Patent Publication Number: US-11396388-B2

Title: Optimized power balanced variable thrust transfer orbits to minimize an electric orbit raising duration

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is related to U.S. patent application Ser. No. 16/228,000, entitled “Autonomous Control of Electric Power Consumption by an Apparatus,” filed the same date and assigned to the same assignee as the present application and is incorporated herein by reference. 
     This application is related to U.S. patent application Ser. No. 16/227,719, now U.S. Pat. No. 11,286,060, entitled “Autonomous Control of Electric Power Supplied to a Thruster During Electric Orbit Raising,” filed the same date and assigned to the same assignee as the present application and is incorporated herein by reference. 
     FIELD 
     The present disclosure relates to spacecraft including satellites and the like, and more particularly to providing optimized power balanced variable thrust transfer orbits to minimize an electric orbit raising duration. 
     BACKGROUND 
     Electric orbit raising involves moving a spacecraft, such as a satellite, from an initial injection orbit after separation from a launch vehicle to a target orbit where the spacecraft will operate during its useful life. Thrusters are used during electric orbit raising to move the spacecraft from the initial injection orbit to the target orbit. The thrusters use a propellant. The amount of thrust generated by each thruster during firing is variable and is controlled by an amount of electric power supplied to the thruster. The amount of propellant used during firing of the thruster is determined by the amount of electric power supplied. The amount of electric power and propellant used will also be dependent upon the duration of firing of the thruster each time the thruster is fired. The thrusters are also used for station-keeping once the spacecraft is in the target orbit. Accordingly, minimizing the duration of electric orbit raising and controlling electric power to the thrusters during electric orbit raising is important to preserve as much propellant as possible for station-keeping. 
     SUMMARY 
     In accordance with an embodiment, an apparatus is configured for providing optimized power balanced variable thrust transfer orbits to minimize an electric orbit raising duration of the apparatus. The electric orbit raising includes a first transfer orbit and a target orbit. The apparatus includes control electronics. The control electronics are configured to transfer the apparatus to a second transfer orbit to reach the target orbit and determine a variable thrust based on a current electric power balance. The control electronics are further configured to execute a thruster maneuver to transfer the apparatus from the first transfer orbit to the second transfer orbit according to the determined variable thrust and a predetermined maneuver plan. The predetermined maneuver plan includes a set of compound steering parameters. The set of compound steering parameters are based on an optimized variable thrust and an associated electrical power balance to the optimized variable thrust. An optimized series of transfer orbits minimize the electric orbit raising duration of the apparatus to the target orbit. The second transfer orbit is one of the optimized series of transfer orbits. 
     In accordance with another embodiment, a method for providing optimized power balanced variable thrust transfer orbits to minimize an electric orbit raising duration of an apparatus includes a process. The process includes determining an estimate of a current transfer orbit of the apparatus and comparing the estimated current transfer orbit to a corresponding optimized transfer orbit of a predetermined optimized transfer orbit profile. The predetermined optimized transfer orbit profile includes an optimized series of transfer orbits. The process also includes maintaining a current maneuver plan to minimize the electric orbit raising duration in response to the estimated current transfer orbit being approximately equal to the corresponding optimized transfer orbit. The process additionally includes determining a variable thrust based on an electric power balance from the current maneuver plan in response to the estimated current transfer orbit being approximately equal to the corresponding optimized transfer orbit. The process further includes executing a thruster maneuver according to a set of compound steering parameters and the variable thrust of the current maneuver plan. 
     In accordance with another embodiment, a method for providing optimized power balanced variable thrust transfer orbits to minimize an electric orbit raising duration of an apparatus includes optimizing a separation orbit of the apparatus from the launch vehicle to minimize the electric orbit raising duration. The separation orbit is optimized by using a multidimensional optimizer. Using the multidimensional optimizer includes using capabilities of the launch vehicle and varying variable parameters of the separation orbit depending upon the capabilities of the launch vehicle used to optimize the separation orbit. 
     In accordance with an embodiment and any of the previous embodiments, the apparatus and method also include optimizing a transfer orbit profile using a geometry of the apparatus, sun, earth, the separation orbit, and a target orbit to maximize available electric power to one or more thrusters of the apparatus to minimize the electric orbit raising duration, wherein the transfer orbit profile comprises an optimized series of transfer orbits. 
     In accordance with an embodiment and any of the previous embodiments, the apparatus and method also include optimizing each transfer orbit of a series of transfer orbits during electric orbit raising. Optimizing each transfer orbit includes determining an electric power balance of each transfer orbit to provide maximum electric power to the one or more thrusters of the apparatus each transfer orbit to minimize the electric orbit raising duration. The apparatus and method additionally include determining a variable thrust from the electric power balance each transfer orbit. The apparatus and method further include executing a thruster maneuver each transfer orbit according to a set of compound steering parameters and the variable thrust to minimize the electric orbit raising duration. 
     In accordance with an embodiment and any of the previous embodiments, the set of compound steering parameters are optimized to minimize the electric orbit raising duration of the apparatus to the target orbit. 
     In accordance with an embodiment and any of the previous embodiments, the associated electrical power balance to the optimized variable thrust is optimized to minimize the electric orbit raising duration of the apparatus to the target orbit. 
     In accordance with an embodiment and any of the previous embodiments, the control electronics are further configured to receive the predetermined maneuver plan. 
     In accordance with an embodiment and any of the previous embodiments, the control electronics are further configured to receive the predetermined maneuver plan from another apparatus. 
     In accordance with an embodiment and any of the previous embodiments, the other apparatus is at least one of a control station, a ground control station, another apparatus, and a repeater station. 
     In accordance with an embodiment and any of the previous embodiments, the control electronics are further configured to determine the predetermined maneuver plan. 
     In accordance with an embodiment and any of the previous embodiments, the control electronics are additionally configured to re-optimize the optimized series of transfer orbits in response to an estimated current transfer orbit not being approximately equal to a corresponding optimized transfer orbit of the optimized series of transfer orbits to minimize the electric orbit duration. The control electronics are also configured to adjust the predetermined maneuver plan according to the re-optimized series of transfer orbits. The control electronics are further configured to execute a thruster maneuver according to a set of adjusted compound steering parameters and an adjusted variable thrust of the adjusted predetermined maneuver plan. 
     In accordance with an embodiment and any of the previous embodiments, the method further includes re-optimizing the optimized series of transfer orbits in response to the estimated current transfer orbit not being approximately equal to the corresponding optimized transfer orbit. 
     In accordance with an embodiment and any of the previous embodiments, re-optimizing the optimized series of transfer orbits includes determining a new optimized series of transfer orbits and predicting a variable thrust based on an electric power balance of each of the new optimized series of transfer orbits. 
     In accordance with an embodiment and any of the previous embodiments, wherein the process further includes adjusting the current maneuver plan according to the new optimized series of transfer orbits to provide a new current maneuver plan to minimize the electric orbit raising duration. 
     In accordance with an embodiment and any of the previous embodiments, wherein the process further includes determining a variable thrust based on the electric power balance from the new current maneuver plan and executing a thruster maneuver according to a set of compound steering parameters and the variable thrust of the new current maneuver plan. 
     In accordance with an embodiment and any of the previous embodiments, the apparatus and method further include determining if the target orbit has been reached and continuing the process until the target orbit is reached. 
     In accordance with an embodiment and any of the previous embodiments, wherein the current maneuver plan includes the set of compound steering parameters and when to fire one or more thrusters and when to shut off the one or more thrusters of the apparatus. 
     In accordance with an embodiment and any of the previous embodiments, wherein the apparatus includes one or more solar arrays and a battery, and wherein the method further includes determining the electric power balance each optimized transfer orbit. 
     In accordance with an embodiment and any of the previous embodiments, wherein determining the electric power balance each optimized transfer orbit includes determining available electric power from the one or more solar arrays during a sunlight duration of each transfer orbit and determining an amount of electric energy drained from the battery during an eclipse of each transfer orbit. The apparatus and method additionally include determining an amount of electric power to recharge the battery during the sunlight duration each transfer orbit. The apparatus and method further include determining remaining electric power available to power the one or more thrusters during the sunlight duration of each transfer orbit using at least the available electric power from the one or more solar arrays, the amount of electric energy drained from the battery during the eclipse of each transfer orbit, and the amount of electric power to recharge the battery during the sunlight duration of each transfer orbit. 
     The features, functions, and advantages that have been discussed can be achieved independently in various embodiments or may be combined in yet other embodiments further details of which can be seen with reference to the following description and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  is a block schematic diagram of an example of an apparatus in accordance with an embodiment of the present disclosure. 
         FIGS. 2A and 2B  are a flow chart of an example of a method for providing optimized power balanced variable thrust transfer orbits to minimize an electric orbit raising duration of an apparatus in accordance with an embodiment of the present disclosure. 
         FIG. 2C  is a flow chart of an example of a method for providing optimized power balanced variable thrust transfer orbits to minimize an electric orbit raising duration of an apparatus where the apparatus is an autonomous apparatus in accordance with another embodiment of the present disclosure. 
         FIG. 2D  is a flow chart of an example of a method for providing optimized power balanced variable thrust transfer orbits to minimize an electric orbit raising duration of an apparatus wherein the apparatus is a non-autonomous or partially autonomous apparatus in accordance with a further embodiment of the present disclosure. 
         FIG. 3  is a flow chart of an example of a method for operation of a multidimensional optimizer to minimize an electric orbit raising duration of an apparatus in accordance with an embodiment of the present disclosure. 
         FIG. 4  is an illustration of an example of electric orbit raising of an apparatus including a transfer orbit profile that minimizes the electric orbit raising duration in accordance with an embodiment of the present disclosure. 
         FIGS. 5A and 5B  are an illustration of an example of optimizing a separation orbit of the apparatus from a launch vehicle that minimizes the electric orbit raising in accordance with an embodiment of the present disclosure. 
         FIG. 6  is an illustration of an example of apsidal rotation of an orbit plane of a transfer orbit of the apparatus during electric orbit raising in accordance with an embodiment of the present disclosure. 
     
