Patent Publication Number: US-11661201-B2

Title: Independent speed variable frequency generator for more electric aircraft using brushless doubly-fed machines (BDFM)

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of U.S. provisional patent application No. 62/991,683, filed on Mar. 19, 2020, and entitled “INDEPENDENT SPEED VARIABLE FREQUENCY GENERATOR FOR MORE ELECTRIC AIRCRAFT USING BRUSHLESS DOUBLY-FED MACHINES (BDFM),” the disclosure of which is expressly incorporated herein by reference in its entirety. 
    
    
     BACKGROUND 
     The existing concept of aviation turboelectric distributed propulsion for passenger aircraft relies on direct current (DC) for power transmission and distribution. The mass and efficiency performance of DC-based propulsion systems becomes challenging as the system power rating increases. 
     Tremendous efforts have been taken towards the more electric aircraft (MEA) over the past few decades. Technological advances of aircraft electrification are making each flight more efficient and eco-friendly. A most recent symbolic innovation of MEA is the replacement of traditional pneumatic power by electrical power on the Boeing 787. 
     The N+X plan is guiding the development of future MEA targeting a 71-dB reduction of noise, an 80% decrease of NOx emissions, and 60% less fuel consumption by 2035. This plan essentially involves a high-power electric drive system in aviation propulsion. Various aviation propulsion architectures have been proposed, including all-electric, hybrid electric, and turboelectric propulsion. An all-electric or a hybrid electric aircraft heavily relies on an onboard energy storage system. While the power density of the existing energy storage system is not high enough to fully decarbonize a single-aisle passenger aircraft, the turboelectric distributed propulsion (TeDP) is believed to be a better solution considering feasibility and reliability. 
     In turboelectric configurations, the mechanical power of the gas turbines is converted to electrical power by generators and then delivered to propulsive motors to drive propellers. The total power rating of a propulsion system for a passenger aircraft can be up to 50 MW. The selection of the propulsion configuration will significantly impact the mass and efficiency of the system. 
       FIG.  1    is an illustration of a DC-based TeDP with permanent magnet machines and full power rated converters. A DC-based TeDP has one or more DC buses between the generators and motors, as shown in  FIG.  1   . A DC-based TeDP requires multiple full power rated converters to transmit the power between alternating current (AC) and DC. It is found that more than 50% of the mass of a DC-based TeDP comes from the converters and protections. As the system voltage level goes up, the overall mass increases, and total system efficiency drops. 
     A TeDP system based on doubly-fed induction machines (DFIMs) is proposed to achieve minimal power conversion and variable speed operation. A DFIM only needs a converter at the rotor winding side to take the slip power of the machine. The stator winding is directly connected to an AC bus. The overall weight and size of power conversion components are reduced. However, a slip ring is required to connect the winding on the rotating rotor for variable speed operation. Therefore, DFIM-based TeDP could be mechanically unstable, considering long-term high-speed operation. 
     It is with respect to these and other considerations that the various aspects and embodiments of the present disclosure are presented. 
     SUMMARY 
     A turboelectric distributed propulsion based on brushless doubly-fed machines (BDFMs) is provided, which minimizes power conversion, enhances mechanical reliability, and strengthens fault-tolerance capability of a DC-based propulsion system. A turboelectric distributed propulsion (TeDP) architecture using BDFMs for aviation applications, and a designed BDFM, inverter, and controller are provided. Simulations and systems are also provided. 
     In an implementation, a turboelectric distributed propulsion (TeDP) system comprises: an engine; a generator comprising a plurality of control windings and a plurality of power windings, wherein the generator is driven by the engine; a plurality of motors, wherein the power windings of the generator are directly connected to power windings of the motors; and a plurality of bidirectional converters, wherein the control windings of the generator and the motors are individually controlled by a respective one of the bidirectional converters. 
     In an implementation, a turboelectric distributed propulsion (TeDP) system comprises: a first brushless doubly-fed machine (BDFM); and a second BDFM, wherein the first BDFM is configured as a generator, and the second BDFM is configured as a motor. 
     In an implementation, a turboelectric distributed propulsion (TeDP) comprises a plurality of brushless doubly-fed machines (BDFMs) configured for use with an aviation application. 
     This summary is provided to introduce a selection of concepts in a simplified form that are further described below in the detailed description. This summary is not intended to identify key features or essential features of the claimed subject matter, nor is it intended to be used to limit the scope of the claimed subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The foregoing summary, as well as the following detailed description of illustrative embodiments, is better understood when read in conjunction with the appended drawings. For the purpose of illustrating the embodiments, there is shown in the drawings example constructions of the embodiments; however, the embodiments are not limited to the specific methods and instrumentalities disclosed. In the drawings: 
         FIG.  1    (prior art) is an illustration of a DC-based TeDP with permanent magnet machines and full power rated converters; 
         FIG.  2    is a cross-section of an implementation of a high-speed BDFM; 
         FIG.  3    is a diagram of an implementation of a BDFM-based TeDP; 
         FIG.  4    is a diagram of an implementation of an example TeDP system comprising one stand-alone generator and one propulsive motor; 
         FIG.  5    is an illustration of inputs, outputs, and interfaces of the modeling of an implementation of a BDFM-based TeDP system; 
         FIG.  6    is a control diagram of an example TeDP (one generator+one motor) in propulsion mode; 
         FIG.  7    is a control diagram of an example TeDP (one generator+one motor) in starting mode; 
         FIG.  8    shows the speed of BDFM 1, 2, 3 in an example TeDP in a simulation containing one generator (BDFM 1) and two motors (BDFM 2 and 3); 
         FIG.  9    shows the torque of BDFM 1, 2, 3; 
         FIG.  10    shows the frequency of the control windings (CWs) of BDFM 1, 2, 3 and the PW; 
         FIG.  11    shows the active power of CW of BDFM 1, 2, 3, and power winding (PW) of BDFM 1; 
         FIG.  12    shows the mechanical power of BDFM 1, 2, 3; 
         FIG.  13    shows the power factor of CW and PW of BDFM 1; 
         FIG.  14    shows the PW voltage of BDFM 1; 
         FIG.  15    shows the PW current of BDFM 1; and 
         FIG.  16    shows the CW current of BDFM 1. 
     
