuration and the remaining P–N dimension represents RA positions keeping all the system constraints into consideration and then evaluated their fitness function value (15). Step 2: Search process ! In SOA, search direction Si ðtÞ ¼ ½Si1; Si2; . . . ; Sip and a step length for each individual i at each iteration t with dimension P, ! ai ðtÞ ¼ ½ai1; ai2; . . . ; aip, where, aiðtÞ P 0 and Si(t) e {1, 0, 1} are computed separately for each seeker i. Then position of ith seeker Xi = [Xi1, Xi2, . . ., Xip] of size N ND where, N ND is the total population size is updated as follows:

Xiðt þ 1Þ ¼ XiðtÞ þ aiðtÞ  SiðtÞ 8i ¼ 1; 2; . . . ; NND

ð19Þ

! ! The evaluation of search direction ðSi ðtÞÞ and step length ð ai ðtÞÞ for each individual i at each iteration t with dimension P are described in detail. Search direction In SOA, each seeker i selects his search direction based on several empirical gradients (EGs) by evaluating the current or historical positions of itself or its neighbours. In this work, the EGs involve egotistic, altruistic, and pro-active behaviours to yield the respective directions. Egotistic behaviour, is completely pro-self and another, altruistic behaviour, which is completely pro-group [28]. Every seeker i, as a single sophisticated agent, is uniformly egotistic, believing that he should go towards his historical best position ~ pi;best ðtÞ. Then an EG from ~ xi ðtÞ to ~ pi;best ðtÞ can be involved where Xi(t) = [Xi1, Xi2, . . ., Xip] is the position of the seeker i at time step t. Hence, the seeker i is associated with an empirical direction vector called as egotistic direction ~ di;ego ðtÞ:

~ di;ego ðtÞ ¼ signð~ pi;best ðtÞ  ~ xi ðtÞÞ

ð20Þ

where the sign() is a signum function on each dimension of the input vector. On the other hand, based on the altruistic behaviour, the seekers in a same neighbourhood region co-operate explicitly,

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D. Kumar, S.R. Samantaray / Electrical Power and Energy Systems 63 (2014) 196–217 Table 9 Performance comparison of the best solutions obtained with different methodologies on 54-bus system. Algorithms

F

f1

f2

f3

f4

Max vol (p.u.)

Min vol (p.u.)

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.13334 0.10413 0.11412 0.10286 0.09849 0.09671 0.09671

0.1986 0.1778 0.1865 0.1766 0.1781 0.1750 0.1750

0.1938 0.1641 0.1880 0.1629 0.1453 0.1442 0.1442

0.07070 0.00110 0.00033 0.00003 0.00156 0.00014 0.00014

0.0074 0.0067 0.0085 0.0050 0.0058 0.0046 0.0046

1 1 1 1 1 1 1

0.9926 0.9933 0.9915 0.9950 0.9942 0.9954 0.9954

communicate with each other and adjust their behaviours in response to others to achieve the desired goal. That means the seekers exhibit entirely pro-group behaviour. For a ‘‘black-box’’ problem in which the ideal global minimum value is unknown, the neighbours’ historical best position ~ g best ðtÞ or the neighbours’ current best position ~lbest ðtÞ is used as the attraction signal source of the self-organized aggregation behaviour. Hence, each seeker di;alt1 ðtÞ and ~ di;alt2 ðtÞ: is associated with two altruistic directions, i.e., ~

~ di;alt1 ðtÞ ¼ signð~ g best ðtÞ  ~ xi ðtÞÞ ~ ~ d ðtÞ ¼ signðl ðtÞ  ~ x ðtÞÞ i;alt2

~ di;pro ðtÞ ¼ signð~ xi ðt1 Þ  ~ xi ðt 2 ÞÞ

ð23Þ

where t1, t2 e {t, t  1, t  2}, and ~ xi ðt1 Þ is better than ~ xi ðt2 Þ.According to the human rational judgement, the actual search direction di ðtÞ, is based on the compromise among the aforeof the ith seeker, ~ di;ego ðtÞ, ~ di;alt1 ðtÞ, mentioned four empirical directions, namely, ~ ~ di;pro ðtÞ. In this work, every dimension j of ~ di is selected di;alt2 ðtÞ, and ~ by applying the following proportional selection rule (Fig. 5):

ð21Þ ð22Þ

i

best

Hence, each seeker i is associated with an empirical pro-activeness di;pro ðtÞ: direction ~

8 9 ð0Þ 0 if r j 6 pj > > > > < = ð0Þ ð0Þ ðþ1Þ dij ¼ 1 if pj < r j 6 pj þ pj > > > > : ; ð0Þ ðþ1Þ 1 if pj þ pj 6 rj 6 1

Moreover, seekers enjoy the properties of pro-activeness behaviour: seekers don’t simply act in response to their environment; they are able to exhibit goal-directed behaviour. In addition, the future behaviour can be predicted and guided by past behaviours. As a result, the seekers may be pro-active to change his search direction and exhibit goal directed behaviour according to his past behaviour.

ð24Þ

ðmÞ

where rj is a uniform random number in [0, 1], pj ðm 2 f0; þ1; 1gÞ di;ego ðtÞ; is the percentage of the number of ‘‘m’’ from the set f~

2 39 RA 40

4

51

F1-19

F1-14 49

F1-26 26

F1-16 17

9

54

41

F1-13 25

F1-25

F1-12

52 15

8

F1-3

F1-11

RA

16

48

F1-4 F1-5 F1-9

47

RA

27

18

19

CB

RA

F1-23

38

24 43

7

F1-24

F2-2

6

5

13

F2-3 42 F2-4 46

F2-5

20

3

RA

37

32

F2-12 12

F2-23

F2-25

22

F2-24

10

F2-22

11

F2-13

33

28

14

36

30

Turkey

34

35 29

Circuit breaker

21

50

F2-21

CB

23

F2-16

F2-14

31

F2-7 F2-6 44

F2-18

53

45

F2-8

Raven Penguin

RA

RA Swan Substation

Fig. 15. Most economical network obtained using SOA for 54-bus system.