    
    
     DETAILED DESCRIPTION 
     The following detailed description of embodiments refers to the accompanying drawings, which illustrate specific embodiments of the disclosure. Other embodiments having different structures and operations do not depart from the scope of the present disclosure. Like reference numerals may refer to the same element or component in the different drawings. 
       FIG. 1  is a block schematic diagram of an example of an apparatus  100  in accordance with an embodiment of the present disclosure. In an example, the apparatus  100  is a spacecraft, such as a satellite or other space vehicle. The apparatus  100  is configured to provide optimized power balanced variable thrust transfer orbits  402  ( FIG. 4 ) to minimize an electric orbit raising  400  duration of the apparatus  100 . Referring also to  FIG. 4 ,  FIG. 4  is an illustration of an example of electric orbit raising  400  of the apparatus  100  including a transfer orbit profile  404  that minimizes the electric orbit raising  400  duration in accordance with an embodiment of the present disclosure. The transfer orbit profile  404  includes a series of transfer orbits  402 . Electric orbit raising  400  is a procedure of transitioning the apparatus  100  from a separation orbit  406  after separation from a launch vehicle  502  ( FIG. 5 ) to a target orbit  408  where the apparatus  100  will operate. Electric orbit raising  400  includes the series of transfer orbits  402  during which variable thrusters  104  are fired to move the apparatus  100  to the target orbit  408 . A position  410  of the sun  504  ( FIG. 5A ) relative to the earth  412  during electric orbit raising  400  is also illustrated in the example of  FIG. 4 . 
     The apparatus  100  includes one or more variable thrusters  104 . In the example in  FIG. 1 , the apparatus  100  includes two variable thrusters  104 . In accordance with an example, the variable thrusters  104  are electric propulsion thrusters, such as Hall Effect thrusters (HET) or similar variable thrusters that are configured to generate variable thrust  108 . The one or more variable thrusters  104  are used for electric orbit raising  400  and station-keeping after the apparatus  100  reaches the target orbit  408 . 
     The variable thrusters  104  use a propellant  106  to generate the variable thrust  108 . The propellant  106  is stored in a tank  110 . In accordance with an example, the propellant  106  is Xenon. In other examples, the propellant  106  is one or more species of propellant or combination thereof. The propellant  106  is fed to the variable thrusters  104  by a feed system  112 . A propellant flow controller  114 , such as a Xenon flow controller (XFC 1  and XFC 2 ) for example, is associated with each variable thruster  104  and couples the feed system  112  to each variable thruster  104  to individually control the supply of propellant  106  to each variable thruster  104 . The amount of variable thrust  108  generated by each variable thruster  104  during firing is controlled by an amount of electric power  116  supplied to each variable thruster  104 . The amount of propellant  106  used during the firing of each variable thruster  104  is determined by the amount of electric power  116  supplied. The amount of electric power  116  and propellant  106  used will also be dependent upon the duration of firing of the variable thruster  104  each time the variable thruster  104  is fired. A power processing unit (e.g., PPU 1  and PPU 2 )  118  is also associated with each variable thruster  104  to control the amount of electric power  116  supplied to each variable thruster  104  during firing. 
     Each variable thruster  104  is mechanically coupled to an exterior  120  of the apparatus  100  by a gimbaled platform mechanism  122 , such as a Xenon gimbaled platform mechanism (GXP 1  and GXP 2 ) for example. The gimbaled platform mechanism  122  provides variable thruster  104  pointing to direct the thrust direction  124 . 
     The apparatus  100  also includes one or more solar arrays  126  (e.g.  126   a  and  126   b ) attached to the apparatus  100  to provide electric power  130  to the apparatus  100 . The solar arrays  126  are configured to convert light energy to electric energy and provide electric power  130  to the components  127  of the apparatus  100 . In the example in  FIG. 1 , the apparatus  100  includes a north solar array  126   a  and a south solar array  126   b . The north solar array  126   a  and the south solar array  126   b  are mounted on opposite sides of the apparatus  100 . The north solar array  126   a  and the south solar array  126   b  are electrically coupled to an integrated power controller (IPC)  128 . The IPC  128  receives electric power  130  from the solar arrays  126  and controls distribution of the electric power  130  to components  127  of the apparatus  100 . In the example illustrated in  FIG. 1 , the IPC  128  provides the electric power  130  to the power processing units  118  to operate the variable thrusters  104  via a first bus  132  which is a 100 volt bus in the example of  FIG. 1 . The IPC  128  also distributes electric power  130  to other components  127  of the apparatus  100  via a second bus  134  which is a 30 volt bus in the example of  FIG. 1 . Other embodiments may include buses of different voltages. The second bus  134  is connected to secondary bus units  136 . 
     The apparatus  100  also includes a battery  138  to power the apparatus  100 . In one embodiment, the battery  138  is a battery pack. The battery  138  is charged by electric power  130  from the solar arrays  126  through the IPC  128 . The IPC  128  controls charging of the battery  138  by the solar arrays  126 . 
     The apparatus  100  further includes control electronics  140 . The control electronics are powered by either the battery  138  or the IPC  128 . In the example of  FIG. 1 , the control electronics  140  are spacecraft control electronics (SCE) if the apparatus  100  is a spacecraft or the like. The control electronics  140  include a processor  142 . In an embodiment where the apparatus  100  is an autonomous apparatus as described in more detail with reference to  FIG. 2C , the control electronics  140  are configured to perform functions, such as providing optimized power balanced variable thrust transfer orbits  402  ( FIG. 4 ) to minimize an electric orbit raising  400  duration of the apparatus  100 . The processor  142  includes an optimizer  144  to provide an optimized series of transfer orbits  402  ( FIG. 4 ) as described in more detail with reference to  FIGS. 2A and 2C . Examples of operation of the optimizer  144  will be described in more detail with reference to block  210  in  FIG. 2A  and blocks  306  and  308  in  FIG. 3 . In an embodiment were the apparatus  100  is an autonomous apparatus, the control electronics  140  are configured to perform the functions of block  210  in  FIG. 2A  and blocks  306  and  308  in  FIG. 3 . 
     In accordance with an embodiment, a ground network  152  includes another processor  143  configured to perform at least some of the functions described with respect to method  200  in  FIGS. 2A-2D  and method  300  in  FIG. 3 . A multidimensional optimizer  145  is embodied in and runs on the processor  143 . The multidimensional optimizer  145  is configured to perform at least a first level  302  of optimization and a second level of optimization  304  of the method  300  in  FIG. 3  using simulated launch data prior to launching the apparatus  100  by a launch vehicle  502  ( FIG. 5A ). As described in more detail with reference to blocks  204 - 208  in  FIG. 2A  and blocks  302  and  304  in  FIG. 3 , the first level (block  302 ) of optimization determines an optimized separation node  530  which is the location where the apparatus  100  separates from launch vehicle  502 . The second level (block  304 ) of optimization determines an optimized separation orbit  406  which is the first transfer orbit  402  of the apparatus  100  after separation from the launch vehicle  502  and start of electric orbit raising  400 . In accordance with another embodiment, the ground network  152  includes a control station and the control station includes a processor similar to processor  143 . 
