Understanding and managing the dynamic evolution of traffic emissions in urban road systems

Pengfei Wang; Xiangyu Wang; Keqiang Li; Limeng Wu; Hanting Li; Tianyu Wang; Peng Liu; Pengfei Wang; Xiangyu Wang; Keqiang Li; Limeng Wu; Hanting Li; Tianyu Wang; Peng Liu

doi:10.48130/dts-0026-0008

Existing literature has extensively explored the day-to-day evolution of traffic flows. However, traffic-flow dynamics and emission dynamics are not equivalent, and research on the day-to-day dynamic evolution of traffic greenhouse gas (GHG) emissions remains insufficient due to their complex nonlinear relationship. This study investigates the evolutionary characteristics and reduction strategies of GHG emissions from urban road traffic. We propose a three-dimensional nonlinear dynamic model that depicts the relationship among energy reserve, vehicle population, and road network area. Subsequently, the existence, uniqueness, and stability of the equilibrium solution for the urban road traffic network are investigated. Finally, from the perspective of macroscopic travel demand restrictions, the optimal authorized vehicle proportion (namely, the vehicles with the right to use the urban road traffic network) that minimizes traffic GHG emissions under several typical scenarios is proposed. Some key findings are obtained: (1) When the urban road traffic reaches a stable state, the three-dimensional nonlinear dynamic system achieves equilibrium, along with stable road network saturation, average vehicle speed, and traffic GHG emissions. (2) Taking a representative first-tier city in China as an example, it is found that adopting the optimal proportion of authorized vehicles significantly reduces GHG emissions from urban road traffic.

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Introduction

The escalation of urbanization and motor vehicle ownership has led to a pervasive issue of traffic congestion in cities worldwide, accompanied by a substantial rise in greenhouse gas (GHG) emissions. This predicament is underscored by authoritative reports, including those by the European Environment Agency^[1], Huang et al.^[2], and the United States Environmental Protection Agency^[3], which highlight the transportation sector's contribution to GHG emissions in the European Union, China, and the United States, accounting for 9%−10%, 27.34%, and 34.95%, respectively. To reduce GHG emissions, many studies have analyzed emission patterns under diverse traffic conditions and policies^[4,5].

The factors behind the evolution of GHG emissions in road transportation systems vary significantly. These factors can be categorized into three groups: traffic-operation factors, such as average speed, acceleration-deceleration behavior, congestion level, and network saturation; infrastructure-related factors, such as road characteristics, road network area, and capacity; and supply-demand factors, such as vehicle population, energy reserves, and vehicle energy-consumption intensity. Nevertheless, the direct comprehension, evaluation, and prognostication of network-wide traffic states and GHG emissions from disaggregated data pertaining to individual links remain challenging. This complexity stems from the intricate and chaotic dynamics of (disaggregated) traffic flows within congested networks, attributable to nonlinear traffic flow phenomena and driver route choice behaviors^[6]. Recently, a macroscopic perspective has shed light on the interrelationship between average traffic flow (traffic production), average traffic density (traffic accumulation), and average vehicle speed through empirical analyses^[7,8].

It is well-acknowledged that vehicle speed profoundly impacts GHG emissions. In this research field, some previous studies^[9,10] have discerned the mathematical association between GHG emissions and average vehicle speed using real-world data, revealing a U-shaped curve characterizing this relationship. This nonlinear relationship implies that similar traffic-flow levels may correspond to different emission outcomes under different congestion structures, speed profiles, and stop-and-go conditions. Therefore, emission dynamics cannot be inferred directly from flow dynamics in a simple one-to-one manner, which indicates that traffic-flow dynamics and emission dynamics are not equivalent. More specifically, a traffic system may approach a stable flow pattern while emissions remain sensitive to congestion structure, driving behavior, acceleration-deceleration patterns, and idling conditions. Consequently, a separate investigation of day-to-day emission evolution is necessary for long-term prediction and policy evaluation. Establishing a model that elucidates the connection among vehicle population (or the number of traveling vehicles under certain policies), road network area (or road network capacity), energy reserve, and GHG emissions at a macroscopic level offers a means to comprehend, forecast, and manage network-wide GHG emissions from urban traffic systems.

