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This paper proposes a novel dual threephase Space Vector Modulation (SVM) for sixphase multilevel inverter to control a Six Phase Induction Machine (SPIM). The main idea is to control the sixphase multilevel inverter as two threephase (1, 3 and 5 phases for the first one and 2, 4 and 6 phases for the second one) multilevel inverters separately by SVM of the Nlevel threephase Separate DC Source (SDCS) inverter. This enables a great the simplification of the control algorithm for six phase multilevel (N level) inverter drive. In SixPhase SVM, N6 vectors are used so if two level N=2, 3 level or 4 level, implies 64 vectors, 729 vectors, 15625 vectors are used respectively. However, in a proposed dual threephase SVM, we use N3 vectors so if 2 level, 3 level or 4 level implies 8 vectors, 27 vectors, 125 vectors are used respectively to control the sixphase multilevel inverters as two threephase multilevel inverter with the same threephase multilevel SVM. Whereas the first threephase inverter is composed by 1, 3, and 5 phases and the second threephase is composed by 2, 4 and 6 phases. This allows to have a new modulation technique for sixphase multilevel inverter and a great simplification of the classical sixphase SVM control algorithm. The simulation results of the Indirect Field Oriented Control (IFOC) of sixphase induction machine drive fed by stacked multilevel inverters are given to highlight the performance of the proposed control structure.
dual threephase induction machine, multiphase machine, multilevel inverter, space vector modulation
During the last decade, the use of multiphase motor drives in high reliability application has substantially increased because of their potential advantages [1, 2]. This rapid surge concerns specifically three particular applications zones: electric ship impetus, “more electric” flying machine, and footing (locomotive, electric vehicles and hybrid electric vehicles) [3]. Two distinct features make multiphase drives an attractive possibility for these applications. There are three possible winding connections of the sixphase machine structures (see Figure 1). In the most famous abundant one, two threephase IM shifted by $\gamma=0^{\circ}$ (Figure 1(a)), asymmetrical sixphase machine, stator winding is composed of two threephase windings shifted in space by $\gamma=30^{\circ}$ (Figure 1(b)). In the other structure, symmetrical sixphase machine, there are two sets of threephase windings that are spatially shifted by $\gamma=60^{\circ}$ (Figure 1(c)), giving rise to spatial displacement between any two successive phases of 60° [3, 6].
The asymmetrical sixphase machine is the customary choice, predominantly for historical reasons associated with PWM of VoltageSource Inverter (VSI) control [4, 6]. This allows a good quality of the current regulation which can be linked to a PWM regulation method suitable for VSI Control at a sufficiently high switching frequency 3xn (n: number of phase) [7, 8]. The asynchronous machines with numbered phases do not have the shape 3xn. For instance, five or seven [9, 10] are infrequently used in practice because they made on demand. On the other hand, the machines with six or nine phases can be gotten two or three available threephase IM respectively. This uses the original frame and stratifications stator/rotor of the machines with threephase and thus allows saving on the making’s cost.
Multilevel converters are able to generate output voltage wave forms consisting of a large number of steps. In this way, high voltages can be synthesized using sources and switching devices with lower voltage values, with the additional benefit of a reduced harmonic distortion and lower dv/dt in the output voltages [11, 12]. Multilevel inverter supplied multiphase drives have been gaining the interest of researchers and industry in recent years [1, 6]. Multilevel multiphase inverter topologies have been widely recognized as a viable solution to overcome current and voltage limits of power switching converters in the environment of highpower mediumvoltage drive systems [13, 15]. Several studies published the news PWM modulations methods that can be used in multiphase applications with several levels [1619]. The first successful implementation of an algorithm of multiphase SVPWM with several levels based on the approach SVM [2026]. Such an approach is considered like a classical approach SVPWM presented in [18]. This presents a general spacevector modulation algorithm for threelevel inverter. The algorithm is to a great extent computationally proficient and independent of the number of converter levels. At the same time, it provides a good insight into the operation of multilevel inverters.
