Control of Spatiotemporal Congested Traffic ... - Semantic Scholar
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Control of Spatiotemporal Congested Traffic Patterns at Highway Bottlenecks Boris S. Kerner
Abstract—It is shown that the probabilistic feature of traffic breakdown at an on-ramp bottleneck leads to great limitations for reliable applications of a free flow control approach in which free flow should be maintained at the bottleneck. Based on these measured features of traffic breakdown at the bottleneck as well as on the Kerner–Klenov microscopic traffic model in the context of the author’s three-phase traffic theory, critical discussions of earlier traffic flow models for freeway control simulations and of ALINEA methods of Papageorgiou et al. for feedback on-ramp metering are made. An alternative congested pattern control approach to feedback on-ramp metering ANCONA introduced by the author in 2004 is numerically studied. In ANCONA, congestion at the bottleneck is allowed to set in. However, ANCONA maintains speeds within a congested pattern higher than about 60 km/h and prevents upstream propagation of the pattern. To reach these goals, after traffic breakdown has occurred spontaneously at the bottleneck, ANCONA tries to return to free flow via reduction of on-ramp inflow. A critical comparison of ANCONA with ALINEA and UP-ALINEA is made. Index Terms—Congested pattern control (ANCONA), critical discussion of on-ramp metering method ALINEA, empirical congested traffic patterns, feedback on-ramp metering, three-phase traffic theory, traffic breakdown at bottlenecks.
I. I NTRODUCTION
T
RAFFIC can be either “free” or “congested.” Congested traffic occurs most at freeway bottlenecks. Freeway congestion is a complex spatiotemporal nonlinear dynamic process [1]–[10]. For this reason, it is not surprising that empirical congested traffic pattern features have only recently been sufficiently understood [1]. Such empirical features are in serious conflict with almost all earlier theoretical and model results reviewed in [2]–[10] (see explanations in Section VI-A). These traffic flow models [2]–[10] are standard ones for the validation of freeway traffic control, management, and assignment (e.g., [3], [4], [6], and [11]–[14]). For this reason, the related simulations of freeway control and management strategies cannot predict many of the freeway traffic phenomena that would occur using a simulated control strategy. Consequently, the author introduced a new traffic flow theory called “three-phase traffic theory,” which can explain these empirical spatiotemporal patterns (see a comprehensive consideration of empirical traffic patterns, three-phase traffic theory,
Manuscript received January 13, 2006; revised July 18, 2006 and December 4, 2006. The Associate Editor for this paper was B. de Schutter. The author is with DaimlerChrysler AG, 71059 Sindelfingen, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TITS.2007.894192
Fig. 1. On-ramp inflow control [14]–[20]. (a) Downstream feedback control. (b) Upstream feedback control.
as well as a critical discussion of earlier traffic flow theories and models in the book [1]). There are a huge number of publications and many regular scientific conferences that are devoted to traffic control (see, e.g., references in [2]–[21]). One of the most used traffic control methods is on-ramp metering at an on-ramp bottleneck. Various models based on a “free flow control approach” to on-ramp metering have been suggested and developed. The basic idea of the free flow control approach is to maintain free flow conditions on the main road at the bottleneck (e.g., [14]–[20]). Free flow conditions at the bottleneck should be maintained at the maximum possible throughput in the free flow downstream of the bottleneck. In other words, the onset of congestion and congested pattern formation at the bottleneck should be prevented by the application of this approach. There are many different methods based on this approach (see [14]–[20]). The basic strategy is carried out through automatic control of flow rate of vehicles that can merge onto the main road from the on-ramp. In some of the methods using detectors for feedback control, traffic variables either downstream [Fig. 1(a)] or upstream of the bottleneck are measured [Fig. 1(b)] [14]–[20]. Results of these measurements are used as feedback for on-ramp inflow control, controlling light signal operation in the on-ramp lane(s). Depending on measurements of current traffic variables, the on-ramp inflow onto the main road from the on-ramp is either restricted or not. However, due to the probabilistic nature of traffic breakdown at bottlenecks, the free flow control approach exhibits a great application limitation: Regardless of control method
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application, in a wide range of the flow rate in free flow at a bottleneck, a traffic breakdown can randomly occur. This leads to congested pattern emergence with subsequent congestion propagation upstream of the bottleneck, which cannot be prevented by free flow control method application. A well-known example of the free flow control approach is the ALINEA model of Papageorgiou et al. [14], [17]–[20]. For these reasons, the author has introduced a qualitatively different “congested pattern control approach” (ANCONA) [1], [22]–[24]. In ANCONA, congestion at the bottleneck is allowed to set in. However, a congested pattern that emerges at the bottleneck should not propagate upstream; rather than propagating upstream, the congested pattern should be localized on the main road in a small neighborhood of the bottleneck. ANCONA should ensure that congestion at the bottleneck does not propagate far upstream of the bottleneck. To achieve this with ANCONA, a feedback control detector [Fig. 1(b)] should register congested patterns at the bottleneck. After traffic congestion has been registered, the on-ramp inflow is automatically reduced via light signal operation in the on-ramp lane(s). This leads to a return phase transition to free flow at the bottleneck, and therefore, traffic congestion on the main road does not propagate continuously upstream. A comparison of ANCONA and ALINEA made in [1] and [24] shows the following benefits of ANCONA: 1) greater throughputs on the main road and onramp; 2) considerably shorter vehicle waiting times at the light signal in the on-ramp lane; and 3) the upstream propagation of congestion that does not occur even if a congested pattern occurs at the bottleneck due to a short-term perturbation in traffic flow. The congested pattern is spatially localized in the vicinity of the bottleneck. The main aim of this paper is a discussion of a problem for freeway control caused by the probabilistic nature of the onset of congestion at bottlenecks. In comparison with previous publications of ANCONA [1], [23], [24], we present new results that explain how ANCONA solves the previously mentioned problem of ramp metering control associated with the probabilistic feature of freeway capacity. In addition, we consider the benefits of ANCONA in comparison with the ALINEA and UP-ALINEA models of Papageorgiou et al. [14], [17]–[20] in more detail than those in [1] and [24]. The paper is organized as follows: The problem for freeway traffic control caused by the probabilistic nature of freeway capacity is discussed in Section II. The Kerner–Klenov microscopic model used for control method simulations is discussed in Section III. Some new results of ANCONA application are presented in Section IV. New results of the comparison of the ANCONA method with the ALINEA and UP-ALINEA methods are considered in Section V. In Section VI, we will make a brief critical analysis of earlier traffic flow models reviewed in [2]–[10], which are widely used for freeway traffic control simulations. However, first, in the remainder of this introductory section, we discuss traffic phase definitions and a double Z-characteristic for phase transitions in three-phase traffic theory [1], the main types of measured (empirical) congested patterns at an on-ramp bottleneck [1], as well as the basic ALINEA [14], [17]–[20] and ANCONA rules [1], [23], [24] that we need for further consideration.
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Fig. 2. Qualitative graph of a double Z-characteristic for phase transitions in three-phase traffic theory [1]. F: free flow; S: synchronized flow; J: wide moving jam (low-speed states within wide moving jams). Arrows symbolize phase transitions between the three traffic phases.
A. Traffic Phase Definition and Double Z-Characteristic in Three-Phase Traffic Theory In three-phase traffic theory [1], there are two phases in congested traffic: 1) synchronized flow and 2) wide moving jam. Thus, there are three traffic phases in three-phase traffic theory: 1) free flow; 2) synchronized flow; and 3) wide moving jam. The synchronized flow and wide moving jam traffic phases are defined through the following spatiotemporal empirical (objective) criteria (definitions) [S] and [J]. The definition [J] for the wide moving jam phase is as follows: A wide moving jam is a moving jam that maintains the mean velocity of the downstream jam front, even when the jam propagates through any other traffic states or freeway bottlenecks. The definition [S] for the synchronized flow phase is as follows: The downstream front of synchronized flow does not exhibit this characteristic feature of wide moving jams; specifically, the latter front is often fixed at a freeway bottleneck. Within the downstream front of synchronized flow, vehicles accelerate from lower speeds in synchronized flow to higher speeds in free flow. In the three-phase traffic theory, the onset of congestion (traffic breakdown) is associated with a local phase transition from free flow to synchronized flow (F → S transition). Thus, F → S transition and traffic breakdown are synonyms. Wide moving jams can emerge spontaneously in synchronized flow only (S → J transition) [1]. Both the F → S and S → J transitions, as well as the associated S → F and J → S return transitions, are first-order local phase transitions, which are accompanied by hysteresis effects and lead to a double Z-characteristic (Fig. 2) [1]. This means that all these transitions exhibit probabilistic nature [1]. The probabilistic nature of traffic breakdown (F → S transition) is discussed in Section II. B. Congested Patterns After an F → S transition has occurred at an isolated onramp bottleneck, different congested patterns can be formed on the main road upstream of the bottleneck. There are two main types of congested patterns at the bottleneck: 1) synchronized flow patterns (SP) and 2) general congested patterns (GP) [1]. An SP consists of synchronized flow only. A typical empirical example of an SP localized at a bottleneck (localized
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Fig. 5. Simulated GP characteristics without on-ramp inflow control. (a) Speed on the main road in space and time. (b) qon (t) [1], [23], [24]. Fig. 3. Empirical average vehicle speed on the main road within an LSP at on-ramp bottleneck in space and time. Measured data from March 26, 1996 [1].
propagate upstream in the case of the LSP (Fig. 3). This means that possible upstream adjacent bottlenecks of a traffic network in the case of the LSP are not affected by congestion at the onramp bottleneck if the LSP width (in the longitudinal direction) is small enough. Second, empirical investigations show that discharge volume from the LSP can be considerably greater than that from the GP. This is because in the GP, the flow rate and speed are mostly strongly limited. The flow rate and speed within an LSP can be considerably greater than those within the GP [1]. A GP is the most frequently observed congested pattern at the on-ramp bottleneck [1]. In an example of GP emergence in microscopic simulations based on three-phase traffic theory [Fig. 5(a)], we suggest that there is no on-ramp inflow control and that the flow rate qin on the main road in free flow upstream of the bottleneck is constant, but the flow rate to the on-ramp qon is a function of time [Fig. 5(b)]. C. ALINEA Method
Fig. 4. Empirical example of GP at on-ramp bottleneck [1]. (a) Average vehicle speed on the main road in space and time. (b) Graph of (a) in space and time. Measured data from January 13, 1997.
SP: LSP) is shown in Fig. 3. In a GP, synchronized flow occurs upstream of the bottleneck, and then, wide moving jams emerge spontaneously in that synchronized flow, i.e., the GP consists of the synchronized flow and wide moving jam traffic phases (Fig. 4).1 There are two differences between the LSP (Fig. 3) and GP (Fig. 4), which are important for a traffic control theory. First, in contrast with the GP (Fig. 4), traffic congestion does not 1 Note that traffic data are measured at a set of road detectors whose spatial arrangement is presented in [1, Fig. 2.1]. Average speeds in Figs. 3 and 4(a) are determined by averaging of speeds of all vehicles passing a detector in a specified time period (1 min). Spatiotemporal pattern overviews in Figs. 3 and 4 are made via data interpolation between detectors. Data interpolation in Figs. 3 and 4(a) is made via a MATLAB function “Shading INTERP.” For interpolation of downstream and upstream front positions of traffic phases between detectors in Fig. 4(b), estimations for front velocities of the traffic phases are made through the use of the FOTO and ASDA models [1, Ch. 21]. Rather than these approximate overviews of spatiotemporal congested patterns (Figs. 3 and 4), for traffic phase distinguishing and a traffic phase dynamics analysis, we used a spatiotemporal correlation analysis of traffic data dynamics measured at adjacent detector locations (e.g., [1, Part II, Figs. 9.6, 9.7, 9.8(b)–9.10(b), 9.13–9.15, 10.1(c), 10.5(a), 10.6(c), 10.7(a), and 10.8(a)]).
In the ALINEA method of Papageorgiou et al. [14], [17]– (cont) [20], the following rule for the flow rate of vehicles qon (ts ), which can merge from the on-ramp onto the main road during the time interval ts ≤ t < ts+1 , is used [17]–[20]: (cont) (cont) (cont) qon (ts ) = qon (ts−1 ) + Ko o(opt) − o (t ) s sum sum
(1)
where Ko is constant; ts = Tav s, s = 1, 2, . . .; Tav (e.g., Tav = 1 min) is the averaging time interval for data measured by (opt) feedback control detectors in Fig. 1(a); osum is a chosen (cont) optimal (target) occupancy; and osum (ts ) is the occupancy measured downstream of the bottleneck using the feedback (cont) control detectors [Fig. 1(a)]; the occupancy osum (ts ) is associated with the time interval ts−1 < t ≤ ts and is averaged during this time interval. Traffic breakdown as a first-order F → S transition is responsible for probabilistic feature of the breakdown and freeway capacity discussed later in the text. This leads to congested pattern emergence with subsequent congestion propagation upstream of the bottleneck, which cannot be prevented by ALINEA appli(opt) cation (see [24, Fig. 9]). Even if target occupancy osum is chosen that is considerably smaller than the maximum occupancy in free flow downstream of the bottleneck, ALINEA cannot prevent GP emergence at the bottleneck when a short-term pulse in the on-ramp inflow occurs (Fig. 6). As found in [1] and [24], these critical conclusions about ALINEA performance
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Fig. 6. ALINEA cannot prevent GP emergence at the bottleneck if a shortterm pulse in the on-ramp inflow occurs [1], [24]. (a) Speed on the main road in space and time. (b) qon (t).
happen, regardless of the location of feedback control detector, as long as the detector is downstream of the bottleneck. D. ANCONA Method The ANCONA approach for on-ramp metering [1], [23], [24] takes into account the probabilistic nature of traffic breakdown at an on-ramp bottleneck. In ANCONA, we used SPs that can be more favorable than GPs in terms of the discharge volume from a congested pattern and of the vehicle time delay due to congestion. In ANCONA, congestion at a bottleneck is allowed to set in. However, average vehicle speed within an SP that emerges at the bottleneck should be maintained using ANCONA at a relatively high level (higher than about 60 km/h). Moreover, ANCONA ensures that congestion does not propagate upstream; rather than propagating upstream, the congested pattern should be localized on the main road in a small neighborhood of the bottleneck. Benefits of the ANCONA approach are considerably greater throughput and considerably shorter travel times in the on-ramp lane(s) than in the ALINEA control method (Section V) [1], [23], [24]. In ANCONA, there is no on-ramp control as long as free flow is measured at the bottleneck. On-ramp inflow control is first realized only after the onset of congestion has occurred due to traffic breakdown (F → S transition) that occurs spontaneously at the bottleneck. In other words, feedback control is performed when the average speed v (det) that is measured at the detectors for feedback control, which is upstream of the bottleneck [Fig. 1(b)], is equal to or drops below a chosen “congestion speed” vcong , i.e., v (det) ≤ vcong .