Automatic recloser Load node

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D. Kumar, S.R. Samantaray / Electrical Power and Energy Systems 63 (2014) 196–217

2 39

40

F1-18 F1-19 F1-16 9

54

41

F1-26

4

F1-8

51

F1-14 F1-15 17 F1-13 25

49 26

F1-7

RA

F1-12

15 8

16

52 F1-4 F1-3

F1-11

48

F1-6 F1-5

F1-9 47 F1-10

27

F1-2 18

F1-23

38

7

19

CB

F1-22 24

43

RA

13

6

5

46 10

F2-11

F2-25

33

23

F2-6 44

21

F2-18 F2-12 12

F2-23 22

F2-22

11

F2-13

28

RA

53

14

34

50 F2-16

F2-14 F2-21 31

F2-7 F2-5

20

37

32 F2-24

45

F2-4

F2-10

RA

F2-8

42 CB

3

RA F2-3

1

35 29

30

36

RA

Fig. 16. Most suitable network obtained using SOA for 54-bus system.

~ di;alt1 ðtÞ; ~ di;alt2 ðtÞ; ~ di;pro ðtÞg on each dimension j of all the four empiriðmÞ cal directions, i.e. pj = number of m/4.

discourse is a given set of numbers, i.e., 1, 2, . . ., s. The expression is presented as (25).

li ¼ lmax  Step length In the continuous search space, there often exists a neighbourhood region close to an extremum point. In this region, the fitness values of the input variables are proportional to their distances from the extremum point. It may be assumed that better points are likely to be found in the neighbourhood of families of good points, and search should be intensified in regions containing good solutions through focusing search [28]. Hence, from the standpoint on human searching, one may find the near optimal solutions in a narrower neighbourhood of the point with lower fitness value and, contrariwise, in a wider neighbourhood of the point with higher fitness value. Fuzzy systems arose from the desire to describe complex systems with linguistic descriptions [28]. According to human focusing search, the uncertain reasoning of human searching could be described by natural linguistic variables and a simple control rule as ‘‘If {fitness value is small} (i.e., the conditional part), Then {step length is short} (i.e., the action part)’’. The understanding and linguistic description of human searching makes fuzzy system a good candidate for simulating human focusing searching behaviour. To design a Fuzzy system to be applicable to a wide range of optimization problems, the fitness values of all the seekers are descendingly sorted and turned into the sequence numbers from 1 to s as the inputs of Fuzzy reasoning. The linear membership function is used in the conditional part since the universe of

s  Ii ðl  lmin Þ s  1 max

ð25Þ

where Ii is the sequence number of ~ xi after sorting the fitness values,

lmax is the maximum membership degree value which is equal to or a little less than 1.0. In this study, the Bell membership function 2 lðxÞ ¼ ex2 =2d is used in the action part. Thus, the membership degree values of the input variables beyond [3d, 3d] are less than 0.0111 (l(±3d) = 0.0111), and the elements beyond [3d, 3d] in the universe of discourse can be neglected for a linguistic atom. Thus, the minimum value lmin = 0.0111 is set. Moreover, the parameter, ~ d of the Bell function is determined by (26).

~ d ¼ w  absð~ xbest  ~ xrand Þ

ð26Þ

where abs() returns an output vector such that each element of the vector is the absolute value of the corresponding element of the input vector.The parameter w is linearly decreasing from wmax to wmin during a run in order to decrease the step length with time step increasing and improve the search precision.  gradually  w min wðtÞ ¼ w max  w max t, Where, t is the iteration number. The t max ~ xrand are the best seeker and a randomly selected seeker xbest and ~ from the same subpopulation to which the ith seeker belongs, respectively. Notice that ~ xrand is different from ~ xbest , and ~ d is shared by all the seekers in the same subpopulation. To introduce the randomicity on each dimension and improve local search capability, (27) is used to change li into a vector ~ li . Then the action part of

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D. Kumar, S.R. Samantaray / Electrical Power and Energy Systems 63 (2014) 196–217

2

RA

39 40

4

51

F1-19

54

41

F1-8

F1-14 17

9 25

F1-7

F1-25

F1-12

16

52

15 8

RA

49

F1-26 26

F1-16

F1-4 F1-3

F1-11

48

F1-6 F1-5

F1-9 47

RA

F1-10

27

18

7

38

19

CB

F1-22 24

43

F1-24 13

6

5

RA

1

CB

F2-3

45

F2-5

F2-6

42

CB

F2-4

F3-1

F2-7

46

10

20 33

RA 3

37

F3-10 32

F3-3

F3-9 22

F3-8

44 F3-18

23

21

34

28

12 F3-17 F3-7

53

11

F3-11 F3-12

14

F3-13

50 F3-14

F3-16

F3-6

35 29

31

36

30

Fig. 17. Most reliable network obtained using SOA for 54-bus system. Table 10 Performance comparison of the most reliable solutions obtained with different methodologies on 54 bus system. Algorithms

F

f1

f2

f3

f4

Max vol (p.u.)

Min vol (p.u.)

