# Very short term forecasting of the Global Horizontal Irradiance using a spatio-temporal autoregressive model

###### tags: `references`

## Introduction

1. In the past, GHI forecasting methods were based on numerical weather forecast(NWF) data. They are the bests methods for forecast horizon higher than 6h.
2. In shorter time scale, satellite based studies have shown more accurate performance.
3. The aim of this study is to combine both the detailed spatio-temporal information from satellite images and the robustness of AR models

## Forecast model
1. Data:
* HelioClim-3 image
* $GHI_{Hc3}(k,p)$: GHI of pixel $p$ at time $k$ from SSI(surface solar irradiance) map
* Resolution: 3(km)x3(km)
* 15 mins 1 image
* $Kc_{Hc3}(k,p)=\frac{GHI_{Hc3}(k,p)}{GHI_{ClearSky}(k,p)}$
* GHI measurement was recorded at the SIRTA site
2. ARX model:
* $\hat{G}(t+h|t)=\alpha_1G(t)+...+\alpha_nG(t-n)+\beta_1u_1(t)+...+\beta_nu_n(t-n)+\gamma_1u_2(t)+...$
* $u_1,...,u_p$ are $p$ exogeneous variables ($Kc$ of pixels)
* time step: 15 mins
* h=1,2,3,4 (15mins, 30mins, 45mins, 60mins)
* $\hat{GHI}(t+h)=\hat{Kc}(t+h)GHI_{ClearSky}(t+h)$
* This model estimates $Kt$ because the dinual pattern in GHI can be removed by divided by $GHI_{ClaerSky}$

3. Persistence model $P_0$:$\hat{Kc}^{P0}(k+h)=Kc(k)$
4. Temporal time series model:
$At=\hat{Kc}^{At}(k+h)=\sum^P_{p=0}(\alpha_pKc(k-p))+\gamma$
Choose order=5, but RMSE only decreases 5W between order 1 and order 21.
5. Spatial model:
* Intercorrelation map: correlation between the difference of the recorded measurements and the HelioClim-3 pixels.
* $C(i,j)_h=corr((Kc(k+1)-Kc(k)), Kc_{Hc3}(i,j))$
* The main regrion of interest is translated to the west as the lag increase. Assume the wind mainly blows from the West on this site

* $Ast$: autoregressive model that takes as spatial input a 3x3 fixed pixels grid centered on the site's location.
* $Ast2$: Autoregressive model includes the precedent information by taking different regions chose from intercorrelation maps.


## Results
The model tends to overestimate low GHI and underestimate higher ones.

* S:
* Predict the stochastic variability of the solar irradiance.
* Ranges the forecasting methods from 0 (persistence performance) to 1 for perfect forecast, negative values indicate methods of which performances are worse than a persistence model.