## 3 Optimization Methods  
As presented in (16), the total cost function of this paper can be expressed as,  
\[\min\quad F=\int_{0}^{t_{total}}\big{(}a_{2}(w_{0},P_{o})\cdot P_{bat}^{2}(t)+a _{1}(w_{0},P_{0})\cdot P_{bat}(t)+a_{0}(w_{0},P_{0})\big{)}e_{on}(t)dt \tag{17}\]  
Figure 5: Fuel-rate validation and approximation. Figure 6: \(a_{2}(w_{0},P_{0})\). Based on analysis of the vehicle model, related boundary constraints can be summarized,  
\[\left\{ \begin{array}{l}P_{eng\_on\_\min}<P_{eng\_on}\leq P_{eng\_on\_\max}\\ P_{bat\_\min}(t)\leq P_{bat}(t)\leq P_{bat\_\max}(t)\\ 0\leq\Delta SOC\leq 0.7\\ T_{not\_\min}\leq T_{not}\leq T_{not\_\max}\end{array} \right. \tag{18}\]  
where \(P_{eng\_on\_\max}\) is the upper limit of engine-on power threshold, which is equal to the maximum output power of engine, \(P_{eng\_on\_\min}\) is the lower limit of engine-on power threshold, \(P_{bat\_\min}\) and \(P_{bat\_\max}\) denote the minimum and maximum values of the battery power commands. \(\Delta SOC\) denotes the range of the battery SOC variation, which is also called the depth of discharge (DOD). In this paper, the ending SOC is set to 0.3, the minimum SOC is 0.2 and thus \(\Delta SOC\) belongs to \([0,0.7]\). In premise of knowing the driveline power requirements, the engine-on power threshold is solved by the SA algorithm, and the battery power command is calculated by the convex optimization algorithm when the engine is on, as shown in Figure 7. Compared with other optimal algorithms, the SA algorithm is faster and more efficient in solving such global optimization problems with boundary constraints. The calculation process can be described as follows. The first step is to acquire the information of the vehicle including speed, driving range and driveline power demand. Based on the sum of the whole power demand and the maximum battery supplied energy, a decision can be made that if the driving range is less than the AER, the vehicle is under pure EV mode, or else the vehicle is under the HEV mode. If the vehicle works under the HEV mode, an initial engine-on power can be supplied by the SA algorithm. Then, based on the setting constraints, the interior point method will be applied to calculate the battery power thereby realizing the power split between the engine and the battery. Now the fuel consumption based on (17) can be calculated. After that, the SA will be iterated to generate a new engine-on power, and accordingly the battery powers and the total fuel consumption will be updated and compared with the previous calculations. The algorithms will continue to iterate until reaching the termination conditions. Finally, the algorithm outputs the engine-on power and the corresponding battery power commands. In (18), the lower bound and upper bound on the engine ON/OFF threshold are set to \(P_{eng\_on\_\min}\) and \(P_{eng\_on\_\max}\). Since the SA algorithm is insensitive to the initial value, the initial engine-on power  
Figure 7: The whole optimization calculation process. SA: simulated annealing. threshold \(P_{eng\_on\_initial}\) is determined randomly as the initial point between 10 kW and 15 kW and a sufficiently large initial temperature \(T\) is randomly selected based on the initial point. After each iteration, an updated state \(P^{*}_{eng\_on\_update}\) is generated, and the temperature increment \(\Delta T\) is calculated comparing with the initial iteration. \[\Delta T=F(P^{*}_{eng\_on\_update})-F(P_{eng\_on\_initial}) \tag{19}\]  
where \(F\) denotes the cost function. If \(\Delta T<0\), \(P^{*}_{eng\_on\_update}\) is accepted as the updated current solution. If \(\Delta T>0\), \(P^{*}_{eng\_on\_initial}\) is accepted as the initial current solution with the probability \(\exp(-\Delta T/T)\). Then, the battery power is programmed by the convex optimization algorithm combined with \(P^{*}_{eng\_on\_update}\), and the cost consumption function is calculated. The temperature value decreases gradually until reaching the termination condition. Finally, the current solution can be achieved.