*3.2. Load-Leveling*

The final strategy considered is a simple method called "Load-Leveling." In this method, the battery is assigned a maximum allowable current for charging and discharging, *Ib*,*max*, which corresponds to minimum and maximum battery powers *Pb*,*min* and *Pb*,*max*. *Pb*,*min* provides the limit on power going into the battery while charging (negative values of *Pbatt*) and *Pb*,*max* provide the limit on discharging. Any power request from the driver that exceeds the allowable amount is handled by the ultracapacitor.

$$P\_{\rm nc} = \begin{cases} P\_{\rm req} - P\_{\rm b,max} & P\_{\rm req} > P\_{\rm b,max} \\ P\_{\rm req} - P\_{\rm b,min} & P\_{\rm req} < P\_{\rm b,min} \\ P\_{\rm reset} & \text{otherwise} \end{cases} \tag{56}$$

where *Preset* is a small amount of power from the battery used to return the ultracapacitor SOC to a target value of *SOCc*,*tgt* = 60%.

$$P\_{\text{reset}} = \begin{cases} 13 \text{ kW}, & \text{SOC}\_{\text{c}} > \text{SOC}\_{\text{c}, \text{tgt}} \\ -13 \text{ kW}, & \text{SOC}\_{\text{c}} < \text{SOC}\_{\text{c}, \text{tgt}} \\ 0 & \text{otherwise} \end{cases} \tag{57}$$

The 13 kW value corresponds approximately to a 0.1 C battery charge or discharge rate, considered sufficiently low to not majorly affect the battery aging. The particular value of *Ib*,*max* is varied to tune the response of the controller.

This controller serves as a lower bound for EMS performance, as it has neither an aging model nor any form of optimal control.

#### **4. Case Study**

The model and developed controllers are, in this section, used for a case-study analysis of aging-aware energy management: simulation is used to determine how the various strategies perform relative to each other.

Each strategy is simulated on the Manhattan Bus Cycle (MBC) drive cycle [48] for an array of different controller tunings.


The range of weights is determined by looking at orders of magnitude of the element of the cost functions. For instance, for Equation (55), the (*SOCc* − *SOCc*,*tgt*)<sup>2</sup> term has an order of magnitude of, at most, 10<sup>−</sup>2, while the *P*<sup>2</sup> *batt* term can have an order of magnitude of up to 1010. Thus, tuning of *Q*3,*<sup>P</sup>* begins at *Q*3,*<sup>P</sup>* = 10−<sup>12</sup> and is varied from that point.

Additionally, a single baseline case that does not use the ultracapacitor is simulated. This corresponds to *Q*1,Δ*<sup>D</sup>* = 0, *Q*2,*SOC* → ∞, *Q*3,*<sup>P</sup>* = 0, or *Pmax* → ∞.

Simulations begin with both the battery and ultracapacitor at the beginning of their life. After each full discharge cycle, the aging of the battery and ultracapacitor are measured, and the capacity, capacitances, and resistances of the HESS are updated. For the purpose of measuring aging, the battery is assumed to recharge at a rate of 0.5 C. Simulations are then repeated until the battery reaches the end of its life, at which point the cycle life, ultracapacitor state-of-aging, and average energy consumption are measured and recorded.

The above is repeated for three HESS designs: a small ultracapacitor unit *Npc* = 10, a middle-sized ultracapacitor unit *Npc* = 40, and a large ultracapacitor unit *Npc* = 100. Both DDP and SDP simulations are performed for the *Npc* = 100 case, in order to establish that the SDP controllers will closely follow the DDP results. For the *Npc* = 10 and *Npc* = 40, only the causal controllers (SDP-B, SDP-EC, SDP-P, and LL) are simulated.

Energy consumption is measured in equivalent miles per gallon (MPGe), while battery aging is measured in capacity loss per mile. For ease of interpretation, the battery cycle life is converted to an approximate lifespan using an estimate of the average number of miles driven per year, *Davg*-*year* = 34,000 [49].

