Use of entropy to analyze level of service of dedicated bike lanes in China

Abstract

Bicycle travel, as an important component of travel modes in China, has gained great support from the government by constructing environmentally friendly bike lanes and other non-motorized facilities. However, there is still lack of effective methods that can be used for assessing operational performance of bikeways and providing practical guide for bikeway planning, design, construction, and operation. This article introduces an approach that can be well suited in China for assessing level of service and operational performance of dedicated bike lanes. The approach is based on the bicyclists’ hindrances experienced. A conception of reactive zone used to express bicyclists’ perceived comfort is highlighted to describe bicyclists’ actions during passing maneuvers. The Entropy of Speed State, deduced from the velocity deviation degree, is related to the density and is used to evaluate the level of service. A dynamic simulation model is developed to describe the bicycle movement characteristics. Controlled experiment and field survey are conducted to collect data for passing maneuvers. Resorting to the simulation method, the bicycle operation process is invested. A five-level classification rating from A to E is defined to evaluate level of service on dedicated bicycle lanes.

Introduction

With the rapid development of urbanization and motorization, traffic congestion and environmental problems (e.g. air pollution and smog caused by motor vehicle emissions) are becoming serious issues in China. Bicycle travel, seen as an environmentally friendly transportation mode, has resurfaced as a valuable means of dealing with the issues. The Beijing Municipal Commission of Transport (BMCT), recognizing the importance of bicycle transportation in its 5-year strategic plan (2016–2020), plans to construct 3200 km bikeways and form a network within the 5th Ring Road. Meanwhile, the first 6.3 km bicycle expressway has been designed in Beijing from Huilongguan to Shangdi. This expressway consists of dedicated bike lanes to serve bike users. It is expected that these facilities will carry 12%–18% more bicycle travel demand by 2020.

Dedicated bike lanes in China are a special type of bikeways. They are physically separated, unidirectional, and uninterrupted bikeways. These bike lanes are wide enough to allow bicyclists to ride side to side at a high speed (approximately 25 km/h or 15 mile/h) or seek sufficient spaces to pass other preceding bicyclists if necessary. These bike lanes are different from those in United States and other countries that only allow bicyclists to ride in a single file due to limited lane width (normally less than 2 m (or 6 ft)). In addition, these bike lanes are different from shared-use bike paths where bicyclists and other users often shared their right of way, causing their travel speed to be low.

There has not been an effective method available for analyzing the unique characteristics of these dedicated bike lanes, assessing the bike lanes’ operational performance (such as level of service (LOS)), and providing practical guide for bike lane planning, design, construction, and operation in China. Previous research studies have been undertaken with focus on finding the statistical relationships between operational performances of bikeways and influencing factors such as width of bike lane, traffic conditions, road geometry. Dixon ¹ introduced a scale of LOS rating system (A through F) for bicycle facilities (including bike lanes) and developed statistical relationships between LOS of bikeways and potential conflicts of bike users, provision of transportation demand management programs, and intermodal connections. Landis, ² using survey responses of bicyclists, introduced a statistically calibrated LOS model and revealed that pavement-surface conditions and lane striping were important factors in quality of bike lane’s service. Petritsch et al. ³ developed a LOS model for arterials which consist of bike lanes. This model was a result of Pearson correlation analyses, stepwise regression, and Probit modeling of real-time feedbacks of bicyclists. Elias ⁴ established a LOS model of road segment by considering influencing factors such as peak hour factor, volume, pavement, width of bicycle lanes, and conflicts per unit segment. Krykewycz et al. ⁵ employed a web-based crowdsourcing approach to improve a bicycle comfort scoring system. Kang and Lee ⁶ developed a LOS model and established a set of LOS criteria in South Korea. The model incorporated level of satisfaction of users and other factors (such as width of bike lanes, road type, number of encounters, and number of lanes approaching to intersection) into consideration. Li et al. ⁷ investigated the impacts of physical environments (e.g. road geometry and surrounding conditions) on perceptions of bicyclists who ride on bike lanes and other bicycle facilities. Willis et al. ⁸ used a university-wide travel survey to examine the effect of built environment’s characteristics, trip characteristics, and seasonal weather on levels of satisfaction of bike lanes. Dai et al. ⁹ used a bicycle compatibility index to optimize a LOS model for urban road segments and identified key factors that influence LOS, including existence of bike lanes, bike lane width, width of vehicle lanes, and illegal parking.

Among previous studies dedicated to LOS of bike lanes, the conception of hindrances encountered during traveling is used to evaluate bicyclists’ perception of comfort. Botma ¹⁰ used number of bicycle passing maneuvers as a function of pedestrian and bicyclist volume to measure LOS in Netherlands. Much of Botma’s work on bike lanes was incorporated into the Highway Capacity Manual (HCM) 2000¹¹ in United States to calculate LOS for both uninterrupted and interrupted bicycle flows based on speed, travel time, freedom to maneuver, traffic interruptions, comfort, and convenience. Hummer et al. ¹² further enhanced Botma’s work and developed a new method to estimate LOS of bike lanes in United States by measuring number of passing maneuvers. The HCM 2010¹³ adopted Hummer’s method and determined LOS of bike lanes on six level evaluation as A-F scale. Shan ¹⁴ suggested the use of maneuvers in bicycle flows to depict bicyclist’s comfort degree and set passing maneuvers as a classification index for LOS. Li et al.^15,16 introduced a LOS evaluation method based on the estimation of bicycle passing maneuvers for physically separated bike lanes.

