# Interdisciplinary Urban Tunnel Control within Smart Cities
## 3 New Road Tunnel Traffic Control Algorithm
### Fundamental Traffic Flow Theory  
In order to develop an optimized traffic control strategy, it is necessary to look at the characteristics of traffic flow. The traffic flow is characterized by three basic variables [20]: flow rate \(q\), density \(k\) and mean speed \(u\). Due to the fundamental relation \(q=k.u\) there are only two independent variables. The relationship among the variables is often depicted using so-called fundamental diagrams (see Figure 9). For the purpose of this paper, we need to focus on the following points and areas of interest in the diagrams:  
Completely free flowing traffic, when vehicles are not impeded by other traffic and travel at a maximum (or desired) speed of \(u_{f}\) (free speed). This free speed depends on the road design speed (based on curvature, transverse slope, road surface quality and other  
Figure 8: Changes in the average speed on selected sensors after opening the tunnel complex Blanka. Figure 9: The fundamental diagram. parameters), the speed restrictions (speed limits), the weather, but also on the driver's characteristics. At free speed, flow rate and density will be close to zero. Saturated traffic, when the flow rate, \(q\), and speed, \(u\), are down to zero. The vehicles are queuing and there is a maximum density of \(\mathbf{k_{j}}\) (jam density). Capacity traffic, with the capacity of a road equal to the maximum flow rate, \(\mathbf{q_{c}}\). The maximum flow rate of \(\mathbf{q_{c}}\) has an associated capacity speed, \(\mathbf{u_{c}}\), and a capacity density, \(\mathbf{k_{c}}\). The diagram shows that the capacity speed, \(\mathbf{u_{c}}\), lies below the maximum speed, \(\mathbf{u_{f}}\). There exist various models trying to find an analytical solution fitting as best as possible the empirical measured data, for example linear model [21], triangular diagram [22], or more realistic high-order macroscopic models addressing certain discontinuity [23]. Here, we focus on the last class of models as they well describe the phenomena we face in tunnels. [PERSON] in [24] uses an approach of homogenizing traffic and addressing the transition among them. As [PERSON] demonstrated in his paper, the measured data confirm two major data clusters (traffic states) with different characteristics as well as transition phase between them. They are called fluid traffic and jam traffic. This is confirmed also by the data measured in the Czech Republic as demonstrated for \(q\)-\(v\) relationship on example in Figure 10. This is in a way similar to [PERSON]'s Three-phase theory [23], where next to the free flow, the jam is further split into two phases, namely synchronized flow and wide moving jam. The critical part which needs an extra focus is the transition from free flow to the jam and back. The traditional models often fail as there is only a limited number of measurements. This is due to the fact that the transition is rather quick (i.e., the flow does not stay in this phase for too long) and there is a clear observable hysteresis (i.e., the recovery from the jam does not follow the same trajectory as falling from the free flow to the jam). The traditional models thus tend to overestimate the road capacity. The principle of this transition is further depicted in Figure 11.  
For the purpose of this paper and the road urban tunnel control system, it is clear from the above-mentioned theory that we need to focus on density. We are mainly interested in the point when the fluid traffic breaks into the jam, that is at density \(\kappa_{KO}\) (i.e., at the state that the average length of the net headways between the vehicles is equal to \(\tau_{KO}\) in the fluid convoy). The jam traffic turns back upwards into the fluid traffic at \(\kappa_{go,\ min}\) (i.e., at the state that the average length of the net headways between the vehicles is equal to \(\tau_{go}\) in the jam convoy). For more details on identifying the critical values of density, refer to [PERSON] [24].