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Managing Congestion and Emissions in Road Networks with Tolls and Rebates Hai YangChair ProfessorDepartment of Civil and Environmental EngineeringThe Hong Kong University of Science and TechnologyClear Water Bay, Kowloon, Hong Kong, PR ChinaIntroductionTraffic congestion and emissions are two main types of travel disutilityCongestionEmissionsIdlingAccelerationBrakingIntroductionThe two objectives, alleviating congestion and reducing emissions, cannot always be achieved at the same time, namely, sometimes conflict with each other.

Such inconsistency naturally leads to a bi-criteria or bi-objective optimization problem.MinimizingCongestionMinimizingEmissionsTraffic ManagementConflictOutline1. Notations and preliminaries2. Pareto system optimum3. Decentralization of Pareto system optimum with increasing emission functions4. Decentralization of Pareto system optimum with non-monotonic emission functions5. Decentralization of Pareto system optimum with revenue-neutral toll and subsidy6. ConclusionsNotations and preliminariesA road network : the set of origin-destination (O-D) pairs : the set of all routes between O-D pair : the flow on link : the flow on route : if route uses link , otherwise : the travel demand between O-D pair , which is given and fixed

Notations and preliminariesThe set of feasible path flow pattern:

The set of feasible link flow patterns:

Separable link travel time functions, , which are assumed to be convex and strictly increasing for all

Notations and preliminariesSeparable emission functions . Example:

which are not always increasing, like:

Notations and preliminariesHomogeneous travelers with an identical value of time (VOT) denoted by : the travel cost of link ; in the absence of any toll scheme, and in the presence of a toll scheme , where is the toll on linkThe user equilibrium conditions:

where represents the minimum path travel cost between OD pair

Pareto system optimumThe system optimum (SO) problem:

The minimum emission (ME) problem:

The bi-objective optimization problem of minimizing total network travel time and total network emissions:

Pareto system optimumLet denote the Pareto-efficient link flow pattern, denote the set of Pareto-efficient solution to the bi-objective optimization problem

Feasible region of System optimumMinimum emissionPareto frontier

Increasing emission functionsThe central problem of interest here is:For any given Pareto-efficient link flow pattern, whether does a toll scheme exist to decentralize it as user equilibrium?Construct the following linear programming (LP) problem:

Increasing emission functionsBy its duality property, we can demonstrate: Any Pareto-efficient flow pattern can be decentralized as user equilibrium by a nonnegative toll scheme when the emission functions are increasing.

The theorem can be extended to:1) Multiclass users in terms of different value of time2) Multiple vehicle types3) Tradable credit scheme

Non-monotonic emission functionsIt is empirically measured or observed that the average vehicle emission on a link decreases at first and then increases as the traffic flow goes up.

Within its realistic range of speed on most arterial roads or freeways with higher speed limits, non-monotonic emission functions are encountered and should be taken into consideration.Non-monotonic emission functionsAn example:

OD demands:Link 2Link 1Link 3231Link 4Emission functions:

Non-monotonic emission functionsThe minimum emission link flow pattern

The minimum emission path flow pattern

Non-monotonic emission functionsUE conditions:

To decentralize the minimum emission flow pattern as user equilibrium, we have to resort to negative link tolls.

Non-monotonic emission functionsAlthough what we concern about is still the Pareto-efficient flow patterns, we should rather consider general feasible link flow patternsIf we adopt both toll and subsidy (negative toll), for a general feasible link flow pattern , set the tolls as , the link travel cost will be . The user equilibrium conditions are definitely satisfiedAny feasible link flow pattern can be decentralized as user equilibrium by a toll and subsidy scheme.Let denote the set of all effective toll and subsidy schemes

Revenue-neutral toll and subsidyRevenue-neutral: the total revenue collected is equal to the total subsidies paid

We will examine the existence of revenue-neutral toll and subsidy scheme in two types of network topology: acyclic networks and cyclic networks

