Energy Management Strategy for Hybrid Multimode Powertrains: Influence of Inertial Properties and Road Inclination

Multimode hybrid powertrains have captured the attention of automotive OEMs for their flexible nature and ability to provide better and optimized efficiency levels. However, the presence of multiple actuators, with different efficiency and dynamic characteristics, increases the problem complexity for minimizing the overall power losses in each powertrain operating condition. The paper aims at providing a methodology to select the powertrain mode and set the reference torques and angular speeds for each actuator, based on the power-weighted efficiency concept. The power-weighted efficiency is formulated to normalize the efficiency contribution from each power source and to include the inertial properties of the powertrain components as well as the vehicle motion resistance forces. The approach, valid for a wide category of multimode powertrain architectures, is then applied to the specific case of a two-mode hybrid system where the engagement of one of the two clutches enables an Input Split or Compound Split operative mode. The simulation results obtained with the procedure prove to be promising in avoiding excessive accelerations, drift of powertrain components, and in managing the power flow for uphill and downhill vehicle conditions.


Introduction
Regarding the increasingly serious issues of energy shortage and environmental pollution, attention from governments, research institutions and automobile companies is shifting from traditional fuel vehicles to new innovative solutions [1][2][3]. To reduce air pollution and emissions of greenhouse gases, many governments incentivize the production of zero emissions vehicles (ZEVs), such as battery electric vehicles (BEVs) and fuel cell electric vehicles (FCEVs). The German Parliament has recently decided to ban internal combustion engine vehicles (ICEVs) by 2030 [4]. Many other countries have announced that they will proceed with a similar policy between 2025 and 2050 [5]. However, most of these bans do not include hybrid electric vehicles (HEVs) and plug-in hybrid electric vehicles (PHEVs), which include ICEs as a power source. While the shift from ICEVs to ZEVs has already started, it may take decades for customers to select ZEVs as a cost effective and convenient choice. Moreover, batteries still require decisive improvements to be considered as unique energy sources, vehicle cost should reduce and infrastructure should be globally adapted to the new changes. In the meantime, thermal efficiency of ICEs is expected to further improve and HEVs and PHEVs are expected to play a significant role. Hybrid electric vehicles represent the best tradeoff between traditional fuel vehicles and pure electric vehicles (EVs), thus becoming the perfect transitional stage. The fuel efficiency of HEV has traditionally improved through new developments in aerodynamics, engine technology, light-weight materials, and innovative concepts for powertrain component 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 powerweighted 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 powerweighted 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.

Multimode Hybrid Powertrain Model
The multimode hybrid powertrains considered within this paper are characterized by a number n of PGSs whose components (called nodes), i.e., the carrier, the ring, and the sun, can be linked together through q rigid connections and/or clutches. The number of the degrees of freedom (dofs) of the powertrain depends on the number of clutches engaged to activate a desired powertrain mode and it is equal to 2n − q. Each powertrain node can be driven by an actuator, i.e., the Internal Combustion Engine (ICE) and the electric Motors/Generators (MGs), as long as their number is equal or greater than the powertrain dofs.
The powertrain architecture selected for this paper is characterized by n = 2 PGSs, one rigid connection between their carriers and two clutches, as depicted in Figure 1.
Appl. Sci. 2021, 11, x FOR PEER REVIEW 3 of driving cycle distribution. The results are compared against the DP algorithm applied the same driving cycle, thus proving the effectiveness of the power-weighted efficienc based design which is also over 10,000 times faster than DP. This technique is introduc as potential fast sizing method for multimode powertrains and determines each actuat operating point map to achieve the best power-weighted efficiency; the later is defined a normalized efficiency that considers the high discrepancy between the engine and ele tric motors efficiency. The paper aims at providing a straightforward and clear procedu to model and select the suitable powertrain mode and the working condition for ea actuator, based on the power-weighted efficiency concept which is resumed and adapt to consider undesirable working conditions where part of the input power is used to a celerate or decelerate one or more powertrain components. Aiming at minimizing t powertrain inertial power, a penalty factor is introduced into the definition of the pow weighted efficiency. The methodology is then applied to the two-mode hybrid powertra presented and analyzed [34][35][36], and the influence of mode selection, inertial powertra parameters, and road inclination on the actuator's operating map is carried out thou simulation results. The paper is organized as follows: Section 2 presents the mathematical model us to describe the powertrain dynamics and the equations adopted to calculate the pow weighted efficiency; the novel procedure to introduce the penalty factor and elaborate t working map for each powertrain actuator as well as the simulation results are shown Section 3. Finally, the conclusions are drawn in Section 4.

