Squirrel search algorithm for economic dispatch with valve-point effects and multiple fuels

ABSTRACT

This paper bestows a new swarm intelligence approach, Squirrel Search Algorithm (SSA) to solve Economic Load Dispatch (ELD) of the thermal unit by addressing the valve point loading effects and multiple fuel options. SSA inspires the foraging behavior of squirrels which is based on dynamic jumping and gliding strategies. The main intention of the ELD problem is to minimize the total generation cost of units while assuring various system constraints. Renovate strategy and selection rules are used in the SSA algorithm to handle the constraints appropriately. The practicability of the proposed algorithm is tested on six different power test systems having different sizes and intricacies. Simulation results ascertain that the proposed SSA approach outperforms the other existing heuristic optimization techniques in terms of solution quality, robustness, and computational efficiency. Consequently, the proposed SSA can be an efficient approach for solving the ELD problems with valve point loading impacts and multi-fuel options.

KEYWORDS:

• Economic load dispatch
• meta-heuristic algorithm
• multi-fuels
• non-convex fuel cost function
• squirrel search optimizer
• valve point effect

Nomenclature

$a_i,b_i,c_i$	=	cost coefficients of generator i
aij, bij, cij,	=	cost coefficients of the unit i for fuel type j
dij and eij	=	cost coefficients of the VPL effect of unit i for fuel type j
Bij	=	line loss coefficients
CD	=	drag coefficient
CL	=	lift coefficient
$d_i,e_i$	=	cost coefficients of the VPL effect of generator i
dij and eij	=	cost coefficients of the VPL effect of unit i for fuel type j
D	=	drag force
dg	=	gliding distance
$F$	=	total fuel cost of the generators
$F(P_i)$	=	fuel cost of ith generating unit
$F(P_i{)}_{max}$	=	cost function of the worst feasible solution in the population
Gc	=	gliding constant
hg	=	gliding height
i and j	=	indices of unit and fuel type respectively
k	=	index of prohibited zone
K	=	number of fuel types for each unit
L	=	lift force
M	=	sample mean
n	=	sample size
ng	=	total number of generating units
nz	=	total number of POZ
PD	=	power demand
Pdp	=	predator presence probability
Pi	=	power generation of ith unit
$P_i,P_i^0$	=	current and previous power output of ith unit respectively
Pij, min and Pij, maxrespectively	=	minimum and maximum power output of unit i with fuel option j
$P_{i,k}^L$, $P_{i,k}^U$	=	lower and upper power outputs of the kth prohibited zone of the ith generator respectively
$P_{i,min},P_{i,max}$	=	minimum and maximum generation of unit i
PL	=	transmission losses
r1, r2 and r3	=	random numbers in the range of [0, 1]
ra and rb	=	randomly distributed numbers in [0, 1]
S	=	surface area of body
Smin	=	minimum value of seasonal constant
SM	=	standard error
SD	=	standard deviation of a sample
t	=	current iteration
tan (ɸ)	=	gliding angle
tmax	=	maximum iteration value
URi, DRi	=	up and down ramp limits of ith unit respectively
V	=	speed
Xh	=	position of squirrel individual which reached the hickory tree
XL, XU	=	lower and upper bounds of squirrel individual
Z	=	Z statistic estimated by confidence level
ρ	=	density of air
β	=	constant
$\mathrm{\Delta }{P}_i$	=	power balance constraint violation