     The processor  143  is also configured to use capabilities  146  of the launch vehicle (LV)  502  ( FIG. 5A ) and to vary variable parameters  148  of the separation orbit  406  ( FIGS. 4 and 5A ), depending upon the capabilities  146  of a particular launch vehicle  502  used, to optimize the separation orbit  406  and minimize the electric orbit raising  400  duration. Launch vehicle capabilities  146  include performance capabilities, such as thrust performance, payload capacity, etc., of a particular launch vehicle  502 . The capabilities  146  may vary based on a particular mission of the launch vehicle  502 . Examples of variable parameters  148  of the separation orbit  406  will be described with reference to blocks  314  and  316  in  FIG. 3 . 
     In accordance with an embodiment where the apparatus  100  is a non-autonomous apparatus or partially autonomous apparatus, the multidimensional optimizer  145  of the ground network  152  is configured to determine a series of optimized transfer orbits  402  ( FIG. 4 ) as described in more detail with reference to blocks  256  and  258  in  FIG. 2D  and blocks  306  and  308  in  FIG. 3 . 
     In accordance with an embodiment, the control electronics  140  are configured to determine an electric power balance  150  on an orbit-by-orbit basis for each transfer orbit  402  to provide maximum electric power  116  to the variable thrusters  104  during a sunlight duration  510  ( FIG. 5A ) of each transfer orbit  402  to minimize the electric orbit raising  400  duration. The sunlight duration  510  of each transfer orbit  402  occurs when the apparatus  100  is not in an eclipse  512 . An eclipse  512  occurs when the earth  412  is between the apparatus  100  and the sun  504  each transfer orbit  402 . However, under some circumstances, for example, if only a single variable thruster  104  is available, firing the available variable thruster  104  during an eclipse  512  may become desirable. The electric power balance  150  is a balance between an amount of electric power  116  supplied to each of the variable thrusters  104  and an amount of electric power  130  supplied by the solar arrays  126  though the IPC  128  to recharge the battery  138  and to provide electric power  130  to other components  127  of the apparatus  100 . As previously described, each variable thrusters  104  generates an amount of variable thrust  108  proportional to the amount of electric power  116  supplied to each variable thruster  104  when the variable thruster  104  is fired. 
     In accordance with an embodiment, the control electronics  140  are configured to transfer the apparatus  100  to a second transfer orbit  402  to reach the target orbit  408  and determine a variable thrust  108  based on a current electric power balance  150 . The control electronics  140  are also configured to execute a thruster  104  maneuver to transfer the apparatus  100  from the first transfer orbit  402  to the second transfer orbit  402  according to the determined variable thrust  108  and a predetermined maneuver plan  156 . The predetermined maneuver plan  156  includes a set of compound steering parameters  158 . The set of compound steering parameters  158  are based on an optimized variable thrust  108  and an associated electrical power balance  150  to the optimized variable thrust  108 . The predetermined maneuver plan  156  also includes an optimized series of transfer orbits  402  to minimize the electric orbit raising  400  duration of the apparatus  100  to the target orbit  408 . The second transfer orbit  402  is one of the optimized series of transfer orbits  402 . The set of compound steering parameters  158  are optimized to minimize the electric orbit raising  400  duration of the apparatus  100  to the target orbit  408 . Examples of compound steering for efficient low thrust transfer orbits are described in U.S. Pat. No. 8,457,810, entitled “Compound Steering Law for Efficient Low Thrust Transfer Orbit Trajectory, issued Jun. 4, 2013, assigned to the same assignee as the present application and incorporated herein by reference. Examples of compound steering are also described in U.S. Pat. No. 8,930,048, entitled “Enhanced Compound Steering Law for General Low Thrust Mission,” issued Jun. 6, 2015, assigned to the same assignee as the present application and incorporated herein by reference. 
     In accordance with an embodiment, the control electronics  140  are further configured to receive the predetermined maneuver plan  156  from another apparatus. Examples of the other apparatus include but are not necessarily limited to at least one of a control station, a ground control station, a spacecraft, and a repeater station. In accordance with another embodiment, the control electronics  140  are configured to determine the predetermined maneuver plan  156 . In one example, the ground network  152  determines the maneuver plan  156 , transmits to one or more intermediary apparatuses, which in turn relays the maneuver plan  156  to the apparatus  100  in question. Examples of intermediary apparatuses include but are not limited to, spacecraft, a control station, satellite or any other apparatus that can communicate with apparatus  100 . In another example, another spacecraft or apparatus that is communicatively connected (directly or indirectly) to apparatus  100  and that is equipped with the multidimensional optimizer  145  receives the location of apparatus  100  and determines the maneuver plan  156  and transmits the maneuver plan  156  to apparatus  100 . 
     In accordance with an embodiment and as described in more detail with reference to  FIG. 2C , the control electronics  140  are configured to re-optimize the optimized series of transfer orbits  402  in response to an estimated current transfer orbit  402  not being approximately equal to a corresponding optimized transfer orbit of the optimized series of transfer orbits  402  to minimize the electric orbit  400  duration. The control electronics  140  are also configured to adjust the predetermined maneuver plan  156  according to the re-optimized series of transfer orbits  402 . The control electronics are also configured to execute a thruster  104  maneuver according to a set of adjusted compound steering parameters  158  and an adjusted variable thrust  108  of the adjusted predetermined maneuver plan  156 . 
       FIGS. 2A and 2B  are a flow chart of an example of a method  200  for providing optimized electric power balanced variable thrust transfer orbits  402  ( FIG. 4 ) to minimize an electric orbit raising  400  duration of an apparatus  100  in accordance with an embodiment of the present disclosure. In accordance with an example, the blocks  204 - 208  of the method  200  are embodied in and performed by the ground network  152 . 
     In block  204 , a separation orbit  406  ( FIG. 4  and  FIG. 5A ) from the launch vehicle  502  ( FIG. 5A ) is optimized to minimize the electric orbit raising  400  duration using the multidimensional optimizer  145 . Using the multidimensional optimizer  145  includes using capabilities  146  of the launch vehicle  502  and varying variable parameters  148  of the separation orbit  406  depending upon the capabilities  146  of a particular launch vehicle  502  used. An example of a method  300  of operation of the multidimensional optimizer  145  will be described with reference to  FIG. 3 . 
