Time-optimal Control Strategies for Electric Race Cars with Different Transmission Technologies

This paper presents models and optimization methods to rapidly compute the achievable lap time of a race car equipped with a battery electric powertrain. Specifically, we first derive a quasi-convex model of the electric powertrain, including the battery, the electric machine, and two transmission technologies: a single-speed fixed gear and a continuously variable transmission (CVT). Second, assuming an expert driver, we formulate the time-optimal control problem for a given driving path and solve it using an iterative convex optimization algorithm. Finally, we showcase our framework by comparing the performance achievable with a single-speed transmission and a CVT on the Le Mans track. Our results show that a CVT can balance its lower efficiency and higher weight with a higher-efficiency and more aggressive motor operation, and significantly outperform a fixed single-gear transmission.Comment: 5 pages, 4 figures, submitted to the 2020 IEEE Vehicle Power and Propulsion Conferenc

A Decomposed Co-design Strategy for Continuously Variable Transmission Design

This paper presents a decomposed co-design optimization framework to jointly design the geometry and the controller of a Continuously Variable Transmission (CVT), accounting for its low-level dynamics. Specifically, we first devise a model of the CVT and the feedback controller, and use it to formulate the optimal co-design problem with the goal of minimizing the transmission's mass as well as the losses that occur in the system, including the lower-level actuation. Second, we divide the resulting nonlinear multi-objective optimization problem into separate hierarchical optimization subproblems and we leverage the concept of Analytical Target Cascading (ATC) to solve the separate optimization subproblems using an interior-point optimization algorithm. Finally, we showcase our framework on a representative drive cycle. Our results demonstrate that the presented co-design method can achieve a more compact CVT design without compromising the desired ratio trajectory and reducing the overall losses by up to 14%

Time-optimal Control Strategies for Electric Race Cars with Different Transmission Technologies

This paper presents models and optimization methods to rapidly compute the achievable lap time of a race car equipped with a battery electric powertrain. Specifically, we first derive a quasi-convex model of the electric powertrain, including the battery, the electric motor, and two transmission technologies: a fixed-gear transmission (FGT) and a continuously variable transmission (CVT) Second, assuming an expert driver, we formulate the time-optimal control problem for a given driving path and solve it using an iterative convex optimization algorithm. Finally, we showcase our framework by comparing the performance achievable with an FGT and a CVT on the Le Mans track. Our results show that a CVT can balance its lower efficiency and higher weight with a higher-efficiency and more aggressive motor operation, and significantly outperform an FGT

Integrated Plant and Control Design of a Continuously Variable Transmission

This paper presents an optimization framework to design the components and the controller of a Continuously Variable Transmission (CVT) in an integrated manner. Specifically, we aim at reducing the mass of the transmission and the leakage losses that occur in the system. To do so, we first formulate the joint plant and control design problem including the corresponding objectives and constraints. Thereafter, we propose a proportional integral structure for the design of the CVT ratio control. The combined plant and control design problem is formulated as a nonlinear multi-objective optimization problem, and is simultaneously solved using an interior point optimization method. We evaluate the obtained design on the Worldwide Harmonized Light Vehicles Test Cycle (WLTC) as well as on more aggressive driving scenarios, and demonstrate that the optimized CVT design is always capable of realizing the required driving performance. Additionally, we study the impact of the plant design parameters on the control performance by analyzing the coupling strength between the subproblems. Thereby, the pulley radius is found to have the strongest influence in the resulting leakage losses that occur at the variator level. Finally, leveraging the presented design framework, we show that up to 13% and 18% reduction in the CVT variator mass and on leakage losses, respectively, can be achieved without compromising the desired ratio trajectory over a representative dynamic driving cycle

A Convex Optimization Framework for Minimum Lap Time Design and Control of Electric Race Cars

Lap time is the most relevant performance indicator in electric racing. In order to minimize it, it is important to carefully design the powertrain of the race car in terms of sizing and technology, and meticulously administer the energy available on-board. To facilitate efficient design, this paper presents a convex optimization framework to rapidly compute the minimum-lap-time control strategies for a battery electric race car. Specifically, we first identify a convex model of the electric powertrain, including the battery, the electric motor, and two transmission technologies: a fixed-gear transmission (FGT) and a continuously variable transmission (CVT). Second, assuming an expert driver, we formulate the minimum-lap-time control problem in a convex fashion for a given driving path, and compute its globally optimal solution with second-order conic programming algorithms. Third, we showcase our framework on the Le Mans track by comparing the performance achievable with an FGT and a CVT, and validate it with nonlinear simulations. Our results show that for the given setup a CVT can balance its lower efficiency and higher weight with a higher-efficiency and more aggressive motor operation, significantly outperforming the lap time achievable with an FGT. Finally, we leverage the computational efficiency of our framework to carry out parameter studies on the components, revealing that optimizing the size of the battery and the motor for the specific scenario can considerably improve the achievable lap time, and that the best transmission strongly depends on the sizing decisions

Integrated Design of a CVT-equipped Electric Powertrain via Analytical Target Cascading

Electric vehicles are gaining momentum as a valid alternative to conventional engine-based cars. However, in order to realize this goal, they must achieve a similar, if not better, performance and driving range. To this end, their powertrain must be carefully designed accounting for the interconnections among the various components in an integrated fashion. In this paper, we present a co-design framework for electric powertrains, whereby we jointly optimize the size of the electric machine (EM) and the geometry of a continuously variable transmission (CVT) together with its ratio trajectory, with the goal of minimizing the energy consumption of the vehicle. Specifically, we first frame the minimum-energy co-design problem in an integrated manner, accounting for the CVT geometry and dynamics, and the EM size. Given the problem complexity, we decompose it into an EM-design and a CVT-design subproblem, whereby we jointly optimize the CVT-ratio trajectory, and leverage analytical target cascading (ATC) to effectively solve the design problem. Finally, we showcase our framework on the New European Driving Cycle (NEDC), highlighting the importance of designing powertrains in an integrated manner: Compared to the case whereby only the EM, the CVT, or the control are optimized, our joint EM-CVT design can improve the energy consumption of the vehicle by 1 to 20%