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This paper presents an analysis of the characteristics of the photovoltaic supercapacitor system that moving fixedly along the longitude. A lot of equations concerning the characteristics are considered including supercapacitor, direct motor, solar radiation from the sun to the photovoltaic module through the atmosphere. Runge-Kutta method is used to predict the time integrations, and Newton-Raphson method ensures the simultaneous solutions at each substeps. The machine flying takes one day trip or one year trip from different latitudes and different seasons. Around solar noon, the photovoltaic generator drives the direct current motor and charges the electrochemical supercapacitor simultaneously. An electrochemical supercapacitor battery is used as a secondary battery. The working ranges of electrochemical supercapacitor battery and direct current motor are found by the simultaneous solution of their characteristics. The thermostat system induces the excess currents and supplies heat energy to keep the photovoltaic module supercapacitor not below the ice point. This study shows the considerable benefits of the supercapacitor used for energy storage, and also finds a performance in which the presented photovoltaic supercapacitor system can continue working for about three and one half years on the trips from the equator and from the latitude of 30? and about four and one half years on the trip from the latitude of 60?.

Photovoltaic (PV) system brings about the trouble taking up a lot of space for lead-acid battery bank in a building or on a flying machine. Another choice for energy storage is electrochemical supercapacitor (ECSC). The ECSC could bear a large number of cycles of charge-discharge without any reduction to its energy storage capacity, and it is under normal operation if the charging voltage does not largely exceed the rated voltage of ECSC battery. In the recent years, the manufacturing technology for ECSC developed greatly, so the photovoltaic supercapacitor (PVSC) that combines PV module with ECSC has been considered. The ECSC may be mounted on the back of PV module and divided up into many ECSC cells. The number of ECSC cells on each module and the area of each ECSC cell can match the demand for driving electric machinery such as direct current (DC) motor.

The flying PVSC system includes the PVSC module, the DC motor, and the thermostat system. The PV generator supplies power to the DC motor for a flight plan during the day, and the energy stored in the ECSC cells makes the flight plan also possible at night, during sunrise, and during sunset. The voltage of the ECSC battery accompanying the output current can match the characteristic of DC motor by the disposition of ECSC cells in series and in parallel. The DC motor transforms electrical energy into mechanical energy for the task of flying all year around, and the separately excited motor is used because it works more approaching to the maximum power point than the series motor at the given levels of radiation [

The ECSC which has the constant capacitance of 5000, the current density of 0.1, the maximum and the minimum charge voltages of 2.70 and 1.70, and the maximum and the minimum discharge voltages of 2.54 and 1.62 is considered under safe conditions. Combining the characteristics of charge with discharge, ECSC can supply the range of voltage between 2.54 (the maximum discharge voltage) and 1.70 (the minimum charge voltage). Note that the two extremes accompany zero currents. The original ECSC mounted on PV module has the same area of 0.633 m^{2}. If it is divided into a lot of ECSC cells, each ECSC cell will have the same characteristics as the original ECSC. But before being divided, the characteristics of the driven electric machinery must be indicated. The present DC motor used on the flying machine has the relationship between current and voltage [

If a flying machine at the altitude of A moves south or north along a fixed longitude and with the velocity of

where

The PVSC system will present different behaviors, if it flies on different dates or takes one year trip from different dates. In the analyses for one year trip, there is no difference in computational method between each day except for initial state of charge in ECSC cells because the flying machine always comes back 24 h later.

