Stator-Current-based MRAS Observer for the ...

Stator-Current-Based MRAS Observer for the Sensorless Control of the Brushless Doubly-Fed Induction Machine Guanguan Zhang*, Jian Yang*, Mei Su*, Weiyi Tang*, Frede Blaabjerg† *School of Information Science and Engineering, Central South University, Changsha, China †Department of Energy Technology, Aalborg University, Aalborg, Denmark [email protected], [email protected], [email protected], [email protected], [email protected] Abstract—Brushless doubly-fed induction machines attract much attention, but the sensorless control method have rarely discussed so far. In this paper, a sensorless control scheme of brushless doubly-fed induction machines is developed based on a simplified mathematical model. The proposed scheme is achieved by designing a model reference adaptive system observer on the basis of the control winding stator current, and a rotor speed estimator is designed by a phase locked loop. In this method, the estimated rotor speed tracks the reference value with satisfactory steady performance and a dynamic tracking response. Moreover, the estimated rotor speed is consistent with the measured value. Experimental results verify the correctness of the proposed method. Keywords—brushless doubly-fed indcution machine;sensorless control; model reference adaptive system observer

I.

INTRODUCTION

Brushless doubly-fed induction machines (BDFMs) [1] are equivalent in function to the doubly-fed induction machines (DFIMs), which are very suitable for applications like Wind Energy Generation systems (WEGs), Marine Propulsion system (MPs) and so on [2], [3]. In order to achieve a good control performance, position encoders are used to obtain rotor position information, which are essential for the coordinate orientation of vector control methods [4-6]. As a result, the system cost and maintenance difficulties increase, and the system reliability decreases. In this case, sensorless control issues of BDFMs deserve to be studied, especially when the BDFMs are working in harsh conditions. Different from brushless doubly-fed reluctance motors (BDFRMs) [7], the physical structure of BDFMs is more complicated. The electromagnetic torque of BDFMs is coupled with the two stator fluxes and the rotor flux [8], thus the conventional sensorless control methods are difficult to utilize directly. In addition, control methods of BDFM are incomplete, only a few studies have been undertaken on the sensorless control schemes [9], [10]. In [9], ignoring the power winding resistance, a sensorless control scheme based on an extended Kalman filter observer is proposed, but the used mathematical model is highly cross-coupled. In [10], a model reference adaptive speed estimator based on the flux linkage of the power winding (PW) is presented, and it is achieved with the assumption that the slip of the power stator is large enough.

However, these methods are only implemented in simulations. In terms of the sensorless control methods, the model reference adaptive system (MRAS) observer scheme is more attractive because of its simplified design rules and implementation [11], [12]. Nevertheless, the complicated BDFM mathematical model is not convenient in order to directly design the MRAS observer. In this case, the simplified mathematical model in [13] provides some ideas for developing the sensorless control based on MRAS for BDFMs. In this paper, a simplified BDFM model is expressed, and a MRAS observer scheme based on the stator current of the control winding (CW) is proposed, where the rotor speed estimator is designed by a phase locked loop (PLL). In the subsynchronous mode and the super-synchronous mode, the estimated rotor speed tracks the reference value well, as well as the measured value in steady state, and a good dynamic performance is achieved when the rotor speed reference changes. Finally, these cases are verified in simulations and experiments. II.

The BDFMs possess two separate stator windings, denoted as power winding (PW) and control winding (CW), and a special designed rotor winding. The power flow in the motor system is realized by properly controlling the CW, and its electrical connection with the grid and the power converter is the same as the DFIM, as shown in Fig. 1. The pole pairs of PW and CW stators are denoted as pp and pc respectively, and the angular frequencies of rotor, PW and CW stators are ωr, ωp and ωc respectively.

This work is supported by the Fundamental Research Funds for the Central Universities of Central South University under Grant 2015zzts057

978-1-5090-5366-7/17/$31.00 ©2017 IEEE

MATHEMATICAL MODEL OF THE BDFM

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Fig. 1. Basic structure of the BDFM system.