(2)
(cont)
When this condition is satisfied, qon is reduced via light (cont) signal operation. Due to the decrease in the flow rate qon , ANCONA tries to achieve a return S → F transition at the bottleneck. As a result of the S → F transition, the speed at the detector v (det) increases above vcong , i.e., v (det) > vcong .
(3) (cont)
is When condition (3) is satisfied, a greater flow rate qon allowed via light signal operation. If under this greater flow (cont) the onset of congestion occurs at the bottleneck rate qon once more, an incipient congested pattern begins to propagate upstream of the bottleneck. As a result, the speed at the detector
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Fig. 7. Suppression of GP emergence via ANCONA under a short-term pulse in qon (t). (a) Speed on the main road in space and time. (b) qon (t) is the same as that in Fig. 6(b) [1], [24].
for feedback control decreases. Therefore, condition (2) is satisfied once more. This leads to a new decrease in the flow (cont) rate qon , and so on. Thus, the basic idea of ANCONA is to allow the congestion condition at the bottleneck at the minimum possible level of traffic congestion. The ANCONA approach has three aims: 1) achievement of greater throughputs in the whole traffic network under very high traffic demand when congestion has to occur somewhere in the traffic network. This is realized due to a more homogeneous distribution of congestion among all bottlenecks in the network; 2) maintenance of shorter travel time of vehicles at the light signal in the on-ramp lane(s); 3) prevention of upstream propagation of a congested pattern that occurs at the bottleneck. For numerical simulations, the following simplified ANCONA on-ramp inflow control rule is used [1], [23], [24]: (cont) (ts ) qon
(cont)
=
qon1 , if v (det) (ts ) ≤ vcong qon2 , if v (det) (ts ) > vcong
(4)
where qon (ts ) is the flow rate of vehicles, which can merge from the on-ramp onto the main road during the time interval ts ≤ t < ts+1 ; the speed v (det) (ts ), which is measured by the feedback control detector [Fig. 1(b)], is averaged during the time interval ts−1 < t ≤ ts ; qon1 and qon2 are chosen model parameters (qon2 > qon1 ). Due to possible complicated time dependences of incoming flow rates qin and qon in real traffic, in real ANCONA installations, qon1 and qon2 should be time functions, which can be chosen in accordance with another ANCONA rule suggested in [24], in which qon1 and qon2 can depend on v (det) (ts ) and on the duration of the free flow phase at the bottleneck, respectively. In this paper, we limit a consideration of the ANCONA rule (4) at a time-independent flow rate qin . When a pulse of the flow rate to the on-ramp appears, ALINEA cannot prevent the formation and dissolution of the GP at the bottleneck (Fig. 6). In contrast, ANCONA leads to the dissolution of this GP (Fig. 7) at the same qin and qon (t) as those in Fig. 6. Rather than a congested pattern, which propagates continuously upstream, either an LSP or a time sequence of moving SP (MSP) and free flows remains at the bottleneck. MSPs dissolve while propagating upstream (see Section IV). LSP widths (in the longitudinal direction) are spatially limited. As a result, the
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influence of these congested patterns at the on-ramp bottleneck on the travel time on the main road is negligible. II. L IMITATIONS OF F REEWAY T RAFFIC C ONTROL C AUSED BY P ROBABILISTIC F EATURE OF F REEWAY C APACITY The probabilistic nature of traffic breakdown is given as follows: At a given flow rate, traffic breakdown can occur, but it should not necessarily occur. This means that on one day, the onset of congestion occurs; however, on another day, at the same flow rate and at the same traffic conditions, the onset of congestion is not observed [25]–[27]. Based on these and other empirical observations [28], in three-phase traffic theory [1], it has been shown that this probabilistic nature of traffic breakdown is associated with a first-order F → S transition. Indeed, traffic breakdown exhibits the following empirical fundamental features of a first-order phase transition: 1) at the same bottleneck, traffic breakdown can be either spontaneous or induced, i.e., the probabilistic feature of the breakdown is associated with its nucleation nature: If a critical speed disturbance at the bottleneck (“nucleus” for traffic breakdown) occurs, traffic breakdown is realized [1]; 2) both in empirical observations [26], [27] and numerical simulations of a microscopic model based on three-phase traffic theory [1], [29], the probability for an F → S transition (traffic breakdown) in free flow at (B) the bottleneck (denoted by PFS ) is an increasing function of the flow rate downstream of the bottleneck denoted by qsum (Fig. 8). Under the free flow condition at the bottleneck, we get qsum = qon + qin if there is no on-ramp metering and (cont) + qin under on-ramp metering application. To qsum = qon avoid confusion, we suggest all flow rates to be the total flow rates. If in some of figures flow rates are presented per freeway lane, it is mentioned in the figure caption (e.g., Fig. 8). There are two important characteristic flow rates on a func(B) tion PFS (qsum ): 1) the critical flow rate for the F → S transition (free B) and 2) the threshold flow rate denoted by denoted by qmax (B) qth (superscript “B” means that values are associated with an F → S transition at a bottleneck). Within the flow rate range [Fig. 8(b)] (B)
(free B) qth ≤ qsum ≤ qmax
(5)
traffic breakdown occurs with probability (B)
PFS > 0.
(6)
The nature of probabilistic traffic breakdown is as follows: Under condition (5), free flow at the bottleneck is in a metastable state with respect to traffic breakdown. This means that small amplitude speed disturbances in the free flow do not lead to traffic breakdown (F → S transition). However, if a random short-time speed disturbance in free flow in a neighborhood of the bottleneck appears whose amplitude (difference between vehicle speed within the disturbance and initial speed) exceeds some critical value, an F → S transition occurs. This random speed disturbance leading to traffic breakdown is called a critical speed disturbance (or critical perturbation in free flow) [1]. The free flow metastability can also be seen in Fig. 2.
(B)
Fig. 8. Probability for traffic breakdown PFS as a function of the flow rate on the main road downstream of the bottleneck. (a) Measured functions in real traffic taken from Persaud et al. [26] for two flow rate averaging intervals of 1 and 10 min. (b) Theoretical function found from numerical simulations (for the interval for observing traffic flow of 15 min) is taken from [29]. Flow rates are shown per freeway lane.
Between states of free flow (F) and synchronized flow (S), there is a gap in speed: Speeds within this gap are related to various speed disturbances that are discussed previously in the text. If in free flow a critical speed disturbance on the main road in a neighborhood of the bottleneck occurs randomly, then the disturbance amplitude self-grows. The duration of this growth until traffic breakdown can be measured and is less than or about 1 min (down arrow from F to S in Fig. 2). This growing critical local speed disturbance can occur randomly at different freeway locations in a neighborhood of the bottleneck, i.e., it is not predictable both in time and space. Until now, one could see the result of critical speed disturbance growth only, i.e., synchronized flow emergence at the bottleneck. Numerical simulations show that this random speed disturbance cannot be registered already at some 100 m downstream of the end of the merging on-ramp region. In particular, no increase in the downstream occupancy appears during the appearance and subsequent growth of the disturbance. Let us explain these common results based on ALINEA applications with downstream feedback control detector locations (B) [Fig. 1(a)]. Traffic breakdown probability PFS is greater than zero under condition (5) (Fig. 8). This means that at a given flow rate qsum within the range (5), free flow at the bottleneck can (B) be kept only with certain probability 1 − PFS < 1. This fundamental traffic feature happens, regardless of whether ALINEA is applied or not. An explanation of this is as follows: Speed and occupancy disturbances at the bottleneck, which can lead to traffic breakdown, do not change traffic variables downstream of the bottleneck as long as no congested pattern has been formed at the bottleneck. Thus, the downstream feedback control detector of ALINEA [Fig. 1(a)] cannot register speed perturbation growth at the bottleneck; therefore, ALINEA cannot
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register resulting traffic breakdown at the bottleneck. This conclusion has been proven in numerical simulations of ALINEA on-ramp metering [1]. To keep free flow at the bottleneck with greater probability, in ALINEA, smaller target occupancy must be chosen, at which smaller flow rate of vehicles is allowed to merge from the on-ramp onto the main road. As a result, the (B) flow rate qsum and breakdown probability PFS are reduced. In the case shown in Fig. 6, ALINEA does not try to maintain the highest possible throughput downstream of the bottleneck. Nevertheless, traffic breakdown can also occur (Fig. 6). Thus, when (5) is satisfied, it can turn out that a control method cannot prevent traffic breakdown at the bottleneck. Only if the flow rate downstream of the bottleneck
(8)
qsum
Gn ∆n = max (−bn τ, min(an τ, v,n − vn ))
(9) (10) (11)
(7)
then probability for traffic breakdown is equal to zero. Hence, only under condition (7) is reliable control method application for maintaining free flow at the bottleneck possible. (B) However, the threshold flow rate qth is considerably smaller (free B) than the maximum flow rate qmax in free flow at the bottleneck (Fig. 8). In other words, the probabilistic nature of traffic breakdown is a great problem for any free flow control approach application. Until now, we have considered the case when a detector for feedback control is downstream of the bottleneck [Fig. 1(a)]. To explain the term “downstream of the bottleneck” used in threephase traffic theory [1], note that after traffic breakdown has occurred at a bottleneck, synchronized flow is formed upstream at the bottleneck. The downstream front of the synchronized flow, within which vehicles accelerate from the synchronized flow to free flow downstream, is usually fixed in a neighborhood of the bottleneck. The location on a freeway at which this downstream front of synchronized flow is spatially fixed is called “the effective location” of the bottleneck [Fig. 1(b)] [1]. In all numerical simulations that follow, a speed disturbance in free flow, within which traffic breakdown occurs at the bottleneck, is localized upstream of the effective bottleneck location. For these reasons, all freeway locations downstream of the effective bottleneck location are associated with the term “downstream of the bottleneck” [Fig. 1(a)]. In contrast, all freeway locations upstream of the effective bottleneck location are associated with the term “upstream of the bottleneck.” Note that in contrast with the case of downstream detector location [Fig. 1(a)], when a feedback control detector is upstream of the effective bottleneck location [Fig. 1(b)] and it is located within speed disturbances at the bottleneck, the detector can measure speed disturbance dynamics. This means that this upstream detector location can be used to suppress traffic breakdown using suitable ramp metering strategy. This is used in ANCONA. III. M ODEL FOR S IMULATIONS OF F REEWAY C ONTROL M ETHODS Here, we consider a stochastic microscopic two-lane model of Kerner and Klenov in the context of three-phase traffic theory [30]. All model results presented are related to numerical
where index n corresponds to the discrete time t = nτ , n = 0, 1, 2, . . .; τ is the time step; vfree is the maximum speed in free flow; gn = x,n − xn − d is the space gap (net distance) between vehicles, where d is the vehicle length; the lower index marks functions (or values) related to the preceding vehicle; vn is the vehicle speed at time step n; xn is the vehicle coordinate; vn is the speed calculated without a noise component ξn ; a is the maximum acceleration; Gn is a synchronization gap; and an ≥ 0 and bn ≥ 0. (safe) (a) , gn /τ + v ) is safe speed, where In (8), vs,n = min(vn (safe) vn is a safe speed of the Krauß model [31], which is a solution of the Gipps equation [32] vn(safe) τ + Xd vn(safe) = gn + Xd (v,n )
(12)
where Xd (u) is the distance travelled by the vehicle with an initial speed u at a time-independent deceleration b until it comes to a stop; in the model with the discrete time Xd (u) = bτ 2 (αβ + α(α − 1)/2) [31]; b is constant; α is the integer (a) part of u/bτ ; β is the fractional part of u/bτ ; and v is an anticipation speed [1, eq. (16.48)]. To simulate driver time delays either in vehicle acceleration or in vehicle deceleration, an and bn in (10) and (11) are taken as the following stochastic functions: an = aθ(P0 − r1 )
(13)
bn = aθ(P1 − r1 ) p0 , if Sn P0 = 1, if Sn p1 , if Sn P1 = p2 , if Sn
(14) = 1 =1 = −1 = −1
(15) (16)
where r1 = rand(0, 1), i.e., this is an independent random value uniformly distributed between 0 and 1; θ(z) = 0 at z < 0 and θ(z) = 1 at z ≥ 0; probabilities p0 (v) and p2 (v) are given functions of speed, and probability p1 is constant; 1 − P0 and 1 − P1 are the probabilities of a random time delay in vehicle acceleration and deceleration, respectively; Sn denotes the state of vehicle motion (Sn = −1 represents deceleration, Sn = 1
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where ξa and ξb are random sources for deceleration and acceleration, respectively, i.e.,
Fig. 9. Hypothetical steady model states [1], [30].
