## 1. Introduction

The increasing spread of medium size HEVs in the market has been led by

CVT architectures equipped with a power-split device (

PSD) able to continuously vary the transmission ratio thanks to the synergy with two motor-generators. This architecture is sketched in

Figure 1, the main components are: internal combustion engine (

ICE), two motor-generators (

MG_{1} and

MG_{2}), planetary gear train, i.e., Sun gear (

S), Ring gear (

R), Carrier gear (

C), and the battery (B).

In HEVs, the alternative to this structure is given by a discontinuously variable transmission (

DVT) equal to that used in conventional vehicles, typically with six different values of transmission ratio. This architecture is sketched in

Figure 2 (note that only one motor-generator is used).

The optimal energy management of the two transmission is fundamental to provide drive comfort and reduce fuel consumption. In scientific literature, one can find various studies on the implementation of optimal control logics, regarding the two architectures described.

In [

1], an optimized control logic for the power split device (

PSD) based architecture is proposed. In [

2], a particular energy management system (EMS) is studied for the

DVT architecture. Both the

CVT structure and the

DVT structure are considered for the optimization logic described in [

3]. Many variables affect the optimization of the entire hybrid powertrain; in particular, the driving style can play a crucial role, as analyzed in [

4]. In [

5,

6], more sophisticated methods based on neural networks and fuzzy logic are proposed.

Thus, it is then important to study optimal control techniques, but to evaluate the energy virtuosity of

CVT based vehicles, analyzing electrical losses is essential. In [

7,

8,

9], experimental tests were conducted with the aim of determining the efficiency of electric drives used in the HEVs available on the market today. Knowing this parameter is a key step in designing control logic aiming at optimizing these components [

10]. Indeed, the

CVT architecture allows an optimal energy management of the

ICE, but at the cost of certain of electrical losses which are absent in the

DVT architecture. Thus, one can find various works on the

PSD based vehicles in scientific literature, however, they do not deal with a combined evaluation of electrical losses and

ICE efficiency improvement deriving from the

CVT system.

The main aim of this paper is to quantify these losses in every possible operating condition and compare them with the ICE energy savings obtainable by continuously regulating the transmission ratio. Firstly, the PSD model is presented; secondly, the electrical losses model is shown; and, finally, the EMS is described. In addition, a brief description is reported of the MATLAB/Simulink (2017b, Mathworks, Natick, MA, USA) DVT and CVT powertrain models.

Particular attention is paid during the results analysis not only to the electrical-loss quantification but also to the ICE management in both architectures.

In fact, to compare the energy efficiency of the two architectures a combined evaluation of electrical losses and ICE efficiency is of primary importance.

## 2. Model

#### 2.1. Power Split Device Energy Model

To mathematically describe the

PSD, the energy convention depicted in

Figure 3 has been chosen.

In

Figure 3, the power flows are positive in the direction shown by the arrows. The analysis carried out in this study, due to the limited influence of the inertial terms of the planetary gear train components on the overall efficiency, does not consider these terms. Thus, the equations shown below are only algebraic equations and are to be considered as a stationary model.

The variable meanings are explained in

Table 1.

It is first necessary to identify the relationship between the different velocity of the components of the

PSD; Willis formula can be applied (valid if the planetary rings are merged into a single one externally toothed together):

where:

While

z_{S} and

z_{R} represent the sun gear and ring-gear-tooth number,

z_{R} is conventionally a negative number because the teeth are internal. By conveniently combining:

Based on the three following equations, once the value of k has been chosen, it is possible to acquire a relationship between the mechanical ring gear power and the mechanical sun gear power:

By analyzing the vehicle energy scheme, it can be immediately understood that the possible configurations in terms of power flows are various; to describe them all, this work would be overburdened. The energy configurations considered in this paper are the ones practically exploited during the vehicle usage. In addition, regenerative braking and pure electric traction are not considered here because the resulting configurations do not have any degree of freedom allowing optimization study.

