Patent Publication Number: US-2020276902-A1

Title: Regenerative Braking Controller for Electric Motors

Description:
CROSS-REFERENCE TO RELATED APPLICATION(S) 
     This continuation application claims priority to U.S. patent application Ser. No. 16/366,263, filed Mar. 27, 2018, which claims priority to U.S. patent application Ser. No. 15/680,261, filed Aug. 18, 2017 (now U.S. Pat. No. 10,286,785), which claims priority to U.S. patent application Ser. No. 14/837,810, filed Aug. 27, 2015 (now U.S. Pat. No. 9,783,063), all of which are hereby incorporated herein by reference in their entirety. 
    
    
     FIELD 
     Disclosed embodiments relate to regenerative braking control for electric motors. 
     BACKGROUND 
     An electric motor is a machine that converts electrical energy into mechanical energy. Electric motors include DC motors and AC motors. One type of AC motor is an AC induction motor. An AC induction motor is typically driven by 3-phase alternating current provided by an electric motor controller coupled to a 3-phase inverter. The AC motor includes an outside stationary stator having coils supplied with alternating current to produce a rotating magnetic field, which induces a current in the rotor windings. As the current flows through the rotor windings, a second magnetic field is generated which interacts with the magnetic field from the stator to produce motion. The rotor is attached to the output shaft that is given a torque by the rotating magnetic field. The interaction of the rotor field and the stator field causes rotation of the rotor which can be used to perform work. 
     Another type of AC motor is a permanent magnet motor (PMM). PMMs have permanent magnets located on the rotor and copper windings located on the stator. The alternating current in the stator windings produces a rotating magnetic field which interacts with the magnetic field from the rotor magnets to produce motion. The frequency at which the stator current oscillates determines the rotor&#39;s angular velocity and the resulting angular position. 
     Electric motors have two mechanical operations, motoring and braking. In a torque vs. speed plot, quadrants I and III of the torque-speed plane represent forward and reverse motoring operations and quadrants II and IV represent forward and reverse braking operations. The braking operation is regenerative when the electric motor is operated as an electric generator such that the kinetic energy of the rotor is converted to electricity and fed back to the power source. Not all operating points in the braking quadrants are regenerative in nature, therefore, regenerative braking is a subset of the braking quadrants. The boundaries of the regenerative braking operation in the braking quadrants need to be determined so that a closed-loop control system does not place an operating point in the non-regenerative braking zone which causes the machine to draw power from the source in order to achieve braking. 
     SUMMARY 
     This Summary is provided to introduce a brief selection of disclosed concepts in a simplified form that are further described below in the Detailed Description including the drawings provided. This Summary is not intended to limit the claimed subject matter&#39;s scope. 
     Disclosed embodiments recognize braking an electric motor can remove kinetic energy from the motor system if done properly, but in conventional electrical braking systems, this kinetic energy is typically not recovered by the system. During a typical electrical braking process, additional electrical energy from the battery is needed to slow down the motor and none of the available kinetic energy is recovered. Disclosed embodiments include an optimal regenerative braking (RB) algorithm which provides a braking torque for electric motors which recovers a significant amount of the kinetic energy when braking from one speed to a lower speed. 
     Disclosed RB algorithms are independent of reference frame transformations and the method of motor control. The electric motor itself is controlled through disclosed software code which functions as the brake so that there is no need for an external mechanical brake (such as hydraulic brakes in an automobile). Disclosed RB algorithms also apply to motor systems having an external mechanical brake. In the presence of an external mechanical brake, disclosed RB algorithms can be used to make sure that the electric motor follows a prescribed torque trajectory such that maximum current is returned to the energy storage system (ESS, such as a battery) during the braking event. Since disclosed RB algorithms are independent of reference frame transformations and the control method, the control scheme utilized with disclosed algorithms can in one particular embodiment be field oriented control (FOC). Disclosed RB algorithms can also be operated in a stationary reference frame which removes the need for FOC. 
     For example, control schemes that can be used include the motor controller operating in the α/β frame, where the voltage/current in each coordinate is independently controlled and then transformed via a Space-Vector Generator (SVG). This converts the desired voltage vector (Vαβ) into pulse width modulation (PWM) time durations to generate motor phase voltages (Vabc) or currents Iabc to provide the desired voltage or current level. The motor controller can also operate in the A/B/C frame to allow control of the voltage/current of each of the motor phases directly. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Reference will now be made to the accompanying drawings, which are not necessarily drawn to scale, wherein: 
         FIG. 1A  depicts a torque vs. speed plot showing representative desired and regenerative braking torques at a nominal rotor speed, according to an example embodiment. 
         FIG. 1B  depicts a flow chart of the logic for an example RB strategy, according to an example embodiment. 
         FIG. 2A  is a block diagram depiction of a controlled motor system without an external brake within a common FOC controller, according to an example embodiment. 
         FIG. 2B  is a block diagram depiction of a controlled motor system including an external brake and an RB controller within a common FOC controller, according to an example embodiment. 
         FIG. 3  is a block diagram depiction of an example microcontroller unit (MCU) chip formed on a substrate implementing the FOC controller shown in  FIG. 2B  including the RB controller, external brake controller and battery management circuit, according to an example embodiment. 
         FIG. 4  shows steps in an example method of RB where each torque vs. speed value generated is designed to maximize the RB current returned to the ESS, according to an example embodiment. 
         FIG. 5  shows a simulated speed response for the case with disclosed RB algorithm disabled. 
         FIG. 6  shows a simulated braking trajectory in the torque-speed plane with a disclosed RB algorithm disabled, according to an example embodiment. 
         FIG. 7  shows the simulated speed response with the disclosed RB algorithm enabled. 
         FIG. 8  shows the simulated braking trajectory on the torque-speed plane with the disclosed RB algorithm enabled. 
         FIG. 9  shows the measured speed response with disclosed RB control disabled. 
         FIG. 10  shows the measured speed response with disclosed RB control enabled. 
     