Figure 1. Winding connection of sixphase machine
In This paper, a novel dual threephase SVM for a multilevel inverter sixphase is proposed to drive a sixphase induction machine. The p^{th} phase (p=1, 2, 3, 4, 5, and 6) of the sixphase multilevel SDCS inverter is shown in Figure 2. The main idea is to use the sixphase inverter as two threephase inverter (p=1, 3, 5) and (p=2, 4, 6) to control the SPIM as two threephase IM (U_{135} and U_{246}) separately. The two multilevel threephase inverter SDCS is controlled by two conventional N level threephase SVM [20]. The first one can control the phases 1, 3, and 5 (U_{a1b1c1}), and the second one can control the phase 2, 4, and 6 (U_{a2b2c2}), according to the type of the sixphase machine with placed windings (two threephase induction machine, asymmetric sixphase or symmetrical respectively). Usually, in SVM of sixphase inverter, we use N^{6} vectors; (for N=2, 3, or 4; there are 64 vectors, 729 vectors or 15625 vectors, respectively), and several transformations and decomposition in two plans. However, in the proposed dual SVM, we use N^{3} vectors; (for N=2, 3 or 4; there are 8 vectors, 27 vectors or 125 vectors respectively) in order to control two multilevel sixphase inverters.
This paper consists of six sections. The first section provides the description and model of the dual stator induction machine in the abc plan. The second section deals with Park’s SPIM model. The third section presents the developed multilevel dual threephase SVM. As for the fourth section, it is devoted to the presentation of indirect field oriented control. The simulation results are discussed in section five, followed by a conclusion in the last section.
Figure 2. General Structure of a multilevel p^{th} phase (p=1… 6) SDCS inverter
All SPIM are strictly being investigated for various high power applications due to their augmented power to weight ratio, augmented frequency and abridged magnitude of torque pulsation, and fault tolerant characteristics. The SPIM with two similar stator three phase windings, shifted by 0, 30, 60 degrees in space and three phase winding in rotor is shown in Figure 3. To put it simply, we will consider that the electrical circuit of the rotor is equivalent to a threephase winding shortcircuit [27, 28]. Figure 3 gives the position of the magnetic axes of the nine windings forming the six phases of the stator and the three phases of the rotor.
Figure 3. Stator and rotor windings of the SPIM for $\gamma=0^{\circ}, 30^{\circ}$ or $60^{\circ}$
2.1 Electrical equation
Electrical equations of the stator, rotor, electromagnetic torque (13) and mechanical equations (4) are expressed as a function of the different currents and the derivative of the flux. These differential equations defined in the real reference frame abc, are written as follows:
The voltage and flux linkage equations are given compactly in matrix form as:
$\left\{\left[\begin{array}{l}{\left[\mathrm{V}_{\mathrm{s}}\right]} \\ {\left[\mathrm{V}_{\mathrm{r}}\right]}\end{array}\right]=\left[\begin{array}{c}{\left[\mathrm{R}_{\mathrm{s}}\right][0]_{6 \times 3}} \\ {[0]_{3 \times 6}\left[\mathrm{R}_{\mathrm{r}}\right]}\end{array}\right]\left[\begin{array}{l}{\left[\mathrm{i}_{\mathrm{s}}\right]} \\ {\left[\mathrm{i}_{\mathrm{r}}\right]}\end{array}\right]+\frac{\mathrm{d}}{\mathrm{dt}}\left[\begin{array}{l}{\left[\varphi_{\mathrm{s}}\right]} \\ {\left[\varphi_{\mathrm{r}}\right]}\end{array}\right]\right.$ (1)