In practical terms, road managers are confronted with a pivotal question: "What factors influence GHG emissions from urban traffic systems, and how can they be reduced from a macroscopic perspective?". As Abbas & Bell^[11] suggested, system dynamics were well-suited to strategic issues and could provide a useful tool for supporting policy analysis and decision-making in the transport field. To address this, our study formulates a three-dimensional nonlinear dynamic model encapsulating the dynamics of energy reserve, vehicle population, and road network area. Subsequently, we conduct an in-depth analysis of the existence, uniqueness, and stability of analytical solutions within this model. This analysis yields valuable strategies for GHG emission reduction, particularly delineating optimal authorized vehicle proportions under varying scenarios. To corroborate our findings, we undertake a case study in Beijing, China, which substantiates the accuracy and validity of our conclusions.

The road usage measures permit a certain proportion of vehicles to travel on roads during specific time periods or under certain conditions. Implemented primarily in cities, these measures aim to mitigate traffic congestion, reduce traffic emissions, and curb environmental pollution. The determination of authorized vehicle proportion or demand restriction intensity may consider various factors, including vehicle license plate numbers, vehicle types, emission standards, or specific time periods. Examples include: (1) License plate restrictions. Some cities, such as Beijing, Mexico City, and Jakarta, regulate road access based on vehicle license plate numbers, determining which vehicles are permitted on the roads on specific days; (2) Alternate-day travel. In response to extreme air pollution, cities like Paris might mandate alternate-day travel, allowing different vehicle groups road access on successive days. The purpose of these measures is to decrease the overall number of vehicles on the road at given times. Cities and regions applying these measures can adjust the proportion of vehicles with the right to use the road network (hereafter referred to as the 'authorized vehicle proportion') and the proportion of vehicles barred from road access (hereafter called 'demand restriction intensity').

Comprehensively reviewing the literature, it becomes evident that research on urban-scale traffic dynamic evolution models considering GHG emission predictions and reductions is notably sparse. Our study aims to fill this gap by providing insights into optimizing urban traffic systems for effective GHG emission mitigation. The primary contributions of this paper are as follows: (1) By constructing a three-dimensional nonlinear dynamic model and conducting equilibrium analysis, this study demonstrates that energy reserve, vehicle population, and road network area, each reach a unique and stable state when the urban road traffic system is in equilibrium, and the system also exhibits unique and stable road network saturation, average vehicle speed, and GHG emissions. (2) We derive the optimal authorized vehicle proportion that minimizes GHG emissions under different time-horizon scenarios. (3) A case study of Beijing reveals that, after 10 years and 40 years, the optimizing authorized vehicle proportion can lead to a reduction of 0.90% and 4.02%, respectively, in GHG emissions from the urban road traffic system.

These findings hold significance for the formulation of sustainable and environmentally-friendly urban traffic policies and practices, providing a tangible strategy towards achieving carbon peak and carbon neutrality in the domain of urban road traffic systems.

Relevant literature

This study is mainly related to two branches of the existing literature mentioned above: the dynamic evolution of the urban road traffic system and its GHG emissions.

Dynamic evolution of transportation systems
In the realm of transportation research, at a macroscopic level, the prevailing consensus among scholars underscores the intricate and dynamic nature of regional traffic and its associated infrastructure systems, characterized prominently by self-organization tendencies and other intricate complexities. Within this context, the application of ecological theory has emerged as a valuable avenue for elucidating the evolutionary dynamics of urban road traffic systems. To exemplify, Abbas & Bell^[11] evaluated the strengths and weaknesses of system dynamics as an approach for modelling in the transportation area. Song et al.^[12] elucidated the dynamics of competition and mutually beneficial symbiosis among distinct modes of transportation. Shepherd^[13] set out a review of over 50 studies since 1994, including the adoption of alternate fuel vehicles, supply chain management affecting transport, highway maintenance, strategic policy, airport infrastructure, airline business cycles, and a range of emerging application areas. Similarly, Yang et al.^[14] delved into the genesis of urban traffic congestion, employing a system dynamics framework, and assessed the efficacy of various extant traffic control policies. Shao et al.^[15] on the other hand, employed a Logistic model to evaluate the evolution of regional transport infrastructure. Sun & Wang^[16] conducted a comprehensive analysis of the developmental trajectories and policy evaluations pertinent to new energy vehicles. Morozov et al.^[17] applied the theories of chaos, dynamic systems, and traffic flows to establish the relationship between road traffic chaos and road accident rates. In the literature, the term 'day-to-day' refers to a dynamic process with one day as the time step, in which the traffic system evolves iteratively from one day to the next and gradually approaches a stable state. This definition covers both short-term daily fluctuations and the longer transition toward equilibrium. At a micro-level, within the framework of the day-to-day dynamic evolution of traffic flow, numerous investigations have predominantly centered on the scrutiny of equilibrium solutions' existence, uniqueness, and stability, with a particular focus on divergent user route choice behaviors. For instance, the works of Li et al.^[18] and Liu et al.^[19] have contributed significantly to this aspect. Meanwhile, analyses have encompassed diverse traffic policies^[20−22]. Lastly, researchers have also considered the amalgamation of traffic flows involving autonomous vehicles and human-driven vehicles^[23].

