The global frame is east-north-up (ENU), with the mirror rotation center at the origin and s pointing toward the Sun. A diagnostic trajectory uses the National Oceanic and Atmospheric Administration (NOAA) fractional-year approximation [12], including its leap-year denominator. Civil time is converted to true solar time with east-positive longitude and local time one hour ahead of Coordinated Universal Time (UTC + 1), giving

Here, ϕ is latitude, δ is solar declination and H is hour angle. Atmospheric refraction is omitted. The main controller and robustness comparisons instead use the Solar Position Algorithm (SPA) of Reda and Andreas [13], implemented independently in pvlib [14]. Section 5.3 defines common and mismatched ephemeris cases; the NOAA cases provide diagnostic comparisons rather than the primary accuracy reference.

The required central reflected direction is t = [ 0 , 1 , 0 ] T . For a general installation, this vector must be obtained from the surveyed optical axis and target geometry. Specular reflection of the incoming direction − s by a mirror of unit normal n gives

The second relation is the heliostat bisector law [2][3]. Substitution into the first gives r = t when the geometry is nonsingular. With azimuth measured from north toward east and elevation measured from the horizontal plane, the actuator reference is

If the mirror instead points directly toward the Sun, n = s and Equation (2) gives r = s : light returns toward the solar direction. This does not deliver the required northward beam in the modeled furnace. Thus, a sensor with four light-dependent resistors (LDRs) can be used only with an appropriate geometric transformation, independent sun-sensing mechanism, or receiver-related optical feedback. End-of-travel switches alone do not resolve the geometry.

Each axis is represented by a rate-limited, first-order velocity response. This is a reduced drive model, not an identified armature circuit or a torque-capacity calculation. For shaft angle θ j , angular velocity v j and held rate command u j ,

The controller is sampled every T s = 1   s . The plant is integrated exactly over each interval for the held command. Defining a = exp ( − T s / τ ) ,

A symmetric play operator approximates total angular lost motion b between the shaft and optical output angle q j :

Hard position limits suppress outward commands and reset velocity at the bounds. Nominal total lost motion is 0.06˚, with encoder offsets of +0.03˚ in azimuth and −0.02˚ in elevation for all active strategies. When disabled, the output stops after velocity decays and then holds its position. This assumes adequate passive retention or a holding mechanism; it is not established by the CAD transmission. Prescribed perturbations applied after the play operator test sensitivity to unobserved output motion, without representing structural wind dynamics.

An explicit ray model avoids treating dish interception as receiver absorption. In the adopted frame, the dish vertex is ( 0 , L , 0 ) and the paraboloid is

Here, D is aperture diameter, d is sag and L is the mirror-to-vertex distance. Rays are launched from points distributed over the actual oriented plane mirror. Substituting each reflected ray into Equation (7) produces a quadratic equation; the nearest positive physical intersection is retained. Rays outside the circular aperture are rejected. The ideal dish normal is proportional to [ x / ( 2 f ) , 1 , z / ( 2 f ) ] T , and a second application of the reflection law gives the outgoing direction from the dish.

The square aperture has a specified orientation in its own plane. Let a and e be the actual output azimuth and elevation after backlash and any imposed angular disturbance. At a = e = 0, the width direction is east, the height direction is up, and the normal points north. Using active, right-handed rotations of column vectors, the rotation is

with the rightmost factor acting first. Equivalently, rotate by −a about global up, then by e about the rotated east axis. The local width and height unit vectors, and the aperture normal, are

There is no additional rotation about the normal. With the mirror center at the origin, width W and height H, ray origins are

Here ξ and η are the centered Halton aperture coordinates. This basis fixes the square edges relative to the circular dish and therefore its clipping. Slope-error perturbations change the reflecting normals only; they do not rotate the aperture or move its sampled points.

The receiver is modeled as a circular disk in the plane y = L − f , centered at the ideal focus. A ray contributes only when it reaches that plane in the forward direction and intersects the disk. Rays that encounter the receiver before reaching the dish are excluded as central obscuration. Frame and support shading, mutual shading by other structures, circumsolar radiation and multiple scattering are omitted. No usable diffuse contribution is added to the concentrator input.

