Usually in modeling torque couplers, no dynamic effects are considered.
The rotational inertia is conveniently attributed to the power and the load shafts, respectively. Moreover, shaft elasticity is usually neglected. Conse- quently, the same quasistatic equations, (4.115)–(4.116) can be used. In partic- ular, they constitutem+ 1 equations in them+ 1 model variables. Therefore, the variablesuj introduced in the previous section are outputs of the model rather than inputs.
4.8 Power Split Devices
Power split devices (PSDs) are widely used in automatic transmissions. A power split device is also found in many hybrid vehicles to combine mechanical power from various (usually, two) power sources to various (usually, two) mechanical loads. In the typical configuration of a combined hybrid vehicle, a PSD connects an engine, a motor, a generator, and the drive train.
The core of most PSDs is a planetary gear set, often referred to also as epicyclic gearing. A basic planetary gear has three main rotating parts (Fig. 4.44). The inner part is thesun, the outer part is thering. The interme- diate part carries rotating elements (planets) and is thecarrier. More complex configurations are possible but they are not considered here. Each of the three parts can be connected to the input shaft, the output shaft, or can be held stationary. Choosing which piece plays which role determines the gear ratio for the gear set.
Common automatic transmissions use a compound planetary gear set in combination with an hydraulic torque converter. In contrast, several PSDs for HEVs use one or two planetary gear sets in combination with two electric machines. The latter case is analyzed in detail in the next section, however, the modeling of these devices can be applied in both cases.
4.8.1 Quasistatic Modeling of Power Split Devices
The causality representation of a power split device in quasistatic simulations is sketched in Fig. 4.45. The input variables are the torque and speed at the load shafts,ωf(t),ωg(t),Tf(t),Tg(t), where the subscriptsf andgstand for
“final driveline” and “generator”, respectively. The output variables are the torque and speed at the power source shafts,ωe(t),ωm(t),Te(t),Tm(t), where the subscriptseandmstand for “engine” and “motor”, respectively. Often in combined hybrids, the roles of the electric machines as generator and motor are interchangeable.
Simple Power Split Devices
The basic equation to consider in analyzing the quasistatic behavior of a planetary gear set is the relationship between the speed of the three main
122 4 Electric and Hybrid-Electric Propulsion Systems
engine generator
motor
to the final drive and the wheels ring gear
planetary carrier
sun gear
Fig. 4.44. Schematics of a planetary gear set arrangement (as in Toyota Prius [250]).
ωm Tm
ω e Te ω f
ω g Tf
Tg
PGS
Fig. 4.45.Power split devices: causality representation for quasistatic modeling.
parts, which can be derived according to the Willis formula. The planetary gear set can be considered as an ordinary gear set in a rotating frame that is attached to the carrier. Thus, the ratio of the relative speeds of the ring and of the sun can be written as
ωr(t)−ωc(t)
ωs(t)−ωc(t) =−z, z= ns nr
, (4.120)
where n is the number of teeth and the subscripts c, s, and r refer to the carrier, the sun, and the ring, respectively. A typical value of the epicyclic gear ratioz is 0.385 [219].
The connection of the machines to the various ports may differ from one system to another. In the Toyota Hybrid System (THS-II) [250], the sun gear is connected to the generator, the planetary carrier to the engine, and the ring gear to the motor shaft. On the output shaft, the power transmitted by the
4.8 Power Split Devices 123 ring gear and the power from the motor are combined. In other hybrid-electric vehicles, the connections can be different [161, 265].9
Assumingωe(t) =ωc(t), ωg(t) =ωs(t),ωf(t) =ωr(t), (4.120) is specified as [219]
ωm(t) =ωf(t), (4.121)
ωe(t) =z·ωg(t) +ωf(t)
1 +z . (4.122)
The balance of power applied to the four ports,
Tf(t)·ωf(t) +Tg(t)·ωg(t) =Te(t)·ωe(t) +Tm(t)·ωm(t), (4.123) combined with the torque balance Tf(t) +Tg(t) = Te(t) +Tm(t) and with (4.121)–(4.122) yields
Te(t) =1 +z
z ·Tg(t), (4.124)
Tm(t) =Tf(t)−Tg(t)
z , (4.125)
which are the remaining two equations.
In contrast to torque couplers, planetary gear sets do not have any available degrees of freedom to control. In other words, planetary gear sets should be regarded as passive elements which are inherently self-controlled. Notice also that if the generator shaft is not connected, i.e., Tg(t) = 0, it follows from (4.124) that alsoTe(t) = 0.
