Defining Highway Node Acceptance Capacity (HNAC): Theoretical Analysis and Data Simulation

A new concept of Highway Node Acceptance Capacity (HNAC) is proposed in this paper inspired by a field data observation. To understand HNAC in microscopic view, boundary condition of successful merging is found using car-following behaviours and lane-changing rules, which could also explain traffic oscillations. In macroscopic view, linear positive relationship between HNAC and background traffic volume is obtained based on moving bottleneck. To determine the explicit form of the relationship, data simulation considering car-following behaviours and traffic flow theory is used. In the results, the synchronization phenomenon of oscillation in on-ramp (with respect to main road) and intersected road is found. The explicit equation of HNAC is determined based on standard deviation and correlation coefficient analysis, and also proved to be accurate with model validation, which is helpful in studies related to propagation mechanism of traffic emergencies on highway network.


Introduction
In our previous work [1], tra c ow eld data was collected on City Ring Road and Lianhuo Expressway, Xi'an. During the process of data collection, an interesting phenomenon was observed on Baqiao Interchange (K450 + 100 of Lianhuo Expressway, G30). Taking northbound tra c merging from City Ring Road to Lianhuo Expressway as an example (Figure 1), tra c condition of this highway node varies between morning and a ernoon peak hours. In Figure 1(a), tra c condition at 8:30 a.m. was depicted. e tra c ow in downstream Lianhuo Expressway had tra c volume of 1830 veh/h * l and density of 26 veh/km * l, respectively. ese two values in upstream City Ring Road were 465 veh/h * l and 11 veh/km * l, and merging volume of on-ramp was 330 veh/h * l. It could be seen, all sections mentioned above were in free ow. is is mainly because during morning peak hour, people go into the city, which means going south. During a ernoon peak hour, people go out northbound, which leads to a di erent situation. In Figure 1(b), tra c condition in downstream Lianhuo Expressway remained the same (volume of 1830 veh/h * l and density of 26 veh/km * l), but congestion was caused in upstream City Ring Road (volume of 272 veh/h * l and density of 91 veh/km * l) with the increase of merging tra c volume (665 veh/h * l) in on-ramp.
Combining the example given above, some terms are de ned here. e rst term is "main road", de ned as road section with tra c merging in of a highway interchange (like Lianhuo Expressway in the example). e second term is "intersected road", de ned as road section with tra c diversion (like City Ring Road in the example). e third one is "highway node", which is a part of an interchange, consists of main road, intersected road and the on-ramp (with respect to main road) connecting them. Inspired by the phenomenon observed above, a problem could be presented here. If there was an oscillation downstream the main road, whether a speci c upper limit exists, when exceeded by merging tra c volume, tra c ow of intersected road will be a ected. is is crucial in research on propagation mechanism of tra c emergencies in highway network, which is very helpful to large scale evacuation and rescue. In 1995, Daganzo [2] provided a quite detailed study on tra c merging problem using cell transmission model. is is a classic research followed by many scholars, and there are still some works remaining to be done, to further improve this eld. e method in his work mainly focused on the topological form of the junction and the logic between adjacent nodes, scilicet a partial mathematical method. erefore, in consideration of practical application, some practical e orts could be done, including microscopic relationship among vehicles and tra c ow's physical characteristics in merging area, e.g. us, to further ameliorate Daganzo's classic work, concept of Highway Node Acceptance Capacity (HNAC) is provided here. HNAC means the upper limit of tra c volume merging into the main road from the intersected road through the on-ramp. If this value was exceeded by merging volume, tra c ow in intersected road will be a ected, showing obvious oscillations and further forming congestion. is parameter shares the same unit with tra c volume, scilicet veh/h ⋅ l. In microscopic view, study of HNAC is directly related to a speci c vehicle's driving behaviour when merging into target lane in main road. Mentioning problems related to microscopic driving behaviour, car-following models and lane-changing rules become crucial because they are basic theories in this eld, and they will be studied in Section 3. In macroscopic view, essence of studying HNAC is calculating the volume of successfully merging vehicles. Typically, a successful merging behaviour depends on enough large spacing between the leading vehicle and the follower in target lane, and the volume of large spacing is related to HNAC. In real tra c stream, spacing between adjacent vehicles is not homogeneous. e existence of slow vehicles will cause moving bottlenecks and reform the tra c stream. erefore, in macroscopic study of HNAC, theories of moving bottleneck should be taken into consideration, and these contents will be studied in Section 4. Furthermore, literature review will be stated in Section 2. e process of data simulation is given in Section 5, and Section 6 provides the results, including the synchronization of oscillation in on-ramp and intersected road, standard deviation and correlation coefcient analysis, and the explicit equation of HNAC. Section 7 provides conclusions and some discussions.

