Modeling Toll Lanes and Dynamic Pricing Control
MModeling Toll Lanes and Dynamic Pricing Control
Elena G. Dorogush and Alex A. KurzhanskiySeptember 19, 2018
Abstract
In this paper we address the problem of dynamic pricing for toll lanes on freeways. The proposed tollmechanism is broken up into two parts: (1) the supply side feedback control that computes the desiredsplit ratios for the incoming traffic flows between the general purpose and the toll lanes; and (2) thedemand side price setting algorithm that aims to enforce the computed split ratios.The split ratio controller is designed and tested in the context of the link-node Cell TransmissionModel with the modified node model of in/out flow distribution. The equilibrium structure of this trafficmodel is presented; and the case, in which the existence of a toll lane is meaningful, is discussed.For the price setting, two alternative approaches are presented. The first one is commonly used, andit relies on the known Value of Time (VoT) distribution. Its shortcoming, however, is in the difficulty ofthe VoT distribution estimation. The second approach employs the auction mechanism, where travelersmake bids on places in the toll lane. The advantage of this approach is that it enables direct control overhow many vehicles will be allowed into the toll lane.
Keywords : congestion pricing, toll lanes, feedback traffic control, value of time, Cell Transmission Model,CTM equilibria
Congestion pricing is an economics concept of using pricing mechanisms to charge the users of public roadsfor the negative effect on others generated by the demand in excess of available supply. It is one of anumber of demand side traffic management strategies that address traffic congestion. Other known demandmanagement strategies include: parking restrictions; park and ride facilities allowing parking at a distanceand continuation by public transport or ride sharing; reduction of road capacity to force traffic onto othertravel modes; road space rationing, where regulatory restrictions prevent certain types of vehicles fromdriving under certain circumstances or in certain areas; and policy approaches, which encourage greater useof existing alternatives through promotion, subsidies or restrictions.Our focus will be on congestion pricing for Express Toll Lanes (ETL) and High Occupancy/ Toll (HOT)lanes. This type of facilities becomes ever more popular in the U.S. In San Francisco Bay Area alone, theMetropolitan Transportation Commission promises the implementation of 550-mile express lane network by2035, and all of it with dynamic pricing strategies.There were numerous studies on congestion pricing over the past two decades. Optimal tolls under stochasticuser equilibria were devised in [29]. Deterministic static equilibrium model for urban transport networkswith elastic demand and capacity constraints was presented in [12]. Optimization of tunnel tolls in HongKong was investigated in [39]. Yang and Bell talked about road pricing in the presence of congestion delay[38]. Other models of congestion toll were discussed in [15]. Wie and Tobin presented two pricing models —1 a r X i v : . [ c s . S Y ] M a y ne static with the same day-to-day demand and road network capacity, and the other one with time varyingtraffic conditions and congestion tolls [37]. Bottleneck model with elastic demand was described in [1]. Themodel of bottleneck congestion was generalized and optimal peak-load toll was derived in [24]. Eliassonsuggested optimal road pricing policy reducing aggregate travel time and distributing toll burden equallyamong travelers, although travelers had utlity functions with constant marginal utilities of time and money,and these marginal utilities were unobservable [11]. The second best pricing policy for a static transportationnetwork, where not all links could be tolled, was presented in [35]. De Palma and Lindsey derived the optimaltolls formulae and analyzed them to reveal the separate influences of traveler heterogeneity, road networkeffects, fiscal effects and equity concerns [8]. In [36] the properties of various types of public and privatepricing on a congested road network were explored, and it was shown that welfare-maximizing pricing wasmore efficient than the revenue-maximizing one. Game theoretical analysis of congestion pricing from itsmicro-foundations, interaction of two or more travelers, was performed in [20]. Simulation based analysis ofvarious road pricing schemes was conducted in [7]. The analytical formulation of the optimal toll calculationfor multi-class traffic was given in [16]. In [10] two dynamic pricing strategies, reactive (feedback) andproactive (feedforward) were considered and compared, favoring the latter. A comprehensive review of thecurrent congestion pricing technologies was offered in [9].It is important to mention the research targeted specifically at the dynamic pricing of managed toll lanes. Alocal feedback strategy for setting tolls is proposed in [42]. The control algorithm is similar to ALINEA [25],where the portion of vehicle flow allowed into the toll lane for a given time step is computed from the tollvehicle flow at the previous time step and the speed observed in the toll and the general purpose lanes. In thesituation when the speed in the toll lane drops below 45 mph, the congestion in the toll lane is assumed, andtolls are increased quicker than otherwise. Proactive dynamic pricing methodology is described in [23]. Tollsare determined based on the cell transmission model (CTM) with stochastic demand and capacity. The roadconfiguration with a single decision point is considered. A distance-based dynamic pricing strategy for tolllanes aimed at maximizing toll revenue is presented in [40]. Here the tolls are set based on the travel timeprediction that is computed using the stochastic variation of the LWR model [3, 21, 27]. The optimizationproblem maximizing the expected toll revenue with constraints keeping the toll lane in free flow is solved.Yin and Lou delivered the proof of concept of a reactive self-learning approach for determining time-varyingtolls in response to the detected traffic arrivals [41]. The approach learns in a sequential fashion motorists’willingness to pay and then determines pricing strategies based on a point-queue model. This result wasextended in [22], where the CTM was used instead of the point-queue model, and tolls are set based on thepredicted traffic state. These papers assume that toll lane should be kept in free flow, and the correspondingconstraints are imposed. The logit model is used to implement the driver’s lane choice.In this paper, we approach dynamic pricing control from both, the supply and the demand sides with themain focus on the supply side. The supply side control is considered in the context of the Cell TransmissionModel (CTM) [6] with the node model proposed in [31]. Here we extend the result of [14] by analyzing theequilibrium structure of the freeway traffic model with a single mainline, similarly to [14], but for a differentnode model. Then, we discuss when it makes sense to split the mainline into two lanes, the general purposeand the toll lanes, where the toll lane is supposed to be less congested than the general purpose one. It isimportant to note that activating a toll lane is not always beneficial for the freeway performance. Obviously,when the traffic density is low, a toll lane does not provide any advantage in terms of travel time and, thus,should be free. On the other hand, when the freeway is too congested, maintaining a toll lane in free flowmay be harmful, as it will lead to large congestion spillback and degrade the throughput. We propose asupply side control algorithm that computes split ratios dividing the incoming flows between the generalpurpose and the toll lanes. Its goal is to keep the toll lane in free flow as long as possible, then, in the caseof continuing excessive demand, allow the toll lane to congest up to the point when the toll lane becomesa general purpose lane in terms of its performance. When the congestion recedes, the control algorithmensures that the toll lane frees up first. The objective is to maintain the same throughput as in the freewaywith all lanes being general purpose. The idea of this control algorithm is inspired by the HERO coordinatedramp metering [26]. While in the case of ramp metering ramps are used as storage for the excess demand,the general purpose lane serves as the