A prediction model for atmospheric pollution reduction from urban traffic

Abstract

In order to reduce the atmospheric pollution in urban areas, an enhanced approach is proposed in this paper for the traffic congestion analysis. The approach is formulated as bi-level optimization program considering additional constraints in the traffic assignment problem. To respect the required eco-friendly threshold constraint, the travel demand between several origin–destination pairs was categorized in two classes: old polluting cars and modern (less) nonpolluting cars. The validity of the formulation was verified by optimality conditions. Two network examples are discussed to explain the properties and advantages of the suggested technique. It is found that for the both examples, the proposed optimal solution displays better results as compared to the common user equilibrium route choice policies. As a result, the enhanced approach leads to traffic network congestion relief with minimum air pollution and maximum use of routes network.

Keywords

Traffic congestion mathematical model travel decisions equilibrium traffic network atmospheric pollution

Introduction

Transport is the basis of economic and social development. Without transport neither economic exchanges nor citizen displacement is possible. However, transport has number of harmful effects on the environment such as air pollution, noise emission, and road accidents. As pointed out by Reynaud (1996), environment and transport have complex relationship. The sustainable development concept helps to find compromise between economic development and environmental compliance.

Road traffic is the most used transport mode and also the major contributor to urban areas air pollution. This causes serious ecologic problems that researchers attempt to solve since the last decades (Johansson et al., 2009; Nasiri et al., 2009). Results helped to raise the awareness of governments and public sectors on the problem urgency.

To assess some toxic effects on the environment, two important horizons have been considered: spatial and temporal. Spatially: locals effect such as smog, and globally like the greenhouse effect. Temporally: short-term effects like peaks of ozone pollution during periods of high temperature and long-term like the destruction of the ozone layer and global warming.

The pollutants emitted by road traffic are produced either directly during the use of vehicles, or during chemical reactions of their gases in the atmosphere. Potential pollution sources are the direct emissions that mainly derive from exhaust gases of combustion, fuel evaporation gases as well as wear of brakes and tires. The main polluting components are: carbon monoxide ( $CO$ ), hydrocarbons ( $HC$ ), nitrogen oxides ( $NOx$ ), sulfur dioxide ( $S O_{2}$ ), lead $Pb$ , fine particles $P M_{2.5}$ and $P M_{10}$ , and benzene (Xia et al., 2015).

Generally, $N O_{x}$ , $P M_{x}$ , and $CO$ are dangerous gases that metropolis citizens breathe daily, Deguen et al. (2015). Road traffic is one of the main factors responsible of air pollution that poses a threat to public health as emphasized in Chen et al. (2017) and Farda and Balijepalli (2018). In 2013, the World Health Organization (WHO) stated that living within 50 m from a congested pathway would increase the risk chronic disease by 7%. Moreover, air pollution causes nearly 13 million deaths per year.

It is obvious that fuel engines are the first responsible of air pollution. Substituting this polluting source would be the ideal and definitive solution to combat this plague (He and Qiu, 2016). Unfortunately, the cost of such radical change prevents its applicability in short term. However, some alternative techniques exist such as travel demand reduction or old vehicles traffic prohibition (OVTP) on congested network axes. The latter has been experienced in France and currently applied throughout Europe as well as in Japan and China. OVTP consists of banning the most polluting vehicles from circulation $(\geq 2 g / km$ of emissions), as old cars for instance.

Since 1992, the European Union regulated the automotive industry according to Euro standards to reduce exhaust emissions levels. The aim of these standards is to respect the pollution emission levels, which are six in number, and expressed it in units of mass per distance (g/km) for light vehicles identified by well-defined driving cycles (Rhys-Tyler et al., 2011). Euro 6 is the last standard, coming into effect since September 2014, with an emission reduction of 97% by mass as compared to that of Euro 1 vehicles (1992). Figure 1 displays the Euro standards evolution for light vehicles according to Fallah et al. (2014).

Figure 1.

Evolution of level emissions according to European standards (Fallah et al., 2014).

The amount of pollutant $p$ emitted per kilometer (g/km) from Euro class vehicles is calculated by the following formula

Q^{p} (v) = \frac{a^{p} + c^{p} v + f^{p} v^{2}}{1 + b^{p} v + d^{p} v^{2}}

(1)

where

a

b

c

d

, and

f

are the vehicle-specific parameters (age, Euro class, energy),

p

the pollutant types and

v

stands for the average vehicle speed in kilometer per hour (Kolak et al., 2013).

Table 1 shows the average emission factors in (g/km) calculated with equation (1), according to the vehicle classifications of the Euro standards.

Table 1.

Emission levels according to Euro standards for passenger vehicles (www.rac.co.uk).

	Date of approval	Gasoline $C O + NOx + P M$	Diesel $C O + NOx + P M$	Average	Average per class
Euro 1	1992	3.69	3.83	3.76	2.598
Euro 2	1996	2.27	1.78	2.025
Euro 3	2000	2.45	1.57	2.01
Euro 4	2005	1.08	1.075	1.0775	1.015
Euro 5	2009	1.165	0.915	1.0575
Euro 6	2014	1.065	0.755	0.91

Several strategies like OVTP have been applied in metropolises to improve the performance of transport systems in term of environmental problems. For instance, the application of the OVTP technique in urban networks is done by sticking on cars environmental vignettes marked “ecological”, basing on EURO standards (Holman et al., 2015). Categories with EURO 1 and EURO 2 stickers are then banned to circulate in high emission zones. However, the strict application of the OVTP in large metropolises is subject to the sine-qua-nonrequirement, inter alia, to make available an expanded public transport network and sufficient parking spaces.

