Table 8 Abbreviations and notations

BEB

Battery electric bus

SP

Sub-problem

WTT

Well-to-tank

LB

Lower bound

GHG

Greenhouse gas

UB

Upper bound

ToU

Time-of-Use

IMP

Inner master problem

SoC

State of charge

ISP

Inner sub-problem

MILP

Mixed integer linear programming

ILB

Inner lower bound

C&CG

Column and constrained generation

IUB

Inner upper bound

NC&CG

Nested column and constrained generation

GTFS

General Transit Feed Specification

MP

Master problem

VKT

Vehicle kilometers traveled

Sets

Description

Index

Description

I

Set of potential charging station locations

i, i′

Potential charging station location index

B

Set of buses

b

bus index

J_b

Set of segments accomplished by bus b

j

Segment index

A^st

Set of power levels of the station charger unit

t

Timeslot index

A^batt

Set of battery capacity levels

q

Failure scenario index

R_b,j,i

Set of timeslots of the recovering time of bus b after accomplished segment j at the location i

r

Inner-level C&CG scenario index

Ξ

Set of the uncertainty of charging station failure

ω

Demand measurement interval index

Ω

Set of daily demand measurement intervals

T

Set of the timeslots within a workday

T_ω

Set of timeslots in demand interval ω

Parameters

Description

Parameters

Description

ρ^st

Construction cost of a charging station ($ per location)

Q^max

Maximum battery size (kWh)

ρ^ch

Charger cost ($ per kW)

λ₁

Upper charging limit related to the battery capacity

ρ^po

Charger pole cost ($ per unit)

\({P}_{i,t}^{\max,{st}}\)

The station’s upper limit of power in timeslot t according to ToU (kW)

ρ^batt

Battery cost ($ per kWh)

ϑ^min

Minimum SoC limit (%)

ρ^bus

Bus cost without battery ($ per bus)

ϑ^max

Maximum SoC limit (%)

δ

Number of workdays (#)

e^base

The constant energy consumption rate (kWh per km)

T_s

Timeslot duration (hour)

e^batt

The energy consumption rate factor relevant to the BEB battery size

\({\rho }_{t}^{{elect}}\)

Electricity rate according to ToU in timeslot t ($ per kWh)

\({P}_{{po}}^{\max }\)

The charger pole’s maximum power (kW)

\({\rho }_{t}^{{em}}\)

WTT GHG emissions rate in the timeslot t ($ per kWh)

d_b,j

Segment j∈ J_b length of bus b (km)

τ

Annualization factor (#)

\({N}_{i}^{\max }\)

Maximum number of charger poles in charging station i (#)

ρ^DC

Electricity daily demand charges ($ per kW)

k

Maximum number of failed stations (#)

η^ch

Charger efficiency (%)

ρ^pen

Penalty cost for the failed bus ($)

\({P}_{t}^{\max,{net}}\)

The network upper limit of power in timeslot t according to ToU (kW)