Capacity Analysis of Intersections When CAVs Crossing in a Collaborative and Lane-Free Order

Connected and autonomous vehicles (CAVs) improve the throughput of intersections by crossing in a lane-free order as compared to the signalised crossing of human drivers. However, it is challenging to quantify such an improvement because the available frameworks to analyse the capacity (i.e., the maximum throughput) of the conventional intersections does not apply to the lane-free ones. This paper proposes a novel theoretical framework to numerically simulate and compare the capacity of lane-free and conventional intersections. The results show that the maximum number of vehicles passing through a lane-free intersection is up to seven times more than a signalised intersection managed by the state-of-the-art max-pressure and Webster algorithms. A sensitivity analysis shows that, in contrast to the signalised intersections, the capacity of the lane-free intersections improves by an increase in initial speed, the maximum permissible speed and acceleration of vehicles.


Introduction
Capacity analysis of intersections is essential for the management of traffic and planning transport systems. Unlike traditional human-driven vehicles (HVs) which are restricted to travel within the road lanes, connected and autonomous vehicles (CAVs) enable a lane-free crossing through intersections.
There are extensive prior works to characterise the capacity of intersections for HVs (e.g. [1,2,3] [1]. The manual introduces a measure to quantify the capacity of the unsignalised two-way and all-way stop-controlled intersections based on, respectively, gap acceptance and queuing theories. Meanwhile, it is recommended in [1] to calculate the capacity of the signalised intersections as the saturation flow rate times the green time ratio. All of these measures assume that headway of each HV in the queue of lanes is known to be around 1.9 s. This assumption makes these measures inappropriate for lane-free intersections where the headway of CAVs is much smaller and almost the same for all the vehicles in the queue [4]. To evaluate capacity of the intersections with CAVs, the authors in [5,4,6] employed the same measure that is defined in [1] for the unsignalised intersections, though with a new headway definition for CAVs. In [5], intersections are assumed as service providers and CAVs headway is redefined as service time (i.e., crossing time) which is derived by applying queuing theory. The service time is based on the safety time gap of CAVs approaching the intersection from the same stream and from the conflicting streams. A similar work is proposed in [4] that employs the M/G/1 queue model to drive a formula for the capacity of the intersections. This model assumes that the intersection capacity is equivalent to the service rate of vehicles. Finally, the authors in [6]  It is also important to note that CAVs are heterogenous in terms of their control strategy [5] and any measure to quantify capacity of intersections must be independent of the performance of these strategies. However, the majority of the abovementioned research measure the capacity of intersections that are controlled by a reservation-based strategy.
The first type of reservation strategies is called intersectionreservation where the controller reserves the whole intersection for one CAV at a time. The authors in [7] formulated an optimal control problem (OCP) to minimise the crossing time of CAVs while reserving the whole intersection to avoid collisions. The references [8] and [9] introduce an intersection crossing algorithm where CAVs are placed into a virtual platoon based on their distance to the centre of the intersection. The algorithm reserves the whole intersection for each CAV in the platoon to pass through without collision. Generally speaking, reservation of the whole intersection reduces the capacity of the intersection.
The second type is a conflict-point-reservation strategy that reduces the area of reservation to just a few conflicting points.
The authors in [10,11] designed optimisation-based algorithms to realise this type of reservation algorithms. A similar work is proposed in [12], where a constraint is added to the optimisation problem for each conflict point to limit the maximum number of crossing vehicles at any time to one. Even though the reservation-based strategies improve the capacity of intersections by nullifying the stop-and-go requirement of the conventional signalised intersections, yet vehicles must follow a set of predefined paths and are not able to fully utilise the intersection area by lane-free manoeuvres.
Alternatively, the authors in [13] developed an OCP to formulate the lane-free crossing problem of intersections. The objective of the developed OCP is to minimise the crossing time of CAVs and therefore the algorithm generates time-optimal trajectories for each CAV. However, the proposed OCP contains a set of highly non-convex constraints to represent the collision avoidance criteria which makes it difficult to solve online. To resolve this issue, Li et al. [13] splits the non-convex formulation into two stages. At stage one which is solved online, CAVs CAVs [13].
In a more recent study, Li et al [14] changed the minimumtime OCP of [13] to a feasibility problem to make the nonconvex formulation tractable. However, this results in a suboptimal solution. The authors in [15] resolved the previous issues by using dual problem theory to convexify the non-convex

