Supercapacitors (also termed electrochemical capacitors or ultracapacitors) store energy in the electric field of an electrochemical double layer. While their specific power is much higher than in batteries, their specific energy is substantially lower. As principal energy storage systems, these devices are be- ing developed for power assist during acceleration and hill climbing, as well as for the recovery of braking energy [213, 144, 106]. Another possible ap- plication is in “mild” hybrids together with an integrated starter/alternator, as a low-voltage (42V) energy buffer that is also capable of high power recu- peration [130, 235]. Supercapacitors are also potentially useful as secondary energy storage systems in HEVs, providing load-leveling power to electro- chemical batteries which may be downsized [166]. Another advantage in this case would be the additional degree of freedom they add to the vehicle energy management, which allows for an optimization of the operating conditions of the main energy storage system.
A supercapacitor differs from conventional capacitors both in the mate- rials of which it is made and in the physical processes involved. In a super- capacitor, the dielectric is an ion-conducting electrolyte interposed between
4.5 Supercapacitors 111 conducting electrodes. The energy is stored by the charge separation taking place in the layers that separate the electrolyte and the electrodes. Since the voltage that can be applied is limited to a few volts by the physical char- acteristics of the electrolyte, the storage capacity is increased by raising the capacitance, i.e., increasing the surface and decreasing the thickness of the electrolyte. The surface is increased by using electrodes made of a porous material. Electrode materials with the required very high specific area are active carbon (103m2/g) and some metallic oxides (ruthenium, iridium). The porous carbon electrodes are connected to metallic plates that collect charge.
The electrodes are separated by an insulating, ion-conducting membrane, re- ferred to as the separator. The separator also has the function of storing and immobilizing the liquid electrolyte. The electrolyte may be an aqueous acid solution or an organic liquid filling the porous electrodes.
Compared to electrochemical batteries, supercapacitors show a very high specific power of 500–2500 W/kg, but a very low specific energy of 0.2–
5 Wh/kg, according to the material of the electrodes (carbon, metallic oxides) and of the electrolyte (aqueous, polymer) [76]. In automotive applications, most of the attention has been focused on carbon-based cells with polymer electrolyte, which seem to offer the best performance at the lowest cost [50].
The future use of this technology seems indeed to be dependent on cost issues in comparison with high-power battery systems [97].
electric conductor electrode electrolyte
porous particle separator
(ion conductor)
energy storage through charge separation
+ + +
+ -+ -
- - + + + +
Fig. 4.34. Schematic of a supercapacitor.
4.5.1 Quasistatic Modeling of Supercapacitors
The causality representation of a supercapacitor in quasistatic simulations is sketched in Fig. 4.35. The input variable is the terminal power P2(t). The output variable is the chargeQ(t).
Similarly to batteries, the state of charge is evaluated from the terminal currentI2 and the nominal capacity Q0. The former may be calculated from the terminal powerP2using the trivial equality
I2(t) =P2(t)
U2(t) (4.95)
112 4 Electric and Hybrid-Electric Propulsion Systems
PA P
U U
I SC Q
2
2
2 2
Fig. 4.35. Supercapacitors: causality representation for quasistatic modeling.
and a relationship between current and voltageU2. Equivalent Circuit
A basic physical model of a supercapacitor can be derived from a description of the system in terms of equivalent circuit. The simplest equivalent circuit consists of a capacitor and a resistor in series [50, 48, 129]. More complex equivalent circuits describe the distributed nature of the resistance and of the charge stored in a porous electrode [167, 187]. The basic equivalent circuit is depicted in Fig. 4.36. Kirchhoff’s voltage law yields
Rsc·I2(t)−Qsc(t) Csc
+U2(t) = 0, I2(t) =−d
dtQsc(t). (4.96)
U2
I2 Rsc
Csc
Qsc
Fig. 4.36. Equivalent circuit of a supercapacitor.
Substitution of (4.95) into (4.96) yields a quadratic equation for the su- percapacitor voltage,
U22(t)−Qsc(t) Csc
·U2(t) +P2(t)·Rsc= 0. (4.97) To let Qsc vanish, both terms of (4.97) are differentiated and the second of (4.96) is used. This leads to the following differential equation forU2
1−Rsc·P2(t) U22(t)
· d
dtU22(t) =−2·P2(t) Csc
. (4.98)
Based on (4.95) the currentI2 and consequently the charge Qsc may be cal- culated.
4.5 Supercapacitors 113 An alternative approach is quite similar to the one discussed for electro- chemical batteries. After having substituted the open-circuit voltageUocwith the ratioQsc/Csc, (4.60) can be used. The resulting equation for the voltage is
U2(t) = Qsc(t) 2·Csc +
s Q2sc(t)
4·Csc2 −P2(t)·Rsc. (4.99) Using numerical integration methods, this equation may be evaluated at any time using the value ofQsc at the previous time step. The maximum power available may be found with the approach used to derive the same quantity for batteries, (4.62)–(4.66).
Supercapacitor Efficiency
The definition of the efficiency of supercapacitors is similar to that of batteries, both elements being energy storage systems rather than energy converters. On the basis of a full charge/discharge cycle, the global (or “round-trip”) efficiency is defined as the ratio of total energy delivered to the energy that is necessary to charge the device [68]. Such a definition is dependent on the features of the charge/discharge cycle, i.e., whether the battery is charged/discharged at constant current (Peukert test) or at constant power (Ragone test) [48, 49].
