Optimal Power Flow using Glowworm Swarm Optimization

Electrical Power and Energy Systems 80 (2016) 128–139

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Optimal Power Flow using Glowworm Swarm Optimization Salkuti Surender Reddy ⇑, Ch Srinivasa Rathnam Department of Railroad and Electrical Engineering, Woosong University, South Korea

a r t i c l e

i n f o

Article history: Received 21 April 2015 Received in revised form 15 December 2015 Accepted 2 January 2016

Keywords: Optimal Power Flow Generation cost Emission Transmission loss Multi-Objective Optimization Evolutionary algorithms

a b s t r a c t An important objective of the Optimal Power Flow (OPF) problem is to minimize the generation cost and keep the power outputs of generators, bus voltages, bus shunt reactors/capacitors and transformer tap settings in their secure limits. Solving this OPF problem using classical methods suffer from the disadvantages of highly limited capability to solve the practical large scale power system problems. To overcome the inherent limitations of conventional optimization techniques, Swarm Intelligence (SI) methods have been developed. However, the environmental concern, dictate the minimization of emissions of the thermal plants. Individually, if one objective is optimized, other objective is compromised. Hence, MultiObjective Optimal Power Flow (MO-OPF) problem has been formulated in this paper. Swarm Intelligence methods, such as Particle Swarm Optimization (PSO) and Glowworm Swarm Optimization (GSO) have been used to solve the OPF problem with generation cost and emission minimizations as objective functions. The effectiveness of the proposed algorithms are tested on IEEE 30 bus and practical Indian 75 bus systems for cost minimization as objective function, and IEEE 30 bus test system for minimization of cost and emission as objectives. The results obtained from both the networks, the PSO and GSO are compared with each other based on different parameters. Ó 2016 Elsevier Ltd. All rights reserved.

Introduction The Optimal Power Flow (OPF) is a highly non-linear and constrained power system optimization problem. The ordinary power flow problem is defined by specifying the real and reactive load demands in megawatts (MWs) and megavars (MVARs) to be supplied at certain bus bars/nodes of a transmission system and by using generated powers and voltage magnitudes at the remaining nodes of this system together with a complete topological description of the system involving its impedances. The objective is to find the complex nodal voltages from which all other quantities like currents, line flows and transmission losses can be calculated. The model of transmission system is given in complex quantities, since an AC system is assumed to generate and supply the powers and loads in MWs and MVARs. Mathematically, the OPF problem can be reduced to a set of non-linear equations, where the real and imaginary components of nodal voltages are variables. The number of equations are equal to twice the number of nodes. The non-linearities can roughly be classified being of a quadratic nature. Gradient and relaxation approaches are the only methods for the solution of these systems. The output of a power flow problem tells the system operator or ⇑ Corresponding author. Tel.: +82 426296737. E-mail address: [email protected] (S. Surender Reddy). http://dx.doi.org/10.1016/j.ijepes.2016.01.036 0142-0615/Ó 2016 Elsevier Ltd. All rights reserved.

system planner of a system in which way the lines in the system are loaded, what the voltages at different buses are, how much of the generated power is lost and where limits are exceeded. The power flow problem is one of the basic problems in which both load and generator powers are given or fixed. The Optimal Power Flow has a long history in its development, and it was first presented by Carpentier in 1962 [1] and the subsequent surveys on OPF in [2–7]. However, it took a long time to become a successful algorithm that could be applied in everyday use. Current interest in the OPF centers on its ability to solve for the optimal solution that takes account of the security of the system. Optimal Power Flow has been applied to regulate the generator active power outputs and voltages, transformer tap settings, shunt reactors/capacitors and other controllable variables to minimize the generator fuel cost, network active power loss, voltage stability index, while keeping all the load bus voltages, generator reactive power outputs, network power flows and all other state variable in the power system in their secure and operational limits. In its most common problem formulation, the OPF is a non-linear, non-convex, static, large-scale optimization problem with both the continuous and discrete control variables. Even in the absence of non-convex generator operating cost functions, prohibited operating zones (POZs) of generating units, and discrete control variables, the OPF problem is a non-convex due to the existence of the non-linear AC power flow equality constraints. The presence

S. Surender Reddy, C. Srinivasa Rathnam / Electrical Power and Energy Systems 80 (2016) 128–139

129

Nomenclature Pgi active power generation of ith bus V gi voltage of ith bus ai ; bi and ci cost coefficients of ith generator n total number of buses in the system NL number of load buses Ntrans transformer taps compensation capacitors Ncap Nobj number of objective functions Ngen number of thermal generators ai ; bi ; ci ; gi and di emission coefficients of ith generator Pgi and Q gi active and reactive power generations of ith generator Ploadi and Q loadi active and reactive load demands x inertia weight V tþ1 velocity vector of ith particle in dimension j at time ‘t’ ij X tij

position vector of ith particle in dimension j at time ‘t’

Ptbest;I

personal best position of ith particle in dimension j found from the initialization through time ‘t’ Gbest global best position of ith particle in dimension found from the initialization through time ‘t’ c1 and c2 positive acceleration constants, and these are used to level the contribution of cognitive and social components, respectively r t1j and r t2j random numbers from uniform distribution at time ‘t’ t max maximum number of iterations

of discrete control variables, such as transformer tap positions, switchable shunt devices, phase shifters, further complicates the problem formulation and solution. The conventional optimization approaches that make use of derivatives and gradients are in general not able to locate or identify the global optimum. Many mathematical assumptions such as convex, analytic and differential objective functions have to be made to simplify the problem. However, the OPF problem is an optimization problem with in general non-convex, nondifferentiable and non-smooth objective functions. Hence, it becomes important to develop the optimization techniques that are efficient to overcome these drawbacks and to handle such difficulties efficiently. Therefore, in this paper heuristic algorithms such as Particle Swarm Optimization (PSO), Glowworm Swarm Optimization (GSO) have been used for solving the OPF problem with different objective functions. The OPF can also solve for an optimal solution with multiple objectives such as minimization of generation cost, emission, and transmission loss minimization, etc. Whenever, we deals with such sort of problems i.e. optimization problem with more than one constraint and objective functions, Swarm Intelligent methods are one of the best method available in the literature. OPF solution approaches are broadly categorized as conventional and intelligent. The conventional approaches include well known methods like Gradient Method, Newton Method, Linear Programming (LP) Method, Quadratic Programming (QP) Method, and Interior Point Method (IPM). The mathematical programming methods, like Non-Linear Programming (NLP) [8,9], QP [10,11] and LP [12,13] have been used for the solution of OPF problem. Many different mathematical methods have been employed for its solution such as Lamda-Iteration, Gradient Method, Newton’s Method and IPM. However, they are not guaranteed to converge to the global optimum of general non-convex problem. Intelligent methodologies include Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and recently developed Glowworm Swarm Optimization (GSO), etc. Ref. [14] combines a

t FðGkbest Þ V min Li V max Li C xK

current iteration number fitness of global optimum solution minimum voltage limit of load demand buses maximum voltage limit of load demand buses chaotic inertia weight factor at iteration k xK inertia weight factor at iteration k chaotic parameter at iteration k Dk q luciferin decay constant (0 < q < 1) c luciferin enhancement constant l control parameters and F max minimum and maximum bounds of ith objective F min i i function bk weight factor for kth objective function m number of non-dominated solutions J j ðtÞ value of objective function at agent j’s location at time t nt threshold parameter to control number of neighbors t time/step index di; jðtÞ Euclidian distance between glowworms i and j at time t r id ðtÞ variable local decision range associated with the glowworm i at time t ljðtÞ luciferin level associated with the glowworm j at time t rs radial range of luciferin sensor b constant parameter

