## 3 Method of Analysis
### Energy Storage Model  
Inverter, transformer and other PV system losses are well known to depend on the received irradiance and system component temperatures. However, these losses depend on the applied technology and are not considered in this study in order to obtain results of general relevance. Further, the PV system is assumed to operate always at the maximum power point (MPP). Also, the energy storage system was assumed to be ideal, i.e., no losses were considered during charging, discharging etc. of the ESS, to obtain generally applicable results.  
It is a common industry standard in utility scale PV systems to dimension the PV generator nominal peak power to be higher than the nominal power of the inverter and transformer. Typical over dimensioning factors are 1.2-1.4 times the nominal power of the inverter. This practice has many benefits, e.g., smoother power output, better energy yield during winter, easier forecasting of power, etc. As PV panels become cheaper compared to the overall balance of system costs, over dimensioning will be more common in the future. However, for the sake of simplicity, it is assumed that inverter and PV generator nominal powers are the same.  
Cloud enhancement is a phenomenon where measured irradiance can temporarily exceed the clear sky irradiance value and it can be observed on days with intermittent clouds [18]. The effect of cloud enhancement can be seen clearly in Figures 1b and 2. In 2015, the highest measured irradiance peak was \(1379\) W/m\({}^{2}\). There were altogether 50 min of instances during 40 days where irradiation was over 1200 W/m\({}^{2}\). The spatial smoothing filter reduces the time to 28, 13 and 2 min and occurrence days to 29, 14 and 3 in 0.1, 1 and 10 MW PV generators, respectively. Due to their very short duration, power spikes caused by cloud enhancement have a negligible effect on the energy capacity requirement for the ESS. The spikes might have a relevant theoretical effect on the power capacity requirements, if the inverter does not curtail the output power. Because PV generator and inverter nominal power are the same, power spikes caused by cloud enhancement are curtailed. In addition to this, the system is set to be in standby mode when energy yield during imbalance settlement period is less than 1% of the nominal value, i.e., late in the evening and early on the morning. Due to these reasons, the total energy yield was \(99.05-99.10\%\) of the optimal value for the studied PV generator sizes and applied imbalance settlement periods. The applied model was more realistic with respect to real PV power production by ignoring the spikes and periods of low irradiance.  
The applied control algorithm for the energy storage system is simple. Power fed to the grid is constant and the energy fed to the grid is equal to the PV generator production for each imbalance settlement period, because an ideal PV power production forecast is used. Operation of the system can be expressed as  
\[P_{\text{ref}}=\frac{E_{\text{PV}}}{t_{\text{ISP}}}\text{ and } \tag{3}\]  
\[P_{\text{grid}}=P_{\text{ref}}=P_{\text{PV}}-P_{\text{bat}}, \tag{4}\]  
where \(P_{\text{ref}}\) is reference power, \(E_{\text{PV}}\) is PV generator energy yield during ISP, \(t_{\text{ISP}}\) is ISP length, \(P_{\text{grid}}\) is power fed to grid, \(P_{\text{PV}}\) is PV generator output power and \(P_{\text{bat}}\) is battery power so that positive value corresponds to discharging and negative value to charging. This means that the energy storage state of charge (SoC) is varying only during the imbalance settlement periods and returns to the base value at the end of each period. This behavior can be seen in Figure 3, where the energy capacity of ESS, i.e., state of charge, returns to the reference value of 0 h at the end of each settlement period of 1 h.  
Accordingly, the total energy generated by the PV plant during each settlement period equals with the energy fed to the grid.