*5.2. Effect of Aging-Aware Control*

Next, all four causal strategies (SDP-B, SDP-EC, SDP-P, and Load Leveling) are compared. The *Npc* = 10 case is shown in Figure 7, the *Npc* = 40 case is shown in Figure 8, and the *Npc* = 100 case is shown in Figure 9.

First, it can be seen that a larger HESS allows for greater improvements to battery lifespan. This is expected—the *Npc* = 10 UC can only reduce current to or from the battery by approximately 0.5 C, while the largest power request from the driver corresponds to 2.5 C. On the other hand, the *Npc* = 100 case can handle much larger power requests and can do much more to limit large battery current. However, these additional improvements come with a monetary cost, which is discussed more at the end of this section.

It is found across all three HESS sizing cases that the SDP-EC does the most to improve battery lifespan, offering a peak lifespan of 4.69 years at 10.15 MPGe, 5.16 years at 9.81 MPGe, and 5.72 years at 9.72 MPGe for the *Npc* = 10, *Npc* = 40, and *Npc* = 100 cases, respectively.

**Figure 7.** Comparison of energy consumption and battery aging for the four causal control methods, *Npc* = 10.

**Figure 8.** Comparison of energy consumption and battery aging for the four causal control methods, *Npc* = 40.

The SDP-B offers substantial lifespan improvements as well, however, not to the degree of SDP-EC. SDP-B exceeds the performance of the the two non-SDP strategies but does not improve lifespan as well as the SDP strategies that include direct aging control. Although SDP-P does substantially increase battery lifespan, it does not "understand" the aging mechanics—such as the different effect of charging and discharging currents, or how damage from large currents is multiplied at high *DoD*—resulting in smaller lifespan increases than the strategies that control aging directly. The performances of SDP-EC and SDP-B relative to SDP-P clearly indicate the power of aging-aware energy management.

Not only does SDP-EC offer the best increase in battery lifespan, it offers the best improvements to the overall energy consumption/battery aging trade-off. That is, for all three HESS sizes and for any given rate of energy consumption, the SDP-EC strategy offers the largest improvements to battery lifespan; further increases to lifespan incur the smallest increases to energy consumption. Not only is the peak lifespan improvements for the SDP-EC method higher than SDP-B and SDP-P, it reaches that peak at a lower MPGe than the peaks of the SDP-B and SDP-P curves. The performance of SDP-EC compared to SDP-B indicates the value of controlling ultracapacitor aging and energy losses in addition to battery aging.

**Figure 9.** Comparison of energy consumption and battery aging for the four causal control methods, *Npc* = 100.

The baseline case, Load Leveling, does not perform well: in the *Npc* = 10 and *Npc* = 40 cases, it offers virtually no lifespan improvement at all. With the large HESS, although it is able to match the SDP performance at low levels of UC usage, it quickly reaches its peak before dropping off. In this case, Load Leveling offers a peak lifespan of only 4.58 years.

The average change in ultracapacitor state of aging, measured at the end of battery life, is plotted in Figure 10 for *Npc* = 40 as a representative case. Nominal aging—the aging of an ultracapacitor that is stored at the target SOC and at the temperature given in Section 2.3, and is otherwise unused—is found to be Δ*SoA* per year = 4.064%. At low degrees of UC usage, all three SDP methods are shown to have UC aging near the nominal. However, as UC usage increases, the SDP-B and SDP-P methods are seen to have the UC aging rate grow—SDP-B, in fact, reaches a peak UC aging rate of 4.415% at 9.19 MPGe. On the other hand, SDP-EC is shown to have measurably less UC aging at high levels of UC usage. This is expected, as SDP-EC seeks to limit UC aging while SDP-B does not.

The *Npc* = 10 and *Npc* = 100 cases are not presented here; however, similar trends in UC aging per controller type are observed.

Taken together, these results indicate a clear benefit to using strategies with predictive power, and that predictive power combined with energy storage aging models incorporated into the control strategy offers the best way to increase battery lifespan. The results of the four causal strategies are summarized in Table 5 for the *Npc* = 10 case, Table 6 for the *Npc* = 40 case, and Table 7 for the *Npc* = 100 case.

