a powerful metaheuristic algorithm to solve static optimal...

International Journal on Electrical Engineering and Informatics - Volume 10, Number 3, September 2018

A Powerful Metaheuristic Algorithm to Solve Static Optimal Power Flow

Problems: Symbiotic Organisms Search

Anulekha Saha, Aniruddha Bhattacharya, Ajoy Kumar Chakraborty, and Priyanath Das

Department of Electrical Engineering, National Institute of Technology, Agartala, India

[email protected]

Abstract: This piece of work deals with implementing a new meta-heuristic algorithm

symbiotic organisms search to address multi-objective optimal power flow (OPF) problems

in power systems considering several operational constraints. The algorithm has been

implemented on IEEE 30 and IEEE 118 bus test systems for various single objective and

bi-objective functions to assess its efficacy in solving the OPF problem and its ability to

handle large systems. A comparative study of the results, predominantly considering those

obtained using quasi oppositional teaching learning optimization(QOTLBO), teaching

learning optimization (TLBO), multiobjective harmony search algorithm (MOHS), non-

dominated sorting genetic algorithm II (NSGA-II) from the literature are detailed in this

paper. Investigation of the results reveal that the algorithm is successful in producing

superior results for both the systems and its performance is also encouraging in solving

conflicting objectives.

Keywords: L-index, multi-objective optimization, optimal power flow, symbiotic organisms

search.

1. Introduction

Modern day power systems are planned to deliver power to the loads in a manner which is

both efficient and economical. Still, with ever increasing load demands, it becomes a necessity

to make the existing systems more robust in order to deal with the ever changing network

parameters. Since its inception, OPF problem has been receiving a lot of attention from the

researchers in the field of power system operation. OPF deals with minimizing the selected

objective function (OF) while satiating different constraints. Equality and inequality constraints

respectively refer to the load flow equations and the bounds of dependent and independent

variables. Literature presents numerous procedures for handling the OPF. Reduced gradient

method, Newton-Raphson, Lagrangian relaxation, linear programming, interior point method

[1]-[3] etc are the few available classical techniques. However, these methods are unable to

handle complex systems with non-convex, non-differentiable, and non-smooth OFs and

constraints. Heuristic algorithms are more sought after, as they are capable to solve the non –

linear problems. [4] – [17] presents various available heuristics which have been used in the

past to solve the complex OPF. These are: evolutionary programming (EP), differential

evolution (DE) [5], hybrid evolutionary programming (HEP) [6], tabu search [7], genetic

algorithm (GA) [8], particle swarm optimization (PSO) [9], bacteria foraging optimization

(BFO) [10], biogeography based optimization (BBO) [11], chaotic ant swarm optimization

(CASO) [12], harmony search algorithm (HSA) [13], teaching learning based algorithm

(TLBO) [14], gravitational search algorithm (GSA) [16] and quasi-oppositional teaching

learning based optimization (QOTLBO) [17]. Many algorithms apart from those mentioned

have also been used for solving the OPF, but have not been mentioned here for brevity. Only

those algorithms have been mentioned with which the results were compared. Many high end

soft computing techniques available in the literature have been applied to multi-objective

optimization (MOO) problems with varied success rates. PSO was applied to address the

problem of MOO by M.A. Abido [15] in 2012. [17]- [20] refers applications of TLBO,

*: Corresponding author

Received: July 2nd, 2017. Accepted: September 23rd, 2018

DOI: 10.15676/ijeei.2018.10.3.10

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QOTLBO, NSGA-II, BBO and MOHS to solve MOO problems. In designing an MOO

problem, conflicting objectives are chosen and a compromising solution is arrived at. In [17],

TLBO and QOTLBO were employed to arrive at the best negotiating result amongst

contradictory objectives. A MOO genetic algorithm, centered on NSGA-II, in [18], was used to

treat the conflicting objectives. BBO algorithm [19], was implemented to solve MOO OPF for

small, medium and large scale test systems. Ref [20] presents multi-objective harmony search

(MOHS) algorithm formulated as a non-linear and constrained multi-objective optimization

problem. The Pareto optimal front was obtained based on fast elitist non-dominated sorting and

crowding distance. A compromising solution from the Pareto set was arrived at applying a

fuzzy based mechanism.

The present paper attempts to solve the OPF problem for various single objective and bi-

objective functions by the relatively new symbiotic organisms search (SOS) algorithm. In

section 2, the problem formulation of OPF is discussed at length. Section 3 provides an insight

into the proposed SOS and its benefits in comparison with available meta-heuristic algorithms.

Formulation of the SOS algorithm for the OPF problem is discussed in Section 4. Section 5

elaborates the simulation results obtained followed by conclusion in Section 6.

2. OPF Problem design

The OPF problem may be designed in the following manner:

min ( , )F a b (1)

Subject to ( , ) 0g a b =

(2)

and ( , ) 0h a b

(3)

where, F represents the OF, and a and b are respectively the vectors representing the dependent

and independent variables. g and h respectively represents the set of equality and inequality

constraints.

a constitutes the slack bus power PG1, the generator delivered reactive power QGi, voltage at

the load bus VLi, and loading on the transmission line SLi:

1 1 1 1, ,...... , ,....... , ,......T

G L LPQ G GPV L LTLa P V V Q Q S S =

(4)

b constitutes the real power outputs excluding the slack bus PGi, shunt VAR compensator

output QCi, generator bus voltage VGi, and transformer tap setting TCi:

2 1 1 1,...... , ,...... , ,...... , ,........G GPV G GPV C CNC NT

Tb P P V V Q Q T T= (5)

where, PV, NT, PQ, TL , and NC represent respectively the total generator buses, transmission

lines , load buses, and tap changing transformers.

g represents the load flow equations as follows:

( )=

+=−

NBUS

j

ijijijijjiLiGi BGVVPP

1

coscos (6)

( )=

+=−

NBUS

j

ijijijijjiLiGi BGVVQQ

1

cossin

(7)

where, i= 1,2,3,…..NBUS.

where, PGi and QGi respectively represent real and reactive power supplied to the network, PLi

and QLi represent respectively the demands of real and reactive power at the ith bus, Gij and Bij

denote the conductance and susceptance, 𝜭ij denotes the voltage phase angle difference of the

ith and jth buses and NBUS represents the total buses constituting the system.

h is represented as follows:

Anulekha Saha, et al.

586

Generator Constraints: maxmin

GiGiGi VVV

(8)

maxmin

GiGiGi PPP

PVi ,....3,2,1=

(9)

maxminGiGiGi QQQ (10)

where, PV signifies the total generator buses inclusive of slack bus.

Transformer Constraints:

maxmin

iii TTT

NTi ,....3,2,1=

(11)

where, NT represents the number of tap changing transformers.

Shunt VAR Compensator Constraints:

maxmincicici QQQ

NCi ,....3,2,1=

(12)

where, NC denotes the total number of shunt compensators connected to the system.

Security Constraints:

maxminLiLiLi VVV

PQi ,.....3,2,1=

(13)

maxLiLi SS

TLi ,......3,2,1=

(14)

where, PQ and TL denote the total number of load buses and transmission lines in the system.

Objective Functions:

Case 1: Generation Cost Minimization

Generation cost is a function of generator real power outputs as follows:

( ) ( ) ( )

++=

== ==

GG N

i

iiiii

N

i

ii PcPbaPFPFG

1

2

1

1 )(min (15)

where, iP represents the power output of the i th generator, ( )ii PF denotes the running cost of the

i th generator, iii cba ,, symbolizes the cost coefficients of the i th generator and NG represents

the number of committed generators.

Case 2: Real Power Loss Minimization

The real power loss ensued in the course of power transmission can be represented as:

( )=

−+==

LN

m

ikkikimL VVVVGPG

1

222 cos2)min( (16)

where, mG , iV and kV represent respectively the conductance of line m linking buses i and k ,

and magnitude of the voltages at buses i and k ; LN and ik represents respectively the number

of transmission lines and the angle difference between the two buses.