Literature Review
As depicted in Section 1, HNAC was de ned as the upper limit of tra c volume merging into main road through on-ramp. If this value was exceeded by merging volume, tra c ow in intersected road will be a ected.
is concept was rstly mentioned in Kerner's classical three phase tra c theory work in 2004 [3], by stating that "in free tra c of merging area, there might be random highway capacities which depend on the ow rate of on-ramp".
is statement is related to the phenomenon observed in Section 1. However, unfortunately, in Kerner's abovementioned and following works [4][5][6], clear de nition of HNAC and explicit formulas were not provided, either in other followers' works [7,8].
is paper began with the de nition of HNAC in Section 1. As depicted in previous part, car-following model is rstly considered. e car-following model developed by Newell [9] showed some signi cant importance and practicality in related works. is model clearly depicts the relationship between the leading vehicle's and the follower's moving condition. It implies the follower will change its velocity and spacing depends on the leader's driving condition with a small spatio-temporal hysteresis. Besides, compared to traditional car-following models, the dynamic process of acceleration and deceleration are considered as instantaneously completed. erefore, this model showed some advantages in explicit tra c ow macroscopic modeling compared to those traditional ones [10][11][12][13][14][15]. Moreover, Newell's model showed the same accuracy and succinct formation with a di erent logic more suitable to modern tra c. Following theoretical and empirical studies of classic car-following model, relaxation phenomenon was discovered, showing that driver will accept shorter spacing and adjust it to more comfortable value in process of lane-changing or merging behaviour [16][17][18][19][20]. is concept and phenomenon could be used as an assumption in microscopic modelling. Tra c oscillations and stop-and-go waves are common phenomena caused by lane-changing behaviours [21][22][23], which could be theoretically explained using car-following models. erefore, this could be used as inspection standard in this paper. In Laval's related study [24], a parsimonious theory explaining the appearance and transformation of tra c oscillation was provided based on Newell's model, and, timid and aggressive driving behaviors were concluded, which could be used in macroscopic modelling of this paper. Set aside the theoretical researches, empirical data were also used in model formulation. In Chen's work [25], a behavioral car-following model was developed based on empirical trajectory data using NGSIM dataset, which revealed the dynamic behaviour pro le of drivers experiencing tra c oscillations. e data characteristics in this work provided some references in data simulation. In recent works, the researches of car-following model were basically related to tra c control strategy and automated vehicle [26][27][28]. Among them, the work by Han provided a novel breakdown probability model based on extending Newell's model, and a tra c control method to obtain uniform spacing was developed considering low passing rate of connected automated vehicle technology.
Based on the concept of HNAC depicted in Section 1, lane-changing rule and model are problems which could not be ignored or avoided in microscopic modelling in this work. e driver's decision to change lanes derived from his or her answers to three questions, whether it is possible, necessary, and desirable to change lanes [29]. In his work, structure of the driver's decision process before changing lanes was modelled. It has been regarded as a procedural and hierarchy basis in lane-changing models. Following lane-changing rules, the method of cellular automata was used in model formation [30,31]. In anterior work, e ects of various rules of lanechanging on characters of tra c ow were studied. e results showed that most e cient rules would be those allow fast vehicles to travel as fast as possible without sacri cing the total throughput. In the latter work, a general scheme of lanechanging rules was proposed based on summary of di erent approaches. In this work, Wagner's gap rules [32] were developed and realistic lane-changing rules were obtained. In the researches of lane-changing model, data simulation was frequently used. e classic study by Hidas [33] should be noticed. e detailed and inspirational lane-changing and merging algorithms were presented in this work, indicating that forced and cooperative lane changing are essential, which could produce realistic volume-velocity relationships during congested conditions. In 2006, a statement that lane-changing behaviours were strictly related to moving bottleneck and tra c volume reduction was proved by Laval [34]. In this work, the mechanism mentioned above was explained by a model that tracks lane-changing vehicles precisely. Besides, two phenomena previously thought to be unrelated were combined by this simpli ed parameters model, passing rate drop of bottleneck at the beginning of congestion, and the relationship between moving bottleneck velocity and its capacity. Safety criteria should be regarded as a boundary condition in lane-changing models. In Kesting's work [35], a general lane-changing model was proposed for discretionary and mandatory lane-changing behaviours. e essence of safety criteria was explored in his paper, in other words, the merging vehicle should keep safe distance with the leader and the follower in targeted lane.
As depicted in previous part, the theories of moving bottleneck could provide a reasonable explanation of vehicle distribution characters in real tra c ow. is phenomenon was systematically studied by Gazis [36], and the widely used model was developed by Newell [9]. In their work, the de nition of moving bottleneck was given, and the model of passing rate, queue behaviour, moving queue growth rate were also provided. ese results were widely used and developed in related researches [28,[37][38][39].

