# Tunnel Traffic Evolution during Capacity Drop Based on High-Resolution Vehicle Trajectory Data
## 4 Analysis of Capacity Drop Based on Phase Transition-Theory
### Identification of Tunnel Capacity Drop  
The advantage of the vehicle oblique accumulation curve is that it can highlight the phase characteristics of the flow change at the section to identify the flow change point. However, the abrupt flow is not always the phenomenon of capacity drop. It is not enough to identify the moment of the capacity drop with the inclined cumulative curve of vehicles. By combining the inclined cumulative curve of time occupancy and recognizing the changing trend of a vehicle passing time at the section, the moment of the capacity drop can be more accurately determined. We draw the cumulative vehicle curve and the cumulative curve of time occupancy in the oblique coordinate system by combining the bottleneck characteristics and state parameters. Then, we enlarge the traffic change characteristics at the section to determine the moment of traffic capacity drop. The basic principle of the oblique accumulation curve is as follows:  
Based on the original cumulative curve of the section, a background value is subtracted. The change characteristics of the curve are visually enlarged, and the change points of vehicle arrival intensity and vehicle delay at the section are identified. The curve formula is as follows:  
\[ \begin{array}{l}V(x,t)=N(x,t)-q_{o}(x)\cdot t\\ O(x,t)=T(x,t)-t_{o}(x)\cdot t\end{array} \]  
where:  
\(V(x,t)\)--the cumulative number of arrivals of vehicles passing through section \(x\) within \(t\) after adjustment;  
\(O(x,t)\)--the cumulative curve of the time occupancy rate of vehicles passing through section \(x\) within \(t\) after adjustment;  
\(N(x,t)\)--the cumulative number of arrivals of vehicles passing through section \(x\) within \(t\);  
\(T(x,t)\)--the cumulative curve of the time occupancy rate of vehicles passing through section \(x\) within \(t\);  
\(q_{o}(x)\)--background flow at section \(x\), usually the maximum capacity at this section;  
\(t_{o}(x)\)--undetermined parameter at section \(x\), unit s/h. Based on the construction method of the oblique cumulative curve of the appealing vehicle and combined with the research division of this area in this paper, the oblique cumulative curve of the critical sections (tunnel entrance and exit, slope change points, etc. ), in the whole region of the tunnel is constructed. The oblique cumulative curves display \"virtual departures,\" such that the vertical displacements between any two are the excess vehicle accumulations (queueing) on the intervening tunnel segment. With the oblique coordinate system, each curve displays the quantity, the virtual vehicle counts to time, minus a background reduction. We conducted the adjustment tests for the background parameters of each section. The original parameter value is the average flow multiplied by the interval extending from the curves' start time to the current time. Then, we tested the parameters varying from \(-10\%\) to \(10\%\) of the original value for final use. After several adjustment tests, the background parameters of each section are determined to significantly enhance the visual effect of the flow rate change of sections. In Figure 2, the oblique accumulation curves of vehicles at the four adjacent sections are all piecewise approximately linear, showing a stepped characteristic. As shown in the figure below, at 123 s, the traffic capacity first drops in the bottleneck section. Secondly, the traffic flow leaving the area downstream of the bottleneck is limited to the merge bottleneck and lags the merge bottleneck. At 131 s, the flow suddenly changes. At the same time, the congestion at the merge bottleneck gradually spreads to the segment at the upstream section. Further, at 177 s, the traffic capacity drops. By calculating the flow rate at the bottleneck section, the capacity before congestion formation is estimated to be 6700 veh/h, while the discharge flow after congestion formation is estimated to be 5850 veh/h. The magnitude of capacity drop at the merge bottleneck is 12.7%.