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Complex Vector Model of the Brushless Doubly Fed Machine in Unified Reference Frame S. ATALLAH D. BENATTOUS M.-S. NAIT-SAID Laboratoire des Systèmes Propulsion-Induction Electromagnétiques LSP-IEโ2000 Batna EL-Oued, Algeria [email protected] Institute of Science Technology, University Center of El-Oued El-Oued, Algeria [email protected] Laboratoire des Systèmes PropulsionInduction Electromagnétiques LSPIEโ2000 Batna, Algeria [email protected] AbstractโThe Brushless Doubly Fed Machine (BDFM) is a machine which incorporates the robustness of the squirrel cage induction machine and the speed and power factor control of a synchronous machine. In this paper, a detailed mathematical derivation of the BDFM unified dโq reference frame model is introduced. The model is based on coupled magnetic circuit theory and complex space-vector notation. Then the obtained dynamic model of the machine will be finally tested in simulation workbench using MATLAB/SIMULINK in order to verify the proposed model. This way, a simple dq model can be established, which could be an interesting tool for control synthesis tasks. Keywords-Brushless Doubly Fed Machine (BDFM), Cross Coupling, Control Winding (CW), Power Winding (PW), Variable Speed Generation, Unified Reference Frame Model I. NOMENCLATURE LIST OF SYMBOLS ๐๐ (๐๐ ): ๐โ๐ (๐โ๐ ): PW (CW) frequency. ๐ ๐ : Stator PW (CW) current vector. Rotor current vector. Stator PW (CW) self-inductance. Rotor self-inductance. Unified frame stator PW (CW) to rotor coupling inductance. Number of rotor nests. Power (control) winding pole pairs Number. Real part. Stator PW (CW) resistance. Rotor resistance. ๐๐๐๐ (๐๐๐๐ ): PW (CW) electromagnetic torque. ๐๐๐ : โโ๐ (๐ โโ๐ ): ๐ Total electromagnetic torque. ๐๐ {โฆ }: Imaginary part. ๐๐๐๐ ๐ (๐๐๐๐ ๐ ): Angle between the PW (CW) and the generic reference ๐โ๐ : ๐ฟ๐ (๐ฟ๐ ): ๐ฟ๐ : ๐๐ (๐๐ ): ๐๐ : ๐๐ (๐๐ ): โ๐ {โฆ }: ๐ ๐ (๐ ๐ ): Stator PW (CW) fed voltage vector. ๐๐ ( ๐๐ ): frame. Rotor shaft displacement between the rotor and the PW reference axis. Synchronous angular frequency of the PW (CW). ๐๐๐ : Angular slip speed of the PW. ฮด: axis. Initial angle between the rotor and the PW references ๐๐ : ๐ โโ๐ (๐ โโ๐ ): ๐ โโ๐ : ฮฉ: ๐พ: Stator PW (CW) flux linkage vector. Rotor flux linkage vector. Rotorโs mechanical angular speed. Angle between the PW and the CW references axis. SUBSCRIPTS p, c, r: ๐๐ (๐๐ ): Power winding, control winding, rotor. Stator power (control) winding phase. SUPERSCRIPTS โ: ๐๐: ๐๐๐ : Complex conjugate. The direct and quadrant component on the power winding flux frame. Generic reference frame of ๐๐ -pole pairs. ๐๐๐ : ๐ฅ๐ฆ๐ : ๐ผ๐ฝ๐ : ๐ผ๐ฝ๐ : Generic reference frame of ๐๐ -pole pairs. Rotor reference frame. Control winding reference frame. Power winding reference frame. ACRONYMS BDFM: CW: PW: WRIM: Brushless Doubly Fed Machine. Control winding Power winding. Wound Rotor Induction Machine. II. INTRODUCTION When wind power generator is connected to the power grid, the output frequency should be identical with the frequency of the power grid. Wind energy capturing and conversion efficiency can be improved by taking advantage of variablespeed constant-frequency (VSCF) method which uses the Wound Rotor Induction Machine (WRIM, old material) [1]. But the main problem is