Microscopic Modelling of HNAC
In microscopic perspective, research on HNAC could be related to speci c situation stated as below. Imagine that, a vehicle traveling from on-ramp at a stable velocity, having the demand to enter the main road. However, the spacing between the leading vehicle and following vehicle in target lane does not satisfy the microscopic lane changing condition. erefore, the vehicle ready to enter the main road has to decelerate and wait for a proper chance. is situation might lead to congestion in on-ramp and further a ects the tra c condition in intersected road.
is situation could be interpreted as: the volume sent by on-ramp exceeds the HNAC of the main road.
From the simpli ed description in previous context, the microscopic modelling should consist of two important parts, car-following behaviour and lane-changing condition.

Car-Following
Behavior. For analytical tractability, vehicles are assumed to follow the classic car-following model by Newell [9]. is model provides two crucial contents, the relationship between the leading vehicle and following vehicle's trajectory in time and space, and, the relationship between a vehicle's spacing and velocity. is model formed the basic laws in car-following research areas, adopted and developed by many researchers [18-20, 24, 25, 28].
In Newell's model, highway is treated as a homogeneous carrier, and spacing is linear related to travel velocity. When the leading vehicle changed its operation velocity, the follower would change its speed according to the leader's, but there is a hysteresis that exists in time and space. is relationship could be depicted in Figure 2.
As depicted in le side of Figure 2(a), the trajectories of leading vehicle and the follower's are represented by n−1 ( ) and n ( ) respectively. At a time point, the leader accelerated from v n−1 to v ὔ n−1 (v n−1 < v ὔ n−1 ). However, the follower did not change its velocity immediately. Instead, a er a time period n , when the spacing of the follower increased from n to ὔ n , the follower also accelerated to v ὔ n . e parameters n and n related to hysteresis phenomenon are independent from v n , they depend on the driver himself. e trajectory in car-following model and the relationship between spacing and velocity depicted in Figure 2 could be expressed as below.

Journal of Advanced Transportation 4
lane-changing vehicle enters the target lane, it should keep a "safe distance" with the leading vehicle and the following one. It is known that the "safe distance", referred to as spacing, is related to operation velocity of the vehicle. erefore, equation (3) should be cooperated in microscopic modelling.
To simplify the modelling process, assuming that merging vehicle completes its lane changing process in a short period without acceleration or deceleration, soon a er it enters the target lane, the vehicle will adjust its velocity according to the leader's driving condition, and the follower will also adjust its velocity according to the lane-changing vehicle's driving condition. is process is depicted in Figure 3.
At time point 0 , a vehicle traveling from on-ramp with velocity v in shows up at any position beside the target lane. In target lane, the follower at position n is traveling at velocity v n , and the leader at position n−1 is traveling at velocity v n−1 . At 0 , the distance between each vehicle are n and n−1 . At time point 0 + Δ , the vehicle has just nished its lanechanging process. e distance between each vehicle are ὔ n and ὔ n−1 . In this process, ὔ n−1 and ὔ n could be calculated using Equations (4) and (5).
In the two equations shown above, n could be de ned as the spacing in totally congested tra c, and n could be de ned as the time consumption by which the kinematic-wave travels a distance of n [37]. As depicted in Figure 2(b), there is a limitation v n ∈ 0, v free , which means equation (1) only established in congested tra c ow (the right side of volume-density curve). To a steady tra c ow, all of the vehicles travel at nearly the same velocity v, then we get: It should be noticed, in real restricted tra c ow, the spacing varied due to the personality of the drivers. Aggressive drivers tend to choose smaller spacing ( n , n ) and vice versa, timid drivers tend to have loose spacing. It should be noted, carfollowing model is used in restricted ow, in which the follower should change its running velocity and spacing according to the leader's. In traditional tra c ow theory, there is a boundary concentration k b between restricted ow and free ow [40]. In free ow condition, the driver operates in free ow velocity v free , and maintains the spacing larger than n,free = n + v free n owning to the low concentration. When the concentration equals to b , the tra c ow is about to enter the restricted condition. At this moment, the driver in free ow gets its smallest spacing equal to n,free .