decoupled quadratic load flow solution with Enhanced Genetic Algorithm (EGA) to solve the OPF problem. A Strength Pareto Evolutionary Algorithm based approach with strongly dominated set of solutions is used to form the Pareto optimal set. In [15], nondominated sorting multi objective opposition based gravitational search algorithm has been proposed to solve different single and multi objective OPF problems. An adaptive real coded biogeography-based optimization approach to solve different objective functions of OPF problems with various physical and operating constraints is proposed in [16]. Ref. [17] investigates the possibility of using recently emerged evolutionary-based approach as a solution for the OPF problems which is based on a teaching learning based optimization algorithm using Lévy mutation strategy for optimal settings of OPF problem control variables. A hybrid algorithm consisting of biogeography based optimization with an adaptive mutation scheme and the concept of predator– prey optimization technique for solving the multi-objective OPF problems is proposed in [18]. Ref. [19] presents the application of non-dominated sorting multi objective based gravitational search algorithm for the solution of different OPF problems. In Genetic Algorithm, the execution time and quality of solution deteriorates as the size of the system increases [20]. Hence, to overcome the above limitations, in this paper an attempt has been made to solve the OPF problem with PSO and GSO techniques. Several approaches have been developed to solve the MultiObjective Optimization (MOO) problems such as, the penalty function method [21], weighted sum method [22], 2-constrained method [23], non-dominated sorting genetic algorithm (NSGA) based approach [24], strength Pareto evolutionary algorithm (SPEA) [25] etc. have been used for solving different MOO problems. But, these techniques have difficulties. For instance, in penalty function approach, choosing proper penalty factors is a difficult task [26], Weighted sum method combines all the objectives to a single objective by using the weight factors. This formulation may lose the significance of the objective function and, moreover, there is no rational basis for finding the weight factors

S. Surender Reddy, C. Srinivasa Rathnam / Electrical Power and Energy Systems 80 (2016) 128–139

Unit no.

Voltage magnitude (in p.u.)

Phase angles (in p.u.)

Scheduled power generation (in MW)

Generation cost (in $/hr)

1 2 3 4 5 6

1.050 1.013 1.024 1.008 1.040 1.074

0.000 0.063 0.182 0.146 0.148 0.172

173.46 47.74 23.77 23.22 11.37 12.25

459.75 123.42 59.08 79.97 37.35 40.51

Total generation cost = 800.08 $/hr Losses = 8.41 MW Time per iteration = 0.0532 s Number of Iterations = 200

COST (in $/hr)

116000 114000 112000 110000 108000

Fig. 3. Variation of cost with iterations of Indian 75 bus systemusing PSO.

Table 2 Scheduled power generation and their production cost for practical Indian 75 bus system using PSO.

840 830 820 810 800 790 1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145 154 163 172 181 190 199

780

ITERATIONS Fig. 1. Variation of generation cost with number of iterations for IEEE 30 bus test system using PSO.

0.126

118000

ITERATIONS

Unit no.

Voltage magnitude (in p.u.)

Phase angles (in p.u.)

Scheduled power generation (in MW)

Generation cost (in Rs/hr)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1.030 1.058 0.980 0.966 1.019 0.968 1.048 1.064 1.012 1.088 0.991 1.075 0.990 1.088 1.068

0.000 0.311 0.620 0.943 0.856 0.902 0.921 0.971 0.228 0.652 0.530 0.272 0.250 0.985 0.685

1776.51 221.87 121.46 136.65 158.17 112.63 91.20 88.37 532.89 125.25 139.70 1132.32 847.32 34.43 275.41

29103.45 5211.92 5213.08 3422.95 709.10 2306.00 384.30 2093.89 13429.40 2550.85 3336.05 21075.19 16192.73 2525.78 4794.78

COST vs ITERATIONS

850

COST vs ITERATIONS

120000

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193

Table 1 Scheduled power generation and generation cost for the IEEE 30 bus test system using PSO.

COST (in Rs/hr)

130

FITNESS vs ITERATIONS

Total generation cost = 112,670 Rs/hr Losses = 243.18 MW Time per iteration = 0.4188 s Number of iterations = 200

0.122 0.12 0.118

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193

0.116

FITNESS

FITNESS

0.124

FITNESS vs ITERATIONS

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193

ITERATIONS

0.09 0.089 0.088 0.087 0.086 0.085 0.084 0.083

Fig. 2. Variation of fitness with number of iterations for IEEE 30 bus test system using PSO.

ITERATIONS

900 880 860 840 820 800 780 760

COST vs EMISSION

0.2385 0.2424 0.2462 0.2509 0.2543 0.2589 0.264 0.268 0.2715 0.2776 0.2803 0.2838 0.2874 0.2896 0.2942 0.298 0.3006 0.3054 0.3094 0.3138 0.3217 0.3264 0.3489 0.356

for non-commensurate objectives [24], ‘‘2-constrained technique avoids the use of weight factors for multiple objectives, but the difficulty with this technique is that it needs repeated run for each relaxed level of” to get the Pareto-optimal set/Pareto optimal front [24], NSGA technique is very sensitive to fitness sharing factor [27]. Moreover, these techniques have not presented the problem of diversity preserving, maximizing the diversity, selection of best compromise solution, etc. The present work emphasizes on the development of MultiObjective Optimal Power Flow (MO-OPF) technique using PSO, which has overcome some of the above mentioned difficulties. In this paper, the proposed Improved PSO (IPSO) algorithm is implemented in order to extract the non-dominated solutions. In this regard, this work uses an external repository to save all the nondominated solutions during the evolutionary process, a fuzzy decision-making approach is used to sort these solutions according

COST (in $/hr)

Fig. 4. Variation of fitness with iterations of Indian 75 bus system using PSO.

EMISSION (in Tons/hr) Fig. 5. Generation cost Vs emission plot for IEEE 30 bus test system using PSO.

S. Surender Reddy, C. Srinivasa Rathnam / Electrical Power and Energy Systems 80 (2016) 128–139

Generation cost weight factor

Emission weight factor

Generation cost (in $/hr)

Emission (in tons/hr)

1 2 3 4

0.2 0.4 0.6 0.8

0.8 0.6 0.4 0.2

883.28 826.43 812.21 807.38

0.2388 0.2953 0.3348 0.3426

850 840 830 820 810 800 790 780 1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145 154 163 172 181 190 199

Sl. no.

COST vs ITERATIONS COST (in $/hr)

Table 3 Results of Multi-Objective Optimal Power Flow (MO-OPF) with different weight factors on IEEE 30 bus test system using PSO.

131

ITERATIONS Table 4 Scheduled power generation and their production cost for IEEE 30 bus test system using PSO. Voltage magnitude (in p.u.)

Phase angle (in p. u.)

Scheduled power generation (in MW)

Generation cost (in $/hr)

1 2 3 4 5 6

1.050 1.018 1.016 1.035 1.044 1.006

0.000 0.064 0.183 0.148 0.156 0.213

174.92 44.15 21.76 25.73 11.12 13.81

464.57 111.36 51.36 89.15 36.45 46.18

FITNESS vs ITERATIONS

0.126 0.124 FITNESS

Unit no.

Fig. 6. Variation of generation cost with number of iterations of IEEE 30 bus system using GSO.

0.122 0.12 0.118 11 21 31 41 51 61 71 81 91 101 111 121 131 141 151 161 171 181 191

1

0.116 ITERATIONS

Total generation cost = 799.06 $/hr Losses = 8.48 MW Time per iteration = 0.0613 s Number of iterations = 200

Fig. 7. Variation of fitness with number of iterations on IEEE 30 bus test system using GSO.

