Study of an Impedance Function for Mixed Traffic Flows Considering the Travel Time–Cost Characteristics of Long-Distance Electric Vehicle Trips

Abstract

To quantify the travel time and cost characteristics of mixed traffic involving electric vehicles (EVs) and fuel-powered vehicles on roads, in this paper, we comprehensively consider three factors affecting road impedance: queue length, waiting time, and service rate. Initially, a time characteristic function and a cost characteristic function for mixed traffic impedance are constructed. From the perspective of travel time, we consider the impact of EV penetration on the actual road capacity and introduce a capacity coefficient to modify the BPR (Bureau of Public Roads) road impedance function. Given that different types of vehicles might need to wait at charging stations, we employ queuing theory to calculate the queuing time at these stations and construct an impedance model that considers travel time. From the perspective of travel costs, we account for the energy consumption costs and road usage fees for different types of vehicles. The energy consumption cost for travel mileage is obtained by multiplying the unit mileage energy consumption cost of mixed traffic by the travel mileage. For road usage fees, we adopt the conventional method of multiplying the per-kilometer rate for each vehicle type by the travel mileage, thus constructing an impedance model that incorporates travel costs. Finally, in the numerical analysis section, based on the vehicle travel mileage, we categorize travel into short-, medium-, and long-distance trips for analysis. With the constructed mixed traffic impedance model, we conduct a detailed analysis of the travel time and cost characteristics of mixed traffic over different travel distances. We explore the specific impacts of the electric vehicle penetration rate, traffic flow volume, and travel mileage on road impedance. The results indicate that as the penetration rate of electric vehicles increases, the total energy consumption of the transportation system significantly decreases. Moreover, at high electric vehicle penetration rates, although an increase in traffic flow leads to higher traffic impedance and longer travel times, the overall travel costs are reduced. This demonstrates that increasing the penetration rate of electric vehicles positively contributes to reducing the energy consumption and costs of transportation systems.

1. Introduction

With the growing global awareness of environmental protection and advancements in technology, the adoption rate of new energy vehicles (NEVs) for travel has significantly accelerated [1]. As a crucial type of NEV, electric vehicles (EVs) have broad development prospects and immense market potential. In the future automotive market, EVs are projected to hold a significant position. EV development has become crucial for addressing the challenges posed by dual-carbon targets. Compared to traditional fuel-powered vehicles, EVs offer advantages such as zero emissions, low noise, and high energy efficiency. However, due to the significant differences between EVs and traditional vehicles in terms of energy consumption, acceleration performance, and charging time, the widespread adoption of EVs will result in substantial changes in future traffic composition [2,3,4]. For the foreseeable future, mixed traffic flows comprising both EVs and fuel vehicles will continue [5]. The coexistence of these two types of vehicles on roads presents new challenges for traffic management. To address this trend, it is urgent and necessary to conduct in-depth research on mixed traffic flows involving traditional vehicles and EVs.

Traffic impedance, as a critical parameter for route planning, directly impacts transportation efficiency and driving experience. Calculating road impedance requires comprehensive consideration of factors such as traffic section, facility, and vehicle characteristics to reflect the operational distance, time, cost, and comfort of travel for different road segments or paths. Traffic impedance includes both general and specific categories; general traffic impedance reflects the resistance effect associated with driving habits, vehicles, and roads, while specific traffic impedance generally refers to the travel time of individuals on roads [6,7].

The importance of traffic impedance cannot be overstated, as it can be used to gain a deep understanding of key information such as traffic flow volumes, speeds, delays, and costs. Through in-depth research on traffic impedance, we can comprehensively evaluate the operational status of road networks, providing a solid decision-making foundation for urban traffic planning and management [8].

In 1964, the US Federal Highway Administration surveyed a large number of road sections and conducted regression analysis on the obtained data to obtain a BPR function model. This model is the most common road resistance function, which is used in the calculation of road section free travel time, and mainly considers the relationship between vehicle flow and road capacity. However, in the context of mixed traffic flows, relying solely on existing models is insufficient for accurately reflecting the dynamic impact of EV penetration on road capacity.

Based on the above factors, an in-depth study of the impact of mixed traffic flows involving fuel-powered vehicles and EVs is performed. Considering the real-time energy consumption differences between traditional vehicles and EVs, we comprehensively consider various factors, such as queue length, waiting time, and service rate, to assess the travel time and cost characteristics of mixed flows.

The structure of the paper is as follows. In the literature review, we summarize and analyze the BPR road impedance function models and their improvements, noting deficiencies.

In the methodology section, we propose mixed traffic flow travel time and cost impedance function models considering the level of EV penetration. In the analysis of travel time characteristics, we consider the effects of EV penetration, traffic volume, travel mileage, and travel time and derive a travel time function that includes the queuing time of vehicles. Based on toll road costs, we introduce the energy replenishment costs required for charging at charging stations and refueling at gas stations, considering the impacts of EV penetration and travel mileage, and derive a travel cost impedance function model.

In the numerical analysis section, we explore the effects of EV penetration, traffic flows, travel mileage, and travel time and cost; analyze the time and cost characteristics of mixed traffic flows in short-, medium-, and long-distance travel scenarios; and examine the impacts of EV penetration, traffic flow, and travel mileage on road segment impedance.

The research results in this paper can provide an improved understanding of the comprehensive impact of mixed traffic involving EVs and traditional vehicles on road traffic, meanwhile can offer scientific evidence and support for traffic management departments in terms of policy formulation and implementation. Thus, our results can be used to establish a more efficient and environmentally friendly traffic system.

