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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

saha.anulekha@gmail.com

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,

Received: July 2nd, 2017. Accepted: September 23rd, 2018 DOI: 10.15676/ijeei.2018.10.3.10

585

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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 g a b( , )=0 (2)

andh a b( , )0 (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 LTL

a = 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

bT= 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

ij ij ij ij j i Li

Gi P V V G B

P

1

cos

cos  (6)

( )

=

+

=

NBUS

j

ij ij ij ij j i Li

Gi Q V V G B

Q

1

cos

sin  (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:

(3)

Generator Constraints:

m ax m in

Gi Gi

Gi

V V

V  

(8)

max min

Gi Gi

Gi P P

P  

i=1,2,3,....PV (9)

max min

Gi Gi

Gi Q Q

Q   (10) where, PV signifies the total generator buses inclusive of slack bus.

Transformer Constraints:

max

min i i

i T T

T  

i=1,2,3,....NT (11)

where, NT represents the number of tap changing transformers.

Shunt VAR Compensator Constraints:

max

min ci ci

ci Q Q

Q  

i=1,2,3,....NC (12) where, NC denotes the total number of shunt compensators connected to the system.

Security Constraints:

max min

Li Li

Li V V

V  

i=1,2,3,...PQ (13)

m ax Li Li S S

i=1,2,3,...TL (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:

( ) ( ) ( )





 + +

=



=

=

 

=

=

G

G N

i

i i i i i N

i i

i P a bP cP

F P

F G

1

2 1

1 min ( ) (15)

where,Pirepresents the power output of the

i

th generator, Fi

( )

Pi denotes the running cost of the

i

th generator, ai,bi,cisymbolizes 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:

( )

=

− +

=

=

NL

m

ik k i k i m

L G V V VV

P G

1

2 2

2 min( ) 2 cos (16) where,Gm

, Vi and Vkrepresent respectively the conductance of line m linking buses

i

and

k

, and magnitude of the voltages at buses iandk; NL and ikrepresents 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:

)

3 min(L G =

(17)

j i N

i ji

j V

F V L

G

=

=

1

1

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

   

11 21

= Y Y

Fji

where,Fjiis the sub matrix found after partially inverting the YBUS matrix.

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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:

=

=

=

NL

i

ref i i V V VD G

1

4 ( )

(18) where, NL denotes the number of load buses;

V

iref 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.

=

=

NG

k k kP G

1

4  (19) where,krepresents 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:

)]

( ),...

( ....

),...

( ), (

[G1x G2 x G x G x

MinG= i m ; x=x1,x2,...xn (20) where,Gi(x)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[ 1 2

1 G G

Gbi =

(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[ 1 3

2 G G

Gbi =

(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[ 1 4

3 G G

Gbi =

(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[ 2 3

4 G G

Gbi = (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:

(5)

] , min[ 3 4

5 G G

Gbi =

(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

( X Mutual Vector BF1 rand

X

Xinew= i+ best− (26) )

* _ (

* ) 1 , 0

( X Mutual Vector BF2 rand

X

Xjnew= j+ best− (27)

_ Xi2Xj

Vector

Mutual +

= (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

( best j

i

inew X rand X X

X = + − − (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 Himin 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]

Generator Real Power Output (MW)

PG1 177.1022 177.1022 177.0817 177.196 177.0177

Shunt Compensator Output (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

Generator Voltage (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.

(15)

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]

Generator Real Power Output (MW)

PG1 1.3725 134.2408 76.788 92.6114

Shunt Compensator Output (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

Generator Voltage (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 Bold font denotes the proposed algorithm and superior performance.

(16)

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]

Generator Real Power Output (MW)

PG1 89.46 x

Shunt Compensator Output (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

Generator Voltage (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 Bold font denotes the proposed algorithm and superior performance.

(17)

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

(18)

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]

Generator Real Power Output (MW)

PG1 126.18 124.179 122.5831 134.5544

Shunt Compensator Output (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

Generator Voltage (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 Bold font denotes the proposed algorithm and superior performance.

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.

(19)

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]

Generator Real Power Output (MW)

PG1 177.04 177.1511 177.3369 175.9384 175.5715

Shunt Compensator Output (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

Generator Voltage (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 Bold font denotes the proposed algorithm and superior performance.

(20)

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]

Generator Real power Output (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

Shunt Compensator Output (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

Generator Voltage (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 Bold font denotes the proposed algorithm and superior performance.

(21)

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]

Generator Real Power Output (p.u)

PG1 0.5125 -

Shunt Compensator Output (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 -

Generator Voltage (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 Bold font denotes the proposed algorithm and superior performance.

References

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