storage for the split ratio control. All proofs are given in the contextof the proposed traffic model. Once the desired flow splits are computed by the split ratio controller, we2chieve them by setting the toll rate — that is already a demand management. The common way of settingthe price is to rely on the Value of Time (VoT) distribution [2], which is assumed to be known. Relyingon the VoT distribution, though, makes our control indirect, as we can only estimate how much travelersare ready to pay. The alternative solution is to use the truthful auction, which we propose as a means ofensuring the desired split ratios.The rest of the paper is organized as follows. Section 2 introduces the traffic model, describes its equilibriumstructure and splitting the mainline into the general purpose and the toll lanes. Section 3 presents the splitratio control algorithm and provides examples. Section 4 is dedicated to pricing mechanisms, discussing theestimation of VoT distribution and proposing the auction algorithm as an alternative. The simulation ofInterstate 680 South in Contra Costa County in California is discussed in Section 5. Ultimately, Section 6concludes the paper. A road network is modeled by a directed graph G = ( E, V ) , where E is the set of edges (links), V is theset of vertices (nodes). A link e ∈ E is an ordered pair of nodes: e = ( u, v ) , u, v ∈ V . In( v ) denotes the setof incoming links of node v , Out( v ) denotes the set of outgoing links of node v . Links without predecessors(whose begin nodes have no inputs) are called entrances . Links without successors (whose end nodes haveno outputs) are called exits . Links other than entrances and exits are referred to as inner links . By E in and E out we denote the set of entrances and the set of exits, and assume that E in ∩ E out = ∅ . Nodes correspondto road intersections, merges, diverges, or subdivide longer links into smaller ones.Every inner link or exit e has the following characteristics, also known as a fundamental diagram: N e maximum number of vehicles in the link, F e capacity (in vehicles per time step), v e free flow speed, w e congestion wave speed.For entrances only capacity and free flow speed need to be defined. The free flow speed and the congestionspeed are measured in link per time step, < v e , w e < .Link e at time t contains n e ( t ) vehicles. Define the outflow demand , or the required outflow, of link e ∈ E at time t as f de ( t ) = min { v e n e ( t ) , F e } . Define the inflow supply , or the maximum inflow, of link e ∈ E \ E in at time t as f se ( t ) = min { w e ( N e − n e ( t )) , F e } . The number of vehicles in link e = ( u, v ) at time t + 1 is n e ( t + 1) = n e ( t ) + f in e ( t ) − f out e ( t ) , (2.1)where f in e ( t ) is the total incoming flow for link e at time t : f in e ( t ) = (cid:88) e ∈ In( u ) f e ,e ( t ) , e ∈ E \ E in , (2.2)the incoming flow f in e ( t ) for every entrance e ∈ E in is given; f out e ( t ) is the total outgoing flow for link e attime t : f out e ( t ) = (cid:88) e ∈ Out( v ) f e,e ( t ) , e ∈ E \ E out , (2.3)the outgoing flow for exits e ∈ E out equals the outflow demand: f out e ( t ) = f de ( t ) .3lows f e ,e ( t ) between adjacent links e , e , are determined by a node model that should satisfy the followingconditions: (1) all flows between adjacent nodes f e ,e ( t ) are non-negative; (2) incoming flows don’t exceedinflow supplies, except for entrances, f in e ( t ) ≤ f se ( t ) , e ∈ E \ E in ; and (3) outgoing flows don’t exceedoutflow demands, f out e ( t ) ≤ f de ( t ) , e ∈ E .Conservation law (2.1) and the flow constraints ensure that if the number of vehicles at time t in eachlink n e ( t ) is positive and less than the maximum number of vehicles N e (that is, n e ( t ) ≥ for all e ∈ E and n e ( t ) ≤ N e for all e ∈ E \ E in ), then the same holds for the next time step t + 1 (namely, n e ( t + 1) ≥ for e ∈ E , and n e ( t + 1) ≤ N e for e ∈ E \ E in ). Moreover, if an exit link e ∈ E out , is in a free flow state attime t , v e n e ( t ) ≤ F e , then it will stay in a free flow state at time t + 1 : v e n e ( t + 1) ≤ F e . We use the node model proposed in [31]. Consider node v with m incoming and n outgoing links, m, n > .Let f di ( t ) be an outflow demand of the i th incoming link at time t , and f sj ( t ) be an inflow supply of the j thoutgoing link at time t as shown in Figure 1. f di ( t ) f sj ( t ) v Figure 1: Node of a road network graph.Let f ij ( t ) be the flow from the i th incoming to the j th outgoing link. A node model determines flows f ij ( t ) from demands f di ( t ) , supplies f sj ( t ) , a split ratio matrix B v ( t ) ∈ R m × n and priorities of the incoming links p i ≥ . [31] proposed that the priorities p i should be proportional to capacities F i of the incoming links.Alternatively, the priorities p i can be proportional to demands f di ( t ) , as suggested in [17] and implementedin [19].Elements of the split ratio matrix B v ( t ) = { β ij ( t ) } j =1 ,...,ni =1 ,...,m are nonnegative, and (cid:80) nj =1 β ij ( t ) = 1 . The splitratio matrix imposes the First-In-First-Out (FIFO) constraint on the flows f ij ( t ) : f ij ( t ) β ij ( t ) = f ij ( t ) β ij ( t ) . (2.4)Priority coefficients of the incoming links, p i , affect flows f ij ( t ) , if a supply of some outgoing link is too smallto accommodate a demand from the incoming links. This will be clarified later.We now describe the algorithm for finding flows f ij . To simplify the notation, the time step t is omitted. The Algorithm
1. Compute oriented demands f dij = f di β ij and directed priorities p ij = p i β ij , i = 1 , . . . , m ; j = 1 , . . . , n .2. Define sets J (1) , V j (1) , and supply residuals ˜ f sj (1) , j ∈ J (1) : J (1) = { j : (cid:80) mi =1 f dij > } ; V j (1) = { i : f dij > } ;˜ f sj (1) = f sj . f ij = 0 if j / ∈ J (1) .Sets V j ( k ) defined at each step k are the sets of incoming links i such that flows f ij are still to bedefined. J ( k ) is a set of outgoing links j such that V j ( k ) (cid:54) = ∅ . The supply residual ˜ f sj ( k ) is the inflowsupply of link j to be distributed among incoming links in V j ( k ) .3. Set k ← .4. If J ( k ) = ∅ , stop.5. For each j ∈ J ( k ) compute a reduction factor a j ( k ) = ˜ f sj ( k ) (cid:80) i ∈ V j ( k ) p ij . Determine the minimum of { a j ( k ) } j ∈ J ( k ) : ˆ a ( k ) = a ˆ ( k ) ( k ) = min j ∈ J ( k ) a j ( k ) .6. Define U ( k ) = { i ∈ V ˆ ( k ) ( k ) : f di ≤ ˆ a ( k ) p i } . It’s a set of incoming links such that their outflow demanddoes not exceed the rightful share in the inflow supply of the remaining outgoing links.(a) If U ( k ) (cid:54) = ∅ , then for all i ∈ U ( k ) , j = 1 , . . . , n , set f ij = f dij , and compute V j ( k + 1) = V j ( k ) \ U ( k ) , j ∈ J ( k ); J ( k + 1) = { j ∈ J ( k ) : V j ( k + 1) (cid:54) = ∅ } ;˜ f sj ( k + 1) = ˜ f sj ( k ) − (cid:80) i ∈ U ( k ) f dij , j ∈ J ( k + 1) . (b) If U ( k ) = ∅ , then for all i ∈ V ˆ ( k ) ( k ) , j = 1 , . . . , n , set f ij = ˆ a ( k ) p ij , and compute V j ( k + 1) = V j ( k ) \ V ˆ ( k ) ( k ) , j ∈ J ( k ); J ( k + 1) = { j ∈ J ( k ) : V j ( k + 1) (cid:54) = ∅ } ;˜ f sj ( k + 1) = ˜ f sj ( k ) − (cid:80) i ∈ V ˆ ( k ) ( k ) ˆ a ( k ) p ij , j ∈ J ( k + 1) . Note that ˆ ( k ) / ∈ J ( k + 1) .7. Set k ← k + 1 , and go to step 4.The first 3 steps are initialization steps. As explained in [31], in each iteration we define the reductionfactors a j (step 5), which the incoming flows competing for the remaining supply of the j th outgoing linkshould be set proportional to, if they were not constrained by the demand of the corresponding incominglink, so f ij = a j p ij . Then, a minimum of a j , denoted by ˆ a , whose index is ˆ , is selected and imposed on allincoming links from V ˆ . If some links are found to be demand-constrained (case 6a), their outgoing flow isset equal to demand, and those demand-constrained links U are removed from every V j . Otherwise (case 6b)the flows from V ˆ are supply-constrained with factor ˆ a , and all links from V ˆ are removed from every V j . Thealgorithm stops when all flows have been defined. Since at least one incoming link is removed from all V j ineach iteration, the algorithm finishes after at most m iterations. The model of a freeway without toll lanes described below is a combination of the Cell Transmission Modelfor networks, similar to [6], and the node model proposed in [31] and explained above.A freeway network consists