06388890.pdf

Steady State Analysis of Brushless Doubly Fed Induction Machine Taking Core Loss into Account

M. N. Hashemnia* and F. Tahami Sharif University of Technology, 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 effect on performance of 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 doublyfed induction machine is similar to the cascaded doubly-fed induction machine in terms of core loss. The energy conservation principle is applied to derive relations for steady state torque of the machine when core loss exists. The steady state characteristics of the machine taking core loss into account are simulated and compared with the model without core loss.

Keywords- Brushless Doubly Fed Machine, Core loss, Equivalent circuit, Slip, Steady State, Torque

1. INTRODUCTION

Wind power is one of the renewable energy sources that have attracted most attention for a long time. Variable speed wind turbines are more popular than fixed speed ones due to advantages such as higher efficiency, less mechanical stress on the turbine, better power quality, etc. Among various kinds of variable speed wind turbine technologies, the 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 [1, 2]. The associated inverter only needs be rated for a fractional power of the machine, depending on rotor slip. Another potential application of the Doubly Fed Induction Machines (DFlM) is adjustable speed drive (ASD) [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 DFIG due to poor magnetic coupling of the rotor, 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 state or dynamic equivalent circuit. The goal of this paper is to take this phenomenon into account and investigate its effect on steady state characteristics of the machine.

978-1-4673-2421-2/12/$31.00 2012 IEEE 2030

IT. THEBDFM

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 pair numbers should be different:

(1) where F] is the number of pole pairs of the power winding and P2 is the number of pole pairs of the control winding. Furthermore, in order to reduce the unbalanced magnetic pull on the rotor, their pole pair numbers difference should be greater than unity [11].

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.

Bidirectional Converter

werWinding Po

Fre quency Cf,)

Supply Voltage

Control Winding Frequency (f ,)

l Control Winding (P2 pole pairs)

J-e--i

\

.,', ... ,','"ro,,_,

_'W"", \ \l:) CP, pole pairs) ""

Fig. 1 The BDFM System

The rotor bars are short circuited and there is no brush or slip ring which is an advantage of BDFM over DFIG. 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 adjacent bars (as illustrated in Fig. 2) should be equal in PI pole pair and the P2 pole pair distributions. Therefore it may be written as [16]:

p'()=P2()+2nq, q=I,2, ...

Hence,

N, = P, P,

q where Nr is the number of rotor bars.

o o

'0 ?I/ '-...---.----o-/ Fig. 2 The location of rotor bars

(2)

(3)

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

(4) 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.

Fig. 3 Nested loop rotor [7]

Ill. 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 machine is modeled as a resistance in parallel with the magnetizing inductance of each phase.

In [19] the iron losses of the DFIM are taken into account in its dynamic model. This could be specifically useful if the

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control is applied to small power DFTMs 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.

Tn [20] a study on core loss of a DFTM 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 a fundamental frequency 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(e, t) = B,Cos(w/ - p,e) + B,Cos(w,t - p,e + If/) (5) where subscripts "1" and "2" refer to power and control windings, respectively, () is the mechanical angle and If/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 [7]: B, (1fJ) = B,Cos(w, t + P,IfJ) + B,Cos(w, t + P,1fJ + If/) (6) where cp is the angle measured from a reference frame fixed

in the rotor and OJ.\ is rotor slip frequency.

Rotor core loss: As there is a single frequency in the rotor circuit of BDFM

at synchronous mode of operation (neglecting all harmonics), calculation of core losses in rotor is much simpler than those in stator. Two major components of core loss are classical eddy current and hysteresis loss. The Eddy current loss is proportional to the square of the induced voltage. The induced voltage in the rotor core by the flux passing the airgap can be expressed by:

E,(t,lfJ) = _ dB,(t,lfJ)

=

dt (7)

Bjw,Sin(w/ + pjlfJ) + B2w,Sin(w/ + P21fJ + If/,) The eddy current loss in an element of rotor core at time t

and position cp is thus in proportion to:

