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Autonomous vehicles in mixed motorway traffic: Capacity
utilisation, impact and policy implications
Andrea Papu Carrone§, Jeppe Rich
1
§, Christian Anker Vandet§, Kun An¤
§Technical University of Denmark, Department of Management
¤Tongji University, College of Transportation Engineering, Shanghai, China
Preprint: Cite as:
Carrone, A.P.V, Rich, J., Vandet, C.A., An, Kun (2021). Autonomous vehicles in mixed traffic: Capacity
utilisation, impact and policy implications. Transportation.
https://link.springer.com/article/10.1007/s11116-020-10154-4.
Abstract
In upcoming years, the introduction of autonomous vehicles (AVs) will reshape the transport system. The
transition from a regular to an autonomous transport system, however, will take place over many years and
lead to a long period with a mixed driving environment where AVs and regular vehicles (RVs) operate side by
side. The purpose of this study is to investigate how the utilisation of the road capacity degrades as a function
of heterogeneity in congested motorways. The analysis is based on a dedicated traffic simulator, which enables
the investigation of complex dynamic spillback from congestion while allowing for different degrees of
heterogeneity. The representation of autonomous vehicles is based on a modified intelligent driver model
(IIDM) presented by Treiber et al. (2000; 2013), while the behaviour of drivers of RVs relies on a stochastic
version of the IIDM. Three main conclusions stand out. Firstly, it is shown that in an idealised environment in
which AVs operate alone, a substantially improved capacity utilisation can be attained. Secondly, when drivers
of RVs are mixed with AVs, capacity utilisation degrades very fast as a function of the share of RVs. Thirdly, it
is shown that the improved capacity utilisation of AVs comes in the form of reduced travel time and increased
throughput, with indications that travel time reductions are the most important. From a strategical planning
perspective, the results underline that dedicated lanes are preferable to attain the positive effects of AVs.
Specifically, we compare a stylised situation with three lanes with a share of 33% AVs to a situation with two
regular lanes and a single dedicated AV lane. The latter represents a tripling in consumer surplus all other
things being equal.
Keywords: autonomous vehicles; microsimulation; consumer surplus, heterogeneous vehicle types; IDM;
road capacity.
1. Introduction
The introduction of autonomous vehicles to the automobile market is expected by many (Fagnant 2014;
Kockelman et al. 2016; Shladover et al. 2012; Tientrakool et al. 2011) to bring a much needed alleviation to
1
Corresponding author rich@dtu.dk.
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the congested road networks around the world. Over the past decades there have been several technological
leaps in sensing technology, wireless communications and data processing which has made the vision of
autonomous driving real and attainable on a large scale within the next car generation (Talebpour and
Mahmassani 2016). This is further backed by evidence suggesting that drivers do exhibit a positive willingness-
to-pay for these new services as demonstrated in Bansal and Kockelman (2018) but also evidence suggesting
that activity patterns and travel behaviour could be influenced more fundamentally (Harb et al., 2018).
The potentially improved capacity utilisation of AVs results from higher densities attained by minimal spacing
while maintaining efficient acceleration and deceleration profiles. A key factor to attain optimality, which is
overlooked, is that of homogeneity among vehicles. This is relevant, not only when mixing AVs with human
drivers but also as a means to understand the effect of different types of AVs. In more recent papers Chen et
al. (2020), Pan et al. (2019) and Hua et al. (2020) find that even small fractions of human vehicles can lead to
an unproportioned reduction of system performance. These papers support the findings in the present paper
although they use a cellular automata approach. Specifically, Hua et al. (2020) consider the influence of lane
policies and conclude that separate lanes for AVs are crucial for obtaining good flow performance.
For year 2050, trips carried out by AVs are expected to represent between 7 and 61 percent of the market
share (Milakis et al. 2017). Although such estimates are subject to great uncertainty, it is clear that the driving
environment for many years to come will be a mixed environment where regular vehicles will interact with AVs.
As the infrastructure in most countries is planned with a horizon of up to 50 years, it is important to analyse,
understand the situation of mixed driving, and investigate the likely changes in capacity utilisation it may cause.
Over the years, there have been many investigations of the effects of autonomous cruise control systems
(ACC) and connected and autonomous vehicles with cooperative autonomous cruise control (CACC) systems
(Hanebutte et al. 1998; Van Arem et al. 2006; Wang et al. 2014; Zhao and Sun 2013). Although it is generally
agreed that a fully automated driving environment will improve the system performance, it is less clear how
this depends on the market share of autonomous vehicles. Kesting et al. (2010) investigated the impacts of
different penetration rates of autonomous and connected vehicles using an advanced version of the Intelligent
Driver Model (IDM). In this model, parameters are adapted continuously according to the traffic situation. Their
results show that even for a 10 percent share of autonomous and connected vehicles, the travel times and the
level of the congestion could be significantly reduced (Kesting et al. 2010). Later, Atkins (2016) implemented
a microsimulation model based on VISSIM in order to investigate the network effects of connected and
autonomous vehicles. This analysis, which was based on the Wiedemann car-following model (Wiedemann
1974) offered two conclusions. Firstly, the great potential of intelligent vehicles as a means to improve the
capacity utilisation, but secondly, that for these vehicles to benefit from their enhanced capabilities, a high
market share of 50-75% was required. Talebpour and Mahmassani (2016) have also studied the effects of
autonomous and connected vehicles on a highway system. They implemented different models to represent
the differences between regular cars, connected vehicles and autonomous vehicles. Their simulation results
only considered the effect of throughput and concluded that a substantial increase in throughput is only
possible under certain positive market penetration scenarios (Talebpour and Mahmassani 2016). Similar
conclusions are presented in the review by Taha et al. (2019) and suggest that improved capacity utilisation
can be expected for different stages of autonomy. More recently Li et al. (2020) presented evidence that
capacity of mixed flow increases convexly with the AV penetration rate and that right-of-away strategies can
be used to further improve performance. These findings are completely in line with the findings in this paper
although based on a slightly different simulator. Also, while Li et al. (2020) focus solely on flow performance
this paper considers the total surplus performance, which in addition includes the time-factor. The surplus
curve is shown to be even steeper than the flow curve as a function of the AV penetration rate.
The purpose of this study is to add to the existing knowledge by quantifying travel time and throughput in
scenarios with different shares of AVs to determine performance in terms of capacity utilisation and total
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surplus and to discuss the policy implications on motorway systems. To this end, we base our analytical model
on the Improved Intelligent Driver Model (IIDM) and allow stochastic preferences for key input parameters
within the simulation framework to represent heterogeneous drivers. This is an extension of (Kesting et al.
2010) where “…human drivers of cars and trucks have been modelled with constant model parameters”.
Although the simulation framework becomes computationally heavy, it has the advantage that the modelling
of vehicle heterogeneity fundamentally refers to the same traffic simulator model but with different
parametrisations. This leaves out potential model bias that may result from comparing different models in the
same Monte-Carlo experiment. Additionally, when modelling heterogeneous vehicles, it is necessary to
consider vehicle-overtaking procedures in order to represent the correct vehicle dynamics. In the paper, two
different overtaking strategies are analysed and it is found that a proposed “shadow lane” approach is
computational efficient and yield results that are almost comparable to a full-fledged discrete model
representation. Further, we provide a complete suite of performance measures from travel time, throughput
and structure of the fundamental diagram under different penetration rates of AVs. This is accomplished by
implementing numerous virtual detectors at strategic positions of the motorway segment under investigation.