of mainline links, entrance links (mostly on-ramps), and exit links (mostly off-ramps). Links are connected to each other by nodes. Nodes correspond to junctions, merge and diverge5oints, and are also used to split long links into shorter ones. When a freeway network is built for simulation,nodes are typically placed so as to keep link lengths in the range between . and . miles.The characteristics of the i th mainline link are: its capacity F i , the maximum number of vehicles it cancontain N i , a free flow speed v i and a congestion speed w i . The capacity is measured in vehicles per timestep, while both speeds are measured in link per time step, < v i , w i < . The freeway contains K links,numbered from 1 to K , the entrance link numbered 0, and the exit link numbered K + 1 . Additionally,there may be an on-ramp with capacity R i and free flow speed v ri < attached to the beginning of mainlinelink i , and an off-ramp with capacity S i at the end of the i th link, i = 1 , . . . , K . If there is no on-ramp atthe beginning of the i th link, then R i = 0 .We now clarify how the model parameters, such as capacities and speeds, are normalized , that is, how thespeeds measured in miles per hour are converted into the speeds measured in link per time step, and howthe capacities measured in vehicles per hour per lane are converted into the capacities measured in vehiclesper time step. Suppose L i is the length of mainline link i measured in miles, τ is the time step measuredin hours, V i is the free flow speed in link i measured in miles per hour, W i is the congestion speed in link i measured in miles per hour, C i is the capacity of link i measured in vehicles per hour per lane, k i is thenumber of lanes in link i . Then v i = V i τ /L i , w i = W i τ /L i , F i = C i k i τ . The speeds v ri and the capacities R i , S i are normalized in the same way. Hence the aforementioned inequalities v i , w i < are equivalentto max { V i , W i } τ /L i < , which is the Courant — Friedrichs — Lewy condition (see [5]). For example, ifthe minimum link length L i is . miles, the maximum free flow speed v i is 60 mph, and the maximumcongestion speed w i is 50 mph, then the maximum time step τ is . / hours, which is 12 seconds. n i − n i q i n i +1 q i +1 Figure 2: Freeway model.Denote n i ( t ) the number of vehicles in link i at time t . We assume that if the i th link is uncongested, that is, v i n i ( t ) ≤ F i , then the number of vehicles it contains doesn’t influence the incoming flow for this link attime t , that is, w i ( N i − n i ( t )) ≥ F i . This assumption is equivalent to the following inequality: F i v i + F i w i ≤ N i . (2.5)Denote d i ( t ) the demand, or desired flow, at the i th on-ramp; r i ( t ) — the actual flow coming from on-ramp i to link i ; and q i ( t ) — the number of vehicles queued at the i th on-ramp, at time t . Both flows, d i ( t ) and r i ( t ) are measured in vehicles per time step. Clearly, q i ( t + 1) = q i ( t ) + d i ( t ) − r i ( t ) . (2.6)Denote f i ( t ) the flow from link i to link ( i + 1) ; and s i ( t ) — the flow from link i to the correspondingoff-ramp. The number of vehicles in mainline links evolves as: n i ( t + 1) = n i ( t ) + f i − ( t ) + r i ( t ) − f i ( t ) − s i ( t ) . (2.7)As for the entrance link 0 and the exit link K + 1 , the same equality holds, except r i ( t ) = s i ( t ) = 0 , i = 0 , K + 1 .Since every exit remains in a free flow state once it’s there, there is no need to take the state of exitsinto consideration: assuming that all exits, including link K + 1 , are initially in a free flow state, and thatinequality (2.5) holds, flows from mainline links to the exits are only constrained by the exit capacities.6ff-ramp flow s i ( t ) is proportional to the flow to the downstream link f i ( t ) : there exist split ratios β fi , β si ,such that β fi > , β si ≥ , β fi + β si = 1 , f i ( t ) β fi = s i ( t ) β si . (2.8)If link i has no off-ramp, let β si = 0 , β fi = 1 .Denote F si = F i , i = 1 , . . . , K + 1 , F d = F , F di = (cid:40) F i , β si = 0 ,β fi min { F i , S i /β si } , β si > , i = 1 , . . . , K + 1 . (2.9) F si is the “inflow capacity” of the i th mainline link, that is, the maximum incoming flow if the link hasenough space for them. F di is the “outflow capacity” of the i th mainline link, that is, the maximum numberof vehicles which can move from this link to link i + 1 per time step if link i + 1 has enough space for them.By introducing the outflow capacity we ensure that the off-ramp flow s i ( t ) = ( β si /β fi ) f i ( t ) never exceeds theoff-ramp capacity S i if the downstream flow f i ( t ) does not exceed the outflow capacity F di : s i ( t ) = β si β fi f i ( t ) ≤ β si β fi F di ≤ S i . (2.10)The incoming flows d ( t ) and f − ( t ) are given, other flows are determined by the node model (see Figure 3).For each node compute outflow demands for the on-ramp r di ( t ) (if present) and the incoming (upstream)mainline link f di − ( t ) , and inflow supply for the outgoing (downstream) mainline link f si ( t ) : f si ( t ) = min { w i ( N i − n i ( t )) , F si } , i = 1 , . . . , K + 1 , (2.11) f di ( t ) = min { β fi v i n i ( t ) , F di } , i = 0 , . . . , K + 1 , (2.12) r di ( t ) = min { v ri q i ( t ) , R i } , i = 1 , . . . , K. (2.13)Let r dK +1 ( t ) = 0 . Note that f sK +1 ( t ) = F sK +1 . f di − , p fi − r di , p ri f si S i − Figure 3: Freeway node.The flow out of exit link K + 1 equals the outflow demand, f K +1 ( t ) = f dK +1 ( t ) . The flow from the lastmainline link K to the exit link K + 1 is a minimum of the outflow demand of the upstream link and theinflow supply of the downstream link: f K ( t ) = min { f dK ( t ) , f sK +1 ( t ) } .Priorities of mainline links p fi , i = 0 , . . . , K − , and priorities of on-ramps p ri , i = 1 , . . . , K , are given,Suppose the priorities are normalized: p fi − + p ri = 1 . For each i = 1 , . . . , K compute f i − ( t ) and r i ( t ) asfollows.1. If the downstream mainline link can accomodate the demand from the on-ramp and the upstream link, f di − ( t ) + r di ( t ) ≤ f si ( t ) , then the actual flows equal the demands: f i − ( t ) = f di − ( t ) , r i ( t ) = r di ( t ) .7. Otherwise, if the outflow demand from upstream mainline link does not exceed its rightful share in theinflow supply of the downstream link, f di − ( t ) ≤ p fi − f si ( t ) , then the flow from the upstream link equalsthe demand, f i − ( t ) = f di − ( t ) , and the flow from the on-ramp takes up the residual of the supply: r i ( t ) = f si ( t ) − f di − ( t ) .3. Otherwise, if the outflow demand from the on-ramp doesn’t exceed its rightful share in the supply, r di ( t ) ≤ p ri f si ( t ) , then, analogously, r i ( t ) = r di ( t ) , f i − ( t ) = f si ( t ) − r di ( t ) .4. Otherwise f di − ( t ) > p fi − f si ( t ) and r di ( t ) > p ri f si ( t ) . In this case the flows are proportional to thepriorities: r i ( t ) = p ri f si ( t ) , f i − ( t ) = p fi − f si ( t ) .These rules are derived from the general node model presented in subsection 2.1. Note that since f i − ( t ) ≤ f di − ( t ) ≤ F di − , the off-ramp flow s i − ( t ) = ( β si − /β fi − ) f i − ( t ) does not exceed the off-ramp capacity S i − for the reasons mentioned after equation (2.9). Thus, both supply constraints — from the downstream linkand from the off-ramp — are satisfied. Suppose the incoming flows f − and d are constant, as well as priorities p fi , p ri and split ratios β fi , β si . We saythat a triple ( n, f, r ) is an equilibrium , if v K +1 n K +1 ≤ F K +1 (that is, the exit link ( K + 1) is uncongested)and there exist queue lengths n , q i , i = 1 , . . . , K , such that if n i ( t ) = n i , i = 1 , . . . , K , n ( t ) = n , and q ( t ) = q , then f ( t + ∆ t ) = f , n i ( t + ∆ t ) = n i , i = 1 , . . . , K , and r ( t + ∆ t ) = r for all ∆ t = 0 , , , . . . . Notethat queue lengths n , q are not included in the equilibrium. Queue lengths either stay constant or grow ata rate of ( f − − f ) or ( d i − r i ) vehicles per time step.Define maximum flows ¯ f = min { f − , F } , ¯ r i = min { d i , R i } , ¯ f i = min { β fi ( ¯ f i − + ¯ r i ) , F di } , i = 1 , . . . , K + 1 . (2.14)It can be shown that inequalities r i ≤ ¯ r i , i = 1 , . . . , K , and f i ≤ ¯ f i , i = 0 , . . . , K + 1 , hold for everyequilibrium ( n, f, r ) .Theorems 2.1, 2.2 fully characterize the set of all equilibria corresponding to the incoming flows f − , d . Theorem 2.1.
