High-Fidelity Solar Power Income Modeling for Solar-Electric UAVs: Development and Flight Test Verification

1 Introduction

This technical report extends previous work on solar power models for Unmanned Aerial Vehicles (UAVs). It is released in conjunction with the development and flight testing of ETH Zurich's AtlantikSolar UAV, which set a world record with an 81-hour continuous flight. Accurate solar power models are critical for both the conceptual design phase—predicting performance metrics like flight endurance ($T_{endur}$) and excess time ($T_{exc}$)—and the operations phase for performance assessment. The quality of the solar power model directly dictates the reliability of these predictions.

1.1 A basic solar power model

Existing literature on solar-powered UAVs often employs simplified models. A common model for instantaneous collected solar power is:

$P^{nom}_{solar} = I_{solar}(\phi_{lat}, h, \delta, t, \vec{n}_{sm}) \cdot A_{sm} \cdot \eta_{sm} \cdot \eta_{mppt}$

Where $I_{solar}$ is the solar radiation (a function of latitude $\phi_{lat}$, altitude $h$, day-of-year $\delta$, time $t$, and module normal vector $\vec{n}_{sm}$), $A_{sm}$ is the module area, $\eta_{sm}$ is the module efficiency (including a camber reduction factor), and $\eta_{mppt}$ is the maximum power point tracker efficiency. While suitable for early design stages, this model lacks the fidelity needed for detailed analysis and troubleshooting during flight tests.

1.2 Contributions of this report

This report addresses the need for higher-fidelity models by: 1) Introducing a comprehensive model accounting for exact aircraft attitude, geometry, and physical effects (temperature, angle-of-incidence). 2) Deriving simplified models suitable for initial design phases. 3) Verifying all models against real flight data from a 28-hour continuous day/night solar-powered flight.

2 High-Fidelity Solar Power Model

The proposed high-fidelity model significantly expands upon the basic formulation. Key enhancements include:

• Dynamic Attitude Integration: The model incorporates the UAV's real-time roll ($\phi$), pitch ($\theta$), and yaw ($\psi$) angles to compute the precise orientation of solar panels relative to the sun, moving beyond the assumption of a horizontal surface.
• Geometric Fidelity: It accounts for the actual 3D geometry and placement of solar cells on the aircraft's wings and fuselage, rather than treating them as a single flat plate.
• Physical Effect Modeling: It integrates factors like cell temperature (which affects efficiency $\eta_{sm}$) and the cosine loss from non-perpendicular sun incidence angles, which are often neglected in simpler models.

The core power calculation becomes a sum over all individual solar cells or panels, each with its own orientation and local conditions: $P_{solar}^{HF} = \sum_{i} I_{solar, i} \cdot A_{i} \cdot \eta_{sm,i}(T) \cdot \cos(\theta_{inc,i}) \cdot \eta_{mppt}$, where $\theta_{inc,i}$ is the angle of incidence for panel $i$.

3 Model Simplification for Conceptual Design

Recognizing that detailed attitude and geometry data are unavailable during early design, the report derives simplified models from the high-fidelity baseline. These models use reduced input sets, such as:

• Time-Averaged Model: Uses average solar irradiation over a day, suitable for very rough sizing.
• Daily-Cycle Model: Incorporates the sinusoidal variation of solar power throughout the day, providing better accuracy for endurance prediction without requiring flight path details.

These models establish a clear trade-off: reduced input complexity for lower predictive accuracy, guiding designers in model selection based on project phase.

4 Flight Test Verification

The models were rigorously tested using flight data from the AtlantikSolar UAV's record-setting missions. A dedicated 28-hour continuous flight provided a complete day/night cycle of data, including:

• Measured solar power income from the UAV's power system.
• High-precision attitude data (roll, pitch, yaw) from the inertial measurement unit (IMU).
• GPS position, altitude, and time data.
• Environmental data (temperature) where available.

This dataset allowed for a direct comparison between the predicted solar power from various models and the actual measured values.

5 Results and Discussion

The verification yielded clear, quantifiable results:

The high-fidelity model's superior accuracy demonstrates the significant impact of incorporating detailed attitude, geometry, and physical effects. The ~18% error of prior models is substantial enough to lead to flawed design decisions, such as undersizing the solar array or overestimating perpetual flight capability.

6 Core Insight & Analyst's Perspective

Core Insight: The solar UAV industry has been flying blind, relying on oversimplified power models that introduce nearly 20% error. This report isn't just an incremental improvement; it's a foundational correction that shifts solar UAV design from guesswork to engineering precision. The sub-5% accuracy benchmark sets a new standard, directly enabling the reliable, multi-day endurance flights that define the field's frontier.

