## 1 Introduction
It has been demonstrated that employment of second life batteries in the grid for peak shaving in China is cost beneficial, especially for the grid companies. The impacts of the second life battery packs with a different state of health (SoH) on the performance of the system was investigated by [PERSON] et al. [15]. The semi-empirical degradation model was used for modelling demonstrating that second life batteries are comparatively more profitable than first life batteries in PV systems. [PERSON] et al. [9] developed a mathematical model for the second life battery sizing and optimization of a stand-alone PV system for a net zero energy residential building. The second life battery sizing was performed to achieve the best load match of the building and the results confirmed the optimum ratio of battery size to PVs total power to achieve the best load match in the residential buildings. Further, [PERSON] et al. [16], modelled a grid-connected residential building equipped with PV and second life ESS considering building demand in various times during the year. The ECM was used to predict the battery parameters at different times and estimated the battery degradation parameters. The results of their work demonstrated that by considering degradation effects on financial parameters, the second life batteries are no longer cost effective for the customers. The technical assessment of integration of second life batteries with grid-connected PV systems for a residential building is demonstrated in [PERSON] et al. [17], by considering a typical European residential building load demand. MATLAB Simulink was used to model the proposed system for three scenarios: without storage, large (Nissan Leaf), and small (Citroen C0) second life battery energy storage system. In the first year, the employment of second life BSS resulted in a reduction of 82.1% and 78.8% in energy exchange between the building and the grid for large and small BSS, respectively. [PERSON] et al. [18] has investigated the integration of second life batteries with an off-grid EV charging station in the United States, where MATLAB SIMULINK has been applied for mathematical modelling of the proposed system. The charging station cost was significantly reduced in some locations, along with the similar performance compared to new batteries in other places. It was evident from the reviewed literature that the integration of the second life battery ESS for a residential building with EV charging station has not been investigated. The main contribution of this study is to reveal the impacts of load increase on the sizing of the second life battery energy storage system. The load applied to the second life battery storage system in this study is the residential building electricity load plus EV charging station.The EV charging stations apply an extra load to the residential building load demand [19]. Accordingly, in this study, the design and sizing of the second life battery ESS applied to a residential building with an EV charging station is investigated. The proposed system is modelled using MATLAB SIMULINK. The performance and stability of the system is assessed in a day in the middle of the winter, with the lowest solar irradiance and highest demand. The assessment considers the second life battery ESS with a different number of packs. It is assumed that the roof area is fully covered with PVs, therefore, the energy supply demand mismatch and the system stability maintenance is accomplished by adjusting the ESS size. The energy assessment and SoH analysis are performed to compare the system energy exchange, degradation, and energy supply demand mismatch in various scenarios.