Analytical models for the power subsystem and the attitude control subsystem of a microsatellite

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(1)UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TECNICA SUPERIOR DE INGENIEROS AERONÁUTICOS. Doctoral Dissertation. ANALYTICAL MODELS FOR THE POWER SUBSYSTEM AND THE ATTITUDE CONTROL SUBSYSTEM OF A MICROSATELLITE Javier Cubas Cano Aeronautical Engineer. Madrid, November 2015.

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(3) UNIVERSIDAD POLITÉCNICA DE MADRID ESCUELA TECNICA SUPERIOR DE INGENIEROS AERONÁUTICOS. Doctoral Dissertation. ANALYTICAL MODELS FOR THE POWER SUBSYSTEM AND THE ATTITUDE CONTROL SUBSYSTEM OF A MICROSATELLITE Author: Co-directors:. Javier Cubas Cano Santiago Pindado Carrión Ángel Sanz Andrés. Madrid, November 2015.

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(5) Tribunal nombrado por el Magfco. y Excmo. Sr Rector de la Universidad Politécnica de Madrid, el día de de 2015. Presidente: Vocal: Vocal: Vocal: Secretario: Suplente: Suplente: Realizado el acto de defensa y lectura de la Tesis el día de 2015 en la E.T.S.I. de Aeronáutica.. de. Calificación: EL PRESIDENTE EL SECRETARIO. LOS VOCALES.

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(7) A mis padres.

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(9) Agradecimientos Me gustaría agradecer en primer lugar a los codirectores de mi tesis, el profesor Santiago Pindado Carrión y el profesor Ángel Sanz Andrés. A Santiago le agradezco su siempre incansable ayuda y desvelos, así como su inagotable energía, confianza y continuo acicate, sin los cuales esta tesis hubiese aspirado a mucho menos. A Ángel le agradezco su maestría y experto consejo en los aprietos matemáticos de la tesis, su presteza y disposición para resolver cualquier problema, así como su atención por el detalle. Quiero mostrar mi gratitud al Instituto Ignacio da Riva, por la oportunidad que me ha dado para desarrollar las investigaciones que han sido la base de esta tesis. Con una mención especial para el profesor José Meseguer cuyo consejo me decidió a emprender una aventura hasta entonces nunca proyectada. Mi agradecimiento al profesor Sebastián Franchini, director de mi proyecto de master que fue germen de esta tesis. Tanto la definición del subsistema de control de actitud del UPMSat-2 como el modelo inicial del simulador del control de actitud en Simulink® son fruto de la dedicación de la doctora Assal Farrahi, a ella quiero agradecerle el compartir conmigo sus conocimientos sobre control de actitud magnético. Si toda construcción necesita unos cimientos firmes, esta tesis (al menos la parte relativa al control de actitud del UPMSat-2) se cimenta sobre su trabajo previo a mi llegada al IDR. A Carlos de Manuel Navío y a Felix Sorribes Palmer, por su colaboración en algunas publicaciones derivadas de este trabajo. Muchas gracias en general a todos los miembros del Instituto (actuales y pasados) por su ayuda durante mi etapa de doctorando, ya que incluso si dicha ayuda no llegó a ser parte substancial de esta tesis, seguro aumentó mis conocimientos o mi tiempo disponible para mejorarla. Inestimables han sido también los medios técnicos y apoyo recibido desde el Instituto de Energía Solar de la UPM. En especial estoy en deuda con la doctora Marta Victoria que me orientó en mis inicios en la energía fotovoltaica y realizó múltiples mediciones que no me hubiese sido posible llevar a cabo por falta de conocimientos y equipo. Le agradezco al profesor Jose Maria Ruiz Pérez la pre-corrección de mi primer artículo derivado de esta tesis, su directa y sincera crítica me ayudaron a mejorar y darle el nivel adecuado a mi investigación, aparte de hacer mucho menos traumática la primera cita con una revisión. La colaboración y ayuda recibida desde la Facultad de Informática de la UPM por el profesor Juan Zamorano y la Facultad de Telecomunicaciones por el profesor Juan de la Puente.

(10) también han sido (y espero serán) muy fructíferas en lo referente a la implementación del control de actitud del UPMSat-2. Al profesor Juan Fernández de la Mora le debo la mención internacional a la que aspiro con esta tesis, y le doy las gracias por la oportunidad que me brindó para trabajar bajo su supervisión no hubiese podido aprender (y sobretodo disfrutar) de un periodo de investigación en el extranjero. Muchas gracias también a lios doctores Ali Ravanbakhsh y Julian Santiago por sacar tiempo en sus apretadas agendas para leerse esta tesis y valorarla. De la ETSIA quiero agradecer a José por su ayuda durante los cursos de doctorado pero sobretodo por hacerlos mas llevaderos. Muchas gracias también a María y María Jesús por ayudarme a sacar adelante toda la documentación de la tesis, a pesar de mi mala memoria y procrastinación en temas burocráticos. Por último, pero no menos importante (es más, puede que lo más importante) a aquellas personas que han hecho posible esta tesis, con su cariño, comprensión y apoyo incondicional. A mi madre, a mi padre, a mi hermana, a mis abuelas y resto de familiares y amigos. Muchas gracias también a ti Carlos, que me has acompañado durante toda tesis y has sido un apoyo constante, pero también el principal perjudicado por algunos malos ratos y el (a veces inagotable) afán de esta tesis por acaparar mi tiempo y dedicación..

(11) Resumen El trabajo realizado en la presente tesis doctoral se debe considerar parte del proyecto UPMSat-2, que se enmarca dentro del ámbito de la tecnología aeroespacial. El UPMSat-2 es un microsatélite (de bajo coste y pequeño tamaño) diseñado, construido, probado e integrado por la Universidad Politécnica de Madrid (España), para fines de demostración tecnológica y educación. El objetivo de la presente tesis doctoral es presentar nuevos modelos analíticos para estudiar la interdependencia energética entre los subsistemas de potencia y de control de actitud de un satélite. En primer lugar, se estudia la simulación del subsistema de potencia de un microsatélite, prestando especial atención a la simulación de la fuente de potencia, esto es, los paneles solares. En la tesis se presentan métodos sencillos pero precisos para simular la producción de energía de los paneles en condiciones ambientales variables a través de su circuito equivalente. Los métodos propuestos para el cálculo de los parámetros del circuito equivalente son explícitos (o al menos, con las variables desacopladas), no iterativos y directos; no se necesitan iteraciones o valores iniciales para calcular los parámetros. La precisión de este método se prueba y se compara con métodos similares de la literatura disponible, demostrando una precisión similar para mayor simplicidad. En segundo lugar, se presenta la simulación del subsistema de control de actitud de un microsatélite, prestando especial atención a la nueva ley de control propuesta. La tesis presenta un nuevo tipo de control magnético es aplicable a la órbita baja terrestre (LEO). La ley de control propuesta es capaz de ajustar la velocidad de rotación del satélite alrededor de su eje principal de inercia máximo o mínimo. Además, en el caso de órbitas de alta inclinación, la ley de control favorece la alineación del eje de rotación con la dirección normal al plano orbital. El algoritmo de control propuesto es simple, sólo se requieren magnetopares como actuadores; sólo se requieren magnetómetros como sensores; no hace falta estimar la velocidad angular; no incluye un modelo de campo magnético de la Tierra; no tiene por qué ser externamente activado con información sobre las características orbitales y permite el rearme automático después de un apagado total del subsistema de control de actitud. La viabilidad teórica de la citada ley de control se demuestra a través de análisis de Monte Carlo. Por último, en términos de producción de energía, se demuestra que la actitud propuesto (en eje principal perpendicular al plano de la órbita, y el satélite que gira alrededor de ella.

(12) con una velocidad controlada) es muy adecuado para la misión UPMSat-2, ya que permite una área superior de los paneles apuntando hacia el sol cuando se compara con otras actitudes estudiadas. En comparación con el control de actitud anterior propuesto para el UPMSat-2 resulta en un incremento de 25% en la potencia disponible. Además, la actitud propuesto mostró mejoras significativas, en comparación con otros, en términos de control térmico, como la tasa de rotación angular por satélite puede seleccionarse para conseguir una homogeneización de la temperatura más alta que apunta satélite y la antena..

