This section describes the experimental setup that was used as a benchmark for this simulation study. The rest of the section presents the model for the simulation of power dynamics and driving behaviors, the network, the algorithm for generation of variability in vehicle dynamics and driving style, the simulation of different vehicle types and the proposed cooperation control logic.
2.1 The car-following model
The proposed car-following model is based on the simplified Lagrangian Godunov scheme, which is described in the work of Leclercq [16] as follows:
$$ X\left(n,t+\varDelta t\right)=\min \left(X\left(n,t\right)+{v}_m\varDelta t,\left(1-\alpha \right)X\left(n,t\right)+\alpha X\left(n-1,t\right)- w\varDelta t\right) $$
(1)
where vm is the desired speed, w is the wave speed, km is the jam density and a = wkmΔt. Equation 1 is the exact solution of the LWR model [17,18,19].
In this work, the lightweight Microsimulation Free-flow aCceleration model (MFC) [18] is incorporated within the framework of the LWR model bounding the free-flow acceleration based on the MFC output. More specifically, based on the definition given, the proposed model is:
$$ {\displaystyle \begin{array}{c}q1=X\left(n,t\right)+\varDelta t\ast \min \left({v}_m,V\left(n,t\right)+{a}_{MFC}\left( DS,G{S}_{th}\right)\varDelta t\right)\\ {}q2=X\left(n-1,t\right)- w\varDelta t\\ {}X\left(n,t+\varDelta t\right)=\mathit{\min}\left(\mathrm{q}1,\mathrm{q}2\right)\end{array}} $$
(2)
where vm is the desired speed, aMFC is the acceleration of the MFC model, w is the wave speed, km is the jam density here set to 0.15veh/m and a = 1. V(n, t) is speed of the vehicle n at time t.
The MFC takes as input common specifications of the vehicle, such as mass, gear ratio, maximum torque etc., which can be found available online and two parameters (DS, GSth) taking values in the range (0, 1] in order to simulate different drivers. Parameter values closer to zero, indicate a timid driver, while when they take values closer to one, an aggressive one. More details can be found in [18].
Figure 1 illustrates the acceleration potential for the same vehicle and three different drivers as they are defined by the different values of DS and GSth parameters. The figure derives from the simulation of a free-flow acceleration from 0 km/h to maximum speed for a commercial vehicle with a 9-speed automatic gearbox. The simulation is performed for three different drivers described by the different (DS, GSth) parameter sets. The figure on the top illustrates the acceleration over speed diagrams per driver from zero to the vehicle’s maximum speed. The behavior of the MFC model can be perceived by the reader intuitively if we consider that the DS parameter adjusts how high the acceleration on the Y-axis will be, while the will the GSth parameter dictates how fast the driver will change gears jumping from a gear to another with higher (upshift) or lower (downshift) acceleration capability (X-axis). The figure on the bottom shows the speed over time trajectory during a free-flow acceleration from zero to a desired speed. The differences of the three drivers can be spotted both regarding the time they need to reach the same speed, and the speed oscillations due to the different gear shifting strategies. From the implementation point of view, the acceleration over speed curves shown in the figure can be precomputed, leaving the proposed model computationally inexpensive.
2.2 Network description
We consider a one-lane road segment with length of 1000 m. Starting on 500 m. we introduce an upgrade for 300 m. The cooperation control logic is applied along a segment with length 500 m from 400 m to 900 m, as shown in Fig. 2. The wave speed is set to 20 km/h, the jam density to 150veh/km. The effect of the upgrade on the vehicle acceleration is dictated by the following equation:
$$ ac{c}_g=\frac{F_g}{m}=\frac{-\Big(9.81\ast \mathit{\sin}(g)\ast m+{F}_0\ast \left(1-\mathit{\cos}(g)\right)}{m} $$
(3)
where g is the gradient of the road, m is the vehicle’s mass and F0 factor is commonly used to characterize the road load of vehicles and expresses the constant part of a vehicle’s resistances (tire rolling resistances). A dedicated module for the calculation of the vehicle’s road loads was developed by Tsiakmakis et al. [31] and was adopted for the needs of this study for the computation of F0 and the free-flow acceleration output of the MFC model [18].