represents acceleration, and Sn = 0 represents motion at nearly constant speed), i.e., −1, Sn = 1, 0,
if vn < vn−1 − δ if vn > vn−1 + δ
(17)
otherwise
where δ is constant (δ aτ ). Equations (9)–(11) describe the speed adaptation effect in synchronized flow. This speed adaptation effect takes place when the vehicle cannot pass the preceding vehicle within the synchronization gap Gn : At gn ≤ Gn , the vehicle tends to adjust its speed to the preceding vehicle without caring what the precise space gap is, as long as it is safe. At a given time-independent speed of the preceding vehicle v,n = v = const, this speed adaptation leads to a car following with vn = v = v at a time-independent space gap gn = g. There is an infinite number of these gaps associated with the same speed v = v . These gaps lie within the gap range gs ≤ g ≤ G (gs is a time-independent safe space gap found from the equation v = vs , where vs = vs,n at gn = g, v,n = v ; G = Gn at vn = v,n = v ), i.e., there is no desired (or optimal) space gap in synchronized flow. The speed adaptation effect within the synchronization gap means that hypothetical steady model states of synchronized flow (in which all vehicles move at the same time-independent speed and at the same space gap to one another) cover a 2-D region in the flow–density plane (Fig. 9), i.e., there is no fundamental diagram for steady speed states of synchronized flow. The boundaries of this 2-D region, i.e., F, L, and U, are associated with free flow, the synchronization gap, and a safe gap determined through the safe speed, respectively. In (10), the synchronization gap that determines the boundary L in steady model states (Fig. 9) is taken as follows: Gn = G(vn , v,n )
G(u, w) = max 0, kτ u + φa−1 u(u − w)
(18)
if Sn+1 = −1 if Sn+1 = 1 if Sn+1 = 0
(21)
ξb = aτ θ(pb − r)
(22)
where pa and pb are probabilities of random acceleration and deceleration, respectively; r = rand(0, 1). Lane-changing rules in a two-lane model are associated with conditions R → L : vn+ ≥ v,n + δ1 and vn ≥ v,n
(23)
L →R :
(24)
vn+
> v,n + δ1 or
vn+
> vn + δ1
for lane changing from the right lane to the left (passing) lane (R → L) (23) and a return change (L → R) (24). The safety conditions for lane changing are
gn+ > min vn τ, G+ (25) n
gn− > min vn− τ, G− (26) n
+ G+ n = G vn , vn
− (27) G− n = G vn , vn . If (23)–(27) are satisfied, then the vehicle changes lane with probability pc . Functions in (27) are associated with the function G(u, w) (18). In (23) and (24), δ1 ≥ 0 is constant. In (23)–(27), superscripts + and − in variables and functions denote the preceding vehicle and the trailing vehicle in the “target” (neighboring) lane, respectively; the speed vn+ or vn− is set to ∞ if gn+ or gn exceeds a look-ahead distance La , respectively; pc , La , and δ1 (δ1 ≥ 0) are constants. In the model, there is an on-ramp on the main two-lane road. Open-boundary conditions are applied. The related on-ramp bottleneck consists of two parts [1, Sec. 16.3.9]: 1) a merging region, where vehicles can merge onto the main road from the on-ramp lane, and 2) a part of the on-ramp lane upstream of the merging region, where vehicles move in accordance with (8)–(22). The model takes merging phenomena explicitly into account. The beginning of on-ramp merging region is at xon = 10 km, and the length of the merging region is Lm = 0.3 km; numerical simulations show that the effective bottleneck location is at xeff ≈ 10.45 km. Quantitative validation of model parameters has been made to have simulation parameters of congested patterns in accordance with measured data [1].
(19)
where k > 1 and φ are constants. Random deceleration and acceleration are applied, depending on whether the vehicle decelerates or accelerates, or else maintains its speed, i.e., −ξb , ξn = ξa , 0,
ξa = aτ θ(pa − r)
(20)
IV. T IME D ELAYS IN ANCONA A PPLICATION For the case of the feedback control detector location x(det) = 9.8 km, numerical simulations of the ANCONA rules (4) have been made in [24]. In this section, time delays in ANCONA application at the same parameters of an on-ramp merging region as those in [24] and various detector locations x(det) are presented. We consider a case of flow rates satisfying (5) when traffic breakdown can occur spontaneously at the bottleneck.
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Fig. 10. Symbolical schema of ANCONA [23], [24]. The spontaneous onset of congestion (F → S transition) is allowed by ANCONA; after congestion occurs, ANCONA tries to achieve an S → F transition labeled by arrow S → F.
In ANCONA, using on-ramp inflow control (4), there can be two nonlinear transitions between two qualitatively different traffic states at the bottleneck (Figs. 2 and 10). One of these states is related to the synchronized flow phase (congestion). The second state is related to the free flow phase. Synchronized flow occurs due to an F → S transition at the bottleneck (arrow F → S in Fig. 10). The F → S transition can occur (cont) = qon2 in (4). As a result of the spontaneously when qon F → S transition, an SP begins to form at the bottleneck, and condition (2) is satisfied. Then, ANCONA tries to achieve a return phase transition from synchronized flow (congestion) to free flow (S → F transition; arrow S → F in Fig. 10). This S → F transition should be forced by light signal operation when (2) is (cont) = qon1 < qon2 . As a result of satisfied, and therefore, qon the S → F transition, the emergent SP dissolves (see the following explanation). Note that in real ANCONA applications, when condition (3) is satisfied during a long enough time interval, there should no longer be on-ramp control. This is realized in an ANCONA rule [24, eq. (44)]. There are several time delays between on-ramp inflow change (4) and the related traffic state change associated with the F → S and S → F transitions at the bottleneck. (cont)
= qon2 is satisfied, there is 1) After the condition qon (B) a random time delay TFS for an F → S transition (Figs. 11 and 12). In the flow–density plane, this F → S transition is accompanied by a drop in speed and a jump (B) in density [arrow labeled by F → S in Fig. 11(e)]. TFS (free B) decreases strongly when qin + qon2 → qmax [1]. Thus, at greater qon2 , free flow can exist at the bottleneck only during short-time intervals. For this reason, for a better ANCONA performance, the value qon2 in (4) should (free B) satisfy condition qin + qon2 < qmax . (cont) = qon1 is satisfied, there is a ran2) After condition qon (B) dom time delay TSF for an S → F transition. In the flow–density plane, this S → F transition is accompanied by a jump in speed and a drop in density [arrow labeled by S → F in Fig. 11(e)]. (B) (B) The time delays TFS and TSF are related to the locations of the F → S and S → F transitions (about 10.3–10.1 km), which are upstream of the effective bottleneck location xeff ≈ 10.45 km.
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3) After the S → F transition has occurred at the bottleneck (item 2), free flow begins to propagate upstream, specifically, an initial SP at the bottleneck transforms into an MSP (see Fig. 11(a); MSPs are considered in [1] in de(B) tail). Consequently, there is a time delay τSF in satisfying the condition (3) at the detector location x(det) , which is upstream of the S → F transition location [Fig. 11(c) and (d)]. 4) If x(det) is far enough upstream from the F → S transition (B) location, then there is a time delay τFS in satisfying the condition (2) after the F → S transition has occurred (x(det) = 9.5 km in Fig. 12). This is because it takes some time for synchronized flow to propagate from the F → S transition location at which synchronized flow has initially occurred to the detector location x(det) . It should be noted that because there is a localized perturbation in free flow at the bottleneck (see Section II), the (B) time delay τFS is remarkable only when the distance between x(det) and xon = 10 km is greater than some critical value, which is about 0.3–0.4 km. For this reason, at x(det) = 9.8 km (B) (B) (Fig. 11), τFS is undistinguishable in comparison with TFS . This explains numerical results in which we did not find some marked difference in ANCONA application if x(det) is within the range 10.3 ≥ x(det) ≥ 9.7 km
(28)
which is associated with the perturbation width (in the longitudinal direction). Condition (28) is related to “optimal” detector locations for ANCONA application. Under condition (28) and chosen ANCONA parameters qon1 and qon2 , congestion does not continuously propagate upstream [Fig. 11(a)]. This result can be explained as follows: Under (B) condition (28), τFS is negligible, and synchronized flow cannot propagate upstream far enough from the bottleneck before (2) (B,mean) is satisfied. Under small qon1 , the mean time delay TSF for S → F transitions is not great (Fig. 11). Thus, free flow returns quickly at the bottleneck: The downstream front of synchronized flow begins to propagate upstream, i.e., an MSP appears, which propagates upstream of the bottleneck as a localized region of synchronized flow: MSP propagation is no longer influenced by the bottleneck. An MSP can exist and continuously propagate upstream only if the MSP inflow rate, which is equal to qin , is greater than a threshold flow rate qth for an F → S transition on a homogeneous road [1]. Otherwise, i.e., at qin < qth , MSP dissolves. The condition for MSP dissolution is easy to satisfy because, as shown in [1], the threshold flow rate qth can be (B) considerably greater than the threshold flow rate qth for an F → S transition at the bottleneck (Figs. 8 and 10). Thus, (cont) decreases to qon1 (4) [Fig. 11(c)], synchronized when qon flow within the MSP dissolves over time, which can clearly be seen in Fig. 11. Note that within MSPs, the average speed is relatively high (about or more than 60 km/h) (Figs. 11 and 12). Under condition (28), no pinch effect occurs, and no wide moving jams are formed spontaneously within this high-speed
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Fig. 11. Study of ANCONA associated with the example of ANCONA general pattern suppression shown in [24, Fig. 11]. (a) Fragment of speed in space (cont) (cont) (t). Fragments of (c) qon (t) and and time that illustrates delay times by occurrence and dissolution of SPs at the bottleneck via ANCONA (4). (b) qon (det) = 9.8 km. (e) Phase transitions in the (left) flow–density plane and on (right) time dependence of (d) speed on the main road at x = 10.3 km and x = x speed associated with (d). Parameters of simulations are the same as those in [24, Fig. 11].
synchronized flow (see explanations about the pinch effect in synchronized flow and wide moving jam emergence in [1]). The opposite case of great flow rates on the main road upstream of the bottleneck qin > qth , at which MSPs propagate upstream continually without dissolution, has been explained and studied in [24, Sec. 8]. At x(det) = 9.5 km, i.e., upstream of the optimal detec(B) tor locations (28), we can distinguish the time delay τFS [Fig. 12(d)]. This time delay leads to a decrease on aver(cont) age value qon . In this case, ANCONA does not perform optimally. Nevertheless, ANCONA can perform because it also prevents upstream congestion propagation in this case [Fig. 12(a)]. However, if condition (28) is not satisfied and the detector location x(det) is markedly less than 9.5 km, the time de(B) lay τFS increases considerably. Then, in contrast with the aforementioned cases, the pinch effect occurs in synchronized flow upstream of the bottleneck: Speed decreases and density increases considerably in synchronized flow. As a result, wide
moving jams can occur in the pinch region of the synchronized flow, i.e., ANCONA cannot perform. Numerical simulations also show that ANCONA application is also not possible if the detector location is downstream of the effective location of the bottleneck [Fig. 1(a)]. This is because in this case, which is characteristic for ALINEA, speed at the detector always satisfies (3), regardless of whether an F → S transition occurs at the bottleneck or not. The speed vcong in the ANCONA rules (4) should satisfy two conditions: 1) If vcong is too low, then synchronized flow of lower speed can be formed within SPs upstream of the bottleneck. This can lead to wide moving jam emergence in the synchronized flow. 2) On the other hand, if vcong is too (cont)
(4) even if high, then ANCONA can reduce the flow rate qon free flow is still at the bottleneck. This is because there can be many random speed disturbances at the bottleneck, which decay over time and do not cause traffic breakdown. Thus, a value of vcong that is too high can lead to a decrease in throughput using ANCONA. In this paper, the optimal range of vcong that
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Fig. 12. Study of ANCONA at x(det) = 9.5 km. (a) Fragment of speed in space and time that illustrates delay times by occurrence and dissolution of SPs at the (cont) (cont) (t). Fragments of (c) qon (t) and (d) speed on the main road at the location of 10.3 km and at the detector bottleneck via ANCONA application (4). (b) qon location (9.5 km). Other parameters are the same as those in Fig. 11.
satisfies both conditions 1 and 2 is determined based on the results of numerical simulations: 70 < vcong < 85 km/h. In real ANCONA applications, the optimal range of vcong should be determined based on measured data. Simulations show that the smaller the flow rate qon1 , the (B) shorter time delay TSF for an S → F transition, i.e., the easier and the quicker the S → F transition is realized. In this case, MSP dissolution also occurs more quickly. In contrast, if qon1 is too great, an S → F transition does not occur. In this case, if qin is also great, rather than an LSP, a widening SP can occur at the bottleneck; moreover, the pinch effect with a wide moving jam formation can be realized later. Thus, there is some “optimal” value of qon1 in (4) at which throughput is great and synchronized flow is localized at the bottleneck (Figs. 11 and 12). To achieve the maximum throughput via ANCONA application in real installations, rather than the simple ANCONA rule (4), a more complex ANCONA rule of [24] should be applied in which qon1 = const: When the speed v (det) (ts ) that is measured at feedback control detector decreases, qon1 should also decrease. V. ANCONA, ALINEA, AND UP-ALINEA: A C OMPARISON Whereas ANCONA tries to achieve an S → F transition after traffic breakdown (F → S transition) has randomly occurred at the bottleneck (Fig. 10), ALINEA tries to achieve the optimal occupancy in free flow downstream of the bottleneck (black point in Fig. 13). Fig. 10 for ANCONA and Fig. 13 for ALINEA illustrate this fundamental difference between ANCONA and ALINEA.
Fig. 13. Symbolical schema of free flow control approach in the flow– occupancy plane. (a) Explanation of theoretical basis of ALINEA in the vicinity of optimal (target) free flow point (F: free flow; C: congested traffic) [20]. (b) Spontaneous traffic breakdown (F → S transition labeled by arrow F → S) occurs with probability greater than zero within the flow rate range (5), regardless of the ALINEA control application [1], [24].
