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REVIEW 4 major objections 6 minor 53 references

Leveraging Neural Networks to Optimize Heliostat Field Aiming Strategies in Concentrating Solar Power Tower Plants

T0 review · 4 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A neural surrogate plus optimization makes heliostat aiming flatter and safer.

desk verdict First real CL+MILP for heliostat aiming, but the safety improvement is partly built into the objective; ask for a 2D check before believing the hotspot claims. read the letter →

arxiv 2412.16995 v1 pith:PFG3SSX7 submitted 2024-12-22 eess.SY cs.SYmath.OC

Antonio Alcántara , Pablo Diaz-Cachinero , Alberto Sánchez-González , Carlos Ruiz This is my paper · ORCID
classification eess.SYcs.SYmath.OC MSC 90C1168T07
keywords concentratingsolarpowertowerheliostataimingstrategyneuralnetworksurrogateconstraintlearningmixed-integerprogrammingfluxuniformitytrustregionreceiverthermalsafety
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

The paper tries to show that heliostat aiming in concentrating solar tower plants can be treated as a data-driven optimization problem: train a neural network to predict a quality score that balances energy collection against flux uniformity, embed that network into a mixed-integer program, and iterate with trust-region sampling. Tested on the Dunhuang 10 MWe plant, the resulting NN+Opt strategy beats the classical k-factor Sweep heuristic at every tested time of day. It reduces the distribution-difference metric by up to 52.3%, cuts peak concentration by roughly 9%, and at 08:00 collects 1.02% more energy with lower spillage. The cost near noon is about 2.2% of collected energy, which the paper presents as a modest price for safer receiver temperatures. If right, the method gives plant operators a tunable way to trade energy against receiver lifetime.

What carries the argument

The central object is the aiming-strategy quality score $\mathrm{QS}$, defined panel-wise as the collected energy $E_p$ minus a penalty $\lambda$ times the average distribution difference $\mathrm{dd}_p$, where $\mathrm{dd}_{p,v} = 1 - S^V_{p,v}$ measures how far the normalized vertical-mean concentration profile deviates from a flat profile over the central vertical nodes. The work is carried by a single-output ReLU neural network that maps the vector of heliostat $k$-factors to $\mathrm{QS}$; the network is then rewritten exactly as big-M mixed-integer linear constraints, so maximizing the surrogate becomes a MILP. An $\varepsilon$-convex-hull trust region confines the search to regions near training data, and progressive sampling around the current optimum retrains the network each iteration. This combination converts a nonconvex, simulation-heavy aiming problem into an iteratively refined tractable optimization.

What would settle it

Go to a real CSPT receiver with a camera-based flux measurement system, apply both the NN+Opt and Sweep aim maps for a clear equinox noon, and compare full two-dimensional flux maps; the central claim would be disproven if the optimized map still contains horizontal hot spots in a panel whose vertical-mean profile is flat, or if measured peak suns exceed the reported 1053 suns by much more than the simulation error.

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Extended reading notes

Core claim

The paper claims that heliostat aiming can be optimized by training a ReLU neural network to predict the aiming-strategy quality score $\mathrm{QS}$, reformulating the trained network as mixed-integer linear constraints, and maximizing $\mathrm{QS}$ inside an $\varepsilon$-convex-hull trust region that is iteratively refined around the current best solution. In the Dunhuang case study, this NN+Opt strategy reduces the distribution difference $\mathrm{dd}$ from the k-factor Sweep by 33.7% to 52.3% across 08:00, 10:00, and solar noon, lowers peak concentration by 7.5% to 9.2% (for example, from 1155.5 to 1053.2 suns at noon), and at 08:00 even collects 1.02% more energy while cutting spillage. The cost at noon is 2.2% of collected energy and 1.8 percentage points more spillage, which the paper frames as a deliberate energy-versus-thermal-safety trade-off.

Load-bearing premise

The load-bearing premise is that the simulation code used to generate all training data and all reported flux maps predicts real receiver fluxes well enough, and that flattening the vertically averaged flux profile rules out local hot spots; if either fails, the safety gain may be smaller than claimed.

