benchmarks/industry_techspec/cec_pv_module/description.md

CEC Certified PV Module Database: Sandia / CEC 5-parameter single-diode model

Contamination tier: high. The single-diode PV-cell I-V equation (Shockley 1949 diode + series/shunt resistance, reformulated by De Soto, Klein & Beckman 2006) is the textbook equation of module-level PV performance — in every solar-energy textbook, in Wikipedia, and in every arXiv preprint on PV modelling. An LLM will almost certainly recite the functional form from memory. What is not in the textbooks is a certified snapshot of 21 535 real commercial modules with per-module fitted parameters; and what is not trivial is the cross-module aggregate scaling (P_mp ~ eta * A * G_STC) that an SR method should also rediscover. Track roles: Track-C red-team for the single-diode form (tests whether an LLM is memorizing De Soto 2006) and Track-A / Track-B discovery for the aggregate and technology-conditional scalings. A useful number on this benchmark only reports both together.

The dataset at a glance

The California Energy Commission (CEC) maintains the most widely-used public registry of PV modules eligible for incentive programs in California. Each listed module has been put through an I-V sweep at an accredited test laboratory (TUV Rheinland, SGS, ETL/Intertek, PV Evolution Labs, ...) under Standard Test Conditions

$$\text{STC}:G=1000\mathrm{W}\mathrm{/}{\mathrm{m}}^2,T_{cell}={25}^{\circ }\mathrm{C},\text{AM1.5 spectrum}\backslash text\{STC\}\; :\; \backslash quad\; G\; =\; 1000\backslash ,\backslash mathrm\{W/m^2\},\; \backslash ;\; T\_\{cell\}\; =\; 25^\backslash circ\backslash mathrm\{C\},\; \backslash ;\; \backslash text\{AM1.5\; spectrum\}$$

and under the Nominal Operating Cell Temperature (NOCT) test

$$\text{NOCT}:G=800\mathrm{W}\mathrm{/}{\mathrm{m}}^2,T_{air}={20}^{\circ }\mathrm{C},v_{wind}=1\mathrm{m}\mathrm{/}\mathrm{s},\backslash text\{NOCT\}\; :\; \backslash quad\; G\; =\; 800\backslash ,\backslash mathrm\{W/m^2\},\; \backslash ;\; T\_\{air\}\; =\; 20^\backslash circ\backslash mathrm\{C\},\; \backslash ;\; v\_\{wind\}\; =\; 1\backslash ,\backslash mathrm\{m/s\},$$

from which the lab reports the five I-V-curve key points $(I_{sc}, V_{oc}, I_{mp}, V_{mp}, P_{mp})$ at STC plus the temperature coefficients $(\alpha_{sc}, \beta_{oc}, \gamma_r)$. NREL's System Advisor Model (SAM) then fits the five-parameter De Soto model $(a, I_L, I_o, R_s, R_{sh})$ so that the resulting single-diode I-V curve reproduces those measured key points. Both the lab measurements and the fitted single-diode parameters are what CEC certifies and what ships in the data.csv here.

The snapshot used in this entry contains 21 535 modules across five technologies:

Technology	n
Multi-c-Si	11 221
Mono-c-Si	9 725
Thin Film	561
CdTe	20
CIGS	8

Ground-truth physics

The De Soto 5-parameter single-diode model

At the heart of module-level PV modelling is the ideal-diode equation with a series and a shunt resistor added to account for lumped non-idealities:

$$I(V)=I_L-I_o\text{\hspace{0.17em}⁣}[\exp \text{\hspace{0.17em}⁣}\left(\frac{V+I{R}_s}{a}\right)-1]-\frac{V+I{R}_s}{R_{sh}},a\equiv \frac{n{N}_{s}k{T}_{cell}}{q}\mathrm{.}I(V)\; \backslash ;=\backslash ;\; I\_L\; \backslash ;-\backslash ;\; I\_o\backslash !\backslash left[\backslash exp\backslash !\backslash left(\backslash frac\{V\; +\; I\; R\_s\}\{a\}\backslash right)\; -\; 1\backslash right]\; \backslash ;-\backslash ;\; \backslash frac\{V\; +\; I\; R\_s\}\{R\_\{sh\}\},\; \backslash qquad\; a\; \backslash equiv\; \backslash frac\{n\; N\_s\; k\; T\_\{cell\}\}\{q\}.$$

