# Optimal Control for a Hydraulic Recuperation System Using Endoreversible Thermodynamics
## 1 Introduction  
For achieving the global goal of reduced fossil fuel consumption and CO\({}_{2}\) emissions the automotive sector plays an important role. In particular, energy recovery systems--electrical, mechanical or hydraulic--are an effective approach to use otherwise wasted energy and improve the vehicles overall efficiency [1, 2, 3, 4, 5, 6, 7, 8, 9]. The starting point of this work is a research project that focuses on hydraulic recuperation systems for medium and heavy commercial vehicles with tipper bodies. For these the use of a hydraulic system for mechanical energy recuperation is particularly promising since these vehicles already have a hydraulic system propelling the tipping body. If this system is complemented by a hydraulic accumulator with its particularly high energy and power density, it is able to absorb high energy fluxes when braking. Then the brake energy, which is absorbed using a hydraulic unit mounted on the cardan shaft, can be stored in the hydraulic accumulator. This not only allows to reuse the energy stored to accelerate the vehicle as well as to operate the auxiliary units, but it also reduces the wear of the conventional disc brakes.  
When investigating and optimizing hydraulic systems, a comprehensive thermodynamic description is necessary to take thermodynamic effects of the fluid movements and heat transfers into account [10]. Therefore, we introduce a model that is build using endoreversible thermodynamics--an approach based on dividing the system in question into reversible acting subsystems and incorporating reversible or irreversible interactions between them [11]. During the last decades, this method has been used for modeling a variety of technical and natural systems such as heat engines [12, 13, 14, 15, 16, 17, 18], thermoelectric refrigerators [19, 20], electrochemical devices [21, 22] and solar thermal heat engines [23, 24, 25, 26, 27]. Even chemical reactions [28, 29], global wind energy production [30] and computation [31] have been modeled and investigated using endoreversible thermodynamics. Control or trajectory optimizations have been carried out, too, e.g., for piston movements of Otto [32, 33],Diesel [34; 35], Miller [36] and Brayton [37; 38] and Stirling [39] cycles as well as light-driven engines [40; 41; 42]. These investigations all show, that it is important to put the focus on dynamical features [43] in order to reach the desired efficiency improvements. This is in particular true for systems in which some of the important input variables to the system are fluctuating. Very recently, the first steps in that regard have been taken in the analysis of stochastic heat engines [44; 45; 46].  
Based on a comprehensive endoreversible model and realistic input data for driving profiles optimal control strategies can be developed such that the recuperation system achieves significant reduction in operating costs.  
A simple control strategy would be based on the presumption that the system supports the conventional power train as soon as possible, either by being switched on in parallel or by replacing the propulsion by the combustion engine. This strategy is typically the baseline strategy and thus the reference point to which control optimization approaches are compared to. Publications often aim at deterministic control strategies that support the combustion engine in its low efficiency areas while taking the efficiency map of the hydraulic unit into account [6; 47; 48; 49]. However, thermodynamic effects such as heat or pressure losses depending on the dynamics of the system are typically neglected.  
In this work, the endoreversible modeling includes the internal combustion engine and the transmission to consider not only power sharing and resulting energy savings, but also the efficiency of the engine and thus the resultant reduction of fuel consumption. Using this model and realistic driving data, an optimization of the control strategy will be carried out, and its potential to reduce fuel consumption will be shown.