Equilibrium flows r and f are uniquely defined. Namely, f K +1 = ¯ f K +1 , and flows f i − , r i are uniquely determined by f i , i = K + 1 , . . . , , as follows.1. If ¯ f i − ≤ p fi − f i /β fi , then f i − = ¯ f i − , r i = f i /β fi − ¯ f i − .2. If ¯ r i ≤ p ri f i /β fi , then r i = ¯ r i , f i − = f i /β fi − ¯ r i .3. Otherwise ¯ f i − > p fi − f i /β fi , ¯ r i > p ri f i /β fi ; in this case f i − = p fi − f i /β fi , r i = p ri f i /β fi . Note that this rule is similar to that in the node model. Again, the off-ramp capacity constraints are notviolated due to the reasons mentioned after equation (2.9) and the fact that f ≤ ¯ f ≤ F d .Theorem 2.1 follows from lemma 2.1 and the node model. Lemma 2.1. If f i < ¯ f i , then f i < f di . r and f are known, the set E of equilibrium density vectors n is defined as follows.Let I = { i : 1 ≤ i ≤ K, f i = F di } ∪ { i : 1 ≤ i ≤ K, f i + r i +1 = F si +1 } . The set I is in fact a set of bottlenecks,since for all i / ∈ I f i < F di and f i < F si +1 − r i +1 . Suppose I = { i , . . . , i M } , i < i < · · · < i M , M = | I | .Assume i = 0 . Let I m = { i : i m − < i ≤ i m } , m = 1 , . . . , M, (2.15) I M +1 = { i : i M < i ≤ K + 1 } . (2.16)If I = ∅ , then I M +1 = I = { , . . . , K + 1 } . The set I m , m ∈ { , . . . , M } , contains all links betweenthe bottleneck i m − and the downstream bottleneck i m , including the downstream bottleneck. The set I M +1 contains links downstream of the last bottleneck i M , the set I contains links upstream of the firstbottleneck i .Let n ui = n ui ( r, f ) = f i β fi v i , n ci = n ci ( r, f ) = N i − r i + f i − w i , i = 1 , . . . , K + 1 . (2.17)Letters u and c here mean uncongested and congested . We claim that n u ≤ n c whenever f i = β fi ( r i + f i − ) ≤ F di . Indeed, this follows from inequality (2.5): n ui = f i β fi v i ≤ F di β fi v i ≤ F i v i ≤ N i − F i w i ≤ N i − F di β fi w i ≤ N i − r i + f i − w i = n ci . (2.18)Define sets U = { i : 1 ≤ i ≤ K − , f i < F di , f i /p fi < r i +1 /p ri +1 } , (2.19) C = { i : 1 ≤ i ≤ K, r i < ¯ r i , f i − + r i < F si } ∪ { , if f < ¯ f , f + r < F s } . (2.20) U is the set of uncongested links: n i = n ui for i ∈ U ; C is the set of congested links: n i = n ci for i ∈ C .These conditions can be obtained directly from the node model. For m = 1 , . . . , M define indices i um = (cid:40) i m − , I m ∩ U = ∅ , max( I m ∩ U ) , I m ∩ U (cid:54) = ∅ , i cm = (cid:40) i m + 1 , I m ∩ C = ∅ , min( I m ∩ C ) , I m ∩ C (cid:54) = ∅ . (2.21)It can be shown that i um < i cm for each m = 1 , . . . , M . Theorem 2.2.
The set E of equilibrium vectors n is a direct product of sets E m corresponding to sets I m : E = M +1 (cid:79) m =1 E m . (2.22) The set E M +1 consists of a single vector, E M +1 = { ( n ui M +1 , . . . , n uK +1 ) } . (2.23) The set E m , m ∈ { , . . . , M } either consists of a single vector, E m = { ( n ui m − +1 , . . . , n ui um , n ci cm , . . . , n ci m ) } , if i um = i cm − , (2.24) or is a union E m = i cm − (cid:91) h = i um +1 E hm , (2.25) where E hm = { ( n ui m − +1 , . . . , n uh − , n h , n ch +1 , . . . , n ci m ) , n uh ≤ n h ≤ n ch } . (2.26)9 i I i I i M i K +1 I M +1 Figure 4: Structure of equilibrium vector n .The proof of theorem 2.2 for U = ∅ and C = ∅ can be found in [14].Figure 4 illustrates theorem 2.2.Note that the equilibrium flows and densities and thus the queue growth depend on priorities p f , p r . Oneextreme case, p ri = 1 , p fi = 0 for all i , is fully studied by [14]. Another extreme case is p ri = 0 , p fi = 1 forall i . In this case all non-bottleneck links are uncongested, whereas the on-ramp just upstream of the lastbottleneck is more congested than for any strictly positive on-ramp priorities p r . Flows f i , i = 1 , . . . , K + 1 , are uniquely defined by flows from entrances f and r i , i = 1 , . . . , K , because theequality f i = β fi ( f i − + r i ) holds for all i = 1 , . . . , K . Namely, f i ( f , r ) = f i (cid:89) k =1 β fk + i (cid:88) j =1 r j i (cid:89) k = j β fk , i = 1 , . . . , K + 1 . (2.27)The incoming flow ( f − , d ) is said to be feasible , if f i ( ¯ f , ¯ r ) ≤ F di , i = 1 , . . . , K + 1 , and infeasible otherwise.The incoming flow is said to be strictly feasible , if f i ( ¯ f , ¯ r ) < F di , i = 1 , . . . , K + 1 .It can be demonstrated that if the incoming flow ( f − , d ) is feasible, then the equilibrium flows equalmaximum flows, r = ¯ r , f = ¯ f = f ( ¯ f , ¯ r ) , and consequently C = ∅ . Moreover, if the incoming flow ( f − , d ) is strictly feasible, then the set E of equilibrium density vectors consists of a single vector, E = { n u } . We model a freeway with toll lanes by splitting each mainline link into two parallel ones (see Figure 5), thefirst one (upper index 1) corresponds to toll lanes, the second one (upper index 2) corresponds to the generalpurpose lanes. Each of these two links has a separate exit. n i − n i − q i n i n i q i +1 n i +1 n i +1 Figure 5: Model of a freeway with toll lanes.The capacities and maximum number of vehicles of the resulting links are proportional to the number oflanes. Let l be the number of toll lanes, and l be the number of general purpose lanes. Then the capacities10 i = F i l / ( l + l ) , F i = F i l / ( l + l ) . The maximum number of vehicles N ξi , the maximum flows F ξ,di , F ξ,si , and priorities p ξ,si , ξ = 1 , , are defined in the same manner.Outflow demands and inflow supplies are defined as in the freeway model: r di ( t ) = min { v ri q i ( t ) , R i } , i = 1 , . . . , K (2.28) f ξ,di ( t ) = min { β fi v i n ξi ( t ) , F ξ,di } , ξ = 1 , , i = 0 , . . . , K + 1 , (2.29) f ξ,si ( t ) = min { w i ( N ξi − n ξi ( t )) , F ξ,si } , ξ = 1 , , i = 1 , . . . , K + 1 . (2.30)Travelers can only choose between the toll and the general purpose lanes when they enter the freeway. Thefreeway state and the toll both influence split ratios of the flow from the entrance into the mainline links.Denote those split ratios by α i , α i . Of course, α i , α i ≥ , α i + α i = 1 . f ,di − , p ,fi − f ,di − , p ,fi − f ,si f ,si r di , p ri S i − S i − Figure 6: Node model of a freeway with toll lanes.Flows f ξ − ( t ) are given, flows out of the exit links equal demands: f ξK +1 ( t ) = f ξ,dK +1 ( t ) . Other flows, namely f ξi − ( t ) , r ξi ( t ) , i = 1 , . . . , K + 1 , are determined by the following node model (see Figure 6). The time step t is implied, but omitted for simplicity. Compute potential flows ψ i , ψ i ψ ξi = min (cid:40) max (cid:40) f ξ,si α ξi p ri α ξi p ri + p ξ,fi − , f ξ,si − f ξ,di − (cid:41) , α ξi r di (cid:41) , ξ = 1 , . (2.31)The potential flow ψ i is the flow r i from the entrance to the toll lane of the i th mainline link (accordingto the general node model from subsection 2.1), if this flow is not constrained by the general purpose lane.Such a constraint may arise, as shown later, since the flows r i , r i should be proportional to the split ratios α i , α i . Similarly, ψ i is the flow from the entrance to the general purpose lane of the i th link, if there are noconstraints from the toll lane.Define λ ξi = (cid:40) , α ξi = 0 ,ψ ξi / ( α ξi r di ) , α ξi > , ξ = 1 , , (2.32)and λ i = min { λ i , λ i } . Clearly, λ ξi ≤ , therefore λ i ≤ . Finally, compute flows r ξi and f ξi − : r ξi = λ i α ξi r di , f ξi − = min { f ξ,di − , f ξ,si − r ξi } , ξ = 1 , . (2.33)If the freeway is in free flow and the demand is