The control variables vector (i.e., [X]) can be expressed as to their importance. Power system decision makers/system operators can then select the desired solution between them by applying the fuzzy decision-making approach to the Pareto optimal solution. The remainder of the paper is organized as follows: Section ‘‘Optimal Power Flow (OPF): problem formulation” describes the Optimal Power Flow (OPF) problem formulation. Sections ‘‘Particle Swarm Optimization (PSO)” and ‘‘Glowworm Swarm Optimization (GSO)” describes Particle Swarm Optimization (PSO) and Glowworm Swarm Optimization (GSO) algorithms. Simulation results and discussion are described in Section ‘‘Results and discussion”. Finally, the contributions with concluding remarks are addressed in Section ‘‘Conclusions”. Optimal Power Flow (OPF): problem formulation An optimization problem is finding the best solution from all feasible solutions. The optimization problem can be a minimization or maximization problem. Maximization of any function is mathematically equivalent to minimization of its additive inverse. Therefore, the term optimization and minimization are used interchangeably [28]. The objective functions for the OPF problem are formulated next. Generation cost minimization objective The commonly used objective in Optimal Power Flow (OPF) problem is the minimization of total cost of active power generation in the system. The individual costs of each generator is assumed to be function of active power generation and they are represented by second order quadratic curves. The objective function can be expressed as the sum of quadratic cost model at each generating unit, and is expressed as,

F 1 ðxÞ ¼

Ngen Xh i¼1

i ai P 2gi þ bi Pgi þ C i $=hr

ð1Þ

½X T  ¼ ½Pg; V g ; TAP; Q c  where Pg ¼ ½P g2 ; Pg3 ; . . . ; PgNgen1 , V g ¼ ½V g1 ; V g2 ; . . . ; V gNgen , TAP ¼ ½TAP1 ; TAP 2 ; . . . ; TAPNtrans , Q c ¼ ½Q c1; Q c2 ; . . . ; Q cNcap . Emission minimization objective The global warming is one of the important concern for the power industry, as it is accountable for the emission of greenhouse gases in the environment. As explained earlier, the amendment of clear air act and environmental friendly policies (i.e., Carbon Credit System (CCS)) creates an interest of power sector towards the reduction of emissions of NOx, SOx and CO2 gases. Various mathematical formulations are used to represent the emission of greenhouse gases in emission dispatch problem. The emission dispatch function can be represented as,

F 2 ðxÞ ¼

Ngen X

½ai P2gi þ bi Pgi þ ci þ gi expðdi Pgi Þ ton=hr

ð2Þ

i¼1

where i ¼ ½1; 2; 3; . . . ; Ngen . The equality and inequality constraints for these optimization problems are presented next. Equality constraints The equality constraints represent the typical load flow equations.

0 ¼ Pgi  Ploadi  V i

n X   V j Gij cosdij þ Bij sindij

ð3Þ

j¼1

0 ¼ Q gi  Q loadi  V i

n X   V j Gij sindij  Bij cosdij

ð4Þ

j¼1

In the above equations, i = 1, 2, . . ., n. Generally, Newton Raphson or Fast Decoupled Load Flow models are used for the solution of equality constraints.

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Table 5 Comparison of control variables and objective function values for IEEE 30 bus system for different algorithms. Control variables and objective function values

GA

Improved GA [44]

EGA [43]

DE [48]

MDE [49]

Gradient method [48]

GAMS [43]

BBO [47]

PSO [46]

GSO

PG1 (MW) PG2 (MW) PG5 (MW) PG8 (MW) PG11 (MW) PG13 (MW) V1 (p.u.) V2 (p.u.) V5 (p.u.) V8 (p.u.) V11 (p.u.) V13 (p.u.) T6,9 (p.u.) T6,10 (p.u.) T4,12 (p.u.) T28,27 (p.u.) Bsh,10 (p.u.) Bsh,12 (p.u.) Bsh,15 (p.u.) Bsh,17 (p.u.) Bsh,20 (p.u.) Bsh,21 (p.u.) Bsh,23 (p.u.) Bsh,24 (p.u.) Bsh,29 (p.u.) Generation cost ($/hr) Loss (MW)

176.1100 49.098 21.718 21.086 11.83 12.218 1.1 1.08 1.064 1.066 1.06 1.087 1.05 0.9375 1.025 1.0 0.05 0.01 0.04 0.05 0.0 0.05 0.0 0.05 0.02 799.52 8.66

177.594 48.722 21.954 20.954 11.768 12.0520 1.0810 1.063 1.034 1.038 1.1 1.055 1.0 0.975 0.975 1.0 0.001 0.007 0.019 0.024 0.015 0.022 0.047 0.047 0.024 800.805 –

177.285 48.93 21.29 20.49 11.93 12.23 1.098 1.080 1.053 1.062 1.08 1.078 0.975 1.05 1.0125 1.0125 0.05 0.01 0.04 0.02 0.03 0.01 0.03 0.02 0.01 799.56 8.755

176.2592 48.5602 21.3402 22.0553 11.7785 12.0217 1.0999 1.089 1.0659 1.0697 1.0965 1.0996 1.0429 0.9179 1.0190 0.9836 0.0455 0.0442 0.0417 0.0252 0.0209 0.0420 0.0255 0.0438 0.0275 799.2891 8.6150

175.974 48.884 21.51 22.24 12.251 12 1.0500 1.0382 1.0113 1.0191 1.0951 1.0837 0.9866 0.9714 0.9972 0.9413 NA NA NA NA NA NA NA NA NA 802.376 9.459

187.219 53.781 16.955 11.288 11.287 13.355 1.1 1.08 1.03 1.04 1.08 1.08 1.072 1.07 1.032 1.068 0.0069 0.0005 0.0029 0.0029 0.0021 0.0 0.0033 0.0094 0.0027 804.853 10.486

177.1 48.8 21.4 21.5 12 12 1.08 1.061 1.032 1.039 1.018 1.047 NA NA NA NA NA NA NA NA NA NA NA NA NA 801.5198 9.4

177.0177 48.641 21.239 21.136 11.944 12.054 1.1 1.0876 1.0614 1.0695 1.0982 1.0998 1.05 0.9 0.99 0.97 0.05 0.05 0.05 0.05 0.05 0.05 0.04 0.05 0.03 799.1116 8.63

176.96 48.98 21.30 21.19 11.97 12 1.0855 1.0653 1.0333 1.0386 1.0848 1.0512 1.0233 0.9557 0.9724 0.9728 0.0335 0.0220 0.0198 0.0315 0.0454 0.0381 0.0398 0.05 0.0251 800.41 9.2169

174.92 44.15 21.76 25.73 11.12 13.81 1.0924 1.0735 1.0429 1.0676 1.0765 1.0829 1.075 0.9125 1.0 0.9875 0.03 0.05 0.04 0.03 0.02 0.02 0.01 0.00 0.02 799.06 8.48

Table 6 Scheduled power generation and their production cost for Indian 75 bus system using GSO.

1.030 0.948 1.019 0.978 1.005 1.013 0.962 1.055 1.041 1.057 1.000 0.960 0.997 1.001 1.025

Phase angle (in p.u.)

0.000 0.429 0.796 0.986 0.883 0.879 0.886 0.959 0.303 0.768 0.548 0.337 0.343 0.482 0.761

Scheduled power generation (in MW)

Generation cost (in Rs/hr)

1688.65 185.95 133.97 45.69 240.00 120.00 100.00 100.00 355.11 250.00 200.00 1291.92 632.67 34.88 404.38

27852.86 4680.27 5392.64 2073.08 709.10 2306.00 384.30 2268.03 11198.36 4419.74 4235.16 22829.02 13187.34 2533.68 6022.60

i ¼ ½1; 2; . . . ; Ngen 

112000 110000 108000

Fig. 8. Variation of generation cost with number of iterations for practical Indian 75 bus system using GSO.

FITNESS vs ITERATIONS

0.92 0.9 0.88 0.86 0.84 0.82

1 9 17 25 33 41 49 57 65 73 81 89 97 105 113 121 129 137 145 153 161 169 177 185 193

ITERATIONS

Generator constraints. Generator active power generation (Pg ), reactive power generation (Q g ) and generator Voltage magnitudes (V g ) are limited by their minimum and maximum bounds.