     Referring also to  FIGS. 5A and 5B ,  FIGS. 5A and 5B  are an illustration of an example of optimizing a separation orbit  406  of the apparatus  100  from a launch vehicle  502  that minimizes the electric orbit raising  400  duration in accordance with an embodiment of the present disclosure. The electric orbit raising  400  duration may also be referred to as transfer orbit  402  duration. In block  206 , optimizing the separation orbit  406  from the launch vehicle  502  includes balancing launch vehicle  502  performance between a lower injection inclination angle  514  ( FIG. 5B ) of a separation orbit plane  516  defined by the separation orbit  406  of the apparatus  100  at separation from the launch vehicle  502  and a higher injection apogee  518  of the separation orbit  406 . In the example in  FIG. 5B , an orthogonal coordinate system is defined with orthogonal axes I, J and K. The injection inclination angle  514  of the separation orbit  406  is defined by the angle between a normal h to the separation orbit plane  516  and the axis K. The separation orbit plane  516  at separation of the apparatus  100  from the launch vehicle  502  is defined by right ascension of an ascending node  520  which is an angle omega, Ω, between axis I and a line of nodes (n)  521  in a plane defined by axes I and J. In block  206 , optimization of the separation orbit  406  of the apparatus  100  from the launch vehicle  502  is achieved by controlling the launch vehicle  502  performance to orient the separation orbit plane  516  of the apparatus  100  to minimize eclipse  512  durations during electric orbit raising  400 . The separation orbit  406  is also optimized to place a perigee  522  of the transfer orbits  402  during electric orbit raising  400  in a location  524  approximately opposite the sun  504  for a longest accumulated time during electric orbit raising  400 . This also minimizes the eclipse  512  durations during electric orbit raising  400  to minimize the electric orbit raising  400  duration. 
     In block  208 , the multidimensional optimizer  145  is used to minimize the electric orbit raising  400  ( FIG. 4 ) by varying the variable parameters  148  of the separation orbit  406 . Referring also to  FIG. 5B , varying the variable parameters  148  of the separation orbit  406  includes varying an injection perigee  517 , an injection apogee  518 , an injection inclination angle  514  and an injection node  526  by controlling performance of the launch vehicle  502 . In one example, the injection node  526  is controlled by the time of day of the launch of the launch vehicle  502  rather than a flight profile  528  ( FIG. 5A ) of the launch vehicle  502 . In another example, the injection node  526 , is controlled by powered flight of the launch vehicle  502 . The injection node  526  is the node of the orbit achieved by the last thruster burn of the launch vehicle  502 . In accordance with an example, the launch vehicle  502  may carry a plurality of apparatuses, such as apparatus  100  in  FIG. 1 . Each apparatus  100  has a separation node  530  ( FIG. 5A ) which is the location where a particular apparatus separates from the plurality of apparatuses. 
     In block  209 , the apparatus  100  is launched using a launch vehicle  502  ( FIG. 5A ). As previously described in the example of  FIG. 1 , the apparatus  100  includes one or more variable thrusters  104 , and a battery  138  to provide electric power  154  to components  127  of the apparatus  100 . The apparatus  100  also includes one or more solar arrays  126  (e.g.  126   a  and  126   b ) to recharge the battery  138 , provide electric power  130  to other components  127  of the apparatus  100 , and to provide electric power  116  to the variable thrusters  104 . 
     In block  210 , a transfer orbit profile  404  ( FIG. 4 ) is optimized using a geometry  506  ( FIG. 5A ) of the apparatus  100 , sun  504 , earth  412 , the separation orbit  406 , and a target orbit  408  to maximize available electric power  116  ( FIG. 1 ) to the variable thrusters  104  to minimize the electric orbit raising  400  duration. Examples of optimizing the transfer orbit profile  404  using the geometry  506  are described in more detail with reference to blocks  212  and  214 . As previously described, in accordance with an embodiment where the apparatus  100  is autonomous, the apparatus is configured to optimize the transfer orbits  402  as described in more detail with reference to  FIG. 2C . In accordance with an embodiment where the apparatus  100  is non-autonomous or partially autonomous, the ground network  152  is configured to optimize the transfer orbits  402  as described in more detail with reference to  FIG. 2D . 
     In block  212 , the transfer orbit profile  404  is optimized to provide a longer accumulated sunlight duration  510  and less accumulated duration in eclipses  512  based on the geometry  506  during electric orbit raising  400 . This provides a higher total electric power  130  generated by the solar arrays  126  to increase the electric power  116  supplied to the variable thrusters  104  to increase variable thrust  108  each transfer orbit  402  and thus decrease the electric orbit raising  400  duration. 
     Referring also to  FIG. 6 ,  FIG. 6  is an illustration of an example of apsidal rotation  600  of an orbit plane  602  of a transfer orbit  402  of the apparatus  100  during electric orbit raising  400  in accordance with an embodiment of the present disclosure. In block  214 , an optimized transfer orbit profile  404  ( FIG. 4 ) is implemented that leverages the apsidal rotation  600  of an orbit plane  602  during the transfer orbit  402  of the apparatus  100  to maximize a coincidence of motion  604  of the orbit plane  602  of the apparatus  100  with an apparent motion  608  of the sun  504  relative to the earth  412 . This also maximizes the amount of time the apparatus  100  is in sunlight during the electric orbit raising  400  to increase the electric power  116  supplied to the variable thrusters  104  and thus decrease the electric orbit raising  400  duration. 
     In block  216 , an electric power balance  150  ( FIG. 1 ) is determined on an orbit-by-orbit basis for each transfer orbit  402  to provide maximum electric power  116  to the variable thrusters  104  during a sunlight duration  510  ( FIG. 5A ) of each transfer orbit  402  to minimize the electric orbit raising  400  duration. The electric power balance  150  is a balance between electric power  116  supplied to each variable thrusters  104  and electric power  130  supplied by the solar arrays  126  via the IPC  128  to recharge the battery  138  and to provide electric power  130  to other components  127  of the apparatus  100 . Each variable thruster  104  provides a variable amount of thrust  108  proportional to an amount of electric power  116  supplied to the variable thruster  104  by the associated power processing unit  118 . 
     An example of determining the electric power balance  150  on an orbit-by-orbit basis for each transfer orbit  402  to provide maximum electric power  116  to each variable thrusters  104  is described in more detail with reference to blocks  218 - 224 . In accordance with an embodiment, the maximum electric power  116  is provided to each variable thruster  104  during a sunlight duration  510  of each transfer orbit  402  to minimize the electric orbit raising  400  duration. However, under some circumstances, for example, if only a single variable thruster  104  is available, firing the available variable thruster  104  during an eclipse  512  may become desirable. 
     In block  218 , an available electric power  130  ( FIG. 1 ) from the solar arrays  126  is determined for each transfer orbit  402  using equation 1:
 
