Impact of the Number of Planetary Gears on the Energy Efficiency of Electrified Powertrains

Planetary gears (PGs) play a critical role in hybrid electric vehicles (HEVs) by combining the output torques of different powertrain components and delivering the resulting torque to the wheels. Whilst previous studies show that the number of planetary gears affects performance of HEVs, there is no prior study to systematically investigate such effects on energy consumption. This paper quantifies the energy efficiency improvement of HEVs due to increasing the number of PGs from one to two, and from two to three. This is done by comparing the minimum energy consumption for different topologies when the rest of the powertrain components – namely electric motors, batteries and engine – are the same. To calculate the minimum energy consumption, the paper proposes an optimal energy management strategy (EMS) for each topology to find the optimum sequence of clutch engagement and torque distribution. The minimum energy consumption of a vehicle with different number of PGs is then evaluated using the automotive simulation models (ASM) from dSPACE. Results show that, for the same electric motors and engine, increasing the number of PGs from one to two and from two to three reduces energy consumption by 5% and 1.5%, respectively.


Introduction
In 2020, the UK government issued a strict 95 g/km limit on the CO 2 emission for all ground vehicles to address the global concern of pollution control [1]. A further 15% decrease is expected by year 2025. Automotive industries have been investing in hybrid electric powertrains to reduce fuel consumption and achieve these ambitious targets. Performance of a hybrid powertrain depends on the efficiency of its components as well as on its topology, number of planetary gears (PGs), and associated energy management strategies (EMSs).
The topology of hybrid powertrains is one of the critical factors that affect fuel consumption [2][3][4]. Topology refers to the way that the powertrain components -namely engine, motors, and differential -are connected to the wheels. There are three well-known topologies which are widely used in hybrid powertrains: series, parallel, and powersplit. The latter is a parallel topology that consists of one or more PGs as power splitters, thus providing better fuel economy [5]. Toyota Prius, Chevrolet Volt, and Ford Fusion Hybrid are among the first to * Corresponding author at: Centre for Future Transport and Cities (CFTC), Coventry University, Coventry CV1 5FB, UK.
launch power-split topologies with one PG. Adding a clutch to the traditional Toyota Prius powertrain with one PG is shown to reduce the total energy consumption by 2.6% [6]. A power-split powertrain that includes clutches is called a multi-mode topology, which provides better fuel economy than the traditional single-mode topologies due to the higher degrees of freedom (DoF) over a wider range of operating conditions [7].
Increasing the number of PGs also appears to reduce energy consumption of a power-split powertrain. For example, Rajput et al. [8] reported a 4% decrease in energy consumption by increasing the number of PGs from one to two. Benefiting from the reduction in total fuel consumption, Toyota and General Motors launched hybrid powertrains with two PGs for their Prius [9] and Chevrolet Volt [10] models, respectively. Later, Zhuang et al. [3] showed that increasing from two to three PGs further reduces the total energy consumption by 1.  for different topologies. Moreover, the engagement and disengagement of clutches are not penalised even though this is known to increase mechanical losses [11]. General Motors (GM) introduced a 3PG powertrain topology, referred to as 2-Mode Hybrid Transmission, for applications involving full acceleration, hill climbing, and towing [12].
The aforementioned studies investigated the performance of commercial hybrid electric vehicles (HEVs) with one, two, and three PGsreferred to as 1PG, 2PG, and 3PG topologies from here forth -and not the explicit effect of increasing the number of PGs on energy consumption. The analysis of the latter requires evaluating the performance of the topologies when the power components are identical and optimally operating. Therefore, the comparison relies on the development of an optimal EMS for each topology to choose the best mode, configuration, and torque distribution over the driving cycle to minimise energy consumption and CO 2 emission.
Each mode of operation of a power-split powertrain represents a type of configuration, which is defined as a topology with a particular order of clutch engagements. A power-split hybrid topology with multiple clutches can generate numerous configurations, some of which may belong to the same mode of operation. For example, a hybrid powertrain may operate at the fully electric mode -where only electric machines contribute to the torque of the wheels -whilst this mode of operation can be realised with different configurations.
The contributions of this paper to address these gaps are as follows: • Evaluation of the impact of number of PGs on the energy consumption of HEVs. It is shown that increasing the number of PGs from one to two and from two to three reduces the energy consumption by 5% and 1.5%, respectively. • Optimal EMS for the multi-mode hybrid electric powertrains with different numbers of PGs. The developed EMS simultaneously controls the clutches of the gearbox as well as energy flows of components to minimise energy consumption over a driving cycle. • Design rules are introduced for HEVs with multiple PGs: 1. Electric motors with higher speed give flexibility to improve the engine efficiency and therefore reduce energy consumption. This can happen by both the 2PG and 3PG topologies. 2. Engine can operate within the optimum range of its rotational speed using the fixed-gear modes and without involvement of electric motors. This is possible in 3PG topologies and reduces the electric losses and hence the energy consumption.
The main motivation of this paper is to develop a test environment to evaluate the sensitivity of the performance of the electrified powertrain with respect to only one design variable that is the number of PGs. The proposed EMS makes the evaluation valid by finding the optimal energy consumption of different topologies, which nullifies the effect of other design variables.
The remainder of this paper is organised as follows: the system components are presented in Section 2; the developed EMS is described in Section 3; the real-time simulation results are discussed in Section 4; whilst conclusions are drawn in Section 5.