A theoretical basis of ALINEA [20] is associated with an assumption of well-known standard traffic flow theories [2], [3], [6] that downstream bottleneck capacity in free flow qcap and the associated critical occupancy ocr are related to the maximum point at the fundamental diagram for downstream traffic flow at the bottleneck [Fig. 13(a)]. Thus, in ALINEA, we suggested that traffic breakdown does not occur as long as (cont) downstream occupancy osum does not exceed critical occupancy ocr . In accordance with this assumption, the target occu(opt) pancy osum should not exceed ocr , and the ALINEA rule (1) (cont) should ensure that measured downstream occupancy osum (ts ) [possible changes of this occupancy are symbolically shown by arrows in Fig. 13(a)] must be in the small neighborhood of the (opt) target one, i.e., osum (labeled by black point in Fig. 13). This theoretical basis of ALINEA is in a very deep contradiction with empirical probabilistic features of traffic breakdown
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Fig. 14. Comparison of ANCONA and ALINEA. Speed in space and time for (a) ANCONA and (b) ALINEA taken from [24, Fig. 12(a) and (b)]. Time dependence of (c) TTS and (d) ATT for (curves 1) ANCONA and (curves 2) ALINEA. Calculations of (c) TTS and (d) ATT are made for the same on-ramp inflows qon for ALINEA and ANCONA.
discussed previously in the text: Under condition (5), with a finite probability, an F → S transition occurs spontaneously at the bottleneck, regardless of ALINEA application [arrow labeled by F → S in Fig. 13(b)]. This explains [24, Fig. 9], in which a GP occurs spontaneously under ALINEA application as a result of this F → S transition at the bottleneck: ALINEA has no influence on GP upstream development. Even if no significant flow and speed disturbances appear at the bottleneck and, therefore, a GP still has not occurred in ALINEA application [see [24, Figs. 6 and 12] and Fig. 14(b)], ALINEA exhibits disadvantages in comparison with ANCONA (Fig. 14). This can be seen from calculations of total time spent (TTS) [Fig. 14(c)] (TTS is defined as TTS by all vehicles on the main road and in the on-ramp lane between x = 7 and 11 km that passed the location of 7 km during 10-min intervals) and of average travel time (ATT) [Fig. 14(d)] (ATT is defined as TTS divided by the number of vehicles that passed the location of 7 km during 10 min). When there is only an upstream detector for feedback control [Fig. 1(b)], UP-ALINEA with the same rules (1) can be used (cont) in which, however, the downstream occupancy osum (ts ) is calculated from the measured upstream occupancy and the flow (cont) rates qon (ts−1 ) and qin [20]. Because, in UP-ALINEA, the feedback control detector is upstream of the effective bottleneck location, UP-ALINEA prevents traffic breakdown. However, congestion prevention via UP-ALINEA is achieved only at the expense of extremely rapid growth of vehicle queue at the light signal and the associated growth of travel time (waiting time) in comparison with travel time growth under ALINEA application at the same target occupancy. For the same reason, UP-ALINEA also exhibits considerable disadvantages in comparison with ANCONA. First, due to speed disturbance at the bottleneck (Section II), UP-ALINEA cannot almost perform at the detector location x(det) = 9.8 km used for ANCONA in [24] and in Fig. 11: The UP-ALINEA rule (1) applied with exact reformulation of downstream occupancy made in [20] leads to an extremely small average flow rate of vehicles, which may merge from the
(cont)
≈ 95 vehicles/h). This result is also valid for on-ramp (qon all other detector locations satisfying (28). This is because occupancy within the speed disturbance is greater than any possible target occupancy in free flow downstream of the bottleneck. Second, if x(det) is chosen upstream of the optimal one for ANCONA (28), nevertheless, ANCONA exhibits great benefits in comparison with UP-ALINEA (Fig. 15). This is because the UP-ALINEA rule (1) is not associated with the probabilistic nature of F → S and S → F transitions at the bottleneck. This can be seen in Fig. 15(c) (curve 2): UP-ALINEA tries to prevent congestion rather than finding an optimal relation between free flow and synchronized flow phase existence at the bottleneck as ANCONA does (curve 1). For this reason, the average flow (cont) rate qon is considerably greater in ANCONA (curve 1) than in UP-ALINEA (curve 2) [Fig. 15(c)]. Correspondingly, the waiting time at the light signal, TTS, and ATT are considerably longer for UP-ALINEA (curves 2) than for ANCONA (curves 1) [Fig. 15(d)–(f)]. As shown in [33], the ALINEA/Q and X-ALINEA/Q methods [20], which limit vehicle queue growth at the light signal, cannot prevent traffic breakdown and the resulting upstream propagation of traffic congestion. VI. D ISCUSSION A. Critical Discussion of Freeway Control Simulations Based on Earlier Traffic Flow Models Simulation results presented previously in the text are based on three-phase traffic theory. These results cannot be derived with the use of earlier traffic flow models for freeway congestion reviewed in [2]–[13]. Indeed, the earlier traffic flow models, which are based on the fundamental diagram hypothesis, can be divided into two classes [1]. The first model class is referred as the classic Lighthill– Whitham–Richards (LWR) model. Examples of this model class are cell transmission models of Daganzo (see references in [6]). However, it has been explained in [1] that this model class cannot predict the following fundamental measured traffic flow
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(cont)
Fig. 15. Comparison of ANCONA and UP-ALINEA at x(det) = 9.5 km. Speed in space and time for (a) ANCONA and (b) UP-ALINEA. (c) qon (t). Time dependence of travel time in the (d) on-ramp lane, (e) TTS, and (f) ATT. In (c)–(f), curves 1 are for ANCONA, and curves 2 are for UP-ALINEA. UP-ALINEA rules (1) and [20, eq. (9)] are used with model parameters in (1) taken from [24, Fig. 6]. In (4), qon1 = 240 vehicles/h; other parameters of ANCONA are the same as those in Fig. 12. Calculations of (e) TTS and (f) ATT are made for the same on-ramp inflows qon for UP-ALINEA and ANCONA.
features: 1) the probabilistic nature of traffic breakdown and 2) the spontaneous moving jam emergence. Thus, the results presented in Figs. 11 and 12, in which probabilistic traffic breakdown (F → S transition) occurs at the bottleneck and later, using ANCONA, a return probabilistic S → F transition with subsequent MSP appearance and dissolution are realized, cannot be found based on the LWR and cell transmission models. The second model class is referred as the classic General Motors model of Herman et al. (see references in [2]–[5] and [7]–[10]). In this model class, beginning at a critical density, there is instability of steady model states on the fundamental diagram. Examples for this model class are the optimal velocity models by Newell et al., the Payne macroscopic model and its variants, the Wiedemann psychophysical model and its variants, the Nagel–Schreckenberg cellular automata model and its variants, the Krauß model, the intelligent driver model by Treiber and Helbing, as well as a huge number of other models. In this model class, the instability of steady model states should explain traffic breakdown. In 1994, Kerner and Konhäuser found out that this instability leads to wide moving jam emergence in free flow (see references in [1] and [8]). As is currently well known, this result is true for all models of this class: Traffic breakdown is explained by a phase transition from free flow to wide moving jam(s) (F → J transition). However, this fundamental model result contradicts empirical results: Rather than an F → J transition, in all measured data, an F → S transition governs traffic breakdown at a bottleneck. Thus, the models of this class cannot also show and explain empirical features of real traffic breakdown as well as real features of congested patterns (for more detail, see [1, Sec. 3.3] and [34]).
Three-phase traffic flow theory is able to show and predict probabilistic traffic breakdown and emergence congested patterns at bottlenecks in accordance with real measured data [1]. The Kerner–Klenov stochastic traffic flow model (Section III), the KKW cellular automata model [29], and a recent deterministic acceleration time delay model [35], [36] can predict all known empirical congested patterns at bottlenecks. For this reason, we can expect that simulation results of freeway traffic control discussed in this paper will also be found in real freeway traffic. VII. C ONCLUSION 1) The probabilistic feature of traffic breakdown at a freeway bottleneck limits the reliability of applications of a free flow control approach in which free flow should be maintained at the bottleneck. This is because this probabilistic feature is associated with free flow metastability at the bottleneck: A random short-time disturbance in free flow whose amplitude is greater than some critical one can self-grow and lead to traffic breakdown, regardless of control method application. As a result, a congested pattern that propagates upstream of the bottleneck is selfformed. In particular, this nucleation of traffic congestion can occur when ALINEA is applied for feedback on-ramp metering. 2) In contrast with ALINEA, ANCONA prevents propagation of the congested pattern upstream. 3) ANCONA exhibits the following benefits in comparison with ALINEA: a) greater throughputs on the main road and on-ramp;
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b) considerably shorter vehicle waiting times at light signal in the on-ramp lane; c) considerably shorter TTS and ATT; d) upstream propagation of congestion that does not occur even if a congested pattern occurs at the bottleneck. 4) UP-ALINEA, which can prevent congestion, almost cannot perform if the location of feedback control detector is within a localized perturbation at the bottleneck. 5) ANCONA exhibits the following benefits in comparison with UP-ALINEA: a) greater throughputs on the main road and on-ramp; b) considerably smaller vehicle waiting times at the light signal in the on-ramp lane; c) considerably shorter TTS and ATT. 6) To prove these simulations results, practical installations of ALINEA should be studied based on measured data; ANCONA application in real installations should be performed.
ACKNOWLEDGMENT The author would like to thank S. Klenov for help in simulations and fruitful discussions. R EFERENCES [1] B. S. Kerner, The Physics of Traffic. Berlin, Germany: Springer-Verlag, 2004. [2] W. Leutzbach, Introduction to the Theory of Traffic Flow. Berlin, Germany: Springer-Verlag, 1988. [3] A. D. May, Traffic Flow Fundamentals. Englewood Cliffs, NJ: PrenticeHall, 1990. [4] M. Cremer, Der Verkehrsfluss auf Schnellstrassen. Berlin, Germany: Springer-Verlag, 1979. [5] N. H. Gartner, C. J. Messer, and A. Rathi, Eds., Special Report 165: Revised Monograph on Traffic Flow Theory. Washington, DC: Transp. Res. Board, 1997. [6] C. F. Daganzo, Fundamentals of Transportation and Traffic Operations. Oxford, U.K.: Elsevier, 1997. [7] D. Chowdhury, L. Santen, and A. Schadschneider, “Statistical physics of vehicular traffic and some related systems,” Phys. Rep., vol. 329, no. 4, pp. 199–329, May 2000. [8] D. Helbing, “Traffic and related self-driven many-particle systems,” Rev. Mod. Phys., vol. 73, no. 4, pp. 1067–1141, Dec. 2001. [9] T. Nagatani, “The physics of traffic jams,” Rep. Prog. Phys., vol. 65, no. 9, pp. 1331–1386, Sep. 2002. [10] K. Nagel, P. Wagner, and R. Woesler, “Still flowing: Approaches to traffic flow and traffic jam modeling,” Oper. Res., vol. 51, no. 5, pp. 681–710, Sep. 2003. [11] M. Papageorgiou, Application of Automatic Control Concepts in Traffic Flow Modeling and Control. Berlin, Germany: Springer-Verlag, 1983. [12] M. Papageorgiou, Concise Encyclopedia of Traffic and Transportation Systems. New York: Pergamon, 1991. [13] Highway Capacity Manual, Transp. Res. Board, Washington, DC, 2000. [14] M. Papageorgiou and A. Kotsialos, “Freeway ramp metering: An overview,” IEEE Trans. Intell. Transp. Syst., vol. 3, no. 4, pp. 271–281, Dec. 2002. [15] K. Bogenberger, “Adaptive fuzzy systems for traffic responsive and coordinated ramp metering,” Dissertation am Fachgebiet Verkehrstechnik und Verkehrsplanung der Technischen Universität München, 2001, Munich, Germany, ISSN 0943-9455. [16] E. D. Arnold, Ramp Metering: A Review of the Literature. Charlottesville, VA: Virginia Transp. Res. Council, 1998. [17] M. Papageorgiou, J.-M. Blosseville, and H. Hadj-Salem, “Modelling and real-time control of traffic flow on the southern part of Boulevard Peripherique in Paris: Part II: Coordinated on-ramp metering,” Transp. Res. A, vol. 24, no. 5, pp. 361–370, 1990.
