134 5 Non-electric Hybrid Propulsion Systems
the power flow of Fig. 5.3 describes also configurations in which the flywheel is mounted on the engine transmission shaft, as shown in Fig. 5.2.
FW CVT T V
E T ω
F ω v
T (a) quasistatic approach
(b) dynamic approach
FW CVT T V
E T ω
F ω v
T P
ω
T
Fig. 5.3. Flow of power factors for a hybrid-inertial propulsion system, with the quasistatic approach (a) and the dynamic approach (b). F: force, P: mechanical power,T: torque,v: speed,ω: rotational speed. Nomenclature of blocks as in Chap. 4, plus CVT: continuously-variable transmission, FW: flywheel.
5.2 Flywheels 135 (a) quasistatic approach
(b) dynamic approach E
L HM T V
ω ω
T T
Q
Q p
p
F v p
Q HP
E
HA L HM T V
ω ω
T T
F P v
P P
HP
HA
Fig. 5.4.Flow of power factors for a hybrid-hydraulic propulsion system, with the quasistatic approach (a) and the dynamic approach (b). F: force, p: pressure,P: mechanical power,Q: flow rate,T: torque,v: speed,ω: rotational speed. Nomencla- ture of blocks as in Chap. 4, plus HA: hydraulic accumulator, HM: hydraulic motor, HP: hydraulic pump, L: hydraulic line.
Mixed inertial/electrical hybrid concepts employ a high-speed flywheel2 to load-level electrochemical batteries [274, 166] or as the only energy storage system [254]. An electric motor/generator is mounted on the rotor shaft both to spin the rotor (charging) and to convert the rotor kinetic energy to elec- trical energy (discharging), while the traction power is provided by a second electric machine. Regenerative braking is possible through the conversion of mechanical energy to kinetic energy through electric energy. Since typically, the built-in motor/generator is an AC machine, it needs a proper converter to interface with the traction motor when that is a DC machine, as it is usual in battery–flywheel applications.
With respect to electrochemical batteries, the potential of flywheels is comparable (but usually lower) in terms of specific energy and higher (up to 10 times or more) in terms of specific power. The advantages of flywheels are that they contain no acids or other potentially hazardous materials, that they are not affected by extreme temperatures, and that they usually exhibit a longer life. So far, flywheel batteries have only been used in some bus applications [171]. For flywheels to be successful in passenger cars, they would need to
2 Often referred to as “electromechanical batteries” in these applications.
136 5 Non-electric Hybrid Propulsion Systems
provide a specific energy higher than the levels currently available. In addition, there are some concerns regarding the complexity and the safety of a device that spins mass at very high speeds. The latter aspect strongly limits the flywheel mass that can be installed on board, thus limiting maximum power and energy capacity.
Flywheels store kinetic energy within a rapidly spinning wheel-like rotor or disk. In order to achieve a sufficient amount of specific energy, modern fly- wheel rotors must be constructed from materials of high specific strength, lead- ing to the selection of composite materials employing graphite fibers rather than metals. This also increases rotor speed, which ranges from typical en- gine speeds up to very high speeds of about 3000 rev/s. The reduction of the aerodynamic losses associated with such high rotor speeds requires spinning the rotor in a vacuum chamber. This in turn leads to additional design re- quirements for the rotor bearing. These bearings must have low losses and must be stiff to adequately constrain the rotor and stabilize the shaft. These requirements frequently lead to choosing magnetic bearings with losses on the order of 2% per hour. In order to reduce the gyroscopic forces transmitted to the magnetic bearings during pitching and rolling motions of the vehicle, a gimbal mount is often adopted [108]. The alternative of using counter-rotating rotors, which do not transmit any gyroscopic forces to the outside, is burdened by the fact that internally they transmit very large forces that are not easily supported by the magnetic bearings.
The specific energy of a flywheel is calculated as the ratio of the energy storedEfto the flywheel massmf. The kinetic energy stored isEf =12·Θf·ω2, with Θf being the moment of inertia of the flywheel andωits rotational speed.
Using the notation of Fig. 5.5 the moment of inertia is calculated as [74]
Θf =ρ·b· Z
r2·2·π·r dr= 2·π·ρ·b· r4 4
d/2
q·d/2
= π
2·ρ·b·d4
16·(1−q4), (5.1) where ρis the material’s density and q the ratio between the inner and the outer flywheel ring. The flywheel mass is given by
mf =π·ρ·b·d2
4 · 1−q2
. (5.2)
Consequently, the energy-to-mass ratio is evaluated as Ef
mf = d2
16 ·(1 +q2)·ω2=u2
4 ·(1 +q2), (5.3) where u= d·ω/2 is the flywheel speed at the outer radius. Equation (5.3) may be written in a more compact way, as
Ef
mf =kf·u2, (5.4)
5.2 Flywheels 137 where the coefficientkf typically ranges from 0.5 to 5, according to the type of construction.
The specific energy is limited by several factors, such as the maximum allowable stress and the maximum allowable rotor speed.3 Small units that are technically feasible today reach 30 Wh/kg, including housing, electronics, etc. Much higher values of up to 140 Wh/kg are predicted by some authors for advanced rotor materials [257].
u = d 2 ω
dw d
q d
.
