In the context of road traffic systems in Japan, let us consider a typical single-lane urban intersection without an auxiliary short right-turning lane or a dedicated turning signal phase, due to a small volume of turning traffic. In a green phase, a vehicle can turn right only if there is a safe gap between a pair of vehicles in the opposite lane or wait until the end of the red phase (i.e., turn in the inter-phase time). When a vehicle fails to turn smoothly at such intersections in relatively dense traffic, it often blocks many straight-going vehicles behind it, leading to severe congestion, as illustrated in Fig. 1. Coordinating a large number of vehicles from a distance optimally is not feasible in such critical scenarios due to the complexity of the required computations and communication, as well as the necessary safety concerns. Therefore, a limited centralized coordination followed by fully distributed local control is the best solution, as considered in this paper. In this study, we assume an upcoming scenario in which all vehicles communicate their states and turning intentions to the ICU upon arrival within the communication range. It is also assumed that the communication is ideal, with no significant delay, and the ICU has a proper sensing method to detect any pedestrian crossing, thereby constraining the turning time in the optimization to avoid a potential accident. The problem considered for the ICU is to optimally determine the safe turning and crossing times of all vehicles, thereby minimizing the overall traffic delay and coordinating vehicles efficiently.

For the development of a new traffic coordination system, the traditional vehicle control with turning decisions at the intersection is modeled first. The dynamic behavior related to the longitudinal motion of a vehicle is the basis of this study, where the position xn, velocity vn, and acceleration an of any vehicle n at discrete time t with step size Δt are expressed as (1)xn(t+1)=xn(t)+vn(t)Δt+0.5an(t)Δt2, (2)vn(t+1)=vn(t)+an(t)Δt.

The primary driving task is determining the control input an(t) based on the traffic flow, signal status, and road conditions. For typical human driving, a time-continuous car-following model called the Intelligent Driver Model (IDM) is employed in this study to simulate traffic conditions under traditional scenarios [20]. Typically, the acceleration of vehicle n can be obtained using the state of the vehicle and its preceding vehicle or traffic signal, which can be given as (3)an′(t)=fcf(xn(t),vn(t),xn−1(t),vn−1(t)).

Using the IDM, the car-following function fcf is defined as (4)fcf(xn,vn,xn−1,vn−1)=a(1−(vnvd)4−(R0+vnT+vn(vn−vn−1)/2abxn-1−xn−ln)2),where variables ln, vd, R0, T, a, and b represent the vehicle length, target velocity, minimum distance between vehicles, target time gap between vehicles, maximum acceleration, and minimum comfortable deceleration, respectively.

The IDM can regulate the acceleration of a vehicle based on its position and velocity; however, it cannot directly incorporate the effects of traffic signals or turning points where a car must stop first. Therefore, for any target stopping point Xstop ahead (e.g., due to a red signal or safety confirmation before turning), a data-driven model is adopted to describe a vehicle motion closely approaching a red signal [21]. Specifically, the velocity as a polynomial function of distance-to-stop lstop for lstop≤Lth is obtained from experimental driving data as (5)vn∗(lstop)=∑o=1𝒪μolstopo,where, based on experimental driving data fittings, the order of polynomial fittings is 𝒪=5, and coefficients μo are obtained as μ5=5.635×10−10, μ4=−3.446×10−7, μ3=+7.925×10−5, μ2=−8.519×10−3, and μ1=0.4805.

Using vn∗(lstop) in (5) and its derivative dvn∗(lstop)/dlstop, the acceleration (or deceleration) of any vehicle n with a velocity vn and stopping distance lstop=Xstop−xn towards a complete stop can be estimated as (6)fstop(xn,vn,Xstop)=vn2vn∗(lstop)×(dvn∗(lstop)dlstop).

Since the model is valid only for a limited distance lstop≤Lth, the acceleration of a vehicle approaching the intersection is obtained as (7)an′′(t)={fstop(xn(t),vn(t),Xstop),if~Xstop−xn(t)≤Lth,fcf(xn(t),vn(t),Xstop,0),otherwise.

In (7), for the case lstop>Lth, the car-following model in (4) is used with a target to stop at the intersection stopping point. In the event of a green signal for a straight-going vehicle, Xstop is assumed to be infinitely large (i.e., Xstop=∞). Finally, the acceleration of a vehicle n for safe driving is given as (8)an(t)=min(an′(t),an′′(t)),which ensures collision-free stopping behavior of vehicle n for any unsafe gap with the preceding vehicle n−1 under all circumstances.

