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

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

arXiv:2610.13588v1stat.APcs.LGstat.ML

TL;DR

EV charging forecasts span multiple aggregation levels and must account for evolving infrastructure, while point forecasts alone miss demand variability. The paper combines reconciliation and conformal prediction for infrastructure-aware intervals; CP-MNR-Nested⋆ produces intervals valid across infrastructure configurations and is more sensitive to infrastructure activity than the alternatives discussed.

  • Problem

    EV charging forecasting spans hierarchical aggregation levels and changing infrastructure, while point forecasts alone fail to capture demand variability.

    Source: Introduction · pp. 1–2

  • Method

    The approach combines forecast reconciliation with conformal prediction and extends CP-MNR to account for the full infrastructure-activity configuration without separate calibration sets for each configuration.

    Source: §4.2 · p. 11, Discussion and future work · p. 22

  • Results

    CP-MNR-Nested⋆ produces valid prediction intervals for different infrastructure configurations, whereas CP-MNR and Adaptive CP-MNR are less sensitive to activity across the whole infrastructure.

    Source: Discussion and future work · p. 22

  • Takeaways & Limitations

    The resulting intervals reflect uncertainty for the current infrastructure state and can serve as inputs to optimization algorithms.

    Source: Discussion and future work · p. 22

    Separate calibration sets for each possible infrastructure configuration are impractical because their number far exceeds available observations.

    Source: §4.2 · p. 11

Abstract

from arXiv · show

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.

Introduction.

EV charging forecasting must quantify uncertainty across aggregation levels while accounting for a changing network. The study examines hierarchical forecasting and uncertainty methods using public charging data from Scotland.

Introduction.

  • Reliable EV-demand forecasts support power-system decisions, while point forecasts alone omit demand variability and operational needs span stations, regions, and national aggregates.Demand at higher aggregation levels is the sum of demand at lower levels, creating a hierarchy across decision scales.Source: Introduction · pp. 1–2
  • Demand forms a hierarchy because each aggregate equals the sum of lower-level demand, allowing information sharing across levels to improve forecast accuracy.Operational needs range from station forecasts to regional and national aggregates.Source: Introduction · p. 2

EV charging energy demand in Scotland.

  • New stations and temporary or permanent removals change which nodes contribute to aggregate demand, creating forecasting challenges for new stations and current infrastructure state.Demand also shows non-stationarity, including heterogeneous responses to tariffs and a decreasing national trend over the study period.Source: Introduction · pp. 2–3

Introduction.

  • The study considers GAM, XGBoost, and TabICL, plus a simple combination, at all three hierarchy levels; reconciliation and conformal prediction provide complementary hierarchy and uncertainty tools.Reconciliation addresses forecasts that are agnostic to hierarchy, while conformal prediction can be applied post hoc to base predictors.Source: Introduction · p. 4

EV charging energy demand in Scotland.

  • 1,398 charging stations across 32 local authorities and one national node form a 1,431-node hierarchy in the ChargePlace Scotland data.Local authorities are a natural intermediate level for planning, reporting, and linking to administrative and demographic covariates.Source: Introduction · p. 2, §2.1 · p. 5

2.1. The ChargePlace Scotland dataset.

  • The forecasting task predicts demand at every hierarchy level from features including weather, calendar effects, lagged demand, and node characteristics.Daily session energy is aggregated by station, and reconciliation projects base forecasts onto the coherent subspace to enforce hierarchy constraints.Source: §2.1 · p. 5

2.2. Modeling the infrastructure activity.

The section uses known infrastructure activity to refine the hierarchy and assess prediction-interval coverage under active-node and neighborhood conditions. It also contrasts fixed and refined reconciliation, illustrating how inactive nodes affect forecasts.