Although there are many methods to determine LOS of bike lanes, these methods do not address decided bike lanes in China where passing maneuvers are critical to the operational performance of bike lanes. The bicycle travel environment (characterized by large bicycle volume, high-speed bike lanes, and unique traffic regulations) in China is different from that in other countries. This unique environment may indicate that the approaches used in HCM and the above-mentioned studies may not be suitable for China. The Chinese highway design manuals have not yet provided specific approaches or guidelines to determine LOS of dedicated bike lanes. As a result, developing a LOS method for dedicated bike lanes that is adaptive to the bicycle travel environment in China is very important. This article proposes a specific approach of using the concept of entropy to determine LOS on dedicated bike lanes in China and gives some recommendations for LOS classification.

The remainder of this article describes characteristics of passing behaviors of bike users on dedicated bike lanes and introduces the new LOS approach. Section “Methodology” introduces the concept of reaction zones, passing maneuvers, entropy of speed states and its relation to LOS evaluation, and a LOS evaluation model. Section “Data collection” describes the data collection procedure used in this article for calibrating the LOS model, while section “LOS analysis” provides a five-level LOS rating system for dedicated bike lanes. In the end, section “Conclusion and future work” concludes the article by summarizing major recommendations of the study and outlining the future work.

Methodology

This article uses the concept of information entropy to analyze LOS of dedicated bike lanes. Before the entropy is introduced in this article, the terms that represent characteristics of bike users on dedicated bike lanes are described in the following.

Reactive zones of bike users

A dedicated bike path is represented as a continuous space Ω ∈ R², in which the bicyclists move from the origins to destinations. Each individual bicyclist and his or her bicycle together is abstracted as an ellipse (refer to Figure 1), which is assumed to follow Newton’s second law. The set theory is employed to describe bicyclist’s movement. A set for all the bicyclists is recorded as B. For giving time t ∈ ζ, the spatial and temporal characteristics of bicyclists in the two-dimensional bike path can be described in B × ζ → R².

Figure 1.

The reactive zone of a bicyclist.

Bike users riding on a dedicated bike lane reserve their territories in order to satisfy their psychological and physiological requirements. Invasion to the territories often results in instinct responses or reactions. These territories are the minimum private space that keeps a bike rider psychologically safe.

A reactive zone for a bicyclist is defined as the range of the personal territory, which is the intrinsic attribution of bicyclist for psychological influences. It describes the range within which obstacles affect movements of the bicyclist. This zone indicates that the bicyclist may take actions to avoid potential collisions with invading obstacles within the zone. In the course of riding, the bicyclist usually pays more attention to the lateral and anterolateral space ahead, and he or she could pass the obstacle aligning to the lateral edge by an extremely small gap. The bicyclist’s movement is considered as anisotropic. Therefore, the shape of the reactive zone is assumed as a gibbous area located in the center of abstracted ellipse (see Figure 1). And the size of reactive zone is a result of the riding speed, the angular field of view (AFV), and the characteristics (or reaction coefficients) of the bicyclist.

The boundary of a bicyclist’s reactive zone R_i(φ_i,θ_ij,t) can be described mathematically by equations (1)–(3) (see Liang et al.’s work ¹⁷ )

$$\begin{array}{c}{R}_i({\phi }_i,{\theta }_{\mathrm{ij}},t)=R_i\hfill \\ =\frac{L_i\left(t\right)b_i\left(t\right)}{\sqrt{L_i^2\left(t\right){\sin }^2({\theta }_{\mathrm{ij}}-{\phi }_i)+b_i^2\left(t\right){\cos }^2({\theta }_{\mathrm{ij}}-{\phi }_i)}}\hfill \end{array}$$

(1)

$$L_i\left(t\right)={\alpha }_i\Vert {\overrightarrow{v}}_i\left(t\right)\Vert +a_i$$

(2)

$$b_i\left(t\right)={\beta }_i\Vert {\overrightarrow{v}}_i\left(t\right)\Vert +b_i$$

(3)

where φ_i is the movement angle of bicyclist i and θ_ij is the angle between bicyclist i and obstacle j. Furthermore, ${\theta }_{\mathrm{ij}}\in [-{\vartheta }_i(t)/{\vartheta }_i(t)2,{\vartheta }_i(t)/{\vartheta }_i(t)2]$ and ${\vartheta }_i(t)$ is the AFV measured by the function of ${\vartheta }_i(t)=179.4\exp (-0.05022v_i(t))$ , where v_i(t) is the bicyclist’s speed at time t; L_i(t) is the longitudinal distance of reactive zone, which is determined by the riding speed v_i(t) of the bicyclist, the length of bicycle/2 or a_i of the bicycle ellipse, and the reaction coefficient α_i of the bicyclist (see equation (2)); b_i(t) is the lateral distance along the semi-minor axis of the reactive zone and is determined by the bicyclist’s riding speed v_i(t) and the lateral reaction coefficient β_i, as well as the width of the bicycle/2 or b_i of the bicycle ellipse (see equation (3)).

