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Spatio-temporal graph neural networks for multi-site PV power forecasting

Jelena Simeunović, Baptiste Schubnel, Pierre-Jean Alet, Rafael E. Carrillo

arXiv:2107.13875v2cs.LGeess.SP

TL;DR

Fine-resolution PV forecasting is challenging because local cloud dynamics create variability and NWP-based methods may be too coarse or costly. This paper models multi-site PV production as graph signals and introduces production-only GCLSTM and GCTrafo architectures. Across Swiss real and synthetic datasets, the proposed models outperform state-of-the-art multi-site methods over six-hour horizons and single-site NWP methods up to four hours ahead.

  • Problem

    PV power forecasting must capture local variability at fine temporal and spatial resolution, while NWP-based approaches may have coarse resolution and costly data requirements.

  • Method

    The paper models multi-site PV production time series as graph signals and introduces production-only GCLSTM and GCTrafo graph neural networks.

  • Results

    The proposed models outperform state-of-the-art multi-site forecasting methods over six-hour horizons and outperform NWP-based single-site methods up to four hours ahead.

  • Takeaways & Limitations

    PV production networks can serve as virtual weather stations for deterministic multi-site forecasting using production data alone.

  • Takeaways & Limitations

    Graph convolutions use only the K closest neighbors, potentially excluding important distant predictors when regional cloud advection dominates.

Abstract

from arXiv · show

Accurate forecasting of solar power generation with fine temporal and spatial resolution is vital for the operation of the power grid. However, state-of-the-art approaches that combine machine learning with numerical weather predictions (NWP) have coarse resolution. In this paper, we take a graph signal processing perspective and model multi-site photovoltaic (PV) production time series as signals on a graph to capture their spatio-temporal dependencies and achieve higher spatial and temporal resolution forecasts. We present two novel graph neural network models for deterministic multi-site PV forecasting dubbed the graph-convolutional long short term memory (GCLSTM) and the graph-convolutional transformer (GCTrafo) models. These methods rely solely on production data and exploit the intuition that PV systems provide a dense network of virtual weather stations. The proposed methods were evaluated in two data sets for an entire year: 1) production data from 304 real PV systems, and 2) simulated production of 1000 PV systems, both distributed over Switzerland. The proposed models outperform state-of-the-art multi-site forecasting methods for prediction horizons of six hours ahead. Furthermore, the proposed models outperform state-of-the-art single-site methods with NWP as inputs on horizons up to four hours ahead.

I. INTRODUCTION

Fine-resolution PV forecasting is difficult because local cloud dynamics create high variability, while NWP-based methods may be too coarse or costly. The paper therefore develops production-only graph models that exploit spatio-temporal relations across PV sites and evaluates them against established multi-site and NWP-based single-site methods.

  • Motivation: Local weather and cloud dynamics make solar generation highly variable, hindering accurate PV power forecasts.The challenge is especially relevant to grid applications such as energy trading and congestion management.
  • Motivation: Existing forecasting approaches use cameras, satellite imagery, or NWP, but their cost, spatial coverage, or short-horizon performance limits applicability.Cameras are expensive and strongest only up to one hour, while satellite and NWP approaches can perform poorly at short horizons and high spatial resolution.
  • Research question: The paper asks whether state-of-the-art forecasting can be achieved using only past PV production data, avoiding additional exogenous inputs.This question follows from the cost and processing burdens associated with local weather information.
  • Research gap: Graph-based modeling is motivated by cloud movement across neighboring stations and by the need to represent spatial relations that image-based methods do not fully exploit.Prior work includes recurrent, convolutional, and attention-based models for temporal or spatio-temporal correlations.
  • Contribution: The proposed GCLSTM and GCTrafo models represent PV time series as graph signals and use graph convolutions with recurrent or attention-based temporal structures.Both models target deterministic multi-site forecasting and rely entirely on production data.
  • Evaluation: Evaluation covers six-hour-ahead forecasting over a year using 304 real and 1000 simulated PV systems in Switzerland, plus NWP-based single-site comparisons.The study compares the proposed models with state-of-the-art multi-site methods and single-site methods at two Swiss sites.