System components
This paper studies the impact of five multi-mode topologies on the energy consumption of a nominal hybrid electric passenger car. This includes four topologies introduced by Toyota, Chevrolet and General Motors, and an additional one proposed in Zhuang et al. [30], on the energy consumption of a nominal hybrid electric passenger car. Throughout this paper, these hybrid topologies are referred to as 1PG-1, 2PG-1, 2PG-2, 3PG-1, and 3PG-2. Table 1 summarises the size of the components and other parameters of the nominal car that hosts the five transmissions. The developed models of the system components are summarised below, whilst more details are provided in Appendices A and B.

Planetary gearset
One or more PGs are in the core of the power-split hybrid powertrains. As shown in Fig. 1, a PG consists of three gears, namely the sun (S), ring (R), and carrier (C) gears. The radii of the ring and sun gears are referred to as and , respectively. However, a PG only has two degrees of freedom because the angular velocities of these three gears are related as follows: Fig. 2 shows a lever diagram of each of the five hybrid topologies considered in this study. Table 2 also summarises all the modes of  [8].

Table 3
Clutch engagements of all topologies in Fig. 2 to realise different modes, where CL +CL means that clutches and are the only ones engaged to generate the mode whilst N.A. means that the corresponding mode-topology pairing is not available.

Modes
Topologies operation that these topologies can generate. Fig. 2(a) shows the 1PG topology, which can realise three modes of operation with different clutch engagements. Figs. 2(b) and 2(c) show 2PG-1 and 2PG-2 topologies, respectively, which can realise up to five modes. Figs. 2(d) and 2(e) show the 3PG-1 and 3PG-2 topologies, respectively, which can generate up to seven modes of operation.
Each mode of operation is modelled by a binary variable indicating whether the mode is selected. For example, 1 = 1 means that Mode 1 is selected. Evidently, only one mode can be selected at any one time.

Electric motors
The hybrid powertrain in this study includes two electric motors which share the same torque characteristics but differ in their power and speed profiles. Motor-Generator 1 (MG1) operates as the engine speed controller in all modes, while it contributes towards the total torque request in 1 and 6 only. Motor-Generator 2 (MG2) is the only source of power in 4 , and the primary source of electric power in all other modes. MG2 is also used for regeneration in all topologies.
The electric motors in this study are modelled with their efficiency maps at different values of mechanical torque and rotational speed. The efficiency map of the electric motor ∈ {1, 2}, which depends on the internal resistance and torque constant of the motor, is represented as in (5).

Engine
The hybrid powertrain of this study uses a gasoline engine which is attached to different gears of the PGs depending on the topology. As shown in Fig. 2, the engine is connected to the carrier gear of the first PG (C 1 ) in both 1PG-1 and 2PG-1 topologies, whereas it is connected to the ring gear of the first PG (R 1 ) in the remaining topologies. In order to make a fair comparison, all topologies use the same engine, which is represented as a map of its brake-thermal efficiency in terms of rotational speed and mechanical torque: The fuel flow rate of the engine ( ) is measured in mm 3 /cycle and modelled as a polynomial of degree five (quintic function) in terms of the engine torque and angular velocity. For further details on this polynomial, refer to Appendix B.

Battery
The state of charge (SoC) of batteries is a function of time, and of their charging and discharging power. This is modelled by the following ordinary differential equation (ODE): where , > 0 shows that the battery is being discharged. The sampling time of the driving cycles is normally one second (i.e. = 1). The battery power is a hybrid algebraic function (i.e. a function of both continuous and discontinuous variables) of torque, angular velocity, and efficiency of MG1 and MG2: where 1 , 2 ∈ {−1, 1} are binary variables that, respectively, indicate whether MG1 and MG2 are in the motoring ( = −1) or generating ( = 1) mode of operation. Both the MG1 and MG2 are used for motoring in mode 1 whereas MG2 is the only source of electric traction in modes 2-8. Without loss of generality, it is assumed that only MG2 performs regeneration.