Equation (4.122) may be regarded as expressing the relationship between engine speed and output axle speed, the generator speed being a parameter.
In other words, the planetary gear set represents a sort of continuously vari- able transmission, where the transmission ratio is regulated by the generator controller. Four typical modes of operation are illustrated in Fig. 4.46, which clearly shows the linear dependency betweenωe andωf [250].
A simple PSD may also have an output stage gearing to combine the torque from the motor to the torque from the ring gear of the planetary gear set. An example of such case is found in the Ford Hybrid System (FHS) [83], for which (4.121)–(4.125) are modified to
ωm(t) =Rm·ωf(t), (4.126) ωe(t) = z·ωg(t) +Rr·ωf(t)
1 +z , (4.127)
Te(t) =1 +z
z ·Tg(t), (4.128)
Tm(t) = 1 Rm
·Tf(t)− Rr Rm
·Tg(t)
z . (4.129)
9 Usually, in combined hybrids, the two electric machines are simply referred to as motor–generator 1 and 2, without distinguishing their prevalent functions.
124 4 Electric and Hybrid-Electric Propulsion Systems
ωe
ωf
ωg
sun carrier ring
0 ω
(a)
(b)
(c)
(d) ωe
ωf
ω ω ωe
ωg
ωg
ωf
ω
0 0 0
ns
nr
Fig. 4.46.Relationship between rotational speeds in four typical operation modes:
(a) purely electric (at start-up, ωe = 0), (b) engine/generator operation (engine start,ωm=ωf = 0), (c) electric assist (normal drive,ωg= 0), (d) acceleration.
where Rr and Rm are the transmission ratios of the motor and ring speeds, respectively, to the final shaft. These ratios are easily calculated as a function of the number of teeth of the three torque-coupling gears.
Compound Power Split Devices
Compound power split devices are increasingly used, especially in large hybrid sport utility vehicle applications, since they have the advantage over simple PSDs of a reduced power of the electric machines.
4.8 Power Split Devices 125 An example of a compound PSD is found in GM’s Two-mode Hybrid System (previously known as AHS-II) shown in Fig. 4.47a [177]. The gears of two planetary gearings are mutually connected and linked to the engine, to the two electric machines, and to the final driveline. Choosing the locking status of a pair of controlled clutches E1 and E2 permits the realization of two different operating modes.
In the first operating mode, used at low loads and relatively low vehicle speeds, E1 is disengaged while E2 is engaged, thus the mechanical connections can be schematized as in Fig. 4.47b. Applying (4.120) and (4.123) to both planetary gear sets, one obtains for the speeds
ωe(t) =−z1·ωg(t) +ωf(t)·(1 +z1) (4.130) and
ωm(t) =1 +z2
z2
·ωf(t), (4.131)
wherez1andz2are the epicyclic ratios of the first and of the second planetary gear set, respectively. The corresponding relationships between the torques are
Te(t) =−1
z1 ·Tg(t) (4.132)
and
Tm(t) = z2 1 +z2
·
Tf(t) +1 +z1 z1
·Tg(t)
. (4.133)
The second operating mode is used for heavy loads and high vehicle speeds.
The clutch E1 is engaged, while E2 is disengaged, see Fig. 4.47c. The rela- tionships between the speeds are
ωe(t) =−z1·ωg(t) +ωf(t)·(1 +z1) (4.134) and
ωm(t) = 1 +z2
z2 ·ωf(t)− 1
z2 ·ωg(t). (4.135) The corresponding relationships between the torques are
Te(t)· 1
z2
−z1
= 1 +z2
z2
·Tg(t) +Tf(t) z2
(4.136) and
Tm(t)·
z1− 1 z2
=z1·Tf(t) + (1 +z1)·Tg(t). (4.137) Understanding the role of mode-switching of AHS-II requires a dynamic analysis that is introduced in the next section. A similar operating principle is used in other compound PSDs, such as the Timken eVT [1] and the Renault e-IVT [262], whose quasistatic modeling is obtained by using the methods illustrated above.
126 4 Electric and Hybrid-Electric Propulsion Systems
engine
to final drive S1
C1 R1
S2 C2 R2 E1
E2
G M
engine
to final drive S1
C1 R1
S2 C2 R2 E1
E2
G M
(a)
(c) engine
to final drive S1
C1 R1
S2 C2 R2 E1
E2
G M
(b)
Fig. 4.47. Schematic of a Two-Mode Hybrid System (a) and of its two modes of operations (b) and (c). The figure show the connections of the engine, the “speeder”
machine or generator (G), the “torquer” machine or motor (M), via two planetary gear sets, for which S: sun gear, C: planetary carrier, R: ring gear. Two control clutches are labelled E1 and E2.
