Probabilistic Forecasting of Electric Vehicle Charging in a Time-varying Hierarchical Infrastructure

Guillaume Principato; Yvenn Amara-Ouali; Yannig Goude; Bachir Hamrouche; Jean-Michel Poggi

stat.APOct 9, 2026

Probabilistic Forecasting of Electric Vehicle Charging in a Time-varying Hierarchical Infrastructure

Authors: Guillaume Principato, Yvenn Amara-Ouali, Yannig Goude, Bachir Hamrouche, Jean-Michel Poggi

Organizations: Laboratoire de Mathematiques d’Orsay, Université Paris-Saclay, CNRS, INRIA · Electricité de France R&D · Université Paris-Cité

Abstract

Uncertainty quantification for electric vehicle (EV) charging demand is essential to turn the increasing pressure of EV adoption on power systems into a source of flexibility. We study a large-scale dataset of public charging stations in Scotland, where demand exhibits a hierarchical structure, organized across multiple aggregation levels, from individual charging points to regional and national scales. The charging infrastructure is continuously evolving, with new stations deployed and existing ones occasionally removed, which calls for forecasting and uncertainty quantification methods tailored to this evolving structure. To address these challenges, we combine hierarchical forecasting with recent advances in conformal prediction with missing data. The conformal procedures considered produce valid prediction intervals at every node of the hierarchy and the algorithm CP-MNR-Nested successfully accounts for the time-varying hierarchical structure of the data.

Figures & tables

Panel of Figure 1 — Fig 1: Hierarchical structure of EV charging energy demand in Scotland.

Panel of Figure 2 — Fig 2: The left panel represents the spatial repartition of the charging stations in Scotland (top) and in Glasgow (bottom). Stations are represented by blue dots, and the size of the dot is proportional to the number of charging points within each station. The right panel shows the corresponding time series of daily energy demand.

Indices	Domain	Description
$t$	Days	time index, on a daily basis
$i$	$[m]$	node index, $i=1,\dots,m$ :
		stations, $i=1,\dots,n$
		local authorities aggregate, $i=n+1,\dots,m-1$
		Scotland aggregate, $i=m$
Variables	Domain	Description

Table 1: Basic notations.

Panel of Figure 4 — Fig 3: Time series of daily energy demand at four charging stations in Glasgow City. The figure illustrates three typical node behaviors: Station 410 is active throughout the entire period, Station 406 becomes inactive at the end of 2023, Stations 403 and 386 both exhibit the effect of predictable temporary closures.

Functions	Domain	Description
$\widehat{A}$	$\mathcal{X}\times\{0,1\}^{m}\to\mathbb{R}^{m}$	base predictor
$\widetilde{A}$	$\mathcal{X}\times\{0,1\}^{m}\to\mathbb{R}^{m}$	reconciled predictor
$\widehat{R}$	$\mathbb{R}^{m}\times\{0,1\}^{m}\to\mathbb{R}^{m}$	reconciliation function
Variables	Formula	Description
$\widetilde{\bm{y}}^{(k)}\in\mathbb{R}^{m}$	$\widetilde{A}\bigl(\bm{x}^{(k)},\bm{a}^{(k)}\bigr)=\widehat{R}\Bigl(\widehat{A}\bigl(\bm{x}^{(k)},\bm{a}^{(k)}\bigr),\bm{a}^{(k)}\Bigr)$	reconciled forecasts at day $k$
$\bm{s}^{(k)}\in\mathbb{R}^{m}$	$\bigl\|\widetilde{\bm{y}}^{\,(k)}-\bm{y}^{(k)}\bigr\|\,\,\,\,=\,s\Bigl(\bigl(\bm{x}^{(k)},\bm{a}^{(k)}\bigr),\,\bm{y}^{(k)};\widetilde{A}\Bigr)$	vectorial non-conformity score at day $k$

Table 2: Advanced notations.

Panel of Figure 6 — Fig 4: The CP procedure, adapted from conformal prediction for hierarchical data.

Fig 5: Rolling-window data partitioning. At the first block, the available data is split into a training set

(80\%)

and a calibration set

(20\%)

. The calibration set of indices is drawn once, combining the week preceding the test period and randomly sampled one-week historical blocks. The same partition is then shifted forward by two weeks across subsequent blocks.

Panel of Figure 8 — Fig 6: Boxplots of the node-wise RMSEs of the base forecasts (a) and reconciled forecasts (b) across aggregation levels. The Global entry (c) corresponds to the square root of the average MSE over all nodes in the hierarchy.

Node type	Level	CP	CP-MNR	CP-MNR Nested ⋆	Adaptive CP-MNR
National	$1-\alpha$	40.0% $\pm$ 16.3%	40.0% $\pm$ 16.3%	90.0% $\pm$ 10.0%	0.0% $\pm$ 0.0%
National	$1-2\alpha$	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%
Regional	$1-\alpha$	51.9% $\pm$ 1.2%	51.9% $\pm$ 1.2%	71.2% $\pm$ 1.2%	47.8% $\pm$ 0.9%
Regional	$1-2\alpha$	98.8% $\pm$ 0.5%	98.8% $\pm$ 0.5%	99.1% $\pm$ 0.5%	100.0% $\pm$ 0.0%
Station	$1-\alpha$	38.1% $\pm$ 0.4%	56.2% $\pm$ 0.4%	73.6% $\pm$ 0.6%	57.6% $\pm$ 0.6%
Station	$1-2\alpha$	67.7% $\pm$ 0.3%	93.6% $\pm$ 0.3%	90.8% $\pm$ 0.4%	98.0% $\pm$ 0.1%

Table 3: Validity rates at two coverage levels: the nominal level

1-\alpha=90\%

and a relaxed level

1-2\alpha=80\%

. Results are grouped by aggregation level/conformal procedure, and reported as

\operatorname{mean}\pm 1.96\times\operatorname{standard\,error}

Panel of Figure 10 — Fig 7: Validity rates at the station-level, respectively marginally and conditionally on the activity regime (Low or High) within the region of the station under scrutiny.