Case 3: Voltage stability index (L-index) minimization

L-index of a node j can be expressed as:

)min(3 LG = (17)

j

i

N

i

jijV

VFL

G

=

−=

1

1

where, j = 1,2,3….NL; NL being the number of load buses;

1

21

1−−

−= YYF ji

where, jiF is the sub matrix found after partially inverting the YBUS matrix.


587

Case 4: Voltage Deviation (VD) Minimization

VD of the load buses is computed as the deviation from the reference voltage of 1 p.u. and may

be expressed as:

=

−==

LN

i

refii VVVDG

1

4 )(

(18)

where, NL denotes the number of load buses; ref

iV is the specified reference value of the

voltage magnitude at the ith load bus and is usually set to be 1.0 p.u.

Case 5: Emission Minimization

This objective considers minimizing the emission of all types of pollutants to the atmosphere.

A linear model for emission minimization as provided in [1] has been considered for the sake

of comparison.

=

=

GN

k

kk PG

1

4 (19)

where, k represents the emission coefficient related to the kth generator.

Multi-objective Optimization:

Multi-objective optimization is done to optimize conflicting OFs in a way that both objectives

are equally compromised to attain the solution. Best feasible solutions for both the objectives

are obtained after satisfying various operational constraints and the solution sets so obtained

are known as the Pareto-optimal set. A multi-objective optimization problem having ‘m’

number of objectives and ‘n’ number of constraints can be defined as follows:

)](),.......(....),........(),([ 21 xGxGxGxGMinG mi= ; nxxxx ,......, 21= (20)

where, )(xGi represents the ith OF and x represents the control variables of the OFs.

Case 6: Simultaneous reduction of fuel cost and transmission loss

The bi-objective function for simultaneously minimizing the fuel cost and the transmission loss

can be represented as follows:

],min[ 211 GGGbi =

(21)

Case 7: Minimization of fuel cost and voltage stability index

The bi-objective function for the simultaneous minimization of fuel cost and voltage stability

index can be represented as follows:

],min[ 312 GGGbi =

(22)

Case 8: Minimization of total fuel cost with the minimization in voltage deviation

The OF for the simultaneous minimization of voltage deviation and the total fuel cost is

represented by the following equation:

],min[ 413 GGGbi =

(23)

Case 9: Simultaneous reduction of transmission loss and voltage stability index

The bi-objective function for the minimization of transmission line loss and voltage stability

index (L-index) is represented below:

],min[ 324 GGGbi = (24)

Case 10: Effect of minimization of voltage deviation on minimization of L-index

To study the effect of the minimization of voltage deviation on minimizing the L-index, the

following bi-objective function is considered:


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],min[ 435 GGGbi =

(25)

(6)-(14) represents the constraints for the above objectives.

3. Symbiotic Organisms Search (SOS) algorithm

Basic concept:

Symbiotic organisms search is a meta-heuristic algorithm applied to numerical and

engineering design problems. It simulates the symbiotic interface schemes embraced by

organisms in order to endure and proliferate in the ecosystem [23]. Almost all meta-heuristic

algorithms available in the literature share some common characteristics of being inspired from

the nature, making use of random variables and several other parameters that need to be

adjusted to the problem. An advantage of SOS algorithm over other meta-heuristic algorithms

lies in the fact that algorithm specific parameters are not required.

Symbiosis refers to the reliance or dependency-based relationship that exists between

organisms in nature. It describes the cohabitation behavior of organisms belonging to different

species. Symbiotic relationships provide at least one of the contributing species with a nutritive

advantage.

The three most common types of interdependent relationships found in nature are:

mutualism, commensalism and parasitism. Mutualism explicates that relationship amongst two

different species where both derive benefit from one another. Commensalism refers to that

where only one species is benefited and the other remains unaffected. Parasitism represents the

symbiosis where only one derives benefits at the expense of the other.

Symbiosis helps organisms to adapt themselves to the ever changing ecosystem thereby

helping the organisms increase their fitness level and chances of survival in the long run.

Figure 1. Examples of Mutualism, Commensalism and Parasitism seen in nature

The SOS Algorithm:

SOS algorithm iteratively uses a population of candidate solutions in the propitious areas in

search space in order to find the global optimal solution. SOS starts with an initial population

known as the ecosystem where a random group of organisms is generated to the search space,

each organism representing a candidate solution to the corresponding problem. The size of the

ecosystem, known as ecosize is defined by the number of constraints to be satisfied. Organisms

in the ecosystem are assigned certain fitness values that reflect their degree of adaptation to the

desired objective. In SOS, new solution sets are generated by mimicking the biological

interactions between two organisms in the ecosystem. The three phases in SOS are namely:

mutation phase, commensalism phase, and parasitism phase. Random interaction between

organisms take place in all the phases until termination criterion are met.

The following are the three phases of the proposed algorithm:

Mutualism Phase:

In mutualism phase in SOS, an organism Xi is matched to the ith member of the ecosystem.

Another organism Xj is randomly selected from within the ecosystem to interact with organism

Xi. Both the organisms get engaged in a mutualistic relation with a common goal of increasing

their chances of survival in the ecosystem. Based on the mutualistic relationship between the

organisms Xi and Xj, new candidate solutions for the organisms are calculated as shown below:


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)*_(*)1,0( 1BFVectorMutualXrandXX bestiinew −+= (26)

)*_(*)1,0( 2BFVectorMutualXrandXX bestjjnew −+= (27)

2

_ji XX

VectorMutual+

= (28)

where, rand(0,1) denote a vector of random numbers. BF1and BF2denote the benefit factors that

each organism has over the other. Mutual_Vector represents the mutualistic relationship that

the two organisms share.

Benefits derived by both the organisms sharing a mutualistic relationship are not the same.

One organism derives greater benefits than the other. Here benefit factors (BF1 and BF2) are

determined randomly as 1 or 2, denoting the level of benefit to each organism i.e. if an

organism is deriving full or partial benefits from the interaction.

Commensalism phase:

In this phase, organism Xj is selected to interact with organism Xi obtained from the

mutualism phase. In this phase, organism Xi tries to derive benefit from the interaction while

the organism Xj remains neutral. Organism Xi is updated only if its present fitness is better

than the previous fitness. Fitness of Xi is calculated as follows:

)(*)1,1( jbestiinew XXrandXX −−+=

(29)

Parasitism phase:

In this phase, a Parasite_Vector is created by duplicating and modifying the dimensions of

the organism Xi with a random number. Organism Xj acts as the host and is selected randomly

from the ecosystem. Both the Parasite_Vector and host Xj try to replace each other from the

ecosystem and eventually, the one having higher fitness value survives and replaces the other

in the ecosystem.

4. SOS algorithm for the OPF problem

The SOS algorithm [23] is effective in handling multiple variables. Initially, a population is

randomly generated in the search space. Each organism in the ecosystem comprises of the

control variables namely: generator real power, generator bus voltages, tap ratio of tap

changing transformers, and reactive power delivered by the shunt compensating transformers.

Initialization of ecosystem: Each element of the ecosystem is randomly initialized within their

operating limits as per (8), (9), (11) and (12). The organisms in the ecosystem are updated in

each phase based on their fitness function. The modified ecosystems obtained from each phase

are used to calculate the OF after computing the dependent variable values employing Newton-

Raphson power flow method. The steps involved in solving the optimal power flow problem

are described below:

Step 1: Choose the size of the ecosystem, ecosize, by selecting the number of generators, tap

changing transformers and shunt compensators. Elements of the ecosystem are known as

organisms whereby each organism represents a candidate solution to the problem. Also

initialize the ecosystem for pre-specified ecosize and maximum function evaluation maxFE.

Step 2: Perform Newton-Raphson load flow using each organism of the ecosystem to obtain

the dependent variables as given in (4), and check whether they satisfy the inequality

constraints as given in equations (10), (13) and (14).

Step 3: Evaluate the fitness function for each organism set obtained. For OPF, fitness function

represents the generation cost, line loss etc.

Step 4: Update ecosystem in every phase as per (26)-(29) and evaluate the fitness function for

the updated ecosystem so obtained.


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Step 5: Obtain the best fitness and best organism. Best fitness is obtained as the minimum of

the fitness function evaluated for each solution set and best organism is obtained as the solution

set for which the best fitness is obtained.