The angular solar source is a uniform disk of radius 4.65 mrad. At each reflection, two independent Gaussian perturbations with nominal standard deviation 2 mrad per tangent component are added to the surface normal and the result is renormalized. These errors describe an assumed angular distribution; they are not a measured, spatially correlated reflector map. The ray-tracing approach follows the geometric principles used in dedicated solar optical tools [15]. Explicit treatment of sunshape, slope error and verification is motivated by the cross-code assessment of Wang etal. [16].

Deterministic Halton coordinates [17], using bases 2, 3, 5, 7, 11, 13, 17 and 19, sample mirror position, solar disk position and the two surface-error pairs. Box-Muller transformations generate the Gaussian components. Reusing the same ray coordinates across strategies reduces sampling differences in paired comparisons. The nominal count is 4096 rays per optical evaluation. Increasing the count is a quadrature check, not a statistical replication of independent weather days.

For mirror area A m , mirror and dish reflectances ρ m and ρ d , receiver absorptance α r , direct normal irradiance (DNI) G b and receiver-hit indicator χ i ,

The projected area uses the nominal planar aperture normal; the perturbed normals affect reflected directions. The reflectances are 0.88 and 0.82, and absorptance is 0.95. These are scenario values. P abs is optical power absorbed by the modeled receiver; it is not useful cooking power. Receiver conduction, convection, radiation loss, thermal inertia and load coupling are not modeled.

Four 10 h windows, from 08:00 to 18:00 local civil time, are evaluated on 21 March, 21 June, 23 September and 21 December 2026. These dates sample seasonal geometry; they are not observations on those dates and are not sufficient for an annual yield estimate. The illustrative clear-input profile is

with zero DNI below the horizon. This smooth profile is an imposed test signal, not a validated Niamey clear-sky model. The intermittent case multiplies it by 0.12 from 10:30 to 11:00, by 0.08 from 13:12 to 13:36, and by 0.25 from 15:09 to 15:18. Abrupt edges stress the supervisor and reacquisition behavior.

The model was executed in base MATLAB R2024b (24.2.0.2712019) on 64-bit Windows (MATLAB platform identifier PCWIN64), without Simulink or specialized toolboxes. A separate Python/NumPy implementation uses the same equations and deterministic rays. The executed sets comprise 16 NOAA reference scenarios with sensitivity and resolution sweeps, and 340 cases covering SPA comparisons, joint uncertainty and disturbances. Section 5.4 consolidates implementation agreement and numerical verification.

The controller period is 1 s, with exact held-input plant propagation. For the NOAA cases, optical power is sampled every 60 s in smooth continuous and hysteretic runs, and every 15 s in periodic and intermittent runs. SPA comparison intervals are specified in Section 6.1; uncertainty and stress resolutions are defined below. Energy uses trapezoidal integration. Optical refinement and changes to the discrete controller period are assessed separately in Section 5.4.

We used ChatGPT (OpenAI, accessed in September-October 2026) to help develop the optical and control models, prepare MATLAB and Python simulation and plotting scripts, design verification and robustness tests, and refine the numerical calculations. It also helped with manuscript writing and language editing. We checked the calculations using the methods described in Section 5.4, including analytical checks, independent calculations, comparisons between implementations, and refinement tests. The authors take full responsibility for the methods, results, and final manuscript.

The reflected central-ray pointing error is the angle between the actual ray and the required dish-axis direction:

Energy is in kWh for power in watts and time in seconds. The root-mean-square error (RMSE) and 95th percentile of pointing error use only times with DNI at least 180 W∙m⁻²; energy includes the full 10 h window. This separates daytime pointing accuracy from large errors during near-dark holds while retaining their optical consequences.