Editorial extensions

If this is right

  • Aiming maps can be generated as continuous per-heliostat $k$-factors rather than discrete fixed zones, so the flux profile can be tailored to the sun position at each time of day.
  • The penalty $\lambda$ gives operators a direct dial for trading collected energy against thermal-peak reduction, so the same algorithm can serve plants with different receiver stress limits.
  • Because the surrogate is trained on simulated flux data, the pipeline transfers to other tower geometries by regenerating the dataset, without changing the optimization formulation.
  • At low sun angles the method can beat the Sweep heuristic on all reported metrics at once, meaning early-morning operation does not have to sacrifice yield for safety.
  • The reported 15-minute-per-iteration cost and iterative refinement make the approach practical for near-real-time re-aiming during changing solar conditions.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the simulation model is faithful, a roughly 9% peak-flux cut for about 2% energy loss could meaningfully extend receiver lifetime, although the paper does not quantify that lifetime gain.
  • The same surrogate-plus-trust-region recipe could be adapted to dynamic aiming over 5-to-15-minute intervals, with thermal stress estimates used as constraints instead of a static quality score.
  • A natural stress test is to evaluate the optimized strategy using the full two-dimensional flux map rather than the vertical-mean profile; if horizontal hot spots still appear, the safety claim would need to be softened.
  • The reported percentage gains are partly metric-dependent, since $\mathrm{dd}$ is computed on a normalized vertical mean; checking against peak suns and panel-level maps would clarify how robust the improvement is.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 6 minor

Summary. The paper proposes a data-driven framework for optimizing heliostat aiming factors in concentrating solar power tower plants. A neural network surrogate is trained on simulated flux maps to approximate a quality score QS that balances collected energy and a uniformity penalty, then embedded as a mixed-integer program with trust-region constraints. An iterative procedure alternates between sampling near the current optimum, retraining the surrogate, and re-optimizing. In a simulated Dunhuang 10 MWe case study, the method (NN+Opt) is compared against the classical k-factor Sweep heuristic at three times of day. The reported results show a 52.3% reduction in the distribution-difference metric at solar noon, a 8.9% reduction in peak concentration, a 2.2% energy loss, and a 1.02% energy gain at 08:00, alongside reduced spillage. The paper claims that the method yields flatter and safer flux distributions than the Sweep baseline.

Significance. If the claimed results are supported, the contribution is significant for the CSP operations community: it is one of the first constraint-learning formulations for heliostat aiming, the iterative trust-region procedure is clearly specified, and the simulation shows consistent improvement over a classical heuristic while remaining computationally tractable (about 15 minutes per iteration). The use of a published Convolution-Projection flux model and the detailed algorithm specifications (Algorithms 1-3) are strengths that aid reproducibility. The central caveat is that the safety/uniformity claims currently rest on a vertical-mean metric that the optimizer directly penalizes, and the neural network surrogate's approximation error is never reported; these gaps weaken the support for the practical conclusions. The paper does not provide code or data, which limits independent verification, but the methodological framework itself is clearly presented.