    
    
     DETAILED DESCRIPTION 
     Example embodiments are described with reference to the drawings, wherein like reference numerals are used to designate similar or equivalent elements. Illustrated ordering of acts or events should not be considered as limiting, as some acts or events may occur in different order and/or concurrently with other acts or events. Furthermore, some illustrated acts or events may not be required to implement a methodology in accordance with this disclosure. 
     Also, the terms “coupled to” or “couples with” (and the like) as used herein without further qualification are intended to describe either an indirect or direct electrical connection. Thus, if a first device “couples” to a second device, that connection can be through a direct electrical connection where there are only parasitics in the pathway, or through an indirect electrical connection via intervening items including other devices and connections. For indirect coupling, the intervening item generally does not modify the information of a signal but may adjust its current level, voltage level, and/or power level. 
     Disclosed methods of RB comprise providing an electric motor (motor) comprising a rotor, a measured or estimated speed of the rotor (rotor speed) and braking torque values as a function of rotor speed, where each torque value is designed to maximize the RB current returned to the ESS that is typically a battery. The general trend is that as the rotor speed decreases, the level of braking torque decreases. A procedure for determining a braking torque that maximizes the RB current is described below using symbolic closed-form expressions or solved using numerical optimization methods. 
     Disclosed braking strategy is described for two different motor system scenarios, one where an external brake is present and one where an external brake is absent. In the presence of an external brake, it may be advisable to decompose the desired braking torque level into two components. The electric motor applies one torque component that provides the maximum RB current to recover energy using the disclosed RB algorithm, and another torque component that is provided by the external brake to meet the desired braking torque command from the speed controller (see speed controller  251 ′ in  FIG. 2B  described below). 
     Therefore, a braking strategy is formulated that uses this decomposition of desired torque concept, as shown in  FIG. 1A  which shows a torque vs. speed plot showing a desired braking torque level (calculated by the speed controller), τ desired , at a desired speed, ω, that is in the non-regenerative braking region. A non-regenerative braking region is a subset of the braking quadrants in the torque vs speed plane where energy from the ESS is used to provide a portion of τ desired . τ desired  can be decomposed into the optimal RB torque, τ regen , that provides the maximum RB current (being in the RB region) and the remaining braking torque that is applied by the external brake, τ ext =τ desired −τ regen . 
     For motor systems having an external brake, since the desired torque level is being applied by the combination of motor braking and external mechanical braking during the braking event, this strategy does not affect the time that it takes to slow down. In the absence of an external brake, disclosed RB braking strategy is designed such that if τ desired  is greater than the optimal RB torque at that rotor speed, τ regen , then the braking torque that is applied by the motor is limited to τ regen  which is a function of rotor speed as described above. This method applies a braking torque that is less than the desired torque level (commanded by the speed controller) thereby increasing the time that it takes to slow the motor down. However, a significant amount of energy is still recovered by the ESS during the braking process. 
     There are a wide variety of motor applications in which braking in the shortest amount of time or the shorted distance is not required or even needed. In these applications, the trajectory can be planned such that the motor operates along the braking torque curve that produces the maximum RB current. A flow chart showing the logic for the overall RB strategy is shown in  FIG. 1B . Step  151  comprises providing a desired braking torque level (τ desired ) that is calculated by the speed controller. Step  152  considers whether the motor system includes an external brake. If the motor system includes an external brake, the method moves to step  153  comprising having the motor controlled to provide a torque τ ref =τ regen , and then step  154  comprises having the external break controlled to provide an external brake torque τ ext =τ desired −τ regen . If the motor system does not include an external break, the method moves to step  155  comprising considering can there be a compromise to brake slowly. If there cannot there be a compromise to brake slowly, the method moves to step  156  comprising having the motor controlled to provide a torque τ ref =τ desired . If there can be a compromise to brake slowly, the method moves to step  157  comprising having the motor controlled to provide a torque τ ref =τ regen . 
     In order implement the disclosed braking strategy, knowledge of the braking torque that corresponds to maximum RB current for essentially every permissible speed is needed for the particular motor design. As described above, this information can be obtained from a closed-form expression for SPMs. A closed-form expression can also be obtained for other motor types, such as switched reluctance motors, DC stepper motors, or induction motors. 
     Obtaining a closed-form expression for the torque that maximizes the RB current is a typically a straightforward task for motors such as surface mounted permanent magnet motors (SPMs) with the core-loss resistance neglected. The RB controller  240  described below relative to  FIG. 2B  is shown using the τ desired  output from the speed controller shown as  251 ′ and the angular velocity estimate {circumflex over (ω)}(n) from the EST block  215  (see  FIG. 2B  where it is shown as “Estimator”  215  described below) to determine the level of external braking and the level of maximum RB current at that particular rotor mechanical speed, which is found from Equation 1 (below): 
     
       
         
           
             
               
                 
                   
                     τ 
                     
                       r 
                        
                       e 
                        
                       g 
                        
                       e 
                        
                       n 
                     
                   
                   = 
                   
                     
                       - 
                       
                         ( 
                         
                           
                             K 
                             b 
                             2 
                           
                           
                             2 
                              
                             
                               R 
                               s 
                             
                           
                         
                         ) 
                       
                     
                      