Acyclic networksAs the network is a directed acyclic graph, there exists a topological ordering, an ordering of the nodes such that the starting endpoint of every edge occurs earlier in the ordering than the ending endpoint of the edge : the order of nodeDefine the topological length of link with the starting endpoint and ending endpoint as

: the topological length of route , which is independent of the nodes other than OD

Acyclic networks can be decentralized as user equilibrium by inThe total revenue is . If , we try to amend the toll scheme to make the total revenue equal to zero. Let denote the parameter for adjusting the tolls, given by

Acyclic networksRevise the toll on each link to be . We can show that can be decentralized by toll and subsidy scheme , and is revenue-neutral.In an acyclic road network, any feasible link flow pattern can be decentralized as user equilibrium by a revenue-neutral toll and subsidy schemeThe theorem applies for the Pareto system optimum link flow pattern directly irrespective of the monotonicity of link emission functions

Cyclic networksStill consider an amendment to . Let denote the parameter for adjusting the tolls, which is given by

exists only whenSet the new toll on each link as The new travel cost of path becomes

Cyclic networks is sufficient to ensure the non-existence of negative cycles

We can show that can be decentralized by toll and subsidy scheme , and is revenue-neutral

Conversely, if can be decentralized as user equilibrium by a revenue-neutral toll and subsidy scheme , then

Cyclic networksIn a road network that may contain cycles, if and only if there exists such that , then can be decentralized as user equilibrium by a revenue-neutral toll and subsidy scheme

With increasing emission functions, for a given Pareto-efficient link flow pattern, we can always find a nonnegative toll scheme such that . Hence we are always able to find a feasible revenue-neutral toll and subsidy scheme

Cyclic networksWhen link emission functions are not monotonic. a revenue-neutral toll and subsidy scheme may not exist to decentralize a Pareto system optimum link flow pattern. Consider the following example:Emission functions:

OD demands:

Link 3Link 1Link 5Link 2132Link 4Link 6

Cyclic networksThe minimum emission link flow pattern

One of the minimum emission path flow pattern

Cyclic networksUE conditions:

Cyclic networksThe minimum emission path flow pattern cannot be decentralized as user equilibrium by a revenue-neutral toll and subsidy schemeWe can further prove that any feasible path flow pattern that can give rise to the unique link flow pattern cannot be decentralized into user equilibrium by a revenue-neutral toll and subsidy schemeA given target link flow pattern, including the Pareto system optimum link flow patterns such as the minimum emission link flow pattern, cannot always be decentralized into user equilibrium by a revenue-neutral toll and subsidy scheme in a network with physical cycles

Comparison of speed limits and road pricing for minimum traffic emissions Speed Limits VS Negative Link Tolls (Subsidy)

PolicyOptimal schemeLink flowTotal timeTotal emissionNo policy interventionN/A(1.0,1.0)2900Minimize emission through toll and subsidy*(3.32, 3.32)8.178341.071Minimize emission through speed limit(1.0,1.0)2.5100

* If only non-negative tolls are allowed, then no emission reduction can be achieved.ConclusionsWe studied the bi-objective optimization problem for simultaneous minimization of congestion and emissions on a road network. We investigated the existence of various pricing cum subsidy scheme to decentralize a given Pareto system optimum link flow patternWe proved that any given Pareto-efficient link flow pattern can be decentralized into user equilibrium by a nonnegative toll scheme if the emission functions are increasing. If, however, the link emission functions are not increasing, negative link toll charges or subsidy may be required in order to decentralize a given Pareto system optimum link flow patternConclusionsWe then investigated the existence of revenue-neutral toll and subsidy scheme for decentralizing a given Pareto system optimum. We found that, for a network without physical cycle, such a revenue-neutral toll and subsidy scheme always exists regardless of the monotonicity of the link emission functions.If, however, for a network with physical cycles, we found that such a revenue-neutral toll and subsidy scheme may not exist.We provided a necessary and sufficient condition for existence of such a desirable revenue-neutral toll and subsidy schemes for a network with cyclesQ&A