    
    
     DETAILED DESCRIPTION 
     This description provides examples not intended to limit the scope of the appended claims. The figures generally indicate the features of the examples, where it is understood and appreciated that like reference numerals are used to refer to like elements. Reference in the specification to “one embodiment” or “an embodiment” or “an example embodiment” means that a particular feature, structure, or characteristic described is included in at least one embodiment described herein and does not imply that the feature, structure, or characteristic is present in all embodiments described herein. 
     Brushless doubly-fed machines (BDFMs) share the same advantage as DFIMs, i.e., a reduced power rating of converters. In comparison, the mechanical performance of BDFMs is more robust than DFIMs, as there is not a slip ring at the rotor side. This makes BDFMs a candidate for aviation propulsion applications. 
     A TeDP based on BDFMs is described herein. 
     This work is based on a high-speed BDFM  200  shown in  FIG.  2   .  FIG.  2    is a cross-section of a designed high-speed BDFM  200 . Its basic parameters are listed in Table I. The BDFM  200  has an 8-pole winding and a 4-pole winding on the stator. The winding fed by the converter is called the control winding (CW). The other winding directly connects the AC bus and can be termed power winding (PW). The rotor has 6 poles. Each pole consists of 4 pieces of flux guides. 
     
       
         
           
               
             
               
                 TABLE I 
               
               
                   
               
               
                 BDFM Characteristics 
               
               
                   
               
             
            
               
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Rated Speed 
                 12,000 
                 rpm 
               
               
                   
                 Rated Power 
                 120 
                 kW 
               
               
                   
                 DC Bus Voltage 
                 540 
                 V 
               
            
           
           
               
               
               
            
               
                   
                 Peak Efficiency 
                 96% 
               
            
           
           
               
               
               
               
            
               
                   
                 PW Self-inductance 
                 250 
                 uH 
               
               
                   
                 CW Self-inductance 
                 400 
                 uH 
               
               
                   
                 Mutual Inductance 
                 220 
                 uH 
               
               
                   
                 PW Resistance 
                 7.6 
                 mΩ 
               
               
                   
                 CW Resistance 
                 9.1 
                 mΩ 
               
               
                   
                   
               
            
           
         
       
     
     The pole combination follows (1), where p r , p p , and p c  are the pole numbers of the rotor, PW, and CW. The frequency follows (2), in which ω re  is the rotor electrical speed, ω rm  the rotor mechanical speed, ω P  and ω C  the electrical speed of the PW and CW, respectively. 
     
       
         
           
             
               
                 
                   
                     p 
                     r 
                   
                   = 
                   
                     
                       
                         p 
                         P 
                       
                       ± 
                       
                         p 
                         C 
                       
                     
                     2 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
             
               
                 
                   
                     ω 
                     re 
                   
                   = 
                   
                     
                       
                         ω 
                         rm 
                       
                       ⁢ 
                       
                         p 
                         r 
                       
                     
                     = 
                     
                       
                         ω 
                         P 
                       
                       ± 
                       
                         ω 
                         C 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     The mathematical model of a BDFM is similar to a DFIM. But the operating principles are significantly different. The two windings of a DFIM have the same pole number. The magnetic fields generated by the two windings can be directly coupled in the airgap. However, in a BDFM, the windings have different pole numbers. The coupling of two magnetic fields is achieved by the field modulation with the help of the rotor. 
     A BDFM operates in natural-synchronous mode if the rotor frequency equals the PW frequency. Depending on the difference between the rotor frequency and PW frequency, a BDFM works in super-synchronous mode or in sub-synchronous mode. The CW excitation compensates for this frequency difference according to (2). As a result, the rotor frequency can be independent of the winding frequencies. In contrast, the rotor speed of a traditional singly-fed machine is fixed given a stator winding frequency. 
     The mechanical power of a BDFM equals the sum of the electrical power through the PW and CW.
 
 P   mech   =P   P   +P   C   (3)
 
where P mech  is the mechanical power, P P  the PW electrical power, and P C  the CW electrical power.
 
     In a BDFM, the frequency ratio approximately equals the active power ratio, as given in (4). This relationship can quickly help estimate the amount of electrical and mechanical power. The CW is supposed to operate at a frequency much lower than the PW frequency to lower the power carried by the converter. 
     
