Mode share equilibrium with tradable credit scheme overdifferent time cycles
L. Balzera,∗ and L. Leclercqa
a Univ Gustave Eiffel, Univ Lyon, ENTPE, LICIT, F-69675 Lyon, Francelouis.balzer@univ-eiffel.fr, ludovic.leclercq@univ-eiffel.fr∗ Corresponding author
Extended abstract submitted for presentation at the 11th Triennial Symposium onTransportation Analysis conference (TRISTAN XI)
June 19-25, 2022, Mauritius Island

March 29, 2022
Keywords: tradable credit scheme; trip-based MFD; user equilibrium; logit; mode choice.

1INTRODUCTION

Most cities around the world are coping with congestion. It induces economic losses, contributes
to global warming, and increases the risks of respiratory diseases. Several demand management
schemes are investigated in the literature to reduce the number of individual cars on the network.
The Tradable Credit Schemes (TCS) require users to have credits to access a given resource, here
traveling by private car. The goal is to foster Public Transportation (PT) usage. Those credits
are issued by the regulating entity and can be freely traded with other users. In opposition topricing, the credit price is not set by the regulator but determined by the market. See Lessan &Fu (2019) for an overview of TCS.
Most of the TCS in the literature assumes the credits have to be used on the day they are
emitted and cannot be stocked. However, a few works investigated TCS in a framework allowing
the credits consumption over several days. In Ye & Yang (2013), the credits are allocated forseveral days, and the price is updated each day based on the number of credits still available.
Tian & Chiu (2015) defines consumption periods for the TCS and the users need to balance theircredit account by the end of the period by using the credit market. If they fail, they need tofill the gap by buying credits at a high price from the regulator. The framework of Miralinaghi& Peeta (2016) allows the users to report credits between periods, subject to a fee. Miralinaghi& Peeta (2019) specifies the multi-period TCS to foster the shift from conventional cars to lowemission ones. In Miralinaghi & Peeta (2020) the authors account for the perception of thefuture prices by the network users.
In most of those works, the congestion model is based on the Bureau of Public Roads (BPR)function. As the BPR function is a static model, the travel time depends only on the numberof cars using the link and does not account for congestion dynamics. As an alternative, we use
here the trip-based Macroscopic Fundamental Diagram (MFD) concept (Mariotte et al. (2017),
Lamotte & Geroliminis (2018)), which considers the congestion dynamics at a large scale andtakes into account the heterogeneity of the trip lengths.
This paper investigates the equilibrium of a trip-based MFD under a TCS over several days
with different validity periods. The demand is elastic as we account for modal choice: car orPT. The need for driving private cars may vary over days, e.g., when picking up someone orbuying groceries. To account for this, riding PT on given days induces a penalty. The users canspend their credits over a validity cycle of several days. The novelty lies in the simultaneous
TRISTAN XI Symposium

Original abstract submittal

L. Balzer and L. Leclercq

2

consideration of the congestion dynamics with an MFD framework, the report of credits, andthe heterogeneity of the days.

2METHODOLOGYThe users are aggregated into N groups. Each group i consists of γi travelers, has a fixed trip
length li and departure time ti . Its degree-of-freedom is the ratio of car users per day xd,i . As
we are using a trip-based MFD framework, the travel time per car Ti of the group i is definedby:Zti +Tili =V (t)dt,(1)tiwhere V (t) is the mean speed on the network at time t. We assume the travel time per PT Ti,PTis independent of the accumulation and depends only on the Origin-Destination (OD) pair. Thecosts for each group for a day d is given by:(Ci,car (d) = αTi (xd ) + (τ − κ)p;(2)
Ci,PT (d) = αTi,PT + µi (d) − κp,where α is the value-of-time (VoT), τ the credit charge: the number of credits needed to drivea car, p the credit price, κ the allocation: the number of credits each traveler gets for free fromthe regulator, and µi (d) the penalty for using PT on day d. It represents a day-specific crucial
need to use the car. The decision process is based on logit. The ratio of group i which wants totake the car is:e−θCi,car (d)ψd,i (xd , p) = −θC (d),(3)
i,care+ e−θCi,PT (d)with θ the coefficient of the logit.
This work focus on the modal shares at equilibrium over a cycle of c days. The users canspend their credits on any day during a given cycle. Then looking at the equilibrium for one day
is not enough. We need to consider credit consumption during the cycle to enforce the credit cap:
the total number of consumed credits cannot exceed the number of allocated credits. The leftover
credits are lost once the cycle is over. We compute the simultaneous equilibrium of several days
forming a cycle. The credit price is defined by the market-clearing condition (MCC): the price iszero or all the issued credits are consumed. The equilibrium over a cycle of c days is formalizedwith the following set of equations:ψdP P
 = xd ∀ d ∈ [1, c];Ncp= 0;(4)
d=1 γi (κ − τ xd,i )i=1
PN Pc γ (τ x − κ)
≤ 0.
i=1d=1 id,iNote that choosing c = 1 is equivalent to forbidding credit carry-over as the credits are thenvalid only for a day.
A cost function is formulated to minimize the gap between the modal shares and logit-based
decisions. At the same time, the MCC should hold. It is added into the cost function to avoid
non-affine constraints for computational purposes. The quadratic cost function is defined as!
cNN XcX
1 XX
1J=(xd,i − ψd,i )2 + η PN
pγi (κ − τ xd,i ) ,(5)2i=1 γii=1 d=1d=1 i=1
with η the coefficient related to the MCC.
The computation of the quadratic function is based on the linearization of the travel times
with respect to the modal shares. One major contribution of this work is quantifying the delay
induced by one user to the users (a.k.a. marginal external cost) in a trip-based MFD framework.
See (Balzer & Leclercq (2022)) for more details.
TRISTAN XI Symposium