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.143216 0.104681 0.115061 0.106036 0.100886 0.097814 0.097814

0.225962 0.191512 0.218623 0.190218 0.191605 0.190605 0.190605

0.192558 0.151961 0.158427 0.159028 0.140169 0.131500 0.131500

0.0799000 0.0014955 0.0012302 0.0013092 0.0012669 0.0010131 0.0010131

0.0084 0.0067 0.0085 0.0050 0.0055 0.0049 0.0049

1 1 1 1 1 1 1

0.9916 0.9933 0.9915 0.9950 0.9945 0.9951 0.9951

the Fuzzy reasoning (shown in Fig. 6) gives every dimension j of step length by (28).

lij ¼ RANDðli ; 1Þ

ð27Þ

qffiffiffiffiffiffiffiffiffiffiffi aij ¼ dij  lnðlij Þ

ð28Þ

Step 3: Modification in existing SOA or Seeker Mutation When solving the mono-objective problems, all seekers aim at the same goal and are synchronously attracted towards the extremum point. As a result, d presented in (26) can adaptively reflect

Table 11 Performance comparison of the most economical solutions obtained with different methodologies on 54-bus system. Algorithms

F

f1

f2

f3

f4

Max vol (p.u.)

Min vol (p.u.)

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.263200 0.167632 0.185284 0.180345 0.173941 0.146182 0.146182

0.195400 0.148748 0.142566 0.153163 0.154094 0.142465 0.142465

0.628000 0.404359 0.462694 0.443701 0.420603 0.340714 0.340714

0.07370 0.00210 0.01033 0.00123 0.00186 0.00114 0.00114

0.0072 0.0064 0.0082 0.0052 0.0058 0.0050 0.0050

1 1 1 1 1 1 1

0.9928 0.9936 0.9918 0.9948 0.9942 0.9950 0.9950

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D. Kumar, S.R. Samantaray / Electrical Power and Energy Systems 63 (2014) 196–217

Fig. 18. Bus voltage profiles for various algorithms on 54-bus system.

Table 12 Computing time details for various algorithms for 54 bus system. Algorithms

Shortest time (s)

Longest time (s)

Average time (s)

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

312.12 385.59 381.41 375.83 382.23 2349.8 312.12

421.54 451.85 450.16 421.14 432.12 3458.2 421.54

386.21 423.41 423.26 413.06 420.12 2465.1 386.21

the aggregation behaviours of the search population. But for the complex problems like PDS planning problem, many seekers may be stuck at a local optimum, and d may deteriorate the algorithms performance because it may sharply decrease with run time increasing and cannot help the seekers escape from a local optimum. In order to overcome from this limitation d can be tuned from the radius of the known region to the diameter of the known region and is expressed as follows: d0j ¼ w  ðX best  X worst Þ where,

Xworst is the worst value of the objective function in the same subpopulation to which the jth seeker belongs. Although the addition of the global search component to the overall search direction reduces the probability of being trapped at local optimum, and also it is beneficial to add an inter sub-population learning mechanism as defined in [28]. According to it, if there are K subpopulations, the K  1 worst seekers in each subpopulation are replaced by the best positions of the remaining K  1 subpopulation as shown in Fig. 7. Step 4: Movement of the search of a new planning solution [29]: After modification in existing SOA, we are getting a set of solution of size N ND . Now we can obtain a new solution from mutated solution by applying certain changes to its topologies (dominated and non-dominated solution). The following feasible changes are allowed to obtain new solutions, (1) remove a feeder by introducing a new feeder, (2) change the size of a selected feeder, (3) change the size of the substation and (4) remove or include a substation. These changes preserve the operation radiality of the new

Fig. 19. Convergence graphs of various algorithms on 54-bus system (objective function versus time).

209

D. Kumar, S.R. Samantaray / Electrical Power and Energy Systems 63 (2014) 196–217 Table 13 Results of overall objective function values for various algorithms on 100-bus system over 20 runs (p.u.). Algorithms

Best

Worst

Mean

Std. Dev.

h

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.24253 0.19569 0.16571 0.13434 0.13019 0.12226 0.12226

0.3204 0.7266 0.5125 0.6075 0.4799 0.4943 0.2691

0.2409 0.4269 0.0733 0.2579 0.2723 0.2942 0.0586

0.0824 0.1987 0.0826 0.1792 0.1039 0.1372 0.0418

1 1 1 1 1 1 –

Table 14 Results of cost function values for various algorithms on 100-bus system over 20 runs (p.u.). Algorithms

Best

Worst

Mean

Std. Dev.

h

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.1986 0.2470 0.2575 0.2581 0.2608 0.2411 0.2411

0.7165 0.9283 0.0489 0.6625 0.9342 0.4893 0.4817

0.51127 0.55630 0.03782 0.24630 0.58402 0.24732 0.24120

0.190480 0.243700 0.017195 0.182800 0.354838 0.101103 0.099100

1 1 1 1 1 1 –

Table 15 Results of reliability function values for various algorithms on 100-bus system over 20 runs (p.u.). Algorithms

Best

Worst

Mean

Std. Dev.

h

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.1969 0.2156 0.1639 0.1478 0.1574 0.1314 0.1314

0.4652 0.9700 0.7239 0.5250 0.6736 0.3061 0.2988

0.09017 0.46720 0.14330 0.26250 0.14199 0.07832 0.07419

0.130231 0.463700 0.199030 0.163800 0.185276 0.069921 0.059790

1 1 1 1 1 1 –

Table 16 Results of power loss values for various algorithms on 100-bus system over 20 runs (p.u.). Algorithms

Best

Worst

Mean

Std. Dev.

h

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.00180 0.00740 0.00048 0.00040 0.00300 0.00058 0.00058