Step 6: Go to step 4 and repeat till the predefined maxFE.

After modifying the ecosystem in step 4, its viability should be verified. They should

satisfy the constraints given by (8), (9), (11) and (12). The organisms are said to be feasible if

each organism and dependent variables satisfy the different operational constraints of the OPF

problem. If the new organisms in the ecosystem are feasible, then the dependent variables are

executed using those organisms. If however, the organism set is found to be infeasible, they

should be mapped to the feasible solution set as follows:

Let Hi be the ith control variable of the problem in hand. If Himax and Hi

min are respectively its

upper and lower limits, then the operating constraints are taken care of in the following

manner:

If output of ith control variable Hi>Himax,

Set Hi=Himax

If output of ith control variable Hi<Himin

Set Hi=Himin

After fixing the independent variables to their individual limits, the Newton-Raphson load

flow is run all over again to achieve the dependent variables. If after performing all the three

phases of the proposed algorithm, the dependent variable values are found to lie beyond their

limits, then that organism set is discarded and all three phases are reapplied to the old value till

all bounds and constraints are satisfied.

5. Simulation results

The machine specifications in which the algorithm was run are as follows:

➢ Processor = Intel Core i7

➢ RAM = 2GB

➢ Clock frequency = 3.4 GHz

The programming was done in MATLAB. IEEE 30 bus and IEEE 118 bus test systems

have been studied considering different objectives. For simulation of the OPF program using

SOS, ecosize of 50 is considered and the algorithm is made to run for 100 iterations in all the

cases. Pareto solution sets for the bi-OFs are also obtained. The results obtained using SOS are

compared with those obtained by basic teaching learning based optimization (TLBO), quasi

oppositional teaching learning based optimization (QOTLBO), non-dominated sorting genetic

algorithm (NSGA-II), as well as multi objective harmony search algorithm (MOHS).

IEEE 30 bus system:

Figure 2. Cost convergence using SOS algorithm


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The SOS algorithm has been applied to the standard IEEE 30 bus test system. Data and

constraints are as given in Ref [4]. The system under consideration has 6 generators positioned

at buses 1, 2, 5, 8, 11 and 13. Four tap changing transformers are placed in lines 6-9, 6-10, 4-12

and 28-27 within an interval (0.9, 1.1). Shunt VAR compensators for reactive power control,

having a capacity of 5 MVAR each are connected to buses 10,12, 15, 17, 20, 21, 23, 24 and 29.

The load bus voltages are to remain within limits of 0.95 - 1.05 p.u. Base MVA is taken as 100

MVA and the system load demand is 2.834 p.u. The algorithm satisfied the constraints as given

by (6)-(14) for this test case.

The simulation results for the single objectives obtained using SOS and their comparison

with other algorithms are listed in Table 1, Table 2, Table 3 and Table 4 respectively.

From the simulation results for cost minimization objective using SOS provided in Table 1,

it can be seen that the optimized fuel cost is 798.9152 $/hr which is the same as that obtained

using QOTLBO [17] as reported in the literature. However the algorithm required fewer than

15 iterations to converge, nearly half of what is required in [17].

Figure 3. Loss convergence characteristic for SOS algorithm

For the transmission loss minimization objective, the proposed algorithm is able to bring

down the loss to 2.8604 MW, which is 0.8 % lower than the previous best result of 2.8834 MW

[17]. For this case also, SOS took less than 25 iterations to convergence which is lower than

that observed in [17].

Figure 4. L-index convergence for the SOS algorithm


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The algorithm’s ability to improve the voltage stability index (L-index) of the system is

analyzed and it is observed to bring down the value of L-index to 0.0958 p.u, down by 3.75 %

from the previous reported minimum value of 0.0994 p.u [17].The plot showed a quicker rate

of convergence for this case too.

Figure 5. Voltage deviation convergence for the SOS algorithm

As for the voltage deviation minimization objective, results obtained using SOS surpassed

the previous obtained value of 0.38 p.u by a high margin of 78.1% thereby lowering the

voltage deviation value to 0.0830 p.u. Also the transmission loss is reduced.

To assess the strength of the proposed algorithm, statistical analysis of SOS has been done

for 50 independent trials. The statistical result comprising of the best, worst, average and

standard deviations for all the four objectives are then compared with those obtained using

QOTLBO [17], TLBO [17], and MOHS[20] and are listed in Table 5. It can be observed from

the above table that the best, worst and average of all the objectives obtained using SOS are

superior to those obtained using QOTLBO[17], TLBO[17], and MOHS[20]. Standard

deviation obtained using SOS is improved to a great extent for all the objectives.

The above case studies prove the superiority of SOS algorithm for optimizing single OFs.

Figure 6. Pareto front for bi-objective cost-loss minimization using SOS


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Figure 7. Pareto front for bi-objective cost-voltage stability index minimization using SOS

Simulation results for simultaneous minimization of fuel cost and L-index show the

superiority of SOS over other algorithms to which it is compared to. It managed to bring down

the total fuel cost by 0.04% and L-index by an even higher margin of 14.96%. Transmission

losses are also simultaneously minimized by 1.11%.

Figure 8. Pareto front for bi-objective cost-voltage deviation minimization using SOS

Figure 9. Pareto front for bi-objective transmission loss and L- index minimization using SOS


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Results obtained for bi-objective optimization of transmission loss and L-index using SOS

are compared with those obtained using PSO [8] as listed in Table 9. It is seen that the

compromising solution set obtained using SOS is much better than that obtained by PSO in

minimizing the objectives. The algorithm is able to effectively reduce the transmission loss by

44.82% and L- index by 19.89%.

IEEE 118 bus system:

In order to observe the performance of SOS in case of large systems, IEEE 118 bus test

system has been considered and three single OFs for fuel cost minimization, emission

minimization and transmission loss minimization are analyzed. Linear emission model as

provided in [17] is considered for emission minimization objective. This test system consists of

14 synchronous condensers, 54 generators, 10 tap changing transformers and 91 loads. System

configuration as provided in [25] has been considered for carrying out the simulations. (6)-(9)

and (11)-(14) are the constraints of this system.

Figure 10. Cost convergence curve for SOS

SOS showed its efficiency in reducing the fuel cost by a large margin of 14.30% and

brought down the cost from a whopping 55,968.14 $/hr [17] to 47,960 $/hr. Also, it effectively

reduced the emission from 410.9816lb/hr [17] to a much lower value of 342.635 lb/hr in case

of single objective optimization itself. Also the proposed algorithm showed rapid convergence

in fewer than 20 iterations to give the optimum result.

Figure 11. Emission Convergence curve for SOS


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Figure 12. Transmission loss convergence curve of SOS algorithm(IEEE 118 bus system)

The results obtained for emission minimization of IEEE 118 bus system have been listed in

Table 12. The algorithm also effectively brought down the emission from 176.1666 lb/hr [17]

to 164.5 lb/hr i.e, by a margin of 6.62%. Also it is able to bring down the fuel cost by 3.29%

from 65,601.64 $/hr [17] and transmission loss by 7.58% from the previous reported best value

of 150.9366 kW [17].

It can be seen from Table 13 that the proposed algorithm gave much better result for loss

minimization as compared to its predecessors. It is proficient in reducing the loss to 16.27 KW

from 35.3191 KW [17]. Simultaneously it is also able to reduce the emission by 6.5127 lb/hr.

From the convergence characteristics of the individual objectives of IEEE 118 bus system, it

can be seen that SOS achieved rapid convergence in fewer than 20 iterations.


