1 Filtering driving cycles for assessment of electrified vehicles Nikolce Murgovski, Xiaosong Hu, Lars Johannesson and Bo Egardt Abstract—We present a method for pre-filtering driving cycles that are to be used for assessment of electrified vehicles. The method ensures that the vehicle may exactly follow the filtered velocity demanded by the driving cycle. Employing convex optimization, the method also allows optimal velocity shaping that minimizes the amount of wasted energy. We illustrate the method by an example of performance assessment of a hybrid electric bus in a series powertrain topology. Index Terms—driving cycle filtering, hybrid electric vehicle, power management, convex optimization, optimal control I. I NTRODUCTION Electrified vehicles are being of major interest in academia and industry due to their potential for improved powertrain efficiency and zero or low level of emissions, compared to conventional vehicles. The improved powertrain efficiency is mainly a result of an additional energy source, e.g. electric bat- tery or supercapacitor, and an electric machine (EM) that may propel the vehicle alongside the internal combustion engine (ICE) (or completely replace the ICE, as in electric vehicles). The cost-effectiveness of the vehicle then strongly depends on the choice and size of powertrain components (electric buffer, ICE, EM), and the control strategy that decides the magnitude of power and energy delivered by these components. Due to many competing powertrain solutions, the cost- effectiveness of the electrified vehicle is typically investigated before the manufacturing phase. Then, the potential of the vehicle is determined by simulating a vehicle model on certified driving cycles, or a set of driving cycles that mimic the typical daily usage of the vehicle (described by e.g. speed and road gradient as a function of time). The theoretically optimal performance is sought considering perfect knowledge of the driving cycle. This generally involves some type of optimization, which is typically not a trivial task. In terms of computational effort, especially challenging is the performance assessment of hybrid electric vehicles (HEVs), which possess both ICE and EM. To lower the computational burden, most of the fast vehicle dynamics are neglected and a backwards simulation model is used. In backwards simulation it is as- sumed that the vehicle exactly follows the demanded velocity, thus removing the velocity state from the problem. This leaves only one state in the problem, the state of energy (SOE) of the electric buffer, allowing the optimal solution to be pursued by dynamic programming, [6, 8, 14], Pontryagin’s maximum principle, [3, 5], or convex optimization, [9–11, 15]. The authors are with the Department of Signals and Systems, Chalmers University of Technology, 41296 Gothenburg, Sweden. E-mail: nikolce.murgovski@chalmers.se, xiaosong@chalmers.se, larsjo@chalmers.se, bo.egardt@chalmers.se The backwards simulation model, however, introduces also some difficulties. One limitation of this model is that it prohibits the usage of some driving cycles (with e.g. high acceleration demands), as these cycles may render the opti- mization problem infeasible. An example are artificial cycles constructed by speed limits changing in a staircase manner. To mitigate this problem, the driving cycles are pre-filtered before using them in optimization. This can be achieved by first measuring the actual speed of another vehicle following a reference velocity, and then optimizing the studied electrified vehicle over the filtered velocity profile. The other vehicle is typically an existing conventional vehicle, or a model of it, for which it is straight forward to obtain the optimal control strategy, by e.g. static optimization. This, however, does not guarantee feasibility, as the electrified vehicle may include downsized powertrain components, thus not being able to deliver the same performance as the conventional vehicle. Moreover, even if the problem is feasible, it may not be optimal to drive the electrified vehicle in the same way as the conventional vehicle. For example, an electrified vehicle may recuperate more braking energy with lower deceleration, and it is therefore beneficial to start braking sooner before reaching the stop. Obviously, the optimal solution can be obtained only when both the velocity shaping and control strategy are optimized simultaneously. The contribution of this paper is a method for potential assessment of electrified vehicles using convex optimization [1]. The method is based on a forward simulation model involving two states, the vehicle velocity and SOE of the electric buffer. We present the method through an example of optimal control of a hybrid electric bus in a series powertrain topology, [5], but the final goal is to extend the method to other types of electrified vehicles and powertrain topologies. The paper is organized as follows: the vehicle model and problem formulation are described in Section II and III; in Section IV the problem is rewritten from sampling in time to sampling in distance; convex remodeling is described in Section V; an example of optimally controlling a city bus is given in Section VI; and the paper is ended with discussions and conclusions in Section VII. II. VEHICLE MODEL The studied vehicle is an HEV in a series topology [5], as illustrated in Fig. 1. This powertrain does not have a direct mechanical link between the ICE and the wheels, but instead, the wheels are driven by an EM that receives energy from a battery or an engine-generator unit (EGU).