0.3387 0.4613 0.0036 0.9300 0.5112 0.0993 0.0972

0.08659 0.09761 0.00543 0.23250 0.11075 0.04053 0.03979

0.102970 0.139730 0.004351 0.327000 0.141260 0.028986 0.028940

1 1 1 1 1 1 –

networks. For example, in Fig. 8 we are going to explain the abovementioned changes (1) and (2). An elementary network is represented in Fig. 8 with a substation at node 1 and six demand

nodes (nodes 2–7). The continuous segments are the existing feeder routes and dashed segments are additional feeder routes that could be part of new solutions. The search of a new solution starts in the aforementioned radial solution. At first, the search process selects a node. Let node 6 be selected. In this case the allowed movements corresponds to node 6 are either to remove the feeder from this route by introducing a new feeder subject to preserve the radiality constraint of a network that supplies power to node 6 (feeder on node (7, 6) or feeder on the route (3, 6)) or change the size of the feeder on the route (7, 6). Since the set of solutions that can be obtained from a given mutated solution is usually composed of large set of solutions, we can select the best seekers of size N ND , according to their fitness function value. Mixed variable handling methods Basic form of the proposed SOA algorithm can only handle continuous variables. However both formation of radial topologies and placement of automatic reclosers are discrete or integer variables in optimal primary distribution system planning problem. In this paper SOA has been extended to handle mixed variables. To handle integer variables, simply truncating the real values to integers to calculate fitness value will not affect the search performance significantly. The truncation is only performed in evaluating the fitness function. That is the seekers will move in a continuous search space regardless of the variable type. For discrete variables of the ith particle zi the most straightforward way is to use the indices of the set of discrete variables with nD elements.

Z Di ¼ ½Z Di;1 ; Z Di;2 ; . . . ; Z Di;nD  Let Z Ci denote the continuous variables with nC elements:

Z Ci ¼ ½Z Ci;1 ; Z Ci;2 ; . . . ; Z Ci;nC  Then particle i is denoted by Z i ¼ ½Z Ci ; Z Di . For particle i, the index value j of the discrete variable Z Di;j is then optimized instead of the variables directly. In the population, the indices of the discrete variables of the ith particles should be the float point variables before truncation. That is, j e [1, nD1], nD is the number of discrete variables. Hence the objective function of the ith particle zi can be expressed as follows: f(zi), i = 1, 2, . . . M where,

Z i ¼ ½Z Ci;1 ; Z Ci;2 ; . . . ; Z Ci;nC ; Z Di;1 ; Z Di;2 ; . . . ; Z Di;nD  (

Z Ci:j ¼ Z i:j ;

Z i:j 2 Z Ci j ¼ 1; 2; . . . ; nC

Z Di:j ¼ Z i:INTðjÞ ; Z i:INTðjÞ 2 Z Di j 2 ½1; nD1  where Z Ci 2 RnC and Z Di 2 RnD denote the feasible subsets of continuous and discrete variables of particle Zi, respectively. INT(x) denotes the greatest integer less than the real value. The basic algorithm of SOA is shown in the form of flow chart in Fig. 9.

Table 17 Performance comparison of the best solutions obtained with various algorithms on 100-bus system. Algorithms

F

f1

f2

f3

f4

Max vol (p.u.)

Min vol (p.u.)

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.24253 0.19569 0.16571 0.13434 0.13019 0.12226 0.12226

0.5820 0.4283 0.2958 0.3061 0.2908 0.2817 0.2817

0.2192 0.2158 0.2519 0.2483 0.2427 0.2124 0.2124

0.00513 0.00740 0.00048 0.01030 0.02290 0.00058 0.00058

0.0057 0.0049 0.0065 0.0098 0.0078 0.0046 0.0046

1 1 1 1 1 1 1

0.9943 0.9951 0.9935 0.9902 0.9922 0.9954 0.9954

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Table 18 Performance comparison of the most reliable solutions obtained with different methodologies. Algorithms

F

f1

f2

f3

f4

Max vol (p.u.)

Min vol (p.u.)

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.24696 0.22605 0.17099 0.21475 0.20928 0.15821 0.15821

0.6165 0.5283 0.3989 0.5625 0.5342 0.3917 0.3917

0.1969 0.2156 0.1639 0.1478 0.1574 0.1314 0.1314

0.00680 0.00749 0.00128 0.00380 0.00320 0.00158 0.00158

0.0079 0.0069 0.0095 0.0045 0.0058 0.0048 0.0048

1 1 1 1 1 1 1

0.9921 0.9931 0.9905 0.9955 0.9942 0.9952 0.9952

Table 19 Performance comparison of the most economical solutions obtained with different methodologies for 100-bus system. Algorithms

F

f1

f2

f3

f4

Max vol (p.u.)

Min vol (p.u.)

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

0.26501 0.37012 0.30232 0.24021 0.28970 0.16329 0.16329

0.1986 0.2470 0.2575 0.2581 0.2608 0.2411 0.2411

0.6652 0.9700 0.7239 0.5250 0.6736 0.2988 0.2988

0.02180 0.01890 0.03158 0.02130 0.04120 0.00168 0.00168

0.0075 0.0062 0.0079 0.0051 0.0057 0.0049 0.0049

1 1 1 1 1 1 1

0.9925 0.9938 0.9921 0.9949 0.9943 0.9951 0.9951

Table 20 Average computing time details for various algorithms for 100 bus system. Algorithms

Shortest time (s)

Longest time (s)

Average time (s)

SGA PSO-w PSO-cf CPSO SPSO GTA SOA

492.12 485.79 481.71 475.86 482.27 2465.4 412.19

621.54 451.85 550.16 521.14 532.12 3501.2 491.54

486.21 483.84 498.26 482.06 493.12 2651.3 423.21

Implementation of SOA for designing advanced power distribution network Step 1: Read input parameters of the system and specify the lower and upper boundaries of each variable. Step 2: Initialize the good acceptable initial set of seekers of size N IS using fundamental loop generator that preserve the radiality operation. Set the time step t = 0. Step 3: Calculate the fitness function value of each seeker using the objective function defined in (15) based on the results obtained using backward/forward load flow program [27] and the proposed mixed variable handling method [28]. Obtain the initial historical best position of each seeker. Set the historical best position as the current best position of each seeker. Step 4: Update the time step t = t + 1. Step 5: Determine the neighbours, search direction and step length of each seeker as defined in Section ‘Seeker optimization algorithm’ (Sub-section ‘Implementation of seeker optimization algorithm’, step 2). Step 6: Update the position of each seeker (19). Step 7: Calculate the fitness function value of new positions using the objective function (15). Update the historical best position of each seeker Step 6: Perform seeker mutation using the inter sub-population learning scheme. Step 7: Perform ‘‘seeker movement’’ operation (step 4 of Subsection ‘Implementation of seeker optimization algorithm’). Step 8: Now select the best seekers of size N ND from the set of combined solution (‘‘seeker mutation’’ and from ‘‘seeker movement’’ operation) depending on their fitness value and consider it as the new population set Xt+1.