Accumulated energized axis time sums the enabled duration of the two axes separately. Thus, two continuously enabled axes over 10 h give approximately 20 axis-hours. The electrical budget is

with time in hours, supervisory power P c = 1   W and enabled-axis power P a = 12   W . A command magnitude above 10⁻⁷˚ s⁻¹ counts as enabled. This numerical criterion assigns the full axis power even to very small commands; it is not a measured current or driver threshold. We report energized time, activation count and absolute angular travel separately from this conditional electrical budget. Optical and electrical energy are not combined into a net-energy or economic optimum.

Activation count is the sum of inactive-to-active command transitions over the two axes. For axis j and held-command interval k, define

There are K = 36,000 intervals for the 10 h window at a 1 s controller period; the terminal sample has no following interval and is excluded. An initially nonzero command is counted, as is each irradiance-driven restart that crosses this threshold after an inactive interval. Prealignment excludes a separate acquisition maneuver, not the first qualifying activation. A direction reversal without an inactive interval adds no activation. Thus, the count measures command-state transitions, not reference updates or hysteresis-state changes alone.

An independent optical calculation uses Newton intersection with the paraboloid and Householder reflection matrices. Twelve configurations combine six times of day, receiver radii of 5 and 25 mm, pointing offsets, finite sunshape and slope errors. It shares the production geometry and rays, but changes the intersection and reflection implementations. Separate encircled-energy tests compare a uniform solar disk and a normally incident Gaussian-slope mirror with exact cumulative distributions, following the simplified-test rationale of Wang etal. [16]. These are verification tests, not a SolTrace comparison or physical validation.

Solar positions are independently generated by the National Renewable Energy Laboratory (NREL) SPA implementation in pvlib 0.15.0 [13][14]. The published 17 October 2003 example is checked before generating four scenario trajectories. Tables contain east-north-up unit vectors every 60 s; the controller uses linearly interpolated, renormalized vectors. Direct 1 s SPA evaluation bounds this interpolation error. Scenario coordinates remain 13.51˚N and 2.11˚E, with UTC + 1, zero reference altitude, an assumed Δ T = 69.2   s and atmospheric refraction disabled. Three cases separate numerical roles: common NOAA input and reference; SPA input with a NOAA controller reference; and common SPA input and reference. Thus, the second case exposes ephemeris mismatch instead of concealing it inside a shared trajectory.

Common-SPA trajectories are used for the primary controller and robustness comparisons. Periodic references update every 60, 300, 900 or 1800 s, with optical evaluation every 5 s. Activation thresholds of 0.06˚, 0.12˚, 0.24˚ and 0.48˚, each releasing at one quarter of activation, are evaluated on all four reference days.

The joint ensemble contains 128 independent draws from the nine uniform ranges in Table 3. Each draw is applied to continuous and hysteretic tracking with identical inputs and rays. The NumPy permuted congruential generator with 64-bit output (PCG64) uses seed 28092026; the generated matrix is archived for identical MATLAB inputs. September screening uses 2048 rays every 60 s, followed by refinement of the six lowest retention ratios. Prefixes of 32, 64 and 128 pairs assess ensemble-size sensitivity. The ranges and their independence are design assumptions; empirical percentiles do not estimate a measured field population or bound the entire parameter domain.

The bounds are engineering sensitivity envelopes around the nominal design assumptions, not measured tolerances or manufacturer specifications. Receiver radii of 10 - 30 mm and slope errors of 1 - 5 mrad bracket the nominal 25 mm and 2 mrad values to test aperture acceptance and optical quality together. The total backlash range extends to 0.16˚, above the 0.12˚ activation threshold, to include commands partly absorbed by transmission play. Encoder offsets of ±0.15˚ exceed both the nominal biases and that threshold, testing residual calibration errors that the same shaft feedback cannot identify.

Table 3. Joint uncertainty assumptions. Inputs are independent within this design ensemble; the distributions are not inferred from measurements.