       
         
           
             
               
                 
                   
                     
                       f 
                       C 
                     
                     
                       f 
                       P 
                     
                   
                   ≈ 
                   
                     
                       P 
                       C 
                     
                     
                       P 
                       P 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
       FIG.  3    is a diagram of an implementation of a BDFM-based TeDP  300 . In the TeDP  300  system, as seen in  FIG.  3   , the engine  310  drives the generator  315 . The mechanical power is converted to three-phase electrical power in the two stator windings of the generator  315 . The PWs of the generator  315  are directly connected to the PWs of the motors  320 ,  322  via an alternating current (AC) bus. The CWs of the generator  315  and motors  320 ,  322  are individually controlled by bidirectional converters  317 ,  325 ,  327 , respectively. The converters  317 ,  325 ,  327  have a common DC bus  330 . It can be a DC bus  330  fed by an energy storage system  340  if it is a hybrid TeDP, or a DC bus of a back-to-back converter, which takes the generator PW as the primary AC power source if it is a regular TeDP. As an example, the DC bus is assumed to be steady at 540 VDC, with neither an energy storage system nor a back-to-back converter considered. 
     The proposed BDFM-based TeDP shows advantages over other TeDP architectures: (1) The mass of the power conversion is significantly minimized as most power is transmitted through the PW and AC bus. The CW converter is downscaled because it only takes a fraction of the machine power; (2) The system is mechanically robust due to the absence of magnets, windings, and slip ring at the rotor side; and (3) The system is safe as it can be immediately de-energized by actively cutting off the CW excitations in fault conditions. 
       FIG.  4    is a diagram of an implementation of a TeDP system  400  comprising one stand-alone generator  410  and one propulsive motor  420 . A TeDP consisting of one generator and one motor, as shown in  FIG.  4   , is taken as an example to illustrate the system modeling and control method. In such a TeDP, the generator  410  (BDFM 1) converts the engine mechanical power to electrical power and is built as a stand-alone generator. The motor  420  converts electrical power to mechanical power and is modeled as a propulsive motor. 
     With respect to TeDP system modeling, the dq-axis voltage and flux linkage of a BDFM in a general reference frame g can be expressed as follows: 
                     v   pd     =         r   p     ⁢     i   pd       +       d   dt     ⁢     λ   pd       -       ω   g     ⁢     λ   pq                 (   5   )                 v   pq     =         r   p     ⁢     i   pq       +       d   dt     ⁢     λ   pq       +       ω   g     ⁢     λ   pd                 (   6   )                 v   cd     =         r   c     ⁢     i   cd       +       d   dt     ⁢     λ   cd       -       (       ω   re     -     ω   g       )     ⁢     λ   cq                 (   7   )                 v   cq     =         r   c     ⁢     i   cq       +       d   dt     ⁢     λ   cq       +       (       ω   re     -     ω   g       )     ⁢     λ   cd                 (   8   )               
where v is the voltage, i the current, λ the flux linkage, r the winding resistance, ω re  the rotor electrical speed, ω g  the electrical speed of the general reference frame, pd, pq, cd, and cq the dq-axes of the PW and CW.
 
λ pd   =l   p   i   pd   +l   m   i   cd   (9)
 
λ pq   =l   p   i   pq   −l   m   i   cq   (10)
 
λ cd   =l   c   i   cd   +l   m   i   pd   (11)
 
λ cq   =l   c   i   cq   −l   m   i   pq   (12)
 
where l p  is the PW self-inductance, l c  the CW self-inductance, l m  the mutual inductance.
 
     In a multi-BDFM TeDP system, the terminal voltage of each PW is the same (i.e., is equal). The PW current vector of the generator is opposite to the sum of PW current vectors of the motors (BDFM 2, 3, . . . , n). This implies that the generator PW delivers power to the motors, and the motor PWs consume the power from the generator.
 
 v   1p   g   =v   2p   g   = . . . =v   np   g   (13)
 
 i   1p   g =−( i   2p   g   + . . . +i   np   g )  (14)
 
where v and i are the voltage and current vectors, and n denotes the n-th BDFM.
 
     The inputs, outputs, and interfaces of a BDFM-based TeDP  500  are illustrated in  FIG.  5   .  FIG.  5    is an illustration of inputs, outputs, and interfaces of the modeling of a BDFM-based TeDP system. The individual machine model is built in the blocks. The CWs of all the machines have voltage as input and current as output. The PW of BDFM 1 takes the opposite of the sum of motor PW currents as input. Then the terminal voltage v lp  is calculated as the input of the PWs of motors. This modeling method allows for an extension to large propulsion systems with multiple propulsive motors. 
     A dynamic torque expression of TeDP is described. 
     For grid-tied power generation and electric drive applications, BDFMs typically rely on field-oriented control (FOC) to achieve active/reactive power control and speed control. FOC selects the PW flux λ p  as the reference frame and locks the d-axis to λ p . Then the q-axis flux of PW is always 0. As a result, the machine torque becomes linearly dependent on the q-axis current of CW. However, FOC is not an optimal solution for aviation propulsion application because: (1) The FOC requires the measurement of PW voltage and current to calculate the orientation of the PW flux linkage. Each motor needs a pair of voltage and current sensors, which increases the cost and computational load, and undermines the overall reliability; (2) The PWM voltages injected into the CWs can easily transfer to the PW. Therefore, severe harmonics exist in the PW voltage. The harmonics would cause inaccuracy in the calculation of flux orientation; and (3) The FOC is sensitive to machine parameters and could be inaccurate in the condition of saturation. 
     A new dynamic torque expression of the TeDP is derived. The control algorithm of the TeDP can be significantly simplified based on the torque expression. Considering the TeDP in  FIG.  4   , the classic torque expression of BDFM 1 in dq-axis form is known as:
 
 T   e1 =3/2 p   r (λ 1pd   i   1pq −λ 1pq   i   1pd )  (15)
 
where T e1  is the electromagnetic torque of BDFM 1.
 