The solar radiation incident on a horizontal plane outside of the atmosphere is given by [

where the solar constant is

The extraterrestrial radiation passes through the atmosphere and contributes beam and diffuse radiations to the surfaces of PVSC modules. The transmittances for beam radiation

where the correction factors

Thus, the beam radiation

Beam radiation and anisotropic diffuse radiation are two components incident on the surfaces of moving PVSC modules. During the day, the cosine of the incidence angle of beam radiation on the horizontal surface at latitude

where

The average incidence angle for diffuse,

where

Considering the effect of the earth’s curvature, the original air mass,

At the reference conditions, the air mass is 1.5 which induces the incidence angle of 48.2˚ by Equation (12) and

According to the equivalent circuit, the electric characteristic of PV module is presented by the relationship between current and voltage:

The five parameters in the model are determined by the effective absorbed solar ratio

Some characteristics of the components in PVSC system are presented in [

When the PVSC modules are under insolation, they supply the system with the voltage of

The simultaneous solutions also can be found under the conditions that the PV generator drives the DC motor and charges the battery at the same time, and that the battery will discharge if the PV generator does not produce enough. Newton method solves the simultaneous problem, and Runge-Kutta method carries the time integration out per 5 minutes.

The PVSC modules all are fixed horizontally on the flying machine. One day trips along fixed longitude from the latitudes of 0˚, 30˚, and 60˚ are considered. As shown in [

The secondary battery dominates the PVSC system at about 72.65 V, and a larger initial charge of secondary battery must be chosen if a larger voltage is requested. For apparent presenting the difference, the fractional state of 0.1 is chosen in the current study. The secondary battery can save the energy and will discharge while the PV generator does not produce enough during sunset. The secondary battery prolongs rest time of primary battery. After one day trip, the battery moving from the equator on the day of WS has the lowest state of charge,

The PV generator moving from the equator on the day of SS supplies energy from 5:10 at the latitude of 25.83˚, but does not produce enough until 6:35. The resistance induces the excess current from battery before 5:15, and induces the excess currents from both battery and PV generator from 5:15 to 6:35. From 6:35 to 7:40, the battery is at rest and the resistance induces the excess current from PV generator which drives alone the DC motor. At 7:40, the PV generator can drive the DC motor and recharge the battery simultaneously. From 9:20 to

13:40, the current through the resistance is larger than the charge current. At 11:55, the photocurrent and resistance current simultaneously present the local minimums of the profiles. At 13:40, the primary battery has reached to the full state of charge and the secondary battery with

The PV generator moving from the equator on the day of AE supplies energy from 6:00 at the latitude of 30˚, and produces enough at 7:35. From 7:35 to 9:25, the battery is at rest. From 9:25 to 11:35, the PV generator can drive the DC motor and recharge the battery simultaneously. From 11:35 to 12:20, the battery is at rest again. From 12:20 to 14:50, the PV generator also can drive the DC motor and recharge the battery simultaneously. From 9:25 to 11:35 and 12:20 to 14:45, the excess current induced by the resistance is small. The resistance current is larger than the motor current with the small amounts from 9:05 to 9:20, 11:35 to 11:50, 12:00 to 12:15, and 14:50 to 15:00. At 11:55, the photocurrent and the resistance current simultaneously have the local minimums of the profiles. From 14:50 to 16:30, the battery is at rest the third time.

Thereafter, the battery works again till midnight ending with F = 0.9638, and works alone after sunset at 18:00 and at the latitude of 30˚. The primary battery can’t reach to the full state of charge, and the secondary battery is always at rest. The heat energy output of the resistive load totals 7.77 MJ.

The PV generator moving from the equator on the day of WS supplies energy from 7:15 to 16:45. The flying machine meets the sunrise and the sunset at the latitude of 36.25˚. The PV generator does not produce enough all day long. The photocurrent is nearly equal to the resistance current during the day, and they have the minimum at 11:55. The battery discharges the whole day, and the final state of charge is 0.9153. The heat energy output of the resistive load totals 5.56 MJ.