Logical Flow: The authors brilliantly deconstruct the problem. They start by exposing the critical flaw in legacy models—their static, geometry-agnostic nature. They then build a physics-grounded, high-fidelity model that dynamically accounts for real-world variables like aircraft wobble and wing curvature. Finally, they don't leave practical users behind; they provide a clear pathway of simplified models, creating a "fidelity ladder" for different design stages. The flight-test validation against a world-record platform (AtlantikSolar) is the masterstroke, providing irrefutable, real-world proof.

Strengths & Flaws: The strength is undeniable: a rigorous, validated framework that closes a major knowledge gap. The methodology is exemplary, mirroring the validation ethos seen in seminal robotics and ML papers, such as those from the Robotics: Science and Systems conference, where simulation-to-real transfer is rigorously tested. However, the flaw is one of scope. The model is heavily tuned for fixed-wing UAVs with wing-mounted panels. The leap to rotary-wing or morphing-wing aircraft, where attitude changes are more violent and rapid, is non-trivial and unaddressed. It also assumes high-quality attitude sensing, which may not be available on ultra-low-cost platforms.

Actionable Insights: For UAV developers: Immediately adopt this high-fidelity model for detailed design and flight test analysis. Use the simplified models for initial sizing, but always budget for the ~18% uncertainty they carry. For researchers: The next frontier is real-time, adaptive modeling. Integrate this with model predictive control (MPC) algorithms—akin to how modern autonomous systems use perception models for planning—to allow UAVs to actively adjust their flight path to maximize solar income, creating truly energy-aware autonomous systems. The work also underscores the need for open-source, validated energy models, similar to the model zoos maintained by institutions like ETH Zurich's Autonomous Systems Lab or MIT's Computer Science and Artificial Intelligence Laboratory (CSAIL), to accelerate industry-wide progress.

7 Technical Details and Mathematical Formulation

The high-fidelity model's mathematical core involves coordinate transformations and efficiency corrections.

1. Solar Vector Transformation: The sun's position vector in the inertial frame ($\vec{s}_{ECEF}$) is transformed into the aircraft's body frame ($\vec{s}_{B}$) using the attitude rotation matrix $R_{B}^{I}$: $\vec{s}_{B} = R_{B}^{I} \cdot \vec{s}_{ECEF}$.

2. Angle of Incidence: For a solar panel with a unit normal vector $\vec{n}_{panel}$ in the body frame, the angle of incidence is: $\theta_{inc} = \arccos(\vec{s}_{B} \cdot \vec{n}_{panel})$. The effective irradiance is then scaled by $\cos(\theta_{inc})$ (Lambert's cosine law).

3. Temperature-Dependent Efficiency: Solar cell efficiency decreases with temperature. A common linear model is used: $\eta_{sm}(T) = \eta_{STC} \cdot [1 - \beta_{T} \cdot (T_{cell} - T_{STC})]$, where $\eta_{STC}$ is efficiency at Standard Test Conditions (STC), $\beta_{T}$ is the temperature coefficient (typically ~0.004/°C for silicon), $T_{cell}$ is the cell temperature, and $T_{STC}=25°C$.

4. Total Power Calculation: The total power is the sum over all $N$ panels/cells: $P_{total} = \eta_{mppt} \cdot \sum_{i=1}^{N} \left( I_{solar} \cdot \cos(\theta_{inc,i}) \cdot A_{i} \cdot \eta_{sm,i}(T) \right)$.

8 Experimental Results and Chart Description

The flight test results are best visualized through a time-series comparison chart (conceptually described):

Chart Title: "Measured vs. Predicted Solar Power During 28-Hour Flight"

Axes: X-axis: Time of Day (over a 28-hour period, showing two sunrises). Y-axis: Solar Power (Watts).

Lines:

• Solid Blue Line: Measured Power. Shows the actual solar power harvested by the UAV, with characteristic sinusoidal peaks at midday, zero during the night, and minor fluctuations due to cloud cover or aircraft maneuvers.
• Dashed Red Line: High-Fidelity Model Prediction. This line closely tracks the Solid Blue Line, with nearly overlapping peaks and valleys. The small gap between them, quantified as the <5% error, is barely perceptible on the chart scale.
• Dotted Green Line: Basic/Previous Model Prediction. This line also shows a sinusoidal shape but consistently runs below the measured power peak, especially in the morning and afternoon. The area between this line and the Measured Power line represents the ~18% average under-prediction. It fails to capture the higher power income when the aircraft's banked attitude presents the wings more favorably to the sun.