2.2. Modeling the infrastructure activity.

  • Infrastructure activity is assumed known at forecast time; inactive nodes are excluded from training and the hierarchy is refined to remove their components from the coherent subspace.This avoids artificial constraints from reconciling against the original hierarchy when inactive nodes do not contribute to aggregate demand.Source: §2.2 · pp. 5–6

2.3. Coverage objectives.

  • Coverage is assessed on active-node days and, more strongly, across low- and high-activity regimes among neighboring nodes in the local authority.Conditioning on every possible infrastructure configuration is impractical with few observations; these requirements are called validity and conditional validity.Source: §2.3 · pp. 7–8

2.4. Efficiency objectives.

  • Prediction intervals should be as narrow as possible while remaining valid, with average length measured only on days when the node is active.Intervals that are overly wide can cover the truth yet misrepresent uncertainty and be unhelpful for decisions.Source: §2.4 · p. 8

3.1. Two forecast reconciliation procedures.

  • OLS projects base forecasts onto the fixed coherent subspace, whereas rOLS projects onto the refined subspace for the infrastructure active on that day.The paper combines reconciliation and conformal prediction as post-hoc frameworks for hierarchical forecasts and associated intervals.Source: §3.1 · p. 8, Point and probabilistic forecasting · p. 8
  • rOLS excludes an inactive station from reconciliation, avoiding the OLS propagation of its nonzero base forecast into other nodes and bringing the remaining forecasts closer to truth in the example.The inactive station’s coordinate is set to zero and the other coordinates are adjusted accordingly.Source: §3.1 · pp. 8–9

3.2. Basics on conformal prediction for hierarchical data.

CP constructs nodewise prediction intervals around reconciled hierarchical forecasts using calibration data, and the paper adapts this framework to evolving infrastructure activity.

3.2. Basics on conformal prediction for hierarchical data.

  • CP splits data into training and calibration sets, fits a base predictor and reconciliation function, then combines them into reconciled forecasts.Source: §3.2 · p. 9
  • Calibration absolute errors yield node-specific empirical quantiles, which set interval radii around each node’s reconciled forecast.Source: §3.2 · pp. 9–10

Conformal prediction for time-varying hierarchical structures.

4.1. Prediction intervals for active nodes.

  • CP-MNR targets validity conditional on node activity by computing non-conformity scores on calibration observations with matching activity status.Source: §4.1 · pp. 10–11

4.2. Infrastructure-aware prediction intervals.

CP-MNR-Nested⋆ extends active-node conditioning to the full infrastructure-activity configuration by treating calibration and test points symmetrically.

4.2. Infrastructure-aware prediction intervals.

  • Whole-infrastructure activity is a source of forecast variability, so conditioning only on a node’s own activity is not sufficient.Source: §4.2 · p. 11
  • Conditioning separately on every activity configuration would require 2^m calibration subsets, vastly more than available observations.Source: §4.2 · p. 11
  • CP-MNR-Nested⋆ uses a shared activity pattern for calibration and test points, enforcing symmetric treatment in non-conformity scoring.For each calibration point, the pattern represents nodes active in both that point and the test point.Source: §4.2 · p. 12
  • CP-MNR-Nested⋆ extends missing-data augmentation guarantees, although some assumptions natural to the original procedure do not hold in this setting.Source: §4.2 · p. 12

4.3. Online extension of the methodology.

The online extension updates conformal coverage levels sequentially to address temporal dependence and distribution shifts, but the paper applies it only to CP-MNR.

4.3. Online extension of the methodology.

  • The base procedures assume calibration and test points are independently drawn from the same distribution, unlike the sequential setting considered here.Source: §4.3 · pp. 12–13
  • Adaptive CP-MNR updates each active node’s coverage level after each outcome: miscoverage raises the level and coverage lowers it.Inactive nodes retain their current level.Source: §4.3 · p. 13
  • Only the adaptive extension of CP-MNR is considered because updating active nodes may hinder CP-MNR-Nested⋆’s stronger conditional-coverage guarantee.Source: §4.3 · p. 13

Forecasting EV charging energy demand with confidence.

