Microscopic Simulation for Coordinated Ramp Metering in Antwerp Ring Road: Combined Feedback Linearization and Model Predictive Control Approach

Abstract

In this paper, an innovative real time coordinated ramp metering based on combined feedback linearization and model predictive control (FLMPC) is implemented and evaluated in a microscopic simulation. The controller employs METANET as a macroscopic traffic flow model to predict the future states of traffic flow. Then, a linear model predictive control (MPC) is designed based on the feedback linearized representation of the METANET model to provide coordinated ramp metering. With the Antwerp, Belgium, ring road as a case study, the controller is applied in a microscopic simulation. This paper presents how this connection was made and used to implement ramp metering in a microscopic simulation. In highway systems, the controllers may create sharp fluctuations in ramp metering signals. Therefore, this paper investigates an improved FLMPC approach to tackle this drawback and evaluates its effectiveness via the microsimulation of the Antwerp, Belgium, ring road. Analyzing the simulation results reveals that FLMPC can generate substantial reductions in congestion, with a small computational cost that enables real time implementation. In addition, the effectiveness of the controller in smoothing the control signals is demonstrated without increasing the delay experienced by drivers or causing excessively long computational time.

Keywords

freeway traffic control ramp metering

Highways are vulnerable to congestion, which harms the economy, increases pollutant emissions, and reduces safety, as vehicles spend more time on the road. Ramp metering is one of the popular approaches to tackle congestion by regulating highway inflow. The benefits of ramp metering include reducing average travel times, the risk of accidents, and fuel consumption ( 1 , 2 ). In addition, journey reliability improves, which means lower variability in travel times ( 3 ). This can be achieved cheaply, with benefit-cost ratios of as high as 15 being possible ( 4 ).

There are many policies for applying ramp metering ( 5 ). Ramp metering is generally categorized into local and coordinated methods. Two popular strategies for local ramp metering are the demand-capacity algorithm and ALINEA. The demand-capacity algorithm, presented by Papageorgiou and Kotsialos, adds enough on-ramp flow to the upstream flow to achieve the capacity of the link ( 6 ). However, this is an open-loop control; these controllers are insensitive to disturbances. To address this issue, the ALINEA algorithm was developed, which applies local feedback control in the calculation of the metering ratio ( 6 ). However, as Papamichail et al. demonstrate, local control mechanisms are inferior to coordinated control schemes, particularly when ramp queues have limited capacity ( 7 ). This concept was later extended by the development of the METALINE algorithm, which provides coordinated control ( 6 , 8 ). An innovative alternative coordinated method to ALINEA was devised and is UP ALINEA ( 9 ). This involves collecting data on traffic characteristics that inform tuning parameters applied to ALINEA. This algorithm gave slight reductions in delay, with particular savings at intersections.

Many coordinated ramp metering algorithms have been considered. For example, using a neural network for traffic flow prediction, real time optimization is possible ( 10 ). Fuzzy logic controllers have also been built—these consider varied congestion levels and have been shown to respond smoothly and rapidly to incidents ( 11 , 12 ). Proactive ramp management analyzes the probability of a flow breakdown and uses this to inform the metering ( 13 ). In contrast, the HERO algorithm centrally coordinates a series of local controllers and is especially beneficial in reducing delay on the mainstream ( 14 , 15 ). A further method, dynamic zone metering, partitions the network into zones to prevent flow breakdown on the mainstream ( 16 ). Another technique is SWARM, which uses the Kalman Filter, a mathematical technique for creating a nonlinear forecast ( 17 ). Research in 2006 combined this with model predictive control (MPC) to produce an enhanced controller ( 18 ). In addition, coordinated ramp metering is greatly improved when combined with variable speed limits ( 19 ).

One of the best ways to test any traffic control algorithm is using the microscopic model METANET ( 1 ). This involves discretization into sections, which results in a nonlinear set of equations. The standard form of METANET is very popular; however, modifications have been proposed, such as ( 20 ). This method proposes alternatives for the convection term of the speed dynamics, which provide good model matching. Another model uses a low-scatter microscopic fundamental diagram, and when combined with hierarchical ramp metering, this produces reductions in congestion ( 21 ). This method was successful in overcoming the discrepancy between the prediction and process model. Further assurance can be sought by testing with a macroscopic model, and some field tests have been conducted, such as where coordinated ramp metering was effective in congested conditions ( 22 ).

Most ramp metering algorithms have been used with high levels of success. Many tests have been conducted to confirm the effectiveness of these algorithms ( 23 , 24 ). Microscopic simulation has also been used to show that ramp metering can reduce fuel consumption, as well as the variance in speeds and the risk of collisions ( 25 , 26 ).

By invoking techniques such as an MPC, the results of coordinated control algorithms can be improved ( 24 , 27 ). Chavoshi and Kouvelas propose a combined feedback linearization and model predictive control (FLMPC) approach for coordinated ramp metering ( 28 – 31 ). This method uses feedback linearization to address the nonlinearity of the macroscopic traffic model (METANET [ 1 ]). Then, a linear MPC is designed for the linearized model to improve traffic conditions on highway systems. To evaluate this controller, Chavoshi et al. applied a macroscopic simulation for the Antwerp ring road in Belgium. Their study found that the controller yielded average travel time savings of about 17%, compared with only 6% with ALINEA. Therefore, the controller can reduce congestion in macroscopic simulations ( 28 ). In addition, compared with nonlinear MPC, FLMPC reduced the average solver time from 90 to 0.6 s, which can be achieved using a convex optimizer ( 32 ). Therefore, the efficiency and real time application of the FLMPC is confirmed by the macroscopic simulation.

