## 3 Benchmark Solutions: ECMS and DP with a Priori Knowledge of the Driving Cycle  
The equivalent consumption minimization strategy (ECMS) is the most frequently used method to address the optimization problem introduced in Section 2. The ECMS is based on using an equivalence factor for the battery power to take into account the potential of currently discharging the batteries and recharging them in the future or vice versa. Assuming that this equivalent factor between battery and fuel energy is constant, the integral problem represented by Equation (1) can be replaced by the minimization of a set of instantaneous objective functions:  
\[c=P_{f}+sP_{b} \tag{13}\]  
where the parameter \(s\) represents an equivalent factor between fuel and battery energy sources and accordingly balances the solution between the fuel and battery usage. The link between the ECMS and optimal control, and more precisely [PERSON]'s minimum principle, is carefully discussed in [33]. From Equation (13), it can be observed that high values of \(s\) impose a high weight to the battery energy in the objective function, then promoting the fuel use and battery charging, while low values of \(s\) entail a low impact of the battery use in the objective function, then favoring fuel savings at the expense of battery depletion. For a given driving cycle, there is an optimal value of \(s\) that leads to the fuel consumption minimization while sustaining the battery energy. Of course, the optimal value of the parameter \(s\) strongly depends on the driving conditions, and the same equivalence factor \(s\) may lead to battery depletion in a given cycle and battery overload in a different one. In cases where the driving cycle is a priori known, the optimal value of \(s\) may be found by iteratively searching the value that satisfies the charge sustainability (\(SoE(t_{f})=SoE(t_{0})\)). In the general case, where the driving cycle is unknown, different methods to calibrate and adapt the value of \(s\) depending on the operating conditions may be found in [18; 21], but essentially, they rely on calibrating the value of \(s\) in a set of representative cycles, and then apply corrections depending on \(SoE\) deviations from the reference value. On the other hand, the optimization problem introduced in Section 2 can be formally cast in the optimal control framework aimed to the minimization of a generic cost function:  
\[J=\psi(x(t_{f}))+\int_{t_{0}}^{t_{f}}L(x(t),u(t),d(t))dt \tag{14}\]  
where \(x\) is a vector containing the system states, \(u\) is the vector of control actions and \(d\) is the vector of disturbances that affect the system evolution. Note that the function \(J\) includes an integral term (\(L\)) and a terminal cost \(\psi\) that penalizes deviations from a desired final state. From Equation (1), one can identify \(L\) in Equation (14) as the power of the fuel consumed (\(P_{f}\)) and \(\psi(x(t_{f}))\) as the deviation from the initial energy stored in the battery. The optimal control problem consists on finding a trajectory \(u(t)\) minimizing the function \(J\), i.e., \(min_{u}J\). The system state evolves from its initial value \(x(t_{0})\) driven by disturbances and control actuations according to a set of first-order differential equations represented by a generic function \(f\):  
\[\dot{x}(t)=f(x(t),d(t),u(t)) \tag{15}\]  
In the case at hand, the states of the system are the battery state of energy (\(SoE\)) and the state of the ICE (on or off), while the control action is the motor torque (see Equation (4)), the disturbance is the vehicle speed and the function \(f\) is defined by Equations (2)-(12).