Microscopic
Modelling Based on Lane-Changing Rules. As described in previous part, the microscopic modelling of HNAC could be transferred to a problem related to lanechanging rules. To be precise, to nd a boundary spacing between two adjacent vehicles in targeted lane of the main road for the vehicle from on-ramp at a speci c velocity to enter.
Lane-changing rules are rst developed by P. G. Gipps in 1985 [29], with the perfections of the followers [21,30,31,33,35]. It is commonly accepted that a er the (3)  Journal of Advanced Transportation 5 merging vehicle to enter the main road at v in , then this spacing could be called an "e ective spacing" (ES). e essence of HNAC macroscopic modelling is to calculate the volume of ES passed the entrance in a time unit, and determine the relationship between ES number (ESN) and tra c volume in main road.

HNAC Macroscopic Modelling in Congested Flow.
As depicted in previous part, congested ow refers to the right side of volume-density curve. In this situation, the velocity of each vehicle in tra c ow is assumed to be the same, v. en limit could be reformed as below.
From equation (9), limit = in + (v) > (v), which means there is no chance for merging vehicle to enter the main road, and this is obviously unrealistic. e truth is, in congested ow, even the velocity is treated as equal, the spacing di ers from each other due to the variance of and . To aggressive drivers, they tend to choose smaller and , as for timid ones, they tend to choose larger values [24], which could be expressed as below. erefore, assuming a vehicle platoon's volume equals to . Among them, the ratio of ESN derived from timid driver is assumed as , then the length of the vehicle platoon could be calculated using equations (12). e time consumption and tra c volume of passing a speci c entrance could be calculated by equations (13) and (14). en, combining equations (12)-(14), the ESN could be expressed as below, which means ESN is linear positive correlated to tra c volume. But, it should be clari ed that the model given above is an ideal situation. Here provides an example to show the reality in some degree, shown in Figure 5. A platoon consists of four vehicles is going to pass an entrance located at = 0, and they are all operated by timid drivers with velocity v. e initial spacing tim = limit . At the time point = 0 , a merging vehicle from on-ramp begins entering the target lane, and adjusts its velocity from a lower value to v. In order to keep the spacing tim , the three followers have to decelerate and lead to an oscil- With the lane-changing rules in the 2nd paragraph of this section, and cooperating equation (3), the boundary spacing limit between two adjacent vehicles in targeted lane could be obtained. e modelling of HNAC is strongly related to the boundary spacing limit . Imaging that, the vehicle traveling from the on-ramp is ready to merge into the target lane in main road. However, the spacing of the tra c ow in target lane of the main road is smaller than limit , meaning there is no chance for this vehicle to merge in. erefore, this vehicle from on-ramp has to slow down or even stop to wait a proper chance. is is the situation for one single merging vehicle, just like an unsuccessful interpolation of two gearwheels. In real tra c condition, merging tra c from the on-ramp is continuous, indicating that if the volume of spacing satis ed limit in main road tra c ow is not enough for the merging tra c volume from on-ramp to consume, a congestion might form on the on-ramp and further a ect the intersected road.
It should be noticed that the merging process varies depending on di erent relationships among v n , v n−1 , and v in . To further study, six situations are established and the merging process in each situation is shown in Figure 4.
In all of the 6 situations, the merging vehicle doesn't need to change its velocity only in situation (2). When v n−1 > v in > v n , the distances among three vehicles will get larger, and this is the most idealized condition. From situation (1) and (4), when v in > v n , whatever the relationship between v n and v n−1 , the driver has to change the velocity according to the leader, which means the merging vehicle will join the downstream platoon and shows no a ection on upstream trafc. From situation (3), (5), and (6), when v in < v n , it is important to notice that an oscillation shows up with a transmission speed equal to / , which means the merging vehicle forms a moving bottleneck that a ects the upstream tra c. Besides, it also proved that the lane-changing behaviour will lead to some tra c oscillations, that is, if intersected road is a ected when the transferred volume exceeds HNAC, it is important to distinguish the a ection of oscillations and from the on-ramp, scilicet HNAC.