major comments (4)
  1. [§3.2, Algorithm 1] The quality score QS used as the optimization objective is computed from the vertical-mean concentration C^V_{p,v} (line 3) and the distribution-difference dd derived from it (lines 4-6). The optimizer therefore cannot perceive horizontal concentration gradients within a panel. The paper's central claim—that the method produces flatter, safer flux distributions and mitigates hotspots—is not supported for horizontal variations. A flat vertical-mean profile is compatible with a persistent horizontal hot spot at the center of a panel, since panel width (1.29 m) is comparable to or smaller than typical beam footprint width. The reported dd reduction is precisely the quantity being optimized, so its improvement is partly by construction; the only independent safety evidence is the peak-concentration reduction (e.g., 8.9% at noon), which does not establish horizontal uniformity. Please add a horizontal-aware uniformity metric (e.g., tube-level peak or per-horizontal-node deviation) to the evaluation, or explicitly qualify the safety claims as referring to vertical-mean uniformity only.
  2. [§3.3-3.6 and §4] The accuracy of the neural network surrogate is never reported. The method's validity relies on the surrogate approximating the true QS well enough within the trust region that the optimized X* improves the true QS when re-evaluated with the Convolution-Projection model. The paper reports final performance metrics and iteration counts but no train/test error (e.g., R^2, MAE, or a parity plot) and no comparison between the surrogate-predicted QS and the true QS at the final solution. Without this information, the reader cannot assess whether the reported improvements come from the optimization of a faithful surrogate or from artifacts of the approximation. Please report the held-out approximation error and the true QS at the final solution for at least the three studied times of day.
  3. [§4.1-4.2, Table 2] The main results are presented for a single run with λ fixed at 5,000, which is selected after inspecting the λ-sweep plot in §4.1 ('this value results in a good trade-off'). This post hoc selection, combined with the absence of replicate runs or error bars for the random sampling procedure (Algorithm 2), leaves the quantitative claims (e.g., 52.3% dd reduction, 2.2% energy loss) without statistical support. The paper should provide a sensitivity analysis over λ and repeated runs with different random seeds to show that the reported margins are not within sampling noise. This is particularly important because the data generation and NN training are stochastic.
  4. [Abstract and §4.3] The term 'real CSPT case study' is used even though all results are obtained from simulation with the Convolution-Projection model; no experimental validation is provided. While comparing two aiming strategies in the same simulation is internally fair, the practical-safety conclusions ('safer thermal conditions', 'prolonging receiver life') require the underlying flux model to be faithful to the real plant. Please state explicitly that the study is purely simulative, or temper the practical implications accordingly. This is currently a load-bearing premise for the paper's real-world relevance.
minor comments (6)
  1. [§2.1, Eq. (3)] The sentence 'For the sunshape standard deviation, σslp = 2.09 mrad is a generally adopted value' mixes the sunshape and mirror slope error; the value 2.09 mrad is attributed to the slope error in Eq. (3), while σ_sun is never defined. Please clarify the notation and give the actual sunshape value used.
  2. [§4] There is a duplicated word in 'Mac Mini M4 pro with with 24GB RAM'.
  3. [§4.3, Table 2] The SPL entries are reported as percent changes (e.g., '0.171 (−12.8%)'), while the text at solar noon says 'increases spillage loss by approximately 1.8 percentage points'. The percentage change is computed relative to the Sweep value, whereas the percentage-point difference is an absolute change; please make the convention explicit to avoid confusion.
  4. [§3.3] The statement 'This integration guarantees the optimal solution of the surrogate problem' should be accompanied by the solver's optimality gap or a statement that the MILP was solved to global optimality; otherwise, the reader does not know the quality of the computed surrogate solution.
  5. [Figure 3 and 4] The schematics in Figures 3 and 4 would benefit from clearer captions; in particular, Figure 3 includes an unlabeled 'Equatorial Aiming (y = 0)' marker and the lower panel-index map is not fully explained in the caption, which makes the figure difficult to interpret without the main text.
  6. [Algorithm 3 and Figure 6] The stopping criterion is referred to as 'Expert-Based Stop' in Figure 6 but is not defined in Algorithm 3 or the main text; please either specify the criterion or remove the label from the figure.

Circularity Check

1 steps flagged · score 4.0 of 10

The 52.3% dd reduction is the optimizer's own objective; peak-suns and energy results remain independent.

  1. self definitional [Algorithm 1, lines 3-8; Section 3.5, Eq. (11a); Section 4.3, Table 2]
    "Line 3 computes the vertical mean concentration C V p,v by averaging the concentration values across all horizontal mesh points h ∈ H... The QS for each panel, QSPanel p , is then obtained by subtracting a penalty term, proportional to ddp weighted by λ, from the total energy Ep (line 8)... (11a) max QS... Table 2 reports a 52.3% reduction in the distribution difference dd for the NN+Opt solution at 12:00 compared to Sweep."

    QS is defined as Ep − λ·ddp (Algorithm 1, line 8) and the optimization problem maximizes QS (Eq. 11a). The NN surrogate is trained to predict QS from aiming factors, so the selected NN+Opt configuration is the one that the objective itself penalizes for large dd. Reporting that NN+Opt reduces dd by 52.3% relative to Sweep is therefore reporting that the optimizer minimized the term it was built to minimize; the direction of improvement is imposed by the objective, not discovered from physics. The magnitude depends on the physical model and the baseline, so the comparison is not vacuous, but the uniformity headline is the objective by construction.

full rationale

The central optimization loop is self-contained and not circular: the NN surrogate is trained on CP-simulation QS labels, embedded as a MILP, and the final metrics are computed with the same physical model, while the Sweep baseline is an external heuristic from the literature. The one partially circular element is the headline dd reduction, because dd enters directly into QS through the λ penalty, so the optimizer is explicitly minimizing the metric later reported as evidence of improved uniformity. This is transparent objective design rather than a hidden equivalence, and the max-suns reductions (1155.5 to 1053.2 suns) and energy/spillage trade-offs come from the physical simulator and are not fitted. The vertical-mean definition of dd is a separate correctness risk: horizontal hot spots could persist under a flat vertical mean, but that is a modeling limitation, not circularity. Overall score 4 reflects partial circularity in the uniformity claim, with independent content in the rest of the comparison.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the fidelity of the CP simulation model, a hand-designed quality score, and several undisclosed hyperparameters. No new physical entities are introduced. Because the baseline and the flux model come from the same research group, independent benchmarking is limited.

