Jacob Beers Dept. of Mechanical Engineering, University of Washington, Seattle.

Computational Techniques in Engineering, ME 535

Optimization of an indefinite solar unmanned aerial flight (UAV) using non-gradient methods • Jacob S. Beers Abstract Keeping a plane aloft requires a delicate balance of the four forces: lift, drag, weight and thrust. During the day, a solar powered unmanned aerial vehicle (UAV) is powered in flight by the energy it collects from the sun through an array of solar cells. To sustain indefinite flight, solar power collected during the day must be used to power the plane at night. Solar airplanes to date have had some success, but the majority of them have been very large and expensive. Improving the plausibility of indefinite flight and reducing the overall costs of a UAV calls for optimizing the energy efficiency of a model. This was achieved by considering the use of energy from solar panels to propel a plane. The goal is to design a solar powered UAV that minimizes the net energy usage by minimizing energy consumption from drag, lift, thrust, and weight while maximizing the solar energy generated. The aim is to focus on how solar panels can affect the design of the aircraft, carefully considering the multiple tradeoffs when coming up with an optimal design. The objective function consists of drag, weight, lift, thrust, and power produced by the solar cells. A non-gradient method will be employed for optimizing the objective function. Drag is dependent upon air resistance that is affected by the varying speed and wing size. The weight is due to the variable size of the wings, battery, motor and solar panel. Lift is affected by the changing wing planform area and speed. The variables include the wing span, wing chord, camber, thickness of foil and location of camber. The wing planform area affects the lift and the drag. The solar panel size affects the weight and energy generated. Tornado, a MATLAB program designed to allow for the input of different wing geometries, was used to compute and obtain the aerodynamic coefficients. The optimal result for the power used is presented in Appendix A. Nomenclature c Chord b Span M Camber P Location of camber (in percent chord from leading edge) XX Thickness of foil 𝐶"# Parasitic drag coefficient 𝐶$ Lift coefficient 𝐶" Drag coefficient 𝜌 Density of air T Thrust 𝑉 Velocity 𝑆 Planform area 𝑊) Total weight

𝐼 Solar irradiance 𝜂,-.. Solar cell efficiency 𝜙𝑠𝑢𝑛 Angle w.r.t. sun 𝐶3- Effective friction coefficient 𝐶3 Skin friction coefficient 𝐾 Induced drag coefficient 𝑒 Oswald span efficiency factor 𝐴𝑅 Aspect Ratio Λ$: Wing sweep angle 𝑅𝑒 Reynolds number WOS Wing loading, weight per area 1 Introduction In the year 2000, NASA and AeroVironment created a plane called Helios, intending to prove that indefinite flight was indeed possible (Dornhiem, 2004). The current world record for the longest unmanned flight is held by a British company, QinetiQ. Their airplane, Zephyr, achieved an astonishing 14 days of continuous flight. Extending the flight time beyond 14 days however, remains a challenge. This could be attributed to multiple factors, such as weather conditions, solar cell, motor, and propeller efficiency. Although technological advances in solar panels have extended the time a solar UAV stays in flight, further improvements can be made when additional modifications to the span, chord, camber, position of camber and the thickness of the airfoil are done. The project consists of optimizing power used by the solar cells of a solar airplane. Solar cells convert energy from the sun to electrical power used to propel the airplane in flight. As more solar cells are mounted on the wings, more power is supplied to the plane, however, more weight is also being added. While the scope of this project takes into account optimizing several elements such as weight, drag, lift, and power; this task also takes into account the weight of the plane being under a certain value, while minimizing the power used by adjusting the following variables:

𝑏 = 𝑆𝑝𝑎𝑛𝑜𝑓𝑡ℎ𝑒𝑤𝑖𝑛𝑔 𝑐 = 𝐶ℎ𝑜𝑟𝑑𝑜𝑓𝑡ℎ𝑒𝑤𝑖𝑛𝑔

𝑀 = 𝑀𝑎𝑥𝑖𝑚𝑢𝑚𝑐𝑎𝑚𝑏𝑒𝑟𝑜𝑓𝑡ℎ𝑒𝑎𝑖𝑟𝑓𝑜𝑖𝑙 𝑃 = 𝑃𝑜𝑠𝑖𝑡𝑖𝑜𝑛𝑜𝑓𝑚𝑎𝑥𝑖𝑚𝑢𝑚𝑐𝑎𝑚𝑏𝑒𝑟𝑜𝑓𝑡ℎ𝑒𝑎𝑖𝑟𝑓𝑜𝑖𝑙

𝑋𝑋 = 𝑇ℎ𝑖𝑐𝑘𝑛𝑒𝑠𝑠𝑜𝑓𝑡ℎ𝑒𝑎𝑖𝑟𝑓𝑜𝑖𝑙

Figure 1 shows a diagram of the variables that are included as part of the optimization problem.

Figure 1. Airfoil geometrical modifications

The chord is the segment joining the leading edge and the trailing edge, which are the points where the circumscribed circle is tangent to the airfoil. The camber line is the line on the cross-section of a wing of an aircraft which is equidistant from the upper and lower surfaces of the wing. The camber is the maximum distance between the chord and the mean camber line. The thickness is the maximum distance between the upper and lower surface of the airfoil, measured perpendicularly to the chord.

When designing and building an airplane’s wing, the shape of its airfoil is considered. For this project, a NACA four-digit series airfoil was found to be optimal. It is defined by the three variables M, P and XX. The first digit M, is the maximum camber as a percentage of the chord. The second digit P, is the position of the maximum camber in tens of percent of the chord. The last two digits XX, is the maximum thickness of the airfoil as a percentage of the chord.

The planform area of the wing (S) is given by the span and chord of the wing. The challenge in this particular assignment lies in finding the objective function. In order to derive it, it is necessary to first determine the variables and parameters governing the power required to lift an aircraft. 2 Derivation of the Objective Function

During steady flight, the thrust, T, from the propeller must balance the drag on the airplane, given by equation 1.1.

𝑇 = 𝐷 = 𝐶"R + 𝐾𝐶$T 0.5𝜌𝑉T𝑆 (1.1)

Power required, Pr, for the flight is found by multiplying the thrust by the velocity, as shown in equation 1.2. Equations 1.2 and 1.3 can be used to predict the thrust and power required for level flight for different flight velocities. It is important to note that the parasitic drag and lift coefficients, CDo and CL, in equations 1.1 and 1.2 are not constant.

𝑃X = 𝑇𝑉 = 𝐶"R + 𝐾𝐶$T 0.5𝜌𝑉Y𝑆 (1.2)

𝐿 = 𝑊R = 0.5𝜌𝑉T𝑆𝐶$ (1.3)

Substituting equation 1.3 into 1.2 gives the equation for power required:

𝑃X = 𝑇𝑉 = [T𝐶"R𝜌𝑉Y𝑆 + 𝑉𝐾𝐶$𝑊R (1.4)

As can be seen in this equation, the power required by the plane changes as the span and chord are adjusted. The power required for the airplane can also be decreased by decreasing𝐶\R, K, and W. The total weight of the plane is also greatly affected by changing the planform area and thus how many cells can be appied. The weight of the payload due to the changing planform area and adding solar cells is given by: 𝑆*WOS, where WOS is the wing loading, or weight per area of solar cells that are on the planform area. WOS was found to be 5.5 𝑁𝑚^T.