feasible, there is no need in toll lane, because it does notprovide any gain in travel time. On the other hand, if the freeway is in the fully congested equilibrium,there is no logic in redistributing the incoming flows r i either: even if the toll lane becomes uncongested, theentrances will accumulate queues, reducing the total output flow of the system and harming everyone. So,in the case of fully congested freeway, even with feasible flow, there is no use for a toll lane. The situationwhen the toll lane is helpful though, is when the freeway segment is partially congested. That is, when11ithout reducing the total output flow of the system, toll lane can provide saving in travel time for thosechoosing to use it. Figure 7 illustrates this case: assume there is a partially congested freeway segment (seealso Figure 4). Once we divide this segment into the toll and the general purpose lanes, it is possible toredistribute vehicles between these two lanes so that the general purpose lane acts as a storage for the extravehicles, while the toll lane frees up, and all of it retaining the original flows. toll lanegeneral purpose lane Figure 7: Redistributing congestion of a partially congested highway.The proposed redistribution can be achieved by controlling the split ratios α i , α i of the on-ramp flows r i .This will be explained in the next section. Here we present an on-ramp flow redistribution control algorithm which aims at bringing the toll lane intoa free flow state and maintaining the free flow state of the toll lane, provided that queues at the entrancesgrow as slowly as possible.For simplicity, assume that the first link does not have an on-ramp, and transform the model in such a way,that the link 0 becomes an on-ramp of link 1. Thus, there is no 0th link and f ξ,s ( t ) ≡ , ξ = 1 , .We assume that all information about the current freeway state, namely, densities n ξi ( t ) and queuelengths q i ( t ) , is available. But we know nothing about the future: at time t no information about incomingflows d i ( t + ∆ t ) , ∆ t = 0 , , . . . , is available.The pricing mechanism for using the toll lane changes split ratios α i ( t ) , α i ( t ) . This dependency will bediscussed in section 4. In this section we assume that the split ratios α i ( t ) , α i ( t ) can be set directly.The algorithm for determining the split ratios of the on-ramp flow r i ( t ) at time step t is described next. Thetime step t is implied, but will be omitted for simplicity. First, we determine a range of split ratios that minimize the queue growth rate. It only makes sense toconsider such links i , where r di > and f ,si + f ,si > , since otherwise r i = r i = 0 .To minimize the queue growth rate, λ ( α i , α i ) = min { λ i ( α i ) , λ i ( α i ) } should be maximized. Denote λ ∗ i = max α i ∈ [0 , λ i ( α i , − α i ) , (3.1) A i = arg max α i ∈ [0 , λ i ( α i , − α i ) . (3.2)Our goal is to find A i .It can be shown that λ ξi is a monotonically decreasing function of α ξi , and λ ξi ( α ξi ) is continuous at least for α ξi ∈ (0 , , ξ = 1 , . Indeed, if α ξi ∈ (0 , , then λ ξi ( α ξi ) = min (cid:40) max (cid:40) f ξ,si r di p ri α ξi p ri + p ξ,fi − , f ξ,si − f ξ,di − α ξi r di (cid:41) , (cid:41) . (3.3)12ote that λ ξi ( α ξi ) ≡ for α ξi ∈ (0 , , if p ri = 0 and f ξ,si ≤ f ξ,di − .Let ¯ α ξi = max { α ξi : α ξi ∈ [0 , , λ ξi ( α ξi ) = 1 } . (3.4)Clearly, ¯ α ξi = min { , max { , a ξi }} , where a ξi = f ξ,si − f ξ,di − r di , p ri = 0 , max (cid:40) f ξ,si − f ξ,di − r di , f ξ,si r di − p ξ,fi − p ri (cid:41) , p ri > . (3.5)Consider the following cases.1. ¯ α i + ¯ α i ≥ . In this case λ i ( α i ) = λ i ( α i ) = 1 and hence λ i ( α i , α i ) = 1 for α i = 1 − α i , α i ∈ [1 − ¯ α i , ¯ α i ] .Thus, A i = [1 − ¯ α i , ¯ α i ] .2. ¯ α i + ¯ α i < . In this case λ i ( α i , α i ) < for all α i , α i ≥ , α i + α i = 1 , therefore, the flow from theon-ramp is inevitably constrained.(a) If p ri = 0 and f ξ,si ≤ f ξ,di − , ξ = 1 , , then ¯ α i = ¯ α i = 0 and λ ( α i , − α i ) = 0 for all α i ∈ [0 , ,therefore A i = [0 , .(b) Assume at least one of the inequalities p ri > , f ξ,si > f ξ,di − , ξ = 1 , , holds.If λ i (1 − ¯ α i ) ≥ lim α → ¯ α i +0 λ i ( α ) , then A i = { − ¯ α i } .If λ i (1 − ¯ α i ) ≥ lim α → ¯ α i +0 λ i ( α ) , then A i = { ¯ α i } .Otherwise there exists exactly one solution α , ∗ i of the equation λ i ( α i ) = λ i (1 − α i ) , and α , ∗ i ∈ (¯ α i , − ¯ α i ) , since λ ξi ( α ξi ) is strictly decreasing for α ξi ∈ (¯ α ξi , , ξ = 1 , . In this case A i = { α , ∗ i } .So the set A i is either a segment or a point. The corresponding flow r i = r i ( α i ) = λ i ( α i , − α i ) α i r di (3.6)belongs to a segment [ r , min i , r , max i ] , where r , min i = r i (min A i ) , r , max i = r i (max A i ) . In order not to redistribute on-ramp flows when the freeway is in a free flow state, introduce the followingcorrection. If ¯ α i + ¯ α i > and ¯ α i > l / ( l + l ) (recall that l and l are the number of toll and generalpurpose lanes), let ¯ α i = max { − ¯ α i , l / ( l + l ) } and recalculate r , max i = ¯ α i r di .Compute the maximum maintainable level of densities for the toll lane n , ∗ . A maintainable level of densitiesis a vector n , ∗ corresponding to a free-flow state ( β fi v i n , ∗ i ≤ F ,di , i = 1 , . . . , K ), such that if n ( t ) ≤ n , ∗ and r ( t ) = 0 , then n ( t + 1) ≤ n , ∗ . The maximum maintainable level of densities n , ∗ is defined asfollows. First, compute flows f , ∗ i , i = K + 1 , . . . , as follows: f , ∗ K +1 = F K +1 , f , ∗ i = min { F ,di , f , ∗ i +1 /β fi +1 } , i = K, . . . , . After that, compute densities n , ∗ i = f , ∗ i / ( β fi v i ) .13ext, define the maximum free-flow equilibrium n ,e . Let M be a number of on-ramps, i < i < · · · < i M be the indices of links with on-ramps, i = 1 by assumption. Let i M +1 = K + 2 . Define f ,ei m = f , ∗ i m , f ,ei = β fi f ,ei − , i = i m + 1 , . . . , i m +1 − , n ,ei = f ,ei / ( β fi v i ) . Is is easily shown that f ,e ≤ f , ∗ and n ,e is anequilibrium corresponding to the on-ramp flow r ,e : r ,ei m = f ,ei m /β fi m − f ,ei m − ≥ . Define N ,e ( i m , i m +1 ) = i m +1 − (cid:88) i = i m n ,ei . (3.7)The controller keeps the total number of vehicles between entrances m and m + 1 less than N ,e ( i m , i m +1 ) ,and the incoming flow f i m − + r i m less than equilibrium flow f ,ei m /β fi m , if the minimum-queue-growth-rateconstraint permits that.If we knew flows r i m , deriving split ratios α i m , α i m would be easy: if r di m = 0 or λ ∗ i m = 0 , then the on-rampflow r i m = 0 regardless of split ratios, otherwise α i m = r i m /r i m , α i m = 1 − α i m , where r i m = λ ∗ i m r di m . So, weneed to determine on-ramp flows r i m ∈ [ r , min i m , r , max i m ] .At every time step t , compute the total number of vehicles between every pair of adjacent on-ramps n ( t, i m , i m +1 ) = (cid:80) i m +1 − i = i m n i ( t ) , estimate flows s i ( t ) , i = 1 , . . . , K , and f i m − ( t ) , m = 1 , . . . , M + 1 , for r i m ( t ) = r , min i m ( t ) . By ˜ s i ( t ) and ˜ f i m − ( t ) denote the estimates.If f ,di m − ( t ) + r , max i m ( t ) > min { f ,si m ( t ) , f ,ei m /β fi m } , then to possibly avoid constraining the flow from the up-stream link f i m − and exceeding the equilibrium inflow f ,ei m /β fi m , set r , max i m ( t ) = max { r , min i m ( t ) , min { f ,si m ( t ) , f ,ei m /β fi m } − f ,di m − ( t ) } .