For each detector, time of passage, speed and number of vehicles are stored in a space-time database. This
is then used to calculate the various micro and macro performance measures, including flow, speed and
density graphs. As a final contribution, we estimate the accumulated consumer surplus of having AVs driving
in dedicated lanes compared to a mixed traffic situation. As stated above, the surplus curve is found to be
significantly different from the flow curve as a function of the penetration rate as it constitutes both flow and
time performance gains. Although derived in an idealised environment, it gives a “ball-park” figure of the
attainable benefits of having separate systems, something that is of value for transport planners and which, to
our knowledge, has not been covered in the existing research literature.
The paper is structured as follows: Section 2 presents the formal representations of AVs and RVs. Section 3
explains the dedicated simulation framework. Section 4 presents the simulation of a section of the Copenhagen
M3 motorway during a high-demand peak and presents findings for this case study. Finally, Section 5 offers
conclusions and provides future research directions.
2. Model formulation
In the following we assume that AVs, which can be machine-controlled, represent a homogenous fleet of cars
with optimal and adaptive speed control according to the properties of the IDM. Regular vehicles (RVs), on the
other hand, are influenced by variations in driving styles ranging from cautious to aggressive and fast to slow.
The modelling of mixed driving preferences creates a number of challenges. However, from a more
fundamental point of view, the need to introduce heterogeneity and to investigate robustness and string stability
in the context of heterogeneous users, rules out a deterministic model formulation. In other words, flow
theoretical kinetic models that are only applicable to homogeneous settings are not suitable in this context.
The application of a microscopic simulation model allows us to characterise every vehicle individually and to
investigate their interaction and collective behaviour. Over the past sixty years, acceleration models with
different levels of complexity have been studied extensively in the literature to capture human driving
behaviour. The most primitive acceleration models are “non-complete” in the sense that they are not able to
describe free-flow traffic regimes or deceleration due to the existence of a static obstacle (Pipes, 1953; Forbes
et al. 1958; Chandler et al. 1958). More sophisticated acceleration models, which are defined in a piecewise
manner according to the traffic regime they operate in, include Bando et al. (1995), Gipps (1981) and
Wiedemann (1974). These models, however, are less capable of representing the transition between regimes.
Furthermore, some models are strongly dependent on the fine-tuning of a large number of parameters that are
generally very difficult to interpret (Bando et al. 1995; Wiedemann 1974).
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In contrast, the IDM is a complete microscopic model that can describe a large number of traffic regimes. This
includes car-following situations, free-flow traffic and stationary traffic. The IDM acceleration is defined by a
continuous function, which ensures that transitions between different traffic regimes are smooth, i.e. the time
derivative of the acceleration function is finite. Moreover, the IDM parameters all have a reasonable
interpretation and are empirically measurable (Treiber and Kesting 2013). In conclusion, the IDM model thus
represents a suitable microscopic model for our context as it allows investigating:
o Heterogeneous vehicle classes (regular vehicles and autonomous vehicles)
o Heterogeneous driving behaviours across a vehicle class
o Realistic acceleration behaviour where vehicles accelerate towards a desired speed if not constrained
by other vehicles
o Collision-free driving (by design)
In the paper, we will start from a pure deterministic and homogenous representation of AVs. Hence, in the
experimental setup the situation of 100% AVs represents an idealised baseline that measures the maximum
attainable capacity utilisation. This is true even when assuming Markov properties of the dynamic system, e.g.
that cars have perfect information only about the cars in front. The fact that the IDM model was developed as
a collision-free model and uses the same input variables as the sensors of ACC systems (Treiber and Kesting
2013) makes it a likely control system for forthcoming AVs.
RVs, on the other hand, are by nature stochastic. Drivers prefer different speeds, acceleration patterns and
safety distances, and have different reaction times. Due to this, the RVs are based on a stochastic version of
the IDM where key driving parameters are drawn from distributions in a Monte-Carlo loop. Depending on the
situation we can draw random distributions across vehicle classes or across drivers within a given vehicle
class. Clearly, it is a challenge to envision how preferences will evolve in the future. However, the important
outcome of this exercise is to measure the “cost” of heterogeneity in the traffic and, in particular, how such
heterogeneity degrades the performance of AVs.
2.1. A model for AVs
The model for AVs (as for RVs) is a microscopic model that describes the movements of the individual vehicles
through the space-time domain on a motorway stretch. Each vehicle is described by the state-space
variables: position, speed and acceleration as a function of the simulation time step, and by the
attribute vehicle length. The vehicle response is a function of vehicle and driver characteristics and their
interaction with the vehicle in front, e.g. the underlying dynamic system adopts Markov properties. The IDM
describes the vehicle response in terms of an acceleration function
and the vehicles trajectories are
simulated following a brute force Monte-Carlo model, where the system state is calculated for each time step,
as will be further detailed in Section 3.
In IDM, the acceleration of vehicle is represented as a continuous function of the current speed, the gap
and the speed difference with the leading vehicle (Treiber et al. 2000). The IDM as presented by Treiber
et al. (2000) is a deterministic acceleration model, which complies with the homogeneity assumption of
autonomous vehicles. Given that where denotes simulation time, simulation time step and
time step size; the acceleration of vehicle at time step is given by Equations (1) and (2) below.
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(1)
(2)
Here are parameters that refer to a deterministic set of values.
The acceleration expression combines the ability to reach a desired speed in a free-flow traffic environment
with the ability to identify the braking required when a vehicle comes too close to the one ahead. The braking
strategy, which plays an essential stabilizing role when approaching congested traffic and avoiding rear-end
collisions, depends on the ratio between the desired safe gap
and the actual gap.
The free acceleration is characterised by the desired speed, the maximum acceleration and the
exponent. The acceleration decreases with increasing speed and goes to zero as the speed approaches
the desired speed, while the exponent controls this reduction (=1 corresponds to a linear decrease and
corresponds to a constant acceleration).
The desired safe gap
for vehicle is formed by: i) a minimum standstill gap, ii) a speed dependent
distance and iii) a dynamic contribution. The gap parameter is only relevant at standstill and at very
low speeds. The speed dependent distance dominates the situation of following a leading vehicle in
stationary traffic. The last term in Equation (2) contributes to non-stationary traffic situations where
and implies an "intelligent" driving behaviour which in normal situations limits the braking deceleration to a
comfortable deceleration. However, the IDM deceleration is in principle unrestricted and may be significantly
higher. This in turn leads to the collision-free property that characterizes the IDM (Treiber et al. 2000).
In the context of continuous-time models, it is efficient to assume a constant acceleration within each time step
update (Kesting and Treiber 2008). This leads to the following updating rules:
(3)
(4)
The update time or time step size is an auxiliary variable of the approximate numerical solution and thus,
for, the result converges to the exact solution.
In stationary traffic when and when the speed of the vehicle is close to its desired speed the
IDM equilibrium gap may be too large compared with the gap
. This causes an unrealistic car-following
behaviour in stationary traffic for platoons of identical vehicles. Furthermore, another unrealistic aspect of the
IDM appears when the speed exceeds the desired speed, and this situation could occur when a vehicle
enters an area with a reduced speed limit. In this case, the deceleration is unrealistically large. Hence, in our
implementation of the IDM we will refer to an adapted version of the model, the Improved Intelligent Driver
Model (IIDM). This model circumvents these limitations but retains the well-tested features of the IDM model
(Treiber and Kesting 2013).