The parasitic drag coefficient𝐶\R, consists primarily of the skin friction, roughness, and pressure drag of the major components. The induced drag coefficient (K) is equal to [

_`a for an elliptical

wing. The Oswald span efficiency factor is added to the equation for non-elliptical wings (Bowman, 2011). With this correction, K is calculated using the equation:

𝐾 = [_-`a

, 𝑒 = 4.61 1 − 0.045𝐴𝑅).gh 𝑐𝑜𝑠Λ𝐿𝐸 ).[j − 3.1 (1.5) AR is the wing aspect ratio and Λ$: is the amount the wing leading edge is swept. The configuration selected is an airplane with a wing sweep angle of zero. The wing sweep angle of zero was not selected arbitrarily, but rather it was selected on the basis that at low subsonic speeds, attachment of plane waves to the airfoil and pressure drag is negligible (Jones, 1946). According to Robert T. Jones, he states that for aerodynamic efficiency, wings designed for flight at high subsonic and supersonic speeds should be swept back. The speed was set right above the stall speed of 8.3 m/s, which is considered to be a low subsonic speed. It is still necessary to figure out the net power after factoring in the power input from the solar panels. The amount of power available from the solar cells depends on the solar irradiance, I, measured in watts per square meter. The amount of power available from the solar cells depends on the solar irradiance as well as the angle at which the sun hits the solar panels. Equation 1.6 can be used to calculate the power available from the solar cells:

𝑃lR.mX = 𝐼𝐴𝜂,-..cos(𝜑lst) (1.6) With the simple assumption that 𝜑lst is zero degrees, the cosine term goes to one. The power used by the solar cells is then defined by equation 1.7:

𝑃sl-\ = 𝑃X − 𝑃lR.mX (1.7) The power produced by the solar cells needs to be as large (and negative) as possible in order for plausible optimization results.

3 Methodology

The objective function to be minimized is:

𝒎𝒊𝒏𝒊𝒎𝒊𝒛𝒆𝑃X =12𝐶"R𝜌𝑉Y𝑆 + 𝑉𝐾𝐶𝐿𝑊𝑜 − 𝐼𝐴𝜂,-..

𝒘𝒊𝒕𝒉𝒓𝒆𝒔𝒑𝒆𝒄𝒕𝒕𝒐𝑏, 𝑐, 𝑀, 𝑃, 𝑋𝑋

𝒔𝒖𝒃𝒋𝒆𝒄𝒕𝒕𝒐0.1 ≤ 𝑏 ≤ 5

0.1 ≤ 𝑐 ≤ 0.5

1 ≤ 𝑀 ≤ 10

1 ≤ 𝑃 ≤ 90

3 ≤ 𝑋𝑋 ≤ 30

𝑏𝑐 𝑊𝑂𝑆 ≤ 13𝑁

Where b is the span, c is the chord, M is the maximum camber, P is the position of the max camber, XX is the thickness, and WOS is the weight per area of the solar panels. The problem involves optimizing the airplane’s power input, with respect to the planform area of the wing and the airfoil shape; and subject to the bounds of the span of the wing, chord of the plane, max camber of the airfoil, position of max camber of the airfoil, thickness of the airfoil, and weight provided by solar cells. 3.1 Tornado – A Vortex Lattice Method Tornado is a vortex lattice method for aerodynamic wing design applications in conceptual aircraft design (Melin, 2001). Tornado gives immediate feedback on design changes, and is very useful to use with applications requiring many iterations. Tornado will give the coefficients for lift and drag that are needed in order to finish computing the objective function. For this reason, it was decided to implement Tornado in MATLAB so that could run through an existing optimizer. Due to the way Tornado asks the user for inputs in the command prompt sequentially and then produces results, The code had to be greatly modified in order to automate the inputs generated by the optimizer. The modification of the code involved stepping through the source code and then finding out the prompts required to define the aircraft geometry, generate the lattice, and output the results. The prompts are extracted, modified, and then inserted into the main function file where the coefficient and lift and drag are outputted. Tornado allows a user to define most types of contemporary aircraft designs with multiple wings both cranked and twisted with multiple control surfaces. Each wing may have taper of both camber and chord. The Tornado solver solves for forces and moments, from which the aerodynamic coefficients are computed (Melin, 2001). For future use or reference, the geometries that were chosen for the configuration of the plane are shown in Appendix A.

