# Optimal Control for a Hydraulic Recuperation System Using Endoreversible Thermodynamics
## 7 Optimized Control  
The numerical results for the introduced endoreversible model are presented in this section. The obtained optimal solution is compared to a baseline strategy with regard to the dynamical behavior of the system and the resultant fuel savings. For the baseline strategy, we assume that the mechanical energy recuperation system supports the conventional power train in parallel, or even replaces the propulsion by the combustion engine as soon as possible. For the initial state the pressure and temperature of the gas were chosen to be \(p_{1}=p_{\text{max}}\) and \(T_{1}=50\,\mathrm{\SIUnitSymbolCelsius}\) which roughly corresponds to a bladder accumulator that has just been fully charged. The velocity of the driving profile and resultant power to accelerate or decelerate the truck is shown in Figure 6. Here, \(P_{\text{truck}}\) was calculated using Equation (22) and the height profile is constantly zero. On the left, in Figure 7 the key variables of the system with the baseline strategy are shown. At the top of the picture the displacement factor \(\gamma\) can be seen, which varies almost only between its extreme values, so that propulsion can immediately be fully supported and the maximum amount of braking energy can be stored. The drive support always lasts until the gas pressure \(p_{1}\) has reached its minimum value, as seen below, which corresponds to the minimum amount of hydraulic fluid within the accumulator, and hence an empty energy storage. Overall, the recuperation system is inactive over very long periods. Underneath, the resulting power distribution between the hydraulic unit and the internal combustion engine is shown, where, again, one can clearly see that the combustion engine is immediately supported as soon as there is energy stored within the bladder accumulator. At the bottom left of Figure 7 the efficiencies of the hydraulic unit \(\eta_{\text{hyd}}\) and the combustion engine \(\eta_{\text{comb}}\) can be seen. Particularly noteworthy here is that the efficiency of the hydraulic unit at low speeds reaches very low values as can bee seen at around \(t=5\,\text{s}\) and \(t=110\,\text{s}\). The resultant diesel fuel savings using the baseline strategy are 19.1%. Having the baseline strategy as a reference we can now look at the optimized control strategies. Figure 6: Velocity profile and corresponding power needed to accelerate or decelerate the truck. Figure 7: From top to bottom: displacement factor, gas pressure, power and efficiencies of hydraulic unit and combustion engine over time for (**left**) baseline strategy and (**right**) optimal strategy. The new feature in the optimized control strategy is that the engine can both propel the vehicle and transfer energy into the recuperation system at the same time. The results are shown on the right in Figure 7. Immediately apparent here is that the displacement factor does not switch only between its extreme values. While energy is still being stored with maximum available power, the displacement factor remains roughly between 0.3 and 0.8 during drive support. Furthermore, the recuperation is now active at almost all time. This leads to a higher mean pressure within the bladder accumulator resulting in higher energy recovering rates. We can now also see that the efficiencies improve slightly in several places and, in addition, up to \(t=10\,\)s the hydraulic unit no longer supports the propulsion since it can operate only with poor efficiency due to the low velocity of the truck and hence the low rotational speeds. On the other hand, while having high efficiency values at higher velocities e.g., at \(t=140\,\)s and \(t=140\,\)s the hydraulic unit supports or replaces the combustion engine which has low efficiency values there. We can see that the possibility of simultaneous energy storing and propulsion is actually used, e.g., at \(t=30\,\)s and \(t=150\,\)s. In these cases, the internal combustion engine operates with improved efficiency due to the increased power output.