This paper analyses the FLMPC ( 28 ). To date, the FLMPC approach has only been applied to macroscopic models, which do not fully capture the stochasticity and dynamics observed in real life; microscopic models are better. Therefore, before this method can be implemented in the real world, it must be assessed using a microscopic model.

A controller may perform well in macroscopic simulation; however, when implemented in a real scenario, sharp fluctuation in the metering ratios often occurs, as happened when testing the FLMPC in this paper. Different approaches are proposed to tackle this problem. For instance, ALINEA, a regulator parameter, has been developed to avoid these fluctuations. However, over-regulating the metering slows the response to changes in traffic conditions ( 33 ). A further method to combat excess fluctuation in metering ratios is the use of an activation switch, which deactivates the controller until the density surpasses a value well below the critical density ( 34 ). This prevents vehicles from being metered when no congestion is present, but does not have any effect when the metering rate is too high. Therefore, this may lead to frequent deactivation and reactivation of the controller. A proposed modification is PI-ALINEA ( 35 ). This method used a proportional-integral regulator and was successful in dampening the behavior of the controller. In this paper, adding a “smoothing” term to the objective function of the MPC controller is proposed to dampen the fluctuations observed from the FLMPC tests. This works in a similar way to the regulator parameter ( 33 ). The main contributions of this paper are summarized as follows.

Modification of the FLMPC based on a smoothing technique to dampen the fluctuations in the metering ratio.

Evaluation of the performance of the FLMPC for coordinated ramp metering via a microscopic simulation of the Antwerp ring road, Belgium.

In this paper, the required tests and analyses are conducted to prove the effectiveness of the FLMPC in microscopic simulation. This provides assurance that this algorithm should perform well in real life, achieving reduced overall congestion while requiring low computational time. In addition, the metering helps to reduce the variability of speeds, improving journey time reliability for drivers, and helps to prevent stop-start traffic. By performing these tests, smoothing is beneficial for providing a practical signal that can be enforced without confusing drivers.

This paper is split into the following sections. In the Methodology section, the underlying theory behind the FLMPC is explained. In the Simulation setup section, how this is applied to simulations of the Antwerp ring road is discussed. The results of these simulations are then discussed in Simulation Results, with Conclusions outlining the key conclusions that can be drawn. Therefore, this paper builds on the existing FLMPC presented in the literature ( 28 ) by demonstrating its effectiveness in microscopic simulation and enhancing its performance using the findings of the experiments.

Methodology

This section outlines the control approach utilized in this paper. By considering the basics of traffic flow theory, METANET can be applied to model traffic flow on highways. Then, an improved version of the FLMPC is presented, and this paper explains how this is used to create an efficient metering algorithm ( 28 ).

The fundamental diagram provides a basis for modeling traffic at a macroscopic level. Several enhanced versions of the fundamental diagram have been proposed; the one used in this paper was the exponential fundamental diagram. This is a development of the fundamental diagram and models the steady-state speed, $V_{s}$ (km/h), on a highway as in the literature ( 28 ).

V_{s} = v_{f} \exp [‒ \frac{1}{α} {(\frac{ρ}{ρ_{cr}})}^{α}]

(1)

where $v_{f}$ (km/h) and $ρ_{cr}$ (veh/km/lane) represent free flow speed and critical density, and $ρ$ (veh/km/lane) represents density. The parameter $α$ indicates the prevalence of higher speeds in the overall distribution.

The fundamental diagram shows that the maximum flow occurs when the density is at the critical value. However, because of the high instability of traffic systems, care must be taken to prevent the critical density from being exceeded and the flow from entering the congested region. Therefore, the goal is to set the density to a point just below the critical value.

With these essentials in place, the macroscopic model is introduced, with which the FLMPC makes its predictions, explaining how the FLMPC uses this model to regulate ramp metering. The macroscopic model used is METANET, a model that splits a highway into segments and introduces a dynamical system to present the evolution of density and speed in a traffic flow. Each segment may have up to one on-ramp, which provides an inflow, and up to one off-ramp, which produces an outflow. The METANET model can be applied to any highway, although difficulties can emerge in cases of abnormal topology, such as when two highways merge. Careful judgment and examination are required in these cases because seemingly small modeling decisions can have a huge effect on the results.

In setting out the METANET model, the METANET model is applied to the study highway system, which is the Antwerp ring road in Belgium ( 28 ).

A macroscopic model takes as input the demand pattern over time and returns, as output, traffic states over time. The demand consists of two parts. First, the arrival rate of vehicles to each on-ramp, and second, the split ratio at each off-ramp. The latter is the average proportion of vehicles that exit at each off-ramp and will remain constant. In this part of the paper, how the METANET model allows us to obtain a complete picture of the traffic states of a highway system is discussed, with the only inputs being on-ramp arrival rates and off-ramp split ratios.

METANET Model

Start by dividing the highway system into $N$ segments. In each segment, the flows and traffic state variables are considered.

The following variables are functions of discrete time $t$ :

q _i–1(t) = flow in vehicles per hour (vph) from segment $i - 1$ into segment $i$

q_i(t) = flow (vph) from segment $i$ into segment $i + 1$

r_i(t) = inflow (vph) from on-ramp in segment $i$

s_i(t) = outflow (vph) to off-ramp in segment $i$

ρ_i(t) = density (veh/km) of traffic in segment $i$

v_i(t) = speed (km/h) of traffic in segment $i$

Density dynamics are modeled by considering the conservation law.

