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Inverse Optimization with Kernel Regression: Application to the Power Forecasting and Bidding of a Fleet of Electric Vehicles

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Pith's one-line read A kernelized inverse-optimization approach forecasts EV-fleet charging and discharging power and derives market bid curves, outperforming support vector and ridge regression on synthetic case studies.

arxiv 1908.00399 v4 pith:A7RKYRXJ submitted 2019-08-01 eess.SY cs.SY

Ricardo Fernández-Blanco , Juan Miguel Morales , Salvador Pineda , Álvaro Porras This is my paper · ORCID
classification eess.SYcs.SY
keywords electricityfleetforecastingmarketmethodproposedaggregatorapplication
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

An electric vehicle aggregator must decide how much power its fleet will draw from or feed into the grid at each hour, and it must submit a price-quantity bid to the market. This paper models the fleet as a simple optimization problem: the fleet chooses power blocks that maximize its surplus given the electricity price. The unknown parts of that model, the allowed power range and the marginal value of each block, are learned from past data using kernel regression. Because the learning problem is hard, the authors split it into two convex optimization steps, one that fits the power bounds and one that fits the marginal values by minimizing the gap between what the fleet should have done and what it did.

The authors test the method on synthetic data they generate from a more detailed simulator of one hundred electric vehicles, using travel patterns from a national survey and Spanish electricity prices. They compare it with support vector regression, kernel ridge regression, a linear inverse optimization variant, and naive benchmarks. The kernel-based inverse optimization method usually has lower forecast error than the linear variant and is comparable to or better than the machine learning baselines. It also produces a monotonically decreasing bid curve for each hour, which the plain forecasting methods cannot provide.

The main caveat is that all results come from simulated fleets. The simplified model that is learned ignores the battery state-of-charge dynamics that the simulator includes, and no confidence intervals are given for the reported error reductions.

Extended reading notes

Core claim

The central claim is that the proposed kernelized two-step inverse-optimization procedure can forecast the aggregate power of a price-responsive EV fleet with V2G capabilities and derive a market-compliant bid/offer curve, with accuracy comparable to or better than support vector regression and kernel ridge regression. The abstract states that the framework 'allows the aggregator to derive a bid/offer curve' and that the tests show benefits against the machine-learning techniques reported to exhibit the best forecasting performance. If true, an aggregator obtains both a forecast and a bidding curve from one model.

Load-bearing premise

The load-bearing modeling assumption is that the static linear program (1), with blockwise marginal utilities and time-varying power bounds, adequately represents the aggregate response of a real EV fleet to electricity prices. This assumption enters in Section 2.1 when (1) is introduced as the forward model. The fleet simulator in Appendix A includes intertemporal battery state-of-charge constraints (A.3), battery degradation (A.8), and charging/discharging efficiencies, none of which appear in the simplified forward LP. If real fleet behavior is dominated by these dynamics, the estimated bounds and marginal utilities will be misspecified and the forecast and bid curve will degrade.

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Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the adequacy of the static LP forward model, on the synthetic data generator, and on the kernel tuning choices; these are the main assumptions. No new physical entities are introduced; H, M, gamma, the block count, and the simulator synchronization penalty are hand-chosen or tuned parameters.