 P   solar_array   =n   circuit ( P   circuit −loss solar_array_time   t   delta )  Equation 1
 
     Where P solar_array  is the available electric power  130  from at least one of the solar arrays  126   a  or  126   b . The baseline is to use both solar arrays  126   a  and  126   b . n circuit  is the number of photocell circuits in the solar arrays  126   a  and  126   b , P circuit  is the power per solar array circuit and loss solar_array_time  t delta  is the power loss in the solar arrays  126   a  and  126   b  over a time period t delta . 
     In block  220 , an amount of electric energy drained from the battery  138  during an eclipse  512  of each transfer orbit  402  is determined from equation 2: 
     
       
         
           
             
               
                 
                   
                     drain 
                     ecl 
                   
                   = 
                   
                     
                       
                         load 
                         ecl 
                       
                       ⁢ 
                       
                         t 
                         ecl 
                       
                     
                     
                       1 
                       - 
                       
                         loss 
                         discharge 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
           
         
       
     
     Where drain ecl  is the amount of electric energy drained from the battery  138  during each eclipse  512 . load ecl  is the load connected to the battery during the eclipse  512  and t ecl  is the duration of the eclipse  512 . 1−loss discharge  takes into account inefficiency of discharging the battery  138  by subtracting loss during discharge (loss discharge ) of the battery  138  from one when the battery  138  is supplying electric power  154  to other components  127  of the apparatus  100 . 
     In block  222 , an amount of electric power  154  to recharge the battery  138  during the sunlight duration  510  of each transfer orbit  402  is determined using equation 3: 
     
       
         
           
             
               
                 
                   
                     load 
                     charge 
                   
                   = 
                   
                     
                       drain 
                       ecl 
                     
                     
                       
                         t 
                         sun 
                       
                       ⁡ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             loss 
                             charge 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   3 
                 
               
             
           
         
       
     
     Where load charge  is the amount of electric charge to recharge the battery  138  during the sunlight duration  510  of each transfer orbit  402 . t sun  is the sunlight duration  510  of each transfer orbit  402  and 1−loss charge  takes into account the charging inefficiency of the battery  138  by subtracting charging loss (loss charge ) from one when the battery  138  is being charged during the sunlight duration  510  of each transfer orbit  402 . 
     In block  224 , the remaining electric power  130  available for the variable thrusters  104  during the sunlight duration  510  of each transfer orbit  402  is determined by equation 4:
 
 P   thrust   =P   solar_array −loss solar_array_instant −load charge −load sun   Equation 4
 