For each iteration, Tornado first asks for an aircraft geometry setup, which is defined by the different values generated by the optimizer. Afterwards, it asks for the flight condition setup, which is essentially the speed (8.3 m/s) and angle of attack (7 degrees). Once the aircraft geometry and flight condition setup are entered the lattice is then generated, after which the computations are ready to be processed. The coefficients for the lift and drag are then obtained and inserted into the objective function. 3.2 Using GODLIKE: Non-gradient Based Optimizer

After implementing the objective function with MATLAB’s fmincon, it was noticed that the Tornado program didn’t feed the objective function with accurate enough results. This made any gradient based optimizer unable to determine a step direction or size, thus keeping fmincon from finding the true optimum. Alternatively, a non-gradient based optimizer package was used, GODLIKE, and the genetic algorithm option was selected. GODLIKE had to handle two objectives: the main objective was to minimize the power used and the secondary was to keep the weight below 13.0 N, if possible. The computation took roughly 5-6 hours with a population size of 250. 4 Results The values that correspond to this optimum design can be found below in Table 1.

Table 1: Optimal Design Results Specification Value

𝒃 4.8633 𝑚 𝒄 .4946 𝑚 𝑴 2.3756%𝑐 𝑷 61.6580 %𝑐 𝑿𝑿 14.6702 %𝑐 𝑷𝒖𝒔𝒆𝒅 -486.7893 W 𝑾cell 13.2296 N

With these optimized values, the foil is equivalent to a NACA 2615 as seen in Figure 2 below.

Figure 2. Optimal Airfoil Shape NACA 2615

The wing shape and size can be seen in Figure 3.

Figure 3. Optimal Wing Shape and Size

As mentioned previously, the wing sweep angle was chosen to be zero. This gave an optimal planform area which was found to be rectangular in shape. It should be noted that the 𝑃sl-\ is negative. This indicates that there was excess power generated by the solar panels which is stored in the batteries. This will be used later when there is not any sunlight, particularly at night; or when there is cloud coverage. The results also show that the bounds and constraints are not necessarily active. The optimizer approaches them, but it stopped just shy of them. This can be attributed to the fact that a multi-objective approach was used. The second objective was the weight constraint, which acted more like a penalty in that it created a compromise. Thus, the results are not exactly on the bound limits. This also explains why the solar cell weight for this design was 13.2296 N even though the constraint was to keep the weight under 13.0 N. Since this constraint was changed to one of the multi-objective functions, it was also compromised slightly to satisfy the main objective function. A benefit of this is that it makes the function more flexible and allows the optimizer to make some trade-offs if it sees that it can get significant gains from relaxing the constraints a bit. In this scenario, the optimizer determined that sacrificing some weight gain was justifiable for the amount of power saved. The relationship between the weight and the power used is shown in Figure 4 below.

0 0.20.4

-4

-3

-2

-1

0

1

2

3

4

-1

-0.5

0

0.5

1

Delta cp distribution

-1.8

-1.6

-1.4

-1.2

-1

-0.8

-0.6

Figure 4. Relationship between Weight and Power Used. Note that the Pareto front could not be produce the same time the program was run to find the optimal value. Re-running the program would be too time costly, since it took around 5-6 hours to converge. Rather than re-running the program at a population size of 250, it was run at a size of 20 so that it did not take as long. As can be seen from the Pareto front above, the relationship between the weight and the power used appears linear. Due to how the weight objective is written; the less negative the objective is, the heavier the weight is. Thus, as the plane becomes heavier, the less amount of power is used within the set bounds.