The experiments assess conformal prediction intervals on a rolling six-month test period in Scotland’s evolving charging hierarchy. CP-MNR-Nested⋆ performs strongly across coverage assessments, while node-level results also reveal trade-offs between validity and interval width.

5.1. Experimental setting.

  • The six-month test period ran from February 5 to August 18, 2024, with training and calibration windows shifted every two weeks and performance averaged across 10 rolling partitions.Calibration used 20% of historical data, including the week before each test block and randomly sampled historical one-week blocks.Source: §5.1 · p. 14, Figure 5 · p. 15

5.2. Choice of the base predictor and reconciliation procedure.

  • rOLS was the best reconciliation procedure nationally and globally; globally, OLS degraded base-forecast performance while rOLS improved it.Combi was selected as the base predictor for robustness, ranking among the top two models at every aggregation level; rOLS also yielded smaller intervals than Direct and OLS while maintaining coverage.Source: §5.2 · p. 16, Figure 6 · p. 16

5.3.1. Validity conditional on the node being active.

  • At nominal 90% coverage, CP-MNR-Nested⋆ validity rates averaged 90% at Scotland, 71.2% regionally, and 73.6% at stations; at relaxed 80%, Adaptive CP-MNR reached 98% versus 90.8% for Nested⋆ at stations.At the nominal level, CP-MNR-Nested⋆ was best at each hierarchy level; CP-MNR also exceeded CP at stations, with 56.2% versus 38.1% nominal validity and 93.6% versus 67.7% relaxed validity.Source: §5.3.1 · p. 17, Table 3 · p. 17

5.3.2. Validity conditional on the activity of the neighboring nodes.

  • In low-activity regimes, CP-MNR-Nested⋆ validity was 81% versus Adaptive CP-MNR’s 53%, while both reached 70% in high-activity regimes.The adaptive extension widened CP-MNR’s low-to-high activity validity gap from 57%–64% to 52%–70%.Source: §5.3.2 · p. 18, Figure 7 · p. 18

5.4. Focus on Scotland, Glasgow, and two stations in Glasgow.

  • The selected cases span national Scotland, urban Glasgow, and two challenging Glasgow stations: Station 406 began operating shortly before testing, while Station 386 reappeared midway through it.Station 386’s calibration data could contain few active observations after its prolonged inactivity.Source: §5.4 · pp. 18–19

5.4.1. Node-wise coverages and lengths.

  • CP-MNR-Nested⋆ was valid at all four selected nodes, with coverage of 91% for Scotland, 97% for Glasgow, and 91% for each station.Adaptive CP-MNR produced valid Glasgow intervals at least 20% shorter, while Nested⋆ produced the widest intervals on average.Source: §5.4.1 · pp. 19–20, Figure 8 · p. 19

5.4.2. Visual assessment of the prediction intervals.

  • At aggregated nodes, two-weekly predictor updates appeared sufficient to track temporal evolution, while Adaptive CP-MNR shortened valid Glasgow intervals when coverage exceeded nominal.CP-MNR and CP-MNR-Nested⋆ intervals appeared overly conservative for Glasgow, especially late in the test period.Source: §5.4.2 · p. 20, Figure 9 · p. 20
  • At stations, CP intervals were too narrow and undercovered; the other three procedures approached nominal coverage, but produced no intervals until nonzero calibration observations appeared for Station 386.The issue was less pronounced for Station 406, which had fewer zero-demand observations than Station 386.Source: §5.4.2 · p. 20, Figure 10 · p. 21, Figure 11 · p. 21

Discussion and future work.

The study combines forecast reconciliation and conformal prediction for Scotland’s time-varying EV charging hierarchy. Its experiments find that CP-MNR-Nested⋆ produces valid intervals across infrastructure configurations, and the authors identify online learning and reconciliation as a future direction.