free parameters (4)
  • Penalty weight lambda = 5000 (main experiments)
    Sets the trade-off between collected energy and flux uniformity in the QS. Chosen as a 'good trade-off' from Figure 7 in Section 4.1; results depend on this choice and no sensitivity analysis is given.
  • NN hidden layer size = 50 neurons (single hidden layer)
    Fixed for all experiments without ablation or justification; the surrogate's accuracy depends on this capacity choice.
  • Trust region radius epsilon = Not reported
    Section 3.6 states the problem is solved for different epsilon values and the best solution kept, but the tested values and the selection criterion are not disclosed, making the exploration strategy unreproducible.
  • Data sampling parameters (sigma, a, b) = Not reported
    Algorithm 2 samples aiming factors from Normal or Uniform distributions, but the actual parameters for mu, sigma, a, b are never specified, so the training dataset cannot be reconstructed.
assumptions (5)
  • domain assumption The Convolution-Projection (CP) model computes flux concentration maps accurately for the Dunhuang plant.
    All training data and all reported results come from this model, cited as [36]; if the model deviates from real plant behavior, the optimized aiming strategy and the Sweep comparison inherit the error.
  • domain assumption The effective error sigma_e from Eq. (3), with sigma_slp=2.6 mrad and sigma_sun from the literature, determines the beam radius in Eq. (2).
    These parameters are taken from prior literature without site-specific calibration and directly affect the flux profiles.
  • ad hoc to paper A single-hidden-layer ReLU network with 50 neurons can approximate the QS well enough within the trust region for optimization to improve the true QS.
    No validation or test error is reported; the optimization's value depends entirely on this surrogate accuracy, yet the architecture is fixed without analysis.
  • domain assumption The k-factor sweep heuristic from [5] is correctly reproduced as the baseline.
    The comparison is only meaningful if the baseline implementation is faithful; [5] is by the same group and the paper gives little detail on the Sweep setup.
  • domain assumption Fixing the aiming factor for all heliostats in a row and sector is a benign simplification.
    Stated in Section 3.4 as aligning with operational constraints, but it reduces the search space and may exclude better strategies that vary within a row.

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Cite this review

Pith. "Pith review of Leveraging Neural Networks to Optimize Heliostat Field Aiming Strategies in Concentrating Solar Power Tower Plants." pith.science (2026). https://pith.science/paper/PFG3SSX7

@misc{pith2026241216995,
  author       = {Pith},
  title        = {Pith review of: Leveraging Neural Networks to Optimize Heliostat Field Aiming Strategies in Concentrating Solar Power Tower Plants},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PFG3SSX7}},
  note         = {Machine review of arXiv:2412.16995}
}
read the original abstract

Concentrating Solar Power Tower (CSPT) plants rely on heliostat fields to focus sunlight onto a central receiver. Although simple aiming strategies, such as directing all heliostats to the receivers equator, can maximize energy collection, they often result in uneven flux distributions that lead to hotspots, thermal stresses, and reduced receiver lifetimes. This paper presents a novel, data-driven approach that integrates constraint learning, neural network-based surrogates, and mathematical optimization to overcome these challenges. The methodology learns complex heliostat-to-receiver flux interactions from simulation data, constructing a surrogate model that is embedded into a tractable optimization framework. By maximizing a tailored quality score that balances energy collection and flux uniformity, the approach yields smoothly distributed flux profiles and mitigates excessive thermal peaks. An iterative refinement process, guided by the trust region and progressive data sampling, ensures the surrogate model improves the obtained solution by exploring new spaces during the iterations. Results from a real CSPT case study demonstrate that the proposed approach surpasses conventional heuristic methods, offering flatter flux distributions and safer thermal conditions without a substantial loss in overall energy capture.

Figures

Figures reproduced from arXiv: 2412.16995 by the authors.