The maximum encoder step, 0.05˚, exceeds the 0.03˚ release threshold and tests discrete measurement effects. The 1 - 5 s response time, 0.10 - 0.50˚ s⁻¹ rate limit and 0.15 - 0.35 s⁻¹ gain bracket the nominal drive settings and include slower or more rate-limited responses. Independent uniform draws give transparent coverage of these selected ranges; they do not assign field probabilities or establish motor and transmission adequacy. Fixed encoder offsets represent residual angular-reference errors only; they do not reproduce the installation-dependent daily and seasonal drift addressed by Moreno-Cruz etal. [8].

Additional paired tests impose cloud interruptions, initial offsets of +5˚ and −3˚, a 0.05˚ encoder step, 0.16˚ lost motion, and unobserved output-angle disturbances. The encoder step uses nearest rounding, with half-integers rounded away from zero. For disturbance amplitude A in degrees and time t in seconds, the two perturbations are

The tested amplitudes are 0.05˚, 0.15˚ and 0.30˚. These kinematic signals combine periodic motion and a smooth gust-shaped pulse; they have no assigned wind-speed or torque equivalence. A combined case adds cloud interruptions, 0.05˚ encoder quantization, 0.16˚ backlash, offsets of +0.15˚ and −0.15˚, a 5 s velocity time constant and A = 0.15 ˚ . Stress cases use 4096 rays every 10 s. Recovery from the initial offset is the first instant after which the central-ray error remains at or below 0.5˚ for 60 consecutive seconds. These tests preserve the same safety bounds, irradiance supervisor and electrical-budget assumptions as the baseline.

Verification covers reflection and ideal-focus identities, the held-input actuator update, axis bounds, irradiance interlocks and the optical-power bound. The six Python checks pass; MATLAB passes three analytical checks and the per-scenario assertions. Maximum reflection and focus residuals are 7.7 × 10⁻¹⁶ and 8.9 × 10⁻¹⁶ m. The actuator recurrence differs from an independently stepped Runge-Kutta calculation by 7.1 × 10⁻¹², within the 10⁻¹⁰ tolerance. These checks establish numerical consistency of the equations and implementation.

All 64 MATLAB-Python metric comparisons pass for the 16 NOAA scenarios. Maximum optical-energy and ray-RMSE differences are 8.9 × 10⁻¹⁶ kWh and 2.9 × 10⁻¹⁴ degrees. For the 340-case set, all 2040 comparisons pass, with maxima of 4.2 × 10⁻¹³ kWh and 2.2 × 10⁻¹¹ degrees. Activation counts agree exactly. Export audits also pass using 10 − 10 + 10 − 12 | x ref | in the respective units. Shared equations can preserve modeling bias despite this implementation agreement.

The independent optical calculation agrees within 5.6 × 10⁻¹⁷ m² in optical factor, with a maximum surface residual of 3.6 × 10⁻¹⁶ m. Encircled-energy discrepancies are below 4.3 × 10⁻⁵ at 131,072 samples. SPA reproduces its published example within 2.1 × 10⁻⁶ degrees; interpolating the four 60 s vector tables differs from direct 1 s evaluation by at most 5.0 × 10⁻⁵ degrees.

Table 4 reports the resolution checks for the NOAA diagnostic cases. Increasing rays from 4096 to 16,384 changes hysteretic energy by 0.078%; halving its optical interval from 60 to 30 s changes it by 0.004%. Periodic tracking is more sensitive: 60 s sampling gives 0.982 kWh, whereas 15 s gives 1.009 kWh. The adopted 15 s result differs from the 5 s check by 0.18%.

Table 4. September NOAA numerical-resolution checks. Controller-period changes alter the discrete control law; they are not plant-integration refinements.

Reducing the controller period from 1 to 0.25 s changes optical energy little but raises the budget from 37.46 to 38.76 Wh (3.45%). This changes event timing in the discrete controller; it is not plant-solver error because propagation is exact for held inputs. The reported actuation results therefore apply to the stated 1 s controller.

