# An Overview of Modelling and Energy Management Strategies for Hybrid Electric Vehicles
## 3 Research and Current Status of Energy Management Strategies for HEVs
### Optimization-Based Energy Management Strategies
#### 3.2.2 Pontryagin's Minimum Principle  
The PMP specifies that the optimal solution to a global optimization problem has to fulfill optimality conditions, which are related to the transient minimal value of the Hamiltonian function [128; 129; 130; 131]. Assuming that the trajectory calculated through PMP is deterministic and satisfies the corresponding constraints and boundary conditions, the  
Figure 9: The torque distribution effects of a vehicle under different driving cycles, (**a**) under UDDS, (**b**) under WLTC.  
optimal trajectory is usually considered to be the global optimal one [132; 133]. Thus, the Hamiltonian function can be described:  
\[H(x(t),u(t),t)=\dot{m}_{f}(x(t),u(t),t)+\lambda(t)f(x(t),u(t),t) \tag{5}\]  
where \(\lambda(t)\) denotes the co-state. The state and co-state of the system must satisfy the following conditions:  
\[\left\{\begin{array}{c}\dot{x}(t)=\frac{\partial H}{\partial\lambda}=f(x(t),u(t),t)\\ \dot{\lambda}(t)=-\frac{\partial H}{\partial x}=-\lambda(t)\frac{\partial f} {\partial x}\end{array}\right. \tag{6}\]  
On the basis of PMP, Onori adopted an adaptive supervision controller to optimize the online energy management of a PHEV [134]. The suggested algorithm depended on the adaptation of the control parameter from the SOC input and incorporates minimum driving information. This new method operated in a distance-based domain and was equipped with rules. The purpose is to prevent the actual SOC from deviating from the reference linear SOC profile, by appropriately resetting the common state.  
In [135], an optimal EMS built on the approximate PMP algorithm was developed for parallel plug-in hybrid electric vehicles (HEVs). The authors were inspired to use a novel piecewise linear approximation strategy, which specified the turning point of the engine fuel rate for Hamiltonian optimization. The optimization process using the A-PMP strategy is presented in the flow diagram shown in Figure 10, and the results are summarized in Table 4.  
[PERSON] et al. presented an energy management approach for power-split PHEVs. A series of quadratic equations were formulated to approximately calculate the fuel rate of the vehicle by analyzing the PHEV powertrain, with the battery current as an input. PMP was introduced to find the battery current command from the solution of the Hamiltonian function [43].  
Strategies based on PMP optimize the vehicle state and simplify the whole vehicle control strategy under vehicle braking and stopping conditions. By defining the instantaneous  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Configuration** & **AE-CS** & **A-PMP** & **PMP** \\ \hline Fuel consumption (L/100 km) & 1.1333 & 1.0577 & 1.0618 \\ Final SOC & 0.3038 & 0.3043 & 0.3050 \\ Corrected fuel consumption (L/100 km) & 1.1333 & 1.0544 & 1.0538 \\ Corrected final SOC & 0.3038 & 0.3038 & 0.3038 \\ Fuel saving rate (\%) & 0 & 6.96 & 7.01 \\ Time step (s) & 0.1 & 0.1 & 0.1 \\ Process time for the whole simulation & 2 min & 4 min & 6 h \\ Real time capability & Yes & Yes & Hardly \\ \hline \hline \end{tabular}
\end{table}
Table 4: Optimization results comparison (Data from the source [135]).  
Figure 10: Flow diagram of A-PMP strategy (source [135]).  
cost function at each moment, the instantaneous optimal control volume for the vehicle is solved and the objective of approximate optimality of the global cost function is achieved.