0018-9545 (c) 2017 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2017.2767201, IEEE Transactions on Vehicular Technology 1 Convex optimization methods for powertrain sizing of electrified vehicles by using different levels of modelling details Mitra Pourabdollah, Bo Egardt, Nikolce Murgovski, and Anders Grauers Chalmers University of Technology, Gothenburg, Sweden mitrapo@gmail.com; bo.egardt, nikolce.murgovski, anders.grauers@chalmers.se Abstract—This study investigates the impact of different levels of modelling details on the problem of optimizing the total cost of ownership of a fuel-cell hybrid electric vehicle. In this optimization, the objective function is a weighted sum of operational and component costs over a driving cycle. The former includes the consumed hydrogen and electrical energy, and the latter includes the sum of the battery, fuel-cell, and electric-motor costs. Three methods with different levels of modelling details are investigated; in the first method, the power split between the two power sources together with component sizes are optimized, while assuming nonlinear loss functions for the components. In the second method, the efficiencies of the components are approximated by constant values. In the third method, the problem is simplified further by considering the energy split between the battery and the fuel-cell. As shown in the results, a more detailed model gives more accurate results at the price of increased computation time. However, the simplified models can give similar results as the detailed model in most cases. In some problems though, the model simplifications lead to results that differ notably from those obtained by using the detailed model. I. Introduction Electrification of vehicles is currently seen as a solution to mitigation of environmental issues such as high fuel consumptions and pollution caused by the ever increasing number of vehicles. One attractive solution is hybrid electric vehicles (HEV), which have a primary energy converter, such as internal combustion engine (ICE) or a hydrogen fuel cell (FC), and a secondary source of energy, such as a battery or an ultra capacitor. HEVs can reduce the fuel consumption and emissions without sacrificing the performance, due to the ability of recuperating the braking energy, stopping the engine at idle, choosing more efficient operating points of the components, and using the grid energy charged in the battery in case of plug-in hybrid electric vehicles (PHEV). Due to the high cost of HEVs and PHEVs, it is important to find the optimal size of the main components to make these vehicles competitive with the conventional vehicles [1, 2, 3, 4, 5, 6, 7, 8]. However, the problem of sizing is complex due to several reasons. First, because the vehicle design is optimized for a given driving cycle, one needs to know the life-time driving of the vehicle, and even the charging pattern in case of PHEVs. Since this is not possible in practice, one solution is to use a long real-life driving cycle, which is representative of the life-time driving. However, extreme driving situations, which are very important to drivers, may not be included in the driving cycle. Therefore, performance requirements should also be added to the problem to guarantee that the designed car is able to deliver power in these situations. Secondly, energy management, which determines the power or energy split between the two power or energy sources, influences the sizing. Ample number of scientific contributions have addressed the problem of optimal energy management and sizing. Energy management can be included into the problem by assuming parametrized rule-based strategies. Pa- rameters of the rule-based control strategy can be fixed as in [9, 10], or can be optimized together with the sizing parameters, using sequential quadratic programming as in [11], or evolutionary programming [12, 13, 14, 15]. Another approach, which relies on optimal control, is to use dynamic programming (DP) to find the optimal energy management [16, 17, 18]. DP can easily be applied to nonlinear, non-convex and mixed-integer problems, however, its computation time increases exponentially with respect to the number of states and sizing variables. Convex optimization can also be used for optimization of hybrid vehicles as in [19, 20, 21, 22, 23]. Compared to dynamic programming, the computation time of convex optimization is not exponentially increasing with respect to the number of states and sizing variables. This enables computa- tionally efficient optimization of problems with more sizing variables or more state variables. Convex optimization may also face computational chal- lenges, when, e.g., powertrain dynamics are modeled with nonlinear relations and long driving cycles are used to resem- ble real-life driving. Moreover, the convex problem of sizing the powertrain may need to be solved iteratively in several loops, in order to study the effect of driving cycles, component models, technologies, energy and components costs on sizing. Hence, it is desirable to reduce the computational burden, by, e.g., simplifying the modeling detail, without loosing accuracy. The aim of this paper is to investigate consequences of model simplifications on the optimal design of a fuel cell plug-in hybrid vehicle, where the energy management and the sizes of the fuel cell system, battery and electric motor are optimization variables. The optimization problem is initially a nonlinear problem, which can be formulated as a convex second order cone program (SOCP) [22]. Then, we investigate simplification steps by abstracting the model to linear relations and thus formulating a linear program (LP) that halves the