In this paper, we propose a new technique, a lighter version of OVTP, which will allow polluting vehicles to circulate in the banned urban areas, but with limitation measures. We will refer to as the “old vehicles traffic limitation (OVTL)” technique. The originality of the OVTL solution is to allow old cars an eco-friendly traveling in urban areas. This way, instead of banning old vehicle drivers from circulation, it will constrain them to use less polluted and less frequented roads. This suggestion breaks down the demand (user sets) into two parts: the recent cars for which drivers have total freedom in their route choices, and old cars for which drivers have the obligation to use the most environmentally friendly roads.

The proposed OVTL solution consists to merge an emission estimation model (Kolak et al., 2013; Wang et al., 2016), with a traffic forecasting model, known as the traffic assignment problem (TAP). The emission estimation model will measure the quantity of gas in a network axis, whereas the TAP model will be the key element in the urban travel demand forecasting process (https://www.caliper.com/glossary/what-is-traffic-assignment.htm), mainly used to estimate the urban networks traffic flows (https://www.caliper.com/transcad/transcadversions.htm).

A mathematical formulation is presented to describe the OVTL technique proposed using an environmental constraint in the TAP (Sheffi, 1985). The formulation is proved and analyzed using the equilibrium principles and optimality conditions. According to Chen et al. (2011), the proposed model’s environmental constraint consists of establishing, for different road of the network, a pollution threshold $P_{s}$ to not exceed. For example, the pollution $g_{a}$ emitted by flows $x_{a}$ through a link $a$ , must be less than the threshold pollution limit $P_{s}$ given by specialists. The mathematical formulation of this statement is as follows

g_{a} (x) \leq P_{s}

(2)

where

g (x)

is a function of the total amount of pollution emitted by the flow

x

and

P_{s}

is the threshold imposed by transport planners. Trends of

g (x)

is discussed in the following section.

Based on the output of this modeling, OVTL will offer an ecological travel plan with many advantages discussed further. The application of OVTL in urban areas will be facilitated by the development of information technology.

This paper is organized as follows: the next section emphasizes some related works modeling traffic emission estimation and introduces the classical models for traffic assignment. Further section presents the mathematical formulation of the proposed technique and the equilibrium characteristics, with a detailed discussion of the proposed model components. Finally, the last section provides some numerical examples to explain the proposed model properties and the advantages of the suggested technique.

Related works

Currently, several researchers use the classical TAP to describe and solve more complex problems. Specifically, environmental studies use TAP to estimate the air pollution levels in urban area. In the following sub-section, a literature review of some relevant works on emission estimations and road traffic distribution models is presented.

Traffic emission models

Emission models can be used as tools to assess the environmental performance of traffic, such as total emissions and concentration based on existing or resulting situations of a predefined policy. Modeling emission estimation and its calculation is a broad area of research. In the first studies, Guensler and Sperling (1994) showed that vehicle emissions are highly dependent on traffic speed. Subsequently, many researchers proposed solid studies demonstrating that traffic emissions depend on several factors such as vehicle type and age (Caserini et al., 2013), speed and travel time (Aziz et al., 2017), and weather conditions (Knittel et al., 2016). In practice, and in terms of simulation, there are several static and dynamic tools that use these emission factors to model or estimate emissions such as Tremove (www.tmleuven.be/), Moves (motor vehicle emission simulator: www.epa.gov/moves), and Copert (computer program to calculate emissions from road transport: http://emisia.com). From a sustainability perspective, Hizir (2006) reported that modeling emissions requires to study the effects of congestion on emission quantities and emissions relative to the flow in a network, which can easily reflect actual amount of emitted gas. So far, Yin and Lawphongpanich (2006) adopted a nonlinear form formula to calculate the emission quantities of $C O$ based on the travel time $t_{a} (x_{a})$ , the distance traveled $L_{a}$ and the traverse speed $V$ . The formula was previously presented by Wallace et al. (1998) as

P_{a} (x_{a}) = 0.2038 t_{a} (x_{a}) \exp (\frac{0.7962 L_{a}}{t_{a} (x_{a})})

(3)

In the same year, Hizir (2006) proposed in his thesis a total emission function in terms of traffic flow and physical capacity of the link, expressing a mathematical relationship between the traffic flow and the speed

P_{a} (x_{a}) = D (C a p_{a}) L_{a} e xp (F (C a p_{a}) x_{a})

(4)

where

D (C a p_{a})

and

F (C a p_{a})

are parameters that depend on the capacity

C a p_{a}

of the link

a

. These parameters are determined by the adaptation software that uses the emission factor tables for the corresponding pollutant (Hizir, 2006).