1. Let ∆ n = 0 , m = M , γ M = 1 .Here ∆ n is an excess of vehicles in the toll lane, m is the number of current entrance, from M downto 1, ≤ γ m ≤ is a reduction coefficient. If γ m is strictly less than 1, it means that some downstreamentrance is unable to reduce congestion in the toll lane on its own. Therefore, we reduce the targetnumber of vehicles between entrances m and m +1 by γ m in order to reduce the flow to the downstreamlinks.2. Compute ∆ n ( t, i m , i m +1 ; γ i m ) = n ( t, i m , i m +1 ) − γ m N ,e ( i m , i m +1 ) , ∆ n ← ∆ n + ∆ n ( t, i m , i m +1 ; γ m ) + ˜ f i m − ( t ) − ˜ f i m +1 − ( t ) − i m +1 − (cid:88) i = i m ˜ s i ( t ) . We estimate the excess number of vehicles in the toll lane in links i m , . . . , i m +1 − .3. Determine r i m ( t ) = max { r , min i m ( t ) , min { r , max i m ( t ) , − ∆ n }} .The on-ramp flow r i m should reduce the excess number of vehicles, if possible.4. If m = 1 , stop: all on-ramp flows for the current time step have been determined.5. Recalculate ∆ n ← max { , ∆ n + r i m ( t ) } .6. Compute the reduction coefficient for the upstream entrance: γ m − = min { , ( f ,si m ( t ) − r i m ( t )) /f ,ei m − } .7. Set m ← m − and go to step 2. 14 .3 Theorems We now prove that the presented controller keeps the toll lane in a free-flow state, if the minimum-queue-growth-rate condition allows it. Moreover, even if the toll lane is initially congested, it becomes almostuncongested in finite time.The toll lane is considered as an independent system, and the upper index ξ = 1 is omitted for simplicity. Theorem 3.1.
Suppose n ( t ) ≤ n e and the inequality r i m ( t ) + f di m − ( t ) ≤ f ei m /β fi m holds for all entrances i m .Then n ( t + 1) ≤ n e .Proof. Note that for m > , n i m − ( t ) ≤ n ei m − and f di m − ( t ) ≤ f ei m − ≤ f ei m /β fi m , therefore the inequality r i m ( t ) + f di m − ( t ) ≤ f ei m /β fi m holds at least for r i m ( t ) = 0 .Clearly, f i − ( t ) = f di − ( t ) for i = 2 , . . . , K + 1 , since f si ( t ) = F si ≥ f ei /β fi ≥ f ei − ≥ f di − ( t ) if there isno on-ramp in link i , and f si ( t ) − r i ( t ) = F si − r i ( t ) ≥ f ei /β fi − r i ( t ) ≥ f di − ( t ) , otherwise. Therefore, f i ( t ) + s i ( t ) = f di ( t ) /β fi = v i n i ( t ) for all i = 1 , . . . , K + 1 . Additionally, f i − ( t ) ≤ f ei − ≤ f ei /β fi if there isno on-ramp in link i , and r i ( t ) + f i − ( t ) ≤ f ei /β fi , otherwise. Thus, if n ( t ) ≤ n e , then n i ( t + 1) = n i ( t ) + f i − ( t ) + r i ( t ) − f i ( t ) − s i ( t ) ≤ n i ( t )(1 − v i ) + f ei /β fi ≤ n ei (1 − v i ) + f ei /β fi = n ei for all i .This theorem means that if the state of the toll lane is in a “target zone” n ( t ) ≤ n e at time t , then it remainsthere at time t + 1 if the on-ramp flow demand r d ( t ) isn’t too high and the general purpose lane isn’t toocongested near the entrances. Lemma 3.1.
Suppose n − ( t ) ≤ n + ( t ) and r − ( t ) ≤ r + ( t ) . Then n − ( t + 1) ≤ n + ( t + 1) . A similar lemma is proved in [14].
Theorem 3.2.
Suppose f e is an equilibrium flow, the inequalities f i m − ( t ) + r i m ( t ) ≤ f ei m /β fi m , f si m ( t ) ≥ f ei m − (3.8) hold for all links with on-ramps i m for all t , and i m +1 − (cid:88) i = i m n i ( t ) ≤ i m +1 − (cid:88) i = i m n ei , m = 1 , . . . , M, (3.9) for all t . Here n ei = f ei / ( β fi v i ) .Then for any ε > there exists a time step T = T ( ε ) such that n i ( t ) ≤ n ei + ε , i = 1 , . . . , K + 1 , for t ≥ T ( ε ) .Proof of Theorem 3.2 (Outline). First transform the system to get rid of the off-ramps. Multiply the “inflowcapacity” of the mainline link F si , the on-ramp capacity R i , the maximum number of vehicles N i , the incomingflow d i ( t ) , the number of vehicles in mainline link n i ( t ) and on the on-ramp q i ( t ) by µ i = (cid:81) i − j =1 ( β fj ) − .15ultiply the “outflow capacity” of the i th link F di by µ i +1 . For the transformed system denoted by ˆ thefollowing equations hold: ˆ f di − ( t ) = min { v i − ˆ n i − ( t ) , ˆ F di − } = µ i f di − ( t ) , ˆ f si ( t ) = min { w i ( ˆ N i − ˆ n i ( t )) , ˆ F si } = µ i f si ( t ) , ˆ r di ( t ) = min { v ri ˆ q i ( t ) , ˆ R i } = µ i r di ( t ) . Therefore, ˆ f i − ( t ) = µ i f i − ( t ) , ˆ r i ( t ) = µ i r i ( t ) , ˆ f i ( t ) = µ i +1 f i ( t ) = µ i f i ( t ) /β fi , thus ˆ n i ( t + 1) = µ i n i ( t + 1) , ˆ q i ( t + 1) = µ i q i ( t + 1) . That is, the transformed system is equivalent to the initial system. Multiply theequilibrium flow f ei by µ i +1 . The inequalities f i m − ( t ) + r i m ( t ) ≤ f ei m /β fi m , f si m ( t ) ≥ f ei m − transform into ˆ f i m − ( t ) + ˆ r i m ( t ) ≤ ˆ f ei m , ˆ f si m ( t ) ≥ ˆ f ei m − , while the inequality (cid:80) i m +1 − i = i m n i ( t ) ≤ (cid:80) i m +1 − i = i m n ei transforms into (cid:80) i m +1 − i = i m µ − i ˆ n i ( t ) ≤ (cid:80) i m +1 − i = i m µ − i ˆ n ei . Hence, it suffices to show that the theorem holds for the freeway withno off-ramps, but the inequality (cid:80) i m +1 − i = i m n i ( t ) ≤ (cid:80) i m +1 − i = i m n ei should be replaced with i m +1 − (cid:88) i = i m α i n i ( t ) ≤ i m +1 − (cid:88) i = i m α i n ei , where α i = µ − i = (cid:81) i − j =1 β fj . Note that α ≥ α ≥ · · · ≥ α K +1 > .Consider the links between two adjacent on-ramps. Consider the clusters of links with n i ( t ) > n ei . It is easyto show that the downstream boundary of a cluster can only move downstream, one cluster cannot splitinto two or more, and new clusters cannot be created. So, starting from some point in time, the number ofclusters remains constant and their downstream boundaries are fixed.Denote h m ( t ) = i m +1 − (cid:88) i = i m max { , n i ( t ) − n ei } . We shall prove that h m ( t ) → as t → ∞ for all m .If there are no clusters between on-ramps m and m + 1 , then, clearly, h m = 0 .It is easily shown that h m ( t ) is decreasing and nonnegative, thus there exists lim t →∞ h m ( t ) ≥ . Sup-pose lim t →∞ h m ( t ) = δ > . Let i ∗ m be the downstream boundary of the most upstream cluster be-tween the on-ramps m and m + 1 . Using lemma 3.1, it can be demonstrated that lim inf t →∞ n i ( t ) ≥ n ei , i = i ∗ m + 1 , . . . , i m +1 − . Consider only those time steps where the number of clusters and their downstreamboundaries are stabilised (that is, do not change anymore) and n ei − η i ≤ n i ( t ) for some small η i > , say, η i = δα i m +1 / (2( i m +1 − i m ) α i ) , i = i ∗ m + 1 , . . . , i m +1 − . We now demonstrate that the most upstream clusterdisappears in finite time. Since h m ( t ) ≥ δ , ≥ i m − (cid:88) i = i m α i ( n i ( t ) − n ei ) == (cid:88) in ei α i ( n i ( t ) − n ei ) + (cid:88) i>i ∗ m ,n i ( t ) ≤ n ei α i ( n i ( t ) − n ei ) ≥≥ α i m (cid:88) i . Therefore, the upstream cluster exists at most (cid:100) h m ( t )( i ∗ m − i m ) /µ (cid:101) time steps. This contradicts our assump-tion that the number of clusters is stabilized. Thus h m ( t ) → as t → , which concludes the proof. Remark 3.1.