The IIDM uses the same set of model parameters as the IDM and produces essentially the same behaviour
except for the previously specified cases. To this end, the IDM acceleration function is separated into two
terms: the free-flow acceleration and the interaction terms (Treiber and Kesting 2013).
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The model implemented here incorporates a reaction time variable, which corresponds to a delayed adaption
of the acceleration. Although it can be assumed that AVs can react almost instantaneously to any stimulus,
incorporating a reaction time variable to the acceleration model allows a direct extension of the proposed
model to represent human driving behaviour. IIDM calculates the acceleration rate at every time step based
on the position and speed of vehicle at previous time step. Hence, the acceleration model for
vehicle at time step is now given by Equation (5) below where is the ‘free road behaviour’ and is
the interaction term that describe behaviour at ‘high approaching rates’.
(5)
Zhu and Zhang (2018) consider a specific model for AVs, where they specifically look at smoothing factors to
balance front and back headway and consider stability criterias. In our model this is handled through the
acceleration and deceleration functions which, when defined for small time steps, will render a smooth flow
execution.
2.2. A model for regular vehicles
RVs differentiate themselves from AVs in a number of ways. Human drivers have a reaction time of the order
of 1 second due to the mental processing and the time it takes to carry out the given action following the
decision (Kesting and Treiber 2008). Following the extended version of the IIDM with built-in reaction time as
described above, a delayed input stimulus can be implemented to calculate the acceleration of a regular
vehicle with a given reaction time. To this end, the reaction time is considered a multiple of the update
time. Another difference between AVs and RVs is the desired speed. Some drive slow and others like to
drive fast. Moreover, the aggressiveness of drivers could lead to different maximum acceleration parameters.
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For all of these random components we will apply truncated normal distributions, e.g. the reaction time cannot
be negative and for the other variables we implement sensible limits to avoid out-of-range values. This leads
to the following general parametrisation of the model with and . The specific
truncation intervals and parameters are to be defined.
As a result, regular vehicles will be based on the formulation in Equation (5) and the only difference between
the acceleration models for autonomous and regular vehicles is given by the definition of the parameters which
in the case of the RVs will be sampled from the distribution . This enables the representation of
heterogeneous driving behaviours based on qualified random distributions for which parameters are based on
cognitive experiments and experience from the literature. To ensure model stability, all distributions are limited
to the parameter domains for which the IIDM has proved to be stable.
Moreover, by exploiting both vehicle classes based on different versions of the IIDM model we avoid potential
modelling bias. This would have been the case if we had applied, for instance a Wiedemann representation of
RVs and an IDM representation of AVs. In that case any benchmarking between the two would have been
difficult. With the current modelling strategy we know that differencies between the two vehicle classes are
solely represented by heterogeniety. Hence, what we essentially measure is the added “cost” of stochasticity
or heterogenity from a system perspective.
3. Simulation framework
This section presents the simulation framework, which we apply to investigate the effects of mixed traffic
(regular and autonomous vehicles) on the capacity utilisation performance, specifically on motorways during
congested periods.
In order to induce congestion into the system, a sub-section of the motorway is classified as a ‘reduced speed
area’. By introducing a location-dependent desired speed parameter that depends on the position of each
vehicle, it becomes possible to measure the effect of a joint reduction in speed. From the simulations, it can
be observed that the local variation of the parameter acts as a bottleneck because of the induced capacity
drop. Traffic breaks down at the bottleneck location and propagates upstream.
3.1. Structure of the simulation model
The microsimulation framework is structured in four modules: demand, car-following, overtaking and
performance indicators. The flow diagram Figure 1 presents the sequence of the processes required to
implement the simulation model.
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Figure 1: Simulation activity flow diagram.
The microscopic simulation is embedded in a Monte-Carlo loop (Loop 1) in order to investigate the results of
the simulations across a distribution of the input parameters. In this case, 100 Monte-Carlo iterations were
carried out.
The microscopic model uses a brute force simulation approach, where the system state is simulated for each
time step (Loop 2). To this end, the simulation is performed assuming a constant update time equal to 0.1
seconds. The underlying assumption is that the acceleration during the time update is constant. The numerical
results of simulations with equal to 0.1 second provide a good approximation to the exact solution (Kesting
and Treiber 2008).
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At each time step of the simulation, all the modelled vehicles must be updated (Loop 3). It is necessary to
maintain a bookkeeping system for registering and storing the position and the speed of each vehicle
at every time step. Knowing the information of the previous time step for the position of vehicle,
and the position of the leading vehicle then makes it possible to calculate the distance gap
of vehicle to its leader at time step according to:
(6)
Analogously, the bookkeeping of speed variables keeps track of the speed of vehicle, and the
difference in speed between vehicle and its leader. Therefore, it is possible to calculate the desired
safe gap
of vehicle to its leader at time step as shown in Equation (2).
The demand input module controls the rate at which vehicles enter the system. The intensity (or frequency) by
which cars arrive to the system is drawn from a truncated normal distribution in order to mimic a 3 hour morning
peak as described in Section 4. In the demand input module of the simulation, at each time step the distance
gap to the precedent vehicle is calculated for vehicles that did not enter the system yet ( ). These
vehicles enter the simulation when the distance gap matches the expected demand intensity for time step . A
warm-up stage is defined to avoid the influence of system loading and vehicles start several kilometres before
they enter the system to reduce noise from starting conditions.
3.2. Car-following and overtake module
The car-following module is a central element in the simulation model. In this module, we implement the
acceleration model with the corresponding parametrisation for each vehicle class, i.e. autonomous vehicles
and regular vehicles. When simulating heterogeneous traffic conditions it is important to consider vehicle
overtaking in order to avoid the unrealistic formation of a long queue behind the slowest vehicle. This is
particularly true when sampling the desired speed. However, the dynamics of overtake manoeuvres in
combination with complex dynamic spillback introduce several computational challenges, e.g. a need for
sideways orientation and corresponding parameters of which little is known in the context of mixed driving
environments.
In the paper different options for calculating the effect of lane-changing were considered. One approach being
a complete lane-changing model and another what will be referred to as a ‘shadow lane approach’. The latter,
is based on a sampling approach to determine overtaking possibilities in a second ‘shadow’ lane and provided
that the two lanes represent an equilibrium state where lane density is approximately similar, this render a
much simpler book-keeping for examining lane-changing dynamics. An elaborate description of the overtaking
mechanism is presented in Appendix A.
3.3. Calculation of performance indicators
Once all the time steps of the microsimulation model are completed, the outputs represent disaggregated
information of positions, speeds and accelerations for all vehicles at each time step. Hence, it is necessary to
define procedures to assess the aggregate performance of the simulated system. To accomplish this, we
implement virtual detectors at several positions of the motorway segment. For each of these we store passage
times, speeds and number of vehicles. The simulations allow us to explore the changes in the following
indicators under different hypothetical scenarios.
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3.3.1. Average travel time
The average travel time is calculated as the mean travel time it takes for all vehicles to complete the road
segment.
(7)
Where denotes the time that vehicle takes to complete the motorway segment and denotes the total
number of vehicles that have travelled through the segment during the simulation period.