     T e1  can be expressed as the magnitude of the cross product of the PW flux vector and the PW current vector in the general reference frame g:
 
 T   e1 =3/2 p   r |λ 1p   g   ×i   1p   g |  (16)
 
     Given that the PW terminal voltages are the same, and ignoring the voltage drop on the resistance, the flux vectors of each PW are the same.
 
λ 1p   g =λ 2p   g = . . . λ np   g   (17)
 
     It can be shown that the torques of the two BDFMs are the same in magnitude but have different signs. 
     
       
         
           
             
               
                 
                   
                     T 
                     
                       e 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                   = 
                   
                     
                       - 
                       
                         T 
                         
                           e 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                     
                     = 
                     
                       
                         
                           3 
                           2 
                         
                         ⁢ 
                         
                           p 
                           r 
                         
                         ⁢ 
                         
                            
                           
                             
                               λ 
                               
                                 2 
                                 ⁢ 
                                 p 
                               
                               g 
                             
                             × 
                             
                               i 
                               
                                 2 
                                 ⁢ 
                                 p 
                               
                               g 
                             
                           
                            
                         
                       
                       = 
                       
                         
                           - 
                           
                             3 
                             2 
                           
                         
                         ⁢ 
                         
                           p 
                           r 
                         
                         ⁢ 
                         
                            
                           
                             
                               λ 
                               
                                 2 
                                 ⁢ 
                                 p 
                               
                               g 
                             
                             × 
                             
                               i 
                               
                                 1 
                                 ⁢ 
                                 p 
                               
                               g 
                             
                           
                            
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
     According to (9) and (10), the PW flux vector of BDFM 2 can be written as:
 
λ 2p   g   =l   2p   i   2p   g   +l   2m   i   2c   g*   (19)
 
or (20) given (14):
 
λ 2p   g   =−l   2p   i   1p   g   +l   2m   i   2c   g*   (20)
 
where i 2c   g*  is the complex conjugate of the CW current space vector of BDFM 2.
 
     The torque expression of BDFM 1 can be written as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           T 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         = 
                         
                           
                             3 
                             2 
                           
                           ⁢ 
                           
                             p 
                             r 
                           
                           ⁢ 
                           
                              
                             
                               
                                 λ 
                                 
                                   2 
                                   ⁢ 
                                   p 
                                 
                                 g 
                               
                               + 
                               
                                 i 
                                 
                                   1 
                                   ⁢ 
                                   p 
                                 
                                 g 
                               
                             
                              
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             3 
                             2 
                           
                           ⁢ 
                           
                             p 
                             r 
                           
                           ⁢ 
                           
                              
                             
                               
                                 ( 
                                 
                                   
                                     
                                       - 
                                       
                                         l 
                                         
                                           2 
                                           ⁢ 
                                           p 
                                         
                                       
                                     
                                     ⁢ 
                                     
                                       i 
                                       
                                         1 
                                         ⁢ 
                                         p 
                                       
                                       g 
                                     
                                   
                                   + 
                                   
                                     
                                       l 
                                       
                                         2 
                                         ⁢ 
                                         m 
                                       
                                     
                                     ⁢ 
                                     
                                       i 
                                       
                                         2 
                                         ⁢ 
                                         c 
                                       
                                       
                                         g 
                                         * 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               × 
                               
                                 i 
                                 
                                   1 
                                   ⁢ 
                                   p 
                                 
                                 g 
                               
                             
                              
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             3 
                             2 
                           
                           ⁢ 
                           
                             p 
                             r 
                           
                           ⁢ 
                           
                             l 
                             
                               2 
                               ⁢ 
                               m 
                             
                           
                           ⁢ 
                           
                              
                             
                               
                                 i 
                                 
                                   2 
                                   ⁢ 
                                   c 
                                 
                                 
                                   g 
                                   * 
                                 
                               
                               × 
                               
                                 i 
                                 
                                   1 
                                   ⁢ 
                                   p 
                                 
                                 g 
                               
                             
                              
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
     Because the PW flux linkage vectors of two BDFMs are the same given (17), (22) is obtained.
 
 l   1p   i   1p   g   +l   1m   i   1c   g*   =l   2p   i   2p   g   +l   2m   i   2c   g*   (22)
 
     A further derivation shows that the PW current vector of BDFM 1 can be expressed as (23). 
     