Multimode Hybrid Powertrain Model
The multimode hybrid powertrains considered within this paper are characteriz by a number of PGSs whose components (called nodes), i.e., the carrier, the ring, a the sun, can be linked together through rigid connections and/or clutches. The numb of the degrees of freedom (dofs) of the powertrain depends on the number of clutch engaged to activate a desired powertrain mode and it is equal to 2 − . Each powertra node can be driven by an actuator, i.e., the Internal Combustion Engine (ICE) and t electric Motors/Generators (MGs), as long as their number is equal or greater than t powertrain dofs.
The powertrain architecture selected for this paper is characterized by = 2 PG one rigid connection between their carriers and two clutches, as depicted in Figure 1. The ICE and the first electric motor MG1 are considered always installed on the ri and the sun of the first PGS, respectively. The second electric motor MG2 and t The ICE and the first electric motor MG1 are considered always installed on the ring and the sun of the first PGS, respectively. The second electric motor MG2 and the transmission output shaft are connected to the sun and the carrier of the last PGS, respectively. A rigid connection constrains the carriers of the two PGS and the presence of two clutches enables the activation of the Input Split (clutch 2 engaged) or the Compound Split (clutch 1 engaged) hybrid modes (see Figure 2). In the Input Split mode, the secondary PGS behaves as an ordinary gear system, while primary PGS operates as a power split device. In the compound split mode, the power split occurs on both PGSs. In both cases, the number of powertrain dofs is always two (n = 2 and q = 2).
The ICE and the first electric motor MG1 are considered always installed on the rin and the sun of the first PGS, respectively. The second electric motor MG2 and the tran mission output shaft are connected to the sun and the carrier of the last PGS, respectivel A rigid connection constrains the carriers of the two PGS and the presence of two clutche enables the activation of the Input Split (clutch 2 engaged) or the Compound Split (clutc 1 engaged) hybrid modes (see Figure 2). In the Input Split mode, the secondary PGS b haves as an ordinary gear system, while primary PGS operates as a power split device. I the compound split mode, the power split occurs on both PGSs. In both cases, the numbe of powertrain dofs is always two ( = 2 and = 2).

Mathematical Model
The mathematical approach adopted to model the two-mode hybrid system is th same described by Tota et al. [36], where an automated routine is described to evalua the dynamic equilibrium equations for a generic multimode powertrain architecture. Th dynamic equilibrium of the two-mode hybrid system shown in Figure 1, with no clutche engaged, is expressed by (see also [36]): includes the angu lar acceleration of the two PGS carriers ( , , , ), suns ( , , , ), and rings ( , , , and the internal forces, and , exchanged between the ring and the pinions gea teeth of the first and the second PGS, respectively; 0 0 0 0 is the generalized torque vector, where represents the equivalent vehicle motion resistance torque (see Equation (9)), is th torque applied by the Internal Combustion Engine (ICE) and and are the to ques applied on the output shaft of the two Motors/Generators (MGs), respectively. Th 8 × 8 matrix includes geometric and inertial parameters of the powertrain compo nents (see also [36]):