Original abstract submittal

L. Balzer and L. Leclercq

33PRELIMINARY RESULTS

0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.050.00

Nb carProbability of PT penalty

The proposed methodology is applied to a case study based on the city of Lyon. The networkof Lyon is represented by one reservoir (Mariotte et al. (2020)). The scenario is based on the
7:00-8:00 demand. It represents 115 628 users aggregated into 1314 groups with 224 different OD
pairs. The PT travel times are retrieved from the navigator HERE. We consider a horizon of tendays. A random draw is generated for each group, and each day to attribute days with a non-zeropenalty. The distribution (common for all groups) is given in Fig. 1a. The equilibriums overthe horizon are computed for different cycle lengths: one, two, five, and ten days for the wholehorizon. It means the horizon consists of several cycles depending on its length. For example,with a cycle length of two days, the horizon of ten days consists of five cycles. The credit charge
τ is set to 200 credits and the allocation κ to 100 credits. It means the regulator authorizes atmost half of the users to drive their cars on average over the cycle. It is equivalent to what is
expected with a license plate rationing scheme.
We define the social cost of the travel as the monetary equivalent of the total travel time plus
the penalty. The number of car users, credit prices, and social costs are represented in Fig. 1.
The TCS reduces the social cost (travel time plus penalty) by about 10%. The credit price is

2

468

Day

90000
85000
80000
75000
70000
65000
60000
55000
110

2

3

40.0145c=1
c=2

c=5c = 100.0130
0.01250.01200.0115
3478

9

10

56Day

78910c=2c=5c = 103700000.0135

26Day

no TCS
c=1Social cost (EUR)

Credit price (EUR)

0.014015(b)(a)0.0110c=5c = 10

no TCSc=1c=23600003500003400003300003200001234(c)56Day

78910(d)Figure 1 – Equilibrium for an horizon of 10 days with different cycle lengths: (a) probability ofPT penalty, (b) number of car users, (c) credit price, and (e) social cost.
acceptable as the toll equivalent p(τ − κ) is around 1.2 EUR. For a cycle of one day, the priceis highly correlated with the penalty for taking PT: the credit price increases by about 30%
between day one and day three. As expected, it is high when many users need to drive theircars. Indeed, the demand is high, and the offer is limited as the issued credits are valid only forthe current day. The TCS is more flexible as the cycle length increases. The number of car users
TRISTAN XI Symposium

Original abstract submittal

L. Balzer and L. Leclercq

4varies. The credit price is more stable across the days and less volatile. However, the number ofcars driving is more variable, but it does not significantly modify the social cost.

4DISCUSSION

We set up a TCS framework with trip-based MFD to study different credit consumption cycle
lengths. On some days, some users suffer from a penalty if they take the PT. We present amethodology to compute the modal equilibrium of a TCS permitting cycle-based credit consumption. We assess the effect of credit cycles on credit price and social costs with a numericalexample based on the morning commute in Lyon. Allowing credit carry-over does not significantly affect the performance of the TCS. It can be an asset when it goes to the acceptabilityof the TCS: the credit price is more stable over days, and the credit consumption is flexible.
Moreover, it makes it possible to use its car for a day without spending a single euro, only bysaving credits from the other days. Future investigations include the comparison of TCS against
other demand management policies.

References
Balzer, Louis, & Leclercq, Ludovic. 2022. Modal equilibrium of a tradable credit scheme with a trip-based
MFD and logit-based decision-making. Accepted for publication at Transportation Research Part C:
Emerging Technologies http://arxiv.org/abs/2112.07277.
Lamotte, Raphaël, & Geroliminis, Nikolas. 2018. The morning commute in urban areas with heterogeneous trip lengths. Transportation Research Part B: Methodological, 117(11), 794–810.
Lessan, Javad, & Fu, Liping. 2019. Credit- and permit-based travel demand management state-of-the-art
methodological advances. Transportmetrica A: Transport Science.
Mariotte, Guilhem, Leclercq, Ludovic, & Laval, Jorge A. 2017. Macroscopic urban dynamics: Analyticaland numerical comparisons of existing models. Transportation Research Part B: Methodological, 101(7),245–267.
Mariotte, Guilhem, Leclercq, Ludovic, Batista, S. F.A., Krug, Jean, & Paipuri, Mahendra. 2020. Calibration and validation of multi-reservoir MFD models: A case study in Lyon. Transportation ResearchPart B: Methodological, 136(6), 62–86.
Miralinaghi, Mohammad, & Peeta, Srinivas. 2016. Multi-period equilibrium modeling planning framework
for tradable credit schemes. Transportation Research Part E: Logistics and Transportation Review,93(9), 177–198.
Miralinaghi, Mohammad, & Peeta, Srinivas. 2019. Promoting zero-emissions vehicles using robust multiperiod tradable credit scheme. Transportation Research Part D: Transport and Environment, 75(10),265–285.
Miralinaghi, Mohammad, & Peeta, Srinivas. 2020. Design of a Multiperiod Tradable Credit Scheme under
Vehicular Emissions Caps and Traveler Heterogeneity in Future Credit Price Perception. Journal of
Infrastructure Systems, 26(3), 04020030.
Tian, Ye, & Chiu, Yi Chang. 2015. Day-to-Day Market Power and Efficiency in Tradable Mobility
Credits. International Journal of Transportation Science and Technology, 4(3), 209–227.
Ye, Hongbo, & Yang, Hai. 2013. Continuous price and flow dynamics of tradable mobility credits.
Transportation Research Part B: Methodological, 57(11), 436–450.

Acknowledgements
This project has received funding from the European Union’s Horizon 2020 research and innovation program under Grant Agreement no. 953783 (DIT4TraM).

TRISTAN XI Symposium

Original abstract submittal