Step 9: If the stopping criteria is satisfied then go to Step 10. Otherwise, go to Step 4 Step 10: Seeker with the minimum fitness value in the last generation is the optimum output Formation of radial configurations using seeker optimization algorithm The proposed methodology creates feasible topologies using topological analysis [30]. It is made to identify the fundamental closed loops of the system in order to originate the radial topologies. When analyzing meshed networks, the number of fundamental loops (NFLs) is

NFL ¼ NB  N þ 1

ð29Þ

where NB is the total number of branches in the system, and N is the total number of nodes available. Eq. (29) also indicates the total number of elements to be disconnected in a meshed distribution network in order to obtain the radial topology.

Loop1 ¼ ½L1 L3 L5  Loop2 ¼ ½L3 L4 L6 L7  Loop3 ¼ ½L2 L4 L7 L8  The NFLs vectors of a network are defined as an group of elements that form a closed loop in a circuit that does not contain any other closed loop. For Fig. 10, the close loops selected are

Loop1 ¼ ½L1 L3 L5  Loop2 ¼ ½L3 L4 L6 L7  Loop3 ¼ ½L2 L4 L7 L8 

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In order to create a radial topology, one should select from the group of fundamental loop vector elements to be disconnected (one for loop). It is important that the vector’s elements are not repeated in a selected topology. The combination of elements can be generated with these vectors, creating all possible radial topologies of the system, which satisfies the radiality constraints, for Fig. 10 (condition 1: The solution must have 5 (total number of load nodes) number of branches. condition 2: the solution must be connected, that means the solution must satisfy the conservation of power flow constraints (it must supply the power demand at every load bus, so that a path between the substation and each other bus exists). Therefore, every bus is linked with the substation bus, forming a connected graph. For the system shown in Fig. 10, the total number of feasible radial topologies, using the proposed method, is 30, while 56 topologies were observed using a random methodology. This is one of the great advantage of the proposed method. These considerations allow the proposed SOA to limit the generation of non-feasible individuals. This also reduces the combinatorial searching space. One topological presentation is shown in Fig. 11, which satisfies the radiality constraint. One radial topology represents one seeker. For a 6 bus system with one source node, the dimension of each seeker is 5. The basic SOA structure for formation of new set of seeker (new feasible radial topologies) is shown in Fig. 12, which forms a radial configuration consist of 5 branches. From, Fig. 12, Start and End nodes represents the line start and end bus numbers. Now, the search direction and step length for each dimension of a seeker is evaluated, as defined in Section ‘Seeker optimization

algorithm’ (step 2, Sub-section ‘Implementation of seeker optimization algorithm’). Now, the position of each dimension of a seeker is updated according to (19), which forms one new radial topology, satisfies the radiality constraint. Now, the backward/forward load flow program is carried out to evaluate the fitness function (15).

Simulation results The proposed research presents, a new multi-objective design of PDS using seeker optimization algorithm. This algorithm can be used for the multi-objective optimal design of contingency based distribution system considering ‘n’ objectives, although the proposed work focuses only on four different sub-objectives (economic cost, overall system reliability, system power losses and voltage deviation) have considered. The performance of the proposed approach is assessed and illustrated on 54-bus and 100bus distribution systems, with a proposed substation capacity of 4 MVA and 18 MVA serving a total load demand of 2.032 MVA and 12.63 MVA, respectively [19]. The proposed method is compared mainly with the three algorithms PSO, GA and their recently modified versions and GTA. PSO have gained a lot of attention in various non-linear, complex multimodal system applications because of its faster convergence, reduced memory requirements, lower computational complexity and easier implementation as compared to other evolutionary algorithms. The proposed design problem does not consider interruption cost into account, as the occurrence of interruption

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of feeder branches are very less in planning problem. In the work of Angeline [31] it is shown that the PSO performs well in early generations than any other evolutionary algorithms, but it degrades as the number of generation increases. Therefore it has a slow finetuning ability of the solution. Since the original PSO is prone to suffer from the so-called ‘‘explosion’’ phenomena [32], two improved versions of PSO, i.e., PSO with adaptive inertia weight (PSO-w) and PSO with a constriction factor (PSO-cf), were proposed by Shi and Eberhart [33] and Clerc and Kennedy [32], respectively. Considering that the PSO algorithm may easily get trapped in a local optimum when solving complex multi-modal problems, Xiao et al. [34] proposed a variant of PSO called Clonal particle swarm optimizer (CPSO), which is adept at complex multi-modal problems in the year of 2007. So, the compared PSOs includes simple PSO (SPSO), PSO-w (learning rate c1 = c2 = 2, inertia weight linearly decreased from 0.9 to 0.4 with run time increasing, the maximum velocity vmax: 20% of the dynamic range of the variable on each dimension), PSO-cf (c1 = c2 = 2.01 and constriction factor = 0.729844), and CPSO (inertia weight linearly decreased from 0.9 to 0.4 with run time increasing). Moreover, a standard binary coded genetic algorithm (SGA) is introduced in [11] are considered for comparison with SOA. Furthermore, a graph theoretical approach (GTA) [29] is also implemented for comparison purpose. The step-by-step process for development of GTA is defined as follows: 1. Store all data for the complete graph of available routes and solve the base power flow equations for the given topology considering maximum tolerable limits for voltage drops and branch loading.

