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REVIEW 4 major objections 6 minor 36 references

Minimising the Levelised Cost of Electricity for Bifacial Solar Panel Arrays using Bayesian Optimisation

T0 review · 4 major / 6 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A Bayesian optimization over panel tilt and row spacing finds large-scale solar farm layouts that produce electricity up to 23 percent cheaper than the usual latitude-tilt, winter-solstice-spacing design rule.

desk verdict A clean, reimplementable computational study showing that rule-of-thumb tilt and spacing for bifacial farms are not LCOE-optimal; the quantitative 23% claim is model-internal and should be framed as such. read the letter →

arxiv 1909.01660 v1 pith:2K5XDWMQ submitted 2019-09-04 physics.app-ph physics.data-an

Peter Tillmann , Klaus Jäger , Christiane Becker This is my paper · ORCID
classification physics.app-phphysics.data-an
keywords bifacialphotovoltaicslevelisedcostofelectricityBayesianoptimisationirradiancemodelmoduletiltrowspacingwintersolsticeruleenergyyield
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

This paper tries to prove that the standard rules for laying out large-scale solar farms—setting the panel tilt to the site latitude and choosing row spacing so rows never shade each other at winter solstice—do not minimize the levelized cost of electricity. Combining a geometric irradiance model with Bayesian optimization over tilt angle and row spacing, the paper maps the cost landscape for bifacial and monofacial arrays at two U.S. locations under five land-cost scenarios. The central finding is that the cheapest configuration depends strongly on land cost and local climate, and can be up to 23 percent cheaper than the rule-of-thumb layout in Seattle when land is expensive. If correct, the result makes site-specific, cost-aware geometry optimization a practical design step for PV plant developers.

What carries the argument

The central engine is a geometric illumination model for a two-dimensional periodic PV field, which decomposes each module's irradiance into four components on front and back—direct and diffuse light from the sky, and direct and diffuse light reflected from the ground—summed as eq. (1), with the ground-reflection terms computed as integrals over geometrical distribution functions in eq. (3). The model yields the annual energy yield of eq. (4), where the module is treated as current-limited by its least-illuminated position, and the cost model collapses to the closed-form LCOE expression eq. (12), namely $\mathrm{LCOE} = (\ell I_P \eta_f c_P + d c_L)/(\ell \cdot EY \cdot T)$. Bayesian optimization with a Gaussian-process surrogate and expected-improvement acquisition then navigates the (tilt, spacing) cost landscape, using the model's LCOE evaluations to identify the global minimum under each cost scenario.

What would settle it

Take a real bifacial test installation at a cloudy high-latitude site, measure the annual energy yield of several (tilt, spacing) configurations spanning the rule-of-thumb and the optimized designs, and compare the measured LCOE differences with the model's predictions; if the model overstates or reverses the yield ranking between configurations, the reported reductions would not carry over to practice.

Watch

Extended reading notes

Core claim

For large-scale bifacial photovoltaic fields, the levelized cost of electricity as a function of module tilt and row spacing is not minimized by the conventional winter-solstice design rule, and the divergence grows with land cost and with diffuse-light fraction. The paper's optimizer finds layouts with up to 23.5% lower LCOE in Seattle at land cost $c_L = 20\,\$/$m^2$ compared with tilt equal to latitude and the no-shading winter-solstice spacing; the improvement in Dallas at the same land cost is 7.4%. The optimized geometries are site-specific: higher land costs push rows closer together and tilt angles down, while bifacial arrays favor larger spacing and higher tilt than monofacial arrays. These results are computed from a two-dimensional periodic-field illumination model with annual TMY3 irradiance data, and they lead the authors to conclude that tilt and spacing should be optimized independently rather than set by rule of thumb.

Load-bearing premise

The predicted savings are only as good as the model's ability to tell how much electricity different panel layouts produce, and that model assumes evenly spaced infinite rows, non-reflective panels, and a fixed 30% ground albedo, without being validated against real field measurements.

Editorial extensions

If this is right

  • At high land costs, the minimum-LCOE row spacing becomes much shorter than the winter-solstice no-shading spacing, so the rule overstates land consumption for a given power output.
  • Bifacial modules are more sensitive to tilt and spacing than monofacial ones because roughly three-quarters of their rear-side illumination comes from ground reflection, so choosing geometry by rule of thumb forfeits more of their potential.
  • In cloudy, high-latitude climates with expensive land, the LCOE landscape is steep, and small deviations from the optimum cost the most; Seattle's 23% reduction illustrates this.
  • Because only the ratio of land cost to total cost enters LCOE, the optimized geometry is stable under proportional scaling of module and land prices, e.g. $c_L = 10\,\$/$m^2$ with $c_P = 1500\,\$/$kWp$ gives the same layout as $c_L = 5\,\$/$m^2$ with $c_P = 750\,\$/$kWp$.
  • The same pipeline can be rerun for any location with TMY3 data, giving site-specific design guidelines rather than a universal tilt/spacing rule.