ρ_{i} (t + 1) = ρ_{i} (t) + \frac{Δ_{t}}{L λ_{i}} (q_{i - 1} (t) - q_{i} (t) + r_{i} (t) - s_{i} (t))

(2)

where $L$ represents the segment length (km), $λ_{i}$ represents the number of lanes, and $Δ_{t}$ (s) represents the time step. Equation (2) describes that the rate of change of density depends on the sources (entering flows) and sinks (exiting flows).

Speed dynamics are more complicated and are modeled using the following equation, which introduces some parameters specific to the METANET model:

\begin{matrix} v_{i} (t + 1) = v_{i} (t) + \frac{Δ_{t}}{τ} (V_{s} (ρ_{i} (t)) - v_{i} (t)) + \frac{Δ_{t}}{L} v_{i} (t) (v_{i - 1} (t) \end{matrix} - v_{i} (t)) - \frac{υ Δ_{t}}{τ L} \frac{ρ_{i + 1} (t) - ρ_{i} (t)}{ρ_{i} (t) + χ} - \frac{δ_{on} Δ_{t}}{L λ_{i}} \frac{r_{i} (t) v_{i} (t)}{ρ_{i} (t) + χ} - ϕ Δ_{t} (λ_{i} - λ_{i + 1}) \frac{v_{i} {(t)}^{2} ρ_{i} (t)}{L λ_{i} ρ_{i}^{cr}} .

(3)

where

$V_{s}$ = function of density and is calculated using Equation (1),

$τ$ = swiftness of drivers to adapt speed,

$υ$ = magnitude of acceleration or deceleration,

$δ_{on}$ = effect on speed dynamics of traffic entering from on-ramps,

$ϕ$ = strength of effect of weaving phenomenon on speed reduction, and

$χ$ = offset to the parameters $ν$ and $δ_{on}$ .

These are the parameters of the METANET model and must be carefully tuned to encapsulate the real-world conditions.

The dynamics of an on-ramp or off-ramp present in a segment are considered. At uncontrolled on-ramps, it is intuitive to understand that the flow entering the mainstream is the minimum of the demand onto the on-ramp, the capacity of the ramp, and the capacity of the mainstream. This is expressed by the following equation, which defines the flow exiting the on-ramp $r_{i} (t)$ :

r_{i} (t) = \min {d_{i} (t) + \frac{w_{i} (t)}{Δ_{t}}, r_{i}^{\max}, q_{i}^{\max} \frac{ρ_{i}^{\max} - ρ_{i} (t)}{ρ_{i}^{\max} - ρ_{i}^{cr}}}

(4)

where $d_{i} (t)$ represents the on-ramp arrival rate vph at segment $i$ and time $t$ , and, as mentioned previously, is an input to the system. The variable $w_{i} (t)$ represents the number of vehicles in the on-ramp queue at time $t$ , $r_{i}^{\max}$ represents the capacity of the on-ramp as a flow (vph), $q_{i}^{\max}$ represents the maximum capacity of the highway segment, and $ρ_{i}^{\max}$ represents the corresponding density.

The case when no control is applied to an on-ramp was considered. However, if an on-ramp is controlled, a modification must be applied in the minimization. This is the metering ratio imposed by the controller $0 \leq c_{i} (t) \leq 1$ times the on-ramp capacity $r_{i}^{\max}$ , and gives the desired flow rate set by the on-ramp. The following is the on-ramp flow equation for the controlled case:

r_{i} (t) = \min {d_{i} (t) + \frac{w_{i} (t)}{Δ_{t}}, c_{i} (t) r_{i}^{\max}, r_{i}^{\max} \frac{ρ_{i}^{\max} - ρ_{i} (t)}{ρ_{i}^{\max} - ρ_{i}^{cr}}} .

(5)

Dynamics at off-ramps are represented by the consideration of a split ratio $β_{i}$ , which denotes the proportion of vehicles that exit the highway at a junction. It follows that the off-ramp outflow is:

s_{i} (t) = \frac{β_{i}}{1 - β_{i}} q_{i} (t) .

(6)

Feedback Linearization Process

Having established the METANET model as a basis for further development, the controller will be introduced, explaining how it works and drawing on the FLMPC design ( 28 ).

METANET is a multiple-input multiple-output nonlinear system, and can be generalized by the following state space model ( 36 ):

\begin{matrix} \overset{\cdot}{X} = f (X) + g (X) U + p (X) D \\ Y = h (X (t)) . \end{matrix}

(7)

where

$X$ = system state variables vector (i.e., the density and speed of segments and on-ramp queues),

$Y$ = output vector (i.e., the density of the segments),

$U$ = control signals vector (i.e., the ramp metering ratios),

$D$ = disturbances vector (i.e., the demand flow entering on-ramps), and

$f (X)$ , $(X),$ $p (X)$ , and $h (X)$ = nonlinear functions.

The FLMPC method employs feedback linearization, which introduces a block into the feedback path of the nonlinear system to cancel the nonlinearities, as shown in Figure 1. This creates a linear representation for the nonlinear model that facilitates the design of the controller (linear MPC). The controller takes the difference $e (t)$ between actual output ( $Y (t))$ and desired output ( $Y_{d} (t)$ ), and calculates the control signal for the linearized model ( $V (t)$ ) to minimize this deviation. Then, $V (t)$ is transformed to the control signal for the original nonlinear system ( $U (t)$ ) by utilizing a control law provided by feedback linearization.