In theorem 3.2 it would be sufficient to require that the inequality 3.9 i m +1 − (cid:88) i = i m n i ( t ) ≤ i m +1 − (cid:88) i = i m n ei holds only asymptotically, that is, lim sup t → + ∞ i m +1 − (cid:88) i = i m n i ( t ) ≤ i m +1 − (cid:88) i = i m n ei . The control algorithm attemps to satisfy the inequalities of theorem 3.2, (cid:80) i m +1 − i = i m n i ( t + 1) ≤ (cid:80) i m +1 − i = i m n ei and f i m − ( t ) + r i m ( t ) ≤ f ei m /β fi m , at step 3. If that is impossible, a reduction coefficient γ m − is introducedat step 6 to reduce the flow from upstream links. If the requirements if the theorem can be met, then the tolllane becomes almost uncongested in finite time, that is, for arbitrarily small ε > the density vector n ( t ) is component-wise less than n ,e + ε , starting from some time step t , regardless of the initial state. Remark 3.2.
The suggested controller calculates only split ratios for on-ramp flow r . However, this schemecan be modified so, that mainline travelers are also allowed to change lanes, paying a toll if they changefrom the general purpose to the toll lane. Since the controller keeps the toll lane less congested than generalpurpose lane, only the travelers from the general purpose lane may want to change lanes. In this model thesplit ratios of on-ramp flow r and the split ratios of flow f from general purpose lane both depend on theprice. The price for entering the toll lane should be the same for those drivers who come from on-ramps andfor those coming from the general purpose lane. For that reason the split ratios for on-ramp flow r i and thesplit ratios for the flow from the general purpose lane f i − are “paired”, since they are both defined by thesame price. The actual flows are defined by the generic node model presented in Subsection 2.1.Some constraints, such as nonnegativity and boundedness, can be imposed on tolls. A toll is admissible if itsatisfies all constraints. To narrow the set of split ratios corresponding to admissible tolls we could introduceadditional requirements, for example, the minimization of queue length growth or the maximization of totalflow through the node. Then an algorithm similar to the one presented above can be applied at each timestep. Consider a freeway with one on-ramp (in the first link) and no exits, except for the last link, ( K + 1) . Alllinks have equal capacities, free flow speeds and congestion speeds, but the last link, K + 1 , is a bottleneckwith a lower capacity: F ξK +1 < F ξK = F ξK − = · · · = F ξ , ξ = 1 , . The initial state of the freeway is anequilibrium. 17 n n n n · · · n K n K n K +1 n K +1 Figure 8: Freeway with one entrance and one exit.
Scenario 1
Both the toll lane and the general-purpose lanes are uncongested from link 1 to some link i and congested from link ( i + 1) to link ( K + 1) . The incoming flow equals the bottleneck capacity: d = F K +1 + F K +1 . The on-ramp flow r is redistributed in order to reduce congestion in the toll lane, but onlyuntil there are uncongested links in the general purpose lane. Scenario 1a : The number of uncongested links is too small for the toll lane to become fully uncongested: i < ( K + 1) / . Figure 9 illustrates this case, the densities (in vehicles per mile) are color-coded.Figure 9: Reducing congestion in the toll lane. Scenario 1a. Densities (in vehicles per mile). Scenario 1b : The number of uncongested links is sufficiently large, i > ( K + 1) / , and the toll lane becomesfully uncongested in finite time (Figure 10). Scenario 2
Initially, the whole freeway is uncongested, the incoming flow d equals the bottleneck ca-pacity F K +1 + F K +1 . At some point in time ( t = 5 minutes), the incoming flow d exceeds the bottleneckcapacity and both the general purpose and the toll lanes become congested, but the congestion grows fasterin the general purpose lane. The toll lane still becomes partially congested, because the general purposelane alone cannot accommodate the whole excess flow: d > F K +1 + F . Then (at time t = 30 minutes) the18igure 10: Reducing congestion in the toll lane. Scenario 1b. Densities (in vehicles per mile).19ncoming flow d drops to the bottleneck capacity and the congestion is transferred from the toll lane to thegeneral purpose lane. Figure 11 illustrates this scenario.Figure 11: Temporarily infeasible incoming flow (scenario 2). Densities.Now consider a freeway with two entrances (at the beginning and in the middle), and two exits (in themiddle and at the end), see Figure 12. The last link, K + 1 , is a bottleneck. The priorities p ξ,fi − , ξ = 1 , and p ri are equal to capacities: p ξ,fi − = β fi − F ξi − , p ri = R i . q n n · · · n i − n i − q i n i n i · · · n K +1 n K +1 Figure 12: Freeway with two entrances and two exits.
Scenario 3
Initially, the toll lane and the general purpose lane are in the same partially congested equi-librium state. The incoming flow is feasible, but not strictly feasible. Flows from both on-ramps (at thebeginning and in the middle) are redistributed, and the toll lane decongests shifting the extra vehicles to thegeneral purpose lane. 20igure 13: Reducing congestion in the toll lane of a freeway with two entrances (scenario 3). Densities.21
Tolls as Actuators
Now that at a given entrance i , the split ratio controller computed the desired values α i and α i , our goal isto enforce them. It is done by setting the price for vehicles entering the toll lane. We consider two ways ofsetting the price: (1) using the known Value of Time distribution; and (2) setting up an auction. These aredescribed next. The Value of Time (VoT) is the marginal rate of substitution of travel time for money in a travelers’ indirectutility function. In essence, this makes it the amount that a traveler would be willing to pay in order to savetime, or the amount he would accept as a compensation for the lost time. The VoT varies considerably fromtraveler to traveler and depends upon the purpose of the journey. The estimation of the VoT was studied in[28, 30, 2, 34].We assume that the VoT distribution is known, ν ( π ) , where π represents the price per time unit. For a givenentrance i , we find the desired price of the time unit, π (cid:63) from the equation (cid:90) π (cid:63) ν ( π ) dπ = α i . (4.1)Figure 14 illustrates the VoT pricing mechanism.Figure 14: Value of Time distribution — finding price per time unit π .To continuously estimate the VoT distribution, we need to know the difference in travel time between thegeneral purpose and the toll lanes, τ ; the number of vehicles in the general purpose lane, n ; the price setfor the toll lane, π (cid:63) τ ; and the amount of money collected, T , every certain time period. This will give usenough information to infer the portion of travelers, whose VoT is higher or equal than π (cid:63) : n = Tπ (cid:63) τ , and (cid:90) ∞ π (cid:63) ν ( π ) dπ = n n + n . (4.2) Time period should be in the order of minutes. π (cid:63) varies, we could eventually, estimate ν ( π (cid:63) ) . It is rather crude approach, since in differenttime of day travelers’ VoT is different, but it conveys the idea.In the case of HOT lanes, where besides paying customers there may be high occupancy vehicles, we wouldneed to know additionally the number of vehicles in the HOT lane, n . Then, equation (4.2) is to be slightlymodified: (cid:90) ∞ π (cid:63) ν ( π ) dπ = Tπ (cid:63) τ ( n + n ) . (4.3)The main shortcoming of the VoT-based price setting is the difficulty of estimating the VoT distribution. Ifthe estimate is too rough, we may end up with underutilized, if the price is too high, or overly congested,if the price is too low, toll lane, which will hinder the performance of the overall system reducing the totaloutput flow.To reduce the inaccuracy of the VoT based toll calculation, the continuing calibration of the willingness topay must be performed. Such calibration would use a discrete choice model. Logit model based calibrationis described in [23, 22]. The alternative method of setting the price for a toll lane is an auction. The auction based scheme applied toa cordon area congestion pricing was described in [32]. Here we describe the auction mechanism for managedtoll lane based on the idea of [13]. At time t , at the entrance i , H = r di ( t ) travelers make bids b , . . . , b H .Without loss of generality, we assume b ≥ · · · ≥ b H . Let h (cid:63) = round ( α i H ) . Vehicles with bids b , . . . , b h (cid:63) will be let into the toll lane, and each of them will pay b h (cid:63) generating b h (cid:63) h (cid:63) total revenue for the entrance i at time t .The main advantage of the auction approach is its deterministic outcome: for each decision point i , we canachieve the desired split ratio coefficients α i , α i exactly.The suggested auction mechanism allows variations. For instance, parameter h (cid:63) may be chosen maximizingthe revenue without compromising the quality of service: h (cid:63) = arg max h ≤ α i H hb h . At the time of this writing, the deployment of an auction as a mechanism for setting tolls presents certaintechnical difficulties. In the coming years, however, the active development of cooperative taxis, connectedand autonomous vehicles and wireless communications will enable passengers to determine the amount theywould be ready to spend reducing their travel time at the start or during the trip. Thus, the auctionmechanism for buying a place in the toll lane guaranteeing the required time saving may become realisticand practical.