Energy Management Strategy (EMS)
For an unbiased study, the energy consumption of hybrid powertrains with different numbers of PGs must be compared operating optimally over the driving cycle. Therefore, we propose here an EMS that performs both mode selection and torque distribution simultaneously to ensure optimal operation of the powertrain over the given cycle. An EMS with simultaneous mode selection and torque distribution leads to a complex mixed-integer optimisation problem due to the existence of discontinuities caused by changing the mode of operation; i.e. due to engaging and disengaging clutches and switching the operation of the electric motors between motoring and generating. This fact, along with the nonlinearity of components and the existence of algebraic constraints, results in the EMS comprising a mixed-integer nonlinear optimal control problem (OCP) with differential and algebraic equations (DAEs): * (.) = arg min where the decision variables are the engine torque, MG2 torque, MG1 angular velocity, modes and , and therefore = is the prediction horizon, which is equal to the length of the driving cycle in seconds since the sampling time is one second. The objectives of the EMS are to minimise (1) energy consumption, (2) number of mode shifts, and (3) deviation of the battery SoC from its initial value (SoC 0 ). This is turned into a single-objective optimisation problem by a convex combination of some quantification of each objective: where [ , , ] = [0.5, 0.1, 0.4] are the optimum convex combination coefficients for 1PG topology, which are calculated by trial and error.
is the rate of energy consumption. The Second and third terms of the cost functions are two soft constraints to smoothen the fluctuation of clutch engagement and to prevent deep discharge of battery.
, =̇e , +̇e qv, (11) and: Minimisinġin (14) leads to minimising the number of mode shifts. Frequent mode shifts in the hybrid powertrain will lead to excessive mechanical losses due to repeated engagement and disengagement of clutches.
where is a weighting factor fixed to 0.02 [31]. The mode change factor = 0 when the next and current modes are the same, whilst = 1 otherwise. Eqs. (9b) and (9c) represent the multi-mode operation of the powertrain. The discontinuities due to the gearbox and electric motor operations are modelled by complementarity constraints (9b), where the operator ⟂ indicates that at least one of the bounds is active (i.e. either , or , is zero at each time ). Eq. (9c) ensures that only one of the eight modes is active at any given .
Eq. (9d) models each powertrain topology as a set of DAEs to characterise the behaviour of the transmission system and constraints of the power flows. Appendix A provides details of these models, where all equations are extracted from the models in Section 2.1 under steadystate conditions. For example, the steady-state equations for the 1PG topology in mode 1 (i.e. wheṅ=̇=̇1 = 0) are as follows: Combining Eqs. (15) and (16) to eliminate 1 , the following algebraic constraint of torques is obtained: which is only valid when the powertrain operates in mode 1 and the electric motors are in traction mode; i.e. 1 = 1 and = 1. Therefore, (17) is rewritten as a mixed-integer algebraic constraint in (18) (see also (19a) in Appendix A).
D. Rajput et al.  The EMS in (9) is discretised and then formulated as a mixed-integer non-linear programming (MINLP) problem using the mathematical programming language (AMPL), which is the modelling language adopted by Neos server as the standard interface for solvers. The formulated problem is then solved with Knitro, available in the Neos server, which achieves a local optimal solution at every sampling time. The reliability of the solution of the EMS is assessed by using a real-time simulator which takes into account a full vehicle model and the environmental disturbances.

Real-time simulation
The real-time simulation models of the five hybrid topologies considered in this paper were developed based on the ASM from dSPACE. As shown in Fig. 3, the models consist of vehicle dynamics, an internal combustion engine, planetary gearbox including clutches, final drive, two motor-generators labelled MG1 and MG2, batteries and electronics, inner controllers, and the EMS developed in Section 3. The developed models are simulated over the United States 06 (US06) drive cycle with 80% initial battery SoC and a one-second sampling time. Different topologies may differ in: 1. The number of PGs in the planetary gearbox block, which varies between one and three; 2. The EMS, which is specific to each topology, solved offline, and stored as a lookup table in the ECU.
The ECU also contains inner controllers for the torques of MG2 and engine, the angular velocity of MG1, and the gearbox clutches.  14.23 m∕s and 12.76 m∕s, respectively. The corresponding tracking errors for the 1PG-1, 2PG-1, and 2PG-2 topologies are 8.22 m∕s, 10.05 m∕s, and 11.65 m∕s, respectively. In other words, the speeds of the vehicle with 3PG topologies deviates more than the other topologies from the desired profile. This is due to the fact that 3PG topologies introduce more number of modes of operation and hence more clutch fluctuations adding the complexity of the soft ECU, clutch, and drivetrain models of the real-time simulation models, which increases the resulting tracking error of speed. More precisely, there are 7 modes in 3PG-1 and 6 modes in 3PG-2 topologies, compared to 4 modes in 1PG and 5 modes in 2PG topologies. Table 4 shows the energy consumption of each of the topologies over the US06 drive cycle. The total fuel consumption is the sum of the fuel consumption by the engine and the equivalent fuel consumption by the electric motors. The equivalent fuel consumption is the amount of burned fuel to recharge the battery through MG1 to its initial SoC (80%) after completing the driving cycle and with the engine operating at its optimal point.