For BDFMs, several different operation modes are achieved like the synchronous mode, cascaded mode and induction mode. Among this modes, the synchronous mode is the most desirable, where the rotor speed is usually around ±30 % of the natural synchronous speed and it is expressed as

r 

 p  c

(1)

p p  pc

where a natural synchronous speed is obtained when ωc is zero. The slip of the two BDFM stators are defined as

sp  sc 

 p  p p r p c  pcr c

xc p  e xr p  e

x dq  e

 j p

x p

(3)

where θp is the unified reference frame position in the PW stator, and x represents the electrical variables, such as voltage, current, and flux. The vector notations dq in this frame are removed to simplify the expressions in the following descriptions. As shown in Fig. 2, the CW current direction is adverse to the conventional one, then a negative symbol is introduced here. The transformations from the αβc reference frame and the αβr reference frame to the αβp reference frame are given as



jp p  r  p



  x c   c 

(4)

xr r

B. Mathematical Model In the dq reference frame, the mathematical model based on the space vector representation are expressed as dφp

 j p φp dt φ p  L p i p  M p ir

v p  Rp i p 

(5) (6)





dφ vc  Rc ic  c  j  p   p p  pc  r φc dt φc  Lc ic  M c ir

where sp and sc are the slips of PW and CW stators respectively.

Then, a dq reference frame with pp-type pole-pair distribution is developed as



where θr is the mechanical rotor-shaft angular position, δp, δc are the initial mechanical rotor-shaft angular position of the PW stator and CW stator respectively, and “*” is the conjugate operator.

(2)

A. Transformations between different reference frames Ignoring the harmonic magnetic fields and their coupling effects, the schematic diagram of the BDFM is shown in Fig. 1, which is similar to the cascaded doubly-fed induction motor (CDFM) in inverse coupling mode [14]. The rotor voltages and currents induced by the PW stator flux and CW stator flux are vr, vrαβrc, irαβr and irαβrc respectively, the current directions are described in red color in Fig. 2, and their relationships are given as vrαβr=(vrαβrc)*, irαβr=-(irαβrc)* respectively. For the stator sides, there are two different static reference frames, which are denoted as αβp and αβc, which correspond to the PW stator and CW stator.

 

j p p  r  p +pc  r  c 

dφ 0  Rr ir  r  j ( p  Ppr )φr dt φr =Lr ir  M p i p  M c ic

Te 

3 p p Im  φp  2

 

*

(7) (8) (9) (10)

3   i p   pc Im φc   ic   (11)    2

d r  Te  Br  TL (12) dt where Rp, Lp, Mp, and Rc, Lc, Mc are resistance, self-inductance, mutual inductance of PW stator and CW stator; Rr and Lr are the rotor resistance and rotor self-inductance; vp, vc, ip, ic, φp and φc are voltage, current and flux of PW stator and CW stator; ir and φr are rotor current and rotor flux; Te and TL are the electromagnetic torque and load torque respectively; J and B are the moment of inertia and friction coefficient, respectively. J

C. Simplification of the Mathematical Model To facilitate the analysis, the d-axis is aligned with the PW stator flux vector, and then φpd=|φp|, φpq=0. The electromagnetic torque in (11) is expressed in steady state as

Te 

3M p M c  p p  pc   p



2 Lp Lr  M p

2



icq 

3M p  p p +pc 



2 Lr L p -M p 2



 p rq (13)

Since the voltage drop on PW stator resistor is a small portion of the grid voltage, the PW stator voltage is given as vpd=0, vpq=ωp|φp|. Then, the rotor flux in steady state is

Fig. 2. Schematic diagram of the brushless doubly-fed induction machine.

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calculated by solving the equations (5)-(10) as

rq 

Rr

M p   p  p p r 

p

(14)

From (14), φrq is reduced as ωp-ppωr increases. According to (13), with the parameters of a 30 kW motor prototype, the corresponding torque component caused by the rotor flux is plotted in Fig. 3. It can be seen that the effects are ignorable when the rotor speed is within the range of ±30 % around the natural speed. In this case, the BDFM mathematical model is simplified by assuming φr=0, and the electromagnetic torque and the reactive power of the PW stator are given as

Te 

3M p M c  p p  pc   p

Qp 

2  L p Lr  M p 2 

3Lr v pq  p

2  L p Lr  M p 2 



icq

3M p M c v pq

2  L p Lr  M p 2 

(15)

icd

From (15), Te is directly controlled by icq, while Qp is accordingly adjusted by icd. Thus, the electromagnetic torque and the reactive power of PW stator are independently

controlled by the CW stator current, and the rotor speed is adjusted by generating a suitable electromagnetic torque. Under this control framework, the sensorless control of the BDFM is discussed. The MRAS observer based on the CW stator current is designed, and the control diagram is shown in Fig. 4. III.

