## 1 Introduction
[16] prove that three or more PGS do not lead to a significant fuel economy improvement with respect to the two PGSs configuration, thus justifying the low number of hybrid powertrains with three PGSs introduced in the automotive market. The second point is the most crucial, since the strategy to control the energy flow among these multiple sources, generally named \"Energy Management Systems\" (EMS), should try to minimize fuel consumption without compromising the smoothness and responsiveness of the vehicle, usually referred to as drivability, as shown by [PERSON] et al. [13]. According to the literature review, the most accredited EMS can be classified into two categories [17; 18]: rule-based and optimization-based strategies. The rule-based strategies decide the power flow among the powertrain actuators based on a set of predefined rules, usually obtained from experience and/or empirical models [19; 20]. Optimization-based algorithms elaborate the best power distribution that minimizes a desired cost function or performance index; they are generally more reliable with respect to rule-based strategies. The optimization process can be solved in real time, such as the Equivalent Consumption Minimization Strategy (ECMS) [21; 22], the Model Predictive Control (MPC) [23; 24; 25] and the intelligent programming [26], or offline by producing maps or algorithms that require a lower computational effort, such as the Pontryagin's Minimum Principle (PMP) [27; 28], the Dynamic Programming (DP) [29; 30; 31], and Genetic Algorithm (GA) [32]. An alternative method, which produces results similar to DP but with lower computational burden, is represented by the so-called Power-weighted Efficiency concept as defined for the first time by [PERSON] et al. [33]. For a given driving cycle, [PERSON] et al. consider all the possible vehicle speeds and load torques combinations into an operative map. Each powertrain actuators' torque and speed is then looped through the map, and the best power-weighted efficiency for hybrid, electric vehicle, and regenerative braking modes is calculated. The powertrain actuators working conditions with the lowest predicted fuel consumption are then selected, according to the battery size and the driving cycle distribution. The results are compared against the DP algorithm applied to the same driving cycle, thus proving the effectiveness of the power-weighted efficiency-based design which is also over 10,000 times faster than DP. This technique is introduced as potential fast sizing method for multimode powertrains and determines each actuator operating point map to achieve the best power-weighted efficiency; the later is defined as a normalized efficiency that considers the high discrepancy between the engine and electric motors efficiency. The paper aims at providing a straightforward and clear procedure to model and select the suitable powertrain mode and the working condition for each actuator, based on the power-weighted efficiency concept which is resumed and adapted to consider undesirable working conditions where part of the input power is used to accelerate or decelerate one or more powertrain components. Aiming at minimizing the powertrain inertial power, a penalty factor is introduced into the definition of the power-weighted efficiency. The methodology is then applied to the two-mode hybrid powertrain presented and analyzed [34; 35; 36], and the influence of mode selection, inertial powertrain parameters, and road inclination on the actuator's operating map is carried out though simulation results. The paper is organized as follows: Section 2 presents the mathematical model used to describe the powertrain dynamics and the equations adopted to calculate the power-weighted efficiency; the novel procedure to introduce the penalty factor and elaborate the working map for each powertrain actuator as well as the simulation results are shown in Section 3. Finally, the conclusions are drawn in Section 4.