Table 5 checks the primary common-SPA September ratio directly, with continuous and hysteretic energy recomputed at every resolution. Geometry, ephemeris, plant parameters, the 1 s controller period and deterministic Halton ray prefixes are shared within each pair. The retained-energy percentage is

These additional refinements use the unchanged Python implementation. Its primary 4096-ray, 30 s pair reproduces the archived native MATLAB energies to within 3 × 10⁻¹⁵ kWh and gives identical activation counts; native MATLAB execution of the additional resolutions is not claimed.

Table 5. Paired resolution check for the primary September common-SPA comparison. Change is relative to the 4096-ray, 30 s ratio, in percentage points (pp); the controller period remains 1 s.

The primary ratio is 99.7482%. Across the tested refinements, retention ranges from 99.7437% to 99.7604% (loss 0.2396% - 0.2563%), with a maximum deviation of 0.0122 percentage points from the primary ratio. At 32768 rays, reducing the optical interval from 5 to 1 s changes retention by +0.0001 percentage points; at 5 s, increasing the count from 32768 to 65536 changes it by +0.0065 percentage points. The approximately 0.25% optical loss is therefore supported at the observed resolution sensitivity, while its last reported decimal is not a certified error bound. The nonmonotone ray-count response is reported explicitly; these deterministic checks do not supply a statistical confidence interval.

Table 6 gives the primary September comparison with SPA defining both sunlight and the controller reference. Hysteretic tracking retains 99.75% of continuous-tracking energy at the primary resolution; Table 5 quantifies the numerical sensitivity of this approximately 0.25% loss. One-minute updates give similar retention (99.62%), but a larger assumed budget (234.7 versus 37.6 Wh) and more activations (1333 versus 1169). Five-minute updates reduce the budget to 60.7 Wh while retaining only 93.40%. Hysteresis therefore offers a favorable optical-actuation compromise for this plant, including against frequent periodic updates. The comparison does not establish dominance over periodic controllers with different stopping rules or drive models.

Table 6. Primary September comparison with common-SPA input and reference. Optical intervals are 5 s for periodic cases and 30 s for continuous and hysteretic cases. Budgets assume 12 W per enabled axis and 1 W supervision.

Figure 3 shows why the deadband cannot be selected from September alone. The 0.12˚ activation threshold retains at least 99.50% on all four nominal SPA days. Increasing it to 0.24˚ remains within 1% loss in March, September and December, but reduces June retention to 98.37%; 0.48˚ gives 94.49% in June. The nominal 0.12˚ setting is thus supported for these four geometries, without being a universal optimum.

Under the joint assumptions in Table 3, median retained energy is 99.64%, with 5th and 95th percentiles of 99.00% and 99.91% (Figure 4). Median energized axis time reduction is 88.95%, with corresponding percentiles of 81.13% and 92.12%. The median budget reduction is 85.40% under Equation (17). The difference reflects the supervisory load, which does not decrease with axis duty.

Seven of 128 screening pairs retain less than 99% of continuous-tracking energy. Median retention changes from 99.71% to 99.66% and 99.64% for 32, 64 and 128 pairs, while the lower-tail percentile is less stable. The ensemble resolves its central tendency more reliably than a tail probability. Ratios marginally above 100% do not demonstrate a superior optical optimum.

Figure 3. Seasonal threshold trade-off using SPA trajectories. The dashed line marks 99% retained optical energy. Lower electrical budgets at wider deadbands are accompanied by season-dependent optical penalties.

Figure 4. The 128 paired uncertainty scenarios. Left: empirical cumulative energy retention. Right: retention versus reduction in energized axis time, colored by receiver radius. Percentiles describe assumed parameter distributions, not field-confidence bounds.

Refining the six lowest screening ratios to 4096 rays every 15 s and 8192 rays every 5 s gives a least favorable retention of 98.27%, compared with 98.33% in the screen. The largest change between refined resolutions is 0.081 percentage points. The nominal sub-1% penalty therefore fails for some adverse combinations; 98.27% is an observed scenario result, not a global lower bound.

