Levy flight movement in the optimization

PEVs (plug-in electric vehicles) which can be recharged from an
external electricity network are generally referred to both PHEVs
(plug-in hybrid electric vehicles) that share electricity and fossil
energy as well as EVs (electric vehicles) which only consume
electricity. They recently have gained an increased interest due to
increasing petroleum price and their opportunity to reduce
greenhouse gas emissions.
In PHEVs, various items can influence over charging performance
of these devices namely type of charger, charging method,
SOC (state of charge) of the battery, charging commencement and
time span, battery capacity and the quantity of batteries being
charged. When considering the charging demand of PHEVs in a MG,
the sum of vehicles load will be unsure due to the uncertain
behaviour of the mentioned charging factors. In order to include
the effects of PHEV charging on the MG, three charging strategies of
uncontrolled, controlled and smart schemes are defined and
considered in this study.
2.1. Charging start time
Among different charging schemes, the easiest charging strategy
is plugging in the vehicle and start charging when arriving
home in the evening. In this “worst-case” scenario, half of privately
owned EVs (privately-owned commuting and non-commuting EVs,
are assumed to charge every two days on average) and all
company-owned EVs, are expected to begin their recharge cycles
every day (excluding weekend) at nearly the same time [21]. Note it
that this model is considering majority of the charging demand by
EVs. It is assumed that the grid loading data is employed to find the
average charging time of EVs in uncontrolled manner. Meanwhile,
in order to consider the uncertainty and inaccuracies of this
modelling, rectangular PDF (probability density function) function
with the average of 6 pme7 pm is used. This model shows that the
charging of PHEVs mainly happens from 6:00pm to 7:00pm with
specific probability levels that are determined according to the PDF
function. Here we have employed the MCS (Monte Carlo simulation)
method to model this uncertainty. It is clear that other statistical
data with probability functions can be considered in the
same manner. Therefore it can be considered that in uncontrolled
charging method, PHEVs will start charging in the range 6:00pm to
7:00pm which mathematically is described using a rectangular
distributed PDF (probability density function) with a small distribution’s
support of charging start time