(13) Abstract The work carried out in the present doctoral dissertation should be considered part of the UPMSat-2 project, falling within the scope of the aerospace technology. The UPMSat-2 is a microsatellite (low cost and small size) designed, constructed integrated and tested for educational and technology demonstration purposes at the Universidad Politécnica de Madrid (Spain). The aim of the present doctoral dissertation is to present new analytical models to study the energy interdependence between the power and the attitude control subsystems of a satellite. First, the simulation of the power subsystem of a microsatellite is studied, paying particular attention to the simulation of the power supply, i.e. the solar panels. Simple but accurate methods for simulate the power production under variable ambient conditions using its equivalent circuit are presented. The proposed methods for calculate the equivalent circuit parameters are explicit (or at least, with decoupled variables), non-iterative and straight forward; no iterations or initial values for the parameters are needed. The accuracy of this method is tested and compared with similar methods from the available literature demonstrating similar precision but higher simplicity. Second, the simulation of the control subsystem of a microsatellite is presented, paying particular attention to the new control law proposed. A new type of magnetic control applied to Low Earth Orbit (LEO) satellites has been presented. The proposed control law is able to set the satellite rotation speed around its maximum or minimum inertia principal axis. Besides, the proposed control law favors the alignment of this axis with the normal direction to the orbital plane for high inclination orbits. The proposed control algorithm is simples, only magnetorquers are required as actuators; only magnetometers are required as sensors; no estimation of the angular velocity is needed; it does not include an in-orbit Earth magnetic field model; it does not need to be externally activated with information about the orbital characteristics and it allows automatic reset after a total shutdown of attitude control subsystem. The theoretical viability of the control law is demonstrated through Monte Carlo analysis. Finally, in terms of power production, it is demonstrated that the proposed attitude (on principal axis perpendicular to the orbit plane, and the satellite rotating around it with a controlled rate) is quite suitable for the UPMSat-2 mission, as it allows a higher area of the panels pointing towards the sun when compared to other studied attitudes. Compared with.

(14) the previous attitude control proposed for the UPMSat-2 it results in a 25% increment in available power. Besides, the proposed attitude showed significant improvements, when compared to others, in terms of thermal control, as the satellite angular rotation rate can be selected to achieve a higher temperature homogenization of the satellite and antenna pointing...

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(17) Table of Contents AGRADECIMIENTOS ............................................................................................ IX RESUMEN ............................................................................................................... XI ABSTRACT ............................................................................................................XIII TABLE OF CONTENTS ........................................................................................... I LIST OF FIGURES ................................................................................................... V LIST OF TABLES .................................................................................................... XI LIST OF ACRONYMS ..........................................................................................XIII 1. 2. INTRODUCTION ............................................................................................. 1 1-1. Problem Statement ........................................................................................................................................ 1. 1-2. Thesis Objectives ........................................................................................................................................... 2. 1-3. Doctoral Dissertation Structure .................................................................................................................. 3. ANALYTICAL MODELS APPLIED TO THE POWER SUBSYSTEM OF A. SPACECRAFT ............................................................................................................ 4 2-1. Introduction .................................................................................................................................................... 4. 2-2. Solar Cell Modeling........................................................................................................................................ 4. 2-2-1. 1-Diode/2-Resistors Equivalent Circuit Model .................................................................................. 9. 2-2-1-1. Analytical parameter calculation based on manufacturer’s data ......................................... 10. 2-2-1-2. Analytical parameter calculation with the slope of the curve at short circuit and open. circuit points ....................................................................................................................................................... 16 2-2-1-3. Analytical parameter calculation with any point of the I-V curve ...................................... 19. 2-2-2. 2-Diode/2-Resistors Equivalent Circuit Model ................................................................................ 21. 2-2-3. Environmental Influence ...................................................................................................................... 25. 2-2-4. Panel Simulation ..................................................................................................................................... 26. 2-3. Power Regulation ......................................................................................................................................... 28. 2-3-1. Maximum Power Point (MPP)............................................................................................................. 30. 2-3-2. Maximum Power Peak Tracking (MPPT) .......................................................................................... 32.

(18) ii 2-3-2-1. Applicability of this work to MPPT comparison .................................................................. 33. 2-3-2-1. MPPT proposal based on this work ........................................................................................ 35. 2-4. Results ............................................................................................................................................................ 36. 2-4-1. Testing of the proposed methodologies ............................................................................................. 37. 2-4-1-1. Analytical parameter calculation based on manufacturers’ data ......................................... 37. 2-4-1-2. Analytical parameter calculation with the slope of the curve at short circuit and open. circuit points ....................................................................................................................................................... 42 2-4-1-3. 3. Analytical parameter calculation for 2-Diode/2-Resistors model ...................................... 46. 2-4-2. Panel measurements and adjust ........................................................................................................... 47. 2-4-1. MPPT method preliminary tests .......................................................................................................... 52. ANALYTICAL MODELS APPLIED TO THE ATTITUDE CONTROL. SUBSYSTEM OF A SPACECRAFT......................................................................... 59 3-1. Introduction .................................................................................................................................................. 59. 3-1-1. Dynamics ................................................................................................................................................. 62. 3-1-2. Reference frames .................................................................................................................................... 62. 3-1-3. Attitude Dynamics ................................................................................................................................. 63. 3-1-4. Earth Magnetic Field ............................................................................................................................. 64. 3-1-4-1. Complex Magnetic Field Models .............................................................................................. 65. 3-1-4-2. Dipole Approximation ............................................................................................................... 67. 3-1-4-3. Polar orbit simplification............................................................................................................ 69. 3-2. Instruments Modeling ................................................................................................................................. 72. 3-2-1. Magnetometers........................................................................................................................................ 72. 3-2-2. Magnetorquers ........................................................................................................................................ 77. 3-3. Control Law Modeling ................................................................................................................................ 82. 3-3-1. Rotation rate correction ........................................................................................................................ 84. 3-3-2. Convergence Time and Gain Value .................................................................................................... 86. 3-3-2-1. Attitude Evolution and Rotation Rate Stabilization ............................................................. 87. 3-3-2-2. Gyroscopic Explanation of Attitude Evolution .................................................................... 89. 3-3-3. 4. Stabilization ............................................................................................................................................. 94. 3-4. Monte Carlo Simulations ............................................................................................................................ 96. 3-5. Analysis of the Proposed Contol Law when Applied to the UPMSat-2 Microsatellite................. 101. ENERGY INTERDEPENDENCE BETWEEN POWER AND ATTITUDE. CONTROL SUBSYSTEMS ..................................................................................... 108 4-1. Orbital attitudes .......................................................................................................................................... 108. 4-1-1. Attitude with nominal control for the UPMSat-2 mission. Axis Z perpendicular to the orbit. plane. 108. 4-1-2. Attitude with pure B-dot control ....................................................................................................... 109.

(19) iii 4-1-3 4-2. Attitude with “Compass” control. Axis Z parallel to the magnetic field .................................... 109 Net energy balance..................................................................................................................................... 110. 4-2-1. Nominal attitude control. Axis Z perpendicular to the orbit plane ............................................. 111. 4-2-2. Attitude with pure B-dot control ....................................................................................................... 114. 4-2-3. Attitude with “Compass” control. Axis Z parallel to the magnetic field .................................... 116. 4-2-4. Conclusion ............................................................................................................................................. 119. 4-3. Further research ......................................................................................................................................... 120. 5. CONCLUSIONS .............................................................................................. 121. 6. BIBLIOGRAPHY............................................................................................. 125. 7. APPENDIX 1: THE UPMSAT-2 PROJECT................................................... 135. 8. APPENDIX 2: EXAMPLE OF APPLICATION TO SIMULATION AND. COMPARISON OF MPPT IN VARIABLE AMBIENT CONDITIONS ............. 139.