     Where P thrust  is the remaining electric power  130  available for the variable thrusters  104  during the sunlight duration  510  of each transfer orbit  402 . loss solar_array_instant  is electric power loss in the solar arrays  126  due to inefficiencies in the solar arrays  126 . load sun  are other loads drawing electric power  130  from the solar arrays  126  during the sunlight duration  510  of each transfer orbit  402 . 
     In block  226 , the remaining available electric power  130  from the solar arrays  126  during sunlight duration  510  of each transfer orbit  402  is supplied to the variable thrusters  104  to minimize the electric orbit raising  400  duration. 
       FIG. 2C  is a flow chart of an example of a method  227  for providing optimized power balanced variable thrust transfer orbits  402  to minimize an electric orbit raising  400  duration of an apparatus  100  where the apparatus  100  is an autonomous apparatus in accordance with another embodiment of the present disclosure. The elements of the exemplary method  227  are separated into those elements that are embodied in and performed by the ground network  152 ; those elements that are embodied in and performed by the launch vehicle  502 ; and those elements that are embodied in and performed by the apparatus  100 . 
     In block  228 , an optimized separation orbit  406  ( FIG. 4 ) is found by using an optimizer, such as for example, multidimensional optimizer  145  in  FIG. 1 . An example of finding an optimized separation orbit  406  is described with reference to blocks  204 - 208  in  FIG. 2A  and blocks  302  and  304  in  FIG. 3 . Finding the optimized separation orbit  406  is performed prior to the launch using simulated launch data. 
     In block  230 , parameters that optimize the separation orbit  406  are received by the launch vehicle  502 . Examples of the parameters that optimize the separation orbit  406  are described with reference to blocks  314  and  316  in  FIG. 3 . In block  232 , the apparatus  100  is launched by the launch vehicle  502  into the separation orbit  406 . The parameters associated with the actual separation orbit  406  may include small errors from the simulated separation orbit  406  found by the ground network  152 . 
     In block  234 , an estimate of a current transfer orbit  402  of the apparatus  100  is determined. In accordance with an example, the estimate of the current transfer orbit  402  of the apparatus  100  is determined by a global positioning system (GPS)  160  ( FIG. 1 ) on the apparatus  100 . 
     In block  236 , the estimated current transfer orbit  402  is compared to a corresponding optimized transfer orbit  402  of a predetermined optimized transfer orbit profile  404 . The predetermined optimized transfer orbit profile  404  includes an optimized series of transfer orbits  402 . In accordance with an embodiment, the predetermined optimized transfer orbit profile  404  is determined prior to launch of the apparatus  100 . An example of determining the predetermined optimized transfer orbit profile  404  by an optimizer, such as for example, multidimensional optimizer  145 , is described with reference to blocks  306  and  308  in  FIG. 3 . Another example of determining the predetermined optimized transfer orbit profile  404  is described with reference to blocks  210 - 214  in  FIG. 2A . A current maneuver plan  156  is based on the optimized transfer orbit profile  404 . A current maneuver plan  156  includes but is not necessarily limited to a set of compound steering parameters  158  and when to fire one or more thrusters  104  of the apparatus  100  and when to shut off the one or more thrusters  104 . 
     If the estimated transfer orbit  402  is approximately equal to the corresponding optimized transfer orbit  402 , the method  227  advances to block  240 . In block  240 , the current maneuver plan  156  is maintained to minimize the electric orbit raising  400  duration. 
     In block  246 , a variable thrust  108  based on an electric power balance  150  is determined from the current maneuver plan  156  in response to the estimated current transfer orbit being approximately equal to the corresponding optimized transfer orbit  402 . The electric power balance  150  is determined each optimized transfer orbit  402 . An example of determining the electric power balance on an orbit-by-orbit basis is described with reference to blocks  216 - 224  in  FIG. 2B . 
     Returning to block  238 , if the estimated transfer orbit  402  is not approximately equal to the corresponding optimized transfer orbit  402  of the predetermined optimized transfer orbit profile  404 , the method  227  advances to block  242 . In block  242 , the optimized series of transfer orbits  402  of the predetermined optimized transfer orbit profile  404  are re-optimized by the apparatus  100 . Re-optimizing the optimized series of transfer orbits  402  includes determining a new optimized series of transfer orbits  402  and predicting a variable thrust  108  based on an electric power balance  150  of each of the new optimized series of transfer orbits  402 . In accordance with an example, the autonomous apparatus  100  includes an optimizer, such as for example optimizer  144  in  FIG. 1 . The optimizer  144  is configured to re-optimize the optimized series of transfer orbits  402  as described herein. 
     In block  244 , the current maneuver plan  156  is adjusted according to the new optimized series of transfer orbits  402  to provide a new current maneuver plan  156  to minimize the electric orbit raising  400  duration. 
     In block  246 , a variable thrust  108  based on an electric power balance  150  is determined from the new current maneuver plan  156  in response to the estimated current transfer orbit  402  not being approximately equal to the corresponding optimized transfer orbit  402  in block  238 . 
     In block  248 , a thruster  104  maneuver is executed according to a set of compound steering parameters  158  and the variable thrust  108  of the current maneuver plan  156  or new current maneuver plan  156 . 
     In block  250 , a determination is made whether the target orbit  408  has been reached by the apparatus  100 . If the target orbit  408  has not been reached by the apparatus  100 , the method  227  returns to block  234  and the process continues similar to that previously described until the target orbit  406  is reached. If the target orbit  406  has been reached in block  250 , the thruster firings end at  252 . 
       FIG. 2D  is a flow chart of an example of a method  253  for providing optimized power balanced variable thrust transfer orbits  402  to minimize an electric orbit raising  400  duration of an apparatus  100  where the apparatus  100  is a non-autonomous or partially autonomous apparatus in accordance with a further embodiment of the present disclosure. The elements of the exemplary method  253  are separated into those elements that are embodied in and performed by the ground network  152 ; those elements that are embodied in and performed by the launch vehicle  502 ; and those elements that are embodied in and performed by the apparatus  100 . Blocks  228 ,  230  and  232  are the same as those described in the method  227  of  FIG. 2C . 
     In block  228 , an optimized separation orbit  406  ( FIG. 4 ) is found by using an optimizer, such as for example, multidimensional optimizer  145  in  FIG. 1 . An example of finding an optimized separation orbit  406  is described with reference to blocks  204 - 208  in  FIG. 2A  and blocks  302  and  304  in  FIG. 3 . Finding the optimized separation orbit  406  is performed prior to the launch. In one implementation, simulated launch data is used by the multidimensional optimizer  145  to determine the optimized separation orbit  406 . 
     In block  230 , parameters that optimize the separation orbit  406  are received by the launch vehicle  502 . Examples of the parameters that optimize the separation orbit  406  are described with reference to blocks  314  and  316  in  FIG. 3 . In block  232 , the apparatus  100  is launched by the launch vehicle  502  into the separation orbit  406 . The parameters associated with the actual separation orbit  406  may include small errors from the simulated separation orbit  406  found by the ground network  152 . 
