- Email: [email protected]
Internalizing Congestion and Emissions Externality on Road Networks with Tradable Credits
Available online at www.sciencedirect.com
ScienceDirect Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
The 9th International Conference on Traffic & Transportation Studies (ICTTS’2014)
Internalizing Congestion and Emissions Externality on Road Networks with Tradable Credits Ge Gao, Huijun Sun* MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, No.3 Shangyuancun, Haidian District, Beijing 100044, P.R. China
Abstract Congestion and emissions are two problems in traffic. In general, they have conflict with each other. Tradable credits scheme is proved an effective method in Environmental Science. Now more and more scholars introduce it into traffic. Because it possesses the advantages that other methods do not have in alleviating congestion and reducing emissions. In this paper we indicate there always exists a tradable credits scheme that induces a traffic flow distribution with minimum emissions. We proved a given target feasible link flow pattern can be decentralized into user equilibrium by a tradable credits scheme, and sought tradable credits schemes for Pareto-efficient control and management of both traffic congestion and emissions on road networks. There are two models which could describe the Pareto-efficient problem between alleviating congestion and reducing emissions, i.e., bi-level congestion pricing model and weighting problem model. When the emission function is an increasing function, any Pareto-efficient flow pattern can be supported as an user equilibrium by the tradable credit scheme. If the emission function is a non-monotonic function, we divided the network into two cases: loops-free network and loops network which satisfies Gallager’s sense. In a loop-free road network, any feasible link flow pattern can be decentralized into user equilibrium by a tradable scheme. In loops network, Pareto-efficient solution may not be decentralized to user equilibrium by tradable credits scheme. However, if the loops network doesn’t satisfy Gallager’s sense, any feasible Pareto-efficient link flow pattern can be decentralized into user equilibrium by a nonnegative toll scheme/tradable credits scheme. On the contrary, it wouldn’t. © 2014 2014 The Elsevier Ltd. This is an open access Ltd. article under the CC BY-NC-ND license © Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of Beijing Jiaotong University (BJU), Systems Engineering Society of China (SESC). Peer-review under responsibility of Beijing Jiaotong University(BJU), Systems Engineering Society of China (SESC). Keyword: Tradable credits; Congestion and emissions; Pareto-efficient; Loops-free; User equilibrium
* Corresponding author. Tel.: +86-(0)10-5168-4265. E-mail address: [email protected].
1877-0428 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of Beijing Jiaotong University(BJU), Systems Engineering Society of China (SESC). doi:10.1016/j.sbspro.2014.07.198
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
215
1. Introduction Traffic congestion and emissions are two global problems we have to face in big cities. Due to congestion, more time is wasted on road; Due to emissions, more and more people have to endure severe air pollution. There is a closed connection between traffic congestion and emissions. Traffic emissions in congested conditions are substantially higher than under conditions of freely traffic flow. However, in general, alleviating congestion conflicts with reducing emissions (Nagurney, 2000; Yin and Lawphongpanich, 2006). In order to alleviate traffic congestion and control emissions, and achieve the sustainability of transportation, for several decades, road pricing has been considered as an effective management instrument. Pigou (1920) firstly proposed the idea of road pricing. Similarly, a road user should pay a charge corresponding to both his own emissions and the increased emissions brought to other users (Jahansson, 1997). Afterward, immense amounts of research on road pricing have been further developed. For recent reviews to congestion pricing, we can see, Yang and Huang (2005), Tsekeris and Voss (2009). However, road pricing has its limitations. The most obvious defect is unfairness. Wu et al. (2012) used an example in which its network located in Seattle regional by Gini coefficient demonstrated the inequity of road pricing. The result showed that none of road pricing policies is progressive (benefiting low-income travelers more than high-income ones). Given the deficiency about road pricing in alleviating congestion and emissions, tradable credits come up. The theory of pollution permit markets was firstly imposed by Coase (1960) on external costs. The Kyoto Protocol (1997) identified and recommended the system of emission permits as an economic tool to be implemented. In traffic, tradable credits are mainly used for two aspects: emissions and congestion. In transportation pollution, Nagurney (2000), Nagurney and Zhang (2001) put apply the emission permit systems to transportation network. Julie (2012) dealt with the feasibility of a tradable emission permit system (TEPs) for urban motorists to develop a new microeconomic theoretical model which reduces urban pollution. In the part of traffic congestion, it was not until 2011, Yang and Wang comprehensively explore the system of tradable travel credits in a general network. Now it has been extended to heterogeneous users (Wang and Yang et al., 2012), transaction costs (Nie, 2012), income effects (Wu et al., 2012), mixed equilibrium behaviors (He et al., 2013), price and flow dynamics (Ye et al., 2013). To our knowledge, tradable travel credits system scarcely ever was applied to alleviating traffic congestion and emissions simultaneously. In this paper, we get the minimum emissions and decentralize general flow patterns with tradable credits. We internalized congestion and emission externalities with tradable credits simultaneous and investigated the existence of tradable credits scheme which decentralizes a given Pareto system optimum link flow pattern. The rest of the paper is organized as follows: Notations and preliminaries are provided in section 2. In section 3, we implemented credits charging to achieve minimum emissions. Tradable credits scheme realized internalizing congestion and emission externality simultaneous and analyzed the Pareto system optimization with credits in section 4. Section 5 presents numerical examples and finally Section 6 concludes the paper. 2. Notation and preliminaries 2.1. The construction of network Consider a general network G=(N,A) with a set of N nodes and a set of A directed links. Let va , t a (va ) and ea (va ) be the total link flow, travel time and emissions on link a A , where t a () is a differentiable, non-negative, strictly increasing and convex function. Let W denotes the set of O–D pairs, and Rw the set of all routes between an O–D pair w W . The travel demand for each O–D pair w W is denoted by d w , d w ! 0 . Let f r ,w denotes the traffic flow on route r Rw between an O–D pair w W . f is a path flow vector f ( f r ,w , r Rw , w W ) . : f denotes the set of feasible path flow patterns defined by (Meng and Yang, 2002)
:f
® f f r ,w t 0, ¦ f r ,w rRw ¯
½ d w , w W ¾ ¿
(1)
216
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
Let v (va , a A) be the link flow vector, and : v represents the set of feasible link flow defined by (Meng and Yang, 2002) :v
Where G a ,r
®v va ¯
½
¦ ¦ f r ,wG a ,r , f : f , a A¾
(2)
¿
wW rRw
1 if route f use link a and 0 otherwise.
