1. Conceptos Relevantes
1.6 Empresa
Ramp metering (RM) has been claimed to be one of the most effective highway traffic control measures [115]. RM improves highway traffic flow by effectively regulating traffic flow onto the highway at an on-ramp, thus increasing mainline throughput and served traffic volume due to the avoidance of capacity loss and blockage of on-ramps due to congestion. RM strategies have been proven effective in both macroscopic as well as microscopic simulation environments, and have been implemented at various locations in the United States of America, France, Italy, Germany, New Zealand, the United Kingdom and the Netherlands [27, 75, 112].
Consider two scenarios of a highway on-ramp, as shown in Figure 3.4, one where RM is not employed (a), and one where RM is employed (b). Let qin denote the upstream highway flow, d
the on-ramp demand, qcon the mainstream outflow in the presence of congestion, and qcap the
highway capacity. It has been shown that the traffic outflow in the presence of congestion is between 5%–10% lower than the highway capacity [115]. It is assumed in Figure 3.4(b) that a
For example, if the total demand exceeds the highway capacity by 20% (i.e. qin+ d = 1.2qcap)
and the capacity drop due to congestion is 5% (i.e. qcon = 0.95qcap), then 4T = 20% results
from (3.10), which demonstrates the power of effective RM strategies [115].
qin qcon qin qcap
d d r (a) (b) highway highway on-ramp on-ramp
Figure 3.4: Two cases of traffic flow onto a section of highway from an on-ramp, (a) without RM and (b) with RM, where shaded areas indicate congestion zones. Adapted from Papageorgiou and Kotsialos [115].
Various RM strategies have been proposed in the literature. The most significant of these are briefly discussed in this section.
Fixed-time strategies
Fixed-time RM strategies are determined in an offline fashion for specific times of day, based on historical demands, without the use of real-time traffic information [115]. This approach was first introduced by Wattleworth [171]. According to this approach, a highway with several on- and off-ramps is partitioned into sections, each containing only one on-ramp. The flow q on a highway section j may then be defined as
qj = j
X
i=1
αijri,
where ri represents the on-ramp flow (in units of veh/h) for section i, and αij ∈ [0, 1] represents
the proportion of vehicles that enter the highway in section i and do not leave the highway upstream of section j. In order to avoid congestion in the network, it is required that
qj ≤ qcap,j,
where qcap,j denotes the capacity of highway section j. Finally, the metering constraint
rj,min≤ rj ≤ min{rj,max, dj}
must hold, where dj is the on-ramp demand and rj,max is the on-ramp capacity at on-ramp
on-ramps, that is to
maximise X
j
rj,
or to minimise the on-ramp queues, i.e.
minimising X
j
(dj− rj)2,
while satisfying the highway and on-ramp capacity constraints. The most prominent fixed-time RM algorithm is AMOC, introduced by Kotsialos et al. [74], in which a second-order macroscopic traffic network model called METANET [100] is employed to solve a nonlinear optimisation problem with the goal of minimising the total travel time. The RM problem is formulated as a dynamic optimal control problem with constrained control variables in the AMOC control strategy. This dynamic control problem may then be solved numerically for given demands di(t)
and turning rates βm
n over a specified time period. The general discrete-time optimal control
problem formulation involves
minimising J = ϑ[T ] + T −1X t=0 ϕ[x(t), u(t), d(t)] (3.11) subject to x(t + 1) = f [x(t), u(t), d(t)], (3.12) x(0) = x0, (3.13)
ui,min ≤ ui(t) for all i∈ {1, . . . , m}, (3.14)
ui(t) ≤ ui,max for all i∈ {1, . . . , m}, (3.15)
where T is the time period under consideration, x ∈ <n is the state vector, u ∈ <m is the
vector of control variables, d is the vector of disturbances acting on the traffic process, and ϑ and ϕ are arbitrary, twice-differentiable, nonlinear cost functions. Gomes and Horowitz [43] presented a similar nonlinear optimisation approach based on a first-order macroscopic traffic simulation model, called the asymmetric cell transmission model (ACTM). First-order models are significantly simpler than second-order models, and as a result, optimisation based on first- order approaches may be solved for much larger problem instances.