Discussion and future work.

  • CP-MNR-Nested⋆ produced valid intervals across infrastructure configurations, while CP-MNR and Adaptive CP-MNR were less sensitive to whole-infrastructure activity.The latter methods’ validity was conditional on the activity of the node under consideration.Source: Discussion and future work · p. 22
  • Increasing adaptivity through online extensions of learning and reconciliation procedures is identified as a future-work direction.The authors point to online reconciliation approaches as potential foundations for this work.Source: Discussion and future work · p. 22

Connection with CP-MDA and Theoretical Results

  • The supplement connects CP-MNR and CP-MNR-Nested⋆ to conformal prediction with missing data, drawing on the CP-MDA framework.It also provides mathematical insights motivating their construction.Source: Connection with CP-MDA and Theoretical… · p. 22

Implementation of the Algorithms

Additional Numerical Results

A. CONNECTION WITH CP-MDA AND THEORETICAL RESULTS

The supplement connects CP-MNR to conformal prediction with missing data by interpreting inactive-node handling as imputation, then states an activity-conditional coverage guarantee for CP-MNR under an exchangeability assumption.

A. CONNECTION WITH CP-MDA AND THEORETICAL RESULTS

  • The supplement frames the CP-MNR and CP-MNR-Nested⋆ analysis as connected to the missing-data augmentation framework of Zaffran et al. (2023, 2024).The connection rests on treating inactive-node base forecasts as deterministically imputed zeros.Source: A. CONNECTION WITH CP-MDA AND… · p. 25, §A.1 · p. 25

A.1. Connection with the conformal prediction with missing data framework.

  • Inactive-node base forecasts are replaced by zero, an imputation strategy that links infrastructure-aware reconciliation to conformal prediction with missing data.The activity vector identifies inactive hierarchy nodes whose zero-valued constraints affect forecasts through reconciliation.Source: §A.1 · p. 25

A.2. Theoretical Result for CP-MNR.

  • Under Assumption 1, CP-MNR provides coverage conditional on the activity of the node under scrutiny for any learning and reconciliation algorithms.The theoretical argument relies on exchangeability, which the text notes may not hold in practice.Source: §A.2 · p. 26

A.3. Theoretical Results for CP-MNR-Nested⋆.

CP-MNR-Nested⋆ is shown to have marginal and, under stronger assumptions, conditional coverage guarantees. The conditional result depends on an independence assumption that the authors say does not hold in their setting.

A.3. Theoretical Results for CP-MNR-Nested⋆.

  • CP-MNR-Nested⋆ has a marginal guarantee under Assumption 1 and a conditional guarantee under stronger assumptions.The first result is weaker than CP-MNR's: it is marginal rather than locally conditional, with coverage level 1−2α rather than 1−α.Source: §A.3 · p. 27
  • The stated conditional guarantee requires Assumptions 2 and 3, including independence of infrastructure activity from observations.The authors say Assumption 3 is very restrictive and fails in their setting because activity is deterministically induced by EV charging demand.Source: §A.3 · p. 28
  • The conditional guarantee holds for any reconciliation algorithm, including one that performs no reconciliation, so the result attributes validity to the theorem's assumptions rather than reconciliation.The authors report that empirical findings indicate the guarantee may still hold in practice when Assumption 3 is violated.Source: §A.3 · pp. 28, 30

B. IMPLEMENTATION OF THE ALGORITHMS

The implementation supplement maps the algorithms' practical details across predictor construction, CP-MNR-Nested⋆, and Adaptive CP-MNR.

B. IMPLEMENTATION OF THE ALGORITHMS

B.1. Construction of the base and reconciled predictors.

Algorithm 4 combines a fitted base predictor with a fitted reconciliation function to produce a reconciled predictor, covering Direct, OLS, and rOLS strategies.

B.1. Construction of the base and reconciled predictors.

