## 1 Introduction
Figure 2: Classification of operation strategies for hybrid electric vehicles. Heuristic operating strategies are implemented as simple rules, maps [12; 13; 14], state machines, or fuzzy controllers [15; 16; 17], where the parametrization is frequently carried out based on the experience of the powertrain's developers and adjusted by means of empirical studies and test driving. It often aspires to reduce fuel consumption by shifting all operating points of the vehicle to an efficient operating area of the ICE. Since the operation of the electric motor is not taken into account optimally, optimal operation in terms of minimal fuel consumption is not possible. A further approach for parametrization is to incorporate the results from an optimization-based method, which is performed using representative driving cycles [18; 19; 20] (see dotted arrow in Figure 2). Usually, heuristic methods provide only suboptimal results, since the parametrization is based on predefined assumptions without considering real driving behavior. Due to this disadvantage, it is generally not possible to obtain the minimal possible fuel consumption. However, the main advantage is the simplicity of the method's implementation, which makes it suitable for real-time application on a vehicle's electronic control units. Moreover, no information on future driving behavior must be known in advance in order to operate the vehicle. Optimization-based methods provide optimal results in terms of fuel economy. Here, fuel consumption is defined to be a cost function, which is minimized for a given driving cycle by using a mathematical optimization. In general, these methods require a powertrain model in order to describe the fuel consumption, depending on the vehicle's operating point. For this purpose, map-based models are frequently used, where the fuel consumption of the ICE and the energy efficiency of the electric drive are considered by means of corresponding maps. When using map-based powertrain models, only numerical optimization methods can be applied. Typical numerical methods for optimization-based operating strategies proposed in the literature are Dynamic Programming (DP) [21; 22], Stochastic Dynamic Programming (SDP) [23], Pontryagin's Maximum Principle (PMP) [24; 25], Particle Swarm Optimization (PSO) [26], and Sequential Quadratic Programming (SQP) [27]. Another approach is to use an analytic powertrain model, where the maps are approximated by convex functions [28]. These functions are often simple polynomials and enable solving the optimization problem of the operating strategy analytically by means of PMP [29; 30; 31]. However, the application is restricted to continuous control variables (e.g., the torque distribution between ICE and the electric drive). In [32], the analytical optimization with PMP was combined with DP in order to consider the gear-shifting command as a discrete control variable. For the same powertrain configuration and control problem, in [33], a combination of DP and an optimization based on interior point methods using the SeDuMi tool was applied on a convex powertrain model. The advantage of an operating strategy based on mathematical optimization is that it provides minimal fuel consumption. However, a driving cycle must be known in advance, and some of the methods require high computational effort. Therefore, these methods are inapplicable for real vehicle operation but are appropriate for powertrain analyses and optimizations. For real driving operation, real-time capable operating strategies are required. An overview of various methods can be found in [34; 35]. Optimization-based methods are preferred in general, since they consider information about the future driving behavior and consequently provide better results than heuristic methods. Since the exact driving behavior is unknown, and due to unforeseeable driving styles and traffic situations, predictions are used (e.g., based on telemetry data).