The set of obstacles influencing bicyclist i at a certain moment can be defined as follows

$$N_i=\{j:\Vert {\overrightarrow{R}}_j\left(t\right)-{\overrightarrow{R}}_i\left(t\right)\Vert \le R_i\}$$

(4)

where ${\overrightarrow{R}}_i(t)$ and ${\overrightarrow{R}}_j(t)$ , respectively, represent the location of bicyclist i and obstacle j at time t in the two-dimensional coordinate system. If N_i = ∅, which means there is no obstacle within the reactive zone of bicyclist i, and the bicyclist can move free. While if N_i ≠ ∅, which indicates that at least one obstacle invades the rider's territory and the rider will initiate defensive responses.

Types of passing maneuvers

In this article, the research framework described in the Dutch manual ¹⁰ and HCM ¹² on bikeways is employed to analyze LOS on dedicated bike lanes. By measuring the quality of operational process, the concept of hindrance is used to determine the bicycle LOS based on bike users’ interactions or maneuvers. In this approach, disturbing events are measured by the frequency of passing and meeting maneuvers. The passing maneuvers indicate that faster bicyclists pass slow-moving travelers, while the meeting maneuvers refer to the events that bicyclists meet opposing riders. Since the dedicated bicycle lanes in China are unidirectional and basically more than 3 m in width, we only need to study the passing type in bicycle operation.

When bicyclists ride on a dedicated bike lane, they intuitively scan their reactive zones, determine the intensity of dynamic and static obstacles invading in their zones, and decide their passing maneuvers to avoid any potential collisions. The intensity of the obstacles in bicyclists’ reactive zones helps classify the passing maneuvers into three types: free pass, adjacent pass, and delayed pass. ¹⁶ This classification of passing movements considers bike users’ perception of bikeways and bike flows. It reflects passing strategies of bicyclists when bicyclists sense their longitudinal and lateral distances to preceding bicyclists. Figure 2 provides the graphical illustrations of these passing maneuvers.

Figure 2.

Graphical illustrations of passing maneuvers: (a) free pass, (b) adjacent pass, and (c) delayed pass.

As shown in Figure 2, the detailed passing behaviors are as follows:

• Free pass. When a passing bicyclist senses that no slower preceding bicyclists are within his or her reactive zone (i.e. the reactive zone is blank (N_i = ∅)), the lateral or the anterolateral space ahead is adequate for the passing bicyclist to pass any slower bicyclists. The passing bicyclist does not need to change his or her speed or direction, while keeping a safe distance to the slower bicyclists. In this case, the passing bicyclist is in free movement state, and his or her riding comfort is unaffected by the passing maneuver (see Figure 2(a)).

• Adjacent pass and delayed pass. When the passing bicyclist senses that slower preceding bicyclists are within his or her reactive zone (i.e. N_i≠∅), the bicyclist then responds to the invasion by altering the speed and/or the moving direction, while working on his or her passing movement. The passing bicyclist has to pay more attention to the slower bicyclists and adjust his or her riding strategies to avoid potential collisions. In doing so, the passing bicyclist may have a declining perception of riding quality and LOS of the dedicated bike lane.

When the density of the bike flow is low than a critical value on the dedicated bike lane, each rider cares about the smoothness of his or her trip and passes slower riders ahead, if necessary, to maintain his or her desired speed (see Figure 2(b)). Therefore, the passing bicyclist needs to change the direction and/or speed to seek more riding spaces and pass the adjacent slower bicyclists safely. This type of adjacent passing maneuvers has certain impacts on the bicyclist’s riding comfort.

When the density of the bike flow increases on the dedicated bike lane, the bicyclists’ passing resistance also increases. When the passing bicyclist senses that there are many preceding obstacles (bicyclists, barricades, and so on) making his or her passing movement impossible (see Figure 2(c)), he or she will slow down and follow the bike flow until the reactive zone is empty again for other passing opportunities. This type of delayed passing movements causes the passing bicyclist to perceive low riding quality of the dedicated bike lane.

The classified standard of pass type, which is related to rider’s comfortable experience and perception during operation, refers to the intensity of the obstacles in bicyclists’ reactive zones. The comfortable and convenient experience can be distinguished in different passing types, which forms the basis of LOS. The clear reactive zone provides more freedom for the passing rider to maintain his or her desired speed and original moving direction. On bike lanes with high dense bike flows, passing maneuvers often bring discomfort, inconvenience, and possible danger to bicyclists. So, the term “hindrance” is embodied in these passing behaviors.