Although formulations (3) and (4) have been proposed to estimate carbon monoxide $C O$ emissions, recent works of Li et al. (2012) and Xu et al. (2015) pointed out that they are also valid for other pollutants. However, they assume that all vehicles are identical in terms of emission and do not consider the Euro classes. In the present study, the linear estimation function proposed by Wang et al. (2016) is employed, which accounts for the emission classes cars (Table 1) and is expressed as follows

P ({x_{a}}^{i}) = {x_{a}}^{i} E^{i} L_{a}

(5)

where

{x_{a}}^{i}

is the flow for the class

i

on the link

a

E^{i}

stands for the emission factor of the class

i

(Table 1), and

L_{a}

is the arc length of

a

Traffic assignment problems

In road traffic networks, the traffic allocation problem for congestion analysis and description of driver behavior attracted much attention in economy, transport planning, and operational research for several decades. A study of a basic model with one origin, one destination and two parallel roads are attributed to Pigout (1920) and was resolved by Knight in 1924 (Sheffi, 1985). Later, in the fifties, an essential concept emerged in this field: the “Wardropean equilibrium” introduced by Wardrop (1952). He stated two descriptive principles of equilibrium situations between supply (network) and demand (travelers). According to these principles, there are two types of equilibria, the first is called user equilibrium (UE) in which travelers seek to minimize their individual travel times, and will be achieved when no traveler has any incentive to change his decision in order to reduce his travel time. The mathematical formulation of the user equilibrium problem was first developed by Beckmann et al. (1956) who proved the equivalence, existence, and uniqueness of the solution. In an editorial note, Boyce (1981) outlined the historical development of the UE concept, including many algorithms and problem formulations. Nowadays, new methods have appeared to improve the UE solution such as the origin-based method (Bar-Gera, 1999; Nie, 2010). The second equilibrium is the social equilibrium or optimal system (OS), and aims to find an optimal distribution to minimize the overall traveling time carried out by all users in the network. The former is objective for decision makers, transportation and road traffic planners, and will be achieved when no road change can reduce overall travel time. OS is used to relieve the network and to reduce congestion. It was studied in Sheffi (1985) who noted that in this equilibrium, vehicles are supposed to be guided, thus the driver has no freedom in his route choice decisions which this is not evident in actual traffic networks.

One of the classes formally related to these two equilibria is the side-constrained class, which have been studied in the literature by (Larsson and Patriksson, 1995) and (Larsson and Patriksson, 1999). Less attention has been paid to the theoretical and practical specifications of various traffic constraints and control policies on a real road in an urban network. In order to fill this gap, Chen et al. (2011) have specified a generalized lateral constraint by considering different types of traffic constraints as traffic management strategies or control policies encountered in most urban networks. Different types of lateral stresses have been imposed. For example, to describe the queue effects at reported intersections or behind bottlenecks, physical capacity constraints may be an appropriate choice. On the other hand, to reflect the environmental requirements imposed by the government, some environmental constraints should be applied. In general, the treatment of different traffic constraints, traffic management strategies, control policies, or environmental constraints requires different secondary constraints depending on discrimination nature (Bagloee and Asadi, 2019). In the literature data, there are several approaches to solve the “SC-TAP” as the dual ascent algorithm, the internal penalty technique (Nie et al., 2004; Prashker and Toledo, 2001) and the augmented Lagrangian (AL) method (Larsson and Patriksson, 1995). However, all previous works focused on a special model formulated as a convex program with capacity constraints fixed on arcs or paths.

In-depth research has been carried out to control the negative externalities of motorized vehicles traffic pollution. During the last two decades, several solutions based on optimization models have been proposed, in order to determine the best network management requirements due to series of environmental objectives and constraints. This provides healthier and more useful tools in the planning process of demand management systems.(Benedek and Rilett, 1998) proposed an equilibrium model by adding to the objective functions of UE and OS a general emission function of $C O$ based on traffic velocity. It was applied on Ottawa (Canada) networks and shown to be effective. Several researchers adopted the general models of equilibrium with side-capacity constraints (Larssonand Patriksson, 1995), by changing the physical constraint (Chen et al., 2011; Xu et al., 2015). A model with environmental constraints has also been applied to describe the traffic equilibrium problem with antipollution traffic permits. On the other hand, some researchers have used the multiobjective aspect by seeking a compromise between the travel time and tolls criteria (Sharma and Mathew, 2011; Szeto et al., 2012).

Another direction was taken in the equilibrium models and pollution reduction in terms of emission pricing, whereby researchers gave a pricing toll according to the emission quantities. A toll system was established and models were proposed as tools for congestion analysis and environmental pricing in networks (see Johari and Haghshenas, 2019; Li et al., 2012).

The bi-level model is generally used in a two-level equilibrium problem with emission considerations. The higher level involves travel decisions to achieve a pollution abatement objective sometimes in the form of congestion or toll pricing. The lower level consists of modeling the individual equilibrium (UE) reflecting the decisions of rational users, hence their decisions will be respected in the higher level, with regard to Hizir (2006), Huang et al. (2009), and Kolak et al. (2013).