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Case 1: Cost Minimization

Table 1. Simulation results for cost minimization using SOS Technique Technique

Control

variables SOS

QOTLBO [17]

TLBO [17]

MOHS [20]

BBO [22]

Control

variables SOS

QOTLBO [17]

TLBO [17]

MOHS [20]

BBO [22]

Gen

erat

or

Rea

l

Po

wer

Ou

tpu

t (M

W)

PG1 177.1022 177.1022 177.0817 177.196 177.0177

Sh

un

t C

om

pen

sato

r

Ou

tpu

t (p

.u)

QC12 0.05 0.05 0.0499 0.0497 0.05

PG2 48.6889 48.6889 48.6814 48.2275 48.641 QC15 0.05 0.05 0.05 0.0478 0.05

PG5 21.3026 21.3026 21.3098 21.2792 21.239 QC17 0.05 0.05 0.0498 0.0466 0.05

PG8 21.0286 21.0286 21.0302 21.2049 21.136 QC20 0.0429 0.0429 0.0431 0.0459 0.05

PG11 11.8621 11.8621 11.8843 11.6715 11.944 QC21 0.05 0.05 0.0499 0.05 0.05

PG13 12 12 12 12.3595 12.054 QC23 0.027 0.027 0.027 0.0423 0.04

Gen

erat

or

Vo

ltag

e

(p.u

)

VG1 1.1 1.1 1.0997 1.0997 1.1 QC24 0.05 0.05 0.0498 0.0497 0.05

VG2 1.0877 1.0877 1.0873 1.0829 1.0876 QC29 0.0231 0.0231 0.0231 0.0238 0.03

VG5 1.0613 1.0613 1.0603 1.0505 1.0614 T6–9 1.0399 1.0399 1.0407 1.0194 1.05

VG8 1.0691 1.0691 1.0685 1.0558 1.0695 T6–10 0.9 0.9 0.9 0.9015 0.9

VG11 1.1 1.1 1.0998 1.0972 1.0982 T4–12 0.9782 0.9782 0.9786 0.9857 0.99

VG13 1.1 1.1 1.0998 1.0978 1.0998 T28–27 0.9614 0.9614 0.9613 0.9558 0.97

QC10 0.05 0.05 0.05 0.0211 0.05 Cost($/h) 798.9152 798.9152 798.9329 798.8 799.1116

Transmission Loss(MW) 8.6076 8.5844 8.5874 8.6541 8.6671

L-index(p.u.) 0.1081 0.1263 0.1264 0.118 0.1075

Voltage Deviation(p.u) 2.0297 2.2153 2.2016 2.0807 2.1665

Bold font denotes the proposed algorithm.


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Case 2: Loss Minimization

Table 2. Simulation results for loss minimization using SOS Technique Technique

Control

variables SOS

QOTLBO [17]

TLBO [17]

MOHS [20]

Control

variables SOS

QOTLBO [17]

TLBO [17]

MOHS [20]

Generator

Real Power

Output

(MW)

PG1 51.25 51.3093 52.1027 52.5327

Shunt

Compensator

Output (p.u)

QC12 0.0349 0.0499 0.0498 0.0486

PG2 80 80 79.9387 79.5432 QC15 0.0276 0.0297 0.0497 0.0493

PG5 50 49.9794 49.9617 49.8152 QC17 0.0137 0.0499 0.0498 0.0488

PG8 35 34.9959 34.5287 34.7403 QC20 0.0277 0.0387 0.0403 0.0442

PG11 30 29.9988 29.9721 29.7884 QC21 0.0495 0.05 0.0496 0.0499

PG13 40 40 39.8304 39.948 QC23 0.0372 0.0273 0.0267 0.0411

Generator

Voltage

(p.u)

VG1 1.1 1.087 1.0798 1.0754 QC24 0.0368 0.05 0.0497 0.0499

VG2 1.0986 1.0825 1.0742 1.0728 QC29 0.0315 0.0207 0.0212 0.0317

VG5 1.0801 1.0632 1.0557 1.054

T6–9 1.0776 1.0309 1.0171 1.0022

VG8 1.0882 1.0707 1.0641 1.0637 T6–10 0.9064 0.9024 0.9 0.9078

VG11 1.0656 1.0998 1.0976 1.0991 T4–12 0.9828 0.9689 0.9681 0.9593

VG13 1.0986 1.0989 1.0989 1.0967 T28–27 0.9798 0.9584 0.9527 0.9533

QC10 0.0371 0.0495 0.0498 0.0499 Cost($/h) 967.0661 967.0371 965.7677 964.5121

Transmission

Loss(MW) 2.8604 2.8834 2.9343 2.9678

L-index(p.u.) 0.1082 0.1262 0.1264 0.1154

Bold font denotes the proposed algorithm and superior performance.


598

Case 3: L-index Minimization

Table 3. Simulation results for L-index minimization using SOS Technique Technique

Control

variables SOS

QOTLBO [17]

TLBO [17]

MOHS [20]

Control variables SOS QOTLBO

[17] TLBO [17]

MOHS [20]

Gen

erat

or

Rea

l

Po

wer

Outp

ut

(MW

)

PG1 1.3725 134.2408 76.788 92.6114

Sh

un

t C

om

pen

sato

r

Ou

tput

(p.u

)

QC12 0.042011 0.0499 0.0487 0.0492

PG2 0.24524 61.8427 63.3618 67.5094 QC15 0.048353 0.0369 0.0497 0.0496

PG5 0.36791 15 45.7092 48.8891 QC17 0.048146 0.05 0.0426 0.0499

PG8 0.32919 10 33.8121 34.8663 QC20 0.0305 0.0187 0.0437 0.05

PG11 0.22393 29.9687 29.9842 29.7139 QC21 0.045927 0.0042 0.0434 0.0497

PG13 0.36286 39.6304 37.4921 14.134 QC23 0.016497 0.0009 0.0193 0.0494

Gen

erat

or

Vo

ltag

e (p

.u) VG1 1.1 1.0832 1.0601 1.0993 QC24 0.029483 0.0005 0.0051 0.0494

VG2 1.075 1.0666 1.0463 1.0986 QC29 0.041407 0.0011 0.0406 0.0496

VG5 1.0349 1.0426 1.043 1.0973

T6–9 1.0927 0.9288 0.9646 0.9027

VG8 1.0478 1.0389 1.0443 1.0998 T6–10 1.0376 0.9 0.9602 0.9001

VG11 1.0386 1.0938 1.0986 1.0984 T4–12 1.0914 0.9442 0.92 0.9036

VG13 0.95604 1.0976 1.0926 1.0996 T28–27 0.90046 0.9082 0.9256 0.9011

QC10 0.05 0.0492 0.0463 0.0499 Cost($/h) 857.4869 844.1237 912.5914 895.6223

Transmission

Loss(MW) 6.7653 7.2826 3.7474 4.3244

L-index (p.u) 0.0958 0.0994 0.1003 0.1006



599

Case 4: Voltage Deviation

Table 4. Simulation results for voltage deviation minimization using SOS Technique Technique

Control variables SOS NSGA-II[18] Control variables SOS NSGA-II[18]

Gen

erat

or

Rea

l

Po

wer

Outp

ut

(MW

)

PG1 89.46 x

Sh

un

t C

om

pen

sato

r

Ou

tput

(p.u

)

QC12 0.00016 x

PG2 79.963 x QC15 0.049962 x

PG5 49.769 x QC17 9.74E-06 x

PG8 34.881 x QC20 0.049496 x

PG11 21.558 x QC21 0.030207 x

PG13 12.909 x QC23 0.049979 x

Gen

erat

or

Vo

ltag

e (p

.u) VG1 1.0079 1.03 QC24 0.05 x

VG2 1.0046 1.03 QC29 0.033662 x

VG5 1.0173 1

T6–9 1.0142 1

VG8 1.0074 1 T6–10 0.98527 1.01

VG11 0.99898 1.02 T4–12 0.97474 1

VG13 1.0074 1.04 T28–27 0.97944 1.04

QC10 0.030257 x Voltage Deviation(p.u.) 0.083 0.38

Transmission loss(MW) 5.1516 5.3513



600

Table 5. Statistical comparison

Objectives SOS QOTLBO

[17]

TLBO

[17]

MOHS

[20]

Cost($/hr)

Best 798.9152 798.9152 798.9329 798.8

Worst 799.1034 801.1229 803.0125 NA

Average 798.9604 799.0037 801.2198 NA

Standard

deviation 0.0812 3.3962 5.6283 NA

Loss Minimization(MW)

Best 2.8422 2.8834 2.9343 2.9678

Worst 2.8427 2.9476 3.1639 NA

Average 2.8424 2.9043 2.9946 NA

Standard

deviation 2.49E-04 3.7528 5.3792 NA

L-index Minimization(p.u)

Best 0.092613 0.0994 0.1003 0.1006

Worst 0.096033 0.1015 0.1096 NA

Average 0.094 0.0999 0.1011 NA

Standard

deviation 0.0017 3.2041 5.0897 NA

Voltage Deviation

Minimization(p.u)

Best 0.07986 NA NA NA

Worst 0.08308 NA NA NA

Average 0.0808 NA NA NA

Standard

deviation 0.0015 NA NA NA


601

Case 5: Simultaneous reduction of fuel cost and transmission loss

Simulation results for the bi-objective optimization of fuel cost and transmission loss are listed in Table 6.