Evaluation index

A new term entitled as velocity migration rate (VMR) is introduced in this article to represent the perceived hindrance of bicyclists’ passing behaviors on a dedicated bike lane. This term is defined as the degree of deviation of bicyclist’s riding speed ${\overrightarrow{v}}_i(t)$ to desired speed $\overrightarrow{V_i}$ and can be calculated by equation (5) as follows

$$\mathrm{VM}{\mathrm{R}}_i=\frac{\Vert {\overrightarrow{v}}_i\left(t\right)-\overrightarrow{V_i}\Vert }{\Vert \overrightarrow{V_i}\Vert }$$

(5)

It is obvious that the greater the hindrance perceived by a bicyclist, the higher the VMR is. Actually, the desired speed $\overrightarrow{V_i}$ is a reference vector by which a bicyclist rides to his or her destination under free moving conditions or very low traffic conditions. In reality, bicyclists encounter traffic, which makes them vulnerable in maintaining their desired speed. Each bicyclist has a small lateral deviation from its direction of desired speed in free moving conditions. VMR represents this lateral deviation.

When a bicyclist is in free pass mode, he or she may not need to change the riding speed or direction during the passing movement. When a bicyclist is in adjacent pass mode, the lateral component of the bicyclist’s speed indicates that the bicyclist needs to change directions in order to avoid potential collisions with preceding bicyclists within the reactive zone. It is noted that this lateral component normally is larger than that of the free pass maneuvers. The longitudinal component of the bicyclist’s speed sometimes increases during the passing maneuvers. When a bicyclist is in delayed pass mode, the bicyclist has a small lateral deviation from its riding trajectory and his or her longitudinal speed normally decreases (see Figure 3).

Figure 3.

Method to analyze the VMR.

In this article, the VMR is measured through a cell-based system. As shown in Figure 3, the cell size is 0.3 m/s × 0.3 m/s, where 0.3 m/s is the maximum lateral deviation observed from the experiments in this research for free movement of bicyclists. The speed is represented through a grid of identically sized cells or speed zones, δv_x × δv_y, normally as 0.3 m/s × 0.3 m/s. The velocities that fall into the same speed zone are in one state. By vectorizing the velocity, the occupied speed zone can be found easily. In Figure 3, we assume a bicyclist rides at the desired speed $\overrightarrow{V_i}=(5.37\mathrm{m}/\mathrm{s},0)$ (shown as a solid line along the biking direction). The double dotted line represents the velocity of free movement ${\overrightarrow{v}}_i^f$ on the dedicated bike lane. As shown in Figure 3, the desired speed and the free movement speed are in the same cell, which indicates that no velocity deviation exists, that is, VMR = 0. When a bicyclist confronts hindrances in his or her reactive zone, he or she has to change the velocity to make a passing maneuver. ${\overrightarrow{v}}_i^a$ and ${\overrightarrow{v}}_i^d$ express the speed during the passing process. As shown in Figure 5, ${\overrightarrow{v}}_i^a$ is always greater than ${\overrightarrow{v}}_i^d$ , which means that the bicyclist is more affected during his or her delayed pass.

The VMR is a microscopic measure of the comfort and the convenience perceived by a bicyclist during his or her riding process. The VMR can be calculated at any time for each bicyclist within the bike flow on a dedicated bike lane. However, a microscopic indicator is required to represent the quality or the LOS of a dedicated bike lane.

This article proposes the use of Entropy of Speed State (ESS) and density to quantity the operational performance of bike users on dedicated bike lanes. The conception of ESS is used as the micro-level parameter, while density is chosen as the macroscopic criterion for bike users.

The ESS, a dimensionless parameter, is defined as shown in formula (6)

$$\mathrm{ESS}=\ln \left(\frac{N}{n_0}\right)$$

(6)

where N is the number of speed zones in which the observed velocity vectors fall at time t, and it is also called Numbers of Speed State (NSS); n₀ is the number of speed zones which is occupied by the desired speeds. For a fixed system, n₀ is a constant.

Generally, ESS refers to disorder or uncertainty. It is a measurement to bicyclist’s systemic confusion extent. Density is the most intuitive indicator to reflect the congestion of a dedicated bike lane. Changes of density are due to the changes of road resources, traffic conditions, and other factors, so the density could largely relate to the LOS of dedicated bike lanes in China. With the relationship established between density and ESS for bike users, we can understand the physical meaning of LOS from the view of ESS.

Dynamic forces applied to bicycles

The validated analytical model ¹⁸ is adopted to describe the movement of bicycles on dedicated bicycle lanes. In this model, the bicycle is treated as a basic element that obeys to long-ranged psychological force ${\overrightarrow{F}}_i^{\mathrm{psy}}(t)$ and contacted physical force ${\overrightarrow{F}}_i^{\mathrm{phy}}(t)$ . The movement is a direct result of the forces acting on the center of the abstracted ellipse. The psychological forces consist of driving force ${\overrightarrow{F}}_i^{\mathrm{drv}}(t)$ , collision avoidance forces ${\overrightarrow{F}}_{\mathrm{ij}}^{\mathrm{ca}}(t)$ , and boundary force ${\overrightarrow{F}}_i^{\mathrm{b}}(t)$ , while the physical forces, happened under high-density situation, can be broken into contact forces ${\overrightarrow{F}}_{\mathrm{ij}}^{\mathrm{cont}}(t)$ and sliding friction forces ${\overrightarrow{F}}_{\mathrm{ij}}^{\mathrm{frc}}(t)$ . The movement of each bicycle is defined by the dynamic equations as follows

$$m_i\frac{d{\overrightarrow{v}}_i}{\mathrm{dt}}={\overrightarrow{F}}_i^{\mathrm{psy}}\left(t\right)+{\overrightarrow{F}}_i^{\mathrm{phy}}\left(t\right)$$