Figure 1.

Block diagram of feedback linearization ( 28 ).

The feedback linearization process works by input–output decoupling to create a precise map between inputs and outputs. Therefore, an analysis of input relative degrees (IRD) is conducted. The IRD is used to measure the influence of an input on an output. A lower value of the IRD indicates a more direct influence.

Because the system is affected by disturbances, a disturbance decoupling is also required. Therefore, the disturbance relative degree (DRD) is used, which, analogously to the IRD, measures the influence of a disturbance on an output. Decoupling of disturbances is only possible if the IRD is less than or equal to the DRD, meaning that disturbances may not affect the outputs more directly than the inputs. The analysis of the IRD and DRD for a generic METANET model was conducted ( 28 ).

The feedback linearization procedure allows the nonlinear system to be transformed into a linear system with new state variables $ξ$ and $η$ .

\begin{matrix} {\overset{\cdot}{ξ}}_{ir d_{i}}^{i} (t) = b_{i} (ξ (t), η (t)) + a_{i} (ξ (t), η (t)) U (t) \\ + m_{i} (ξ (t), η (t)) D (t) = V_{i} (t) \\ y_{i} (t) = ξ_{1}^{i} (t) = h_{i} (X), \end{matrix}

(8)

where

$b_{i} (ξ (t), η (t)) = \sum_{j = 0}^{ir d_{i}} L_{f}^{j} h_{i} (X),$

$a_{i} (ξ (t), η (t)) = L_{g} L_{f}^{ir d_{i} - 1} h_{i} (X), and$

$m_{i} (ξ (t), η (t)) = {\begin{matrix} 0 if ir d_{i} \leq dr d_{i} \\ L_{p} L_{f}^{ir d_{i} - 1} h_{i} (X) otherwise \end{matrix} \cdot$

In these equations, $L$ represents the lie derivative operator.

Rearrangement of Equation (8) gives the following control law, which describes the relationship between the input to the original nonlinear system, $U_{i} (t)$ , and the input to the feedback linearized system, $V_{i} (t)$ :

U_{i} (t) = \frac{V_{i} (t) - b_{i} (ξ (t), η (t)) - m_{i} (ξ (t), η (t)) D (t)}{a_{i} (ξ (t), η (t))} .

(9)

Improving the Controller

After creating the feedback linearized model for the METANET, a linear MPC was designed for the coordinated ramp metering on highway systems ( 28 ). A common real-world observation of ramp metering is abrupt changes in control signals. The frequent changes would be confusing for drivers and would threaten to disrupt the flow of vehicles, which also increases the risk of accidents. Therefore, an attempt is made to improve the FLMPC presented in the literature ( 28 ) by addressing the abruptness with which the signals change over time. To mitigate this, a term was added to the objective function of the MPC to encourage smaller changes in the control signal from one time step to the next. Therefore, the controller must take the previous control signal as input. Mathematically, the objective function of the MPC is,

min \sum_{i = 1}^{N} \sum_{k = t}^{t + k_{p} - 1} [100 {(ρ_{i} (k) - ρ_{i}^{cr})}^{2} + ω ε_{i}^{2} (k)],

(10)

where $k_{p}$ represents the prediction horizon for the MPC. The auxiliary variable $ε_{i}$ is defined as $ε_{i} = U_{i} (k) -$ $U_{i} (k - 1)$ , with $U_{i} (k)$ written as a function of $V_{i} (k)$ based on the control law presented in Equation (9).

The parameter $ω$ is a weighting factor that determines the magnitude of smoothing applied. Choosing a very high value for $ω$ would replicate the uncontrolled case, as keeping the metering ratio at one would be highly incentivized. In contrast, setting $ω$ to zero would produce the same outcome as that of the original FLMPC.

In reality, there is no information about the demand for future time steps. Therefore, in implementing this scheme, the disturbances $D (k) \forall k \in [t, t + k_{p} - 1]$ must be assumed to be the same as at time $t$ . This is an acceptable assumption because the demand does not fluctuate rapidly, but it may be a small source of error.

After modification by a mapping technique presented in the literature ( 28 ), the constraints for the MPC become the following:

h_{\min, i} \leq h_{i} (k + j) \leq h_{\max, i}, \forall j, 1 \leq j \leq k_{p}

(11)

V_{\min, i} \leq V_{i} (k + j | k) \leq V_{\max, i}, \forall j, 0 \leq j < k_{p}

(12)

The MPC forms a set of optimization problems. To solve these, CVXPY was used, a Python package designed for convex optimization ( 32 ). CVXPY works by converting the problem into conic form. It then uses disciplined convex programming to verify convexity, which constructs problems from a library of functions with known properties.

Simulation Setup

Having outlined the method, this section elaborates on how ramp metering was applied in Aimsun.

In the online control scenarios, the controller (implemented in Python) reads traffic state data from Aimsun and uses this, coupled with the METANET model, to calculate the control signal. The METANET model was calibrated using data from the simulation ( 37 ). The FLMPC knows the current traffic state in the microscopic simulation from Aimsun data and uses the METANET model to predict subsequent states over the prediction horizon as a function of the control signal applied. With this, the CVXPY solver then finds the optimal control signal.

To connect the FLMPC in Python to Aimsun, Python must read the data, pass this data to the FLMPC and then relay the control signals to Aimsun. This procedure is performed using the Aimsun Application Programming Interface.

