Model for bus holding control of electric buses that can be solved in real-time. The initial model is reformulated as a convex mathematical program.

1. UNIVERSITY OF TWENTE.
Bus Holding of Electric Vehicles: An Exact Optimization
Approach
Paper no. 20-01738
Dr. Konstantinos Gkiotsalitis
Assistant Professor, University of TWENTE, Netherlands.
99th TRB Annual Meeting

2. UNIVERSITY OF TWENTE.

3. UNIVERSITY OF TWENTE.
INTRODUCTION

4. UNIVERSITY OF TWENTE.
Electric Bus Sales Projection

5. UNIVERSITY OF TWENTE.
Motivation
Bunching might increase the waiting times of passengers and delay the arrival of the bus at the
charging point resulting in:
1. charging with an increased energy price;

6. UNIVERSITY OF TWENTE.
Motivation
Bunching might increase the waiting times of passengers and delay the arrival of the bus at the
charging point resulting in:
1. charging with an increased energy price;
2. delay or refuse of charging because the charging point is occupied by another bus;

7. UNIVERSITY OF TWENTE.
Motivation
Bunching might increase the waiting times of passengers and delay the arrival of the bus at the
charging point resulting in:
1. charging with an increased energy price;
2. delay or refuse of charging because the charging point is occupied by another bus;
3. disruption of future bus dispatches/crew schedules;

8. UNIVERSITY OF TWENTE.
Motivation
Bunching might increase the waiting times of passengers and delay the arrival of the bus at the
charging point resulting in:
1. charging with an increased energy price;
2. delay or refuse of charging because the charging point is occupied by another bus;
3. disruption of future bus dispatches/crew schedules;
4. and excess strain on the power grid.

9. UNIVERSITY OF TWENTE.
Dual objective of this work
(a) minimize the deviation between the actual and planned headways (regularity),
(b) adhere to the pre-planned charging schedule

10. UNIVERSITY OF TWENTE.
Research Gap
there is a lack of bus holding studies that consider
• the improvement of the service regularity and, at the same time,
• cater for the charging requirements of electric buses.

11. UNIVERSITY OF TWENTE.
Contribution
• bus holding control model applicable to electric buses;

12. UNIVERSITY OF TWENTE.
Contribution
• bus holding control model applicable to electric buses;
• a reformulated program which is proven to be convex;

13. UNIVERSITY OF TWENTE.
Contribution
• bus holding control model applicable to electric buses;
• a reformulated program which is proven to be convex;
• the generation of reliable bus holding solutions that account for the interstation travel time
uncertainty.

14. UNIVERSITY OF TWENTE.
Bus Holding Decision
Time
Control stop, s
𝐻0
𝑑𝑖−1,𝑠 𝑇𝑖,𝑠
𝑖 − 1
𝑖
𝑑𝑖,𝑠
Bus trajectory
realized trajectory
expected trajectory
𝑖 − 1
𝑖
Space
Figure 1: Illustration of bus trajectories in a time-space diagram

15. UNIVERSITY OF TWENTE.
MATHEMATICAL MODEL

16. UNIVERSITY OF TWENTE.
Departure Constraint
Trip i cannot depart prior to Ti,s which is the time when trip i has completed the boardings/alighting
at stop s.
This yields:
Ti,s ≤ di,s (1)
and is a physical (hard) constraint.

17. UNIVERSITY OF TWENTE.
Charging Constraint
If bus trip i needs to reach its charging location before time ρi for charging as planned, then:
di,s + E[ti,s] ≤ ρi (2)
where:
di,s = the departure time of trip i from stop s, including its holding time
E[ti,s] = the expected travel time of trip i from stop s to the last stop
ρi = the scheduled charging time of trip i at the charging location

18. UNIVERSITY OF TWENTE.
Charging Constraint
If bus trip i needs to reach its charging location before time ρi for charging as planned, then:
di,s + E[ti,s] ≤ ρi (2)
where:
di,s = the departure time of trip i from stop s, including its holding time
E[ti,s] = the expected travel time of trip i from stop s to the last stop
ρi = the scheduled charging time of trip i at the charging location
Note: if bus trip i arrives late at stop s, the charging constraint cannot be satisfied even if we do not
apply holding.

19. UNIVERSITY OF TWENTE.
Objective Function
f(di,s) = µi,s
(
di,s − (di−1,s + H0)
)2
+ (1 − µi,s)
(
di,s − Ti,s
)2
(3)
which strives to minimize the squared difference between the planned (ideal) and the actual
headway between trips i − 1 and i.
If this is achieved, regularity is maintained.

20. UNIVERSITY OF TWENTE.
Mathematical Program
(ˆQi,s) : min
di,s
f(di,s) + M max(di,s + E[ti,s] − ρi, 0)
s.t.: Ti,s ≤ di,s
di,s ∈ R+
(4)
• our charging constraint, which cannot be always satisfied, is added in the objective function
with the penalty M max(di,s + E[ti,s] − ρi, 0).
• the objective function is not convex because of the max term.

21. UNIVERSITY OF TWENTE.
Reformulation with the use of slack variable, ν
(˜Qi,s) : min
ν,di,s
f(di,s) + Mν
s.t.: di,s ≥ Ti,s
ν ≥ 0
ν ≥ di,s + E[ti,s] − ρi
(5)
• the reformulated program does not have the max term any longer.
• it can be solved to global optimality and has a unique solution, as proven in our central
Theorem (see manuscript).