2.3 Driving profile generation
A driving profile is constructed and randomly assigned to each vehicle entering the network. Each individual driving profile is developed using the car technical characteristics and the two parameters (DS, GSth) of the car-following model taking values in the range (0, 1] in order to simulate different drivers. Figure 3 illustrates the generation of different driving profiles (vehicle and driving parameters). The top row corresponds to five vehicle with different power specifications, the second row correspond to five GSth values that regulate how fast the driver changes gears (X-axis) and the third row correspond to five DS values that regulate the maximum desired acceleration (Y-axis). This work uses five different vehicles from car segments B and C (representative small and medium cars) [30] to cover the most common passenger cars segments in use in Europe in terms of mass, power, engine technology, and transmission. The driver characteristics derive from randomly selected values for the parameters (DS, GSth). In general drivers with (DS, GSth) values closer to 0 have more timid driving behavior, while drivers with (DS, GSth) values closer to one, are more aggressive.
Consequently, each driving profile can be defined as follows:
$$ D{P}_i=\left\{{C}_k,G{S}_{th,l},D{S}_m\right\} $$
where the i driving profile refers to car k, with gear shifting value l and driving style value m.
2.4 Vehicle types
All the simulation experiments involve three different types of vehicles, manually-driven (manual), automated vehicles (AVs) and cooperative automated vehicles (Coop-AVs). All types are simulated using the same car-following model described by Eq. 2. The difference lies in the variation of different driving styles that enter the network and the existence of cooperation or not with the infrastructure. More specifically:
2.4.1 Human-driven vehicles (CVs)
In simulation tests with CVs, a random driving profile is assigned to each vehicle entering the network among all the possible combinations of {Ck, GSth, l, DSm} values as shown in Fig. 3 (125 in total).
2.4.2 AVs
In simulation tests with AVs, a fixed pair of assigned (DS, GSth) values is assigned to each of the five vehicles shown in the first row of Fig. 3. Consequently, each AV will have one of the five different driving profiles (green vertical lines).
2.4.3 Coop-AVs
Cooperative automated vehicles have exactly the same driving profiles as AVs. The difference lies that they behave according to the proposed cooperation control logic (see Section 2.5) for the network segment 400-900 m as illustrated in Fig. 2.
2.5 Cooperation control logic
In the literature, there are references that drivers underestimate high driving speeds under freeway conditions [32]. Variable Speed Limit (VSL) strategies have been used as a freeway metering mechanism or a homogenization scheme to reduce speed differences, which most probably derive from the variability in the vehicle specs and the driving style [27]. Finally, in uphill areas it has been observed that the drivers slow down reducing their speed [33]. An important factor of traffic congestion uphill is that most drivers do not accelerate enough and consequently, they do not compensate instantaneously for the increase in resistance force resulting from the increase in gradient, which limits the acceleration of their vehicles. Considering that each vehicle has different capabilities and each driver different response times, gives a good explanation of the periodic formulation of stop-and-go waves when the traffic demand is sufficiently high.
Here, a simple control logic in cooperation with the infrastructure is proposed. More specifically, we assume a central controller over an area, which imposes a unified DS parameter value (DScoop) for all the vehicles within range. DScoop parameter regulates how much of the vehicle potential the driver will use (see Fig. 1), i.e. how hard the driver pushes the gas pedal. Since the vehicle models have different power specifications, it can be derived that each vehicle will have a different desired acceleration, proportional to the vehicle’s individual capabilities. However, by asking each automated vehicles to use the same DScoop values, the controller unifies the driving profiles without setting explicit desired acceleration values per vehicle. This is a realistic strategy since some vehicle might not be physically able to comply with a suggested desired acceleration value due to power limitations.
Two cooperation strategies have been implemented here, which corresponds to two DScoop values, 0.8 and 0.6. The first value corresponds to more aggressive driving and thus higher acceleration values, while the second corresponds to a more conservative strategy for the vehicles entering the central controller’s area.