Case Study

The Antwerp ring road in Belgium is a high-speed road, with a speed limit of 120 km/h. At its southern and easternmost points, the road passes close to the city center. The more urban nature of these sections means that the speed limit is reduced to 100 km/h. The simulation case study is the anticlockwise direction of this ring road. In total, this is over 48 km of road and around 150 km of lanes. In addition, there are 23 on-ramps and 20 off-ramps to model. In addition, lane drops must be considered, as the number of lanes on the mainstream varies between one and six.

In controlled simulations, the coordinated FLMPC ramp metering was applied to the designated on-ramps. To achieve this, locations were chosen where ramp metering is imposed. Then, metering was implemented in the simulations.

The ramp metering locations selected were those where the most congestion was observed in the base scenario. Out of the 23 on-ramps in the study highway, seven were assigned ramp meters; Figure 2 shows where the metering was applied.

Figure 2.

Ramp metering locations (segments in italics, and on-ramp locations in red) in the Antwerp ring road.

The METANET and Aimsun models simulated 4 h, three of which were loaded with demand. This meant that about 37,000 vehicles were generated in each simulation, a value that was checked to ensure that both models had the same demand. Figure 3 shows a typical demand pattern over time entering the on-ramp at Segment 23. The demand pattern is well represented by the arrival rate of vehicles willing to enter the system in every segment.

Figure 3.

On-ramp arrival rate of vehicles loaded into simulations at Segment 23.

The first hour slowly loads the system, and the system then experiences congestion during the second hour. In the third hour, this congestion slowly dissipates, allowing us to determine whether the controls enable the congestion to clear faster. In the fourth hour, no more vehicles enter the system, enabling the system to be unloaded completely.

The demand input into METANET and Aimsun was split into 10-min blocks. In METANET, this was divided uniformly over each 10-s time step, but in Aimsun, arrivals were set as exponentially distributed, because this better reflects reality.

Implementing the METANET Model for Antwerp Ring Road

The METANET model was implemented using Python, based on a MATLAB script created for the FLMPC ( 28 ). The study highway system was divided into 96 segments of 500 m. Each segment may have at most one on-ramp and one off-ramp. These constraints create a challenge when capturing the Antwerp ring road in a METANET model. Therefore, the lengths of these segments in the real highway may be 500 ± 300 m.

In this simulation, a time step of 10 s was selected. Therefore, for the 4-h simulation, there were 1,440 discrete time steps. As mentioned previously, the METANET model has internal parameters. Different sets of parameters were experimented with, and a set that best fitted the observations in Aimsun was found (Table 1). These parameters were used for all simulations.

Table 1.

Values Assigned to METANET Parameters

Parameter	Value	Units
$τ$	20.16	seconds
$ν$	16.0	km²/h
$δ_{on}$	1.4	-
$ϕ$	2.0	-
$χ$	13.0	veh/km

Note: $τ$ = swiftness of drivers to adapt speed; $υ$ = magnitude of acceleration or deceleration; $δ_{on}$ = effect on speed dynamics of traffic entering from on-ramps; $ϕ$ = strength of effect of weaving phenomenon on speed reduction; $χ$ = offset to the parameters $ν$ and $δ_{on}$ ; parameters with no units are marked with a hyphen.

Other parameters that had to be estimated were the flow capacity and critical density of a traffic lane, and the fundamental diagram parameter $α$ . Since the lane capacity was observed in the Aimsun simulations to vary between 2,000 and 2,500 vphpl, a value in between these two was used in METANET. This value was 2,300 vphpl and was taken to be the same in every segment. For $α$ , when a value of 1.4 was used, the models were well aligned with their traffic state variables (see Figure 4); therefore, this value was adopted. With these, the critical density could be estimated in each segment from the fundamental diagram.

ρ_{cr} = \frac{Q_{\max}}{v_{f} \exp [- \frac{1}{α}]},

(13)

where $Q_{\max}$ represents the capacity flow of a lane.

Figure 4.

Traffic states (flow, density, and harmonic speed) compared between Scenarios 1 and 2.

Therefore, the critical density depends on the capacity, free flow speed, and the shape parameter of the fundamental diagram.

Microscopic Modeling Using Aimsun

An Aimsun model was then made to represent the Antwerp ring road. This was an adaptation of a previously used model ( 37 ). The model was designed to resemble the real highway system as closely as possible.

The Aimsun model was segmented into sections consistent with the METANET model. This was to allow data collected by Aimsun to be compared directly with results from METANET. Because of the mentioned constraints on the METANET topology, there was some variation in segment lengths in Aimsun, which were 500 ± 300 m.

Some METANET segments corresponded to multiple links in Aimsun, sometimes as many as four links. In these cases, the Aimsun data for the corresponding links was averaged before comparison with METANET. Therefore, the METANET segments are different from the Aimsun links.

To represent lane-changing behavior more realistically, the two-lane vehicle following model was enabled. The default values for the two-lane following model were then used, with the following two exceptions:

Maximum speed difference = 10 km/h

Maximum speed difference with on-ramp = 10 km/h

These amendments were made to ensure that vehicles did not unduly linger in the nearside lanes, which had been observed to hinder the outflow from on-ramps.

The simulation step in Aimsun was set to 0.4 s. This parameter represents the frequency of update of vehicle trajectories. The goal in choosing the simulation step is to achieve the best balance between speed and accuracy. Other related parameters in Aimsun are reaction time at a stop and at traffic lights, for which the default values were 0.8 and 1.6 s, respectively. It is essential to ensure that all reaction times selected in Aimsun are multiples of the simulation step. In addition to the changes described in this section on parameters, all other simulation parameters were left at their default values.