22. UNIVERSITY OF TWENTE.
Reliable Holding Solution
The average travel time E[ti,s] in program ˜Qi,s can be substituted by the y-th percentile since it will
only exceed that value at (100-y)% of the cases.
With this substitution, the reliable solution is found by solving the following program:
(˜Pi,s) : min
ν,di,s
f(di,s) + Mν
s.t.: di,s ≥ Ti,s
ν ≥ 0
ν ≥ di,s + ty
s − ρi
(6)

23. UNIVERSITY OF TWENTE.
NUMERICAL EXPERIMENTS

24. UNIVERSITY OF TWENTE.
Solution with CPLEX
In an idealized scenario, trip i arrives at control stop s and completes its boardings/alightings at time
Ti,s = 1500s. The parameter values of our scenario are: di−1,s = 1000 s, Ti,s = 1500 s, H0 = 600 s,
and E[ti,s] = 3000s.
Table 1: Globally Optimal Holding decisions for different values of ρi
ρi Solving ˜Qi,s with CPLEX
ν di,s Comp. time
4800 s 0 s 1600 s 0.02 s
4600 s 0 s 1600 s 0.02 s
4550 s 0 s 1550 s 0.02 s
4500 s 0 s 1500 s 0.02 s
4200 s 300 s 1500 s 0.02 s

25. UNIVERSITY OF TWENTE.
Demonstration of the objective function
1500 1600 1700 1800
0
2
4
·104
di,s (s)
˜f(di,s,ν)
.
=f(di,s)+Mν
ρi = 4800 (s)
˜f(di,s)
optimal
Figure 2: Performance of the objective function in the
region F = { ν = 0, di,s ≥ Ti,s } for ρi = 4800 s.
1500 1520 1540 1560 1580 1600
0
0.5
1
·104
di,s (s)
˜f(di,s,ν)
.
=f(di,s)+Mν
ρi = 4600 (s)
˜f(di,s)
optimal
Figure 3: Performance of the objective function in the
region F = { ν = 0, di,s ≥ Ti,s } for ρi = 4600 s.

26. UNIVERSITY OF TWENTE.
Simulation
• We consider a short time period of the day with 10 bus trips.
• Trips start from stop 1 and complete their service at the same stop, which happens to be a
charging point.
• Buses can be held at the control point stop, s = 2.
• The target headway is H0 = 6 min
• the realization of each interstation travel time from stop j to stop j + 1 is ti,j ∼ N(E(ti,j), Var(ti,j))
𝑠1

27. UNIVERSITY OF TWENTE.
Parameter Values of the Simulation
Table 2: Parameter values of the simulated line
Parameter Value
E(ti,1),
√
Var(ti,1) 1700 s, 100 s
E(ti,s),
√
Var(ti,s) 1000 s, 100 s
tmin
i,1 1500 s
tmin
i,s 800 s
H0 360 s
t
y
s 1200 s
c 1.0

28. UNIVERSITY OF TWENTE.
Scheduled Dispatching and Charging Times
Table 3: Dispatching and scheduled charging times of the 10 simulated trips
Trip, i Dispatching time Scheduled charging time, ρi
1 0 s 2900 s
2 360 s 3260 s
3 720 s 3980 s
4 1080 s 4340 s
5 1440 s 4700 s
6 1800 s 5060 s
7 2160 s 5420 s
8 2520 s 5780 s
9 2880 s 6140 s
10 3240 s 6500 s

29. UNIVERSITY OF TWENTE.
Comparative Analysis
Then, we perform a comparative analysis using as a benchmark the control logic of Fu and Yang
(2002). The logic of Fu and Yang (2002) is summarized as:
di,s =
{
di−1,s + H0 if Ti,s < di−1,s + cH0
Ti,s otherwise
(7)

30. UNIVERSITY OF TWENTE.
Results (1/2)
Table 4: Average performance of the two control logics in 1,000 simulation scenarios
Key Performance Control Logic of Proposed
Indicator Fu and Yang (2002) Control Logic
Average Passenger
Waiting Time (s) 182.4 184.4
Average Trip
Travel Time (s) 4996 4887
Missed
Chargings 4 1
Overall Charging
Delay (s) 96.59 63.52

31. UNIVERSITY OF TWENTE.
Results (2/2)
Average
Passenger
Waiting Time
Average Trip
Travel Time
Overall Charg-
ing Delay
0
20
40
−1.08
2.18
34
Improvement(%)
Performance of Key Performance Indicators
Figure 4: Average improvement (deterioration) when applying the proposed control logic.

32. UNIVERSITY OF TWENTE.
Conclusions
• the charging time delay(s) can be reduced by up to 34% with a minimal trade-off of 1.08%
increase in passenger waiting times;
• restraining the holding times due to the charging constraints can improve the total trip travel
times and limit schedule sliding effects.
Future research
• expand our method to railway operations that operate under regularity-based schemes.
• expand towards using a two-headway-based logic to consider the headway with the preceding
and the following bus when making a holding decision,
• expand to incorporate the time-varying charging costs in the objective function.

33. UNIVERSITY OF TWENTE.
Thank you for attending
Questions?
Contact:
Dr. Konstantinos Gkiotsalitis
E: k.gkiotsalitis@utwente.nl
www.researchgate.net/profile/Konstantinos_Gkiotsalitis

34. UNIVERSITY OF TWENTE.
References
Fu, Liping, and Xuhui Yang. 2002. “Design and implementation of bus-holding control strategies with
real-time information.” Transportation Research Record: Journal of the Transportation Research Board
(1791): 6–12.