(7)

$${\overrightarrow{F}}_i^{\mathrm{psy}}\left(t\right)={\overrightarrow{F}}_i^{\mathrm{drv}}\left(t\right)+\sum {\overrightarrow{F}}_i^{\mathrm{ca}}\left(t\right)+\sum {\overrightarrow{F}}_i^{\mathrm{b}}\left(t\right)$$

(8)

$${\overrightarrow{F}}_i^{\mathrm{phy}}\left(t\right)=\sum {\overrightarrow{F}}_i^{\mathrm{cont}}\left(t\right)+\sum {\overrightarrow{F}}_i^{\mathrm{frc}}\left(t\right)$$

(9)

$${\overrightarrow{F}}_i^{\mathrm{drv}}\left(t\right)=\{\begin{array}{cc}{m}_i[\overrightarrow{V_i}-{\overrightarrow{v}}_i\left(t\right)]/{\tau }_i\hfill & N_i=\varnothing \hfill \\ \multicolumn{1}{c}{-m_i{\overrightarrow{v}}_i\left(t\right)/{\tau }_i^0}& N_i\ne \varnothing \hfill \end{array}$$

(10)

$${\overrightarrow{F}}_i^{\mathrm{ca}}\left(t\right)=\{\begin{array}{cc}{A}_i\exp [-\frac{d_{\mathrm{ij}}\left(t\right)-r_j}{R_i}]\Delta v_{\mathrm{ij}}^n\left(t\right){\overrightarrow{t}}_{\mathrm{ij}}\hfill & N_i\ne \varnothing \hfill \\ \multicolumn{1}{c}{0}& N_i=\varnothing \hfill \end{array}$$

(11)

$${\overrightarrow{F}}_i^{\mathrm{b}}\left(t\right)=A_i^b\exp [-\frac{d_{\mathrm{ib}}\left(t\right)-b_i}{R_i^b}]{\overrightarrow{n}}_{\mathrm{ib}}{d}_{\mathrm{ib}}\left(t\right)-b_i<R_i^b$$

(12)

$${\overrightarrow{F}}_i^{\mathrm{cont}}\left(t\right)=K\Theta (r_i+r_j-d_{\mathrm{ij}}){\overrightarrow{n}}_{\mathrm{ij}}$$

(13)

$${\overrightarrow{F}}_i^{\mathrm{frc}}\left(t\right)=k\Theta (r_i+r_j-d_{\mathrm{ij}})\Delta v_{\mathrm{ji}}^t\left(t\right){\overrightarrow{t}}_{\mathrm{ij}}$$

(14)

where m_i is the mass of rider and bicycle. τ_i and ${\tau }_i^0$ are the relaxation times and ${\tau }_i^0$ is slightly larger than that of τ_i. A_i is a constant denoted as the strength of collision avoidance and $A_i^b$ is the strength from the boundary that is much less than A_i. d_ij(t) represents the distance between bicycle i and j, while the corresponding d_ib(t) is the distance between bicycle i and the boundary. r_i and r_j are defined as the radius of the abstract elliptical i and j, respectively, which lie within the angle between i and j. $R_i^b$ is the critical distance for the boundary force at work. The parameters K and k determine the obstruction effects in cases of physical contacts. The step function Θ(r_i + r_j − d_ij) can guarantee the contact force to be zero if individuals i and j do not come into contact. $\Delta v_{\mathrm{ij}}^n(t)$ and $\Delta v_{\mathrm{ij}}^t(t)$ denote the normalized and tangential relative velocity. ${\overrightarrow{n}}_{\mathrm{ij}}$ and ${\overrightarrow{t}}_{\mathrm{ij}}$ are the normalized vector and the tangential vector respectively pointing from bicycle j to i, while ${\overrightarrow{n}}_{\mathrm{ib}}$ indicates the normalized vector from boundary to bicycle i, all the three vectors also refer to the directions of different kinds of forces.

Driving force makes a bicyclist to move in a certain direction at a given desired speed. Collision avoidance forces and boundary force are imposed on bicyclists from other obstacles nearby (such as other bicycles, boundaries, and roadblocks) to prevent possible collisions. Contact forces and sliding friction forces are trigged by physical contact between target bicycle and other obstacles under high-density conditions. Contact forces provide an impenetrable potential barrier between bicycles to prevent them overlap from elementary mechanics and mathematics. Sliding friction forces protect the bicycles from bumping into obstacles and quickly passing within a short distance.