3.3.2. Throughput and capacity
The inflow and the throughput or outflow of the system are computed as well. The detectors located
at the borders of the system measure these. Hereby it is possible to calculate the capacity drop as:
(8)
The capacity drop is measured over a period of time after a warm-up stage. Throughout the paper we have
applied a one hour period. The capacity drop refers to the phenomenon that the maximum throughput observed
at the downstream exit point is usually smaller than the flow observed before the bottleneck and before the
congestion builds up.
3.3.3. Queue
The queue length is registered across the entire simulation period in time intervals. The queue length is
measured as the total number of vehicles before the bottleneck which drive at a lower speed than 35 km/h at
a given .
The area of the queue can be calculated as:
(9)
Where denotes the time interval length and the number of time intervals for the whole simulation period.
is the length of the queue at time interval measured in number of vehicles.
3.3.4. Fundamental diagrams
The fundamental diagrams describe the statistical relationship between the macroscopic traffic flow variables,
e.g. flow, density and speed. Flow and density relate to the average time and distance gap, respectively.
Hence, the underlying assumption of the fundamental diagrams is that vehicles will behave in a similar way
when traffic conditions are also assumed similar (Hoogendoorn and Knoop 2013).
In order to obtain the simulated fundamental diagrams, we consider a stationary period of one hour of the
simulation period. The data is collected with time intervals of 1 minute by the virtual detectors. Based on the
detector data we determine the flow, the space mean speed and the density.
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4. Case study
In the following, we apply the model to a standard motorway around Copenhagen (M3). The simulation
experiment is set up for a 13 km segment of the M3 motorway during a morning peak hour period with traffic
spillback from an artificial bottleneck.
The demand variation during the simulated period is based on a realistic morning peak commuting profile. This
profile has been estimated on the basis of the Danish Transport Survey, from 6:00 a.m. to 9:00 a.m. The
demand profile fitted a normal distribution with mean 7:30 a.m. and standard deviation of 40 minutes. In order
to initiate the system, we consider a warm-up period of 30 minutes.
The desired speed along the segment is 110 km/h and identical to the actual speed limit. A one kilometre
stretch with reduced speed acts as a bottleneck. The bottleneck is located 10 kilometres ahead of the system
entry point to make sure that this boundary is unaffected by congestion spillbacks (see Figure 2). Moreover,
vehicles are launched several kilometres before they enter the system during a warm up stage.
Figure 2: Layout of the experiment setup.
4.1. Model inputs and calibration
The definition of the parameters for the RVs are based on existing literature as shown in Table 1,
which we will discuss in more details below.
Parameter
Unit
Symbol
RVs
AVs
Literature
bounds
Maximum acceleration
(m/s2)
3
3
[1.7 ; 4.5]
Comfortable deceleration
(m/s2)
1.67
1.67
[0.5 ; 3.1]
Standstill minimum distance
(m)
1
1
[0 ; 2]
IDM acceleration exponent
4
4
4
Vehicle length
(m)
4.5
4.5
Time gap
(s)
1.3
1
[1 ; 1.6]
Reaction time
(s)
N ~ (0.5, 0.12)
0
Desired speed
(m/s)
N ~ (110, 132)
110
Safe deceleration when overtaking
(m/s2)
1.67
0
Maximum deceleration
(m/s2)
7.5
7.5
[7.5 ; 9.5]
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Table 1: Model parametrisation of autonomous and regular vehicles.
We assume that AVs can react almost instantaneously to any stimulus from the driving environment and
therefore their reaction time is set to 0. Consequently, the gap time is also reduced. Specifically, we set the
gap time parameter to be 1 second for autonomous vehicles and 1.3 seconds for regular vehicles. In both
cases the parameter values are consistent with the literature and render a realistic and dynamically stable IDM
model.
For regular vehicles the reaction time is generally considered to be in the order of 1 second. However, as
pointed out in Kesting and Treiber (2008) this varies significantly between drivers. In the IIDM model, in order
to achieve robust string stability with the current set of parameters, the reaction time cannot be higher than
0.65 seconds. This is due to the delayed responses to the input stimulus (Kesting and Treiber 2008). Therefore,
for this stochastic representation of the IIDM framework the reaction time was drawn from a normal distribution
N (0.5, 0.12) and truncated between 0.3 and 0.65 seconds. As the simulation parameters are therefore lower
than the expected experimental values, we have put a significant effort into studying this matter and its possible
consequences. This has led to several sensitivity tests of the reaction time up to the limit of 0.65 seconds and
we are confident about the following conclusions. Although it may slightly underestimate the effects of
heterogeneity and thereby slightly overestimate the capacity utilisation of regular vehicles, effects are indeed
small. In any case, it will further strengthen the conclusion of the paper, namely that the cost of heterogeneity
is significant and that a large share of AVs is required to attain substantial gains in network utilisation. On a
side note, we have no hesitation in applying the model in a benchmark with AVs since the simulation model
mimics the fundamental diagram of the current traffic situation quite closely and with very little calibration effort.
In Section 2.2 we suggested that it would be possible to use a random distribution for the maximum
acceleration. However, in the following we will simplify the experiment by applying a constant rate of
maximum acceleration equal to 3 m/s2. There are three motivations for doing so. Firstly, this is only relevant
close to the entry point (upstream of the bottleneck). As soon as vehicles experience congestion resulting from
spillback, is largely irrelevant. Secondly, introducing a distribution for would force one to consider the
correlation between the desired speed and . It is likely that drivers with an aggressive acceleration profile
would also have a higher desired speed. Hence, the covariance matrix in Section 2.2 would have off-diagonal
elements. As we currently have very little information about this correlation we prefer the simpler assumption
of independence. Finally, introducing a distribution for would not change the overall conclusion of the paper
although it would likely amplify it somewhat.
While autonomous vehicles can be considered a homogeneous group with the same desired speed that
corresponds to the speed limit, the desired speed of regular vehicles will vary. Some drivers are more
aggressive and tend to exceed the motorway speed limits, while others choose to drive slower. The desired
speed profile of regular vehicles is assumed to follow a normal distribution V N (110, 132) km/h truncated
between 80 and 145 km/h based on the available data for motorways maximum speed in Denmark (Spaabaek
et al. 2007), UK (National Statistics UK 2015) and US (National Highway Traffic Safety Administration US
Department of Transportation 2012).
During an overtaking manoeuvre, a vehicle will overtake as long as the new following vehicle is able to
decelerate with rate equal to and avoid a collision. For regular vehicles, is considered equal to. It
means that as long as the new following vehicle is able to safely decelerate at a comfortable deceleration rate
of 1.67 m/s2 overtaking will take place. Instead, for autonomous vehicles the safe deceleration rate is
assumed equal to zero. This implies that AVs will behave optimally from a system perspective and that
overtaking will only occur when the vehicle does not affect the following vehicles.
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The maximum deceleration does not appear explicitly in the IDM formulation. As previously said, the best
well-known characteristic of the IDM is that the model is collision-free which is due to its dynamic updating but
also to the fact that the deceleration is unrestricted. Hence, in a critical situation the IDM can render high values
of deceleration, which could be physically unrealistic. In order to avoid such unrealistic situations, the IIDM is
implemented with a maximum deceleration. In this case the maximum deceleration was restricted to 7.5
m/s2, a conservative choice based on the existing literature.