                     ω 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where K b  is the back-EMF constant of the motor, R s  is the resistance of the stator windings, and ω is the rotor&#39;s mechanical speed. 
     As the rotor speed w decreases, less braking torque is commanded during operation along the curve of maximum regenerative braking current. Therefore, the rotor speed approaches zero asymptotically and reaches zero speed in infinite time. Accordingly, there is a minimum speed, ω regen,min  below which the RB controller will disable operating along the optimal braking torque curve. The operating point is only modified for points in the braking quadrants (4 th  and 2 nd ) that lie below (or above for the 2 nd  quadrant) the braking torque curve that provides the maximum RB current. 
       FIG. 2A  is a block diagram depiction of a controlled motor system  200  without an external brake within a common FOC controller  220 , according to an example embodiment. The motor  210  can be a permanent magnet (PM) motor or an AC (alternating current) induction motor. The FOC controller  220  includes a non-volatile (NV) memory, such as a read only memory (ROM) or static random access memory (SRAM), and a processor such as a central processing unit (CPU) that implements in software all the blocks in the encircled region shown as software as shown in  FIG. 2A . Hardware components of system  200  such as implemented by a CPU and NV memory are shown in an encircled region labeled hardware to distinguish them from components of system  200  implemented in software. 
     System  200  includes analog circuitry  230  between the FOC controller  220  and the motor  210  comprising power driver  231 , 3-phase inverter  232 , and sense/measurement circuits (shown as sense circuits)  233 . The FOC controller  220  includes analog-to-digital converters (ADC&#39;s)  243   a - c  coupled to receive outputs from the sense circuits  233 . 
     The current measurements for the respective phases Iabc output by ADC  243   a  is shown coupled to an input of the Clarke transform block (Clark block)  241   a , and the voltage measurements for the respective phases Vabc output by ADC  243   b  is shown coupled to an input of the Clarke block  241   b , where the Clark blocks perform the known Clarke transformation which takes the measured current or voltage in the abc frame and transforms it into the α/β coordinate system to generate Iαβ which is coupled to an input of the Park transform block (Park block)  246   a , and Vαβ which is coupled to an input of the Park transform block  246   b . Park blocks  246   a ,  246   b  also receive B [n]. Park block  246   a , performs the known Park transformation to provide outputs Id and Iq being measured phase current values from the stator terminals of the motor  210 , and Park block  246   a  performs the known Park transformation to provide outputs Vd and Vq being measured phase voltage values from the stator terminals of the motor  210 . Id, Iq Vd and Vq are all input to the EST block  215  labeled as “Estimator”  215 . 
     The EST block  215  is shown outputting an angular velocity estimate {circumflex over (ω)}(n). The estimated angular velocity {circumflex over (ω)}(n) is subtracted from the reference velocity, ω ref , with the result of the difference provided to the speed controller  251  as shown. The {circumflex over (θ)}[n+1] angular estimate from the EST block  215  is also coupled to an input of the inverse Park (iPARK) block  253 , which outputs Vαβ that is coupled to space-vector generator (SV generator) block  254 , where the output of SV generator block  254  is coupled to a PWM driver  255 . The SV generator  254  computes the PWM time durations for each phase of the motor  210  to produce the desired Vαβ voltage values. The output of the PWM driver  255  is coupled to an input of a power driver  231  that has an output coupled to drive an input of the 3-phase inverter  232 . In this voltage mode control embodiment, the 3-phase inverter  232  forces voltage onto the stator terminals associated with each of the three phases of the motor  210 . 
     The reference current generator  256  receives from the speed controller  251  and {circumflex over (ω)}(n) from the EST  215  as inputs, and outputs I d,ref  and I q,ref  reference currents. I d  from the Park block  246   a  is subtracted from I d,ref  with the result coupled to an input of a D-axis current controller  257  which provides a V d  output. I q  from Park block  246   a  is subtracted from I q,ref  with the result coupled to an input of a Q-axis current controller  258  which provides a V q  output. The V d  output and V q  output are both coupled to an iPARK block  253  which receives the V q  output from Q-axis current controller  258 , the V d  output from the D-axis current controller  257 , and as described above the {circumflex over (θ)}[n+1] angular estimate output by the EST block  215 . 