Figure 1
Figure 1. Schematic representation of a CSPT plant. [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. Heliostat field layout at Dunhuang 10 MW [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. Receiver divided into P panels, 2D representation of the panels in the receiver, and schematic representation of vertical mean concentration calculation. Algorithm 1 illustrates the calculation of the QS. It begins by iterating over each panel p ∈ P in the receiver (line 1). For each panel, the flux concentration values are computed using the CP method described in [36], which considers the heliostat aiming factors … view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: Schematic representation of aiming strategy quality score calculation: (a) Area calculation [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: NN structure for optimizing heliostat field aiming strategies. [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: Iterative approach for improving the optimal solution using refined NN training data. [PITH_FULL_IMAGE:figures/full_fig_p019_6.png]
Figure 7
Figure 7. Figure 7: Collected energy and performance metrics versus distribution different penalty ( [PITH_FULL_IMAGE:figures/full_fig_p020_7.png]
Figure 8
Figure 8. Figure 8: illustrates how the QS and related performance metrics evolve over a series of iterative optimization steps conducted at three different times of day: 8:00, 10:00, and solar noon. Each iteration consists of generating new training data around the previously found optim…
Figure 9
Figure 9. Figure 9: NN+Opt (up) and Sweep (down) at solar noon. Maps of concentration and aim points, [PITH_FULL_IMAGE:figures/full_fig_p023_9.png]
Figure 10
Figure 10. Figure 10: NN+Opt (up) and Sweep (down) at 10:00, solar time. Maps of concentration and aim [PITH_FULL_IMAGE:figures/full_fig_p024_10.png]
Figure 11
Figure 11. Figure 11: NN+Opt (up) and Sweep (down) at 08:00, solar time. Maps of concentration and aim [PITH_FULL_IMAGE:figures/full_fig_p025_11.png]

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Works this paper leans on

53 extracted references · 45 canonical work pages

  1. [1]

    L. L. Vant-Hull, Central tower concentrating solar power (CSP) systems, in: Concentrating solar power technology: principles, developments and applications, Woodhead Publishing - Elsevier, 2012, Ch. 8. URL http://www.sciencedirect.com/science/article/pii/ B978184569769350008X

    work page 2012

  2. [2]

    Sánchez-González, M

    A. Sánchez-González, M. R. Rodríguez-Sánchez, D. Santana, Aiming strategy model based on allowable flux densities for molten salt central receivers, Sol. En- ergy 157 (2017) 1130–1144.doi:10.1016/j.solener.2015.12.055. URL http://linkinghub.elsevier.com/retrieve/pii/S0038092X16001468

    work page doi:10.1016/j.solener.2015.12.055 2017

  3. [3]

    Watkins, M

    M. Watkins, M. Wagner, J. Hinze, S. Sullivan, E. Vollnogle, J. Boxleitner, He- liostat aiming strategies for arbitrary flux profiles for high temperature gas-based receivers, in: SolarPACES 2019, AIP Conference Proceedings, 2020, p. 160008. doi:10.1063/5.0032149. URL http://aip.scitation.org/doi/abs/10.1063/5.0032149

    work page doi:10.1063/5.0032149 2019

  4. [4]

    Wang, C.-A

    S. Wang, C.-A. Asselineau, W. R. Logie, J. Pye, J. Coventry, MDBA: An accurate and efficient method for aiming heliostats, Sol. Energy 225 (2021) 694–707.doi: 10.1016/j.solener.2021.07.059. URL https://linkinghub.elsevier.com/retrieve/pii/S0038092X21006307

    work page doi:10.1016/j.solener.2021.07.059 2021

  5. [5]

    Sánchez-González, M

    A. Sánchez-González, M. R. Rodríguez-Sánchez, D. Santana, Aiming factor to flatten the flux distribution on cylindrical receivers, Energy 153 (2018) 113–125. doi:10.1016/j.energy.2018.04.002. URL https://linkinghub.elsevier.com/retrieve/pii/S0360544218305929 27

    work page doi:10.1016/j.energy.2018.04.002 2018

  6. [6]

    D. J. Erasmus, A. Sánchez-González, T. W. von Backström, Blossaim, a determin- istic aiming strategy for circular aperture receivers, in: SolarPACES 2020, AIP Conference Proceedings, 2022, p. 120012.doi:10.1063/5.0085654. URL http://aip.scitation.org/doi/abs/10.1063/5.0085654

    work page doi:10.1063/5.0085654 2020

  7. [7]