Different vehicle types can be simulated in Aimsun. In this case, two types of vehicles were included for demand: (1) small passenger cars; and (2) heavy goods vehicles. Of the total demand, 80% were passenger cars and 20% were heavy goods vehicles.

Simulation Scenarios

Once the highway system had been loaded with vehicles, the simulations were run. There are four scenarios of interest, numbered 1–4. The details of these scenarios are outlined in Table 2.

Table 2.

List of Simulation Scenarios

Scenario	Control type	Simulator	Description
1	Uncontrolled (base)	METANET	No control applied
2	Uncontrolled (base)	Aimsun	No control applied
3	Online FLMPC	Aimsun	Controller uses Aimsun data
4	Improved online FLMPC	Aimsun	Controller uses Aimsun data

Note: FLMPC = feedback linearization and model predictive control.

The first two scenarios are uncontrolled, and for convenience, the uncontrolled case is often referred to as the base scenario. These scenarios demonstrate what happens in the macroscopic and microscopic models, respectively, if no control is applied. The third and fourth scenarios are online, where the controller receives real time traffic state data from Aimsun, calculates metering ratios, and then enforces these in the Aimsun simulation. Then, how the online FLMPC (used in Scenario 3) might be improved was considered, implementing these modifications in the fourth scenario.

Each Aimsun scenario was run with three random seeds, then averaged to obtain a more reliable result.

In all FLMPC simulations, any metering ratio between zero and one was allowed at all seven metered locations.

Evaluation Criteria

Two important metrics for assessing the congestion in each scenario are vehicle hours traveled (VHT) and vehicle kilometers traveled (VKT). For any two scenarios being compared, the VKT should be similar. This is a necessary condition for both scenarios to have the same demand. The VHT then indicates the total travel time of all vehicles in the simulation; a higher VHT means more congestion. However, only if the VKT values are close to each other is the VHT total meaningful.

Aimsun automatically saves data, including VHT and VKT statistics. However, with METANET, this data is not so readily available; therefore, these statistics were calculated using the traffic state variables.

The formulas used were as follows ( 28 ):

VHT = Δ_{t} \sum_{t = 1}^{t_{end}} (\sum_{i = 1}^{N} ρ_{i} (t) L λ_{i} + \sum_{i = 1}^{N} w_{i} (t)),

(14)

VKT = Δ_{t} \sum_{t = 1}^{t_{end}} \sum_{i = 1}^{N} q_{i} (t) L,

(15)

where $t_{end}$ represents the number of simulation time steps (1,440 in this case).

The time mean speed (TMS) and space mean speed (SMS) were analyzed. The TMS is defined for a time step $t$ as,

TM S_{t} = \frac{\sum_{i = 1}^{N_{j}} {\bar{v}}_{i, t}}{N_{j}},

(16)

where

$N_{j}$ = number of METANET segments in the network (96 in this paper), and

${\bar{v}}_{i, t} =$ harmonic speed in segment $i$ during time step $t$ (km/h).

In contrast, the SMS is defined for a segment $i$ , as follows:

SM S_{i} = \frac{\sum_{t = 1}^{t_{end}} {\bar{v}}_{i, t}}{t_{end}}

(17)

In addition, the standard deviation of these mean speeds will be used to measure journey reliability. A lower degree of deviation is desirable, as this means more predictability for drivers.

Finally, the mean number of stops per vehicle will be considered to indicate fuel economy. This is defined as ( 38 ),

N S_{s} = \frac{\sum_{j = 1}^{N_{s}} \frac{TN S_{j}}{D_{j}}}{N_{s}},

(18)

where

$N_{s}$ = total number of simulated vehicles,

$TN S_{j}$ = total number of stops made by the $j$ th vehicle over the simulation, and

$D_{j}$ = total distance traveled by the $j$ th vehicle over the simulation (km)

Simulation Results

This section of the paper presents the findings of the simulations. The traffic states are illustrated in contour plots of time in one dimension and space in the other. All contour plots are laid out in the same way, showing the traffic states of one simulation on the left and another on the right. The first row is flow, the second is density, and the third is harmonic speed. Then, the VHT and VKT bar plots show whether overall congestion has improved or worsened. For the controlled simulations, plots were produced of the ramp metering ratios over time, as well as plots indicating how density in metered segments compares with the critical density.

Scenarios 1 and 2: Uncontrolled

The results of the first two scenarios allow the microscopic and macroscopic models to be compared, ensuring that the METANET parameters were well-tuned. First, the traffic states were compared over time $t$ and space $x$ in Figure 4.

The overall pattern is similar for both models. The main difference is that METANET generally predicts lower densities. However, the models agree that the main traffic bottleneck occurs around Segment 41, just upstream of where the two highways merge.

Both models indicate that another key area of congestion is Segment 58, where congestion results from a joining on-ramp followed by a two-lane off-ramp, prompting frequent weaving. In addition, a small amount of congestion occurs around Segment 21. This is a three-way junction between highways, and the mainstream experiences a lane drop to just one lane before connecting with another highway.

Scenario 3: Online FLMPC in Aimsun

Then, there is the most important scenario, in which the FLMPC is run in Aimsun, with full communication between the two models enabled. This will provide insight into the effectiveness of the controller in microscopic simulation. To enhance this analysis, ALINEA was implemented using the built-in functionality in Aimsun, and the results were compared. To begin with, the traffic states in a space-time diagram are shown in Figure 5.