Table 6. Simultaneous minimization of fuel cost and transmission loss using SOS

Technique

Technique

Control

variables

SOS QOTLBO

[17]

TLBO

[17]

NSGA-

II[24]

Control

variables

SOS QOTLBO

[17]

TLBO

[17]

NSGA-

II[24]

Gen

erat

or

Rea

l

Po

wer

Outp

ut

(MW

)

PG1 126.18 124.179 122.5831 134.5544

Sh

un

t C

om

pen

sato

r

Ou

tput

(p.u

)

QC12 0.05 0.0498 0.05 0.0491

PG2 51.92 51.7127 52.1608 46.2891 QC15 0.0499 0.0489 0.0444 0.0442

PG5 30.18 30.8638 31.2324 32.936 QC17 0.05 0.0498 0.05 0.0466

PG8 35 34.9488 35 30.1163 QC20 0.0419 0.0441 0.0397 0.0384

PG11 25.34 26.1841 26.5497 18.735 QC21 0.0499 0.05 0.05 0.047

PG13 20.15 20.7843 21.1623 26.5392 QC23 0.0185 0.0273 0.025 0.0393

Gen

erat

or

Vo

ltag

e (p

.u) VG1 1.1 1.0985 1.0969 1.0999 QC24 0.0395 0.0497 0.05 0.041

VG2 1.0907 1.0876 1.0771 1.0892 QC29 0.024 0.0232 0.0216 0.0376

VG5 1.0679 1.0627 1.0627 1.0723 T6–9 0.9943 1.0103 1.0327 0.9845

VG8 1.0779 1.0741 1.0729 1.0746 T6–10 1.0067 0.9241 0.9 1.0476

VG11 1.0891 1.0851 1.098 1.0884 T4–12 0.9763 0.9688 0.9736 1.0299

VG13 1.1 1.0978 1.1 1.0864 T28–27 0.9656 0.9586 0.9597 1.0096

QC10 0.0023 0.05 0.05 0.0296 Cost ($/hr) 823.8467 826.4954 828.53 823.8875 Loss (MW) 5.3782 5.2727 5.2883 5.7699

L-index (p.u) 0.1077 0.1255 0.1259 NA


From the above table, it is observed that SOS managed to bring down the cost from 826.4954 $/hr [17] by 0.32 % to 823.8467 $/hr, but at the same

time, increased the transmission loss by 1.96 % to 5.3782 MW from previous reported best result of 5.2727 MW [17]. It can be observed that L-index

value has improved simultaneously increasing the voltage stability margin.


602

Case 6: Minimization of fuel cost and voltage stability index (L-index)

Table 7. Minimization of fuel cost and voltage stability index (L-index) using SOS Technique Technique

Control

variables SOS

QOTLBO [17]

TLBO [17]

NSGA-II[24]

MOHS [20]

Control

variables SOS

QOTLBO [17]

TLBO [17]

NSGA-II[24]

MOHS [20]

Gen

erat

or

Rea

l

Po

wer

Outp

ut

(MW

)

PG1 177.04 177.1511 177.3369 175.9384 175.5715

Sh

un

t C

om

pen

sato

r

Ou

tput

(p.u

)

QC12 0.05 0.0415 0.0499 0.0443 0.049

PG2 48.7 48.6412 48.6998 48.2385 48.9863 QC15 0.05 0.0435 0.047 0.0461 0.0495

PG5 21.3 21.2199 21.2113 21.2667 21.7973 QC17 0.05 0.0422 0.0439 0.0488 0.0496

PG8 21.06 21.3907 21.0798 19.3039 21.7515 QC20 0.0463 0.0396 0.0306 0.0456 0.0464

PG11 11.9 11.7021 11.9314 13.7654 11.4848 QC21 0.05 0.0316 0.034 0.0493 0.0498

PG13 12 12 12 13.7048 12.5533 QC23 0.0212 0.0075 0.0015 0.0446 0.0492

Gen

erat

or

Vo

ltag

e (p

.u) VG1 1.1 1.0991 1.099 1.0999 1.0998 QC24 0.0412 0.0242 0.0018 0.0485 0.0495

VG2 1.0879 1.0873 1.0778 1.0937 1.0892 QC29 0.0233 0.0232 0.0002 0.0463 0.049

VG5 1.0619 1.0566 1.0463 1.0705 1.0673

T6–9 0.9987 0.9535 0.9359 0.9234 0.901

VG8 1.0694 1.0605 1.0529 1.0904 1.0757 T6–10 1.0109 0.942 0.9269 0.916 0.906

VG11 1.0885 1.0985 1.0909 1.0982 1.0977 T4–12 0.97 0.9603 0.9583 0.9105 0.9062

VG13 1.1 1.0978 1.0974 1.1 1.0999 T28–27 0.9589 0.9342 0.9303 0.9522 0.9232

QC10 0.05 0.0288 0.0019 0.0497 0.0459 Cost($/h) 799.0124 799.3415 799.8564 800.317 799.9401

Transmission

Loss(MW) 8.6079 8.705 8.8592 NA NA

L-index (p.u) 0.1068 0.1256 0.127 0.1083 0.1075



603

Case 7: Minimization of total fuel cost with the minimization in voltage deviation

Table 8. Minimization of total fuel cost with the minimization in voltage deviation using SOS Technique Technique

Control

variables SOS BBO[22] PSO[7] DE[8] Control variables SOS BBO[22] PSO[7] DE[8]

Gen

erat

or

Rea

l

po

wer

Outp

ut

(p.u

) PG1 1.775 1.734298 1.7368 1.831277

T6–10 1.0372 0.9 0.9 0.923

PG2 0.4874 0.4906 0.491 0.474435 T4–12 1.0473 1 0.9954 0.9345

PG5 0.2133 0.2177 0.2181 0.187281 T28–27 1.0213 0.9708 0.9703 0.9616

PG8 0.2088 0.2327 0.233 0.161515

Sh

un

t C

om

pen

sato

r

Ou

tput

(p.u

)

QC10 0 0.0414 0.0403 0.036479

PG11 0.1186 0.1384 0.1388 0.118855 QC12 0.0448 0.0355 0.0369 0.003806

PG13 0.1201 0.1198 0.12 0.16505 QC15 0.0455 0.05 0.05 0.040931

Gen

erat

or

Vo

ltag

e (p

.u) VG1 1.1 1.0272 1.0142 1.049 QC17 0.0073 0 0 0.029372

VG2 1.0796 1.0088 1.0022 1.0335 QC20 0.0497 0.05 0.05 0.047958

VG5 1.0444 1.0145 1.017 1.0117 QC21 0.05 0.05 0.05 0.044684

VG8 1.0493 1.0092 1.01 1.0043 QC23 0.0354 0.05 0.05 0.038162

VG11 0.9888 1.051 1.0506 1.0432 QC24 0.05 0.049 0.05 0.042009

VG13 1.0138 1.017 1.0175 0.9931 QC29 0.0269 0.0265 0.0259 0.012597

T6–9 1.0515 1.0722 1.0702 1.0439 Cost($/h) 800.0484 804.9982 806.38 805.2619

Transmission

Loss(MW) 8.9314 9.95 NA 10.4412

L-index(p.u.) 0.1278 NA NA NA

Voltage Deviation(p.u) 0.3471 0.102 0.089 0.1357



604

Case 9: Simultaneous minimization of transmission loss along with voltage stability index

Table 9. Simultaneous minimization of transmission loss along with voltage stability index using SOS Technique Technique

