Diffusion Models for High-Resolution Solar Forecasts

Diffusion models are generative in that they describe a probability distribution $p(\mathbf{x})$, where $\mathbf{x}$ can be a high-dimensional random variable, for example, an image or the state of a physical system. In general, learning a probability distribution over a high-dimensional space from data is extremely challenging because the number of required samples grows exponentially with dimensionality. However, learning is possible when the data lives on low-dimensional manifolds, such as natural images and other data with spatial and temporal structures.

The last decade has seen a number of innovative deep-learning approaches to address the problem of learning a generative model in high dimensions. These include autoregressive models (Uria et al., 2016), GANs, VAEs, and flow-based models (Rezende & Mohamed, 2015; Dinh et al., 2016), all of which have been used heavily in scientific applications. Score-based diffusion models are a special case of reversible generative models, which parameterize a one-to-one mapping between a known distribution and the data distribution. Previous reversible generative models such as NICE (Dinh et al., 2014), Real NVP (Dinh et al., 2016), and GLOW (Kingma & Dhariwal, 2018) can be trained by directly maximizing the data likelihood, but require significant computation for each parameter training update and scale poorly to large datasets. Score-based diffusion models provide an efficient stochastic gradient learning algorithm by approximating the score function (the gradient of the log probability density $\nabla_{\mathbf{x}}\log{p(\mathbf{x})}$) rather than the probability density function $p(\mathbf{x})$. The score function is easier to learn because local updates can be made during training without the need to ensure the probability density function integrates to one.

A score-based diffusion model consists of a neural network representing a time-dependent function with the same input and output dimensions, $\mathbf{s}_{\theta}(\mathbf{x},t):(\mathbb{R}^{D},[0,T])\rightarrow\mathbb{R}^{D}$, parameterized by $\theta$. For image data, this is typically implemented as a U-Net to model the local structure in images. The data-generating process can be defined by sampling $\mathbf{x}_{T}$ from a standard multivariate normal distribution and then solving the following ordinary differential equation in the reverse time direction to generate a sample at $t=0$, $\mathbf{x}_{0}$:

where $a(t)$ and $\mathbf{b}(\mathbf{x},t)$ are pre-specified drift and diffusion coefficients that determine the shape of the prior. Because the data-generating process is reversible, we can compute the likelihood $p_{\theta}(\mathbf{x})$ of any data point $\mathbf{x}$. But rather than maximizing the training data likelihood directly, it is more efficient to optimize a score-based objective such as the “denoising” objective suggested by Song et al. (2021):

where the outer expectation is over the training data set, the second expectation is over time $t\sim\mathcal{U}(0,T)$, and the inner expectation is over a distribution of sample corruptions in which the training sample $\mathbf{x}_{0}$ is propagated through a stochastic noising process for time $t$ (in practice, this is just $\mathbf{x}_{0}$ plus some Gaussian noise which increases with $t$). The objective is efficient to optimize using stochastic gradient descent. At each iteration, samples are drawn from each of the three distributions, and parameters are updated with gradient descent. The result is a simple, intuitive, training algorithm: Gaussian noise is added to data samples and the parameterized model is trained to denoise the samples. This objective has been shown to upper bound the negative log-likelihood of the training data (Song et al., 2021) and thus the model learns to approximate a high-dimensional joint distribution. In this work, we generate many samples from such a model to provide a probabilistic forecast.

Solar irradiance is a particularly interesting forecasting application since better forecasting can reduce the risks to electricity grid stability that are associated with increasing grid penetration of non-dispatchable solar photovoltaics (PV). The last few years have seen a number of applications of computer vision techniques to solar irradiance forecasting, including optical flow models (Wood-Bradley et al., 2012) or more recently deep learning (Zhang et al., 2018; Sun et al., 2018; Hart et al., 2020; Berthomier et al., 2020; Kellerhals et al., 2022), including GANs (Nie et al., 2021). However, none of these approaches provide fully-probabilistic predictions that have the ability to estimate the risk of many PV sites being cloudy at once; this work is the first to propose a method for estimating this risk in a tractable way.