[18] M. Papageorgiou, H. Hadj-Salem, and J.-M. Blosseville, “ALINEA: A local feedback control law for on-ramp metering,” Transp. Res. Rec., no. 1320, pp. 58–64, 1991. [19] M. Papageorgiou, H. Hadj-Salem, and F. Middleham, “ALINEA local ramp metering: Summary of field results,” Transp. Res. Rec., no. 1603, pp. 98–99, 1997. [20] E. Smaragdis and M. Papageorgiou, “Series of new ramp metering strategies,” Transp. Res. Rec., no. 1856, pp. 74–86, 2003. [21] D. Srinivasan, M. C. Choy, and R. L. Cheu, “Neural networks for realtime traffic signal control,” IEEE Trans. Intell. Transp. Syst., vol. 7, no. 3, pp. 261–272, Sep. 2006. [22] B. S. Kerner, Control of Spatial–Temporal Congested Traffic Patterns at Highway Bottlenecks, 2003. cond-mat/0309017. [Online]. Available: http://arxiv.org/abs/cond mat/0309017 [23] B. S. Kerner, “Control of spatial–temporal congested traffic patterns at highway bottlenecks,” presented at the TRB 83rd Annual Meeting, Washington, DC, 2004, TRB Paper No. 04-3062. [24] B. S. Kerner, “Control of spatial–temporal congested traffic patterns at highway bottlenecks,” Physica A, vol. 355, no. 2–4, pp. 565–601, 2005. [25] L. Elefteriadou, R. P. Roess, and W. R. McShane, “Probabilistic nature of breakdown at freeway merge junctions,” Transp. Res. Rec., no. 1484, pp. 80–89, 1995. [26] B. Persaud, S. Yagar, and R. Brownlee, “Exploration of the breakdown phenomenon in freeway traffic,” Transp. Res. Rec., no. 1634, pp. 64–69, 1998. [27] M. Lorenz and L. Elefteriadou, “A probabilistic approach to defining freeway capacity and breakdown,” Trans. Res. Circ., no. E-C018, pp. 84–95, 2000. [28] B. S. Kerner and H. Rehborn, “Experimental properties of phase transitions in traffic flow,” Phys. Rev. Lett., vol. 79, no. 20, pp. 4030–4033, Nov. 1997. [29] B. S. Kerner, S. L. Klenov, and D. E. Wolf, “Cellular automata approach to three-phase traffic theory,” J. Phys. A, Math. Gen., vol. 35, no. 47, pp. 9971–10 013, 2002. [30] B. S. Kerner and S. L. Klenov, “A microscopic theory of spatial–temporal congested traffic patterns at highway bottlenecks,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 68, no. 3, p. 036 130, 2003. [31] S. Krauß, P. Wagner, and C. Gawron, “Metastable states in a microscopic model of traffic flow,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 55, no. 5, pp. 5597–5602, May 1997. [32] P. G. Gipps, “A behavioral car-following model for computer simulation,” Transp. Res. B., vol. 15, no. 2, pp. 105–111, 1981. [33] B. S. Kerner, “On-ramp metering based on three-phase traffic theory II,” Traffic Eng. Control, vol. 48, no. 2, pp. 66–75, 2007. [34] B. S. Kerner, “On-ramp metering based on three-phase traffic theory I,” Traffic Eng. Control, vol. 48, no. 1, pp. 28–35, 2007. [35] B. S. Kerner and S. L. Klenov, “Deterministic microscopic three-phase traffic models,” J. Phys. A, Math. Gen., vol. 39, no. 8, pp. 1775–1809, 2006. [36] B. S. Kerner and S. L. Klenov, “Deterministic microscopic three-phase traffic models,” J. Phys. A, Math. Gen., vol. 39, no. 23, p. 7605, 2006.
Boris S. Kerner was born in Moscow, Russia, on December 22, 1947. He received the degree in electronics from the Technical University MIREA, Moscow, the Ph.D. degree in physics and mathematics in 1979, and the Dr.Sc. degree (habilitation) in physics and mathematics from the Russian Academy of Science, in 1986. In 1972, he was a Semiconductor Electronics Engineer at the Semiconductor Research Institute “Pulsar,” Moscow. From 1980 to 1992, he was with the Research Institute “Orion,” Moscow, as a Senior Scientist and later as a Head of a theoretical department in semiconductor engineering and nonlinear physics. In 1990–1991, he was a Professor at the Technical University MIREA. Since 1992, he has been with DaimlerChrysler Research, Stuttgart, Germany, in various fields of intelligent transportation systems (ITS). He is currently the Head of the research field “Traffic” at the Group Research and Advanced Engineering, DaimlerChrysler AG, Sindelfingen, Germany. He has published three books and more than 150 articles and reviews on traffic flow theory and its applications on ITS, nonlinear science, semiconductors, solid-state physics, and gas plasma physics. He is the inventor of about 70 patents on ITS.
IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS, VOL. 8, NO. 2, JUNE 2007
Control of Spatiotemporal Congested Traffic Patterns at Highway Bottlenecks Boris S. Kerner
Abstract—It is shown that the probabilistic feature of traffic breakdown at an on-ramp bottleneck leads to great limitations for reliable applications of a free flow control approach in which free flow should be maintained at the bottleneck. Based on these measured features of traffic breakdown at the bottleneck as well as on the Kerner–Klenov microscopic traffic model in the context of the author’s three-phase traffic theory, critical discussions of earlier traffic flow models for freeway control simulations and of ALINEA methods of Papageorgiou et al. for feedback on-ramp metering are made. An alternative congested pattern control approach to feedback on-ramp metering ANCONA introduced by the author in 2004 is numerically studied. In ANCONA, congestion at the bottleneck is allowed to set in. However, ANCONA maintains speeds within a congested pattern higher than about 60 km/h and prevents upstream propagation of the pattern. To reach these goals, after traffic breakdown has occurred spontaneously at the bottleneck, ANCONA tries to return to free flow via reduction of on-ramp inflow. A critical comparison of ANCONA with ALINEA and UP-ALINEA is made. Index Terms—Congested pattern control (ANCONA), critical discussion of on-ramp metering method ALINEA, empirical congested traffic patterns, feedback on-ramp metering, three-phase traffic theory, traffic breakdown at bottlenecks.
I. I NTRODUCTION
T
RAFFIC can be either “free” or “congested.” Congested traffic occurs most at freeway bottlenecks. Freeway congestion is a complex spatiotemporal nonlinear dynamic process [1]–[10]. For this reason, it is not surprising that empirical congested traffic pattern features have only recently been sufficiently understood [1]. Such empirical features are in serious conflict with almost all earlier theoretical and model results reviewed in [2]–[10] (see explanations in Section VI-A). These traffic flow models [2]–[10] are standard ones for the validation of freeway traffic control, management, and assignment (e.g., [3], [4], [6], and [11]–[14]). For this reason, the related simulations of freeway control and management strategies cannot predict many of the freeway traffic phenomena that would occur using a simulated control strategy. Consequently, the author introduced a new traffic flow theory called “three-phase traffic theory,” which can explain these empirical spatiotemporal patterns (see a comprehensive consideration of empirical traffic patterns, three-phase traffic theory,
Manuscript received January 13, 2006; revised July 18, 2006 and December 4, 2006. The Associate Editor for this paper was B. de Schutter. The author is with DaimlerChrysler AG, 71059 Sindelfingen, Germany (e-mail: [email protected]). Digital Object Identifier 10.1109/TITS.2007.894192
Fig. 1. On-ramp inflow control [14]–[20]. (a) Downstream feedback control. (b) Upstream feedback control.
as well as a critical discussion of earlier traffic flow theories and models in the book [1]). There are a huge number of publications and many regular scientific conferences that are devoted to traffic control (see, e.g., references in [2]–[21]). One of the most used traffic control methods is on-ramp metering at an on-ramp bottleneck. Various models based on a “free flow control approach” to on-ramp metering have been suggested and developed. The basic idea of the free flow control approach is to maintain free flow conditions on the main road at the bottleneck (e.g., [14]–[20]). Free flow conditions at the bottleneck should be maintained at the maximum possible throughput in the free flow downstream of the bottleneck. In other words, the onset of congestion and congested pattern formation at the bottleneck should be prevented by the application of this approach. There are many different methods based on this approach (see [14]–[20]). The basic strategy is carried out through automatic control of flow rate of vehicles that can merge onto the main road from the on-ramp. In some of the methods using detectors for feedback control, traffic variables either downstream [Fig. 1(a)] or upstream of the bottleneck are measured [Fig. 1(b)] [14]–[20]. Results of these measurements are used as feedback for on-ramp inflow control, controlling light signal operation in the on-ramp lane(s). Depending on measurements of current traffic variables, the on-ramp inflow onto the main road from the on-ramp is either restricted or not. However, due to the probabilistic nature of traffic breakdown at bottlenecks, the free flow control approach exhibits a great application limitation: Regardless of control method
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application, in a wide range of the flow rate in free flow at a bottleneck, a traffic breakdown can randomly occur. This leads to congested pattern emergence with subsequent congestion propagation upstream of the bottleneck, which cannot be prevented by free flow control method application. A well-known example of the free flow control approach is the ALINEA model of Papageorgiou et al. [14], [17]–[20]. For these reasons, the author has introduced a qualitatively different “congested pattern control approach” (ANCONA) [1], [22]–[24]. In ANCONA, congestion at the bottleneck is allowed to set in. However, a congested pattern that emerges at the bottleneck should not propagate upstream; rather than propagating upstream, the congested pattern should be localized on the main road in a small neighborhood of the bottleneck. ANCONA should ensure that congestion at the bottleneck does not propagate far upstream of the bottleneck. To achieve this with ANCONA, a feedback control detector [Fig. 1(b)] should register congested patterns at the bottleneck. After traffic congestion has been registered, the on-ramp inflow is automatically reduced via light signal operation in the on-ramp lane(s). This leads to a return phase transition to free flow at the bottleneck, and therefore, traffic congestion on the main road does not propagate continuously upstream. A comparison of ANCONA and ALINEA made in [1] and [24] shows the following benefits of ANCONA: 1) greater throughputs on the main road and onramp; 2) considerably shorter vehicle waiting times at the light signal in the on-ramp lane; and 3) the upstream propagation of congestion that does not occur even if a congested pattern occurs at the bottleneck due to a short-term perturbation in traffic flow. The congested pattern is spatially localized in the vicinity of the bottleneck. The main aim of this paper is a discussion of a problem for freeway control caused by the probabilistic nature of the onset of congestion at bottlenecks. In comparison with previous publications of ANCONA [1], [23], [24], we present new results that explain how ANCONA solves the previously mentioned problem of ramp metering control associated with the probabilistic feature of freeway capacity. In addition, we consider the benefits of ANCONA in comparison with the ALINEA and UP-ALINEA models of Papageorgiou et al. [14], [17]–[20] in more detail than those in [1] and [24]. The paper is organized as follows: The problem for freeway traffic control caused by the probabilistic nature of freeway capacity is discussed in Section II. The Kerner–Klenov microscopic model used for control method simulations is discussed in Section III. Some new results of ANCONA application are presented in Section IV. New results of the comparison of the ANCONA method with the ALINEA and UP-ALINEA methods are considered in Section V. In Section VI, we will make a brief critical analysis of earlier traffic flow models reviewed in [2]–[10], which are widely used for freeway traffic control simulations. However, first, in the remainder of this introductory section, we discuss traffic phase definitions and a double Z-characteristic for phase transitions in three-phase traffic theory [1], the main types of measured (empirical) congested patterns at an on-ramp bottleneck [1], as well as the basic ALINEA [14], [17]–[20] and ANCONA rules [1], [23], [24] that we need for further consideration.
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Fig. 2. Qualitative graph of a double Z-characteristic for phase transitions in three-phase traffic theory [1]. F: free flow; S: synchronized flow; J: wide moving jam (low-speed states within wide moving jams). Arrows symbolize phase transitions between the three traffic phases.
A. Traffic Phase Definition and Double Z-Characteristic in Three-Phase Traffic Theory In three-phase traffic theory [1], there are two phases in congested traffic: 1) synchronized flow and 2) wide moving jam. Thus, there are three traffic phases in three-phase traffic theory: 1) free flow; 2) synchronized flow; and 3) wide moving jam. The synchronized flow and wide moving jam traffic phases are defined through the following spatiotemporal empirical (objective) criteria (definitions) [S] and [J]. The definition [J] for the wide moving jam phase is as follows: A wide moving jam is a moving jam that maintains the mean velocity of the downstream jam front, even when the jam propagates through any other traffic states or freeway bottlenecks. The definition [S] for the synchronized flow phase is as follows: The downstream front of synchronized flow does not exhibit this characteristic feature of wide moving jams; specifically, the latter front is often fixed at a freeway bottleneck. Within the downstream front of synchronized flow, vehicles accelerate from lower speeds in synchronized flow to higher speeds in free flow. In the three-phase traffic theory, the onset of congestion (traffic breakdown) is associated with a local phase transition from free flow to synchronized flow (F → S transition). Thus, F → S transition and traffic breakdown are synonyms. Wide moving jams can emerge spontaneously in synchronized flow only (S → J transition) [1]. Both the F → S and S → J transitions, as well as the associated S → F and J → S return transitions, are first-order local phase transitions, which are accompanied by hysteresis effects and lead to a double Z-characteristic (Fig. 2) [1]. This means that all these transitions exhibit probabilistic nature [1]. The probabilistic nature of traffic breakdown (F → S transition) is discussed in Section II. B. Congested Patterns After an F → S transition has occurred at an isolated onramp bottleneck, different congested patterns can be formed on the main road upstream of the bottleneck. There are two main types of congested patterns at the bottleneck: 1) synchronized flow patterns (SP) and 2) general congested patterns (GP) [1]. An SP consists of synchronized flow only. A typical empirical example of an SP localized at a bottleneck (localized
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Fig. 5. Simulated GP characteristics without on-ramp inflow control. (a) Speed on the main road in space and time. (b) qon (t) [1], [23], [24]. Fig. 3. Empirical average vehicle speed on the main road within an LSP at on-ramp bottleneck in space and time. Measured data from March 26, 1996 [1].
propagate upstream in the case of the LSP (Fig. 3). This means that possible upstream adjacent bottlenecks of a traffic network in the case of the LSP are not affected by congestion at the onramp bottleneck if the LSP width (in the longitudinal direction) is small enough. Second, empirical investigations show that discharge volume from the LSP can be considerably greater than that from the GP. This is because in the GP, the flow rate and speed are mostly strongly limited. The flow rate and speed within an LSP can be considerably greater than those within the GP [1]. A GP is the most frequently observed congested pattern at the on-ramp bottleneck [1]. In an example of GP emergence in microscopic simulations based on three-phase traffic theory [Fig. 5(a)], we suggest that there is no on-ramp inflow control and that the flow rate qin on the main road in free flow upstream of the bottleneck is constant, but the flow rate to the on-ramp qon is a function of time [Fig. 5(b)]. C. ALINEA Method
Fig. 4. Empirical example of GP at on-ramp bottleneck [1]. (a) Average vehicle speed on the main road in space and time. (b) Graph of (a) in space and time. Measured data from January 13, 1997.