Control

variables SOS PSO[7] Control variables SOS PSO[7]

Gen

erat

or

Rea

l

Po

wer

Outp

ut

(p.u

)

PG1 0.5125 -

Sh

un

t C

om

pen

sato

r

Ou

tput

(p.u

)

QC12 0.0499 -

PG2 0.8 - QC15 0.0499 0.05

PG5 0.5 - QC17 0.05 -

PG8 0.35 - QC19 0 0.05

PG11 0.3 - QC20 0.0483 -

PG13 0.4 - QC21 0.05 -

Gen

erat

or

Volt

age

(p.u

)

VG1 1.1 1.062 QC23 0.021 -

VG2 1.098 1.037 QC24 0.0455 0.1

VG5 1.0807 1.035 QC29 0.0188 -

VG8 1.0875 1.037

T6–9 1.0232 0.97

VG11 1.0629 1.046 T6–10 0.998 0.96

VG13 1.1 1.072 T4–12 0.988 0.99

QC10 0.0498 0.2 T28–27 0.9684 1.01

Cost ($/h) 967.0661 -

Transmission loss (MW) 2.847 0.0516

L-index (p.u.) 0.1047 0.1307



605

Case 10: Cost Minimization

Table 10. Simulation results for cost minimization of IEEE 118 bus system using SOS Technique Technique Technique

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS

QOTLB

O [17]

TLBO

[17]

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

PG1 29 5.0513 5.0374 PG100 100.05 114.770 113.6624 VG76 0.95683 1.0327 1.0271

PG4 5 5.0223 5.0318 PG103 8.0138 8.0188 8.0586 VG77 1.009 1.0423 1.0343

PG6 5.0254 5.0187 5.1436 PG104 25.084 25.0423 25.3061 VG80 1.0496 1.0886 1.0892

PG8 150.01 150.3032 150.7002 PG105 25.001 25.2536 25.1527 VG85 1.0316 1.0199 1.02

PG10 100.06 169.3889 171.3829 PG107 8.0026 8.0091 8.0143 VG87 1.0619 1.0897 1.0898

PG12 10.001 10.0213 10.0116 PG110 25 25.0222 25.2862 VG89 1.0423 1.0366 1.0266

PG15 25.055 25.2916 25.1637 PG111 25.123 25.0547 25.0384 VG90 1.0231 1.0305 1.0185

PG18 5.0034 5.0674 5 PG112 25.368 25.0117 25.2232 VG91 1.0311 1.0363 1.0256

PG19 5.0016 5 5.0182 PG113 25.003 25.1642 25.2035 VG92 1.0348 1.0386 1.0286

PG24 100.01 120.1963 120.4126 PG116 25.001 25.0107 25.0643 VG99 0.96978 1.0545 1.0456

PG25 349.91 349.5982 349.7829 VG1 1.083 1.0607 1.0496 VG100 0.99934 1.0402 1.0296

PG26 8.0025 8.0623 8.0728 VG4 0.9625 1.0546 1.0413 VG103 0.9971 1.0346 1.023

PG27 9.6979 8.0846 8.1045 VG6 0.9661 1.0857 1.0841 VG104 1.0084 1.0297 1.019

PG31 25.003 25.0722 25.1863 VG8 0.9945 1.0895 1.09 VG105 1.0113 1.0278 1.0161

PG32 8.0045 8.0192 8.1232 VG10 1.1 1.0524 1.0419 VG107 1.0315 1.0215 1.0087

PG34 99.985 25.1262 25.1527 VG12 0.9735 1.0467 1.0352 VG110 0.97009 1.0346 1.0243

PG36 8.0117 8.0206 8.0528 VG15 0.9531 1.0505 1.0385 VG111 0.98105 1.0411 1.0298

PG40 8.0038 8.0448 8.2236 VG18 0.9384 1.0452 1.0343 VG112 0.92616 1.0345 1.0256

PG42 25.325 25.0537 25.3548 VG19 0.9459 1.0712 1.0658 VG113 0.95195 1.0615 1.0501

PG46 50.015 249.6018 249.0325 VG24 1.0839 1.09 1.0897 VG116 1.1 1.0679 1.0677

PG49 50.514 249.9137 248.1637 VG25 1.0419 1.09 1.0896 QC34 0.09012 0.0663 0.2668

PG54 25 25.2048 25.1607 VG26 1.0278 1.0596 1.0518 QC44 0.2995 0.0516 0.0507

PG55 25 25.0768 25.4524 VG27 1.0825 1.0512 1.0406 QC45 0.12034 0.2169 0.2236

PG56 50.004 199.9312 198.7935 VG31 1.0494 1.0575 1.049 QC46 0.22851 0.0086 0.2494

PG59 50 199.7859 199.8001 VG32 1.0588 1.0394 1.0329 QC48 0.2964 0.0846 0.081

PG61 25.002 25.042 25.5916 VG34 0.9608 1.0389 1.03 QC74 0.26788 0.1156 0.1583

PG62 100.01 327.0837 326.3556 VG36 0.9537 1.0323 1.0271 QC79 3.13E-06 0.2996 0.298

PG65 420 319.6931 314.8521 VG40 0.9163 1.0366 1.0374 QC82 0.29271 0.2997 0.2998

PG66 30.043 30.1746 30.2394 VG42 0.9370 1.0291 1.0417 QC83 0.017622 0.0979 0.1099

PG69 29 110.9331 115.4795 VG46 1.0096 1.0412 1.0551 QC105 0.2996 0.0989 0.1729

PG70 10.003 10.0207 10.2128 VG49 0.9627 1.0237 1.0453 QC107 0.1983 0.1616 0.0879

PG72 5.0124 5.0423 5.1262 VG54 0.9388 1.0197 1.0424 QC110 0.18926 0.1008 0.1388

PG73 5 5.0288 5.0132 VG55 0.9347 1.0206 1.0431 T8–5 0.93692 1.0042 1.0121

PG74 25.003 25.3524 25.1628 VG56 0.9368 1.0241 1.046 T26–25 1.0599 1.0999 1.0998

PG76 25.125 25.0485 25.1246 VG59 0.9439 1.0309 1.0533 T30–17 1.0922 1.0182 1.0275

PG77 299.95 150.3722 150.6738 VG61 1.01 1.0288 1.0513 T38–37 1.0731 1.0265 1.0315

PG80 25.054 25.0812 25.0102 VG62 0.9990 1.0699 1.0696 T63–59 1.0534 1.0378 1.0159


606

Technique Technique Technique

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS

QOTLB

O [17]

TLBO

[17]

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

PG85 10.002 10.0242 10.1632 VG65 1.0506 1.0431 1.061 T64–61 0.9342 1.0311 1.0086

PG87 100.04 183.8538 183.7419 VG66 0.9829 1.038 1.0362 T65–66 1.0379 0.9002 0.9

PG89 50.084 97.2006 95.8227 VG69 0.9426 1.053 1.0525 T68–69 0.9 0.9961 0.9945

PG90 8.0001 8 8.0211 VG70 0.9884 1.0572 1.0542 T81–80 0.9 1.0042 1.0113

PG91 47.302 20.0123 20.1013 VG72 1.0406 1.042 1.0399 Fuel

cost($/hr) 47,960 55,968.14 55,989.87

PG92 214.8 103.6475 104.3625 VG73 1.0245 1.0243 1.0217 Emission

(lb/hr) 342.635 410.9816 410.5538

PG99 100 100.0325 100.1091 VG74 0.9591 1.0194 1.0163 Transmission

loss(kW) 173.83 69.8886 69.4617

Bold fonts denote the proposed algorithm and superior performance. Generator real power outputs (PG) are in MW, generator voltages (VG) and shunt

compensator outputs (QC) are in p.u.