Mathematical Model
The mathematical approach adopted to model the two-mode hybrid system is the same described by Tota et al. [36], where an automated routine is described to evaluate the dynamic equilibrium equations for a generic multimode powertrain architecture. The dynamic equilibrium of the two-mode hybrid system shown in Figure 1, with no clutches engaged, is expressed by (see also [36]): where the vector x 0 = . ω c,2 . ω r,1 . ω r,2 ) and the internal forces, F 1 and F 2 , exchanged between the ring and the pinions gears teeth of the first and the second PGS, respectively; u 0 = T load T ICE T MG1 T MG2 0 0 0 0 T is the generalized torque vector, where T load represents the equivalent vehicle motion resistance torque (see Equation (9)), T ICE is the torque applied by the Internal Combustion Engine (ICE) and T MG1 and T MG2 are the torques applied on the output shaft of the two Motors/Generators (MGs), respectively. The 8 × 8 matrix A 0 includes geometric and inertial parameters of the powertrain components (see also [36]): where r r,1 and r r,2 are the ring gears radii and r s,1 and r s,2 the solar gears radii of the two PGSs, respectively; J r,1 , J c,1 , and J s,1 are the torsional moments of inertia of the ring, the carrier and the solar gears of the first PGS, meanwhile J r,2 , J c,2 , and J s,2 are the corresponding torsional moments of inertia of the second PGS; J ICE , J MG1 , and J MG2 represent the torsional moments of inertia of the ICE, the MG1 and MG2, respectively; The presence of rigid connections or clutch engagements between the component i and j of the two PGSs, modifies Equation (2) by summing up the j th row to the i th row and removing the j th row, thus reducing the number of system equation. Due to the presence of a grounding clutch that links the k th component of the two PGSs to the chassis, i.e., clutch 2 in Figure 1, the number of system equations in Equation (2) is further reduced by removing the k th row. For both Input Split and Compound Split modes, the system in Equation (2) is reduced to (see also [36]): ω MG1 , and . ω MG2 are the angular accelerations of the ICE, MG1 and MG2, respectively, meanwhile . ω out is the transmission output angular acceleration. It is important to remark that this mathematical model does not include the transmission nonlinearities such as the gear backlashes, whose presence may affect the powertrain torsional dynamics in specific conditions [37,38].
Finally, by numerically inverting Equation (3) and considering the useful part of vector x, named in the following T , the inverse dynamic equation is obtained (see also [36]): where T = T load T ICE T MG1 T MG2 T represents the torque distribution vector and A * is a 4 × 4 matrix.

Power-Weighted Efficiency
The main task for the design of a hybrid powertrain is to search for the optimal torque distribution that satisfies a desired target. The approach adopted within this paper is to solve an energy loss minimization problem through the criterion of the power-weighted efficiency proposed by Zhang et al. [15]. The power-weighted efficiency η PW is evaluated as (see also [36]): where P out,tr = T out,tr ω out represents the output power of the transmission with a transmission output torque T out,tr = T load + (I out + I c1 + I c2 ) . ω out . In the definition of η PW , two energy sources are considered for satisfying the demanded power at the transmission output shaft: the battery and the fuel tank. The transmission efficiency is considered constant and equal to 1 within the paper, but its influence can be introduced to enhance the model as described by Galvagno [39].
In the generic scheme of Figure 3, P load = T load ω out is the loading power (positive or negative) meanwhile the engine output power P e = T e ω e can be split into three contributions: the engine power P e1 that flows to the battery through the electric generators, the engine power P e2 that flows to the electric motors through the generators and the engine power P e3 that directly flows to the transmission output shaft (P e3 = P e − P e1 − P e2 ). A correct elaboration of engine power flow among P e1 , P e2 , and P e3 requires the calculation of the electric power produced by the generators P G , and the electric power absorbed by the motors P M , as described in Figure 4.
contributions: the engine power 1 that flows to the battery through the electric gen tors, the engine power 2 that flows to the electric motors through the generators the engine power 3 that directly flows to the transmission output shaft ( 3 = 1 − 2 ). A correct elaboration of engine power flow among 1 , 2 , and 3 requ the calculation of the electric power produced by the generators , and the electric po absorbed by the motors , as described in Figure 4.  If > , the additional generated electric power is used to charge the bat Otherwis < , the battery energy is consumed: and represent the absorbed battery and fuel power, res tively. The battery efficiency is evaluated by considering a simple circuit model the model described by Serrao et al. [17]): is the current through the battery circuit: contributions: the engine power 1 that flows to the battery through the electric ge tors, the engine power 2 that flows to the electric motors through the generator the engine power 3 that directly flows to the transmission output shaft ( 3 = 1 − 2 ). A correct elaboration of engine power flow among 1 , 2 , and 3 re the calculation of the electric power produced by the generators , and the electric p absorbed by the motors , as described in Figure 4.  If > , the additional generated electric power is used to charge the ba < , the battery energy is consumed: and represent the absorbed battery and fuel power, re tively. The battery efficiency is evaluated by considering a simple circuit mod the model described by Serrao et al. [17]): is the current through the battery circuit: If P G > P M , the additional generated electric power is used to charge the battery: . P batt and P f uel represent the absorbed battery and fuel power, respectively. The battery efficiency η batt is evaluated by considering a simple circuit model (see the model described by Serrao et al. [17]): where I batt is the current through the battery circuit: The resistance of the battery circuit R batt and the battery open circuit voltage V oc are considered constant quantities with the hypothesis of a constant State of Charge (SOC = 70%) thus mostly relying on the electric energy stored in the battery (charge depleting).
The power-weighted efficiency allows to normalize the efficiencies from different type of power sources, without penalizing the engine operation even if the maximum ICE efficiency η e,max is much lower than η G,max , and η M,max .