that the slip rings and wound rotor arrangement which limit its application in harsh environment. Among the possible solutions for these shortcomings is the introduction of the so-called Brushless Doubly Fed Machine (BDFM), which can be seen as an advanced version of the (WRIM), because it is based on the same principle of the slip energy recovery used for the output control [2]. Fig.1. BDFM system The Brushless Doubly-Fed Machine (BDFM) has the potential to be employed as a variable speed generator such as in wind power applications or as a motor in adjustable speed drives [1] [2] [4]. The BDFM consists of two electrically independent balanced three phase windings which have no mutual couplings and wound on the same core in the stator. The rotor circuit in BDFM is considered as a nested-loop type which couples to both fields of the stator and is actually the most well known one for BDFM. It consists of nests equally spaced around the circumference whose number is equal to the sum of the stator windings pole pairs Fig 1 [3]. A BDFM model was derived assuming that the machine was composed of two superposed subsystems [5] [6]. Each subsystem contained the dynamics of one of the two stator windings (PW or CW) and the corresponding rotor dynamics. The set of equations of the PW or CW subsystem were written in two different synchronous reference frames related to each pole-pair distribution. This leads to a couple of equations describing the dynamics of two independent rotor currents which correspond to two different synchronous reference frames. The electromagnetic torque depends on the current and the flux of both subsystems, as well as the so-called โsynchronous angleโ between the two reference frames. But the existence of multiple reference frames related to the two stator windings and the rotor makes it difficult to exploit the well known standard induction machine control strategies. The main aim of this paper is to develop a mathematical dynamic model for the BDFM based on the complex space vector notation, leading to a unified dq reference frame model. III. CONCEPT OF THE CROSS COUPLING EFFECT For the BDFM the major interest is the operation in synchronous mode, the essential feature of synchronous operation is the electromagnetic coupling of one stator winding system with the other, exclusively through the rotor. Since the stator windings which can be assumed to be sinusoidally distributed for different pole-pair numbers, there is (intentionally) no direct coupling between both stator windings [7]. However, each stator winding can be coupled directly with the rotor. The induced rotor currents from both stator windings should be have an appropriate sequence and frequency. It results that the rotor current creates the appropriate fields Fig.2. BDFM coupling mechanism schematic which can induce voltages in the power windings (stator) initially due to control windings currents, and vice versa. This indirect induction mechanism is referred to as cross coupling effect which is illustrated in Fig 2 [8]. This concept development assumes a linear magnetic circuit and deals solely with synchronous operation of the BDFM in which the two windings act in a complementary manner. Asynchronous operation, in which the system consists of two conflicting induction machines, is undesirable and is