0.05 dβ
.
d.
ω
.
Fig. 5.5.Flywheel accumulator for duty-cycle operation and regenerative braking.
5.2.1 Quasistatic Modeling of Flywheel Accumulators
The causality representation of a flywheel accumulator in quasistatic simula- tions is sketched in Fig. 5.6. The input variable is the powerP2(t) required at the output shaft. A positive value ofP2(t) discharges the flywheel, a nega- tive value ofP2(t) charges it. The output variable is the flywheel speedω2(t), sometimes regarded as the “state of charge” of the flywheel.
P 2 FW ω2
Fig. 5.6.Flywheel accumulators: causality representation for quasistatic modeling.
Models of flywheel accumulators may be derived on the basis of Newton’s second law for a rotational system. The resulting equation is
Θf·ω2(t)· d
dtω2(t) =−P2(t)−Pl(t), (5.5)
3 This is usually limited by the first critical speed of the flywheel [257].
138 5 Non-electric Hybrid Propulsion Systems
from which the flywheel speed can be calculated. The termPl describes the power losses.
For flywheel accumulators, two main loss contributions are usually consid- ered, namely air resistance and bearing losses, Pl(t) =Pl,a(t) +Pl,b(t). Both terms are functions ofω2(t), i.e., of the peripheral velocity u(t). The general expression for the air resistance force is proportional to the air density ρa, to u2(t), and tod2, through a coefficient that is a function of the Reynolds number and of the geometric ratio β = b/d. For Reynolds numbers above 3·105, an expression for the power losses due to air resistance [74] is
Pl,a(t) = 0.04·ρ0.8a ·η0.2a ·u2.8(t)·d1.8·(β+ 0.33), (5.6) whereηa is the dynamic viscosity of air.
For the bearing losses, a general expression frequently used [74] is Pl,b(t) =µ·k·dw
d ·mf ·g·u(t), (5.7) where the physical quantities involved are a friction coefficientµ, a corrective force factorkthat models unbalance and gyroscopic forces, etc., and the ratio of the shaft diameterdwto the wheel diameterd.
A first estimation of the power losses may be obtained using the values for the physical parameters listed in Table 5.1. Figure 5.7 shows the variation of the bearing losses and the air resistance losses as a function of the flywheel rotational speed for an optimized flywheel construction [74]. The figure clearly shows the dependency of the power losses on the speed as given by (5.6). With the same data it is possible to obtain a value for the time range of the flywheel, i.e., the time that the flywheel speed remains above a certain threshold without any external torque. Typical values are about 10 min.
The main design task in designing flywheels consists of assigning values to the flywheel dimensionsb,d in order to obtain the desired kinetic energy at a given rotational speed (thus, a given moment of inertia), while minimizing weight and power losses. A typical value for the stored energy may be esti- mated considering the kinetic energy of a vehicle with a mass of 910 kg and a speed of 80 km/h. The energy to be recuperated in the flywheel during a braking until stop is 247 kJ, having assumed a first tentative value of the fly- wheel mass equal to 10% of the vehicle mass. The moment of inertia is related to dimensions and mass by (5.1)–(5.2). Figure 5.8 shows that the dimension b of the flywheel increases asm2f, while the diameter ddecreases as 1/√
mf. Consequently, the power losses and the peripheral speed u decrease as well (the rotational speed is 100 rev/s). A possible compromise between flywheel weight and power losses can be obtained with a flywheel mass of about 50 kg.
5.2.2 Dynamic Modeling of Flywheel Accumulators
The physical causality representation of a flywheel accumulator is sketched in Fig. 5.9. The model input variable is the rotational speed at the output
5.2 Flywheels 139
0 1000 2000 3000 4000 5000 6000
0 50 100 150 200 250 300 350
Pl [W]
n [rev/s]
Pl,tot Pl,b Pla
Fig. 5.7. Power losses as a function of the rotational speed. Flywheel data:d = 0.36 m,b= 0.108 m,β= 0.3,q= 0.6,mf = 56.33 kg.
30 35 40 45 50 55 60 65 70
0 0.1 0.2 0.3 0.4 0.5
d, b [m]
mf [kg]
d b
30 35 40 45 50 55 60 65 70
0 200 400 600 800 1000
Pl [W]
mf [kg]
Pl,tot Pl,b
Fig. 5.8.Dimensions and losses of a flywheel as a function of its mass, for a given energy Ef = 247 kJ, speed n = 100 rev/s, and thus a moment of inertia Jf = 1.24 kg m2.
140 5 Non-electric Hybrid Propulsion Systems
Table 5.1.Numerical values for the flywheel parameters.
µ 1.5·10−3
k 4
dw/d 0.08 ρa 1.3 kg/m3 ηa 1.72·10−5Pa s ρ 8000 kg/m3
or downstream shaft, ω2(t). The model output variable is the torque at the output shaft,T2(t).
FW T 2
ω2
Fig. 5.9. Flywheel accumulators: physical causality for dynamic modeling.
The equations developed in the previous section are suitable also for dy- namic modeling. The only dynamic term is related to the derivative of the flywheel speed. It can be easily estimated sinceω2(t) is an input variable.