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(21) v. List of Figures FIGURE 1. TOP: CURRENT-VOLTAGE (I-V) CURVE OF DIFFERENT SOLAR CELLS: SILICON (SI) CELL FROM R.T.C. FRANCE (MEASURED WITH CBM8096 MICROCOMPUTER AT 33°C [11]), TNJ TRIPLE-JUNCTION (GAINP2/GAAS/GE) CELL FROM SPECTROLAB (MEASURED AT AM0 135.3 MW/CM² AND 28°C), AND ZTJ TRIPLE-JUNCTION (INGAP/INGAAS/GE) CELL FROM EMCORE (MEASURED AT AM0 135.3 MW/CM² AND 28°C). IN EVERY CURVE SHORT CIRCUIT (TRIANGLES), MAXIMUM POWER (CIRCLES), AND OPEN CIRCUIT. (SQUARES) POINTS ARE INDICATED. BOTTOM: POWER CURVE OF THE AFOREMENTIONED SOLAR CELLS. DATA FROM TNJ AND ZTJ SOLAR CELLS EXTRACTED FROM THE MANUFACTURER DATASHEETS. .....................................6. FIGURE 2. DIFFERENT CIRCUIT MODELS TO STUDY THE BEHAVIOR OF SOLAR CELLS. (A) 1-DIODE; (B) 1-DIODE/1RESISTOR; (C) 1-DIODE/2-RESISTORS; (D) 2-DIODES/2-RESISTORS. .....................................................................7. FIGURE 3. SKETCH OF A SOLAR PANEL EQUIVALENT CIRCUIT MODEL, COMPOSE BY NS CELLS IN SERIES IN EACH ARRAY AND NP ARRAYS IN PARALLEL. ......................................................................................................................................... 27. FIGURE 4. GENERAL SCHEME OF A PHOTOVOLTAIC SYSTEM FORMED BY THE SOLAR PANELS, DC-DC CONVERTER CONTROLLED BY A MPPT METHOD, AND LOAD. .......................................................................................................... 29. FIGURE 5. SKETCH OF THE IDEAL BOOST CONVERTER. ........................................................................................................... 29 FIGURE 6. EFFECT OF BOOST CONVERTER DUTY CYCLE Α (I.E., THE EQUIVALENT RESISTOR) ON THE OPERATIONAL POINT OF A SOLAR PANEL. ................................................................................................................................................ 30. FIGURE 7 . LEFT AXIS: OIL PRICES PER BARREL (BRENT AND WEST TEXAS INTERMEDIATE). RIGHT AXIS: NUMBER OF PAPERS RELATED WITH MAXIMUM POWER POINT TRACKING (MPPT) PUBLISHED BETWEEN 1968 AND. AUGUST 2005 ACCORDING TO ESRAM AND CHAPMAN (OPEN CIRCLES). ................................................................ 32 FIGURE 8 EXPERIMENTAL (MANUFACTURER SUPPLIED) I-V CURVE. KC200GT KYOCERA SOLAR CELLS. ................... 38 FIGURE 9. LEFT SCALE: EXPERIMENTAL (MANUFACTURER SUPPLIED) I-V CURVE, ESTIMATED CURVES WITH PRESENT METHOD AND METHOD FROM VILLALVA ET AL FOR A = 1.3; RIGHT SCALE: ABSOLUTE DIFFERENCES BETWEEN EXPERIMENTAL AND ESTIMATED I FOR BOTH METHODS. ZTJ EMCORE SOLAR CELLS. ....................................... 40. FIGURE 10. ERROR  OF THE EQUIVALENT CIRCUIT, CALCULATED WITH PRESENT METHOD, DEPENDING ON THE ESTIMATED VALUE FOR THE PARAMETER A. ZTJ EMCORE SOLAR CELLS. ............................................................. 41. FIGURE 11. LEFT SCALE: EXPERIMENTAL (MANUFACTURER SUPPLIED) I-V CURVE, ESTIMATED FOR A = 1.1794; RIGHT SCALE: ABSOLUTE DIFFERENCE BETWEEN EXPERIMENTAL AND ESTIMATED I. ZTJ EMCORE SOLAR CELLS. .................................................................................................................................................................................. 42. FIGURE 12. DIFFERENCE BETWEEN THE CALCULATED AND THE EXPERIMENTAL I-V CURVES WITH REGARD TO THE PHOTOWATT-PWP 201 SOLAR MODULE. (*) SOME POINTS WITH REGARD TO THE PHANG METHOD ARE OUT OF SCALE.............................................................................................................................................................................. 45. FIGURE 13. LEFT SCALE: EXPERIMENTAL (MANUFACTURER SUPPLIED) I-V CURVE, ESTIMATED FOR 2-DIODES METHOD; RIGHT SCALE: ABSOLUTE DIFFERENCE BETWEEN EXPERIMENTAL AND ESTIMATED I. ZTJ EMCORE SOLAR CELLS. ...................................................................................................................................................................... 47. FIGURE 14. EXPERIMENTAL AND ESTIMATED I-V CURVE OF SELEX GALILEO PHOTOVOLTAIC ASSEMBLY (PVA) [84] FOR IRRADIATION OF 700W. .......................................................................................................................................... 49.

(22) vi FIGURE 15. EXPERIMENTAL AND ESTIMATED I-V CURVE OF SELEX GALILEO PHOTOVOLTAIC ASSEMBLY (PVA) [84] FOR IRRADIATION OF 850W. .......................................................................................................................................... 49. FIGURE 16. EXPERIMENTAL AND ESTIMATED I-V CURVE OF SELEX GALILEO PHOTOVOLTAIC ASSEMBLY (PVA) [84] FOR IRRADIATION OF 1367W. ........................................................................................................................................ 49. FIGURE 17. EXPERIMENTAL AND ESTIMATED PV CURVE OF SELEX GALILEO PHOTOVOLTAIC ASSEMBLY (PVA) [84] FOR IRRADIATION OF 700W. .......................................................................................................................................... 50. FIGURE 18. EXPERIMENTAL AND ESTIMATED PV CURVE OF SELEX GALILEO PHOTOVOLTAIC ASSEMBLY (PVA) [84] FOR IRRADIATION OF 850W. .......................................................................................................................................... 50. FIGURE 19. EXPERIMENTAL AND ESTIMATED PV CURVE OF SELEX GALILEO PHOTOVOLTAIC ASSEMBLY (PVA) [84] FOR IRRADIATION OF 1367W. ........................................................................................................................................ 50. FIGURE 20. ABSOLUTE DIFFERENCE BETWEEN EXPERIMENTAL AND ESTIMATED CURRENT FOR RADIATION OF 700 W.......................................................................................................................................................................................... 51 FIGURE 21. ABSOLUTE DIFFERENCE BETWEEN EXPERIMENTAL AND ESTIMATED CURRENT FOR RADIATION OF 850 W.......................................................................................................................................................................................... 51 FIGURE 22. ABSOLUTE DIFFERENCE BETWEEN EXPERIMENTAL AND ESTIMATED CURRENT FOR RADIATION OF 1367W. .............................................................................................................................................................................. 51 FIGURE 23. PREDICTION OF MAXIMUM POWER VOLTAGE (VMP) DEPENDING ON WHICH OPERATING VOLTAGE POINT (VI) IS USED IN CALCULATIONS. ESSAY AT 700 W AND 22.4ºC................................................................................ 54 FIGURE 24. PREDICTIONS OF MAXIMUM POWER POINTS AND EXPERIMENTAL POINTS FOR 700 W AND 22.4ºC. .... 54 FIGURE 25. PREDICTION OF MAXIMUM POWER VOLTAGE (VMP) DEPENDING ON WHICH OPERATING VOLTAGE POINT (VI) IS USED IN CALCULATIONS. ESSAY AT 700 W AND 28.5ºC................................................................................ 55 FIGURE 26. CALCULATED MAXIMUM POWER POINTS AND EXPERIMENTAL POINTS FOR 700 W AND 28.5ºC. .......... 55 FIGURE 27. PREDICTION OF MAXIMUM POWER VOLTAGE (VMP) DEPENDING ON WHICH OPERATING VOLTAGE POINT (VI) IS USED IN CALCULATIONS. ESSAY AT 700 W AND 35.6ºC................................................................................ 56 FIGURE 28. CALCULATED MAXIMUM POWER POINTS AND EXPERIMENTAL POINTS FOR 700 W AND 35.6ºC. .......... 56 FIGURE 29. PREDICTION OF MAXIMUM POWER VOLTAGE (VMP) DEPENDING ON WHICH OPERATING VOLTAGE POINT (VI) IS USED IN CALCULATIONS. ESSAY AT 700 W AND 35.6ºC................................................................................ 57 FIGURE 30. CALCULATED MAXIMUM POWER POINTS AND EXPERIMENTAL POINTS FOR 700 W AND 45.7ºC. .......... 57 FIGURE 31. PREDICTION OF MAXIMUM POWER VOLTAGE (VMP) DEPENDING ON WHICH OPERATING VOLTAGE POINT (VI) IS USED IN CALCULATIONS. ESSAY AT 700 W AND 55.8ºC................................................................................ 58 FIGURE 32. CALCULATED MAXIMUM POWER POINTS AND EXPERIMENTAL POINTS FOR 700 W AND 55.8ºC. .......... 58 FIGURE 33. EARTH MAGNETIC FIELD IN ONE POINT OF A POLAR ORBIT (I.E., INCLINATION I = 90º) WITH AXES PARALLEL TO. FRAME DRAWN IN THAT POINT.. IS PARALLEL TO. AND. ARE INCLUDED IN ORBIT PLANE. . ............................................................................................................................................ 71. FIGURE 34. SKETCH INDICATING THE ANGLE OF MISALIGNMENT, Α, BETWEEN LINES OF ACTION OF AND. AND. VECTOR. AXIS DURING A PERIOD OF 24 HOURS, FOR THE NEAR-POLAR ORBIT OF THE UPMSAT-2 (SUN-. SYNCHRONOUS ORBIT: I = 98º INCLINATION, AT 700 KM ALTITUDE)...................................................................... 71.