avoided through proper control of the control winding voltage (or current). When excited, each of the stator winding systems produces a traveling flux wave in the airgap. Each of these can be expressed in the form, ๐๐ (๐, ๐ก) = ๐ต๐ ๐๐๐ (๐๐ ๐ก โ ๐๐ ๐) ๐๐ (๐, ๐ก) = ๐ต๐ ๐๐๐ (๐๐ ๐ก โ ๐๐ ๐) (1) (2) In order to obtain the desired cross-coupling effect, the PW and the CW currents induce at the rotor bars must evolve with the same frequency [2]. This operating restriction leads to the so-called synchronous rotor speed, which is equal to, ๐บ= ๐๐ +๐๐ (3) ๐๐ +๐๐ The stator produced flux densities can be written in terms of a particular rotor observer angle, p, and time by subsisting (4) into (1) and (2). ๐ = ๐บ๐ก + ๐ฬ (4) Which yields, ๐๐ (๐, ๐ก) = ๐ต๐ ๐๐๐ [๐๐๐ ๐ก โ ๐๐ ๐ฬ] ๐๐ (๐, ๐ก) = ๐ต๐ ๐๐๐ [๐๐๐ ๐ก โ ๐๐ ๐ฬ ] (5) (6) In which, ๐๐๐ = ๐๐ ฯ๐ โ๐๐ ฯ๐ ๐๐ +๐๐ = โ๐๐๐ (7) Thus, the effect of the stator operating at the speed given by (7) which is the mean of the two stator-produced field speeds is to produce flux velocities which are equal but in opposite directions when viewed from the rotor. These flux waves can be viewed collectively according to (8) where ๐๐1 1 ๐โ๐๐ โฆ โฆ ๐ โ(๐โ1)๐๐ ๐๐2 ๐ โ๐๐๐ (๐๐ +๐ฟ) ( ๐(1 ๐โ๐๐ โฆ โฆ ๐โ(๐โ1)๐๐ ))} โฎ โฎ ๐2 (1 ๐โ๐๐ โฆ โฆ ๐โ(๐โ1)๐๐ ) (๐๐๐ ) ฯrp = ฯr , ๐๐ (๐, ๐ก) = ๐ต๐ ๐๐๐ [๐๐ ๐ก โ ๐๐ ๐ฬ] + ๐ต๐ ๐๐๐ [๐๐ ๐ก + ๐๐ ๐ฬ] IV. (8) DYNAMIC MODEL OF THE BDFM 2๐ The objective of this section is to develop the unified dq reference frame model of the BDFM based on the space vector notation. A. Stator Model The PW flux linkage can be written as the contribution of three components as, Sp Sp Sp (9) ฯSp = ฯSp + ฯSc + ฯR The first term can be expressed as, ๐๐ ๐๐๐1 ๐๐ ๐๐๐ ๐๐ ๐๐ = (๐๐๐2 ) = (๐๐ ๐๐ ๐๐ ๐ ๐๐3 ๐๐ ๐๐ ๐๐ 0 ๐ฟ๐ 0 (10) 0 ๐๐๐1 ๐๐๐1 0 ) (๐๐๐2 ) = ๐ฟ๐ (๐๐๐2 ) ๐๐๐3 ๐๐๐3 ๐ฟ๐ (11) โฆโฆ โฆโฆ โ ๐๐ [(๐๐ + ๐ฟ) + ๐ผ๐ ]] โฆ โฆ โ ๐๐ (๐๐ + โฆโฆ โ ๐๐ (๐๐ + ๐ฟ)] cos [2ฯ โ ๐ [(๐ + ๐ฟ) + ๐ผ ]] ๐ ๐ ๐ โฆโฆ 3 ๐ ๐1 [(๐ (๐ ] cos ๐๐ ๐ + ๐ฟ) + โ 1)๐ผ๐ ๐๐2 cos [2ฯ โ ๐๐ [(๐๐ + ๐ฟ) + (๐ โ 1)๐ผ๐ ]] โฎ 3 โฎ [(๐ (๐ ]] cos [4ฯ โ ๐ + ๐ฟ) + โ 1)๐ผ ๐ ๐ ๐ 3 ) (๐๐๐ ) 2ฯ ๐0๐ (cos[ 3 4ฯ cos[ 3 ๐๐ ๐๐ = cos ๐๐ [(๐๐ + ๐ฟ) + ๐ผ๐ ] ๐0๐ ๐๐ (๐ +๐ฟ) { ๐ ๐ ๐ [1 ๐๐๐ ๐2๐๐ โฆ โฆ ๐(๐โ1)๐๐ ] + 2 ๐๐1 ๐๐2 โ๐ โ2๐๐ ๐ โ๐๐๐(๐๐ +๐ฟ)[1 ๐ ๐ ๐ โฆ โฆ ๐โ(๐โ1)๐๐ ]} โฎ โฎ (๐๐๐ ) ๐๐ ๐0๐ 2 (๐โ1)๐๐ 1 ๐ โฆโฆ ๐ { ๐๐๐๐ (๐๐ +๐ฟ) (๐2 (1 ๐๐๐ โฆ โฆ ๐(๐โ1)๐๐ )) + ๐(1 ๐๐๐ โฆ โฆ ๐(๐โ1)๐๐ ) (16) yields, ๐ โโ๐ ๐ = ๐ฟ๐ ๐โ๐ ๐ + ๐พ ๐ โโ๐ ๐ = ๐ฟ๐ ๐โ๐ ๐ + ๐พ ๐ 2 ๐ ๐ฅ๐ฆ๐ ๐0๐ ๐ ๐๐๐ (๐๐ +๐ฟ) ๐โ๐ ๐ฅ๐ฆ๐ ๐ ๐ ๐๐๐ [(๐๐ +๐ฟ)โ๐พ] ๐โ๐ 2 0๐ (17) (18) The power and control windings voltage equation may be done as followed, (19) Where, ๐โ๐ ๐ = 23(๐๐๐1 + ๐๐๐๐2 + ๐2 ๐๐๐3 ) ๐ฅ๐ฆ๐ (12) ๐โ๐ = 2 ๐พ๐ (๐๐1 + ๐๐2 ๐๐๐ + โฏ + ๐๐๐ ๐(๐โ1)๐๐ ) ๐โ๐ ๐ = 23(๐๐๐1 + ๐๐๐๐2 + ๐2 ๐๐๐3 ) ๐ฅ๐ฆ๐ ๐โ๐ = 2 ๐๐ (๐๐1 + ๐๐2 ๐๐๐ + โฏ + ๐๐๐ ๐(๐โ1)๐๐ ) (13) (20) (21) (22) (23) B. Rotor Model The rotor flux can be divided into three components, ๐ ๐ ๐๐ = ๐๐ ๐ + ๐๐๐ + ๐๐๐ In the same way for the second and third components, results, ๐๐ = (15) 1 ๐๐๐ โฆ โฆ ๐(๐โ1)๐๐ ๐๐๐1 ๐0๐ 2 [(๐ ๐๐ +๐ฟ)โ๐พ] ๐ ๐ ๐๐๐ = ๐ฟ๐ (๐๐๐2 ) + {๐ (๐ (1 ๐๐๐ โฆ โฆ ๐(๐โ1)๐๐ )) + 2 ๐๐๐3 ๐(1 ๐๐๐ โฆ โฆ ๐(๐โ1)๐๐ ) ๐๐1 1 ๐โ๐๐ โฆ โฆ ๐โ(๐โ1)๐๐ ๐๐2 ๐ โ๐๐๐[(๐๐ +๐ฟ)โ๐พ] ( ๐(1 ๐โ๐๐ โฆ โฆ ๐โ(๐โ1)๐๐ ))} โฎ (16) โฎ ๐2 (1 ๐โ๐๐ โฆ โฆ ๐โ(๐โ1)๐๐ ) (๐๐๐ ) ๐๐ก ๐๐ ๐1 = ๐๐ ๐๐๐1 1 ๐๐๐ โฆ โฆ ๐(๐โ1)๐๐ ๐0๐ ๐๐๐(๐๐ +๐ฟ) ๐ 2 (1 ๐ ๐๐ โฆ โฆ ๐ (๐โ1)๐๐ ) ๐ ๐๐๐ = ๐ฟ๐ ( ๐๐2 ) + {๐ ( )+ 2 ๐๐๐3 ๐(1 ๐๐๐ โฆ โฆ ๐(๐โ1)๐๐ ) ๐๐1 1 ๐โ๐๐ โฆ โฆ ๐โ(๐โ1)๐๐ ๐๐2 โ๐ โ(๐โ1)๐ ๐) ๐ โ๐๐๐ (๐๐ +๐ฟ) ( ๐(1 ๐ ๐ โฆ โฆ ๐ )} โฎ โฎ ๐2 (1 ๐โ๐๐ โฆ โฆ ๐โ(๐โ1)๐๐ ) (๐๐๐ ) โโโ๐ ๐,๐ ๐ โโ๐ ๐,๐ ๐ = ๐ ๐ ๐,๐ ๐ ๐โ๐ ๐,๐ ๐ + ๐๐ ๐ 2ฯ ๐ฟ)] cos [ 3 So we can write, for example, for the first component of last flux vector, ๐ So (9) becomes, Applying the three-phase space-vector definition to (15) The second term equal zero because the two stator winding sets have different numbers of poles (pp โ pc ) [8]. and the last term is the contribution of the rotor current the cage rotor having n rotor nest can be assumed as a system of n phases [9]. The total stator flux linkage due to the rotor currents can be derived as, cos ๐๐ (๐๐ + ๐ฟ) Where, ๐ = ๐๐๐ผ๐ , ๐ผ๐ = 2๐ , ๐ = ๐๐ 3 ๐ In the same manner for the CW flux linkage, we can write, ๐๐ ๐๐๐1 ๐๐ ) (๐๐๐2 ) ๐๐๐3 ๐๐ So, (10) becomes, ๐ฟ๐ ๐๐ ๐๐๐ = ( 0 0 (14) (24) One due to the rotor currents and two due to the PW and CW currents, more detailed explanations of each term will be developed separately in the following subsections. Rotor flux linkage in the rotor nest due to rotor nest currents ๐๐ ๐ ๐11 ๐21 =( โฎ ๐๐1 ๐12 โฆ โฆ ๐1๐ ๐๐1 ๐ ๐22 โฆ โฆ ๐2๐ ) ( ๐2 ) โฏ โฆโฆโฆ โฎ โฎ โฏ โฆ โฆ ๐๐๐ ๐๐๐ (25) Applying the space-vector theory definition to (25) by multiplying the first term of by (1, b Pp , โฆ , b (nโ1)Pp ) yields the following serial equations, ๐ ๐๐ 1 ๐ ๐ ๐๐ 2 ๐11 ๐ ๐2๐๐ ๐๐ 3 ๐ = {(1, ๐๐๐ , ๐2๐๐ , โฆ , ๐(๐โ1)๐๐ ) ( 21 โฎ โฎ โฎ ๐๐1 โฎ ๐ (๐ (๐โ1)๐๐ ๐๐ ๐ ) ๐๐ ๐12 โฆ โฆ ๐22 โฆ โฆ โฏ โฆโฆโฆ โฏ โฆ โฆ ๐๐1 ๐๐2 ๐1๐ ๐๐3 ๐2๐ )} โฎ โฎ โฎ ๐๐๐ โฎ (๐๐๐ ) With (1 + b 2Pp + โฏ + b 2(nโ1)Pp = 0) , the rotor nests flux linkage due to the power winding current becomes, ๐ ๐ โโ๐๐ = 3 ๐0๐ 2 ๐พ ๐ผ๐ฝ๐ ๐ โ๐๐๐ (๐๐ +๐ฟ) ๐โ๐ ๐ Flux linkage in the rotor nest due to control winding currents (26) Using the identity formulation ๐(๐+๐)๐๐ = ๐๐๐๐ , we can get ๐ ๐๐ 1 ๐11 ๐ ๐๐๐ ๐๐ 2 ๐21 ๐ ๐31 ๐2๐๐ ๐๐ 3 = (1, ๐๐๐ , ๐2๐๐ , โฆ , ๐ (๐โ1)๐๐ ) โ โฎ โฎ โฎ โฎ โฎ โฎ (๐๐1 ) ๐ (๐ (๐โ1)๐๐ ๐๐ ๐ ) ๐๐1 ๐๐2 ๐๐3 (๐โ1)๐ ๐) (1, ๐๐๐ , ๐2๐๐ , โฆ , ๐ โฎ โฎ โฎ { (๐๐ใ ณ )} And therefore, ๐ฅ๐ฆ๐ ๐ โโ๐ ๐ = ๐ฟ๐ ๐โ๐ (28) With, ๐11 ๐21 ๐31 ๐ฟ๐ = (1, ๐๐๐ , ๐2๐๐ , โฆ , ๐(๐โ1)๐๐ ) โฎ โฎ โฎ (๐๐1 ) = ๐11 + ๐๐๐ ๐12 + ๐2๐๐ ๐13 + โฏ + ๐(๐โ1)๐๐ ๐1๐ (29) Flux linkage in the rotor nest due to power winding currents 1 1 1 ๐๐๐1 ๐0๐ ๐๐๐ ๐๐๐ ๐๐๐ ๐๐๐ (๐๐ +๐ฟ) = {๐ ( ) (๐2 ๐๐๐2 ) + โฎ โฎ โฎ 2 ๐๐๐๐3 ๐(๐โ1)๐๐ ๐(๐โ1)๐๐ ๐(๐โ1)๐๐ 1 ๐ โ๐๐๐ (๐๐ +๐ฟ) ( ๐ 1 ๐โ๐๐ โฎ โ(๐โ1)๐๐ ๐โ๐๐ โฎ ๐ โ(๐โ1)๐๐ ๐ (30) Applying the space-vector theory definition to (30) and after multiplying each component of rotor flux successively by (1, b Pp , b 2Pp , โฆ , b (nโ1)Pp ), we can get, ๐ 1 ๐๐๐ 1 1 1 ๐ 2 ๐๐๐1 ๐ ๐๐๐ ๐0๐ ๐2๐๐ ๐ 2๐๐ ๐2๐๐ = { ๐ ๐๐๐ (๐๐ +๐ฟ) ( ) (๐2 ๐๐๐2 ) + โฎ โฎ โฎ โฎ 2 ๐๐๐๐3 โฎ ๐2(๐โ1)๐๐ ๐2(๐โ1)๐๐ ๐2(๐โ1)๐๐ (๐โ1)๐๐ ๐ ๐ ๐ ๐ ( ๐๐ ) ๐๐ ๐ โ๐๐๐ (๐๐ +๐ฟ) ( 1 1 โฎ 1 1 1 โฎ 1 1 ๐๐๐1 1 ) ( ๐๐๐๐2 )} โฎ ๐2 ๐๐๐3 1 2 {๐ ๐๐๐ [(๐๐ +๐ฟ)โ๐พ] โ๐๐๐ [(๐๐ +๐ฟ)โ๐พ] 1 1 1 ๐๐๐1 ๐๐๐ ๐๐๐ ๐๐๐ ( ) (๐2 ๐๐๐2 ) + โฎ โฎ โฎ ๐๐๐๐3 ๐(๐โ1)๐๐ ๐(๐โ1)๐๐ ๐(๐โ1)๐๐ 1 1 1 ๐๐๐1 ๐โ๐๐ ๐โ๐๐ ๐โ๐๐ ( ) ( ๐๐๐๐2 )} โฎ โฎ โฎ ๐2 ๐๐๐3 ๐โ(๐โ1)๐๐ ๐โ(๐โ1)๐๐ ๐โ(๐โ1)๐๐ (33) The calculation of this magnetic coupling effect is vital to determine the machine operation, since its existence produces a cross coupling being well indicated by Fig.2 between both stator windings through the rotor. Once the cross coupling is