2. Literature Review

The BPR impedance function is a classic traffic flow equation that provides a concise and efficient description of the relationship between road traffic flow and travel time. Consequently, this model has extensive application prospects in traffic planning, congestion management, and traffic flow optimization. Initially, proposed by the BPR in 1964 [9], the BPR impedance function is used to evaluate traffic congestion levels and road operating efficiency. The primary advantage of the BPR impedance function lies in its accurate depiction of road impedance and its excellent mathematical properties, making it one of the most widely used traffic flow models internationally.

$${\mathrm{T}}_{\mathrm{i}}={\mathrm{T}}_{\mathrm{i}0}\times (1+\mathsf{\alpha }{({\displaystyle \frac{\mathrm{Q}}{\mathrm{C}}})}^{\mathsf{\beta }})$$

(1)

In Formula (1), T_i represents the actual travel time through segment i, T_i0 represents the free-flow travel time on segment i, Q is the traffic volume passing through the segment, measured in passenger car units per hour (pcu/h), and C is the actual capacity of the segment, also measured in pcu/h. The parameters α and β are obtained by fitting the model to actual data and adjusting the relationship between traffic flow and travel time. The BPR recommends values of 0.15 and 4 for α and β, respectively. These recommended values are highly applicable in the United States, and with proper calibration, the delay discrepancy between the model results and observed values is greatly reduced [10]. The BPR function curve is shown in Figure 1.

The BPR impedance function conveys the simple idea that as traffic volume increases, road congestion also increases, leading to longer travel times. When the traffic volume is far below the road capacity, the travel time is almost independent of the traffic volume and is close to the basic travel time. However, as traffic volume approaches or exceeds road capacity, congestion intensifies, and travel time increases sharply. With accelerating urbanization and rapid growth in traffic demand in China, traffic problems have become increasingly severe. Therefore, quantifying the impact of road impedance on travelers’ route choices is particularly important.

As a simple and convenient road impedance equation, the BPR function has been widely used both domestically and internationally. However, its simple form and limited number of parameters make it inapplicable in many cases, prompting many scholars to make various improvements. Spiess [11] explored the relationship among travel time, distance, and flow on road segments by modifying the parameter β in the BPR function. Wang et al. [12] improved the BPR function by adding a traffic volume reduction in relation to the road segment capacity, thus enhancing the simplicity of the model form and calculation method. He et al. [13] revised the free-flow travel time and saturation in the BPR function based on measured traffic flow data and calibrated parameters for various urban roads in Dalian. Liu et al. [14] introduced major factors such as intersection density, road speed limits, bus stop density, and saturation into the BPR function and established a heuristic road impedance function. Fu et al. [15] provided methods for calibrating network flow speed conversion relations and traffic delay function parameters based on spatiotemporal statistical methods and determined the conversion relationships between BPR parameters and traffic characteristic parameters. Zhou et al. [16] proposed an improved impedance function that retains some properties of the BPR function and has a shorter runtime. Wang et al. [17] improved the classic BPR impedance function model to more accurately calculate road traffic impedance and a long short-term memory (LSTM) neural network was established to predict undetermined coefficients in the improved function. Davidson [18] proposed a new road impedance model based on queue theory, but it had inconsistencies in terms of parameter definitions. Akcelik [19] corrected the Davidson function and used coordinate transformation to derive a time-dependent Davidson function, thus overcoming parameter definition inconsistencies. He et al. [20] proposed an asymptotic impedance function based on queue theory, where the travel time approaches infinity as the traffic volume reaches the road capacity, and conducted fitting analysis research. Han [21] used a discrete-time method to associate cost with the flow, improved the impedance function, and established a method suitable for dynamic network loading, proposing a new path-based solution algorithm. Li et al. [22] re-established and calibrated urban road segment vehicle travel time models and downstream signal-controlled intersection traffic delay models by analyzing the traffic flow characteristics of road segments and downstream signal-controlled intersections. Liu et al. [23] derived the relationship between segment flow and travel time under congested and noncongested conditions based on the Edie traffic flow model and proposed an impedance function expression. Pan et al. [24] divided urban road traffic impedance into segment impedance and node impedance, optimized the BPR function using the relationship between traffic volume and density to obtain segment impedance and selected different models based on intersection road saturation to obtain node impedance. Wang et al. [25] derived the relationship between segment flow and travel time, compared the differences between the BPR function and the derived relationship, and performed exponential parabolic fitting for the derived relationship.

Some scholars have also improved the theoretical model of traffic flow for mixed traffic. For example, Yang et al. [26] discussed urban road segment travel time function models under mixed traffic conditions in China using traffic flow theory to establish travel time function models for different road widths. Si et al. [27] analyzed the operating characteristics of major traffic modes (such as cars, buses, and bicycles) in urban mixed-traffic networks, developed road segment impedance functions for these networks, and introduced methods for calibrating relevant parameters. Zhao et al. [28] proposed an improved mixed-traffic road impedance function that explicitly considers the interactions between motorized and nonmotorized traffic. Li et al. [29] proposed a new fuzzy neural network self-organizing measurement model to simulate impedance changes for mixed-traffic roads. Muller et al. [30] considered the impact of trucks on the travel time of other vehicles on highways and found that an increase in the number of trucks significantly affects the travel time of passenger cars, thereby greatly increasing road impedance. Lu et al. [31] considered the effects of the traffic volume and the proportion of various vehicles (trucks, passenger cars, etc.) on highways on road impedance, proposed an improved model, and performed Vissim microscopic simulations for parameter estimation. The authors concluded that the traffic composition has a significant impact on the traffic volume and travel time. Wang et al. [32] incorporated transportation costs and tolls into impedance functions and considered delays in time. The parameters were calibrated using the maximum likelihood estimation method, and the model was suitable for fully enclosed highways in China and able to fit actual traffic volumes well.

Through the review of existing studies, it is evident that significant progress has been made in optimizing road impedance and travel time models through various methods. However, with the gradual popularization of EVs and the increasing complexity of traffic systems, current models still have deficiencies in several key areas that urgently require further improvement. Below, potential improvements are described in the context of the key points of this study.