The main drawback of fixed-time strategies is that the optimal solutions found are specific to the historic period taken into account when solving the problem [115]. Thus real-time traffic fluctuations are not taken into account when RM strategies are defined. This may lead to congestion or underutilisation of the highway even though RM strategies have been determined and implemented. Traffic-responsive metering strategies are required to remedy this deficiency [115].
Traffic-responsive ramp metering strategies
Typically, the aim of traffic-responsive RM strategies is to keep the conditions on a highway close to a set of pre-specified values, based on real-time traffic information [115]. The simplest traffic- responsive RM controllers are local or independent controllers, which rely on measurements taken directly in the vicinity of the on-ramp in order to determine the metering rate [130]. One of the earliest local RM controllers is the demand-capacity algorithm introduced by Masher et
where qcaprepresents the highway capacity downstream of the on-ramp, qin is the highway flow
upstream of the on-ramp, ρout is the density of vehicles downstream of the on-ramp, ρcrit is
the critical highway density at which maximum flow occurs, and rmin is the minimum allowable
metering rate. The demand-capacity algorithm is considered to be an open-loop or feed-forward control approach due to the output of the system not being directly employed in the determi- nation of the control signal [115]. Due to the open-loop nature of the control algorithm, this control measure is prone to model deficiencies and its performance will degrade if the target value qcap is not accurate [130].
As a closed-loop alternative, Papageorgiou et al. [112] introduced the well-known Asservissement Lin´eaire d’Entre´e Autorouti`ere (ALINEA) RM strategy. The aim of the ALINEA algorithm is to regulate the downstream density so as to achieve maximum outflow. The metering rate according to ALINEA is given by
r(t + 1) = r(t) + Kr[ˆρ− ρout(t + 1)], (3.17)
where Kr > 0 is a control parameter, and ˆρ represents the desired downstream density (typically
ˆ
ρ = ρcrit), at which the highway outflow becomes close to qcap. This control structure is known
as integral-control, and is one of the simplest linear time-invariant controllers. A comparison of the functional structures of these algorithms is illustrated graphically in Figures 3.5 (a) and (b).
qin ρout qin ρout
r(t) r(t)
(a) (b)
Demand-capacity strategy r(t) = r(t− 1) + Kr[ˆρ− ρout(t)]
Feedforward (open loop) ALINEA (closed loop)
qcap ρˆ
Figure 3.5: Functional structure of (a) the demand-capacity algorithm and (b) the ALINEA algorithm. Adapted from Papageorgiou et al. [115].
The simplicity and effectiveness of the ALINEA algorithm have made it the best-known RM controller, with validated real-world performance through field implementations [113]. Various alterations of the ALINEA algorithm have been proposed. Zhang and Ritchie [178], for example, proposed the use of an artificial neural network in the place of the control parameter Kr. The
aim of such a neural network is to replace the constant parameter with one that varies according to the downstream density in order to provide improved traffic regulation based on the density.
Another well-known extension of the ALINEA RM control strategy is the so-called PI-ALINEA RM control strategy introduced by Wang et al. [167]. In this extension, a shortcoming of the ALINEA control strategy, namely that ALINEA is unable to take into account bottlenecks further downstream than the direct lane merge is addressed. This is achieved by adding an integral control loop to the feedback controller, which works in conjunction with the existing proportional control loop in the original ALINEA controller. The metering rate to be applied at the on-ramp is then determined according to
r(t) = r(t− 1) − Kp[ρout(t)− ρout(t− 1)] + Kr[ˆρ− ρout], (3.18)
where Kp and Kr denote the integral and proportional controller gain parameters, respectively.
Following a theoretical analysis of the proposed controller, as well as extensive numerical ex- periments, Wang et al. [167] concluded that using extensive parameter tuning, PI-ALINEA would perform at least as well as ALINEA in situations with only the immediate downstream bottleneck, while PI-ALINEA was able to outperform ALINEA in situations where a distant downstream bottleneck had to be taken into account.
