# Generation of Data-Driven Expected Energy Models for Photovoltaic Systems
## 3 Results and Discussion  
Data processing activities generally increased the correlations between the predictor variables (i.e., irradiance and capacity) and the response variable (i.e., energy) (Table A1). The processed data were inputted into a total of 13 trained models--ranging in model complexity (pre-lasso) from 3 parameters for the'simple additive' model pre-lasso to 151 parameters for the 'third-order-hour' (see Table 3). Generally, the number of parameters were lower for all models post-lasso fit, except for the'simple additive' and 'additive interaction' models, likely indicating the already sparse construction of these models.  
Initially, the various models were trained using data across all system sizes. However, this approach demonstrated systemic underperformance for low-capacity systems (\(<\)1000 kW DC capacity). Specifically, the best trained models (i.e., 'third-order interactions' and 'additive interaction') outperformed the IEC model in terms of \(\log RMSE\) when tested on every single system above 1300 kW DC capacity; however, 12 of 34 systems below a 1300 kW DC capacity underperformed relative to the IEC model. This result likely reflects the varying relationships between the site DC capacity and the energy generated; systems of higher capacity tend to receive a higher maximum energy generated per DC capacity (Figure A1). To better deal with this varying linearity, two separate sets of models were trained: one for models under 1000 kW DC capacity and another over.  
Across both high-capacity and low-capacity systems, models with the \(I\times C_{DC}\) interaction term perform better than those without the interaction term (i.e., 'hour','month','seasonal', and'simple additive' models). For example, two of the top-performing models (across both high-capacity and low-capacity systems) are the'simple additive' and the 'third-order interactions' models, both of which contain this interaction term (Table 3 and Figure 5). The 'additive interaction' trained (AIT) model has four parameters:  
\[\dot{e}=\begin{cases}0.07+0.69i+0.65c+0.42 ic&C_{DC}<1000\text{ kW}\\ -0.06+0.29i+0.76c+0.40 ic&C_{DC}\geq 1000\text{ kW}\end{cases} \tag{8}\]  
where \(i\) and \(c\) define the standardized irradiance and capacity variables, respectively, as defined in Table 1. The 'third-order interactions' model, on the other hand, contains these four terms as well as higher-order interactions (e.g., irradiance\({}^{2}\times\) capacity, capacity\({}^{3}\)). Although the variables within both of these two models are similar to the IEC standard, the inclusion of the interaction term, which highlights that the linear relationship between \(I\) and \(E\) is moderated by \(C_{DC}\) (Figure 4), likely explains the superior performance of these models relative to that standard. The heatmaps of the \(\log RMSE\) values highlight the evaluation metric's dependence on site capacity (Figure 5a,c), while the adjusted \(R^{2}\) heatmaps show consistent performance across site capacity (Figure 5b,d). The vertical concentration of dark bars likely reflects data quality issues not addressed by data preprocessing steps. A comparison of the associated partner generated expected energy estimates to those predicted from models also demonstrates that the AIT-derived estimates have lower average percent errors than the other models and the IEC (Table A2). Further evaluation of 2 years of records at a single site demonstrates that the AIT-derived estimates have a lower standard deviation and do not overestimate as much as the IEC-derived estimates (Figure A3). Given its parsimonious nature, the AIT model is subjected to further evaluation for both high- and low-capacity systems.