(23) vii FIGURE 35. ANGLE OF MISALIGNMENT, Α, BETWEEN LINES OF ACTION OF. VECTOR AND Z 0 AXIS. DURING A PERIOD OF 24 HOURS, FOR THE NEAR-POLAR ORBIT OF THE UPMSAT-2 (SUN-SYNCHRONOUS ORBIT: I = 98º INCLINATION, AT 700 KM ALTITUDE). DASHED LINE INDICATES THE AVERAGE VALUE  = 6.05º. ...... 72. FIGURE 36. MAGNETIC FIELD MEASURED DURING TEST 1 WITH MAGNETOMETER 3....................................................... 74 FIGURE 37. MAGNETIC FIELD MEASURED DURING TEST 2 WITH MAGNETOMETER 3....................................................... 74 FIGURE 38. MAGNETIC FIELD MEASURED DURING TEST 1 WITH MAGNETOMETER 1....................................................... 75 FIGURE 39. MAGNETIC FIELD MEASURED DURING TEST 2 WITH MAGNETOMETER 1....................................................... 75 FIGURE 40. MAGNETIC FIELD MEASURED DURING TEST 1 WITH MAGNETOMETER 2....................................................... 76 FIGURE 41. MAGNETIC FIELD MEASURED DURING TEST 2 WITH MAGNETOMETER 2....................................................... 76 FIGURE 42. ALGORITHM IN SIMULINK THAT REPRESENTS THE MAGNETOMETER OPERATION. ...................................... 77 FIGURE 43. MAGNETIC MOMENT RESPONSE OF A MAGNETORQUER DEPENDING OF THE APPLIED CURRENT/VOLTAGE [113] .................................................................................................................................................................................. 78 FIGURE 44. SKETCH OF THE EXPERIMENT TO MEASURE THE MAGNETIC FIELD GENERATED BY THE MAGNETOMETER. .............................................................................................................................................................................................. 79 FIGURE 45. MAGNETORQUER’S GENERATED MAGNETIC FIELD DURING TESTS FOR A VOLTAGE VARYING FROM -22V TO 22V. ............................................................................................................................................................................... 79. FIGURE 46. RESPONSE WITH THE TIME AFTER A STEP SIGNAL IN THE VOLTAGE FOR ZARM MAGNETORQUER (FROM MANUFACTURER DATASHEET) [114]. ........................................................................................................................... 80. FIGURE 47. EXCITATION AND RESPONSE SIGNALS DURING THE STEP RESPONSE EXPERIMENTS ON MAGNETORQUERS ZARM . ............................................................................................................................................................................... 81 FIGURE 48. ALGORITHM IN SIMULINK THAT REPRESENTS MAGNETORQUER OPERATION................................................ 82 FIGURE 49. MAXIMUM, MINIMUM AND AVERAGED VALUES OF THE MISALIGNMENT ANGLE, Α, BETWEEN THE DIRECTION OF VECTOR OF. AND Z0 AXIS, FOR ORBITS AT 700 KM ALTITUDE, AS A FUNCTION. OF THE INCLINATION, I. ..................................................................................................................................................... 89. FIGURE 50. SKETCH OF THE SATELLITE REGARDING THE EULER ANGLES........................................................................... 90 FIGURE 51 EVOLUTION OVER TIME, T, OF ANGLE Θ, STARTING FROM Θ = 90° (UPPER) AND Θ = 45° (LOWER) ........ 93 FIGURE 52. EVOLUTION OVER TIME, T, OF ANGLE Θ, STARTING FROM Θ = 90° (UPPER) AND Θ = 45° (LOWER). .... 93 FIGURE 53. NON-DIMENSIONAL DIFFERENCE BETWEEN THE STABILIZATION ANGULAR VELOCITY AND THE TARGET ROTATION RATE,. , PLOTTED ALONG TWO ORBITS FOR A SATELLITE EQUIPPED WITH THE. PROPOSED CONTROL STRATEGY. DATA FROM A SUN-SYNCHRONOUS ORBIT AT 700 KM ALTITUDE AND I = 98.. THE RESULTS FROM DIFFERENT VALUES OF THE GAIN CONSTANT, K, ARE INCLUDED, TOGETHER WITH THE ONES CORRESPONDING TO THE LIMITS K→ 0 AND K→ ∞. .......................................................................................... 95. FIGURE 54. MAXIMUM AND MINIMUM PEAKS OF THE NON-DIMENSIONAL DIFFERENCE BETWEEN THE STABILIZATION ANGULAR VELOCITY AND THE TARGET ROTATION RATE,. , AS A FUNCTION OF THE ORBIT’S. INCLINATION. THESE PEAKS ARE THE MAXIMUM AND MINIMUM VALUES FOR THE LIMIT K→ ∞ , SEE FIGURE 53.. THE VALUES CORRESPONDING TO THE LIMIT K→ 0 AT EACH INCLINATION ARE ALSO PLOTTED IN THE GRAPH. .............................................................................................................................................................................................. 96.

(24) viii FIGURE 55. RESULTS OF THE FIRST MONTE CARLO ANALYSIS, CARRIED OUT TO STUDY THE CHARACTERISTIC TIME, TANG, NEEDED TO STABILIZE THE ROTATION RATE. OPEN CIRCLES (WITH THEIR STANDARD DEVIATION BARS) STAND FOR THE AVERAGE VALUES OBTAINED FROM THE 1000 CASES SIMULATED FOR THE SELECTED CONTROL GAIN VALUES K = 5·105 , 1·106, 2.5·106 , 5·106, 7.5·106, 1·107, 1·108 AND 1·109. THE DASHED LINE INDICATES THE APPROXIMATION DERIVED IN EQUATION (121). ............................................................................. 97. FIGURE 56. RESULTS OF THE SECOND MONTE CARLO ANALYSIS, CARRIED OUT TO STUDY THE SITUATION ONCE THE ROTATION RATE IS STABILIZED FOR CONTROL GAIN CONSTANTS K = 1·107 , 1·108 AND 1·109 (K’ = 1·106 IN THE GRAPHS’ LEGENDS). AVERAGED VALUE OF THE MISALIGNMENT ANGLE, Γ, BETWEEN THE DIRECTION OF THE SATELLITE’S ROTATION AXIS, ZS, AND THE NORMAL DIRECTION TO THE ORBIT’S PLANE, Z0, ALONG ONE ORBIT, AS A FUNCTION OF THE ORBIT’S INCLINATION, I. MAXIMUM, MINIMUM AND AVERAGED VALUES OF THE MISALIGNMENT ANGLE, Α, BETWEEN THE DIRECTION OF VECTOR OF. AND Z0 AXIS HAVE BEEN. ADDED TO THE GRAPHS..................................................................................................................................................... 99. FIGURE 57. RESULTS OF THE SECOND MONTE CARLO ANALYSIS, CARRIED OUT TO STUDY THE SITUATION ONCE THE ROTATION RATE IS STABILIZED FOR CONTROL GAIN CONSTANTS K = 1·107 , 1·108 AND 1·109 (K’ = 1·106 IN THE GRAPHS’ LEGENDS). NON-DIMENSIONAL DIFFERENCE BETWEEN THE STABILIZATION ANGULAR VELOCITY AND THE TARGET ROTATION RATE,. , AS A FUNCTION OF THE ORBIT’S INCLINATION, I.. MAXIMUM AND MINIMUM PEAKS OF THE ANALYTICAL SOLUTION FOR K→ ∞, AND THE ANALYTICAL SOLUTION FOR K→ 0 ARE INCLUDED IN THE GRAPH. ................................................................................................................... 100. FIGURE 58 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT FOR UPMSAT-2. INITIAL CONDITIONS:. AND. . .......................... 103. FIGURE 59 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT FOR UPMSAT-2. INITIAL CONDITIONS:. AND. . ......................... 104. FIGURE 60 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT FOR UPMSAT-2. INITIAL CONDITIONS:. AND. . .............................. 104. FIGURE 61 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT Θ FOR UPMSAT-2. INITIAL CONDITIONS:. AND. . .......................... 105. FIGURE 62 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT Θ FOR UPMSAT-2. INITIAL CONDITIONS:. AND. . ......................... 105. FIGURE 63 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT Θ FOR UPMSAT-2. INITIAL CONDITIONS:. AND. . ............................. 106. FIGURE 64 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT FOR UPMSAT-2. INITIAL CONDITIONS:. AND. .......................... 106. FIGURE 65 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT FOR UPMSAT-2. INITIAL CONDITIONS:. AND. . ......................... 107. FIGURE 66 EVOLUTION OVER TIME, T, OF ANGULAR VELOCITY COMPONENTS AND ANGLE OF MISALIGNMENT FOR UPMSAT-2. INITIAL CONDITIONS:. AND. . ............................. 107. FIGURE 67. ORBIT WITH THE AXIS Z PERPENDICULAR TO ORBIT. ..................................................................................... 109 FIGURE 68. ORBIT WITH THE AXIS Z PARALLEL TO THE MAGNETIC FIELD....................................................................... 110 FIGURE 69. SIMULATOR OF UPMSAT-2 MISSION. .............................................................................................................. 111.