Q gimin 6 Q gi 6 Q gimax

114000

ITERATIONS

Inequality constraints Inequality constraints represent the system operating limits, and they are presented below.

i ¼ ½1; 2; . . . ; Ngen 

116000

106000

Total generation cost = 110092.16 $/hr Total system losses = 221.42 MW Time per iteration = 0.4818 s Number of iterations = 200

Pgimin 6 Pgi 6 P gimax

COST vs ITERATIONS

118000

1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145 154 163 172 181 190 199

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Voltage magnitude (in p.u.)

FITNESS

Unit no.

COST (in Rs/hr)

Bold letters represents the optimum objective function values by using different optimization algorithms.

ð5Þ ð6Þ

Fig. 9. Variation of fitness with number of iterations of Indian 75 bus system using GSO. max V min gi 6 V gi 6 V gi

i ¼ ½1; 2; . . . ; Ngen 

ð7Þ

Transformer constraints. Transformer taps have the lower and upper limits, and they are represented as,

T imin 6 T i 6 T imax

i ¼ ½1; 2; . . . ; Ntran 

ð8Þ

Switchable VAR sources. The switchable VAR sources are restricted by,

133

S. Surender Reddy, C. Srinivasa Rathnam / Electrical Power and Energy Systems 80 (2016) 128–139 Table 7 Results of MO-OPF with different weight factors on IEEE 30 bus system using GSO. Sl. no.

Generation cost weight factor

Emission weight factor

Generation cost (in $/hr)

Emission (in tons/hr)

1 2 3 4

0.2 0.4 0.6 0.8

0.8 0.6 0.4 0.2

852.89 815.23 811.60 806.70

0.2446 0.2900 0.3028 0.3320

Table 11 Comparison of GSO and PSO. GSO

PSO

1 2

Memory not required Agent movement along line-of-sight with a neighbor in the local decision range

3

Local decision domain based on varying range of neighbors Maximum range hard limited by finite sensor range Effective detection of multiple sources/peaks in addition to numerical optimization tasks

Uses a memory element Direction of movement based on previous best positions Dynamic neighborhood based on k nearest neighbors Neighborhood range covers the entire search space Limited to numerical optimization models

4

COST vs EMISSION

900 880 860 840 820 800 780 760

850

PSO vs GSO

EMISSION (in Tons/hr) Fig. 10. Generation cost Vs emission plot for IEEE 30 bus test system using GSO.

COST (in $/hr)

840 0.2332 0.2342 0.2344 0.2365 0.2375 0.2392 0.2421 0.2442 0.2456 0.2496 0.252 0.2545 0.2581 0.262 0.2664 0.2743 0.2776 0.2797 0.2828 0.2855 0.2895 0.2935 0.3028 0.3115 0.3198 0.3687

COST (in $/hr)

5

830 820 810

PSO

800

GSO

780 Table 8 Comparision of generation cost, losses, slack bus power and time per iteration for IEEE 30 bus test system for generation cost minimization as objective function.

Generation cost best (in $/hr) Generation cost worst (in $/hr) Generation cost mean (in $/hr) Total system losses (in MW) Time per iteration (in seconds) Slack bus power (in MW) Iterations for convergence

PSO algorithm

GSO algorithm

800.05 800.81 800.08 8.41 0.0532 173.46 150

799.05 799.91 799.06 8.48 0.0613 174.92 80

ITERATIONS Fig. 11. Variation of generation cost with the number of iterations for IEEE 30 bus system using PSO and GSO.

max V min Li 6 V Li 6 V Li

SLi 6 Smax Li

Table 9 Comparision of generation cost, system losses, slack bus power and time per iteration for Indian 75 bus system for generation cost minimization as objective function.

Generation cost best (in Rs/hr) Generation cost worst (in Rs/hr) Generation cost mean (in Rs/hr) System losses (in MW) Time per iteration (in seconds) Slack bus power (in MW) Iterations for convergence

1 10 19 28 37 46 55 64 73 82 91 100 109 118 127 136 145 154 163 172 181 190 199

790

PSO algorithm

GSO algorithm

112590.00 114620.00 112670.00 243.18 0.4288 1776.51 185

110046.00 113735.00 110092.16 221.42 0.4818 1688.65 150

i ¼ 1; 2; . . . ; NL

ð10Þ

i ¼ 1; 2; . . . ; Nline

ð11Þ

P gi min and Pgi max are minimum and maximum active power generations at ith generating bus, respectively. Q gi min and Q gi max are minimum and maximum reactive power generations at ith generating bus. Q ci min and Q ci max are minimum and maximum shunt capacitor limits. T i max and T i min are maximum and minimum values of transformer tap settings.

Multi-Objective Optimization problem

Minimize ½F 1 ; F 2 T Q cimin 6 Q ci 6 Q cimax

i ¼ ½1; 2; . . . ; Ncap 

ð9Þ

Security constraints. These include the constraints on load bus voltage magnitudes and thermal limits of lines.

ð12Þ

Subjected to the equality and inequality constraints as defined in Sections ‘‘Equality constraints” and ‘‘Inequality constraints”, respectively. where F 1 and F 2 are the objectives as defined in Eqs. (1) and (2) are to be minimized over the set of admissible decision vector Pg.

Table 10 Results for different weight factors using PSO and GSO for multi-objective function. Sl. no.

1 2 3 4

Generation cost weight factor

0.2 0.4 0.6 0.8

Emission weight factor

0.8 0.6 0.4 0.2

PSO

GSO

Generation cost ($/hr)

Emission (tons/hr)

Generation cost ($/hr)

Emission (tons/hr)

883.28 826.43 812.21 807.38

0.2388 0.2953 0.3348 0.3426

852.89 815.23 811.60 806.70

0.2446 0.2900 0.3028 0.3320

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Table 12 Generator data for the IEEE 30 bus system. Bus number

50 20 15 10 10 12

P max G (MW) 200 80 50 35 30 40

Q min G (MVar) 20 20 15 15 10 15

Q max G (MVar) 200 100 80 60 50 60

a ($/ hr) 0 0 0 0 0 0

b ($/ MW hr) 2.0 2.75 1.0 3.25 3.0 3.0

c ($/ MW2 hr) 0.0037 0.0175 0.0625 0.0083 0.0250 0.0250

118000 116000 114000 112000 110000 108000 106000

(

Ptþ1 best;i

¼

  if f xtþ1 > P tbest;i i  tþ1  if f xi > P tbest;i

P tbest;i xtþ1 i

ð13Þ

where f: Rn ! R is the fitness function. The global best position (Gbest ) at time step is computed as,

Gbest ¼ minðPtbest;i Þ; where i 2 ½1; 2; . . . n and n > 1

ð14Þ

Hence, it is very important to note that Pbest;i is the best position that the individual particle has visited since the first time step. On the other hand, Gbest is the best position discovered by any of the particles in the entire swarm [32]. The velocity and position of particle is computed as,

PSO vs GSO

PSO GSO 1 12 23 34 45 56 67 78 89 100 111 122 133 144 155 166 177 188 199

COST (in Rs/hr)

1 2 5 8 11 13

P min G (MW)

Considering minimization problems, then the personal best position Pbest,i at the next time step, t + 1, where t 2 ½1; 2; . . . ; N is calculated as,

ITERATIONS Fig. 12. Variation of generation cost with number of iterations for practical Indian 75 bus system.

Particle Swarm Optimization (PSO)

V tþ1 ¼ xV tij þ c1 r t1j ½Ptbest;i  X tij  þ c2 r t2j ½Gbest  X tij  ij

ð15Þ

¼ X tij þ V tþ1 X tþ1 ij ij

ð16Þ

Suppose, if a particle’s velocity goes beyond its specified maximum velocity (i.e., V max ), then this velocity is set to the maximum value (V max ) and then adjusted before the position update by,

  V tþ1 ¼ min V 1;tþ1 ; V max ij ij

ð17Þ

where V ij1;tþ1 is calculated using Eq. (15).