Finally, the cost-benefit of the HESS is determined. First, the value of the HESS is determined based on the industrial average price per kWh of \$300/kWH for lithium ion batteries and \$15, 000/kWh ultracapacitors from [46] and on the battery and ultracapacitor size given in Section 2.1. This gives a battery value of *Vbatt* = \$38,760 and an ultracapacitor value of *Vuc* = \$13,021. Additionally, an average electrical energy price of *Vnrg* = \$0.1065/kWh for the U.S. is obtained from [47]. Costs and benefits are normalized by mile driven for a fair comparison between configurations. Then, the battery cost per mile (*BCPM*) is determined from the miles driven over the life of the battery, denoted as "battery lifetime miles driven" (BLMD).

$$BCPM = \frac{V\_{batt}}{BLMD} \tag{58}$$

The ultracapacitor cost per mile (*UCCPM*) is similarly determined, this time including a term for the ultracapacitor state of aging at the battery end-of-life (BEOL), *SoABEOL*.

$$LICCPM = SoA\_{BEOL} \times \frac{V\_{\text{nc}}}{BLMD} \tag{59}$$

Finally, the energy costs per mile (*ECPM*) are given as

$$ECPM = \frac{V\_{nr\text{\%}} \times GGE}{MPGe} \tag{60}$$

where *GGE* is the gasoline gallon equivalent to convert from gallons of gasoline to kWh, *GGE* = 33.41 kWh/gal. gasoline.

Then, the cost or benefit of the HESS can be determined by comparing the result to the nominal case where no UC is present. Letting the subscript *nom* denote the nominal case and (*k*) denote any particular simulation, the benefit per mile (*BPM*) is given by

$$\text{BPM}(k) = \left(\text{BCPM}\_{\text{nom}} - \text{BCPM}(k)\right) - \text{LIC} \text{CPM}(k) + \left(\text{ECPM}\_{\text{nom}} - \text{ECPM}(k)\right) \tag{61}$$

where a positive benefit per mile indicates that value is being added to the system, while a negative value indicates that the cost of the UC outweighs the benefit it adds.

Finally, the payback time *Tpayback* (in years) for the HESS can be estimated from the UC value, average miles driven per year, and the benefit per mile.

$$T\_{\text{payback}} = V\_{\text{nc}} \times \frac{1}{BPM(k)} \times \frac{1}{D\_{\text{avg\% year}}} \tag{62}$$

Payback time assumes a positive benefit per mile. If the *BPM* is zero or negative, then a payback time does not exist.

#### **5. Results**

The simulation results are analyzed as follows: first, it is verified that the SDP controllers closely follow the DDP controllers. Next, the impact of aging-aware control on the causal controllers is assessed. Then, the effect of overuse of the ultracapacitor is discussed. Finally, the cost-benefit of the HESS is analyzed and discussed.

#### *5.1. Verification of DP Controllers*

First, the DDP and SDP methods are compared for the *Npc* = 100 case for each of the three DP cost functions, as given in Equations (52)–(55). These simulation results are shown in Figure 6. In the case of the DP methods with an incorporated aging model, the SDP controller closely tracks the global optimal DDP controller. For the cases that use a battery aging model, the lifespan of the SDP-controlled battery is typically within 1% of the DDP result for a given MPGe, while the difference is greater for the controller that only limits battery power, especially near the peak. These results demonstrate that the causal SDP controllers are able to closely match the DDP global optima and indicate that the SDP controllers behave as intended.

**Figure 6.** Energy consumption and battery aging for the DP-based methods for *Npc* = 100.

The DP results are summarized in Table 4. Note that the value for "Mean Life Difference" is the average difference in lifespan of an SDP controller compared to a DDP controller at any given operating MPGe value between the nominal point and the DDP peak. Again, the key result of these data is that the SDP controllers follow the DDP controllers within 1.7%, and within 1.0% for the aging-aware control specifically. Comparisons of the different cost functions are discussed next.


**Table 4.** Comparison of DP Controllers for *Npc* = 100.
