# Speed Advisor for Fuel Consumption Minimisation Under Real Driving Conditions
## 2 Vehicle Model  
A vehicle model is required in order to compute the vehicle speed sequence that minimises the fuel consumption in the route. The vehicle should combine accuracy, to have prediction capabilities, and low computation burden, to allow the optimisation of a realistic route with long distance. For the scope of this work, the engine quasi-static modelling approach, usually followed in vehicle speed optimisation strategies, has been used. Models of increasing complexity lead to harder optimisation problems so increased computation time. In this sense, the model considers longitudinal forces acting on the chassis by the following ordinary differential equation (ODE) representing the [PERSON]'s Second Law of motion:  
\[\dot{v}=\frac{-F_{a}-F_{r}-F_{g}-F_{b}+F_{t}}{m_{v}} \tag{1}\]  
where \(v\) is the vehicle speed and accordingly, its time derivative (\(\dot{v}\)) is acceleration, \(m_{v}\) represents an equivalent vehicle mass including the inertia of powertrain elements, and the terms \(F_{j}\) represent different forces acting on the vehicle, namely, aerodynamic drag:  
\[F_{a}=\frac{1}{2}\rho AC_{d}v^{2} \tag{2}\]  
where \(\rho\) is the air density, \(A\) the front area of the vehicle and \(C_{d}\) the drag coefficient; rolling resistance:  
\[F_{r}=\mu mg\cos\beta \tag{3}\]  
where \(\mu\) is the road friction coefficient (in the case at hand 0.012 [23]), \(m\) the vehicle mass, \(g\) the gravity acceleration and \(\beta\) the road inclination angle; effect of gravity:  
\[F_{g}=mg\sin\beta \tag{4}\]  
the brake force:  
\[F_{b}=u_{b}\hat{F}_{b} \tag{5}\]with \(u_{b}\) is braking actuation ranging from full (\(-1\)) to no braking (\(0\)) and \(\hat{F}_{b}\) is the maximum braking force. and finally engine torque contribution to the wheels:  
\[F_{t}=\frac{T_{w}}{r_{w}} \tag{6}\]  
where \(r_{w}\) is the wheel radius and \(T_{w}\) is the engine torque at the wheels. The transmission from the engine to the wheels is modelled as an ideal set of gears. In this sense, the relations between engine torque (\(T\)) and wheel torque as well as vehicle speed and engine speed are given by the following expressions [24]:  
\[T_{w}=\eta_{gb}R_{gb}(u_{gb})T \tag{7}\]  
\[N=\frac{30R_{gb}(u_{gb})v}{\pi r_{w}} \tag{8}\]  
where \(\eta_{gb}\) is the transmission efficiency and \(R_{gb}\) is the transmission ratio, both including gearbox and differential and therefore depending on the selected gear \(u_{gb}\). The multiplication factor \(\frac{30}{\pi}\) converts the engine speed from radians per second to revolutions per minute. The gear is chosen following a vehicle speed based policy that keeps engine speed in the range of 1000 to 2750 rpm. The engine torque is modelled as a non-linear function of engine speed (\(N\)) and pedal position (\(u_{p}\)) [25]. In particular, torque is calculated from the pedal position and the maximum and minimum torque curves depending on engine speed (\(T_{max}(N)\) and \(T_{min}(N)\) respectively):  
\[T=u_{p}(T_{max}-T_{min})+T_{min} \tag{9}\]  
Meanwhile, fuel consumption \(\dot{m}_{f}\) is calculated as a function of engine speed \(N\) and pedal position \(u_{p}\) obtained from experiments. Figure 1 shows the engine maps for torque and fuelling and in the vehicle model. Both variables are calculated by simple interpolation of experimental data as a function of engine speed (\(N\)) and throttle (\(u_{p}\)). Note that, in line with other works in the literature [24], the engine dynamics are completely neglected, the main reasons for such a simplification are:  
* A detailed engine model including dynamics substantially increases the complexity of the optimisation problem with its corresponding increase in computational time. * Vehicle dynamics are much slower than the main engine dynamics. Figure 1: Quasi-steady engine model obtained from experiments: Engine map for torque (left) and fuel consumption (right) depending on engine speed and pedal position.