Simulation Results
The definition of the power-weighted efficiency-based methodology is then evaluated by considering a set of vehicle and powertrain parameters, as reported in Table 1.  The ICE and the two MGs torque and efficiency characteristics are shown in Figure 5 and obtained from a commercial software usually adopted for modelling and simulating the behavior of hybrid vehicles. The MGs considered within the paper have the typical architecture adopted for hybrid vehicle powertrains, i.e., AC electric motors for the twomode hybrid system from General Motors, whose steady-state power characteristics is a linear function of output shaft angular speed up to a specific speed limit above which the power saturates. The powertrain model presented in this paper does not include the effect of environment/external influences on powertrain components characteristics, but the methodology can be also extended to powertrain characteristics that are variable with external parameters, such as the temperature. An interesting experimental study is reported by Zhang et al. [40], where a complete energy flow test platform is designed and built to understand the influence of the environmental temperature on steady-state and transient plug-in hybrid electric vehicle energy transmissions at different states of charge. The powertrain architecture, shown in Figure 1, allows the activation of two modes: the Input Split mode, where the ring of the second PGS is connected to the chassis through the ground clutch 2, and the Compound Split mode, where the solar of the first PGS is linked to the ring of the second PGS through clutch 1.
To evaluate the efficacy of the methodology based on the power-weighted efficiency, a simulation setup is implemented in Matlab ® environment, by considering the following procedure: • The angular velocities of the remaining actuators, i.e., the two MGs angular speeds ω MG1 and ω MG2 , are calculated from the two PGS kinematic relations: ω s,1 r s,1 + ω r,1 r r,1 = ω c,1 (r s,1 + r r,1 ) ω s,2 r s,2 + ω r,2 r r,2 = ω c,2 (r s,2 + r r,2 ) • Evaluation of the transmission output resistance torque T load : where f 0 is the constant tire rolling resistance coefficient, g is the gravitational acceleration, ρ is the air density, C x is the aerodynamic drag coefficient, S F the vehicle frontal area; It is important to remark that the vehicle braking is only entrusted to the electric generators, thus not considering the presence of a conventional hydraulic braking system (see the results presented by Galvagno et al. [41] for further details about hydraulic braking system modelling).