$\left[\begin{array}{l}{\left[\varphi_{s}\right]} \\ {\left[\varphi_{r}\right]}\end{array}\right]=\left[\begin{array}{l}{\left[L_{s}\right]\left[M_{s r}\right]} \\ {\left[M_{r s}\right]\left[L_{r}\right]}\end{array}\right] \cdot\left[\begin{array}{l}{\left[i_{s}\right]} \\ {\left[i_{r}\right]}\end{array}\right]$ (2)
2.2 Electromagnetic torque equation
The electromagnetic torque can be expressed in the following equations:
$\left[C_{e}\right]=\left[i_{s}\right]^{t} \frac{\left[M_{r s}\right]}{d \theta}\left[i_{r}\right]$ (3)
2.3 Mechanical equation
$\frac{\left[\Omega_{m}\right]}{d t}=\frac{1}{J}\left(C_{e}C_{r}\right)$ (4)
With:
$\left[V_{s}\right]=\left[\begin{array}{l}{\left[V_{s, a b c 1}\right]} \\ {\left[V_{s, a b c 2}\right]}\end{array}\right] ;\left[i_{s}\right]=\left[\begin{array}{l}{\left[i_{s, a b c 1}\right]} \\ {\left[i_{s, a b c 2}\right]}\end{array}\right]$
$\left[\varphi_{s}\right]=\left[\begin{array}{l}{\left[\varphi_{s, a b c 1}\right]} \\ {\left[\varphi_{s, a b c 2}\right]}\end{array}\right] ;\left[V_{r}\right]=\left[V_{r, a b c}\right] ;\left[i_{r}\right]=\left[i_{r, a b c}\right]$
$\left[R_{s}\right]=\left[\begin{array}{c}{\left[R_{s 1}\right][0]_{3 \times 3}} \\ {[0]_{3 \times 3}\left[R_{s 2}\right]}\end{array}\right] ;\left[\varphi_{s}\right]=\left[\left[\begin{array}{c}{\left[\varphi_{s, a b c 1}\right]} \\ {\left[\varphi_{s, a b c 2}\right]}\end{array}\right]\right.$
$\left[L_{s}\right]=\left[\begin{array}{l}{\left[L_{s 1, s 1}\right]\left[M_{s 1, s 2}\right]} \\ {\left[M_{s 2, s 1}\right]\left[L_{s 2, s 2}\right]}\end{array}\right] ;\left[M_{s r}\right]=\left[\begin{array}{l}{\left[M_{s 1, r}\right]} \\ {\left[M_{s 2, r}\right]}\end{array}\right]$
$\left[M_{r s}\right]=\left[M_{s r}\right]^{t} ;\left[M_{s 1, s 2}\right]=\left[M_{s 2, s 1}\right]^{t}$
After the replacement of the stator and rotor flows and the different inductances in equation (2), we obtain [21]:
$\left[\begin{array}{c}\varphi_{s, a b c 1} \\ \varphi_{s, a b c 2} \\ \varphi_{r, a b c}\end{array}\right]=\left[\begin{array}{ccc}L_{s 1, s 1} & M_{s 1, s 2} & M_{s 1, r} \\ M_{s 2, s 1} & L_{s 2, s 2} & M_{s 2, r} \\ M_{r, s 1} & M_{r, s 2} & L_{r, r}\end{array}\right]\left[\begin{array}{c}i_{s, a b c 1} \\ i_{s, a b c 2} \\ i_{r, a b c}\end{array}\right]$ (5)
where, $\left[\mathrm{v}_{\mathrm{s}}\right],\left[\mathrm{i}_{\mathrm{s}}\right],\left[\mathrm{v}_{\mathrm{r}}\right],\left[\mathrm{i}_{\mathrm{r}}\right],\left[\varphi_{\mathrm{s}}\right]$ are the stator and rotor voltage, current and flux vectors of the sixphase IM. We find that the machine sixphase is composed of two times the machine threephase asynchronous. This allows for the study and synthesis of control laws of two threephase induction machines using the methods already developed instead of ordering directly the six phase machine. In the next part, we will present Park's transformation, which leads to a simplified model of SPIM that is easy to solve and more suitable for the study of dynamic regimes and control.
Two park models are presented. The first obtained by transforming the stator voltages (6/2x3) is used for the SVM control. The second is obtained for a transformation (6/2) that will be used as a twophase system for Indirect Field Oriented Control.
3.1 Dual threephase model of stator voltages
In this section, the association of two threephase machines. By applying the Park transformation of the twostator stars, we obtain twice the twophase models equivalent to the threephase model (Figure 4) [28, 29]. In which we use for this the usual transformation of Concordia and Park.