(25) ix FIGURE 70. SIMULATOR OF UPMSAT-2. ............................................................................................................................... 111 FIGURE 71. ERROR ANGLE BETWEEN Z AXIS AND THE PERPENDICULAR OF THE ORBIT WITH NOMINAL CONTROL. 112 FIGURE 72. ERROR ANGLE BETWEEN THE Z AXIS AND THE DIRECTION OF THE MAGNETIC FIELD WITH NOMINAL CONTROL.......................................................................................................................................................................... 112 FIGURE 73. ANGULAR VELOCITY WITH NOMINAL CONTROL. ............................................................................................. 113 FIGURE 74. POWER PRODUCTION WITH NOMINAL CONTROL. ........................................................................................... 113 FIGURE 75. POWER CONSUMED BY THE MAGNETORQUERS USING NOMINAL CONTROL................................................ 114 FIGURE 76. ERROR ANGLE BETWEEN Z AXIS AND THE PERPENDICULAR OF THE ORBIT WITH B-DOT CONTROL. ..... 114 FIGURE 77. ERROR ANGLE BETWEEN THE Z AXIS AND THE DIRECTION OF THE MAGNETIC FIELD WITH B-DOT CONTROL.......................................................................................................................................................................... 115 FIGURE 78. ANGULAR VELOCITY WITH B-DOT CONTROL. .................................................................................................. 115 FIGURE 79. POWER PRODUCTION WITH B-DOT CONTROL. ................................................................................................ 116 FIGURE 80. POWER CONSUMED BY THE MAGNETORQUERS USING B-DOT CONTROL. .................................................... 116 FIGURE 81. ERROR ANGLE BETWEEN Z AXIS AND THE PERPENDICULAR OF THE ORBIT WITH COMPASS CONTROL. 117 FIGURE 82. ERROR ANGLE BETWEEN THE Z AXIS AND THE DIRECTION OF THE MAGNETIC FIELD WITH COMPASS CONTROL.......................................................................................................................................................................... 117 FIGURE 83. ANGULAR VELOCITY WITH COMPASS CONTROL. ............................................................................................. 118 FIGURE 84. POWER GENERATED WITH THE SOLAR PANELS USING “COMPASS” CONTROL............................................ 118 FIGURE 85. POWER CONSUMED BY THE MAGNETORQUERS USING “COMPASS” CONTROL. ............................................ 119 FIGURE 86 SKETCH OF UPMSAT-2 ........................................................................................................................................ 135 FIGURE 87 DEPLOYED SKETCH OF UPMSAT-2 .................................................................................................................... 136 FIGURE 88. CALCULATED VALUES OF THE EQUIVALENT CIRCUIT PARAMETERS RS, RSH, IPV, AND I0, FOR YL280C-30B MONOCRYSTALLINE PANEL AS A FUNCTION OF THE TEMPERATURE, T. CALCULATED POINTS ARE INDICATED WITH SYMBOLS WHEREAS THE POLYNOMIAL APPROXIMATIONS FITTED TO THOSE DATA (EQUATIONS (139) HAVE BEEN INCLUDED IN EACH CASE AS SOLID LINES). ............................................................................................ 140. FIGURE 89. AMBIENT CONDITIONS FOR MAY THE 13TH 1971 (TOP) AND MAY THE 14TH 1971 (BOTTOM). SOLAR IRRADIANCE IS REPRESENTED IN LEFT SIDE, AND TEMPERATURE AND WIND VELOCITY IN RIGHT SIDE. ......... 141. FIGURE 90. SOLAR PANEL TEMPERATURES ON MAY THE 13TH 1971 (TOP) AND MAY THE 14TH 1971 (BOTTOM), AT THE GODDARD SPACE FLIGHT CENTER (GSFC). THESE GRAPHS WERE CALCULATED ACCORDING TO AMBIENT CONDITIONS FOR THOSE DAYS, SEE FIGURE 89 AND EQUATION (141). ............................................................... 142. FIGURE 91. EVOLUTION OF THE I-V CURVE OF YL280C-30B MONOCRYSTALLINE SOLAR PANEL ACCORDING TO AMBIENT CONDITIONS, THAT IS, SUN IRRADIANCE AND PANEL TEMPERATURE LEVELS, ON MAY THE 13TH 1971. (TOP), AT THE GODDARD SPACE FLIGHT CENTER (GSFC). EVOLUTION OF CURRENT (BOTTOM-LEFT) AND VOLTAGE (BOTTOM-RIGHT) AT MAXIMUM POWER POINT (MPP). ...................................................................... 143. FIGURE 92. EVOLUTION OF THE I-V CURVE OF YL280C-30B MONOCRYSTALLINE SOLAR PANEL ACCORDING TO AMBIENT CONDITIONS, THAT IS, SUN IRRADIANCE AND PANEL TEMPERATURE LEVELS, ON MAY THE 14TH 1971. (TOP), AT THE GODDARD SPACE FLIGHT CENTER (GSFC). EVOLUTION OF CURRENT (BOTTOM-LEFT) AND VOLTAGE (BOTTOM-RIGHT) AT MAXIMUM POWER POINT (MPP). ...................................................................... 144. FIGURE 93. BLOCK-DIAGRAMS OF THE DIFFERENT MPPT................................................................................................. 146.

(26) x FIGURE 94. POWER PRODUCED BY YL280C-30B MONOCRYSTALLINE SOLAR PANEL CALCULATED WITH THE STUDIED MPPT METHODS, FOR AMBIENT CONDITIONS MEASURED AT THE GODDARD SPACE FLIGHT CENTER (GSFC) ON MAY THE 13TH 1971 (CLOUDY DAY) AND MAY THE 14TH 1971 (SUNNY DAY). MAXIMUM EXTRACTABLE POWER (IDEAL) HAS BEEN ALSO INCLUDED. .............................................................................................................. 148.

(27) xi. List of Tables TABLE 1. I-V CURVE DATA (SHORT CIRCUIT –V = 0, I = ISC–, OPEN CIRCUIT –V = VOC, I = 0–, AND THE MAXIMUM POWER –V = VMP, I = IMP– POINTS; THE SLOPES OF THE I-V CURVE AT THE OPEN CIRCUIT AND SHORT CIRCUIT POINTS, RS0 AND RSH0) OF SEVERAL SOLAR CELLS. (CDS –CADMIUM SULFIDE SOLAR CELL–, BSC AND GSC – SILICON SOLAR CELLS–, Q6LMTM – SILICON CELL FROM Q-SOLAR–). ........................................................................9. TABLE 2. 1-DIODE/2-RESISTORS CIRCUIT MODEL PARAMETER VALUES FROM DIFFERENT SOLAR CELLS.................... 10 TABLE 3. VALUES OF EVERY TERM FROM EQUATION (6), OBTAINED FOR DIFFERENT SOLAR CELLS (CALCULATED WITH COEFFICIENTS FROM TABLE 1 AND TABLE 2). .................................................................................................. 11. TABLE 4. VALUES OF EVERY TERM FROM EQUATION (8), OBTAINED FOR DIFFERENT SOLAR CELLS (CALCULATED WITH COEFFICIENTS FROM TABLE 1 AND TABLE 2). .................................................................................................. 11. TABLE 5. VALUES OF EVERY TERM FROM EQUATION (27), OBTAINED FOR DIFFERENT SOLAR CELLS (CALCULATED WITH COEFFICIENTS FROM TABLES 1 AND 2). ............................................................................................................. 17. TABLE 6. MANUFACTURER’S DATA FROM A KC200GT SOLAR ARRAY [8], [9]. THE CORRESPONDING VALUES FROM THE EQUIVALENT ELECTRIC CIRCUIT CALCULATED WITH THE PROPOSED METHOD HAVE ALSO BEEN INCLUDED.. .............................................................................................................................................................................................. 37 TABLE 7. PARAMETERS OF 1-DIODE/2-RESISTORS CIRCUIT MODEL ADJUSTED WITH THE PRESENTED METHODOLOGY TO THE KC200GT SOLAR ARRAY. THE RESULTS FROM REFERENCES [8], [9], HAVE ALSO BEEN INCLUDED IN THE TABLE. ......................................................................................................................................................................... 38. TABLE 8. CHARACTERISTICS OF THE SOLAR CELL ZTJ (29.5% EFFICIENCY) FROM EMCORE. ........................................ 39 TABLE 9. EMCORE ZTJ (29.5% EFFICIENCY) SOLAR CELL EQUIVALENT CIRCUIT PARAMETERS FOR PRESENT METHOD AND METHOD FROM REFERENCES [8], [9] . .................................................................................................................. 40. TABLE 10. ZTJ EMCORE CELL EQUIVALENT CIRCUIT PARAMETERS FOR ESTIMATED A = 1.1794. ................................ 42 TABLE 11. I-V CURVE FROM A PHOTOWATT-PWP 201 SOLAR MODULE COMPOSED OF 36 SOLAR CELLS IN SERIES [11]. .................................................................................................................................................................................... 44 TABLE 12. VALUES OF THE CURRENT AND VOLTAGE AT SHORT CIRCUIT, OPEN CIRCUIT, AND MAXIMUM POWER POINTS (ISC, VOC, VMP, IMP), CALCULATED FOR A PHOTOWATT-PWP 201 SOLAR MODULE (SEE [11]) FROM EXPERIMENTAL DATA........................................................................................................................................................ 44. TABLE 13. PARAMETERS OF 1-DIODE/2-RESISTORS CIRCUIT MODEL ADJUSTED WITH THE PRESENTED METHODOLOGY TO THE PHOTOWATT-PWP 201 SOLAR MODULE BEHAVIOR (TABLE 11). THE RESULTS FROM OTHER AUTHORS HAVE ALSO BEEN INCLUDED IN THE TABLE..................................................................................... 45. TABLE 14. NON DIMENSIONAL STANDARD DEVIATION, , CALCULATED, USING THE PROPOSED METHOD, WITH RESPECT TO THE EXPERIMENTAL VALUES OF THE I-V CURVES OF THE R.T.C. FRANCE SILICON CELL, AND THE. PHOTOWATT-PWP 201 SOLAR MODULE. RESULTS FROM OTHER NUMERICAL METHODS FROM THE AVAILABLE LITERATURE HAVE ALSO BEEN INCLUDED. SEE ALSO FIGURE 12. ............................................................................. 46. TABLE 15. PARAMETERS OF THE 2-DIODE/2-RESISTORS EQUIVALENT CIRCUIT MODELS TO THE EMCORE ZTJ TRIPLE-JUNCTION. ............................................................................................................................................................. 46.