**Figure 10.** Comparison of energy consumption and UC aging for the four causal control methods, *Npc* = 40.

**Table 5.** Comparison of Causal Controllers, small HESS (*Npc* = 10).


**Table 6.** Comparison of Causal Controllers, mid-sized HESS (*Npc* = 40).


**Table 7.** Comparison of Causal Controllers, large HESS (*Npc* = 100).


#### *5.3. Ultracapacitor Overuse*

In Section 2.4, it was established that there is necessarily a trade-off between battery aging and energy consumption when using a HESS to limit battery aging. A consequence of this is seen in every simulated controller, shown in Figures 6–9, at the tail end of each curve: as the ultracapacitor is used more and more extensively, energy consumption increases as more energy is lost from the ultracapacitor internal resistance. These losses must be made up for by discharging the battery more deeply—depth of discharge being a key aging factor, per the model presented in Section 2.2. At some point, increases to the *DoD* aging factor outweigh the impact of decreases in the other aging factors, and battery lifespan eventually begins to decrease rather than increase. Thus, in cases where the ultracapacitor is used very heavily, attempts to control battery aging can have the opposite of the intended effect.

The *Npc* = 100 SDP-P results for energy losses are shown in Figure 11 as a representative example. Energy losses increase with increasing ultracapacitor usage, eventually leading to a decreased lifespan. This behavior emphasized the importance of tuning the energy management strategy properly. With poor tuning, it is possible for the HESS to do more harm than good. Similar trends are seen with the other control methods and other HESS sizes, but are not shown here.

**Figure 11.** Comparison of energy losses (left axis) to battery lifespan (right axis) versus the *Q*3,*<sup>P</sup>* weighting parameter, where increasing *Q*3,*<sup>P</sup>* increases ultracapacitor usage.

#### *5.4. Cost-Benefit Analysis*

The cost–benefit of each simulated point is computed per Equations (58)–(61) and plotted versus MPGe in Figure 12 for the *Npc* = 10 case, in Figure 13 for the *Npc* = 40 case, and in Figure 14 for the *Npc* = 100 case. Positive values indicate that value is added to the system, while negative values indicate a cost.

In general, it is observed that the SDP-EC method offers clear value over the other methods: for any HESS sizing and for any given MPGe, the SDP-EC method offers the highest benefit per mile. In the *Npc* = 10 case, it was shown to be the only method that offered a positive return on investment. In the *Npc* = 40 case, the SDP-P and SDP-B methods did offer a positive return: the SDP-EC's maximum benefit per mile was over 50% greater than SDP-B and over 120% greater than SDP-P. The *Npc* = 100 case is similar to the *Npc* = 10 case in terms of relative performance: only the SDP-EC offers notable benefit. Although the SDP-B does offer a small positive return for some tunings, the maximum benefit of the SDP-B method is less than a quarter of the maximum benefit of the SDP-EC method.

**Figure 12.** Cost–benefit analysis for the four causal control methods, *Npc* = 10.

**Figure 13.** Cost–benefit analysis for the four causal control methods, *Npc* = 40.

Another takeaway from Figure 12 is that the peak economic benefit occurs at a higher MPGe (lower ultracapacitor utilization) than the peak lifespan increase. This makes intuitive sense: lifespan improvements level off near the peak while energy consumption continues to grow. Therefore, near the lifespan peak, the marginal improvement batterycost-per-mile is less than the marginal decrease in fuel economy. For the *Npc* = 100 SDP-EC method, there is not much difference between the peak lifespan increase (occurring at 9.71 MPGe) and the peak benefit (occurring at 9.80 MPGe); however, for a smaller HESS or for weaker strategies, the difference can be substantial: in the *Npc* = 100 case, the SDP-B peak lifespan increase is at 9.51 MPGe, which is effectively break-even in terms of value, while the peak benefit occurs at 9.75 MPGe. SDP-P has its peak benefit at 10.06 MPGe and peak lifespan at 9.32 MPGe; looking at the the SDP-EC method in the *Npc* = 10 case, the benefit at the peak lifespan increase is approximately half of the maximum possible benefit. Clearly, the economic factors should be considered when deciding on the controller tuning.

**Figure 14.** Cost-benefit analysis for the four causal control methods, *Npc* = 100.

Finally, the estimated payback time is computed for the maximum benefit of the SDP-EC method for all three cases using Equation (62). It is found that, for the given UC, battery, and energy costs, the small HESS (*Npc* = 10) has a payback time of 11.3 years, the mid-sized HESS (*Npc* = 10) has a payback time of 15.6 years, and the large HESS (*Npc* = 100) has a payback time of 21.6 years.

In order to observe the full trend of the payback period for different ultracapacitor sizes, additional simulations are run for *Npc* equal to 2, 5, and all increments of 10 between 10 and 100. The optimal EMS and optimal *Q*2,*SOC* are recomputed and the vehicle is simulated again for the new EMS and new UC size. The payback time and battery lifespan at the most cost-effective tuning for each *Npc* are then plotted in Figure 15. This shows that, although increasing the HESS does does improve the battery lifespan, the cost of the extra UCs exceeds the savings of that extra lifespan.

