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arxiv: 2605.02334 · v1 · submitted 2026-05-04 · 📡 eess.SY · cs.SY

Efficient Multi-Market Scheduling of Virtual Power Plants via Spectral Representation of Uncertainty

Lorenzo Zapparoli , Blazhe Gjorgiev , Giovanni Sansavini This is my paper

Pith reviewed 2026-05-08 19:28 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords virtual power plantpolynomial chaos expansionmulti-market biddingstochastic schedulingdistributed energy resourcesuncertainty quantificationspectral methods
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The pith

Intrusive polynomial chaos expansion reformulates stochastic multi-market virtual power plant scheduling into a compact deterministic problem that matches scenario-based accuracy at far lower cost.

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

The paper develops a scheduling method for virtual power plants that participate in multiple electricity markets while facing uncertain prices, renewable output, and customer loads. It represents these uncertainties through intrusive polynomial chaos expansion, which expands the random variables in a polynomial basis and projects the entire stochastic optimization into a single deterministic system. The resulting model stays small even as uncertainty complexity grows, unlike scenario-based approaches that suffer from rapid growth in problem size. When tested on a realistic Swiss low-voltage network with distributed energy resources, the method produces bidding decisions nearly identical to a high-fidelity scenario benchmark yet solves up to 137 times faster. The authors also release an open-source tool that automates the same spectral reformulation for other stochastic programs.

Core claim

The central claim is that intrusive polynomial chaos expansion converts a multi-stage stochastic program for virtual power plant bidding into a low-dimensional deterministic counterpart whose optimal solution preserves the probabilistic structure of the original problem and yields market decisions of comparable quality to scenario-based methods while requiring substantially fewer computational resources.

What carries the argument

Intrusive polynomial chaos expansion, which expands uncertain parameters as polynomials in a chosen basis and substitutes the expansion into the original stochastic constraints to obtain an equivalent deterministic system of equations.

If this is right

  • The spectral reformulation keeps problem size manageable even when the number of uncertain parameters or markets increases.
  • Virtual power plants can obtain near-optimal bids across day-ahead, intraday, and reserve markets without enumerating thousands of scenarios.
  • The same projection technique extends to any single- or two-stage stochastic program once the open-source tool is used to generate the deterministic equivalent.
  • Computational effort drops enough to allow more frequent re-optimization as new forecasts arrive.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Operators could embed the spectral model inside rolling-horizon controllers that update bids every few minutes using fresh price and weather data.
  • The approach may scale to larger aggregations of distributed resources if the polynomial basis is chosen to match the dominant uncertainty sources in each market.
  • Integration with distribution-system operators could let the same framework enforce network constraints inside the bidding problem without adding scenario explosion.

Load-bearing premise

The uncertainties in prices, generation, and loads must admit an accurate low-order polynomial representation that keeps the key statistical dependencies intact for the resulting bidding decisions.

What would settle it

Apply the method to a test case whose price or generation uncertainty has strong skewness or multimodality that low-order polynomials cannot capture, then check whether the obtained bids differ materially from those of a large scenario-based benchmark on the same data.

Figures

Figures reproduced from arXiv: 2605.02334 by Blazhe Gjorgiev, Giovanni Sansavini, Lorenzo Zapparoli.

Figure 1
Figure 1. Figure 1: High-level description of the methodology for VPP multi-market scheduling under uncertainty. view at source ↗
Figure 2
Figure 2. Figure 2: Case study network map showing the 77 buses, lines, and substation view at source ↗
Figure 3
Figure 3. Figure 3: Accuracy–runtime comparison between intrusive PCE and scenario approximation (SA). RMSE with respect to the 2000 samples SA reference is view at source ↗
Figure 4
Figure 4. Figure 4: First-stage bid trajectories obtained with intrusive PCE compared with the 2000-scenario SA reference. Solid blue lines denote the reference mean, view at source ↗
read the original abstract