GHG emissions from road transportation systems
In the field of transportation research, the primary focus of scholarly investigations has revolved around the precise prediction and efficacious mitigation of GHG emissions. Several seminal studies that are closely aligned with the purview of our study are delineated below: Previous studies^[9,24] have undertaken an empirical exploration into the interrelationship between traffic congestion, specifically vehicle speed, and GHG emissions, employing real-world datasets. Subsequently, they formulated a specific function for computing the GHG emission factor. Liu & Cirillo^[25] introduced a comprehensive model system aimed at forecasting GHG emissions emanating from household transportation activities. This model was subsequently employed to assess the ramifications of three distinct tax schemes: vehicle ownership tax, purchase tax, and fuel tax. Bharadwaj et al.^[26] conducted an exhaustive analysis of GHG emissions within the road transport sector of the Mumbai Metropolitan Region, utilizing methodologies grounded in fuel consumption and vehicle km-traveled metrics. Guzman & Orjuela^[27] presented an integrated model for reliable estimation of passenger transport emissions of local pollutants and CO₂ using an integrated Land-Use and Transport Interaction (LUTI) model in Bogota (Colombia) between 2010 and 2026. Zhang et al.^[28] provided a comprehensive overview of measurement methodologies (top-down and bottom-up), influencing factors (supply-side, demand-side, and environmental factors), and reduction strategies (administrative control, economic stimulus, and technology promotion) pertaining to carbon emissions arising from transportation activities. Alfaseeh et al.^[29] pioneered the development of a deep learning framework designed for the prediction of link-level GHG emission rates. This framework leveraged crucial predictors, including speed, traffic density, and historical GHG emission rates. Wang et al.^[30] presented an optimal dynamic strategy for the allocation of investments in the construction of intelligent transportation infrastructure and road maintenance, with the overarching objective of minimizing environmental costs. Wang et al.^[31] developed a comprehensive bottom-up GHG emissions assessment model that accounts for GHG emissions from newly produced commercial vehicles as well as those already in circulation. Although existing studies have investigated eco-driving in scenarios such as signalized intersections, hilly roads, and curved roads^[32−34], this study adopts a macroscopic dynamic perspective to analyze emission evolution and travel-demand restriction. In addition, research on multimodal transport emission reduction^[35−37] has provided important references for the low-carbon management and control of urban multimodal transportation systems.

The existing literature above has extensively explored the day-to-day evolution of traffic flow. However, given the complex nonlinear relationship between traffic flow and emissions, there is a lack of research on the day-to-day dynamic evolution of traffic GHG emissions. For that, from a macroscopic travel demand restriction perspective, the optimal authorized vehicle proportion that minimizes traffic GHG emissions under several typical scenarios is proposed.

Modelling the dynamic evolution of traffic emissions

A macroscopic strategy for managing traffic emissions

Specific expressions for parameters

Road network saturation

The parameter of road network saturation plays a pivotal role in determining the average vehicle speed, a relationship corroborated by macroscopic empirical studies^[7,8]. Consequently, we assert that road network saturation is a crucial determinant of the GHG emission factor. Building upon the work of Chen et al.^[40], it is worth noting that various definitions of road network capacity exist; However, several representative definitions have direct links to the road network area z. In general, road network capacity bz tends to expand proportionally with the road network area z, which is consistent with what we described in system (1).

From a traffic management perspective, it is necessary to consider the effect of parking occupancy on the road network capacity. Then, we define road network saturation s as follows:

$ s=\dfrac{\theta y}{b\left[z-e\left(1-\theta \right)\right]}=\dfrac{\sqrt{\theta {r}_{1}K}}{\sqrt{ab}\left[K-e\left(1-\theta \right)\right]}. $

(14)

Here, e signifies the adverse effect of non-traveling parked vehicles on the effective road network area due to travel demand management policies, hence it is referred to as the 'parking impact coefficient'. Notably, (illegal) on-street parking exerts a substantial influence on vehicle speed, a factor that cannot be disregarded^[41,42].