II. PROBLEM FORMULATION

The paper formulates PV production across stations as graph signals and uses localized spectral graph convolutions to process spatial dependencies efficiently. The formulation extends graph convolution to multivariate node features and learnable neural-network weights.

  • A. Graph convolution: A PV network is represented as a weighted undirected graph whose nodes are stations and whose adjacency matrix encodes inter-station relationships.The graph Laplacian is L := D − A, with D containing node degrees.
  • A. Graph convolution: A graph signal assigns a scalar value to each node; here, it is the vector of PV production values across stations at one time.The graph Fourier transform is defined using the Laplacian eigenvectors.
  • A. Graph convolution: Spectral graph convolution filters signals in the graph Fourier domain using a function of the Laplacian eigenvalues.The paper introduces spectral and spatial graph convolution as two broad categories.
  • A. Graph convolution: Chebyshev polynomial expansion parameterizes the spectral filter using coefficients θ_k and scaled Laplacian eigenvalues.The scaled eigenvalue matrix is ˜Λ = 2Λ/λmax − I_N, and T_k(˜Λ) contains the corresponding Chebyshev polynomials.
  • A. Graph convolution: The polynomial filter is evaluated through the scaled Laplacian ˜L = 2L/λmax − I_N, avoiding explicit eigendecomposition in the practical computation.The formulation uses the graph Laplacian directly.
  • A. Graph convolution: Polynomial order K makes the graph convolution spatially localized to nodes within K hops of the central node.The same localization reduces computation from O(N^2) to O(K|ε|).
  • A. Graph convolution: For multivariate graph signals, scalar filter coefficients are replaced by learnable weights W ∈ R^(K×f_out×f_in) connecting input and output features.The dimensions f_in and f_out denote the numbers of input and output features.
  • A. Graph convolution: These graph-convolution operations are used as neural-network layers with learnable parameters analogous to standard CNN weights.The operation supports multiple feature channels at each graph node.

B. Multi-site time-series forecasting on graphs

The forecasting problem treats simultaneous PV production across stations as a graph-valued time series. Future multi-site production is predicted from past observations by minimizing error over the full forecast horizon.

  • B. Multi-site time-series forecasting on graphs: Clouds act as a dynamic mask, allowing dense PV networks to provide information about regional diffusion and advection through past production data.Stations become graph nodes, and edge weights measure expected correlations between sites.
  • B. Multi-site time-series forecasting on graphs: Each PV station is a graph node carrying a temporal production signal, with edges representing spatial proximity or other inter-site relationships.Typical edge-weight choices include Pearson correlation, distance correlation, and kernel-based measures.
  • B. Multi-site time-series forecasting on graphs: The model forecasts the next H production vectors from M past observations across all N stations.The target is the sequence ˆp(t), …, ˆp(t + H − 1) produced by a parametric estimator f_β.
  • B. Multi-site time-series forecasting on graphs: Training estimates parameters β by minimizing prediction error over the entire forecast horizon.The fitting times are drawn from the training set T of past observations.

III. GRAPH CONVOLUTIONAL FORECASTING MODELS

The forecasting models use encoder-decoder sequence-to-sequence architectures with spectral graph convolutions to combine temporal modeling and spatial information. GCLSTM uses recurrent graph-convolutional cells, while its inputs include production and clear-sky irradiance features.