(28) xii TABLE 16. PARAMETERS OF THE EQUIVALENT CIRCUIT AND ERROR IN SIMULATION OF EXPERIMENTAL CONDITIONS FOR ESSAYS OF SELEX GALILEO PHOTOVOLTAIC ASSEMBLY (PVA) [84] WITH DIFFERENT TEMPERATURE AND IRRADIATION CONDITIONS. ...................................................................................................................................... 52. TABLE 17. LIST OF CONTROLS THAT PROPOSE B-DOT MODIFICATIONS, ITS IMPROVEMENTS AND ADDITIONAL NEEDS. .............................................................................................................................................................................................. 61 TABLE 18. WMM2010. SPHERICAL-HARMONIC MAIN FIELD MODEL FOR 2010.0 IN UNITS OF NT AND SECULAR VARIATION MODEL FOR THE PERIOD 2010.0 TO 2015.0 IN UNITS OF NT/YEAR.................................................. 66. TABLE 19. OUTPUTS OF THE MAGNETOMETER 3 (BARTINGTON) DURING TESTS 1 AND 2. ............................................ 73 TABLE 20. OUTPUTS OF THE MAGNETOMETER 1 (SSBV FM-002) DURING TESTS 1 AND 2 . ....................................... 74 TABLE 21. OUTPUTS OF THE MAGNETOMETER 1 (SSBV FM-003) DURING TESTS 1 AND 2. ........................................ 75 TABLE 22. INTERNAL RESISTANCE AND TIME RESPONSE OF THE MAGNETORQUERS ZARM........................................... 81 TABLE 23. ENERGY BALANCE OF DIFFERENT ATTITUDE CONTROLS. ................................................................................ 119 TABLE 24 UPMSAT-2 MISSION CHARACTERISTICS. .................................................................................................... 136 TABLE 25 UPMSAT-2 PAYLOADS........................................................................................................................................... 137 TABLE 26. CHARACTERISTIC POINTS OF YL280C-30B SOLAR PANEL (YINGLI SOLAR), INCLUDED IN THE MANUFACTURER DATASHEETS [123] AT STANDARD TEST CONDITIONS (STC): 1000 W/M2 IRRADIANCE, 25. °C CELL TEMPERATURE, AM1.5G SPECTRUM ACCORDING TO EN 60904-3 [123]. ......................................... 140 TABLE 27. POWER OBTAINED FROM YL280C-30B MONOCRYSTALLINE SOLAR PANEL CALCULATED WITH AMBIENT CONDITIONS MEASURED AT THE GODDARD SPACE FLIGHT CENTER (GSFC) ON MAY THE 13TH 1971. (CLOUDY DAY) AND MAY THE 14TH 1971 (SUNNY DAY). ...................................................................................... 147.

(29) xiii. List of Acronyms a. =. Quality factor of diode in single diode model.. a1/2. =. Quality factor of diode in double diode model.. s 0. =. Coordinates transformation matrix from. B. =. Geomagnetic-field vector, T.. B. =. Rate of variation of the geomagnetic-field vector, T/s.. Ekin. =. Kinetic energy, J.. A. eˆx , eˆ y , eˆz. 0. to. s. .. = Unitary vectors of a frame.. E. =. ECEF, Earth Centered, Earth Fixed frame.. i. =. ECI, Earth-centered inertial frame.. s. =. Spacecraft fixed-frame.. 0. =. Auxiliary Reference frame parallel to orbit plane.. 1. =. First auxiliary frame.. 2. =. Second auxiliary frame.. G. =. Solar irradiation, W.. Gr. =. Reference solar irradiation, W.. h. =. Angular momentum vector, kg·m2·s-1·rad.. I. =. Current, A.. I. =. Spacecraft matrix of inertia along its principal axes, kg·m2.. I. =. Inertia moment of the spacecraft around. xs. I ||. =. Inertia moment of the spacecraft around. zs. I mp. =. Current at maximum power point, A.. I sc. =. Short circuit current, A.. and. ys. axis, kg·m2.. axis, kg·m2..

(30) xiv. I pv. =. Photocurrent delivered by the constant current source in the solar panel equivalent circuit, A.. I0. =. Reverse saturation current of diode in single diode model, A.. I 01/2. =. Reverse saturation current of diode in double diode model, A.. i. =. Inclination of the orbit, rad.. k. =. Control gain, A·m2·T-1·s.. k. =. Boltzmann constant, m2·kg·s-2·K-1.. m. =. Magnetic dipole moment vector of the control, A·m2.. mE. =. Magnetic dipole moment vector of the Earth, A·m2.. N. =. Number of solar cells in a solar cells.. NP =. Number of arrays in parallel in a solar panel.. NS. =. Number of cells in series in a solar panel.. P. =. Power, W.. Pmp. =. Maximum power, W.. q. =. Charge of electron, C.. R. =. Vector position respect to the Earth center, m.. R. =. Distance to the Earth center, m.. RE. =. Earth average radius, m.. Ri. =. Slope of I-V curve at point i, Ω.. Rs. =. Series resistance in the solar panel equivalent circuit, Ω.. Rs 0 =. Slope of I-V curve at open circuit point, Ω.. Rsh =. Shunt resistance in the solar panel equivalent circuit, Ω.. Rsh 0 =. Slope of I-V curve at short circuit point, Ω.. R0. Orbit radius, m.. =. RMSE. = Root-mean-square deviation..

(31) xv. S() =. Skew symmetric matrix.. SD =. Standard deviation.. T. =. Temperature, ºC.. T. =. External torque acting on the spacecraft, N·m.. Tcoil. =. Magnetic control torque, N·m.. Tdist. =. Disturbance torque, N·m.. u. =. Argument of latitude in a circular orbit, rad.. V. =. Voltage, V.. Vmp. =. Voltage at maximum power point, V.. Voc. =. Open circuit voltage, V.. VT. =. Thermal voltage, V.. . =. Angle between. . =. Angle between. T. =. Rotation angle between. . =. Second Euler angle, rad..  'm. =. Co-elevation angle of the dipole direction in. . =. Non-dimensional standard deviation.. . =. Attitude correction torque, N·m.. . =. First Euler angle, rad..  'm. =. East longitude angle of the dipole direction in. . =. Third Euler angle, rad.. o. =. Orbital rotation rate, rad/s..  02. =. Angular velocity of. 0. B(t )  0 B(t ). and. Z 0 , rad.. z s and Z 0 , rad.. 2. E. frame and. respect to. 0. i. , rad.. , rad/s.. E. , rad.. E. , rad..

(32) xvi. . =. Satellite angular velocity vector, rad/s.. d. =. Desired angular velocity, parameter of the control, rad/s.. Left Superscript. 0. =. Vector components expressed in. 0. 1. =. Vector components expressed in. 1. 2. =. Vector components expressed in. 2. frame.. E. =. Vector components expressed in. E. frame.. i. =. Vector components expressed in. i. frame.. s. =. Vector components expressed in. s. frame.. frame. frame..

