Short-term solar irradiance forecasting using convolutional neural networks and cloud imagery

We used geostationary satellite imagery captured by NOAA's Geostationary Operational Environmental Satellite 8 (GOES8) infrared channels. The satellite imagery data is maintained by the Satellite Data Services (SDS) group at the University of Wisconsin-Madison Space Science and Engineering Center. The data was collected from Multi-format Client-agnostic File Extraction Through Contextual HTTP, a web-based API, maintained by SDS (SSEC UW-Madison 2020). Each GOES 8 image has a 2125 × 825 pixel resolution per channel. In this study, 4 infrared channels are used: channels 2–5, which operate on wavelengths: 3.9, 6.8, 10.7, and 12 µm. The spatial resolutions vary between 4 and 8 km. Figure 1 shows a sample of the input data from all 4 channels.

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We collected solar irradiance data for 12 different stations in 7 states throughout the CONUS (figure 2): Florida (FL), Georgia (GA), Mississippi (MS), New Mexico (NM), North Carolina (NC), Texas (TX), and West Virginia (WV). Details of the 12 stations are given in table 1. The irradiance data have a 5 min temporal resolution, with the exception of Cape Canaveral FL data, with a 6 min temporal resolution. The geographic characteristics of the study sites are summarized in table 1 and figure 2, respectively. Moreover, the correlation of GHI between different case study sites are depicted in figure 3.

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Data is available from the start of 1996 until the end of 2012. To demonstrate the applicability of our proposed approach, we train and validate our models using data from 1999 to 2001 as it is the time period with the most complete data across all the sites. However, the model can easily be extended to more recent years, which will likely result in higher predictive accuracy owing to improved resolution of satellite imagery over time.

Both GOES8 images and GHI data are preprocessed to facilitate statistical modeling. GOES8 images are downscaled into 256 × 256 resolution using bicubic interpolation to reduce computation time. All 4 image channels are then converted into one 256 × 256 × 4 tensor (i.e. matrix)—each entry in the tensor corresponds to one pixel in each of the 4 channels. Input data are also normalized to values in [0,1]. Missing data—caused by technical failure, regular maintenance of GOES8 systems, or missing/cracked imagery—are removed. In preprocessing of GHI data, night-time values—GHI values less than 10 W m⁻²—are removed. Similarly, unreliable irradiance measurements are removed based on quality control flags ⁴ provided by NREL's CONFRRM data (SERI 1988).

ANNs are a statistical learning technique inspired by the way neurons work in the human brain. They consist of a series of neurons (i.e. nodes) which activate each other in parallel or in sequence. Each neuron and its connections are called a perceptron. Mathematically, this consists of a combination of multivariate linear regressions followed by a non-linear transformation. The non-linear transformation—called an activation function—is typically applied on the output of a perceptron.

A 'vanilla' version of a neural network algorithm consists of multiple layers of perceptrons. A neural network can learn the complex non-linear mapping between input and target variables by stacking multiple non-linear functions. A mathematical representation of a layer of a fully connected network can be written as:

$$\begin{eqnarray} &&f_{i+1}(X) = a(W_{i+1}\times f_i(X) + b_{i+1}) \end{eqnarray} \tag{ 1 }$$

where f_i (X) is the output of ith layer given input X; W_i and b_i are the weight tensor and bias of ith layer respectively; and a(·) is a non-linear activation function.

The output of the last layer (f_N (X)) is the prediction value given an input X, where N is the number of hidden layers of a model. A neural network with more than 2 hidden layers is considered a 'deep' network. The training of a neural network is done by finding the weights which minimize the distance between the prediction values and the ground-measured observations (distance is represented by a loss function).

Convolutional neural networks (CNNs) are a class of deep neural network commonly used for image analysis (Bishop 2006). CNNs take in an input image, assign importance to visual features in the image, and ultimately make predictions about the image's contents. They are typically used for image classification, object detection and segmentation (LeCun et al 2015), and have been used in several environmental applications including urban expansion detection (He et al 2019), biophysical modeling of crop growth (Lin et al 2020), and environmental and water management (Sun and Scanlon 2019).

CNNs differ from ANNs in the way in which the layers connect. Contrary to ANN, in a CNN, a neuron of a convolutional layer takes only a portion of the output of the previous layer as its input; this portion is called a receptive field. Within a layer, each neuron has a similar sized receptive field generated with the same weights and biases, called a kernel. Convolutional layers generally utilize a linear kernel followed by a non-linear activation. To generate the output of the layer, it slides the kernel throughout the input. This sliding is called a convolution operation. A formula of a 2D convolutional layer can be written as:

$$\begin{align} f_{i+1}(x, y) & = a((f_i * h)(x,y)) \nonumber\\ & = a\Big(\sum_{t_x}\sum_{t_y} f_i(x-t_x, y-t_y)h(t_x,t_y)\Big) \end{align} \tag{ 2 }$$

where (x, y) is one entry in the output tensor of (i + 1)th layer, h is a two-dimensional kernel consisting of $|\{t_x\}|\times|\{t_y\}|$ weights (which are shared in the layer for every (x, y)), f_i is the output of ith layer, and a(·) is an activation function.