  • III. GRAPH CONVOLUTIONAL FORECASTING MODELS: Both proposed models encode M past observations and decode H future observations in a shared sequence-to-sequence structure.The encoder processes historical data, and the decoder predicts the future sequence.
  • A. Graph convolutional long-short term memory neural network: GCLSTM combines recurrence and spectral graph convolution in both encoder and decoder to model temporal and spatial correlations jointly.The decoder is initialized with the encoder’s final state and an MLP maps decoder outputs to predicted production.
  • A. Graph convolutional long-short term memory neural network: In GCLSTM cells, graph convolutions replace matrix multiplications in the LSTM gates, allowing signals to diffuse across neighboring nodes.The cell and hidden states are represented across N nodes and the latent dimensions.
  • A. Graph convolutional long-short term memory neural network: The graph-convolutional recurrent updates use learnable filters whose dimensions depend on input features, latent dimensions, and Chebyshev order K.The graph-convolution operation and Hadamard product define the gate computations.
  • A. Graph convolutional long-short term memory neural network: Figure 1 depicts the encoder-decoder graph-convolutional LSTM architecture.Its role is to show how the historical encoder and forecasting decoder are arranged.
  • A. Graph convolutional long-short term memory neural network: The graph adjacency is initialized with k-nearest neighbors, while nonzero sparse-Laplacian entries are also learned during training.This lets the model capture very local site-specific relationships.
  • A. Graph convolutional long-short term memory neural network: Encoder inputs combine current production, rolling mean power, and deterministic clear-sky irradiance signals.Clear-sky irradiance is computed from node geographic coordinates using the Ineichen and Perez model.

B. Graph convolutional transformer

GCTrafo combines temporal convolutions, graph convolutions, and attention to embed each site's variation features in spatio-temporal context. Its decoder omits graph convolution and uses clear-sky irradiance and rolling-mean inputs.

  • Overview: The architecture is the Graph Convolutional Transformer presented for spatio-temporal PV forecasting.The paper identifies this model as GCTrafo and illustrates its architecture in Figure 2.
  • Encoder: GCTrafo's encoder applies 1D convolutions to the input sequence before graph-convolutional attention embeds node features in spatio-temporal context.Queries, keys, and values are created by graph convolutional layers before softmax dot-product attention.
  • Encoder: Multi-head attention duplicates the convolution and attention layers so different heads can focus on different input patterns.The head outputs are concatenated and passed to a final linear layer to produce the encoded sequence.
  • Decoder: The decoder resembles the encoder but removes graph convolution because its clear-sky irradiance and rolling-mean inputs are not expected to benefit from neighboring-node propagation.The decoder directly uses the encoded vector as the attention value.

IV. EXPERIMENTAL RESULTS

The experimental-results section evaluates GCLSTM and GCTrafo on two datasets for both multi-site and single-site forecasting, after describing the experimental setting.

  • Experimental scope: GCLSTM and GCTrafo are evaluated on two datasets for multi-site and single-site forecasts.The section first describes the experimental setting and then presents the results.
  • Section organization: The experimental setting is described before the forecasting results are presented.
  • Forecasting tasks: The evaluation covers both multi-site and single-site forecasting tasks.

A. Datasets

The study uses real and synthetic PV datasets distributed across Switzerland and compares its models with graph, convolutional, and NWP-based forecasting baselines. Dataset preparation includes site-wise normalization and interpolation of benchmark weather data.

  • Datasets: The real dataset contains 304 PV plants across Switzerland, while the synthetic dataset contains 1000 statistically matched simulated plants.The synthetic plants match the real dataset in location density, size, orientation, and pitch angles.
  • Baselines: Multi-site forecasting is benchmarked against STAR, a graph-based autoregressive model, and STCNN, a non-graph convolutional model using geographic plant rearrangement.STAR selects relevant plants with group Lasso, whereas STCNN applies 2D convolutions to geographically ordered plants.
  • Baselines: Single-site comparisons use SVR and EDLSTM, both incorporating NWP inputs such as irradiance and temperature.EDLSTM also uses clear-sky global irradiance and past weather and PV-site observations.
  • Spatial distributions: Figure 3 colors encode each site's peak production across the spatial distributions of the datasets.
  • Preprocessing: Power data for GCLSTM, GCTrafo, and STCNN are normalized at each node by that node's maximum training-year production.STAR uses a separate normalization scheme to extract daily mode profiles.
  • Preprocessing: Benchmark NWP data are interpolated to 15-minute resolution without gaps and normalized with min-max scaling.Polynomial interpolation is used for GFS data, while sample-and-hold interpolation is used for Meteotest data.

D. Training

Models are trained and evaluated using year-based or rolling-window procedures, with NRMSE and NMAE measuring site-level forecast errors while excluding nighttime periods.