•
For each set of boundary conditions (α, V, a v ), there exist infinite solutions for the distribution torque vector T, which satisfy Equation (4) (number of actuators larger than the number of dofs). By ranging the two MGs torques, T MG1 and T MG2 , between their minimum and maximum values at ω MG1 and ω MG2 , respectively, the system of equations in Equation (4)  ω ICE through the PGS kinematic relations in Equation (8). Finally, the procedure computes a discrete number of solutions, identified by a combination of ω ICE , T MG1 , and T MG2 , that satisfy the boundary conditions of road slope, vehicle speed, and acceleration. An admissible solution is accepted, to calculate the output power and efficiency of each actuator, only if the following constraints are satisfied: The classic definition of the power-weighted efficiency in Equation (5) is only valid for steady-state working conditions of the powertrain components (negligible influence of inertial parameters) and cannot be directly adopted as discriminant factor to select the optimal combination of ω ICE , T MG1 , and T MG2 , for a given road slope, vehicle speed, and acceleration. Consequently, a modified version of power-weighted efficiency, η p PW , is defined to penalize the admissible solutions with a high inertial power P I of the powertrain components: where the inertial power is defined as: Appl. Sci. 2021, 11, 11752 9 of 17 The inertial power enters as a penalty factor in the definition of the power-weighted efficiency but it physically represents the variation of kinetic energy accumulated or released to equilibrate the power distribution of the transmission in Figure 3: P I = P e2 η G η M + P e3 + µP batt η batt η M − P load . The admissible solution with a combination of ω ICE , T MG1 and T MG2 that guarantees the lowest value of η p PW is then selected as the optimal choice, in terms of actuators power distribution, for that boundary conditions of road slope, vehicle speed, and acceleration. The optimal solution is then represented through 2D maps reporting the value of speed, torque and power for each powertrain component as function of vehicle speed and acceleration, calculated for a fixed road slope, as shown in Figure 6 for the Input Split mode and α = 0. • For 2 dofs powertrain configurations, such as the Input Split and the Compound S modes, the angular speed of one actuator, i.e., the ICE speed , is also var within its admissible range; • The angular velocities of the remaining actuators, i.e., the two MGs angular spe and , are calculated from the two PGS kinematic relations: where is the constant tire rolling resistance coefficient, is the gravitational accele tion, is the air density, is the aerodynamic drag coefficient, the vehicle fron area; It is important to remark that the vehicle braking is only entrusted to the elec generators, thus not considering the presence of a conventional hydraulic braking syst (see the results presented by Galvagno et al. [41] for further details about hydraulic br ing system modelling).

•
For each set of boundary conditions ( , , ), there exist infinite solutions for distribution torque vector , which satisfy Equation (4) (number of actuators lar than the number of dofs). By ranging the two MGs torques, and , betw their minimum and maximum values at and , respectively, the system equations in Equation (4) can be numerically discretized and solved to elaborate resulting ICE torque and angular acceleration . The angular accelerati of the two MGs, and , are then calculated from and throu the PGS kinematic relations in Equation (8). Finally, the procedure computes a d crete number of solutions, identified by a combination of , , and , t satisfy the boundary conditions of road slope, vehicle speed, and acceleration. The inertial power enters as a penalty factor in the definition of the power-weigh efficiency but it physically represents the variation of kinetic energy accumulated or leased to equilibrate the power distribution of the transmission in Figure 3: The admissible solution with a combination , 1 and 2 that guarantees the lowest value of is then selected as the timal choice, in terms of actuators power distribution, for that boundary conditions road slope, vehicle speed, and acceleration. The optimal solution is then represen through 2D maps reporting the value of speed, torque and power for each powertr component as function of vehicle speed and acceleration, calculated for a fixed road slo as shown in Figure 6 for the Input Split mode and = 0. If there is not even one combination of , 1 , and 2 that satisfies the wh set of physical constraints described by Equation (10), then there is no solution for t boundary conditions of road slope, vehicle speed and acceleration and the correspond point on the 2D map is not displayed. If there is not even one combination of ω ICE , T MG1 , and T MG2 that satisfies the whole set of physical constraints described by Equation (10), then there is no solution for that boundary conditions of road slope, vehicle speed and acceleration and the corresponding point on the 2D map is not displayed.