As a way to maintain data dimensions, padding (usually zero padding) and stride values can be adjusted. Padding refers to adding extra (non-informative) pixels around the input images to keep size of output after a convolution, and stride refers to a step size that a kernel slides. A pooling layer, which performs a similar operation to a convolution but applies predefined summary function instead of convolution kernel, may also be applied after a convolutional layer to reduce dimension and reduce over-fitting. Two most common pooling methods are max-pooling and average-pooling. Max-pooling takes a maximum function for the summary function and average-pooling takes average function. As a result of this convolution operation, the (x, y)'s value of the output tensor can hold the information of multiple neighboring pixels in the input image (i.e. receptive field). By stacking multiple convolutional layers, each neurons of the output of the last convolutional layer encompasses a very large receptive field with respect to the input image.

The CNN model structure used in our proposed C-GHI model is based on the VGG16 model (Simonyan and Zisserman 2014), with adjustments made for the number of layers and regularization parameters for each study site. We build separate models for each region and time horizon, using the same network architecture briefly described below. The convolutional layers receive the input image in series, each of which with filter size 32, 64, 64, 128, 128, and, 128. The convolutional kernel size is 3, and the stride is 1, in all layers. Max-pooling is applied after every other layer with kernel size 3, 3, and 2 respectively. After the convolutional layers, the data is passed through three fully connected layers—with 1024, 1024, and 1 nodes respectively. Rectified linear unit, a type of non-linear activation function, is also applied for every layer except for the last fully connected layer. Additional L2-regularization with respect to all weights is added to the loss function to reduce the risk of over-fitting. In short, the final loss function is $MSE + \lambda \sum\|W\|_2$, where MSE stands for mean squared error. Lastly, Adam Optimizer is utilized (Kingma and Ba 2015) to tune the model weights, given the input dataset.

As a posterior analysis, we apply feature visualization techniques to the fitted models to visualize model performance at different spatial and temporal scales. Specifically, we harness saliency maps for the model inference. Saliency maps help visualize the gradient of the cost function with respect to each pixel of the input (Simonyan et al 2013). In other words, saliency maps involve calculating $[\partial E^j/\partial X]_{X = X^j}$, where E^j is the prediction error of jth observation, X is the variable indicating an image, and X^j is the jth image. The gradient is then projected onto the original image plane. Saliency maps (figure 4) show the effect of a unit change in a pixel on the prediction accuracy. If the gradient of a pixel is higher, it indicates that the pixel is more informative for the particular prediction.

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In our experiments, two model performance metrics are used: mean-squared error (MSE), and root mean-squared error (RMSE). For parameter tuning in fitting each model, we use MSE, mathematically represented as:

$$\begin{eqnarray} &&MSE = \frac{1}{N}\sum_{i = 1}^{N} (y^i-\hat{y}^i)^2 \end{eqnarray} \tag{ 3 }$$

where yⁱ and $\hat{y}^i$ are the ground-measured and predicted GHI of ith observation respectively.

RMSE is used in comparing the predictive performance of different models. We report RMSE as it is a widely applied metric for solar irradiance prediction, particularly in exogenous predictive models. Additional performance metrics of MAPE, R², and nRMSE are also calculated and included in the appendix tables A3–A5.

Since two different GHI prediction models cannot be directly compared if the prediction regions, time, and temporal horizons do not match, and there is no agreed upon benchmark dataset, we compare our models' performance against the persistence model which is represented as:

$$\begin{eqnarray} &&\hat{y}(t) = y(t - T) \end{eqnarray} \tag{ 4 }$$

where $\hat{y}(t)$ is the predicted GHI at time t and y(t) is the observed GHI at time t, and T is the prediction interval. In essence, the persistence model assumes all future predictions will be equal to the last observed value at T time before.

We hypothesize that the model prediction quality is the result of the C-GHI locating the prediction region and adjusting the size and scope of the input data. That is, given the whole US satellite imagery and historical GHI data for a specific point, the learning process encourages the C-GHI to seek out the prediction location and a sufficient surrounding area to optimally map an input to an output.

To test this hypothesis, we leverage saliency maps (introduced in section 3.3) to understand the geographic regions which most contribute to the GHI prediction. An example is illustrated in figure 4. In these maps, lighter colors indicate regions of higher gradients with respect to the prediction and thus a greater contribution to GHI prediction. In each location, the brighter region becomes wider and more diffuse around the correct region as the prediction horizon grows.

