An Equivalent Circuit Model for Brushless Doubly Fed ... - IEEE Xplore

An Equivalent Circuit Model for Brushless Doubly Fed Induction Machine Considering Core Loss M. N. Hashemnia, F. Tahami Sharif University of Techno l ogy , Electrical Engineering Department Tehran, Iran [email protected]

Abstract-

Brushless

doubly

fed

induction

machines

show

promising results for wind power applications. Due to their poor rotor magnetic coupling and relatively high value of slip, core loss is an important factor which affects the steady state and dynamic performance. The core loss in brushless doubly fed induction machines has not been extensively studied in the literature. In this paper, a steady state equivalent circuit taking core loss into account is introduced. Simple relationships are derived which show that the brushless doubly-fed induction machine is similar to the cascaded doubly-fed induction machine in terms of core loss. The proposed equivalent circuit is simulated to derive the steady state characteristics of the machine. Keywords- Brushless Doubly Fed Machine; Cascade Doubly Fed Induction Machine; Core loss; Equivalent circuit; Slip

I.

INTRODUCTION

Wind power is one of the renewable energy sources that have attracted most attention for a long time. Due to higher efficiency,variable speed wind turbines are more popular than fixed speed ones. Among various kinds of variable speed wind turbine technologies,Doubly Fed Induction Generator (DFIG) is the most common. DFIGs are capable of decoupled active and reactive power control in both sub-synchronous and super-synchronous speed ranges. The associated inverter only needs be rated for a fractional power of the machine, the fraction depending on rotor slip. Additional advantages expected from these systems include; precise synchronous operation over a wide speed range, adjustable displacement power factor, possibility of working as mains fed induction machine in case of power converter failure,and low harmonic distortion. Potential applications include wind power generation [1,2] and adjustable speed drives (ASDs) [3]. The main disadvantages of DFIGs are their brushes and slip rings which increase maintenance costs and fault rate. Many studies have been carried out in order to develop a machine which combines the great advantages of DFIGs with high reliability and low maintenance. Among other solutions, the use of the so-called Brushless Doubly Fed Machine (BDFM)

(also known as self-cascaded induction machine) could overcome this problem. Although BDFM is slightly larger in size than DFrG due to poor magnetic design,the aforementioned advantage makes it a good candidate for use in wind power generation [4], particularly in offshore wind turbines as well as in pump drives [5]. The first attempts of creating such machine can be traced down to the machine proposed by Hunt (1907) [6] where wound rotor machines were used. But in 1970's, Broadway and Burbridge proposed a new squirrel cage rotor for the BDFIM, the nested loop rotor, which is very similar to the ones used nowadays [7]. There are a lot of papers considering steady state and dynamic modeling of the BDFM. Several investigators have taken a BDFM virtually as the connection of two induction motors with different pole pairs with their rotors electrically and mechanically connected [8-13]. Modeling of this system (known as Cascade Doubly Fed Machine (CDFM» can be undertaken by appropriate connection of two induction motor equivalent circuits. It should be noted that there are two simultaneous stator fields in the same airgap at BDFM while there is one field in each airgap of a CDFM. Modeling of BDFM in its real form is therefore more complex than derivation of the model based on separation of the system into two induction motor subsystems. Although there are some researches on modeling stator and rotor core losses of CDFM, the modeling of core losses of BDFM has not been studied in the literature. Due to existence of rotor field spatial harmonics, poor magnetic design and relatively high rotor electrical frequency (it may be as high as 30 Hz),the core loss of BDFM is more than the conventional induction machines with similar nominal power. Furthermore, time harmonics of the rotor current increase the rotor copper loss. It is thus necessary to consider stator and rotor core loss in equivalent circuit model of BDFM in order to evaluate its steady state behavior with good precision. Moreover,core loss also affects flux and torque dynamic responses which should be taken into account in dynamic modeling if a high performance control is required. Until now,there has been no contribution on modeling core loss of BDFM in its steady

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348

state or dynamic equivalent circuit. The goal of this paper is to propose a conceivable approach for modeling of core loss in BDFM steady state model. II.

adjacent bars (as illustrated in Fig. 2) should be equal in PI pole pair and the P2 pole pair reference frames. Therefore it may be written as [16]:

THEBDFM

(2)

The BDFM comprises of two, three-phase winding sets in the same stator. The winding sets are excited independently and actively participate in the electro-mechanical energy conversion process. To avoid direct transformer coupling between the two windings, their pole numbers should be different. Furthermore, in order to reduce the unbalanced magnetic pull on the rotor,their pole pair numbers difference should be greater than unity [11]: PI * P2 (1) where � is the number of pole pairs of the power winding and P2 is the number of pole pairs of the control winding. As shown in Fig,1,the Power Winding (PW) is connected directly to the grid and therefore works at grid frequency. Most of the power is transferred between the BDFM and grid through this winding. The PW produces a field in the air gap rotating at the grid frequency. The control winding (CW) is connected to the grid via a bidirectional frequency converter and controls the rotor speed and reactive power supplied or absorbed by the machine [14, 15]. The frequency converter usually consists of two back to back voltage source converters. The machine side converter controls the CW current and due to the coupling between the CW and the PW through the rotor circuit,it also controls the PW current. The other converter is connected to the grid (grid side converter) and controls the DC Link voltage.