     System  200  also includes a battery management circuit  260  and an ESS  265  such as a battery. The battery management circuit  260  is coupled to an output of the inverter  232 . The battery management circuit controls  260  a maximum level of RB current flowing into the ESS  265  using a current regulation circuit, wherein the maximum RB current level can be based on the maximum charging current the ESS  265  can support. 
     A disclosed RB controller can be seamlessly integrated into the common FOC system  200  shown in  FIG. 2A  to provide the controlled motor system (system)  250  shown in  FIG. 2B  having a RB controller  240  with a motor controller  220 ′ (FOC controller) for controlling a motor  210  including an external brake  259  shown as a three-phase motor, according to an example embodiment. In this embodiment the speed controller shown as  251 ′ receives the difference of the reference velocity ω ref  and the estimated angular velocity {circumflex over (ω)}(n) from EST block  215 , with this difference provided to the speed controller  251  as shown. Speed controller  251 ′ outputs a desired torque τ desired  to an input of the RB controller  240 . The RB controller  240  also receives the angular velocity estimate {circumflex over (ω)}(n) from the EST block  215 , and provides two outputs shown as τ ext  and τ ref . The τ ref  output from the RB controller  240  is shown coupled to an input of the reference current generator shown as  256 ′ which also receives {circumflex over (ω)}(n) from EST  215  as another input, and τ ext  is coupled to the input of an external brake controller  245  which is coupled to drive the external brake  259 . 
     As with system  200 , the reference current generator  256 ′ outputs I d,ref  and I q,ref  reference currents. I d  from the Park block  246   a  is subtracted from I d,ref  with the result coupled to an input of a D-axis current controller  257  which provides a V d  output. I q  from Park block  246   a  is subtracted from I q,ref  with the result coupled to an input of a Q-axis current controller  258  which provides a V q  output. The V d  output and V q  output are both coupled to an iPARK block  253  which receives the V q  output from Q-axis current controller  258  and the V d  output from the D-axis current controller  257 . The Vαβ output of iPARK block  253  is coupled to a SV generator block  254 , where the output of SV generator block  254  is coupled to a PWM driver  255 . The SV generator  254  computes the PWM time durations for each phase of the motor  210  to produce the desired Vαβ voltage values. The output of the PWM driver  255  is coupled to an input of the power driver  231  that has an output coupled to drive an input of the 3-phase inverter  232 . In this voltage mode control embodiment, the 3-phase inverter  232  forces voltage onto the stator terminals associated with each of the three phases of the motor  210 . 
     The FOC controller  220  or  220 ′ can be a sensorless controller or can be a controller having a position sensor. FOC controllers having position sensors can include encoders and sensors that measure position directly and then estimate the angular speed therefrom. FOC controllers  220 ,  220 ′ can be implemented by a MCU, such as the MCU chip  300  shown in  FIG. 3  described below. Circuitry other than a MCU can also be used to realize FOC controller  220  or  220 ′, such as implementing a disclosed RB controller block  240 , external brake controller  245  and an EST block  215  using software stored in memory implemented by a computing device, and the battery management circuit  260  using hardware/circuitry, such as coprocessors or accelerators built using application-specific integrated circuit (ASIC) logic gates or field programmable gate arrays (FPGAs). 
       FIG. 3  is a block diagram depiction of an example MCU chip  300  formed on a substrate  305  implementing the FOC controller  220 ′ shown in  FIG. 2B  including the RB controller  240 , external brake controller  245 , and battery management circuit  260 , according to an example embodiment. Although not shown, the MCU chip  300  typically includes other integrated circuit modules, for example, a Universal Serial Bus (USB) controller and a transceiver. MCU chip  300  is shown including NV memory  272 , volatile data memory  273  shown as “data memory volatile”, digital I/O (interface)  274 , central processing unit (CPU)  275 , and clock (or timer)  276 . Core for implementing the EST block  215 , RB controller  240 , external brake controller  245  are shown collectively as  282  stored in NV memory  272 , although it is possible one or more of these blocks or the controller can be implemented in hardware. MCU chip  300  is also shown including a digital data bus  278  and an address bus  279 . 