Figure 5.

Traffic states (flow, density and harmonic speed) are compared between Scenarios 2 and 3.

As Figure 5 shows, the controller succeeds in significantly reducing the overall density and increasing the mean speed. Looking at Table 3, the time savings on the mainstream outweigh the increase in queueing time on the ramps, resulting in a lower total VHT. Therefore, the controller has performed well, reducing the VHT by 7.57% with little change in the VKT.

Table 3.

Vehicle Hours Traveled (VHT) Compared Between Scenarios 2 and 3

Simulation	VHT total(vph)	VHT in queue(vph)
Uncontrolled (base)	5,441.9	26.1
Online	5,029.8	1,110.8

Therefore, the controller is effective in the Aimsun microscopic model. This result seems promising; when the computed metering ratios were inspected, foreseen issues with fluctuating signals were observed. Figure 6 shows the metering ratios and the 10-min moving average of the normalized demand in Aimsun. The normalized demand is the arrival flow rate onto the on-ramp divided by the capacity of the on-ramp; a normalized demand of one would mean that the arrival rate equals the capacity of the ramp.

Figure 6.

Ramp metering ratio at each ramp meter and 10-min moving mean of normalized demand set by Aimsun in Scenario 3. Segments are those with ramp metering (19, 41, 42, 53, 54, 58, and 61).

From Figure 6, ramp metering is required in only three locations: Segments 41, 42, and 58, because the metering ratio falls below the normalized demand. For the other on-ramps, the metering ratio does not significantly drop below the normalized demand, which means that no vehicles are held by a red signal; therefore, the metering is not activated by the controller. In these cases, the noise observed is because of the distribution in demand, which does not indicate an issue with the controller, as drivers will continually see a green signal.

The online FLMPC was compared to ALINEA. For this simulation, a regulator value $K_{R} = 70$ , occupancy of 26%, and a time interval of 60 s were used. These are close to the standard values defined in the literature ( 33 ). To ensure optimal performance of ALINEA, an override had to be implemented whenever the flow dropped below 1,000 vphpl. As Table 4 highlights, the FLMPC performs better than ALINEA. In addition, the VKT was about 0.7% lower.

Table 4.

VHT in Online FLMPC Ramp Metering in Scenario 3 Compared with ALINEA Ramp Metering

Simulation	VHT total(vph)	VHT in queue(vph)
Online	5,029.8	1,110.8
ALINEA	5,151.4	1,035.4

Note: VHT = vehicle hours traveled; FLMPC = feedback linearization and model predictive control.

Crucially, as Figure 10 shows, the FLMPC successfully maintains the density close to the critical density. The previous results suggest that the FLMPC does this more effectively than ALINEA. However, an issue is that fluctuation was observed while ramp metering is activated. The difference that will be made by implementing the smoothing technique in the following control scenario will be discussed.

Scenario 4: Improved Online FLMPC

Figure 7 shows the analysis for segments where congestion occurred and where ramp metering was necessary, one in each column. Each row represents a different value of the smoothing parameter $ω$ .

Figure 7.

Ramp metering ratios for first random seed in various scenarios in three metered segments, with different smoothing parameters. Segments 41, 42, and 58. Scenarios are online ( $ω = 0)$ , online ( $ω = 3 \times 10^{3})$ , online ( $ω = 1 \times 10^{5})$ , and online ( $ω = 3 \times 10^{6})$ .

As Figure 7 shows, without smoothing, changes to the signal are generally sharp. This is particularly an issue in Segments 42 and 58. The lightest smoothing has little effect, but this is improved if a smoothing factor $ω$ = $1 \times 10^{5}$ is applied, for which the metering in Segment 58 reacts swiftly to the easing of congestion, with few oscillations. With $ω$ = $3 \times 10^{6}$ , a very smooth signal is seen, especially for Segment 58.

The VHT were then compared; Figure 8 and Table 5 demonstrate that the controller is effective for all the smoothing parameters analyzed, all with similar VHT. Of note, the VKT is roughly the same in all cases. Therefore, a well-tuned smoothing parameter can allow the FLMPC to be beneficial for total travel time and in reducing the confusion experienced by drivers.

Figure 8.

Vehicle hours traveled (VHT) and vehicle kilometers traveled (VKT) in various scenarios, with different smoothing parameters. Scenarios are base, online ( $ω = 0)$ , online ( $ω = 3 \times 10^{3})$ , online ( $ω = 1 \times 10^{5})$ , and online ( $ω = 3 \times 10^{6})$ .

Table 5.

Comparison Between Online Simulations Based on Vehicle Hours Traveled (VHT) and Computation Times for Different Weighting Factors

Simulation	VHT total(vph)	VHT in queue(vph)	% VHT improvement (on uncontrolled) (vph)	Computation time(s)
Uncontrolled (base)	5,441.9	26.1	-	-
$ω = 0$	5,029.8	1,110.8	7.57	2.387
$ω = 3 \times 10^{3}$	5,121.2	1,182.3	5.89	7.393
$ω = 1 \times 10^{5}$	5,119.4	1,185.9	5.93	1.715
$ω = 3 \times 10^{6}$	5,112.2	1,175.7	6.06	5.706

Note: $ω$ = smoothing parameter; VHT = vehicle hours traveled; parameters with no units are marked with a hyphen.