5 Discussion and Conclusion

According to this model, these results for the optimal design were more than adequate for indefinite flight. Given the output of 486 W, it yeilds a conservative average of 10.5e6 Joules charged per day. This would charge, in excess, a large enough battery that it could run off of during a 12 hour night. It is also found that as MATLAB was driving the variables in Table 1 to an optimal, it was seen that a lower parasitic drag and a higher lift coefficient was found. This agrees with the results obtained, confirming that the more efficient model would require less power to take off and remain in flight. Since the chord has seen very little change, the Reynolds number has little or no effect at all on the lift and drag coefficients. One of the takeaways from this project was that the Tornado program doesn’t work with any gradient based optimizer, as discussed in section 3.2. Thus, a non-gradient based optimizer with a GA option was used. It was found that while this works when gradient based optimizers don’t it was inefficient. As stated in section 3.2, with a population size of 250 it took over five hours to converge. As the project moves forward, it is planned to explore in more detail the different types of solar panels that would provide the best power input per unit square area. This is so that it can accurately mimic a real-world result, also taking into careful consideration factors such as climb rate, aerodynamics and the stall speed of the aircraft. Some other factors that would be considered for adding to the model are the cost and battery size. Cost would most likely be a constraint while battery size would be a variable that would affect weight size and nocturnal flight duration. Also for future improvements, it would be ideal to consider designing the UAV for suboptimal environments such as stormy weather conditions. In order to do this added factors of safety would be needed along with using feasibility robustness, sensitivity robustness and dynamic optimization. Overall, the project gave me much insight into non-gradient based optimization and fluid dynamics. A better understanding of flight mechanics and airfoil geometries gained through hours of research and much effort allowed me to overcome challenges in formulating the objective function. The other challenge I faced was getting Tornado to integrate with the existing code I had. The challenges were overcome by reading the documentation from Tornado and MATLAB, and spending much time going through each line of the code in order to understand how it works.

Appendix A: Configuration of aircraft geometry for Tornado

Number of panels on flap chords (2d array): 0 Number of panels in span (2d array): 5 Number of panels on chord (2d array): 5 Flap deflection symmetry boolean bit (2d array): 0 Flap chord in percent of wingchord (2d array): 0 Flapped partition(wing part) boolean bit (2d array): 0 Partition twist (3d array)<1 inboard, 2 outboard>: 0 Partition airfoils (3d array): {NACA} (variable) Taper ratio (2d array): 1 Sweep (2d array): 0 Root chord (2d array): Chord (variable) Dihedral (2d array): 0 Span(distance root->tip chord) (2d array): Span (variable) Wing symmetry boolean bit (2d array): 1 Partition starting coordinate (2d array): 0 Number of wings (scalar): 1 Number of partitions on each wing (1d array): 1 Flap deflection vector: 0 System reference point: [0,0,0] System center of gravity (around which all rotations occur): [0,0,0] Meshtype: Linear Airspeed: 8.3 m/s Angle of attack: 7 degrees Angle of sideslip: 0 Roll angluar rate: 0 Pitch angular rate: 0 Yaw angular rate: 0 Angle of attack time derivative: 0 Angle of sidesliptime derivative: 0 Altitude, meters: 3350m Air density: 1.225 kg/m3 Prandtl-Glauert compressibillity correction: 0 Panel corner matrix (2d array): 0 Collocation point matrix: 0 Vortex sling cornerpoint position matrix: 0 Airfoil collocation point normal direction matrix: 0 Reference area: 0 mean aerodynamic chord: 0 Start position of mac: 0 Mean geometric chord: 0 Reference span: 0

References Bowman, J (2011) Beginning Aircraft design. Provo, UT. Bowman, J (2011) Design and Optimization of a Solar Aircraft. San Diego, CA. Dornhiem, MA (2004) Helios breakup review. Aviation Week & Space Technology 12(5):59-60 Jones, R (1946) Wing Plan Forms for High-Speed Flight. Langley Field, VA. Melin, T (2001) Tornado, the Vortex Lattice Method. http://www.redhammer.se/tornado/TBG.html. Accessed 9 April 2015