$\left\{\begin{array}{c}X_{(\alpha \beta 0) i}=[T]^{1} X_{(a b c) i} \\ X_{(d q 0) i}=[\rho(\theta)]^{1} X_{(\alpha \beta 0) i}\end{array}\right.$ (6)
where, X can represent the current, the voltage or the flux in the machine, and i can represent 1 for the first stator and 2 for the second stator.
with:
$[T]^{1}=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\cos (0) & \cos \left(\frac{2 \pi}{3}\right) & \cos \left(\frac{4 \pi}{3}\right) \\ \sin (0) & \sin \left(\frac{2 \pi}{3}\right) & \sin \left(\frac{4 \pi}{3}\right) \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right]$ (7)
$[\rho(\theta)]^{1}=\left[\begin{array}{ccc}\cos (\theta) & \sin (\theta) & 0 \\ \sin (\theta) & \cos (\theta) & 0 \\ 0 & 0 & 1\end{array}\right]$ (8)
Figure 4. Representation of the SPIM windings in the Park
By applying the Park transformation (6) to the equations of the voltages (1), the real equation system of the SPIM will be decomposed into two diphasic subsystems (d1q1) and (d2q2).
$\left\{\begin{array}{l}V_{s d i}=R_{s} I_{s d i}+\frac{d \varphi_{s d i}}{d t}\omega_{s} \varphi_{s q i} \\ V_{s q i}=R_{s} I_{s q i}+\frac{d \varphi_{s q i}}{d t}+\omega_{s} \varphi_{s d i} \\ V_{s o i}=R_{s} I_{s o i}+\frac{d \varphi_{s o i}}{d t}\end{array}\right.$ (9)
with: (i=1 or 2)
This shows that we have two stators, and the voltages are decoupled [32]. This property of the stator voltages will operate to use two SVM, each one will control three phases separately.
3.2 Sixphase model in the system (d, q) (x, y) (o_{1}, o_{2})
In this section, a biphase command model of SPIM is presented. Therefore, the matrix of stator inductors [Ls] is diagonalized by the stator decoupling matrix $\left[\mathrm{T}_{\mathrm{s}}(\gamma)\right]^{1}$ [30], [31] and the rotor matrix [Tr]^{1} is follow:
$\left[T_{s}(\gamma)\right]^{1}=\frac{1}{\sqrt{3}}\left[\begin{array}{cccccc}\cos (0) &\cos \left(\frac{2 \pi}{3}\right) &\cos \left(\frac{4 \pi}{3}\right)& \cos (\gamma) & \cos \left(\gamma+\frac{2 \pi}{3}\right) &\cos \left(\gamma+\frac{4 \pi}{3}\right) \\ \sin (0) &\sin \left(\frac{2 \pi}{3}\right) &\sin \left(\frac{4 \pi}{3}\right) &\sin (\gamma) & \sin \left(\gamma+\frac{2 \pi}{3}\right) &\sin \left(\gamma+\frac{4 \pi}{3}\right) \\ \cos (0) &\cos \left(\frac{4 \pi}{3}\right) &\cos \left(\frac{2 \pi}{3}\right) &\cos (\pi\gamma) &\cos \left(\frac{\pi}{3}\gamma\right) &\cos \left(\frac{5 \pi}{3}\gamma\right) \\ \sin (0) &\sin \left(\frac{4 \pi}{3}\right)& \sin \left(\frac{2 \pi}{3}\right) &\sin (\pi\gamma) & \sin \left(\frac{\pi}{3}\gamma\right) &\sin \left(\frac{5 \pi}{3}\gamma\right) \\ 1 & 1&1&0&0&0 \\ 0 & 0 & 0&1&1&1\end{array}\right]$ (10)
$[T r]^{1}=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}1 & \frac{1}{2} & \frac{1}{2} \\ 0 & \frac{\sqrt{3}}{2} & \frac{\sqrt{3}}{2} \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right]$ (11)
with, $\gamma=0^{\circ} ; 30^{\circ} ;$ or $60^{\circ}$ two threephase induction machine, asymmetric sixphase or symmetrical SPIM respectively. By applying the transformation matrix $\left[\mathrm{T}_{\mathrm{s}}(\gamma)\right]^{1}$ to the equations of voltages (1) and fluxes (2), the system of real sixdimensional stator equations will be decomposed into three decoupled subsystems of dimension two: the systems (dq), (xy) et (o_{1}o_{2}). Geometrically, we will "project" the stator variables on three orthogonal "planes". The transformation matrix has the property of separating the harmonics into several groups, and projecting them into each subsystem. For rotor variables, the usual Concordia transformation, denoted by [Tr]^{1} (11), is used. (See Figure 5).