Reading between the lines

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

  • Inference: if the geometric model's ranking of configurations is validated against field measurements, the same optimization could be extended to additional free parameters such as mounting height, variable ground albedo, or row segmentation, where the sweet spot is even harder to guess.
  • Inference: the paper's strongest regime—cloudy high-latitude sites with expensive land—is also the regime where its fixed-albedo and isotropic-sky assumptions are most likely to matter, so the 23% figure should be treated as a model-based estimate until field data confirm the energy-yield differences.
  • Inference: a practical design tool could precompute one design map per general climate class, letting developers choose tilt and spacing from land price and latitude without running an optimization themselves; the paper's cost-ratio invariance makes such a map two-dimensional rather than four-dimensional.
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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 develops a two-dimensional, periodic illumination model for bifacial PV modules in large fields, computes annual energy yields from TMY3 data for Dallas and Seattle, and combines the yield model with a simple levelized-cost-of-electricity (LCOE) expression to optimize module tilt and row spacing by Bayesian optimization. Five land-cost scenarios are considered. The central quantitative claim is that the usual design rules, namely module tilt equal to site latitude and row spacing with no mutual shading at the winter solstice, are suboptimal: the optimizer finds configurations with up to 23% lower LCOE in Seattle at cL = 20 $/m2, with the optimized geometries depending strongly on location and cost scenario.

Significance. If the illumination model is accurate, the paper makes a useful contribution by showing, within a single internally consistent framework, that site-specific simultaneous optimization of tilt and spacing can matter more than traditional heuristic rules, and that land cost shifts the optimum toward denser layouts. The internal consistency is a genuine strength: the cost algebra in Eqs. (6)-(12) is correct, the baseline and optimized LCOE values are computed from the same forward model, and no parameter is fitted to reproduce the headline result. The series-connected current-limit assumption in Eq. (5) is also conservative for dense layouts. However, the quantitative significance is conditional on the irradiance model, which is not validated against field data or independent benchmark models, so the specific 23% figure should be treated as a model prediction rather than an established result.

major comments (4)
  1. [Section 2, Eqs. (4)-(5); Table 4] The central claim of up to 23% lower LCOE in Seattle at cL = 20 $/m2 is computed entirely with the Section 2 illumination model, which assumes an isotropic diffuse sky, optically black modules, and a Lambertian ground with fixed albedo A = 30%. No numerical comparison is made with measured yields or with the published models cited as related work (Marion et al., ref. 19; Kreinin et al., ref. 17). At the relevant Seattle optimum the geometry is dense (d = 3.6 m, Table 5), so rear-side ground-reflected irradiance and inter-row shading dominate the comparison. A systematic bias in these components would shift both the optimized geometry and the magnitude of the LCOE gap. To establish the headline number, the authors should benchmark the model against at least one measured system or against an independent published model, or alternatively provide a quantified sensitivity/uncertainty analysis showing how the optimized geometries and the 23% reduction change under plausible variations in the model assumptions.
  2. [Section 2, Eq. (3)] The illumination model is not specified in enough detail to be reproduced. Eq. (3) leaves the ground distribution functions gamma_dir(xg) and gamma_diff(xg) and the integration limits alpha1(s) and alpha2(s) undefined; the treatment of shadowed and sunlit ground fractions is described only verbally. Since the entire optimization result is generated by this model, the authors should either provide explicit closed-form definitions of these functions and integration bounds, or make a reference implementation available as supplementary code.
  3. [Section 4.2-4.3, Figs. 6-7] The manuscript repeatedly uses the term "global minimum" for the Bayesian optimization results, but expected-improvement Bayesian optimization is a heuristic method with no guarantee of global optimality. No iteration count, kernel choice, acquisition-function optimization settings, or convergence diagnostics are reported. The red dots in Figs. 6 and 7 should be described as the best configurations found by the optimizer, and the claims in the abstract and conclusions should be softened accordingly unless convergence to the true global optimum is demonstrated.
  4. [Section 3.2 and Table 5] All results fix the module height at h = 0.5 m and the albedo at A = 30%. The paper states that the bifacial gain saturates above h = 0.5 m and cites agreement with Kreinin et al., but no quantitative comparison is shown. In the high-land-cost optima of Table 5 (d = 3.2-3.6 m), rear irradiance is sensitive to both h and A. A short sensitivity sweep over these two parameters is needed to establish that the design guidelines and the reported LCOE reductions are robust rather than artifacts of the chosen fixed values.
minor comments (6)
  1. [Section 4.1 heading] The heading contains a typo: "Levelied cost of electricity" should be "Levelized cost of electricity" (or "Levelised" in British spelling).
  2. [References] References 16 and 35 are the same paper by Patel et al. in Applied Energy; the duplicate should be merged and the citation list cleaned up.
  3. [Abstract and Conclusions] The claim that the algorithm enables design guidelines "for most regions on Earth" is stronger than what is demonstrated, since only two locations and one TMY per location are used; this should be qualified.
  4. [Section 4.3, baseline definition] The rule-of-thumb baseline is defined with the 9 am winter-solstice no-shading rule. Since variants such as noon or 9 am-3 pm windows also appear in the literature, the choice should be justified and its influence on the reported LCOE reductions noted.
  5. [Eq. (4)-(5)] Hourly TMY3 values are treated as instantaneous irradiance when evaluating the current-limiting position in Eq. (5). Sub-hourly irradiance variability could affect the minimum-current position, and it would be helpful to state why this effect is expected to be small for annual energy totals.
  6. [Figs. 6-7] The color maps saturate at 4.5 and 5.4 cents/kWh respectively, which makes it difficult to read the LCOE landscape near the rule-of-thumb geometries; using a perceptually uniform scale without clipping would improve interpretability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the LCOE landscape is computed by forward illumination and cost equations, the optimizer minimizes that objective, and the baseline rule is an external benchmark; the only self-citation is a background textbook.