       
         
           
             
               
                 
                   
                     i 
                     
                       1 
                       ⁢ 
                       p 
                     
                     g 
                   
                   = 
                   
                     
                       
                         
                           l 
                           
                             2 
                             ⁢ 
                             m 
                           
                         
                         ⁢ 
                         
                           i 
                           
                             2 
                             ⁢ 
                             c 
                           
                           
                             g 
                             * 
                           
                         
                       
                       - 
                       
                         
                           l 
                           
                             1 
                             ⁢ 
                             m 
                           
                         
                         ⁢ 
                         
                           i 
                           
                             1 
                             ⁢ 
                             c 
                           
                           
                             g 
                             * 
                           
                         
                       
                     
                     
                       
                         l 
                         
                           1 
                           ⁢ 
                           p 
                         
                       
                       + 
                       
                         l 
                         
                           2 
                           ⁢ 
                           p 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
     Then (21) can be written as (24), in which the torque of the generator becomes a function of the mutual inductance, PW self-inductance, and the CW currents of the two BDFMs. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           T 
                           
                             e 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             1 
                           
                         
                         = 
                         
                           
                             3 
                             2 
                           
                           ⁢ 
                           
                             p 
                             r 
                           
                           ⁢ 
                           
                             l 
                             
                               2 
                               ⁢ 
                               m 
                             
                           
                           ⁢ 
                           
                              
                             
                               
                                 i 
                                 
                                   2 
                                   ⁢ 
                                   c 
                                 
                                 
                                   g 
                                   * 
                                 
                               
                               × 
                               
                                 
                                   
                                     
                                       l 
                                       
                                         2 
                                         ⁢ 
                                         m 
                                       
                                     
                                     ⁢ 
                                     
                                       i 
                                       
                                         2 
                                         ⁢ 
                                         c 
                                       
                                       
                                         g 
                                         * 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       l 
                                       
                                         1 
                                         ⁢ 
                                         m 
                                       
                                     
                                     ⁢ 
                                     
                                       i 
                                       
                                         1 
                                         ⁢ 
                                         c 
                                       
                                       
                                         g 
                                         * 
                                       
                                     
                                   
                                 
                                 
                                   
                                     l 
                                     
                                       1 
                                       ⁢ 
                                       p 
                                     
                                   
                                   + 
                                   
                                     l 
                                     
                                       2 
                                       ⁢ 
                                       p 
                                     
                                   
                                 
                               
                             
                              
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                         
                           
                             - 
                             
                               3 
                               2 
                             
                           
                           ⁢ 
                           
                             p 
                             r 
                           
                           ⁢ 
                           
                             
                               
                                 l 
                                 
                                   1 
                                   ⁢ 
                                   m 
                                 
                               
                               ⁢ 
                               
                                 l 
                                 
                                   2 
                                   ⁢ 
                                   m 
                                 
                               
                             
                             
                               
                                 l 
                                 
                                   1 
                                   ⁢ 
                                   p 
                                 
                               
                               + 
                               
                                 l 
                                 
                                   2 
                                   ⁢ 
                                   p 
                                 
                               
                             
                           
                           ⁢ 
                           
                              
                             
                               
                                 i 
                                 
                                   2 
                                   ⁢ 
                                   c 
                                 
                                 
                                   g 
                                   * 
                                 
                               
                               × 
                               
                                 i 
                                 
                                   1 
                                   ⁢ 
                                   c 
                                 
                                 
                                   g 
                                   * 
                                 
                               
                             
                              
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
     Converting the space vector form to the dq-axis form, the torque of BDFM 1 can be written as follows: 
     
       
         
           
             
               
                 
                   
                     T 
                     
                       e 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     
                       3 
                       2 
                     
                     ⁢ 
                     
                       p 
                       r 
                     
                     ⁢ 
                     
                       
                         
                           l 
                           
                             1 
                             ⁢ 
                             m 
                           
                         
                         ⁢ 
                         
                           l 
                           
                             2 
                             ⁢ 
                             m 
                           
                         
                       
                       
                         
                           l 
                           
                             1 
                             ⁢ 
                             p 
                           
                         
                         + 
                         
                           l 
                           
                             2 
                             ⁢ 
                             p 
                           
                         
                       
                     
                     ⁢ 
                     
                       ( 
                       
                         
                           
                             i 
                             
                               2 
                               ⁢ 
                               cd 
                             
                           
                           ⁢ 
                           
                             i 
                             
                               1 
                               ⁢ 
                               cq 
                             
                           
                         
                         - 
                         
                           
                             i 
                             
                               2 
                               ⁢ 
                               cq 
                             
                           
                           ⁢ 
                           
                             i 
                             
                               1 
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                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     The above expression is useful for control purposes involving the dq currents of the CWs of the BDFM 1 and 2. If the reference frame is aligned with the CW current vector of BDFM 1, there will not be any q-axis current component. In such a manner, the expression can be simplified as: 
     
       
         
           
             
               
                 
                   
                     T 
                     
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                       ⁢ 
                       
                           
                       
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                   = 
                   
                     
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                     ⁢ 
                     
                       
                         
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                         ⁢ 
                         
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                           l 
                           
                             1 
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                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     T e1  is the torque of the generator, which is negative. This can be achieved by applying a positive i 2cq . This implies that the torque of the TeDP system can be controlled by the d-axis CW current of BDFM 1 (generator) and q-axis CW current of BDFM 2 (motor). In other words, when the motor CW current and the generator CW current is 90-degrees apart, the torque of the TeDP system can be maximized. 
     A similar torque expression of BDFM 1 in a TeDP with one generator and two motors can be derived based on the same idea, as shown in (27). The torque of the generator is opposite to the sum of the torque of each motor. 
     
