Solar Power Can Substantially Prolong Maximum Achievable Airtime of Quadcopter Drones

Abstract Sunlight energy is potentially excellent for small drones, which can often operate during daylight hours and fly high enough to avoid cloud blockade. However, the best solar cells provide limited power, compared to conventional power sources, making their use for aerial vehicles difficult to realize, especially in rotorcraft where significant lift ordinarily generated by a wing is already sacrificed for the ability to hover. In recent years, advances in materials (use of carbon‐fiber components, improvement in specific solar cells and motors) have finally brought solar rotorcraft within reach. Here, the application is explored through a concise mathematical model of solar rotorcraft based on the limits of solar power generation and motor power consumption. Multiple solar quadcopters based on this model with majority solar power are described. One of them has achieved an outdoor airtime over 3 hours, 48 times longer than it can last on just battery alone with the solar cells carried as dead weight and representing a significant prolongation of drone operation. Solar‐power fluctuations during long flight and their interaction with power requirements are experimentally characterized. The general conclusion is that solar cells have reached high enough efficiencies and can outperform batteries under the right conditions for quadcopters.


Derivation of quadcopter power requirements
To clearly evaluate quadcopter power consumption and production from any power source, including solar power, it will be of great help to formulate a relationship between power and aircraft weight from fundamental physical principles, as will be shown. The lifting force in a rotorcraft is provided by downward airflow, with the thrust produced by motor-driven propellers. Such an airflow carries a certain amount of momentum and downward kinetic energy per unit time. Assuming that the area of the air flow is A, the air density is , and the average downward speed of an air molecule is v, within a time durationthen the total mass of the air propelled downwards over this interval is Av. The downward kinetic energy KE that this air mass acquires is given by KE = (1/2) (Avv 2 and the power P which must be provided to acquire this energy is P = KE/ v 3 . Also, the momentum acquired, p, when this air mass is accelerated from an average speed of zero to v, is equal to (Avv, leading to a lifting force F given by F = p/ = (Av 2 (S1) Equation (S1) is the same as Equation (1) in Reference [1], but different from a derived formula, v = √ 2 , based on reference [2]. From Equation (S1), a simple relation can be given: Also, from the relation of F and v, we are able to obtain the relation between the power P and the lifting force F of the downward air flow: P = 1 2 1 √ ρ 3/2 , which is slightly different by a proportional constant from the formula given in reference [2].
In electrical copters, the power is provided by propellers that are further driven by the motors through the input of electrical power. Several routes of losses, including motor stator and rotor core loss, [3] copper loss, [4] propeller efficiency loss, [5] and so on, lead to a conversion efficiency of input power to downward air-flow power (thrust) less than 100%. In addition, the electrical motor power is usually provided by an electronic speed controller (ESC), which does not have a 100% efficiency of converting input electrical power to the waveform needed by the motors, [6] either. In any case, taking into account all losses, we can simply write P = P e , where P e is the input electrical power to the system and  is the overall conversion efficiency. Then the relation between the input electrical power P e and the lifting force F of the air flow is given by This lifting force F is equal to the drone weight, in terms of gravitational force, if the drone is in static hover. If the drone is unbalanced or moving, then more power than that needed to overcome gravity would be needed, so required ≥

Confirmation of the relation between the input electrical power and the lifting force with a real motor
According to Equation (S2), the required input electrical power super-linearly increases with the lifting force, which cannot be less than the total weight of the copter. Figure S1 shows a fitted theoretical Equation (S2) curve and a measurement of a commercial MAD5005 motor.
The assumed air density is 1.225 kg m -3 . For Figure S1, the propeller used was 18 inches in diameter with an area swept of 0.164 m 2 , and the efficiency  was determined to be 47 %. In the measurement, the spin rate of the motor was controlled through the ESC with its power input from a 6S battery. The applied voltage and current, monitored through a voltage meter and a current meter, to the ESC were recorded and the power was calculated accordingly. Because a large-capacity battery was used, the voltage could maintain at 23.0 V during the measurement period of time, which was not long. The lifting force, measured using a weight scale attached to the motor with the propeller, was also recorded simultaneously at a certain voltage and current.
The measurement curve and the theoretical curve match well with a constant efficiency, so we did not use a voltage-dependent efficiency. There is only a small discrepancy between the two curves, indicating this simple model can predict power requirements pretty well. The measured power consumption increases at a slightly lower rate than the theoretical prediction. When the power increases, two opposing effects may be happening. One is that motors fast-spinning motors propel air molecules to a higher ultimate velocity, making the air stream more directional and hence increasing the overall efficiency. [7] The other is that extra loss could be induced through an increased motor temperature at high-power consumption levels, decreasing the overall efficiency. [8] According to Figure S1, the second effect plays less of a role than the first one, probably due to the excellent thermal properties of the motor. Figure S1. Comparison of required power vs. lifting force between an experimental and theoretical evaluation.
For multi-copters, there is more than one motor. Assuming n motors and n corresponding propellers, the total lifting force is F t = nF and the total power consumption is P t = nP e , while the total area swept by the propellers A t = nA, leading to the following equation: With Equation (S3), a battery-powered quadcopter can first be considered. For a battery, the stored energy E is proportional to its weight W b , E = kW b , with k around 0.2 Wh g -1 in a lithium battery. The airtime that the battery can provide is given by E/P t , where P t is the required power shown in Equation (S3). The total weight of the copter is the fuselage weight W f plus the battery weight W b . The lifting force is no less than the total weight. The maximum airtime T is obtained when the total lifting force F t is equal to the copter weight, F t = W f +W b = W f +E/k.
Thus the maximum flight time T is given by Based on Equation (S4), the flight time of a quadcopter using an ordinary battery can be estimated. As shown in Figure S2, the flight time does not always increase with battery mass, because the consumed power super-linearly increases with total copter weight. This is a major limitation of batteries. However, if the power is provided from sunlight, the flight time can be greatly increased. including its solar module is then given by where W f is the fuselage weight. and other solar cell loss mechanisms. Also, the solar modules might not operate at their maximum power points and their orientation will not be optimal due to the attitude changes of the copters 9 and the position of the sun during flights. Hence the actual power is substantially reduced, but with healthy margin to account for such loss, we expect the quadcopters to be substantially solar powered. The quadcopters were built with a 5S or 6S voltage, so a power module is used to convert this to 5 V for the electronics, including the pixhawk flight controller, the rf receiver, the GPS module, the telemetry data link, etc. The motors are brushless. A 4-in-1 electronic speed controller (ESC) is used to control the four motors.

Estimation of solar quadcopter flight time
To stabilize the operation voltage of a solar quadcopter, a battery is used together with the solar module. Then the power supply can be viewed as a hybrid system. Here let's evaluate the flight time with this hybrid energy source. Assume that the total current required for the copter is Y amperes and the battery has X amp-hours of useful charge, so the battery-only flight time would be T b = X/Y. If the solar module fractionally provides z of the total current, the current consumed from the battery would be (1-z)Y amperes. Then the battery will last longer, to a total time of T h = X/[(1-z)Y], extending flight by a factor of EF = T h /T b = 1/(1-z). Figure S3 shows the relation between the extended factor (EF) and the ratio of current provided by the solar module. The extended factor becomes significant only when the ratio is above 90%. Figure S3. Extended factor (EF) vs. ratio of current provided by the solar module.