    Ashley, E

    T. Ashley, E. Carrizosa, E. Fernández-Cara, Optimisation of aiming strategies in Solar Power Tower plants, Energy 137 (2017) 285–291.doi:10.1016/j.energy. 2017.06.163. URL http://linkinghub.elsevier.com/retrieve/pii/S0360544217311581

    doi:10.1016/j.energy 2017

  8. [8]

    Sánchez-González, D

    A. Sánchez-González, D. J. Erasmus, M. C. Georgiou, Aiming Strategy for a Circular Aperture Receiver: Experimental Validation at PROTEAS, SolarPACES Conference Proceedings 1 (jan 2024).doi:10.52825/solarpaces.v1i.734. URL https://www.tib-op.org/ojs/index.php/solarpaces/article/view/ 734

    work page doi:10.52825/solarpaces.v1i.734 2024

Show all 53 references
  1. [9]

    S. M. Besarati, D. Yogi Goswami, E. K. Stefanakos, Optimal heliostat aiming strategy for uniform distribution of heat flux on the receiver of a solar power tower plant, Energy Conv. Manag. 84 (2014) 234–243.doi:10.1016/j.enconman.2014. 04.030. URL http://linkinghub.elsevier.co...

    2014 doi

  2. [10]

    Q. Yu, Z. Wang, E. Xu, Analysis and improvement of solar flux distribution inside a cavity receiver based on multi-focal points of heliostat field, Appl. Energy 136 (2014) 417–430. doi:10.1016/j.apenergy.2014.09.008. URL http://linkinghub.elsevier.com/retrieve/pii/S0306261914009556

    2014 doi

  3. [11]

    Flesch, C

    R. Flesch, C. Frantz, D. Maldonado Quinto, P. Schwarzbözl, Towards an optimal aiming for molten salt power towers, Sol. Energy 155 (2017) 1273–1281. doi: 10.1016/j.solener.2017.07.067. URL http://dx.doi.org/10.1016/j.solener.2017.07.067

    2017 doi

  4. [12]

    Milidonis, M

    K. Milidonis, M. J. Blanco, V. Grigoriev, C. F. Panagiotou, A. M. Bonanos, M. Constantinou, J. Pye, C. A. Asselineau, Review of application of AI tech- niques to Solar Tower Systems, Sol. Energy 224 (June) (2021) 500–515. doi: 10.1016/j.solener.2021.06.009. URL https://doi.org...

    2021 doi

  5. [13]

    A. I. Turja, M. M. Hasan, M. M. Ehsan, Y. Khan, Multi-objective performance optimization & thermodynamic analysis of solar powered supercritical co2 power 28 cycles using machine learning methods & genetic algorithm, Energy AI 15 (2024) 100327

    2024

  6. [14]

    Ouyang, M

    T. Ouyang, M. Pan, Y. Huang, X. Tan, P. Qin, Thermodynamic design and power prediction of a solar power tower integrated system using neural networks, Energy 278 (PA) (2023) 127849.doi:10.1016/j.energy.2023.127849. URL https://doi.org/10.1016/j.energy.2023.127849

    2023

  7. [15]

    Z. Zeng, D. Ni, G. Xiao, Real-time heliostat field aiming strategy optimiza- tion based on reinforcement learning, Appl. Energy 307 (September 2021) (2022) 118224. doi:10.1016/j.apenergy.2021.118224. URL https://doi.org/10.1016/j.apenergy.2021.118224

    2022

  8. [16]

    S. Wu, D. Ni, A method for real-time optimal heliostat aiming strategy generation via deep learning, Eng. Appl. Artif. Intell. 127 (PB) (2024) 107279.doi:10.1016/ j.engappai.2023.107279. URL https://doi.org/10.1016/j.engappai.2023.107279

    2024

  9. [17]

    J. A. Carballo, J. Bonilla, N. C. Cruz, J. Fernández-Reche, J. D. Álvarez, A. Avila- Marin, M. Berenguel, Reinforcement learning for heliostat aiming: Improving the performance of Solar Tower plants, Appl. Energy 377 (PB) (2025) 124574.doi: 10.1016/j.apenergy.2024.124574. URL ...