       
         
           
             
               
                 
                   
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                   ( 
                   27 
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     The above derivation has been verified by the TeDP mathematical model and an FEA-circuit co-simulation. A further investigation reveals that the equation holds regardless of the initial rotor positions of the machines. However, encoders are still needed for system control. 
     In propulsion mode, the system operates BDFM 1 as the generator and BDFM 2 as a motor. The control diagram of the same TeDP  600  is illustrated in  FIG.  6   .  FIG.  6    is a control diagram of a TeDP  600  (one generator+one motor) in propulsion mode. 
     It is known that the active power ratio approximately equals the frequency ratio, and the CW is supposed to carry a small portion of the total machine power. Therefore, the CW frequency command of BDFM 1, f* 1c  can be obtained by multiplying the rotor frequency f 1re  by a factor of f* 1c /f 1re . Theoretically, f 1c  can be commanded as 0 Hz at low speeds unless the PW frequency is too high or a wide-range variable speed is needed. 
     Once f 1c  is known, the angular position of BDFM 1 CW, θ 1c  is obtained. Because the PW frequency of both machines is the same, the angular position of BDFM 2 CW, θ 2c  can be calculated using:
 
θ 2c =θ 2re −(θ 1re −θ 1c )  (28)
 
     The motor operates in speed mode. A PI controller of the speed control loop keeps the motor speed on track by outputting CW current commands, i* 1d  of BDFM 1 and i* 1q  of BDFM 2. According to (26), the CW currents of both machines determine the torque level and, therefore, the amount of active power that can be extracted from the engine. 
     The TeDP system has the capability of self-starting. In self-start mode, the system functions in reverse compared to the propulsion mode. The generator operates as a motor to drive the turbo engine to a certain speed level until the engine can work by itself. The motor is in generating mode. This can be achieved by commanding a CW current of BDFM 2 90-degree lagging the CW current of BDFM 1. 
     BDFM 2 does not have any mechanical power input, and its shaft speed is 0 rpm. To have PW 2 deliver power to the PW 1, CW 2 should be excited by a high-frequency current. The shaft of BDFM 2 should be fixed. In such a manner, the BDFM 2 can be considered as a transformer with CW 2 as the primary side and PW 2 the secondary side. 
       FIG.  7    is a control diagram of a TeDP  700  (one generator+one motor) in starting mode. As shown in  FIG.  7   , a frequency command of CW 2, f* 2c  is firstly given. The angular position of each CW, θ 1c  and θ 2c  can be subsequently determined. BDFM 1 operates in speed mode. The output of the PI controller of the speed loop becomes the current command of CWs. A negative gain is needed to keep the current of CW 2 90-degree lagging the current of CW 1. 
     The control system can be easily expanded when more motors are added to the propulsion system. All the motor CWs can use the same angular position θ 2c  keeping all the motor CW currents perpendicular to the generator CW current. The system control is significantly simplified because only CW currents and rotor positions are measured. One pair of voltage and current sensor is enough to monitor the PW for safety consideration. 
     A simulation of a flight profile is described. A flight profile is simulated to demonstrate the capability of an implementation of a TeDP system described herein. The TeDP in the simulation contains one generator (BDFM 1) and two motors (BDFM 2 and 3). The flight profile lasts 36 seconds.  FIG.  8    shows the speed of BDFM 1, 2, 3.  FIG.  9    shows the torque of BDFM 1, 2, 3.  FIG.  10    shows the frequency of the CWs of BDFM 1, 2, 3 and the PW.  FIG.  11    shows the active power of CW of BDFM 1, 2, 3, and PW of BDFM 1.  FIG.  12    shows the mechanical power of BDFM 1, 2, 3.  FIG.  13    shows the power factor of CW and PW of BDFM 1.  FIG.  14    shows the PW voltage of BDFM 1.  FIG.  15    shows the PW current of BDFM 1.  FIG.  16    shows the CW current of BDFM 1. 
     As marked in  FIG.  8   , the flight profile consists of 5 stages, i.e., 1. Startup, 0-4 s; 2. Taxiing, 4-8 s; 3. Taking-off and climbing, 8-15 s; 4. Cruising, 15-28 s; 5. Descending and landing, 28-36 s. The following data are recorded:  FIG.  8   , machine speed;  FIG.  9   , machine torque; 
       FIG.  10   , frequency of CWs and PW;  FIG.  11   , mechanical power of the machines;  FIG.  12   , the active power of the CWs and PW; and  FIG.  13   , the power factor of BDFM 1 CW and PW. The voltage and current of BDFM 1 PW, and current of BDFM 1 CW are given as additional information in  FIGS.  14 ,  15 , and  16   . 
     More particularly, in starting mode, a large load, i.e., the engine inertia and friction is applied to the generator. The generator provides a positive torque while the motors have negative torque. The CWs of BDFM 2 and 3 are both excited by 100 Hz currents to induce a −100 Hz excitation in the PW, given that the motors are at a standstill. The current amplitude and frequency of BDFM 1 CW are controlled to drive the engine to 2000 rpm. 
     The system switches to the propulsion mode in the 4th second. The generator is then driven by the engine. The two motors speed up to 2000 rpm with light load until the plane finishes taxiing at the 8th second. Starting from second 8, the plane takes off. The generator and motors accelerate to 12,000 rpm with a full load. It can be seen that the torque of the generator is twice that of each motor. 
     During the cruising stage, the speed of all machines reduces to 8,000 rpm. As known, the torque and speed requirements of each propulsive motor can be different. Therefore, a differential operation capability is considered to be necessary for an aircraft based on distributed propulsion. The differential operation is demonstrated between second 20 and 26. The generator speed keeps constant at 8000 rpm. The load of each motor is different. Each motor is able to follow an independent speed reference that is changing back and forth. During this differential operation, the frequency of the PW is constant. The variable-speed operation of motors is achieved by controlling the frequency of the CW of each motor. In the descending and landing stages, all machines decelerate with a light load until the speed goes back to 2000 rpm. 