Figure 5. Equivalent model of SPIM in (dq) plan
By applying the transformation matrix (10) to equation (2) the system becomes:
$\left[\begin{array}{c}\varphi_{s d} \\ \varphi_{s q} \\ \varphi_{s x} \\ \varphi_{s y} \\ \varphi_{s o 1} \\ \varphi_{s o 2}\end{array}\right]=\left[\begin{array}{cccccc}L_{s} & 0 & 0 & 0 & 0 & 0 \\ 0 & L_{s} & 0 & 0 & 0 & 0 \\ 0 & 0 & L_{s 1} & 0 & 0 & 0 \\ 0 & 0 & 0 & L_{s 1} & 0 & 0 \\ 0 & 0 & 0 & 0 & L_{s 1} & 0 \\ 0 & 0 & 0 & 0 & 0 & L_{s 1}\end{array}\right]\left[\begin{array}{c}i_{s d} \\ i_{s q} \\ i_{s x} \\ i_{s y} \\ i_{s o 1} \\ i_{s o 2}\end{array}\right]+M\left[\begin{array}{cccc}\cos \theta_{r, s} & \sin \theta_{r, s} & 0 & \\ \sin \theta_{r, s} & \cos \theta_{r, s} & 0 & \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0\end{array}\right]\left[\begin{array}{c}i_{r d} \\ i_{r q} \\ i_{r 0}\end{array}\right]$ (12)
$\left[\begin{array}{c}\varphi_{r d} \\ \varphi_{r d} \\ \varphi_{r 0}\end{array}\right]=M\left[\begin{array}{ccc}{\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & \cos \theta\end{array}\right]} & {[0]_{2 * 4}} \\ {[0]_{1 * 4}} & {[0]_{4 * 4}}\end{array}\right]\left[\begin{array}{c}i_{s d}\\i_{s q} \\ i_{s x} \\ i_{s y} \\ i_{s 01} \\ i_{s 02}\end{array}\right]+\left[\begin{array}{ccc}L_{r} & 0 & 0 \\ 0 & L_{r} & 0 \\ 0 & 0 & L_{r 1}\end{array}\right]\left[\begin{array}{c}i_{r d} \\ i_{r q} \\ i_{r 0}\end{array}\right]$ (13)
The next step is to eliminate the dependence of θ of the matrix of inductances. For this, we use a rotation matrix [⍴(θ)] to express the rotor variables in the stator reference:
$\left\{\begin{array}{l}X_{\alpha \beta 0}=\left[T_{s}(\gamma)\right]^{1} X_{a b c} \\ X_{d q 0}=[\rho(\theta)]^{1} X_{\alpha \beta 0}\end{array}\right.$ (14)
where, X can represent the current, the voltage, or the flux in the sixphase machine, dq xy 0_{1} 0_{1}.
with:
$[\rho(\theta)]^{1}=\left[\begin{array}{ccc}{\left[\begin{array}{cc}\cos \theta & \sin \theta \\ \sin \theta & \cos \theta\end{array}\right]} & {[0]_{2 * 4}} \\ {[0]_{4 * 2}} & {[0]_{4 * 4}}\end{array}\right]$ (15)
where the models are:
We use the common Park transformation matrix for the rotor size.
