*2.1. Problem Formulation*

Figure 1 shows the power system of PHEV with a serious topology, in which the battery pack is the main onboard energy storage system (ESS) to provide the power load of the electric motor and an engine generator group is used as the assistance power unit (APU). The electric motor drives the front axle of the vehicle through an automatic mechanical transmission (AMT). The electrical power demand of the motor is split between the ESS and APU, as commanded by the energy management system. The parameters of these main components and the modeling have been described in our previous research presented in Reference [27,28], that is the foundation of this paper.

**Figure 1.** The structure diagram of plug-in hybrid electric vehicles (PHEV) power system.

The energy management issue is described as finding the optimal power allocation for minimizing the cost function while meeting the constraints. The objective function is minimizing the energy consumption during a certain driving cycle, described by

$$\text{Minimize}: J(u) = \Phi\_{\mathbb{B}} + \sum\_{k=1}^{N} \left\{ \upsilon\_{fud} P\_E(k) \Delta t \right\} \tag{1}$$

where *u* denotes the control policy, Φ*<sup>B</sup>* is the cost of electricity, *N* is the total number of steps, *P*<sup>E</sup> is the engine power, υ*fuel* is the price of fuel (CNY-¥/kWs), *t*<sup>0</sup> and *t*<sup>f</sup> denote the time at the beginning and ending the driving cycle, respectively, *t* denotes the discrete time instant, *t* = *t*<sup>0</sup> + *k*Δ*t*, *k* = 1,2,...,*N*.

For a certain driving condition, the electricity consumption is directly calculated according to the total battery power consumption, described by

$$\begin{split} \Phi\_{B} &= \upsilon\_{\text{clcc}} \cdot \Delta E\_{\text{Ratt}} \\ &= \upsilon\_{\text{clcc}} \cdot \frac{\sum\_{k=1}^{N} \{P\_{\text{Ratt}}(k) \Lambda t\}}{\eta\_{b}}; \end{split} \tag{2}$$

in which

$$
\eta\_b = \begin{cases}
\begin{array}{c}
\eta\_{\text{discharge}} \text{ at discharging} \\
\frac{1}{\eta\_{\text{obs}}} & \text{at charging}
\end{array}
\end{cases}
$$

where Δ*EBatt* is the battery electricity consumption, υ*elec* is the price of electricity (CNY-¥/kWs), *PBatt* is power of the battery, η*<sup>b</sup>* is the efficiency of the battery, η*cha* and η*discha* denote the efficiencies at charging and discharging, respectively.

If the battery aging is not considered, the cost function is only impacted by the power allocation. Thus, the control variable can be defined as *u* = [*PE(k) PBatt(k)]T.* The optimization problem is to find

out the optimal *u*\* to minimize the above cost function. To ensure the optimization results conform to feasible solutions, the control variable is subject to some constraints below.

$$P\_{Batt, \text{min}}(z(k)) \le P\_{Batt}(k) \le P\_{Batt, \text{max}}(z(k)) \tag{3}$$

$$0 \le P\_E(k) \le P\_{E,\text{max}}\tag{4}$$

$$
\!\!\!\!\!\/} \!\!\!\!\!\/) \le \!\!\!\!\/(k) \le \!\!\!\/) \tag{5}
$$

where *PBatt*,max and *PBatt*,min are power limitations of the battery pack, *PE*,max is the maximum power of the engine, ϕ<sup>1</sup> and ϕ<sup>2</sup> are limitations of battery state of charge (SOC).

Based on the power balance relation, equality constraints are given as:

$$P\_{\text{Req}}(k) = P\_E(k)\eta\_{APIL} + P\_{\text{Ratt}}(k)\eta\_{\text{Batt}} \tag{6}$$

in which

$$\eta\_{Batt} = \begin{cases} \begin{array}{c} \eta\_{dis} \eta\_{Inv} \quad \text{if } P\_{Batt}(k) \ge 0\\ \frac{1}{\eta\_{cls} \eta\_{Inv}} \quad \text{if } P\_{Batt}(k) < 0 \end{array} \end{cases}$$

where *PReq* is the power demand of the electric motor, η*APU* and η*Batt* are efficiencies of APU and battery system, respectively, η*dis* is the discharging efficiency of battery, *PBatt* is positive while the battery is discharging and negative while the battery is charging.

In addition, the battery SOC at the initial time of the optimization horizon should be pre-set by

$$z(t\_0) = z\_0 \tag{7}$$

where *z*<sup>0</sup> is initial SOC of the battery pack.