2. Obtain the current passing through the each branch of a given meshed network. Set count = 1. 3. Form the weight matrix (W) of dimensionDn  Dn, where, Dn: Set of nodes. Weight of each branch is defined as (15). 4. Arrange edges of a given topology in a queue in increasing order of their weight. 5. Select from the graph, the smallest weight edge e e E in an empty matrix T  G. 6. Select the next smallest weight in a matrix T and repeat the same procedure for (n  1) number of edges, where, (n  1) is the total number of load nodes available in the network. 7. During the formation of radial network if any loop gets formed while selecting any branch, that particular branch is removed and the next smallest weight edge is considered such that no loops get formed. 8. Once the radial topology is obtained, optimized the position of RAs using binary coded genetic algorithm (BCGA) (population size: 100, crossover probability: 0.8, mutation probability: 0.02, maximum generation: 50, independent run: 20). Now, repeat the same procedure from steps 4 to 7 for ‘50’ number of counts. 9. Now, perform the radial power flow simulation for each of the resulted topology using backward/forward load flow program and obtain the objective function value defined in (15). The solution which possesses the best fitness value while satisfying all the system constraints, such as feeder and substation capacity limit, voltage limit constraints of different load points etc., will be selected as the minimum weight resulted network topology.

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All the algorithms were implemented on a compatible PC (CPU Intel core i5-2400, 3.10 GHz and 8GB of RAM) with 32 bit operating system (Windows 7 professional) and the program were developed on MATLAB R2010a platform. In the experiments, the same population size of 100 for the 54 and 100-bus system is considered; total of 20 independent runs and the maximum generations of 100 are considered. The value of g (yearly capital recovery rate) is considered to be 0.05 for both the systems. For SOA, other parameters such as lmax: 0.95, lmin: 0.0111, xmax: 0.8, xmin: 0.2 and the penalty factor ðkV : 500; kS : 500; kI : 500Þ are considered for both the test systems.

The 54-node network The results of 54-bus system are presented for a one year time range. The possible fundamental loops obtained is 52 according to Eq. (29). Total of 3000 numbers of feasible radial topologies have been obtained. Now, best 100 seekers, depending on their fitness function (15) have been selected for formation of new seekers. Total of four sub-populations have been created with equal sizes. Maximum of six RA’s have been used in the network to enhance the overall system reliability. Fig. 13 shows the location of load points on X–Y coordinate. Fig. 14 shows all the possible paths between the nodes. The substation is numbered as node-1. Table 1 show the power demands and X–Y coordinate of the distribution network nodes. The installation cost of substation and each RA are considered to be 750,000 $ and 15384.61 $, respectively. Different conductor size specifications are given in Table 2. Parameters

used for different methodologies are given in Table 3. CB’s specification is given in Table 4. The modal parameters are set as: ploss_max: 0.8 MW, ploss_min: 0.000001 MW, cost_max: 7.2E+06$, cost_min: 0.6E+06$, CLLI_max: 1.0, CLLI_min: 0.0, w1: 0.3, w2: 0.3, w3: 0.2, w4: 0.2. The system loads are Pload = 0.015225 p.u., Qload = 0.013398 p.u. For GTA, store all data for the complete meshed topology (Fig. 14) with conductor type of Penguin have selected (Table 2). To compare the proposed method with other evolutionary algorithms, the concerned performance indices including the best, worst, mean and standard deviation (Std.) of the overall and sub-objective function values are summarized in Tables 5–8. In order to determine whether the results obtained by SOA are statistically different from the results generated by other algorithms, the T-tests [35] are conducted. An h value of one indicates that the performances of the two algorithms are statistically different with 95% certainty, whereas h value of zero implies that the performances are not statistically different. The corresponding h values for overall fitness function, cost, reliability and active power losses are presented in Tables 5–8, respectively. The most suitable planning solutions from 20 independent runs for various algorithms are tabulated in Table 9, where DVL denotes the maximum voltage deviations of the proposed method in comparison with other algorithms. Table 5 indicates that SOA has the smallest Best, Mean, Worst and Std. values of overall fitness function (15) than all the other listed algorithms and the h values for SOA in comparison with others are equal to one except GTA. The best value obtained with GTA is same as obtained with the proposed approach. Table 6 demonstrates that SOA has the smallest Best, Mean, Worst and Std. values of overall

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Fig. 24. Bus voltage profiles for various algorithms on 100-bus system.

system cost than all the listed other algorithms except that SOA has the little poor best value than that of CPSO, little larger standard deviation value than SGA and the same best value than GTA with h = 1. Tables 7 and 8 results show that SOA performance for the two sub-objective values (overall system reliability, and system power losses respectively) is almost same or superior than all the other listed algorithms. Table 7 demonstrates that the SOA has the smallest Best, Mean, Worst and Std. values (overall sys-

tem reliability) than all the listed other algorithms except that SOA has the little larger standard deviation value than PSO-w and PSOcf, and the same best value than GTA. Table 8 demonstrates that the results obtained using SOA which shows the highest reduction of power loss than the other distribution networks that is obtained by the other approaches except GTA over 20 runs. The performance of GTA is almost same as obtained using SOA. The network obtained (a single feeder network) using

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215

Fig. 25. Convergence graphs of various algorithms on 100-bus system (objective function versus time).