     MCU chip  300  is shown as a monolithic integrated circuit (IC). The substrate  305  may comprise silicon, such as bulk silicon or silicon epi on a bulk silicon substrate. The substrate  305  may also generally comprise other materials, such as elementary semiconductors besides silicon including germanium. Substrate  305  may also comprise a compound semiconductor. 
       FIG. 4  shows steps in an example method  400  of RB where each torque vs. speed value generated is designed to maximize the RB current returned to the ESS, according to an example embodiment. Step  401  comprises providing an electric motor (motor) comprising at least one stator having a winding with a stator terminal, and a rotor, a measured or estimated speed of the rotor (rotor speed), a motor controller that regulates a current level in the winding, and a power inverter which controls an energy flow to the stator terminal. An ESS is also provided which exchanges energy with the motor, a battery management circuit is coupled between the ESS and the power inverter, and a processor has an associated memory storing a RB algorithm, where the processor is programmed to implement the RB algorithm during braking to cause the motor controller to execute steps  402  and  403 . 
     Step  402  comprises determining a value for RB torque (RB torque value) from the rotor speed that maximizes a level of RB current (maximum RB current). Step  403  comprises using the power inverter to redirect the maximum RB current to maximize a power transfer from the motor to the ESS. 
     The battery management circuit can control the maximum RB current flowing into the ESS using a current regulation circuit based on a maximum charging current level supported by the ESS. In some embodiments the motor includes an external brake, where the method further comprises determining a value for an external braking torque from the rotor speed to provide additional mechanical braking, and decomposing a desired torque level into two components wherein the motor controller applies the RB torque value as one torque component to provide the maximum RB current to recover energy and a second torque component is provided by the external brake to the motor to meet the desired torque command from the speed controller. 
     Benefits of disclosed algorithms include disclosed RB algorithms can be implemented for a variety of motor controllers, for example for the Texas Instruments&#39; INSTASPIN-FOC sensorless motor control technology, such as for PICCOLO and POTENZA MCU chip-based motor controllers. In the case of INSTASPIN-FOC technology and related technology, code for the disclosed RB algorithms can be embedded in the ROM of the MCU chip (see NV memory  272  in  FIG. 3 ). 
     Uses for disclosed embodiments include for electric automobiles such as the Tesla Model S P85+. Other example applications include any system that uses electric motors that operate at times with slowing or stopping, such as manufacturing and assembly lines, white appliances, elevators, cranes, and quadcopters. 
     Any equation described herein may be implemented by either hardware or by software. Regarding hardware-based implementations, a disclosed equation can be converted to a logic gate pattern, such as using VHDL which can then be realized such as using FPGA or application-specific integrated circuit (ASIC) to implement the logic gate pattern. VHDL is an acronym which stands for VHSIC (Very High Speed Integrated Circuits) Hardware Description Language. For example, a software-based implementation can be realized using a math library provided by a conventional MCU chip. 
     EXAMPLES 
     Disclosed embodiments are further illustrated by the following specific Examples, which should not be construed as limiting the scope or content of this Disclosure in any way. 
     Simulation Results 
     The disclosed RB algorithm was applied to a battery powered converter-controlled electric machine. The electric machine model was the Anaheim Automation Brushless DC motor (P/N: BLY172S-24V-4000). The battery pack comprised six A123 Racing AR26650M1B Li-Ion Nano Cells with each cell having a nominal voltage of 3.3V. The battery pack had a nominal voltage of 19.8V and an internal resistance of 36 mOhm. The parameters of the system are listed in Table 1 below. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Parameters of the motor system used for Simulations 
               