Most current impedance models are based on traditional fuel vehicles and insufficiently consider the specific needs of EVs. The penetration rate of EVs also impacts the actual road capacity. As EV penetration increases, the layout of charging facilities and the time spent charging significantly impact travel time and road impedance. Systematic research that incorporates the time spent by vehicles at energy replenishment stations into travel time models is lacking. Ignoring these factors results in models that do not accurately reflect actual traffic conditions, thus affecting the effectiveness of traffic planning and management decisions. Although some studies have introduced transportation costs and tolls, they fail to fully consider the differences in energy replenishment costs between EVs and traditional fuel vehicles. With the changing energy structure, the impacts of EV charging costs and travel mileage energy consumption costs on overall travel expenses are becoming increasingly significant. There is a lack of sufficient research on existing models in this area. Formulating a comprehensive travel cost model that includes charging costs, refueling costs, and travel mileage energy consumption costs for different types of vehicles can aid in accurately evaluating the costs of various travel modes and optimizing traffic planning and resource allocation.

This paper introduces the waiting time and energy replenishment costs for EVs at charging stations and combines them with the refueling costs for traditional fuel vehicles to comprehensively analyze the time and cost characteristics of different travel scenarios.

3. Impedance Function Model of Mixed Traffic Flows

On the basis of the above review, the existing model is broadened and improved in this section. First, the travel time model is improved. In the traditional BPR travel time model, the impact of the electric vehicle penetration rate on the actual road capacity is considered, and the time spent by EVs at recharge stations is also included. A travel time function containing the time required by a vehicle in the queuing system is derived. This improvement allows the model to more accurately reflect the impact of EV charging on the overall travel time. Additionally, a travel cost model is formulated. Combining the travel mileage energy consumption cost and road use cost for different types of vehicles, we construct a comprehensive travel cost model. In particular, based on road toll information, the cost of energy supplied by charging stations and refueling stations is introduced, the effects of the penetration rate and driving range of electric vehicles are considered, and a travel cost impedance function model is derived.

3.1. Improvement of the Travel Time Model

The travel times are based in part on the US Federal Highway Administration’s (FHWA) BPR resistance function [9]. The BPR function is widely used in traffic engineering to estimate the impact of road congestion on travel time. In this model, the consumption time of different types of vehicles at recharge stations is considered, especially the charging time of EVs at charging stations.

$$\mathrm{T}={\mathrm{T}}_{\mathrm{B}\mathrm{P}\mathrm{R}}+{\mathrm{T}}_{\mathrm{W}}$$

(2)

In Formula (2), T_BPR is the time spent by a vehicle driving on a road section, which is included in the BPR function proposed by the US FHWA, and T_W is the time spent by the vehicle at a recharge station, that is, during a recharge operation at a gas station or charging station.

It is assumed that the traffic on the selected road is composed of EVs and traditional vehicles, regardless of the impact of other types of vehicles.

In the BPR base function, a fixed value C₀ in Formula (3) is used to represent the actual capacity of the road section. However, in practice, different percentages of EVs will result in different impacts on the capacity of the road. Through the simulation of data [33], it was found that there is a certain relationship between the penetration rate of EVs and C₀. The simulation results show that as the penetration rate of EVs increases, the road capacity also increases accordingly. We fit this relationship based on a function diagram, as shown in Figure 2, where the red dotted line represents the fitted trend line, and the Bev-C₀ function is shown in Formula (3).

$$\mathrm{C}={\mathrm{C}}_0\times (1+0.0017{{\mathrm{B}}_{\mathrm{E}\mathrm{V}}}^2+0.0049{\mathrm{B}}_{\mathrm{E}\mathrm{V}}-0.0037)$$

(3)

According to the functional relationship between the proportion of EVs and the road capacity, the C parameter in the BPR function is improved to obtain T_BPR, as expressed in Formula (7).

$$\mathrm{Q}={\mathrm{Q}}_{\mathrm{E}\mathrm{V}}+{\mathrm{Q}}_{\mathrm{G}\mathrm{V}}$$

(4)

$${\mathrm{B}}_{\mathrm{E}\mathrm{V}}={\displaystyle \frac{{\mathrm{Q}}_{\mathrm{E}\mathrm{V}}}{\mathrm{Q}}}$$

(5)

$${\mathrm{B}}_{\mathrm{G}\mathrm{V}}={\displaystyle \frac{{\mathrm{Q}}_{\mathrm{G}\mathrm{V}}}{\mathrm{Q}}}=1-{\displaystyle \frac{{\mathrm{Q}}_{\mathrm{E}\mathrm{V}}}{\mathrm{Q}}}$$

(6)

$${\mathrm{T}}_{\mathrm{B}\mathrm{P}\mathrm{R}}={\mathrm{T}}_0\times (1+\mathsf{\alpha }{\left({\displaystyle \frac{{\mathrm{Q}}_{\mathrm{E}\mathrm{V}}+{\mathrm{Q}}_{\mathrm{G}\mathrm{V}}}{\mathrm{C}}}\right)}^{\mathsf{\beta }})$$

(7)

where B_EV is the proportion of EVs in the traffic flow, B_GV is the proportion of GVs in the traffic flow, Q is the mixed traffic volume on the road, Q_EV is the traffic volume of EVs on the road, and Q_GV is the traffic volume of fuel-consuming vehicles on the road.

Queuing theory provides a useful framework for considering the effect of the charging time of EVs and the refueling time of traditional vehicles on the impedance of nodes in traffic networks. Queueing theory is a branch of mathematics in which queueing systems are studied, and this theory can be applied to analyze and optimize systems with random arrival rates and service rates. In this context, we consider a charging station or gas station as a queuing system with several key parameters: the arrival rate, service rate, queue length, and average waiting time.

(1)

The arrival rate λ refers to the rate at which vehicles arrive at the charging station or gas station, depending on the traffic flow.

(2)

The service rate μ refers to the rate at which the station provides services to vehicles, which is affected by the performance and efficiency of the charging or refueling equipment.

(3)

The queue length L_q refers to the number of vehicles waiting for service at a station, which depends on the difference between the arrival rate and the service rate.