The PVSC system has two initial directions, flying north and flying south, if it moves from the latitude of 30˚. The results for initial south flying are presented in brackets following the results for initial north flying. The PV generator moving from the latitude of 30˚ on the day of SS supplies energy from 3:55 (6:00) at the latitude of 49.58˚ (0.0˚), but does not produce enough until 5:50 (7:15). From 5:50 (7:15) to 7:35 (8:10), the primary battery is at rest. From 7:35 (8:10) to 12:55 (14:50), PV generator recharges the primary battery until it is at full state, in addition to driving the DC motor. From 9:05 (9:10) to 16:00 (16:40), the resistance current is larger than the motor current. From 11:10 (11:00) to 12:50 (12:45), the resistance current is over two times of motor current. The maximum of the profile of photocurrent is presented at 12:05 (11:50). The secondary battery with

The PV generator moving from the latitude of 30˚ on the day of AE supplies energy from 6:00 (6:00) at the latitude of 60.0˚ (0.0˚), and produces enough at 7:55 (7:15). From 7:55 (7:15) to 9:10 (8:15), the primary battery is at rest. The resistance current is larger than the motor current from 9:00 (16:20) to 9:05 (16:35) and from 10:25 (9:20) to 14:40 (13:30). The maximum resistance current and the maximum photocurrent are presented simultaneously at 12:35 (11:25). From 9:10 (8:15) to 15:50 (16:00), the PV generator can drive the DC motor and recharge the battery simultaneously. From 15:50 (16:00) to 16:50 (16:20), the primary battery is at rest again. Thereafter, the primary battery discharges till midnight ending with F = 0.9728 (0.9743), and bears alone driving the DC motor after sunset at 18:00 (18:00) and at the latitude of 0.0˚ (60.0˚). The heat energy output of the resistive load totals 15.40 MJ (14.96 MJ).

The PV generator moving from the latitude of 30˚ on the day of WS supplies energy from 8:05 (6:00) at the latitude of 49.58˚ (0˚), and produces enough at 9:40 (7:35). From 9:40 (7:35) to 11:00 (8:55), the primary battery is at rest. From 11:00 (8:55) to 15:15 (13:00), the PV generator can drive the DC motor and charge the primary battery simultaneously. The resistance current is larger than the motor current from 10:50 (8:45) to 10:55 (8:50). At 13:05 (10:55), the photocurrent reaches to the maximum and the resistance current presents the local maximum. From 15:15 (13:00) to 16:35 (14:20), the primary battery is at rest again. Thereafter, the primary battery discharges till midnight ending with F = 0.9564 (0.9551), and bears alone driving the DC motor after sunset at 18:00 (15:55) and at the latitude of 0.0˚ (49.58˚). The heat energy output of the resistive load totals 7.18 MJ (7.30 MJ).

The PV generator moving from the latitude of 60˚ on the day of SS supplies energy from 4:45 at the latitude of 36.25˚, and produces enough at 6:40. From 6:40 to 7:55, the primary battery is at rest. From 7:55 to 17:10, the PV generator can drive the DC motor and charge the battery simultaneously. From 14:00 to 17:10, the primary battery is at full state and the secondary battery is at the state of being charged. At 11:55, the photocurrent and the resistance current have the local minimums respectively. The resistance current is getting to the maximum at 14:00, larger than the motor current from 14:00 to 16:20, and over 2 times of the motor current from 14:00 to 15:20. The motor current is decreasing from 14:00 to 18:25 reaching to 4.23 A, and the primary battery charges at about 1.47 A from 14:00 to 17:15. From 17:15 to 17:40, all batteries are at rest and PV generator alone drives the DC motor. From 17:40 to 18:25, the secondary battery discharge at −6.17 A. Thereafter, the primary battery works till midnight ending with F = 0.9804, and bears alone driving the DC motor after sunset at 19:15 and at the latitude of 36.25˚. The heat energy output of the resistive load totals 22.22 MJ.

