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328 Renewable Energy Devices and Systems with Simulations in MATLAB and ANSYS ®
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FIGURE 13.7 Examples of Li-ion ultracapacitors of 2200 F, 2300 F, and 3300 F.

1.17 m

1.03 m
0.93 m

FIGURE 13.8 Ultracapacitor bank in the UWM lab. The approximate overall dimensions are 1.03 × 0.93 ×
1.17 m. Rated at 720 V and 0.5 Ah.

13.3.2 Ultracapacitor Modeling
Models for ultracapacitors may be categorized into three main groups: physical/electrochemical,
equivalent circuit, and behavioral neural network types. Physical models rely on electrochemical
reactions and corresponding high-order differential equations, and therefore, they are the most phe-
nomenological, relevant and, in principle, also the most accurate [31, 32].
The equivalent circuit models for ultracapacitors typically employ several passive circuit ele-
ments, such as resistors and capacitors [33–36]. A simplified equivalent circuit model for Li-ion
ultracapacitors, based on the concept proposed in [29], is shown in Figure 13.9, where the self-
discharging characteristic of an ultracapacitor is modeled by R , the cell and the junction resistance
sd
are totally quantized by series resistance R , and the RC elements, R and C , model the transient
s
ss
ss