Optimal Control of Electriﬁed Powertrains in Ofﬂine and Online Application Concerning Dimensioning of Li-Ion Batteries

: Various energy management systems (driving strategies) have been developed to improve the efﬁciency of electriﬁed vehicle drives. These include strategies from the ﬁeld of ofﬂine optimization to determine the theoretical optimum for a given system, as well as online strategies designed for an on-board application in the vehicle. In this paper, investigations are performed on an SUV electriﬁed by a 48 V hybrid system in P14 topology regarding both ofﬂine and online strategies. To calculate the global optimum, the performance of Dynamic Programming (DP) compared to an Equivalent Consumption Minimization Strategy (ECMS) with an iteratively determined equivalence factor is shown. Furthermore, with regard to online energy management strategies (EMS), it is presented how a predictive Online ECMS achieves additional fuel savings compared to a robust, non-predictive implementation. The simulation-based vehicle development allows detailed investigations regarding interactions between battery requirements and EMS. In this context, it is shown how various battery capacities are exploited by the discussed EMS.


Introduction
Increasingly stringent emission limits and the overall rise in environmental awareness have led to the development of a wide range of alternative drive systems. In addition to purely electric vehicles, these also include 48 V hybrid electric vehicles (HEVs), which offer the major advantage of significantly reducing the CO 2 emissions with comparably low system expenditure, especially for inner-city driving.
The hybrid drive is characterized through component dimensioning, topology, and an energy management strategy (EMS) [1]. In the development phase for a 48 V hybrid system, besides the design of the electric motor with the associated power electronics, especially the sizing of the Li-ion battery is challenging. The aim here is to achieve the greatest reachable energy saving potential with the lowest possible battery capacity. On the one hand, this results from the fundamental economic interest in a battery that is as cost-effective as possible, since the battery is one of the main cost drivers of a 48 V system [2,3]. On the other hand, the increasing number of other electric energy consumers, including preheating of the exhaust after treatment [4], an electrical compressor [5], or chassis systems [6], means that less energy is available for the optimal EMS.
The EMS itself must ensure the optimal operation of the system functions in various driving scenarios. EMS have been the subject of intensive research for many years, with a wide variety of approaches being the subject of associated work.
In this article, a comprehensive analysis of both online and offline EMS is performed on a 48 V system using real driving cycles. Therefore, in a first step, two offline EMS are presented to find the theoretical optimum for a given system configuration. In contrast to offline EMS, there exist online strategies for on-board use in the vehicle. Accordingly, a comparison of both a robust non-predictive and a predictive Equivalent Consumption Minimization Strategy (ECMS) is elaborated. During investigations, a special focus is placed on the interactions between the EMS and the reduction of the capacity of a 48 V Li-ion battery.

Related Work
EMS for HEVs have been widely investigated during the last few years. An overview of the most common methods is provided by [1,[7][8][9][10][11][12][13]. There are different approaches to subdividing the strategies. One possible distinction is made between rule-based, optimizationbased, and learning-based strategies, whereby mixed forms also exist. Furthermore, a distinction can be made between offline and online strategies: offline strategies are characterized by the requirement that the entire driving profile has to be known in advance. With the help of the respective optimizer, a specific hybrid architecture is characterized for a specific cycle-for example, regarding potentials of reduction in fuel consumption. In other words, for a given driving cycle, the global optimum is identified, which is calculated for benchmark analysis. In online optimization, however, only restricted a priori knowledge about the future within a driving cycle is necessary. This is why online strategies aim to be utilized on-board the vehicle [14].
The focus of this work is on the comparison of optimization-based concepts, both for the online and offline field. For determination of the global optimum (offline strategies), Dynamic Programming (DP) is considered to be unrivaled, since, apart from the errors resulting from the necessary discretisation of the state and control variable space, it always determines the global optimum. On the other hand, it requires very high computational effort. For the application of DP to HEVs, special reference should be made to [15][16][17] and the corresponding further development [18].
In addition to the DP, Pontryagin's minimum principle (PMP) can be utilized to determine the global optimum. Comparisons between DP and PMP can be found in particular in [19,20]. The ECMS, which was first published by Paganelli et al. [21], can in turn be derived from the PMP.
