Modern Swarm Intelligence based Algorithms for Solving Optimal Power Flow Problem in a Regulated Power System Framework

This paper presents artificial swarm intelligent based algorithms viz., Firefly Algorithm (FFA), Dragonfly Algorithm (DA) and Moth Swarm Algorithm (MSA) to take care of the issues related to optimal power flow (OPF) problem in a power system network. The optimal values of various decision variables obtained by swarm intelligent based algorithms can optimize various objective function of OPF problem. This article is focused with four objectives such as minimization of total fuel cost (TFC) and total active power loss (TAPL); improvisation of total voltage profile (TVD) and voltage stability index (VSI). The effectiveness of various swam intelligent algorithms are investigated on a standard IEEE-30 bus. The performance of distinct algorithms is compared with statistical measures and convergence characteristics.


Introduction
The optimal power flow (OPF) problem is a familiar problem in an interconnected power system network. The OPF solution gives optimal values of generators, transformers, reactive compensating devices through which different objectives can be optimized. The objective function of OPF problem has to meet out constraints in the form of hard (equality) and soft (inequality) constraints which make the problem to become more complex, multi-modal and non-convex problem. The traditional methods such as linear programming, gradient based methods and quadratic programming are suffered with local convergence in order to obtain global optimal value [1]. So as to defeat the disadvantages of traditional methods, artificial swarm intelligent based algorithms have been used in this article for solving OPF problem. The swarm intelligent based algorithms viz., Firefly algorithm, Moth swarm algorithm and Dragonfly algorithm can handle multi-modal problem effectively and has fast convergence rate [2].
The Firefly Algorithm (FFA) is one of the swarm intelligent based algorithms developed by Xin-She-Yang based on the attracting behaviour of fireflies [3]. All the fireflies are viewed as unisexual and their fascination is straight-forwarded to their glimmering light illumination. The firefly with lower illumination moves towards the firefly with higher light illumination. In the event that there are no fireflies close by or the fireflies have same illumination, the firefly will move an irregular way. The illumination of blaze is related with the fitness value. The light force likewise complies with the reverse square law [4].
The Moth Swarm Algorithm (MSA) is suggested by Al-Attar Ali Mohammad in 2016 is dependent on the direction of moths in an uproarious climate towards light of the moon [5]. Moth swarm comprises of three groups of moths, namely, pathfinders, prospectors and onlookers. Pathfinders' is a little gathering of moths that can find new regions over the improvement space. This is to segregate the optimal positions of the light source to control the development of the principle swarm. Next, prospectors' is a gathering of moths that will roam in an irregular way inside the neighborhood of the light sources and are set apart from pathfinders. Onlookers are a group of moths that can float straightforwardly toward the optimal arrangement that has been acquired by prospectors [6].
The dragonfly algorithm (DA) is another artificial swarm intelligent algorithm and is based on the behaviour of dragonflies described by Mirjalili in 2016 [7]. The population of DA has five behaviours, namely separation (staying away from impacts among the dragonflies), alignment (keeping up the trip among the dragonfly gathering), cohesion (moving close to each other individual), foraging and eluding enemies [8].

1). Case-1:
The target element is to limit the total fuel cost which is expressed as eq.5 (5) where Fi : fuel cost of generator-i, ai, bi, ci are the cost coefficients of generator-i 2). Case-2: The target element is to limit the total active power loss which is expressed as eq.6 (6) Where Gij : conductance of transmission line of bus i and bus j, δij : voltage phase difference between bus i and bus j.
3). Case-3: The target element is to limit the total voltage deviation of load bus from specified voltage which is expressed as eq.7

4). Case-4:
The target element is to limit the voltage stability index (L) value, thereby keep the system far away from voltage collapse. The objective function is expressed as eq.8 OF_4 = min(L max ) = min(max(L n )) n = 1 … , NPQ

Equality constraints
(10) where PGi :MW at bus i, QGi : MVAR at bus i, PDi : MW load demand at bus i, QDi : MVAR load demand at bus i, Bij : susceptance of transmission line from bus i to bus j, Vi : voltage at bus i, Vj : voltage at bus j.

Methodology
This section presents pseudo-codes for FFA, DA and MSA algorithms.
where I0 is unique light force, γ is assimilation (absorption) co-effective, r is separation between fireflies. Assess the attractiveness of firefly.
( ) = 0 exp (− . 2 ) (12) Where ( ) is monotonically decreasing function and 0 is attractiveness at r=0 Assess the development of firefly towards brightness Where being the step number varied (random moment), rand is contingent number varied between [0,1], ri,j is Cartesian separation from firefly-i to firefly-j.
Increment the iteration

Results and Discussions
The swarm intelligent algorithms discussed in Section III are applied for standard test system(IEEE-30 bus system) and practical test system (62-bus Indian system) and the simulation is carried out in a PC with 64-bit Windows 7 OS having i5 Intel processor operating at 3.2 GHz with RAM of 6GB. The results are obtained by using MATPOWER 7.0 in MATLAB 2013b. A maximum number of iterations are set as 200. A number of search agents are taken as 50 and 25 individual runs have been conducted to obtain best optimal value for each objective function.

