FA-93576 / Solar tracker geometry / Open access
Fixed-tilt row pitch for a design sun: relative azimuth reference · case 01
Rows facing south get their pitch from a northern sun.
ROOT CAUSE
The row azimuth is treated as the direction the rows face away from.
VERIFIED REPAIR
Use cos(sun_az - row_az).
Unsuccessful approach: Ignoring the row azimuth only works for north-facing rows.
Case contract
Rows of slant length length (mm) tilted by tilt degrees face row_az. The shadow reaches forward height*max(cos(sun_az - row_az), 0)/tan(sun_el). Minimum unshaded pitch = length*cos(tilt) + shadow. Return [pitch rounded 3, length/pitch rounded 4]; None 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(length, tilt, sun_el, sun_az, row_az):
if sun_el <= 0:
return None
t = math.radians(tilt)
rel = math.cos(math.radians(sun_az - row_az - 180))
depth = length * math.cos(t)
height = length * math.sin(t)
shadow = height * max(rel, 0.0) / math.tan(math.radians(sun_el))
pitch = depth + shadow
return [round(pitch, 3), round(length / pitch, 4)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [1700, 10, 15, 340, 200], [1674.173, 1.0154]],
['regression: relative azimuth reference (partial repair)', [2300, 30, 30, 0, 180],
[1991.858, 1.1547]],
['control 1', [1700, 30, 30, 90, 200], [1472.243, 1.1547]],
['control 2', [1700, 30, 0, 90, 180], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [1700, 20, 10, 300, 180], [1597.477, 1.0642]],
['regression: relative azimuth reference (partial repair)', [1700, 25, 10, 60, 180],
[1540.723, 1.1034]],
['control 1', [1700, 30, 0, 225, 170], None],
['control 2', [2000, 30, 30, 135, 180], [2956.796, 0.6764]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2300, 35, 20, 200, 200], [5508.593, 0.4175]],
['regression: relative azimuth reference (partial repair)', [1700, 25, 30, 340, 170],
[1540.723, 1.1034]],
['control 1', [2000, 0, -2, 315, 0], None], ['control 2', [2000, 30, 0, 30, 0], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2000, 10, 15, 200, 180], [3187.577, 0.6274]],
['regression: relative azimuth reference (partial repair)', [1000, 25, 10, 135, 200],
[1919.234, 0.521]],
['control 1', [2300, 35, 30, 60, 0], [3026.533, 0.7599]],
['control 2', [1000, 30, 20, 250, 180], [1335.872, 0.7486]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2300, 30, 30, 60, 200], [1991.858, 1.1547]],
['regression: relative azimuth reference (partial repair)', [1700, 10, 20, 100, 180],
[1815.012, 0.9366]],
['control 1', [2000, 35, -2, 225, 170], None],
['control 2', [1700, 20, 20, 90, 170], [1874.877, 0.9067]]]]
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: winter noon design sun | [1732.051, 1.1547] | [5464.102, 0.366] | Failed |
| boundary: flat rows | [2000.0, 1.0] | [2000.0, 1.0] | Passed |
| boundary: sun behind the rows | [5464.102, 0.366] | [1732.051, 1.1547] | Failed |
| regression: relative azimuth reference | [2518.131, 0.6751] | [1674.173, 1.0154] | Failed |
| regression: relative azimuth reference (partial repair) | [3983.717, 0.5774] | [1991.858, 1.1547] | Failed |
| control 1 | [1975.78, 0.8604] | [1472.243, 1.1547] | Failed |
| control 2 | None | None | Passed |
SHA-256 / 4a7b12ab913f5f3a7ffcad9dc1f2588e022beb92b8828e0069ab49749402129c
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(length, tilt, sun_el, sun_az, row_az):
if sun_el <= 0:
return None
t = math.radians(tilt)
rel = math.cos(math.radians(sun_az))
depth = length * math.cos(t)
height = length * math.sin(t)
shadow = height * max(rel, 0.0) / math.tan(math.radians(sun_el))
pitch = depth + shadow
return [round(pitch, 3), round(length / pitch, 4)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [1700, 10, 15, 340, 200], [1674.173, 1.0154]],
['regression: relative azimuth reference (partial repair)', [2300, 30, 30, 0, 180],
[1991.858, 1.1547]],
['control 1', [1700, 30, 30, 90, 200], [1472.243, 1.1547]],
['control 2', [1700, 30, 0, 90, 180], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [1700, 20, 10, 300, 180], [1597.477, 1.0642]],
['regression: relative azimuth reference (partial repair)', [1700, 25, 10, 60, 180],
[1540.723, 1.1034]],
['control 1', [1700, 30, 0, 225, 170], None],
['control 2', [2000, 30, 30, 135, 180], [2956.796, 0.6764]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2300, 35, 20, 200, 200], [5508.593, 0.4175]],
['regression: relative azimuth reference (partial repair)', [1700, 25, 30, 340, 170],