  • Training: All methods except STAR train on 2016 data and evaluate on 2017 data, whereas STAR refits every two weeks using a rolling procedure.STAR coefficients are fitted over two months and used to predict the following two weeks.
  • Evaluation and metrics: The evaluation uses peak normalized root mean-squared error and averaged normalized mean absolute error at each site and forecast horizon.
  • Evaluation and metrics: NRMSE and NMAE use ground-truth and predicted site power, normalized by each site's maximum evaluation-period power.The evaluation interval is the 2017 year, and T denotes its number of time steps.
  • Evaluation and metrics: Nighttime observations are excluded from error computations.

F. Results

Across multi-site and single-site evaluations, the proposed graph-based models perform strongly at short and intermediate horizons, while their relative advantage depends on forecast horizon, dataset, and weather-data availability. Performance remains stable across site isolation up to 40 km, but long-horizon forecasts can exhibit sunrise bias and NWP-based methods can become preferable.

  • Multi-site forecasting: GCLSTM achieves the lowest error across the full six-hour horizon on the real dataset, while GCTrafo leads from four to six hours on the synthetic dataset.On the synthetic dataset, GCLSTM has the lowest error through four hours.
  • Multi-site forecasting: GCTrafo has a lower error slope than other methods beyond one hour, despite STAR performing better during the first hour.The authors attribute this longer-horizon behavior to attention over spatial and temporal information rather than recurrent fading memory.
  • Metric comparison: At six hours on the synthetic dataset, GCLSTM has lower NMAE while GCTrafo has lower NRMSE.NRMSE weights large errors more heavily because it uses squared errors, whereas NMAE expresses error as a percentage of yearly production.
  • Spatial robustness: For real-data sites, NRMSE remains comparable across neighbor distances: 3-hour values are 10%-19% below 5 km and 11%-17% at 30-43 km.The analysis reports no performance drop for sites up to 40 km from their closest neighbor.
  • Single-site forecasting: For Bern and Bätterkinden, GCLSTM outperforms other methods from one to four hours, while EDLSTM and SVR using NWP are better at five to six hours.Higher temporal resolution of weather data is associated with higher forecast accuracy at Bätterkinden.
  • Illustrative forecasts: Graph-based methods outperform EDLSTM during cloudy days at one hour, but can show a six-hour sunrise bias because nighttime inputs contain only zero and recent average production.Once non-zero daytime production enters the graph models, their forecasts begin to adjust.

V. CONCLUSIONS

The paper introduces GCLSTM and GCTrafo for deterministic multi-site PV forecasting and compares them with state-of-the-art methods. Across real and synthetic datasets, the models achieve strong six-hour forecasting performance, while remaining scalable to larger networks.

  • GCLSTM and GCTrafo are two graph convolutional neural network architectures for deterministic multi-site PV generation forecasting.
  • 8.3% (GCLSTM) and 8.4% (GCTrafo) average NRMSE errors were obtained on the synthetic dataset over the six-hour horizon.
  • The evaluation used real production from 304 PV plants and synthetic production from 1000 PV plants distributed across Switzerland.
  • 12.6% (GCLSTM) and 13.6% (GCTrafo) average NRMSE errors were obtained on the real dataset over the six-hour horizon.
  • Both architectures were trained on a single GPU for 1000 nodes and can scale to more nodes through multi-GPU computing.
  • Future work targets limited information diffusion through nearest-neighbor graph convolutions, incomplete datasets, unseen nodes, and probabilistic forecasting.

APPENDIX

The appendix reports the principal architecture hyperparameters and optimization settings for GCLSTM, GCTrafo, STCNN, and EDLSTM.

  • GCLSTM used 32 hidden dimensions in encoder and decoder cells and an MLP sized [8, 48, 48].
  • GCTrafo used a 1D-convolutional kernel of size 4, latent spaces of size 8, and 8 attention heads.
  • STCNN used three 2D convolutional layers with channel sizes [128, 64, 32] and kernel size 11.
  • EDLSTM used a latent representation size of 64 followed by an MLP sized [64, 32].
  • STCNN, GCLSTM, and GCTrafo were trained with stochastic gradient descent and Adam optimizer without regularization.
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