(33) 1. 1 Introduction 1-1. Problem Statement. The work carried out in the present doctoral dissertation should be considered part of the UPMSat-2 project, falling within the scope of the aerospace technology. The UPMSat-2 is a microsatellite (low cost and small size) designed, constructed integrated and tested for educational and technology demonstration purposes at the Universidad Politécnica de Madrid (Spain). One of the defining characteristics of space technology is the reduced possibility of maintenance or changes during the operational life of a spacecraft, in order to correct possible errors or to optimize the operation. Therefore, simulation becomes one of the main instruments for space technology development. This doctoral dissertation is focused on improving the analytical modeling of power and attitude control subsystems. Analytical modeling of an engineering problem, compared with numerical modeling, has the advantage of maintaining physical properties in equations governing the model. This allows a better understanding of the physical phenomena, which provides, in the present case, the opportunity to develop specific strategies of control in relation to both mentioned subsystems (power and attitude control). The simulation of a power subsystem deals with the correct modeling of the solar panels operation, the influence of environmental conditions in its energy production, and its possible optimization. All these factors need to be properly included in the analysis tools, (no matters if they are equations or a simulator), for reproducing the entire subsystem. On the other hand, the attitude control that will be analyzed in this work is completely magnetic. It comprises the modeling of the instruments, sensors and actuators; the modeling of the magnetic field of the earth and finally the control law implementation. Magnetic attitude control is a perfect example of the benefits of facing a complex problem under an analytical point of view. Magnetic control systems only have two degrees of freedom (as it is not possible to produce torques perpendicular to the magnetic field, see section 3), so most numerical control laws are not suitable in this case as they suppose control in the three axes. Consequently, a deep study and knowledge of the peculiarities of.

(34) 2 this kind of control system, based on an analytical approach, is essential for designing proper control laws. Due to the strong dependence between subsystems in aerospace technology, the design (or optimization) of a subsystem cannot be faced as an independent problem (in fact this design is carried out nowadays in concurrent design facilities). In the present work, analytical models are presented to study both the power and the attitude control subsystems, taking into account their interaction. This is particularly interesting during the predesign of the satellite, as, including the aforementioned interaction, the power budget can be more precisely estimated. The power subsystem in a satellite is affected by the control subsystem, as the instrumentation of the latter and the control law will affect the total amount of energy consumed; furthermore, the orientation of the satellite defined by the control subsystem will determine the total amount of energy produced by the solar cells. These factors define the interdependence within the power and the control subsystem. Finally, it should be pointed out that the power production will be also affected by other subsystems such as the thermal control subsystem, but the analysis of this interaction falls out of the scope of this doctoral dissertation.. 1-2. Thesis Objectives. The aim of the present doctoral dissertation is to develop new analytical models to study the energy interdependence between the power and the attitude control subsystems of a satellite. To do this, the first step is to analyze all the environmental factors that affect both subsystems and its modeling. Then, a thorough analysis of elements and parts that compose the subsystems is needed. This analysis is carried out by comparison of the analytical models with performance data, either experimental or provided by the manufacturer. The performance data has to be reproduced with adequate models that behave like the real instruments. Therefore, among the objectives of this doctoral dissertation is to simulate the instruments of the attitude control subsystem and the elements of power production, as a function of the ambient conditions that affect both subsystems..

(35) 3. 1-3. Doctoral Dissertation Structure. The doctoral dissertation document includes a general introduction of the studied problem in Chapter 1. Chapter 2 includes the work done in relation to the power subsystem. Following the introduction, in this chapter the study focuses on the analytical modeling of the solar panels. The solar panels are the most important part of the power subsystem from the standpoint of controlling the power production. Thus, the different stages to model a complete solar panel and the various proposed methods are described. After that, the influence of the environmental effects is included in the analytical modeling. The next part of the chapter describes the control of the power generation, and the possible use of the analytical models proposed to analyze its performance. This section ends with simulations carried out and the analysis of the results. In Chapter 3, the work carried out to study the attitude control subsystem of a satellite is included. This doctoral dissertation focuses on a purely magnetic control subsystems, similar to the one included in the UPMSat-2. The different environmental conditions that influence the performance of this kind of control subsystem are described in the introduction of the chapter. Next, a description of the models used to reproduce the instruments behavior is included. In subsection 3.3 of the chapter a control law is proposed, making special emphasis on its operation and demonstrating its physical foundations. Finally simulations and results are included. In Chapter 4 the energy interdependence of the two subsystems is analyzed. The proposed control law is compared with other potential control laws considered for the UPMSat-2 Finally, the conclusions in relation to the present work are summarized in Chapter 5. At the end of this document, after the literature, two appendixes can be found. Appendix 1 describes the characteristics of the mission UPMSat-2 that serves as a reference for the subsystems described in this dissertation. Appendix 2 includes an application of the panel simulation methods presented in this doctoral dissertation to an on-earth simulation of solar panels. This on-earth example was selected as the ambient conditions were well known, and also in order to demonstrate further applications of this work..

(36) 4. 2 Analytical Models Applied to the Power Subsystem of a Spacecraft 2-1. Introduction. The use of renewable energy is a big concern in modern societies, and among the different sources, photovoltaic energy is one of the most relevant in terms of increase of installed power. In space, photovoltaic energy is even more important, because it is the most convenient energy source for long orbiting periods of time and the only one renewable. These facts have led scientists to study the behavior of photovoltaic cells and the methods to optimize their power generation. From the middle of the twentieth century, descriptions of the mechanisms that rule the conversion of solar radiation into electric power have been published [1–6]. Also, a great effort has been done to develop equivalent electrical/mathematical models to analyze the behavior of solar cells under different conditions, mainly different radiation levels and cell temperatures. An electrical model consists in a simple circuit whose behavior fits the real behavior of a solar cell (see Figure 1 and Figure 2). The use of these circuit models, together with the correct definition of the electric parameters involved, is extremely important in order to maximize the extracted power from the cell working under real conditions. Also, the use of equivalent circuit models makes the simulation of more complex power systems that include solar cell panels possible. It should be pointed out that sometimes these power systems can have a very complicated behavior; in space applications these systems include batteries and programmed power consumptions, with important temperature gradients and different radiation levels affecting the output voltage of the solar panels, and must be very well optimized to ensure the survival of the satellite/spacecraft.. 2-2. Solar Cell Modeling. The photoelectric effect is responsible for transforming the sun radiation on the solar cells into electric energy. In general, the easiest way to characterize a solar cell is considering a current source connected in parallel to an ideal diode (see Figure 2-a) [1], [2], [5]. The equation that describes the behavior of the solar cell is then composed of two terms, one.

(37) 5 related to the source and the other to the p-n junction (which is, in fact, Shockley’s ideal diode equation) [5]:   qV I  I pv  I 0 exp   kT .     1 .  . (1). The first term of the expression above, Ipv, is the photocurrent delivered by the constant current source, the second term is the ideal recombination current from the diffusion and recombination of electrons and holes in p and n sides of the cell (Shockley diffusion theory), where I0 is the reverse saturation current corresponding to it, T is the temperature and k is the Boltzmann constant. Finally, q is the charge of the electron. The last three constants are usually grouped into the so called thermal voltage, VT:. VT . kT . q. (2). In order to improve Expression (1) and have a better fitting to the cell behavior, two resistors are usually added to the circuit (see Figure 2-c). One resistor (the shunt resistor, Rsh), represents the current leakage through the high conductivity shunts across the p-n junction and is added in parallel with the source and the diode. The other one (the series resistance, Rs), is connected in series and represents the losses in cell solder bonds, interconnection, junction box, etc. [4], [7]. Also, a non-dimensional constant, a, is added to the term of the recombination current in the p- and n- sides. This constant is called ideality or quality factor (or sometimes emission coefficient), and it takes into account the deviation of the diodes from the Shockley diffusion theory (the value of this factor, a, is assumed to be constant and between 1 and 1.5 for one-junction cells [8], [9], although some authors suggest that it depends on the ratio between the current, I, and voltage, V, of the cell [10]). The 1-Diode/2-Resistors circuit model is then defined with the expression:.   V  IRs   V  IRs I  I pv  I 0 exp  .   1  Rsh  aVT   . (3).

