# Generation of Data-Driven Expected Energy Models for Photovoltaic Systems
## Appendix A
### Top-Performing Trained Models  
Mathematical equations associated with the trained regression models. In general, these model formulations contain more parameters and do not perform as well as the additive interaction model. High-capacity refers to systems with greater than or equal to 1000 kW in \(C_{DC}\) while low-capacity refers to systems smaller than 1000 kW in \(C_{DC}\).  
High-capacity second-order seasonal:  
\[e =0.309 iI_{\text{winter}}+0.292 iI_{\text{spring}}+0.281 iI_{\text{ Summer}}+0.287 iI_{\text{fall}}\] \[+0.762 cI_{\text{winter}}+0.734 cI_{\text{spring}}+0.724 cI_{\text{ Summer}}+0.730 cI_{\text{fall}}\] \[-0.003i^{2}I_{\text{spring}}-0.009i^{2}I_{\text{Summer}}-0.007i^{ 2}I_{\text{fall}}+0.430 iI_{\text{winter}}\] \[+0.397 icI_{\text{spring}}+0.380 icI_{\text{Summer}}+0.384 icI_{ \text{fall}}+0.008c^{2}I_{\text{winter}}\] \[+0.008c^{2}I_{\text{spring}}-0.003c^{2}I_{\text{fall}}-0.054\]  
High-capacity third-order interactions:  
\[e =0.293i+0.769c-0.019i^{2}+0.394 ic+0.021i^{2}c+0.004 ic^{2}+0.001c^{3}-0.039\]  
High-capacity third-order seasonal:  
\[e =0.265 iI_{\text{winter}}+0.270 iI_{\text{spring}}+0.252 iI_{\text{ Summer}}+0.243 iI_{\text{fall}}\] \[+0.749 cI_{\text{winter}}+0.740 cI_{\text{spring}}+0.722 cI_{\text{ Summer}}+0.730 cI_{\text{fall}}\] \[-0.006i^{2}I_{\text{Summer}}-0.388 icI_{\text{winter}}+0.378 icI_{ \text{spring}}+0.365 icI_{\text{Summer}}\] \[+0.346 icI_{\text{fall}}+0.019c^{2}I_{\text{winter}}+0.005c^{2}I_{ \text{spring}}-0.002c^{2}I_{\text{fall}}\] \[+0.016i^{3}I_{\text{winter}}+0.008i^{3}I_{\text{spring}}+0.014i^{ 3}I_{\text{Summer}}+0.017i^{3}I_{\text{fall}}\] \[-0.013c^{2}I_{\text{winter}}+0.008i^{2}I_{\text{spring}}+0.007i^{ 2}I_{\text{Summer}}+0.019i^{2}I_{\text{fall}}\] \[-0.002c^{3}I_{\text{winter}}-0.054\]  
Figure 11: A 2-year comparison of observed energy with expected energy estimates from the trained additive interaction trained (AIT) model and the IEC standard at a single site. The IEC-derived estimates overestimate much more so than AIT-derived estimates. Slightly higher values of AIT-derived estimates relative to observed values likely indicate local failures at the site.  
High-capacity second-order interactions:  
\[e=0.295i+0.745c-0.018i^{2}+0.404 ic+0.004c^{2}-0.042\]  
Low-capacity second-order seasonal:  
\[e =0.711 iI_{\text{winter}}+0.686 iI_{\text{spring}}+0.680 iI_{\text{ Summer}}+0.714 iI_{\text{fall}}\] \[+0.617 cI_{\text{winter}}+0.576 cI_{\text{spring}}+0.605 cI_{\text{ Summer}}+0.636 cI_{\text{fall}}\] \[-0.046i^{2}I_{\text{winter}}-0.028i^{2}I_{\text{spring}}-0.046i^{2 }I_{\text{ Summer}}-0.038i^{2}I_{\text{fall}}\] \[+0.401 icI_{\text{winter}}+0.381 icI_{\text{spring}}+0.382 icI_{ \text{ Summer}}+0.427 icI_{\text{fall}}\] \[+0.001c^{2}I_{\text{winter}}-0.006c^{2}I_{\text{ Summer}}-0.104\]  
Low-capacity third-order interactions:  
\[e =0.746i+0.642c-0.043i^{2}+0.426 ic-0.019i^{3}-0.034i^{2}c-0.017c^{3 }+0.115\]  
Low-capacity third-order seasonal:  
\[e =0.757 iI_{\text{winter}}+0.684 iI_{\text{spring}}+0.667 iI_{\text{ Summer}}+0.712 iI_{\text{fall}}\] \[+0.577 cI_{\text{winter}}+0.486 cI_{\text{spring}}+0.567 cI_{\text{ Summer}}+0.619 cI_{\text{fall}}\] \[-0.023i^{2}I_{\text{winter}}-0.027i^{2}I_{\text{spring}}-0.050i^{ 2}I_{\text{ Summer}}-0.033i^{2}I_{\text{fall}}\] \[+0.413 icI_{\text{winter}}+0.382 icI_{\text{spring}}+0.381 icI_{ \text{ Summer}}+0.428 icI_{\text{fall}}\] \[-0.028i^{3}I_{\text{winter}}-0.007i^{3}I_{\text{ Summer}}-0.02i^{2}cI_{\text{winter}}-0.004i^{2}cI_{\text{fall}}\] \[-0.047c^{3}I_{\text{winter}}+0.068c^{3}I_{\text{spring}}+0.027c^{ 3}I_{\text{ Summer}}+0.016c^{3}I_{\text{fall}}+0.099\]  
Low-capacity second-order interactions:  
\[e=0.719i+0.630c-0.063i^{2}+0.411 ic-0.124\]