Note that the input to the systems are: ${\omega}_{w},\text{}{P}_{w}\left({T}_{w}\right),\text{}k$

Case 1 (MG_{1} motor, MG_{2} generator): $k<0,\text{}{P}_{e}0,\text{}{P}_{Sm}0,\text{}{P}_{L1}0,\text{}{P}_{L2}0$

Combining the following equations:

Case 2 (MG_{1} generator, MG_{2} motor): $k>0,\text{}{P}_{e}0,\text{}{P}_{Sm}0,\text{}{P}_{L1}0,\text{}{P}_{L2}0$

Combining the following equations:

Repeating the same calculations as above, one has:

Case 3 (

MG_{1} generator—only from a mechanical point of view,

MG_{2} generator):

This particular configuration is rather frequent and it occurs in low sun gear velocity situations; in this case, the mechanical power delivered by the sun gear is not enough to overcome electrical losses in

MG_{1}, hence

MG_{2} has to provide the remaining power. Thus:

Since the torque of the two motor generators is determined in each configuration, and their speed is known thanks to the Willis formula, once the ratio

k is found (goal of the optimization process, see

Section 2.3), the electrical losses can be computed (as shown in

Section 2.2), and all the other variables of interest can easily be obtained.

#### 2.2. Electrical Losses and Internal Combustion Engine Modeling

The aim of this paper being to evaluate losses in the

CVT system to implement a control logic able to consider these losses, similar to that proposed in [

10], the modeling of electric machines and their converters is of primary importance.

As regards electrical machines, electrical losses were modeled exploiting the results presented in [

11]. These loss data are available for a specific size permanent magnet synchronous motor, in order to use these results for machines of different size (torque, speed), and losses were put in per unit, as suggested in [

12]. The losses in the inverters were estimated starting from the results shown in [

13], which were interpolated in accordance to that proposed in [

14] and finally put in per unit.

By doing so, these maps can be used in the simulation environment for machines of different (but comparable) size.

Hence, a look up table was created for each contour map; these look up tables are formed by almost 10,000 points. To obtain a halfway loss value, linear interpolation is used. This was implemented thanks to available interpolation blocks in Matlab/Simulink environment.

The electrical losses of each electric drive are presented in per unit in

Figure 4.

Please note that the speed value 1 is not for the maximum speed but for the base speed instead.

To evaluate the

ICE efficiency, the spark ignition engine presented in [

15] was considered; the fuel consumption data available in [

15] were elaborated to obtain the efficiency map. This contour map is shown in

Figure 5.

#### 2.3. CVT Powertrain Efficiency Optimization

The goal of this study is to obtain for each couple (wheel speed and wheel torque) the optimal transmission ratio k able to minimize fuel consumption. This process is not a mere search for the k value which maximizes the

ICE efficiency, but it is a broader study aiming at the maximization of the entire powertrain. In fact, given the same power delivered to the wheels, the electrical losses must be compensated by a higher

ICE power generation with its consequent consumption increase. To carry out this optimization process, a MATLAB/Simulink model in accordance to that proposed at

Section 2.1 and

Section 2.2 was created. A sufficiently numerous finite set of couples (wheel speed and wheel torque) was created to consider the variable

k a continuous

k_{opt} =

k_{opt}(

ω_{w};

T_{w}). The

PSD model and the

ICE model were necessary to perform the optimization. Therefore, a combination of different value of

ω_{w} and

T_{w} were provided as input to the model, together with all the possible

k values to evaluate the fuel consumption in every operating condition. The

k value, corresponding to the minimum value of fuel consumption for that particular operating condition, was then selected as a

k_{opt} value. The logical scheme of this process is represented in

Figure 6.

Thanks to the code implemented, the optimal transmission ratio function of wheel speed and wheel torque was found. In particular, due to the model implemented, this function is a “collection” of steady state points. The optimization result is shown in

Figure 7.

The

ICE work points deriving from the optimization process are shown in

Figure 8a. In

Figure 8b, the

ICE work points declared by Toyota are shown [

16]. It is easy to notice how the optimization result is the same as that declared by Toyota, and it basically corresponds to the

ICE efficiency maximization; in other terms, the electrical losses do not count in the optimization process (although they are not negligible), if compared to the losses in the

ICE power generation. Please note that this conclusion could not have been drawn a priori.