Panel of Figure 11 — Fig 8: Coverage and average length of the prediction intervals for Scotland (top left), Glasgow (top right), Station 406 (bottom left), and Station 386 (bottom right). The results are reported as $\operatorname{mean}\pm 1.96\times\operatorname{standard\,error}$ .

Panel of Figure 12 — Fig 9: Prediction intervals for Scotland (top) and Glasgow (bottom).

Panel of Figure 13 — Fig 10: Prediction intervals for Station 406.

Panel of Figure 14 — Fig 11: Prediction intervals for Station 386.

Node type	Reconciliation	CP	CP-MNR	Adaptive CP-MNR
National	OLS	50.0% $\pm$ 16.7%	50.0% $\pm$ 16.7%	80.0% $\pm$ 13.3%
National	rOLS	60.0% $\pm$ 16.3%	60.0% $\pm$ 16.3%	90.0% $\pm$ 10.0%
Regional	OLS	55.9% $\pm$ 2.2%	55.9% $\pm$ 2.2%	46.9% $\pm$ 1.5%
Regional	rOLS	61.6% $\pm$ 2.4%	61.6% $\pm$ 2.4%	47.5% $\pm$ 1.5%
Station	OLS	48.0% $\pm$ 0.4%	42.4% $\pm$ 0.3%	37.7% $\pm$ 0.3%
Station	rOLS	81.0% $\pm$ 0.2%	59.4% $\pm$ 0.3%	57.4% $\pm$ 0.2%

Table 4: Win rate of OLS and rOLS with respect to Direct in terms of prediction interval average lengths. Results are reported as

\operatorname{mean}\pm 1.96\times\operatorname{standard\,error}

Node type	Reconciliation	CP	CP-MNR	Adaptive CP-MNR
National	Direct	10.0% $\pm$ 10.0%	10.0% $\pm$ 10.0%	0.0% $\pm$ 0.0%
	OLS	30.0% $\pm$ 15.3%	30.0% $\pm$ 15.3%	0.0% $\pm$ 0.0%
	rOLS	40.0% $\pm$ 16.3%	40.0% $\pm$ 16.3%	0.0% $\pm$ 0.0%
Regional	Direct	67.2% $\pm$ 0.8%	67.2% $\pm$ 0.8%	62.8% $\pm$ 1.4%
	OLS	53.8% $\pm$ 1.7%	53.8% $\pm$ 1.7%	49.1% $\pm$ 1.6%
	rOLS	51.9% $\pm$ 1.2%	51.9% $\pm$ 1.2%	47.8% $\pm$ 0.9%

Table 5: Validity rates at level

1-\alpha=90\%

for the different reconciliation procedures. Results are reported as

\operatorname{mean}\pm 1.96\times\operatorname{standard\,error}

. The best-performing reconciliation procedure is shown in bold.

Node type	Reconciliation	CP	CP-MNR	Adaptive CP-MNR
National	Direct	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%
	OLS	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%
	rOLS	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%	100.0% $\pm$ 0.0%
Regional	Direct	97.2% $\pm$ 0.7%	97.2% $\pm$ 0.7%	100.0% $\pm$ 0.0%
	OLS	98.8% $\pm$ 0.5%	98.8% $\pm$ 0.5%	100.0% $\pm$ 0.0%
	rOLS	98.8% $\pm$ 0.5%	98.8% $\pm$ 0.5%	100.0% $\pm$ 0.0%

Table 6: Validity rates at the relaxed level

1-2\alpha=80\%

for the different reconciliation procedures. Results are reported as

\operatorname{mean}\pm 1.96\times\operatorname{standard\,error}

. The best-performing reconciliation procedure is shown in bold.

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Accurate forecasting of electric vehicle (EV) charging demand is critical for grid management and infrastructure planning. Yet the field continues to rely on legacy benchmarks; such as the Palo Alto (2020) dataset; that fail to reflect the scale and behavioral diversity of modern charging networks. To address this, we introduce a novel large-scale longitudinal dataset collected across Scotland (2022 2025), which release it as an open benchmark for the community. Building on this dataset, we formulate EV charging demand as a spatio-temporal latent Gaussian field and perform approximate Bayesian inference via Integrated Nested Laplace Approximation (INLA). The resulting model jointly captures spatial dependence, temporal dynamics, and covariate effects within a unified proba bilistic framework. On station-level forecasting tasks, our approach achieves competitive predictive accuracy against machine learning baselines, while additionally providing principled uncertainty quan tification and interpretable spatial and temporal decompositions properties that are essential for risk-aware infrastructure planning.

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