Hence, ,

N

PI ±P2

(3)

=

q

where Nr is the number of rotor bars. 2;z-

---

�

i�\ \ )

0

�o.....-/

Fig. 2 The location of rotor bars

Few rotor bars result in a very high rotor slot-leakage reactance. The large magnitude of unwanted harmonics also impairs the machine's performance [17]. Therefore, q is set equal to unity and the positive sign in the numerator is chosen:

� = A+�

� It can also be shown that the slot-leakage reactance decreases drastically as the number of rotor slots is increased [17]. Therefore,each rotor nest is distributed in several slots. The most popular structure is called "nested loop" rotor as shown in Fig. 3.

CONTROLLED FREQUENCY fe CONVERTER bidiredionol

froctionol t'Q1ino; �_-+---.

VTlLIT'( SUPPLY

\ \

POWER WINDINGS \ ' Pp pole-pain Fig. 1 The BDFM System

Ill.

The rotor bars are short circuited and there is no brush or slip ring. The number of bars is determined so that an indirect cross coupling between the PW and the CW is produced. To achieve this, the rotor should produce a P2 pole pair field in response to the PI pole pair field produced by the PW; moreover, rotor should produce a PI pole pair field in response to the P2 pole pair field produced by the CWo To satisfY these requirements,the electrical distance between two

CORE LOSS MODELING

In an ordinary squirrel cage induction machine under normal operating conditions the slip is relatively low and therefore the rotor core loss may be neglected. Stator core loss is usually included as part of rotational losses. Some references compensate core loss detuning effect in vector control of an induction motor by taking it into account by two resistances in the d-q circuit model [18]. The iron loss of the

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349

machine is modeled as a resistance in parallel with the magnetizing inductance of each phase. In [19] the iron losses of the OFiM are taken into account in its dynamic model. This could be specifically useful if the control is applied to small power OFIMs where the iron losses cannot be normally neglected, comparing to higher power machines. When the deviation from the synchronous speed is higher,more attention to the power loss should be paid. In [20] a study on core loss of a DFIM with both stator and rotor fed with variable frequency is carried out. A method for correct selection of stator and rotor frequency is given in order to run the motor at an optimum efficiency with different loading conditions. It is also shown that core losses constitute a considerable amount of the total losses and hence should not be neglected in the evaluation of efficiency. Accurate modeling of core losses is very complicated and a simple approach is used here which is adequate for an equivalent circuit model of core loss in BDFM. Neglecting all harmonics, the airgap field due to the two stators can be expressed in a stationary reference frame as: B(B,t)

B,Cos(w,t - p,B)

+

B,Cos(w,t - p,B + Ij/)

(5) where subscripts "1" and "2" refer to power and control windings,respectively, () is the mechanical angle and /f/s is a constant phase shift between the two fields. The airgap field as seen in the rotor reference frame can be simply shown to be as following: =

(6) B, (rp) B,C (w,t + p, rp ) + B,C (w,t + p, rp + /f/) where qJ is the angle measured from a reference frame fixed in the rotor and

())s

is rotor slip frequency.

Rotor core loss:

As there is a single frequency in the rotor circuit of BOFM at synchronous mode of operation (neglecting all harmonics), calculation of core losses in its rotor is much simpler than its stator. Two major components of core loss are classical eddy current and hysteresis loss. Eddy current loss arises from flowing current in the core due to the induced voltage. Hence, Eddy current loss is proportional to the square of this voltage. The induced voltage in the rotor core by the flux passing the airgap can be expressed by: E,(t,rp)

_

dB,(t,rp)

=

=

dt

(7)

B,w,Sin(w,t + p,rp) + B2w,Sin(w,t + p2rp + Ij/,)

The eddy current loss in an element of rotor core at time t and position qJ is thus in proportion to:

2 2 p" (t, qJ) E; (t, qJ) BI ());Sin (())/+ PlqJ) + 2 Bi ());Sin (())i+ P2qJ+/f/r)+ 2BlB2());Sin(())i+ PlqJ)Sin(())/+ P2qJ+/f/r) ex