The user-environmental equilibrium formulation

Throughout this study, the proposed technique OVTL is formulated, by giving a bi-level program for describing the optimal allocation of the old cars prohibited to circulate in the existed OVTP technique. Subject to an imposed environmental thresholds, the result of this formulation is in term of optimal old cars flow distribution, taking into account the recent car drivers choices. In other words, given two classes of vehicles, polluting and nonpolluting, what is the optimal distribution for polluting vehicles in the urban network, considering that drivers of nonpolluting vehicles have the freedom to choose their shortest paths leading to their destinations.

The road network is represented by a graph denoted $G = (N, A)$ where $N$ is a set of consecutively numbered nodes representing the street crosses and $A$ is a set of links $a$ also consecutively numbered representing the portions of streets.

Let $O$ be the set of origins and $D$ the set of destinations.

Let $q_{o d} = \underset{i = 1, 2}{\cup} q_{o d}^{i}$ denotes the travel demand between the origin $o \in O$ and destination $d \in D$ , $i = 1, 2$ stands for recent or old vehicles respectively. The travel demand for each class is represented by a matrix containing the travel rate between each $(o, d)$ couple, so-called origin–destination matrices.

Note by $x_{a} = \sum_{i = 1}^{2} ‍ x_{a}^{i}$ the total flow on arc $a \in A$ and $C a p_{a}$ is the capacity of the link.

$t_{a}$ is the travel time based on the so-called link performance function $a$ , giving the link traverse time based on the flow denoted by $t_{a} = t (x_{a})$ . Generally, this function is called performance function (BPR, 1964), and given as follows

t_{a} (x_{a}) = t_{a} (0) [1 + α {(\frac{x_{a}}{C a p_{a}})}^{β}]

(6)

where $α$ and $β$ are constants of congestion measurements. The commonly

used value are $(α, β) = (0.15, 4)$ .

$f_{k}$ is the flow traversing the path $k \in K$ given by $f_{k} = \sum_{i = 1}^{2} ‍ x_{k}^{i}$ where $K$ is the set of all paths, $K_{o d}$ is the set of paths connecting the $(o, d)$ couple

Let $C_{k}$ be the cost of the path $k$ : $C_{k} = \sum_{a} ‍ t_{a} (x_{a}) δ_{a, k}$ , $a \in A$ , where $δ_{a, k}$ is the link-path incidence matrix

δ_{a, k} = \{\begin{array}{l} 1, & i f a \in k \\ 0, & Otherwise \end{array}

A flow on an arc $a$ , i.e. $x_{a}$ , can be decomposed into the sum of the flow along the different paths $k$ , i.e. $f_{k}$ which pass through this arc. This relationship is defined as follows

x_{a} (f_{k}) = \sum_{k} ‍ f_{k} δ_{a, k} \forall a \in A

(7)

Hypothesis

To simplify the problem, the following assumptions were adopted:

Static cases have been considered, i.e. the travel request between the origin–destination pair $q_{d}^{o}$ is a fixed quantity and the variation in the ratio of pollution caused by vehicles belonging to the same class is fixed to zero.

The vehicles classes are distinguished by grouping in the first class of old vehicles (Euro1, Euro2, Euro3) and new vehicles (Euro4, Euro5, Euro6) in the second class (see Table 1).

Three gases $CO$ , $N O_{x}$ , and $PM$ were considered. The emission factors of the three gases will be grouped into a single factor since we can measure them in grams per kilometer as illustrated in Table 1.

A single average emission factor for each vehicle class is calculated and taken into account (see Table 1).

The study focuses only on passenger vehicles for two reasons: (i) the heavy vehicle emissions are negligible compared to the passenger vehicles (EPA, 2017; Muncrief, 2016); (ii) for simplicity, the heavy vehicles can be converted to a number of passenger vehicles in terms of produced emissions.

The OVTL technique can be formulated as a bi-level program by extending the standard traffic assignment TAP and the constrained traffic assignment SC-TAP models as follows

\min Z (x_{a}^{2}) = \sum_{a} ‍ x_{a}^{2} t_{a} (x_{a}^{1}, x_{a}^{2})

(8)

s . t . \sum_{k} ‍ f_{k, o d}^{2} = q_{o d}^{2} \forall o \in O, d \in D

(9)

g_{a} ((x_{a}^{1})^{*}, x_{a}^{2}) \leq P_{s}, \forall a \in A

(10)

f_{k}^{2} \geq 0 \forall k \in K

(11)

where

{((x_{a}^{1})^{*})}_{a \in A} = argminT (x^{1}) = \sum_{a} ‍ \int_{0}^{x_{a}^{1}} ‍ t_{a} (w) d w

(12)

\sum_{k} ‍ f_{k, o d}^{1} = q_{o d}^{1} \forall o \in O, d \in D

(13)

f_{k}^{1} \geq 0 \forall k \in K

(14)

The proposed formulation (8–14) is a bi-level model so that the upper level designates the optimal allocation of old cars while respecting the results of the lower level in terms of an optimal distribution of new cars.

In the upper level, the objective function minimizes the total travel time used by all old vehicles drivers in the network. The old cars optimal distribution (flow pattern) that minimizes this objective function will satisfy:

The flow conservation constraints (9), which states that the sum of flows on all paths $k \in K_{o d}$ connecting the $o d$ pair is equal to the travel rate (for older cars) between $o$ and $d$ .

The constraints (10) that represent the general form of environmental constraints on arcs (Xu et al., 2015).