[1540.723, 1.1034]],
['control 1', [2000, 0, -2, 315, 0], None], ['control 2', [2000, 30, 0, 30, 0], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2000, 10, 15, 200, 180], [3187.577, 0.6274]],
['regression: relative azimuth reference (partial repair)', [1000, 25, 10, 135, 200],
[1919.234, 0.521]],
['control 1', [2300, 35, 30, 60, 0], [3026.533, 0.7599]],
['control 2', [1000, 30, 20, 250, 180], [1335.872, 0.7486]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2300, 30, 30, 60, 200], [1991.858, 1.1547]],
['regression: relative azimuth reference (partial repair)', [1700, 10, 20, 100, 180],
[1815.012, 0.9366]],
['control 1', [2000, 35, -2, 225, 170], None],
['control 2', [1700, 20, 20, 90, 170], [1874.877, 0.9067]]]]
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: winter noon design sun | [1732.051, 1.1547] | [5464.102, 0.366] | Failed |
| boundary: flat rows | [2000.0, 1.0] | [2000.0, 1.0] | Passed |
| boundary: sun behind the rows | [5464.102, 0.366] | [1732.051, 1.1547] | Failed |
| regression: relative azimuth reference | [2709.441, 0.6274] | [1674.173, 1.0154] | Failed |
| regression: relative azimuth reference (partial repair) | [3983.717, 0.5774] | [1991.858, 1.1547] | Failed |
| control 1 | [1472.243, 1.1547] | [1472.243, 1.1547] | Passed |
| control 2 | None | None | Passed |
SHA-256 / ee60d57edd73cbde8659ad3fca17c199c263ffa6a05343e19136628863b3bb0d
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
N = 1
observations = []
def solve(length, tilt, sun_el, sun_az, row_az):
if sun_el <= 0:
return None
t = math.radians(tilt)
rel = math.cos(math.radians(sun_az - row_az))
depth = length * math.cos(t)
height = length * math.sin(t)
shadow = height * max(rel, 0.0) / math.tan(math.radians(sun_el))
pitch = depth + shadow
return [round(pitch, 3), round(length / pitch, 4)]
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [1700, 10, 15, 340, 200], [1674.173, 1.0154]],
['regression: relative azimuth reference (partial repair)', [2300, 30, 30, 0, 180],
[1991.858, 1.1547]],
['control 1', [1700, 30, 30, 90, 200], [1472.243, 1.1547]],
['control 2', [1700, 30, 0, 90, 180], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [1700, 20, 10, 300, 180], [1597.477, 1.0642]],
['regression: relative azimuth reference (partial repair)', [1700, 25, 10, 60, 180],
[1540.723, 1.1034]],
['control 1', [1700, 30, 0, 225, 170], None],
['control 2', [2000, 30, 30, 135, 180], [2956.796, 0.6764]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2300, 35, 20, 200, 200], [5508.593, 0.4175]],
['regression: relative azimuth reference (partial repair)', [1700, 25, 30, 340, 170],
[1540.723, 1.1034]],
['control 1', [2000, 0, -2, 315, 0], None], ['control 2', [2000, 30, 0, 30, 0], None]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2000, 10, 15, 200, 180], [3187.577, 0.6274]],
['regression: relative azimuth reference (partial repair)', [1000, 25, 10, 135, 200],
[1919.234, 0.521]],
['control 1', [2300, 35, 30, 60, 0], [3026.533, 0.7599]],
['control 2', [1000, 30, 20, 250, 180], [1335.872, 0.7486]]],
[['normal: winter noon design sun', [2000, 30, 15, 180, 180], [5464.102, 0.366]],
['boundary: flat rows', [2000, 0, 15, 180, 180], [2000.0, 1.0]],
['boundary: sun behind the rows', [2000, 30, 15, 0, 180], [1732.051, 1.1547]],
['regression: relative azimuth reference', [2300, 30, 30, 60, 200], [1991.858, 1.1547]],
['regression: relative azimuth reference (partial repair)', [1700, 10, 20, 100, 180],
[1815.012, 0.9366]],
['control 1', [2000, 35, -2, 225, 170], None],
['control 2', [1700, 20, 20, 90, 170], [1874.877, 0.9067]]]]
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: winter noon design sun | [5464.102, 0.366] | [5464.102, 0.366] | Passed |
| boundary: flat rows | [2000.0, 1.0] | [2000.0, 1.0] | Passed |
| boundary: sun behind the rows | [1732.051, 1.1547] | [1732.051, 1.1547] | Passed |
| regression: relative azimuth reference | [1674.173, 1.0154] | [1674.173, 1.0154] | Passed |
| regression: relative azimuth reference (partial repair) | [1991.858, 1.1547] | [1991.858, 1.1547] | Passed |
| control 1 | [1472.243, 1.1547] | [1472.243, 1.1547] | Passed |
| control 2 | None | None | Passed |
SHA-256 / 90266539dac2ee1392d42a316f7b6de566a088bb0b703e55e749e404662c92ea
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 2026-09-29T14:51:56.365007+00:00.
Case digest / 8893da6fae291db381c59e72a622b7a2eefa4298c6f11037adce23f026d41671