            
           
           
               
               
               
               
            
               
                 Parameter 
                 Description 
                 Value 
                 Units 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                 E s   
                 Nominal Open Circuit Battery 
                 19.8 
                 V 
               
               
                   
                 Voltage 
                   
                   
               
               
                 R s   
                 Internal Resistance of Battery 
                 0.036 
                 Ohms 
               
               
                 N 
                 Number of Pole-Pairs of PMSM 
                 4 
                 — 
               
               
                 R p   
                 Per-Phase Stator Winding 
                 1.2 
                 Ohms 
               
               
                   
                 Resistance of PMSM 
                   
                   
               
               
                 R c   
                 Per-Phase Core Loss Resistance  
                 150 
                 Ohms 
               
               
                   
                 of PMSM 
                   
                   
               
               
                 L d   
                 D-axis Inductance of PMSM 
                 1.8 
                 mH 
               
               
                 L q   
                 Q-axis Inductance of PMSM 
                 1.8 
                 mH 
               
               
                 Λ 
                 Magnitude of Permanent Magnet 
                 0.010998 
                 V/rad/s 
               
               
                   
                 Flux 
                   
                   
               
               
                 I max   
                 Maximum Length of Current  
                 5.091 
                 A 
               
               
                   
                 Vector in the d-q plane 
                   
                   
               
               
                   
