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270 CHAPTER 8 Hybrid PV/Batteries Bank/Diesel Generator

(because of high prices of batteries and their short lifetime) to achieve lower overall
system cost.
The second phase of optimization strategy is applied at the nighttime where the
DG supplies the load when the energy stored in the batteries bank is not sufficient
to meet the load needs. The batteries dod parameter is the main criterion used to
decide when to stop using the batteries bank and use the DG instead. The maximum
dod value typically used at which the DG is turned ON is 80%. However, the
minimum typical value of dod parameter at which the DG is turn OFF is 20%.
The maximum and minimum dod values affect the hybrid energy system overall
cost. Thus, optimal range of dod values is crucial to optimally size the hybrid
renewable energy system. Generally, minimizing CO 2 emission requires maxi-
mizing the number of batteries to minimize the operational time of the DG.
Thus, the optimal number of batteries found in the first two phases is contradictory.
The same number of batteries cannot be used to achieve minimum cost and min-
imum pollution emission rates at the same time. Hence, in the third and last phase
of optimization process, the results obtained in the first two phases are used to get
optimal values of the number of PV panels, the number of batteries, the DG output
power, and operational time over 24 h. It is clear that the optimization-sizing pro-
cess is a nonlinear optimization problem.

3.2.4 Optimization Process Using Particle Swarm Optimization
The PSO algorithm is a stochastic technique that was developed in 1995 by Kennedy
and Eberhart [20e22]. It is based on a set of population of birds called swarm that is
generated randomly. Each particle of the swarm is considered as a potential solution
for the optimization problem. It is characterized by its position vector
! !
x i ¼ðx i1 ; x i2 ; .; x in Þ and its velocity vector v i ¼ðv i1 ; v i2 ; .; v in Þ, where n rep-
resents the number of population in the swarm and i represents the number of
variables of the optimization process. Each particle remembers its own best position
ƒƒƒ!
Pbest i ¼ðPbest 1 ; Pbest 2 . Pbest in Þ, the best swarm global experience vector
ƒƒƒ!
gbest i ¼ðgbest 1 ; gbest 2 . gbest in Þ and its previous velocity vector as described
in the following equations:

v tþ1 ¼ wv t þ c 1 r t pbest t x t þ c 2 r t
i; j i; j 1i; j i; j i; j 2i; j
(8.24)

t t
gbest i; j x i; j ; j˛f1; 2; .ng
x tþ1 ¼ x t þ v tþ1 ; j˛f1; 2; .ng (8.25)
i; j i; j i; j
where t is the iteration index, the variables i and j are indices of the optimization vec-
tor, r t and r t are random variables with values in the range [0,1], the variables
1i; j 2i; j
c 1 and c 2 are acceleration constants with value usually equal to 2, w is the inertia
weight, w max and w min are the initial and final inertia weights, and max iter is the
maximum number of iterations. To improve the algorithm conversion process, the