From Eq. (14), several key conclusions can be drawn: (1) Road network saturation s remains independent of vehicle scrapping rate r₂, the growth rate of road network area r₃, the increment coefficient of energy reserves on vehicle growth c, and the decrement coefficient of road network area occupied by vehicles d. (2) Road network saturation s increases with the magnitude of on-street parking impact e, since a reduction in the effective road network area available for vehicle travel leads to this outcome. (3) Enhanced road network planning and road design all contribute to an augmentation of conversion coefficient b and a reduction in road network saturation s, promoting more efficient traffic flow. (4) Road network saturation s increases (resulting in decreased average vehicle speed) as the growth rate of energy reserves r₁ escalates, signifying a larger vehicle population. (5) Conversely, road network saturation s decreases (resulting in increased average vehicle speed) with higher average energy consumption of vehicles a.

Average energy consumption

The non-negative energy consumption parameter a is widely acknowledged to be primarily contingent upon a set of determinants, including engine characteristics, traffic congestion levels, loading, and travel mileage. The first factor falls outside the purview of our present discussion, while the second factor has already been addressed in Eq. (1). Consequently, our focus here narrows down to an examination of the third and fourth factors.

Regarding the third factor, empirical investigations utilizing real-world data reveal a linear increase in energy consumption with vehicle loading^[43,44]. It is reasonable to posit that individuals unable to travel independently will opt for ride-sharing, as their initial mode choice likely centered on private cars. Simultaneously, we must account for detours incurred during ride-sharing arrangements. As for the fourth factor, the detour effect introduced by ride-sharing demands consideration. Hence, we define the average energy consumption as follows:

$ a=\left[{a}_{1}\left(1-\theta \right)+{a}_{2}\right]\dfrac{m}{\theta }, $

(15)

where, a₁ signifies the increment in energy consumption per vehicle with travel demand restriction intensity (1 − θ), namely, a₁ means maximal loading with θ = 0; a₂ represents the minimal energy consumption per vehicle under minimal loading; m encapsulates the detour coefficient which captures the additional ride-sharing and detour burden associated with restricted mobility, m/θ diminishes with the authorized vehicle proportion θ.

Notably, the socio-economic implications of reducing the authorized vehicle proportion are two-sided. On the positive side, moderate restriction can relieve congestion, reduce GHG emissions, improve air quality, and increase system-wide travel efficiency. It may also promote a shift toward public transport, ride-sharing, and other low-carbon travel options. On the negative side, excessive restriction may reduce accessibility for groups that rely heavily on private cars, especially suburban commuters and travelers with limited modal alternatives. These negative effects are represented in a stylized way, with mobility-loss effects represented indirectly via m and e, rather than through a fully monetized welfare-loss term.

A pertinent issue arises concerning the relative magnitudes of a₁ and a₂: (1) When a₁ − a₂ ≥ 0, it implies a substantial divergence in energy consumption between unladen and fully laden vehicles. In practice, this suggests that a significant proportion of heavy-duty vehicles, such as large buses and heavy goods vehicles, are present in the vehicle composition^[45]. (2) When a₁ − a₂ < 0, it indicates a small increase in energy consumption with changes in vehicle loading. In practical terms, this suggests that a substantial portion of the vehicle composition comprises private and light vehicles. It is generally observed that a₂ exceeds a₁, especially within urban road traffic systems. However, it's conceivable for a₁ to surpass a₂ within the highway logistics sector.

GHG emissions

The fundamental objective of road management is typically the minimization of GHG emissions arising from urban road traffic systems. Building upon the preceding analyses, apart from the influence of engine technology, GHG emissions are predominantly shaped by several key factors: the number of vehicles in operation (in accordance with traffic demand policies), network-wide traffic congestion levels (reflected by road network saturation), the GHG emission factor, vehicle loading, and travel distances. As such, we introduce the GHG emission factor (with unit travel distance) and define the GHG emissions function g(θ) as follows:

$ g\left(\theta \right)=\theta ysh\dfrac{m}{\theta }\dfrac{n}{\theta }=\dfrac{hKn{r}_{1}}{\left[{a}_{1}\left(1-\theta \right)+{a}_{2}\right]\left[K-e\left(1-\theta \right)\right]}. $