Depending on the specific implementation, the ECMS is assigned to the offline or online strategies. Due to the equivalence of the ECMS compared to the PMP, it can be used to find the global optimum for time-invariant systems. In this case, the so-called equivalence factor is determined iteratively [22,23]. Especially due to the low computational effort, this so-called Offline ECMS is widely used to determine the global optimum in offline application [24]. For an investigation of topologies with several traction motors, the 2D-ECMS has been developed [25].
Regarding the use of predictive information, it was shown that a non-predictive Online ECMS concept for online application can be improved considering predictive information [44]. Apart from this, there also exist predictive Online ECMS approaches, where predictions are mandatory for the basic functionality of the ECMS [28,45]. Such control systems, in which the optimal control is determined taking into account the corresponding prediction horizon, are generally referred to as model predictive control (MPC). For an MPC, also several alternative optimizers other than ECMS can be applied, as presented in [46][47][48]. Here, for example, also approximated approaches of the DP (ADP) are used, whereby, due to the significantly reduced computational effort, an online capability of the DP can be realized [49,50]. In recent years, however, a variety of other approaches have been developed in which the strategy is based purely on Artificial Intelligence [51][52][53].
It should be clearly noted that there are countless other studies on EMS for HEVs. The authors would like to limit themselves here to the approaches most relevant to this work.
The analysis of the state of the art reveals that a comprehensive investigation of both online and offline EMS focusing on dimensioning of Li-ion batteries in 48 V HEVs using real driving cycles has not been undertaken yet. In particular, the proposed comparison of a robust non-predictive in contrast to an intuitive predictive Online ECMS should be emphasized here, whereby the main idea is based on the papers [35,44]. The basic procedure of the investigations proposed in this paper is shown in Figure 1. The variables will be explained in more detail in the following sections.

Modeling
In the field of vehicle simulation, a distinction is made between forward and backward simulation. Forward simulation models represent the physical causality of the system: with the help of a driver model (e.g., a PID controller), the desired speed is compared with the actual vehicle speed. Based on the acceleration resulting from the control input of the driver model, a speed can then be determined in each time step. In contrast, a backward simulation model assumes that the vehicle follows a predefined speed and acceleration profile. Thus, no driver model is required. The individual advantages and disadvantages can be found in [9]. In this paper, the validated backward calculation model of a 48 V HEV including an Offline ECMS with an iteratively determined λ from the work by [24] is applied. The 48 V battery is represented by a simple inner resistance model. Hereby, Equations (1) and (2) are used both to compute the battery voltage under load U bat and the corresponding battery current I bat . Therefore, the battery power P em , the battery losses P em,loss , and the power from auxiliary consumers P aux are considered. Moreover, the opencircuit voltage U OCV and the inner resistance R i are required. In addition, as a measure of energy deviation from the starting conditions, an energy deviation dE from reference SOC is calculated (Equation (3)). It is used as a criterion for a neutral energy balance.
The battery is of a nickel-mangan-cobalt/graphite cell type, whereby R i and U OCV are calculated using SOC-specific component data. However, to ensure time-invariant properties, a constant absolute state of charge (SOC = 70%) is used for the calculation of battery parameters. Other effects, such as degradation of the battery and its impact on CO 2 emissions, are neglected [24]. The model is coupled with the dpm function provided by ETH Zurich [15][16][17]. The studies are performed on an SUV hybridized in a P14 topology including two electric motors on positions P1 and P4. Figure 2 shows common topologies of HEVs in parallel configuration. The most important parameters for the analyzed HEV are summarized in Table 1.  The investigations are based on 12 real driving cycles from [54], which represent the statistical totality of 1,000,000 km. With regard to the design of an online-capable ECMS, a particularly promising approach from [35] is being further developed. Ideas from [44] are further considered also. The goal of the investigations is to minimize fuel consumption, whereby there is a proportional relationship between fuel consumption and CO 2 emissions. The CO 2 values representing fuel consumption in this work are calculated with the relation 1 l/100 km = 23.2 g CO 2 /km. The main features of the two approaches considered in this work, DP and ECMS, are described below.