The base scenario for regular cars is based on the stochastic version of the IIDM. This situation has been
calibrated to represent the actual traffic flow behaviour on the M3 motorway. With the parameters defined as
in Table 1 and the demand represented as a typical morning peak, the model renders very comparable results
as to what can be observed on the motorway. The following Figure 3(a) shows the observed fundamental
relation between speed and density (Brems and Nielsen 2012). The plot presents two fundamental curves,
namely pre-queue and queue discharge. Our experiment is essentially an attempt to reproduce the two first
parts, namely the “insignificant congestion” and “great congestion” area of the queue discharge fundamental
curve. Figure 3(b) shows the corresponding simulated speed and density relationship for this area. As we do
not enforce a complete stand-still, our curve will not reproduce the critical congestion area.
The simulation exercise, however, is an artificial representation of the M3 motorway system in that we do not
consider effects from ramps and intersections. This would complicate the evaluation and calibration of the
system considerably. This simplification could underestimate local congestion that might ‘jam’ the system
locally and change the vehicle-to-vehicle dynamics locally. It could lead to even stronger support for AVs as
the true environment is likely to be more heterogeneous compared to the single-bottleneck stylised system.
However, it is not expected to change overall conclusions of the paper.
(a)
P a g e | 14
(b)
Figure 3: Traffic fundamental relation between travel speed and traffic density. (a) Observed data based on GPS traces
on Copenhagen motorways (Brems and Nielsen 2012). (b) Base scenario simulated speed-density relation for one lane.
The colour gradient defines the number of observations for a given point, where each point represent a measurement in
the (x, y) space to form a heat-map. Yellow: 0-12, Green: 13-50, Pink: 50-150, Red: 150-250, Blue: 250-500, Black: 500-.
The colouring can therefore be considered as a measure of density from low (yellow) to high (black).
At the general level the simulation of the base scenario was validated looking at the traffic flow properties:
capacity drop and speed propagation of congested traffic waves. The capacity drop due to traffic breakdown
is reported to be between 10 to 30% (Wang et al. 2016). The simulated capacity drop was 15.1% for the base
scenario. Moreover, congested traffic waves propagate against the driving direction with a characteristic speed
in the order of 10 to 20 km/h (Wang et al. 2016). The simulation shows that the speed of the propagating traffic
wave is approximately 19 km/h. As the modelled situation does not consider a situation of hyper-congestion
with a complete stand-still (for which the waves are slower), it is expected that the wave propagation should
be close to the maximum.
At the local level we examined how the model replicated observed behaviour. Specifically, we measured the
average travel time (16.1 minutes) that vehicles took to complete the 13-kilometre segment of the motorway
in the base scenario simulation. This agreed well with findings in a report from Rambøll for the same motorway
(Rambøll 2015).
4.2. Results
4.2.1. Analysis of different driving behaviours of autonomous vehicles
Table 1 showed the parameters applied to represent the autonomous driving style. In the following we test the
sensitivity of the overall performance of the capacity utilisation by using different desired speeds and time
gaps for AVs. The sensitivity is tested in a mixed traffic environment with 50% AVs.
P a g e | 15
The sensitivity of the desired speed is analysed by keeping all the other parameters unchanged. We
investigate the effect of AVs having a desired speed of 80, 110 and 145 km/h. In the same way, we analyse
the sensitivity of the time gap by testing values equal to 0.5, 1 and 1.3 seconds.
The following Figure 4 (a) and (b) shows the sensitivity of the average travel time () with respect to the
desired speed and the time gap parameters whereas Figure 4 (c) and (d) shows the corresponding
sensitivity of the throughput (). Figure 4 (e) and (f) visualises the effect of the desired speed and the
time gap parameters on the capacity drop (). The base scenario reference values for 0% AVs are
represented as the “black frame”.
Average travel time sensitivity of desired speed
(a)
Average travel time sensitivity of time gap (b)
Throughput sensitivity of desired speed (c)
Throughput sensitivity of time gap (d)
P a g e | 16
Capacity drop sensitivity of desired speed (e)
Capacity drop sensitivity of time gap (f)
Figure 4: Sensitivity analysis with 50% AVs with different desired speed profiles and time gaps. The “black frame”
represents the situation of 0% AVs. Values refer to one lane.
In mixed systems (50% share of AVs and 50% share of RVs), introducing AVs with a higher desired speed
than the mean speed of regular ones (145 versus 110 km/h) has only negligible effects. Conversely, in mixed
systems if the desired speed of AVs is lower than the mean speed of regular vehicles (80 versus 110 km/h),
the performance is similar to the case of 100% RVs where the desired speed is around 110 km/h. In Figure
4-a the experiment where “slow” (desired speed of 80km/h) AVs are added to a mixed system, the RVs which
are in general faster than the AVs will overtake the “slow” AVs and therefore, in terms of travel time a similar
performance to the 100% RV scenario is observed. However, AVs have a shorter safe gap to the precedent
vehicle and drive in denser traffic conditions. Therefore, the mixed system with “slow” AVs still has a slightly
higher throughput than the scenario of 100% RVs, as observed in Figure 4-c.
The average travel time in a mixed traffic environment is dependent of the time gap , which characterises
the driving style of autonomous vehicles. If AVs represent 50% of the vehicles, the time savings compared to
the 0% base scenario could range from 3.7% to 17.0% when AVs’ time gaps vary from 1.3 to 0.5 seconds,
respectively.
The throughput and the capacity drop indicators are also more sensitive to the time gap parameter than to the
desired speed. For different values of the desired speed parameter, the throughput is approximately constant
and the capacity drops by 13.5%. For the time gap parameter on the other hand, a variation in the throughput
from 1800 to 2200 vehicles per hour and a change in capacity drop from 15.1% to 9.8% is observed.
4.2.2. Analysis of different penetration rates of autonomous vehicles
The effect of a mixed traffic environment is investigated by systematically varying the proportion of autonomous
vehicles. The results are presented for the base scenario with 0% of AVs and for six other scenarios with 10%,
25%, 50%, 75%, 90% and 100% AVs.
The assessment of the impacts of AVs is based on the following performance indicators
- average travel time of all the vehicles
- throughput from the congested area and the capacity drop that is induced by the traffic breakdown
- the queue that builds up as a consequence of the bottleneck and the increasing demand
- the speed reduction due to congestion and propagation of the congested traffic wave backwards
- fundamental diagrams
The average travel time () to complete the 13-kilometre motorway segment is presented in Figure 5 and
varies from 16.1 minutes in the base scenario without any AVs to 8.6 minutes in a scenario with 100% AVs.
However, as can be observed, the variation of travel time with respect to the share of AVs is non-linear. Hence,
the marginal travel time saving of adding more AVs to the system is much lower when the share of AVs is low
compared to a situation where the share is high. To obtain significant time savings, the proportion of AVs
should be at least 50%.
The average hourly throughput () is an indicator of the effective capacity of the motorway. Figure 6 shows
that the throughput approximately follows a linear function of the share of AVs. When going from a 0% AV
P a g e | 17
environment to a 100% AV environment, the throughput increases from 1800 vehicles per hour per lane to
almost 2300 vehicles per hour per lane.
However, the capacity drop () shows a different pattern. The capacity drop decreases as the rate of AVs
increases but significant reductions are only observed when the proportion of AVs is greater than 90%.
Moreover, the variance of the capacity drop across simulations is high. At higher AV penetration rates the
capacity drop decreases whereas its variance increases.
