Day-Ahead Probabilistic Forecast of Solar Irradiance: A Stochastic Differential Equation Approach

Abstract

In this work, we derive a probabilistic forecast of the solar irradiance during a day at a given location, using a stochastic differential equation (SDE for short) model. We propose a procedure that transforms a deterministic forecast into a probabilistic forecast: the input parameters of the SDE model are the AROME numerical weather predictions computed at day $D-1$ for the day D. The model also accounts for the maximal irradiance from the clear sky model. The SDE model is mean-reverting towards the deterministic forecast and the instantaneous amplitude of the noise depends on the clear sky index, so that the fluctuations vanish as the index is close to 0 (cloudy) or 1 (sunny), as observed in practice. Our tests show a good adequacy of the confidence intervals of the model with the measurement.

Appendix: Proof of Well-Posedness of the SDE Model (4.3), When \(\frac{ 1 }{2 }\le \alpha \) and \(\frac{ 1 }{2 }\le \beta \)

Because the exponents $\alpha$ and $\beta$ are possibly non integers in the definition of (4.3), the signs of $X_t$ and $1-X_t$ may be an issue. Therefore, we start with a modification of the SDE model (4.3) avoiding the sign problems:

$$\begin{aligned} \mathrm{d}X_t=-a(X_t-x^{\mathtt{forecast}}_t)\mathrm{d}t+\sigma {\mathbf {1}_{X_t\in [0,1]}X_t^\alpha (1-X_t)^\beta } \mathrm{d}W_t, \end{aligned}$$

(4.10)

where $X_0\in [0,1]$ is a given deterministic initial value.

Existence/uniqueness. A direct application of [23, Chapter IX, Theorem 3.5-(ii), p. 390 and Theorem 1.7 p. 368] shows that the model (4.10) is well-posed, in the sense that there is an unique strong solution on the probability space $(\varOmega , {\mathscr {F}}, \mathbb {P})$ where the filtration is the natural filtration of the Brownian motion completed as usually with the $\mathbb {P}$-null sets. In [23, Chapter IX, Theorem 3.5-(ii), p. 390] we have used $\frac{ 1 }{2 }\le \alpha$ and $\frac{ 1 }{2 }\le \beta$.

The solution (4.10) takes values in [0, 1]. We invoke a comparison theorem for SDEs. Denote $b^X(t,x)=-a(x-x^{\mathtt{forecast}}_t)$ the drift coefficient of X and now, consider the solution to

$$\begin{aligned} \mathrm{d}Y_t=-a Y_t \mathrm{d}t+\sigma {\mathbf {1}_{Y_t\in [0,1]} Y_t^\alpha (1-Y_t)^\beta } \mathrm{d}W_t, \qquad Y_0=0. \end{aligned}$$

(4.11)

Its initial condition fulfills $X_0\ge Y_0$, its drift $b^Y(t,y)=-a y$ is globally Lipschitz in space (and $b^X$ too) and last, we have $b^Y(t,x)-b^X(t,x)=-ax^{\mathtt{forecast}}_t\le 0$. Therefore, [15, Chapter V, Proposition 2.18, p. 293] shows that $X_t\ge Y_t$ for any t with probability 1. But since the solution to (4.11) is 0, the above proves that X remains positive.

Similarly, set

$$\begin{aligned} \mathrm{d}Y_t=-a (Y_t-1) \mathrm{d}t+\sigma {\mathbf {1}_{Y_t\in [0,1]} Y_t^\alpha (1-Y_t)^\beta } \mathrm{d}W_t, \qquad Y_0=1. \end{aligned}$$

Clearly, $Y_0\ge X_0$, $b^X(t,x)-b^Y(t,x)=-a(1-x^{\mathtt{forecast}}_t)\le 0$, $Y_t=1$ and we conclude that $X_t\le 1$. This justifies why we can remove the indicator function in (4.10) to get (4.3).

$\square$