Table 5 Nomenclature for sets, parameters, and variables

Sets

N

Set of nodes that describes the trips

D

Set of nodes that describes start and end depots

C

Set of nodes that describes charging actions at a terminal station

W

Set of nodes that describes dead-heading and charging at the depot

B

Set of bus types

V

Set of all nodes

\(\mathcal {P}\)

Electric bus indices

\(\mathcal {T}\)

Set of time steps

Parameters

\(T_{i,j}\)

Sum of the planned service time of trip i and relocation time from destination node of i to start node of j

\(t^s_i\)

Planned start time of trip i

\(E_{i,j}\)

Planned energy consumption of serving trip i and relocation to the starting node of trip j.

\(\Delta e_{i}\)

Stochastic deviation of energy consumption related to trip i.

\(\psi ^i\)

Energy required to recharge bus i

u

Ramp rate

Variables

\(x^b_{i,j}\)

Bus of type b reaches node j after serving node i.

\(\Gamma _{i,j}\)

Sum of the actual service time of trip i and relocation time from destination node of i to start node of j

\(\gamma _{i}\)

Actual start time of trip i

\(soc_i\)

State of charge at node i

\(\Delta soc_i\)

State of charge change corresponding to the charging event at node i

\(c_{bat}\)

Battery size

\(\zeta _{i}\)

Charging time at node i

\(p_i^c\), \(p_i^d\)

Charging and depot charging power at node i

\(\kappa\)

Total trip delay time

\(\lambda\), \(\lambda _e\)

Total number of buses and electric buses.

\(\chi _i\)

Decision variable corresponding to delayed trips.

\(p^i_t\)

Overnight charging power of bus i at time step t

\(y_t\)

Total overnight charging power at time step t

\(\Phi\)

Maximum power demand