               
               
                 max(U dq ) 
                 Maximum Assignable Length of the Duty-Cycle Vector in the d-q plane 
                 
                   
                     
                       
                         3 
                         
                           2 
                            
                           
                             √ 
                             2 
                           
                         
                       
                     
                   
                 
                 — 
               
               
                   
               
               
                 ω max   
                 Maximum Mechanical Speed 
                 9000 
                 RPM 
               
               
                   
                 of the Machine 
                   
                   
               
               
                 J m   
                 Rotor Inertia 
                 4.8 × 10 −6   
                 Kg-m 2   
               
               
                 β m   
                 Rotor Viscous Friction Co- 
                 5.0 × 10 −6   
                 Nm/rad/s 
               
               
                   
                 efficient 
               
               
                   
               
            
           
         
       
     
     The motor system was loaded with an external inertia and viscous friction load for the simulations. The system was assumed to start from a non-zero equilibrium point (non-zero speed and currents) and was slowed down to zero speed using closed-loop speed control with RB disabled (as a control) and enabled. The performance metrics are measured in terms of stopping time and energy recovered at the terminals of the battery. The energy recovered at the terminals of the battery is the time integral of the product of DC bus current and DC bus voltage t=0 to t=t stop . 
         E   elec =∫ t=0   t     stop     i   dc ν dc   dt  
 
     If the value of E elec  is negative, then it means that energy was returned to the ESS (regenerative braking) during the braking event and if the value of E elec  is positive, then energy was removed from the ESS (non-regenerative braking) during the braking event. If the initial speed of the combined rotor-load combination is ω 0 , then the initial kinetic energy stored in the rotor is: 
         E   mech,init =½( J   m   +J   L )ω 0   2  
 
     The ratio if energy is recovered is expressed as: 
     
       
         
           
             
               E 
               
                 rati 
                  
                 o 
               
             
             = 
             
               { 
               
                 
                   
                     
                       
                         - 
                         
                           
                             E 
                             elec 
                           
                           
                             E 
                             
                               
                                 m 
                                  
                                 e 
                                  
                                 c 
                                  
                                 h 
                               
                               , 
                               
                                 i 
                                  
                                 n 
                                  
                                 i 
                                  
                                 t 
                               
                             
                           
                         
                       
                        
                       
                           
                       
                       , 
                     
                   
                   
                     
                       
                           
                       
                        
                       
                         
                           if 
                            
                           
                               
                           
                            
                           
                             E 
                             elec 
                           
                         
                         &lt; 
                         0 
                       
                     
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                           
                       
                        
                       otherwise 
                     
                   
                 
               
             
           
         
       
     
     The value for ω regen,min  was set to zero for the simulations. The value of rotor speed was initially set to ω 0 =100 rad/s. The value of the initial kinetic energy stored in the rotor-load combination was 1013.6 mJ. The results of the simulation for the scenario with regenerative braking disabled and enabled are shown in Table 2 below. The speed controller gains are not changed between the RB disabled and enabled cases. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Simulation Results with  
               
               
                 Regenerative Braking Disabled and Enabled 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Viscous 
                   
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Friction 
                 Regenerative  
                 Regenerative  
               
               
                   
                 Co- 
                 Braking 
                 Braking 
               
               
                 Load  
                 efficient, 
                 Disabled 
                 Enabled 
               
            
           
           
               
               
               
               
               
               
            
               
                 Inertia, 
                 β L    
                 Stopping 
                 Electric 
                 Stopping 
                 Electric 
               
               
                 J L    
                 (Nm/ 
                 Time  
                 Energy, 
                 Time  
                 Energy, 
               
               
                 (kg-m 2 ) 
                 rad/s) 
                 (ms) 
                 E elec  (mJ) 
                 (ms) 
                 E elec  (mJ) 
               
               
                   
               
               
                 1.98 × 10 −4   
                 3.16 × 10 −5   
                 203.7 
                 550.51 
                 1193.9 
                 −432.4 
               
               
                   
               
            
           
         
       
     