Although independent RM controllers are often effective and easily implemented, problems may arise when several on-ramps are located in close proximity, as then equity cannot be achieved in respect of all on-ramps. Furthermore, performance is often severely degraded when the on-ramp queue storage space is limited [130]. In order to deal with limited on-ramp storage space, a second algorithm is often implemented in order to determine a minimum metering rate so as to prevent the maximum permissible queue length being exceeded. This minimum metering rate may, in turn, lead to degraded RM performance. Coordinated RM approaches attempt to remedy this situation by simultaneously controlling the available queueing space, as well as the metering rate at multiple adjacent on-ramps [130].
Two early examples of coordinated RM algorithms are BOTTLENECK [65] and ZONE [80]. The BOTTLENECK algorithm has two major components, a local RM algorithm, determining metering rates at a local level based on occupancy, and a coordination algorithm for determin- ing system-level metering rates, based on system capacity constraints. The local-level controller functions similarly to the demand-capacity algorithm. For the system level controller, the high- way is partitioned into a number of sections. For each of these sections, the number of vehicles stored in that section are determined by monitoring vehicle inflows and outflows. If the number of vehicles stored in a section is positive, the metering rate for that section is reduced. Finally, the more restrictive of the local- and system-level metering rates is applied at each of the on- ramps. Similar to BOTTLENECK, the ZONE algorithm also consists of a local-level controller, and a system-level controller. RM at the local level is again determined based on occupancy, typically using a variation on the demand-capacity algorithm. At the system level, volume con- trol is employed to ensure that the total traffic volume flowing through a pre-defined zone is not exceeded. Chu et al. [27] compared the performance of BOTTLENECK and ZONE with ALINEA, and showed that ALINEA performed better than both coordinated RM strategies. When the local-level controllers of BOTTLENECK and ZONE were replaced with the ALINEA controller, however, the coordinated strategies could be improved significantly, outperforming ALINEA in its original form.
The first attempt at a coordinated generalisation and extension of ALINEA was the MET- ALINE multivariable regulator strategy proposed by Papageorgiou et al. [110]. RM volumes are calculated as
r(t + 1) = r(t)− K1[ρ(t + 1)− ρ(t)] + K2[ ˆρ− ρ(t)], (3.19)
where r= [r1, . . . , rm]T is the vector of m controllable on-ramp metering rates, ρ= [ρ1, . . . , ρm]T
the simple ALINEA algorithm.
In an attempt to find a simpler coordinated RM strategy based on ALINEA, Papamichail and Papageorgiou [118] proposed a linked RM strategy with the aim of equalising the queue length of each on-ramp with the on-ramp downstream of its location. In this algorithm, three metering rates are calculated, the first being the local ramp flow r(t), calculated from (3.17). The second ramp flow, called the queue override ramp flow rw(t), is given by
rw(t) =−1 Tc
[wmax− w(t)] + d(t − 1), (3.20)
where Tc is the control cycle, wmax is the maximum allowable queue length, w is the current
queue length, and d is the on-ramp demand. The aim of this control law is to maintain an on-ramp queue that does not exceed the maximum allowable queue length. The third control law links the coordinates of each on-ramp with the on-ramp downstream of its location in an attempt to ensure that they have queues of similar length. This linked control ramp flow rLC is
determined by
rLC(t) =−Kw[wmin− w(t)] + d(t − 1), (3.21)
where Kw is a control parameter for managing the smoothness of response, and wmin is the
desired minimum queue length determined according to the queue present at the on-ramp sit- uated downstream of the current on-ramp location. The value of wmin is initially set to zero, and is only changed once congestion forms at the downstream on-ramp, and the queue length at that on-ramp exceeds some threshold value. Then the minimum queue length wmin is enforced
at the upstream on-ramp so as to provide more space on the highway for vehicles entering the highway at the downstream on-ramp. The minimum queue length is reset to zero once the queue length at the downstream on-ramp has fallen below the pre-specified threshold value. The final metering rate may then be calculated as
r(t) = max{min[r(t), rLC(t)], rw(t)}. (3.22)
The linked RM algorithm has been evaluated within the context of a macroscopic traffic simu- lation model, and it has been shown that its performance is significantly better than that of the conventional ALINEA algorithm when ramp queue limits are imposed [118].
