FA-93551 / Solar tracker geometry / Open access
Tracker row-to-row shaded fraction: coverage ratio placement · case 01
Densely packed rows are reported as less shaded than sparse ones.
ROOT CAUSE
The coverage ratio multiplies the numerator instead of dividing it.
VERIFIED REPAIR
Divide by gcr * cos(rot - proj).
Unsuccessful approach: Dropping gcr models rows with no spacing at all.
Case contract
rot is the row rotation and proj the sun zenith projected on the plane perpendicular to the axis, both signed degrees. |proj| >= 90 or cos(rot - proj) <= 0 means no direct light: [1.0, True]. Otherwise shaded fraction = clip(1 - cos(proj)/(gcr*cos(rot - proj)), 0, 1). Return [fraction rounded 4, fraction > 0].
Why this case matters
Single-axis and dual-axis solar trackers turn a sun direction into actuator commands; a sign, frame or limit mistake points a whole plant away from the sun or into a mechanical stop.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(rot, proj, gcr):
if abs(proj) >= 90:
return [1.0, True]
c = math.cos(math.radians(rot - proj))
if c <= 0:
return [1.0, True]
fs = 1 - math.cos(math.radians(proj)) * gcr / c
fs = min(1.0, max(0.0, fs))
return [round(fs, 4), fs > 0]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [-30, -80, 0.5], [0.4597, True]],
['regression: coverage ratio placement (partial repair)', [45, 70, 0.5], [0.2452, True]],
['control 1', [45, -45, 0.5], [0.0, False]], ['control 2', [45, 80, 0.6], [0.6467, True]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, -80, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [-10, -70, 0.4], [0.0, False]],
['control 1', [-10, -45, 0.6], [0.0, False]], ['control 2', [30, 45, 0.5], [0.0, False]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, -30, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [-10, -30, 0.3], [0.0, False]],
['control 1', [-30, 0, 0.5], [0.0, False]], ['control 2', [30, -80, 0.6], [1.0, True]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, 70, 0.6], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [45, 70, 0.6], [0.371, True]],
['control 1', [-45, -30, 0.6], [0.0, False]], ['control 2', [-20, 45, 0.6], [0.0, False]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [-10, 45, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [30, 60, 0.4], [0.0, False]],
['control 1', [0, -95, 0.5], [1.0, True]], ['control 2', [-10, -70, 0.4], [0.0, False]]]]
for label, args, expected in fixtures[N-1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| normal: rows facing the sun at noon | [0.6, True] | [0.0, False] | Failed |
| boundary: steep morning sun | [0.8264, True] | [0.3054, True] | Failed |
| boundary: sun below horizon | [1.0, True] | [1.0, True] | Passed |
| regression: coverage ratio placement | [0.8649, True] | [0.4597, True] | Failed |
| regression: coverage ratio placement (partial repair) | [0.8113, True] | [0.2452, True] | Failed |
| control 1 | [0.0, False] | [0.0, False] | Passed |
| control 2 | [0.8728, True] | [0.6467, True] | Failed |
SHA-256 / 82ab7fb352e3b33940d696c9f123fa16ada93029fe0926c8d85822c5dade3eee
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(rot, proj, gcr):
if abs(proj) >= 90:
return [1.0, True]
c = math.cos(math.radians(rot - proj))
if c <= 0:
return [1.0, True]
fs = 1 - math.cos(math.radians(proj)) / c
fs = min(1.0, max(0.0, fs))
return [round(fs, 4), fs > 0]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [-30, -80, 0.5], [0.4597, True]],
['regression: coverage ratio placement (partial repair)', [45, 70, 0.5], [0.2452, True]],
['control 1', [45, -45, 0.5], [0.0, False]], ['control 2', [45, 80, 0.6], [0.6467, True]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, -80, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [-10, -70, 0.4], [0.0, False]],
['control 1', [-10, -45, 0.6], [0.0, False]], ['control 2', [30, 45, 0.5], [0.0, False]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, -30, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [-10, -30, 0.3], [0.0, False]],