     It can be seen that operation with the RB algorithm disabled stops very fast but consumes energy from the battery to do so (E ratio =0). In contrast, operation with the RB algorithm enabled has recovered about 42.65% of the initial kinetic energy (E ratio =0.4265), however it took about six times longer to come to a stop. 
     The simulation plots for the scenarios in Table 2 are described below. The speed response for the case with the RB algorithm disabled is shown in  FIG. 5 . The braking trajectory in the torque-speed plane is shown in  FIG. 6 . It can be seen that most of the braking trajectory is in the non-RB region (below the maximum regenerative braking boundary) which explains why there was no charge returned to the battery. 
       FIG. 7  shows simulation results for scenario with the RB algorithm enabled. The speed response as seen in  FIG. 7  approaches zero asymptotically. A faster braking event can be achieved by setting the value of ω regen,min  to a non-zero value at the cost of less energy being recovered by the RB algorithm. The braking trajectory as seen in  FIG. 8  approaches and stays on the curve of maximum regenerative braking current defined by equation 1 shown above. This results in maximum current flowing back into the battery at every rotor speed. 
     The motor system described in Table 1 above was set up on the laboratory bench with an aluminum disc as the inertial load. The machine was controlled using the Multi-Axis Motor Control Kit with the TMS320F28069 Piccolo series microcontroller from Texas Instruments. The values of inertial load and viscous friction co-efficient correspond to those of case 4 of Table 2. The inertial load was accelerated to a constant speed of 100 rad/s and then brought to a stop with the RB algorithm disabled and enabled. The results of the experiment are shown in Table 3 below and the measured speed responses for the RB algorithm disabled and enabled are shown in  FIG. 9  and  FIG. 10 , respectively. 
     The experimental results show that the stopping time with the RB algorithm enabled is about 3.5 times that with the RB algorithm disabled. With the RB algorithm disabled, the battery expends energy in order to bring the inertia to a stop resulting in a net positive electric energy as seen in Table 3. However with the RB algorithm enabled, there is a conversion of mechanical energy from the rotating inertial load to electrical energy that charges the battery. The energy recovered in the case with the RB algorithm enabled was about 41.38% of the initial kinetic energy that was stored in the rotating rotor-load combination. When comparing the simulation results to the experimental results, the stop time in the experiments is shorter than the simulations. The primary reason for this difference is due to unmodeled dynamics in the system simulation such as inverter loss. This unmodeled effect is also visible in the amount of energy stored back into the battery, which is less for the experiments than for the simulations. 
     
       
         
           
               
             
               
                 TABLE 3 
               
             
            
               
                   
               
               
                 Experimental Results with  
               
               
                 Regenerative Braking Disabled and Enabled 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Viscous 
                   
                   
                   
               
            
           
           
               
               
               
               
            
               
                   
                 Friction 
                 Regenerative  
                 Regenerative  
               
               
                 Load 
                 Co- 
                 Braking Disabled 
                 Braking Enabled 
               
            
           
           
               
               
               
               
               
               
            
               
                 Inertia, 
                 efficient, 
                 Stopping 
                 Electric 
                 Stopping 
                 Electric 
               
               
                 J L    
                 β L    
                 Time  
                 Energy, 
                 Time  
                 Energy, 
               
               
                 (kg-m 2 ) 
                 (Nm/rad/s) 
                 (ms) 
                 E elec  (mJ) 
                 (ms) 
                 E elec  (mJ) 
               
               
                   
               
               
                 1.98 × 10 −4   
                 3.16 × 10 −5   
                 192.0 
                 427.2 
                 712.0 
                 −419.5 
               
               
                   
               
            
           
         
       
     
     CONCLUSION 
     From the simulations and experiments, it can be concluded that by using a disclosed RB algorithm there is a significant increase in charging power going into the ESS such as a battery at the expense of an increase in average stopping time for a motor system with no external mechanical brake. However, as described above, disclosed embodiments can also benefit motor systems having an external mechanical brake. 
     Those skilled in the art to which this disclosure relates will appreciate that many other embodiments and variations of embodiments are possible within the scope of the claimed invention, and further additions, deletions, substitutions and modifications may be made to the described embodiments without departing from the scope of this disclosure.