As the penetration of distributed energy resources increases, harnessing their flexibility becomes critical for power system operations. Virtual power plants (VPPs) offer a promising solution. However, existing VPP market scheduling tools exhibit a tradeoff between economic performance and tractability. Stochastic formulations provide probabilistically optimal decisions but are computationally intractable for large systems due to scenario explosion. Robust approaches are more tractable but often yield conservative decisions. This paper addresses this gap by proposing a stochastic multi-market VPP scheduling framework that represents uncertainty in the spectral domain via intrusive Polynomial Chaos Expansion (PCE). The resulting reformulation yields a low-dimensional deterministic spectral counterpart that preserves the stochastic structure and can be solved efficiently with standard optimization tools. The proposed spectral approach is demonstrated on a DER-based VPP operating on a realistic Swiss low-voltage grid and benchmarked against a state-of-the-art scenario-based solution. Results show that intrusive PCE achieves solution quality comparable to the scenario-based benchmark, with up to a 137 times reduction in computational effort, while yielding highly accurate bidding decisions. Finally, to facilitate adoption and reproducibility, we release an open-source, application-agnostic projection tool that automates the spectral reformulation for generic single- and two-stage stochastic programs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a multi-market VPP scheduling framework that applies intrusive polynomial chaos expansion (PCE) to represent joint uncertainty in prices, DER generation, and loads. This converts the two-stage stochastic program into a low-dimensional deterministic spectral counterpart that is solved with standard solvers. The method is tested on a realistic Swiss low-voltage grid VPP, where it produces bidding decisions of quality comparable to a scenario-based benchmark while achieving up to 137× computational speedup; an open-source projection tool for generic stochastic programs is also released.

Significance. If the accuracy claims hold, the approach offers a scalable route to probabilistically optimal VPP decisions that avoids both the conservatism of robust methods and the scenario explosion of full stochastic programs. The release of an application-agnostic open-source tool for intrusive PCE reformulation is a clear strength for reproducibility and reuse across other two-stage stochastic programs in power systems.

major comments (2)
  1. [§5 (Numerical Results) and Table 2] §5 (Numerical Results) and Table 2: The claim that intrusive PCE yields 'highly accurate bidding decisions' and 'solution quality comparable to the scenario-based benchmark' is supported only by a single fixed-order comparison. No convergence study (e.g., first-stage bids and objective values for PCE degrees 2–5) or a posteriori error bound on the optimal first-stage decisions is provided, leaving open the possibility that truncation error in the tails of price or load distributions distorts the bids.
  2. [§3.3 (Intrusive PCE Reformulation), Eq. (12)–(15)] §3.3 (Intrusive PCE Reformulation), Eq. (12)–(15): The two-stage structure requires that the spectral expansion correctly propagates uncertainty through the recourse problem. It is not shown whether the polynomial basis is applied to second-stage variables or only to the expectation operator; without this detail or a small-scale verification against Monte Carlo, it is unclear whether the stochastic structure is fully preserved.
minor comments (2)
  1. [Abstract and §1] The abstract and §1 cite a 137× speedup; the exact number of scenarios used in the benchmark and the PCE order should be stated explicitly in the main text for reproducibility.
  2. [Figure 4] Figure 4 (bidding curves): Axis labels and legend entries could be enlarged for clarity when printed.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the detailed and insightful comments, which have helped us identify areas for improvement in the manuscript. We address each major comment below and will incorporate revisions to enhance the clarity and rigor of our presentation.

read point-by-point responses

  1. Referee: [§5 (Numerical Results) and Table 2] §5 (Numerical Results) and Table 2: The claim that intrusive PCE yields 'highly accurate bidding decisions' and 'solution quality comparable to the scenario-based benchmark' is supported only by a single fixed-order comparison. No convergence study (e.g., first-stage bids and objective values for PCE degrees 2–5) or a posteriori error bound on the optimal first-stage decisions is provided, leaving open the possibility that truncation error in the tails of price or load distributions distorts the bids.

    Authors: We appreciate this observation. While the manuscript demonstrates comparable performance for the chosen PCE order, we agree that a convergence analysis would strengthen the claims. In the revised manuscript, we will include a study varying the PCE degree from 2 to 5, reporting the resulting first-stage bids and expected objective values, along with a comparison to the scenario-based benchmark. Additionally, we will discuss the truncation error and its impact on the tails of the distributions based on the Swiss grid data. This will provide evidence that the selected order is sufficient for the accuracy claimed. revision: yes

  2. Referee: [§3.3 (Intrusive PCE Reformulation), Eq. (12)–(15)] §3.3 (Intrusive PCE Reformulation), Eq. (12)–(15): The two-stage structure requires that the spectral expansion correctly propagates uncertainty through the recourse problem. It is not shown whether the polynomial basis is applied to second-stage variables or only to the expectation operator; without this detail or a small-scale verification against Monte Carlo, it is unclear whether the stochastic structure is fully preserved.