Here we explain a vehicle’s typical right-turning decision process using the illustration in Fig. 2, where vehicle n (shown in red) from the West has to turn right, and others are straight-going vehicles. It approaches the intersection stop point Xstop and stops completely if a safe gap with vehicles from the East is insufficient. However, when the vehicle is in the turning decision zone (tdz) of a length Wtdz (from Xstop), it can estimate the safe passage between the opposite vehicle pair (m−1 and m from the East as shown in Fig. 2) and executes turning by following the standard right turn path (rtp) shown by the arrow of a length Wrtp at the intersection. We assume a binary variable θn(t)=0 represents the vehicle position at xn(t)∈[Xstop−Wtdz,Xstop] is not in a feasible or safe state for turning execution over the intersection, which is defined as (9)θn(t)={1if~(τ¯em−τ¯wn>τsafe)~and~(τ¯wn−τ¯e(m−1)>τsafe),0if~xn(t)<Xstop−Wtdz~or~xn(t)>Xstop+Wrtp,where τ¯c, c∈{wn,em,e(m−1)} is the estimated time to cross the point Xstop+Wrtp and τsafe is a safety threshold for crossing. For the right turning vehicle n with θn(t)=0 approaching to stop at Xstop, change its path to execute the turning process when θn(t)=1, which is reset again once it passes beyond Xstop+Wrtp. At a signalized intersection, the inter-phase time (at the end of the green signal) often provides a sufficient gap (for a large τ¯em), and a vehicle can execute the turning. However, any vehicles waiting behind must wait for at least the red period before passing through the intersection in the next cycle. In dense traffic conditions, such a right-turning vehicle at the leading part of a queue may trigger spillback congestion. Therefore, it is desired to coordinate all vehicles to ensure smooth traffic flows, which is described in the next section.

Fig. 3 illustrates the schematic diagram of the proposed optimal coordination system under a cyber-physical framework. We assume a connected vehicle environment under a VANET around the intersection with a communication zone of range dcz, where vehicles transmit their states and turning intention to the roadside ICU using V2I communication and receive individualized coordination signals using I2V communication. At the ICU cyber level, edge processing is carried out to determine the optimal turning and crossing times for all vehicles from both directions, minimizing an objective function while satisfying signal-relevant rules, ensuring safety, and promoting user comfort. In particular, vehicle states, turning information, and signal-phase-and-timing (SPaT) information are used to predict the natural trajectory of the vehicles using some models. Then, the optimization of trajectories incorporating turning time is performed at the start of a green signal. The optimal timing for each vehicle is fed back to them through I2V communication, as shown in Fig. 3. It is assumed that an automated driving system or a driver with an assistance system controls each vehicle adequately.

To reduce total delay time for all vehicles at an intersection, right-turning vehicles and incoming traffic from opposite directions should adjust their spacing and velocity efficiently. The optimal right-turn coordination system receives necessary information from vehicles approaching the intersection, computes the best driving strategy considering the right-turn vehicles, and communicates this strategy to individual vehicles. Then, the local controller of each automated vehicle executes this driving strategy, ensuring it satisfies safety constraints on the road. The optimal right-turn coordination system plans when to execute the right turn.

