Comparative Study Of Back EMF Based Sensorless Control Methods For Dual Three-Phase PMSM

Multiphase electrical machines have gained increased attentions recently due to its fault tolerance capability which is of great importance for more-electric aircraft application. This paper studies sensorless control of a high-speed dual three-phase electrical machine for turboprop aircraft green taxiing motor and engine generator applications. After introduction of a detailed mathematical model of a dual three-phase permanent magnet machine, two different types of back EMF based sensorless control are compared in this paper. The first method uses a phase-locked-loop (PLL) type speed and rotor position estimator and the other uses a Luenberger observer estimator. The effectiveness of these two different methods is demonstrated using simulations in the Matlab/Simulink environment. The comparison studies show that Luenberger type estimator has better dynamic performance but suffers high frequency noise in the estimated speed error and requires the use of machinal parameters. In contrast, PLL type estimator has inferior but acceptable performance. Its low-pass characteristics frees it from the high frequency noise. Moreover, it does not require the use of mechanical parameters.


I. INTRODUCTION
Over the last few decades, there has been tremendous progress in the efforts to move toward more electric aircraft (MEA). Many subsystems that previously used hydraulic, mechanical, and pneumatic power have been fully or partially replaced by electrical systems. Multiphase machine has been widely used in MEA [1] since it can provide notable improvements in various aspects of performance when compared with the use of conventional three-phase machine [2]. Multiphase machine exhibits outstanding advantages, such as reduced phase current rating as well as torque ripple, less DC link harmonic current; smooth magneto motive force (MMF), improved efficiency, and excellent fault tolerant characteristics [2][3][4][5]. In addition, the system reliability is greatly improved with multiphase machines since the machine can operate continuously even if one or several phases (in some cases) are lost [1]. One of the various multiphase drive solutions is the dual three-phase (DTP) machine that has two identical star-connected three-phase stator windings with isolated neutral points [6]. Considering the spatially shifted angle between two sets of three-phase windings, DTP machines can be classified as symmetrical (shifted by 60 electrical degrees) machines and asymmetric (shifted by 30 electrical degrees) machines. According to [7], symmetric DTP-PMSM is preferred to satisfy the severe fault-tolerant requirements that are imposed by the specific aerospace application. Hence, in this study, a symmetric DTP-PMSM is investigated to try to develop a mechatronic device for turboprop aircraft applications. The electrical machine is used for green taxiing application and driving propellers during motoring mode. During the generation mode, the electrical machine is driven by the engine and runs as an electrical generator. The schematic diagram of the machine is shown in Fig. 1.
Sensorless control is required in this study because the space in the gear-box is too small to install mechanical speed/position sensors. Besides, system reliability is improved since vulnerable components such as mechanical sensors in entire system are removed. Generally, sensorless control methods can be divided into two categories [8]. One is rotor saliency based methods such as high frequency signal injection approaches [9,10]. The other one is a back EMF based method. In this study, the characteristics of the estimated back EMF signals are analyzed. The differences between a phase-locked-loop (PLL) type estimator and a Luenberger observer type speed and position estimator are compared. Sensorless control design for symmetric DTP-PMSMs has not been studied in details in existing publications and this paper undertakes the attempt to fill this gap.

II. DUAL THREE-PHASE PMSM MODEL
In this section, mathematic models of DTP-PMSM are built in three-phase coordinate frame, dq rotating frame and ߙߚ stationary frame.

A. Mathematical Model in Three-Phase Coordinate Frame
The voltage and flux equations of a DTP-PMSM in original phase coordinate frame can be written as is the inductance coefficient matrix which can be presented as where ‫ܮ‬ ௭ is the stator leakage inductance, ‫ܮ‬ ௗ and ‫ܮ‬ are stator inductances in d-axis and q-axis respectively. I is a unity matrix with six columns and six rows. and are detailed in Appendix II. The system parameters used in this study are given in Appendix I.
The electromagnetic torque of the PMSM is obtained by taking the partial derivative of the system co-energy with respect to the rotor position angle, where ܲ is the number of pole pairs.

B. Model Representation in dq Rotating Frame
By ignoring zero sequence component, DTP-PMSM can be represented as two single three-phase machines with mutual coupling through transformation matrix ܶ ௦ .
where ݂stands for variable such as voltage, current or flux.