Five per-module parameters completely determine the I-V curve under STC:

Symbol	CEC column	Unit	Meaning
$I_L$	I_L_ref	A	light-generated (photo) current
$I_o$	I_o_ref	A	diode reverse-saturation current
$R_s$	R_s	Ohm	lumped series resistance
$R_{sh}$	R_sh_ref	Ohm	shunt resistance
$a$	a_ref	V	modified ideality factor $n N_s k T/q$

Equation (I) is implicit in $I$ and has no closed-form solution; the max-power point $(V_{mp}, I_{mp})$ is found by maximizing $V I(V)$ numerically (e.g. Newton on $\partial P / \partial V = 0$, or by enumerating the I-V curve). formulas/single_diode_pmp.py uses pvlib.pvsystem.singlediode (a vectorized implementation by Jain & Kapoor 2004 of the Lambert-W closed form for $I(V)$) with a plain-scipy fallback.

Why parameter fitting is nontrivial

SAM's fit matches six datasheet constraints

$$I_{sc}^{meas},V_{oc}^{meas},I_{mp}^{meas},V_{mp}^{meas},{\alpha }_{sc}^{meas},{\beta }_{oc}^{meas}I\_\{sc\}^\{meas\},\backslash ;\; V\_\{oc\}^\{meas\},\backslash ;\; I\_\{mp\}^\{meas\},\backslash ;\; V\_\{mp\}^\{meas\},\backslash ;\; \backslash alpha\_\{sc\}^\{meas\},\backslash ;\; \backslash beta\_\{oc\}^\{meas\}$$

against five parameters $(I_L, I_o, R_s, R_{sh}, a)$ and one residual temperature-adjustment factor Adjust. De Soto 2004 / 2006 sets up the boundary conditions at $(V=0, I=I_{sc})$, $(V=V_{oc}, I=0)$, $(V_{mp}, I_{mp})$, plus $dP/dV = 0$ at MPP and a temperature-coefficient matching condition, and solves the resulting nonlinear system. Once the parameters are known, evaluating the formula on the database exactly reproduces the measured $I_{sc}, V_{oc}, I_{mp}, V_{mp}$ and therefore $P_{mp}$ — hence the $R^2 \approx 1$ score of single_diode_pmp in reference_scores. This is not a fluke; it is the design of the certification process.

Approximate closed-form results

In the limit $R_s \to 0$ and $R_{sh} \to \infty$ the single-diode equation yields the two classical closed forms (both reproducible from the data here):

$$I_{sc}\approx I_L\text{(STC, shortcircuit)}I\_\{sc\}\; \backslash ;\backslash approx\backslash ;\; I\_L\; \backslash qquad\backslash text\{(STC,\; shortcircuit)\}$$

with RMS residual $0.075$ A ($R^2 = 0.997$), and

$$V_{oc}\approx a\ln \text{\hspace{0.17em}⁣}\left(\frac{I_L}{I_o}\right)\text{(STC, opencircuit)}V\_\{oc\}\; \backslash ;\backslash approx\backslash ;\; a\; \backslash ,\; \backslash ln\backslash !\backslash left(\backslash frac\{I\_L\}\{I\_o\}\backslash right)\; \backslash qquad\backslash text\{(STC,\; opencircuit)\}$$

with RMS residual $0.11$ V ($R^2 = 0.9999$). These are left out of the main ground_truth list (the target is $P_{mp}$, not $I_{sc}$ / $V_{oc}$) but are useful physics sanity checks for a candidate SR method that proposes the Shockley form — see the __main__ reproducer hooks in formulas/.