Results and discussion
The total fuel consumption of the 1PG-1 topology is 596 g whereas that of the 2PG-1 topology is 566 g (i.e. 5.0% lower). Moving from D. Rajput et al.  2PG to 3PG can reduce the total fuel consumption by a further 1.5% (i.e. 6.5% lower than that of the 1PG topology). As shown in Table 4, the 2PG-2 and 3PG-1 topologies also reduce fuel consumption with respect to the 1PG-1 topology by 1.3% and 2.8%, respectively. Engine fuel consumption by every topology with multiple PGs is lower than that consumed by the 1PG topology at the expense of a small increase in the equivalent fuel consumption. In other words, for a maximum of 1.6% higher discharged battery, the topologies with multiple PGs consume up to 67 g less engine fuel. Fig. 5 summarises the best total fuel consumption, fuel economy, and reduction in total fuel consumption by 1PG, 2PG, and 3PG topologies. As 2PG-1 and 3PG-2 are the best topologies with respect to fuel consumption, the remaining analysis in this section focuses on these cases. As shown in Fig. 5, the rate of reduction of fuel consumption decreases when the number of PGs increases from two to three. It is hypothesised that further increasing the number of PGs from three to four would lead to a saturation in the reduction of the total fuel consumption. Fig. 6(a) shows the evolution of the total fuel consumption of the 2PG-1 and 3PG-2 topologies over the US06 drive cycle as compared to 1PG-1 topology. Fig. 6(b) shows that the fuel consumed by the engine is independent of the topology during the 200 − 500 s time interval, whilst it is much lower for the 2PG-1 and 3PG-2 topologies (especially the former) during the 0-180 s and 500-600 s time intervals. On the other hand, equivalent fuel consumption of 1PG-1 and 2PG-1 topologies are similar and higher than that of the 3PG-2 topology during the 200-500 s time interval. This is a similar trend to the one observed for the battery SoC in Fig. 6(c).
The main reasons for the reduction of fuel consumption of the 2PG and 3PG topologies are: (i) the speed of electric motors is higher with multiple PG topologies, leading to lower power demand from the engine; and (ii) fixed-gear ratio topologies save the electric power losses resulting from regulating the engine speed. These reasons are explained in more details in the remainder of this section.
As shown in Table 2, 1PG-1 and 2-PG-1 topologies can generate up to four modes whilst the 2PG-2, 3PG-1 and 3PG-2 topologies can generate five, seven and eight modes, respectively. 3PG-2 topology realise mode 1, mode 4, and mode 3 by engaging only one clutch (CL 1 ), which significantly reduces frequent clutch engagement and disengagement compared to other topologies. Fig. 7 shows that the EMS never chooses modes 8 or 9 for any of the topologies. Mode 8 (series mode) can only be selected when the battery ≤ 40%, which never happens for the entire US06 drive cycle. The EMS does not select mode 9 because MG1 is used as a speed controller for the engine, and therefore MG1 and engine are not allowed to simultaneously provide torque to the wheels. Fig. 7 shows that the EMS chooses mode ∈ {1, 3, 4} for 1PG-1 and 2PG-1 topologies and mode ∈ {1, 2, 3, 4, 6, 7} for 3PG-2 during the 0-180 s and 500-600 s time intervals. Modes 1 and 4 are EV modes (see Table 2), where the former uses both MGs and the latter only MG2 to provide torque to the wheels. Conversely, all drivetrains contribute to the wheel torque in input-split mode 3, with the engine speed being regulated by the speed of MG1. Modes 2, 6 and 7 represent parallel topologies with a fixed ratio between the engine speed and the vehicle speed. However, unlike mode 4, mode 2 utilises only one MG.
Unlike in 1PG topology, the speed of MG2 in all modes of 2PG-1 and 3PG-2 other than mode 7 for 3PG-2 topologies are larger than the speed of final drive, with a coefficient of 2 2 > 1 for 2PG-1 and 1 + 3 3 > 1 for 3PG-2 topologies. For example, given the gear ratio in Table 1, the speed of MG2 in 2PG-1 topology over the US06 drive cycle is up to 2.6 times higher than the speed of MG2 in 1PG-1 topology (see Fig. 8(a)). Moreover, Fig. 8(b) shows that the engine in 2PG-1 and 3PG-2 topologies rotates closer to its optimal speed (2050 rpm, as stated in Table 1) because of the higher speed of MG2, whilst engine speed D. Rajput et al. Fig. 7. The optimal mode selection by the EMS for different topologies. is too high in 1PG-1 topology. Fig. 8(c) shows that there is not much difference in MG1 operation across the different topologies. For more details, please refer to Appendix A.
The higher speed of MG2 enables the engine to operate at high power with higher efficiency due to the shape of the efficiency map in Fig. 8(a). Therefore, the EMS utilises MG2 at higher power and efficiency for the 2PG and 3PG topologies than for the 1PG topology, as can be observed in Figs. 9(a) and 9(b). This will also reduce the required power from the engine and hence the direct fuel consumption (see Fig. 9(c)). This is shown in Fig. 8(b), where the brake specific fuel consumption (bsfc) of the engine of the 2PG-1 and 3PG-2 topologies is compared against that of the 1PG topology. This is at the expense of degrading the efficiency of MG1 for a similar delivered power (see Fig. 9(f)). However, MG1 does not contribute to the wheel torques during the 0-180 s and 500-600 s time intervals, and hence such efficiency reduction comprises an insignificant drawback.
As discussed before and shown in Fig. 5, increasing the number of PGs from two to three further reduces the total fuel consumption by an extra 1.5%. This is partially because the speed of MG2 in 3PG-2 topology can be even larger than in 2PG-1 (see Fig. 8(a)), resulting in MG2 being able to provide more power with high efficiency than 2PG-1.
However, the main reason for this is the availability and selection of the fixed-gear modes (i.e. modes 2 and 6), as shown in Fig. 7. The fixedgear modes reduce equivalent energy consumption by not requiring MG1 to regulate the engine speed, hence excluding the corresponding losses [12]. During the 80-100 s, 170-180 s and 295-345 s time intervals, the EMS of the 3PG-2 topology chooses these fixed-gear modes whereas D. Rajput et al.