The PV generator moving from the latitude of 60˚ on the day of AE supplies energy from 6:00 at the latitude of 30.0˚, and produces enough at 7:20. From 7:20 to 8:20, the primary battery is at rest. From 8:20 to 16:35, the PV generator can drive the DC motor and charge the primary battery simultaneously. The resistance current is larger than the motor current from 9:15 to 14:40, and over 2 times of the motor current from 10:50 to 13:10. At 12:05, the photocurrent and the resistance current have the maximums respectively. From 15:00 to 16:35, the primary battery is at full state and the secondary battery is at the state of being charged. The motor current is decreasing to 4.23 A from 15:00 to 17:15, and the secondary battery charges at about 1.46 A from 15:00 to 16:35. The secondary battery is at rest for 15 minutes from 16:35. Thereafter, the primary battery works till midnight ending with F = 0.9766, and bears alone driving the DC motor after sunset at 18:00 and at the latitude of 30.0˚. The heat energy output of the resistive load totals 20.80 MJ.

The PV generator moving from the latitude of 60˚ on the day of WS supplies energy from 6:50 at the latitude of 25.83˚, and produces enough at 8:05. From 8:05 to 9:00, the primary battery is at rest. From 8:50 to 8:55, from 10:00 to 14:00, and at 15:05 the resistance current is larger than the motor current. At 11:5, the photocurrent and the resistance current have the maximums respectively. From 9:00 to 15:00, the PV generator can drive the DC motor and recharge the primary battery simultaneously. From 15:05 to 16:00, the primary battery is at rest again. Thereafter, the primary battery works till midnight with F = 0.9670, and bears alone driving the DC motor after sunset at 17:10 and at the latitude of 25.83˚. The heat energy output of the resistive load totals 15.11 MJ.

Based on the analytical methods for one day trips, one year trips start from spring equinox (SE), summer solstice, autumn equinox, and winter solstice to next year are considered. The profiles of the surrounding temperatures at the equator and the latitudes of 30˚ and 60˚ for the whole year are determined by the interpolation of two Lagrange quadratic equations, one for dates before SS and one for dates after SS. The two quadratic equations are determined by the following method. The average surrounding temperature on the first day of the year is equal to that on the last day. The surrounding temperature on the last day of the year is determined by the condition that the value on the day of WS derived from the interpolation is just equal to the value on the day of WS presented in [

Moreover, the schematic presentations to the profiles for 12 cases, flying from three latitudes and from four dates, are shown in

The PVSC system works at any time according to the simultaneous solutions of the characteristics of four components, the PV generator, the ECSC battery, the DC motor, and the resistive load. The characteristics of ECSC indicate that the range of charge is different from the range of discharge. The possible working range of PVSC system is between the maximum discharge voltage and the minimum charge voltage. The simultaneous solution of the discharge characteristic of ECSC battery and the characteristic of driving DC motor is chosen as the actual

maximum voltage of working range for ensuring that the current from DCSC battery is always larger than the current to DC motor. At the actual maximum voltage the fractional state of charge is 1.0, and furthermore the actual minimum of working range is the voltage at which the fractional state of charge is 0.1 because the minimum charge voltage of the possible working range accompanies zero charge current. The PVSC system has 50 ECSC batteries mounted on 25 PV modules, and 49 ECSC batteries with full state of charge are primary battery and one ECSC battery with the lowest fractional state of charge is secondary battery.

The flying machine usually takes the routes combining the paths along fixed latitudes and along fixed longitudes. The characteristics of PVSC system flying along fixed longitude were found in this work. Newton method and Runge-Kutta method complete the time integration per 5 min. The detail results for 1 day trips from three latitudes and on the days of SS, AE, and WS are presented for the design engineers. The flight plans considered total 24 cases for 1 year trips. The demands for primary battery are the same for the flight plans from the latitudes of 30˚ and 60˚ and from any seasons. The 49 ECSC batteries with full charge can work for about three and one half years on the trips from the equator and from the latitude of 30˚ and about one half and four years on the trip from the latitude of 60˚.

This research was supported by National Science Council at Taiwan, Contract No. NSC 100-2221-E-151-058.