*3.1. Optimization Algorithm for Sizing Components*

We used Autonomie to simulate the vehicle performance of a fuel cell-powered truck. Autonomie is a vehicle system simulation tool for the energy consumption and performance [18]. We optimized its onboard hydrogen storage and battery pack size to minimize the ownership costs. This ensures that all performance requirements are met within a 2% tolerance. Figure 1 shows a sizing process. Input variables to be optimized are hydrogen tank, battery capacity, and fuel cell power. When the three input variables change, the vehicle model developed performs the three-vehicle performance test while checking their constraints. Through the optimization process, the input parameters are modified using previous results. Then the feedback process runs until the algorithm finds the optimal value for the objective. It will find the tradeoff relationship between hydrogen storage and battery power. We propose optimization problem as follows:

$$\min\_{r\_i} \{ rco(r\_i) : l\_i \le r\_i \le u\_{i\prime} c\_j(r\_i) \le 0, \ i, j = 1, 2, 3 \},\tag{1}$$

where *rco* is the RCO. It is minimized over *ri* whose range is from lower limit *li* to upper limit *ui*. The value of *cj* represents the vehicle performance constraints. The subscript *i* represents different components and the *j* represents additional performance constraints.

**Figure 1.** Sizing process with vehicle simulation and POUNDERS (practical optimization using no derivatives for sums of squares).

We used the optimization algorithm for components as the POUNDERS. The POUNDERS is a derivative-free optimization to seek local solutions to a potentially multimodal problem, which is a bound-constrained augmented Lagrangian problem [10]:

$$\min\_{r\_i} \left\{ h(r, s) = r \circ \alpha(r) - \sum\_{j \in \mathcal{I}} \lambda\_j \left( c\_j(r) + s\_j \right) + \frac{\mu}{2} \sum\_{j \in \mathcal{I}} \left( c\_j(r) + s\_j \right)^2 : s \ge 0; \, 0 \le r \le u \right\}, \tag{2}$$

where *h* is a cost function value, *s* represents slack variables, *λ<sup>j</sup>* is an estimate of Lagrangian multipliers, and *μ* is the penalty parameter for the constraints. If the constraints are not satisfied, the slack variables increase the cost function value, *h*.

The optimization process is conducted based on Autonomie including the POUNDERS, which developed using MATLAB/Simulink. We update the component cost estimate depending on the component sizes, and define the cost function, performance tests and constraints, as explained further in more detail. The cost is calculated by running the performance tests for each vehicle according to the powertrain components. Because the vehicle weight is different and the component performances change themselves, different powertrain components can affect the vehicle performances through the optimization process. Although the algorithm is one objective optimization, it is difficult to optimize powertrain components because each constraint of the vehicle performance has its own objective. The process optimizes the component sizes to minimize the cost, while ensuring the performance. In this study, the process iterates more 31 times, sufficient to find an optimal value.

## *3.2. Relevant Cost of Ownership*

RCO is the net present cost to own and operate the vehicle. It includes the investment cost with the purchase price and any fees, taxes, and incentive or disincentives. It also includes all operating costs, maintenance costs, and a resale or residual value. The following equation is a way to calculate the RCO. For more detail, see reference [16]:

$$
\tau \alpha = \alpha \text{ost}\_{\text{inv}} + \text{cost}\_{\text{pv\\_energy}} + \text{cost}\_{\text{pv\\_main}} + \text{cost}\_{\text{pv\\_what\\_replace}} - \text{cost}\_{\text{residual}}.\tag{3}
$$

where *costinv* is total investment cost for purchase, initial registration, home electric vehicle service equipment, and vehicle incentive. The *costpv*\_*energy* value is the present value energy cost while considering vehicle fuel efficiency and the current costs of fossil fuel and hydrogen. The *costpv*\_*maint* is the present value of maintenance costs, repairs, and so on. The *costpv*\_*batt*\_*replace* is the present cost of battery replacement. The *costresidual* is the residual value of function with initial cost, the vehicle's annual vehicle miles travelled (VMT), and a discount rate. Purchase price and fuel (or energy) costs are the primary variables for RCO. All other factors are either constants or a function of the purchase price.

We assume that the depreciation is 5%, and vehicle life is 15 years. We assume that the ownership period is 5 years but the actual value will depend on class and vocation. The Federal Highway Administration states that the average delivery truck travels an average of 21,108 km annually [19]. We assume that VMT is 22,531 km for class 4. However, class 8 drives more, so we assume 160,934 km per year.

To calculate the energy cost, we assume that the fuel price is \$3 per gallon and the hydrogen price is \$4 or \$12 per gasoline gallon equivalent (gge). We estimate the manufacturing cost based on component costs, which is based on 2017 FCTO/VTO Benefit Analysis Assumptions [20]. The purchase price is set at 1.5 times the cost of manufacturing. Battery cost is estimated using energy and power, which is \$243 per kilowatt-hour (kWh) and \$20 per watt (W). The fuel cell system is calculated using simplified cost calculations based on the peak efficiency of the stack and weight ratio for the tank. The values are based on the assumptions used for the Fiscal Year 2016 fuel cell technology analysis [21]. The cost of the fuel cell tank is estimated to be \$595 per kilogram of usable hydrogen at a 4.4% storage weight ratio. We estimate that the cost of the fuel cell system is \$50.69 per kilowatt (kW) at 59.5% efficiency.

Component size also affects vehicle weight, which affects specific power. We assume that the specific power of the fuel cell system is 659 W/kg [22,23]. Motor specific power is 1.9 kW per kilogram (kW/kg), based on U.S. Department of Energy (DOE) estimates [24].