  • The learning algorithm is fitted on the training subset and outputs the base predictor, while the reconciliation algorithm is fitted on that same subset.The resulting reconciliation function is combined with the base predictor to form the reconciled predictor.Source: §B.1 · pp. 30–31
  • Direct leaves base forecasts unchanged, whereas OLS projects them onto the coherent subspace Im(H).Source: §B.1 · p. 30
  • rOLS projects base forecasts onto a refined coherent subspace.Source: §B.1 · p. 30

B.2. Implementation of CP-MNR-Nested⋆.

The implementation makes CP-MNR-Nested⋆ prediction-set computation practical and discusses calibration choices, computational cost, and the fact that the resulting sets need not be intervals.

B.2. Implementation of CP-MNR-Nested⋆.

  • The conformal set is computed by finding where an interval-overlap count exceeds a threshold, sorting its breakpoints, and making one pass to recover connected components.The construction uses overlapping intervals I_k = [c_k − r_k, c_k + r_k].Source: §B.2 · p. 31
  • Reconciled forecasts can be computationally expensive because each calibration and test point requires a projection matrix.The prediction-set computation itself is not computationally intensive.Source: §B.2 · p. 31
  • Practitioners may select a subset of calibration indices to mitigate sparsity in augmented infrastructure activity patterns.In the experiments, calibration observations had at most 600 activity-vector components different from the test observation.Source: §B.2 · p. 31
  • CP-MNR-Nested⋆ need not produce intervals, so the experiments use the convex hull of its prediction set.Source: §B.2 · p. 31

B.3. Implementation of Adaptive-CP-MNR.

Adaptive-CP-MNR applies CP-MNR sequentially with node-specific miscoverage levels that update over time. Its updates occur when a node is active, and the procedure inherits an asymptotic coverage guarantee.

B.3. Implementation of Adaptive-CP-MNR.

  • Adaptive-CP-MNR combines CP-MNR with adaptive conformal inference and processes test points chronologically using sequentially updated, node-specific coverage levels.The sequence starts at nominal level α; the method also updates the dataset and training and calibration index sets for the next round.Source: §B.3 · p. 32
  • The miscoverage update occurs only when the node is active, and the step size γ controls adaptation speed.The experiments set γ = 0.02; smaller values adapt more slowly to temporal drift, while larger values may cause oscillations outside [0,1].Source: §B.3 · p. 32
  • Adaptive CP-MNR inherits the classic asymptotic coverage guarantee of adaptive conformal inference.The stated result is a direct application of that framework’s guarantee.Source: §B.3 · p. 32

C. ADDITIONAL NUMERICAL RESULTS

The additional results favor rOLS for shortening prediction intervals across hierarchy levels while preserving validity. The comparison also highlights that interval-length gains must be considered alongside coverage.

C.1. Comparison of the reconciliation procedures in terms of efficiency.

  • At the station level, CP had an 81% win rate, but this efficiency gain came with failure to achieve nominal coverage.OLS shortened intervals at national and regional levels but was not consistently beneficial at station level; CP-MNR-Nested⋆ was excluded because it is specific to rOLS.Source: §C.1 · p. 33

C.2. Comparison of the reconciliation procedures in terms of validity.

  • At nominal 90% and relaxed 80% levels, Tables 5 and 6 show that rOLS does not compromise validity; the stated result assumes i.i.d. elliptical non-conformity scores.CP is the exception: its lack of activity-based calibration subsampling can impair coverage conditional on a node being active at test time.Source: §C.2 · p. 33, Table 5 · p. 34, Table 6 · p. 34

C.1. Comparison of the reconciliation procedures in terms of efficiency.

  • 60%, 61.6%, and 59.4%: CP-MNR rOLS win rates against Direct at national, regional, and station levels, respectively.The win rate compares prediction-interval average lengths; a rate above 50% indicates shorter intervals than Direct.Source: §C.1 · p. 33, Table 4 · p. 34
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