We assume that among vehicles 𝒱West={1,2,…,N} from the West, the n-th vehicle has to turn right, considering a safe gap between a suitable pair of m−1 and m vehicles among 𝒱Eest={1,2,…,M}, or after the M-th one, i.e., M+1 arriving from the East. At any initial vehicle state at the start of a green phase, the natural or unconstrained arrival times of vehicles for crossing the intersection from the East and the West are defined as τem=[τe1,τe2,…,τeM]⊤∈RM and τwn=[τw1,τw2,…,τwN]⊤∈RN, respectively. The aim is to find the optimal p∗∈{1,2,…,M,M+1} as the relative position (concerning one of m vehicles) where a vehicle q∈𝒱West from the West should execute turning safely based on the optimal crossing times τwn∗,τem∗ by minimizing an objective function and typical driving safety constraints. Specifically, p∗=2 indicates that vehicle q is allowed to turn before vehicle m=2 (after vehicle m=1), and p∗=M+1 implies that vehicle q should turn after vehicle m=M+1 from the East. We consider the following quadratic objective function in terms of the anticipated delay (τ¯⋅⋅−τ⋅⋅) of each vehicle for a given turning path defined by p as (10)J(p,τwn,τem)=ωEast∑m=pM(τ¯em−τem)2+ωWest∑n=qN(τ¯wn−τwn)2,where ωEast and ωWest are constant weights used to softly adjust the priority by tuning the values to reflect the relative cost of delay for cars from the East and the West. Note that vehicles 1 to q−1 from the West and 1 to p−1 from the East move without interruption and are excluded from the objective function since their delay time is zero. The car-following vehicles on the same path must have a safe gap with others, and the turning vehicle (from the West) must find a sufficient safe gap between the two vehicles from the East. Considering such constraints, the optimal coordination task is defined as the solution p∗,τw∗,τe∗ of the objective function given by (11)minp,τwn,τemJ(p,τwn,τem)subject to (12)τem=τ¯em, m=1,2,…,p−1,τwn=τ¯wn, n=1,2,…,q−1,τwq=max(τe(p-1)+τsafe, τ¯wq),τep=max(τwq+τsafe, τ¯ep),τem=max(τe(m−1)+htime, τ¯em),m=q+1,q+2,…,M,τwn=max(τw (n−1)+htime, τ¯wn),n=p+1,p+2,…,N,where htime is the safe car-following time headway, which is determined considering a typical saturation flow rate, as htime=2 s. Note that the above optimization problem can also be extended to multiple right-turning vehicles per cycle by applying it successively to optimize the first turning vehicle, then the second, considering only those natural arrivals that are affected, and so on. Such an approach only requires the selection of those vehicles in the optimization framework, without changing this formulation. This optimization problem is solved at the beginning of each green phase, considering all vehicles anticipated to arrive at the intersection before the onset of the red phase. The individual vehicles are given the target crossing and turning time using I2V communication to adjust their speed accordingly. After this initial coordination, the vehicles are controlled in a distributed manner, enabling safe operation in accordance with local conditions.

In the above optimization problem, the free flow arrival times, τ¯em,m∈𝒱East, and τ¯wn,n∈𝒱West for crossing the intersection need to be estimated once at the beginning of a green phase, when the vehicles may have different position and velocity along the road, i.e., traveling at a constant speed is very unlikely. Therefore, an interactive approach to estimating vehicle arrival times is considered. Specifically, with a discrete time frame k=0,1,2,3,…,K with a step size Δτ, the arrival time of a vehicle i∈{m,n} from ψ∈{e, w} at stop point Xstop of the intersection is given as (13)τ¯ψi=max(KΔτ,τ¯ψ(i−1)+htime),where K is the final step in the iterative process, with the relationship given as (14)z¯i(K−1)<Xstop≤z¯i(K),subject to, with z¯i(0)=xi(0),ν¯i(0)=vi(0), (15)z¯i(k+1)=z¯i(k)+ν¯i(k)Δτ+0.5ai(k)Δτ2,ν¯i(k+1)=ν¯i(k)+ai(k)Δτ,for k=0,1,2,…,K. The acceleration ai(k) is determined based on (3) if the vehicle goes straight without slowing down, and when the vehicle needs to slow down or stop due to a right turn, a data-driven acceleration model is used according to [21] as (16)ai(k)={fss(vi,Xstop−xi)for stoppingfcf(xi,vi,x∞,vd)otherwise.

Finally, the overall functional flowchart is shown in Fig. 4, including the safe driving process of individual vehicles and the cooperating process with the ICU.

To evaluate the proposed right turn coordination system, a microscopic traffic simulation framework is built in MATLAB, as in Fig. 2, where a 1 km road features an intersection at 500 m, and has a speed limit of 50 km/h. The route is single-lane and allows vehicles to turn right at the intersection. Traffic lights are set at 40 s each for red and green signals, with an 80 s cycle. The traffic flow volume is set at a moderate level, and the parameters of the car-following model are chosen typically to reflect typical human driving behavior. The safe car-following time gap htime and right turning gap are set as 2 and 4 s, respectively. For slowing down due to turning or stopping at a red signal, the maximum length is set at Lth=180 m from the intersection, based on the experimental driving data fitting model. The weights ωEast and ωWest are set to 1 for equal priority in this study.