Data collection

To assess the ESS method described in section “Methodology” and determine its practical value, behavioral investigations and field studies were conducted in this research. The ESS values under different conditions were determined through the following controlled experiments and field surveys:

Controlled experiments

The goal of the controlled experiments is to determine rider’s reactive zone in real situation. ¹⁰ Under controlled experiments, the hindrance perceived by testing riders were observed and described. A separated 75 m × 6 m flat dedicated bike lane within Beijing Jiaotong University was selected as the experiment site (see in Figure 4). As the bike lane is 6 m wide, which is nearly 8.6 times larger than the abstract ellipse, the boundary impacts are negligible.

Figure 4.

The experimental area exhibition.

Three young riders were recruited for the experiments. The bicycle selected with 26-in rim in the tests is the one commonly used in China for commuting purpose. This bicycle is marked with a brightly colored ball placed in the middle of the bicycle’s handlebar. The riders were instructed to ride the bicycle without stopping from the initial position (in the midpoint of the lane as shown in Figure 4) to the target point located 70 m away from the original point. A system of coordinates with the origin at the initial point is established, and the y-axis is the direction of the tester’s desired speed.

The experiment process has two parts:

1. The testers were instructed to ride the bicycle at three speeds (natural/fast/slow) from static to reach to their destination without any obstacles on the dedicated bike lane. Trajectories which contain information about desired speed, maximum lateral deviation, and relaxation time τ_i (or the time taken by the testers to change their speed from 0 to the desired speed) can be recorded.

2. A cylinder, with the height of 155 cm and the diameter of 4.7 cm, was set at (50 m, 0) from the initial position perpendicular to y-axis. In this round of tests, the cylinder was placed randomly along the line between (50 m, −1 m) and (50 m, 1 m) at the interval of 10 cm, so 21 trajectories for each tester were recorded. The testers were informed to go to the destination without bumping into the cylinder. Through the testing runs, adjacent passing maneuvers were observed (see Figure 5), and the starting point of the passing maneuvers for each run was recorded. The starting point and the cylinder were used to define the reactive zone (see Figure 6). The parameters, A_i, ${\tau }_i^0$ , and VMR, were also obtained.

Figure 5.

Adjacent passing trajectories for different obstacle positions: (a) obstacles position from −1 to 0 m and (b) obstacles position from 0 to 1 m.

Figure 6.

Cross-section of the reactive zone for one tester.

The trajectories of bicycles were recorded by video camera. Each trajectory extracted from the video clip contains the position information of the tester at the interval of 0.1 s. The speed of the bicycle at each time interval was calculated. As the cylinder was a narrow subject, it was treated as a particle in this research.

Figure 5 shows the adjacent passing trajectories for different scenarios. The term “−2” in the legend means the cylinder located at (50 m, −0.2 m). From the figures, we can have the following findings:

• When the clearance between the tester and the cylinder is larger than 70 cm, the cylinder will have no impacts on passing maneuvers.

• The closer the cylinder is placed to the midpoint of the dedicated bike lane, the greater the impact of the cylinder on the passing maneuvers. The cylinder placed at (50 m, 0 m) has the highest impact on passing maneuvers. The tester starts its passing maneuver at 3.72 m before the cylinder.

• The lateral clearance is the distance between the cylinder’s position and the bicycle’s position at the passing point (as shown in Figure 2(b)). The recorded lateral distances range from 0.62 to 0.89 m, with an average of 0.75 m. Furthermore, no significant correlation has been found between lateral clearance and cylinder’s position.

• The recorded passing maneuvers show that the testers chose their efficient direction to pass the cylinder. Each tester perceived the left and right traffic conditions ahead of him or her and selected the side (with less traffic congestion) to pass the cylinder. It is noted that more passing maneuvers were on the left side, which indicates that the testers intended to pass the preceding cylinder following the left-hand rule.

Figure 6 demonstrates the reactive zone in which the rider is set at the original point, and the direction along the positive y-axis is the bicycle’s moving direction. It is noted that the reactive zone, marked by the solid or dotted line, is an angular dependence. When the transverse distance is closer from the cylinder to the tester on the riding direction, the tester needs a larger longitudinal distance to pass the cylinder. Moreover, the reactive zone is a symmetrical shape along the bicycle’s moving direction, which indicates that the influences of the cylinder are equivalent when the cylinder is placed on the left or right side. In addition, Figure 6 shows that the reactive zone is dependent on the moving speed of the tester. The length of the reactive zone for the tester at the natural, fast, and slow speed is defined by (3.98 m, 0), (5.26 m, 0), and (3.28 m, 0), respectively. With the process of moving, the faster bicycle moves, the larger reactive zone rider will maintain. And the changes of longitudinal direction are much higher than that of the later direction along with move speed. Unlike the shape of the reactive zone for pedestrians,^19,20 the reactive zone of bicycle riders is sharp at the peak. That may be a result of the moving speed.

Field survey

To apply the dynamic model as described in section “Methodology,” this research conducted a field survey to collect information about the density and instantaneous speed of bike flows. A dedicated bicycle lane on Weijin Road in Tianjin, China, was selected for data collection. Video cameras were placed to record bike traffic along a lane segment which is far from any intersections. The width of the lane segment is 3.5–5.8 m (as shown in Figure 7). Passing maneuvers were also recorded within this survey.

Figure 7.

Weijin Road, survey location in Tianjin, China.