free parameters (5)
  • H (feasibility penalty) = 0.64 (naive-ch), 0.82 (sync), 0.94 (non-sync)
    Controls the width of the estimated power bounds; chosen by grid search on the validation set in Section 2.3.3.
  • M (regularization weight) = 0.0002 (naive-ch), 0.0001 (sync), 0.002 (non-sync)
    Regularizes the kernel coefficients in the feasibility problem; tuned by grid search.
  • gamma (Gaussian kernel width) = 0.1 or 0.01
    Scale of the Gaussian kernel in the feasibility regression; selected by grid search.
  • NB (number of energy blocks) = 6
    Number of charging and discharging blocks in the forward model; chosen by the authors without sensitivity analysis.
  • CS (synchronization penalty in the simulator) = 0, 520, or 52 e/MWh^2 depending on scenario
    Data-generation parameter used to create the different synthetic fleet behavior patterns; the reported results depend on these chosen scenarios.
assumptions (5)
  • domain assumption The static linear program (1) with blockwise marginal utilities and time-varying power bounds can represent the aggregate price response of an EV fleet with V2G capabilities.
    This is the forward model in Section 2.1; it ignores intertemporal state-of-charge constraints, driver heterogeneity, and battery degradation that the simulator in Appendix A includes.
  • domain assumption Synthetic data from simulator (A.1)-(A.9), calibrated with NHTS travel patterns and one six-week Spanish price series, is a valid stand-in for real EV aggregator data.
    Section 4.1 uses no real aggregator data; the paper acknowledges this and relies on the simulator for all numerical comparisons.
  • standard math Strong duality for the forward LP (1) holds at the estimated parameters, justifying the duality-gap formulation of the optimality problem.
    Used in constraints (8b)-(8d) in Section 2.3.2; requires primal feasibility and boundedness, which the feasibility problem enforces for a chosen percentage of training points.
  • ad hoc to paper Gaussian kernel in the feasibility problem and linear kernel in the optimality problem is the right modeling choice.
    Section 3 states this combination exhibited the best trade-off in experiments, i.e., it is selected on validation data without a separate principled justification.
  • domain assumption The two-step feasibility-then-optimality procedure recovers parameters acceptably despite the split from the original bilevel problem.
    Section 2.3 introduces the two-step procedure to avoid bilevel optimization; no consistency result or error bound is proven.

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Pith. "Pith review of Inverse Optimization with Kernel Regression: Application to the Power Forecasting and Bidding of a Fleet of Electric Vehicles." pith.science (2026). https://pith.science/paper/A7RKYRXJ

@misc{pith2026190800399,
  author       = {Pith},
  title        = {Pith review of: Inverse Optimization with Kernel Regression: Application to the Power Forecasting and Bidding of a Fleet of Electric Vehicles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A7RKYRXJ}},
  note         = {Machine review of arXiv:1908.00399}
}
read the original abstract

This paper considers an aggregator of Electric Vehicles (EVs) who aims to learn the aggregate power of his/her fleet while also participating in the electricity market. The proposed approach is based on a data-driven inverse optimization (IO) method, which is highly nonlinear. To overcome such a caveat, we use a two-step estimation procedure which requires solving two convex programs. Both programs depend on penalty parameters that can be adjusted by using grid search. In addition, we propose the use of kernel regression to account for the nonlinear relationship between the behaviour of the pool of EVs and the explanatory variables, i.e., the past electricity prices and EV fleet's driving patterns. Unlike any other forecasting method, the proposed IO framework also allows the aggregator to derive a bid/offer curve, i.e. the tuple of price-quantity to be submitted to the electricity market, according to the market rules. We show the benefits of the proposed method against the machine-learning techniques that are reported to exhibit the best forecasting performance for this application in the technical literature.

Figures

Figures reproduced from arXiv: 1908.00399 by the authors.

Figure 1
Figure 1. Three-block stepwise offer (bid) price function of the EVs’ aggregator. In this [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Values of the Gaussian kernel for each time period [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Charging power for cases naive-ch, sync, and non-sync in the left y-axis and the corresponding electricity prices in the right y-axis. 0 200 400 Aggregate power (kW) 20 40 60 80 Price ( /MWh) (a) 0 200 400 Aggregate power (kW) 20 40 60 80 (b) 0 50 100 Aggregate power (kW) 20 40 60 80 (c) [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figures from the paper (4 more)
Figure 4
Figure 4. Figure 4: Power versus price for cases (a) naive-ch, (b) sync, and (c) non-sync. The sizes of the training, validation, and test sets are 672 h, 168 h, and 168 h, in that order [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Estimated power bounds as well as forecast and observed power for case [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Results for case sync: (a) Estimated marginal utility price per block (in grey) and electricity price (in black) and (b) estimated power bounds as well as forecast and observed power. Note that the inset plot represents the bid price function and the corresponding elec…
Figure 7
Figure 7. Figure 7: Results for case non-sync: (a) Estimated marginal utility price per block (in grey) and electricity price (in black) and (b) estimated power bounds as well as forecast and observed power. Note that the inset plot represents the bid price function and the corresponding …

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