The fuel consumption rate f for a typical vehicle i can be obtained using an arbitrary velocity vi and acceleration ai as follows [22] (17)f=b0+b1vi+b2vi2+b3vi3+a¯i(c0+c1vi+c2vi2),where a¯i=ai for ai>0 otherwise a¯i=0 (for braking) and the fuel consumption parameters for a typical car are set as b0=0.1569, b1=2.450×10−2, b2=−7.415×10−4, b3=5.975×10−5, c0=0.07224, c1=9.681×10−2, c2=1.075×10−3. It is presumed that no fuel is consumed while braking, i.e., ai≤0 from a high speed.

First, we demonstrate the estimation of arrival/crossing times of vehicles at the intersection using (16), to show that for any initial velocity and distance (to the crossing point of the non-turning vehicle), it can estimate the required arrival time (in the absence of a nearby preceding vehicle). Particularly, we choose arbitrary distances to the intersection and velocity (represented by a black circle) to estimate a vehicle’s predicted natural arrival time and associated velocity patterns. Fig. 5a depicts the sample velocity profiles with estimated arrival/crossing times of vehicles at the intersection for freely traveling in a green signal, and Fig. 5b depicts the same in the case of slowing or stopping (due to turning or a red signal). Specifically, the time is calculated in seconds using the velocity curves from the black circle to the intersection crossing point at 500 m, and the corresponding velocity profile is displayed for each case. It is found that the estimated arrival times mostly closely match the measured arrival times for uncontrolled cases. These estimated values in τ¯ψi are utilized to estimate a vehicle’s delay and evaluate parameters to optimize the objective function and determine the optimal turning coordination.

For simplicity, we consider that the signal becomes green at simulation time 0. Eleven vehicles from the West and eight vehicles from the East approach the intersection from some random positions. The fifth one from the West is set as a turning vehicle. In this case, the uncoordinated traditional traffic is first simulated, and the trajectories of the East and the West cars are displayed in Fig. 6a,b, respectively. For clarity, the traffic light is depicted with a white dashed line on each side, indicating the precise center of the intersection, and the vehicles are numbered in order of arrival. In Fig. 6, the 5th vehicle, indicated in red, slows down when approaching the intersection and eventually stops while looking for a safe gap. The vehicles from the East (represented in blue solid lines) do not create a gap for the turning vehicle from the West because they have a greater presence or priority. Finally, it executes the turning maneuver when all eight vehicles from the East cross the intersection. The 6th to 11th vehicles from the West take extra time to cross afterward. The trajectory of the West vehicles (Fig. 6b) shows that the 7th to 11th vehicles have to stop at the red signal for about 40 to 80 s.

This is because the vehicles to the East (Fig. 6a) do not consider the right-turning vehicle and pass through the intersection without slowing down. This extra delay of some vehicles may create spill-back congestion with the same green duration. To avoid such a delay, the signal cycle is typically made longer in the traditional system, resulting in extra idling of other phase vehicles during the red phase.

In contrast, Fig. 7 shows the trajectory of the vehicles under the proposed coordination scheme. Based on optimal coordination at the beginning of green time, each vehicle is assigned its desired optimal crossing or turning time, which reduces the total delay in a quadratic sense, as defined by the objective function. The optimal p∗=5 is found to be a solution that creates a gap between the 4th and 5th vehicles from the East, and the 5th vehicle from the West arrives in a way that allows it to turn at a comparatively higher speed without waiting. Consequently, the rest of the vehicles from the West also cross the intersection within the green time, with little impact on the East vehicles. Therefore, such coordination can accommodate all vehicles within the given 40 s green time. Comparing the velocity of the 5th vehicle in both cases (Figs. 6 and 7), with coordination, the right-turning vehicle turns the intersection at a velocity of about 24 km/h, much higher than about 4 km/h without coordination. Such a low velocity drop also indicated better fuel-efficient driving.

Fig. 8a shows the average speed of these vehicles for both sides, and the corresponding mean speed using solid lines. The 5th and its follower from the East side slightly reduce their speed under coordination to facilitate the smooth turning of the 5th vehicle from the West side. Such a slight drop in velocity significantly increases the speed of many vehicles in the West. Considering only the affected vehicles from both sides (i.e., the 5th–8th from the East and 5th–11th from the West), the average speed for the proposed coordinated system is 35.62 km/h, significantly higher than that of traditional systems, which average 24.76 km/h.