Macroscopic Modelling of HNAC
In this part, the range of macroscopic study should be rstly de ned. Microscopic study mentioned in Section 3 depicted a research scope concentrate on single vehicle's moving condition. Compared to concept of microscopic study, macroscopic modeling concentrates on tra c ow in a certain highway node, especially the tra c ow in main road. In Section 3, and were adopted as average values, which means limit only depends on velocity. If a spacing in target lane is larger than limit , which means that it's e ective for a ὔ n ≥ n = + ⋅ v n , Journal of Advanced Transportation 6

HNAC Macroscopic Modelling in Free Flow.
e vehicle distribution character in free ow is di erent from congested ow. Due to the existence of moving bottleneck, it is unreasonable to regard every vehicle's velocity as free ow speed v free . In the part without moving bottleneck, each vehicle travels with velocity v free . Taking the description in last part lation. e actual spacing passing the entrance were 1 , 2 , and 3 . From the gure depicted this process given below, it is easy to conclude that 1 = 3 = limit , 2 < limit . erefore, the actual ESN turns from three to two. at is to say, in real tra c ow, the actual ESN will be lower than the value given by equation (15).  In moving bottleneck region, v b < v free , then we have: For simpli cation, discrepancy among the drivers is ignored. erefore, it is reasonable to assume that ES only exists in the part of free driving. In the part of moving bottleneck, every spacing does not satisfy the merging condition. Assuming that the length of a stable moving bottleneck is b , the distance between the leading vehicle and the entrance is free . In a time period , the last vehicle in moving bottleneck passed the entrance, and the background tra c ow parameters counted by an observer on the entrance is average volume , average speed v, average concentration . e parameters in free ow part and bottleneck part are represented by v free , free and v b , b , respectively. en ESN could be calculated by equation (20).
In equation (20), free free represents the vehicle number in free ow part, and Tq r stands for vehicle number escaping from queue region and it entered the target lane. It is known, b + free = QT/ , combined with equation (18), we have: By equation (21), we can draw the conclusion that ESN is linear positive correlated to tra c volume and linear inversely proportional to tra c concentration.

Data Simulation
According to results obtained from Section 4.1 and 4.2, HNAC (ESN) is theoretically linear correlated to the background tra c volume of the main road. Besides, from analysis in the last part of Section 4.1 and equation (21), it is known that the real ESN in free ow is lower than the value calculated by equation (15), and the equality relationship is only obtained in congested tra c ow. Moreover, in equation (21), ESN could not be directly obtained because the values of parameters free , , , and v b are undetermined.
of Section 3.1, the distance between adjacent vehicles n and limit could be expressed as below.
In the part of a stable moving bottleneck, the tra c condition becomes complex. Moving bottleneck is caused by slow moving vehicles in free tra c ow. Free running vehicles have the demand to pass the slow one (moving bottleneck), and a queue would form behind the bottleneck until it reached a stable length. e tra c characteristics in moving bottleneck could be expressed by Figure 6 [36]. In queue region, it could be seen that the travel velocity and concentration in all of the lanes are nearly the same, expressed as v b and b . While in downstream, operation speed basically equals to v free . However, concentration blo in blocked lane is smaller than free in upstream region, concentration un in unblocked lane is smaller than blo . is could be interpreted as the vehicles escaped from queue region didnot spread in each lane in average.
Moreover, when getting back to the calculation of HNAC, the tra c volume escaped from queue region and entered the target lane should be taken into consideration. When the capacity of downstream road is known as , the portion of volume enters the target lane is , then the volume referred in previous r could be calculated as below [38]. (16) n > free = + v free , (17) limit = in + + v free = in + free . Journal of Advanced Transportation 8 last part of Section 3.2, the oscillation derived from exceeding HNAC shows synchronization with the tra c condition in on-ramp. erefore, in Figure 7, a detector is also deployed to observe the tra c condition in on-ramp. To nd HNAC, the merging tra c volume also varies, shown in Table 2.

