# Hierarchical Model Predictive Control for Hydraulic Hybrid Powertrain of a Construction Vehicle
## 2 System Description and Control Problem
\tag{5}\]  
where \(p_{0}\), \(V_{0}\) denote the precharge pressure and volume of the accumulator, respectively; \(p_{1}\) is the minimum working pressure; \(V_{1}\) is the corresponding volume; \(p_{acc}\) and \(V_{acc}\) are the pressure and volume of the accumulator at any time; \(n\) is polytrophic exponent, here \(n=1.4\)[23]; and \(SOC^{*}\) is the energy of the accumulator. It can be seen from Equation (5) that when \(p_{1}\) is determined, the \(SOC^{*}\) depends on hydraulic oil volume flowing into the accumulator \(\Delta\mathbf{V}\). Ignoring the loss of gas flow, the volume change in the accumulator is calculated as:  
\[-\frac{dV}{dt}=Q_{pm}+Q_{b} \tag{6}\]  
Thus, considering Equations (4) to (6), the accumulator power is characterized as:  
\[SOC^{*}=2\pi T_{pm}n_{e}t_{pm}^{sgn(-T_{pm})}+\eta_{b}\frac{p_{pm}V_{b}v}{2 \pi r_{w}} \tag{7}\]  
It is worth mentioning that for simplifying the computation of the control design, the \(SOC^{*}\) is normalized as the state of charge (SOC) by defining  
\[SOC=\frac{SOC^{*}-SOC^{*,\min}}{SOC^{*,\max}-SOC^{*,\min}} \tag{8}\]  
with \(SOC^{*,min}=-\int_{V_{0}}^{V_{min}}p_{pm}dV\), \(SOC^{*,max}=-\int_{V_{0}}^{V_{max}}p_{pm}dV\), where \(V_{min}\) and \(V_{max}\) are the minimum and maximum volume of accumulator corresponding to the accumulator being empty and the accumulator being full [25], respectively. Accordingly, the SOC of the accumulator can be regarded as the ratio of the  
Figure 2: Brake-specific fuel consumption, \(\mathbf{b_{e}}\), map of the diesel engine. variation of instantaneous fluid volume to the variation of maximum fluid capacity in the accumulator under certain pressures, and \(SOC^{min}=0\) and \(SOC^{max}=1\). In the powertrain structure, as shown in Figure 1, hydraulic energy is converted into mechanical energy through a hydraulic PM coupled with the energy output from the diesel engine to form a torque coupling structure similar to a parallel HEV. Thus, by regulating the transfer of energy to/from the accumulator, the engine operates at its best operating zone to improve its fuel efficiency when it works. To achieve this, an appropriate EMS is necessary to allocate the demand of the load torque, and a satisfying tracking controller is necessary to ensure the desired output torque of the engine. However, during the actual driving cycle of the construction vehicle, the driving style of the driver and the external environment, such as road type, slope, and other information, cause uncertain driving conditions with some randomness, which should be considered in the design of the EMS. Meanwhile, the uncertainty and nonlinearity of the engine dynamic and external disturbance should be taken into account in the design of the engine tracking controller. Consequently, a hierarchical control scheme, which can effectively deal with the randomness, uncertainty, and nonlinearity, are presented in this paper. The upper layer performs energy management to determine the energy distribution between the engine and the accumulator. The control objective is to minimize fuel consumption and ensure the control range of the SOC of the accumulator. The lower-layer controller is to ensure that the engine and the auxiliary power source operate at the desired operating zone based on the energy distribution result of the upper-layer.