IEEE Power Electronics Magazine - December 2014 - 32

approximations, it can be shown that
C2
C n = C +b C
b
s
Cb Cs
Cp = C + C
b
s
R t (C b + C s) 2
C 2b
R p = Rd + Rt .
Rn =

Hybrid Models to Consider Mass
Transport Effects

are required. In such situations, more accurate electrolyte
transport models are used [14]. Equivalent circuit commencing from the Randles model is extended to show the
(5)
effect of 1) generalized capacitance elements, 2) a generalized dc voltage source, and 3) a parallel current path for
(6)
gassing, as shown in Figure 7(d). The use of EIS for
dynamic modeling of a lead-acid battery to estimate this
(7)
equilibrium voltage that depends on the SOC is discussed
in [15]. An analysis of the uncertainties in an EIS measure(8)
ment for a lead-acid battery is given in [16], including the
case of Randles parameter variations
over service life. In general, while
resistances in a Randles' equivalent
In addition to the
circuit keep increasing, the doubleautomotive area,
layer capacitance keeps decreasing
lead-acid chemistry
over the service life [16].

Fractional SOC

Voc = Vo + Vh

The commonly used impedancebased models assume that the battery is in a quasi-stationary state during measurements. However, battery
packs such as those in EVs are subis still frequently
jected to very high continuous, quasiniMH Battery Models
used in EVs.
continuous, or medium-term disAn NiMH cell is a dual-intercalation
charge processes. In these situations,
electrochemical system, where comto accurately estimate the SOC/SOH,
plex proton insertion and the hydroimproved impedance-based nonlinear simulation models
gen deinsertion electrodes occur during discharge and vice
versa during charge. In addition, discharge or charge of
an NiMH cell is controlled by a number of factors,
including the finite rate of solid-state diffusion inside the
+ Vdl+ Vdiffactive material particles of either one or both electrodes,
Cdiff
Cdl1
Cdl2
charge transfer kinetics at the electrode/electrolyte interface, and ohmic drop through the electrolyte phase. Any
RX
of these factors may result in too low a cell potential
before the active materials loaded in the cell are completely used up. As a result, the inefficient utilization of
Vo
active materials becomes a critical issue, particularly for
Ret1
Ret2
Rdiff
electric and hybrid-EV batteries [17].
Double-Layer
Effect
Diffusion
Effect
Vh
Figure 8 schematically illustrates these processes in
+
V
an NiMH equivalent circuit form [17]. Modeling of the
NiMH devices uses different approaches, such as an
(a)
impedance model [18], a combination of numerical and
1
experimental approaches [15], a nonisothermal model
0.9
[19], isothermal models [20], fuzzy logic modeling [21],
0.8
and artificial neural networks [22]. A summary of these
0.7
is provided in [24].
0.6
In [20], a lumped model for NiMH cell is developed
with
attention to many important characteristics,
0.5
including
the hysteresis potential behavior. While bat0.4
tery models and equivalent circuits similar to the case
0.3
of lead-acid chemistry are developed with various com0.2
plexities in these publications, one important charac0.1
teristic of NiMH cells for EVs is the hysteresis behavior [23]. As discussed in [23], using a simplified model,
0
11
12
13
14
15
16
10
such as that shown in Figure 8(b), NiMH hysteresis can
Voc (V)
be developed to include both the double-layer effect
(b)
and the diffusion effect.
fig 8 An NiMH battery equivalent circuit for hysteresis estimation: (a) the circuit used and (b) the measured hysteresis of a
12-V NiMH battery module (COBASYS Series 1,000) at C/20 rate.
(Figure adapted from [23].)

32

IEEE PowEr ElEctronIcs MagazInE

z	December 2014

Conclusions
Electronic engineers who wish to develop useful
equivalent circuits for design work should be able to



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