## 2 Modelling of PV System's Components
### Modelling of Battery System
Energy stored in battery is depend on inverter efficiency, cable losses, battery's charging and discharging efficiency, load demand and energy generated from PV system. Besides that, there are several other methods that can be used to model a battery system. A simulation model for grid tied residential PV battery system was done in [38], and evaluated using energy assessment criteria. The authors modelled their battery system using AC coupled battery system layout, multi-crystalline PV module technologies and lithium-based battery system. Energy assessment criteria, in term of self-consumption rate, \(s\) and self-sufficiency, \(d\) of battery were evaluated, using the following equations,  
\[s=\frac{E_{DU}+E_{BC}}{E_{PV}} \tag{17}\]  
\[d=\frac{E_{DU}+E_{BD}}{E_{L}} \tag{18}\]where \(E_{DU}\) is energy directly used by load, \(E_{BC}\) is energy to charge battery, \(E_{BD}\) is energy discharged by battery, \(E_{PV}\) is energy generated by PV panel, and \(E_{L}\) is load demand. Energy assessment is important to monitor battery usage, in order to prevent energy saturation. However, in this battery model the authors neglect several factors that affect a battery's efficiency, like self-discharge power, temperature effect and battery age. Besides, there is no explanation for the method to model PV generated power, valuation of battery's state of charge, and energy provided from/to grid. Another study for optimization of battery capacity in GCPV was done, by considering battery's operation scheduling and ageing effect on economic analysis [39]. Discrete dynamic equation for battery in day's time step, (\(i\), \(t\)), can be given as,  
\[\frac{E_{B}(i,t)-E_{B}(i,t-\Delta t)}{\Delta t}=P_{Bdc}(i,t) \tag{19}\]  
where \(P_{Bdc}\) is power stored in battery, \(E_{B}\) is energy stored in battery, and \(t\) is time. If \(P_{Bdc}\) > 0, it is charging since energy stored in battery is increase, and vice versa. Power feed to AC bus from battery supply, \(P_{Bac}\) is given by,  
\[P_{Bac}(i,t)=\left\{ \begin{array}{c}\eta_{Bi}P_{Bdc}(i,t),\;if\;P_{Bdc}(i,t) <0\\ \frac{P_{Bdc}(i,t)}{\eta_{Bi}}\;otherwise\end{array} \right. \tag{20}\]  
where \(\eta_{Bi}\) is battery bidirectional converter's efficiency. Besides that, the equation for battery's ageing model is described as,  
\[\Delta C(i,t)=\left\{ \begin{array}{c}\Delta C(i,t-\Delta t)-Z\frac{P_{Bac}(i,t)}{\eta_{B}}\Delta t,\;if\;P_{Bac}(i,t)<0\\ \Delta C(i,t-\Delta t),\;if\;P_{Bac}(i,t)>0\end{array} \right. \tag{21}\]  
where \(C\) is usable battery capacity, \(\eta_{B}\) is conversion capacity of battery, and \(Z\) is ageing coefficient. A study to compare two type of storage types for grid tied PV generation application, which are lead acid storage and hydrogen storage model are simulated in [40]. Equation (21) below represent voltage of lead acid battery, \(V_{b}\), where battery current, \(I_{b}\) is positive during charging, negative during discharge, \(R\) is internal resistance, and \(V_{o}\) is equilibrium voltage. \[V_{b}=V_{o}+RI_{b} \tag{22}\]  
Meanwhile, in [41], the authors proven that CIEMAT (Copetti) model is able to present dynamic and complex lead acid battery operation. In this paper, the Copetti battery model was simulated and compared with two experimental PV system models. The model was presented by discharge, charge and overcharge process. However, the result was slightly over approximated charging. A model of PV pumping system, including PV array, battery and electric motor were modelled in [42; 43]. However, instead of using state of charge, the authors implemented state of voltage (SOV) to model the battery system. It is because, since battery's state of charge is dependent on voltage.