FA-93421 / Solar tracker geometry / Open access
Fixed panel angle of incidence: incidence formula terms · case 01
Incidence is wrong even for a flat panel.
ROOT CAUSE
The zenith sine and cosine are swapped between the vertical and horizontal terms.
VERIFIED REPAIR
Use cos(zen)cos(tilt) + sin(zen)sin(tilt)cos(dAz).
Unsuccessful approach: Dropping the azimuth factor treats the sun as always in front of the panel.
Case contract
Panel tilted by tilt degrees from horizontal, facing pan_az (clockwise from north). zenith = 90 - sun_el. cos(aoi) = cos(zen)cos(tilt) + sin(zen)sin(tilt)cos(sun_az - pan_az), clamped to [-1, 1]. Return [aoi degrees rounded 3, max(cos aoi, 0) rounded 4, 'front' or 'back'], or [None, 0.0, 'night'] when sun_el <= 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(tilt, pan_az, sun_az, sun_el):
if sun_el <= 0:
return [None, 0.0, 'night']
zen = math.radians(90 - sun_el)
t = math.radians(tilt)
c = math.sin(zen) * math.cos(t) + math.cos(zen) * math.sin(t) * math.cos(math.radians(sun_az - pan_az))
c = max(-1.0, min(1.0, c))
aoi = math.degrees(math.acos(c))
side = 'front' if c >= 0 else 'back'
return [round(aoi, 3), round(max(c, 0.0), 4), side]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 135, 0, 10], [101.408, 0.0, 'back']],
['regression: incidence formula terms (partial repair)', [10, 225, 60, 89],
[10.969, 0.9817, 'front']],
['control 1', [0, 180, 225, -5], [None, 0.0, 'night']],
['control 2', [0, 135, 150, 45], [45.0, 0.7071, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [20, 225, 270, 3], [73.101, 0.2907, 'front']],
['regression: incidence formula terms (partial repair)', [90, 180, 250, 45],
[76.005, 0.2418, 'front']],
['control 1', [25, 180, 60, 10], [92.907, 0.0, 'back']],
['control 2', [60, 225, 270, 20], [41.716, 0.7465, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 270, 225, 10], [60.095, 0.4986, 'front']],
['regression: incidence formula terms (partial repair)', [10, 135, 30, 45],
[48.35, 0.6646, 'front']],
['control 1', [60, 90, 135, 30], [38.709, 0.7803, 'front']],
['control 2', [45, 270, 60, 89], [45.868, 0.6963, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 135, 100, 10], [56.376, 0.5537, 'front']],
['regression: incidence formula terms (partial repair)', [20, 225, 270, 3],
[73.101, 0.2907, 'front']],
['control 1', [45, 180, 315, 0], [None, 0.0, 'night']],
['control 2', [30, 90, 225, 45], [68.754, 0.3624, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [10, 180, 100, 30], [58.767, 0.5185, 'front']],
['regression: incidence formula terms (partial repair)', [30, 225, 270, 60],
[22.062, 0.9268, 'front']],
['control 1', [45, 180, 0, 75], [60.0, 0.5, 'front']],
['control 2', [45, 225, 270, -5], [None, 0.0, 'night']]]]
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 |
|---|---|---|---|
| boundary: sun on horizon | [None, 0.0, 'night'] | [None, 0.0, 'night'] | Passed |
| normal: sun normal to panel | [30.0, 0.866, 'front'] | [0.0, 1.0, 'front'] | Failed |
| boundary: sun behind vertical panel | [110.0, 0.0, 'back'] | [160.0, 0.0, 'back'] | Failed |
| regression: incidence formula terms | [37.676, 0.7915, 'front'] | [101.408, 0.0, 'back'] | Failed |
| regression: incidence formula terms (partial repair) | [98.657, 0.0, 'back'] | [10.969, 0.9817, 'front'] | Failed |
| control 1 | [None, 0.0, 'night'] | [None, 0.0, 'night'] | Passed |
| control 2 | [45.0, 0.7071, 'front'] | [45.0, 0.7071, 'front'] | Passed |
SHA-256 / f411b453df8812eade78f28c07c59047874d9ca2062e8f9c0d70e1e485ab236f
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(tilt, pan_az, sun_az, sun_el):
if sun_el <= 0:
return [None, 0.0, 'night']
zen = math.radians(90 - sun_el)
t = math.radians(tilt)
c = math.cos(zen) * math.cos(t) + math.sin(zen) * math.sin(t)
c = max(-1.0, min(1.0, c))
aoi = math.degrees(math.acos(c))
side = 'front' if c >= 0 else 'back'
return [round(aoi, 3), round(max(c, 0.0), 4), side]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 135, 0, 10], [101.408, 0.0, 'back']],
['regression: incidence formula terms (partial repair)', [10, 225, 60, 89],
[10.969, 0.9817, 'front']],
['control 1', [0, 180, 225, -5], [None, 0.0, 'night']],
['control 2', [0, 135, 150, 45], [45.0, 0.7071, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [20, 225, 270, 3], [73.101, 0.2907, 'front']],
['regression: incidence formula terms (partial repair)', [90, 180, 250, 45],
[76.005, 0.2418, 'front']],
['control 1', [25, 180, 60, 10], [92.907, 0.0, 'back']],
['control 2', [60, 225, 270, 20], [41.716, 0.7465, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 270, 225, 10], [60.095, 0.4986, 'front']],
['regression: incidence formula terms (partial repair)', [10, 135, 30, 45],