produced, the current of each stator winding will not solely depend on its own supply voltage, but it will also vary according to the voltage of the other stator winding. On the other hand, if the cross coupling does not produce the electrical machine would operate like two independent asynchronous machines with the same axis. ๐ 1 ๐๐๐ 1 1 1 ๐ 2 ๐๐๐1 ๐๐๐ ๐๐๐ ๐0๐ ๐๐๐ +๐๐ ๐๐๐ +๐๐ ๐๐๐ +๐๐ = { ๐ ๐๐๐ [(๐๐ +๐ฟ)โ๐พ] ( ) (๐2 ๐๐๐2 ) + โฎ โฎ โฎ โฎ 2 ๐๐๐๐3 โฎ ๐ (๐โ1)(๐๐ +๐๐ ) ๐ (๐โ1)(๐๐ +๐๐ ) ๐ (๐โ1)(๐๐ +๐๐ ) ๐ ๐ (๐(๐โ1)๐๐ ๐๐๐ ) ๐ โ๐๐๐ [(๐๐ +๐ฟ)โ๐พ] ( 1 ๐๐๐ โ๐๐ โฎ 1 ๐๐๐ โ๐๐ โฎ ๐(๐โ1)(๐๐ โ๐๐ ) ๐(๐โ1)(๐๐ โ๐๐ ) ๐ 1 ๐๐๐ โ๐๐ โฎ ๐๐๐1 ) ( ๐๐๐๐2 )} ๐2 ๐๐๐3 (๐โ1)(๐๐ โ๐๐ ) (34) As shown in (34), the rotor flux vector due to CW currents depends on the selected values of pp and pc pole-pairs. By analyzing different combinations, there are two possible cases [8], ๏ Possibility 1 ๐ ๐ โโ๐๐ = 0 โน ๐ (๐๐ +๐๐ ) & ๐ (๐๐ โ๐๐ ) โ 1 โน Inexistence of a cross 1 ๐๐๐1 ๐โ๐๐ )} ( ๐๐๐๐2 ) โฎ ๐2 ๐๐๐3 โ(๐โ1)๐๐ = ๐0๐ From (33), after multiplying each component of rotor flux successively by (1, b Pp , b 2Pp , โฆ , b (nโ1)Pp ) , which defines the cross coupling, we can obtain, The proportionality constant Lr corresponds on the equivalent rotor self inductance. Note that its value is expressed only in terms of rotor nestโs dimension. ๐ ๐๐๐ ๐ ๐๐๐ ๐ (27) (32) (31) coupling between the both stator windings through the rotor current. ๏ Possibility 2 ๐ ๐ โโ๐๐ โ 0 โน ๐ (๐๐ +๐๐ ) ๐๐ ๐ (๐๐ โ๐๐ ) = 1 โน The existence of a cross coupling between the two stator windings through the rotor current. There are two possible configurations, 2๐ ๏ท configuration1: ๐(๐๐โ๐๐) = ๐ ๐ ๐ (๐๐โ๐๐) = 1 โน ๐ = ๏ท configuration2: ๐(๐๐+๐๐) = ๐ 2๐ ๐ ๐ (๐๐+๐๐ ) Where, ๐ = 0, ±1, ±2 โฆ โฆ โฆ โฆ โฆ =1โน ๐= (๐๐โ๐๐ ) ๐ (๐๐ +๐๐ ) ๐ (35) (36) So, to ensure this cross coupling effect, we should chose the second configuration (36) which maximizes the number of rotor nestโs i.e (๐ = 1). So, ๐ = ๐๐ + ๐๐ implying that, ๐ฅ๐ฆ๐ ๐โ๐ 2 = ๐๐ (๐๐1 + ๐๐2 ๐๐๐ + โฏ + ๐๐๐ ๐(๐โ1)๐๐ ) (37) We know that ๐๐ = ๐ โ ๐๐ โน ๐ฅ๐ฆ๐ ๐โ๐ 2 = ๐๐ = = So, (๐๐1 + ๐๐2 ๐๐โ๐๐ + โฏ + ๐๐๐ ๐(๐โ1)(๐โ๐๐) ) 2 ๐๐ (๐๐1 + ๐๐2 ๐ โ๐๐ + โฏ + ๐๐๐ ๐ (๐โ1)(โ๐๐ ) ) โ๐ฅ๐ฆ ๐โ๐ ๐ ๐ฅ๐ฆ ๐โ๐ ๐ = โ๐ฅ๐ฆ ๐โ๐ ๐ (38) So we conclude that, in last configuration, one of the current vectors behaves as the conjugate of the other. According to this relation, it becomes straightforward to change from a pp โ type reference frame to a pc โ type one or vice versa. This one constitutes the key step for the derivation of the unified ๐๐ reference frame model. Replacing with ๐ = ๐๐ + ๐๐ in (34), yields ๐ 1 ๐๐๐ 1 1 ๐ 2 ๐๐๐ ๐๐๐ ๐ 1 1 = 0๐ { ๐ ๐๐๐[(๐๐ +๐ฟ)โ๐พ] ( โฎ ๐ โฎ โฎ โฎ 1 1 (๐โ1)๐๐ ๐ ๐ ๐๐๐ ) (๐ 1 ๐๐๐1 1 ) (๐2 ๐๐๐2 )} โฎ ๐๐๐๐3 1 (39) And so, ๐ ๐ โโ๐๐ = ๐ ๐0๐ ๐ ๐ฒ โ๐ผ๐ฝ๐ ๐ ๐๐๐[(๐๐ +๐ฟ)โ๐พ] ๐โ๐ ๐ (40) ๐๐ , ๐๐ pole-pairs (45) ๐0(๐,๐) As it can be observed the initial set of (44) is referred to three different reference frames and tow possible pole-pairs distributions may be considered, the goal is to get a set of equations with a unified reference frame with a given polepairs distribution (e.g. ๐๐ ) which form the main aim of the follows part. V. UNIFIED REFERENCE FRAME MODEL OF THE BDFM We can easily to write the previous system in a unified reference frame model if we followed the steps given in appendix IX.1. By means of these vector