     In the whole flight profile, the CW frequency of BDFM 1 is commanded as 25% of the rotor frequency. Therefore, the ratio between the rotor frequency to the CW frequency is 4, and the ratio between the PW frequency to the CW frequency is 3. The frequency ratio of 3 can be observed in  FIG.  10   . The frequency ratio approximately equals the active power ratio. As a result, the ratio of 3 can also be observed in  FIG.  11    by comparing the PW active power and the CW active power. In the comparison of  FIG.  11    and  FIG.  12   , the active power carried by the BDFM 1 CW is only 25% of the total generator mechanical power. So the power rating of the CW converter can be much lower compared to a DC-based TeDP system using traditional singly-fed machines. 
     The power factor of the windings, especially the CW, is of interest because a too low power factor would undermine the advantage of using BDFMs for electric propulsion. The power factor is calculated based on a low-pass filter, as shown in  FIG.  13   . The power factor of BDFM 1 CW and PW is negative because both windings are generating power. The spikes are caused by transients and are negligible. It can be seen that the generator PW power factor is close to −1. The PW power factor of generator and motors are symmetric about 0 because the PW voltages are the same, and the currents are 180-degree shifted. 
     The peak power factor of the BDFM 1 CW is −0.42, which is lower than traditional singly-fed machines. It is because of the large leakage inductance of the machine. The low CW power factor might be a limiting factor in using BDFMs for TeDP. Nevertheless, a CW power factor of 0.42 can still be competitive. Considering a DC-based TeDP using permanent magnet synchronous generator with a power rating of 1 in per-unit value and power factor of 0.85 in flux-weakening operation at high speed, the apparent power through the converter is 1.176. For a BDFM generator with the same power rating, the CW only takes up to 25% of the mechanical power, which is 0.25 in per-unit value. Given a power factor of 0.42, the power rating of the converter is 0.595. It is almost reduced by half compared to the converter of a permanent magnet generator. It is worth noting that the power factor can be improved from an electromagnetic design perspective. Other than that, the generator CW power factor increases as the load increases. A better CW power factor can be expected when more propulsive motors operate simultaneously. 
     The BDFM-based TeDP described herein has benefits of reduced size and rating of the power converter, robust mechanical performance, and high fault-tolerance capability compared to conventional TeDP concepts. 
     A mathematical model of the TeDP system is described to simulate the behavior of the proposed TeDP. The model can be expanded when more BDFMs join the system. A dynamic torque equation of the TeDP provided herein is derived based on the space vector theory. It is analyzed that the torque is a function of the CW currents of the generator and motors, and can be maximized when the CW current of the generator and CW currents of motors are 90-degrees apart. 
     Based on the derived torque equation, a control algorithm of the BDFM-based TeDP is described. The system control can be achieved by only measuring the CW currents and rotor positions. The control method is feasible for both the propulsion mode and the starting mode. 
     The capability of the proposed BDFM-based TeDP and the developed control method are verified using a simulation of a flight profile. The flight profile showcases the transition from the starting mode to the propulsion mode and demonstrates the differential operation of individual BDFM, which is one of the requirements of a TeDP. The CW power factor might become a limiting factor; however, it is still competitive in comparison with a DC-based TeDP. 
     Moreover, the proposed aircraft generation system using BDFMs has the following additional advantages over the state of the art: 1) one device provides the same functions as the state of the art 3-stage synchronous generator; 2) much simplified structure; 3) no electronics on the rotating parts; and 4) no need for rare earth materials. 
     In an implementation, three 120-kW BDFMs and three SiC-based two-level 75-kW three-phase inverters are designed and prototyped. Vector control and field-oriented control (FOC) are implemented to operate BDFMs in generating and motoring mode, respectively. A designed controller can fulfill the communication and control requirements of the TeDP system. The TeDP using BDFMs as described herein is a solution to aviation variable-frequency power generation and can be a potential candidate for future aviation propulsion. 
     An implementation of a 120-kW BDFM is provided. The BDFM has 8-pole PW in the outer two layers and 4-pole CW in the inner two layers. It has a 6-pole rotor with four pieces of flux guides on each pole. The rotor mechanical design takes advantage of a three-dimensional concept that allows a rotational speed up to 12,000 rpm while minimizing the flux leakage and enhancing the electromagnetic coupling between the two windings. The BDFM can provide 95-NM peak torque, 95% efficiency at 12,000-rpm rated speed. The mechanical capability has been verified by the over-speed test at 12,500 rpm. 
     The control winding of an implementation of a BDFM is fed by a SiC-based two-level 75-kW three-phase inverter. The input 540-V DC bus voltage is supplied by a bidirectional DC power supply. The fundamental frequency of the inverter (control winding) varies from 0 to 800 Hz corresponding to 0 to 12,000-rpm rotor speed. 
     A controller is designed for the control of the TeDP system. In a three-BDFM TeDP system, 6 PW voltages, 6 PW currents, and 6 CW currents are sampled and converted to digital signals at the sensor conditioning board. The digital signals and encoder (QEP) signals are transmitted through fiber optics and collected at the fiber optic board. The FPGA on the motherboard decodes and sends all digital signals to DSP through SPI (Serial Peripheral Interface). The control algorithm is implemented in the DSP. The reference voltage signals are output to the FPGA through UPP (Universal Parallel Port). 18 PWM signals are generated in the FPGA and then sent to the gate drives for machine operation. This controller has full capability of implementing sensing, ADC, communication, and machine control at 10K switching frequency for three-machine system, or over 20K switching frequency for single machine operation. 