If the supercapacitor is represented by the equivalent circuit model of (4.96) with constant capacitance and internal resistance, the global efficiency can be evaluated in both “Peukert” and “Ragone” cases.
At constant-current discharge, the supercapacitor is depleted in a time tf = Q0/I2. The charge varies linearly with time, Qsc(t) =Q0−I2·t. The terminal voltage thus varies according to (4.96). The discharge energy is there- fore
Ed= Z tf
0
U2(t)·I2dt=I2·
Q20
2·Csc·I2−Rsc·Q0
. (4.100)
Charging the supercapacitor with a current of the same magnitude, i.e., I2=−|I2|, the charge varies asQsc=|I2|t. The charge energy is evaluated as
|Ec|= Z tf
0
U2(t)· |I2|dt=|I2| ·
Q20
2·Csc· |I2|+Rsc·Q0
. (4.101) By definition the ratio of Ed to Ec is the global efficiency, which is a function ofI2:
ηsc= Ed
Ec =Q0−2·Rsc·Csc· |I2|
Q0+ 2·Rsc·Csc· |I2| . (4.102) At constant-power discharge (Ragone test), the current varies with time, thus (4.100)–(4.101) cannot be used. Instead, (4.98) may be solved for constant power, yielding an implicit dependency of the terminal voltage on time:
114 4 Electric and Hybrid-Electric Propulsion Systems t= Csc
2·P2 · Rsc·P2·ln U2
U0 2
+U02−U22(t)
!
. (4.103)
The initial voltage U0 follows from (4.99) with Qsc = Q0. From (4.99), the discharge ends when Qsc = 2·Csc·√
P2·Rsc which, based on (4.96), corresponds to a terminal voltage Uf = √
P2·Rsc. Note that, in contrast with the Peukert discharge, the supercapacitor voltage is not zero at the time tf. From (4.103), the final time is calculated as
tf = Csc 2·P2
·
Rsc·P2·ln
Rsc·P2 U02
+U02−Rsc·P2
. (4.104)
The discharge energy is given by Ed=tf·P2= Csc
2 ·
−Rsc·P2·ln U02
Rsc·P2
+U02−Rsc·P2
. (4.105) For a charge with a constant power of the same intensity,P2=−|P2|, the initial voltage is p
Rsc· |P2|, while the final voltage equals the value of U0
calculated previously. The charge energy is evaluated as
|Ec|=tf· |P2|= Csc 2 ·
Rsc· |P2| ·ln
U02 Rsc· |P2|
+U02−Rsc· |P2|
. (4.106) The efficiency is therefore calculated from
ηsc= Ed
Ec =
U02−Rsc· |P2| −Rsc· |P2| ·ln
U02 Rsc· |P2|
U02−Rsc· |P2|+Rsc· |P2|ln
U02 Rsc· |P2|
. (4.107)
A third possibility is to charge and discharge the supercapacitor with max- imum power. Equation (4.99) states that discharge power is limited by the state of charge,P2(t)< Q2sc(t)/4/Csc2/Rsc. This limit is thus varying in time, in contrast to what happens during charge, when the power is limited only by the constant maximum current. The efficiency cannot be evaluated analyti- cally in this case, thus a numerical integration is needed. Examples of super- capacitor discharges at constant current, at constant power, and at maximum power are illustrated in Fig. 4.37.
The local definition of supercapacitor efficiency is based on a power ratio rather than an energy ratio. If the discharge and charge powers are expressed in terms of charge and current, then the local efficiency is evaluated as
ηsc(I2) = P2,d
|P2,c| =Qsc−Rsc·Csc· |I2|
Qsc+Rsc·Csc· |I2| . (4.108) Note that, if in (4.108) an average charge Qsc = Q0/2 is used, the re- sult is the efficiency of (4.102). The advantage of the local definition is that it
4.5 Supercapacitors 115
0 20 40 60 80 100
0 200 400 600 800
t [s]
Qsc [C]
0 20 40 60 80 100
0 100 200 300 400
t [s]
I2 [A]
0 20 40 60 80 100
0 10 20 30 40 50 60 70
t [s]
U2 [V]
0 20 40 60 80 100
0 2000 4000 6000 8000 10000 12000 14000
t [s]
P2 [W]
Fig. 4.37. Calculated discharge tests for a supercapacitor (Csc = 12.5 F, Rsc = 0.08 Ω,Q0= 800 C). Solid lines: discharge with constant currentI2= 60 A. Dashed lines: discharge with constant powerP2= 1500 W. Dashdot lines: discharge at max- imum power.
does not require any assumption on the type of charge/discharge, while global expressions like (4.102) or (4.107) are strictly valid for Peukert and Ragone cycles, respectively. Similarly to batteries and energy converters, the local su- percapacitor efficiency can be easily represented in terms of “efficiency maps”
as well.
4.5.2 Dynamic Modeling of Supercapacitors
The physical causality representation of a supercapacitor is the same as that of a battery and is sketched in Fig. 4.38. The model input variable is the terminal currentI2(t). A positiveI2(t) discharges the supercapacitor, a negativeI2(t) charges it. The model output variables are the terminal voltageU2(t) and the chargeQ(t).
The basic equivalent circuit model, (4.96) can be used to calculate the terminal voltage as a function of the terminal current. Integrating the current yields the supercapacitor charge and thus the non-dimensional state of charge.
116 4 Electric and Hybrid-Electric Propulsion Systems
U SC I 2 Q
2
Fig. 4.38.Supercapacitors: physical causality for dynamic modeling.