transformations the machine model (44) is expressed in a common dq-generic reference frame where dq symbol indices have been removed to simplify resulting expressions which are given as follows, ๐๐ก ๐๐ก ๐๐ก 3 ๐0๐ 2 ๐พ ๐ผ๐ฝ๐ ๐ โ๐๐๐ (๐๐ +๐ฟ) ๐โ๐ ๐ + 3 ๐0๐ 2 ๐พ โ๐ผ๐ฝ๐ ๐ ๐๐๐[(๐๐ +๐ฟ)โ๐พ] ๐โ๐ ๐ (41) ๐ 3 ๐0(๐,๐) 2 2 ๐พ (42) The normalizing gain is identified as, 3 (43) ๐ ๐ โโ๐ = ๐ฟ๐ ๐โ๐ + ๐๐ ๐โ๐ โโโ๐ โโ๐ = ๐ ๐ ๐โ๐ + ๐๐ ๐ + ๐[๐๐๐๐ ๐ โ ๐๐ ฮฉ]๐ โโ๐ (46) ๐๐ก In (17), (18) and (41) in order to obtain the same equivalent mutual inductance from rotor to stator as from stator to rotor, the following constraint must be fulfilled, ๐พ=โ 2 a ๐๐ pole-pairs โโโ๐ โโ๐ = ๐ ๐ ๐โ๐ + ๐๐ ๐ + ๐[๐๐๐๐ ๐ โ (๐๐ + ๐๐ )ฮฉ]๐ โโ๐ โโโ๐ โโ๐ = ๐ ๐ ๐โ๐ + ๐๐ ๐ ๐พ ๐0(๐,๐) = โ3๐ a pp pole-pairs ๐ โโ๐ = ๐ฟ๐ ๐โ๐ + ๐๐ ๐โ๐ So the rotor nestโs voltage equation is given by ๐ โโ๐ = ๐ฟ๐ ๐โ๐ + ๐๐,๐ = assuming that the โโโ๐ โโ๐ = ๐ ๐ ๐โ๐ + ๐ ๐ ๐ + ๐๐๐๐๐ ๐ ๐ โโ๐ With, ๐โ๐ ๐โ๐ผ๐ฝ๐ = (๐๐๐1 + ๐2 ๐๐๐2 + ๐๐๐๐3 ) = (๐๐๐1 + ๐โ1 ๐๐๐2 + ๐โ2 ๐๐๐3 ) { The system defined by (44) is given in following nomenclature, ๐ผ๐ฝ๐ ๏ ๐โ๐ ๐ โก ๐โ๐ ๐ : PW reference frame in distribution. ๏ ๐โ๐ ๐ โก ๐โ๐ ๐๐ผ๐ฝ๐ : CW reference frame in distribution. ๐ฅ๐ฆ ๏ ๐โ๐ โก ๐โ๐ ๐ : Rotor references related to distribution. With, Taking into account the obtained value from (43), we can write the equations system from (17), (18), (19) and (41) as follows, โโโ๐ ๐ โโ๐ ๐ = ๐ ๐ ๐ ๐โ๐ ๐ + ๐๐ ๐ โโ๐ = ๐ฟ๐ ๐โ๐ + ๐๐ ๐โ๐ + ๐๐ ๐โ๐ {๐ This model is similar to the vector model of the induction machine in presence of two stator winding. The expressions related to stator power winding are the same as that of the induction machine. In rotor flux equation, the influence of the two stator currents is well represented. In stator control winding, the factor [๐๐๐๐ ๐ โ (๐๐ + ๐๐ )๐บ] characterizes the relative angular velocity between the reference frames dq and ๐ผ๐ฝ๐ [10]. VI. TORQUE CALCULATION The power absorbed by the machine caused by three excitations PW, CW and rotor is given by, ๐๐ก โโ๐ . ๐โโ๐ } + โ๐ {๐ โโ๐ . ๐โโ๐ } + โ๐ {๐ โโ๐ . ๐โโ๐ } ๐๐๐๐ = โ๐ {๐ ๐ โโ๐ ๐ = ๐ฟ๐ ๐โ๐ ๐ + ๐๐ ๐๐๐๐(๐๐ +๐ฟ) ๐โ๐ โโ๐ ๐ = ๐ ๐ ๐ ๐โ๐ ๐ + ๐ โโโ๐ ๐ ๐๐ ๐๐ก ๐ โโ๐ ๐ = ๐ฟ๐ ๐โ๐ ๐ + ๐๐ ๐ ๐๐๐[(๐๐ +๐ฟ)โ๐พ] ๐โ โ๐ โโโ๐ โโ๐ = ๐ ๐ ๐โ๐ + ๐๐ ๐ ๐๐ก โโ๐ = ๐ฟ๐ ๐โ๐ + ๐๐ ๐ โ๐๐๐ (๐๐ +๐ฟ) ๐โ๐ ๐ + ๐๐ ๐ ๐๐๐[(๐๐ +๐ฟ)โ๐พ] ๐โ โ๐ ๐ { ๐ (44) (47) Multiplying the voltage equations of (46) by ๐โโ๐ , ๐โโ๐ , ๐โโ๐ respectively and taking the real part we can write, โโโ๐ โ โโ๐ . ๐โโ๐ } = ๐ ๐ ๐๐2 + โ๐ {๐๐ โ๐ {๐ . ๐โ๐ } + โ๐ {๐๐๐๐๐ ๐ ๐ โโ๐ . ๐โโ๐ } ๐๐ก (48) โโโ๐ โ โโ๐ . ๐โโ๐ } = ๐ ๐ ๐๐2 + โ๐ {๐๐ โ๐ {๐ . ๐โ๐ } + โ๐ {๐[๐๐๐๐ ๐ โ (๐๐ + ๐๐ )ฮฉ]๐ โโ๐ . ๐โโ๐ } ๐๐ก โโโ๐ ๐๐ 2 โ โโ๐ . ๐โโ๐ } = ๐ โ๐ {๐ โ๐ {๐[๐๐๐๐ ๐ โ ๐๐ ฮฉ]๐ โโ๐ . ๐โโ๐ } โ ๐ ๐๐ + โ โ๐ { ๐๐ก . ๐โ๐ } + โ ๐๐ฝ๐๐๐ ๐๐๐๐๐ (49) (50) ๐๐๐๐๐๐ With, ๐๐๐๐๐๐ = ๐๐๐๐ ๐ โ๐ {๐๐ โโ๐ . ๐โโ๐ } + [๐๐๐๐ ๐ โ (๐๐ + ๐๐ )ฮฉ]โ๐ {๐๐ โโ๐ . ๐โโ๐ } + [๐๐๐๐ ๐ โ ๐๐ ฮฉ]โ๐ {๐๐ โโ๐ . ๐โโ๐ } (51) By definition, the torque may be obtained from the relationship of the total electromagnetic power at the rotor shaft speed ฮฉ, ๐๐๐ = ๐๐๐๐๐๐ (52) ฮฉ A simple identity shows that, โ๐ {๐๐โ๐ด . ๐โ๐ตโ } = ๐๐ {๐โ๐ต . ๐โ๐ดโ } = โ๐๐ {๐โ๐ด . ๐โ๐ตโ } (53) So, Pempcr becomes, ๐๐๐๐๐ = ๐๐๐๐ ๐ ๐๐ {๐โ๐ . ๐ โโ๐โ } + [๐๐๐๐ ๐ โ (๐๐ + ๐๐ )ฮฉ]๐๐ {๐โ๐ . ๐ โโ๐โ } + ๐๐๐๐ (54) Where, ๐๐๐๐ = [๐๐๐๐ ๐ โ ๐๐ ฮฉ]๐๐ {๐โ๐ . ๐ โโ๐โ } (55) The Conjugate of โฯ โโr is, ๐ โโ๐โ = ๐ฟ๐ ๐โโ๐ + ๐๐ ๐โโ๐ + ๐๐ ๐โโ๐ (56) Fig3. Open loop speed scalar control scheme presents open loop speed scalar control scheme from CW. Note that two cases will be considered: CW short circuited and CW fed controlled while PW is always grid supplied, relevant parameter employed for simulation tasks are collected in appendix IX.2.? A. Singly fed induction mode operation In this mode the PW is connected to the grid and the CW is short-circuited. The existence of a single power supply in the machine facilitates enormously the synchronization of the both windings stator currents. Fig4.a shows the simulated BDFM start-up speed-time response under no-load condition, the obtained curve resembles very closely to that of an induction motor. It will be observed that once the synchronous speed is reached (ฮฉ = Replacing (56) in (55) yields, ๐๐๐๐ = [๐๐๐๐ ๐ โ ๐๐ ฮฉ]๐๐ {๐๐ ๐โ๐ . ๐โโ๐ + ๐๐ ๐โ๐ . ๐โโ๐ )} (57) โโโโ and ๐ โโโโ , we get also, From the equation of ๐ ๐ ๐ ๐ ๐ ๐โ๐ = โโโ๐ ๐ ๐๐ ๐โ๐ = โ โโโ๐ ๐ ๐๐ ๐ฟ๐ ๐โ ๐๐ ๐ โ (58) ๐ฟ๐ ๐โ ๐๐ ๐ (59) (58) and (59) in (57) conduct to, ๐๐๐๐ = [๐๐๐๐ ๐ โ ๐๐ ฮฉ]๐๐ {๐ โโ๐ . ๐โโ๐ + ๐ โโ๐ . ๐โโ๐ } (60) Replacing (60) in (54) and after arrangement we get, ๐๐๐๐๐๐ = ๐๐ ฮฉ๐๐ {๐ โโ๐โ . ๐โ๐ } + ๐๐ ฮฉ๐๐ {๐ โโ๐ . ๐โโ๐ } (61) From (52) the electromagnetic torque, which is given by the contribution of PW and CW, can be expressed as, ๐๐๐ = ๐๐๐_๐ + ๐๐๐_๐ (62) Whereas, ๐๐๐ = ๐๐ ๐๐ {๐ โโ๐โ . ๐โ๐ } + ๐๐ ๐๐ {๐ โโ๐ . ๐โโ๐ } (63) In addition, we can express the electromagnetic torque by the PW, CW and rotor currents as follows ๐๐๐ = ๐๐ ๐๐ ๐๐ {๐โ๐ . ๐โโ๐ } + ๐๐ ๐๐ ๐๐ {๐โ๐ . ๐โโ๐ } (64) VII. SIMULATIONS RESULTS To test the BDFM, the model has been implemented using MATLAB/SIMULINK package as shown in Fig3. Which Fig4. Simulation results of singly fed induction mode operation 77.58 rad/๐ ๐๐), the frequency of CW is quite small near zero as shown in Fig4.b. Initially, the machine was running synchronously at 750 rpm (78.5rad/sec) with unload torque. Fig4.c and Fig4.d shows the Temporal values of the currents for both stator windings. At t=3.5 seconds, the CW excitation voltage is applied. The BDFM Speed decreases from synchronous to subsynchronous regimes and the electromagnetic torque is remained after the transient unchanged. The starting torque-speed characteristic is also of great interest. Simulation results are shown in Fig4.e and Fig4.f as would be expected, the BDFM follows the torque-speed characteristic of induction motor. Note that the total electromagnetic torque, Tem produced by the machine is composed of two components, Temp and Temc . Temp , is produced by the PW pole-pairs system whileTemc , is due to the CW pole-pairs system. Interaction between both torques can be clearly observed. B. Doubly-fed synchronous operation mode In this case the controllability of the system is tested when an external voltage is applied on the CW side which follows a conventional Volt/Hertz law (Vc โfc = constant). Fig.5.a shows the variation of time speed response. Switching from the short-circuit case to the step of fc = โ4Hz at t=3.5s, and after it increases once more until the step fc = โ2Hz at t = 6s. Oscillations of the first transition in Fig.5 at t = 3.5s are very high relatively to the second one which occurs at t=6s. Step change for the first step explains the moment of CW connection