Let k ^kw , w W ` denote a general O–D based credit distribution scheme, where k w is the amount of credits distributed to each traveler over the O–D pair w W . K represents the total amount of credits. Then we have the equation K ¦ wW kw d w . κ (N a , a A) denotes the charge scheme, where N a is the credit charging for using link a A . < denotes the feasible set of credit schemes where (Yang and Wang, 2011) <
^(K , N ) f :
f
such that ¦ aA N a va d K , v :v
`
(3)
Let p denotes the unit credit price in the credit market. 2.2. The equilibrium of network Given a feasible scheme with tradable credits, ( K , N ) < , the user equilibrium (UE) conditions can be written as(Yang and Wang, 2011):
¦ aA (ta (va ) pN a )G a,r uw , if f r ,w ! 0, r Rw , w W
(4)
¦ aA (ta (va ) pN a )G a,r t uw , if f r ,w 0, r Rw , w W
(5)
The credit market equilibrium conditions are given by (Yang and Wang, 2011):
¦ aAN a va K , if p ! 0
(6)
¦ aAN a va d K , if p 0
(7)
3. Emission charging with credits 3.1. Decentralization of minimum emission flow pattern with credits The minimum emission problem is to find the minimum emission flow distribution v ME :
¦ ea (va )va min v: aA
(8)
v
Now we have the following definition: Definition 1. A tradable credit charge scheme vector, N (N a , a A) , is said to decentralize a given minimum emissions flow pattern v :v , if it makes v satisfies the following user equilibrium condition in the form of a variational inequality (VI): (t (v ME ) IpN )T (v v ME ) t 0 v :v , N <
(9)
217
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
Where t () is a vector of link travel time, I is a parameter that transfers money into equivalent time value. Theorem 1. If ea () is an increasing function for all a A and ¦ aA ea va is strictly convex, then there exists a non-negative tradable credits charge scheme which could decentralize the minimum emission flow pattern. Proof. We know that vaME (the minimum flow on link a ) solves the BUE problem (Yin and Lawphongpanich, 2006). To obtain a contradiction, assume that vaME does not solve the problem, let va' solves the BUE problem instead and vaME z va' . That is to say, vaME d va' for all a A and vaME va' for some a A . Because ea (va ) is increasing, then we have ea (va' )va' d ea (vaME )vaME for all a A and ea (va' )va' ea (vaME )vaME for some a A . Because vaME is the minimum emission flow pattern. It is contradictory with our assumption. Thus, the assumption is false and vaME solves the BUE problem. 3.2. Decentralization of general flow patterns with credits For congestion toll, if pure rebate scheme exists, when the rebate scheme satisfies: W a Eta where E ! 0 , and ta ta (va ) , the link travel cost ca becomes 0. So any feasible link flow pattern v can be decentralized into equilibrium by a rebate scheme. To tradable credits, we have the following theorem. Theorem 2. A given target feasible link flow pattern v can be decentralized into user equilibrium by a tradable credits scheme. Similar with Yang and Huang (2004), we have the following proof: Proof. Let va , a A be the feasible link flow, and va ta (va ) , a A be the corresponding link travel time. Under our aforementioned assumption of increasing convex link cost functions, link flows v a and hence link travel time t a , a A , are uniquely given. We now begin to seek a tradable credit scheme to decentralize the given link flow as UE. Consider the following linear programming (LP) problem:
s.t.
min ¦ aA E t a v a
(10)
¦ ¦ f r ,wG aw,r va , a A
(11)
¦ rRw f r ,w d w , w W
(12)
f r ,w t 0
(13)
wW rRw
The dual formulation of the above LP problem is given by max ¦ v a Oa O ,u
aA
¦ d wu w
(14)
wW
s.t uw ¦ OaG aw,r d ¦ E t aG a ,r r Rw , w W wW
(15)
aA
Where the dual variables, O and u are associated with the equality constrains (11) and (12), respectively, and Oa and u w are unrestricted in sign. Let pN a Oa . Consider a path r Rw with nonzero flow (i.e. f r ,w ! 0, r Rw , w W ). By the primal complementary slackness conditions, for every path r , we have
^¦
aA
`
(E t a pN a )G aw,r uw f r ,w
Because f r ,w ! 0 , that is to say, (E t a pN a )G aw,r uw same for all r Rw . By inequality (15), we have
0, r Rw , w W
(16)
0 . This means that the utility of the agents using path r is
218
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
¦ aA (E t a pN a )G aw,r uw t 0, r Rw , w W
(17)
Therefore, agents do not have an incentive to switch their paths which achieve user equilibrium, and complete the proof. 4. Decentralization of Pareto-efficient link flow pattern with credits From previous statement, we know in general, alleviating congestion conflicts with reducing emissions. Such inconsistency naturally leads to a bi-criteria or bi-objective optimization problem for simultaneously minimizing the two conflicting objectives. When the emission function’s form is not same, it will produce different results. So we divided emission function into two categories, i.e., increasing function and non-monotonic function. 4.1. ea (va ) is an increasing function Bi-level congestion pricing model is proposed by Yang and Lam (1996). The model can be described as following: ¦ aA t a (va )va ½ min ® ¾ v:v ¯¦ aA ea (va )va ¿
(18)
This is a typical Pareto-efficient problem which is described as following: Definition 2. A link flow pattern v* :v is said to be Pareto-efficient if there does not exist another flow pattern v :v such that E (v) d E (v* ) , T (v) T (v* ) , and at least one of E(v) E(v* ) and T (v) T (v* ) holds. When the emission function is an increasing function, and ¦ aAea (va )va is strictly convex, any Pareto-efficient flow pattern v* :*v can be supported as an user equilibrium by the tradable credit scheme. (Chen et al., 2011) The Pareto-efficient solution can also be obtained by solving the following weighting problem:
