*4.2. Determining the Number of Buses Needed to Drive the Trips*

The number of buses needed is calculated in two steps. First, the number of buses required to drive the trips are calculated. This will be determined by the highest number of buses in traffic during the peak periods, and it will be equal to the number of conventional buses required. After that, there is a calculation of how many extra buses are needed in order to have time to charge electric buses. This number can be zero or larger, depending on the timetable and charging strategy. The number of extra buses is calculated in the next section.

We start by looking in detail at each bus needed to drive the trips from one of the two end stops, and later we derive the formulas needed to calculate the number of buses from the detailed analysis. The use of the buses is illustrated in Figure 4. There, we can see that bus 1 starts the first trip at time *T*start according to the timetable, and before that, it has used some time driving from the depot to the start of the route, illustrated by the light blue bar. bus 1 drive the first trip during the time shown by the green bar, and there is a need for layover time at the end of it. One headway time after bus 1, bus 2 starts the second trip, followed by bus 3 and 4 after each additional headway time. Thus, the number of buses initially increases by one bus for each headway time that passes. The increase in number of buses stops after the gross trip time, because at this point, the buses which have been driving the route in the other direction have arrived and had their layover time, and they are ready to drive the next trip as a return trip. Therefore, after the gross trip time, the number of buses in traffic does not need to be increased as long as the headway is constant. Figure 4 shows that the buses alternate driving the route in both directions, as indicated by the green and blue bars.

**Figure 4.** Example of buses needed to drive the trips during early morning traffic.

The headway is reduced during the morning rush hours after some time in the early morning. There are no longer enough of buses returning from earlier trips to start all the trips. If the headway during rush hour is half of the headway during the early morning, the number of buses will need to increase, as shown in Figure 5, in which the morning rush hours starts at 06:00 and ends at 09:00. Note that these times show when the headway changes for the departures from the end stop. Further down the line, the reduced headway will occur later, as it takes some time for the buses to drive from the end stop. Just like at the start of the traffic in early morning, there will be a need for more buses at

the beginning of the rush hours. In this example, every second bus starting a trip from the end stop must be an additional bus coming from the depot. As before, the number of buses increases, now by one every second headway time. This continues for a time equal to the gross trip time when enough buses arrive from the other direction of the route.

**Figure 5.** Example of buses needed to drive the trips during early morning traffic and the morning rush hours.

At the end of the rush hour, when the headway is increased, not all buses arriving from the other direction are needed, so after the end of the morning peak, some of the buses are taken out of traffic and return to the depot.

Based on the previous analysis, we can determine the number of buses needed in traffic during the whole day. Note that we previously showed which bus is driving which trip, so that each row in the diagram is the schedule for one particular bus during the day. In the following analysis, we will derive a diagram that looks very similar, but it only shows how many buses are occupied by different activities during the day, without showing which bus is doing what. This way, we can simplify the analysis a lot, and do not need to plan the schedules of the buses. On the other hand, this analysis cannot capture all the small details involved in planning bus schedules, and some of the details in the scheduling are instead included as factors to take into account that it is not possible to plan bus schedules completely without slack for the bus and drivers.

The number of buses required for the traffic will vary during the day, as shown in the diagram in Figure 6, and it is derived from the timetable and data regarding driving time and layover time for the route. We will later use this diagram to determine how much time is available for charging during different parts of the day. Right now, we only need to know the number of buses required to drive during the off-peak period, during the peak times and in the evening. Note that despite being similar to the timetable diagram in Figure 2, this shows the total number of busses in traffic, while the timetable diagram shows the frequency of departures. How many buses are needed will not only depend on the timetable but also on the time it takes to drive the route. A short route of course needs fewer buses to follow a certain timetable than a longer bus route with the same timetable.

The number of buses required for driving all trips during peak traffic:

$$N\_{\text{busPeak}} = 2 \times T\_{\text{TripGrossPeak}} \times \eta\_{\text{DepPerrfloorPeak}} \tag{1}$$

where the gross trip time in the peak is:

$$T\_{\text{TripGrosakPeak}} = T\_{\text{TripNetPeak}} + T\_{\text{LaysoverPeak}} \tag{2}$$

**Figure 6.** Number of buses in traffic during the day. Derived from the timetable and route parameters.

As stated earlier we do not round this off to the nearest higher integer, but instead analyse the TCO based on a non-integer number of buses. This way of calculating the number of buses assumes that the gross trip time is shorter than the peak periods. That is the case for most routes in cities, at least in Sweden, since the peak period in the morning and afternoon are typically 2 h and 3 h or more, respectively, while very few routes have more than a 2-h trip time. The number of buses in traffic during the midday off-peak period can be calculated in the same way:

$$N\_{\text{busOfPeak}} = 2 \times T\_{\text{TripCorosOfPeak}} \times \text{rt}\_{\text{DepPerflowOfPeak}}\tag{3}$$

where the gross trip time off-peak is:

$$T\_{\text{TripCorossOfPeak}} = T\_{\text{TripNetOfPeak}} + T\_{\text{LayowerOfPeak}} \tag{4}$$

Finally, the number of buses in traffic during the evening is:

$$N\_{\text{busEvering}} = 2 \times T\_{\text{TripCoros} \text{Evering}} \times \eta\_{\text{DepPerflowEvering}} \tag{5}$$

where the evening gross trip time is:

$$T\_{\text{TripGrossEvering}} = T\_{\text{TripNetEvering}} + T\_{\text{LaycoverEvering}} \tag{6}$$