Case 11: Emission Minimization

Table 11. Simulation results for emission minimization of IEEE 118 bus system using SOS. Technique Technique Technique

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

PG1 29 30 13.1937 PG100 246.91 213.1142 175.3218 VG76 0.95109 1.0386 1.0291

PG4 11.361 29.9302 28.7162 PG103 11.754 19.4834 15.8526 VG77 1.0113 1.0629 1.0351

PG6 29.983 29.8782 18.4882 PG104 74.106 80.2192 95.8003 VG80 1.0464 1.0899 1.0842

PG8 150 299.8738 270.3844 PG105 67.72 73.9628 83.2438 VG85 1.0087 1.0146 1.0199

PG10 100.64 142.7617 264.8307 PG107 14.257 19.7113 18.4581 VG87 1.063 1.0898 1.0871

PG12 29.714 22.0728 29.1476 PG110 48.915 48.6485 46.4433 VG89 1.0419 1.0428 1.0291

PG15 91.021 44.0345 35.9218 PG111 31.689 85.3081 99.9002 VG90 1.0971 1.0415 1.0244

PG18 5 5 5 PG112 83.836 83.2133 90.2355 VG91 1.0322 1.0524 1.0381

PG19 5 5.0111 5 PG113 33.722 67.1145 95.4081 VG92 1.0337 1.0537 1.0424

PG24 100 100.4435 100.4172 PG116 34.93 41.9238 29.7641 VG99 0.97643 1.0281 1.0883

PG25 100 100.4082 100 VG1 1.0597 1.0327 1.0191 VG100 1.0075 1.0706 1.0706

PG26 8.1201 26.9576 29.3547 VG4 0.9347 1.0254 1.0222 VG103 1.0086 1.0791 1.0743

PG27 8.7412 12.112 28.2438 VG6 0.95933 1.072 1.0606 VG104 1.0215 1.0631 1.043


607


Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS

PG31 67.54 97.9324 83.8513 VG8 1.0014 1.0788 1.0775 VG105 1.0206 1.062 1.0447

PG32 8 8.0141 8.0142 VG10 1.0859 1.0208 1.0258 VG107 1.0566 1.0513 1.0255

PG34 25 25.0213 25.0202 VG12 0.98318 1.0373 1.0333 VG110 0.97574 1.0824 1.0256

PG36 8 8 8.0134 VG15 0.96741 1.0342 1.0374 VG111 1.0213 1.0798 1.0492

PG40 8 8 8.0283 VG18 0.95698 1.0319 1.036 VG112 0.91073 1.0865 1.0417

PG42 32.46 99.6537 26.8744 VG19 0.96208 1.0449 1.0446 VG113 0.96626 1.0522 1.053

PG46 249.95 179.4485 126.7132 VG24 1.0996 1.0529 1.0583 VG116 1.0521 1.0805 1.0739

PG49 50 50.0503 50.0483 VG25 1.1 1.0766 1.078 QC34 0.29995 0.1784 0.29

PG54 25 25.0728 25.1737 VG26 1.0156 1.0319 1.0311 QC44 0.18116 0.0462 0.0421

PG55 25 25.0137 25.4218 VG27 1.0882 1.0272 1.0426 QC45 0.10321 0.2032 0.1844

PG56 50 50.0234 50.3506 VG31 1.0442 1.0338 1.0469 QC46 0.002995 0.057 0.0233

PG59 199.99 123.5961 132.1043 VG32 1.0657 1.0395 1.0453 QC48 0.043801 0.0766 0.0679

PG61 38.07 84.2513 33.5485 VG34 0.98492 1.0375 1.0397 QC74 0.11167 0.1794 0.2857

PG62 420 173.6788 254.15 VG36 0.97803 1.0249 1.0496 QC79 0.033682 0.2993 0.297

PG65 80 80 80 VG40 0.93486 1.0278 1.0571 QC82 0.3 0.2925 0.0962

PG66 30 30.331 32.3293 VG42 0.92039 1.037 1.0694 QC83 1.57E-06 0.2999 0.287

PG69 29 80.0753 107.9949 VG46 0.97081 1.0373 1.0714 QC105 0.06274 0.0487 0.197

PG70 10 10.2342 10.1963 VG49 0.99474 0.9948 1.0702 QC107 0.14288 0.2225 0.2103

PG72 5 5.4008 10.5836 VG54 0.9451 1.049 1.0877 QC110 0.000793 0.0043 0.1421

PG73 5 5.6437 5.3809 VG55 0.94718 1.0261 1.0736 T8–5 0.987 1.0119 1.017

PG74 25 27.8556 25.5065 VG56 0.94714 1.0526 1.0845 T26–25 0.92439 1.0989 1.0992

PG76 25 27.6722 30.1247 VG59 0.9598 1.0403 1.0883 T30–17 1.0706 1.0134 1.0133

PG77 240.34 272.0213 237.3293 VG61 1.012 1.031 1.0674 T38–37 1.0145 1.0021 0.989

PG80 84.397 70.0576 63.4527 VG62 1.0163 1.0648 1.0663 T63–59 0.96107 0.9761 1.0046

PG85 13.542 28.1374 19.2557 VG65 1.0287 1.0456 1.0851 T64–61 0.9181 1.0315 0.9929

PG87 179.01 181.63 299.4339 VG66 0.9994 1.0304 1.0203 T65–66 1.0442 0.9612 0.9003

PG89 113.86 144.6645 155.9003 VG69 0.94544 1.039 1.0349 T68–69 0.94171 1.0321 1.0927


608


Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS

PG90 15.976 16.9528 19.6379 VG70 0.96819 1.0406 1.042 T81–80 0.90039 1.0116 1.0368

PG91 45.96 44.9324 41.5361 VG72 1.053 1.0333 1.0233 Fuel cost

($/hr) 63,441 65,601.64 65,037.34

PG92 196.58 147.4587 114.5003 VG73 1.0027 1.0143 1.0106 Emission

(lb/hr) 164.5 176.1666 182.9609

PG99 265.61 172 131.9728 VG74 0.94244 1.0193 1.0072 Transmission

Loss (kW) 139.49 150.9366 188.5034



Case 12: Loss Minimization

Table 12. Simulation results for transmission loss minimization objective using SOS Technique Technique Technique

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

Variables SOS

QOTLBO

[17]

TLBO

[17]

Control

variables SOS QOTLBO[17]

TLBO

[17]

PG1 29.728 29.8931 14.1548 PG100 100.07 107.3321 105.7234 VG76 1.0049 1.0107 0.989

PG4 17.57 20.7667 23.0473 PG103 12.4 11.9027 8.1071 VG77 1.0061 1.0191 1.0036

PG6 5.0015 23.5384 25.2495 PG104 25 27.5847 40.6412 VG80 0.99582 1.0069 0.9844

PG8 150 160.8306 161.4631 PG105 99.627 28.8382 45.4382 VG85 1.0382 1.0126 1.0103

PG10 298.92 105.1021 202.4847 PG107 11.383 13.3147 18.1017 VG87 1.0998 1.0818 1.0797

PG12 10.387 29.1323 29.1093 PG110 50 31.3725 29.1604 VG89 1.0539 1.0387 1.0511

PG15 41.455 87.5141 94.3115 PG111 25 25.2184 25.2835 VG90 1.0393 1.0382 1.0438

PG18 5.8168 8.1933 23.9002 PG112 74.902 35.8435 31.7249 VG91 1.038 1.0407 1.0497

PG19 5 23.0788 15.1346 PG113 59.328 56.6809 80.6904 VG92 1.0418 1.0357 1.0369

PG24 100.01 147.7714 113.9182 PG116 49.523 49.2005 44.257 VG99 0.98987 1.0393 1.051

PG25 100.09 126.8438 100.3799 VG1 0.98098 1.051 1.0202 VG100 0.99156 1.0273 1.043

PG26 8.0177 22.9912 22.1332 VG4 1.005 1.0434 1.0214 VG103 0.98944 1.0193 1.0661

PG27 8.0197 28.5314 27.1458 VG6 0.99514 1.0701 1.0499 VG104 0.98184 1.0113 1.0361

PG31 54.483 78.2105 87.9738 VG8 0.97352 1.0755 1.0557 VG105 0.98766 1.0094 1.0323

PG32 18.793 20.1445 29.6566 VG10 1.0416 1.0367 1.0148 VG107 0.96808 1.0078 1.0318

PG34 43.155 83.2007 78.1281 VG12 0.98244 1.0221 1.0188 VG110 1.0205 1.0155 1.0311

PG36 29.929 24.6842 28.9505 VG15 0.95509 1.0265 1.0159 VG111 1.0208 1.0198 1.0368

PG40 29.853 20.4124 29.0852 VG18 0.94316 1.0181 1.0133 VG112 1.0421 1.0219 1.0349


609


Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

Variables SOS

Control

variables SOS

QOTLBO

[17]