Figure 5: Performance indicator: average travel time (mean and standard deviation) for different AV market penetration
rates.
Figure 6: Performance indicators: throughput and capacity drop (mean and standard deviation) for different AV market
penetration rates.
P a g e | 18
As the demand varies over time, it is possible to observe how the queue builds up across the simulation period
and how it dissipates when the demand is reduced. From Figure 7 and Table 2 we can assess the relation
between the length of the queue () and the AV penetration rate. Similarly, as we saw for the capacity drop,
the length of the queue is barely reduced at low AV penetration rates. The highest change in the length of the
queue is when changing from a 90% AVs environment to a completely automated environment. This indicates
that there could be substantial benefits of enforcing AVs to operate in separate or closed autonomous systems.
Figure 7: Performance indicator: mean length of the queue () for different AV market penetration rates.
Penetration rates
AVs
Queue area of vehicles ()
%
Mean
95% CI
0
31664
[29095 ; 37244]
10
30926
[28176 ; 35872]
25
29644
[26971 ; 34891]
50
27104
[24015 ; 31660]
75
23716
[17633 ; 29464]
90
19557
[13616 ; 23572]
100
117
[117 ; 117]
Table 2: Performance indicator: queue area (mean and standard deviation) during 180 minutes of simulation time for
different AV market penetration rates.
Data collected by means of virtual loop detectors located at strategical locations enable us to picture the
fundamental diagrams of the system. These diagrams characterise the relationship between traffic flow,
density and speed. The data is collected for 17 virtual detectors of which most are located just before and after
the bottleneck in order to measure spillback effects and maximum flow.
By comparing the structure of the flow-density diagrams shown in Figure 8 and the rest of the fundamental
diagrams, which are included in Appendix B, it is visualised how the flow and the speed increase with the AV
share. However, the density level at which the traffic starts to break down is not very different across scenarios.
Speed-density and speed-flow curves (Appendix B) show that at higher shares of AVs, the speed before the
P a g e | 19
congested regime increases and equals the maximum allowed speed. The fully automated scenario is a
deterministic simulation and therefore represents a diagram with almost no scatter.
0% AVs
25% AVs
50% AVs
75% AVs
90% AVs
100% AVs
Figure 8: Flow-density diagrams, for different scenarios of AV market penetration rates for a single lane. The colour
gradient defines the number of observations for a given point, where each point represent a measurement in the (x, y)
P a g e | 20
space to form a heat-map. Yellow: 0-12, Green: 13-50, Pink: 50-150, Red: 150-250, Blue: 250-500, Black: 500-. The
colouring can therefore be considered as a measure of density from low (yellow) to high (black).
As can be observed, there is a continuous increase of the maximum flow as the AV share increases in the
scenarios from 0% to 90% AVs. In these scenarios there is a degree of heterogeneity among the vehicles,
which causes the system break down at the bottleneck. As a result, vehicles concentrate upstream of the
bottleneck and when they get through it, they head a relative empty motorway with normal speed conditions.
One of the virtual detectors is positioned just after the bottleneck and will record the maximum flow in the
system. As expected, the higher the share of AVs, the higher the maximum flow.
On the other hand, the homogeneous scenario with 100 % AVs never breaks down. Instead, AVs tend to
platoon with a minimum safe gap distance (which depends on the speed conditions at every different position
in the motorway). This means that the vehicles do not concentrate upstream of the bottleneck. As a result, the
virtual detector located just after the bottleneck will not record an equal high flow as observed in the mixed
scenario. Instead, the 100% AVs scenario presents a theoretical diagram with maximum flow of approximately
2500 vehicles/hour and little scatter. This is because the AVs arrange smoothly into platoons and ‘stop and
go’ traffic is avoided.
4.3. Socio-economic impact of separate AV lanes
To investigate how autonomous vehicles may affect the planning of our infrastructure, we present a simple yet
relevant cost-benefit analysis in order to benchmark a system with separated AV lanes compared to a system
for mixed traffic. As we have seen previously, autonomy has an effect, not only the throughput of vehicles but
also on the direct savings with respect to travel time. Hence, to capture the joint effect, a cost-benefit analysis
is required.
The computation of the travel time savings relies on the average travel time savings from the simulation
results presented in Figure 5. This is then converted to a monetary value by using the official social value-of-
time (VoT) as published by the Ministry of Transport of Denmark (Transportministeriet 2017). Below we
present the annual surplus for a three-hour peak period. It is not trivial how to convert these values to annual
benefits as it would require to simulate over the entire 24 hours rather than the specified period. Also,
because of the stylised setup and specific nature of the bottleneck design, the absolute values are of less
importance. Rather, the importance is the relative difference between the different shares of AVs.
Share of AVs
Direct benefits
New benefits
Total benefits
%
M DKK / lane / 3-
peak hour/ year
M DKK / lane / 3-
peak hour/ year
M DKK / lane / 3-
peak hour/ year
0
-
10
1.60
0.01
1.61
25
4.05
0.06
4.11
33.3
5.17
0.10
5.28
50
8.49
0.28
8.77
75
14.13
0.67
14.80
90
20.46
1.17
21.64
100
42.50
2.88
45.38
P a g e | 21
Table 3: Total benefits calculated for different scenarios of AV market penetration rates, measured in M
DKK/lane/3-peak hour/year.
Results indicate that most of the benefit is due to travel time savings and to a less extent throughput. This is a
consequence of the congestion level in the system (over the three hour period). The cars in the system is not
experiencing a high degrees of congestion and this is why the throughput effect is small and the direct time
effect is large. Put differently, over the three hour period, cars will eventually pass through the system whether
these are autonomous or conventional and the total throughput will be largely invariant because cars before
and after the peak period make it up for the capacity loss in the peak. However, their speed is affected because
of the idealised performance of the IDM model. Had we considered a future situation with additional demand
and more congestion, a higher share of the benefits would arrive in the form of new benefits and we should
also see a relative improvement of scenarios with a high share of AVs.
Results also illustrate the non-linear relationship between the share of AVs the net total benefits of the system.
Benefit more than doubles when the share of AVs increases from 90% to 100%.
Needless to say, there are several uncertainties underlying these results and caution should be taken when
using the absolute numbers. This will be elaborated in more details in Section 4.4. However, relatively
speaking, the impact of an AV penetration rate represents a qualified “ball-park” figure for what can be
expected in the future based on a set of assumptions.
A straightforward analysis based on the above calculations is a comparison between a three-lane motorway
on which no dedicated lanes are enforced, i.e. a 33.3% of autonomous vehicles in all lanes, and a situation of
a single dedicated lane for autonomous vehicles and two other lanes for regular cars. The results are shown
below in Table 4.
Mixed lanes
Dedicated lanes
(3x33%)
(2x0% + 1x100%)
Total year benefits (M DKK/ 3-
peak hour/ year)
15.9
45.38
Vehicle throughput (vehicles/3h)
Regular vehicles
9171
8810
AVs
4585
4745
Total
13756
13555
Average travel time (minutes)
Regular vehicles
15.2
16.1
AVs
15.2
8.6
Weighted average TT
15.2
13.5
Table 4: Comparative evaluation of motorway planning with mixed traffic lanes or with dedicated lanes for AVs.