min D ¦ aA (ta (va ) (1 D )ea (va )va ) v:v
(19)
The marginal cost based tradable credit scheme ( K msc , N msc )of problem(19)can be obtained as:
N amsc
va*
wt a (va* ) 1 D 1 D * wea (va* ) ea (va* ) v wva D D a wva
K msc
¦ aAN amscva
(20) (21)
4.2. ea (va ) is an non-monotonic function Here we want to analyze whether a tradable credits scheme can decentralize the Pareto-efficient solutions of the bi-objective optimization problem into user equilibrium when the link emission functions are not monotonically increasing. The problem is examined in two types of network topology: loop-free networks and loops networks, respectively. The concrete definitions about loop-free networks and loops networks, we can refer Gallager (1977). (1) Loop-free networks From Hagstrom and Abrams (2010), it has the following theorem: in a loop-free road network, any feasible link flow pattern v can be decentralized into user equilibrium by a nonnegative toll scheme. Through above theorem, we can readily conclude the existence of nonnegative tradable credit schemes for supporting a Pareto-efficient link flow pattern, based on the result generated by Yang and Wang (2011). They
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
showed that a tradable credit scheme can emulate a congestion pricing scheme in supporting various target link flow patterns as an user equilibrium. So we have the following corollary. Corollary 2. In a loop-free road network, any feasible link flow pattern v can be decentralized into user equilibrium by a tradable credits scheme. (2) loops networks If the network contains loops, and the emission function is non-monotonic, Pareto-efficient solution may not be decentralized to user equilibrium. About loops networks, we have the following Corollaries and Theorems: If there is no flow on the loop, that is to say the links’ cost is big enough that travelers don’t choose the routes which contain the links on the loop. We can “subtract out” the loops on the network, it becomes loop-free network. According to Corollary 2, we have Corollary 3. Corollary 3. In loops networks, if the flow on the loops is zero, any feasible link flow pattern v can be decentralized to user equilibrium by a nonnegative toll scheme/tradable credits scheme. Theorem 3. In loops networks, if and only if loops of flow in the Gallager’s sense do not exist, a feasible link flow pattern v can be decentralized into user equilibrium by a nonnegative toll scheme/ tradable credits scheme. Proof. Suppose there exists loops of flow in the Gallager’s sense, and a feasible link flow pattern v can be decentralized into user equilibrium by a nonnegative toll scheme. Let x be a feasible flow vector, possibly containing single-or-multi-commodity loops. Let K be the maximal set of links used by single-or-multi-commodity loops. Now H is defined to be the set of links k such that ck 0 for every c which satisfies [ x, c] is an equilibrium and c t 0 . Here ca is the cost of link a . It follows from Theorem 2 that K and H are not empty. For each link in K contract the link by identifying the head and tail as a single node. The resulting network with same flow between origin and destination pairs will not have a multi-commodity loop and hence Theorem 3 applies. Thus there exists a strictly positive c on the contracted network. For each contracted link k , set the cost ck equal to zero. Because every path from an origin to destination has the same length as in the contracted network, we have a cost vector c that yields an user equilibrium for the original flow x . Thus for every k K , this construction gives a positive ck , implying k H . Hence H is contained in K . Now let k K as pointed out at the beginning of this section, in an equilibrium with nonnegative costs, the cost of traversing any link k contained in a loop must be zero, that is, ck 0 . Hence K is contained in H . We also know ck Etk W k . Because tk ! 0, E ! 0, ck 0, Etk W k 0,W k Etk 0 . That is to say, toll schemes in K are negative, and K is contained in H . It is contradicted with our hypothesis, the same as tradable credits scheme, which completes the proof. 5. Numerical examples The following loops network satisfies Gallager’s sense. We want to check whether exists the Pareto-efficient solution in the network. For simplicity, the emission functions are quadratic form which is used by Sbayti et al. (2002) to express the NOx emission factors. There are four O-D pairs: 3ė1, 3ė6, 2ė4 and 2ė5. With O-D demands d3ė1=2 and d3ė6=2, d2ė4=2 and d2ė5=2. In the network there exist two loops 2ė5ė3 and 6ė8ė9. Emission functions:
/LQN
/LQN
/LQN
/LQN
/LQN
/LQN
/LQN
/LQN /LQN
/LQN
e1 (v1 ) (v1 2) 2 0.2 e2 (v 2 ) (v2 1) 2 0.1
e3 (v 3 )
(v3 1) 2 0.1
e4 (v 4 )
(v4 2) 2 0.2
e5 (v 5 )
(v5 1) 2 0.1
e6 (v 6 )
(v6 1) 2 0.1
e7 (v 7 )
(v7 2) 2 0.2
e8 (v 8 )
(v8 1) 2 0.1
e9 (v 9 )
(v9 1) 2 0.1
e10 (v10 )
(v10 2) 2 0.2
Fig. 1. A six node-network which has loops
219
220
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
We can obtain the minimum total emissions link flow pattern, it is listed in the Table 3, and its corresponding path flow is listed in Table 4. If the link flow is decentralized into user equilibrium, every O-D pairs’ paths’ costs are equal. So, according to UE conditions, for O-D pairs 3ė1, c1 c3 c4 , O-D pairs 3ė6, c4 c1 c2 c5 . Combining the two equations leads to c2 c5 c3 , at the same time, ca Eta W a is for tolling scheme, ca Eta IpN a is for tradable credits scheme. It is equivalent to: E (t2 t5 ) W 2 W 5 or E (t2 t5 ) Ip(N 2 N 5 ) Et3 IpN 3 , that is,