Capacity evaluation of intersections
HVs A measure based on gap acceptance theory with CAV's follow-up and critical gap time [6] CAVs A measure based on queuing theory with CAV's service time [4,5] The proposed unified measure in this paper A measure based on queuing theory with HVs service time [1] A measure based on gap acceptance theory with HV's follow-up and critical gap time [1] A measure using saturation flow rate & green time ratio [1] Unsignalised intersections two-way stopcontrolled Unsignalised intersections all-way stopcontrolled

Reservation-based intersections
Lane-free intersections   intersections with respect to different values of speed and acceleration is presented in Section 5 followed by a conclusion in Section 6. assumes that there is a centralised coordinator for this purpose that is placed at the intersection.

System Description
The formulated collision avoidance constraints for the red and black CAVs after applying the dual problem theory [15] are also shown in Fig. 2 represents the dual form of the distance between the two CAVs where S i j is the separating hyperplane placed between them. Similarly, the expression −b ⊤ i λ ir − b ⊤ r λ ri represents the dual form of the distance between the green CAV and the highlighted road boundary and S ir is the separating hyperplane. For more details on the convexification of collision avoidance constraints, refer to [15].
It is worth noting that capacity of intersections is defined in this study as the maximum number of vehicles that can continuously cross the intersection within a unit of the time and under predefined conditions (e.g, intersection geometry, roadway and distribution of traffic) [5,17]. This definition is equivalent to the service rate in queuing theory and it is not the same as the one provided in HCM [1].

A Novel Framework to Quantify Capacity of Intersections
Conventionally, capacity of intersections (both the signalised and unsignalised) are measured using a set of collected data from either real-time observation of vehicles [1] or running a micro-simulation [18,19].  Figure 2: A lane-free and signal-free intersection. S i j and S ir are the separating hyperplanes between, respectively, two CAVs and a CAV and road boundary. However, such real-time data are not available for CAVs (especially for lane-free crossing of CAVs) because of the lack of real infrastructure or realistic simulators that consider enough number of heterogeneous CAVs crossing an intersection.

The proposed novel measure of capacity of intersections
Intersections can host limited number of vehicles at the same time and if the intersection capacity exceeds, either the waiting time of crossing vehicles will significantly increase or collisions will happen. Therefore, to evaluate the capacity of intersections a suitable measure must consider the maximum number of vehicles and the time that it takes for those vehicles to pass through the intersection without any collisions. In effect, the following measure is proposed to calculate the capacity of any intersec-tion: where C is the capacity (veh/h) of the intersection, N denotes the number of crossing HVs/CAVs (veh) and T represents the time (s) that takes for those vehicles to fully cross the intersection.
It is already shown in [15] that the minimum crossing time

Calculation of T min and N max for lane-free intersections
The time-optimal crossing algorithm proposed in [15] for the lane-free intersections is employed to calculate the minimum crossing time T min and the maximum number of CAVs N max that can cross the intersection within T min without collision. The work formulates a centralised convexified OCP to generate global optimum crossing time of CAVs. The formulated time-optimal OCP is solved for gradually increasing number of CAVs to determine the N max . For the sake of complete-ness of this paper, a summary of the algorithm, including the considered dynamics of the vehicles, collision avoidance constraints and the resulting OCP, is also explained here.