The survey was conducted in 23 October 2015, from 7:00 a.m. to 9:00 a.m. and from 4:00 p.m. to 6:00 p.m. during bicycle’s peak hours. The trajectories of bicycle’s front wheel were recorded every 0.1 s to determine the location of the recorded bicycle. The speed and density were calculated and the ESS was determined using the method described in section “Methodology.”

Determination of model parameters

With the help of the controlled experiment and the field survey, trajectories were recorded and the parameters for the model were determined. These parameters in the model can be classified into three groups. The first group consists of semi-major axis a_i and semi-minor axis b_i of the abstract ellipse. The second group contains desired speed $v_i^0$ , mass of rider $m_i^r$ and bicycle $m_i^b$ , as well as normal and tangential constant K and k. ²¹ The third group consists of collision avoidance strength A_i, relaxation time τ_i, longitudinal response coefficient α_i, personal sensory coefficient β_i, and incidence of boundary B_i. Through the controlled experiment, the parameters of α_i, β_i, τ_i, and A_i were determined. Through the help of the field survey, the range of B_i and $A_i^b$ was determined. Table 1 shows the parameters and their values used in the model for simulation.

Table 1.

Parameters determination.

Parameter	Description	Unit	Values used
ai	Semi-major axes of individual	m	0.88
bi	Semi-minor axis of individual	m	0.35
αi	Longitudinal response coefficient	s	Uniform[0.76, 0.9]
βi	Personal sensory coefficient	s	Uniform[0.15, 0.18]
$m_i^b$	Mass of bicycle	kg	Uniform[10, 20]
$m_i^r$	Mass of rider	kg	Normal[45, 85]
Vi	Desired speed	m/s	Normal[4.17, 5.56]
τi	Relaxation time	s	Uniform[2.0, 2.5]
${\tau }_i^0$	Deceleration relax time	s	Uniform[2.8, 3.2]
Ai	Collision avoidance strength	kg/s	Random[120, 150]
$A_i^b$	Boundary avoidance strength	kg/s	Ai/3
Bi	Incidence of boundary	m	0.4
K	Coefficient of contact force	kg/m/s	12,000
k	Coefficient of sliding friction	kg/s2	24,000

LOS analysis

Essentially, the macroscopic state of bicyclists on a dedicated bike lane is a response to the microscopic interactions among bicyclists. In this article, we first combined the microscopic and macroscopic levels of operational performances on dedicate bike lanes through the model described in section “Methodology.” Then we used the simulations to implement the model for LOS analysis.

The simulation model was developed in C# language. The number of bicyclists was kept constant for each simulation, within a range [0, 200] bicycle/min. The simulation interval was 0.05 s. The positions and velocities of the bicycles were simulated at an interval of 0.05 s. The NSS, ESS, and density were calculated as the simulations are completed. Figure 8 shows the NSS values under different density scenarios of bicyclists on a simulated bike lane (lane width = 5 m). Sixty bicycle riders were simulated in the model.

Figure 8.

NSS under density with 60 bicycles: (a) n₀ = 6, (b) ρ = 0.08 bicycle/m², NSS = 8, (c) ρ = 0.25 bicycle/m², NSS = 26, and (d) ρ = 0.45 bicycle/m², NSS = 11.

With the velocity component divided by δv_x and δv_y, the desired speed in simulations from 4.17 to 5.56 m/s indicates that six speed zones are considered as the benchmark ESS (see Figure 8(a)). Since the density is lower, the velocity vectors concentrate on the desired speed with slight lateral deviations for adjacent passing maneuvers. In these circumstances, the riders have sufficient spaces for free passing. During the medium-dense situations, as ρ = 0.25 bicycle/m², the longitudinal velocity intervals are more decentralized and the concentration shifts to the left. This indicates that there is a high proportion of the riders involved in the influenced situations. These riders want to maintain at a higher speed for their passing maneuvers. As a result, the adjacent passing maneuvers are dominant. Furthermore, in congested traffic regimes, the velocity vectors have small migration to the lateral direction and are more concentrated on lower speed zones. This indicates that the riders’ movements are constrained by each other and the delayed pass becomes the primary type of passing maneuvers.

The relationship between the ESS and the density of the bike flows is the basis for the LOS classification of dedicated bike lanes. The simulation results show that the range of desired speed for bicyclists is [4. m/s, 5.56 m/s], when the speed interval is 0.3 m/s × 0.3 m/s, n₀ = 6. The value of ESS can be calculated by equation (6). Figure 9 shows the relationship between ESS and the average density.

Figure 9.

Relationship between ESS and average density.

From Figure 9, we can summarize that the microscopic ESS has a close relationship with the macroscopic density. According to this relationship, this research defines five levels of service for dedicated bicycle lanes, which range from A to E. LOS A represents the best operating conditions from the bicyclist’s perspective and LOS E represents the worst operating conditions.

When the density (ρ) is less than 0.10 bicycle/m², the ESS will increase sharply from approximate n₀ to 15. In this case, the density is very low and the spacing between riders is large. Bicyclists can ride freely at their desired speed, and free passing maneuvers become the main passing mode. As a result, bicyclists feel comfortable in such traveling environment. Therefore, we define LOS A with the density ρ < 0.10 bicycle/m².