e Synchronization of Oscillation in On-Ramp and Intersected Road.
To make the simulation feasible, number of simulation groups should be relatively limited. erefore, 11 large groups were divided according to heavy vehicle mixing ratio , seen in Table 2, = 5% was settled as step distance. In each large group, 10 sub-groups were divided according to concentration in main road, based on step distance of Δ main = 4. So, the concentration in main road varied from relatively free ow to congested ow, representing the most common tra c conditions in daily life. T aking the data from four groups when = 10% as examples, which numbered 10-2-1 (v main = 54.9 km/h, main = 34 veh/km * l, main = 1868 veh/ h * l, in = 100 veh/h * l), 10-2-3 (v main = 54.9 km/h, main = 34 veh/km * l, main = 1868 veh/h * l, in = 300 veh/h * l), 10-2-7 (v main = 54.9 km/h, main = 34 veh/km * l, main = 1868 veh/h * l, in = 700 veh/h * l), 10-2-10 (v main = 54.9 km/h, main = 34 veh/km * l, main = 1868 veh/h * l, in = 1000 veh/h * l), respectively. e synchronization of oscillation in on-ramp and intersected road could be easily observed in Figure 8. e gures depicted in Figure 8 come from a simulation group consisting of 18 simulation tests, in which the merging tra c volume in varies from 100 veh/h * l to 1800 veh/h * l (Δ in = 100 veh/h * l). In Figure 8(a), in = 100 veh/h * l, which is a relatively very low volume, caused no oscillation in both the intersected road and on-ramp. In Figure 8(b), in = 300 veh/h * l, higher than previous volume, caused some oscillations in intersected road. It should be noticed that there is no oscillation that appears in on-ramp, considering the analysis in the last part of Section 3.2, this is due to the lane-changing behaviours in intersected road. In Figure 8(c), in = 700 veh/h * l, there is also no oscillation appears in on-ramp. But the frequency of oscillation in the intersected road is obviously larger than that in Figure 8(b), which means the increasing number of lane-changing behaviours lead to severe velocity dispersion in the intersected road. In Figure 8(d), in = 1000 veh/h * l, when ∈ [0 s, 1500 s], the tra c condition in intersected road and on-ramp showed the To determine free , , , and v b based on theoretical analysis is unrealistic and unworthy because they are assumed to be random. erefore, to obtain the explicit form of equation (21), method of data simulation is adopted in this paper. In the process of data simulation, tra c model in our previous work [1,40] is adopted, shown in equation (22), depicts the relationship among tra c volume , concentration and heavy vehicle mixing ration .
In this paper, the simulation platform VISSIM is adopted, which provides a high level of details and exibility in highway design, vehicle performance and driver's behaviours. According to the user manual, three detailed aspects should be noticed in the process of building HNAC (ESN) simulation model [41]: vehicle movement at highway merging area, velocity adjusting area in highway on-ramp, and car-following behaviours.
In aspect of vehicle movement in highway merging area, routes of the merging vehicles from on-ramp should extend beyond the whole weaving area, which ensures that vehicles from highway on-ramp successfully complete their merging movement. In consideration of second aspect, a velocity adjusting area should be de ned in simulation model to provide a space for the drivers to decelerate or accelerate to v in when approaching the entrance to identify if a suitable gap (ES) was available in target lane. e length of this area adopted in simulation model is 30 m setup 10 m upstream of the entrance according to previous eld data collection. Taking car-following behaviors, Wiedemann 99 model is adopted since it is suitable for interurban tra c, and the microscopic parameters related to driving behaviours are adopted in Table  1 [42], based on the large amount eld data collected on West 3rd Ring Expressway, Beijing. Simulation model derived from the contents above is shown in Figure 7.
As depicted in previous contents, the purpose of data simulation is to obtain the explicit form of HNAC (ESN). Considering the concept of HNAC in Section 1, the manifestation of exceeding HNAC is an oscillation in intersected road conducted through on-ramp from the main road. erefore, a detector should be deployed upstream from the ramp in intersected road to observe the tra c condition a ected by exceeding HNAC, shown in the below part of Figure 7. In order to catch the oscillation, the background tra c input in intersected road should possess a relatively lower robustness, which is the demarcation point of free ow and congested ow (v inter = 81.8 km/h, inter = 33 veh/km * l, inter = 2699 veh/h * l) [1].
In the upper part of Figure 7, a detector is deployed downstream at the terminal of the ramp to supervise the tra c condition of the main road. Besides, as depicted in Section 4.2, HNAC (ESN) is linearly related to tra c volume in main road. erefore, the background tra c input in the main road should vary in tra c volume and heavy vehicle mixing ratio. e speci c values of main , v main , and main are shown in Table 2 (the concentration varies regularly, the volume and speed varies according to equation (22). As depicted in the  Journal of Advanced Transportation 9 background tra c condition. In all of the 110 simulation groups in this study, the fuzzy boundary in every group could be found. However, to determine the explicit equation of HNAC, some further studies are still needed.