SP: LSP) is shown in Fig. 3. In a GP, synchronized flow occurs upstream of the bottleneck, and then, wide moving jams emerge spontaneously in that synchronized flow, i.e., the GP consists of the synchronized flow and wide moving jam traffic phases (Fig. 4).1 There are two differences between the LSP (Fig. 3) and GP (Fig. 4), which are important for a traffic control theory. First, in contrast with the GP (Fig. 4), traffic congestion does not 1 Note that traffic data are measured at a set of road detectors whose spatial arrangement is presented in [1, Fig. 2.1]. Average speeds in Figs. 3 and 4(a) are determined by averaging of speeds of all vehicles passing a detector in a specified time period (1 min). Spatiotemporal pattern overviews in Figs. 3 and 4 are made via data interpolation between detectors. Data interpolation in Figs. 3 and 4(a) is made via a MATLAB function “Shading INTERP.” For interpolation of downstream and upstream front positions of traffic phases between detectors in Fig. 4(b), estimations for front velocities of the traffic phases are made through the use of the FOTO and ASDA models [1, Ch. 21]. Rather than these approximate overviews of spatiotemporal congested patterns (Figs. 3 and 4), for traffic phase distinguishing and a traffic phase dynamics analysis, we used a spatiotemporal correlation analysis of traffic data dynamics measured at adjacent detector locations (e.g., [1, Part II, Figs. 9.6, 9.7, 9.8(b)–9.10(b), 9.13–9.15, 10.1(c), 10.5(a), 10.6(c), 10.7(a), and 10.8(a)]).
In the ALINEA method of Papageorgiou et al. [14], [17]– (cont) [20], the following rule for the flow rate of vehicles qon (ts ), which can merge from the on-ramp onto the main road during the time interval ts ≤ t < ts+1 , is used [17]–[20]: (cont) (cont) (cont) qon (ts ) = qon (ts−1 ) + Ko o(opt) − o (t ) s sum sum
(1)
where Ko is constant; ts = Tav s, s = 1, 2, . . .; Tav (e.g., Tav = 1 min) is the averaging time interval for data measured by (opt) feedback control detectors in Fig. 1(a); osum is a chosen (cont) optimal (target) occupancy; and osum (ts ) is the occupancy measured downstream of the bottleneck using the feedback (cont) control detectors [Fig. 1(a)]; the occupancy osum (ts ) is associated with the time interval ts−1 < t ≤ ts and is averaged during this time interval. Traffic breakdown as a first-order F → S transition is responsible for probabilistic feature of the breakdown and freeway capacity discussed later in the text. This leads to congested pattern emergence with subsequent congestion propagation upstream of the bottleneck, which cannot be prevented by ALINEA appli(opt) cation (see [24, Fig. 9]). Even if target occupancy osum is chosen that is considerably smaller than the maximum occupancy in free flow downstream of the bottleneck, ALINEA cannot prevent GP emergence at the bottleneck when a short-term pulse in the on-ramp inflow occurs (Fig. 6). As found in [1] and [24], these critical conclusions about ALINEA performance
KERNER: CONTROL OF SPATIOTEMPORAL CONGESTED TRAFFIC PATTERNS AT HIGHWAY BOTTLENECKS
Fig. 6. ALINEA cannot prevent GP emergence at the bottleneck if a shortterm pulse in the on-ramp inflow occurs [1], [24]. (a) Speed on the main road in space and time. (b) qon (t).
happen, regardless of the location of feedback control detector, as long as the detector is downstream of the bottleneck. D. ANCONA Method The ANCONA approach for on-ramp metering [1], [23], [24] takes into account the probabilistic nature of traffic breakdown at an on-ramp bottleneck. In ANCONA, we used SPs that can be more favorable than GPs in terms of the discharge volume from a congested pattern and of the vehicle time delay due to congestion. In ANCONA, congestion at a bottleneck is allowed to set in. However, average vehicle speed within an SP that emerges at the bottleneck should be maintained using ANCONA at a relatively high level (higher than about 60 km/h). Moreover, ANCONA ensures that congestion does not propagate upstream; rather than propagating upstream, the congested pattern should be localized on the main road in a small neighborhood of the bottleneck. Benefits of the ANCONA approach are considerably greater throughput and considerably shorter travel times in the on-ramp lane(s) than in the ALINEA control method (Section V) [1], [23], [24]. In ANCONA, there is no on-ramp control as long as free flow is measured at the bottleneck. On-ramp inflow control is first realized only after the onset of congestion has occurred due to traffic breakdown (F → S transition) that occurs spontaneously at the bottleneck. In other words, feedback control is performed when the average speed v (det) that is measured at the detectors for feedback control, which is upstream of the bottleneck [Fig. 1(b)], is equal to or drops below a chosen “congestion speed” vcong , i.e., v (det) ≤ vcong .
(2)
(cont)
When this condition is satisfied, qon is reduced via light (cont) signal operation. Due to the decrease in the flow rate qon , ANCONA tries to achieve a return S → F transition at the bottleneck. As a result of the S → F transition, the speed at the detector v (det) increases above vcong , i.e., v (det) > vcong .
(3) (cont)
is When condition (3) is satisfied, a greater flow rate qon allowed via light signal operation. If under this greater flow (cont) the onset of congestion occurs at the bottleneck rate qon once more, an incipient congested pattern begins to propagate upstream of the bottleneck. As a result, the speed at the detector
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Fig. 7. Suppression of GP emergence via ANCONA under a short-term pulse in qon (t). (a) Speed on the main road in space and time. (b) qon (t) is the same as that in Fig. 6(b) [1], [24].
for feedback control decreases. Therefore, condition (2) is satisfied once more. This leads to a new decrease in the flow (cont) rate qon , and so on. Thus, the basic idea of ANCONA is to allow the congestion condition at the bottleneck at the minimum possible level of traffic congestion. The ANCONA approach has three aims: 1) achievement of greater throughputs in the whole traffic network under very high traffic demand when congestion has to occur somewhere in the traffic network. This is realized due to a more homogeneous distribution of congestion among all bottlenecks in the network; 2) maintenance of shorter travel time of vehicles at the light signal in the on-ramp lane(s); 3) prevention of upstream propagation of a congested pattern that occurs at the bottleneck. For numerical simulations, the following simplified ANCONA on-ramp inflow control rule is used [1], [23], [24]: (cont) (ts ) qon
(cont)
=
qon1 , if v (det) (ts ) ≤ vcong qon2 , if v (det) (ts ) > vcong
(4)
where qon (ts ) is the flow rate of vehicles, which can merge from the on-ramp onto the main road during the time interval ts ≤ t < ts+1 ; the speed v (det) (ts ), which is measured by the feedback control detector [Fig. 1(b)], is averaged during the time interval ts−1 < t ≤ ts ; qon1 and qon2 are chosen model parameters (qon2 > qon1 ). Due to possible complicated time dependences of incoming flow rates qin and qon in real traffic, in real ANCONA installations, qon1 and qon2 should be time functions, which can be chosen in accordance with another ANCONA rule suggested in [24], in which qon1 and qon2 can depend on v (det) (ts ) and on the duration of the free flow phase at the bottleneck, respectively. In this paper, we limit a consideration of the ANCONA rule (4) at a time-independent flow rate qin . When a pulse of the flow rate to the on-ramp appears, ALINEA cannot prevent the formation and dissolution of the GP at the bottleneck (Fig. 6). In contrast, ANCONA leads to the dissolution of this GP (Fig. 7) at the same qin and qon (t) as those in Fig. 6. Rather than a congested pattern, which propagates continuously upstream, either an LSP or a time sequence of moving SP (MSP) and free flows remains at the bottleneck. MSPs dissolve while propagating upstream (see Section IV). LSP widths (in the longitudinal direction) are spatially limited. As a result, the
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influence of these congested patterns at the on-ramp bottleneck on the travel time on the main road is negligible. II. L IMITATIONS OF F REEWAY T RAFFIC C ONTROL C AUSED BY P ROBABILISTIC F EATURE OF F REEWAY C APACITY The probabilistic nature of traffic breakdown is given as follows: At a given flow rate, traffic breakdown can occur, but it should not necessarily occur. This means that on one day, the onset of congestion occurs; however, on another day, at the same flow rate and at the same traffic conditions, the onset of congestion is not observed [25]–[27]. Based on these and other empirical observations [28], in three-phase traffic theory [1], it has been shown that this probabilistic nature of traffic breakdown is associated with a first-order F → S transition. Indeed, traffic breakdown exhibits the following empirical fundamental features of a first-order phase transition: 1) at the same bottleneck, traffic breakdown can be either spontaneous or induced, i.e., the probabilistic feature of the breakdown is associated with its nucleation nature: If a critical speed disturbance at the bottleneck (“nucleus” for traffic breakdown) occurs, traffic breakdown is realized [1]; 2) both in empirical observations [26], [27] and numerical simulations of a microscopic model based on three-phase traffic theory [1], [29], the probability for an F → S transition (traffic breakdown) in free flow at (B) the bottleneck (denoted by PFS ) is an increasing function of the flow rate downstream of the bottleneck denoted by qsum (Fig. 8). Under the free flow condition at the bottleneck, we get qsum = qon + qin if there is no on-ramp metering and (cont) + qin under on-ramp metering application. To qsum = qon avoid confusion, we suggest all flow rates to be the total flow rates. If in some of figures flow rates are presented per freeway lane, it is mentioned in the figure caption (e.g., Fig. 8). There are two important characteristic flow rates on a func(B) tion PFS (qsum ): 1) the critical flow rate for the F → S transition (free B) and 2) the threshold flow rate denoted by denoted by qmax (B) qth (superscript “B” means that values are associated with an F → S transition at a bottleneck). Within the flow rate range [Fig. 8(b)] (B)
(free B) qth ≤ qsum ≤ qmax
(5)
traffic breakdown occurs with probability (B)
PFS > 0.
(6)
The nature of probabilistic traffic breakdown is as follows: Under condition (5), free flow at the bottleneck is in a metastable state with respect to traffic breakdown. This means that small amplitude speed disturbances in the free flow do not lead to traffic breakdown (F → S transition). However, if a random short-time speed disturbance in free flow in a neighborhood of the bottleneck appears whose amplitude (difference between vehicle speed within the disturbance and initial speed) exceeds some critical value, an F → S transition occurs. This random speed disturbance leading to traffic breakdown is called a critical speed disturbance (or critical perturbation in free flow) [1]. The free flow metastability can also be seen in Fig. 2.
(B)
Fig. 8. Probability for traffic breakdown PFS as a function of the flow rate on the main road downstream of the bottleneck. (a) Measured functions in real traffic taken from Persaud et al. [26] for two flow rate averaging intervals of 1 and 10 min. (b) Theoretical function found from numerical simulations (for the interval for observing traffic flow of 15 min) is taken from [29]. Flow rates are shown per freeway lane.
Between states of free flow (F) and synchronized flow (S), there is a gap in speed: Speeds within this gap are related to various speed disturbances that are discussed previously in the text. If in free flow a critical speed disturbance on the main road in a neighborhood of the bottleneck occurs randomly, then the disturbance amplitude self-grows. The duration of this growth until traffic breakdown can be measured and is less than or about 1 min (down arrow from F to S in Fig. 2). This growing critical local speed disturbance can occur randomly at different freeway locations in a neighborhood of the bottleneck, i.e., it is not predictable both in time and space. Until now, one could see the result of critical speed disturbance growth only, i.e., synchronized flow emergence at the bottleneck. Numerical simulations show that this random speed disturbance cannot be registered already at some 100 m downstream of the end of the merging on-ramp region. In particular, no increase in the downstream occupancy appears during the appearance and subsequent growth of the disturbance. Let us explain these common results based on ALINEA applications with downstream feedback control detector locations (B) [Fig. 1(a)]. Traffic breakdown probability PFS is greater than zero under condition (5) (Fig. 8). This means that at a given flow rate qsum within the range (5), free flow at the bottleneck can (B) be kept only with certain probability 1 − PFS < 1. This fundamental traffic feature happens, regardless of whether ALINEA is applied or not. An explanation of this is as follows: Speed and occupancy disturbances at the bottleneck, which can lead to traffic breakdown, do not change traffic variables downstream of the bottleneck as long as no congested pattern has been formed at the bottleneck. Thus, the downstream feedback control detector of ALINEA [Fig. 1(a)] cannot register speed perturbation growth at the bottleneck; therefore, ALINEA cannot
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register resulting traffic breakdown at the bottleneck. This conclusion has been proven in numerical simulations of ALINEA on-ramp metering [1]. To keep free flow at the bottleneck with greater probability, in ALINEA, smaller target occupancy must be chosen, at which smaller flow rate of vehicles is allowed to merge from the on-ramp onto the main road. As a result, the (B) flow rate qsum and breakdown probability PFS are reduced. In the case shown in Fig. 6, ALINEA does not try to maintain the highest possible throughput downstream of the bottleneck. Nevertheless, traffic breakdown can also occur (Fig. 6). Thus, when (5) is satisfied, it can turn out that a control method cannot prevent traffic breakdown at the bottleneck. Only if the flow rate downstream of the bottleneck
(8)
qsum
Gn ∆n = max (−bn τ, min(an τ, v,n − vn ))
(9) (10) (11)
(7)
then probability for traffic breakdown is equal to zero. Hence, only under condition (7) is reliable control method application for maintaining free flow at the bottleneck possible. (B) However, the threshold flow rate qth is considerably smaller (free B) than the maximum flow rate qmax in free flow at the bottleneck (Fig. 8). In other words, the probabilistic nature of traffic breakdown is a great problem for any free flow control approach application. Until now, we have considered the case when a detector for feedback control is downstream of the bottleneck [Fig. 1(a)]. To explain the term “downstream of the bottleneck” used in threephase traffic theory [1], note that after traffic breakdown has occurred at a bottleneck, synchronized flow is formed upstream at the bottleneck. The downstream front of the synchronized flow, within which vehicles accelerate from the synchronized flow to free flow downstream, is usually fixed in a neighborhood of the bottleneck. The location on a freeway at which this downstream front of synchronized flow is spatially fixed is called “the effective location” of the bottleneck [Fig. 1(b)] [1]. In all numerical simulations that follow, a speed disturbance in free flow, within which traffic breakdown occurs at the bottleneck, is localized upstream of the effective bottleneck location. For these reasons, all freeway locations downstream of the effective bottleneck location are associated with the term “downstream of the bottleneck” [Fig. 1(a)]. In contrast, all freeway locations upstream of the effective bottleneck location are associated with the term “upstream of the bottleneck.” Note that in contrast with the case of downstream detector location [Fig. 1(a)], when a feedback control detector is upstream of the effective bottleneck location [Fig. 1(b)] and it is located within speed disturbances at the bottleneck, the detector can measure speed disturbance dynamics. This means that this upstream detector location can be used to suppress traffic breakdown using suitable ramp metering strategy. This is used in ANCONA. III. M ODEL FOR S IMULATIONS OF F REEWAY C ONTROL M ETHODS Here, we consider a stochastic microscopic two-lane model of Kerner and Klenov in the context of three-phase traffic theory [30]. All model results presented are related to numerical
where index n corresponds to the discrete time t = nτ , n = 0, 1, 2, . . .; τ is the time step; vfree is the maximum speed in free flow; gn = x,n − xn − d is the space gap (net distance) between vehicles, where d is the vehicle length; the lower index marks functions (or values) related to the preceding vehicle; vn is the vehicle speed at time step n; xn is the vehicle coordinate; vn is the speed calculated without a noise component ξn ; a is the maximum acceleration; Gn is a synchronization gap; and an ≥ 0 and bn ≥ 0. (safe) (a) , gn /τ + v ) is safe speed, where In (8), vs,n = min(vn (safe) vn is a safe speed of the Krauß model [31], which is a solution of the Gipps equation [32] vn(safe) τ + Xd vn(safe) = gn + Xd (v,n )
(12)
where Xd (u) is the distance travelled by the vehicle with an initial speed u at a time-independent deceleration b until it comes to a stop; in the model with the discrete time Xd (u) = bτ 2 (αβ + α(α − 1)/2) [31]; b is constant; α is the integer (a) part of u/bτ ; β is the fractional part of u/bτ ; and v is an anticipation speed [1, eq. (16.48)]. To simulate driver time delays either in vehicle acceleration or in vehicle deceleration, an and bn in (10) and (11) are taken as the following stochastic functions: an = aθ(P0 − r1 )
(13)
bn = aθ(P1 − r1 ) p0 , if Sn P0 = 1, if Sn p1 , if Sn P1 = p2 , if Sn
(14) = 1 =1 = −1 = −1
(15) (16)
where r1 = rand(0, 1), i.e., this is an independent random value uniformly distributed between 0 and 1; θ(z) = 0 at z < 0 and θ(z) = 1 at z ≥ 0; probabilities p0 (v) and p2 (v) are given functions of speed, and probability p1 is constant; 1 − P0 and 1 − P1 are the probabilities of a random time delay in vehicle acceleration and deceleration, respectively; Sn denotes the state of vehicle motion (Sn = −1 represents deceleration, Sn = 1
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where ξa and ξb are random sources for deceleration and acceleration, respectively, i.e.,
Fig. 9. Hypothetical steady model states [1], [30].