The heuristic adaptive RM control approaches presented above are generally easy to implement. They do, however, often require extensive tuning of parameters, which may result in subpar performance in a practical environment [130]. In another approach found in the literature, the aim is to overcome the problem of parameter tuning. This approach is called model predictive control (MPC). Following this approach, metering rates for multiple on-ramps are determined by solving nonlinear optimisation models with the goal of minimising the total travel time of vehicles. In order to provide traffic-responsive solutions, this optimisation problem is solved in a rolling horizon scheme. At each time instant t, a new optimisation is performed over the prediction horizon Np, and only the first cycle of the solution is applied. This procedure is
repeated at each time period. In order to reduce complexity, a control horizon Nc≤ Np is often
introduced. After the control horizon has passed, the control signal is assumed to be constant. As a result, there are effectively two loops: The rolling horizon loop repeated at each time step t, and the optimisation loop inside the controller. The optimisation loop is repeated as often as
Traffic demand Traffic system Control input: ramp metering speed limits Controller Prediction (Np, Nc) Optimisation Expected demand Traffic state: speed, flow, density Rolling horizon (each k) Control signals Perfor- mance
Figure 3.6: Schematic illustration of the MPC structure for traffic control problems. Adapted from Hegyi et al. [53].
required in order to find an optimal solution for the control signal at a time instant t, given the values of Np and Nc, the current traffic state and the expected demand [53]. The structure of
MPC is illustrated graphically in Figure 3.6.
Bellemans et al. [15] and Hegyi et al. [53] have successfully applied MPC for optimal, traffic- responsive RM. In both cases, the highway was modelled in the context of the second-order macroscopic traffic simulation model METANET. In both cases, only one on-ramp was consid- ered due to the large computational overhead of solving the nonlinear optimisation problem. In order to overcome the problem of the computational overhead, Ghods et al. [42] proposed a decentralised solution approach based on the game theoretic concept of fictitious play for solv- ing the nonlinear optimisation model. The decentralisation approach allows the computation to be handled by multiple nodes, thereby rendering the approach applicable to larger problem instances. Optimal fixed-time strategies provide optimal performance based on the premise that there are no disturbances, and that the forecast input data are accurate. MPC approaches may mitigate the performance drop caused by disturbances, but are limited to small networks due to the large computational overhead. As a trade-off, Papamichail et al. [117] proposed a hierarchical approach with the aim of providing semi-optimal control for large networks. The hierarchical control approach is illustrated in Figure 3.7. As may be seen in the figure, the hier- archical control approach consists of three layers. Real-time traffic measurements, together with historical data, are provided to the estimation/prediction layer, which are then used to provide an estimate of the state of the traffic, and provide predictions of future demands. The optimal control engine of the optimisation layer then solves the nonlinear optimisation problem in order to determine optimal RM values for each of the local regulators. This optimisation is carried out every ten minutes. Due to disturbances, the traffic flow does not remain stable during the ten-minute intervals and, as a result, the metering rates become suboptimal. Finally, instead of direct application of the optimal metering rates, ALINEA is employed at the direct control layer, in order to vary the metering rate around the point set by the optimisation layer [117].
disturbances
Optimal control
(AMOC) Optimisation layer Current state estimation
Predicted disturbance trajectories
Local regulator Local regulator Local regulator
Open-loop optimal solution
Freeway network traffic flow process
Incidents On-ramp demand Weather conditions Routing behaviour . . . . . . Computer Real world Direct control layer Total time spent
Figure 3.7: A hierarchical MPC control structure with distributed controllers. Adapted from Pa- pamichail et al. [117].