The in-equation (11) which is a non-negativity constraint and means that the flows on arcs are physically realistic.

The generalized side-constraints (10) can be formulated as

g_{a} (x_{a}^{1}, x_{a}^{2}) = x_{a}^{1} P_{a} (x_{a}^{1}) + x_{a}^{2} P_{a} (x_{a}^{2})

(15)

It prevents, in terms of pollution, users with old vehicles from crossing saturated arcs in relation to a certain pollution threshold on arcs i.e. $P_{s}$ , where $P_{a} (x_{a}^{1})$ and $P_{a} (x_{a}^{2})$ are the quantities of pollution on arc $a$ caused by new cars and old cars. This is calculated according to the flows and distance traveled between the ends of arc $a$ . The pollution threshold on arcs $P_{s}$ will be fixed as a constant defined by organizations specialized in the environment (see, for example, the Approved Air Quality Monitoring Associations (AASQA) include 26 associations (www.lcsqa.org/aasqa), which are responsible for monitoring air quality in the different French regions); Chen et al., 2011).

The superior model is similar to traffic assignment with side-capacity constraints “SC-TAP” that aims to affect travel demands between $o, d$ pairs in the traffic network with capacity threshold constraints on the arcs. Caserini et al. (2013) provided more details on the “SC-TAP” study. For the resolution of “SC-TAP”, the adaptation of the AL algorithm is the most efficient (Larsson and Patriksson, 1995; Nie et al., 2004).

The lower level is the well-known formulation of Beckmann et al. (1956) for the UE problem (Bar-Gera, 1999; Sheffi, 1985). The objective function minimizes the average travel time over the entire network. The recent cars optimal distribution (flow pattern) that minimizes this objective function will satisfy:

The flow conservation constraints (13), that verify that the sum of the flows on all paths $k \in K_{o d}$ connecting the $o d$ pair is equal to the travel rate (for recent cars) between $o$ and $d$ .

The in-equation (14) is a non-negativity constraints which means that the flows on arcs are physically realistic

As mentioned in the earlier section, UE is based on the first Wardrop principle (Wardrop, 1952), such that each user wishes to minimize his travel time, so that travel times on all used paths between each $o d$ pair are equal. And the travel time on an unused path is equal to or greater than the travel time used. In the equilibrium state, no user can reduce travel time by unilaterally changing the path. The mathematical expression for UE is

f_{k, o d} (C_{k, o d} - C_{\min, o d}) = 0, \forall k, o, d

(16)

C_{k, o d} - C_{\min, o d} \geq 0, \forall k, o, d

(17)

These two formulae (16) and (17) guarantee that travel time $C_{k, o d}$ on all used paths $k \in K_{o d}$ from origin $o$ to destination $d$ are equal to the minimum $c_{\min}^{o d}$ and the travel time on any unused path $f_{k, o d}^{1} = 0$ is equal to or greater than the minimum travel time. Therefore, the lower problem finds the optimal distribution for recent vehicles in the network, and estimates the quantity of gas emitted by this class in the network s main roads. This problem may be solved by the Frank and Wolfe (1956). This algorithm is a first order iterative optimization algorithm for constrained convex optimization problems.

Lemma. The upper-level solution is a user-environmental equilibrium (UEE), which ensure that older cars will be assigned to the shortest roads with respect to an additional cost depending on the environmental threshold.

Proof. This demonstration is based on (Larsson and Patriksson, 1999). To show that the upper level solution is an optimal distribution in terms of environmental equilibrium, one passes by the verification of the conditions of Karush–Kuhn–Tucker “K.K.T”. The constraint (10) is replaced by its general form, the model with an environmental constraint

\min Z (x_{a} (f_{k})) = \sum_{a} ‍ x_{a} t_{a} (x_{a})

(18)

s . t . \sum_{k} ‍ f_{k, o d} = q_{o d} \forall o \in O, d \in D

(19)

G_{a} (x_{a}) \leq 0, \forall a \in A

(20)

f_{k} \geq 0 \forall k \in K

(21)

The associated Lagrange equation of this model is depicted as follows

L (x, λ, μ) = Z (x (f)) + \sum_{o d} ‍ λ_{o d} (q_{o d} - \sum_{k} ‍ f_{k, o d} δ_{k, o d}) - \sum_{k} ‍ μ_{k} f_{k} + \sum_{a} ‍ π_{a} (G_{a} (x_{a}))

(22)

Where, $δ_{k, o d}$ is the path-pair incidence matrix $(o, d)$ , $λ_{o d}$ , $π_{a}$ , and $μ_{k}$ are the Lagrange multipliers associated with equations (19) and inequalities (20) and (21) respectively.