['control 1', [-30, 0, 0.5], [0.0, False]], ['control 2', [30, -80, 0.6], [1.0, True]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, 70, 0.6], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [45, 70, 0.6], [0.371, True]],
['control 1', [-45, -30, 0.6], [0.0, False]], ['control 2', [-20, 45, 0.6], [0.0, False]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [-10, 45, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [30, 60, 0.4], [0.0, False]],
['control 1', [0, -95, 0.5], [1.0, True]], ['control 2', [-10, -70, 0.4], [0.0, False]]]]
for label, args, expected in fixtures[N-1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| normal: rows facing the sun at noon | [0.0, False] | [0.0, False] | Passed |
| boundary: steep morning sun | [0.6527, True] | [0.3054, True] | Failed |
| boundary: sun below horizon | [1.0, True] | [1.0, True] | Passed |
| regression: coverage ratio placement | [0.7299, True] | [0.4597, True] | Failed |
| regression: coverage ratio placement (partial repair) | [0.6226, True] | [0.2452, True] | Failed |
| control 1 | [0.0, False] | [0.0, False] | Passed |
| control 2 | [0.788, True] | [0.6467, True] | Failed |
SHA-256 / 7bb96a35ea35a8309c9ebb96d623a4ee7a59e9767334e2c449a3891a0a5195ce
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(rot, proj, gcr):
if abs(proj) >= 90:
return [1.0, True]
c = math.cos(math.radians(rot - proj))
if c <= 0:
return [1.0, True]
fs = 1 - math.cos(math.radians(proj)) / (gcr * c)
fs = min(1.0, max(0.0, fs))
return [round(fs, 4), fs > 0]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [-30, -80, 0.5], [0.4597, True]],
['regression: coverage ratio placement (partial repair)', [45, 70, 0.5], [0.2452, True]],
['control 1', [45, -45, 0.5], [0.0, False]], ['control 2', [45, 80, 0.6], [0.6467, True]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, -80, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [-10, -70, 0.4], [0.0, False]],
['control 1', [-10, -45, 0.6], [0.0, False]], ['control 2', [30, 45, 0.5], [0.0, False]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, -30, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [-10, -30, 0.3], [0.0, False]],
['control 1', [-30, 0, 0.5], [0.0, False]], ['control 2', [30, -80, 0.6], [1.0, True]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [0, 70, 0.6], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [45, 70, 0.6], [0.371, True]],
['control 1', [-45, -30, 0.6], [0.0, False]], ['control 2', [-20, 45, 0.6], [0.0, False]]],
[['normal: rows facing the sun at noon', [0, 0, 0.4], [0.0, False]],
['boundary: steep morning sun', [-60, -70, 0.5], [0.3054, True]],
['boundary: sun below horizon', [0, 95, 0.4], [1.0, True]],
['regression: coverage ratio placement', [-10, 45, 0.5], [0.0, False]],
['regression: coverage ratio placement (partial repair)', [30, 60, 0.4], [0.0, False]],
['control 1', [0, -95, 0.5], [1.0, True]], ['control 2', [-10, -70, 0.4], [0.0, False]]]]
for label, args, expected in fixtures[N-1]:
check(label, solve(*args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| normal: rows facing the sun at noon | [0.0, False] | [0.0, False] | Passed |
| boundary: steep morning sun | [0.3054, True] | [0.3054, True] | Passed |
| boundary: sun below horizon | [1.0, True] | [1.0, True] | Passed |
| regression: coverage ratio placement | [0.4597, True] | [0.4597, True] | Passed |
| regression: coverage ratio placement (partial repair) | [0.2452, True] | [0.2452, True] | Passed |
| control 1 | [0.0, False] | [0.0, False] | Passed |
| control 2 | [0.6467, True] | [0.6467, True] | Passed |
SHA-256 / e4cf5fc4786c574cd57c24e4e1c4208c58f6892cf67ec1d0cedbcc3f1551b8ae
Verification & scope
Deterministic stipulated toy contract for teaching; no claim of conformance with any standard, vendor protocol or production controller. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.
Observations recorded using Python 3.12.14 at 2026-09-29T14:51:56.196096+00:00.
Case digest / bcbf4bcfa85c6c41cdf40ef3a7bfb0a736b049ba473a4d8264eaf00638e80b39