The C-GHI demonstrates high predictive accuracy when the entire CONUS imagery is utilized to predict local GHI values. In order to understand which areas of the country contribute to a particular regional GHI prediction, we perform the same analysis, but crop the input images to a specific window around the prediction location. We apply fixed-location deterministic cropping as a data pre-processing step, given the target prediction region's location. By fitting and comparing models for different cropping sizes and prediction horizons, we aim to understand the extent to which increased regional information improves prediction accuracy across different prediction horizons.

What follows is an example of the cropping procedure at the AU location for one channel. First, we calculate the location of the AU site (lat, lon) = (30.29^∘ N, 97.74^∘ W) within the satellite imagery. Each image is then centered around the AU site and cropped into squares with side lengths of: 10, 20, 40, 60, 100, 140, 180, 220, 260, 300, 360, 420, and 512 pixels. All cropped images are resized to 256 × 256 using bicubic interpolation to preserve the network architecture and maintain the number of parameters within the C-GHI. Finally, each model is separately trained per cropping window size and time horizon using the same dataset. We show the results of the predictions in figure 5.

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Results indicate that prediction error decreases rapidly in 5 min and 1 h prediction as a result of increasing cropping window size when window size is small, and prediction error levels off as the window size becomes very large. Additionally, the effect of increasing the cropping window size become weaker in longer-term prediction. In figure 5, for 5 min ahead prediction in AU, the cropped-image model surpasses the error of full-size-image model at approximately 30 000 pixels—the result of a cropping window of 180 × 180. For 1 h ahead prediction, this threshold increases to 150 000 pixels (roughly 380 × 380). For 1 d ahead prediction, the threshold is approximately 230 000 pixels (480 × 480). As all cropped-image models surpass the predictive quality of full-size-image models, we conclude that regional image cropping is beneficial to GHI prediction, particularly in short-term prediction.

Additionally, we find that the predictive performance of models with noisy saliency maps (i.e. 5 min ahead prediction in AU and CL) is substantially improved when input images are cropped. We believe this is due to the model identifying the prediction location more accurately in longer temporal horizons and the the wider regional cloud information surrounding the target point which can be incorporated. For example, the 5 min ahead prediction in AU had an 8.2% improvement by the method at window size 300 × 300, while the 5 min ahead prediction in BC had only a 4.0% improvement even at 512 × 512. We can see that in figure 4 the saliency map of AU in the 5 min ahead prediction is noisier than that of BC. The ability of the CNN-based model to maintain predictive accuracy at increased time horizons combined with the benefits to model accuracy gained by cropping the input image reveals that the C-GHI is accurately capturing the regional meteorological patterns which contribute to GHI values. The cropping window size required to achieve similar performance to an uncropped model increases as the prediction horizon increases.

Lastly, we test the sensitivity of C-CHI's prediction performance to the degree of cloud cover, focusing on the best performing regions of ED and BC. ED, located in McAllan, Texas, has a semi-arid climate under the Köppen climate classification; it has very hot and humid summers and short and warm winters. BC, located in Daytona Beach, Florida, has a humid subtropical climate with warm and wet summers and cooler and drier winters. These two regions have similar climatological characteristics, but ED is drier and hotter.

To assess the sensitivity of model performance to degree of cloud cover, we first label the days in the model as 'sunny', 'partly cloudy', and 'cloudy'. The labels are chosen based on data from the Weather Underground (The Weather Underground 2020). We looked at daily rather than hourly weather data, following the convention in the solar irradiance prediction literature. The cloud cover labels are selected based on the following criteria: (i) 'sunny' represents days with more than 90% clear sky conditions, or with more than 80% clear skies, with the remaining being partly cloudy; (ii) 'cloudy' represents days with either no clear sky conditions, or when more than 70% of the day is mostly cloudy and/or cloudy; and (iii) 'partly cloudy' represent days that are neither completely sunny nor cloudy.

Table 4 shows the test set performance changes over different cloud cover conditions and prediction time horizons. Results indicate that as the prediction time horizon increases, C-GHI performs better than the persistence model, especially in cloudy and partly cloudy days. We find that persistence model performs well in day-ahead predictions during sunny days, possibly due to high auto-correlation in consecutive sunny days. We also find that predictions with longer lead times tend to overestimate irradiance in cloudy days and underestimate irradiance in sunny days to minimize the total sum of squared error (figure 6). We hypothesize that this helps C-CHI to better fit partly cloudy days which are most frequent in both regions, thus the overall performance benefits.

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Assessing the C-CHI's prediction performance as a function of cloud types is not within the scope of this paper. However, it should be highlighted that future work analyzing the effect of cloud types on CNN-based solar irradiates prediction performance will be of great value to renewable energy planners, operators and regulators.