The analysis suggests that the potential benefit of planning dedicated lane(s) for AVs could be beneficial
although the analysis hinges on several premises. As noted, the main benefits emerge in the form of saved
travel time. Hence, the dedicated lane alternative causes the AVs to fully benefit from their enhanced
capabilities.
P a g e | 22
4.4. Planning and policy implications
As illustrated in the previous section, a physical separation of AVs and regular vehicles, possibly in the form
of separated lanes, is likely to lead to better capacity utilisation. As such, the result provides a valuable input
for the discussion of how an effective future AV infrastructure is to be designed. However, the analysis is
‘stylised’ and simple in its measurement of throughput and time but also with respect to its underlying
assumptions. In reality, the implementation of a separate lane infrastructure is likely to be less tempting for a
number of reasons, which we will be elaborate in the following.
The first implication of the findings is that planners and politicians should aim for a separate AV infrastructure.
As we saw, if it can be assumed that the level of AVs can “supply” a single lane to perfection, this will potentially
render substantial benefits in the form of higher throughput and faster travel time. However, in reality, at least
in the first phases of automation, this will not be possible and the likely under-utilisation of the AV lane will
lower the performance accordingly. Therefore, it is relevant to consider policy incentives that may seek to
utilise the supply of dedicated lanes up to the maximum capacity. This in turn also implies that a separate
infrastructure needs to be put in place, only when the penetration rates reaches a certain threshold where the
gains from the improved capacity utilisation offset the possible underutilisation of AV lanes.
A second implication that derived from the above discussion relates to the challenges of how to access and
exit dedicated lanes. In the example put forward in our paper, cars are assumed to exit the system without any
consideration where these cars end up. Hence, we do not consider how AVs get from the ramp to the dedicated
lane and back again. If AV’s are driving in the inner-lane, as it is often suggested, these cars needs to be
routed to the outer lane when exiting the motorway and this will imply that the cars are mixed temporarily with
regular cars. If cars are driving in the outer-lane a mixing is also inevitable because regular cars need to pass
the AV lane. This suggests that from a planning perspective, it is important to focus on the entry and exit
problem and possibly formulate separate exit and entry infrastructures if possible. A mean to accomplish a
smoother exit and entry is to ‘connect’ cars with the infrastructure. It may be possible to have dedicated signals
for specific vehicles.
A somewhat related issue is that AV’s operating in complicated urban environments, with traffic lights,
crossings and alternative soft modes mixed with cars, will have much lower capacity utilisation gains compared
to the situation where these cars operate on motorways. Therefore, it will be important to first focus on the
motorway system and subsequently on other types of non-urban roads in a second stage.
“Platooning technology was tested in other simulations not reported in this paper. These involved bundling of
groups of vehicles with even lower time gaps and it showed to be effective as a mean to improve the density
of the system. These tests were only carried out for a dedicated AV lane as platooning in mixed traffic involves
a more complicated overtaking setup.” This is consistent with other finding in Guo and Ma (2019). However,
platooning brings above other challenges that related to the aforementioned. The size of platoons may hinder
effective lane crossings and if the density increases substantially at exits it will amplify some of the problems
mentioned before. In particular, it will call for connectivity between the infrastructure and the cars and platoons
to make prioritisation schemes.
Finally, another policy implication relates to legislation. It is important to make sure that AV’s are indeed
homogenous and comply to common standards as regard desired speed, accepted safety gaps, acceleration
and de-acceleration patterns. The more heterogeneity is allowed, the less effective the AV’s will be. It is a non-
trivial question, which common standards should be preferred for others and which should come first?
5. Summary and conclusion
P a g e | 23
The aim of this study is to investigate the effects of autonomous vehicles at different levels of market
penetration for a congested motorway. We present a microscopic simulation framework that is able to model
mixed vehicle classes combined with human heterogeneity. To this end, the model formulation for autonomous
vehicles is based on the deterministic IIDM while a stochastic version of the IIDM is developed to model
heterogeneity resulting from regular vehicles.
We propose a way to model the congestion and the resulting spillback for a single representative lane a virtual
representation of the Copenhagen M3 motorway. This approach is computationally simple and led us to
circumvent a great deal of computational complexity while maintaining realism in the way equilibrium dynamics
between lanes were modelled. This led to a more robust traffic simulator model and with the possibility of
investigating a large number of scenarios.
Simulation results revealed that a significant increase in capacity utilisation is possible when market
penetration rates of AVs exceed 50%. The penetration rate of AVs and the resulting benefits do not follow a
linear relationship. Hence, higher marginal benefits are observed at high penetration rates. In summary, our
results indicate the following:
o Relative travel time savings with reference to the current baseline are 9% for a 50% penetration rate
of AVs and 16% for 75% AVs. In a completely automated driving environment, the travel time is
reduced by as much as 49%.
o The throughput increases as well. With respect to the current baseline, the relative increase is 8% for
a 50% penetration rate of AVs and 14% for 75% AVs. The maximum throughput is attained in a
completely automated driving environment and is 30% higher than observed for the baseline.
Clearly, the results of the paper open up a discussion of how to plan our existing motorway infrastructure.
Based on our results, it is suggested that, from a system optimum perspective and for early stages of
implementation dedicated lanes for AVs could be beneficial.
At a more fundamental level, our study suggests that although AVs have the potential of benefitting the traffic
system by extending its capacity it could be difficult to perceive these benefits at early stages of
implementation. At least there are several challenges related to the design of transport systems that need
consideration. A real and sizeable challenge is how AVs, should we decide to introduce dedicated lanes, can
access and exit these lanes with minimal interference with the rest of the system. Another challenge is that the
optimal utilisation of dedicated lanes is based on a perfect match between supply and demand. Hence, the
findings in our paper in many ways represent the “maximum” perceivable potential. In reality, there is no such
thing as a constant share of AVs as it will vary during the day and over time. This underlines that the experiment
is indeed stylised. As it is unlikely that all future AVs will be completely aligned and homogenous and that we
can attain the 100% autonomy, our estimates are an upper bound for the expected effects on motorways. On
smaller roads and in particular on roads in urban areas, the potential is much lower unless we can synchronise
signals and cars, which is another interesting yet different research path. Another path of research is to study
how AVs will perform in more complex multi-bottleneck systems, possibly with ramps and intersections. While
this involves a more complicated setup and calibration, it could be used to investigate local bottleneck effects
of an increasing share of AVs. Other sensitivity tests could also be introduced to investigate the effect of
increasing variance. In this setup, we investigate the cost of variation by changing the share of AVs. However,
it could also be investigated by changing the variance of certain input parameters, e.g. such as desired speed.
The finding of the paper are in line with Atkins (2016), who finds that capacity utilisation can be significantly
affected by AVs and that a large share of these AVs are required to attain good performance. The paper is
also in line with Taha et al. (2019) who suggest that improved capacity utilisation can be expected for different
P a g e | 24
stages of autonomy and Li et al. (2020) who presented evidence that capacity of mixed flow increases convexly
with the AV penetration rate and that right-of-away strategies.
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Author information
Affiliations
Transport Division, Department of Management, Technical University of Denmark, Bygningstorvet 1,
Kgl. Lyngby 2800, Denmark.