W 2 W 3 W 5 E (t2 t3 t5 ) or Ip(N 2 N 3 N 5 ) E (t2 t3 t5 ) . We know E ! 0, I ! 0, p ! 0, ta ! 0 , therefore W 2 W 3 W 5 0 or N 2 N 3 N 5 0 , at least one of W 2 ,W 3 and W 5 or one of N 2 , N 3 and N 5 must be negative. It shows if loops of flow satisfy Gallager’s sense, a feasible link flow pattern v can’t be decentralized into user equilibrium by a nonnegative toll scheme or by tradable credits scheme, but not vice versa. Table 1. The minimum emission link flow Link vaME
ea
ea (vaME )vaME (g/h)
1
2
0.2
0.4
2
1
0.1
0.1
3
1
0.1
0.1
4
2
0.2
0.4
5
1
0.1
0.1
6
1
0.1
0.1
7
2
0.2
0.4
8
1
0.1
0.1
9
1
0.1
0.1
10
2
0.2
0.4
Total emissions
2.2
Table 2. The corresponding path flow of Table 3 O-D 3ė1 3ė6 2ė4 2ė5
Path
Flow
3ė1
1
3ė6ė1
1
3ė6
1
3ė1ė5ė6
1
2ė4
1
2ė5ė4
1
2ė4
1
2ė4ė6ė5
1
6. Conclusion We realized the minimum emission and internalized congestion and emission externalities with tradable credits simultaneously. At the same time, we investigated the existence of tradable credits scheme which decentralizes a given Pareto system optimum link flow pattern. With the results established for general feasible flow patterns, we proved that any given Pareto-efficiency system optimum link flow pattern can be decentralized into user equilibrium by a tradable credits scheme if the link emission functions are increasing. However, if the link emission functions are not increasing, the existence of such schemes depends on the topology of the network. If there is a loop-free network, any feasible link flow pattern v can
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
be decentralized into user equilibrium by a tradable credits scheme. If it is a loops network, a Pareto-efficient solution may not be realizable as user equilibrium by a tradable credits scheme. Acknowledgements This paper is partly supported by National Basic Research Program of China (2012CB725400), NSFC (71322102), the Program for New Century Excellent Talents in University (NCET-12-0764) and Fundamental Research Funds for the Central Universities (2012JBZ005). Reference Chen, L. X. (2011). Managing congestion and emissions on road networks. MPhil Thesis. The Hong Kong University of Science and Technology. Regan, D. H. (1972). The problem of social cost revisited. Journal of Law and Economics, 15, 427-437. Gallager, R.G. (1977). Loops in multicommodity flows. In: Proceedings of the Tenth IEEE Conference on Decisions and Control. New Orleans. 819–825. Hagstrom, J.N., & Abrams, R.A. (2010). Loops and the existence of equilibrium-inducing tolls. In: Presented at the Workshop and Conference on “Innovations in Pricing of Transportation Systems”, 13–14 May 2010, Orlando, Florida. Han, D., Yang, H., & Wang, X.L. (2010). Efficiency of the plate-number-based traffic rationing in general networks. Transportation Research Part E: Logistics and Transportation Review , 46, 1095–1110. He, F., Yin, Y.F., Shirmohammadi, N., & Nie, Y. (2013). Tradable credit schemes on networks with mixed equilibrium behaviors. Transportation Research Part B: Methodological, 57, 47–65. Johansson, O. (1997). Optimal road-pricing: simultaneous treatment of time losses, increased fuel consumption, and emissions. Transportation Research part D: Transport and Environment, 2, 77-87. Julie, B. (2012). Tradable emission permit system for urban motorists: The neo-classical standard model revisited. Research in Transportation Economics 36, 101–109. Nagurney, A. (2000). Congested urban transportation networks and emission paradoxes. Transportation Research Part D: Transport and Environment, 5, 145–151. Nagurney, A., & Zhang, D. (2001). Dynamics of transportation pollution permit system with stability analysis and computation. Transportation Research Part D: Transport and Environment, 6, 243-268. Pigou, A. C. (1920). The Economics of Welfare. MacMillan, London. Raux, C., & Marlot, G. (2005). A system of tradable CO2 permits applied to fuel consumption by motorists. Transport Policy, 12, 255–265. Sbayti, H., El-Fadel, M., & Kaysi, I. (2002). Effect of roadway network aggregation levels on modeling of trafficinduced emission inventories in Beirut.Transportation Research Part D: Transport and Environment, 7, 163–173. Tsekeris, T., & Voss, S. (2009). Design and evaluation of road pricing: state-of-the-art and methodological advances. Netnomics, 10, 5–52.
221
222
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
Wang, X. L., Yang, H., Zhu, D. L., & Li, C. M. (2012). Tradable travel credits for congestion management with heterogeneous users. Transportation Research Part E: Logistics and Transportation Review, 48, 426–437. Wu, D., Yin, Y., Lawphongpanich, S., & Yang, H. (2013). Design of more equitable congestion pricing and tradable credit schemes for multimodal transportation networks. Transportation Research Part B: Methodological, 46 (9), 1273–1287. Yang, H., & Huang, H. J. (2004). The multi-class, multi-criteria traffic network equilibrium and systems optimum problem. Transportation Research Part B: Methodological, 38, 1–15. Yang, H., & Huang, H. J. (2005). Mathematical and Economics Theory of Road Pricing. Elsevier Ltd., Oxford. Yan, H., & Lam, W. H. (1996). Optimal road tolls under conditions of queueing and congestion. Transportation Research Part A: Policy and Practice, 30, 319-332. Yang, H., & Wang, X. L. (2011). Managing network mobility with tradable credits. Transportation Research Part B: Methodological, 45, 580–594. Ye, H. B., & Yang, H. (2013). Continuous price and flow dynamic of tradable mobility. Transportation Research Part B: Methodological, in press. Yin, Y., & Lawphongpanich, S. (2006). Internalizing emission externality on road networks. Transportation Research Part D: Transport and Environment, 11, 292–301.
