# Hybrid Model-Based Traffic Network Control Using Population Games
## 3 Mathematical Description of a Urban Traffic Network  
According to [28], a vehicular traffic network can be represented as a directed graph \(\mathcal{G}=(\mathcal{V},\mathcal{E},\psi)\), where \(\mathcal{V}=\{1,2,3,\ldots,m\}\) and \(\mathcal{E}=\{1,2,\ldots,n\}\) are the sets of \(m\) nodes and \(n\) edges of the network, respectively. \(\psi\) is the incidence function that matches each edge \(i\in\mathcal{E}\) to an ordered pair \((p,r)\in\mathcal{V}\),  
\[\psi(i)=  
represents the red-yellow transition, and one that represents the yellow light after the green one. The total idle time is denoted as \(L_{p}\). During the period composed of the idle and green light times in (1,2), the traffic light in (3,2) must be in red. The same scheme applies for (3,2) in the second half of the cycle time \(C_{p}\). Considering the above definitions, the following constraint applies to node \(p\):  
\[\sum_{i\in\mathcal{I}_{p}}g_{i}+L_{p}=C_{p}, \tag{4}\]  
where \(g_{i}\) is the green light time for link \(i\).  
One _stage_ is defined as the time at which an input link to a node \(p\) has the right of way (ROW), whereas the set of stages for a node \(p\) is denoted as \(F_{p}\). The _saturation flow_\(o_{i}\), with \(i\in\mathcal{I}_{p}\), represents the output vehicular flow capacity for the link \(i\) while the traffic light is green. The set of values \(\mathbf{o}=[o_{1},\dots,o_{n}]^{T}\) is normally known as \(\forall i\in\mathcal{E}\). The _turn rate_ on each node, denoted by \(\tau_{(i,j)}\in[0,1]\), with \(i\in\mathcal{I}_{p}\) and \(j\in\mathcal{O}_{p}\), represents the distribution of vehicles turning from link \(i\) to \(j\). In other words, \(\tau_{(i,j)}\) is the turn rate from link \(i\) to \(j\), with \(\psi(i)=(p,r)\) and \(\psi(j)=(l,p)\).  
Figure 1: Urban traffic network. (**a**) A traffic network case including intersections and traffic lights. (**b**) Urban traffic network graph representation. (**c**) Traffic lights’ plan for node 2.  
The _traffic light state_ in a link \(i\in\mathcal{I}_{p}\), denoted as \(s_{i}\in\{0,1\}\), can be equal to \(1\) when the traffic light is green (ROW) and \(0\) in a different case (yellow or red). In general, the set of network states is \(\mathbf{s}=[s_{1},\ldots,s_{n}]^{T}\), \(\forall i\in\mathcal{E}\).  
The expressions \(c_{i}\) and \(w_{i}\), with \(i\in\mathcal{I}_{p}\), represent the _external vehicular input flow_ and the _vehicular storing capacity_ of \(i\), respectively. In general terms, \(\mathbf{c}=[c_{i},\ldots,c_{n}]^{T}\), \(\Omega=[w_{i},\ldots,w_{n}]\), and \(\forall i\in\mathcal{E}\) represent the vehicular input flows and storing capacity of all the network links.  
Network RestrictionsThe geometry restriction on each link is given by \(0\leq l_{i}\leq\overline{l}_{i}\), and the control input restriction is represented as \(\underline{g}_{i}\leq g_{i}\leq\overline{g}_{i}\), where \(l_{i}\) and \(\overline{l}_{i}\) are the current and average queue lengths of link \(i\), respectively, and \(\underline{g}_{i}\) is the minimum value in seconds for the green light, which depends on the system properties (e.g., it can be equal to zero when there are no vehicles in link \(i\)). Finally, the term \(\overline{g}_{i}\) represents the addition of all green times in intersection \(p\) and is given by \(\overline{g}_{i}=\sum\limits_{i\in\mathcal{I}_{p}}g_{i}\leq C_{p}-L_{p}\). Observe that this value must not exceed the difference between the cycle time and the total idle time.