     A BDFM in motoring mode is controlled using FOC, according to an implementation. FOC selects the PW winding flux λ PW  as the reference frame and locks the d-axis to λ PW . Then the q-axis flux of PW is always 0. As a result, the machine torque becomes linearly dependent on the q-axis current of CW. The stand-alone generator in a TeDP provides the desired power and PW frequency to the motors. The control of the generator is achieved by vector control. The active power sent to the motor through PW is controlled using q-axis current of generator CW. The reactive power is regulated by d-axis current of CW. The frequency of the PW can be stabilized by controlling the CW angular speed. 
     In an implementation, a turboelectric distributed propulsion (TeDP) system comprises: an engine; a generator comprising a plurality of control windings and a plurality of power windings, wherein the generator is driven by the engine; a plurality of motors, wherein the power windings of the generator are directly connected to power windings of the motors; and a plurality of bidirectional converters, wherein the control windings of the generator and the motors are individually controlled by a respective one of the bidirectional converters. 
     Implementations may include some or all of the following features. The plurality of bidirectional converters have a common direct current (DC) bus. The DC bus is fed by an energy storage system when it is a hybrid TeDP, or a DC bus of a back-to-back converter which takes the generator power windings as the primary AC power source when it is a regular TeDP. Mechanical power is converted to three-phase electrical power in stator windings of the generator. The power windings of the generator are directly connected to power windings of the motors via an alternative current (AC) bus. 
     In an implementation, a turboelectric distributed propulsion (TeDP) system comprises: a first brushless doubly-fed machine (BDFM); and a second BDFM, wherein the first BDFM is configured as a generator, and the second BDFM is configured as a motor. 
     Implementations may include some or all of the following features. The generator is a stand-alone generator and the second BDFM is a propulsive motor. Each of the BDFMs comprises a power winding and a control winding. A terminal voltage of each power winding is equal. When in self-start mode, the generator operates as a motor, and the motor operates in generating mode. 
     In an implementation, a turboelectric distributed propulsion (TeDP) comprises a plurality of brushless doubly-fed machines (BDFMs) configured for use with an aviation application. 
     Implementations may include some or all of the following features. The TeDP further comprises an inverter and a controller. The TeDP further comprises three 120-kW BDFMs and three SiC-based two-level 75-kW three-phase inverters. Each BDFM comprises an 8-pole power winding (PW) in the outer two layers and 4-pole control winding (CW) in the inner two layers, and a 6-pole rotor with four pieces of flux guides on each pole. Each BDFM is configured to allow a rotational speed up to 12,000 rpm while minimizing the flux leakage and enhancing the electromagnetic coupling between the two windings. Each BDFM is configured to provide 95-NM peak torque, 95% efficiency at 12,000-rpm rated speed. The control winding of each BDFM is fed by a SiC-based two-level 75-kW three-phase inverter, wherein the input 540-V DC bus voltage is supplied by a bidirectional DC power supply, and wherein the fundamental frequency of the inverter (control winding) varies from 0 to 800 Hz corresponding to 0 to 12,000-rpm rotor speed. The TeDP further comprises vector control and field-oriented control (FOC) configured to operate the BDFMs in generating and motoring mode, respectively. The TeDP is further configured for aviation variable-frequency power generation. The TeDP is further configured for aviation propulsion. 
     As used herein, the singular form “a,” “an,” and “the” include plural references unless the context clearly dictates otherwise. As used herein, the terms “can,” “may,” “optionally,” “can optionally,” and “may optionally” are used interchangeably and are meant to include cases in which the condition occurs as well as cases in which the condition does not occur. 
     Ranges can be expressed herein as from “about” one particular value, and/or to “about” another particular value. When such a range is expressed, another embodiment includes from the one particular value and/or to the other particular value. Similarly, when values are expressed as approximations, by use of the antecedent “about,” it will be understood that the particular value forms another embodiment. It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint, and independently of the other endpoint. It is also understood that there are a number of values disclosed herein, and that each value is also herein disclosed as “about” that particular value in addition to the value itself. For example, if the value “10” is disclosed, then “about 10” is also disclosed. 
     Although exemplary implementations may refer to utilizing aspects of the presently disclosed subject matter in the context of one or more stand-alone computer systems, the subject matter is not so limited, but rather may be implemented in connection with any computing environment, such as a network or distributed computing environment. Still further, aspects of the presently disclosed subject matter may be implemented in or across a plurality of processing chips or devices, and storage may similarly be effected across a plurality of devices. Such devices might include personal computers, network servers, and handheld devices, for example. 
     Although the subject matter has been described in language specific to structural features and/or methodological acts, it is to be understood that the subject matter defined in the appended claims is not necessarily limited to the specific features or acts described above. Rather, the specific features and acts described above are disclosed as example forms of implementing the claims.