After derivation, the “K.K.T” conditions will be as follows

\forall (o d) \in (O D), k \in K_{o d}, a \in A

\{\begin{array}{l} f_{k, o d} ({\tilde{C}}_{k, o d} + \sum_{a \in A} ‍ δ_{a, k} (\sum_{a \in A} ‍ π_{a} \frac{\partial G_{a} (x_{a})}{\partial x_{a}}) - λ_{o d}) = 0 \\ {\tilde{C}}_{k, o d} + \sum_{a \in A} ‍ δ_{a, k} (\sum_{a \in A} ‍ π_{a} \frac{\partial G_{a} (x_{a})}{\partial x_{a}}) - λ_{o d} \geq 0 \\ f_{k, o d} \geq 0 \\ - π_{a} G_{a} (x_{a}) = 0 \\ - G_{a} (x_{a}) \geq 0 \\ - π_{a} \geq 0 \end{array}

(23)

with

{\tilde{C}}_{k, o d}

being the generalized marginal path cost associated with optimum system equilibrium (OS).

The two first equations in the system (23) indicate the user equilibrium with a marginal cost ${\tilde{\tilde{C}}}_{k, o d}$

{\tilde{\tilde{C}}}_{k, o d} = {\tilde{C}}_{k, o d} + \sum_{a \in A} ‍ δ_{a, k} (\sum_{a \in A} ‍ π_{a} \frac{\partial G_{a} (x_{a})}{\partial x_{a}})

(24)

That is, when $f_{k, o d} = 0$ then ${\tilde{\tilde{C}}}_{k, o d} \geq λ_{o d}$ and if $f_{k, o d} \geq 0$ then ${\tilde{\tilde{C}}}_{k, o d} = λ_{o d}$

The additional cost in (24) i.e. $δ_{a, k} (\sum_{a \in A} ‍ π_{a} \frac{\partial G_{a} (x_{a})}{\partial x_{a}})$ is interpreted as a marginal cost for pollution reduction (environmental criteria). For example, pricing on the arc $a$ that travelers must pay if they want to use the saturated link in terms of pollution.

Finally the Lagrange multiplier $λ_{o d}$ can be interpreted as a generalized minimum cost (taking into account the environmental threshold) of paths connecting a couple $(o d)$

λ_{o d} = m i n_{k \in K_{o d}} [{\tilde{\tilde{C}}}_{k, o d}]

(25)

Hence, equation (25) indicates that the UEE will be achieved if all users (old cars) are assigned to the shortest roads “ $λ o d ”$ with respect to the environmental threshold. And no user can improve the quality of his travel or traffic state by changing this road.

In practice, this model can be used to predict the distribution of travel demand (categorized into two classes) in a real urban network, where the recent vehicle class has the freedom to choose their routes (minimum time). With regard to older vehicle class, the additional cost caused by environmental constraints can be transformed into a crossing toll in a congested network links, evaluation studies must be performed for the toll measurements (Li et al., 2012; Sharma and Mathew, 2011). Contrary to existing models in the literature (e.g. Hizir, 2006), the particularities of the proposed model are first, the consideration of the environmental constraint linearity by using function (5), this ensures the model convexity and consequently, it guarantees the uniqueness of the optimal solution and increases the speed and precision of the solution. Second, the categorization of vehicles, which ensures that the quantities of emissions are accurately estimated.

Numerical examples

To show the benefits of the proposed technique OVTL, the resolution of the proposed mathematical formulation have been made using the adaptation of the Lagrangian augmented (AL; Larsson and Patriksson, 1995) and the Frank–Wolf method (Sheffi, 1985) for the upper and lower levels, respectively. We discuss the two following cases.

First example

The first example is a two-route directed graph shown in Figure 2, which consists of seven nodes and seven arcs. The physical capacity of each arc is $C a p_{a} = 100$ and the length $L_{a} = 1 km$ . There is a pair $(o d) = (1, 7)$ and the travel demand between $(1, 7)$ is broken down into two classes: $q_{1, 7}^{1} = 70$ and $q_{1, 7}^{2} = 50$ . The environmental threshold on each arc $P_{s} = 120 g$ . The travel time on empty arcs equals $1$ unit time. There are two paths connecting the pair $(1, 7)$ , $k_{1} = a - c - e$ and $k_{2} = b - d - f - g$ . The costs of the paths when the flow is absent are $C_{1} = 3$ and $C_{2} = 4$ respectively.

Figure 2.

The two routes network.

Following these inputs, we applied the previous procedure to find the resulting user distribution in the network using the proposed technique. The results were compared to the distribution according to user equilibrium UE model. Table 2 displays the distribution of vehicles of both classes based on UE model criteria. i.e. the two classes seek to minimize travel time without taking into account the pollution threshold.

In Table 2, $x_{a}$ and $P (x_{a})$ indicate the link flow (of both classes) and the resulting pollution of this link, respectively.

Table 2.

The user equilibrium optimal assignment results.

Link	$x_{a}^{1}$	$P_{a}^{1} (x_{a}^{1})$	$x_{a}^{2}$	$P_{a}^{2} (x_{a}^{2})$	Overall $P_{a}$	$P_{s}$
a	70	71	50	130	201	120
b	0	0	0	0	0	120
c	70	71	50	130	201	120
d	0	0	0	0	0	120
e	70	71	50	130	201	120
f	0	0	0	0	0	120
g	0	0	0	0	0	120

It is clearly seen that the total demand is assigned to the shortest path $a - c - e$ . Consequently, this path is heavily congested and the pollution threshold is exceeded $P_{s}$ (excess environmental threshold with 67% more pollution). Whereas, the longer second path $b - d - f - g$ is remained empty. This distribution is not significant because the concentration of pollution on the congested route produces a human health risk and, in addition, the inequitable use of roads creates an unbalanced economy in urban areas.