B,(t,rp)

=

B,Cos(w,t + p,rp)

B,Cos(w,t + p,rp)

+

B2Cos(w,t + p2rp)

+ B 2Cos(m,t +

p,rp + r)

=

(10)

where: (11) r (P2 - pJrp Rotor hysteresis loss in an element of rotor core at time t and position qJ is thus in proportion to: (12) P, oc w,(B,' + B: + 2B,B,Cosy) The total rotor hysteresis loss is calculated by averaging the above relationship. As PI "* P2 ' the average of the third term over r will be zero. It is again apparent that BDFM can be virtually considered as CDFM as long as rotor hysteresis loss is a matter of interest. =

os

os

=

The total rotor eddy current loss is calculated by averaging the above relationship. As the pole pair numbers are chosen unequal, the third term has zero average. The rotor eddy current loss becomes: p. oc (Bl2 ())� + B� ()): ) / 2 (9) The above relationship is worth taking into consideration; as long as calculating rotor eddy current loss is concerned, BDFM resembles CDFM. Rotor hysteresis loss is dependent on rotor frequency and the amplitude of rotor flux density. It has been proved in [21] that core loss is independent of the constant phase shift (/f/r ) between the two fields. It is thus chosen zero for the sake of simplicity. The rotor field can thus be expressed as:

=

(8)

Stator core loss:

The induced voltage in the stator core due to the airgap flux can be expressed as:

dB (t,B) BSin(wt+ pB)+ dt ' " B,Sin(w,t+ p,B+ /f/,) E ,(t,B)

'

=

-

=

(13)

The eddy current loss in an element of stator core at time t and position e is thus in proportion to:

P. (t,B)

oc

E: (t,B)

=

B,'w,'Sin' (w,t+ pB , )+ B:w:Sin' (w,t+ p,B+ /f/)+ 2BB , ,w,w,Sin(w,t+ p,B)Sin(w,t+ p,B+ /f/)

(14)

The total stator eddy current loss is calculated by averaging the above equation. Again, the difference in the number of stator pole pairs forces the third term to become zero after averaging over (). The stator eddy current loss becomes: (15) It means that the two fields are decoupled in stator hysteresis loss, just as rotor eddy current and hysteresis loss. Unfortunately, much more complexity arises when stator hysteresis loss is to be considered; this is mainly due to existence of two fields with different frequencies in the stator of BDFM. This subject has been studied in [21] using the concepts of dissipation and restoring functions. The stator


350

Fig. 4 Proposed model of BDFM including core loss

hysteresis loss due to the two fields can't be decoupled. It this paper, this fact is neglected and it is supposed that all the elements of core loss (eddy current and hysteresis losses of both stator and rotor) can be decoupled with respect to the two stator fields. It should be emphasized that the circuit is nonlinear (otherwise, there would be no core loss) and the principle of superposition can't be generally applied. It is just claimed in this paper that core loss effects of the two stator fields (which are themselves decoupled) can be decoupled. IV.

MODELING CORE Loss IN THE EQUIVALENT CIRCUIT

It was shown in the last section that BDFM can be approximately considered as a CDFM as long as the core loss is the matter of interest. Therefore,the same circuit elements as those of CDFM can be used to model stator and rotor core losses due to power and control windings. In this paper, the core losses are modeled classically by using resistors in parallel with magnetizing branch of power and control windings. As it was shown in [22], two factors should be taken into account when dealing with rotor core loss (this also applies to modeling stator core loss due to control winding); the first one is the dependence of rotor core loss on the variable slip and the second one is the scaling needed to preserve power at the rotor side. For core loss calculation, the slip of rotor and control winding with respect to power winding are important. The justification for rotor core loss of power winding is presented hereinafter. The proof for stator and rotor core loss of control winding is similar. Rotor core loss is proportional to rotor frequency which is equal to power winding frequency multiplied by rotor slip: (16) fr = sri p where f. and Ip are rotor and power winding frequencies respectively and sr is rotor slip relative to power winding. As hysteresis and eddy current losses are proportional to rotor frequency and the square of rotor frequency,respectively,the total rotor core loss can be expected to be approximately in proportion to the absolute value of slip to a power of an exponent coefficient between 1 and 2. In this paper the coefficient is assumed to be 1.3 after [22]. It should be noted that the model should be derived referred to the power winding stator. Therefore a scaling is necessary to refer the

equivalent core loss resistor to the stator side. For this purpose,the airgap power relationship is considered: P