Then, the mean number of stops per km per vehicle was assessed and used as an indicator of fuel economy ( 38 ). Figure 9 shows that, on average across the ring road, vehicles brake to a halt less frequently when metering is applied. This change is pronounced, with smoothing yielding a decrease of roughly 50% relative to the uncontrolled case. Greater smoothing also reduces the frequency of stops. This means less wasted energy, which reduces fuel consumption and emissions.

Figure 9.

Mean number of stops per km per vehicle. Scenarios are base, online ( $ω = 0)$ , online ( $ω = 3 \times 10^{3})$ , online ( $ω = 1 \times 10^{5})$ , and online ( $ω = 3 \times 10^{6})$ .

Finally, as shown in Figure 10, how the density changes in the different smoothing instances and the base scenario is investigated.

Figure 10.

Densities in various scenarios, with different smoothing parameters, compared with critical density. Scenarios are base, online ( $ω = 0)$ , online ( $ω = 3 \times 10^{3})$ , online ( $ω = 1 \times 10^{5})$ , and online ( $ω = 3 \times 10^{6})$ .

Figure 10 shows that the density reaches much closer to the critical value with the smoothing parameters $ω = 3 \times 10^{3}$ and $ω = 1 \times 10^{5}$ . Therefore, the smoothing helps the system to converge to an optimal state. However, with the highest smoothing, the controller effectively becomes inert, taking too long to react to changes in density; therefore, the density pattern resembles that of the uncontrolled scenario.

Variability in Speeds

The results show that the improved FLMPC significantly reduces speed variability. Figure 11 shows that during the most congested period of the simulation, when ramp metering is activated, the standard deviation of speeds decreases massively. Note that in Figure 11, a 10-min moving average is considered to remove noise.

Figure 11.

Standard deviation of harmonic speed over time (10-min moving averages).

Figure 12 shows the standard deviation $σ$ over the simulation. Using a smoothing factor of $ω = 1 \times 10^{5}$ , all online simulations gave similar improvements of about 30%.

Figure 12.

Standard deviation of harmonic speed over the simulation.

Therefore, applying this ramp metering control reduces speed variability, increasing the reliability of journey times in the system.

Computational Times

One of the advantages of the FLMPC approach is its real time implementation. Therefore, this property of the controller must be verified by analyzing the required computational time for the controller implemented in a microscopic simulation. In addition, the computational times were compared for all the discussed variations of the online controller, as part of the evaluation of each variant.

As the FLMPC calculation is repeated every 10 s in these simulations, the average running time of the controller must be much less than this for such an operation to be feasible. Table 5 compares the running times of the four proposed variations of the online control.

This confirms the effectiveness of the controller in administering ramp metering for a microscopic simulation in real time. With the middle smoothing parameter, the running time of the controller decreases, which indicates that the smoothing adds complexity to the optimization problem to be solved. Nonetheless, the average running times are all below 10 s.

Selection of Smoothing Parameters

For the highest smoothing parameter, the controller was insensitive to density changes; therefore, this option was not the best for the Antwerp ring road in Belgium. All smoothing parameters gave almost identical VHT, and the best performing case was that with $ω = 1 \times 10^{5}$ , because this helped to remove fluctuations from the control signal with the lowest computational time, and still reduced the VHT by 5.93%.

Figure 13 shows the differences between the uncontrolled case and the two controlled cases by plotting the TMS and SMS. These two smoothing parameters yield practically the same speed distribution when compared with the uncontrolled case. Furthermore, in Segment 41, the parameter $ω = 1 \times 10^{5}$ was marginally more effective at raising the mean speed than the parameter $ω = 3 \times 10^{3}$ . Therefore, choosing $ω = 1 \times 10^{5}$ successfully removes smoothing with almost no difference in average travel times.

Figure 13.

Time mean speed (TMS) and space mean speed (SMS) observed in Scenarios 2 and 4, with different smoothing parameters.

Therefore, the results show that with these modifications in place, the FLMPC could be successfully implemented in the Antwerp ring road with benefits to travel times and journey time reliability.

Conclusions

This paper has confirmed the efficacy of FLMPC as a real time coordinated ramp metering approach in reducing congestion, by conducting online microscopic simulations in Aimsun for the Antwerp ring road in Belgium. The efficiency of the method is evaluated by the VHT measurements, which showed that the online control scenarios could reduce the total time vehicles spend traveling by over 7%, without a significant change in the total distance traveled. In addition, the average number of stops per vehicle halved and the standard deviation of speeds dropped by over a quarter, which confirms the positive effect of the improved FLMPC on fuel economy and journey reliability, respectively. In addition, the computational times of the FLMPC were quite reasonable, which means that real time implementation of this technique is feasible.

When initially implementing the FLMPC, heavy fluctuations were observed in the control signals, which is a common issue with ramp metering mechanisms. However, by modifying the objective function of the MPC controller, large fluctuations in the computed metering ratios were penalized, smoothing the signals. This did not lead to an increase in congestion; therefore, the FLMPC can execute ramp metering that is comprehensible for road users without the confusion of abrupt changes between green and red signals. This will lead to reduced journey times, as well as a drop in fuel consumption and emissions.

Footnotes

Acknowledgements

The authors express their gratitude for the support of Dr. Michalis Makridis, who created the microscopic model of the Antwerp ring road in Belgium and provided advice on the modeling process.

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: K. Chavoshi, A. Kouvelas; data collection: R. Chuka; analysis and interpretation of results: R. Chuka, K. Chavoshi, A. Kouvelas; draft manuscript preparation: R. Chuka, K. Chavoshi. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Rene Chuka

Kimia Chavoshi

Anastasios Kouvelas

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