## 3 Problem Description and Cost Model
### Savings Algorithm for Generating Initial Solutions
We specifically investigate the impact of considering complete and partial charging procedures on solutions. ### Experimental Design

Our computational experiments are performed on 10 real-world instances that are inspired by real-world public transport data. The instances are characterized by different numbers of stop points and service trips as well as different distributions of service trips over a day. To simplify the analysis, the instances' labels reflect the numbers of service trips and stop points. The instances' distributions of cumulative service trips over the day are presented by Figure 2. The figure shows that the instances differ substantially with regard to the distributions. It is worth mentioning that the last five instances consist of subsets of the service trips taken from instance t3067_s209 for runtime reasons. In the case of instances t1580_s209 and t1487_s209 the original set of service trips was halved randomly, and in the case of instances t1060_s209, t1074_s209, and t933_s209 the set was divided into three parts also in a random way. Within our experiments, we presume a single vehicle depot, a single vehicle type, and a single charging system. Accordingly, each timetabled service trip can be executed by every available BEB. Additionally, each BEB can charge its battery at every charging station. With regard to the practical implementations of BEBs, we assume that three buses at most can be charged at a charging station at the same time. This is because building sites for charging systems are usually restricted, especially in urban areas. In our study, we distinguish between complete and partial charging procedures. In order to incorporate battery aging, we presume that a battery's SoC ranges between 20% and 80% of a battery's capacity as indicated by [PERSON] et al. [35]. In our experiments, we first presume that a vehicle is always charged up to a SoC of 80%. After that, we consider partial chargings. In that regard, the threshold until a battery is charged may vary depending on the idle times at charging stations. Irrespective of the threshold until a battery is charged during its rotation, we assume that a vehicle always begins its rotation with a fully charged battery. Following [PERSON] and [PERSON] [36], charging modern batteries is a nonlinear and therefore complex procedure. The current during a charging process is of particular importance. As demonstrated by [PERSON] and [PERSON] [34], the current decreases quickly

Figure 2: Profiles of cumulative service trips. when a battery is charged to over 80% of its capacity. Below this threshold, the current is almost constant. For that reason, we assume a constant current and thus linear charging times for vehicle batteries within this paper. We assume that 5 kWh can be transferred into a vehicle battery per minute. In our study we consider chargings before the start or after the end of service trips. To reflect the lower consumption of BEBs on deadhead trips we therefore assume a consumption of 1.5 kWh per kilometer and of 1.8 kWh per kilometer driving on service trips. These parameters are inspired by the data of the previously introduced project at the Schiphol Airport in Amsterdam. At present, there is a wide range of battery capacities offered on the market that range between approximately 60 and 300 kWh. Based on this, we consider different battery capacities of 60, 120, 300 and 500 kWh within our experiments. A battery capacity of 500 kWh may be considered as a future development of battery research. Since we consider only one vehicle type at the same time, we conduct our study for each capacity. Based on [PERSON] and [PERSON] [36], a BEB in use and equipped with a 60-kWh battery causes fixed costs of about 350,000 monetary units. Measured by the battery sizes this results to fixed costs for the other vehicles of 365,000, 405,000, and 450,000 monetary units. With regard to the operational costs, we presume 0.5 units per driven kilometer and 50 units per hour of operation.