    Authors: We thank the referee for highlighting this point of potential ambiguity. In the intrusive PCE approach for two-stage problems, the decision variables in both stages are expanded in the polynomial chaos basis, and the constraints and objective are projected onto the basis functions, resulting in a deterministic system of equations. The second-stage variables are indeed represented spectrally to capture the recourse decisions under uncertainty. To clarify this, we will revise Section 3.3 to explicitly state that both first- and second-stage variables are expanded, with the projection applied to the full stochastic program. Furthermore, we will add a small-scale verification example comparing the PCE reformulation against Monte Carlo sampling on a simplified two-stage problem to demonstrate preservation of the stochastic structure. revision: yes

Circularity Check

0 steps flagged

No circularity: standard PCE reformulation applied to VPP scheduling with external benchmarking

full rationale

The derivation applies intrusive PCE to obtain a deterministic spectral counterpart of the two-stage stochastic program. This is a standard projection technique whose output is not defined in terms of the bidding decisions or accuracy metrics being claimed; the paper instead benchmarks the resulting first-stage bids against an independent scenario-based solver on a Swiss LV grid instance. No self-citation is load-bearing for the central reformulation, no fitted parameters are relabeled as predictions, and the open-source projection tool simply automates an existing method without creating self-referential loops. The speedup and quality claims rest on numerical comparison rather than construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract only; no explicit free parameters, axioms, or invented entities are described. The approach relies on standard PCE properties and stochastic programming assumptions not detailed here.

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Works this paper leans on

45 extracted references · 45 canonical work pages · 1 internal anchor

  1. [1]

    Grid codes for renewable powered systems,

    IRENA, “Grid codes for renewable powered systems,” International Renewable Energy Agency, Tech. Rep., 2022. 10

    work page 2022

  2. [2]

    Unlocking the potential of distributed energy resources,

    IEA, “Unlocking the potential of distributed energy resources,” International Energy Agency, Tech. Rep., 2022, [online]. Available: https://www.iea.org/reports/ unlocking-the-potential-of-distributed-energy-resources. Accessed: May 26, 2025

    work page 2022

  3. [3]

    On feasibility and flexibility operating regions of virtual power plants and TSO/DSO interfaces,

    S. Riaz and P. Mancarella, “On feasibility and flexibility operating regions of virtual power plants and TSO/DSO interfaces,” in2019 IEEE Milan PowerTech, 2019, pp. 1–6

    work page 2019

  4. [4]

    Risk averse optimal operation of a virtual power plant using two stage stochastic program- ming,

    M. A. Tajeddini, A. Rahimi-Kian, and A. Soroudi, “Risk averse optimal operation of a virtual power plant using two stage stochastic program- ming,”Energy, vol. 73, pp. 958–967, 8 2014

    work page 2014

  5. [5]

    Risk assessment of virtual power plants offering in energy and reserve markets,

    S. R. Dabbagh and M. K. Sheikh-El-Eslami, “Risk assessment of virtual power plants offering in energy and reserve markets,”IEEE Transactions on Power Systems, vol. 31, pp. 3572–3582, 9 2016

    work page 2016

  6. [6]

    Day-ahead self-scheduling of a virtual power plant in energy and reserve electricity markets under uncertainty,

    A. Baringo, L. Baringo, and J. M. Arroyo, “Day-ahead self-scheduling of a virtual power plant in energy and reserve electricity markets under uncertainty,”IEEE Transactions on Power Systems, vol. 34, pp. 1881– 1894, 5 2019

    work page 2019

  7. [7]

    Risk-averse optimal energy and reserve scheduling for virtual power plants incorporating demand response programs,

    M. Vahedipour-Dahraie, H. Rashidizadeh-Kermani, M. Shafie-Khah, and J. P. Catal ˜ao, “Risk-averse optimal energy and reserve scheduling for virtual power plants incorporating demand response programs,”IEEE Transactions on Smart Grid, vol. 12, pp. 1405–1415, 3 2021

    work page 2021

  8. [8]

    Stochastic optimization of trading strategies in sequential electricity markets,

    E. Kraft, M. Russo, D. Keles, and V . Bertsch, “Stochastic optimization of trading strategies in sequential electricity markets,”European Journal of Operational Research, vol. 308, pp. 400–421, 7 2023

    work page 2023

  9. [9]