(t , 1fJ) oc E ,"(t , 1fJ) = B j2 W,

2Sin

2 (w/ + pjlfJ) +

B:W,2Sin\W/ + P21fJ + If/,)+

2B jB 2w,2Sin(w/ + pjlfJ)Sin(w/ + P21fJ + If/,)

(8)

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: oc (BI2OJ +B:OJ)/2 (9)

The above relationship deserves more 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 proven in [21] that core loss is independent of the constant phase shift (If/ r ) between the two fields. It is chosen zero for the sake of simplicity. The rotor field can thus be expressed as: B, (t, 1fJ) = BjCos(m/ + pjlfJ) + B,Cos(m/ + p,lfJ) =

BjCos(m/ + pjlfJ) + B,Cos(m/ + PjlfJ + y) where:

(10)

(11)

Rotor hysteresis loss averaged by time in an element of rotor core at position cp is thus in proportion to:

p" oc w,CB,' + B: + 2B,B,Cosy) (12) The total rotor hysteresis loss is calculated by averaging the

above relationship. As PI ;;j:. 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.

Stator core loss: The induced voltage in the stator core due to the airgap flux

can be expressed as: dB (t, e)

E (t, e) = - ' = B Sinew t + P e) + , dt ' " (13)

B,Sin(w/ + p,e + Ijf) The eddy current loss in an element of stator core at time t and position () is thus in proportion to:

p" (t, e) oc E,' (t, e) =

(14)

2B,B,w,w,Sin(w,t + p,e)Sin(w/ + p,e + Ijf) 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 eddy current loss, just as rotor eddy current and hysteresis losses. 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 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

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+

Fig. 4 Proposed model of BDFM including core loss

nonlinear due to presence of hysteresis effect (otherwise, there would be no core loss). Therefore, 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 derivation for rotor core loss of power winding is presented hereinafter. The similar procedure can be used for the case of stator and rotor core loss of control winding. Rotor core loss is proportional to rotor frequency which is equal to power winding frequency multiplied by rotor slip:

f =s! (16) r p where f and II' are rotor and power winding frequencies

respectively and S J 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 unity and two. In this paper the coefficient is assumed to be l.3 after [22]. It should be noted that the model is to be referred to the power winding stator side. 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:

l' =1"m =F +1' ag sync !oss, rotor mech

m rotor

l' ag

=(J-s)m ..... 1' sync !OSS, rotor

l' !OSS, rotor

S r

=F +1' +1"m CU, rotor Fe, rotor rotor

=F +1' = s l' ..... (17) r CU, rotor Fe, rotor ag

The last relationship shows that when the rotor loss is 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 S r 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 their 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 power loss is associated with

resistor R;. / I S r 113 Thus there is some amount of electrical power converted to mechanical power, or vice versa, which is:

p _ 3 I E 121 S r 113 -1 mech .pr - R Ie ( ) (18) . S PI' r

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.

V. DERIVATION OF TORQUE EQUATION

In this section, we will derive statements for steady state torque equation of BDFM in consideration of core loss for stator/rotor of power/control windings. Let's derive the general torque equation taking stator and rotor core losses modeled in power and control winding sides.

By applying the principle of power conservation to the equivalent circuit:

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* s 2 * 2 Rr '2 "s 2 "2 3Re(V/l )+3Re(V/2)3Rllfll +3lfr I +3R2lf2 1 + '\1 '\1 '\1 IV 12 IV 121 s (3 IV 121 s 113 IV 121 s 113 3_r1_+3 rl 1 +3 r2 1 +3---,-r,,-2 __ Rfeps Rfeprsl Rfecrsl Rfecss

(19)

At the same time, applying the same principle to the physical system yields: 3 Re(VI/; )+3 Re(V 2/; ) =3R1 1/1 12 +3R 1< 12 +3R; II; 12 +