(16)

In formulating the GHG emission factor, we introduce the parameter h, signifying a non-negative conversion coefficient representing the relationship between road network saturation and the GHG emission factor. It's crucial to acknowledge two significant statements in theoretical analysis: (1) Prior research^[7,8] has consistently shown that average vehicle speed exhibits a monotonically decreasing trend as road network saturation (or traffic density, average traffic density) increases. (2) Notably, empirical evidence indicates that when the average vehicle speed falls below a certain threshold (e.g., 60 km/h), GHG emissions show a steady decline as a function of average vehicle speed^[9,10]. In practice, during peak hours, the average vehicle speed in nearly all cities worldwide tends to be below 40 km/h^[46].

Therefore, for theoretical analysis, in urban transportation systems at a macro-level, the GHG emission factor can be represented as the product of road network saturation and the non-negative conversion coefficient h. It's worth noting that, in numerical experiments (see the numerical experiment section), the GHG emission factor is calculated using a comprehensive GHG emission factor function that includes average vehicle speed, which is derived from real-world data^[9]. Additionally, n signifies the loading coefficient, and n/θ decreases with the authorized vehicle proportion θ. Drawing from the foregoing analyses, it becomes evident that an optimal value for the authorized vehicle proportion θ* exists, one that minimizes GHG emissions. In relation to Eq. (16), we outline the following pivotal considerations: (1) θ functions as a control parameter, thus optimization of its value is instrumental in achieving GHG emission minimization. (2) The values of a₁ and a₂ fluctuate in response to variations in vehicle composition and vehicle type. Therefore, it is essential to assess how changes in these parameters impact the optimal authorized vehicle proportion θ*. (3) Certain variables, such as r₁, b, e, and K, are challenging to alter within short timeframes due to various factors and have no direct relationship with the control variable. Consequently, they are often considered exogenous variables.

Optimal authorized vehicle proportion

Prior to determining the optimal authorized vehicle proportion θ* that minimizes GHG emissions, it is essential to conduct a thorough analysis of the stationary points, non-differentiable points, and boundary points of the function g(θ). Initially, we compute the first-order derivative of the function g(θ) with respect to θ and set it equal to zero.

$ \dfrac{\partial g\left(\theta \right)}{\partial \theta }=\dfrac{hKn{r}_{1}\left\{-{a}_{2}e+{a}_{1}\left[K-2e\left(1-\theta \right)\right]\right\}}{{\left[{a}_{1}\left(1-\theta \right)+{a}_{2}\right]}^{2}{\left[K-e\left(1-\theta \right)\right]}^{2}}=0. $

(17)

This process yields a unique stationary point, expressed as follows:

$ {\theta }^{S}=1+\dfrac{1}{2}\left(\dfrac{{a}_{2}}{{a}_{1}}-\dfrac{K}{e}\right). $

(18)

It is important to note that its value may fall outside the interval [0, 1]. Furthermore, we present the following proposition to establish the conditions for the existence of local (minimal/maximal) values of the function g(θ) at the stationary point θ^S:

Proposition 2. Function g(θ) reaches its local minimum when θ = θ^S.

Proof. To substantiate this claim, we initially derive the second derivative of the function g(θ) with respect to the authorized vehicle proportion θ:

$\begin{split} &\dfrac{{\partial }^{2}g\left(\theta \right)}{\partial {\theta }^{2}}=\\ &\dfrac{2hKn{r}_{1}\left\{a_{2}^{2}{e}^{2}-{a}_{1}{a}_{2}e\left[K-3e\left(1-\theta \right)\right]+a_{1}^{2}\left[{K}^{2}-3eK\left(1-\theta \right)+3{e}^{2}{\left(1-\theta \right)}^{2}\right]\right\}}{{\left[{a}_{1}\left(1-\theta \right)+{a}_{2}\right]}^{3}{\left[K-e\left(1-\theta \right)\right]}^{3}}. \end{split}$

(19)

Subsequently, substituting Eq. (18) into Eq. (19), we can obtain:

$ {\left.\dfrac{{\partial }^{2}g\left(\theta \right)}{\partial {\theta }^{2}}\right| }_{\theta ={{\theta }^{S}}}=\dfrac{32a_{1}^{3}{e}^{3}hKn{r}_{1}}{{\left({a}_{2}e+{a}_{1}K\right)}^{4}}. $