The 98.27% result is uncertainty pair 001 on the clear 23 September common-SPA case. Its parameters are a receiver radius of 12.2411 mm, slope error of 1.72126 mrad per tangent axis at each reflector, total backlash of 0.139983˚, azimuth and elevation encoder offsets of −0.128763˚ and −0.078782˚, respectively, a velocity time constant of 3.71454 s, gain of 0.201938 s⁻¹, maximum rate of 0.486628˚ s⁻¹ and encoder step of 0.033626˚. The activation and release thresholds remain 0.12˚ and 0.03˚.

At 8192 rays and 5 s optical sampling, with the 1 s controller unchanged, hysteretic and paired continuous energies are 2.350063 and 2.391462 kWh, respectively, giving 98.2689% retention. There is no imposed cloud interruption, initial misalignment or output-angle forcing in this pair. The penalty belongs to this joint parameter combination; it is not a separately identified effect of any one parameter.

Table 7 pairs each hysteretic case with continuous control under identical defects. Both controllers recover from the +5˚ and −3˚ initial offsets in 15 s under the 60 s persistence criterion. Quantization and backlash affect the actuation sequence and optical yield. The combined-defect case retains 99.79% of the equally degraded continuous result, not 99.79% of an undegraded clear day.

Table 7. Paired stress cases. Retention and axis-time reductions use the continuous controller under the same defect and weather inputs as the denominator.

Refining the largest angular-forcing case and the combined case from 4096 rays every 10 s to 8192 rays every 5 s changes absolute energies by less than 0.07%. Perturbations act after shaft feedback and cannot be directly rejected by this controller. Their limited optical effect for the 25 mm receiver reflects aperture tolerance, not demonstrated wind compensation.

The NOAA diagnostic cases explain the optical mechanisms behind the comparison. The exact bisector identity has a maximum vector residual of 7.7 × 10⁻¹⁶. Direct solar alignment at 09:00, 12:00, 15:00 and 17:00 instead produces 95.7˚ - 103.0˚ errors relative to the northward dish axis and no captured receiver rays. The numerical range is installation-specific; the distinction between solar alignment and target alignment is general.

Correct aiming does not make dish interception equivalent to receiver absorption. Continuous tracking sends 3.369 kWh to the dish after the first reflection, but the receiver absorbs 2.593 kWh after the second. Periodic tracking still illuminates the dish while focal-plane displacement and off-axis aberration cause greater receiver losses.

Table 8 and Figure 5 show the September NOAA diagnostic response. The 30-minute controller has a 4.331˚ ray RMSE, compared with 0.070˚ for continuous and 0.171˚ for hysteretic control. Its held reference becomes obsolete between updates, causing receiver power to fall before the next correction.

Table 8. Clear September NOAA baseline over 08:00 - 18:00. Electrical values are estimates from assumed powers; activations are summed across axes.

Hysteretic tracking retains 99.74% of continuous energy (2.586 versus 2.593 kWh) while reducing energized axis time by 88.6%. The associated budget falls from 250.0 to 37.5 Wh under the fixed-power assumptions. This agrees qualitatively with the primary SPA comparison; the two ephemeris datasets are kept separate.

Figure 5. September NOAA diagnostic response: absorbed optical power and central-ray error. Periodic30 uses 15 s optical samples; other strategies use 60 s samples.

Absolute angular travel is almost unchanged: 172.95˚ for hysteretic versus 173.18˚ for continuous tracking. Hysteresis traverses this path in shorter enabled intervals, with 1165 activations rather than four. Reduced duty therefore does not establish reduced wear or mechanical work. The fixed-mirror result of 0.103 kWh is a limiting reference specific to this narrow receiver.

The focal-plane samples at 14:24 in Figure 6 explain the receiver losses. With its reference held since 14:00, the periodic strategy places much of the spot beyond the 25 mm aperture. Continuous and hysteretic tracking keep it substantially centered despite broadening by sunshape and slope errors.

Figure 6. Focal-plane samples at 14:24 on the clear September day, using 16,384 launched rays. The black circle marks the 25 mm receiver radius. Samples outside the plotting window are not shown.

