# Model-Based Dynamic Toll Pricing: An Overview
## 2 Overview of Dynamic Toll Price Definition Methods
### Control-Based Algorithms  
The general topic of traffic modeling and control has been studied at length in the literature (ex. [67]). Broadly speaking, traffic control can be classified as either road-based or vehicle-based. Road-based traffic control includes ramp management via methods such as ramp metering [68; 69], posting of variable speed limits so as to homogenize speeds and avoid stop-and-go waves [70; 71], average speed enforcement [72; 73], lane control redirecting vehicles to different lanes [74], Route Guidance and Information Systems (RGIS) communicated to drivers through various messaging methods [75; 76]. A fairly large literature exists on vehicle-based control, which pertain to Vehicle-to-Infrastructure (V2I) based communication strategies and Connected and Automated Vehicles (CAVs) based coordination strategies [77]. In addition to the above methods, one can consider a dynamic toll-pricing based approach, which may be viewed as either road-based or vehicle-based, where the toll price is messaged to the driver (either through a message posted on the road or directly to the vehicle), and the driver makes a decision in turn whether to enter the highway or not [17; 22]. Viewing real-time traffic information data as a sensor and the toll based on these data as a control input, the underlying traffic congestion problem is posed as a control problem, where the driver can be viewed as an actuator. The concept of transactive control, a feedback control strategy enabled through economic contracts to incentivize and enable flexible consumption, is eliciting significant attention of late [78], with multiple applications in dynamic toll pricing. Figure 4: The distribution of the analyzed studies categorized by scope (inner circle) and main price definition method (outer circle). Labels: ML = Managed Lanes, NW = Networks, TF = Tolled Facilities, OPT = Optimization, C = Control, OTH = Other. A simple feedback digital control law for on-ramp metering called ALINEA was proposed in [79] as in (1):  
\[r(k)=r(k-1)+k_{p}\cdot[o_{ref}-o(k)]. \tag{1}\]  
The metering rate at time \(k\), \(r(k)\) (in equivalent passenger vehicles/hour, or veh/h), is defined by the product of a proportional gain \(k_{p}\) with the difference between the desired value of an occupancy parameter \(o_{ref}\) and its current value \(o(k)\) (in % km/km), which, in turn, is related to traffic density \(\rho(k)\) (in equivalent passenger vehicles/km, or veh/km) by Equation (2):  
\[\rho(k)=\eta\cdot o(k), \tag{2}\]  
where \(\eta\) is defined as in (3):  
\[\eta=\frac{\
u}{100\cdot l}, \tag{3}\]  
with \(\
u\) equal to the number of lanes on the mainstream and \(l\) (in km/veh) equal to the mean effective vehicles length in km. A principle similar to ALINEA was applied in [13] to build a toll pricing controller, given by Equation (4):  
\[\pi(k+1)=\pi(k)+k_{p}\cdot[o(k)-o_{ref}], \tag{4}\]  
where the toll price \(\pi(k)\) (in monetary units) is defined by a controller, being the control variable an occupancy parameter. A similar concept was applied in [11], where the optimal HOT flow ratio to set the dynamic toll is defined by a control-based piecewise linear function, in [50] and in [52]. Also in [14; 15; 16] the control mechanism is based on a piecewise linear function but with variable revenue-maximizing optimal gains. In [17], desired flow into the HOT at time \(t\) (in veh/h), indicated with \(y(t)\), is computed through a proportional derivative (PD) analog controller with a feedforward additive component (5):  
\[y(t)=k_{p}\cdot e(t)+k_{d}\cdot e(t)+k_{ff}\cdot\rho_{ref}, \tag{5}\]  
where \(k_{p}\), \(k_{d}\) and \(k_{ff}\) represent proportional, derivative and feedforward gains, respectively, and \(e(t)\) is an error signal, given by the difference between the desired density \(\rho_{ref}\) and the current density \(\rho(t)\) (in veh/km).