The proposed model results are presented in Table 3. The results include recent and old vehicles optimal distribution in terms of flow pattern, taking into consideration of the environmental criterion.

Table 3.

The OVTL model optimal assignment results.

Link	$x_{a}^{1}$	$P_{a}^{1} (x_{a}^{1})$	$x_{a}^{2}$	$P_{a}^{2} (x_{a}^{2})$	Overall $P_{a}$	$P_{s}$
a	70	71	18	46.5	117.5	120
b	0	0	32	82.8	82.8	120
c	70	71	18	46.5	82.8	120
d	0	0	32	82.8	82.8	120
e	70	71	18	46.5	117.5	120
f	0	0	32	82.8	82.8	120
g	0	0	32	82.8	82.8	120

Table 3 shows that adopting the OVTL policy is advantageous because there are no crowded paths and no pollution excess. Drivers with recent vehicles are assigned to the shortest route following the criterion of travel time minimization (results of the lower level), and drivers with old or polluting vehicles are directed to paths dependent on an additional generalized route (defined by equation (24)) and less polluted with respect to the environmental pollution threshold (upper level). The latter solution can benefit in terms of environmental measurements. In fact, 2% less pollution was observed as compared to the environmental threshold. In addition, due to the offered fluidity, the overall travel time in the network resulting from the proposed model distributions is better of the UE model results, as shown in Table 4.

Table 4.

Overall system travel time comparison UE vs OVTL.

Assignment	UE	OVTL
T	7.933	7.276

UE: user equilibrium; OVTL: old vehicles traffic limitation.

Second example

In this example, a larger network (Figure 3) have been considered to a better view of the practical feasibility of the OVTL model, and to show more the advantages of OVTL technique in a real network. The network is oriented, includes $11$ nodes, $40$ links, and four $(o, d)$ pairs: $\{(1, 10), (1, 11), (11, 1), (11, 10)\}$ and the travel demand between $(o, d)$ is ${3000, 4000, 4000, 3000}$ , respectively. This travel demand $q_{o d}$ is decomposed into two equal classes of vehicles (old and recent): $q_{o d}^{i} = \frac{1}{2} q_{o d}$ . Finally, the length of each link in the network is $L_{a} = 1$ unit.

Figure 3.

Prashker and Toledo (2001) network.

The link characteristics are shown in Table 5.

Table 5.

Network characteristics.

From	To	$Cap$	$t_{0}$	From	To	$Cap$	$t_{0}$
1	2	4000	30	6	10	2000	6
1	3	4000	30	7	3	4000	30
1	5	3000	20	7	8	4000	20
1	6	3000	20	7	11	4000	30
2	1	4000	30	8	6	3000	15
2	4	4000	30	8	7	4000	20
2	5	4000	20	8	10	2000	6
3	1	4000	30	8	11	3000	20
3	6	4000	20	9	4	4000	20
3	7	4000	30	9	5	3000	15
4	2	4000	30	9	10	2000	6
4	9	4000	20	9	11	3000	20
4	11	4000	30	10	5	2000	6
5	1	3000	20	10	6	2000	6
5	2	4000	20	10	8	2000	6
5	9	3000	15	10	9	2000	6
5	10	2000	6	11	4	4000	30
6	1	3000	20	11	7	3000	30
6	3	4000	20	11	8	3000	20
6	8	3000	15	11	9	3000	20

In Table 6, the link-flow emissions results of both models UE and OVTL were presented. The link-flow emissions resulting from the UE policy were calculated by function (5) only. However, the link-flow emissions resulting from the OVTL policy have been performed by functions (5) and (15), and by using the assignment procedure. These findings are compared with the link environmental capacity (threshold), which is defined according to the physical link capacity of Table 5.

Table 6.

Results of OVTL model.

From	To	UE	OVTL	Threshold
1	2	0	3237.5	6000
1	3	0	3237.5	6000
1	5	6308.75	3071.25	4500
1	6	6308.75	3071.25	4500
2	1	0	3237.5	6000
2	4	0	3237.5	6000
2	5	0	0	6000
3	1	0	3237.5	6000
3	6	0	0	6000
3	7	0	3237.5	6000
4	2	0	3237.5	6000
4	9	0	0	6000
4	11	0	3237.5	6000
5	1	3605	367.5	4500
5	2	0	0	6000
5	9	1478.05	1296.75	4500
5	10	4830.7	1774.5	3000
6	1	3605	367.5	4500
6	3	0	0	6000
6	8	1478.05	1296.75	4500
6	10	4830.7	1774.5	3000
7	3	0	3237.5	6000
7	8	0	0	6000
7	11	0	3237.5	6000
8	6	1478.05	1296.75	4500
8	7	0	0	6000
8	10	4830.7	1774.5	3000
8	11	3605	367.5	4500
9	4	0	0	6000
9	5	1478.05	1296.75	4500
9	10	4830.7	1774.5	3000
9	11	3605	367.5	4500
10	5	2126.95	0	3000
10	6	2126.95	0	3000
10	8	2126.95	0	3000
10	9	2126.95	0	3000
11	4	0	3237.5	6000
11	7	0	3237.5	4500
11	8	6308.75	3071.25	4500
11	9	6308.75	3071.25	4500

UE: user equilibrium; OVTL: old vehicles traffic limitation.