Powertrain Mode Selection
The concept of power-weighted efficiency represents an important discriminant factor to evaluate and select the most suitable powertrain mode and satisfy the minimum energy consumption criterion, for each vehicle operating condition (vehicle speed and acceleration, road slope). In particular, the hybrid system analyzed within this paper, whose architecture is depicted in Figure 1, can work with the Input Split or the Compound Split modes whose resulting maps, for a fixed road slope α = 0, are reported in Figures 6 and 7

Powertrain Mode Selection
The concept of power-weighted efficiency represents an important discriminant tor to evaluate and select the most suitable powertrain mode and satisfy the minim energy consumption criterion, for each vehicle operating condition (vehicle speed and celeration, road slope). In particular, the hybrid system analyzed within this paper, wh architecture is depicted in Figure 1, can work with the Input Split or the Compound S modes whose resulting maps, for a fixed road slope = 0, are reported in Figure 6   The first main difference between the two modes is that the Input Split mode manage higher vehicle accelerations while the Compound Split mode is able to run up vehicle at higher speeds. All maps are obtained by limiting the maximum speed to km/h. The output power requested to the three actuators at the maximum speed with Compound Split mode is lower than the corresponding value for the Input Split mo since the vehicle can reach higher velocities with the Compound Split mode. For h request of output power, in terms of vehicle speed and accelerations, the MG1 tend work mainly as an electric motor during both Input Split and Compound Split mod The MG2 operating range is fully exploited during the Input Split mode, meanwhile output power is nearly for the Compound Split mode. The ICE mainly provides h torque at lower engine speeds, thus working within its most efficient condition. torque and angular speed maps of each actuator are overlapped on their correspond efficiency characteristics, as shown in Figure 8, thus proving that the power-weighted ficiency methodology tends to bind the working conditions within a high efficiency ran The first main difference between the two modes is that the Input Split mode can manage higher vehicle accelerations while the Compound Split mode is able to run up the vehicle at higher speeds. All maps are obtained by limiting the maximum speed to 150 km/h. The output power requested to the three actuators at the maximum speed with the Compound Split mode is lower than the corresponding value for the Input Split mode, since the vehicle can reach higher velocities with the Compound Split mode. For high request of output power, in terms of vehicle speed and accelerations, the MG1 tends to work mainly as an electric motor during both Input Split and Compound Split modes. The MG2 operating range is fully exploited during the Input Split mode, meanwhile its output power is nearly for the Compound Split mode. The ICE mainly provides high torque at lower engine speeds, thus working within its most efficient condition. The torque and angular speed maps of each actuator are overlapped on their corresponding efficiency characteristics, as shown in Figure 8, thus proving that the power-weighted efficiency methodology tends to bind the working conditions within a high efficiency range.
The procedure here proposed relies on the minimum energy consumption criterion, which leads to a charge depleting of the battery. Indeed, a large contribution of battery power, shown in Figure 9 together with ICE fuel consumption map, is requested during high speed acceleration. The procedure here proposed relies on the minimum energy consumption criter which leads to a charge depleting of the battery. Indeed, a large contribution of bat power, shown in Figure 9 together with ICE fuel consumption map, is requested du high speed acceleration. Finally, the selection between Input Split and Compound Split modes is carried by considering the one that guarantees the highest value of power-weighted efficienc the whole range of the vehicle operating condition, as shown in Figure 10.  The procedure here proposed relies on the minimum energy consumption criteri which leads to a charge depleting of the battery. Indeed, a large contribution of batt power, shown in Figure 9 together with ICE fuel consumption map, is requested dur high speed acceleration. Finally, the selection between Input Split and Compound Split modes is carried by considering the one that guarantees the highest value of power-weighted efficiency the whole range of the vehicle operating condition, as shown in Figure 10. Finally, the selection between Input Split and Compound Split modes is carried out by considering the one that guarantees the highest value of power-weighted efficiency in the whole range of the vehicle operating condition, as shown in Figure 10.

