# Model-Based Optimization of a Plug-In Hybrid Electric Powertrain with Multimode Transmission

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Powertrain Model

## 3. Optimization: Number of Speeds

_{g}) can be determined without considering any specific test cycle. Therefore, the optimal choice of the gear s

_{g}is determined according to the following:

_{loss}is minimized for all of the possible final drive torques T

_{FD}and vehicle speeds v

_{veh}. As a result, a map that defines the optimal choice of the gear ${s}_{\mathrm{g}}^{\ast}$(T

_{FD}, v

_{veh}) is obtained. Since the number of possible solutions for Equation (7) is equal to the number of speeds, a feasible and still fast method for solving Equation (7) is to check all of the possibilities and to choose the best one.

**i**

_{ICE,EM}, the shift strategy is adapted and the powertrain model is executed, where the resulting fuel consumption V

_{fuel}or energy consumption E represents the cost function for the optimization. In order to consider a wide range of vehicle speeds and accelerations, the test cycles WLTP (Worldwide Harmonized Light Vehicles Test Procedure), Urban Dynamometer, and FTP 75 are considered. A scalar cost function is obtained by averaging the results of these three test cycles. Figure 4b shows the energy savings, which can be obtained by multi-speed transmissions for the electric motor operation. The diagram shows the energy savings as a function of the transmission ratios for a two-speed transmission, and the table contains the optimized energy consumptions and the corresponding transmission ratios for a one- to four-speed transmission. In order to ensure that the electric motor can be operated up to a vehicle speed of 180 km/h, the transmission ratio of the highest gear is constrained to values smaller than 10.37. The table shows that the benefit of a two-speed transmission is up to 3.5% energy saving, whereas the benefit of each additional speed becomes drastically smaller.

## 4. New Transmission Concept

## 5. Optimization: Transmission Ratios

#### 5.1. Calculus of Variations (CoV)

_{ICE}, because it is directly affected by the input $\tilde{u}$. In order to have an analytic equation for the Hamiltonian function, Equation (14), the power losses of the electric motor are also approximated by a convex function, as follows:

#### 5.2. Dynamic Programming (DP)

**H**containing the Hamiltonian function values for all of the possible states

**s**

_{m}and inputs

**u**

_{m}. Equation (22) is solved backwards in time from k = N − 1 to k = 0, with the initial costs

**J**

_{DP,N}= 0. The result in each time step k is an optimal control vector ${\mathbf{u}}_{\mathrm{m},\mathrm{k}}^{\ast}$ and the minimal accumulated costs

**J**

_{DP,k}for each state within the state grid vector

**s**

_{m,k}. The optimal inputs ${\mathbf{u}}_{\mathrm{m},\mathrm{k}}^{\ast}$ are also allocated to the state grid vector and are saved in a new matrix, as follows:

_{m,k}, as follows:

_{m,k+1}is calculated according to Equation (21) by means of the result of Equation (24).

#### 5.3. DP and CoV Combined

_{0}is determined in step seven (electric driving is the initial operation state S

_{m,0}= 5). If the deviation regarding the desired value $So{C}_{0}^{\ast}$ becomes small enough, the resulting fuel consumption is calculated in step eight by means of the forward calculation and the matrices that were saved in step five.

#### 5.4. Optimization Results

## 6. Concept Evaluation

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Multimode transmission with fixed gear ratios for the internal combustion engine (ICE) and electric motor.

**Figure 4.**(

**a**) Optimization procedure of the transmission ratios in terms of a minimal fuel and energy consumption and (

**b**) optimized energy savings and transmission ratios of a multi-speed transmissions for the electric motor operation.

**Figure 5.**Optimization results of multi-speed transmissions for the ICE operation. Left and middle: visualization of the relation between transmission ratios and fuel savings. Right: optimized fuel savings and transmission ratios.

**Figure 6.**New transmission concept and the switching pattern of the clutches and brakes for activating the available operation modes.

**Figure 7.**(

**a**) Dynamic programming (DP) and calculus of variations (CoV) algorithm for the optimization of the plug-in hybrid electric vehicles (PHEV) operation; (

**b**) schematic representation of the algorithm; and (

**c**) example of the relation between the constant Lagrange multiplier λ and the initial state SoC

_{0}(above) and optimal state trajectory for a given λ (below).

**Figure 8.**Visualization of the fuel savings achieved with the new transmission concept in dependency of the transmission ratios. For comparison, the average value fuel consumption of the three above mentioned test cycles is considered and as reference (0%), the fuel consumption obtained with the parameters in Section 3, is chosen. The black dot indicates the optimum with approximately 1.4% benefit, compared to the parametrization in Section 3.

**Figure 9.**(

**a**) Reference powertrain concept, (

**b**) powertrain with new transmission concept (see Figure 6), and (

**c**) simulation results, namely: comparison of the fuel consumptions of both powertrains.

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Geng, S.; Meier, A.; Schulte, T. Model-Based Optimization of a Plug-In Hybrid Electric Powertrain with Multimode Transmission. *World Electr. Veh. J.* **2018**, *9*, 12.
https://doi.org/10.3390/wevj9010012

**AMA Style**

Geng S, Meier A, Schulte T. Model-Based Optimization of a Plug-In Hybrid Electric Powertrain with Multimode Transmission. *World Electric Vehicle Journal*. 2018; 9(1):12.
https://doi.org/10.3390/wevj9010012

**Chicago/Turabian Style**

Geng, Stefan, Andreas Meier, and Thomas Schulte. 2018. "Model-Based Optimization of a Plug-In Hybrid Electric Powertrain with Multimode Transmission" *World Electric Vehicle Journal* 9, no. 1: 12.
https://doi.org/10.3390/wevj9010012