DESIGN OF THE SENSORLESS CONTROL SCHEME

Usually, two models are necessary for designing the MRAS observer: the reference model and the adaptive model. The former one is used to provide the reference for the relative variables; while the latter one is adjusted by the estimated rotor speed and rotor position in order to drive the position error to zero. Therefore, a proper mathematical model is indispensable, which affects the observation accuracy and subsequently influence the control performance of the BDFM system. In this paper, the two models are established based on the CW stator current, and the reference model is obtained by the flux equations of the BDFM in (6) and (10), while the adaptive model is constructed by the measured CW stator current and the estimated rotor position information. A. Reference Model Since the PW stator of the BDFMs is directly connected to the power grid, the PW stator voltage and its angular frequency are considered to be constant. Thus, the PW stator flux is estimated by the stator voltage equation (5), where a low pass filter (LPF) replaces the pure integral in order to overcome the zero drift problem [15]. By ignoring the effects of the rotor flux and assuming φr=0 in (10), the CW stator current is calculated from (6) and (10), and it is expressed in the dq reference frame as

ic   Fig. 3. Effects of the rotor flux φr on the electromagnetic torque.

Fig. 4. Sensorless control diagram of the brushless doubly-fed induction machine.

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Lr L p  M p 2 Lr φp  ip McM p McM p

(16)

In (16), the CW stator current is calculated by the estimated PW stator flux and the measured PW stator current and it does not need the rotor position information, then the reference model is selected. B. Adaptive Model From (3) and (4), the measured CW stator current is transformed to the dq reference frame by the estimated angle, and it is expressed as  j  ˆ  iˆc  e p ic c







(17)

where γ is the estimated angle and γ=(pp(θr+δp)+ pc(θr+δc)), the symbol “^” represents the estimated value. C. Implementation of the MRAS Obeserver From (17), the expression of the estimation error is easy to obtain as it is calculated by the cross product between the reference model and the adaptive model, which is given as

Fig. 5. Schematic diagram of the model reference adaptive system observer.

  iˆc  ic  iˆcd icq  icd iˆcq  ic iˆc sin(  ˆ ) (18) where, ε represents the error signal caused by the estimation error , and “  ” is the cross product operator. In (18), i  iˆ c

c

and they are equal to the measured amplitude, then the observer gain is denoted as KM and KM=|ic|2. Usually, ic is nonzero under normal working conditions, thus the estimated angle is close to the true value when ε converges to zero. In this case, the sensorless control is achieved by making the error ε to be zero. The estimation error is denoted as γ and γ=γ- ̑γ, which is small and then satisfy sin(γ)≈γ. In this case, the MRAS observer is designed by a PLL and (18), and the estimator is expressed as

d ˆ e  k1 K M  dt d ˆ =ˆ e  k2 K M  dt

(19)

IV.

SIMULATION AND EXPERIMENTAL RESULTS

In this part, a BDFM system as shown in Fig. 1 is evaluated by the Matlab/Simulink platform and then validated experimentally, BDFM parameters are listed in Table I, and the experimental platform is shown in Fig. 6. The PW stator of the BDFM is directly connected to the power grid (380 V/50 Hz), and the CW stator of the BDFM is fed by a back-to-back PWM converter. In the experiments, the BDFM is mechanically coupled with a 30 kW induction machine that works as a load machine, the MRAS observer and control scheme are implemented in a floating-point DSP controller, and the driver pulses are transmitted to the gate driver of the IGBTs by a FPGA. A. Simulation Verification In order to verify the effectiveness of the proposed method in the sub-synchronous mode and super-synchronous mode of

where ωe represents the angular frequency corresponding to γ, k1 and k2 are the controller parameters. The estimated angle is obtained as the integral of ωe and corrected by the term k2KMγ, while the rotor speed is estimated by ͡ωr=͡ωe/( pp+pc). The estimation method is illustrated in Fig. 5, where the error ε is fed to the input of the PI controller, and the output of PI controller is used to correct the adaptive model and obtain the estimated rotor speed. In this method, the gain KM is a measured quantity, and it affects the estimation performance instead of the machine parameters. For the differential equations in (19), the characteristic polynomial is s2+k2KMs+k1KM=0, and the poles are placed at s=-ρ(ρ>0) in order to obtain robustness and avoid oscillations. Thus, the parameters are selected as k1=ρ2/KM, k2=2ρ/KM, where ρ is selected to be larger than the bandwidth of the speed controller to guarantee the dynamic response of the observer, and the stability analysis can refer to [16].