**Figure 15.** Estimated payback time for optimal SDP-EC controller with varying HESS size.

Finally, the authors note the sensitivity of the benefit per mile and the payback time period to assumptions about component pricing, energy pricing, and aging mechanisms. For instance: this research assumes that energy is priced at the US average across all sectors, *Vnrg* = \$0.1067/kWh [47]. If, instead, energy was priced at the transportation sector average for the state of Illinois (such as for a Chicago Transit Authority bus), the energy price of *Vnrg* = \$0.0632/kWh, also from [47], would reduce the payback time by approximately 30%. On the other hand, the California price *Vnrg* = \$0.1280/kWh would increase the payback time by 30%.

Alternatively, we can consider that battery and UC components use the pricing of reference [50] rather than [46] while maintaining *Vnrg* = \$0.1067/kWh; the increased battery and ultracapacitor prices from [50] result in a payback time of 6.9 years for the *Npc* = 10 UC and 13.7 years for the *Npc* = 100 UC. On the other hand, the component prices of [51] would indicate that the HESS is not beneficial under any circumstance.

A different battery aging model in the literature [52], used in an array of battery and HESS control literature such as [12,15,19,23,53,54], models lithium ion phosphate batteries as aging at up to 3× the rate of the model used in this research. If the battery ages even 1.5× the modeled rate, then we would see a payback period of 4.8 and 8.8 years for *Npc* = 10 and *Npc* = 100, respectively.

Finally, although battery end-of-life can be considered a hard limit for battery use based on range constraints, the ultracapacitor, on the other hand, can continue to be used beyond 80% capacitance fade. This would not be unreasonable, considering how Figure 15 shows that the effectiveness of the proposed control method is maintained as the number of cells (and over UC pack capacitance) is decreased. Therefore, if, for instance, the UCs were used until 70% capacitance fade, then we would see a payback period of 7 and 12 years for *Npc* = 10 and *Npc* = 100, respectively. One could, alternatively, assume that the UC does not need to be replaced at all (setting *UCCPM* to 0), as UC life exceeds the 12 year lifespan of an individual transit bus [55]. However, there is still value to considering these replacement costs from the perspective of an entire vehicle fleet.

All this is to say: an engineer must take caution that a HESS is economically appropriate for a given application; there may be circumstances where a HESS is highly beneficial, and others where it may be impractical. With that said, this research has demonstrated that, for any HESS sizing and for any given MPGe, the SDP-EC method offers a larger increase to battery lifespan and a higher benefit per mile than the other considered methods. The important takeaway of this analysis is how proper control of the HESS is critical for maximizing both battery lifespan and HESS value, and that joint control of battery aging, UC aging, and energy losses is the most effective method to manage the HESS.

#### **6. Conclusions**

This paper develops controllable battery and ultracapacitor aging models for a HESS. Various energy management strategies are developed for the purpose of minimizing battery aging. As a case study, these models and control strategies are applied to a simulated electric bus to determine the battery lifespan and energy consumption of each strategy. An array of different HESS sizes and controller tunings are simulated in order to determine the trade-off between battery aging and energy consumption for each strategy. Additionally, the cost–benefit of the HESS is analyzed to determine the relative economic benefit of the proposed control strategies.

Simulation results showed that the SDP-EC method, which controls a weighted combination of battery aging, ultracapacitor aging, and energy losses, offers the biggest improvement to the aging–energy consumption trade-off across all considered HESS sizes. At its peak, this strategy offered a 28.2% increase in battery lifespan and required only a 7.0% decrease in MPGe. The SDP-B method, which controls battery aging but neither ultracapacitor aging nor energy losses, was the next most effective controller, indicating the importance of including an aging model directly in the control.

Simulation results also demonstrated that excessive use of the ultracapacitor can, in fact, be detrimental to the lifespan of the battery. Ultracapacitor use incurs additional energy losses and, if the ultracapacitor is heavily used, then these losses can result in additional battery aging. Furthermore, the cost–benefit analysis showed that only the strategies that included direct aging control would reliably add value to the system; the SDP-EC method was the most proven manner of adding economic value to the HESS. These points, taken together, indicate the importance of control strategy selection and design.

Future work for this research includes the optimization of component sizing, given the proposed new methods of energy management. Additionally, work is ongoing to investigate the robustness of the control strategies for uncertainty in the battery and ultracapacitor models. Finally, other energy management strategies should be considered and compared to the methods here, such as DDP formed into a rule base or the Equivalent Consumption Minimization Strategy applied to aging control.

**Author Contributions:** K.M. and F.A. conceived and designed the experiments; K.M. developed the bus model performed the numerical experiments; K.M. and F.A. analyzed the data; K.M. wrote the paper. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Acknowledgments:** This work was supported by the University of California, Davis, Department of Mechanical and Aerospace Engineering.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

The following abbreviations are used in this manuscript:


#### **References**