Standard Deviation and Correlation Coe cient Analysis.
From the analysis in Section 6.1, it could be known that with the increasing of merging tra c volume, velocity dispersion in intersected road also increased gradually. When exceeding the fuzzy boundary (HNAC), velocity dispersion in on-ramp shows a steep increase and the synchronization also appears. In mathematical language, standard deviation (Std) of the tra c velocity in intersected road is increasing with merging tra c volume. In a moment around the steep increase point, Std from the intersected road will equal to that from same characters in Figures 8(b) and 8(c). When coming to ∈ [1500 s, 3500 s], it is important to notice that velocity collapse and velocity synchronization appeared in both the intersected road and on-ramp, meaning that the merging tra c volume exceeded HNAC, leading oscillations conducting to intersected road through on-ramp. In all of the 18 simulation tests in the group of v main = 54.9 km/h, main = 34 veh/km * l, main = 1868 veh/h * l, a fuzzy boundary in = 1000 veh/h * l is found. When the merging tra c volume is less than 1000 veh/h * l, tra c condition in the intersected road is only a ected by lane-changing behaviors. When the merging tra c volume is larger than 1000 veh/h * l, tra c condition in the intersected road would be a ected by the exceeding HNAC. is result corresponds to the theoretical analysis in Section 3.2, moreover, providing a fuzzy value of HNAC in a speci c  Figure 9. In these gures, trends of Std in intersected road tra c velocity and on-ramp velocity are depicted in upper parts. SCCs between intersected road tra c velocity and on-ramp velocity are depicted in lower parts. From the gures shown above, discontinuity point of Std and SCC implies a same fuzzy boundary (HNAC) of the merging tra c volume. In le side of this point, both Std and SCC stayed in a very low stable value. In right side of this point, they stayed in a relatively high stable value. It could be seen in upper parts that there exists a point where the Std of the intersected road tra c velocity equals to that in on-ramp. is could provide us the accurate value of the fuzzy boundary (HNAC). Moreover, with the value of main decreased, which means the tra c condition of the main road became more severe, the fuzzy boundary (HNAC) also went down, corresponding to the theoretical result in Section 4.2. Figure 9, the point where Std of intersected road tra c velocity equals to that in on-ramp refers to the ccurate value of the fuzzy boundary (HNAC). In that, 10 HNAC values could be obtained in di erent tra c conditions of the main road, when heavy vehicle mixing ratio equals to 10%. erefore, 110 HNAC values in tra c stream of di erent heavy vehicle mixing ratios on-ramp, caused by synchronization. Moreover, the synchronization could be expressed by correlation coe cient. When merging tra c volume is less than the boundary (HNAC), the coe cient remains in a relatively low level, and when the merging tra c volume is larger than HNAC, the coe cient remains in a high level.