Dynamic Programming (DP)
DP is based on Bellman's principle of optimality, which states that the optimal trajectory for a discrete decision problem is also optimal for the corresponding subproblem [55]. Based on this principle, the DP is a numerical method with which the global optimal solution is found by operating backwards in time. For the application of DP, the state of charge x as well as the torque distribution u must be discretized, leading to u k ∈ U k and x k ∈ Ω k ; see Figure 3 (left). When considering u = {u 0 , u 1 , . . . , u N−1 } as the torque split strategy and defining the initial state as x(0) = x 0 , then the cost function J is defined as follows [9,16].
g N (x N ) stands for the final costs. These are zero, in the case that SOC(N) = SOC(0), to guarantee charge-sustaining (CS) operation, and infinite otherwise. The cost function g k is the fuel consumption of the combustion engine. The optimal solution J * is written as: As in offline optimization, the entire driving cycle is known in advance, the algorithm can proceed backward, to find the sequence of controls which generates the optimal control starting from the final step N. For more detailed information, the reader is referred to [9].

Equivalent Consumption Minimization Strategy (ECMS)
For an ECMS, as well as with the DP algorithm, the control input u has to be discretized. A discretization of x, however, is not needed. The optimal solution is found by calculating the equivalent fuel consumption considering the lower heating value of the fuel Q lhv and an equivalence factor λ, which transforms the battery power into fuel power at each time step. Using the resulting fuel equivalent, a cost function J is defined, which is minimized at each time step to obtain the optimal torque split. The corresponding optimization problem P is written as [24]: The determination of λ depends on the individual implementation of the ECMS. For determination of the global optimum, λ has to be chosen constant for a time-invariant system according to the PMP. This constant λ is usually determined iteratively, as visualized in Figure 3 (right). . Principle DP (left) and ECMS with iteratively determined λ 0 (right) regarding SOC, based on [9,56]. For DP, the entire state space must be discretized to calculate the costs of any control and state combination. When applying an ECMS, no state space discretization is needed. The iteratively determined λ 0 finally leads to an optimal control ensuring CS operation.

Global Optimum-Dynamic Programming vs. Offline ECMS Unlimited Battery Capacity
Since the system has time-invariant properties, a constant equivalence factor λ 0 is sufficient according to PMP to determine the theoretical optimum (Equation (8)).
λ 0 is iteratively calculated using the shooting method to ensure CS operation of the vehicle. Hereby, in a first step, an initial value is determined, which is based on engineering experience. After the simulation run, the actual final SOC value is compared with the desired final SOC value. Depending on whether the SOC of the battery is too high or too low, λ 0 is adjusted accordingly and the cycle is run again, until a suitable λ 0 is found [9,56]. Table 2 lists the minimum CO 2 emissions for the 12 real driving cycles for both the DP and the Offline ECMS including λ 0 . λ 0 enables the Offline ECMS to calculate the optimum trajectory for the respective cycle, while at the same time ensuring CS operation.
The presented CO 2 emissions result for the respective fuel-optimal trajectory. The deviations are <1%, which is why the results are considered equivalent at this point. The remain-ing deviations result from numerical errors and the fundamentally different functionality of the two EMS; see Section 3.

Limited Battery Capacity
As mentioned in the Introduction, this work pays special attention to the interactions between the selected EMS and the sizing of the battery. Therefore, in Figure 4, Offline ECMS and DP are compared regarding the calculation of the global optimum for different battery capacities. As can be seen, the Offline ECMS cannot ensure optimal operation of the HEV when reducing the energy content of the battery. As a result of the reduced energy content of the battery, time-variant effects occur due to the frequent reaching of the respective SOC limits. With a battery capacity of 100 Wh, the calculated trajectory of the ECMS can deviate by 30% from the optimum; for 25 Wh, the deviation rises up to 45%. It can be concluded that the optimum battery size should be determined with the DP approach, since an Offline ECMS with iteratively determined λ is less accurate.  When applying DP, as in in Figure 5, the fundamental usefulness of considering batteries with low energy content is proven. It becomes clear that partly similar savings effects (extra fuel consumption < 2%) are achieved for a 25 Wh battery compared to the large battery-for example, for highway aggressive and urban low mild cycles.
