3.2 Highway Traffic Control Measures
3.2.2 Variable Speed Limits
Variable speed limits (VSLs) are another popular control measure implemented on highways in response to the prevailing weather conditions. VSL installations were first implemented in Germany during the 1980s. Today, numerous VSL installations are encountered throughout Europe and the United States of America [114]. Initially, the main goal of VSL was improved traffic safety achieved by lowering the speed limits upstream of congested areas. More recently, however, attempts have been made to increase traffic flow through the use of VSLs [52]. These are the two main approaches towards employing VSLs in the literature, the first emphasising the homogenisation effect, while the focus in the second approach is on preventing traffic breakdown by controlling the flow by means of VSLs [53].
The idea behind the homogenisation effect is that the reduced speeds due to the newly imple- mented speed limits result in a reduction of the differences in speed of vehicles travelling in the
same lane, as well as vehicles travelling in adjacent lanes [114]. Increased traffic flow homogenisa- tion has a positive impact on traffic safety, and a correlation between VSLs and reduced accident probabilities has been demonstrated, with multi-year evaluations of the effect of VSLs on traffic safety showing reductions of up to 30% in accident numbers after VSL installation [23].
q0 Flo w (q ) qmax ρ0 Density (ρ) ρjam VSL 1 2 3 No speed control
Figure 3.8: The effect of VSLs on the fundamental diagram. Adapted from Hegyi et al. [53].
The focus in the traffic breakdown prevention approach is on preventing overcritical densities. Typically, this is achieved by reducing the speed limit before a bottleneck area, or an on-ramp, thereby altering the fundamental diagram of traffic flow of that section, as shown in Figure 3.8. When traffic on the highway is in state 1, it is nearly unstable and even small disturbances or on-ramp flows may cause traffic breakdown. Adjusting the speed limit will change the state from 1 to somewhere between 2 and 3, changing the shape of the fundamental diagram from the grey line to the dashed black line. The resulting decrease in flow stabilises the traffic flow and allows more space for traffic entering the highway from the on-ramp [53]. The gradient of the straight line forming part of the altered fundamental diagram is directly proportional to the magnitude of the newly imposed speed limit. By resolving these high-density areas (bottlenecks), higher flow rates may be achieved due to the prevention of traffic breakdown [52].
One of the earliest examples of VSL control was introduced by Smulders [151]. The VSL control problem was formulated as an optimal control problem, based on a macroscopic simulation model, with the aim of finding the maximum expected time until congestion. This could be achieved by maximising the expected value of the time until congestion sets in, given by
V (ρ0) = E Z τ 0 (`ρtuit− δIi=1) dt| ρ0 , (3.23)
where ` represents the number of lanes of the highway section under consideration, ρt and uit
represent the density and velocity at time t, respectively, i is a binary variable which indicates whether VSLs are operational, δ is a control variable specific to the stretch of highway, and I{.}
is an indicator function for determining whether congestion has formed on the highway stretch considered. Finally,
τ = inf{t ≥ 0 : ρt= ρjam} (3.24)
represents the time to congestion. Initially, a one-switch control policy was considered, taking the form
i(ρ) = I(ρ>¯ρ),
which means that control is applied only for densities exceeding a pre-defined value ¯ρ. This did, however, lead to frequent switching on and off of the VSL control, and as a result, hysteresis control was introduced. In the resulting hysteresis control policy, a single variable speed sign is
¯
ρoff (veh/km/lane) 56 26 2 5 8 12
¯
ρon (veh/km/lane) 70 42 28 28 29 31
As may be seen in the table, the VSL control is switched on at specific densities, ¯ρon, based on
the current traffic flow q0, and then only switched off again once the density has reached a lower
limit ¯ρoff. This prevents frequent switching of the VSL control, and thus results in a more stable traffic control policy. It is also evident from the results of the table that for VSL control it is important to detect an increase in the traffic flow above 3 500 vehicles per hour, as the control policy hardly changes for these values, and the control is most effective at these high densities [151].
Another early example of VSL control is the sliding-mode approach proposed by Lenz [83], who defined a control law for adjusting the speed limit based on the current traffic density. According to this control law, the speed limit ulimit is adjusted according to
ulimit= 120 if ρ≤ 14 100 if 14≤ ρ ≤ 17.5 80 if 17.5≤ ρ ≤ 23 60 if ρ≥ 23, (3.25)
where all speeds are expressed in kilometres per hour, and all densities are expressed in vehicles per kilometre. It was found, however, that this control law led to so-called standing waves, or shock waves propagating downstream of the speed limit sign, and as a result, a predictive element was introduced by Lenz et al. [84], where the density measure ρ in (3.25) is replaced by ¯ρ = ψρi + (1− ψ)ρi+1 in order to take into account, as a predictive measure, the density
of downstream traffic when adjusting the speed limit in order to prevent standing waves from forming.
