Realistic scenario planning for solar power

A framework that constrains solar power projections to physically achievable limits provides a more reliable basis for future energy planning. The uncertainty quantification framework developed by a KAUST-led team uses mathematically robust numerical methods to implement time-varying physical constraints^[1]. This framework could support energy producers in scheduling reserves, planning trades, and designing reliable strategies.

“Grid operators and energy traders must make critical planning decisions in the face of variability as solar is integrated at scale,” says Raúl Tempone, who leads the Stochastic Numerics Research Group at KAUST. “Traditional uncertainty ranges for an official forecast can ignore physical upper and lower limits that occur over a day, which means that operators may be modelling unrealistic scenarios.”

“Our work puts physics back into the picture while keeping the math transparent and verifiable,” adds Tempone.

Scenario modelling enables operators to examine evolving production and reserves, helping them plan for realistic best- and worst-case futures. The primary objective is not to artificially narrow the range of futures, but to accurately characterize it so it is physically consistent and data-driven.

Traditional ranges based solely on historical data can be unrealistic, as they may dip below zero or exceed the maximum feasible production for a given irradiance. To address this, the team developed a sophisticated mathematical model – known as a nonlinear, time-inhomogeneous stochastic differential equation (SDE) model – specifically tailored to solar photovoltaic (PV) forecasts.

The model uses two key inputs: the official forecast and a time-varying physical upper bound derived from irradiance and system capacity. A Jacobi-type diffusion keeps all simulated trajectories within feasible limits: never below zero and never above the time-dependent bound.

Crucially, the researchers proved the existence and uniqueness of a strong solution under a mild condition on a time-varying mean-reversion term, ensuring the model is mathematically well-posed.

“We don’t change the forecast; instead, we quantify its uncertainty within the space of possible paths,” Tempone explains. “This gives operators physically consistent scenarios that build on the official forecast and capture the correlation pattern seen in past forecast errors.”

The approach refines probability estimates by using likelihood-based calibration and a tailored kernel-smoothed estimator of the transition density enhanced by a control-variate coupling. In practice, the team compares the transition density using two statistical proxies – beta and truncated normal – which enables the researchers to select the better fit.

The team tested the approach on Uruguay’s 2019 national solar production and forecast data. The model generated complete scenario paths of possible solar output that were consistent with historical error correlations, strictly centred on the official forecast, and that respect physical constraints. The case study also estimated the daily maximum PV production in a way that aligns with the model’s time-dependent upper bound.

“Focusing on uncertainty quantification rather than forecast improvement makes the tools flexible and robust for the energy industry,” says Tempone. “Confidence bands around the forecast help schedule reserves, reduce blackout risk, and integrate solar more effectively, while traders and policymakers gain a clearer, unbiased picture of risk.”

This solar study is part of the group’s numerical analysis–driven program. The group previously applied the same derivative-tracking SDE framework to wind power uncertainty, as well as in other domains such as medical imaging. “The portability of the mathematics across technologies is a defining strength of the approach,” explains Tempone.