Effect of the Inertial Penalty Factor
The main original contribution of this paper is the introduction of the inertial penalty factor P I into the definition of the dynamic power-weighted efficiency to discriminate the most suitable torque distribution. This contribution allows to penalize all the admissible solutions where a considerable amount of input power would be exploited to accelerate one or more powertrain components instead of accelerating the vehicle and/or overcome its motion resistances. To proof the efficacy of this effect, the resulting maps of ICE, MG1 and MG2 angular accelerations are compared against the case where the actuators torque distribution is chosen based on the lower power-weighted efficiency η PW instead of η p PW . The simulation results, obtained for a null road slope, are shown in Figure 11. Appl. Sci. 2021, 11, x FOR PEER REVIEW 12 o Figure 10. Resulting powertrain mode map for the two-mode hybrid system with road slope = 0.

Effect of the Inertial Penalty Factor
The main original contribution of this paper is the introduction of the inertial pena factor into the definition of the dynamic power-weighted efficiency to discriminate most suitable torque distribution. This contribution allows to penalize all the admissi solutions where a considerable amount of input power would be exploited to acceler one or more powertrain components instead of accelerating the vehicle and/or overco its motion resistances. To proof the efficacy of this effect, the resulting maps of ICE, M and MG2 angular accelerations are compared against the case where the actuators torq distribution is chosen based on the lower power-weighted efficiency instead of The simulation results, obtained for a null road slope, are shown in Figure 11.

Effect of the Inertial Penalty Factor
The main original contribution of this paper is the introduction of the inertial pen factor into the definition of the dynamic power-weighted efficiency to discriminate most suitable torque distribution. This contribution allows to penalize all the admiss solutions where a considerable amount of input power would be exploited to accele one or more powertrain components instead of accelerating the vehicle and/or overc its motion resistances. To proof the efficacy of this effect, the resulting maps of ICE, M and MG2 angular accelerations are compared against the case where the actuators tor distribution is chosen based on the lower power-weighted efficiency instead of The simulation results, obtained for a null road slope, are shown in Figure 11.  The color bar scale has been adapted for each map, in order to center the null angular acceleration within a green zone between the maximum (red zone) and the minimum (blue zone) angular accelerations for each powertrain actuator. Both red and blue zones are considered as undesirable since part of the input power would be wasted to increase the kinetic energy of one of the three actuators inertia. The presence of the penalty factor P I automatically discards the admissible but undesirable solutions by reducing their corresponding η p PW , as it is evident by the larger green zone for the top maps compared to the bottom ones, especially for ICE and MG1. Instead, MG2 angular acceleration is not affected by the introduction of the penalty factor since it is kinematically correlated to the vehicle acceleration a v for the Input Split mode (clutch 2 engaged).
A drawback correlated to the penalty factor is that the resulting map of the powerweighted efficiency η PW assumes a different shape, as shown in Figure 12: when the penalty factor is activated there are some regions where the power-weighted efficiency is smaller. the bottom ones, especially for ICE and MG1. Instead, MG2 angular acceleration is n affected by the introduction of the penalty factor since it is kinematically correlated to vehicle acceleration for the Input Split mode (clutch 2 engaged). A drawback correlated to the penalty factor is that the resulting map of the pow weighted efficiency assumes a different shape, as shown in Figure 12: when the pe alty factor is activated there are some regions where the power-weighted efficiency smaller. The penalty factor discharges some admissible solutions with higher pow weighted efficiency values since it does not fully exploit the kinetic energy stored with the powertrain components. However, this drawback does not affect in a significant w the power-weighted efficiency, as proved by Figure 12. A similar consideration can drawn with the Compound Split mode as reported by Figures 13 and 14. The penalty factor discharges some admissible solutions with higher power-weighted efficiency values since it does not fully exploit the kinetic energy stored within the powertrain components. However, this drawback does not affect in a significant way the power-weighted efficiency, as proved by Figure 12. A similar consideration can be drawn with the Compound Split mode as reported by Figures 13 and 14.