(20)

It is evident that the second derivative of the function g(θ) at the stationary point θ^S remains positive, as all parameters involved are positive. This concludes the proof. □

Next, we proceed to identify the non-differentiable points, denoted as θ^N. To achieve this, we set the denominator of Eq. (17) to be zero, mathematically that is:

$ {\left[{a}_{1}\left(1-\theta \right)+{a}_{2}\right]}^{2}{\left[K-e\left(1-\theta \right)\right]}^{2}=0. $

(21)

Upon solving Eq. (21), we determine the following two non-differentiable points:

$ \left\{\begin{aligned} &\theta _{1}^{N}=\dfrac{{a}_{1}+{a}_{2}}{{a}_{1}};\\ &\theta _{2}^{N}=1-\dfrac{K}{e}. \end{aligned}\right. $

(22)

Given that a₁ + a₂ > a₁ and K > e inherently hold, we find that $ \theta _{1}^{N}\geq 1 $ and $ \theta _{2}^{N}< 0 $. It is worth noting that the values of these two points fall outside the theoretical domain of θ.

Finally, we analyze the boundary points, denoted as θ^B changing from 0 to 1. Nevertheless, in practice, we discover that the most reasonable range for θ is [0.5, 1]. Specifically, $ \theta _{1}^{B}=0.5 $ signifies the odd-and-even license plate rule, while $ \theta _{2}^{B}=1 $ signifies the absence of travel demand management,

$ \left\{\begin{aligned} &g\left(\theta _{1}^{B}\right)=\dfrac{4hKn{r}_{1}}{\left({a}_{1}+2{a}_{2}\right)\left(2K-e\right)};\\ &g\left(\theta _{2}^{B}\right)=\dfrac{hn{r}_{1}}{{a}_{2}}. \end{aligned}\right. $

(23)

As mentioned above, it's important to emphasize that the function g(θ) and all parameters involved are inherently positive. Consequently, it is difficult to make a definitive judgment about the validity of $ g\left(\theta _{1}^{B}\right)\geq g\left(\theta _{2}^{B}\right) $ or $ g\left(\theta _{1}^{B}\right)< g\left(\theta _{2}^{B}\right) $ based solely on Eq. (23).

In summary, drawing upon the analyses conducted above, we derive significant conclusions regarding the optimal authorized vehicle proportion $ {\theta }^{*} $ that minimizes GHG emissions, elucidated across three distinct cases. Case 1: θ* = θ^S holds if $ {\theta }^{S}\in \left[0.5{,}1\right] $. Case 2: $ {\theta }^{*}=\theta _{1}^{B} $ holds if $ {\theta }^{S}\in \left[0\right.,\left.0.5\right) $. Case 3: $ {\theta }^{*}=\theta _{2}^{B} $ holds if $ {\theta }^{S}\in \left(1,+\infty \right) $. These outcomes can be concisely summarized as follows:

$ {\theta }^{*}=\left\{\begin{aligned} & 0.5,&&if\;{\theta }^{S}\in \left[0\right.,\left.0.5\right);&\\ &1+\dfrac{1}{2}\left(\dfrac{{a}_{2}}{{a}_{1}}-\dfrac{K}{e}\right),&&if\;{\theta }^{S}\in \left[0.5{,}1\right];&\\ &1,&&if\;{\theta }^{S}\in \left(1,+\infty \right).& \end{aligned}\right. $

(24)

In practice, road management aims to mitigate GHG emissions through the formulation of effective policies, with a particular emphasis on determining and implementing the optimal authorized vehicle proportion θ*. In light of the analytical solutions presented above, several key insights emerge: (1) The precise determination of the optimal authorized vehicle proportion $ {\theta }^{*} $ is contingent upon the specific values of a₁, a₂, e, and K. Therefore, before making policies, t is essential to conduct a comprehensive assessment of various factors, including vehicle energy consumption, vehicle composition, the availability of different types of parking facilities, and the maximum land area. At the same time, an evaluation of existing policies is necessary. (2) The accurate categorization of cases (i.e., case 1, case 2, and case 3) holds significant importance. A misjudgment in this regard could exacerbate traffic congestion in response to travel demand management efforts (1 − θ). (3) Addressing issues related to excessive on-street parking and illegal parking assumes added importance as these factors can impede road network capacity. Consequently, the promotion of shared parking policies and the development of three-dimensional parking facilities are warranted strategies.