PSO was originally designed and proposed by Kennedy and Eberhart [29,30]. PSO is a population based search algorithm and its basic idea was originally inspired by the simulation of social behavior of animals such as bird flocking, fish schooling, etc. As compared to other optimization techniques, PSO is faster, cheaper and more efficient. In addition, there are only few parameters to adjust in PSO. Therefore, PSO is an ideal optimization problem solver in optimization problems. PSO is best suited to solve the nonconvex, non-linear, continuous, discrete and integer variable type optimization problems. In PSO, each individual particle, i = [1, 2, 3, . . ., n], where n > 1 has a current position in the search space Xi, a current velocity Vi, and a personal best position in search space, Pbest,i. The personal best Pbest,i position corresponds to a position in the search space, where particle had the smallest value as determined by the objective function, considering a minimization problem. In addition, the position yielding the lowest value amongst all the personal best Pbest,i is called the global best position (Gbest) [31]. The following Eqs. (13) and (14) explains how the personal best (Pbest;i ) and global best (Gbest ) values are updated, respectively.

This problem can be solved, when V max is calculated by a fraction of domain of the search space on each dimension by subtracting the minimum bound from the maximum bound, and is expressed by

V max ¼ lðX max  X min Þ

ð18Þ

where X min and X max are minimum and maximum values of X, and

l 2 ð0 1Þ. V max was introduced to prevent the divergence and explosion. But, it has become unnecessary for the convergence because of the use of inertia weight (i.e., x) [27]. To control the balance between local and global exploration, to get the quick convergence, and to reach an optimum, the inertia weight whose value reduces linearly with iteration number is set according to the below equation [28,33].



xtþ1 ¼ xmax 

xmax  xmin t max

 t

xmax > xmin

ð19Þ

where xmin and xmax are the initial/minimum and final/maximum values of the inertia weight, respectively. Generally, the inertia weight reduces linearly from 0.9 to 0.4 over the entire run.

Table 13 Generator data for the Indian 75 bus system. Generator number

(MW) P min G

P max (MW) G

(MVar) Q min G

Q max (MVar) G

a (Rs/hr)

b (Rs/MW hr)

c (Rs/MW2 hr)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

100 100 40 40 2 1 1 20 60 30 40 80 50 10 20

1500 300 200 170 240 120 100 100 570 250 200 1300 900 150 454

1.50 0.25 0.50 0.20 0.50 0.25 0.10 0.10 1.00 0.10 0.20 2.00 1.50 1.00 0.80

4.00 0.96 0.83 0.60 0.80 0.50 0.25 0.68 2.50 0.56 0.50 3.44 2.80 0.84 1.50

3855 1928 3470 1395 709.1 2306 384.3 771 6746 674.6 1253 8674 4337 1926 2174

1419 1480 1435 1484 0 0 0 1497 1253 1498 1491 1092 1398 1742 951.3

0.1258 0.0629 0.1132 0.0671 0 0 0 0.0252 0.2202 0.0220 0.0409 0.2831 0.1415 0.2400 0.1049

S. Surender Reddy, C. Srinivasa Rathnam / Electrical Power and Energy Systems 80 (2016) 128–139

135

PSO parameters

Dk ¼ lDk1 ð1  Dk1 Þ

There are some parameters in PSO algorithm that may affect the search performance. For any given optimization problem, some of these parameter’s values and choices have large impact on the efficiency of the PSO algorithm, and the other parameters have small impact or no effect [13]. The description and selection of swarm size (n), number of iterations and the acceleration coefficients (c1 and c2) is presented in [32,33].

The value of l is selected between [0, 4]. The variation of l is highly influenced on Dk that is able to evaluate the Dk in a constant size, oscillates randomly or oscillates between a limited sequence of size. Above Eq. (23) is without any stochastic pattern in the value of l and indicating the chaotic dynamic when l = 4 and D0 – f0; 0:25; 0:50; 0:75; 1g [36]. The searching efficiency and capability of proposed algorithm can be increased greatly using the chaotic x.

Multi-Objective Optimization (MOO) The MOO problem is the optimization of two or more objectives that are required to be optimized simultaneously. PSO for Multi-Objective Optimal Power flow (MO-OPF) The performance of the original PSO greatly depends on C1, C2 and x. In order to increase the efficiency of PSO, a chaos algorithm is used to calculate the inertia weight (x), a self-adaptive method to calculate learning factors (C1 and C2). Also, an external repository has been used to save all non-dominated solutions during the evolutionary process, and a fuzzy decision making method is applied to sort these solutions according to their importance. Power system decision makers/system operators can select the desired solution between them by applying the fuzzy decision making method/fuzzy min–max approach to the Pareto optimal front/Pareto optimal solution. Self-adaptive method for computing the learning factors. The learning factors C1 and C2 determine the influence of the personal best Gkbest ,

P ki

and the global best respectively. They have a crucial effect on the convergence of PSO. When the global fitness is large, the particles are far away from the optimum point. Hence, a large velocity is required for global search in the search space of the problem, and so C1 and C2 must be larger values. For the local search, a small movement is required, therefore C1 and C2 must set to small values. According to above statement, learning factors in the proposed method are adjusted as a function of the global optimum fitness. Therefore, a self-adaptive method is developed to adjust the learning factors, which is expressed as follows [34]. k

C J ¼ 1 þ 1=½1 þ expðaxFðGkbest Þ Þ J ¼ ½1; 2

ð20Þ

The a value can be considered as the inverse of best compromise objective value in the first iteration [35], and is expressed by,

a¼

1=FðG1best Þ

ð21Þ

Chaotic formula for inertia weight. As explained, inertia weight factor is utilized to control the impact of the previous experience of velocities on the current velocity. A large value of velocity leads to a global search, whereas a small value of velocity tends to facilitate local search. As mentioned, this factor usually reduces linearly from 0.9 down to 0.4 [36]. Despite many important benefits of PSO such as its simple concept, minimal storage requirement, easy implementation and the easy incorporation with other optimization techniques, it may experience inappropriate convergence and fall into the local optima. Hence, proper tuning of the PSO parameters is crucial for calculating the global optimum solution. Here, inertia weight factor is tuned dynamically in every iteration, and is expressed as,

C xK ¼ xK D k K

ð22Þ k

x is reduced linearly from 0.9 down to 0.4. The value of D can be determined using the iterator chaotic system, i.e., the logistic map [37].

ð23Þ

Membership function for MOO Problem [14,38]. The MOO gives not a single optimum solution, but a set of Pareto optimal solutions, in which one objective cannot be improved without sacrificing the other objectives. For real life applications, however, we require to select one solution, which will satisfy the various goals to some extent. One of the important factors of the trade-offs decision is the imprecise nature of system operator’s/decision maker’s judgement. Therefore, the membership functions is presented to represent the goals of the each objective function [24]. In this paper, a simple linear membership function has been selected for every objective function. The membership function includes maximum and minimum boundary values together with a strictly continuous and monoton  ically decreasing function. The membership function li ðxÞ is calculated under given constraints for each objective function (i.e., F i ðxÞ). The ith membership function is expressed by [14,38],

li ðxÞ ¼

8 1 > > < max Fi

if F i ðxÞ < F min i F i ðxÞ

F max F min > i > : i 0

if F min < F i ðxÞ < F max i i

ð24Þ

if F i ðxÞ > F max i

where F min and F max are obtained from the optimization of each i i objective independently. Decision maker/system operator. In the proposed method, a repository is defined to save non-dominated solutions in every iteration. The solutions that are saved in this repository in all iterations are sorted by a type of decision-making function. Hence, it is possible to select the best solution by selecting the top solutions in this collection. A decision making function [39] that is utilized is explained below,