    A multi-stage stochastic programming model for the unit commitment of conventional and virtual power plants bidding in the day-ahead and ancillary services markets,

    A. Fusco, D. Gioffr `e, A. F. Castelli, C. Bovo, and E. Martelli, “A multi-stage stochastic programming model for the unit commitment of conventional and virtual power plants bidding in the day-ahead and ancillary services markets,”Applied Energy, vol. 336, p. 120739, 4 2023

    work page 2023

  10. [10]

    Scheduling and operation of res- based virtual power plants with e-mobility: A novel integrated stochastic model,

    D. Falabretti, F. Gulotta, and D. Siface, “Scheduling and operation of res- based virtual power plants with e-mobility: A novel integrated stochastic model,”International Journal of Electrical Power & Energy Systems, vol. 144, p. 108604, 1 2023

    work page 2023

  11. [11]

    Risk-Aware Multi-Market Scheduling of Virtual Power Plants with Dynamic Network Tariffs

    L. Zapparoli, P. F ¨ath, B. Gjorgiev, and G. Sansavini, “Risk-aware multi-market scheduling of virtual power plants with dynamic network tariffs,” 2026. [Online]. Available: https://arxiv.org/abs/2604.26424

    work page internal anchor Pith review Pith/arXiv arXiv 2026

  12. [12]

    Strategic bidding for a virtual power plant in the day-ahead and real-time markets: A price-taker robust optimization approach,

    M. Rahimiyan and L. Baringo, “Strategic bidding for a virtual power plant in the day-ahead and real-time markets: A price-taker robust optimization approach,”IEEE Transactions on Power Systems, vol. 31, pp. 2676–2687, 7 2016

    work page 2016

  13. [13]

    Robust optimization based bidding strategy for virtual power plants in electricity markets,

    L. Zheming and Y . Guo, “Robust optimization based bidding strategy for virtual power plants in electricity markets,”IEEE Power and Energy Society General Meeting, vol. 2016-November, 11 2016

    work page 2016

  14. [14]

    Research on bidding strategy of virtual power plant considering carbon-electricity integrated market mechanism,

    X. Liu, “Research on bidding strategy of virtual power plant considering carbon-electricity integrated market mechanism,”International Journal of Electrical Power & Energy Systems, vol. 137, p. 107891, 5 2022

    work page 2022

  15. [15]

    Data- driven interval robust optimization method of vpp bidding strategy in spot market under multiple uncertainties,

    Y . Ma, Z. Li, R. Liu, B. Liu, S. S. Yu, X. Liao, and P. Shi, “Data- driven interval robust optimization method of vpp bidding strategy in spot market under multiple uncertainties,”Applied Energy, vol. 384, p. 125366, 4 2025

    work page 2025

  16. [16]

    Single- level flexible robust optimal bidding of renewable-only virtual power plant in energy and secondary reserve markets,

    H. Nemati, P. S ´anchez-Mart´ın, A. Baringo, and ´Alvaro Ortega, “Single- level flexible robust optimal bidding of renewable-only virtual power plant in energy and secondary reserve markets,”Energy, vol. 328, p. 136421, 8 2025

    work page 2025

  17. [17]

    Two-stage robust optimization strategy for vpp participation in the energy and reserve markets considering intertemporal carbon trading,

    H. Nemati, ´Alvaro Ortega, and P. S ´anchez-Mart´ın, “Two-stage robust optimization strategy for vpp participation in the energy and reserve markets considering intertemporal carbon trading,”International Journal of Electrical Power & Energy Systems, vol. 172, p. 111093, 11 2025

    work page 2025

  18. [18]

    Flexible robust optimal bidding of renewable virtual power plants in sequential markets under asymmetric uncertainties,

    H. Nemati, P. S ´anchez-Mart´ın, ´Alvaro Ortega, L. Sigrist, E. Lobato, and L. Rouco, “Flexible robust optimal bidding of renewable virtual power plants in sequential markets under asymmetric uncertainties,” Sustainable Energy, Grids and Networks, vol. 43, p. 101801, 9 2025

    work page 2025

  19. [19]

    Xiu,Numerical Methods for Stochastic Computations: A Spectral Method Approach

    D. Xiu,Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, 2010

    work page 2010

  20. [20]

    Power systems optimization under uncertainty: A review of methods and applications,