#### 2.4. DVT Powertrain Effciency Optimazation

The

DVT powertrain optimization is a simpler process compared to the previous one; indeed, there is only one variable to optimize, i.e., the

ICE efficiency (equivalent to the fuel consumption). It is sufficient to find for each couple (wheel speed and wheel torque) the transmission ratio able to satisfy the

ICE range of functioning and to maximize its efficiency. The result of this optimization process is shown in

Figure 9.

#### 2.5. MATLAB/Simulink Model

#### 2.5.1. Power Split-Device Vehicle MATLAB Simulink Model

To quantify the variable of interest, the implementation of the two powertrain models in MATLAB/Simulink was necessary. In

Figure 10, the

CVT powertrain model is shown while the

DVT powertrain model is shown in

Figure 11. The battery and its control system were neglected because they do not intervene in the computation of the electrical losses due to the realization of the transmission ratio

k. Indeed, the role played by this component is the same in the two architectures; the same control logics can be applied and the benefits deriving from regenerative braking and electric traction are identical in both configurations.

Implementing control logic for the storage system management and its accurate modeling are important, for instance, to determine fuel consumption, emission evaluation, etc. It is not necessary in this study but would be to evaluate the electrical losses associated with the realization of the transmission ratio (in the CVT architecture) and to determine the average ICE efficiency.

The red blocks in

Figure 10 represent the model of physical components, the light blue block computes the resistant torque and the orange blocks implement the control logic. The complete model is necessary to simulate the powertrain response to the various road missions with respect to the reference speed and road slope. On the contrary, the light blue block is not necessary to calculate electrical losses and the

ICE efficiency; therefore, the inputs in this case are the vehicle speed and the wheel power directly, which are provided as ramp signals to compute the desired parameters in each operating condition.

#### 2.5.2. DVT MATLAB/Simulink Model

Similar to that implemented for the CVT architecture, one model for the DVT powertrain was created. The same considerations previously mentioned, as far as the role of each block is concerned, remain valid.

#### 2.6. Vehicle Features

The vehicle features are reported in

Table 2; the parameters common to the two architectures are characterized by the same values to allow a fair comparison.

#### 2.7. Road Mission Features

As previously described, the electrical-loss evaluation is firstly carried out considering every possible operating condition of the powertrain. However, some operating conditions are much more important and more frequent than others during the real usage of the vehicle, hence the importance of the simulation of the powertrain response over actual road missions.

The vehicle speed and road slope data (inputs to the vehicle models) were experimentally collected via GPS, in particular these missions were distinguished in: urban mission, extra urban mission and a highway mission. In addition, simulations were also carried out on three American type approval tests (HWFET, US06, and UDD6).

The main characteristics of these three missions are listed in

Table 3.

Detailed results are only reported for real missions; however, the powertrain efficiency is evaluated on the totality of the described missions.

## 4. Conclusions

The PSD energy model and losses in the electric drives are modeled in this paper. Subsequently, the MATLAB/Simulink model for the CVT and DVT architectures are described. The CVT electrical-loss contour map is then presented. To estimate the impact of the two different transmissions on the ICE operating conditions, contour maps of the ICE as a function of road variables are obtained for both the parallel configurations by knowing the transmission ratio associated for each work point (acquired by the fuel consumption minimization process).

Finally, three real road missions were simulated to locate the work points on the different contour maps, evaluating strengths and weaknesses of both architectures.

The results obtained confirmed what was previously speculated: the CVT configuration allows the engine to work at higher efficiency rate, but at the cost of consistently high electrical losses.

Again, the CVT architecture is characterized by an 8% ICE efficiency increase in the urban mission (28.6% vs. 26.4%), 9% efficiency increase in the extra-urban mission and 16% in the highway mission.

Nevertheless, this improvement is not sufficient to compensate for the large amount of electrical losses generated by the continuously variable transmission on the three missions: 13%, 15% and 21% of the ICE generated energy.

Although the CVT based hybrid electric vehicles are widely spread in the medium size car market sector, they do not prove to be the best energy solution.

Further developments of this study will include the experimental validation of the results achieved via simulations by means of prototyping in collaboration automotive companies.