4.8.2 Dynamic Modeling of Power Split Devices
The physical causality representation of a power split device is sketched in Fig. 4.48. The model input variables are the torque at the power source shafts, Te(t) andTm(t), and the rotational speeds at the load shafts,ωf(t) andωg(t).
The model output variables are the rotational speed at the power source shafts,ωe(t) andωm(t), and the torques at the load shafts,Tf(t) andTg(t).
ωm Tm
ω e Te
ω f ω g
Tf
Tg
PGS
Fig. 4.48.Planetary gear sets: physical causality for dynamic modeling.
4.8 Power Split Devices 127 Dynamic models of PSDs include the inertia effects of the gears. Equa- tions (4.121)–(4.122) are still valid and already in the required causality rep- resentation. Conversely, (4.124)–(4.125) are substituted by the more general equations
Tf(t) =Tm(t) + 1
1 +z·Te(t)−ω˙g(t)· z·Θc
(1 +z)2 −ω˙f(t)· Θc
(1 +z)2 + Θr
(4.138) and
Tg(t) = z
1 +z·Te(t)−ω˙g(t)·
z2·Θc (1 +z)2 + Θs
−ω˙f(t)· z·Θc
(1 +z)2 , (4.139) where Θc, Θs, and Θrare the moment of inertia of the carrier, the sun, and the gear ring, respectively. Despite the generality of (4.138)–(4.139), in planetary gear sets the dynamic effects are normally not considered, as in the case of torque couplers. The rotational inertia of the gears is indeed negligible when compared with those of the machines and shafts connected.
However, the physical causality representation is still useful to describe the role and the operation of a PSD within a generic “variator” architecture, i.e., a system to split an input power between a main mechanical path and a parallel path that may not be mechanical. Indeed several variator technologies exist, including hydraulic, toroidal [87], and belt/chain CVT [264]. A hybrid-electric powertrain as a whole can be seen as an electric variator [169, 262, 54].
Since PSDs are full four-port converters, their system equations can be expressed in the general form
ωe(t) ωm(t)
!
=
"
A B C D
#
· ωf(t) ωg(t)
!
=M· ωf(t) ωg(t)
!
(4.140) and
Tf(t) Tg(t)
!
=MT · Te(t) Tm(t)
!
=
"
A C B D
#
· Te(t) Tm(t)
!
, (4.141)
where the parameters A, B, C, D depend on the system configuration. For example, the simple planetary arrangement of THS-II has A = 1/(1 +z), B=z/(1 +z),C= 1, andD= 0. Compound PSDs usually are designed such thatA >0,B <0,C >0. The fourth termDcan be zero as in the first mode of AHS-II, negative as in its second mode, or positive.
The PSD operation is described by a few parameters, among which the speed transmission ratio K(t) = ωf(t)/ωe(t) and the variator speed ratio Kv(t) =ωm(t)/ωg(t). The relationship between these two parameters is cal- culated from (4.140)–(4.141) as
Kv(t) =D·
1−K(t) K1
1−K(t) K2
(4.142)
128 4 Electric and Hybrid-Electric Propulsion Systems whereK1= (A−B·C/D)−1 andK2= 1/A.
The relationship between K(t) and Kv(t) is shown in Fig. 4.49a. If the system allows Kv varying in a wide range, from negative to positive values, enabling a generator or motor speed equal to zero, then at a given input speed (engine speed), the output speed (vehicle speed) can vary continuously from reverse to forward speed passing through the zero speed. In other words,K may vary “infinitely”10 without using a torque converter or any equivalent device.
The variator ratioKv not only controls the speed transmission ratio, but also the split of power between the mechanical path and the electrical path. In variator theory, the power split ratio r(t) =Pg(t)/Pe(t) is conveniently used to describe this effect. In the ideal case of unitary efficiency of the electrical path and no power supply from the battery, Pm(t) = Pg(t).11 After some manipulation of (4.140)–(4.141), the power split ratio thus can be expressed as a function ofK(t) as
r(t) =
1−K(t) K2
·
1−K(t) K1
K(t)· 1
K2 − 1 K1
. (4.143)
The dependency betweenr(t) andK(t) is shown in Fig. 4.49b forK1< K2. The figure clearly shows that K1 and K2 are particular values of the speed transmission ratio called nodes, such that r(t) = 0, a condition that corre- sponds to a purely mechanical transmission. Between the nodes, the electric power exhibits a finite maximum. Outside of this range, however, the elec- tric power would have nonphysically high levels [262]. Therefore, variators are designed to operate properly for values ofK between nodes.