Fig. 8b shows the corresponding fuel consumption of the vehicles and the average values for each side under both schemes. The fuel consumption of the 5th and its followers from the East under the coordinated scheme tends to increase compared to the traditional case because they adjust their acceleration to allow more space for the right turn, which increases fuel consumption, albeit only slightly. However, the fuel consumption of the vehicles in the West (Fig. 8 right side) is significantly reduced compared to the East vehicles. The total fuel consumption of the vehicles in both lanes affected by the right-turning vehicles was 228.5 mL for the traditional driving system and 171.1 mL for the proposed coordinated driving system, resulting in a 25.12% reduction in overall fuel consumption.

Fig. 8 shows the velocity and fuel consumption of the first four vehicles from the West, and four from the East are unchanged in both cases, as they are not affected by the right turning of the 5th vehicle. Therefore, the overall performance is investigated as it varies depending on the position of the turning vehicle from the West. Fig. 9 shows the average traffic velocity and fuel consumption of all vehicles from both sides using markers under the schemes for the position of the turning vehicle in the queue. When it is in the 1st position, the impact is very high, whereas when the turning vehicle is after the 9th position, the impact is low or negligible. Considering equally likely positions from 1 to 11 for the turning vehicle in the same case, the expected performances are shown using solid lines. This reflects the long-term performance of a vehicle when it makes a right turn per cycle. In this case, the expected fuel consumption per vehicle is 12.29 mL under the coordinated system, a significantly lower value than the traditional system, which consumes 15.10 mL, resulting in an 18.56% reduction by the proposed scheme. Furthermore, the expected speed increases to 41.75 km/h under the proposed coordination scheme, which is significantly higher than the traditional one, yielding 35.82 km/h, resulting in a 16.55% speed improvement.

Finally, the performance of vehicles under various traffic flow conditions (moderate to high volume, without exceeding capacity) from both sides with asymmetric flows is investigated. In this case, the probabilistic position of a turning vehicle, along with its corresponding expected velocity and fuel consumption, is considered. Fig. 10 compares the average velocity for different West traffic volumes while keeping a fixed East traffic volume at 315 and 405 veh/h, respectively. In each, fixing the traffic volume on one side and varying it on the other side is an easy way to better understand the influences of traffic demand from each side. As the traffic flow volume increases, the traditional scheme shows a worsening trend, whereas the proposed scheme shows a similar velocity profile. Even for the highly asymmetric flow cases, e.g., 315 veh/h from the East with 540 veh/h from the West, a similar pattern is observed. Overall, as shown by the solid line, the performance improvement varies from approximately 8% to 15% for low to high traffic volumes.

Fig. 11 shows similar trends for fuel consumption improvement. Although the difference in average fuel consumption is low at low traffic volumes between the traditional and coordinated traffic, it becomes significant at higher traffic volumes. Specifically, higher traffic from the East increases the likelihood of blocking many vehicles from the West, thereby worsening the fuel consumption performance. In such a high East traffic scenario, coordination improves fuel consumption by 9%–12%.

The computation time of any optimization scheme is crucial. In a brute-force approach, all possible distinct candidate solutions are evaluated to determine the best one without repetitions. However, since the number of constraints depends on the relative positions of the turning vehicles, the computation time primarily depends on the number of affected vehicles, i.e., those that arrive at the same time or later. For the core optimization, which requires the estimated arrival times of all cars, the computation takes only about 4 to 8 ms for the problems considered in this study. In practical applications, vehicles transmit their estimated arrival times directly to the Intersection Control Unit (ICU). Considering a latency of about 50 to 300 ms (based on our Local 5G V2X communication experiments) and an optimization computation time of less than 10 ms, the entire decision-making process can be completed within 500 ms, and feedback to the vehicles can be completed within one second.

Another notable advantage of the proposed scheme is that it requires coordination only once per cycle. After this initial coordination, the vehicles operate independently using their local controllers to ensure safety. Each vehicle is controlled in a distributed manner, allowing for safe operation. Since coordination occurs only once, in unavoidable situations, such as lane blockages, pedestrian crossings, or the need to prioritize emergency vehicles, the system defaults to uncoordinated local driving regulations.

Finally, the limitations of this work should be acknowledged, as we primarily focus on developing an optimal coordination mechanism under ideal traffic and communication conditions, i.e., with minimal driving uncertainty and no communication delay or packet losses. For a human-driven vehicle with connectivity, it may be possible to provide a speed advisory to adjust the arrival at the intersection. Still, such a system may provide better performance than an uncoordinated system, as the vehicle may slow down early instead of going fast and idling. Therefore, such a study is left for future work in this paper.