In 2013-14 Caltrans conducted a Corridor System Management Plan (CSMP) study for the Interstate 680corridor in Contra Costa County [4]. Certain improvements were considered, including ramp metering, California Department of Transportation Modeling of improvementscenarios was done at UC Berkeley PATH using simulation Tools for Operational Planning (TOPL) [33].In this Section we present the simulation for a 10-mile segment of I-680 South corridor extending fromMartinez through Concord and Pleasant Hill to Walnut Creek, shown in Figure 15. It goes from postmile56 to postmile 46. An HOV lane spans most of this segment. The start and end of the HOV lane are shownon the map. Figure 15: Map of the simulated I-680S corridor.The data collection effort of the CSMP project revealed a severe bottleneck during the AM peak hourslocated near the end of the HOV lane. This bottleneck generates large congestion in the GP lane, while theHOV lane stays in free flow and is underutilized.A macroscopic CTM-based simulation model for the I-680 corridor was built and calibrated using TOPLand measurement data collected in the CSMP study. Three scenarios were simulated:1. base case that reproduces existing conditions; HOV lane is a lane for High Occupancy Vehicles. Typical minimum vehicle occupancy level for HOV lanes in the U.S. is 2(2+HOV) or sometimes 3 (3+HOV). Presently, I-680 corridor has 2+HOV lanes. HOT stands for High Occupancy or Tolled.HOT lane is free for HOVs, others must pay a toll.
HOV lane as GP that shows how the corridor would perform if the HOV lane were treated as an extraGP lane; and3.
HOV lane as HOT , where the HOT lane was modeled instead of the HOV lane using the controllerdescribed in this paper.Whereas the first scenario represents the reality, scenarios 2 and 3 are hypothetical and were explored aspart of the planning exercise. The input demand was assumed the same in all three scenarios. Morning peakhours were simulated: from 5 to 10 AM. This is when travelers currently experience large delays at thatsegment of I-680S.Speed contour maps resulting from the three simulations are shown in Figure 16. Contours on the left showthe speed dynamics in the GP lane, and contours on the right correspond to the special lane, which inscenario 1 is HOV, in scenario 2 is GP and in scenario 3 is HOT. Black stripes on the left and on the rightof the special lane contours correspond to locations, where there is no special lane. Tables 1 and 2 containVehicle Miles Traveled (VMT) and delay values from the simulated scenarios.25ase GP lane VMT Special lane VMT Total VMTBase case with existing HOV lane 22502 2742 25244Special lane is treated as GP 21087 4157 25244HOV lane is converted to HOT 21925 3319 25244Table 1: Vehicle Miles Traveled in the period from 5 to 10 AM in the three simulated cases.Case GP lane delay Special lane delay Total delayBase case with existing HOV lane 250 0 277Special lane is treated as GP 77 14 91HOV lane is converted to HOT 75 2 77Table 2: Vehicle-hours of delay in the period from 5 to 10 AM in the three simulated cases. Total delay inthe last column includes delay from on-ramp queues, which exists only in the base case.In the base case scenario we observe large congestion in the GP lane, while the HOV lane has no congestionat all. Opening the HOV lane to everyone (scenario 2) helps a lot, as is evident from the delay table(Table 2), but congestion is not eliminated completely, and now both GP and HOV lanes have the samecongestion pattern, so travelers have no mode choice. Not surprisingly, the HOT lane scenario shows thebest performance of the three both in congestion mitigation and keeping the special lane as free as possible.One interesting detail to notice is the slight reduction of the GP lane delay in scenario 3 compared to scenario2. That happens despite the fact that the GP lane has more vehicles in scenario 3 than in scenario 2 (Table1). This happens because the HOT controller try to keep the flow in the GP lane as close to capacity aspossible, and thus more vehicles travel with higher speed.
In this paper we described the toll lane control algorithm for freeways in the context of the link-node CTMwith the modified node model. Its design was presented in two stages: (1) the supply control computing thedesired split ratios for the incoming flows with the goal of keeping the toll lane in free flow as long as and asmuch as possible, yet fully utilized; and (2) the demand control, setting the price so that the flows betweenthe toll and the general purpose lanes were distributed according to the computed split ratios.We described the equilibrium of the traffic model and pointed out that activating the toll lane is meaningfulonly in the partially congested equilibrium, when the general purpose lane has enough storage space toaccommodate extra vehicles from the toll lane without reducing the total output flow of the system. In freeflow state with feasible demand there is obviously no need for the toll lane, as it does not provide the benefitof the shorter travel time. When the freeway is in the fully congested equilibrium, the deployment of a tolllane is not recommended either, as it can only harm the overall system performance by oversaturating thegeneral purpose lane, creating queues at entrances and reducing the total output flow of the system.The split ratio control algorithm presented in the paper behaves as follows: • in free flow state with feasible demand it is essentially non-active letting the mainline operate as onepiece; • in the case of infeasible demand, the excess flow is directed into the general purpose lane, letting it tocongest first; • once the general purpose lane is fully congested, the algorithm lets the toll lane to congest;26 if it comes to the point when the toll lane is fully congested, the algorithm is deactivated, as there isnothing it can improve at this point, so the mainline returns back to one piece; • once the demand drops again, and the freeway starts to decongest, the algorithm first brings the tolllane into the free flow state, and then the general purpose lane.All these manipulations with flows are done through split ratios at the end nodes of the entrance links. Thecontrol algorithm computes these split ratios.The split ratio control algorithm can be extended to the case of HOT lanes. For that, however, one has tointroduce and deal with multiple vehicle types, as described in [18]. In this case, two vehicle types will beneeded: High Occupancy Vehicles (HOVs) and Single Occupancy Vehicles (SOVs). The HOV flow won’t becontrolled, but will be governed by the algorithm of finding the path of the least resistance, presented in[18], whereas the SOVs will be subject to the toll control, and the available space in the general purpose andthe HOT lanes will be computed accounting for both vehicle types.Finally, we discussed two mechanisms for price setting — the VoT distribution and the auction. The formeris commonly used, but provides only rough estimates of the actual traveler behavior, resulting mostly in theunderutilization or oversaturation of the toll lane. The latter mechanism is hardly deployable at current timedue to technical difficulties, but in the foreseeable future with the emerging technologies, especially, those ofautonomous vehicles and ride shares, it looks promising. Acknowledgement
This work is supported by the California Department of Transportation (Caltrans) under the ConnectedCorridors program and by the Russian Foundation for Basic Research (grants 13-01-90419 and 12-01-00261-a).
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