after its short circuit regime. Fig.6 shows time responses of speed and torque corresponding to the synchronous the subsynchronous and the fault tolerant behavior of the BDFM system. At t=5.0 seconds, load torque is applied up from zero to 2 Nm. Similar to conventional synchronous machines, in this case, the rotor speed remains after the transient to its initial value. Thus, the machine presents a synchronous operation at speed of 750 rpm (78.5rad/sec), in which the rotor speed depends only on the supply frequencies. At t=7 seconds, we can see from Fig 6 the dynamic responses for a sudden loss of the CW excitation when a short circuit is applied to the CW terminals accompagned with unload machine. An advantage of the BDFM drive system is that a loss of synchronism does not lead to a catastrophic situation and the machine can remain connected to the grid. As a result, the drive system still operates in the singly-fed induction mode and can be re-synchronized again. VIII. CONCLUSION This paper has provided the detailed analysis of the BDFM principle operations, in which its dynamic model has been developed both in separate and in unified references frame. The second one has been based on the generic dq reference frame that will be used in vector control strategies. This model has been validated in MATLAB/SIMULINK packages where the BDFM has been controlled in open loop Volt/Hertz. The simulation results attest the BDFM literature assertion. As expected, the speed of BDFM can be controlled through adjusting the voltage applied to the CW. The model discussed above is an important part of this work, which offers the basis of differ control strategy for the BDFM. IX. APPENDIX IX.1. Transformation between different Reference Frames The resulting model (44) referred in three initial reference frames and two possible pole pair distribution (shown in Fig.b-1): Fig.5 Rotor speed and electromagnetic torque Fig6. Speed and electromagnetic torque time response under load torque Fig.IX.1. Unified reference frames (mechanical angle) ๏ Coupling Relation ๐ฅโ ฮฑฮฒ๐ = ๐(๐ฅโ ฮฑฮฒ๐ ) It is assumed that the rotor of the BDFM fulfils the equation (3) and maximizes the number of nests i.e. ๐๐ + ๐๐ = ๐, which implies that, ๐ฅโ ๐ฅ๐ฆ๐ = ๐ฅโ ๐ฅ๐ฆ๐ โ IX.1.1 From Fig.IX.1. it can be deduced that: ๐ฅโ xy๐ = ๐ฅ๐๐๐๐ ๐ผ ๐ฅโ xy๐ = ๐ฅ๐ IX.1.2 ๐๐๐ ๐ผ IX.1.3 ๐ฅโ ฮฑฮฒ๐ = ๐๐๐๐ (ฮธr +ฮด) ๐ฅโ xy๐ IX.1.4 ๐ฅโ ฮฑฮฒ๐ = ๐๐๐๐(ฮธr +ฮดโฮณ) ๐ฅโ xy๐ IX.1.5 Combining IX.1.1, IX.1.4, IX.1.5 we get, โ ๐ฅโ ฮฑฮฒ๐ = ๐ฅโ ฮฑฮฒ๐ ๐๐ฮธa IX.1.6 With: ฮธa = (๐๐ + ๐๐ )(ฮธr + ฮด) โ ๐๐ ฮณ IX.1.7 ๏ vector transformations from original reference frames to generic ๐๐๐ reference frame We can define a generic ๐๐๐ reference frame with a Pp pole-pair distribution and located at any mechanical position (ฮธobsp /pp ) from ฮฑฮฒp , the vector transformation is defined as, ๐ฅโ ฮฑฮฒ๐ = ๐๐ฮธobsp ๐ฅโ ๐๐๐ IX. 1.6 & IX. 1.8 โ ๐ฅโ ฮฑฮฒ๐ = ๐ฅโ IX.1.8 ๐๐๐ โ ๐(ฮธa โฮธobsp) . ๐ IX.1.9 ๐ โ๐๐๐ (ฮธr +ฮด) ๐ฅโ ฮฑฮฒ๐ IX.1.10 IX. 1.8 in IX. 1.10 โ ๐ฅโ xy๐ = ๐๐[ฮธobsp โ๐๐(ฮธr +ฮด)] ๐ฅโ ๐๐๐ IX.1.11 IX. 1.4 โ ๐ฅโ xy๐ โ๐๐๐ (ฮธr +ฮดโฮณ) ฮฑฮฒ๐ IX.1.12 IX. 1.9 In IX. 1.12 โ ๐ฅ xy๐ = ๐ โ๐[ฮธobsp โ๐๐ (ฮธr +ฮด)] ๐ฅโ โ๐๐๐ IX.1.13 IX. 1.5 โ ๐ฅโ xy๐ = =๐ ๐ฅโ In this way any machine variable can be defined in a generic dqp reference frame. 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