# The Impact of Automated Vehicles on Road and Intersection Capacity
## 3 The Homogeneous Road Traffic Flow-Density Relationship  
### Theoretical Derivation  
Based on the basic diagram model of traffic flow, the equilibrium \"speed-time headway\" function relationship of traffic flow, and the vehicle-following model, the flow-density relationship of homogeneous traffic flow of different vehicle types in a single environment, can be deduced. When the traffic flow is in an equilibrium state, the speed of all vehicles on the road is consistent, the speed difference between all vehicles is 0, and the speed of all vehicles is 0, as shown in Equation (16). \[\left\{ \begin{array}{l}v_{n}(t)=v_{b}\\ \Delta v=0\\ a_{n}(t)=0\end{array} \right., \tag{16}\]  
Therefore, the relationship between the headspace of the human-driven vehicle can be deduced using Formulas (9), (10), (11) and (16), as shown in Equation (17). \[h_{dI}=h_{n}=\frac{s_{0}+v_{b}T}{\sqrt{1-\left(\frac{v_{b}}{v_{0}}\right)^{4}}} +l, \tag{17}\]  
where: \(h_{dI}\) and \(h_{n}\) represent the distance between the locomotive in the balanced state; \(v_{b}\) represents the equilibrium state velocity, and the value range is \([1,v_{0})\); \(v_{0}\) represents the (expected) velocity in the free flow; \(s_{0}\) represents the minimum expected (safety) workshop distance; \(T\) represents the expected (minimum) headway; and \(l\) represents the length of the vehicle. Then, through the relationships between the three parameters of traffic flow, it can be seen that there is a reciprocal relationship between the density and the distance between the heads, and the \"speed-density relation\" of fully manual driven vehicles can be obtained, as shown in Equation (18). \[K_{I}=\frac{1000\sqrt{1-\left(\frac{v_{b}}{v_{0}}\right)^{4}}}{s_{0}+v_{b}T+l \sqrt{1-\left(\frac{v_{b}}{v_{0}}\right)^{4}}}, \tag{18}\]  
where \(K_{I}\) represents lane density. In the equilibrium state, the time headway can be obtained by comparing the headway with the vehicle speed, and then the traffic volume can be obtained, as shown in Equations (19) and (20). \[h_{tI}=\frac{h_{dI}}{v_{b}}, \tag{19}\]  
\[Q_{I}=\frac{3600}{h_{tI}}, \tag{20}\]  
where \(h_{tI}\) is the headway of the artificially driven vehicle in the equilibrium state, \(Q_{I}\) is the traffic volume of manually driven vehicle in equilibrium state. The above represents the three-parameter relationship of traffic flow for artificially driven vehicles. Similarly, the basic graph model relationships of ACC and CACC autonomous vehicles can be deduced, respectively, through Equations (12)-(16). The basic map model of an ACC autonomous vehicle is as follows:  
\[h_{dA}=h_{n}=l+s_{0}+t_{a}v_{b}, \tag{21}\]  
\[K_{A}=\frac{1000}{h_{dA}}, \tag{22}\]  
\[h_{tA}=\frac{h_{dA}}{v_{b}}, \tag{23}\]\[Q_{A}=\frac{3600}{h_{tA}}, \tag{24}\]  
where \(h_{dA}\) and \(h_{n}\) represent the distance between the heads of ACC autonomous driving vehicles in the balanced state, \(v_{b}\) represents the balance state speed of ACC autonomous vehicles with a value range of \([1,v_{0})\), \(s_{0}\) represents the minimum expected (safety) workshop distance of ACC autonomous vehicles, \(t_{a}\) represents the expected (minimum) headway of ACC autonomous vehicles, \(l\) represents vehicle length, \(K_{A}\) represents lane density at equilibrium speed, \(h_{tA}\) represents the time headway of a balanced state, and \(Q_{A}\) represents the traffic volume of ACC autonomous vehicles in an equilibrium state. The basic diagram model of the CACC autonomous vehicle is as follows.