[48.35, 0.6646, 'front']],
['control 1', [60, 90, 135, 30], [38.709, 0.7803, 'front']],
['control 2', [45, 270, 60, 89], [45.868, 0.6963, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 135, 100, 10], [56.376, 0.5537, 'front']],
['regression: incidence formula terms (partial repair)', [20, 225, 270, 3],
[73.101, 0.2907, 'front']],
['control 1', [45, 180, 315, 0], [None, 0.0, 'night']],
['control 2', [30, 90, 225, 45], [68.754, 0.3624, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [10, 180, 100, 30], [58.767, 0.5185, 'front']],
['regression: incidence formula terms (partial repair)', [30, 225, 270, 60],
[22.062, 0.9268, 'front']],
['control 1', [45, 180, 0, 75], [60.0, 0.5, 'front']],
['control 2', [45, 225, 270, -5], [None, 0.0, 'night']]]]
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 |
|---|---|---|---|
| boundary: sun on horizon | [None, 0.0, 'night'] | [None, 0.0, 'night'] | Passed |
| normal: sun normal to panel | [0.0, 1.0, 'front'] | [0.0, 1.0, 'front'] | Passed |
| boundary: sun behind vertical panel | [20.0, 0.9397, 'front'] | [160.0, 0.0, 'back'] | Failed |
| regression: incidence formula terms | [50.0, 0.6428, 'front'] | [101.408, 0.0, 'back'] | Failed |
| regression: incidence formula terms (partial repair) | [9.0, 0.9877, 'front'] | [10.969, 0.9817, 'front'] | Failed |
| control 1 | [None, 0.0, 'night'] | [None, 0.0, 'night'] | Passed |
| control 2 | [45.0, 0.7071, 'front'] | [45.0, 0.7071, 'front'] | Passed |
SHA-256 / dd9e635262c2157c454d903a4eed6908021a7f7204fbf7d61fc594a97bda9a16
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(tilt, pan_az, sun_az, sun_el):
if sun_el <= 0:
return [None, 0.0, 'night']
zen = math.radians(90 - sun_el)
t = math.radians(tilt)
c = math.cos(zen) * math.cos(t) + math.sin(zen) * math.sin(t) * math.cos(math.radians(sun_az - pan_az))
c = max(-1.0, min(1.0, c))
aoi = math.degrees(math.acos(c))
side = 'front' if c >= 0 else 'back'
return [round(aoi, 3), round(max(c, 0.0), 4), side]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 135, 0, 10], [101.408, 0.0, 'back']],
['regression: incidence formula terms (partial repair)', [10, 225, 60, 89],
[10.969, 0.9817, 'front']],
['control 1', [0, 180, 225, -5], [None, 0.0, 'night']],
['control 2', [0, 135, 150, 45], [45.0, 0.7071, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [20, 225, 270, 3], [73.101, 0.2907, 'front']],
['regression: incidence formula terms (partial repair)', [90, 180, 250, 45],
[76.005, 0.2418, 'front']],
['control 1', [25, 180, 60, 10], [92.907, 0.0, 'back']],
['control 2', [60, 225, 270, 20], [41.716, 0.7465, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 270, 225, 10], [60.095, 0.4986, 'front']],
['regression: incidence formula terms (partial repair)', [10, 135, 30, 45],
[48.35, 0.6646, 'front']],
['control 1', [60, 90, 135, 30], [38.709, 0.7803, 'front']],
['control 2', [45, 270, 60, 89], [45.868, 0.6963, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [30, 135, 100, 10], [56.376, 0.5537, 'front']],
['regression: incidence formula terms (partial repair)', [20, 225, 270, 3],
[73.101, 0.2907, 'front']],
['control 1', [45, 180, 315, 0], [None, 0.0, 'night']],
['control 2', [30, 90, 225, 45], [68.754, 0.3624, 'front']]],
[['boundary: sun on horizon', [30, 180, 180, 0], [None, 0.0, 'night']],
['normal: sun normal to panel', [30, 180, 180, 60], [0.0, 1.0, 'front']],
['boundary: sun behind vertical panel', [90, 180, 0, 20], [160.0, 0.0, 'back']],
['regression: incidence formula terms', [10, 180, 100, 30], [58.767, 0.5185, 'front']],
['regression: incidence formula terms (partial repair)', [30, 225, 270, 60],
[22.062, 0.9268, 'front']],
['control 1', [45, 180, 0, 75], [60.0, 0.5, 'front']],
['control 2', [45, 225, 270, -5], [None, 0.0, 'night']]]]
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 |
|---|---|---|---|
| boundary: sun on horizon | [None, 0.0, 'night'] | [None, 0.0, 'night'] | Passed |
| normal: sun normal to panel | [0.0, 1.0, 'front'] | [0.0, 1.0, 'front'] | Passed |
| boundary: sun behind vertical panel | [160.0, 0.0, 'back'] | [160.0, 0.0, 'back'] | Passed |
| regression: incidence formula terms | [101.408, 0.0, 'back'] | [101.408, 0.0, 'back'] | Passed |
| regression: incidence formula terms (partial repair) | [10.969, 0.9817, 'front'] | [10.969, 0.9817, 'front'] | Passed |
| control 1 | [None, 0.0, 'night'] | [None, 0.0, 'night'] | Passed |
| control 2 | [45.0, 0.7071, 'front'] | [45.0, 0.7071, 'front'] | Passed |
SHA-256 / f605aec9306c323658cc6ad87583e70c74fb619dce993eb47d6f81e831236e51
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 2026-09-29T14:51:54.945595+00:00.
Case digest / e024c82e7946a0e5da6bfd0bde0c19da7082392da58327c6f912cf814e3f4950