    L. A. Roald, D. Pozo, A. Papavasiliou, D. K. Molzahn, J. Kazempour, and A. Conejo, “Power systems optimization under uncertainty: A review of methods and applications,”Electric Power Systems Research, vol. 214, p. 108725, 1 2023

    work page 2023

  21. [21]

    A generalized framework for chance-constrained optimal power flow,

    T. M ¨uhlpfordt, T. Faulwasser, and V . Hagenmeyer, “A generalized framework for chance-constrained optimal power flow,”Sustainable Energy, Grids and Networks, vol. 16, pp. 231–242, 12 2018

    work page 2018

  22. [22]

    Chance-constrained ac optimal power flow: A polynomial chaos ap- proach,

    T. Muhlpfordt, L. Roald, V . Hagenmeyer, T. Faulwasser, and S. Misra, “Chance-constrained ac optimal power flow: A polynomial chaos ap- proach,”IEEE Transactions on Power Systems, vol. 34, pp. 4806–4816, 11 2019

    work page 2019

  23. [23]

    Efficient polynomial chaos expansion for uncertainty quantification in power systems,

    D. M ´etivier, M. Vuffray, and S. Misra, “Efficient polynomial chaos expansion for uncertainty quantification in power systems,”Electric Power Systems Research, vol. 189, p. 106791, 12 2020

    work page 2020

  24. [24]

    General polynomial chaos in the current–voltage formulation of the optimal power flow problem,

    T. V . Acker, F. Geth, A. Koirala, and H. Ergun, “General polynomial chaos in the current–voltage formulation of the optimal power flow problem,”Electric Power Systems Research, vol. 211, p. 108472, 10 2022

    work page 2022

  25. [25]

    Stochastic optimal power flow for hybrid ac/dc grids considering continuous non- gaussian uncertainty,

    K. Yurtseven, A. Koirala, H. Ergun, and D. V . Hertem, “Stochastic optimal power flow for hybrid ac/dc grids considering continuous non- gaussian uncertainty,”International Journal of Electrical Power & Energy Systems, vol. 170, p. 110828, 9 2025

    work page 2025

  26. [26]

    Congestion management through stochastic optimal transmission switching in hybrid ac/dc grids considering continuous non-gaussian uncertainty,

    ——, “Congestion management through stochastic optimal transmission switching in hybrid ac/dc grids considering continuous non-gaussian uncertainty,”IEEE Transactions on Power Systems, pp. 1–14, 2025

    work page 2025

  27. [27]

    Chance-constrained optimization based pv hosting capacity calculation using general polynomial chaos,

    A. Koirala, T. V . Acker, M. U. Hashmi, R. D’Hulst, and D. V . Hertem, “Chance-constrained optimization based pv hosting capacity calculation using general polynomial chaos,”IEEE Transactions on Power Systems, vol. 39, pp. 2284–2295, 1 2024

    work page 2024

  28. [28]

    Day-ahead dynamic operating envelopes using stochastic unbalanced optimal power flow,

    A. Koirala, F. Geth, and T. V . Acker, “Day-ahead dynamic operating envelopes using stochastic unbalanced optimal power flow,”Sustainable Energy, Grids and Networks, vol. 40, p. 101528, 12 2024

    work page 2024

  29. [29]

    Empowering prosumers: Incentive design for local electricity markets under generalized uncertainty and grid constraints,

    P. F. Austnes, M. Jacobs, L. Wang, and M. Paolone, “Empowering prosumers: Incentive design for local electricity markets under generalized uncertainty and grid constraints,” 2025. [Online]. Available: https://arxiv.org/abs/2510.12318

    work page arXiv 2025

  30. [30]

    Polychaos.jl — a julia package for polynomial chaos in systems and control,

    T. M ¨uhlpfordt, F. Zahn, V . Hagenmeyer, and T. Faulwasser, “Polychaos.jl — a julia package for polynomial chaos in systems and control,”IFAC- PapersOnLine, vol. 53, pp. 7210–7216, 1 2020

    work page 2020

  31. [31]

    StochasticPowerModels.jl repository,

    “StochasticPowerModels.jl repository,” [Online]. Available: https://github.com/Electa-Git/StochasticPowerModels.jl

    work page

  32. [32]

    Spectralstochopt,

    L. Zapparoli, “Spectralstochopt,” https://github.com/lorenzozapparoli/ SpectralStochOpt, 2026

    work page 2026

  33. [33]