It is shown that the vehicles categorization in two classes has provided a valuable advantage. Indeed, the UE distribution without categorization is not significant (Figure 4), the links belonging to the shortest paths are severely congested, very polluted, and some paths like $1 - 3 - 7 - 11$ and $1 - 2 - 4 - 11$ are empty, as shown in Figure 4.

Figure 4.

The result of UE assignment-link pollution.

However, the OVTL technique has good results in terms of optimal link flow distribution. On the one hand, the pollution has been reduced and health safety maximized. On the other hand, the crossing of unused axes in the network is maximized. For example, the allocation of link (1,2), (2,4)(1,3), (3,7), and (7,11) have been used in the UE assignment which equilibrate the economy. In addition, the congested or saturated links are relieved, such as links (1,6), (1,5), (11,8), and (11,9) that improve the traffic management as displayed in Figure 5.

Figure 5.

The result of OVTL assignment-link pollution.

A histogram of link flow emissions displayed in Figure 6 shows more clearly the improvements bringing by the proposed technique.

Figure 6.

Comparison of the links pollution: OVTL, UE, and Threshold.

In each section (link) of the network, the resulting emission quantity of EU and OVTL are shown in red and green respectively. Both are compared with the pollution threshold (purple). The emissions resulting from EU exceed the threshold in the majority of the links. On the other hand, the pollution of OVTL respects this threshold in all the links of the network.

Moreover, due to the network relief by OVTL, speed has increased. Consequently, the overall travel in the network time has decreased by 6% (Table 7).

Table 7.

Overall system travel time comparison UE vs OVTL.

Assignment	UE	OVTL
T (unit)	860,451	802,333

UE: user equilibrium; OVTL: old vehicles traffic limitation.

Conclusion

Eco-friendly considerations are getting a growth attention due to its importance in planning a sustainable ecological transportation system. An imposed environmental threshold is a useful means to explicitly reflect various environmental protection requirements imposed by governments. In this work, a mathematical model is proposed describing vehicle distribution controlled by a new technique OVTL in urban networks. Vehicles are categorized into two classes: recent cars for which drivers have a total freedom in their choice of routes, and old cars for which drivers are forced to take less polluted paths. The formulation has been proved and analyzed through the use of the traffic assignment problem with the environmental side-constraints, and the optimality conditions. Two network examples are presented and discussed to explain the properties of the proposed model and the advantages of the suggested technique. Many good results have surfaced as, the relief of network congestion, the reduction of pollution and the use of the maximal number of links. Although the numerical results for small and middle networks straightforwardly illustrate the essential advantages of our technique, we recognize that case studies of large and lifelike networks are needed to auxiliary confirm the outcomes of the numerical examples and the performance of the proposed model. Finally, OVTL can be considered as an essential instrument for modeling and evaluating environmental protection requirements, and a tool for studying congestion and environmental reducing in urban road networks. For this, as a perspective, we intend to process this technique through the categorization of demand into several classes according to the “six euro norms” by priority and to take into account the uncertainty aspect for several parameters such as uncontrollable overemission of starting and stopping, speed, climate change, fuel quality, etc. in order to effectively provide good and real descriptions for sustainable transport planning policies.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

Abdelfettah Laouzai is currently a PhD student at the University of Science and Technology Houari-Boumediene,and works as a researcher at SONATRACH-Central Direction of Research and Development. Algeria. Abdelfettahperforms research on transportation science and operations research.

Rachid Ouafi currently works at the Department of Operations Research , University of Science and TechnologyHouari Boumediene. Rachid performs research in Applied Mathematics.

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Link	$x_{a}^{1}$	$P_{a}^{1} (x_{a}^{1})$	$x_{a}^{2}$	$P_{a}^{2} (x_{a}^{2})$	Overall $P_{a}$	$P_{s}$
a	70	71	50	130	201	120
b	0	0	0	0	0	120
c	70	71	50	130	201	120
d	0	0	0	0	0	120
e	70	71	50	130	201	120
f	0	0	0	0	0	120
g	0	0	0	0	0	120

Link	$x_{a}^{1}$	$P_{a}^{1} (x_{a}^{1})$	$x_{a}^{2}$	$P_{a}^{2} (x_{a}^{2})$	Overall $P_{a}$	$P_{s}$
a	70	71	50	130	201	120
b	0	0	0	0	0	120
c	70	71	50	130	201	120
d	0	0	0	0	0	120
e	70	71	50	130	201	120
f	0	0	0	0	0	120
g	0	0	0	0	0	120

Link	$x_{a}^{1}$	$P_{a}^{1} (x_{a}^{1})$	$x_{a}^{2}$	$P_{a}^{2} (x_{a}^{2})$	Overall $P_{a}$	$P_{s}$
a	70	71	50	130	201	120
b	0	0	0	0	0	120
c	70	71	50	130	201	120
d	0	0	0	0	0	120
e	70	71	50	130	201	120
f	0	0	0	0	0	120
g	0	0	0	0	0	120