# Model-Based Dynamic Toll Pricing: An Overview
## 3 Overview of Simulations
### Traffic Simulation  
The traffic model is an essential part of dynamic tolling design studies; it aims to quantify some traffic characteristics corresponding to each transport alternative, depending on the traffic volumes choosing that particular transport solution. A comprehensive survey on traffic state estimation can be found in [97]. According to the scale, traffic flow models can be classified as macroscopic, mesoscopic or microscopic, with an increasing level of detail:  
* macroscopic models are usually based on the analogy of traffic with fluid dynamics, thus traffic is described by the value of a few synthetic variables (flow, density and speed);
* microscopic models focus on the single vehicles' trajectories, and mesoscopic models share both the previously mentioned families' elements. In terms of formulation in a macroscopic model, traffic state is described by vector (8):  
\[H(x,t)= \begin{bmatrix}v(x,t)\\ \rho(x,t)\\ q(x,t)\end{bmatrix} , \tag{8}\]  
where \(v(x,t)\) is velocity (in km/h), \(\rho(x,t)\) is density (in veh/km) and \(q(x,t)\) is flow (in veh/h), all function of time \(t\) and location \(x\) on the infrastructure. Traffic flow is modeled locally through a system of partial differential equations inspired by hydrodynamics, as the system given by (9):  
\[ \begin{cases}\frac{\partial q(x,t)}{\partial x}+\frac{\partial p(x,t)}{ \partial t}=0\\ q(x,t)=q(\rho(x,t))\end{cases} . \tag{9}\]  
The first equation of the system is a conservation law, whereas the second equation, expressing flow rate in function of density, is called fundamental diagram. The Lighthill-Whitham-Richards (LWR) kinematic wave model [98; 99], characterized by the fundamental relation (10):  
\[q(x,t)=v(x,t)\cdot\rho(x,t). \tag{10}\]  
This relation is considered pioneering in macroscopic traffic flow modeling [97]. This model is used to characterize traffic flow evolution on the HOT lane in [20]. A stochastic macroscopic traffic flow model built by [47] on the classical LWR model is adopted in [46] for managed lanes. An accumulator model consisting in an aggregate, lumped-parameter model, based on the LWR model, is proposed in references [17; 22]. The Cell Transmission Model (CTM) [100; 101] and its derivatives, which are based on the discretization of the differential equations of the LWR model, are broadly used in traffic flow modeling for dynamic tolling design. In [40; 48], traffic dynamics on managed lanes is modeled through CTM. [PERSON] [8] adopts a multi-lane hybrid CTM proposed by [32] to describe traffic dynamics in HOT facilities. The Macroscopic fundamental diagram, also called Network Fundamental Diagram, introduced by [102] to model networks traffic dynamics on an aggregate level by considering relations between space-mean traffic flow and density in urban regions, is adopted in [18; 19; 36; 50; 51]. In microscopic models, on the contrary, traffic is described at a disaggregated level by single-vehicle trajectories as given by (11):  
\[x_{i}=x_{i}(t). \tag{11}\]  
In (11), \(x_{i}(t)\) identifies the position of vehicle \(i\) at time \(t\). Microscopic modeling is at the base of simulation platforms that have been often used in dynamic toll pricing studies at the network level: car-following-based VISSIM [11; 14; 15; 16; 23; 36], Paramics [21; 27], CORSIM [29; 52] and MITSIM [30]. A mesoscopic modeling-based simulator, DynusT,is employed in [53] and a regional mesoscopic dynamic traffic assignment simulation environment is adopted in [54; 56]. DynaMIT, a mesoscopic dynamic traffic simulator is applied in [30] toghether with microscopic simulation. Usually, these platforms can model both traffic and driver behavior. In some studies, where the considered situation is analyzed as a bottleneck, traffic modeling is limited to a delay operator: in [13; 49] traffic dynamics are described through the point-queue concept [103], assuming that the average travel time is given by the sum of a congestion-independent cruise time and a queuing delay.