     In block  254 , an estimate of a current transfer orbit  402  of the apparatus  100  is determined. The estimate of the current transfer orbit  402  is determined by the apparatus  100  if the apparatus  100  includes a GPS  160 . The estimate of the current transfer orbit  402  is transmitted to the ground network  152 . Transfer orbit  402  data is received by the ground network  152  at preset time intervals. For example, the transfer orbit  402  data is transmitted to the ground network  152  daily. If the apparatus  100  is not equipped with a GPS  160 , the ground network  152  is configured to determine the estimate of the current transfer orbit  402  using ranging data. 
     In block  256 , optimized transfer orbits  402  are determined by the ground network  152 . For example, the ground network  152  includes an optimizer, such as multidimensional optimizer  145  ( FIG. 1 ) for example, configured to determine a series of optimized transfer orbits  402 . An example of determining the optimized transfer orbits  402  by the multidimensional optimizer  135  is described with reference to blocks  306  and  308  in  FIG. 3 . Another example of determining the optimized transfer orbits  402  is described in blocks  210 - 214  of  FIG. 2A . Optimizing the series of transfer orbits  402  includes predicting a variable thrust  108  based on an electric power balance  150  of each of the optimized series of transfer orbits  402  using previous knowledge and future estimates. A predetermined maneuver plan  156  is updated based on differences between the estimated transfer orbit and a corresponding optimized transfer orbit  402  of the optimized series of transfer orbits  402 . 
     In block  258 , any update to a predetermined maneuver plan  156  is sent from the ground network  152  to the apparatus  100 . Updates to the predetermined maneuver plan  156  may be sent at periodic time intervals, for example, once every two weeks or any other periodic time interval depending upon circumstances and how often corrections may need to be made to the series of optimized transfer orbits  402 . Updates to the predetermined maneuver plan  156  may be sent directly to the apparatus  100  or relayed to apparatus  100  through intermediary apparatuses or other control stations of the ground network  152 . The predetermined maneuver plan  156  includes a set of compound steering parameters  158  and thruster maneuver times, when the one or more thrusters  104  are fired and when the one or more thrusters  104  of the apparatus  100  are shut down. The set of compound steering parameters  158  are based on an optimized variable thrust  108  and an associated electrical power balance  150  to the optimized variable thrust  108 . 
     In block  260 , a variable thrust  108  is determined by the apparatus  100  based on the electric power balance  150 . As previously described, an example of determining the electric power balance is described with reference to blocks  216 - 224  in  FIG. 2B . The current maneuver plan  156  will be continued to be performed in response to no new update to the current maneuver plan  156  being received from the ground network  152 . The updated maneuver plan  156  will be performed in response to an update to the current maneuver plan  156  being received by the apparatus from the ground network  152 . 
     In block  262 , a thruster  104  maneuver is executed according to the received set of compound steering parameters  158  of the updated maneuver plan  156  and variable thrust  108  determined onboard the apparatus  100 . 
     In block  264 , a determination is made whether the target orbit  406  has been reached. The method  253  will return to block  254  in response to the target orbit  406  not having been reached. The process will continue similar to that previously described. If the target orbit  406  has been reached in block  264 , the thruster firings end at  266 . 
       FIG. 3  is a flow chart of an example of a method  300  of operation of a multidimensional optimizer  145  ( FIG. 1 ) to minimize an electric orbit raising  400  duration of an apparatus  100  in accordance with an embodiment of the present disclosure. In accordance with the example in  FIG. 3 , the method  300  includes four optimization levels  302 ,  304 ,  306  and  308 . A first level  302  of optimization includes optimizing the separation node  530  ( FIG. 5A ). A second level  304  of optimization includes optimizing the separation orbit  406  ( FIG. 4 ,  FIG. 5B ) subject to separation mass constraints of the apparatus  100  as described below. A third level  306  of optimization includes optimizing each of the transfer orbits  402  ( FIG. 4 ,  FIG. 6 ). A fourth level  308  of optimization includes optimization of the transfer from the separation orbit  406  to the target orbit  408  ( FIG. 4 ). At each of the optimization levels  302 ,  304 ,  306  and  308 , the electric orbit raising  400  is optimized assuming the variable parameters (blocks  314 ,  316  and  326 ) at other higher optimization levels are fixed, obtaining a local minimum of electric orbit raising  400 . 
     In one embodiment, only a subset of the optimization levels  302 ,  304 ,  306  and  308  is performed. For example, the variable parameters in block  326  are preselected and fixed, and thus the electric orbit raising  400  for level  306  is not minimized and block  328  is either bypassed or put a default yes (no optimization at level  306 ). Variable parameters at blocks  314  and  316  are however varied. A local electric orbit raising at levels  304  and  302  are determined despite level  306  optimization not being performed. In other examples, the optimization at  304  or  302  is bypassed and variable parameters at those levels are fixed or preselected. 
     In accordance with an example, any known mathematical optimization algorithm, such as a gradient descent optimization or similar optimization, is used to perform the optimizations in each optimization level  302 ,  304 ,  306  and  308 . While  FIG. 3  shows an example where electric orbit raising  400  is minimized in stages, in another embodiment, a single overall optimization using at least one of a subset of the variable parameters in optimization levels  302 ,  304  and  306  is applied and the electric orbit raising  400  is minimized in a single stage. When all variable parameters in optimization levels  302 ,  304  and  306  are optimized at once, an optimal value of the electric orbit raising is obtained. 
     Going back to the example shown in  FIG. 3 , the first level  302  of optimization includes orienting the separation orbit plane  516  ( FIG. 5B ) via selection of the right ascension of the ascending node  520  to minimize the electric orbit raising  400  duration. The first level  302  of optimization corresponds to block  206  of  FIG. 2A . Input variables  310  are received and the optimization is started in the first level  302 . The input variables  310  correspond to the variable parameters  148  ( FIG. 1 ) associated with each optimization level  302 - 308  that are variable in each optimization level  302 - 308  to achieve an optimized solution (block  312  in  FIG. 3 ) to minimize the electric orbit raising  400  duration. Examples of the variable parameters used for each optimization level  302 - 308  are described below with the associated optimization level  302 - 308 . 
     In block  314 , the separation node  530  ( FIG. 5A ) is optimized by varying variable parameters associated separation of the apparatus  100  from the launch vehicle  502 . For example, the variable parameters include the separation orbit plane  516  ( FIG. 5B ) being oriented via selection of the right ascension of the ascending node  520  in  FIG. 5B  to minimize the electric orbit raising  400  duration. 
     In the second level  304  of optimization, the separation orbit  406  ( FIG. 5B ) is optimized subject to the separation mass (e.g. propellant  106  mass and apparatus  100  dry mass) constraints, and for a given set of variable parameters at block  314 . The apparatus dry mass is the mass of the apparatus  100  without propellant  106 . The second level  304  of optimization corresponds to block  208  in  FIG. 2A . In block  316 , the separation orbit  406  is optimized by varying variable parameters including the injection inclination angle  514  ( FIG. 5B ), injection apogee  518 , and injection perigee  517  in the presence of a constraint, block  318 . The constraint in block  318  is defined by equation 5:
 