SOA is the network with minimum cost is shown in Fig. 15. The most suitable planning solution obtained (a two feeder network) with SOA is shown in Fig. 16. The best possible reliable solution obtained (three feeder network) with SOA is shown in Fig. 17. Evaluation of all the objective functions f1 (the cost function), f2 (the reliability function: CLLI), f3 (active power loss function), and f4 (voltage deviation function), have been done using Eqs. (11)– (14), respectively. It is observed from Tables 9–11, the most suitable solution (Fig. 16) has a total system cost of $ 1,755,000 (f1: 0.1750), which is 7.06% less than most reliable network having total system cost of $ 1,857,960 (f1: 0.1906), and 18.84% higher than most economical network having total system cost of $ 1,467,900 (f1: 0.1315). The CLLI of most suitable solution is 0.1442 (f2: 0.1442), which is 9.66% higher than most reliable network (f2: 0.1315), and 57.59% lower than the most economical network (f2: 0.340714). Also, from Table 9 it is observed that the results obtained using SOA for most suitable solution has a total system cost of 1,755,000 $ (f1: 0.1750), which is 1.15% (1,775,460 $ (f1: 0.1781)), 0.59% (1,765,560 $ (f1: 0.1766)) , 1.04% (1,773,480 $ (f1: 0.1778)), 4.15% (1,830,900 $ (f1: 0.1865)), and 8.15% (1,910,760 $ (f1: 0.1986)) lower than most suitable networks obtained using SPSO, CPSO, PSO-w, PSO-cf and SGA respectively. The CLLI index of the most suitable solution is 0.1442 which is 0.76% (0.1453 (f2: 0.1453)), 11.48% (0.1629 (f2: 0.1629)), 12.13% (0.1641 (f2: 0.1641)), 23.29% (0.1880 (f2: 0.1880)), and 25.59% (0.1938 (f2: 0.1938)) lower than most suitable networks obtained using SPSO, CPSO, PSO-w, PSO-cf and SGA respectively. Also, while comparing with other system performance indices such as system power losses and maximum voltage deviation, results obtained using SOA is superior than the other listed algorithms. Similar observations can be made for the most reliable network (Table 10). From Table 10, it is observed that the results obtained using SOA for most reliable solution has a total system cost of 1,857,960 $ (f1: 0.190605), which is 0.36% (1,864,593 $ (f1: 0.191605)), 1.04% (1863979.2 $ (f1: 0.191512)), 9.05% (2,042,911.8 $ (f1: 0.218623)), and 11.16% (2091349.2 $ (f1: 0.225962)) lower than most reliable networks obtained using SPSO, PSO-w, PSO-cf and SGA respectively. While in comparison with CPSO, total system cost incurred by SOA is more than 0.14% (1855438.8 $ (f1: 0.190218)). The CLLI index of the most reliable solution obtained using SOA is 0.1312 which is 6.39% (0.140169

(f2: 0.140169)), 17.49% (0.159028 (f2: 0.159028)), 13.66% (0.151961 (f2: 0.151961)), 17.19% (0.158427 (f2: 0.158427)), and 31.86% (0.192558 (f2: 0.192558)) lower than most suitable networks obtained using SPSO, CPSO, PSO-w, PSO-cf and SGA, respectively. As, we can observe from Table 10, the cost of SOA is more than by 0.14% than CPSO, but the CLLI index result show that SOA results better reliability than CPSO. While comparing the power loss and voltage deviation objective functions, SOA outperforms all the other algorithms. The performance of GTA is same as obtained using SOA for most reliable solution. Similar observation can be made for most economical network (Table 11), which clearly shows the superiority of the results obtained using SOA in comparison with the other listed algorithms except GTA. The performance of GTA is same as obtained using SOA. The corresponding bus voltages for the most suitable solutions obtained using SOA and other listed evolutionary algorithms are illustrated in Fig. 18. Furthermore, from Fig. 18, it can be seen that all the bus voltages optimized by SOA are kept within acceptable limits. In this proposed work, the computing time at every function evaluation is recorded for various algorithms. The total time of each algorithm is summarized in Table 12. The average convergence curves for overall function value vs. computing time for all the algorithms are depicted in Fig. 19. From Table 12, it can be observed that the computing time of SOA is less than that of the other listed evolutionary algorithms. Fig. 19 shows that while comparing with other listed evolutionary algorithms, SOA has faster convergence speed and, on the contrary, needs less time to achieve the best fitness function value. Although CPSO has faster convergence speed at the earlier search phase, but the CPSO rapidly get trapped in premature convergence or search stagnation with the bigger reduction in fitness function value than that of SOA. In all the listed algorithms, GTA is more computationally intensive as obtained from Table 12. The drawback associated with GTA is that it can’t optimize the RA position simultaneously while forming the radial topology. It is a step-by-step approach, as defined in results section (Section ‘Simulation results’). We have optimized the RA position for the obtained radial topology through GTA by using BCGA. Although the results obtained are same for most economical, most balanced, and most reliable solution, but GTA cannot guarantee that the solution will always converge to the global