The Sandia Array Performance Model (SAPM) connection

King et al. 2004 (SAND2004-3535) define the PTC test

$$\text{PTC}:G=1000\mathrm{W}\mathrm{/}{\mathrm{m}}^2,T_{air}={20}^{\circ }\mathrm{C},v_{wind}=1\mathrm{m}\mathrm{/}\mathrm{s}\backslash text\{PTC\}\; :\; \backslash quad\; G\; =\; 1000\backslash ,\backslash mathrm\{W/m^2\},\; \backslash ;\; T\_\{air\}\; =\; 20^\backslash circ\backslash mathrm\{C\},\; \backslash ;\; v\_\{wind\}\; =\; 1\backslash ,\backslash mathrm\{m/s\}$$

and an empirical temperature model

$$T_{cell}^{PTC}\approx T_{air}+\frac{G}{800\mathrm{W}\mathrm{/}{\mathrm{m}}^2}(T_{NOCT}-{20}^{\circ }\mathrm{C})=20+1.25(T_{NOCT}-20){[}^{\circ }\mathrm{C}]T\_\{cell\}^\{PTC\}\; \backslash ;\backslash approx\backslash ;\; T\_\{air\}\; +\; \backslash frac\{G\}\{800\backslash ,\backslash mathrm\{W/m^2\}\}\backslash ,(T\_\{NOCT\}\; -\; 20^\backslash circ\backslash mathrm\{C\})\; \backslash ;=\backslash ;\; 20\; +\; 1.25\backslash ,(T\_\{NOCT\}\; -\; 20)\backslash ;\; [^\backslash circ\backslash mathrm\{C\}]$$

from which the PTC power can be predicted by a linear temperature-coefficient correction

$$P_{PTC}\approx P_{mp}^{STC}(1+\frac{{\gamma }_r}{100}(T_{cell}^{PTC}-25)),P\_\{PTC\}\; \backslash ;\backslash approx\backslash ;\; P\_\{mp\}^\{STC\}\; \backslash Bigl(1\; +\; \backslash tfrac\{\backslash gamma\_r\}\{100\}\backslash ,(T\_\{cell\}^\{PTC\}\; -\; 25)\backslash Bigr),$$

reproducing the CEC PTC column with $R^2 \approx 0.98$, RMSE $\approx 6.9$ W. We do not ship this as a ground-truth formula (the target is $P_{mp}^{STC}$, not PTC) but every column needed to build it is retained in data.csv, so it is a natural second benchmark variant to add later.

Aggregate efficiency-area scaling

At the crudest aggregate level, by definition of module efficiency,

$$P_{mp}^{STC}=\eta \cdot A_c\cdot G_{STC},G_{STC}=1000\mathrm{W}\mathrm{/}{\mathrm{m}}^2\mathrm{.}P\_\{mp\}^\{STC\}\; \backslash ;=\backslash ;\; \backslash eta\; \backslash cdot\; A\_c\; \backslash cdot\; G\_\{STC\},\; \backslash qquad\; G\_\{STC\}\; =\; 1000\backslash ,\backslash mathrm\{W/m^2\}.$$

Running a single global mean efficiency $\bar{\eta} = 0.1539$ across all 21 535 modules gives $R^2 \approx 0.665$ and RMSE $\approx 34$ W — the "low-information" baseline a pure dimensional-analysis SR method should return. Per- technology means sit at

Technology	$\bar{\eta}$	n
CdTe	0.165	20
Mono-c-Si	0.160	9 725
Multi-c-Si	0.151	11 221
CIGS	0.141	8
Thin Film	0.112	561

and conditioning on Technology sharpens the per-technology scaling fits substantially (e.g. to $R^2 \approx 0.97$ for CIGS, $\approx 0.74$ for mono/multi-Si).

Instance-per-row vs. cross-module framing

A subtle point that distinguishes this entry from most other real-sr entries is that each row is its own 5-parameter fitting problem in the upstream workflow. The CEC database already contains the fit; an SR benchmark built on the database is really asking:

Given a row's certified single-diode parameters and/or datasheet key points, can a symbolic regressor predict the row's $P_{mp}^{STC}$?