The power split ratio of simple PSDs is calculated as a special case of (4.143) withD= 0. This condition applies also to represent the first operating mode of a compound PSD such as the AHS-II, see (4.130)–(4.133). In both cases,
r(t)(simple)= 1−K(t) K2
, (4.144)
that is to say, simple PSDs exhibit only one nodeK2, whileK1→0 asD→0.
ForK= 0 the system works as a series hybrid, withr= 1. In Fig. 4.49b the functionr(t) =r(K(t)) of a simple PSD is compared with those of a compound PSD characterized by the sameK2. In the nominal operating range ofK, i.e., betweenK1andK2, the power split ratio of the simple PSD is always greater than that of the compound PSD. In other terms, the use of a compound
10Whence the name Infinitely Variable Transmission (IVT) often given to these systems.
11This assumption is obviously not realistic and is used here only for illustrating the general operation of PSDs; however, it has to be handled carefully since, for instance, it leads to the conclusion that forK→0,Tm→ ∞.
4.8 Power Split Devices 129 PSD permits the reduction of the electric power for the same speed span.
Consequently, the sizing of the electric machines can be reduced.
0 0.1 0.2 0.3 0.4 0.5
!100 0 100
Kv [!]
(a)
0 0.1 0.2 0.3 0.4 0.5
!2 0 2
r [!]
K [!]
(b)
Fig. 4.49. Variator ratio (top) and power split ratio (bottom) as a function of the speed transmission ratio. Dashed curve is representative of a simple PSD with D= 0. Solid curves are representative of a compound PSD withK1 < K2. Numerical values:K1= 0.05,K2= 0.25.
The analysis above also explains how dual-mode PSDs allow increasing the range of K without increasing the power split ratio. Some systems, like for instance the e-IVT system, have two operating modes with two nodes each.
The two modes share the same node K2, while the other node K1 switches fromK1(1)< K2toK1(2)> K2. Consequently, the curver(t) =r(K(t)) results from the superimposition of the two curves labeled A and B in Fig. 4.50a, one for each mode, with the switching occurring atK2. In this way, large values of Kare obtained without any excessive increase ofr. The operation of a system such as the AHS-II is described as in Fig. 4.50a with K1(1) → 0, thus with curve A given by (4.144).
The nodes of the PSD also play the role of vehicle speed thresholds at which the directions of energy flows between the linked machines are inverted and the system operation mode is changed [1, 262]. This behavior is observed in Fig. 4.50b, which shows ωg and ωm as a function of K for a dual-mode compound PSD. For K < K1(1), the motor and the generator rotate in the same negative direction and they operate in their reverse power direction, with the “generator” working as a motor and the “motor” as a generator. As the vehicle speed increases,K1(1)< K < K2, the machines rotate in opposite
130 4 Electric and Hybrid-Electric Propulsion Systems
0 0.1 0.2 0.3 0.4 0.5
!2 0 2
r [!]
(a)
A
B
0 0.1 0.2 0.3 0.4 0.5
!1 0 1
!m/!e, !g/!e [!]
K [!]
(b)
!m/!e
!g/!e
Fig. 4.50.Power split ratio (top), generator and motor speed (bottom) as a function of the speed transmission ratio. In (a) the solid curve is the resulting superimposition of curve A for the first mode and curve B for the second mode, with a switching occurring atK2. Numerical values:K(1)1 = 0.05,K2= 0.25,K1(2)= 0.5.
directions and in their positive power direction. Thus, the “generator” works as a generator and the “motor” as a motor. ForK2< K < K1(2)both machines rotate in the same positive direction; their power directions are inverted again.
Finally, forK > K1(2), the two machines counter-rotate again, while keeping their positive power direction.
5
Non-electric Hybrid Propulsion Systems
The introductory section of this chapter describes several devices that may all be classified as short-term storage systems. The use of short-term storage systems in powertrain applications and the possible powertrain configurations are discussed first.
The next sections analyze the modeling approaches of hybrid-inertial, hybrid-pneumatic, and hybrid-hydraulic powertrains. Specific components such as flywheel accumulators, continuously-variable transmissions, pneumatic and hydraulic accumulators, pumps/motors are all described in accordance with the quasistatic and the dynamic modeling approach.