As presented in Figure 5, it should also be mentioned that there is a small increase in fuel consumption at 1000 Wh compared to the 100 Wh variant (negative values). This results from the coarser discretization due to the constant amount of interpolation points in combination with the increased battery capacity; see principles of the DP demonstrated in Figure 3. To sum up, for powertrain design, DP is preferable to Offline ECMS, especially in terms of battery sizing. It is shown that even smaller batteries of 25 Wh or 100 Wh show savings potentials similar to those of a 1000 Wh battery for certain driving cycles and should therefore be considered in powertrain design. Since the previous studies focused on purely offline investigations, the following will examine how the different battery sizes behave in an on-board implementation.

Online ECMS (Non-Predictive)
In the following, the parameterization of a robust, non-predictive Online ECMS is presented. This also includes discussions of the chosen battery size and the parametrization of the robust non-predictive Online ECMS.
Unlimited Battery Capacity As shown in Table 2, the suitable equivalence factors for the 12 real driving cycles considered range between λ 0 = 2.92 and λ 0 = 3.21. Based on the determined equivalence factors for the different driving cycles, an average equivalence factor is determined, which can serve as an initial value for a non-predictive Online ECMS. Assuming that real-world operation is represented by equal weighting of all cycles, the arithmetic mean of all cycles is determined for a first calculation of a non-predictive Online ECMS (λ 0,avg = 3.01). Without further measures, this does not ensure CS behavior in online operation. The SOC trajectories lead to excessive charging or discharging of the battery depending on the cycle (see principles of the ECMS demonstrated in Figure 3). The simplest measure here is the introduction of an additional penalty term in the calculation of λ (Equation (8)). Hereby, the value of the energy (=λ) is either increased or decreased depending on the deviation dSOC of the actual SOC from the reference SOC. According to [35,56], the penalty via a trigonometric function is preferable to a proportional one: it permits small deviations from the reference SOC but large deviations are severely penalized. Therefore, the penalty term consists of the penalty factor kp SOC multiplied by the cubic derivation of SOC dSOC 3 (Equation (9)).
As already mentioned, it is assumed that real-world operation is represented by an equal weighting of all given cycles. Furthermore, expert discussions define the requirement for the EMS that the final deviation of the energy content of the battery at the end of cycle is restricted to ±100 Wh regarding CS operation. Based on these assumptions, an appropriate kp SOC for the non-predictive Online ECMS is determined. Table 3 reveals that kp SOC has to be raised to up to 191.92, to keep the energy deviation dE end from the reference SOC within the allowed range of ±100 Wh for the extra-urban average cycle (λ 0 = 3.01). The extra consumption is up to 2.90% higher compared to the global optimum determined from the DP. With the aid of these preliminary investigations using an unlimited battery capacity, basic knowledge regarding a suitable choice of λ 0 and kp SOC was gained. In the following, an optimal battery-specific parametrization of λ 0 and kp SOC should be found. This also takes into account factor interactions of both parameters.
Limited Battery Capacity Figure 6 reveals the CO 2 emissions for a 25 Wh battery for different parametrizations of the Online ECMS over the selected lambda and the selected kp SOC . It is shown that the CO 2 emissions over all cycles depend almost exclusively on the selected λ 0 , which is noticeable by the strip-like structure. The influence of kp SOC , on the other hand, is almost negligible.
In contrast to the results of a 25 Wh battery (Figure 6), for the 100 Wh battery, a high dependence of fuel consumption on both kp SOC and λ 0 can be observed (Figure 7). It can be seen that a higher kp SOC reduces the influence of λ 0 on the achievable low fuel consumption (dark blue region). On the other hand, kp SOC should not be chosen to be too high, since this restricts hybrid operation. Nevertheless, for both analyzed battery capacities, the lowest CO 2 emissions are achieved with an equivalence factor of λ 0 ≈ 3.