represents acceleration, and Sn = 0 represents motion at nearly constant speed), i.e., −1, Sn = 1, 0,
if vn < vn−1 − δ if vn > vn−1 + δ
(17)
otherwise
where δ is constant (δ aτ ). Equations (9)–(11) describe the speed adaptation effect in synchronized flow. This speed adaptation effect takes place when the vehicle cannot pass the preceding vehicle within the synchronization gap Gn : At gn ≤ Gn , the vehicle tends to adjust its speed to the preceding vehicle without caring what the precise space gap is, as long as it is safe. At a given time-independent speed of the preceding vehicle v,n = v = const, this speed adaptation leads to a car following with vn = v = v at a time-independent space gap gn = g. There is an infinite number of these gaps associated with the same speed v = v . These gaps lie within the gap range gs ≤ g ≤ G (gs is a time-independent safe space gap found from the equation v = vs , where vs = vs,n at gn = g, v,n = v ; G = Gn at vn = v,n = v ), i.e., there is no desired (or optimal) space gap in synchronized flow. The speed adaptation effect within the synchronization gap means that hypothetical steady model states of synchronized flow (in which all vehicles move at the same time-independent speed and at the same space gap to one another) cover a 2-D region in the flow–density plane (Fig. 9), i.e., there is no fundamental diagram for steady speed states of synchronized flow. The boundaries of this 2-D region, i.e., F, L, and U, are associated with free flow, the synchronization gap, and a safe gap determined through the safe speed, respectively. In (10), the synchronization gap that determines the boundary L in steady model states (Fig. 9) is taken as follows: Gn = G(vn , v,n )
G(u, w) = max 0, kτ u + φa−1 u(u − w)
(18)
if Sn+1 = −1 if Sn+1 = 1 if Sn+1 = 0
(21)
ξb = aτ θ(pb − r)
(22)
where pa and pb are probabilities of random acceleration and deceleration, respectively; r = rand(0, 1). Lane-changing rules in a two-lane model are associated with conditions R → L : vn+ ≥ v,n + δ1 and vn ≥ v,n
(23)
L →R :
(24)
vn+
> v,n + δ1 or
vn+
> vn + δ1
for lane changing from the right lane to the left (passing) lane (R → L) (23) and a return change (L → R) (24). The safety conditions for lane changing are
gn+ > min vn τ, G+ (25) n
gn− > min vn− τ, G− (26) n
+ G+ n = G vn , vn
− (27) G− n = G vn , vn . If (23)–(27) are satisfied, then the vehicle changes lane with probability pc . Functions in (27) are associated with the function G(u, w) (18). In (23) and (24), δ1 ≥ 0 is constant. In (23)–(27), superscripts + and − in variables and functions denote the preceding vehicle and the trailing vehicle in the “target” (neighboring) lane, respectively; the speed vn+ or vn− is set to ∞ if gn+ or gn exceeds a look-ahead distance La , respectively; pc , La , and δ1 (δ1 ≥ 0) are constants. In the model, there is an on-ramp on the main two-lane road. Open-boundary conditions are applied. The related on-ramp bottleneck consists of two parts [1, Sec. 16.3.9]: 1) a merging region, where vehicles can merge onto the main road from the on-ramp lane, and 2) a part of the on-ramp lane upstream of the merging region, where vehicles move in accordance with (8)–(22). The model takes merging phenomena explicitly into account. The beginning of on-ramp merging region is at xon = 10 km, and the length of the merging region is Lm = 0.3 km; numerical simulations show that the effective bottleneck location is at xeff ≈ 10.45 km. Quantitative validation of model parameters has been made to have simulation parameters of congested patterns in accordance with measured data [1].
(19)
where k > 1 and φ are constants. Random deceleration and acceleration are applied, depending on whether the vehicle decelerates or accelerates, or else maintains its speed, i.e., −ξb , ξn = ξa , 0,
ξa = aτ θ(pa − r)
(20)
IV. T IME D ELAYS IN ANCONA A PPLICATION For the case of the feedback control detector location x(det) = 9.8 km, numerical simulations of the ANCONA rules (4) have been made in [24]. In this section, time delays in ANCONA application at the same parameters of an on-ramp merging region as those in [24] and various detector locations x(det) are presented. We consider a case of flow rates satisfying (5) when traffic breakdown can occur spontaneously at the bottleneck.
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Fig. 10. Symbolical schema of ANCONA [23], [24]. The spontaneous onset of congestion (F → S transition) is allowed by ANCONA; after congestion occurs, ANCONA tries to achieve an S → F transition labeled by arrow S → F.
In ANCONA, using on-ramp inflow control (4), there can be two nonlinear transitions between two qualitatively different traffic states at the bottleneck (Figs. 2 and 10). One of these states is related to the synchronized flow phase (congestion). The second state is related to the free flow phase. Synchronized flow occurs due to an F → S transition at the bottleneck (arrow F → S in Fig. 10). The F → S transition can occur (cont) = qon2 in (4). As a result of the spontaneously when qon F → S transition, an SP begins to form at the bottleneck, and condition (2) is satisfied. Then, ANCONA tries to achieve a return phase transition from synchronized flow (congestion) to free flow (S → F transition; arrow S → F in Fig. 10). This S → F transition should be forced by light signal operation when (2) is (cont) = qon1 < qon2 . As a result of satisfied, and therefore, qon the S → F transition, the emergent SP dissolves (see the following explanation). Note that in real ANCONA applications, when condition (3) is satisfied during a long enough time interval, there should no longer be on-ramp control. This is realized in an ANCONA rule [24, eq. (44)]. There are several time delays between on-ramp inflow change (4) and the related traffic state change associated with the F → S and S → F transitions at the bottleneck. (cont)
= qon2 is satisfied, there is 1) After the condition qon (B) a random time delay TFS for an F → S transition (Figs. 11 and 12). In the flow–density plane, this F → S transition is accompanied by a drop in speed and a jump (B) in density [arrow labeled by F → S in Fig. 11(e)]. TFS (free B) decreases strongly when qin + qon2 → qmax [1]. Thus, at greater qon2 , free flow can exist at the bottleneck only during short-time intervals. For this reason, for a better ANCONA performance, the value qon2 in (4) should (free B) satisfy condition qin + qon2 < qmax . (cont) = qon1 is satisfied, there is a ran2) After condition qon (B) dom time delay TSF for an S → F transition. In the flow–density plane, this S → F transition is accompanied by a jump in speed and a drop in density [arrow labeled by S → F in Fig. 11(e)]. (B) (B) The time delays TFS and TSF are related to the locations of the F → S and S → F transitions (about 10.3–10.1 km), which are upstream of the effective bottleneck location xeff ≈ 10.45 km.
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3) After the S → F transition has occurred at the bottleneck (item 2), free flow begins to propagate upstream, specifically, an initial SP at the bottleneck transforms into an MSP (see Fig. 11(a); MSPs are considered in [1] in de(B) tail). Consequently, there is a time delay τSF in satisfying the condition (3) at the detector location x(det) , which is upstream of the S → F transition location [Fig. 11(c) and (d)]. 4) If x(det) is far enough upstream from the F → S transition (B) location, then there is a time delay τFS in satisfying the condition (2) after the F → S transition has occurred (x(det) = 9.5 km in Fig. 12). This is because it takes some time for synchronized flow to propagate from the F → S transition location at which synchronized flow has initially occurred to the detector location x(det) . It should be noted that because there is a localized perturbation in free flow at the bottleneck (see Section II), the (B) time delay τFS is remarkable only when the distance between x(det) and xon = 10 km is greater than some critical value, which is about 0.3–0.4 km. For this reason, at x(det) = 9.8 km (B) (B) (Fig. 11), τFS is undistinguishable in comparison with TFS . This explains numerical results in which we did not find some marked difference in ANCONA application if x(det) is within the range 10.3 ≥ x(det) ≥ 9.7 km
(28)
which is associated with the perturbation width (in the longitudinal direction). Condition (28) is related to “optimal” detector locations for ANCONA application. Under condition (28) and chosen ANCONA parameters qon1 and qon2 , congestion does not continuously propagate upstream [Fig. 11(a)]. This result can be explained as follows: Under (B) condition (28), τFS is negligible, and synchronized flow cannot propagate upstream far enough from the bottleneck before (2) (B,mean) is satisfied. Under small qon1 , the mean time delay TSF for S → F transitions is not great (Fig. 11). Thus, free flow returns quickly at the bottleneck: The downstream front of synchronized flow begins to propagate upstream, i.e., an MSP appears, which propagates upstream of the bottleneck as a localized region of synchronized flow: MSP propagation is no longer influenced by the bottleneck. An MSP can exist and continuously propagate upstream only if the MSP inflow rate, which is equal to qin , is greater than a threshold flow rate qth for an F → S transition on a homogeneous road [1]. Otherwise, i.e., at qin < qth , MSP dissolves. The condition for MSP dissolution is easy to satisfy because, as shown in [1], the threshold flow rate qth can be (B) considerably greater than the threshold flow rate qth for an F → S transition at the bottleneck (Figs. 8 and 10). Thus, (cont) decreases to qon1 (4) [Fig. 11(c)], synchronized when qon flow within the MSP dissolves over time, which can clearly be seen in Fig. 11. Note that within MSPs, the average speed is relatively high (about or more than 60 km/h) (Figs. 11 and 12). Under condition (28), no pinch effect occurs, and no wide moving jams are formed spontaneously within this high-speed
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Fig. 11. Study of ANCONA associated with the example of ANCONA general pattern suppression shown in [24, Fig. 11]. (a) Fragment of speed in space (cont) (cont) (t). Fragments of (c) qon (t) and and time that illustrates delay times by occurrence and dissolution of SPs at the bottleneck via ANCONA (4). (b) qon (det) = 9.8 km. (e) Phase transitions in the (left) flow–density plane and on (right) time dependence of (d) speed on the main road at x = 10.3 km and x = x speed associated with (d). Parameters of simulations are the same as those in [24, Fig. 11].
