# Speed Advisor for Fuel Consumption Minimisation Under Real Driving Conditions
## 3 Problem Description  
The aim of this work is to explore the potential of speed profile optimisation in real driving conditions by means of a driving advisor application. In order to provide advice to the driver, a sensible possibility is to compute what is the velocity profile to minimize a given cost function and then provide this information to the driver. Regarding the speed profile computation, the fuel consumption is an intuitive cost function while the time spent during the trip seems a sensible constraint. Other restrictions that should be taken into account are the speed limits during the route or the limits of the vehicle and the powertrain. In any case, a generic control problem consisting in finding the control policy to drive a  
Figure 2: Vehicle model performance. (**Left**): comparison between measured and modelled fuel consumption during the set of driving cycles tested. (**Right**): Percentage of cycles with modelling error (\(e_{V_{f}}\)) below certain limit \(v_{V_{f}}\). White circles represent default driving conditions while black circles are associated to assisted ones. dynamic system from a given initial state to a defined final state with minimum cost and fulfilling a set of constraints fits in the field of Optimal Control. The formulation of an Optimal Control Problem (OCP) considers a cost index and constraints expressed in terms of system states (\(x\)), system actuators (\(u\)), problem disturbances (\(w\)) and domain. Taking into account that, in the case at hand, the route is a priori known, the slope and speed limits (which are the problem disturbances) are space (\(s\)) dependent, it seems more convenient to consider \(s\) as the problem domain. In this case, the cost index can be defined as:  
\[\mathcal{J}=\int_{0}^{L}\frac{\dot{m}_{f}}{v}ds \tag{12}\]  
where \(L\) is the route length and for the sake of readability, the dependence of the fuel consumption on the actuator (\(u_{p}\)) has been omitted. The system, i.e the vehicle model described in Section 2 can be condensed in the generic dynamic system:  
\[\dot{x}=f(x,u,w,t) \tag{13}\]  
where the dependence of the state (velocity) evolution on the state itself, the control actions (in the case at hand \(u\)), disturbance (road slope) and time is described. Therefore the state evolution with the problem domain (space) becomes:  
\[\frac{dx}{ds}=\frac{f}{v} \tag{14}\]  
where the dependence of \(f\) on the system state, actions and disturbances has been omitted for the sake of clarity. The vehicle should cover the defined distance (\(L\)) within a given time (\(\tau\)), which leads to the following integral constraint:  
\[\int_{0}^{L}\frac{1}{v}ds\leq\tau \tag{15}\]  
Road speed limits have to be respected which leads to the following path constraint:  
\[v(s)\leq\vartheta(s) \tag{16}\]  
Limits on the engine (speed range and maximum power output) and brakes (maximum braking force) are also considered. Finally, the initial speed is also imposed. Despite the vehicle starts from rest a minimum vehicle speed is imposed to avoid divisions by 0 in the model (boundary constraint):  
\[v(s)=\underline{v} \tag{17}\]  
Regarding final speed, no restrictions are imposed beyond the maximum speed of the road. In the present case, to analyse the performance of the speed advisor, a 60 km daily commute route between two cities in Spain (A7 route between Canals and Valencia) has been chosen. In this sense, Figure 3 shows the evolution of road slope and maximum vehicle speed along the route. Both variables have been considered in the optimisation problem as disturbance and path constraint respectively. The route was covered daily (during consecutive working days) with an instrumented vehicle to obtain a set of 45 tests. The experiments where carried out between 6:00 a.m. and 7:00 a.m. to avoid the rush hour and therefore traffic jams that complicate the optimization. The occupancy in the road did not exceed 5% according to the measurements of the traffic agency.