Moreover, as seen in Figure 8, for the 1000 Wh battery, the lowest consumptions are also found around λ 0 ≈ 3. Parametrizations in which CS cannot be observed are clearly indicated by corresponding white areas. For these parameterizations, kp SOC is chosen to be too low, so the boundary condition of ±100 Wh final SOC cannot be guaranteed. Consequently, λ 0 must be selected perfectly for each cycle in order to guarantee CS operation.     Next, the battery-specific ideal parametrization of the Online ECMS is selected. This parametrization is characterized by the smallest mean deviation from the global optimum over all 12 cycles. Final parameters are listed in Table 4. Along the lines of the Offline ECMS in Section 4.1, the increase in fuel consumption compared to the global optimum (DP solution) decreases with a larger battery (3.41%->0.88%). The cycle-specific results are presented in Figure 9. For the variant with 1000 Wh and 100 Wh battery, the deviation of the non-predictive Online ECMS from the DP solution is max. 2%. For the 25 Wh battery, the fuel consumption is raised and shows a clear downward trend over the cycles. An analysis of the SOC trajectories clearly indicates that for cycle urban low-mild (+6.6%), the selected λ 0 tends to be lower compared to the iteravely determined one (λ 0,Online = 2.90 vs. λ 0,O f f line = 3.14). As a result, the SOC is permanently at the lower SOC limit, which in turn significantly restricts the hybrid functionalities. For the extra-urban aggressive cycle, on the other hand, the lambda value from the online implementation is closer to that from the Offline ECMS (λ 0,Online = 2.90 vs. λ 0,O f f line = 3.06; see Table 2). The small difference of λ 0 = 3.06 and λ 0 = 3.14 underlines the high sensitivity of the ECMS with regard to the selected λ 0 .
Deviations λ 0,Online to λ 0,O f f line in the highway cycles are even significantly greater with up to λ 0,O f f line = 3.21. However, since the 48 V system can only achieve a very limited reduction in consumption on the motorway due to the high power levels [24], even a not ideally set Online ECMS only leads merely to a slight increase in fuel consumption. To sum up, with regard to online implementation on-board, a non-predictive implementation of an Online ECMS already approaches the global optimum when choosing a large battery within powertrain design. However, if smaller batteries in the order of 25 Wh or 100 Wh are considered, the fuel consumptions deviate significantly from the global optimum. Thus, a high potential for improvement by incorporating predictive information is expected. Therefore, it is demonstrated in the following, by the example of a 25 Wh battery, how an additional reduction in consumption is possible by taking into account predictive information.

Online ECMS (Predictive)
In the context of this work, the objective is to fulfill the premise of a predictive Online ECMS that is understood intuitively. While the optimizer from energy management basically determines the best possible operation from a number of potential operating modes [56], there are a few operating modes that are not further determined by the optimizer and are therefore not part of the optimization problem. One of these is the recuperation mode. As long as the component limits are not exceeded and there is residual capacity in the battery, recuperation is carried out when possible. The exact amount of energy that is recuperated can vary depending on the specific application. Further influences on recuperation are neglected within the presented studies. This also includes the influence of dynamic wheel load changes.
The developed approach of a predictive Online ECMS is based on the following idea: using a simple longitudinal dynamics model, the resulting recuperation potential at the wheel is determined from the predicted torque demand within the prediction horizon. Neglecting the efficiency of the drivetrain, this results in a measure of the guaranteed SOC increase within the prediction horizon. According to this predicted energy increase, the value of the electrical energy (=the value of the equivalence factor λ) is preventively lowered by the introduction of an additional term kp pred · p recu (Equation (10)). p recu represents the recuperation potential in the prediction horizon in Wh; kp pred is a proportionality factor to be parameterized.
As described, a lower λ means a reduction in the value of the electrical energy in the ECMS. In consequence, the battery is rather discharged in advance. Especially in the case of small batteries, the benefit of recuperation potential from the driving cycle is raised by less reaching the SOC limits in recuperation phases. Hereby, consequently, additional fuel saving potentials can be achieved.
In the case of large batteries, the predictive Online ECMS has a similar effect. However, for large batteries, individual recuperation phases rarely lead to reaching the limits. To achieve savings potential, a reduction in the penalty of the SOC deviation kp SOC can be realized without violating the defined boundary conditions for a robust CS operation (|dE end |< 100 Wh). A reduction in the penalty of the SOC deviation kp SOC in turn results in more degrees of freedom in the EMCS. Since the effects of a reduction in battery capacity are the focus in this work, merely an additional term is added to the existing parametrization. Further adjustments of the robust non-predictive implementation from the section above are not carried out.
