Component sizing of a plug-in hybrid electric powertrain via convex optimization Nikolce Murgovski 1,a , Lars Johannesson a,b , Jonas Sj ¨ oberg a , Bo Egardt a a Department of Signals and Systems, Charmers University of Technology, 41296 Gothenburg, Sweden b Viktoria Institute, 41756 Gothenburg, Sweden Abstract This paper presents a novel convex modeling approach which allows for a simultaneous optimization of battery size and energy management of a plug-in hybrid powertrain by solving a semidefinite convex problem. The studied powertrain belongs to a city bus which is driven along a perfectly known bus line with fixed charging infrastructure. The purpose of the paper is to present the convexifying methodology and validate the necessary approximations by comparing with results obtained by Dynamic program- ming when using the original nonlinear, non-convex, mixed-integer models. The comparison clearly shows the importance of the gear and engine on/off decisions, and it also shows that the convex optimization and Dynamic Programming point toward similar battery size and operating cost when the same gear and engine on/off heuristics are used. The main conclusion in the paper is that due to the low computation time, the convex modeling approach enables optimization of problems with two or more state variables, e.g. allowing for thermal models of the components; or to include more sizing variables, e.g. sizing of the engine and the electric machine simultaneously. Keywords: plug-in hybrid electric vehicle, slide-in electric vehicle, battery sizing, power management, convex optimization 1. Introduction Hybrid Electric Vehicles (HEVs) are widely regarded as one of the most promising means for achieving a near-term reduc- tion of emissions and energy consumption from the transporta- tion sector. HEV powertrains include an internal combustion engine (ICE), one or more electric machines (EMs), and an en- ergy buffer, typically a battery and/or a super capacitor, which depending on their configuration are commonly divided in three different topologies: series, parallel and series-parallel. The powertrain topologies mainly differ in the available degree of freedom in choosing the ICE operating point, but their capabil- ity to improve energy consumption can be generally described by 1) the possibility to recover braking energy by using the EMs as generators and storing the energy in the buffer, 2) ability to shut down the ICE during idling and low load demands and 3) the possibility to run the ICE at more efficient load conditions while storing the excess energy in the buffer. For a detailed overview on hybrid vehicles, see e.g. [1]. The so-called plug-in HEVs (PHEVs) are equipped with a charging connector (typically an on-board charger), which al- lows the PHEVs to charge the electric buffer from the grid. PHEVs are designed to be charged either by a standard house- hold electric power infrastructure or at stations installed on, e.g. parking lots, shopping malls, or other locations. Email addresses: nikolce.murgovski@chalmers.se (Nikolce Murgovski), larsjo@chalmers.se, lars.johannesson@viktoria.se (Lars Johannesson), jonas.sjoberg@chalmers.se (Jonas Sj ¨ oberg), bo.egardt@chalmers.se (Bo Egardt) 1 Corresponding author, Tel: +46 31 7724800, Fax: +46 31 7721748 In recent years, PHEVs have been considered for use in pub- lic transportation by equipping high traffic bus lines with a charging infrastructure [2, 3], offering a flexible crossbreed be- tween an HEV city bus and a trolley-bus. In [2] the PHEV city bus is, while driving, inductively charged from underground ca- bles that have been buried along sections of the bus line. In [3] the PHEV is equipped with a super capacitor which is charged at bus stops through a docking station. Since the PHEVs are to be charged at relatively high power, the energy buffer makes it possible to drive a significant part of the bus line on elec- tric power even though the charging infrastructure might be sparsely distributed. The cost optimal sizing of the energy buffer, i.e. determining power rating and energy capacity of these PHEVs, is heavily dependent on the charging infrastructure, the drive pattern and the topography along the bus line. The solution to this opti- mization problem, however, depends not only on the city bus system configuration and the cost of the on-board electric com- ponents, but also on changing factors such as fuel and electricity prices. Moreover, a complicating issue when evaluating HEV city buses is that the energy efficiency of the powertrain depends on how well adapted the energy management strategy is to the bus line [4]. For PHEV city buses the energy management strat- egy decides the operating point of the ICE and thereby when and at which rate the energy buffer is to be discharged. When optimizing the PHEV public transportation system based on a dynamic model of the powertrain, a badly tuned energy man- agement may lead to a non-optimal sizing of the energy buffer [5]. There are two main approaches to the problem of optimal siz-