The NOAA seasonal diagnostics in Figure 7 and Table 9 give hysteretic energies of 1.824 - 2.960 kWh, with losses of 0.11% - 0.51% relative to continuous tracking. Budgets range from 29.8 to 52.4 Wh as the axis trajectories change. These common synthetic irradiance inputs isolate seasonal geometry; they are not a measured climatology.

Table 9. Seasonal NOAA baseline scenarios with the same synthetic DNI expression.

Figure 7. Four synthetic seasonal scenarios. Optical energy and the assumed electrical budget are shown separately. The electrical comparison assumes constant power while each axis is enabled.

In the intermittent NOAA case (Figure 8), hysteretic and continuous energies are 2.317 and 2.324 kWh, with budgets of 34.8 and 228.4 Wh. Both drives stop during the two deepest irradiance reductions. Misalignment grows during the hold and decreases after re-enabling; hysteretic daytime RMSE is 0.223˚. The supervisor therefore reduces low-irradiance actuation but introduces reacquisition excursions.

Figure 8. Intermittent-irradiance response with 15 s optical evaluation. Shading identifies the two deepest imposed cloud intervals. Ray error rises during the low-irradiance hold and falls after reacquisition.

This behavior assumes a deterministic solar reference and a perfect DNI signal. Local shading, soiling, diffuse illumination and LDR electronic errors are not simulated. Receiver feedback, as used by Kribus etal. [5], could reveal optical misalignment, but is absent from the present controller.

The NOAA threshold sweep (Figure 9) illustrates the aperture dependence of the compromise. For a 25 mm receiver, widening activation from 0.12˚ to 0.24˚ reduces the budget from 37.5 to 23.4 Wh and energy from 2.586 to 2.572 kWh. This remains within 1% loss on the September case, but the SPA seasonal results in Section 6.2 show why that setting cannot be adopted for all dates.

At 10 mm receiver radius, widening the threshold from 0.12˚ to 1.00˚ reduces energy from 2.325 to 1.464 kWh; the corresponding 25 mm values are 2.586 and 2.303 kWh. A control deadband must therefore be chosen jointly with receiver acceptance.

Figure 10 shows the interaction with surface quality. Raising slope error from 2 to 10 mrad reduces energy from 2.586 to 1.913 kWh for a 25 mm receiver, and from 1.776 to 0.258 kWh at 5 mm. A larger receiver intercepts more scattered rays but increases obscuration: under nearly ideal surfaces, 50 mm does not necessarily outperform 25 mm. Thermal losses are not included in this aperture comparison.

Figure 9. Hysteresis threshold sensitivity for receiver radii of 10 and 25 mm. The release threshold is one quarter of the activation threshold. Reduced motor duty comes with increased optical loss.

Figure 10. Sensitivity to prescribed slope-error standard deviation at both reflectors and receiver radius. The trade-off includes central receiver obscuration; thermal losses are outside the model.

Increasing opposite-sign encoder biases from ±0.1˚ to ±1.0˚ reduces energy from 2.581 to 1.813 kWh with essentially unchanged energized time. A shaft servo cannot identify its own static calibration bias from the same encoder signal. Motor activity is consequently an incomplete indicator of optical alignment.

NOAA-to-SPA direction discrepancies are 0.436˚, 0.111˚, 0.426˚ and 0.053˚ RMS in March, June, September and December. In September, using NOAA commands with SPA sunlight raises hysteretic ray RMSE from 0.169˚ to 0.451˚ and lowers energy from 2.575 to 2.544 kWh (Figure 11). Ephemeris error can compound or partly cancel encoder bias and lag, so the effect varies by date. This motivates the common-SPA comparisons in Sections 6.1 - 6.3 and shows why agreement between implementations sharing one trajectory is insufficient.

Figure 11. Separation of solar input and controller ephemeris. Left: hysteretic central-ray RMSE. Right: optical-energy change when NOAA drives the controller while SPA defines the incoming rays, relative to common-SPA tracking.