This study considers passenger travel costs, operation costs, and train capacity equity in VC. There are constraints on available carriages, load rates, and other factors to determine the optimal combination of train departure frequency, train composition, and turn-back stations.
3.1.1. Passenger Travel Costs
To accurately calculate passenger travel costs, we divide passengers in
Table 3. Class 1 represents passengers who can reach their destination directly without transfer, while Class 2 represents passengers who need to transfer to reach their destination. Additionally, passengers are further subdivided based on their origin-destination (OD) pairs, accounting for varying train frequencies in different zones. For instance, in Class 1, passengers traveling online 1 with their OD not entirely within zone II are categorized as Case 1a. Consequently, we further classify Class 1 into five cases, denoted as
P1 = {1
a, 1
b, 1
c, 1
d, 1
e}, as illustrated in
Figure 3. Similarly, Class 2 is subdivided into three cases, denoted as
P2 = {2
a, 2
b, 2
c}, as illustrated in
Figure 4.
Generally, passenger travel costs mainly include the resistance time for type-
p passengers
, in-vehicle time for type-
p passengers
, and additional VC time for type-
p passengers
. Among them,
includes the waiting time and walking transfer time; the waiting time is set as half of the train departure interval as the passenger arrival is uniformly distributed, i.e., 30/
. Notably, when trains from different lines enter collinear zones, VC is performed, wherein two trains operate in the form of one train. Consequently, for passengers who can travel on both local and cross-line trains, the passenger waiting time is calculated using the greater of the two trains’ departure frequencies.
is mainly influenced by the origin and destination stations of passengers. In addition, in CO, the uncertainty of the train’s arrival time may lead to additional VC time for the trains of different lines during the VC process. Thus,
represents additional VC time for type-
p passengers. The critical point will be the first station where the VC occurs (including cross-line turn-back station 2,
a, 1,
b; transfer station 1
p, which is represented by
a,
b,
tra). Additional VC time applies to passengers who either originate from or pass critical points. Additional VC time is proportional to the number of VC and is determined by lower train departure frequency between the local line and the cross-line services. The passenger travel cost is calculated using Equation (1).
The detailed calculation process for each type of passenger is as follows.
- (1)
The travel costs of the direct passengers
Case 1a: The passengers in line 1, excluding those whose ODs are both within zone II (see
Figure 3a). The resistance time for type-1
a passengers
, in-vehicle time for type-1
a passengers
, and additional VC time for type-1
a passengers
can be calculated as follows. In addition,
qm,n represents the number of passengers from
m to
n, and
tm,n represents the running time of the train from
m to
n.
Case 1b: The OD of these passengers both within zone II. The resistance time for type-1
b passengers
, in-vehicle time for type-1
b passengers
, and additional VC time for type-1
b passengers
can be calculated by Equations (5)–(7).
Case 1c: Passengers traveling on the cross-line and OD within zones II and IV. resistance time for type-1
c passengers
, in-vehicle time for type-1
c passengers
, and additional VC time for type-1
c passengers
can be calculated by Equations (8)–(12). Among them, type-1
c passengers not only need additional VC time at the transfer station
tra for train 1 and train 3. Part passengers also need additional VC time at station
a for train 2 and train 3. Therefore, the calculation is divided into two parts, shown in Equations (11) and (12).
Case 1d: Passengers traveling on line 2, excluding those whose OD is within zone V (see
Figure 3b). The resistance time for type-1
d passengers
, in-vehicle time for type-1
d passengers
, and additional VC time for type-1
d passengers
can be calculated as follows:
Case 1e: The OD of these passengers both in zone IV. The resistance time for type-1
e passengers
, in-vehicle time for type-1
e passengers
, and additional VC time for type-1
e passengers
can be calculated as follows:
- (2)
The travel costs of the transfer passengers
Case 2a: Passengers that need to transfer and the OD within zones I, III, and IV (see
Figure 4a). For instance, consider passengers traveling from zone I to zone IV: these passengers must first take train 1 in zone I to reach the transfer station, then board either train 2 or train 3 to complete their travel to zone IV.
of Case 2a passengers is affected by frequency of train 1, train 2, and train 3, i.e.,
,
,
. Thus, the resistance time for type-2
a passengers
, in-vehicle time for type-2
a passengers
, and additional VC time for type-2
a passengers
can be calculated as follows. For type-2
a passengers, additional VC time is required at the transfer station for train 1 and train 3. Furthermore, some passengers require additional VC time at station
a for train 2 and train 3. Consequently, the calculation is divided into two components, as represented in Equations (22) and (23).
Case 2b: Passengers are in zones I, III, and V (see
Figure 4b). The resistance time for type-2
b passengers
, in-vehicle time for type-2
b passengers
, and additional VC time for type-2
b passengers
can be calculated as follows. For type-2
b passengers, additional VC time is required at the transfer station for train 1 and train 3. Furthermore, some passengers require additional VC time at station
a for train 2 and train 3. Consequently, the calculation is divided into two components, as represented in Equations (27) and (28).
Case 2c: Passengers requiring transfers and the OD within zones II and V. The resistance time for the type-2
c of passengers
, in-vehicle time for type-2
c passengers
, and additional VC time for type-2
c passengers
can be calculated as follows. For type-2
c passengers, additional VC time is required at the transfer station for train 1 and train 3. Furthermore, some passengers require additional VC time at station
a for train 2 and train 3. Consequently, the calculation is divided into two components, as represented in Equations (32) and (33).
3.1.3. Operation Costs
This study examines two primary components of operation costs, the train operating cost and train usage cost, as represented in Equation (36), where
C1 represents the train operating cost, and
C2 represents the train usage cost.
The train operating cost
C1 represents the expenses incurred for the distance traveled during train service. In CO systems, the train composition of line
l , operation frequency of line
l , operating distance of line
l , and train operation cost per kilometer
can be calculated using Equation (37).
The train usage cost
C2 consist of wear and tear, depreciation, etc. The train usage cost is related to the number of carriages in service and can be calculated by the train composition of line
l , operation frequency of line
l , the full cycle time of line
l , and the train depreciation cost per min
. Unlike traditional trains, VC trains are equipped with multiple sensors, communication equipment, and other technologies to facilitate coupling operations. The train usage cost is calculated using Equation (38).