Modelling of CAVs and road boundaries
CAVs are represented in this study with their two degree-offreedom (DoF) bicycle models [20].
where the control inputs and states of CAV i are presented as u = and a i (t) are the wheel steering angle (rad) and acceleration (m/s 2 ) of CAV i . The constants m and I z represent, respectively, the mass (kg) of the vehicle and its moment of inertia (kg.m 2 ) around axis z. The vehicle parameters Y r , Y β , Y δ , N r , N β and N δ are calculated as in [20].
To ensure CAVs drive within their admissible range, the following constraints are imposed for each CAV i : where . and . are the lower and upper boundaries respectively.
Each CAV i is presented as a rectangular polytope β i where it is composed of intersection of half-space linear inequality Road boundaries are also modelled with convex polytopic sets O r where r ∈ {1..N r } and N r denotes the total number of road boundaries which is 4 for a four-legged intersection.
where λ i j , λ ji , s i j , λ ri , λ ir , s ir are dual variables and the subscripts i, j and r refer to, respectively, CAV i , CAV j and rth road bound-ary. A i and b i represent the size and location of CAV i which are functions of the CAV i 's pose z i (t). (6c)-(6h) constrain CAVs to avoid collisions with each other and with road boundaries [15].
Problem (6) is nonlinear and is solved using CasADI [21] and IPOPT [22] for any given number N and initial location of

Measurement of the capacity of signalised intersections
This study employs the Webster [23] and max-pressure [24] algorithms as the adaptive traffic controllers of the signalised intersections. Webster in [23] derived a formulation that calculates the cycle length of traffic lights. The derived cycle length is used to find the green time of each phase to allow vehicles to cross the intersection. Similarly, the max-pressure algorithm calculates the signal timings, however, the green time of each phase is calculated based on the number of vehicles in the incoming and outgoing lanes [24].
Whilst Webster is a well-known algorithm for timing control of traffic lights, it is already shown that the max-pressure algorithm yields the lowest travelling time, queues length and crossing delays among all the state-of-the-art controllers, including the algorithms based on the self organising [25,26], deep Q-network [27], deep deterministic policy gradient [28] and Webster [23] methods [29].
To calculate the capacity of signalised intersections, this study simulates both the max-pressure and Webster algorithms in SUMO for gradually increasing number of HVs based on the works in [29]. As the number of vehicles increases, through- Safe margin between CAVs (m) 0.1 put of the intersection also increases until reaching its capacity (i.e., C in (3)) where adding more vehicles causes a drop in throughput due to a sharp rise in the crossing time.

Capacity of the Lane-Free Intersections as Compared to the Signalised Intersections
The uniform definition of capacity as in (2)   It is worth noting that this different behaviour has no effect on the calculation of the maximum throughput. Fig. 3 shows that the maximum number of crossing vehicles through the intersection is the same and equals 21 for both the lanefree and signalised with the max-pressure cases. However, it is slightly lower (i.e., 18 vehicles) for the signalised intersection with the Webster algorithm.

Sensitivity Analysis of the Intersection Capacity
This section shows that capacity of the lane-free and signalised intersections varies by changes in the initial speed, maximum speed and maximum acceleration of vehicles. The crossing scenario is the same for both the intersections that makes the results comparable.    Fig. 5a and Fig. 5b show that the crossing time is almost fixed to a constant value when the maximum speed of CAVs exceeds 18 (m/s) and 25 (m/s) for initial speed of, respectively, 5 (m/s) and 10 (m/s). In other words, even though CAVs are allowed to drive with a faster speed, yet they are not able to cross the intersection faster than a limit.  Fig. 3 shows that the number of crossing vehicles N at which throughput of the intersection reaches its capacity is almost fixed for a given intersection (e.g., that is 21 for the intersection in Fig. 2).

Signalised intersections
Hence, referring to (2), capacity of the signalised intersections is insensitive to the parameters.   This work also provides a benchmark to evaluate the performance of the algorithms that will be developed to collaboratively cross CAVs through intersections. Future work will extend the provided analysis to the case with multiple intersections that consider more factors such as passenger comfort into the measurement of capacity.