TLBO

[17]

Control

Variables SOS

PG42 99.967 94.9384 87.2005 VG19 0.94615 1.0468 1.0063 VG113 0.95155 1.0347 1.0207

PG46 75.876 58.6024 127.8746 VG24 1.0096 1.0541 1.0088 VG116 1.0138 1.0505 1.072

PG49 173.17 236.2374 93.3376 VG25 0.9889 1.0897 1.0756 QC34 0.27182 0.117 0.105

PG54 99.992 80.436 59.2742 VG26 0.96862 1.0413 0.994 QC44 0.003608 0.0404 0.0019

PG55 68.862 97.7809 71.2112 VG27 0.94772 1.0344 0.9982 QC45 0.29979 0.2089 0.2249

PG56 168.52 77.1155 136.8644 VG31 0.94728 1.0399 0.9874 QC46 0.22726 0.274 0.1187

PG59 199.99 179.9903 97.9237 VG32 0.94767 1.0247 1.0123 QC48 0.005322 0.0744 0.0558

PG61 25.004 76.5748 58.1055 VG34 0.97135 1.025 1.0166 QC74 0.17685 0.0844 0.1757

PG62 100 185.5016 336.6485 VG36 0.96532 1.0138 1.0136 QC79 0.23106 0.2919 0.2922

PG65 419.48 192.6547 142.6236 VG40 0.95865 1.0387 1.0146 QC82 0.02353 0.2949 0.245

PG66 30.345 78.8729 79.8481 VG42 0.97136 1.0371 1.0554 QC83 0.11192 0.0931 0.1293

PG69 29.728 80.0818 95.1218 VG46 0.99009 1.0337 1.0439 QC105 0.002739 0.155 0.1657

PG70 29.898 10.9715 10.1947 VG49 0.98744 1.039 1.0544 QC107 0.29809 0.1093 0.0988

PG72 5.001 20.1172 19.8263 VG54 0.9954 1.0454 1.0525 QC110 0.008781 0.0965 0.0926

PG73 18.818 19.9734 10.6108 VG55 0.99326 1.0399 1.0529 T8–5 0.948-46 0.9941 1.0113

PG74 99.348 89.9006 97.7145 VG56 0.99336 1.0382 1.0199 T26–25 1.0418 0.9026 1.0515

PG76 99.571 88.5942 31.0182 VG59 0.96351 1.0544 1.0123 T30–17 0.96495 1.0364 1.0292

PG77 274.26 227.4576 185.5746 VG61 1.0046 1.0527 1.008 T38–37 1.0303 1.0087 1.0112

PG80 100 30.4116 44.2247 VG62 0.99878 1.0538 1.0512 T63–59 1.0664 1.0081 1.0128

PG85 26.926 14.0215 11.3845 VG65 1.0354 1.0462 1.027 T64–61 0.96749 0.9929 1.0674

PG87 100.32 100.1543 107.1638 VG66 0.98545 1.0162 1.0134 T65–66 1.0576 0.9045 1.0588

PG89 50.911 57.9964 61.6745 VG69 1.0152 1.0277 1.0175 T68–69 0.96828 0.9881 1.033

PG90 19.989 12.4839 10.6637 VG70 1.0413 1.0367 1.0061 T81–80 0.98847 1.0138 1.0538

PG91 44.355 24.0536 23.5206 VG72 1.0281 1.0244 1.0065 Fuel cost

($/hr) 70,963 63,693.91 63,515.12

PG92 273.51 100.3382 115.7448 VG73 1.0693 1.0039 1.0032 Emission

(lb/hr) 373.2768 379.7895 318.0176

PG99 100.04 105.0007 119.2147 VG74 1.0232 1.0077 1.0056 Transmission

loss(MW) 16.27 35.3191 36.8482




610

Statistical Comparison of the Objectives

Table 13. Statistical Comparison of the algorithms for different single objective functions Objectives SOS QOTLBO TLBO

Cost Minimization ($/hr)

Best 47,960 55968.14 55989.87

Worst 47,969 55975.13 56112.46

Average 47,961 55970.24 55998.12

Standard deviation 2.7274 2.9841 5.2844

Emission Minimization (lb/hr)

Best 164.5 176.1666 182.9609

Worst 172.75 177.4792 187.1342

Average 165.5313 176.628 184.2006


Loss Minimization (MW)

Best 16.27 35.3191 36.8482

Worst 17.594 36.2125 41.6643

Average 16.6407 35.6723 38.0047


To check the robustness of the proposed algorithm in case of a large IEEE 118 bus test system, statistical analysis of the results for 50 independent

trials are done. In all the cases of cost, emission and loss minimization objectives, it is found that SOS gives superior results in terms of best, worst,

average and standard deviation values as compared to QOTLBO and TLBO.


611

6. Conclusion

The main objective of this paper is to introduce a new powerful meta-heuristic algorithm

named symbiotic organisms search algorithm to be applied to the OPF problem. Successful

implementation of the proposed method for different objectives has been done IEEE 30 and

IEEE 118 bus test systems. Results achieved by SOS are compared to those obtained by other

algorithms in literature. Analysis of the results show that SOS effectively handles varied

objectives for both small and large systems and also provides better compromising solutions

for bi-objective functions too. Another observation is that this algorithm acquires very fast

convergence in all the cases. Hence it may be resolved that SOS algorithm is very promising

and further research might be conducted with this algorithm in other areas of power system.

7. References

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[9]. Abido, M. A. "Optimal power flow using particle swarm optimization." International

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problem." Handbook of swarm intelligence. Springer, Berlin, Heidelberg, 2011. 241-268.

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[17]. Mandal, Barun, and Provas Kumar Roy. "Multi-objective optimal power flow using

quasi-oppositional teaching learning based optimization." Applied Soft Computing (2014):

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IEEE118.doc

Anulekha Saha received her B.Tech degree in Electrical Engg. from

Birbhum Institute of Engineering and Technology, India and M.Tech from

the department of Electrical Engg., NIT Agartala in 2010 and 2012

respectively. Presently, she is a research scholar in the department of

electrical engineering, NIT Agartala. Her research area deals in power

system optimization using various soft-computing techniques.

Aniruddha Bhattacharya did his B.Sc. in Electrical Engg. from the

Regional Institute of Technology, Jamshedpur, India, in 2000, and M.E.E. &

Ph.D in Electrical power system from Jadavpur University, Kolkata, India, in

2008 and 2011 respectively. His experience is diversified in industries as

well as academics. He has worked with Siemens Metering Limited, India;

Jindal Steel & Power Limited, Raigarh, India; Bankura Unnayani Institute of

Engineering, Bankura, India; Dr. B. C. Roy Engineering College, Durgapur,

India.

Presently he is an Assistant Professor in the electrical engineering department, NIT Agartala,

India. His areas of interest include power system load flow, optimal power flow, economic

load dispatch, soft computing applications to power system problems.


613

Ajoy Kumar Chakraborty received his L.L.E from the State Council of

Engg. and Technical Education, West Bengal in 1979, B.E.E from Jadavpur

University in 1987, M.Tech (Power System) from IIT Kharagpur in 1990

and Ph.D (Engg) in 2007 from Jadavpur University respectively. His areas

of interest include Application of soft computing techniques to different

power system problems, Power Quality, FACTS & HVDC and Deregulated

Power System. He is presently working as an Associate Professor in the

Department of Electrical Engineering, NIT Agartala, India.

Priyanath Das obtained his B.Tech and M.Tech in Electrical Engineering in

1994 and 2002 respectively. He completed his PhD from Jadavpur

University in 2013. He is presently working as an Associate Professor in the

Department of Electrical Engineering, NIT Agartala, India. His areas of

interest include Application of FACTS & HVDC, Deregulated Power

System and High Voltage Engineering.


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