ScienceDirect Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
The 9th International Conference on Traffic & Transportation Studies (ICTTS’2014)
Internalizing Congestion and Emissions Externality on Road Networks with Tradable Credits Ge Gao, Huijun Sun* MOE Key Laboratory for Urban Transportation Complex Systems Theory and Technology, Beijing Jiaotong University, No.3 Shangyuancun, Haidian District, Beijing 100044, P.R. China
Abstract Congestion and emissions are two problems in traffic. In general, they have conflict with each other. Tradable credits scheme is proved an effective method in Environmental Science. Now more and more scholars introduce it into traffic. Because it possesses the advantages that other methods do not have in alleviating congestion and reducing emissions. In this paper we indicate there always exists a tradable credits scheme that induces a traffic flow distribution with minimum emissions. We proved a given target feasible link flow pattern can be decentralized into user equilibrium by a tradable credits scheme, and sought tradable credits schemes for Pareto-efficient control and management of both traffic congestion and emissions on road networks. There are two models which could describe the Pareto-efficient problem between alleviating congestion and reducing emissions, i.e., bi-level congestion pricing model and weighting problem model. When the emission function is an increasing function, any Pareto-efficient flow pattern can be supported as an user equilibrium by the tradable credit scheme. If the emission function is a non-monotonic function, we divided the network into two cases: loops-free network and loops network which satisfies Gallager’s sense. In a loop-free road network, any feasible link flow pattern can be decentralized into user equilibrium by a tradable scheme. In loops network, Pareto-efficient solution may not be decentralized to user equilibrium by tradable credits scheme. However, if the loops network doesn’t satisfy Gallager’s sense, any feasible Pareto-efficient link flow pattern can be decentralized into user equilibrium by a nonnegative toll scheme/tradable credits scheme. On the contrary, it wouldn’t. © 2014 2014 The Elsevier Ltd. This is an open access Ltd. article under the CC BY-NC-ND license © Authors. Published by Elsevier (http://creativecommons.org/licenses/by-nc-nd/3.0/). Peer-review under responsibility of Beijing Jiaotong University (BJU), Systems Engineering Society of China (SESC). Peer-review under responsibility of Beijing Jiaotong University(BJU), Systems Engineering Society of China (SESC). Keyword: Tradable credits; Congestion and emissions; Pareto-efficient; Loops-free; User equilibrium
* Corresponding author. Tel.: +86-(0)10-5168-4265. E-mail address: [email protected].
1877-0428 © 2014 Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Peer-review under responsibility of Beijing Jiaotong University(BJU), Systems Engineering Society of China (SESC). doi:10.1016/j.sbspro.2014.07.198
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
215
1. Introduction Traffic congestion and emissions are two global problems we have to face in big cities. Due to congestion, more time is wasted on road; Due to emissions, more and more people have to endure severe air pollution. There is a closed connection between traffic congestion and emissions. Traffic emissions in congested conditions are substantially higher than under conditions of freely traffic flow. However, in general, alleviating congestion conflicts with reducing emissions (Nagurney, 2000; Yin and Lawphongpanich, 2006). In order to alleviate traffic congestion and control emissions, and achieve the sustainability of transportation, for several decades, road pricing has been considered as an effective management instrument. Pigou (1920) firstly proposed the idea of road pricing. Similarly, a road user should pay a charge corresponding to both his own emissions and the increased emissions brought to other users (Jahansson, 1997). Afterward, immense amounts of research on road pricing have been further developed. For recent reviews to congestion pricing, we can see, Yang and Huang (2005), Tsekeris and Voss (2009). However, road pricing has its limitations. The most obvious defect is unfairness. Wu et al. (2012) used an example in which its network located in Seattle regional by Gini coefficient demonstrated the inequity of road pricing. The result showed that none of road pricing policies is progressive (benefiting low-income travelers more than high-income ones). Given the deficiency about road pricing in alleviating congestion and emissions, tradable credits come up. The theory of pollution permit markets was firstly imposed by Coase (1960) on external costs. The Kyoto Protocol (1997) identified and recommended the system of emission permits as an economic tool to be implemented. In traffic, tradable credits are mainly used for two aspects: emissions and congestion. In transportation pollution, Nagurney (2000), Nagurney and Zhang (2001) put apply the emission permit systems to transportation network. Julie (2012) dealt with the feasibility of a tradable emission permit system (TEPs) for urban motorists to develop a new microeconomic theoretical model which reduces urban pollution. In the part of traffic congestion, it was not until 2011, Yang and Wang comprehensively explore the system of tradable travel credits in a general network. Now it has been extended to heterogeneous users (Wang and Yang et al., 2012), transaction costs (Nie, 2012), income effects (Wu et al., 2012), mixed equilibrium behaviors (He et al., 2013), price and flow dynamics (Ye et al., 2013). To our knowledge, tradable travel credits system scarcely ever was applied to alleviating traffic congestion and emissions simultaneously. In this paper, we get the minimum emissions and decentralize general flow patterns with tradable credits. We internalized congestion and emission externalities with tradable credits simultaneous and investigated the existence of tradable credits scheme which decentralizes a given Pareto system optimum link flow pattern. The rest of the paper is organized as follows: Notations and preliminaries are provided in section 2. In section 3, we implemented credits charging to achieve minimum emissions. Tradable credits scheme realized internalizing congestion and emission externality simultaneous and analyzed the Pareto system optimization with credits in section 4. Section 5 presents numerical examples and finally Section 6 concludes the paper. 2. Notation and preliminaries 2.1. The construction of network Consider a general network G=(N,A) with a set of N nodes and a set of A directed links. Let va , t a (va ) and ea (va ) be the total link flow, travel time and emissions on link a A , where t a () is a differentiable, non-negative, strictly increasing and convex function. Let W denotes the set of O–D pairs, and Rw the set of all routes between an O–D pair w W . The travel demand for each O–D pair w W is denoted by d w , d w ! 0 . Let f r ,w denotes the traffic flow on route r Rw between an O–D pair w W . f is a path flow vector f ( f r ,w , r Rw , w W ) . : f denotes the set of feasible path flow patterns defined by (Meng and Yang, 2002)
:f
® f f r ,w t 0, ¦ f r ,w rRw ¯
½ d w , w W ¾ ¿
(1)
216
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
Let v (va , a A) be the link flow vector, and : v represents the set of feasible link flow defined by (Meng and Yang, 2002) :v
Where G a ,r
®v va ¯
½
¦ ¦ f r ,wG a ,r , f : f , a A¾
(2)
¿
wW rRw
1 if route f use link a and 0 otherwise.