In the following, the influence of the prediction horizon is presented. Moreover, an additional parameter is introduced, which specifies the minimum amount of energy from which the recuperated energy is considered.
Prediction Horizon t horizon Figure 10 shows the influences of a small (10 s) and a large (50 s) prediction horizon illustrated on two sequences from a real driving cycle. In both cases, the SOC curve and speed were plotted. Additionally, in the case on the left, the wheel torque was plotted. In the case on the right, the torque of the combustion engine is shown.
As presented in Sequence 1 , a consideration of the recuperation potential with a time horizon of 10 s at t = 770 s is limited. The battery is not discharged properly; in consequence, SOC Max is already reached at an early point in time of the recuperation phase (red line). In order to take into account a higher amount of recuperation potential from deceleration, a larger prediction horizon of 50 s should be considered (blue dotted line in SOC). On the other hand, regarding Sequence 2 , a prediction horizon of 50 s can also lead to recuperation potentials being taken into account too early in the predictive Online ECMS. At t = 280 s, for example, due to an insufficient state of charge resulting from the future recuperation potential, the ICE is activated. In consequence, this leads to an increase in fuel consumption.
The wheel torque is not shown in Sequence 2. The decrease in speed at t = 290 s can be seen on the lines of a negative wheel torque. It can be concluded that the selected prediction horizon majorly influences the EMS.
Threshold E min In the following, the introduction of a threshold is discussed. Therefore, in Figure 11, λ and SOC are plotted over time. On the left, only λ is plotted for both a predictive Online ECMS and a non-predictive Online ECMS without a threshold. On the right, the same plot is presented with a threshold. Additionally, for the case with the threshold, the SOC is introduced on the right.
The consideration of the recuperation potential by introducing the term kp pred · p recu (Equation (10)) leads to a continuous reduction in λ compared to the non-predictive Online ECMS without further measures in Figure 11 (left). This is problematic, as explained in the following: according to [49,56], in the case of time-invariant properties of the battery, a constant λ leads to a global optimal control, when it is determined as described in Section 3.2. The proposed non-predictive Online ECMS includes an average value for λ 0 over the 12 real driving cycles. The resulting λ is further adjusted via kp SOC in the nonpredictive implementation to ensure CS operation in the Online ECMS. Although global optimal operation cannot be reached using this non-predictive Online ECMS, the best possible alignment to the λ 0 values that lead to an optimal solution for the individual cycles is achieved. In the case of a continuous reduction, λ deviates significantly from the iteratively determined values. Due to the resulting always lower λ (=low value of the electrical energy), the battery is always discharged sooner, which leads to an overall significantly lower SOC trajectory. This results in an increase in fuel consumption due to the restricted hybrid functionality. To avoid such continuous reduction in λ, a threshold is introduced. A minimum amount of recuperable energy E min is necessary for an intervention in the lambda calculation (see Figure 11, middle plot). An intervention to λ should be only allowed if as much recuperation potential is expected that the additional deviation from the averaged λ 0 value is overcompensated by the fuel savings potential resulting from kp pred · p recu . Overall, the resulting SOC trajectories remain quite similar; see Figure 11, right. However, the influence of the predictive Online ECMS is clearly visible in the SOC plot at t = 1600 s (see Figure 11, zoom view right plot). Finally, in the context of the procedure for the non-predictive Online ECMS, parameter studies are performed. The boundaries of the three parameters to be examined are listed in Table 5. From these parameter studies, the ideal parameterization for the predictive Online ECMS is specified. Results show a significant reduction potential compared to the nonpredictive Online ECMS; see Table 6. A robust parametrization results in up to 1.65% saving potentials in the urban-low aggressive cycle. In individual cycles, a slight increase in fuel consumption is observed for the robust parametrization. An average saving potential of 0.4% can be expected when all saving potentials are equally weighted. In the case of cycle-specific parametrization, the saving potentials rise up to 2.32% compared to the purely non-predictive Online ECMS. On average, a savings potential of 0.7% can be expected here. An individual parametrization of the ECMS must not be completely ruled out, but further algorithms are necessary in order to be able to adapt the parametrization of the predictive Online ECMS to on-board use.