(4)

The average waiting time W_q refers to the average time vehicles spend in the queue waiting for service at a station, which is jointly affected by the arrival rate, service rate, and queue length.

The time cost is obtained by using the queuing theory model, and the corresponding expression is shown in Formula (8).

$${\mathrm{T}}_{\mathrm{W}}={\mathrm{t}}_1+{\mathrm{t}}_2+{\mathrm{t}}_{\mathrm{q}}-{\mathrm{t}}_0$$

(8)

$${\mathrm{t}}_1={\displaystyle \frac{{\mathrm{V}}_0}{3.6{\mathrm{a}}_1}}$$

(9)

$${\mathrm{t}}_1={\displaystyle \frac{{\mathrm{V}}_0}{3.6{\mathrm{a}}_1}}$$

(10)

where t_q is the time spent by a vehicle in a queuing system during a replenishment operation, t₁ is the time spent by the vehicle decelerating at the replenishment station, a₁ is the deceleration rate, t₂ is the time from the time when the vehicle leaves the replenishment station to the time when it accelerates to normal speed, a₂ is the acceleration rate, and t₀ is the time when the vehicle has no replenishment demand and normally passes through the road section with the replenishment station.

The input process refers to the distribution of vehicle arrival times. The source of traffic is infinite, and the arrivals of vehicles are independent of each other, conforming to the Poisson distribution with parameter λ. The queuing rule is established for vehicles waiting for service, using the principle of First-In-First-Out (FIFO). The service organization includes the number of service desks (represented by c), the service mode, and the service time distribution. The service time of each vehicle is independent and subject to a fixed-length distribution. The queuing rule is adopted for a mixed system, which means that there are multiple service queues. In this study, Poisson-distributed traffic flow is considered, and the average arrival rate is λ. The average service rate per service desk is μ, and the number of vehicles is n. The average service rate for the entire service organization can be expressed as nμ (when n < c) or cμ (when n ≥ c). $\mathsf{\rho }$ is the service intensity of the system. When $\mathsf{\rho }$ > 1, queuing will occur.

Through the analysis of the queuing system, the waiting time and departure time of a vehicle at a charging station can be accurately calculated.

$$\mathsf{\rho }={\displaystyle \frac{\mathsf{\lambda }}{\mathrm{c}\mathsf{\mu }}}$$

(11)

L_q is the average queue length, which is the average number of vehicles waiting for service in the queue system, and L_s is the average number of vehicles in the queuing system.

$$\begin{array}{cc}{\mathrm{L}}_{\mathrm{q}}& ={\displaystyle \sum _{\mathrm{n}=\mathrm{c}+1}^{\mathrm{\infty }}}(\mathrm{n}-\mathrm{c}){\mathrm{P}}_{\mathrm{n}}={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{\infty }}}\mathrm{k}{\mathrm{P}}_{\mathrm{k}+\mathrm{c}}={\displaystyle \sum _{\mathrm{k}=1}^{\mathrm{\infty }}}{\displaystyle \frac{\mathrm{k}}{\mathrm{c}!{\mathrm{c}}^{\mathrm{k}}}}(\mathrm{c}\mathsf{\rho }{)}^{\mathrm{k}+\mathrm{c}}{\mathrm{P}}_0={\displaystyle \frac{(\mathrm{c}\mathsf{\rho }{)}^{\mathrm{c}}\mathsf{\rho }}{\mathrm{c}!(1+\mathsf{\rho }{)}^2}}{\mathrm{P}}_0\end{array}$$

(12)

$${\mathrm{L}}_{\mathrm{S}}={\mathrm{L}}_{\mathrm{q}}+{\displaystyle \frac{\mathsf{\lambda }}{\mathsf{\mu }}}$$

(13)

P₀ is the probability that there are no vehicles in the queuing system waiting for service; W_q is the average waiting time, the average time a vehicle waits in line in the system; and W_s is the average dwell time, the average amount of time a vehicle stays in the system.

$${\mathrm{P}}_0=[\sum _{\mathrm{k}=0}^{\mathrm{c}-1}{\displaystyle \frac{1}{\mathrm{k}!}}({\displaystyle \frac{\mathsf{\lambda }}{\mathsf{\mu }}}{)}^{\mathrm{k}}+{\displaystyle \frac{1}{\mathrm{c}!}}{\displaystyle \frac{1}{1-\mathsf{\rho }}}({\displaystyle \frac{\mathsf{\lambda }}{\mathsf{\mu }}}{)}^{\mathrm{c}}{]}^{-1}$$

(14)

$$\begin{array}{c}{\mathrm{W}}_{\mathrm{q}}={\displaystyle \frac{{\mathrm{L}}_{\mathrm{q}}}{\mathsf{\lambda }}}={\displaystyle \frac{(\mathrm{c}\mathsf{\rho }{)}^{\mathrm{c}}\mathsf{\rho }}{\mathrm{c}!(1-\mathsf{\rho }{)}^2\mathsf{\lambda }}}{\mathrm{P}}_0.\end{array}$$

(15)

$${\mathrm{W}}_{\mathrm{S}}={\mathrm{W}}_{\mathrm{q}}+{\displaystyle \frac{1}{\mathsf{\mu }}}$$

(16)

${\mathrm{t}}_{\mathrm{q}}$ is the time spent by a vehicle in a mixed traffic flow in the replenished queuing system. Considering the difference between fuel-powered vehicles and electric vehicles, the value of ${\mathrm{t}}_{\mathrm{q}}$ is determined according to the following formula considering the combined influence of the volume of traffic and the penetration rate of EVs.

$${\mathrm{t}}_{\mathrm{q}}=\mathrm{m}\mathrm{a}\mathrm{x}\left\{{\mathrm{W}}_{\mathrm{S},\mathrm{E}\mathrm{V}},{\mathrm{W}}_{\mathrm{S},\mathrm{G}\mathrm{V}}\right\}$$

(17)