Andrea Papu Carrone
Jeppe Rich
Christian Anker Vandet
College of Transportation Engineering, Tongji University, 4800 Cao’an Road, Shanghai, 201804, R.P.
of China
Kun An
Contributions
The authors confirm contributions to the paper as follows: Andrea and Jeppe developed a first version of the
simulator, designed the different scenarios. Andrea developed and coded the different performance
indicators under the supervision of Jeppe. Christian coded and tested alternative overtaking models and
helped validating these. Andrea and Jeppe prepared a first draft and received valuable comments and
suggestions from Kun in this process.
P a g e | 25
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Appendix A: Lane changing models
The dynamics of overtake manoeuvres in combination with complex dynamic spillback introduce several
challenges, e.g. a need for sideways orientation and corresponding parameters of which little is known in the
context of mixed driving environments.
In the paper, a true lane-changing model was implemented and tested. It was based on a probabilistic model
for the choice of overtaking conditional on spacing and speed conditions. More specifically, it consisted of a
probabilistic choice (logit model) of making an overtake conditional on the distance to the leader in both
lanes and by comparing with the desired launch gap. Conditions for making an overtake is that that the
leader in the current lane is driving slower than the overtaking vehicle and that the leader in the opposite
lane is driving faster than the leader in the current lane and that the current gap is larger than the desired
gap. The lane-changing model however, requires a complex book-keeping and introduce computational
challenge when exploring many scenarios and when linking with the complex IDM model. To be able to
explore a wider set of scenarios an alternative ‘shadow lane approach’ was implemented. The approach is
described in greater details below, but is based on the same principles as an overtaking model but in a way
that makes calculations much simpler.
Shadow lane approximation
The “shadow lane” approach makes it possible to extent any single lane model in a very simple and efficient
way and to deal with overtake manoeuvres in a way which is realistic in congested networks. It means that the
model by design simulates the traffic flow of a representative lane for which overtaking is possible rather than
the entire dynamic pattern across all lanes.
More specifically, overtaking manoeuvres are modelled as a discrete choice at a given time step between
two alternatives: no overtake and overtake. A vehicle can choose to overtake; however, the required space in
the contiguous lane in which the manoeuvre is supposed to take place may not always be available. Hence,
when vehicles select the overtaking alternative as the “best” one, they also need to check for available space
to perform the manoeuvre. To accomplish this, a shadow lane is modelled. The shadow lane is assumed to
have the same traffic density as the modelled lane, e.g. resemble an equilibrated system of lanes. The
probability of finding the required space to perform the overtake manoeuvre is modelled as a function of the
traffic density in the shadow lane. For a road segment with high traffic density, i.e. congested regimes, the
probability of an overtaking being viable is low while in free-flow regimes the probability of being able to
overtake is high.
The overtaking procedure is formulated based on the following assumptions:
o The first vehicle in the system cannot be overtaken and therefore its desired speed is the highest in
the system
o The speed of a vehicle before and after an overtaking is the same (
o The “best” alternative is the one that allows a vehicle to maximise its speed, therefore, it is the
alternative for which a greater acceleration can be performed
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Figure 9: Definition of notation in an overtaking situation.
Figure 9 illustrate the situation before and after the overtaking and presents the notation used in this
subsection. Note that the notation of the vehicles changes in the “after overtake” situation. Hence, vehicle in
the “before overtake” situation becomes vehicle in the “after overtaking” situation, and becomes ,
whereas becomes . To simplify notation the time step sub index is suppressed.
A vehicle that overtakes fulfils the following conditions:
o The vehicle must have entered the system
o The speed of the vehicle must be lower than its desired speed (it overtakes to be able to drive at its
desired speed)
o The desired speed of the vehicle must be greater than the speed that is being performed by its leader
o The vehicle must be as close as possible to its leader (the ratio of the desired safe gap and the gap
must be approximately equal to 1)
A safe braking condition ensures that the distance between the position in which the vehicle enters after
overtaking and the position of the new following vehicle is wide enough for the follower to decelerate at
a safe deceleration rate and avoid a collision. The minimum required gap among vehicles is derived
from the IIDM acceleration function.
(10)
A vehicle will only perform an overtaking manoeuvre if it can accelerate to a higher speed compared to the
non- overtaking situation. In this case, an acceleration threshold of 0.5 m/s2 has been considered. Hence,
the required gap between the position in which the vehicle enters after overtaking and the position of its
new leading vehicle should be large enough to enable this increase in acceleration. The required gap
is derived from the IIDM acceleration function. Hence,
P a g e | 29
(11)
In order to fulfil the previous conditions stated in Equations (10) and (11), the distance gap of the vehicle
that is overtaken should satisfy the following:
(12)
It is possible that a vehicle selects overtaking as the "best" alternative, i.e. all the previous conditions are
fulfilled, but in the end does not carry out the overtaking. When overtaking is selected as the "best" alternative,
the vehicle must check for space availability in the shadow lane. To this end, the average spacing among
vehicles in the shadow lane is calculated as the average spacing in the vicinity of the vehicles’ location.
Furthermore, the probability of finding the required space to perform the overtaking manoeuvre is modelled
as a linear function of the average spacing in the shadow lane.
To simulate if a vehicle is able to find the required space for overtaking, we then draw a number from a uniform
distribution. If the random draw is lower or equal to the probability of finding the required space to overtake,
the vehicle performs the overtaking manoeuvre. If not, the vehicle will stay in its position until the next time
step in which it can check for overtaking.
The overtaking procedure assumes that a vehicle can instantaneously move from its position before
overtaking to the position after overtaking (see Figure 10). Therefore, vehicle jumps the distance
difference between these positions, which is equal to the sum of its gap, the safe gap and its length.
All vehicles engaged in overtaking activities are therefore faced with a reduced travel time. Consequently, the
time it would take for a given vehicle to cover this distance at the given speed is stored and added to the
travel time of the vehicle at the end of the simulation. The purpose of the overtaking simulator is not to mimic
the entire overtaking manoeuvre at each time step at all positions but to provide a good approximation of the
lane density, the flow, the speed and flow for the examined lane under the assumption of lane equilibrium. This
is indeed possible and render a much simpler calculation setup which enable us to explore many more
scenarios.
P a g e | 30
Appendix B: Fundamental diagrams
In Figure 10 to Figure 12 the colour gradient (as before) defines the number of observations for a given point,
where each point represent a measurement in the (x, y) space to form a heat-map. Yellow: 0-12, Green: 13-
50, Pink: 50-150, Red: 150-250, Blue: 250-500, Black: 500-. The colouring can therefore be considered as a
measure of density from low (yellow) to high (black).
0% AVs
25% AVs
50% AVs
75% AVs
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90% AVs
100% AVs
Figure 10: Speed-flow diagrams, for different scenarios of AV market penetration rates: (a) 0% AVs, (b) 25% AVs, (c)
50% AVs, (d) 75% AVs, (e) 90% AVs, (f) 100% AVs.
0% AVs
25% AVs
50% AVs
75% AVs
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90% AVs
100% AVs
Figure 11: Speed-density diagrams, for different scenarios of AV market penetration rates: (a) 0% AVs, (b) 25% AVs, (c)
50% AVs, (d) 75% AVs, (e) 90% AVs, (f) 100% AVs.
0% AVs
25% AVs
50% AVs
75% AVs
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90% AVs
100% AVs
Figure 12: Speed-spacing diagrams, for different scenarios of AV market penetration rates: (a) 0% AVs, (b)
25% AVs, (c) 50% AVs, (d) 75% AVs, (e) 90% AVs, (f) 100% AVs.
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