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Fig. 6. Experimental platform of the BDFM. TABLE I. Rated voltage (V)

THE BDFM PARAMETERS 380

Rated power (kW) Pole-pairs Resistance (Ω) Self-inductance (H) Mutual inductance (H) Self-inductance (H)

30 PW

CW

Rotor

1 0.403 0.710 0.706 0.710

3 0.268 0.0476 0.0462 0.0476

— 0.785 0.760 — 0.760

the BDFM, the reference rotor speed changes from 600 r/min and 800 r/min at t=3 s. The load torque is 100 N∙m, and the reference reactive power of PW stator is 0 Var. Parameters of MRAS observer are set as k1=1.3, k2=0.08, PI parameters of the rotor speed controller and the current controller are kωP =1, kωI =0.3, kiP =1.8 and kiI =15 respectively.

where icd is a constant value in order to keep the reactive power constant, and icq is used to generate a suitable electromagnetic torque. Thus, the estimation performance of the rotor speed is good at the sub-synchronous and super-synchronous operations.

The simulation results are shown in Fig. 7, where the estimated and measured rotor speed, reactive power of the PW stator, dq components of CW stator currents are illustrated from the top to the bottom. It can be seen that the estimated and measured rotor speed tracks the reference value quickly, and the rise time is about 0.5 s. When the reference rotor speed changes, icq increases immediately in order to generate a large electromagnetic torque, and it turns into a steady state after the acceleration process. While icd has a small fluctuation as well as the reactive power, they maintain constant values in steady state. Furthermore, the tracking performance of the rotor speed and the reactive power are good when the BDFM works in steady state. B. Experimental Verification To further investigate the proposed optimal control strategy, an accompanying experiment is carried out. The reference rotor speeds are 600 r/min and 1000 r/min, and experimental results are shown in Fig. 8. The estimated rotor speeds track the reference values and the measured values well, and the speed errors are within 2 % of the reference values, as shown in Fig. 8.a and Fig. 8.d. The reactive power are within ±200 Var, as shown in Fig. 8.b and Fig.8.e. The dq components of CW stator currents are illustrated in Fig. 8.c and Fig. 8.f accordingly,

Fig. 8. Experimental results. (a) ωr and ω ͡ r at 600 r/min. (b) the reactive power at 600 r/min. (c) CW stator currents at 600 r/min. (d) ωr and ω ͡ r at 1000 r/min. (e) the reactive power at 1000 r/min. (f) CW stator currents at 1000 r/min.

Fig. 7. Simulation results when the rotor speed reference changes from 600 r/min to 800 r/min. (a) Measured rotor speed and estimated rotor speed (b) the reactive power. (c) CW stator currents.

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Fig. 9 shows the rotor speed and CW stator currents when the reference rotor speed changes from 600 r/min to 800 r/min. The rotor speed tracks the reference value quickly and the rise time is about 1 s as shown in Fig. 9.a. The reactive power has some fluctuations in the dynamic state as shown in Fig. 9.b, while icq changes accordingly to adjust the electromagnetic torque to increase the rotor speed as shown in Fig. 9.c. Also, icq shows a tendency toward stabilization after the acceleration, while icd has some small fluctuations when the reference rotor speed is changed, but it enters into steady state quickly in order to keep the expected reactive power. Compared with the simulation results, the tracking error of the rotor speed and reactive power are larger in experiments, which are mainly caused by the non-ideal power sources, inaccurate motor parameters and the converter losses. In sum, the correctness of the proposed MRAS observer based on CW stator current is verified, and the control scheme is effective in order to estimate the rotor speed in steady state and dynamic state. V.

CONCLUSION

speed is good when the reference value is changed, and a quick response of the control system is guaranteed. Simulation and experimental results verified the correctness of the proposed method. REFERENCES [1] [2]

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[6]

In this paper, a CW stator-current-based MRAS observer scheme is proposed for the sensorless control of BDFMs. The rotor speed and reactive power of PW stator are adjusted independently, and the estimated rotor speed is consistent with the measured one for the sub-synchronous mode and the supersynchronous mode. The estimation performance of the rotor

[7] [8] [9]

[10] [11]

[12] [13] [14]

[15] [16]

Fig. 9. Measured rotor speed, estimated rotor speed and CW stator currents when the rotor speed is changed from 600 r/min to 800 r/min. (a) ωr and ͡ωr. (b) the reactive power. (c) CW stator currents

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