## 3 Optimization-Based Strategies
### Global Information-Driven Optimization
#### 3.2.1 Pontryagin's Minimum Principle (PMP)
The error between the terminal SOC value and the initial SOC value can be calculated, and then the error can be brought within the target range (\(\delta\)) by readjusting the value of the covariant variable, that is, \(\left|SOC\Big{(}t_{f}\Big{)}-SOC(t_{0})\right|\leq\delta\)[74]. The method of linear interpolation or the Newton method can be used to correct the value. On the other hand, the value of the state variable must satisfy the boundary condition \(SOC(\cdot)\), which can be defined as the hard boundary. To avoid missing the optimal value on the boundary, [50] proposed the concept of soft bounds inside the hard bounds, defined as \(SOC_{min}^{soft}\leq SOC(t)\leq SOC_{max}^{soft}\). The value of SOC can exceed the soft bounds by an allowable amount; however, exceeding those bounds will be penalized by increasing or decreasing \(\lambda\). Compared with the rule-based strategy, the PMP-based strategy has a significant improvement in fuel economy and the battery SOC at the final moment is unchanged [74]. The PMP-based strategy combining with the intelligent algorithms can be developed to improve the optimal performance of control strategy. In [75], the PMP-based strategy combining with SA algorithm is applied to determine the battery current command and engine-on power using a series of quadratic equations to approximate fuel-rate when the engine is operating. Moreover, the proposed algorithm is validated with by considering the battery's state of health (SOH) to extend the application. In addition, the selection of covariant variable \(\lambda\) can directly affect the optimal power distribution between the power sources, meaning that it will affect the fuel economy and driving performance of HEVs. Therefore, the influence of the co-state variable on the optimal solution of the PMP-based strategy and its determination in terms of certain driving cycles and uncertain driving cycles should be discussed, as detailed in [76]. According to the restriction of the state variable, the periodically updated co-state variable is used for uncertain driving cycles, and the initial \(\lambda\) is given in advance. The Hamiltonian function is a complicated function of the control variables, and the process of seeking an optimal control variable needs to traverse the whole domain of the control variables, which leads to a large computational burden and difficulty in implementing real-time control. To overcome the challenge of optimizing the Hamiltonian function, the quadratic performance index is widely used, which is an explicit solution for Hamiltonian optimization by applying quadratic fitting to both engine and electric powertrains. Generally speaking, the consideration of engine on/off control is not necessary for a HEV because the engine needs to be on due to the limited electrical energy. However, the engine on/off control should be considered in PHEV because the engine can be kept off for a relatively long period of time due to the sufficient electrical energy obtained from the extra grid. Aiming to solve the problem mentioned above, [77] introduces the energy management of PHEVs based on an approximate PMP (A-PMP) while utilizing a piecewise linear approximation, which was determined by specifying the turning point of the engine fuel rate. In the A-PMP strategy, the engine on command is separated from the engine torque command, and the engine state with the smaller Hamiltonian function is considered to be the engine on command. Based on the above research, the main research areas of the PMP-based strategy are described in Figure 6.  
Figure 6: Depiction of the main research areas of the PMP-based strategy. Although the results of the PMP-based strategy are very close to DP, and it has a lower computational burden compared to the DDP-based strategy, there are still some challenges of PMP. On the one hand, the quadratic fitting to the engine fuel rate is used to simplify the engine fuel map, which changes some important features of the engine.