
Due to the non-linearity, multi-variable and multi-modal
features of solar cell models, it is difficult for these tech-
niques to accurately extract the unknown parameters. As
a result, arti ficial intelligence (AI) techniques, such as arti -
ficial neural network (ANN) or fuzzy logic based intellige nt
modeling techniques (AbdulHadi et al., 2004) and meta-
heuristic algorithms based parameter extraction techniques
(Hachana et al., 2013), have been developed and proposed
for this problem. The recent developments on this topic as
well as different parameter extraction techniques are
described in the following section.
Chaos optimization algorithm (COA) (Li and Jiang,
1998; Yang et al., 2007; Yuan and Wang, 2008) is a novel
global optimization technique, which employs numerical
sequences generated by means of chaotic maps. Due to
the distinctive properties of chaotic sequences, such as sen-
sitive dependence on initial conditions, stochasticity and
ergodicity, many existing application results have demon-
strated that COA escapes from local minima more easily
than algorithms such as genetic algorithm (GA), simulated
annealing (SA), particle swarm optimization (PSO), differ-
ential evolution (DE) and harmony search algorithm
(HSA) (Okamoto and Hirata, 2013; Yang et al., 2014). In
this article, a novel mutative-scale parallel chaos optimiza-
tion algorithm (MPCOA) is proposed to extract parame-
ters of solar cell models. In this article, the parameter
extraction of solar cell models is posed as an optimization
process which makes the difference minimum between the
real data and estimated values. In order to evaluate the
ability of the MPCOA, different solar cell models, i.e., dou-
ble diode, single diode and PV module, have been applied.
Compared with other optimization algorithms based
parameter extraction techniques, the proposed MPCOA
has the superiority of global optimization ability and good
performance. Comparison results with other parameter
extraction techniques are in favor of the MPCOA.
The rest of this article is arranged as follows. Section 2
briefly surveys recent developments in parameters extrac-
tion of solar cell models. The optimization problem for
parameters extraction of solar cell models is introduced
in Section 3. Section 4 gives presentation of the proposed
MPCOA technique. Simulation and experiment results
have been used to verify the ability of the MPCOA in Sec-
tion 5. Conclusions are presented in Section 6.
2. Survey on recent developments in parameter extraction of
solar cell models
Accurate parameter values are extremely crucial in the
simulation, evaluation, control and optimization of PV sys-
tems. Conse quently, a lot of techniques have been devel-
oped and proposed for this problem. A brief review of
these techniques is as follows.
Firstly, conventional nonlinear fitting algorithms, such
as least-squares method (LSM) and its variations, have been
put forward to solve this problem. In Easwarakhantha n
et al. (1986), Newton’s method based modified nonlinear
LSM was proposed to extract solar cell models parameters
from measured current and voltage (I–V) data, and a
reduced LSM initialization routine was used to overcome
the difficulties in initializing parameters. In Kim and Choi
(2010), the diode I–V curve was constructed by the LSM
based correlation between estimated values and measured
ones. The major defect of LSM fitting is its sensitivity to
outliers. Outliers usually have a great impact on the fitting
performance because the effect of these extreme data points
is magnified by squaring the residuals.
Second, analytical solution methods have also been
widely used in this field. According to different mathemati-
cal representations, analytical solution methods in this field
can be classified to two main types: one type is approximate
analytical methods which are commonly expressed in terms
of elementary functions, the other type is exact analytical
methods based on the Lambert W-function which cannot
be expressed in terms of elementary functions. Karmalkar
and Haneefa (2008) proposed a simple explicit current–den-
sity–voltage (J–V) model, which simplified the prediction of
J–V curves. Saleem and Karmalkar (2009) proposed analyt-
ical parameters extraction method which avoided the diffi-
cult measurements of dJ =dV and ðJ
p
; V
p
Þ. The parameters
of the single exponential J–V model were extracted simulta-
neously from measurements of open c ircuit voltage
V
oc
, short circuit current density J
sc
; V j
J¼0:6J
sc
, and
Jj
V ¼0:6V
sc
. Das (2012) developed an analytical explicit J–V
solar cell model from the physics based on implicit J–V
equation, which was obtained by using Taylor’s series
expansion based algebraic manipulations. Lun et al.
(2013) used Pad
e approximant to represent the exponential
function term of I–V equation, both basic and modified
Pad
e approximants models were considered. Lun et al.
(2013) developed Taylor’s series expansion based explicit
analytical model, which described the entire singl e-diode
solar cell I–V characteristic using two modified five-param-
eter models. In a few words, the main disadvantage of these
approximate analytical methods is that the availability is
under certain conditions, such as continuity, differentiabil-
ity and convexity ( AlRashidi et al., 2011). Moreover, the
implementation of these methods is not easy because they
typically involve heavy computation and tedious algebraic
manipulation.
Nowadays, the Lambert W-function based exact analyt-
ical methods have already demonstrated useful. In mathe-
matics, the Lam bert W-function is a set of functions,
namely the branches of the inverse function with the defi ni-
tion as: x ¼ W ðxÞe
W ðxÞ
, for any complex number x. Many
researches have demonstrated that Lambert W-function
can distinctly represent the relationship between voltage
V, current I and resistance R in a diode (Jain and
Kapoor, 2004; Jain and Kapoor, 2005). More importantly,
solutions based on Lambert W-function are not only exact
and explicit but also easily differentiable. Jain and Kapoor
(2004) pro posed Lambert W-function based exact analyti-
cal solutions of real solar cells parameters, which were
obtained by the W-function based explicit solutions. Jain
X. Yuan et al. / Solar Energy 108 (2014) 238–251 239