From the point of view of queuing theory, we can assess the effects of charging time and refueling time differences and the penetration rate of EVs on the node impedance in a transportation network. Considering factors such as queue length, wait time, service rate, and flow control, transportation planners can develop strategies to optimize the layout and management of charging facilities to mitigate the impact of node impedance in transportation networks. Combined with the above formula, the time impedance function for mixed traffic flows can be obtained as follows:

$$\mathrm{T}={\mathrm{T}}_{\mathrm{B}\mathrm{P}\mathrm{R}}+{\mathrm{T}}_{\mathrm{W}}={\mathrm{T}}_0\times (1+\mathsf{\alpha }{({\displaystyle \frac{{\mathrm{Q}}_{\mathrm{E}\mathrm{V}}+{\mathrm{Q}}_{\mathrm{G}\mathrm{V}}}{\mathrm{C}}})}^{\mathsf{\beta }})+{\mathrm{t}}_1+{\mathrm{t}}_2+{\mathrm{t}}_{\mathrm{q}}-{\mathrm{t}}_0$$

(18)

3.2. Creation of the Travel Cost Model

Travel cost is an inevitable factor to be considered in the process of travel, especially in the current economic situation, and financial management has received increasing attention in the literature. In this paper, the travel cost M includes the toll cost ${\mathrm{M}}_{\mathrm{a}}$ incurred by a vehicle driving on the toll road and the cost ${\mathrm{M}}_{\mathrm{t}}$ of the supplementary energy required for charging at a charging station or refueling at a gas station, while excluding the costs incurred due to other consumption behaviors. Specifically, the drivers of fuel vehicles mainly consider fuel consumption costs and road tolls, and fuel consumption costs are related to vehicle fuel efficiency and fuel prices. The drivers of EVs mainly consider the cost of electricity consumption and charging costs. The cost of electricity consumption is related to the energy efficiency of a vehicle (i.e., electricity consumption per kilometer) and the price of electricity. Charging costs include the cost of charging at public charging stations, which may vary depending on the charging speed and the location of the charging station.

$$\mathrm{M}={\mathrm{M}}_{\mathrm{t}}+{\mathrm{M}}_{\mathrm{a}}$$

(19)

$${\mathrm{M}}_{\mathrm{t},\mathrm{i}}=\mathrm{A}\times {\mathrm{L}}_{\mathrm{i}}$$

(20)

where A is the average energy consumption per mile in a mixed traffic flow, in yuan/km, and ${\mathrm{L}}_{\mathrm{i}}$ is the mileage traveled by vehicle i, in km.

$$\mathrm{A}={\mathrm{B}}_{\mathrm{E}\mathrm{V}}\times {\mathrm{D}}_{\mathrm{E}\mathrm{V}}+{\mathrm{B}}_{\mathrm{G}\mathrm{V}}\times {\mathrm{D}}_{\mathrm{G}\mathrm{V}}={\mathrm{D}}_{\mathrm{G}\mathrm{V}}+{\displaystyle \frac{{\mathrm{Q}}_{\mathrm{E}\mathrm{V}}}{\mathrm{Q}}}({\mathrm{D}}_{\mathrm{E}\mathrm{V}}-{\mathrm{D}}_{\mathrm{G}\mathrm{V}})$$

(21)

where D_EV is the unit mileage energy consumption cost of EVs, in yuan/km, and D_GV is the unit mileage energy consumption cost of GVs, in yuan/km.

$${\mathrm{M}}_{\mathrm{t}}=\sum _{\mathrm{i}=1}^{\mathrm{Q}}{\mathrm{M}}_{\mathrm{t},\mathrm{i}}=\sum _{\mathrm{i}=1}^{\mathrm{Q}}\mathrm{A}\times {\mathrm{L}}_{\mathrm{i}}$$

(22)

Toll roads are a type of infrastructure, and vehicles need to pay road use fees when traveling on these roads. Costs associated with road construction, maintenance and management can vary based on the type of vehicle and the distance the vehicle travels. The travel distance of a vehicle will directly affect the cost. In addition, different types of vehicles contribute to different degrees of road wear, so there may be different rates accordingly.

$${\mathrm{M}}_{\mathrm{a}}=\sum _{\mathrm{i}=1}^{\mathrm{Q}}{\mathrm{M}}_{\mathrm{a},\mathrm{i}}=\sum _{\mathrm{i}=1}^{\mathrm{Q}}\mathrm{m}\times \mathsf{\theta }\times {\mathrm{L}}_{\mathrm{i}}$$

(23)

where ${\mathrm{M}}_{\mathrm{t},\mathrm{i}}$ is the energy consumption cost generated by vehicle i during its journey, ${\mathrm{M}}_{\mathrm{a},\mathrm{i}}$ is the road usage cost generated by vehicle i when traveling on a toll road, θ is the toll coefficient and is set to 1, and m is the toll rate per vehicle type per kilometer of travel on the toll road.

In summary, the formula for calculating travel expenses is as follows:

$$\mathrm{M}=\sum _{\mathrm{i}=1}^{\mathrm{Q}}\mathrm{A}\times {\mathrm{L}}_{\mathrm{i}}+\sum _{\mathrm{i}=1}^{\mathrm{Q}}\mathrm{m}\times \mathsf{\theta }\times {\mathrm{L}}_{\mathrm{i}}=\sum _{\mathrm{i}=1}^{\mathrm{Q}}({\mathrm{D}}_{\mathrm{G}\mathrm{V}}+{\mathrm{B}}_{\mathrm{E}\mathrm{V}}({\mathrm{D}}_{\mathrm{E}\mathrm{V}}-{\mathrm{D}}_{\mathrm{G}\mathrm{V}})+\mathrm{m}\times \mathsf{\theta })\times {\mathrm{L}}_{\mathrm{i}}$$

(24)

4. Numerical Analysis

To analyze the time and cost characteristics of mixed traffic flows in different travel scenarios and to explore the effects of EV penetration, traffic flows, and driving distance on road impedance, a numerical analysis is carried out in this section. The variables are traffic flow (Q), mileage (L), and EV penetration rate (B_EV), with traffic flows between 0 and 2000 pcu/h, mileages between 0 and 1000 km, and EV penetration rates between 0 and 1. With a two-way four-lane road section as an example, the main objective is to assess the impacts of Q, L, and B_EV on T. The values of some basic parameters in the impedance function are shown in

Table 1. Basic parameter values.