(38) 6. 0.8. I [A]. RTC France (Si). 0.7. TNJ Spectrolab. 0.6. ZTJ Emcore. 0.5 0.4 0.3 0.2 0.1 0.0 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. V [V] 1.2. P [W]. RTC France (Si). 1.0. TNJ Spectrolab. ZTJ Emcore. 0.8 0.6 0.4 0.2 0.0 0.0. 0.5. 1.0. 1.5. 2.0. 2.5. 3.0. V [V] Figure 1. Top: Current-voltage (I-V) curve of different solar cells: silicon (Si) cell from R.T.C. France (measured with CBM8096 microcomputer at 33°C [11]), TNJ triple-junction (GaInP2/GaAs/Ge) cell from Spectrolab (measured at AM0 135.3 mW/cm² and 28°C), and ZTJ triple-junction (InGaP/InGaAs/Ge) cell from Emcore (measured at AM0 135.3 mW/cm² and 28°C). In every curve short circuit (triangles), maximum power (circles), and open circuit (squares) points are indicated. Bottom: Power curve of the aforementioned solar cells. Data from TNJ and ZTJ solar cells extracted from the manufacturer datasheets..

(39) 7. Figure 2. Different circuit models to study the behavior of solar cells. (a) 1-diode; (b) 1-diode/1-resistor; (c) 1-Diode/2-Resistors; (d) 2-Diodes/2-Resistors.. Another change to the solar cell model was proposed in 1961 by Wolf and Rauschenbach [4]. These authors suggested that the I-V characteristics of silicon solar are more accurately represented by a double exponential expression (see Figure 2-d), the second exponential standing for the current from the recombination of electrons and holes in the depletion region, and being dominant at lower forward-bias voltages. The behavior of the solar cell can be then translated into the following equation:.    V  IRs    V  IRs   V  IRs I  I pv  I 01 exp  ,   1  I 02 exp    1  Rsh  a1VT    a2VT    . (4). where I01 and I02 are the saturation currents of each diode, and a1 and a2 the ideality factors that take into account the deviation of the diodes from the Shockley diffusion theory, a1 is close to 1 while a2 is frequently greater than 2. The mentioned 2-Diode/2-Resistors electrical model has been used by some authors to study solar cell performances [7], [12–15]. However, it is a quite complicated model as up to seven constants (only six if, as said, the hypothesis a1~1 is assumed, or five if a2~2 is also assumed) must be firstly defined by means of experimental testing at defined levels of irradiance and temperature. On the other hand, it should be said that the more simple 1Diode/2-Resistors model is commonly used as it correctly represents the behavior of the solar cell around the maximum power point, that is, at high voltage levels [16], [17]. This model simplifies the analysis of the solar cell behavior as a function of the different circuit.

(40) 8 variables [18–21], and has been used to analyze the effect of the irradiance and the temperature on the cell behavior [22], [23]. An even more simplified version of the solar cell circuit, the 1-diode/1-resistor model (Figure 2-b), has been proposed by some authors to study a specific aspect of the solar cell, like the series resistor or the behavior under particular conditions [4], [24]. Finally, it should be also mentioned that each model can be the best option depending on the solar cell [14], [25]. The aforementioned theoretical approximations to the solar cell behavior are widely accepted, nevertheless, some new models have arisen describing the solar cell as a multiplezone element, with different electrical behavior within these zones [26], [27]. Once the circuit model has been chosen to study a particular solar cell (or solar cell array), it has to be adjusted (that is, the value of the circuit parameters must be estimated as accurately as possible). These calculations can be based on the calibration results of the cell, that is, once the I-V curve has been measured under certain irradiance and temperature conditions in a lab, the parameters of the model can be adjusted to reach the best possible fit to the mentioned curve [11], [28–33]. However, sometimes the only information available to adjust the selected circuit model comes from the manufacturer, and it is limited to only certain points of the I-V curve (short circuit, open circuit, and maximum power points, see Figure 1) [8], , [18], [34–37]. Finally, with regard to the methods developed to adjust the parameters of the selected circuit model, some of them are numerical [14], [15], [19], [20], [38], [39] whereas some others are analytical [2], [40–42]. Analytical methods have the advantage of being simple, reliable and fast. However, such methods are normally based on experimental characteristics of the I-V curve, that is, they require extensive testing results [43]. On the other hand, some authors have developed numerical methods to adjust the electric circuit parameters to the mentioned characteristic point of the curve [8], [9], [37]. This approach is quite interesting, as it uses only few data to allow final users to analyze the performance of photovoltaic systems. In this section, a new analytical method for photovoltaic equivalent electric circuit parameters extraction is proposed. This methodology is based only on manufacturers’ data. As far as the author know, and according to a recent review [43], this approach to the parameter extraction problem does not seem to have been studied yet..

(41) 9. 2-2-1. 1-Diode/2-Resistors Equivalent Circuit. Model As said, the 1-Diode/2 Resistors circuit model is one of the most commonly used to study the behavior of solar cells and photovoltaic systems. The equation that describes the solar performance (that is, the relationship between electric current, I, and voltage V) of this model is the aforementioned Expression (3) [20], reproduced here for convenience:.   V  IRs   V  IRs I  I pv  I 0 exp  ,   1  Rsh  aVT   . (5). where Ipv is the photovoltaic current, I0 is the saturation current of the diode, Rs is the series resistance, Rsh is the shunt resistance, and a the ideality factor of the diode. Finally, the thermal voltage of the cell, VT, is a known quantity defined with Expression (2). Obviously, prior to the use of this model to simulate the cell behavior it is necessary to identify its five parameters: Ipv, I0, Rs, Rsh and a. To calculate them, five boundary conditions extracted from the solar cell I-V curve are needed. These boundary conditions can be obtained either from the manufacturer’s data, or by testing the solar cell. As an example of the data normally available, the values of the most representative points (short circuit: V = 0, I = Isc ; open circuit: V = Voc , I = 0; and maximum power: V = Vmp , I = Imp points) of the measured I-V curve from different solar cells are included in Table 1, together with the temperature during the test. These data have to be translated into boundary conditions in order to get the five parameters of the model. The values of the 1-Diode/2-Resistors circuit model parameters related to the cells from Table 1 are included in Table 2. These parameters were obtained by the authors both numerically or analytically depending on each case. Table 1. I-V curve data (short circuit –V = 0, I = Isc–, open circuit –V = Voc, I = 0–, and the maximum power –V = Vmp, I = Imp– points; the slopes of the I-V curve at the open circuit and short circuit points, Rs0 and Rsh0) of several solar cells. (CdS –Cadmium sulfide solar cell–, BSC and GSC –silicon solar cells–, Q6LMTM – silicon cell from Q-Solar–). Reference. Cell. Voc [A]. Isc [A]. Vmp [A]. Imp [A]. Rsh0 [Ω]. Rs0 [Ω]. Kennerud,1969 [20]. CdS. 0.420. 0.804. 0.316. 0.698. 20. 0.08. Charles, 1981 [21]. BSC. 0.536. 0.1023. 0.437. 0.0925. 1000±30. 0.45±0.01. 300. Charles, 1981 [21]. GSC. 0.524. 0.561. 0.390. 0.481. 25.9±0.8. 0.162±0.005. 307. Lo Brano 2010 [42]. Q6LM. 0.608. 7.665. 0.513. 7.174. 9.967. 0.00443. 298. T [K].

(42) 10. Table 2. 1-Diode/2-Resistors circuit model parameter values from different solar cells. Reference. Cell. Rs [Ω]. Rsh [Ω]. Ipv [A]. Kennerud,1969 [20]. CdS. 0.03. 20.3. 0.805. Charles, 1981 [21]. BSC. 0.070±0.009. 1000±50. 0.1023±0.0005. Charles, 1981 [21] Lo Brano 2010 [42]. GSC Q6LM. 0.08±0.01 7.7315·10. 5. 26±1 9.9672. 0.5625±0.0005 7.65549. I0 [A] 1.84·10. a. 5. 1.37. (1.10±0.05)·107 (6±3)·10. 6. 7.87236·10. 8. 1.51±0.07 1.72±0.08 1.28670. 2-2-1-1 Analytical parameter calculation based on manufacturer’s data Most solar cells manufacturers include in the specifications datasheet, at least, information with regard to the most representative points of the I-V curve, that is, short circuit, open circuit, and the maximum power point. With these three points (indicated in Figure 1 for three different solar cells), it is possible to derive four boundary conditions for Equation (5). Then, if no other information with regard to the I-V curve is available either from the manufacturer or from a testing campaign, it is possible to obtain a solution that represents an approximation to the solar cell performance, as a function of the ideality factor, a. For single junction solar cells, it is suggested to give an initial value of this factor in the bracket [1, 1.5], in order to reduce the number of parameters to four [8], [9], [37]. As the curvature of the I-V curve is affected by this parameter, its value could be adjusted once the other parameter values have been calculated [39]. The short-circuit conditions once introduced in Equation (5), lead to the following expression [20]:.  I R   I R I sc  I pv  I 0 exp  sc s   1  sc s .  aVT   Rsh . (6). In Table 3 the values of each term of this expression are included. These values, obtained from different references (see Table 1 and Table 2), can be used to evaluate the relative importance of each term. As a result, the second term of the right side of (6) can be neglected [20], [44], and the expression above can be rewritten as:.