full rationale

The derivation chain is self-contained and no fitted parameter is used to reproduce the headline result. The illumination model (Sec. 2, Eqs. 1-5) converts DNI/DHI, geometry, and albedo into front/back irradiance distribution functions; Eq. (4) sums these over TMY3 hours to annual energy yield; Eq. (12) combines the resulting EY with cost inputs cP and cL to LCOE. Bayesian optimization minimizes this forward-computed objective, so the optima and Table 4 reductions are consequences of the model, not inputs to it. The comparative baseline is the external winter-solstice/latitude rule, which is not derived from the model. The self-citation (Ref. 24, a textbook co-authored by K. Jäger) is used only for standard solar-position background and a generic LCOE discussion; it does not supply the illumination model, the cost model, or the optimization. The agreement with Kreinin et al. for bifacial-gain saturation is cited external work and is not a fit. The model assumptions (isotropic diffuse sky, black modules, fixed 0.3 albedo, current-limit Eq. 5) are strong and unvalidated, but they are assumptions, not circular definitions; validation concerns belong to correctness risk, not circularity. Therefore no circular step can be exhibited.

Assumptions & free parameters 7 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a forward irradiance model with several hand-set inputs (efficiency, albedo, costs, lifetime, mounting height, module length) and on standard domain assumptions about periodicity, isotropy, and series-connected current limiting. No new entities are introduced. The model is not validated against measured yields, so the free parameters are assumptions rather than fitted values.