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solution as it depends on the initial topological structure (Fig. 14). Hence, from the simulation results, SOA is synthetically superior to the other algorithms with respect to computational complexity and convergence rate. The 100-node network To further evaluate the performance of the proposed method, a large 100-bus test system is used. In this case, each seeker has 109 dimensions, (99 branches, and 10 automatic reclosers). Fig. 20 shows all the possible paths between the nodes. The possible fundamental loops obtained using (29) is 81. Total of 3200 numbers of feasible radial topologies have been obtained. Now, best 100 seekers depending on their fitness function value (15), have been selected for formation of new seekers. Total of four subpopulations have created with equal sizes. The modal parameters are set as: ploss_max: 0.8 MW, ploss_min: 0.000001 MW, cost_ max: 9.2E+06$, cost_min: 0.2E+06$, w1: 0.3, w2: 0.3, w3: 0.2, w4: 0.2. The system loads are given as follows: Pload: 0.1084 p.u, Qload: 0.0676 p.u. For GTA, store all data for the complete meshed topology (Fig. 20) and conductor of type Penguin have selected (Table 2). The corresponding simulation results over 20 independent runs are summarized in Tables 13–16. The best planning solutions from 20 independent runs for various algorithms are tabulated in Table 17. From Tables 13–16, it can be seen that SOA has the smallest Best, Mean, Worst and Std. values for overall fitness function (Table 13) and reliability (Table 15) than all the other listed algorithms, and all the h values are equal to one except GTA (which shows the same best value for both overall fitness function and reliability function as compared with SOA). Table 14 indicates that SOA has the smallest Best, Mean, Worst and Std. values of overall system cost than all the other listed algorithms except that SOA has the same best value than GTA. Power loss function (Table 16) shows that SOA has the little larger worst and best value than PSO-cf, a little larger best value than CPSO, and same best value that of GTA. The network obtained (a single feeder network) is the network with minimum cost as shown in Fig. 21. The most suitable planning solution obtained (a two feeder network) with SOA is shown in Fig. 22 and the best possible reliable solution obtained (three feeder network) with SOA is shown in Fig. 23. It is observed from Tables 17–19 that the most suitable solution (Fig. 22) has a total system cost of 2,735,300 $ (f1: 0.2817), which is 26.58% higher than most reliable network having total system cost of 3,725,300 $ (f1: 0.3917), and 15.42% lower than most economical network having total system cost of 2,369,900 $ (f1: 0.2411). The CLLI of most suitable solution is 0.2124, which is 61.64% higher than most reliable network (f2: 0.1314), and 29.82% lower than the most economical network (f2: 0.2988). Also, from Table 17 it is observed that the results obtained using SOA for most suitable solution has a total system cost of 2,735,300 $ (f1: 0.2817), which is 2.91% (2,817,200 $ (f1: 0.2908)), 7.43% (2,954,900 $ (f1: 0.3061)) , 32.54% (4,054,700 $ (f1: 0.4283)), 4.43% (2,862,200 $ (f1: 0.2958)), and 49.70% (5,438,000 $ (f1: 0.5820)) lower than most suitable networks obtained using SPSO, CPSO, PSO-w, PSO-cf and SGA respectively. The CLLI index of the most suitable solution is 0.2124 which is 12.48% (0.2427 (f2: 0.2427)), 14.46% (0.2483 (f2: 0.2483)), 1.58% (0.2158 (f2: 0.2158)), 15.68% (0.2519 (f2: 0.2519)), and 3.10% (0.2192 (f2: 0.2192)) lower than most suitable networks obtained using SPSO, CPSO, PSO-w, PSO-cf and SGA respectively. While comparing with other system performance indices such as system power losses and maximum voltage deviation, results obtained using SOA is superior than the other listed algorithms. Similarly, the most reliable network (Table 18) and the most economical network (Table 19) results, which clearly show the

superiority of the results, obtained using SOA in comparison with the other listed algorithms except GTA. Although the results obtained are same for most economical, most balanced, and most reliable solution, but GTA cannot guarantee that the solution will always converge to the global solution because it depends on the initial topological structure (Fig. 20), and also it is more computationally intensive than others. The corresponding bus voltages for the most suitable solutions obtained using SOA and other listed evolutionary algorithms are illustrated in Fig. 24. From Fig. 24, it can be seen that all the bus voltages optimized by SOA are acceptably kept within the limits. As the voltage profile for both SOA and GTA are same for most suitable solution obtained. Thus GTA performance has not been shown in Fig. 24. The average convergence curves for overall function value vs. computing time for all the algorithms are depicted in Fig. 25. Fig. 25 shows that SOA has faster convergence speed and needs less time to achieve a better overall function value than all the other listed evolutionary algorithms. The computing time of each algorithm is summarized in Table 20, which clearly shows that SOA is computationally less intensive as compared to other listed algorithms. In all the listed algorithms, GTA is more computationally intensive. Hence, from the simulation results, conclusion can be drawn that SOA is better than, or comparable to, all the other listed algorithms in terms of global search ability and convergence speed. Conclusion In this paper, a new methodology based on SOA is proposed to design advanced power distribution system which considers weighted aggregation of (total system economic cost, overall system reliability, system power losses and voltage deviation) as an objective function. The SOA uses an original structure of ‘movement operation’ on network nodes of the planning solutions, enabling the feasibility of the searched solutions to be maintained. Extensive tests are carried out in order to find the efficacy of the proposed algorithm to design a real PDS including distribution automation devices such as automatic reclosers. The simulation results show that SOA has better performance in balancing global search ability and convergence speed than the other evolutionary algorithms. Nevertheless, the method presented in this paper can be effective and helpful to system planners for obtaining typical designs of an advanced power distribution system. Acknowledgement This work is fully supported by Inspire Programme (DST/ INSPIRE Fellowship/2012/224), DST, Ministry of Science and Technology, New Delhi, India. References [1] Gonen T, Ramirez-Rosado IJ. Review of distribution system planning models: a model for optimal multi-stage planning. IEE Proc 1986;133(7):397–408. [2] Wall DL, Thompson GL, Northocote-Green JED. An optimization model for planning of radial distribution networks. IEEE Trans Power Apparat Syst 1979;PAS-98(3):1061–8. [3] El-Kady MA. Computer-aided planning of distribution substation and primary feeders. IEEE Trans Power Apparat Syst 1984;PAS-10(6):1183–9. [4] Boardman JT, Meckiff CC. A branch and bound formulation to an electricity distribution planning problems. IEEE Trans Power Apparat Syst 1985;PAS104(8):2112–8. [5] Youssef HK, Abu-El-Magd MA. Novel optimization model for long range distribution planning. IEEE Trans Power App Syst 1985;PAS104(11):3195–202. [6] Ponnavaikko M, Rao KSP, Venkata SS. Distribution system planning trough a quadratic mixed integer programming approach. IEEE Trans Power Del 1987;PD-2(4):1157–63. [7] Kumar D, Samantaray SR, Joos G. A reliability assessment based graph theoretical approach for feeder routing in power distribution networks

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