synchronized flow (see explanations about the pinch effect in synchronized flow and wide moving jam emergence in [1]). The opposite case of great flow rates on the main road upstream of the bottleneck qin > qth , at which MSPs propagate upstream continually without dissolution, has been explained and studied in [24, Sec. 8]. At x(det) = 9.5 km, i.e., upstream of the optimal detec(B) tor locations (28), we can distinguish the time delay τFS [Fig. 12(d)]. This time delay leads to a decrease on aver(cont) age value qon . In this case, ANCONA does not perform optimally. Nevertheless, ANCONA can perform because it also prevents upstream congestion propagation in this case [Fig. 12(a)]. However, if condition (28) is not satisfied and the detector location x(det) is markedly less than 9.5 km, the time de(B) lay τFS increases considerably. Then, in contrast with the aforementioned cases, the pinch effect occurs in synchronized flow upstream of the bottleneck: Speed decreases and density increases considerably in synchronized flow. As a result, wide
moving jams can occur in the pinch region of the synchronized flow, i.e., ANCONA cannot perform. Numerical simulations also show that ANCONA application is also not possible if the detector location is downstream of the effective location of the bottleneck [Fig. 1(a)]. This is because in this case, which is characteristic for ALINEA, speed at the detector always satisfies (3), regardless of whether an F → S transition occurs at the bottleneck or not. The speed vcong in the ANCONA rules (4) should satisfy two conditions: 1) If vcong is too low, then synchronized flow of lower speed can be formed within SPs upstream of the bottleneck. This can lead to wide moving jam emergence in the synchronized flow. 2) On the other hand, if vcong is too (cont)
(4) even if high, then ANCONA can reduce the flow rate qon free flow is still at the bottleneck. This is because there can be many random speed disturbances at the bottleneck, which decay over time and do not cause traffic breakdown. Thus, a value of vcong that is too high can lead to a decrease in throughput using ANCONA. In this paper, the optimal range of vcong that
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Fig. 12. Study of ANCONA at x(det) = 9.5 km. (a) Fragment of speed in space and time that illustrates delay times by occurrence and dissolution of SPs at the (cont) (cont) (t). Fragments of (c) qon (t) and (d) speed on the main road at the location of 10.3 km and at the detector bottleneck via ANCONA application (4). (b) qon location (9.5 km). Other parameters are the same as those in Fig. 11.
satisfies both conditions 1 and 2 is determined based on the results of numerical simulations: 70 < vcong < 85 km/h. In real ANCONA applications, the optimal range of vcong should be determined based on measured data. Simulations show that the smaller the flow rate qon1 , the (B) shorter time delay TSF for an S → F transition, i.e., the easier and the quicker the S → F transition is realized. In this case, MSP dissolution also occurs more quickly. In contrast, if qon1 is too great, an S → F transition does not occur. In this case, if qin is also great, rather than an LSP, a widening SP can occur at the bottleneck; moreover, the pinch effect with a wide moving jam formation can be realized later. Thus, there is some “optimal” value of qon1 in (4) at which throughput is great and synchronized flow is localized at the bottleneck (Figs. 11 and 12). To achieve the maximum throughput via ANCONA application in real installations, rather than the simple ANCONA rule (4), a more complex ANCONA rule of [24] should be applied in which qon1 = const: When the speed v (det) (ts ) that is measured at feedback control detector decreases, qon1 should also decrease. V. ANCONA, ALINEA, AND UP-ALINEA: A C OMPARISON Whereas ANCONA tries to achieve an S → F transition after traffic breakdown (F → S transition) has randomly occurred at the bottleneck (Fig. 10), ALINEA tries to achieve the optimal occupancy in free flow downstream of the bottleneck (black point in Fig. 13). Fig. 10 for ANCONA and Fig. 13 for ALINEA illustrate this fundamental difference between ANCONA and ALINEA.
Fig. 13. Symbolical schema of free flow control approach in the flow– occupancy plane. (a) Explanation of theoretical basis of ALINEA in the vicinity of optimal (target) free flow point (F: free flow; C: congested traffic) [20]. (b) Spontaneous traffic breakdown (F → S transition labeled by arrow F → S) occurs with probability greater than zero within the flow rate range (5), regardless of the ALINEA control application [1], [24].
A theoretical basis of ALINEA [20] is associated with an assumption of well-known standard traffic flow theories [2], [3], [6] that downstream bottleneck capacity in free flow qcap and the associated critical occupancy ocr are related to the maximum point at the fundamental diagram for downstream traffic flow at the bottleneck [Fig. 13(a)]. Thus, in ALINEA, we suggested that traffic breakdown does not occur as long as (cont) downstream occupancy osum does not exceed critical occupancy ocr . In accordance with this assumption, the target occu(opt) pancy osum should not exceed ocr , and the ALINEA rule (1) (cont) should ensure that measured downstream occupancy osum (ts ) [possible changes of this occupancy are symbolically shown by arrows in Fig. 13(a)] must be in the small neighborhood of the (opt) target one, i.e., osum (labeled by black point in Fig. 13). This theoretical basis of ALINEA is in a very deep contradiction with empirical probabilistic features of traffic breakdown
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Fig. 14. Comparison of ANCONA and ALINEA. Speed in space and time for (a) ANCONA and (b) ALINEA taken from [24, Fig. 12(a) and (b)]. Time dependence of (c) TTS and (d) ATT for (curves 1) ANCONA and (curves 2) ALINEA. Calculations of (c) TTS and (d) ATT are made for the same on-ramp inflows qon for ALINEA and ANCONA.
discussed previously in the text: Under condition (5), with a finite probability, an F → S transition occurs spontaneously at the bottleneck, regardless of ALINEA application [arrow labeled by F → S in Fig. 13(b)]. This explains [24, Fig. 9], in which a GP occurs spontaneously under ALINEA application as a result of this F → S transition at the bottleneck: ALINEA has no influence on GP upstream development. Even if no significant flow and speed disturbances appear at the bottleneck and, therefore, a GP still has not occurred in ALINEA application [see [24, Figs. 6 and 12] and Fig. 14(b)], ALINEA exhibits disadvantages in comparison with ANCONA (Fig. 14). This can be seen from calculations of total time spent (TTS) [Fig. 14(c)] (TTS is defined as TTS by all vehicles on the main road and in the on-ramp lane between x = 7 and 11 km that passed the location of 7 km during 10-min intervals) and of average travel time (ATT) [Fig. 14(d)] (ATT is defined as TTS divided by the number of vehicles that passed the location of 7 km during 10 min). When there is only an upstream detector for feedback control [Fig. 1(b)], UP-ALINEA with the same rules (1) can be used (cont) in which, however, the downstream occupancy osum (ts ) is calculated from the measured upstream occupancy and the flow (cont) rates qon (ts−1 ) and qin [20]. Because, in UP-ALINEA, the feedback control detector is upstream of the effective bottleneck location, UP-ALINEA prevents traffic breakdown. However, congestion prevention via UP-ALINEA is achieved only at the expense of extremely rapid growth of vehicle queue at the light signal and the associated growth of travel time (waiting time) in comparison with travel time growth under ALINEA application at the same target occupancy. For the same reason, UP-ALINEA also exhibits considerable disadvantages in comparison with ANCONA. First, due to speed disturbance at the bottleneck (Section II), UP-ALINEA cannot almost perform at the detector location x(det) = 9.8 km used for ANCONA in [24] and in Fig. 11: The UP-ALINEA rule (1) applied with exact reformulation of downstream occupancy made in [20] leads to an extremely small average flow rate of vehicles, which may merge from the
(cont)
≈ 95 vehicles/h). This result is also valid for on-ramp (qon all other detector locations satisfying (28). This is because occupancy within the speed disturbance is greater than any possible target occupancy in free flow downstream of the bottleneck. Second, if x(det) is chosen upstream of the optimal one for ANCONA (28), nevertheless, ANCONA exhibits great benefits in comparison with UP-ALINEA (Fig. 15). This is because the UP-ALINEA rule (1) is not associated with the probabilistic nature of F → S and S → F transitions at the bottleneck. This can be seen in Fig. 15(c) (curve 2): UP-ALINEA tries to prevent congestion rather than finding an optimal relation between free flow and synchronized flow phase existence at the bottleneck as ANCONA does (curve 1). For this reason, the average flow (cont) rate qon is considerably greater in ANCONA (curve 1) than in UP-ALINEA (curve 2) [Fig. 15(c)]. Correspondingly, the waiting time at the light signal, TTS, and ATT are considerably longer for UP-ALINEA (curves 2) than for ANCONA (curves 1) [Fig. 15(d)–(f)]. As shown in [33], the ALINEA/Q and X-ALINEA/Q methods [20], which limit vehicle queue growth at the light signal, cannot prevent traffic breakdown and the resulting upstream propagation of traffic congestion. VI. D ISCUSSION A. Critical Discussion of Freeway Control Simulations Based on Earlier Traffic Flow Models Simulation results presented previously in the text are based on three-phase traffic theory. These results cannot be derived with the use of earlier traffic flow models for freeway congestion reviewed in [2]–[13]. Indeed, the earlier traffic flow models, which are based on the fundamental diagram hypothesis, can be divided into two classes [1]. The first model class is referred as the classic Lighthill– Whitham–Richards (LWR) model. Examples of this model class are cell transmission models of Daganzo (see references in [6]). However, it has been explained in [1] that this model class cannot predict the following fundamental measured traffic flow
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(cont)
Fig. 15. Comparison of ANCONA and UP-ALINEA at x(det) = 9.5 km. Speed in space and time for (a) ANCONA and (b) UP-ALINEA. (c) qon (t). Time dependence of travel time in the (d) on-ramp lane, (e) TTS, and (f) ATT. In (c)–(f), curves 1 are for ANCONA, and curves 2 are for UP-ALINEA. UP-ALINEA rules (1) and [20, eq. (9)] are used with model parameters in (1) taken from [24, Fig. 6]. In (4), qon1 = 240 vehicles/h; other parameters of ANCONA are the same as those in Fig. 12. Calculations of (e) TTS and (f) ATT are made for the same on-ramp inflows qon for UP-ALINEA and ANCONA.
features: 1) the probabilistic nature of traffic breakdown and 2) the spontaneous moving jam emergence. Thus, the results presented in Figs. 11 and 12, in which probabilistic traffic breakdown (F → S transition) occurs at the bottleneck and later, using ANCONA, a return probabilistic S → F transition with subsequent MSP appearance and dissolution are realized, cannot be found based on the LWR and cell transmission models. The second model class is referred as the classic General Motors model of Herman et al. (see references in [2]–[5] and [7]–[10]). In this model class, beginning at a critical density, there is instability of steady model states on the fundamental diagram. Examples for this model class are the optimal velocity models by Newell et al., the Payne macroscopic model and its variants, the Wiedemann psychophysical model and its variants, the Nagel–Schreckenberg cellular automata model and its variants, the Krauß model, the intelligent driver model by Treiber and Helbing, as well as a huge number of other models. In this model class, the instability of steady model states should explain traffic breakdown. In 1994, Kerner and Konhäuser found out that this instability leads to wide moving jam emergence in free flow (see references in [1] and [8]). As is currently well known, this result is true for all models of this class: Traffic breakdown is explained by a phase transition from free flow to wide moving jam(s) (F → J transition). However, this fundamental model result contradicts empirical results: Rather than an F → J transition, in all measured data, an F → S transition governs traffic breakdown at a bottleneck. Thus, the models of this class cannot also show and explain empirical features of real traffic breakdown as well as real features of congested patterns (for more detail, see [1, Sec. 3.3] and [34]).
Three-phase traffic flow theory is able to show and predict probabilistic traffic breakdown and emergence congested patterns at bottlenecks in accordance with real measured data [1]. The Kerner–Klenov stochastic traffic flow model (Section III), the KKW cellular automata model [29], and a recent deterministic acceleration time delay model [35], [36] can predict all known empirical congested patterns at bottlenecks. For this reason, we can expect that simulation results of freeway traffic control discussed in this paper will also be found in real freeway traffic. VII. C ONCLUSION 1) The probabilistic feature of traffic breakdown at a freeway bottleneck limits the reliability of applications of a free flow control approach in which free flow should be maintained at the bottleneck. This is because this probabilistic feature is associated with free flow metastability at the bottleneck: A random short-time disturbance in free flow whose amplitude is greater than some critical one can self-grow and lead to traffic breakdown, regardless of control method application. As a result, a congested pattern that propagates upstream of the bottleneck is selfformed. In particular, this nucleation of traffic congestion can occur when ALINEA is applied for feedback on-ramp metering. 2) In contrast with ALINEA, ANCONA prevents propagation of the congested pattern upstream. 3) ANCONA exhibits the following benefits in comparison with ALINEA: a) greater throughputs on the main road and on-ramp;
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b) considerably shorter vehicle waiting times at light signal in the on-ramp lane; c) considerably shorter TTS and ATT; d) upstream propagation of congestion that does not occur even if a congested pattern occurs at the bottleneck. 4) UP-ALINEA, which can prevent congestion, almost cannot perform if the location of feedback control detector is within a localized perturbation at the bottleneck. 5) ANCONA exhibits the following benefits in comparison with UP-ALINEA: a) greater throughputs on the main road and on-ramp; b) considerably smaller vehicle waiting times at the light signal in the on-ramp lane; c) considerably shorter TTS and ATT. 6) To prove these simulations results, practical installations of ALINEA should be studied based on measured data; ANCONA application in real installations should be performed.
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Boris S. Kerner was born in Moscow, Russia, on December 22, 1947. He received the degree in electronics from the Technical University MIREA, Moscow, the Ph.D. degree in physics and mathematics in 1979, and the Dr.Sc. degree (habilitation) in physics and mathematics from the Russian Academy of Science, in 1986. In 1972, he was a Semiconductor Electronics Engineer at the Semiconductor Research Institute “Pulsar,” Moscow. From 1980 to 1992, he was with the Research Institute “Orion,” Moscow, as a Senior Scientist and later as a Head of a theoretical department in semiconductor engineering and nonlinear physics. In 1990–1991, he was a Professor at the Technical University MIREA. Since 1992, he has been with DaimlerChrysler Research, Stuttgart, Germany, in various fields of intelligent transportation systems (ITS). He is currently the Head of the research field “Traffic” at the Group Research and Advanced Engineering, DaimlerChrysler AG, Sindelfingen, Germany. He has published three books and more than 150 articles and reviews on traffic flow theory and its applications on ITS, nonlinear science, semiconductors, solid-state physics, and gas plasma physics. He is the inventor of about 70 patents on ITS.