Standard Test System: (IEEE-30 Bus System)
The parameters that show effect on FFA are mutation co-efficient (α), initial attractive co-efficient (β), absorption co-efficient (γ), mutation co-efficient damping ratio (δ). The best value of the parameters at which the test system gives optimal value are considered as α=0.2, β=2, γ=1, δ=0.98. The specification that has significance on MSA is the number of pathfinders which should be greater than 4 and lesser than 20% of the search agents. Therefore, the number of pathfinders is 8. DA is controlled by parameters like weights of inertia (w), enemy distraction (e), separation (s), alignment (a), cohension (c) & food attraction (f). Inertia weight is varied from 0.4 to 0.9. The inertia weight is considered as w = 0.9. Enemy distraction weight is varied in between 0 and 0.9 and the best value is taken as e = 0.1. The remaining specifications s, a, c, f are random values between 0 and 1. The parameter settings of swarm intelligence algorithm are given below Table 1. The best parameter values for the FFA are given in the Table 2. The best value is noticed at which the objective function has given best fit value. The minimization of total fuel cost for standard test system is considered as the objective function. The best fit value for the objective function is 802.1310 given at α=0.2, β=2, γ=1, δ=0.98. α is varied from 0.1 to 0.8 and the optimal value is 0.2. β is varied from 0.5 to 4 and the optimal value is 2. The optimal value is obtained at γ=1 by varying from 0.4 to 1.4. δ is varied with the interval of 0.02 from 0.92 to 1.04 and the best value is given at 0.98. The best parameter values are obtained by conducting 100 iterations with 20 search agents and 10 individual trials.   The best fit value of objective function with variation of α is shown in Figure 1. The best fit value of objective function with variation of β is shown in Figure 2. The best fit value of objective function with variation of γ is shown in Figure 3. The best fit value of objective function with variation of δ is shown in Figure 4.   Table 3 shows the best fit value of the objective function with different values of δ, β, γ, δ. The best fit value for the objective function is 802.1310 is obtained with the values of α=0.2, β=2, γ=1, δ=0.98. The best fit value of objective function with different values of δ, β, γ, δ as P1, P2, P3, P4, P5, P6, P7 is shown in Figure 5.
The optimal values of control variables for each objective function of standard test system using FFA is given in Table 4. The range of voltage magnitude is 0.95 and 1.1. The minimum and maximum transformer tap setting ratio is 0.9 and 1.1. The variation of shunt compensator is from 0-5 MVAR. About 25 variables are taken as decision variables. The decision variables are generated MVA at bus no 1,2,5,8,11,13, generated MV at bus no 1,2,5,8,11,13, transformer tap setting ratio at branch no 11,12,15,36 and MVAR shunt compensators at bus no 10,12,15,17,20,21,23,24,29. A comparison of the best value for each objective function using DA and MSA with FFA is given in Table 5. By comparison, the FFA gives the minimum optimized value for all objective functions. The convergence characteristic curves of four cases for Standard test system are shown in Figure 6- Figure 9. The best, mean, worst values of each objective function with FFA, MSA, DA for Standard test system are given in Table 6. The best optimal value given by FFA is very close to mean value for all objective functions for standard test system. FFA is more efficient than MSA and DA for standard test system in the view of success rate, global optimal value as well as iterations.

Conclusion
The performances of various swarm intelligent algorithms viz., FFA, DA and MSA for solving OPF problem are presented in this paper. Four different objective functions such as minimization of total fuel cost, minimization of total active power loss, minimization of voltage magnitude deviation and improvisation of voltage stability index are considered subject to various hard and soft constraints. The investigations are carried on a standard IEEE 30 bus system. Based on the simulation results, it is acquired that the best values for OPF problem is from FFA. The outstanding parameter values for FFA is considered at α=0.2, β=2, γ=1, δ=0.98. The best value contributed by FFA is exceptionally near the mean value among 25 trial runs. FFA is much more efficient in finding the global optima with higher success rate and less iteration for four objective functions when compared with MSA and DA. Due to its cognitive nature, FFA outperforms the other algorithms for this test framework. Based on the convergence characteristics and statistical measures such as best, mean and the worst values of various objectives of OPF, FFA has better ability to give optimal solutions than MSA and DA for solving OPF problem.