We evaluate our approach on the Hawaiian island of Oahu (population 1,000,000 and the site of ICML 2023), where the problems and opportunities of renewable energy are amplified. The isolation of each Hawaiian island’s electrical grid and a dependence on imported petroleum result in high electricity prices, with Hawaii consumers paying $0.30/kWh in 2021, twice that of customers in California (geopolitical events in May 2022 caused this to increase 34% to $0.42/kWh (Bureau of Labor Statistics, U.S. Department of Labor, 2022)). On the other hand, Hawaii’s low latitude makes solar an attractive alternative energy source, and Honolulu has more PV production per capita than any other city in the United States, with PV systems installed on nearly a third of single-family residential homes (Penn & Fremson, 2022; Hawaiian Electric, 2022). This results in large spikes in energy demand when clouds pass over solar generation sites and dense population areas. Predicting these events could enable grid operators to mitigate their impact through demand response strategies such as starting new generators, charging batteries, and turning off water heaters and HVAC systems.

In experiments we use two cascaded diffusion models (Ho et al., 2021), where the first model generates $64\times 64$ pixel images, then a second model performs super-resolution to produce $128\times 128$ pixel images (Fig. 3). Both diffusion models are conditioned on the same coarse-resolution atmospheric variables output by numerical simulations. At training time, the model is conditioned on atmospheric variables from ERA5 reanalysis data. The trained model is then evaluated for two different applications: (1) historical cloud cover prediction by conditioning on held-out reanalysis data; (2) future cloud cover prediction by conditioning on forecasts from GFS. GFS and ERA5 use the same underlying physics models, so the same atmospheric variables are available at approximately the same resolution.

A U-Net architecture (Ronneberger et al., 2015) is used for each diffusion model (Figure 4). It utilizes the skip connections to preserve the context at different spacial levels, while the network progressively encodes the image to capture information on wider range. The specific U-net configurations largely follow details provided in Ho et al. (2021) for the 64$\times$64$\rightarrow$128$\times$128 ImageNet model. Briefly, each downsampling and upsampling block adjusts the spatial dimensionality by a factor of 2 while the channel count is modified by a multiplier $M_{n}$. The atmospheric conditioning information is injected as a flat vector into each block in the network. Our full cascaded diffusion model implementation is based on the publicly available code by Wang (2022).

Three types of data are used for experiments. The model is trained to predict cloud cover images derived from GOES satellite data from reanalysis data. Then this trained model is used to super-resolve GFS forecasts. Model training was performed on data from January 2019 – June 2021, and all model evaluations were performed on data from July 2021 – June 2022.

On the held out test set, we evaluated predictions from the diffusion model using the total cloud cover derived from satellite data as ground-truth. Diffusion model predictions were produced by sampling many cloud cover images for a given timestep (N=45), conditioning on the atmospheric variables provided by ERA5, and taking the mean image. The root mean squared error (RMSE) was used to measure prediction error.

The diffusion model prediction error was compared to two baseline prediction methods: (1) the coarse resolution total cloud cover from ERA5, and (2) a high resolution mean over all satellite-derived cloud cover images in the training set from the same time of day; we refer to this as the persistence model. The diffusion model takes the former as an input, but can improve upon this prediction by incorporating learned patterns that account for local topography. This can be clearly seen in Figure 6, where the samples tend to have clouds accumulated at the mountainous regions of the island. The persistence model also includes these features, but is unable to account for any variations in weather. The diffusion model accounts for both of these factors and has lower RMSE (Table 1). A paired t-test between the RMSEs of the diffusion model and those of the persistence model was conducted because the mean RMSE values were close; the differences were statistically significant ($p<.01$) in both experiments, since the test dataset is a large sample.

The diffusion model was then conditioned on GFS data to generate high-resolution forecasts for 11 am local time each day. In this experiment, we used N=90 samples from the diffusion model, as additional samples continued to increase prediction accuracy (Figure 5), and samples are computationally inexpensive, taking only seconds to generate. Once again the mean of the diffusion samples had lower RMSE than both the coarse resolution forecast it was conditioned on (GFS), and the time-conditioned persistence model.