When ρ ∈ (0.10, 0.16] bicycle/m², the ESS begins to increase gradually. In this case, due to the low density and large spacing between different individuals, bicyclists can move under less restricted flow and change lateral positions frequently to get more spaces. The passing maneuvers mainly focus on free passing and adjunction passing. Although little interruption exists in the flow, bicyclists can ride their bicycles in this free riding mode for a long time. As a result, bicyclists have a pleasant perception of comfort. Therefore, we define this condition as LOS B, with ρ ∈ (0.10, 0.16] bicycle/m².

When ρ ∈ (0.16, 0.28] bicycle/m², ESS reaches a maximum and keeps stable. With the increase of density, the spacing between different riders has become smaller gradually. Almost every bicyclist is disturbed and could no longer move forward at the desired speed. However, each individual has a tendency to obtain more spaces, which maximizes ESS and makes it remain stable. Under such case, adjacent passing is the dominating passing mode. There are many interruptions experienced by riders. The bike flow on the lane is unstable. Bicyclists may feel stressful and uncomfortable in such conditions. Based on the above results, we define LOS C with the density (0.16, 0.28] bicycle/m².

When ρ ∈ (0.28, 0.39] bicycle/m², the ESS presents a slowly decreasing tendency. Within such density range, different behaviors among riders have emerged gradually. Some riders stop looking for free riding spaces and turn to form a single file with other bicyclists. At this time, the passing maneuvers are adjacent passing and delayed passing. The bike flow on the lane has come to an influenced state with crowed bicyclists. There are significant interruptions in the flow. Bicyclists’ perception of comfort declines obviously. We define LOS D for ρ ∈ (0.28, 0.38] bicycle/m².

When the density is more than 0.39 bicycle/m², the value of ESS is less than 1.2 and decreases rapidly. In this case, the bicycle movement has been constrained greatly by other individuals. Delayed passing becomes the main passing mode in this case. There are enormous interruptions in the flow. As a result, the bicycle flow on the lane has been in a heavily congested state. Bicyclists are nervous, stressful, and feel extremely uncomfortable. Therefore, we define LOS E with the density larger than 0.39 bicycle/m².

Generally, LOS A and B represent high service level of bike lanes on which the bicycle flow moves smoothly. LOS C represents the service level within a bicyclist’s acceptable range. While LOS D is the acceptable service limit and bicyclists have been in an unstable state under this level. However, LOS E is considered to be beyond bicyclist’s tolerance, which is usually caused by over-saturated bike flow.

Conclusion and future work

Due to the special bicycle traffic environment, high bicycle travel demand and unidirectional bikeways in China, the bicycle's traffic characteristic may be unique. In most cities, the bicycle road was built on the right side of the motorized vehicular lanes and usually separated from vehicles and pedestrians with physical entities. Therefore, the unidirectional dedicated bicycle path is highlighted as the research subject, and an approach to determine the LOS is proposed to measure the quality of operational process. This article adopts the research framework of Botma and Hummer and defines the hindrance by the concept of reactive zone for bicyclists. The reactive zone is an effective concept that can describe the intrinsic interactions among riders on the dedicated bike lanes. Based on the occupied conditions within the reactive zone, passing events can be classified into free passing, adjacent passing, and delayed passing. The changing degree of the speed parameter VMR is used to express the hindrance experienced. Moreover, the ESS is calculated from VMR for every time point. In this article, we establish a relationship between the ESS and the density of bike flows and develop a multi-scale mechanism for LOS evaluation.

In order to define the shape of reactive zone, this research conducted a series of controlled experiments to measure the behavioral effects of interactions among bicyclists. It is demonstrated that the reactive zone is bilateral symmetry and has an angular dependence, which indicates that the movement of bicyclists is considered as anisotropic. In addition, the testing results show that the reactive zone is controlled by rider’s riding speed. When the rider rides at a higher speed, the reactive zone will grow in size.

Computer simulations are used to build the relationship between the ESS and the density. A five-level LOS classification ranging from A to E is defined in this article for dedicated bicycle lanes, in which LOS A represents the best operating conditions from the bicyclist’s perspective and LOS E represents the worst operating conditions.

Although the rider’s comfort and perception are considered in this research by the use of the reactive zone concept, the article still has some limitations. First, the model that describes the passing maneuvers has a great set of parameters. This article uses the controlled experiment and the field survey (with limited observations) to determine some parameters. However, these parameters may be different from different urban settings. Future studies should be considered to use various controlled experiments to define these parameters.

Second, the simulation model should be validated with observed data. Also the model needs to be updated so that the transitions among the three passing maneuvers (free passing, adjacent passing, and delayed passing) should be considered in the future.

The method described in this article is suitable for homogeneous bicycle flow. Future research could be conducted to see whether the model can be revised to consider for heterogeneous traffic flow (such as mixed flows of bikes and pedestrians).

Finally, the relationship between ESS and density is established in this article using data from simulations. It is recommended that in the future, this relationship should be validated using data from real life cases.