e Explicit Equation of HNAC. From
In this part, Std is calculated by equation (23), and Spearman correlation coe cient ( ) (SCC) is adopted because it is suitable to randomly distributed data set, calculated by equation (24).
Taking the data test groups of = 10% as examples, main ∈ [30, 66](veh/km * 1), main is calculated by equation (22). e total number of simulation groups of = 10% is 10, and information concluded from each simulation group is basically of same kind. erefore, there is no need to list all the simulation results. Std and SCC in group v main = 54.9 km/h, main = 1868 veh/h * l, in ∈ [100, 1800]veh/h * l and group    Table 2 and di erent main road's tra c conditions could be obtained. e theoretical conclusion in Section 4.2 that HNAC is linear positive correlated to tra c volume of the main road main has been proved by simulation data. With our previous work [1], the tra c volume is also linearly related to heavy vehicle mixing ratio . us, the assumption of the explicit equation of HNAC could be reasonably presented as below.
Using the 110 HNAC values obtained in Section 6.2 to linear t equation (25), the result is shown in Figure 10. e values of ( ) and ω( ) in tra c conditions with di erent values are also obtained, provided in Table 3.
From the linear tting results given above, the explicit equation of relationship (26) and (27)   It should be declared that the microscopic merging process is simpli ed, by bypassing the process of deceleration or acceleration of the merging vehicle. Compared to related works, this consideration largely reduced the complexity of microscopic modelling based on lane-changing rules. Moreover, the microscopic explain of tra c oscillations, the result of macroscopic model in Section 4.2, and data simulation also proved its rationality. e macroscopic model of HNAC is built using the boundary condition mentioned above. Macroscopic models were built in congested ow and free ow based on di erent assumptions respectively, providing the function of HNAC and also explained the production of oscillations. And this is an advantage which made it stay closer to actual tra c situations. Besides, it provided a thought of connecting microscopic problem and macroscopic problem. e data simulation method is chosen instead of eld data because it is di cult to collect enough real road data with di erent heavy vehicle mixing ratios and merging tra c volume in one speci c highway node. erefore, the simulation model is built based on rigid car-following behaviors organization and also controlled by macroscopic tra c ow model, to guarantee not only the typical tra c ow phenomenon such as oscillations and tra c waves, but also the comprehensive background tra c condition needed in this paper. However, it should be noticed, driving characters were simpli ed and idealized in theoretical models, and for instance, vehicle acceleration and deceleration processes were neglected. is might cause some di erences. From Figure 8 in Section 6.1, though synchronization phenomenon could be signi cantly observed, lines depicting velocity in on-ramp and intersected road were not closely matched, and volatility of velocity curves were obvious. is phenomenon was mainly caused by the gap between theoretical models and real tra c conditions represented by simulation. Furthermore, from Figure 10 and Table   mentioned in Section 6.1 happens, merging volume should be recorded and compared to theoretical HNAC obtained from equation (30), to verify the accuracy of the model.
In order to prove the accuracy of HNAC model introduced in this article, Maoerliu interchange of Xi'an Ring Expressway (G3001) was chosen as the supervision objective. Taking northbound section as the main road, and westbound as the intersected road, detectors were deployed as shown in Figure 11. Besides, all detectors have avoided the merging and diversion areas, to reduce the impact of weaving tra c. Tra c volume and velocity of the three detection points were supervised during 2019.09.15-2019.09. 26. In eld detection, the oscillation mentioned in previous could only be caught in peak hours, and 28 oscillations in intersected road caused by exceeding HNAC were observed. From equation (30), it is known that HNAC varied with tra c volume and heavy vehicle mixing ratio ( ) in main road. In selected part of G3001, remains in about 12-14%, taking 13% as average, the HNAC distribution curve could be obtained. Moreover, merging volume and tra c volume of 28 oscillations were collected. Comparing the theoretical values with the supervised values, the accuracy of the model could be observed, shown in Figure 12.
From Figure 12, it can be seen, 28 supervised HNACs spread around the theoretical curve obtained from equation (30), basically in range of [1100, 1650]veh/h * l. e distribution of 28 supervised HNACs is random and homogeneous, and the average relative error of supervised HNACs is 14.6%, which is an acceptable level, proving the accuracy of the model introduced in this article. Moreover, owning to the lack of related studies, it is unfortunately that the model in this paper could not be compared to other similar ones. Unfortunately, accsurate number or proportion of timid and aggressive drivers could not be obtained. erefore, to make HNAC more accurate and practical, large amount of real tra c data are needed, including detailed information of drivers in real road, geometric conditions, roadside facilities and tra c rules. e de nition of HNAC provided in Section 1 has implied that this concept lies in the macroscopic aspect, which is more related to a speci c highway node than a road section. is concept derived from a simple observation towards the tra c condition of a speci c highway node, and it is very likely to be noticed by other researchers. However, directly related works were not found, owning to the contribution of car-following models and moving bottleneck theory, which provide su cient and e cient methods to solve many important existing tra c problems, lane-changing behaviours, tra c oscillations and breakdown, stop-and-go waves, relaxation phenomenon, etc. But, moving the consideration to large scale macroscopic problems, especially the propagation mechanism of tra c emergencies on highway network, which is very helpful to large scale evacuation and rescue, the systematic research of HNAC becomes important. In policy and planning aspects, when an emergency happens downstream a highway node, the tra c condition in road section could be obtained through kinetic wave models. In this situation, if the tra c volume in on-ramp exceeded HNAC of the main road, congestion will form in on-ramp and further a ects the tra c ow in intersected road, meaning that the emergency e ect will spread into the intersected road. Based on the judgement mentioned above, the approximate range of emergency e ect in road network could be obtained, providing a base line to tra c management departments in dealing tra c emergency.

Data Availability
Data in this article are available only with permission of corresponding author.

Conflicts of Interest
e authors declare that they have no con icts of interest.
3 provided in Section 6.3, accuracy obtained from simulation data were signi cantly higher than that from the eld test, depicted in Figure 12. is might be caused by ignorance of timid and aggressive drivers mentioned in Section 4. When timid drivers took the majority part of on-ramp tra c ow, they might lose some opportunities of merging into the tra c ow in main road, and HNAC of this situation might be smaller than theoretical value. Points below theoretical curve in Figure 12 stand for the situation mentioned above, and points above the curve represent the opposite situation.