Key Takeaway from Chart: The visual clearly demonstrates the high-fidelity model's superior tracking ability, especially during non-noon hours where attitude effects are most pronounced, while highlighting the consistent inaccuracy of the simpler model.

9 Analysis Framework: A Case Study

Scenario: A solar UAV team is analyzing a disappointing flight test where the aircraft ran out of battery 2 hours before sunset, despite clear skies.

Step 1 – Problem Definition with Basic Model: Using the legacy model ($P^{nom}_{solar}$), they input average irradiation, horizontal panel area, and nominal efficiency. The model predicts sufficient power. It offers no root cause, only indicating a generic "performance shortfall."

Step 2 – Investigation with High-Fidelity Framework:

1. Data Ingestion: Import flight logs: GPS, IMU (attitude), power system data, and aircraft CAD model (for panel normals).
2. Model Execution: Run the high-fidelity model retrospectively. The model reconstructs the expected power minute-by-minute.
3. Comparative Analysis: The software generates the comparison chart (as in Section 8). The team observes that the predicted power from the high-fidelity model also matches the low measured values, unlike the optimistic basic model.
4. Root Cause Isolation: Using the model's modularity, they disable specific effects:
   • Disabling attitude correction causes only a minor change.
   • Disabling the temperature-dependent efficiency correction ($\eta_{sm}(T)$) causes the prediction to rise significantly above the measurement.
5. Conclusion: The analysis pinpoints excessive solar cell heating as the primary culprit. The cells, mounted on a dark composite wing with poor thermal management, were operating at 70°C instead of the assumed 45°C, causing a ~10% efficiency drop. The basic model, blind to temperature, missed this entirely.

Outcome: The team redesigns the panel mounting for better heat dissipation, leading to successful subsequent flights. This case demonstrates the framework's value as a diagnostic tool, not just a predictor.

10 Future Applications and Directions

The implications of high-fidelity solar modeling extend beyond fixed-wing UAVs:

• Rotary-Wing and VTOL UAVs: Adapting the model for drones with complex, time-varying geometries is a key challenge. This requires dynamic mapping of panel exposure during hover, transition, and forward flight.
• Energy-Aware Path Planning: Integrate the model into flight control algorithms for real-time, optimal path planning. The UAV could autonomously adjust its heading and bank angle to maximize solar gain, similar to how sailboats tack to harness wind.
• Swarm and Persistent Networks: For swarms of solar UAVs acting as communication nodes, accurate individual power models are essential for predicting network lifetime and optimizing relay schedules.
• Planetary Exploration: This modeling approach is directly applicable to Mars or Venus aerial vehicles (e.g., NASA's Mars Helicopter "Ingenuity"), where understanding solar income in thin atmospheres and with different solar constants is critical.
• Digital Twin Integration: The model forms a core component of a UAV's "digital twin," enabling high-fidelity simulation for training AI pilots, testing mission plans, and predictive maintenance.
• Standardization and Open Source: The field would benefit from an open-source library implementing these models (in Python or MATLAB), similar to ROS for robotics, allowing for community validation and extension.

11 References

1. Oettershagen, P. et al. (2016). [Previous work on solar power models].
2. Oettershagen, P. et al. (2017). Design of a small-scale solar-powered unmanned aerial vehicle for perpetual flight: The AtlantikSolar UAV. Journal of Field Robotics.
3. Duffie, J. A., & Beckman, W. A. (2006). Solar Engineering of Thermal Processes. Wiley.
4. Stein, J. S. (2012). Photovoltaic Power Systems. Sandia National Laboratories Report.
5. Noth, A. (2008). Design of Solar Powered Airplanes for Continuous Flight. ETH Zurich.
6. Klesh, A. T., & Kabamba, P. T. (2009). Solar-powered aircraft: Energy-optimal path planning and perpetual endurance. Journal of Guidance, Control, and Dynamics.
7. Zhu, J., et al. (2017). Unpaired Image-to-Image Translation using Cycle-Consistent Adversarial Networks (CycleGAN). IEEE International Conference on Computer Vision (ICCV). [Cited as an example of a rigorous, influential methodology paper in a related field of applied machine learning].
8. Autonomous Systems Lab, ETH Zurich. (n.d.). Official Website and Publications. [Cited as an authoritative source for robotics and UAV research].