Alessandri et al. [1, 2] introduced a nonlinear optimisation model based on a macroscopic traffic flow model. The optimal control problem formulated by Alessandri et al. [2] involves a stretch of highway partitioned into K + 1 sections. For the macroscopic model, the state vector is defined as
xt= [ρ0(t), ρ1(t), . . . , ρK(t), u0(t), u1(t), . . . , uK(t)], (3.26)
where ρi(t) and ui(t) represent the traffic density and average traffic velocity in section i ∈
{1, . . . , K} during time interval t, respectively. The performance measurement vector is given by
yt= [q0(t), q1(t), . . . , qK(t), w0(t), w1(t), . . . , wK(t)], (3.27)
where qi(t) and wi(t) represent the exit flow from section i to section i + 1, and the harmonic
mean speed of vehicles coming from section i and moving into section i + 1 during the time interval t, respectively. Furthermore, the vectors
rt= [r0(t), r1(t), . . . , rK(t)] (3.28)
and
represent the ramp in- and outflow values for section i ∈ {1, . . . , K} during time interval t. These vectors are updated at each model time interval t, based on the output of the underlying macroscopic traffic model. Finally, the control vector
ct= [b0(t), b1(t), . . . , bK(t)] (3.30)
captures the speed limit control commands for time interval t, where a value of 0.5 ≤ bi < 1
indicates various levels of speed restrictions being applied at section i∈ {1, . . . , K}. Finally, the objective function J = T X t=0 g(xt, ct) (3.31)
is to be minimised, where g is a function of penalising arguments based on the state vector xt and the control vector ct, respectively. This control vector also takes the form of hysteresis
control, similar to that employed by Smulders [151]. This optimal control problem was solved using Powell’s method for minimising an objective function approximately, and the results were implemented in the macroscopic simulation environment [2].
Kang et al. [67] employed a linearised traffic model, based on a linear speed-density relationship, such as the one shown in Figure 3.1, for example, in order to determine optimal VSLs for work zone operations on a highway adopting an MPC approach. This linear relationship is updated continually using real-time data from the microscopic simulation environment. The linearisation of the speed-density relationship allows the optimal control problem to be formulated as a linear programming (LP) problem in terms of macroscopic traffic flow variables. This LP problem is then reformulated at each MPC control time step using the latest traffic information from the simulation environment, and was solved using Lindo c [88], after which the new VSLs were ap-
plied in the simulation environment [67]. Another application of VSLs in an MPC context was demonstrated by Hegyi et al. [53], who extended their macroscopic traffic model to include con- trol structures for both RM, as explained in§3.2.1, and VSLs in an integrated control approach. A second attempt at integrating the RM and VSL control approaches was demonstrated by Carl- son et al. [23], who extended the AMOC strategy for finding optimal fixed-time RM strategies, introduced in §3.2.1, to include VSL control as well. Again the formulation entails minimising a nonlinear objective function, based on the underlying macroscopic traffic model, built in the METANET modelling environment.
In an attempt at simplifying the VSL control problem, Carlson et al. [25] proposed a feedback controller which takes as input real-time traffic flow and density measurements in order to calculate, in real time and within a closed loop, appropriate speed limits so as to maintain a stable traffic flow which is close to a pre-specified reference value, with the aim of achieving maximal throughput for any appearing demand. The control system developed takes the form of a cascade controller comprising two nested control loops, as may be seen from the controller structure shown in Figure 3.9.
The secondary controller is employed to adjust the VSL rate b which determines the outflow qc, which is, in turn, compared to the reference value for the outflow ˆqc. The outflow qc is
measured immediately downstream of the application area. The aim of the primary control loop is to control the measured density ρout with respect to the user specified reference density ˆρout
which, should be set to the critical density for the highway stretch under consideration in order to maximise the throughput. As may be seen in Figure 3.9, the secondary controller is designed as an integral (I) controller, whose transfer function in the time domain is given by
Figure 3.9: An MTFC feedback cascade controller structure using VSLs as actuator. Adapted from Carlson et al. [25].
where KI is the integral gain of the controller and eq(t) = ˆqc− qc is the flow control error. In
the primary loop, a proportional-integral (PI) controller is employed. The transfer function in the time domain for this PI controller is given by
ˆ
qc(t) = ˆqc(t− 1) + (KP0 + K 0
I)eρ(t)− KP0 eρ(t− 1),
where K0
I and KP0 represent the integral and proportional gains of the controller, respectively,
and eρ(t) = ˆρout−ρout(t) is the density control error [25]. The relationship between the difference
in the applied speed limit and the resulting difference in flow is modelled as a linear discrete-time transfer function given in the frequency domain by
4qc(z)
4b(z) = K
z− α z− β,
where α, β and K are model parameters which have to be tuned appropriately. In the time domain, this transform yields the difference equation
4qc(t + 1)− β ,c(t) = K4b(t + 1) − α4b(t).
In the absence of congestion, the transform from the flow at the application area qcto the flow at
the bottleneck qout is modelled as a first-order system with a time delay, given in the frequency
domain by
4qout
4qc
= τ
z + τ − 1,
where τ is again a model parameter. Finally, the transform of the bottleneck flow qout to the
bottleneck density ρoutis enabled by a linearisation of the fundamental flow-density relationship,
as shown in the fundamental diagram, around the critical density, and thus may be achieved simply through a proportional gain, given by K0 [25]. This controller was implemented, and the parameters tuned within a METANET macroscopic simulation model.
Another, and arguably simpler feedback-based VSL controller was designed by M¨uller et al. [105]. The controller takes a form very similar to ALINEA, as it is also an integral controller. Therefore, there is only a single controller parameter which requires empirical parameter tuning. In this controller, the speed limit is adjusted according to a VSL metering rate b ∈ [0.2, 0.8] which is calculated as
b(t) = b(t− 1) + KI[ˆρ− ρout], (3.32)
where KI is the integral gain of the controller, ˆρ denotes the target density at the bottleneck
that the controller aims to maintain, and ρout denotes the measured density at the bottleneck
location during control interval t. The metering rate is then rounded to the nearest tenth and the VSL to be applied is determined by
which resulted in the set of speed limits VSL∈ {20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120} in the original implementation. This controller was evaluated within the context of a simple highway network consisting of a dual carriageway and a single on-ramp joining the highway. The network and the VSL controller were implemented within the Aimsun microscopic traffic simulation environment.