Separation Mass=Apparatus Dry Mass+Electric Propellant Mass  Equation 5
 
     Where the total apparatus mass (Separation Mass) at separation from the launch vehicle  502  equals the apparatus dry mass (Apparatus Dry Mass) without the propellant  106  mass plus the propellant  106  mass (Electric Propellant Mass). 
     In block  318 , the constraint forces a propellant  106  mass and an apparatus  100  dry mass to be compatible with the launch vehicle capabilities  146  to deliver the propellant  106  mass and the apparatus  100  dry mass to a given separation orbit  406 . The goal of the second level  304  of optimization is to find the optimized combination of separation orbit variable parameters that minimizes the electric orbit raising  400  duration. 
     If the constraint (block  318 ) is not satisfied in block  320 , the method  300  returns to block  316  and the variable parameters are varied to find a different combination of launch vehicle variable parameters that satisfy the constraint in block  318 . If the constraint is satisfied in block  320 , the method  300  advances to block  322 . In block  322 , if the electric orbit raising  400  duration at  304  level is not minimized, the method  300  returns to block  316  and the method  300  continues similar to that previously described. If the electric orbit raising  400  duration is minimized in block  322 , the method  300  advances to block  324  to check if the electric orbit raising  400  has been minimized at the  302  level given the set of variable parameters already selected in block  314  while optimizing in level  304 . 
     The third level  306  of optimization includes optimizing each of the transfer orbits  402  ( FIG. 4 ) to minimize electric orbit raising  400  duration from a given separation orbit  406  to the target orbit  408  given an already selected set of variable parameters in blocks  314  and  316 . The third level  306  of optimization corresponds to block  212  in  FIG. 2A . In accordance with an example, optimizing each transfer orbit  402  includes making modifications to each transfer orbit  402 . The modifications to each transfer orbit  402  include modifications to an orientation of the transfer orbit plane  532  ( FIG. 5A ) relative to the earth  412  and sun  504  throughout the electric orbit raising  400  to maximize sunlight durations  510 , minimize eclipse  512  durations, and minimize the electric orbit raising  400  duration. 
     In block  326 , each transfer orbit  402  is optimized by varying variable parameters  148  including a mass of the propellant  106  and global compound steering weight factors. Examples of compound steering laws for efficient low thrust transfer orbits and global compound steering weight factors are described in U.S. Pat. No. 8,457,810, entitled “Compound Steering Law for Efficient Low Thrust Transfer Orbit Trajectory, issued Jun. 4, 2013, assigned to the same assignee as the present application and incorporated herein by reference, and U.S. Pat. No. 8,930,048, entitled “Enhanced Compound Steering Law for General Low Thrust Mission,” issued Jun. 6, 2015, assigned to the same assignee as the present application and incorporated herein by reference. Briefly, global compound steering and global compound steering weight factors include firing the variable thrusters  104  at a particular variable thrust  108  and in a particular compound steering direction  534  ( FIG. 5A ), as determined for each variable thrust transfer orbit  402 , to minimize the electric orbit raising  400  duration at level  306  given the variable parameters in blocks  314  and  316 . 
     In block  328 , a determination is made whether the electric orbit raising  400  duration is minimized by the optimization in the third level  306  of optimization. If not, the method  300  will return to block  326  and the variable parameters  148  in block  326  are varied until an optimized transfer orbit  402  that minimizes the electric orbit raising  400  duration in block  328  is found. If the determination is made in block  328  that the electric orbit raising  400  duration is minimized by the optimized transfer orbit  402  at the third level  306  and given the selected variable parameters at blocks  314  and  316  and the optimized parameters in  326 , the method  300  returns to block  320  and the method  300  proceeds similar to that previously described. 
     The fourth level  308  of optimization includes optimizing the variable thrust  108  ( FIG. 1 ) generated by the variable thrusters  104  during each transfer orbit  402  ( FIG. 4 ) to make progress from the separation orbit  406  toward the target orbit  408 . The fourth level  308  of optimization corresponds to block  216  in  FIG. 2B . Variable thrust  108  and compound steering are used in the fourth level  308  of optimization. In block  330 , the apparatus  100  propagates to a next transfer orbit  402  for a next variable thruster maneuver. 
     In block  332 , a variable thrust  108  is determined from the electric power balance  150 . In accordance with an embodiment, the electric power balance  150  is determined from block  216  in  FIG. 2B . In block  336 , compound steering parameters are optimized. Examples of compound steering parameters are described in U.S. Pat. Nos. 8,457,810 and 8,930,048. Briefly described, the compound steering parameters define the compound steering direction  534  and amount of variable thrust  108  each transfer orbit  402  and are optimized to minimize the electric orbit raising  400  duration. 
     In block  338 , a determination is made whether the target orbit  408  ( FIG. 4 ) has been reached. If not, the method  300  returns to block  330  and the apparatus  100  propagates to a position in the next transfer orbit  402  to perform a variable thruster  104  maneuver and the method  300  will proceed as previously described. If the target orbit  408  has been reached in block  338 , the method  300  will advance to block  328  and the method  300  will proceed as previously described. If all optimization levels  302 ,  304 ,  306  and  308  have been successfully performed, the method  300  advances to block  312  and an optimized solution that minimized the electric orbit raising  400  duration is achieved. 
     The flowchart and block diagrams in the Figures illustrate the architecture, functionality, and operation of possible implementations of systems, methods, and computer program products according to various embodiments of the present disclosure. In this regard, each block in the flowchart or block diagrams may represent a module, segment, or portion of instructions, which comprises one or more executable instructions for implementing the specified logical function(s). In some alternative implementations, the functions noted in the block may occur out of the order noted in the figures. For example, two blocks shown in succession may, in fact, be executed substantially concurrently, or the blocks may sometimes be executed in the reverse order, depending upon the functionality involved. It will also be noted that each block of the block diagrams and/or flowchart illustration, and combinations of blocks in the block diagrams and/or flowchart illustration, can be implemented by special purpose hardware-based systems that perform the specified functions or acts or carry out combinations of special purpose hardware and computer instructions. 
     The terminology used herein is for the purpose of describing particular embodiments only and is not intended to be limiting of embodiments of the disclosure. As used herein, the singular forms “a”, “an” and “the” are intended to include the plural forms as well, unless the context clearly indicates otherwise. It will be further understood that the terms “include,” “includes,” “comprises” and/or “comprising,” when used in this specification, specify the presence of stated features, integers, steps, operations, elements, and/or components, but do not preclude the presence or addition of one or more other features, integers, steps, operations, elements, components, and/or groups thereof. 
     The corresponding structures, materials, acts, and equivalents of all means or step plus function elements in the claims below are intended to include any structure, material, or act for performing the function in combination with other claimed elements as specifically claimed. The description of the present embodiments has been presented for purposes of illustration and description, but is not intended to be exhaustive or limited to embodiments in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art without departing from the scope and spirit of embodiments. 
     Although specific embodiments have been illustrated and described herein, those of ordinary skill in the art appreciate that any arrangement which is calculated to achieve the same purpose may be substituted for the specific embodiments shown and that the embodiments have other applications in other environments. This application is intended to cover any adaptations or variations. The following claims are in no way intended to limit the scope of embodiments of the disclosure to the specific embodiments described herein.