0885-8950 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TPWRS.2018.2850049, IEEE
Transactions on Power Systems
1
Lead-Acid Battery Modeling over Full State of
Charge and Discharge Range
Ilario Antonio Azzollini, Valerio Di Felice, Francesco Fraboni, Lorenzo Cavallucci, Marco Breschi, Senior
Member, IEEE , Alberto Dalla Rosa, and Gabriele Zini
Abstract—The discharge behavior of electrochemical solid state
batteries can be conveniently studied by means of electrical
analogical models. This paper builds on one of the best known
models proposed in literature for lead-acid electrochemistry
(the Ceraolo’s model) by formulating an alternative third-order
model and implementing a methodology to compute all model
parameters (using particle swarm and non-linear programming
optimization) with increased precision and full usability over the
whole range of possible states of charge and discharge currents.
The developed methodology is used efficiently to model all
commercial lead-acid batteries and enable their integration into
simulation software for the optimized design of energy systems
using energy storage.
Index Terms—Battery modeling, energy storage, lead-acid
battery, non-linear programming, optimization, particle swarm
optimization.
I. INTRODUCTION
L
EAD-ACID batteries are the most widespread recharge-
able electrochemical devices, used in many different
applications to guarantee quality of service even in absence
of a power source for reasonably long periods of time. To
better design and evaluate complex systems resorting to lead-
acid energy storage for correct functioning, reliable and precise
battery models are needed.
A number of different modeling strategies can be adopted
for energy storage characterization [1]. Natural models aim
at devising an accurate, usable, and reproducible physical-
chemical framework that explains the natural phenomena
occurring during the functioning of the batteries; such models
are translated into sets of equations that are used to evaluate
the variation of the battery parameters during functioning.
Empirical modeling is based on descriptions of the observed
data, frequently employing regression analysis or artificial
intelligence techniques to model the response of the battery
from a set of input variables. Abstract models use analogies
(normally with electrical circuits or stochastic process models)
that simplify the real behavior of the system but improve
the capability to interpret and understand the way the battery
works. Mixed models combine the advantages of the previous
strategies to obtain more refined results while retaining a
reasonably simplified model.
Natural models are normally developed by experts in elec-
trochemistry [2]–[9]; the complexity that is inherent in this
modeling technique is hampering its efficient use for commer-
cial purposes. This gives way to abstract or mixed models [10],
[11] which have proved to be more readily usable, although
less stringent from a scientific point of view. In particular,
mixed models based on electrical circuit parametrization can
be more easily implemented in electrical simulation codes
that are already designed to deal with the input and output
variables typical of battery technology (namely, voltage and
current). As a matter of fact, the third-order dynamical model
developed in [11] for lead-acid batteries has been implemented
in a package of a well-known commercial mathematical soft-
ware [12]. Electrochemical batteries are indeed conveniently
modeled by means of electrical analogy, i.e. using networks of
well-known electrical components (resistors, capacitors, elec-
tromotive forces, etc.); in literature, two different approaches
can be found: modeling every single part of the battery with
a corresponding electrical element [13], [14], or modeling the
battery behavior by using a black box approach interpreting
what is the output at the terminals of the battery [15]–[17].
An example of abstract battery modeling by means of
artificial intelligence techniques can be found in [18], where
fuzzy logic is employed to characterize the discharge of lead-
acid batteries by modeling the relationship between the battery
open-circuit voltage, the state of charge, and the discharge
currents.
A general model structure for lead-acid batteries was de-
fined by Ceraolo in [11], from which specific models can be
inferred, and in particular, the implementation of the third-
order model was developed in detail. In the context of the
general framework proposed by Ceraolo, an alternative third-
order model formulation is presented in this paper, showing
an extended validity and usability over all the State Of Charge
(SOC) and discharge current ranges. Compared to Ceraolo’s
third-order model formulation, a resistance is added to the
main branch, whose characteristic equation is designed for
fitting as close as possible the lead-acid battery behavior at
the beginning of the discharge process. Then, as the model
is characterized by a significant number of parameters to be
identified, an optimization methodology is proposed and tested
on a battery commercial data-sheet. The overall result is a
complete, fully functional, and easily usable methodology that
can be implemented as a library for further integration into
other advanced energy systems’ simulation software.
The paper is organized as follows. Section II describes
the model modification and the parameter evaluation for
the correct characterization of the battery at all current rate
discharging over the full SOC range with comparison between
modeled and real curves. Section III discusses the main results
and draws the main conclusions. The Appendix describes the
original third-order model adopted for this study.
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