PNobj bk X lk ðjÞ Nuj ¼ Pm k¼1 PNobj j¼1 k¼1 bk X lk ðjÞ

ð25Þ

The weight values bk can be considered by the operator based on the importance of objective function. The solution with maximum membership function Nuj is the most preferred compromise solution based on the adopted weight factors. Finding Pi best and Gbest. In the first iteration, initial population is considered as P kibest; but in the second to final iteration, if P ikþ1 could dominate the P ki then, the Pki is replaced by Pikþ1 ; and if could not dominate P ki , then P ki is remained as the best solution in the next generation/iteration. If none of them dominates each other, then Pki is updated by min–max approach in the next generation/ iteration. By computing the following equation for all objectives, lDi is calculated for each individual is as follows,

lDi ¼ maxðminðl1 ; l2 ÞÞ

ð26Þ

Þ; lD ðX kmut Þ; lD ðPki Þ Pikþ1 ¼ max½lD ðX kþ1 i

ð27Þ

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Gbest ¼ Best of ½Pbest 

ð28Þ

GSO algorithm [40] is a new swarm optimization algorithm, which is proposed by K.N. Krishnanad and D. Ghose. The basic idea of this algorithm is explained from the natural glowworm’s activities in the night, the Glowworms exercise in group in nature, they interaction and inter-attraction with each other by one’s luciferin. If the glowworm emits luciferinmore light, it can attract more glowworms move toward it. Through simulate this natural phenomena, combined with the characteristics of natural glowworm populations, each glowworm at the owns field of view in search for the glowworm, which release the strongest luciferin, also move to the strongest glowworm. Currently, the GSO algorithm has been successfully applied to the multi-modal function optimization, multi-source tracking, multi-source position, collective robotics and harmful gas leak location, and so on [41,42]. In this paper, the GSO algorithm is considered to solve the OPF problem including the equality and inequality constraints of IEEE 30 bus system [43] and practical Indian 75 bus system. Further, GSO algorithm is used to solve the Multi-Objective Optimal Power Flow (MO-OPF) problem on IEEE 30 bus test system. Also, comparison of PSO and GSO algorithms has been made. Description of GSO algorithm The GSO algorithm starts by placing the glowworms randomly in the workspace, so that they are well dispersed. Initially, all the glowworms contain an equal quantity of luciferin. Each iteration consists of a luciferin-update phase followed by a movementphase based on the transition rule. Luciferin update phase The luciferin update phase depends on the function value at the glowworm position and so, even though all glowworms start with the same luciferin value during the initial iteration, these values change according to the function values at their current positions. During this phase, each glowworm adds, to its previous luciferin level, a luciferin quantity proportional to the measured value of the sensed profile (fitness) at that point. In the case of a function optimization problem, this would be value of the objective function at that point. Also, a fraction of the luciferin value is subtracted to simulate the decay in luciferin with time. The luciferin update rule is expressed by using,

ð29Þ

Movement phase During this phase, every glowworm decides, using a probabilistic mechanism, to move towards a neighbor that has a luciferin value more than its own. This means that they are attracted to neighbors who are growing brighter. For every glowworm i, the probability of moving towards a neighbor j is represented by,

Pj ðt Þ ¼ P

lj ðt Þ

ð30Þ

k2Ni ðtÞ lk ðt Þ

where k 2 Ni ðtÞ; Ni ðtÞ ¼ fj : di; jðtÞ < r id ðtÞ; li ðtÞ < lj ðtÞg. Let, the glowworm i select a glowworm j 2 NiðtÞ with pj(t) is expressed by Eq. (30). Then, the discrete-time model of glowworm movements can be defined as,



xi ðt þ 1Þ ¼ xi ðtÞ þ s

xj ðt Þ  xi ðtÞ jjxi ðtÞ  xi ðt Þjj



s¼

Glowworm Swarm Optimization (GSO)

lj ðt þ 1Þ ¼ maxf0; ð1  qÞlj ðtÞ þ cJ j ðt þ 1Þg

where



ð31Þ

d

if dij ðtÞ P d

dij ðtÞ otherwise

ð32Þ

Local-decision range update rule When the glowworms depend on only local information to decide their movements, it is expected that the number of peaks captured would be a strong function of radial sensor range. For instance, if the sensor range of each agent covers the entire workspace, all the agents move to the global optimum point, and the local optima are ignored. Since, we have considered that a prior information about the objective function is not available, in order to detect multiple peaks, the sensor range must be made a varying parameter. For this purpose, we associate each agent i with a localdecision domain whose radial range rid is dynamic in nature (0 < rid 6 r is ). A suitable function is selected to adaptively update the local-decision domain range of each glowworm and is expressed by,



rid ðt þ 1Þ ¼ min r s ; max 0; r id ðt Þ þ bðnt  jNi ðtÞjÞ

ð33Þ

The comparison of GSO and PSO [40] is described in Table 11 in Appendix A.1. Glowworm Swarm Optimization (GSO) algorithm 1. Read the input data including the generator real powers, bus voltages, generator fuel cost coefficients, emission coefficients of generators, reactive power of switchable VAR capacitors/sources, and tap settings of transformer. 2. Read GSO algorithm parameters. 3. Initialize initial luciferin value lo , local decision range r o . 4. Initialize the glowworm within the limits of each variable. 5. Run load flows and find the luciferrin value of all glowworms using Eq. (29). 6. Find the neighborhood glowworms having brighter glow and are in the local decision range. 7. Find the probability of glowworm moving towards a neighbor using Eq. (30). 8. Update the glowworm movement using Eq. (31) and check the limits. 9. Update the local decision range of all glowworms using Eq. (33). 10. Repeat the above steps (5)–(9) until maximum iterations are attained. 11. Display the results. GSO algorithm of MO-OPF 1. Read the input data including generator real powers, bus voltages, generator fuel cost coefficients, emission coefficients of generators, reactive power of switchable VAR

2. 3. 4. 5. 6. 7. 8.

capacitors/sources, tap settings of transformers, F min and i F max . i Initialize the agents. Initialize initial luciferin value, circular sensor range (rs) and local decision range (rd). Run load flows. Find the objective functions values and normalize the values using membership function. For every individual (Xi), compute the membership values for various objective functions. Apply Pareto method and add non-dominated solutions to the repository. Compute the luciferin value for all the agents using objective function values.

S. Surender Reddy, C. Srinivasa Rathnam / Electrical Power and Energy Systems 80 (2016) 128–139

9. Find the neighborhod agents which are in the local decision range. 10. Find the probability of agent moving towards the brighter agent within local decision range and having maximum probability using probability mechanism. 11. Update the agent movement using movement formula. 12. Update the local decision range using local decision range update rule. 13. If any element of each individual violates its limit, then the position of individual is fixed to its minimum/maximum operating point. 14. Find the non-dominated solutions between the repository members using the Pareto method. In each iteration, some non-dominated solutions are added to the repository and they might dominate solutions which had existed in the repository. Hence, it is necessary to apply the Pareto method again to find the real non-dominated solutions. 15. Check for iterations limits. If not reached then goto Step 4. 16. Apply decision maker formula with different weights, and display the results.

137

minimum (wmin) inertia weights are 0.9 and 0.1, respectively. Minimum cost fcmin = 802.00, Maximum cost fcmax = 950.00, Minimum emission femin = 0.2, Maximum emission femax = 0.35. The generation cost and emission values for various weight factors are depicted in Table 3. From this Table 3, it can be observed that generation cost will vary based on the weight factor. The operator has to decide the weight factor based on the operating condition. The generation cost is optimum when the generation cost weight factor is more, whereas emission is minimum when the emission weight factor is more. Fig. 5 depicts the graph between the generation cost and emission for IEEE 30 bus system using PSO. Simulation results using GSO The parameters used in GSO parameters are: Luciferin decay constant (q) is 0.95, Luciferin enhancement constant (c) is 0.95, Constant parameter (b) is 0.0005; Neighborhood threshold (nt) is 4; Radial range of Luciferin sensor (rs) is 0.005; and the local decision domain range (rd) is 0.0005.