Road Slope Influence
The last analysis presented to validate and emphasize the potential of the pow weighted efficiency toolbox, is represented by the influence of the road slope on the m generation as well as on the powertrain mode selection. The map obtained on a flat ro is compared against the uphill ( = 10%) and downhill ( = −10%) scenarios; the cor spondent power flow distribution among the three actuators is reported in Figure 15.

Road Slope Influence
The last analysis presented to validate and emphasize the potential of the powerweighted efficiency toolbox, is represented by the influence of the road slope on the map generation as well as on the powertrain mode selection. The map obtained on a flat road is compared against the uphill (α = 10%) and downhill (α = −10%) scenarios; the correspondent power flow distribution among the three actuators is reported in Figure 15. Passing from downhill to uphill conditions, all maps tend to shift toward lowe celerations due to the presence of the road inclination. The MG1 mostly works as an tric generator, while MG2 provides extra power during extreme accelerations maneu The presence of a road inclination (positive or negative) also contributes to modify acceleration threshold for switching between Input Split and Compound Split mode shown in Figure 16, with the speed threshold kept fixed at 50 km/h. Passing from downhill to uphill conditions, all maps tend to shift toward lower accelerations due to the presence of the road inclination. The MG1 mostly works as an electric generator, while MG2 provides extra power during extreme accelerations maneuvers. The presence of a road inclination (positive or negative) also contributes to modify the acceleration threshold for switching between Input Split and Compound Split modes, as shown in Figure 16, with the speed threshold kept fixed at 50 km/h.
Passing from downhill to uphill conditions, all maps tend to shift toward lower celerations due to the presence of the road inclination. The MG1 mostly works as an e tric generator, while MG2 provides extra power during extreme accelerations maneuv The presence of a road inclination (positive or negative) also contributes to modify acceleration threshold for switching between Input Split and Compound Split modes shown in Figure 16, with the speed threshold kept fixed at 50 km/h.

Conclusions
The paper aims at providing a straightforward and clear procedure to model and select the suitable powertrain mode and the operating condition for each actuator, based on the power-weighted efficiency concept. The main conclusions can be summarized in the following points: • Hybrid powertrains can be controlled by actuators whose nature is extremely different in terms of efficiency and dynamic characteristics. An energy management strategy relying only on the individual efficiency map of each actuator does not provide a satisfying power distribution, since it limits or completely excludes the actuators with lower maximum efficiency, i.e., the ICE. The power-weighted efficiency provides a unique parameter that normalize the overall efficiency contribution of each actuator based on the power flow requested during a specific vehicle working condition which is not correlated to a specific driving cycle.

•
There are multiple solutions, in terms of actuator torque distribution, for a given set of boundary conditions in terms of road slope, vehicle speed, and acceleration. The power-weighted efficiency could represent a valid parameter to discriminate and select the best working map for each actuator that satisfies the minimum energy loss principle. However, this solution would not prevent from undesirable working conditions where part of the power is addressed to accelerate or decelerate one or more powertrain components. Aiming at minimizing the variation of the powertrain kinetic energy, a penalty factor is introduced into the definition of the power-weighted efficiency that always guarantees high efficiency, though slightly lower than the maximum possible value, with a more targeted power flow towards the vehicle and/or the battery. • This methodology also considers the effect of the road inclination, which modifies the admissible vehicle acceleration range. This approach is able to regulate the operative condition of each actuator and to switch the two electric machine modes between generator and motor, thus taking advantage or compensating the road inclination influence.
This paper represents an initial step towards the validation of the effectiveness of the presented methodology which would require a further investigation in terms of comparison against other conventional strategies as well as a fair experimental campaign to assess the efficiency and dynamic performance of the new formulation, based on the power-weighted efficiency.