C. Model Representation in ߙߚ Stationary Frame
The transformation of variable between dq rotating frame and ߙߚ stationary frame can be presented as:

A. Estimation of Back EMF
The voltage equation in dq rotating frame can be derived from (1)-(2) and (5) e is defined as the extend EMF which presented as: Since the two sets of three-phase windings are exactly symmetric, if the current controllers of these two dq frames share same parameters, (12) can be expressed as: Voltage equation in ߙߚ stationary frame can be deduced combining (9)- (11) and (14)- (15): where ‫ܯ‬ ଷ is detailed in Appendix II.
According to (16), two sets of three-phase windings can be completely decoupled and voltage equation of DTP-PMSM in ߙߚ stationary frame can be simplified as According to [12], a PI type Back EMF estimator consists of two parts: the DTP-PMSM model (17) without the extended EMF ( ݁ ఈఉ భǡమ ) and a proportional-integral (PI) compensator. Since the extended EMF is unmodeled, it is inherently estimated by the PI compensator. The schematic diagram of this estimation is shown in Fig. 2. The voltage equation in Fig. 2

B. Estimation of Speed and Position
In this section, two different types of speed and position estimator will be introduced, i.e. PLL type and Luenberger observer type.

1) PLL Type Speed and Position Estimator
The PLL type speed and position estimator is shown in Fig. 3.

From (16), it can be deduced that
According to (23) and (24), the error signal ο݁ in Fig. 3 can be presented as: The following approximation exists if ߠ െ ߠ is small enough [12]: Thus, a PI controller can be utilized to correct the position error ο݁ and make the estimated position converge to the reference one.  According to (26), Fig. 3 can be simplified equivalently to Fig. 4. The transfer function from ߠ to ߠ in Fig. 4 can be given by: The characteristic equation of the standard second-order system can be written as: where ߦ is damping ratio and ߱ is natural frequency.
By assuming that the denominator of (27) is the same as that of (28), ‫ܭ‬ and ‫ܭ‬ can be deduced as follows:

 
The transient response of the PLL type estimator can be improved by adding a double integral term into the PI controller as follows: The transfer function from ߠ to ߠ is given by: The three gains in (31) are determined to satisfy the following condition: where ߦ and ߱ are the same values as in the foregoing simulations.

2) Luenberger Observer Type Speed and Position Estimator
A Luenberger observer type speed and position estimator can also be used for the estimation of rotor speed and position [13] as shown in Fig. 6. The transfer function of the Luenberger observer type position estimator, shown in Fig. 6 is given by (37) [14]  where J is rotational inertia, B is viscous friction.
The gains of the estimator in Fig. 6 can be selected such that the characteristic equation of (33) has the same roots as the followings [14]: where ߙ is the root of the characteristic equation. Fig. 5 shows the bode plots of (30) and (33) where ߦ and ߱ are set to 0.5 and 100 rad/s, respectively. As shown in Fig.  5, when the double integral term is added, the phase delay declines and the performance of transient state is improved. Hence, a double integral pulsed PI controller is utilized as PLL estimator.

IV. SIMULATION RESULTS
The block diagram of the sensorless control system is shown in Fig. 7. Simulation results of PLL based back EMF method and Luenberger observer based back EMF method are shown in Fig. 8. The comparison of these two methods is made on the same scenario which can be divided into two stages. The first stage continues from beginning to 2.5 s while the second stage lasts from 2.5 s to the end. In the first stage, there is no load torque and rotor speed increases steadily from 0 rpm to 18 krpm during the first two seconds and retains at 18 krpm afterwards. In the second stage, a torque load is applied. The torque load grows steadily from 0 N • m to 100% load (14.8 N • m) between 2.5 s and 3.5 s and then stays at 14.8 N • m for 1 s. After unloading from 100% load to 0 N • m between 4 s and 5 s, it keeps at 0 N • m to the end of simulation.
The aims of using sensorless control in this study is achieved according to Fig. 8. Satisfactory performance of current is observed and flux weakening control is thereby verified after calculation. System keeps at stable status as variable load torque is applied. The tracking of speed reference (߱ * ) is satisfactory. Yet, there are some differences of speed and position estimation between control performance obtained using PLL type estimator and Luenberger type estimator. The Luenberger type estimator has a better dynamic performance which can be observed especially in the partial enlargement figure during dwell 0 s to 0.2 s. This advantage is due to include the demanded torque as a feed forward term [15] as shown in Fig. 5. Luenberger observer type estimator is parameter dependent although it provides the best dynamic response [16]. Besides, high frequency noise is observed in the estimated speed obtained using Luenberger observer type estimator. In contrast, the high frequency component in the estimated speed error is filtered by the PLL type estimator since the estimator and the low-pass filter share a same frequency.

CONCLUSIONS
Two back EMF based estimators for the sensorless control of a DTP-PMSM are investigated in this study. Sdomain simulations are conducted to compare the effectiveness of these two estimators. Luenberger type estimator has better dynamic performance but suffers high frequency noise in the estimated speed error and requires the use of machinal parameters. In contrast, PLL type estimator has inferior but acceptable performance. Its low-pass characteristics frees it from the high frequency noise. Moreover, it does not require the use of mechanical parameters. PLL type estimator is preferred in further study and potential applications due to its simpler structure and independence from mechanical parameters.