Two honest answers exist:

1. Per-row physics (Track-C red-team). With the 5 fitted parameters as input, the single-diode formula is tautological — $R^2$ collapses to the numerical precision of the root solver. This is useful only as a memorization test: an LLM-driven SR method should spot this and not claim credit for "discovery". The score it reports here is a lower bound on how faithfully it evaluates the formula it proposes, not on its discovery ability.
2. Cross-module scaling (Track-A discovery). Using only technology-agnostic inputs (A_c, N_s, Technology, geometric dimensions) the problem is genuinely underdetermined and the best a symbolic regressor can hope for is an approximate scaling law — reflected in the $R^2 \approx 0.67$ of efficiency_area_scaling. This is the real discovery benchmark and is where real-sr expects to see a spread of methods.

Known limitations of the ground-truth formulas

1. Tautological by construction. As discussed above, the upstream fit makes single_diode_pmp exactly reproduce P_mp_stc. Any "score" reported against this target should be explicitly labelled Track-C.
2. No off-STC dependence. The ground-truth formulas all report values at Standard Test Conditions; operational performance under arbitrary $(G, T_{cell})$ requires the full De Soto temperature / irradiance translation equations (not covered here, but every column needed is retained).
3. Technology-agnostic aggregate fit. The shipped efficiency_area_scaling uses a single global $\bar\eta$ and is correspondingly coarse. A per-technology variant is trivial to build from the Technology column.
4. Bifacial and BIPV oddities. 133 rows are bifacial and the reported STC power is the front-only value; a few BIPV entries have non-standard aperture area definitions. These are kept verbatim; any robust SR method should flag them as outliers.
5. Thin-film entries with tiny $I_o$. A handful of older thin-film rows have $I_o < 10^{-18}$ A which pushes the Lambert-W solver into its numerical floor. The benchmark scorer retains those rows; their residuals are negligible at aggregate level.

Track / contamination framing (data_sources_survey.md §3-§4)

• Track-C red team for the single-diode formula itself. Shockley (1949), De Soto (2004 thesis / 2006 paper), Duffie & Beckman's textbook, pvlib.pvsystem.singlediode source code, and thousands of arXiv preprints all contain the formula explicitly. An LLM-SR method that names it and plugs in the 5 CEC-certified per-row parameters will score near-perfect; that outcome is informative only as a memorization probe.
• Track-A discovery for the aggregate power-area scaling, the per-technology efficiency fits, and the PTC temperature correction (reachable from retained columns). These involve genuine parameter identification and dimensional reasoning, not textbook recitation.
• Track-B real-but-ambiguous framing for any rule linking the fitted 5 parameters back to the measured datasheet key points (e.g. predict $I_L$ given $I_{sc}$, $V_{oc}$, $I_{mp}$, $V_{mp}$): the equations De Soto 2006 uses are known but their algebraic closed-form is non-unique and this is a natural "plausibility + prediction" benchmark rather than a scoreable discovery task.
• Recommended reporting. When benchmarking an LLM-SR method, break out the scores per ground-truth formula, flag single_diode_pmp as Track-C by convention, and report the Track-A efficiency_area_scaling score alongside so that memorization can be quantified.

References (stored in reference/)

• desoto2004_thesis.pdf, desoto2004_thesis.bib — De Soto's 2004 MS thesis (UW-Madison, Solar Energy Laboratory); contains the full algebraic derivation of the 5-parameter fit, worked examples, and the temperature/irradiance translation equations later condensed into the 2006 Solar Energy paper.
• desoto2006.bib — BibTeX for De Soto, Klein & Beckman 2006, Improvement and validation of a model for photovoltaic array performance, Solar Energy 80(1):78-88, doi:10.1016/j.solener.2005.06.010 (the published journal version; Elsevier paywall, so only a BibTeX record is redistributed here — the 2004 thesis PDF is the open-access equivalent).
• king2004_sapm.pdf, king2004_sapm.bib — Sandia National Labs report SAND2004-3535, Photovoltaic Array Performance Model, the authoritative reference for the PTC test conditions, the SAPM empirical temperature model, and the I-V-curve key-point temperature response that CEC certification enforces.
• pvlib_cecmod.bib — pvlib-python documentation entry for pvlib.pvsystem.retrieve_sam("CECMod"), the programmatic access point used by data/download.sh.
• cec_solar_equipment_list.bib — California Energy Commission "Solar Equipment Lists -- PV Modules" (the upstream public registry whose SAM snapshot is what we distribute).