    2025

  10. [18]

    A. O. Fajemisin, D. Maragno, D. den Hertog, Optimization with constraint learn- ing: A framework and survey, Eur. J. Oper. Res. 314 (1) (2024) 1–14

    2024

  11. [19]

    Maragno, H

    D. Maragno, H. Wiberg, D. Bertsimas, c. I. Birbil, D. den Hertog, A. O. Fajemisin, Mixed-integer optimization with constraint learning, Oper. Res. (2023)

    2023

  12. [20]

    Fahmi, S

    I. Fahmi, S. Cremaschi, Process synthesis of biodiesel production plant using ar- tificial neural networks as the surrogate models, Comput. Chem. Eng. 46 (2012) 105–123

    2012

  13. [21]

    J. L. Cremer, I. Konstantelos, S. H. Tindemans, G. Strbac, Data-driven power system operation: Exploring the balance between cost and risk, IEEE Trans. Power Syst. 34 (1) (2018) 791–801

    2018

  14. [22]

    Halilbašić, F

    L. Halilbašić, F. Thams, A. Venzke, S. Chatzivasileiadis, P. Pinson, Data-driven security-constrained AC-OPF for operations and markets, in: 2018 power systems computation conference (PSCC), IEEE, 2018, pp. 1–7. 29

    2018

  15. [23]

    Y. Chen, Y. Shi, B. Zhang, Input convex neural networks for optimal voltage regulation, arXiv preprint arXiv:2002.08684 (2020)

    2020 arXiv

  16. [24]

    Venzke, S

    A. Venzke, S. Chatzivasileiadis, Verification of neural network behaviour: Formal guarantees for power system applications, IEEE Trans. Smart Grid 12 (1) (2020) 383–397

    2020

  17. [25]

    E. Prat, S. Chatzivasileiadis, Learning active constraints to efficiently solve linear bilevel problems: Application to the generator strategic bidding problem, IEEE Trans. Power Syst. 38 (3) (2022) 2376–2387

    2022

  18. [26]

    Alcántara, C

    A. Alcántara, C. Ruiz, A neural network-based distributional constraint learning methodology for mixed-integer stochastic optimization, Expert Syst. Appl. 232 (2023) 120895

    2023

  19. [27]

    Dolányi, K

    M. Dolányi, K. Bruninx, J.-F. Toubeau, E. Delarue, Capturing electricity mar- ket dynamics in strategic market participation using neural network constrained optimization, IEEE Trans. Power Syst. 39 (1) (2023) 533–545

    2023

  20. [28]

    Y. Song, G. Chen, H. Zhang, Constraint learning-based optimal power dispatch for active distribution networks with extremely imbalanced data, CSEE J. Power Energy Syst. (2023)

    2023

  21. [29]

    Alcántara, C

    A. Alcántara, C. Ruiz, Optimal day-ahead offering strategy for large producers based on market price response learning, Eur. J. Oper. Res. 319 (3) (2024) 891– 907

    2024

  22. [30]

    Alcántara, C

    A. Alcántara, C. Ruiz, C. Tsay, A Quantile Neural Network Framework for Two- stage Stochastic Optimization, arXiv preprint arXiv:2403.11707 (2024)

    2024 arXiv

  23. [31]

    J. Zhou, T. Shi, J. Ren, C. He, Accelerating operation optimization of complex chemical processes: A novel framework integrating artificial neural network and mixed-integer linear programming, Chem. Eng. J. 481 (2024) 148421

    2024

  24. [32]

    Allowable Flux Density

    L. L. Vant-Hull, The Role of "Allowable Flux Density" in the Design and Oper- ation of Molten-Salt Solar Central Receivers, J. Sol. Energy Eng. Trans.-ASME 124 (2) (2002) 165.doi:10.1115/1.1464124. URL http://solarenergyengineering.asmedigitalcollection.asme.org/ article.aspx?...

    2002 doi

  25. [33]

    Biggs, C

    F. Biggs, C. N. Vittitoe, Helios model for the optical behavior of reflecting solar concentrators, Tech. rep., Sandia National Laboratories, SAND-76-0347 (1979). 30

    1979

  26. [34]

    Schmitz, P

    M. Schmitz, P. Schwarzbözl, R. Buck, R. Pitz-Paal, Assessment of the potential improvement due to multiple apertures in central receiver sys- tems with secondary concentrators, Sol. Energy 80 (1) (2006) 111–120. doi:10.1016/j.solener.2005.02.012. URL http://www.sciencedirect.c...

    2006 doi

  27. [35]

    W. R. Logie, J. D. Pye, J. Coventry, Thermoelastic stress in concentrating solar receiver tubes: A retrospect on stress analysis methodology, and comparison of salt and sodium, Sol. Energy 160 (November 2017) (2018) 368–379.doi:10.1016/ j.solener.2017.12.003. URL http://linkin...