A closer look at the columns on the right in Table 6 (-> individual parametrization) indicates that λ 0 = 2.90 of the Online ECMS for cycles urban low mild, urban high average, urban high mild tends to be lower compared to the iteratively determined values ( Table 2). Since the battery SOC hardly reaches the upper SOC limit, the predictive Online ECMS in the form described offers only very limited saving potentials, since only a reduction in λ is possible. In other cycles, significantly greater savings effects are observed, as the non-predictive Online ECMS often reaches SOC Max . The analyses reveal that cycles must be checked individually to understand the resulting fuel saving potentials and take appropriate measures. It must be considered when recuperation potentials occur while the SOC trajectory is at SOC min and how to handle this. Many recuperation potentials can only be taken into account to a limited extent. In other cycles, the non-existent savings potentials simply result from relatively low recuperation potentials in the cycle overall. Moreover, in the context of the non-predictive variant, in the case of high power requirements, there is a limited influence of the hybrid functionality through the 48 V system overall (e.g., on highway cycles). In this case, a predictive control will not lead to any measurable additional fuel savings.
Consequently, in a follow-up work, a dependency of kp pred on SOC is introduced. Thus, if the battery state of charge at time t k is already at SOC min , no additional reduction in the value of the electrical energy (λ) is allowed. Furthermore, a dependence of kp pred on the occurrence of the recuperation potential in the predicted horizon can be implemented. If the recuperation occurs early in the time horizon, a large influence is aimed at; if it occurs late in the horizon, a small influence should be realized. Moreover, the predictive Online ECMS algorithm is to be further developed by taking stationary phases or pure e-drive phases into account. Here, the demand for the ICE is smaller than the minimum threshold for activation of the ICE, which is why it is expected that the auxiliary consumers or traction are supplied purely from the electrical energy storage.
In this section, additional findings are proposed with regard to an implementation on-board. It is proven that in the case of choosing a small battery in powertrain design, a predictive Online ECMS can achieve significant reduction potentials in fuel consumption compared to a robust, non-predictive implementation. The suggested predictive Online ECMS, which takes into consideration the future recuperation potential, offers savings potentials of up to 2.32% when compared to the robust, non-predictive Online ECMS with a 25 Wh battery.

Conclusions
A research study into both online and offline energy management strategies (EMS) concentrating on the dimensioning of Li-ion batteries in 48 V HEVs by using real driving cycles is presented in this publication. This includes, in particular, a comparison of a robust nonpredictive Online ECMS to an intuitive predictive Online ECMS. Therefore, investigations were performed on an SUV electrified by a 48 V hybrid system in P14 topology.
It is demonstrated how two widespread approaches from the field of offline strategies perform for determining the global optimum as a function of battery size (capacity). It can be seen that with the 100 Wh battery, there is up to +30% deviation from the global optimum when applying an Offline ECMS compared to DP. For the battery of 25 Wh, the deviation increases up to +50%. Thus, DP is identified as a suitable optimizer for the proposed investigations. Using DP, it is presented that even a smaller battery of 25 or 100 Wh shows similar savings potentials in certain cycles as when using the 1000 Wh battery. This motivates deeper investigations regarding reduced battery capacity.
Furthermore, in the field of online strategies , it is worked out how a predictive Online ECMS can achieve additional savings compared to a purely robust, non-predictive implementation. Therefore, it is demonstrated in a first step that the proposed non-predictive Online ECMS leads merely to an additional consumption of ≈1%, compared to the global optimum in the case of a 1000 Wh battery. With a 25 Wh battery, the additional consumption increases to ≈5%. It is concluded that with a large battery, the non-predictive variant of an Online ECMS already closely approaches the global optimum. In other words, the potential for improvement through predictive implementation is very limited for the 1000 Wh battery. For a 25 Wh battery, however, it is revealed that the proposed predictive Online ECMS, in which the future recuperation potential is taken into account, shows savings potentials of up to 2.32% compared to the robust, non-predictive Online ECMS variant.
The predictive Online ECMS algorithm will be improved in the future by taking into consideration stand phases or pure e-drive phases. It is also possible to make kp pred SOC-dependent or dependent on the exact appearance of the recuperation potential in the predicted horizon.