Variable	Value	Unit
$\mathsf{\alpha }$	0.15	——
$\mathsf{\beta }$	4	——
$\mathrm{m}$	0.45	yuan/km
$\mathsf{\theta }$	1	——
${\mathrm{D}}_{\mathrm{E}\mathrm{V}}$	0.19	yuan/km
${\mathrm{D}}_{\mathrm{G}\mathrm{V}}$	0.56	yuan/km
${\mathrm{B}}_{\mathrm{E}\mathrm{V}}$	0~1	——
${\mathrm{B}}_{\mathrm{G}\mathrm{V}}$	0~1	——
${\mathsf{\mu }}_{\mathrm{E}\mathrm{V}}$	1/30	pcu/min
${\mathsf{\mu }}_{\mathrm{G}\mathrm{V}}$	1/4	pcu/min

.

The values for α and β are set at 0.15 and 4, respectively, as recommended by the US Highway Administration; the toll rate for a four-lane highway is set at 0.45 yuan per standard vehicle kilometer; the vehicles under study in this paper are all ordinary 5- or 7-seat family sedans. Based on the “Classification of Toll Road Vehicle Types,” they are categorized as Class 1 passenger cars; hence, θ is 1; D_EV, D_GV, μ_EV, and μ_GV are all hypothetical values derived from practical considerations.

${\mathsf{\lambda }}_{\mathrm{E}\mathrm{V}}$ and ${\mathsf{\lambda }}_{\mathrm{G}\mathrm{V}}$ represent the average arrival rates of EVs and GVs at recharge nodes, respectively, according to Formulas (25) and (26).

$${\mathsf{\lambda }}_{\mathrm{E}\mathrm{V}}={\displaystyle \frac{{\mathrm{B}}_{\mathrm{E}\mathrm{V}}\times \mathrm{Q}}{60}}$$

(25)

$${\mathsf{\lambda }}_{\mathrm{G}\mathrm{V}}={\displaystyle \frac{{\mathrm{B}}_{\mathrm{G}\mathrm{V}}\times \mathrm{Q}}{60}}={\displaystyle \frac{{(1-\mathrm{B}}_{\mathrm{E}\mathrm{V}})\times \mathrm{Q}}{60}}$$

(26)

4.1. Analysis of Travel Time Characteristics

To intuitively show the impacts of Q, L, and B_EV on T, we apply the 3D mapping function in Python to produce a heatmap. The x-axis of the graph represents the traffic flow Q, the y-axis represents the mileage L, the z-axis represents the penetration rate of electric vehicles B_EV, the color of the data points represents the travel time T, and a warmer (colder) color represents a longer (shorter) travel time, as shown in Figure 3.

As illustrated in Figure 3, as Q increases, T increases nonlinearly. This indicates that the impact of traffic congestion on travel time is most significant for high traffic flows. The impact of B_EV on T depends on Q. At low traffic volumes, the penetration rate of EVs has little influence on travel time. However, under high traffic conditions, an increase in the penetration rate of EVs leads to an increase in the charging time, thus extending the travel time. The influence of L on travel time is linear, and the travel time increases linearly with increasing mileage.

This comprehensive analysis method provides a new perspective for understanding the impedance characteristics in different travel situations, overcomes the shortcomings of existing models by utilizing a mixed traffic impedance function, and provides accurate and comprehensive decision support for traffic planning and management.

• The time spent by a vehicle in the replenishment queuing system t_q

t_q represents the time spent at a recharge node, which is calculated according to the queuing system Formula (16) in Section 3.1, and the results for 10 kinds of EVs with different penetration rates are calculated, as shown in Figure 4.

According to the analysis of Figure 4, the relationship between traffic volume (Q) and recharge time (t_q) under different B_EV values can be described as follows.

The vertical axis (t_q) represents the time spent at a recharge node, and the horizontal axis (Q) represents the traffic volume. The curves in different colors show the penetration rates (B_EV) of different electric vehicles, ranging from 0 to 1 with an interval of 0.1.

(1)

Low EV penetration (B_EV = 0 to 0.3)

When B_EV is low (e.g., B_EV = 0 to 0.3), the replenishment time shows a relatively slow upward trend with increasing traffic volume. In the small range of Q, the replenishment time slowly increases. For example, when Q < 500, t_q increases slowly. This shows that for a low penetration rate of EVs, the number of EVs on the road is small, the vast majority of vehicles are still fuel-powered vehicles, the number of charging piles is sufficient relative to the number of EVs, the usage rate of charging piles is low, and a vehicle does not need to wait for a long time to complete the charging process; therefore, the time spent at a recharge node is relatively short.

(2)

Moderate EV penetration (B_EV = 0.4 to 0.7)

As B_EV increases to moderate levels (B_EV = 0.4 to 0.7), the increase in the replenishment time becomes increasingly significant. When Q < 500, tq increases rapidly, indicating that the use frequency of charging piles increases at this time, resulting in an increase in the average waiting time. When Q > 500, the increase in the number of EVs leads to a significant increase in charging demand, and more vehicles need to be replenished at charging piles, leading to an increase in the frequency of use of charging piles at recharge nodes; additionally, the average waiting time also increases. When Q > 1000, charging piles begin to approach their maximum service capacity. Although the traffic volume continues to increase, the use of the charging pile approaches saturation; consequently, the replenishing time tends to stabilize, and the upward trend slows.