Sensitivity analysis

Let S(θ*,l) represent the elasticity of θ* to l, denoted as $ \left(\partial {\theta }^{*}/\partial l\right)\cdot \left(l/{\theta }^{*}\right) $, where l can be the parameter a₁, a₂, e, or K. By conducting a sensitivity analysis concerning the optimal authorized vehicle proportion θ*, we have established the following proposition:

Proposition 3. Equations $ \left| \mathrm{S}\left({\theta }^{*},{a}_{1}\right)\right| =\left| \mathrm{S}\left({\theta }^{*},{a}_{2}\right)\right| $ and $ \left| \mathrm{S}\left({\theta }^{*},e\right)\right| = \left| \mathrm{S}\left({\theta }^{*},K\right)\right| $ hold.

Proof. Synthesizing the definition of S(θ*,l) and Eq. (24), we can obtain:

$ \mathrm{S}\left(\theta^*,a_1\right)=\dfrac{\partial\theta^*}{\partial a_1}\dfrac{a_1}{\theta^*}=-\dfrac{a_2e}{2a_1e+a_2e-a_1K}, $

(25)

$ \mathrm{S}\left(\theta^*,a_2\right)=\dfrac{\partial\theta^*}{\partial a_2}\dfrac{a_2}{\theta^*}=\dfrac{a_2e}{2a_1e+a_2e-a_1K}, $

(26)

$ \mathrm{S}\left(\theta^*,e\right)=\dfrac{\partial\theta^*}{\partial e}\dfrac{e}{\theta^*}=-\dfrac{a_1K}{2a_1e+a_2e-a_1K}, $

(27)

$ \mathrm{S}\left({\theta }^{*},K\right)=\dfrac{\partial {\theta }^{*}}{\partial K}\dfrac{K}{{\theta }^{*}}=\dfrac{{a}_{1}K}{2{a}_{1}e+{a}_{2}e-{a}_{1}K}. $

(28)

Based on the above comparison, we obtain Proposition 3. □

We have the following conclusions from Proposition 3: (1) The determination of the optimal authorized vehicle proportion θ* can be grounded on an evaluation of the incremental energy consumption parameter a₁ and the minimal energy consumption parameter a₂ for an entire city. In practical contexts characterized by a diversification of energy sources used by vehicles, road managers should remain flexible and prepared to adapt the optimal authorized vehicle proportion θ* to effectively minimize GHG emissions. (2) Within the confines of a city, factors such as the parking impact coefficient e and the maximal land area K are generally characterized by limited variability, exerting minimal influence on the determination of the optimal authorized vehicle proportion θ*. This underscores the lesser role played by these parameters in this particular context.

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Parameters/ variables	Meanings	Values
T_max	Maximum cycle number	3,650−36,500
r₁	Energy growth rate	0.005
r₂	Vehicle scrapping rate	0.0007
r₃	Growth rate of road network area	0.0002
a₁	Average increment of energy consumption per vehicle	1 × 10⁻⁵, 2.5 × 10⁻⁴
a₂	Minimal energy consumption per vehicle	1 × 10⁻⁷, 1.5 × 10⁻⁵
b	Conversion coefficient	15
b₀	Coefficient of function of GHG emission factor	7.613534994965560
b₁	Coefficient of function of GHG emission factor	−0.138565467462594
b₂	Coefficient of function of GHG emission factor	0.003915102063854
b₃	Coefficient of function of GHG emission factor	−0.000049451361017
b₄	Coefficient of function of GHG emission factor	0.000000238630156
c	Increment coefficient	0.15
d	Decrement coefficient area	0.00001
e	Parking impact coefficient	0.0001
K	Maximal land area for building road network	100
m	Detour coefficient	1
n	Loading coefficient	0.63
x₀	Initial value of energy reserve	10
y₀	Initial value of vehicle population	640.7474 × 10⁴ veh
z₀	Initial value of road network area	90.24696 km²
$ \theta $	Authorized vehicle proportion	[0.5, 1]
$ \Delta \theta $	Calculation interval of $ \theta $	0.001
$ \Delta {a}_{1} $	Calculation interval of a₁	1 × 10⁻⁷
$ \Delta {a}_{2} $	Calculation interval of a₂	1 × 10⁻⁷

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Understanding and managing the dynamic evolution of traffic emissions in urban road systems

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Understanding and managing the dynamic evolution of traffic emissions in urban road systems