Results and discussion In this paper, IEEE 30 bus and practical Indian 75 bus systems are used to test the effectiveness of the proposed GSO algorithm for the OPF problem. The IEEE 30 bus, 41 branch test system [44] includes 6 generator buses, 21 load buses and 41 lines, of which 4 branches/lines are tap setting transformer branches. The generator data of IEEE 30 bus system is presented in Table 12 in Appendix A.2. The practical Indian 75 bus [45], 97 branch system includes of 15 generating units, 24 transformers, 97 lines and 12 shunt reactors. The generator data of Indian 75 bus system is presented in Table 13 in Appendix A.3. Simulation results using PSO PSO parameters considered are as follows: Swarm size is 30, Acceleration constants C1 = C2 = 2, The maximum number of iterations are 200, minimum and maximum inertia weights are 0.1 and 0.9, respectively. Case 1: OPF with generation cost minimization as objective function on IEEE 30 Bus Test System using PSO Table 1 shows the scheduled power generation, generation cost, voltage magnitudes and phase angles for the generation cost minimization objective function for IEEE 30 bus test system using PSO. The total generation cost obtained is 800.08 $/hr. The total losses obtained in this case is 8.41 MW. Figs. 1 and 2 depicts the convergence characteristics (generation cost Vs number of iterations), and fitness Vs number of iterations for the IEEE 30 bus system. Case 2: OPF with generation cost minimization as objective on practical Indian 75 bus system using PSO Table 2 shows the scheduled power generations, voltages and optimum objective function values for the generation cost minimization objective for the practical Indian 75 bus system using PSO. The optimum generation cost obtained using PSO is 112,670 Rs/hr. Fig. 3 depicts the convergence characteristics using PSO on Indian 75 bus system. The variation of fitness with respect to number of iterations is presented in Fig. 4. Case 3: Multi-Objective Optimal Power Flow (MO-OPF) on IEEE 30 bus test system using PSO The PSO parameters considered in MO-OPF are as follows: Swarm size is 30, Acceleration constants are C1 = C2 = 2, maximum number of iterations are 200, maximum (wmax) and

Case 1: OPF with generation cost minimization as objective on IEEE 30 bus test system using PSO Table 4 presents the optimum generation schedules, voltages and objective function values for IEEE 30 bus test system using GSO. The optimum generation cost obtained using PSO on IEEE 30 bus test system is 799.0642 $/hr. Fig. 6 shows the convergence characteristics of GSO on IEEE 30 bus system. Fig. 7 depicts the fitness Vs number of iterations graph using GSO on IEEE 30 bus system. Table 5 presents the control variables and the objective function values for generation cost minimization objective function using different algorithms. The optimum cost obtained usning proposed GSO is 799.06 $/hr. The optimum value obtained uisng GSO is also compared with different algoritms in the literature like Genetic Algorithm (GA), Improved GA (IGA), Enhanced GA (EGA), Differential Evolution (DE), Modified DE (MDE), Gradient method, Biography Base Optimization (BBO), Particle Swarm Optimization (PSO) and General Algebric Modeling System (GAMS). From this it can be observed that the optimum cost obtained using GSO is better than all other algorithm presented in the literature. Case 2: OPF with generation cost minimization as objective function on practical Indian 75 bus system using GSO Table 6 presents the optimum generation cost, generator voltages, and optimum objective function values for the practical 75 bus system using GSO. The optimum cost obtained in this case is 110092.16 $/hr, and the total system losses are 221.42 MWs. The time per iteration in this case is 0.4818 s. Figs. 8 and 9 depicts the convergence characteristics and fitness characteristics of GSO on practical Indian 75 bus system. Case 3: Multi-Objective Optimal Power Flow (MO-OPF) on IEEE 30 bus system using GSO The parameters used in GSO are: Luciferin decay constant (q) is 0.95, Luciferin enhancement constant (c) is 0.95, Constant parameter (b) is 0.0005, Neighborhood threshold (nt) is 4, Radial range of Luciferin sensor (rs) is 0.005, Local decision domain range (rd) is 0.0005, Minimum cost (fcmin) is 802.00, Maximum cost (fcmax) is 950.00, Minimum emission (femin) is 0.2, and maximum emission (femax) is 0.35. Table 7 presents the optimum generation cost and emission values using MO-OPF using GSO. The optimum generation cost obtained with generation weight factor of 0.8 is 806.6973 $/hr,

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and the emission is 0.332 tons/hr. Whereas emission is optimum with the emission weight factor of 0.8. Therefore, the operator has to decide the weight factors based on the operating conditions. Fig. 10 depicts the multi-objective generation cost and emission characteristics of IEEE 30 bus system using GSO.

Appendix A

Comparison of PSO and GSO algorithms

A.3. Generator data for the Indian 75 bus system

Case 1: Comparison of PSO and GSO on IEEE 30 bus system Table 8 presents the comparison of PSO and GSO on IEEE 30 bus test system. The best, worst and mean generation costs obtained using PSO are 800.0532 $/hr, 800.8054 $/hr and 800.0831 $/hr, respectively. Whereas using GSO are 799.0519 $/hr, 799.9132 $/ hr and 799.0642] $/hr, respectively. The time per iteration for PSO and GSO are 0.0532Sec and 0.0613Sec, respectively. Fig. 11 presents the variation of generation cost Vs total number of iterations for IEEE 30 bus system using PSO and GSO.

References

Case 2: Comparison of PSO and GSO on Indian 75 bus system Table 9 presents the best, worst and mean generation costs of PSO are 112,590 Rs/hr, 114,620 Rs/hr and 112,670 Rs/hr, respectively. Table 9 also presents the best, worst and mean generation costs of GSO are 110,046 Rs/hr, 113,735 Rs/hr and 110092.16 Rs/ hr, respectively. Time per iteration using PSO and GSO are 0.4288 s and 0.4818 s, respectively. Fig. 12 presents the comparison of convergence characteristics of PSO and GSO on the practical Indian 75 bus system.

Case 3: Comparison of PSO and GSO for MO-OPF on IEEE 30 bus system Table 10 presents the simulation results of MO-OPF using PSO and GSO on IEEE 30 bus test system. From this table, it is observed that the simulation results obtained for MO-OPF using GSO are optimum compared to PSO. From the above studies, it is observed that cost based OPF solutions are not effective solutions from environmental point of view. Therefore, these optimization problems (cost and emission minimization) should not be treated as independent objectives. OPF problem with these conflicting objectives should be treated using Multi-Objective Optimization algorithms. In this paper, Glowworm Swarm Optimization (GSO) is proposed for the single and MultiObjective Optimization problem, and the simulation results obtained using GSO are also compared with Particle Swarm Optimization (PSO).

Conclusions In this paper, the Optimal Power Flow (OPF) problem has been solved using the Particle Swarm Optimization (PSO) and Glowworm Swarm Optimization (GSO) algorithms. Here, the OPF problem is solved taking minimization of generation cost and emission into consideration. These methods are studied on IEEE 30 bus and practical Indian 75 bus systems. PSO and GSO algorithms are also used to study the Multi-Objective Optimal Power Flow (MO-OPF) problem considering generation cost and emission as objectives on IEEE 30 bus system. The results show that GSO provides a better solution for OPF problem. Further, GSO takes less number of iterations for converge compared with PSO and also less memory is sufficient for GSO algorithm in compared with PSO, since best particles (Pbest and Gbest) are required to be memorized in PSO. However, computational time for iteration in PSO is less compared with GSO. In case of MO-OPF problem, GSO algorithm gives better results than PSO algorithm.