Let k ^kw , w W ` denote a general O–D based credit distribution scheme, where k w is the amount of credits distributed to each traveler over the O–D pair w W . K represents the total amount of credits. Then we have the equation K ¦ wW kw d w . κ (N a , a A) denotes the charge scheme, where N a is the credit charging for using link a A . < denotes the feasible set of credit schemes where (Yang and Wang, 2011) <
^(K , N ) f :
f
such that ¦ aA N a va d K , v :v
`
(3)
Let p denotes the unit credit price in the credit market. 2.2. The equilibrium of network Given a feasible scheme with tradable credits, ( K , N ) < , the user equilibrium (UE) conditions can be written as(Yang and Wang, 2011):
¦ aA (ta (va ) pN a )G a,r uw , if f r ,w ! 0, r Rw , w W
(4)
¦ aA (ta (va ) pN a )G a,r t uw , if f r ,w 0, r Rw , w W
(5)
The credit market equilibrium conditions are given by (Yang and Wang, 2011):
¦ aAN a va K , if p ! 0
(6)
¦ aAN a va d K , if p 0
(7)
3. Emission charging with credits 3.1. Decentralization of minimum emission flow pattern with credits The minimum emission problem is to find the minimum emission flow distribution v ME :
¦ ea (va )va min v: aA
(8)
v
Now we have the following definition: Definition 1. A tradable credit charge scheme vector, N (N a , a A) , is said to decentralize a given minimum emissions flow pattern v :v , if it makes v satisfies the following user equilibrium condition in the form of a variational inequality (VI): (t (v ME ) IpN )T (v v ME ) t 0 v :v , N <
(9)
217
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
Where t () is a vector of link travel time, I is a parameter that transfers money into equivalent time value. Theorem 1. If ea () is an increasing function for all a A and ¦ aA ea va is strictly convex, then there exists a non-negative tradable credits charge scheme which could decentralize the minimum emission flow pattern. Proof. We know that vaME (the minimum flow on link a ) solves the BUE problem (Yin and Lawphongpanich, 2006). To obtain a contradiction, assume that vaME does not solve the problem, let va' solves the BUE problem instead and vaME z va' . That is to say, vaME d va' for all a A and vaME va' for some a A . Because ea (va ) is increasing, then we have ea (va' )va' d ea (vaME )vaME for all a A and ea (va' )va' ea (vaME )vaME for some a A . Because vaME is the minimum emission flow pattern. It is contradictory with our assumption. Thus, the assumption is false and vaME solves the BUE problem. 3.2. Decentralization of general flow patterns with credits For congestion toll, if pure rebate scheme exists, when the rebate scheme satisfies: W a Eta where E ! 0 , and ta ta (va ) , the link travel cost ca becomes 0. So any feasible link flow pattern v can be decentralized into equilibrium by a rebate scheme. To tradable credits, we have the following theorem. Theorem 2. A given target feasible link flow pattern v can be decentralized into user equilibrium by a tradable credits scheme. Similar with Yang and Huang (2004), we have the following proof: Proof. Let va , a A be the feasible link flow, and va ta (va ) , a A be the corresponding link travel time. Under our aforementioned assumption of increasing convex link cost functions, link flows v a and hence link travel time t a , a A , are uniquely given. We now begin to seek a tradable credit scheme to decentralize the given link flow as UE. Consider the following linear programming (LP) problem:
s.t.
min ¦ aA E t a v a
(10)
¦ ¦ f r ,wG aw,r va , a A
(11)
¦ rRw f r ,w d w , w W
(12)
f r ,w t 0
(13)
wW rRw
The dual formulation of the above LP problem is given by max ¦ v a Oa O ,u
aA
¦ d wu w
(14)
wW
s.t uw ¦ OaG aw,r d ¦ E t aG a ,r r Rw , w W wW
(15)
aA
Where the dual variables, O and u are associated with the equality constrains (11) and (12), respectively, and Oa and u w are unrestricted in sign. Let pN a Oa . Consider a path r Rw with nonzero flow (i.e. f r ,w ! 0, r Rw , w W ). By the primal complementary slackness conditions, for every path r , we have
^¦
aA
`
(E t a pN a )G aw,r uw f r ,w
Because f r ,w ! 0 , that is to say, (E t a pN a )G aw,r uw same for all r Rw . By inequality (15), we have
0, r Rw , w W
(16)
0 . This means that the utility of the agents using path r is
218
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
¦ aA (E t a pN a )G aw,r uw t 0, r Rw , w W
(17)
Therefore, agents do not have an incentive to switch their paths which achieve user equilibrium, and complete the proof. 4. Decentralization of Pareto-efficient link flow pattern with credits From previous statement, we know in general, alleviating congestion conflicts with reducing emissions. Such inconsistency naturally leads to a bi-criteria or bi-objective optimization problem for simultaneously minimizing the two conflicting objectives. When the emission function’s form is not same, it will produce different results. So we divided emission function into two categories, i.e., increasing function and non-monotonic function. 4.1. ea (va ) is an increasing function Bi-level congestion pricing model is proposed by Yang and Lam (1996). The model can be described as following: ¦ aA t a (va )va ½ min ® ¾ v:v ¯¦ aA ea (va )va ¿
(18)
This is a typical Pareto-efficient problem which is described as following: Definition 2. A link flow pattern v* :v is said to be Pareto-efficient if there does not exist another flow pattern v :v such that E (v) d E (v* ) , T (v) T (v* ) , and at least one of E(v) E(v* ) and T (v) T (v* ) holds. When the emission function is an increasing function, and ¦ aAea (va )va is strictly convex, any Pareto-efficient flow pattern v* :*v can be supported as an user equilibrium by the tradable credit scheme. (Chen et al., 2011) The Pareto-efficient solution can also be obtained by solving the following weighting problem:
min D ¦ aA (ta (va ) (1 D )ea (va )va ) v:v
(19)
The marginal cost based tradable credit scheme ( K msc , N msc )of problem(19)can be obtained as:
N amsc
va*
wt a (va* ) 1 D 1 D * wea (va* ) ea (va* ) v wva D D a wva
K msc
¦ aAN amscva
(20) (21)
4.2. ea (va ) is an non-monotonic function Here we want to analyze whether a tradable credits scheme can decentralize the Pareto-efficient solutions of the bi-objective optimization problem into user equilibrium when the link emission functions are not monotonically increasing. The problem is examined in two types of network topology: loop-free networks and loops networks, respectively. The concrete definitions about loop-free networks and loops networks, we can refer Gallager (1977). (1) Loop-free networks From Hagstrom and Abrams (2010), it has the following theorem: in a loop-free road network, any feasible link flow pattern v can be decentralized into user equilibrium by a nonnegative toll scheme. Through above theorem, we can readily conclude the existence of nonnegative tradable credit schemes for supporting a Pareto-efficient link flow pattern, based on the result generated by Yang and Wang (2011). They
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
showed that a tradable credit scheme can emulate a congestion pricing scheme in supporting various target link flow patterns as an user equilibrium. So we have the following corollary. Corollary 2. In a loop-free road network, any feasible link flow pattern v can be decentralized into user equilibrium by a tradable credits scheme. (2) loops networks If the network contains loops, and the emission function is non-monotonic, Pareto-efficient solution may not be decentralized to user equilibrium. About loops networks, we have the following Corollaries and Theorems: If there is no flow on the loop, that is to say the links’ cost is big enough that travelers don’t choose the routes which contain the links on the loop. We can “subtract out” the loops on the network, it becomes loop-free network. According to Corollary 2, we have Corollary 3. Corollary 3. In loops networks, if the flow on the loops is zero, any feasible link flow pattern v can be decentralized to user equilibrium by a nonnegative toll scheme/tradable credits scheme. Theorem 3. In loops networks, if and only if loops of flow in the Gallager’s sense do not exist, a feasible link flow pattern v can be decentralized into user equilibrium by a nonnegative toll scheme/ tradable credits scheme. Proof. Suppose there exists loops of flow in the Gallager’s sense, and a feasible link flow pattern v can be decentralized into user equilibrium by a nonnegative toll scheme. Let x be a feasible flow vector, possibly containing single-or-multi-commodity loops. Let K be the maximal set of links used by single-or-multi-commodity loops. Now H is defined to be the set of links k such that ck 0 for every c which satisfies [ x, c] is an equilibrium and c t 0 . Here ca is the cost of link a . It follows from Theorem 2 that K and H are not empty. For each link in K contract the link by identifying the head and tail as a single node. The resulting network with same flow between origin and destination pairs will not have a multi-commodity loop and hence Theorem 3 applies. Thus there exists a strictly positive c on the contracted network. For each contracted link k , set the cost ck equal to zero. Because every path from an origin to destination has the same length as in the contracted network, we have a cost vector c that yields an user equilibrium for the original flow x . Thus for every k K , this construction gives a positive ck , implying k H . Hence H is contained in K . Now let k K as pointed out at the beginning of this section, in an equilibrium with nonnegative costs, the cost of traversing any link k contained in a loop must be zero, that is, ck 0 . Hence K is contained in H . We also know ck Etk W k . Because tk ! 0, E ! 0, ck 0, Etk W k 0,W k Etk 0 . That is to say, toll schemes in K are negative, and K is contained in H . It is contradicted with our hypothesis, the same as tradable credits scheme, which completes the proof. 5. Numerical examples The following loops network satisfies Gallager’s sense. We want to check whether exists the Pareto-efficient solution in the network. For simplicity, the emission functions are quadratic form which is used by Sbayti et al. (2002) to express the NOx emission factors. There are four O-D pairs: 3ė1, 3ė6, 2ė4 and 2ė5. With O-D demands d3ė1=2 and d3ė6=2, d2ė4=2 and d2ė5=2. In the network there exist two loops 2ė5ė3 and 6ė8ė9. Emission functions:
/LQN
/LQN
/LQN
/LQN
/LQN
/LQN
/LQN
/LQN /LQN
/LQN
e1 (v1 ) (v1 2) 2 0.2 e2 (v 2 ) (v2 1) 2 0.1
e3 (v 3 )
(v3 1) 2 0.1
e4 (v 4 )
(v4 2) 2 0.2
e5 (v 5 )
(v5 1) 2 0.1
e6 (v 6 )
(v6 1) 2 0.1
e7 (v 7 )
(v7 2) 2 0.2
e8 (v 8 )
(v8 1) 2 0.1
e9 (v 9 )
(v9 1) 2 0.1
e10 (v10 )
(v10 2) 2 0.2
Fig. 1. A six node-network which has loops
219
220
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
We can obtain the minimum total emissions link flow pattern, it is listed in the Table 3, and its corresponding path flow is listed in Table 4. If the link flow is decentralized into user equilibrium, every O-D pairs’ paths’ costs are equal. So, according to UE conditions, for O-D pairs 3ė1, c1 c3 c4 , O-D pairs 3ė6, c4 c1 c2 c5 . Combining the two equations leads to c2 c5 c3 , at the same time, ca Eta W a is for tolling scheme, ca Eta IpN a is for tradable credits scheme. It is equivalent to: E (t2 t5 ) W 2 W 5 or E (t2 t5 ) Ip(N 2 N 5 ) Et3 IpN 3 , that is,
