# Hierarchical Model Predictive Control for Hydraulic Hybrid Powertrain of a Construction Vehicle
## 2 System Description and Control Problem  
In a common spray-painting vehicle power system, the diesel engine directly matches the load torque demand of the main pump. The operation points of the engine probably enter the area with low fuel efficiency due to the frequent fluctuations of load. For fuel efficiency, a torque coupling structure using a secondary element hydraulic pump motor to adjust the engine torque output is a feasible power hybrid mode. For this study, the powertrain structure of a spray-painting vehicle is studied,in which a single accumulator and a hydraulic bridge are used to achieve two-way energy collection instead of the hybrid structure of most HHVs using high and low pressure with two accumulators, as shown in Figure 1. In addition, for the secondary adjustment, the pressure coupling control commonly used in industrial control is introduced, therefore, the hydraulic pump/motor (PM) of the secondary adjustment component can respond quickly and achieves the purpose of rapid adjustment and energy saving. The diesel engine, hydraulic PM, and hydraulic main pump are connected to the transfer case with the transmission ratio 1:1:1, respectively. In the diesel hydraulic hybrid system, hydraulic energy is converted into mechanical energy through the hydraulic pump/motor and coupled with the energy output from the diesel engine to form a \"torque coupling structure\", and therefore it is a parallel HHV. The engine dynamic is described as follows:  
\[J_{e}\dot{\omega}_{e}=T_{e}-T_{pm}-\frac{T_{L}}{\eta_{p}}-\mu\omega_{e} \tag{1}\]  
where \(J_{e}\) is the rotational inertia, \(\omega_{e}\) is the engine angular velocity, \(\eta_{p}\) is the efficiency of the main pump, \(\mu\) is the friction coefficient, and \(T_{e}\), \(T_{pm}\), \(T_{L}\) represent engine torque, PM torque, and load torque, respectively, which are described as:  
\[\left\{ \begin{array}{c}T_{e}=f_{e}(\alpha_{e},n_{e})\\ T_{pm}=\frac{p_{pm}Q_{pm}}{2\pi n_{e}}\eta_{pm}^{sgn(Q_{pm})}\\ T_{L}=\left(\frac{C_{e}\mu\omega^{2}}{21.15}+f_{r}mg\ cos\theta+mg\ sin\theta+ \delta m\dot{v}\right)\end{array} \right. \tag{2}\]  
where \(n_{e}=\frac{30\omega_{e}}{\pi}\), \(\alpha_{e}\) is the throttle opening angle of the engine; \(f_{e}(\cdot,\cdot)\) is a nonlinear and empirical function of \(\alpha_{e}\) and \(n_{e}\), which is obtained by the surface fitting in Matlab toolbox; \(p_{pm}\), \(Q_{pm}\), \(\eta_{pm}\) are the pressure, flow, and efficiency of PM, respectively; \(C_{d}\), \(f_{r}\) are air and rolling resistance coefficients, respectively; \(A\), \(m\), \(v\) are the frontal area, mass, and speed of vehicle, respectively; \(\theta\) is the climbing angle; \(\delta\) is a conversion factor of rotation mass; and \(r_{w}\) is the tire radius. The fuel consumption rate of the diesel engine is calculated by the following relationship:  
\[\dot{m}_{f}=\frac{T_{e}n_{e}}{9550}\cdot\frac{b_{e}(T_{e}n_{e})}{3600} \tag{3}\]  
where \(b_{e}\) is the brake-specific fuel consumption, which can be obtained by a steady-state map of the engine speed and torque, shown in Figure 2 for a certain diesel engine. Figure 1: Powertrain system of a spray-painting vehicle: (**a**) A spray-painting vehicle and (**b**) sketch of the powertrain structure of a hybrid hydraulic vehicle (HHV). When \(T_{L}>0\), the hydraulic unit is in an energy output state, and the traction torque is provided by the main pump. When \(T_{L}<0\), the vehicle is in a braking state, and the hydraulic unit does not provide energy. The flow of the axial piston pump when braking is:  
\[Q_{b}=\eta_{b}\frac{V_{b}\cdot v}{2\pi r_{w}} \tag{4}\]  
where \(\eta_{b}\) is the transmission ratio and \(V_{b}\) is the displacement of the axial piston pump. According to [PERSON]'s gas formula, the accumulator is modeled as follows:  
\[\left\{ \begin{array}{c}p_{0}V_{0}^{n}=p_{1}V_{1}^{n}=p_{axc}V_{acc}^{n}= \text{const}\\ SOC^{*}=-\int_{V_{0}}^{V_{axc}}p_{pm}dV=\frac{p_{1}V_{1}^{n}}{1-n}[V_{1}^{1-n} -(V_{1}-\Delta V)^{1-n}]\end{array} \right.