Improving Spatial Allocation for Energy System Coupling with Graph Neural Networks

Abstract: In energy system analysis, coupling models with mismatched spatial resolutions is a significant challenge. A common solution is assigning weights to high-resolution geographic units for aggregation, but traditional models are limited by using only a single geospatial attribute. This paper presents an innovative method employing a self-supervised Heterogeneous Graph Neural Network to address this issue. This method models high-resolution geographic units as graph nodes, integrating various geographical features to generate physically meaningful weights for each grid point. These weights enhance the conventional Voronoi-based allocation method, allowing it to go beyond simply geographic proximity by incorporating essential geographic information.In addition, the self-supervised learning paradigm overcomes the lack of accurate ground-truth data. Experimental results demonstrate that applying weights generated by this method to cluster-based Voronoi Diagrams significantly enhances scalability, accuracy, and physical plausibility, while increasing precision compared to traditional methods.

• The paper introduces a heterogeneous graph neural network that learns normalized grid-cell allocation weights from OSM land-use data, population, and regional GVA without substation-level labels.
• Combined with CIVD, the method reduces RMSE by an average of 4.87% across training regions, improves 3 of 4 held-out regions, and achieves a best regional gain of 11.36%.
• The results show that downstream compatibility matters: GNN-generated weights can worsen rigid Voronoi allocation, highlighting the need to model target-facility geography and account for proxy-data limitations.

Problem and motivation

Coupled energy system models frequently operate at mismatched spatial resolutions, and converting aggregate quantities (e.g., regional electricity demand) into inputs for models defined over discrete nodes such as substations remains a persistent "granularity gap" problem. Unlike classical spatial disaggregation, which produces continuous high-resolution rasters, this allocation task must distribute a total volume across a defined set of target entities. Prior work by the same group combined Grid Point Modeling (GPM) with Clustering Induced Voronoi Diagrams (CIVD) to disaggregate macro-regional totals onto fine grid points before assigning them to substations. That approach, however, computes grid point weights statically from a single dominant land use type, so all points of the same class receive identical weights, and it cannot integrate multi-source heterogeneous geospatial information or capture nonlinear relationships between geographic features and demand.

Method

The paper casts weight generation as a heterogeneous graph learning problem with four components: a Graph Builder, a Graph Encoder, an Edge Weight Predictor, and a self-supervised loss.

Graph construction: Source nodes represent macro-regions, featurized by population and Gross Value Added (GVA) by industry; agent nodes are the centers of roughly 50,000 uniform grid cells per region, featurized by land use area percentages and one-hot encodings of the dominant land use type derived from OpenStreetMap (OSM). Edges connect agent nodes to the source region whose polygon contains them, with bidirectional directed edges enabling information flow in both directions.

Encoding and prediction: Node features are projected into a shared latent space via type-specific linear layers, then processed by stacked Heterogeneous Graph Transformer (HGT) layers, which learn relation-specific attention weights. The Edge Weight Predictor computes a gated distance cost $c(s,a) = g(s,a)\cdot\|h_s - h_a\|_2$, where $g(s,a)$ is a sigmoid-bounded MLP gating score over concatenated embeddings, followed by temperature-scaled grouped softmax normalization so that weights per source sum strictly to 1. The full pipeline is differentiable end-to-end.

Self-supervised objective: Because ground-truth loads at micro-unit resolution are unavailable, training relies on a domain assumption: the sectoral composition of GVA and population in a macro-region should correlate with the distribution of energy consumption across functional land use categories. The model reconstructs each region's normalized socioeconomic distribution vector by weighted summation of agent-node one-hot land use vectors using predicted weights $w_{sa}$, and minimizes the summed Kullback–Leibler divergence against the true macro-distribution. No substation-level data are used as supervision.

Allocation stage: The learned weights feed the CIVD method, which clusters substations into load centers, assigns each weighted grid point to its nearest load center, and distributes cluster totals evenly among member facilities.

Results

Experiments use the Great Britain primary substation peak-demand dataset across 16 ITL regions (12 training, 4 held-out test), with London treated separately as an ultra-dense stress case. Baselines are plain Voronoi Diagram (VD), VD-GPM, CIVD, and CIVD-GPM.

Method	Training regions improved	Test regions improved	Best single-region gain
CIVD-GNN-GPM vs. CIVD-GPM	12/12	3/4	−11.36% RMSE (TLC1)
VD-GNN-GPM vs. VD-GPM	11/12	0/4	−5.55% RMSE (TLG2)

On the training set, CIVD-GNN-GPM reduces RMSE relative to CIVD-GPM in every region, averaging 4.87%, with the largest improvement in TLC1 (8.882 → 7.873). On the test set, it improves three of four unseen regions (e.g., TLE3: 7.479 → 6.870, −8.14%), while TLD4 shows a marginal degradation of 0.43%. Notably, these gains are achieved without any ground-truth supervision, indicating that the model learns physically meaningful allocation patterns purely from open-source OSM features and macro-socioeconomic statistics.

A key negative result concerns module compatibility: when paired with rigid geometric VD partitioning, GNN-generated weights can hurt performance. In London, GNN-GPM under VD raises RMSE from 16.234 to 17.013 (−4.80%); on the entire test set, VD-GNN-GPM degrades in all four regions. The authors attribute this to error amplification—sharp, accurate weight distributions misaligned with rigid Voronoi boundaries assign high-load areas to suboptimal facilities, whereas fuzzy average allocation is more forgiving. In TLD4 specifically, both GNN and static GPM assign weight to a large "other" land area lacking a central substation; the GNN's sharper distribution amplifies the resulting error. This demonstrates that hybrid pipeline performance depends on inter-module compatibility, not merely on the sophistication of individual components.

Qualitative heat maps support physical plausibility: in London, weights concentrate in the commercial/residential urban core; in TLC1, weights concentrate in industrial and residential areas of the east while agricultural land receives negligible weight.

Limitations and open questions

The authors identify several constraints. Performance depends heavily on input data quality—OSM coverage and accuracy—and effectiveness may degrade in data-sparse regions. More fundamentally, the self-supervised objective uses macro-level socioeconomic proxies to supervise micro-level electricity allocation, which introduces bias when consumption decouples from those indicators, e.g., for energy-intensive industrial loads or data centers whose demand does not track GVA or population shares. The TLD4 failure mode exposes a second structural gap: because the graph contains no information about target facility locations, the model cannot recognize geographic outliers such as large unpopulated areas without nearby substations. The stated future work—integrating energy-intensity proxies, uncertainty-aware weighting, and incorporating target-node geography directly into the GNN—remains unrealized in this paper, leaving open whether proxy-mismatch errors can be systematically corrected within a self-supervised framework.

Conclusion

This work extends GPM-based spatial allocation by replacing static, single-attribute weighting with a heterogeneous GNN that fuses multi-source geospatial features, trained entirely self-supervisively through reconstruction of macroeconomic distributions. Combined with CIVD, the approach yields consistent RMSE reductions (average 4.87% on training regions, up to 11.36%) and generalizes to most held-out regions without labeled data. The results also carry a cautionary finding: sharper learned weights can amplify error when downstream geometric allocation is incompatible, underscoring that multi-stage allocation frameworks must be designed for inter-module coherence rather than component-wise advancement alone.