Conclusions
This work systematically analysed the effects on energy consumption of the number of planetary gears in the transmission system of hybrid electric powertrains. Thus, the minimum energy consumption of five hybrid topologies with one, two, and three PGs were compared over the US06 drive cycle. For each topology, an EMS ensured that the combination of clutch engagements, torque distributions and components' speeds is such that it minimises the total energy consumption. The latter is calculated as the sum of the engine fuel consumption and the equivalent fuel consumption of the battery's discharge energy.
The incorporation of mode selection via clutch engagements into the nonlinear EMS led to a mixed-integer problem, which was then converted into a complementarity constrained problem, and solved using KNITRO within Neos server.
Real-time simulation results showed that increasing the number of PGs from one to two and from two to three reduces the total fuel consumption by 5% and 1.5%, respectively. The primary reason for lower fuel consumption is the higher power provided by the primary electric motor (MG2). Additional PGs help MG2 to operate at higher speed, where it can provide more power with higher efficiency. This also reduces the power required from the engine, resulting in lower engine fuel consumption. 3PG topologies also benefit from the availability of fixed gear modes, in which MG1 is not involved in regulating the engine speed, therefore avoiding the corresponding power loss.
To summarise, increasing the number of PGs reduces total fuel consumption due to the higher rotational speed of MG2 and to the availability of fixed gear modes. However, it is important to note that the rate at which total fuel consumption reduces seems to stagnate as the number of PGs increases.
The methodology developed for this systematic analysis can be applied to different types of electrified powertrains and drive cycles. However, it should be noted that mechanical losses, frictional losses, and the added cost and weight of the powertrains due to increasing the number of PGs were not considered in this study.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgement
This work is funded by both coventry university and FEV, Europe GmbH, grant number 90440-01. All authors have read and agreed to the published version of the manuscript.

Appendix B. Mass flow rate of fuel injection to engine per cycle
The mass of fuel from engine is calculated by Eq. (12). The variable q used in that equation refers to mass flow rate of fuel per cycle which is modelled as a polynomial function of order 5 of engine torque (T ) and engine speed ( ): where: