Table 3 Constraints

x_i,t≤atDepot_i,t ∀i,t

Charge only if vehicle at depot

y_i,t≤atDepot_i,t ∀i,t

Discharge to grid only if vehicle at depot (V2G)

x_i,t+y_i,t≤1 ∀i,t

Discharge to grid only if vehicle not charging (V2G)

soc_i,t=arrival=SOC_Arr,i ∀i

SOC on arrival constrained to be equal to the calculated parameter, known from calculation based on the known schedule

soc_{i,t=departure}≥SOC_Dep,i ∀i

The SOC of a bus at departure must be equal to a predefined level SOC_Dep,i, set to 100%

soc_i,t=soc_i,t−1+ΔSOC_i×(x_i,t−1−y_i,t−1) ∀i,t

SOCBalance: The SOC at time t for bus i is equal to the SOC in the previous time slot plus/minus the amount charged/discharged in the current time slot.

z_i,t≤x_i,t ∀i,t

Constraints employed for ‘No-Preemption Condition’

z_i,t≤x_i,t−1 ∀i,t

z_i,t≥x_i,t−1+x_i,t−1 ∀i,t

By default, x_i,t variables are independent of each other with respect to the time slot. This allows the bus recharging process to be interrupted and restarted later (preemption).

\(\sum _{t}{x_{i,t}} - \sum _{t}{z_{i,t}} \leq 1 \ \forall i,t\)

If there is a requirement to prevent the interruption of the charging phase, the following constraints are introduced. The first 3 equations derive from a common trick used in optimization to assign the value 1 to a variable (z_i,t in this case) if and only if both other variables are equal to 1 (in this case the charging command in the time slot t and (t−1)). The last equation constrains the number of occasions in which the vehicle is being charged at t but not in (t−1) to be 1. Effectively, this leads to the optimization charging a bus in a single charging process across multiple time slots until the battery is fully charged.