ag

w

=Tw

rotor

.\ync =Ploss, rotor +Pmech =P rotor +PFe, rotor +Twrotor CU,

= (1 � s)w

�ync

�

P

loss, rotor

=P

CU,

+P

rotor Fe, rotor

= .1·rP

(17)

ag

The last relationship shows that when the rotor loss is to be referred to the stator side it should be multiplied by a factor of 1 / Sr . Therefore, the rotor copper resistance should be divided by sr as it is a series element and the rotor core loss resistance should be multiplied by sr as it is a parallel element. The complete steady state model is shown in Fig. 4. A point is worthy of attention regarding the proposed model; The model gives stator core loss of power winding directly, but for the other three resistors, part of its power is transferred to mechanical power while the rest of it shows the corresponding core loss. Consider rotor core loss of power winding as an example. The correct value of power loss is developed in resistor R;; / 1sr 113 . Thus there is some amount of electrical power converted to mechanical power, or vice versa,which is:

31EI21sr 113 (�-1) mech.pr = Rfpre sr

p

(18)

where E is the voltage across the magnetizing reactance of power winding. The validity of this model has been verified experimentally for CDFM in [22] where it has been observed that there is a small torque component estimated at 2-3 Nm with open rotor circuits created only due to the existence of the rotor core losses. It seems that this torque (which is due to rotor core hysteresis and eddy current losses) is developed in a similar manner to hysteresis motors,although hard magnetic cores are used there to amplifY this effect. It is important at this point to note that there will be a component of stator current supplying stator core loss and another component contributing to electrical torque. It is therefore necessary to subtract the stator core loss component from the total stator current in calculating the torque. The same principle applies to the rotor current when rotor core loss in being taken into account in the equivalent circuit model.

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V.

SIMULATION RESULTS

The steady state equivalent circuit in Fig. 4 has been simulated using Matlab® Simulink. The stator power and control windings have 4 and 8 poles, respectively. Other parameters of the machine are also the same as the Cambridge University prototype [7]. In the foregoing simulations, all the core losses of BDFM have been represented by a constant resistance at the power winding side for the sake of simplicity. The value of this resistance can be computed using several steady state measurements, in a manner somewhat similar to the no load test for conventional induction machines. A low value of core loss resistance has been assigned in the simulations so that the two curves can be easily discriminated. Figure 5 shows the simulated torque speed characteristics of the machine when the power winding is fed by 90 V/50HZ voltage and the control winding is short circuited. This operating mode is referred to as cascade induction mode. The circuit has been simulated with and without taking core loss effect into account.

7

___

-l- __

__

___

-1 _

_

1

__ ___

%!;------�=�--��--�,��--�,�=��,�=��,� Rotor Speed (rpm)

Fig. 6 Steady state power winding current magnitude at cascade induction mode with and without core loss

1o,----,--��---,----,----,----r=��

Fig. 7 Steady state rotor current magnitude at cascade induction mode with and without core loss

-10 O ----0,"' 20"'0 ----0,-:- 1 50!;---' --4"" OO,-------- 6""00 -; ,------ --8�00;------;-,;!;; 40"'0 -----; ;! OO" 2""00--� 1600 Rotor Speed (rpm)

Fig. 5 Steady state torque-speed characteristic of BDFM at cascade induction mode with and without core loss

It is evident that there are three speeds at which the developed torque is zero, among which two speeds are independent of the parameters of equivalent circuit,including core loss. The highest of these speeds is equal to synchronous speed of an induction machine with stator frequency of WI and stator and rotor poles equal to PI . The lowest speed is equal to synchronous speed of an induction machine with stator frequency of WI and stator and rotor poles equal to PI + P2 . The middle one is dependant on equivalent circuit parameters. It was observed that changing the value of core loss resistance does not have any effect on this speed. It is also evident that taking core loss into account results in lower absolute values for the developed torque of the machine, in comparison to the same machine without core loss. Power winding and rotor current have also been simulated with and without taking core loss into account. The results are depicted in figures 6 and 7,respectively. It is visible that the core loss effect is generally an increment of power winding current which is due to its core loss component. Moreover,the rotor current in the model with core loss is lower which makes the steady state torque lower in tum.

VI.

CONCLUSION

In this paper an equivalent circuit of Brushless Doubly Fed Machine was introduced which includes core loss. In this model both stator and rotor core losses are taken into account. It was shown that BDFM is similar to CDFM as long as core loss is considered. This is mainly due to the absence of direct coupling between the two stator fields, which is itself guaranteed by proper choice of pole pair numbers of the two stator windings. The proposed model is useful for steady state analysis of BDFMs. There is ongoing research to determine the values of the core loss resistances experimentally and verify the proposed model. REFERENCES [I]

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[2]

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[3]

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