W 2 W 3 W 5 E (t2 t3 t5 ) or Ip(N 2 N 3 N 5 ) E (t2 t3 t5 ) . We know E ! 0, I ! 0, p ! 0, ta ! 0 , therefore W 2 W 3 W 5 0 or N 2 N 3 N 5 0 , at least one of W 2 ,W 3 and W 5 or one of N 2 , N 3 and N 5 must be negative. It shows if loops of flow satisfy Gallager’s sense, a feasible link flow pattern v can’t be decentralized into user equilibrium by a nonnegative toll scheme or by tradable credits scheme, but not vice versa. Table 1. The minimum emission link flow Link vaME
ea
ea (vaME )vaME (g/h)
1
2
0.2
0.4
2
1
0.1
0.1
3
1
0.1
0.1
4
2
0.2
0.4
5
1
0.1
0.1
6
1
0.1
0.1
7
2
0.2
0.4
8
1
0.1
0.1
9
1
0.1
0.1
10
2
0.2
0.4
Total emissions
2.2
Table 2. The corresponding path flow of Table 3 O-D 3ė1 3ė6 2ė4 2ė5
Path
Flow
3ė1
1
3ė6ė1
1
3ė6
1
3ė1ė5ė6
1
2ė4
1
2ė5ė4
1
2ė4
1
2ė4ė6ė5
1
6. Conclusion We realized the minimum emission and internalized congestion and emission externalities with tradable credits simultaneously. At the same time, we investigated the existence of tradable credits scheme which decentralizes a given Pareto system optimum link flow pattern. With the results established for general feasible flow patterns, we proved that any given Pareto-efficiency system optimum link flow pattern can be decentralized into user equilibrium by a tradable credits scheme if the link emission functions are increasing. However, if the link emission functions are not increasing, the existence of such schemes depends on the topology of the network. If there is a loop-free network, any feasible link flow pattern v can
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
be decentralized into user equilibrium by a tradable credits scheme. If it is a loops network, a Pareto-efficient solution may not be realizable as user equilibrium by a tradable credits scheme. Acknowledgements This paper is partly supported by National Basic Research Program of China (2012CB725400), NSFC (71322102), the Program for New Century Excellent Talents in University (NCET-12-0764) and Fundamental Research Funds for the Central Universities (2012JBZ005). Reference Chen, L. X. (2011). Managing congestion and emissions on road networks. MPhil Thesis. The Hong Kong University of Science and Technology. Regan, D. H. (1972). The problem of social cost revisited. Journal of Law and Economics, 15, 427-437. Gallager, R.G. (1977). Loops in multicommodity flows. In: Proceedings of the Tenth IEEE Conference on Decisions and Control. New Orleans. 819–825. Hagstrom, J.N., & Abrams, R.A. (2010). Loops and the existence of equilibrium-inducing tolls. In: Presented at the Workshop and Conference on “Innovations in Pricing of Transportation Systems”, 13–14 May 2010, Orlando, Florida. Han, D., Yang, H., & Wang, X.L. (2010). Efficiency of the plate-number-based traffic rationing in general networks. Transportation Research Part E: Logistics and Transportation Review , 46, 1095–1110. He, F., Yin, Y.F., Shirmohammadi, N., & Nie, Y. (2013). Tradable credit schemes on networks with mixed equilibrium behaviors. Transportation Research Part B: Methodological, 57, 47–65. Johansson, O. (1997). Optimal road-pricing: simultaneous treatment of time losses, increased fuel consumption, and emissions. Transportation Research part D: Transport and Environment, 2, 77-87. Julie, B. (2012). Tradable emission permit system for urban motorists: The neo-classical standard model revisited. Research in Transportation Economics 36, 101–109. Nagurney, A. (2000). Congested urban transportation networks and emission paradoxes. Transportation Research Part D: Transport and Environment, 5, 145–151. Nagurney, A., & Zhang, D. (2001). Dynamics of transportation pollution permit system with stability analysis and computation. Transportation Research Part D: Transport and Environment, 6, 243-268. Pigou, A. C. (1920). The Economics of Welfare. MacMillan, London. Raux, C., & Marlot, G. (2005). A system of tradable CO2 permits applied to fuel consumption by motorists. Transport Policy, 12, 255–265. Sbayti, H., El-Fadel, M., & Kaysi, I. (2002). Effect of roadway network aggregation levels on modeling of trafficinduced emission inventories in Beirut.Transportation Research Part D: Transport and Environment, 7, 163–173. Tsekeris, T., & Voss, S. (2009). Design and evaluation of road pricing: state-of-the-art and methodological advances. Netnomics, 10, 5–52.
221
222
Ge Gao and Huijun Sun / Procedia - Social and Behavioral Sciences 138 (2014) 214 – 222
Wang, X. L., Yang, H., Zhu, D. L., & Li, C. M. (2012). Tradable travel credits for congestion management with heterogeneous users. Transportation Research Part E: Logistics and Transportation Review, 48, 426–437. Wu, D., Yin, Y., Lawphongpanich, S., & Yang, H. (2013). Design of more equitable congestion pricing and tradable credit schemes for multimodal transportation networks. Transportation Research Part B: Methodological, 46 (9), 1273–1287. Yang, H., & Huang, H. J. (2004). The multi-class, multi-criteria traffic network equilibrium and systems optimum problem. Transportation Research Part B: Methodological, 38, 1–15. Yang, H., & Huang, H. J. (2005). Mathematical and Economics Theory of Road Pricing. Elsevier Ltd., Oxford. Yan, H., & Lam, W. H. (1996). Optimal road tolls under conditions of queueing and congestion. Transportation Research Part A: Policy and Practice, 30, 319-332. Yang, H., & Wang, X. L. (2011). Managing network mobility with tradable credits. Transportation Research Part B: Methodological, 45, 580–594. Ye, H. B., & Yang, H. (2013). Continuous price and flow dynamic of tradable mobility. Transportation Research Part B: Methodological, in press. Yin, Y., & Lawphongpanich, S. (2006). Internalizing emission externality on road networks. Transportation Research Part D: Transport and Environment, 11, 292–301.