    2018

  28. [36]

    Sánchez-González, D

    A. Sánchez-González, D. Santana, Solar flux distribution on central receivers: A projection method from analytic function, Renew. Energy 74 (2015) 576–587.doi: 10.1016/j.renene.2014.08.016. URL http://linkinghub.elsevier.com/retrieve/pii/S0960148114004753

    2015 doi

  29. [37]

    F. J. Collado, A. Gómez, J. Turégano, An analytic function for the flux density due to sunlight reflected from a heliostat, Sol. Energy 37 (3) (1986) 215–234. doi:10.1016/0038-092X(86)90078-2. URL http://linkinghub.elsevier.com/retrieve/pii/0038092X86900782

    1986

  30. [38]

    S. I. Gallant, Neural Network Learning and Expert Systems, MIT press, 1993

    1993

  31. [39]

    Wadsworth Intern, Group, Belmont, California (1984)

    L.Breiman, J.H.Friedman, R.A.Olshen, C.J.Stone, Classificationandregression trees. Wadsworth Intern, Group, Belmont, California (1984)

    1984

  32. [40]

    Breiman, Random forests, Mach

    L. Breiman, Random forests, Mach. Learn. 45 (2001) 5–32

    2001

  33. [41]

    Goodfellow, Y

    I. Goodfellow, Y. Bengio, A. Courville, Deep Learning, MIT Press, 2016

    2016

  34. [42]

    Huchette, G

    J. Huchette, G. Muñoz, T. Serra, C. Tsay, When deep learning meets polyhedral theory: A survey, arXiv preprint arXiv:2305.00241 (2023)

    2023 arXiv

  35. [43]

    Bonami, A

    P. Bonami, A. Lodi, A. Tramontani, S. Wiese, On mathematical programming with indicator constraints, Math. Program. 151 (2015) 191–223

    2015

  36. [44]

    Grimstad, H

    B. Grimstad, H. Andersson, ReLU networks as surrogate models in mixed-integer linear programs, Comput. Chem. Eng. 131 (2019) 106580. 31

    2019

  37. [45]

    Anderson, J

    R. Anderson, J. Huchette, W. Ma, C. Tjandraatmadja, J. P. Vielma, Strong mixed- integer programming formulations for trained neural networks, Math. Program. 183 (1) (2020) 3–39

    2020

  38. [46]

    C. Tsay, J. Kronqvist, A. Thebelt, R. Misener, Partition-based formulations for mixed-integer optimization of trained ReLU neural networks, Advances in neural information processing systems 34 (2021) 3068–3080

    2021

  39. [47]

    Lomuscio, L

    A. Lomuscio, L. Maganti, An approach to reachability analysis for feed-forward relu neural networks, arXiv preprint arXiv:1706.07351 (2017)

    2017 arXiv

  40. [48]

    Tjeng, K

    V. Tjeng, K. Xiao, R. Tedrake, Evaluating robustness of neural networks with mixed integer programming, arXiv preprint arXiv:1711.07356 (2017)

    2017 arXiv

  41. [49]

    Fischetti, J

    M. Fischetti, J. Jo, Deep neural networks and mixed integer linear optimization, Constraints 23 (3) (2018) 296–309

    2018

  42. [50]

    Idris, NumPy: Beginner’s Guide, Packt Publishing Ltd, 2015

    I. Idris, NumPy: Beginner’s Guide, Packt Publishing Ltd, 2015

    2015

  43. [51]

    Kronqvist, B

    J. Kronqvist, B. Li, J. Rolfes, S. Zhao, Alternating mixed-integer programming and neural network training for approximating stochastic two-stage problems, in: International Conference on Machine Learning, Optimization, and Data Science, Springer, 2023, pp. 124–139

    2023

  44. [52]

    Gimelshein, L

    A.Paszke, S.Gross, F.Massa, A.Lerer, J.Bradbury, G.Chanan, T.Killeen, Z.Lin, N. Gimelshein, L. Antiga, et al., Pytorch: An imperative style, high-performance deep learning library, Advances in neural information processing systems 32 (2019)

    2019

  45. [53]

    URL https://www.gurobi.com 32

    Gurobi Optimization, LLC, Gurobi Optimizer Reference Manual (2024). URL https://www.gurobi.com 32

    2024

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.