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

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

and under the Nominal Operating Cell Temperature (NOCT) test

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

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:

$$
I(V) \;=\; I_L \;-\; I_o\!\left[\exp\!\left(\frac{V + I R_s}{a}\right) - 1\right] \;-\; \frac{V + I R_s}{R_{sh}},
\qquad a \equiv \frac{n N_s k T_{cell}}{q}.
$$

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

$$
I_{sc}^{meas},\; V_{oc}^{meas},\; I_{mp}^{meas},\; V_{mp}^{meas},\;
\alpha_{sc}^{meas},\; \beta_{oc}^{meas}
$$

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):

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

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

$$
V_{oc} \;\approx\; a \, \ln\!\left(\frac{I_L}{I_o}\right) \qquad\text{(STC, opencircuit)}
$$

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

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

and an empirical temperature model

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

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

$$
P_{PTC} \;\approx\; P_{mp}^{STC} \Bigl(1 + \tfrac{\gamma_r}{100}\,(T_{cell}^{PTC} - 25)\Bigr),
$$

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,

$$
P_{mp}^{STC} \;=\; \eta \cdot A_c \cdot G_{STC}, \qquad G_{STC} = 1000\,\mathrm{W/m^2}.
$$

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).