(3)

High EV penetration (B_EV = 0.8 to 1.0)

When B_EV is high (i.e., B_EV = 0.8 to 1.0), EVs account for almost all the vehicles on the road, which means that almost all vehicles need to use charging piles. Therefore, the burden on the charging pile increases, and the queuing phenomenon at recharge nodes becomes common, resulting in a significant extension of the recharge time. When Q exceeds 500, the recharge time tends to stabilize because the charging piles approach their maximum service capacity. Although the traffic volume continues to increase, due to the limited number of charging piles and queuing space, the replenishment time will not increase indefinitely, reaching a saturated state and plateauing. Due to the saturated charging demand, the charging piles become the bottlenecks of the system, and the efficiency of the overall traffic system is reduced. In this case, even if the traffic volume increases, the replenishment time will not be significantly extended because the number of charging piles and queuing space can no longer accommodate additional charging needs. However, the lack of power for EVs can cause vehicles to slow or even stop, causing traffic paralysis.

2.

The total time spent by a vehicle at a recharge station ${\mathrm{T}}_{\mathrm{W}}$

${\mathrm{T}}_{\mathrm{W}}$ represents the total time spent at a recharge node, which is calculated according to Formula (8), and the corresponding results for 10 kinds of EVs are calculated, as shown in the figure below.

Figure 5 indicates that the magnitude of T_W is closely related to that of t_q. Similarly, BEV has a significant impact on T_W. At low electric vehicle penetration rates (B_EV = 0 to 0.3), T_W increases slowly with increasing traffic volume, the number of charging stations is sufficient, and the waiting time is short. At medium penetration rates (B_EV = 0.4 to 0.7), T_W increases significantly as the demand for charging increases and charging stations approach saturation. At high penetration rates (B_EV = 0.8 to 1.0), almost all vehicles require charging, the burden on charging stations is high, T_W increases and tends to stabilize, and charging stations become the bottleneck of the system, leading to a decline in overall traffic efficiency.

4.2. Travel Cost Function Analysis

Similarly, to intuitively show the impacts of Q, L, and B_E) on the travel cost (M), the drawing function in Python is used to produce a three-dimensional thermal map. The x-axis of the graph represents Q, the y-axis represents B_EV, the z-axis represents L, and the color of the data points represents M. As shown in Figure 6, the color shading indicates the total cost.

• The impact of traffic flow Q on travel cost M

To intuitively explain the impacts of Q and B_EV on M, the following specific data points are selected for further analysis, as shown in

Table 2. Changes in the travel cost M with the traffic flow Q.

Q	M (BEV = 0.5, L = 500)
0	0
200	35,500
400	71,000
600	106,500
800	142,000
1000	177,500

.

As shown in Figure 7, in the case of B_EV = 0.5 and L = 500, M increases linearly as Q increases from 0 to 1000. Notably, M is proportional to Q.

Fitting function:

$$\mathrm{M}=177.50\mathrm{Q}+0.00$$

The fitting function shows that the effect of the total number of vehicles Q on the total cost M is linear, with a coefficient of 177.50. This means that for each additional vehicle, the total cost increases by 177.50 yuan. This shows that the total number of vehicles directly affects the total cost, and the more vehicles there are, the higher the total cost. Because each vehicle incurs a certain cost while driving, an increase in the total number of vehicles will increase the total cost.

2.

Influence of the EV penetration rate B_EV on the travel cost M

As shown in

Table 3. The change in the travel cost M with the penetration rate of EVs B_EV.

BEV	M (Q = 500, L = 500)
0	126,250
0.2	111,250
0.4	96,250
0.6	81,250
0.8	66,250
1	51,250

and Figure 8, in the cases of Q = 500 and L = 500, the total cost M gradually decreases as B_EV increases from 0 to 1.

Fitting function:

$$\mathrm{M}=-\mathrm{75,000.00}{\mathrm{B}}_{\mathrm{E}\mathrm{V}}+\mathrm{126,250.00}$$

The fitting function shows that the effect of B_EV on M is linear, with a coefficient of −75,000.00. This means that for every 1 unit increase in the proportion of electric vehicles, the total cost is reduced by 75,000.00 yuan. The energy consumption per mile by an EV D_EV is lower than the energy consumption per mile of a GV D_GV. As the proportion of electric vehicles increases, the total cost decreases. Therefore, increasing the proportion of electric vehicles will help reduce the overall cost of travel.

3.

The influence of mileage L on travel cost M

As shown in

Table 4. The change in travel cost M with miles traveled L.

L	M (Q = 500, BEV = 0.5)
0	0
200	35,500
400	71,000
600	106,500
800	142,000
1000	177,500

and Figure 9, in the case of Q = 500 and B_EV = 0.5, M increases linearly as L increases from 0 to 1000.

Fitting function:

$$\mathrm{M}=177.50\mathrm{L}+0.00$$

The fitting function shows that the influence of L on M is linear, and the coefficient is 177.50. This means that for each additional kilometer of driving, the total cost increases by 177.50 yuan. The greater the mileage is, the greater the energy consumption and usage costs of the vehicle, resulting in an increase in the total cost. This suggests that the distance traveled has a direct impact on the total cost; the longer the distance traveled is, the greater the total cost. At the same time, the impacts of the total number of vehicles and mileage on the total cost have the same slope and intercept, which is reasonable because the increase in cost is the same in both cases given the proportional relationships between these parameters. An increase in the proportion of EVs will significantly reduce the total cost, which is also expected.

In summary, Q directly affects the total cost, and an increase in Q will linearly increase the total cost. Increasing the penetration rate of EVs can effectively reduce the total cost because the energy cost per unit of EV travel is lower than that for GVs. The total cost is proportional to the distance traveled, and the longer the distance is, the greater the total cost. Effective strategies to reduce the total cost of travel include controlling the total number of vehicles on the road, increasing the proportion of EVs, and rationally planning mileage. These strategies can be used to better control the total travel cost and improve traffic efficiency in traffic management and policymaking.