A.1. Comparison of GSO and PSO [40] A.2. Generator data for the IEEE 30 bus system

[1] Carpentier J. Contribution to the economic dispatch problem. Bull Soc Franc Electr 1962;3(8):431–47. [2] Chowdhury B, Rahman S. A review of recent advances in economic dispatch. IEEE Trans Power Syst 1990;5(4):1248–59. [3] Huneault M, Galiana FD. A survey of the optimal power flow literature. IEEE Trans Power Syst 1991;6(2):762–70. [4] Momoh JA, El-Hawary ME, Adapa R. A review of selected optimal power flow literature to 1993. Part i: Nonlinear and quadratic programming approaches. IEEE Trans Power Syst 1999;14(1):96–104. [5] Momoh JA, El-Hawary ME, Adapa R. A review of selected optimal power flow literature to 1993. Part ii: Newton, linear programming and interior point methods. IEEE Trans Power Syst 1999;14(1):105–11. [6] Pandya KS, Joshi SK. A survey of optimal power flow methods. J Theor Appl Inform Technol 2008;4(5):450–8. [7] Carlisle A, Dozier G. An off-the-shelf PSO. In: Workshop particle swarm optimization, Indianapolis; 2001. [8] Alsac O, Scott B. Optimal load flow with steady state security. IEEE Trans Power Appar Syst 1974;93(3):745–51. [9] Shoults RR, Sun DT. Optimal power flow based on P–Q decomposition. IEEE Trans Power Appar Syst 1982;101:397–405. [10] Reid GF, Hasdorf L. Economic dispatch using quadratic programming. IEEE Trans Power Appar Syst 1973;92:2015–23. [11] Burchett RC, Happ HH, Wirgau KA. Large-scale optimal power flow. IEEE Trans Power Appar Syst 1982;101:3722–32. [12] Stott B, Marinho JL. Linear programming for power systems network security applications. IEEE Trans Power Appar Syst 1979;98:837–48. [13] Momoh JA. Electric power system applications of optimization. Markel Dekker; 2001. [14] Sailaja Kumari M, Maheswarapu S. Enhanced genetic algorithm based computation technique for multi-objective optimal power flow solution. Int J Electr Power Energy Syst 2010;32(6):736–42. [15] Bhowmik AR, Chakraborty AK. Solution of optimal power flow using non dominated sorting multi objective opposition based gravitational search algorithm. Int J Electr Power Energy Syst 2015;64(January):1237–50. [16] Kumar AR, Premalatha L. Optimal power flow for a deregulated power system using adaptive real coded biogeography-based optimization. Int J Electr Power Energy Syst 2015;73(December):393–9. [17] Ghasemi M, Ghavidel S, Gitizadeh M, Akbari E. An improved teaching– learning-based optimization algorithm using Lévy mutation strategy for nonsmooth optimal power flow. Int J Electr Power Energy Syst 2015;65 (February):375–84. [18] Christy AA, Vimal Raj PAjayD. Adaptive biogeography based predator–prey optimization technique for optimal power flow. Int J Electr Power Energy Syst 2014;62(November):344–52. [19] Bhowmik AR, Chakraborty AK. Solution of optimal power flow using nondominated sorting multi objective gravitational search algorithm. Int J Electr Power Energy Syst 2014;62(November):323–34. [20] Vaisakh K, Srinivas LR. Genetic evolving ant direction HDE for OPF with nonsmooth cost functions and statistical analysis. Expert Syst Appl 2011;38 (3):2046–62. [21] Arya LD, Choube SC, Kothari DP. Emission constrained secure economic dispatch. Int J Electr Power Energy Syst 1997;19(4):279–85. [22] Rosehart WD, Canizares CA, Quintana VH. Multi-objective optimal power flows to evaluate voltage security costs in power networks. IEEE Trans Power Syst 2003;8(2):578–87. [23] Yokoyama R, Bae SH, Morita T, Sasaki H. Multiobjective optimal generation dispatch based on probability security criteria. IEEE Trans Power Syst 1988;3 (1):317–24. [24] Abido MA. A novel multi-objective evolutionary algorithm for environmental/ economic power dispatch. Electric Power Syst Res 2003;65(1):71–81. [25] Elbeltagi E, Hegazy T, Grierson D. Comparison among five evolutionary-based optimization algorithms. Adv Eng Inform 2005;19(1):43–53. [26] Ramesh VC, Li X. A fuzzy multi-objective approach to contingency constrained OPF. IEEE Trans Power Syst 1997;12(3):1348–54. [27] Abraham A, Konar A, Das S. ; 2008. [28] Rao SS. Engineering optimization theory and practice. 4th ed. John Wiley and Sons; 2009. [29] Eberhart RC, Kennedy J. A new optimizer using particle swarm theory. In: Proc. of the 6th international symposium on micro machine and human science, Nagoya, Japan; 1995. p. 39–43. [30] Kothari DP, Dhillon JS. Power system optimization. 2nd ed. PHI learning Private limited; 2011.

S. Surender Reddy, C. Srinivasa Rathnam / Electrical Power and Energy Systems 80 (2016) 128–139 [31] Engelbrecht AP, van den Bergh F. A new locally convergent particle swarm optimizer. In: IEEE conference on systems, man and cybernetics, Tunisia; 2002. [32] Engelbrecht AP. Computational intelligence: an introduction. John Wiley and Sons; 2007. chapter 16. p. 289–358. [33] Lee KY, Park JB. Application of particle swarm optimization to economic dispatch problem: advantages and disadvantages. IEEE; 2010. [34] Niknam T, Narimani MR, Aghaei J, Azizipanah-Abarghooee R. Improved particle swarm optimization for multi-objective optimal power flow considering the cost, loss, emission and voltage stability index. IET Gener Trans Distrib 2012;6(6):515–27. [35] Niknam T, DoagouMojarrad H, ZeinoddiniMeymand H. A novel hybrid particle swarm optimization for economic dispatch with valvepoint loading effects. Energy Convers Manage 2011;52(4):1800–9. [36] Park JB, Jeong YW, Kim HH, Shin JR. An improved particle swarm optimization for economic dispatch with valve-point effect. Int J Innov Energy Syst Power 2006;1(1):1–7. [37] Caponetto R, Fortuna L, Fazzino S, Xibilia MG. Chaotic sequences to improve the performance of evolutionary algorithms. IEEE Trans Evol Comput 2003;7 (3):289–304. [38] Surender Reddy S, Abhyankar AR, Bijwe PR. Reactive power price clearing using multi-objective optimization. Energy 2011;36(5): 3579–89. [39] Niknam T, Meymand HZ, Mojarrad HD. An efficient algorithm for multiobjective optimal operation management of distribution network considering fuel cell power plants. Energy 2011;36(1):19–132.

139

[40] Krishnand KN, Ghose D. Glowworm swarm optimisation for simultaneous capture of multiple local optima of multimodal functions. Swarm Intell 2009;3 (2):87–124. [41] Krishnand KN, Ghose D. Glowworm swarm optimisation: a new method for optimizing multimodal functions. Int J Comput Intell Stud 2009;1(1):93–119. [42] Krishnand KN, Ghose D. Glowworm swarm based optimization algorithm for multimodal functions with collective robotics applications. Multiagent Grid Syst 2006;2(3):209–22. [43] Surender Reddy S, Bijwe PR, Abhyankar AR. Faster evolutionary algorithm based optimal power flow using incremental variables. Int J Electr Power Energy Syst 2014;54:198–210. [44] Lai LL, Ma JT, Yokoyama R, Zhao M. Improved genetic algorithms for optimal power flow under both normal and contingent operation states. Int J Electr Power Energy Syst 1997;19:287–92. [45] Raglend IJ, Padhy NP. Solutions to practical unit commitment problems with operational power flow and environmental constraints. Proc IEEE 2006:1–6. [46] Abido MA. Optimal power flow using particle swarm optimization. Int J Electr Power Energy Syst 2002;24:563–71. [47] Bhattacharya A, Chattopadhyay PK. Application of biogeography based optimization to solve different optimal power flow problems. IET Gener Transm Distrib 2011;5:70–80. [48] Abou El Ela AA, Abido MA, Spea SR. Optimal power flow using differential evolution algorithm. Electric Power Syst Res 2010;80:875–85. [49] Sayah S, Zehar K. Modified differential evolution algorithm for optimal power flow with non-smooth cost functions. Energy Convers Manage 2008;49: 3036–42.