**1. Introduction**

With the deepening of environmental deterioration and energy crisis issues, developing high efficient and clean automobiles has been recognized as a matter of global significance [1]. In the recent years, electric vehicles (EVs) are widely recognized as the development tendency of automobile industries all over the world [2,3]. The benefits of EVs highly depend upon the onboard high-capacity battery pack that can be recharged by the power grid. The frequently discharge/charge cycles during the vehicular utilizations will inevitably cause the degradation of the power battery [4]. Battery aging, which implies a complex electrochemical evolution process of the gradually loss of lithium inventory and active material, has been fully discussed in many literatures [5,6]. In order to monitor the battery health condition and improve the battery performance, many efforts have been exerted on estimation of the battery state of health (SOH) or remaining useful life (RUL) [7–9]. However, battery aging not only influences the performance of the battery itself, but it also has an impact on the vehicle performance, like reducing maximum power, driving range, energy economy, etc., of the vehicle. Especially in hybrid electric vehicles (HEVs), battery aging will seriously affect the overall energy efficiency since the

engine has to contribute more power than expected, resulting in the increased energy consumption and emissions. Therefore, the energy management strategy should be adjusted to maintain the optimality during the entire lifespan of lithium-ion batteries.

Energy management strategy (EMS) is integral part of improving the fuel economy of both the traditional HEVs and plug-in hybrid electric vehicles (PHEVs), which have drawn attentions from many researchers [10–13]. Nevertheless, the current studies mainly focus on the optimization methods towards how to maximize the hybrid system's advantages, without enough concerns of the impacts of battery aging. Normally, the existing methods of EMS can be divided into two categories: the rule-based method and the optimization-based method. The rule-based strategies mainly depend upon some predefined control rules, containing deterministic rules and fuzzy logic rules, to operate the power units at high efficiency [14]. For example, Gao et al. [15] proposed a deterministic rule-based energy management strategy for PHEV focused on all electric range and charge depletion range operations, which has been verified by an example passenger car in a typical urban driving cycle; Schouten et al. [16] presented a fuzzy logic-based energy management strategy to improve the fuel economy of the parallel hybrid electric vehicle; Ali et al. [17] proposed a fuzzy logic control for electric vehicles, the presented method can achieve an efficient and fast-charging of the lithium-ion batteries. The rule-based strategies have been widely used in real-time control because they are simple, easy to be online implemented, and have good robustness.

The optimization-based strategies are designed to achieve the optimal control performance by using advanced optimization algorithms, such as dynamic programming (DP) [18,19], genetic algorithm (GA), particle swarm optimization (PSO) [20], etc. These algorithms adopt a common cost function, namely, to minimize the fuel consumption (or maximum the fuel conversion efficiency) of the vehicle during a certain time horizon. For example, Larsson et al. [21] investigated the DP-based energy management strategy to minimize the fuel consumption of a hybrid electric vehicle and discussed how much computational demand can be reduced. The drawback of these global optimization algorithm-based strategies is that they can barely be implemented in real-time control due to their dependence on an a priori known speed profile. Therefore, they are often implemented offline as a reference or a benchmark for other algorithms [22]. In addition, there is another kind of optimization approach, namely, the instantaneous minimization algorithm, which is to minimize the cost function at each time step. Most representative one of this kind is equivalent consumption minimization strategy (ECMS). Although ECMS can only provide a near-optimal solution, it can be implemented online because it does not rely on an a priori known speed profile. The specific descriptions about ECMS can be found in References [23–26].

The above investigations have achieved a great progress of resolving the energy management issues; however, most of these studies are based on the characteristics of the fresh battery. Although the battery aging induces a significant impact on energy consumption, it is still unknown how much the extra energy consumption can be caused by the battery aging in PHEVs and there are few studies to deal with battery aging from the perspective of energy management. In this study, we expect to reveal the maximum influence that battery aging can produce on the vehicle energy consumption and to present a global optimal control strategy over the entire lifespan of onboard batteries. The main target of the presented strategy is to maintain the optimal energy efficiency even after the serious aging of the battery and partially compensate for the negative impact of battery aging from the system level. The remainder of this paper is organized as follows: the model, energy management scheme and optimization method for PHEVs concerning the impacts of battery aging are described in Section 2; the battery model and the mathematical expression of the aging characteristics are proposed in Section 3; the impacts of the battery aging on energy consumption and the results of the EMS are illustrated in Section 4 while the conclusions are summarized in Section 5.