IV 12 IV 121 s (3 IV 121 s (3 IV 121 s IIJ 3 -r-I -+3 rl I +3 r2 I +3 r2 +Tw

R R R R r jeps jepr jeer jees

(20)

where T is the torque generated by the machine (positive for motoring mode) and slips are defined as following:

WI - plwr SI = WI

S W S = --2. = _ _ 2 SI WI

Subtracting (19) from (20) and after manipulation it can be proved that:

T = 3PILmllm ( (II - Ilep, )(( - II:pJ) + 3 P 21

'm 2 1m ( (I -Ilees )(( + II:C,)

)

(21)

considerable

(22)

This equation has an interesting interpretation; the torque is developed due to the interaction of the two currents the some of which flow in the magnetizing branch (both for power and control windings). These currents are namely II - Ifeps and

I ,: - IlePr for the power winding side and I - Ilees and ( + I;cI for the control winding side.

VI. SIMULATION AND EXPERIMENTAL RESULTS

The steady state equivalent circuit in Fig. 4 has been simulated using Matlab Simulink. The stator power and control windings are both delta connected and have 4 and 8 poles, respectively. Other parameters of the machine are also the same as the Cambridge university prototype which have been summarized in Table I.

TABLE L BDFM ELECTRICAL PARAMETERS

PW CW Rotor (referred to

PW) Resistance 7.28 4.81 5

(0) Self 17.1 15.2 99.4

Inductance (mH)

Magnetizing 1125 333 -Inductance

(mH)

Tn the foregoing simulations, two resistances out of four, standing for stator core loss have been assumed in the equivalent circuit model. There is therefore one constant resistance at the power winding side and a resistance dependant on control winding slip relative to the power winding at the control winding side. The values of these two resistances have been identified using the experimental results of the simple induction mode of BDFM operation, where one winding is energized by supply voltage and frequency and the other one is left open. The values used for stator power winding and stator control winding core loss resistances are 1400 and 1800 ohms respectively. Figure 5 shows the simulated torque speed characteristics of the machine when the power winding is fed by 200 V 150HZ 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.

It is evident that the developed torque is zero at three rotor speeds, among which two speeds are totally independent of the parameters of the equivalent circuit, including core loss. The highest speed is equal to the synchronous speed of an induction machine with stator frequency of OJI and stator and rotor poles equal to PI . The lowest speed is equal to the synchronous speed of an induction machine with stator frequency of OJI and stator and rotor poles equal to PI + P2 . The middle one is generally dependent on equivalent circuit parameters. It was observed that changing the value of core loss resistance does not affect this speed. It is also evident that taking core loss into account results in lower absolute values for the developed torque of the machine (for the same speed), in comparison to the same machine without core loss.

The power winding line current 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 turn. It should be clarified that no measurements were available for rotor current and therefore just the simulation results have been shown.

30,----,----,----,----,----,-----.========;] -Simulated (No Core Loss) -Simulated (With Core Loss) 20i .y-:l\r t r f-'* ... E"'E' '"""me""'m a"-' -,--_Jj

E , : :oS rj' ., ......... .:ir-= .... , ......... + .. "-: ... , .......... -1 Ql : : -10 --------+--------+-----\ ... , ....... /1;/ ....... , ........... , ......... , .......... ':: "

: ::j:j -40 200 400 600 800 1000 1200 1400 1600 1800

Rata r Speed (R pm)

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

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12

10 :5: c 8 8 '" 6 c '6

4

"-

DO 1800

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

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

VII. CONCLUSION

In this paper an equivalent circuit of Brushless Doubly Fed Machine including both stator and rotor core losses was proposed. 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. Torque equation was derived based on the energy conservation principle.

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[18] E. Levi, "Impact of iron loss on behaviour of vector controlled induction machines", 1994 IEEE Industry Applications Society Annual Meeting, Oct 1994, pp. 74.

[19] GONZALO ABAD BLAIN, Predictive Direct Control Techniques of the Doubly Fed Induction Machine for Wind Energy Generation Applications, PhD Thesis, University of Mondragon, July 2008.

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