free parameters (7)
  • back-side efficiency eta_b = 0.18
    Chosen from literature as a bifaciality factor of 0.9 relative to the front; directly scales the back-side energy yield and the bifacial gain.
  • ground albedo A = 0.30
    Fixed for all simulations; ground-reflected light is the dominant contributor to rear-side irradiance, so this assumption materially affects optimal tilt and spacing.
  • system cost cP = 1000 $/kWp
    Assumed lifetime cost excluding land; sets the absolute LCOE scale and the trade-off against land cost.
  • land cost cL = 1, 2.5, 5, 10, 20 $/m2
    Five scenarios chosen to cover land-cost regimes; the largest LCOE reductions versus the rule of thumb occur at the highest values.
  • mounting height h = 0.5 m
    Fixed because the bifacial gain saturates above 0.5 m, citing Kreinin et al.; height is not optimized and its effect is only qualitatively discussed.
  • lifetime T = 25 years
    Typical module power-warranty period; LCOE scales inversely with T.
  • module length l = 1.96 m
    Geometric input used in the LCOE formula (eq. 12) and in the example geometry figures; the module width cancels out.
assumptions (6)
  • domain assumption The PV field is infinite and periodic, so boundary effects are neglected (Section 2).
    Boundary rows receive more light and would have lower LCOE; the paper targets large fields where this is a reasonable approximation.
  • domain assumption Solar modules are optically black and do not reflect light to other modules (Section 2).
    Simplifies the radiosity problem but omits inter-row reflections that can matter for bifacial modules and high-albedo ground.
  • domain assumption Diffuse sky radiation is isotropic at every timestamp (Section 3.1).
    TMY3 provides only DHI, so no angular distribution is used; real skies have circumsolar and horizon brightening.
  • domain assumption The series-connected module current is limited by the minimum irradiance point on the module (eq. 5, Section 3.1).
    Stated as a lower bound on module performance; bypass diodes would improve output, so the model is conservative.
  • domain assumption LCOE is computed with no discounting or capital cost, with constant annual energy output over the 25-year lifetime (Section 4.1, eqs. 6-12).
    The authors acknowledge more involved financial models exist; this simplification affects absolute LCOE but not necessarily the ranking of geometries.
  • domain assumption A Typical Meteorological Year represents long-term climate at each location (Section 3.1).
    Standard practice, but the quantitative conclusions are tied to the specific TMY3 datasets for Dallas/Fort Worth and Seattle.

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Cite this review

Pith. "Pith review of Minimising the Levelised Cost of Electricity for Bifacial Solar Panel Arrays using Bayesian Optimisation." pith.science (2026). https://pith.science/paper/2K5XDWMQ

@misc{pith2026190901660,
  author       = {Pith},
  title        = {Pith review of: Minimising the Levelised Cost of Electricity for Bifacial Solar Panel Arrays using Bayesian Optimisation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2K5XDWMQ}},
  note         = {Machine review of arXiv:1909.01660}
}
read the original abstract

Bifacial solar module technology is a quickly growing market in the photovoltaics (PV) sector. By utilising light impinging on both, front and back sides of the module, actual limitations of conventional monofacial solar modules can be overcome at almost no additional costs. Optimising large-scale bifacial solar power plants with regard to minimum levelised cost of electricity (LCOE), however, is challenging due to the vast amount of free parameters such as module inclination angle and distance, module and land costs, character of the surroundings, weather conditions and geographic position. We present a detailed illumination model for bifacial PV modules in a large PV field and calculate the annual energy yield exemplary for two locations with different climates. By applying the Bayesian optimisation algorithm we determine the global minimum of the LCOE for bifacial and monofacial PV fields at these two exemplary locations considering land costs in the model. We find that currently established design guidelines for mono- and bifacial solar farms often do not yield the minimum LCOE. Our algorithm finds solar panel configurations yielding up to 23 % lower LCOE compared to the established configuration with the module tilt angle equal to the latitude and the module distance chosen such that no mutual shading of neighboring solar panels occurs at winter solstice. Our algorithm enables the user to extract clear design guidelines for mono- and bifacial large-scale solar power plants for most regions on Earth and further accelerates the development of competitively viable photovoltaic systems.

Figures

Figures reproduced from arXiv: 1909.01660 by the authors.

Figure 1
Figure 1. Illustrating the geometrical configuration of a (periodic) PV field and the illumination components, which [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. An example for (a) a module configuration and (b) the corresponding diffuse and direct geometrical distribution [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Geometrical distribution functions on the module for light the module receives (a) from the sky and (b) the [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: Annual radiant exposure for bifacial modules and the contributions from front and back sides in a large PV [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: (left) Different annual radiant exposure components for a bifacial solar cell in Dallas. (right) Detailed picture [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: Results of the Bayesian optimisation for minimising LCOE of (a) [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]
Figure 7
Figure 7. Figure 7: Results of the Bayesian optimisation for minimising LCOE of (a) [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]
Figure 8
Figure 8. Figure 8: Results of the optimisation for different land cost scenarios in Dallas (red lines) and Seattle (blue lines): (a) [PITH_FULL_IMAGE:figures/full_fig_p012_8.png]
Figure 9
Figure 9. Figure 9: Climate diagrams for (a) the Dallas/Fort Worth area and (b) Seattle. These charts were generated on [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: 30 days moving average of diffuse light share (blue curve) and daily global horizontal irradiance (orange [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: 30 days moving average of diffuse light share (blue curve) and daily global horizontal irradiance (orange [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: (left) Different annual radiant exposure components for a bifacial solar cell in Seattle. (right) Detailed [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.