5. Analysis of the Travel Time-Cost Characteristics of Mixed Traffic Flows

The purpose of the numerical analysis is to analyze the temporal and financial characteristics of mixed traffic flows in various travel scenarios, as well as to investigate the impacts of electric vehicle penetration, traffic flows, and driving distance on road impedance. In this section, the time and cost characteristics of mixed traffic flows are assessed from three perspectives: short trips in urban areas, intermediate trips between cities, and long-distance trips across provinces and regions. Through a numerical analysis of these scenarios, a comprehensive understanding of the characteristics of mixed traffic flows in different travel scenarios can be obtained, providing robust decision support for traffic planning and management.

5.1. Short Trips within a City

First, we explore the above trends for short trips in urban areas, typically covering distances of 0–100 km. The primary factors to consider are traffic congestion and the adoption of EVs.

Based on the data presented in Figure 3, a value of L = 100 is utilized to generate Figure 10. Figure 10 clearly shows that the color representing the travel time for short-distance travel is relatively light, with minimal overall color variation. This indicates that for short-distance travel, both EVs and GVs are influenced primarily by the traffic flow rather than the energy replenishment demand, aligning with the fundamental road resistance function.

Based on the information provided in Figure 6, a value of L = 100 is used to produce Figure 11. Figure 11 shows the correlation between the total cost and B_EV. As B_EV increases, there is a gradual decrease in total cost, attributed to reduced energy consumption per unit mileage of EVs. Specifically, when B_EV approaches zero and GVs constitute a significant portion of the total traffic flow, the total cost is comparatively higher. Conversely, as B_EV approaches one and EVs account for a larger proportion of the total traffic flow, the total cost decreases. Further examination of the color gradient suggests that an increase in the total number of vehicles coupled with an increase in the proportion of EVs results in a relatively slow growth rate for the total cost. This underscores the substantial impact of promoting EV usage for reducing overall costs.

In summary, for short-distance travel scenarios, travel time is predominantly influenced by traffic flow, while total cost is significantly impacted by the proportion of EVs present. Increasing this proportion effectively reduces overall costs while managing traffic flows contributes to optimizing travel time.

5.2. Travel between Cities

Moderate-distance travel between cities, usually within 100–200 km, is analyzed in this section. The main variables are traffic flow and EV penetration rate in this case.

On the basis of the data in Figure 3, L = 200 is used to obtain Figure 12. According to the results in Figure 12, the travel time shows a slight red trend with increasing traffic flow and the penetration rate of EVs during intercity travel. This is because, in the process of intercity travel, the demand for supplementary energy from EVs gradually increases, resulting in a significant increase in the travel time of EVs at high penetration rates.

Similarly, Figure 13 shows that the color of the travel cost results is still dark, indicating that the penetration rate of EVs has no obvious impact on travel costs during intercity travel, but it has a greater impact than during short-distance travel. Therefore, travel time is still the main factor affecting people’s travel choices.

In summary, in intercity travel, the travel time is mainly affected by the demand for the replenishment of EVs, and although the travel cost changes, the impact is not as significant as that of the travel time. Therefore, travel time is still a key factor affecting travel choice.

5.3. Long-Distance Travel across Provinces and Regions

On the basis of the data in Figure 3, L = 500 is used to obtain the change in travel time during long-distance travel, as shown in Figure 14. The analysis shows that when L = 500, the travel time under high-traffic and high-EV-penetration conditions gradually changes from light red to dark red. The closer the color is to dark red, the longer the travel time, and traffic congestion and other traffic issues may occur at this time.

However, as shown from the travel cost image shown in Figure 15, the higher the penetration rate of electric vehicles is, the lower the travel cost for high traffic flows. Therefore, when users travel long distances across provinces, it is ideal to choose EVs.

In summary, in cross-provincial long-distance travel, travel time is affected mainly by the traffic flow and the demand for supplementary energy for EVs, while travel costs decrease with the increasing penetration rate of EVs. Therefore, choosing an EV for long-distance travel is more cost-effective than operating a GV.

6. Conclusions

In this paper, the characteristics of hybrid traffic flows with GVs and EVs are studied in detail, and travel time and cost characteristics are analyzed. Considering the queue length, waiting time, service rate, and flow control, a travel time and cost impedance function model of mixed traffic flows is developed. Numerical analysis results show that the penetration rate, traffic volume, and travel mileage of EVs have significant effects on the travel time and cost of mixed traffic flows.

• In the analysis of travel time characteristics, we incorporate the penetration rate of EVs and the traffic flow to derive a travel time function, which includes the vehicle queuing time during travel. The findings indicate that an increase in EV penetration leads to an increase in overall travel time, particularly under high traffic flow conditions. Furthermore, there is a notable disparity in travel times for different driving distances, with long-distance travel being the most affected.
• In the cost analysis, we comprehensively consider the additional energy costs associated with charging and refueling stations and develop a model for the cost impedance function. The findings indicate that the adoption rate and driving distance of EVs have a significant impact on travel expenses. A high adoption rate of EVs leads to low overall travel costs, particularly for short-distance and mid-range trips, highlighting the advantages of EVs.
• Through numerical analysis, we further investigated the impacts of EV penetration, traffic volume, and driving distance on road impedance. An increase in EV penetration can effectively reduce total energy consumption within the transportation system while also significantly influencing travel time and cost characteristics. Specifically, at high EV penetration rates and under heavy traffic conditions, there is a corresponding increase in traffic impedance, leading to extended travel times but reduced travel costs.

In this paper, we explore the travel time and cost characteristics of GVs and EVs in mixed traffic flows; however, the results are mainly based on theoretical models and numerical analyses and lack verification and support with actual traffic data. These results should be combined with actual traffic flow data to conduct empirical analyses to verify the accuracy and applicability of the models. In addition, the differences among different cities and regions are not considered. In future studies, these considerations can be accounted for to provide a more comprehensive and in-depth understanding of the mixed traffic flows of GVs and EVs.