    Polynomial chaos expansion for sensitivity analysis of model output with dependent inputs,

    T. A. Mara and W. E. Becker, “Polynomial chaos expansion for sensitivity analysis of model output with dependent inputs,”Reliability Engineering & System Safety, vol. 214, p. 107795, 2021

    work page 2021

  34. [34]

    T. J. Sullivan,Introduction to Uncertainty Quantification, 1st ed., ser. Texts in Applied Mathematics. Cham: Springer, 2015

    work page 2015

  35. [35]

    Large-scale generation of geo-referenced power distribution grids from open data with load clustering,

    A. Oneto, B. Gjorgiev, F. Tettamanti, and G. Sansavini, “Large-scale generation of geo-referenced power distribution grids from open data with load clustering,”Sustainable Energy, Grids and Networks, vol. 42, p. 101678, 2025

    work page 2025

  36. [36]

    Future deployment and flexibility of distributed energy resources in the distribution grids of switzerland,

    L. Zapparoli, A. Oneto, M. P. Herrera, B. Gjorgiev, G. Hug, and G. Sansavini, “Future deployment and flexibility of distributed energy resources in the distribution grids of switzerland,”Scientific Data, vol. 12, no. 1, p. 1491, 2025

    work page 2025

  37. [37]

    Transparency platform,

    ENTSO-E, “Transparency platform,” https://newtransparency.entsoe.eu/ market/energyPrices, 2024, accessed: 2025-8-10

    work page 2024

  38. [38]

    Ancillary services tenders and auction results,

    Swissgrid AG, “Ancillary services tenders and auction results,” https://www.swissgrid.ch/en/home/customers/topics/ancillary-services/ tenders.html, 2025, accessed: 2025-08-06

    work page 2025

  39. [39]

    (2025) Regelleistung.net – data center: afrr capacity market data

    50Hertz Transmission GmbH, Amprion GmbH, TenneT TSO GmbH, and TransnetBW GmbH. (2025) Regelleistung.net – data center: afrr capacity market data. Accessed: 2025-08-18. [Online]. Available: https://www.regelleistung.net/apps/datacenter/tenders/

    work page 2025

  40. [40]

    Stromtarife 2024,

    EWZ, “Stromtarife 2024,” https://ewz.ch/dam/ewz/Privatkunden/Strom/ Tarife/Dokumente/Uebersicht Tarife ewz 2024.pdf, Tramstrasse 35, 8050 Z ¨urich, 2024, accessed: 2024-10-23

    work page 2024

  41. [41]

    Power reserve capacity from virtual power plants with reliability and cost guarantees,

    L. Zapparoli, B. Gjorgiev, and G. Sansavini, “Power reserve capacity from virtual power plants with reliability and cost guarantees,”IEEE Transactions on Power Systems, pp. 1–12, 2026

    work page 2026

  42. [42]

    Short-term residential load forecasting based on lstm recurrent neural network,

    W. Kong, Z. Y . Dong, Y . Jia, D. J. Hill, Y . Xu, and Y . Zhang, “Short-term residential load forecasting based on lstm recurrent neural network,” IEEE Transactions on Smart Grid, vol. 10, no. 1, pp. 841–851, 2019

    work page 2019

  43. [43]

    Forecasting day- ahead electricity prices: A review of state-of-the-art algorithms, best practices and an open-access benchmark,

    J. Lago, G. Marcjasz, B. De Schutter, and R. Weron, “Forecasting day- ahead electricity prices: A review of state-of-the-art algorithms, best practices and an open-access benchmark,”Applied Energy, vol. 293, p. 116983, 2021

    work page 2021

  44. [44]

    Deep learning-based fore- casting of the automatic frequency reserve restoration band price in the iberian electricity market,

    J. Cardo-Miota, E. P ´erez, and H. Beltran, “Deep learning-based fore- casting of the automatic frequency reserve restoration band price in the iberian electricity market,”Sustainable Energy, Grids and Networks, vol. 35, 2023

    work page 2023

  45. [45]

    Deep learning approaches for predicting the upward and downward energy prices in the spanish automatic frequency restoration reserve market,

    J. M. Failing, J. Cardo-Miota, E. P ´erez, H. Beltran, and J. Segarra- Tamarit, “Deep learning approaches for predicting the upward and downward energy prices in the spanish automatic frequency restoration reserve market,”Energy, vol. 320, 2025

    work page 2025