$\left[\begin{array}{c}x_{\mathrm{sc}} \\ x_{s \beta} \\ x_{\mathrm{sx}} \\ x_{\mathrm{sy}} \\ x_{\mathrm{so1}} \\ x_{\mathrm{so} 2}\end{array}\right]=\left[T_{s}(\gamma)\right]^{1}\left[\begin{array}{c}x_{\mathrm{sa} 1} \\ x_{\mathrm{sb} 1} \\ x_{\mathrm{sc} 1} \\ x_{\mathrm{sa} 2} \\ x_{\mathrm{sb} 2} \\ x_{\mathrm{sc} 2}\end{array}\right]$ (16)
$\left[x_{r \alpha} x_{r \beta} x_{r o}\right]^{t}=\left[T_{r}\right]^{1}\left[x_{r a} x_{r b} x_{r c}\right]^{t}$ (17)
$\left[T_{r}\right]^{1}=\sqrt{\frac{2}{3}}\left[\begin{array}{ccc}\cos (0) & \cos \left(\frac{2 \pi}{3}\right) & \cos \left(\frac{4 \pi}{3}\right) \\ \sin (0) & \sin \left(\frac{2 \pi}{3}\right) & \sin \left(\frac{4 \pi}{3}\right) \\ \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}}\end{array}\right]$ (18)
By expressing the model of the SPIM in a reference linked to the stator, we obtain:
$\left[\begin{array}{c}V_{s \alpha} \\ V_{s \beta} \\ V_{r \alpha} \\ V_{r \beta}\end{array}\right]=\left[\begin{array}{cccc}r_{s} & 0 & 0 & 0 \\ 0 & r_{s} & 0 & M \\ 0 & \omega M & r_{r} & \omega L_{r} \\ \omega M & 0 & \omega L_{r} & r_{r}\end{array}\right]\left[\begin{array}{c}i_{s \alpha} \\ i_{s \beta} \\ i_{r \alpha} \\ i_{r \beta}\end{array}\right]+\left[\begin{array}{cccc}L_{s} & 0 & M & 0 \\ 0 & L_{s} & 0 & M \\ M & 0 & L_{r} & 0 \\ 0 & M & 0 & L_{r}\end{array}\right] \frac{d}{d t}\left[\begin{array}{c}i_{s \alpha} \\ i_{s \beta} \\ i_{r \alpha} \\ i_{r \beta}\end{array}\right]$ (19)
$\left[\begin{array}{l}\mathrm{V}_{\mathrm{sx}} \\ \mathrm{V}_{\mathrm{sy}}\end{array}\right]=\left[\begin{array}{ll}\mathrm{r}_{\mathrm{s}} & 0 \\ 0 & \mathrm{r}_{\mathrm{s}}\end{array}\right]\left[\begin{array}{l}\mathrm{i}_{\mathrm{sx}} \\ \mathrm{i}_{\mathrm{sy}}\end{array}\right]+\left[\begin{array}{ll}\mathrm{L}_{\mathrm{s} 1} & 0 \\ 0 & \mathrm{L}_{\mathrm{s} 1}\end{array}\right] \frac{\mathrm{d}}{\mathrm{dt}}\left[\begin{array}{l}\mathrm{i}_{\mathrm{sx}} \\ \mathrm{i}_{\mathrm{sy}}\end{array}\right]$ (20)
$\left[\begin{array}{c}V_{s o 1} \\ V_{s o 2} \\ V_{r o}\end{array}\right]=\left[\begin{array}{ccc}r_{s} & 0 & 0 \\ 0 & r_{s} & 0 \\ 0 & 0 & r_{r}\end{array}\right]\left[\begin{array}{c}i_{s 01} \\ i_{s 02} \\ i_{r o}\end{array}\right]+\left[\begin{array}{ccc}L_{s 1} & 0 & 0 \\ 0 & L_{s 1} & 0 \\ 0 & 0 & L_{r 1}\end{array}\right] \frac{d}{d t}\left[\begin{array}{c}i_{s 01} \\ i_{s o 2} \\ i_{r 0}\end{array}\right]$ (21)
The mechanical equation and electromagnetic torque are as follows:
$C_{e m}=p . M\left(i_{s \beta} i_{r \alpha}i_{s \alpha} i_{r \beta}\right)$
$C_{e m}C_{r}K_{f} \Omega_{m}=J_{1} \frac{d \Omega_{m}}{d t} ; \frac{d \theta}{d t}=\Omega_{m}$ (22)
where, J_{1} denotes the moment of inertia of the sixphase machine and C_{r} the resistive torque.
We note that the new SPIM model has three completely decoupled submodels (dq), (xy) and (o_{1}o_{2}). The submodel (dq) is exactly similar to that of a threephase asynchronous machine whose variables are responsible for the electromechanical conversion of energy in the SPIM. On the other hand, the equations of the submodel (xy) are totally decoupled from the other equations, and in particular from the rotor equations. The variables in this subsystem represent the circulation currents in the SPIM and therefore do not contribute to the electromechanical conversion of the energy. Finally, the subsystem (o_{1}o_{2}) which contains the classical homopolar components [30]. We saw previously that the electromechanical conversion of the energy is only related to the two components i_{sd}, i_{sq}, for that we will interest only to the reference (dq), in which the six quantities of the SPIM will be represented by two sizes equivalents in quadrature, and which called biphase control model. The SPIM model can be formatted as a state equation as follows:
$\left\{\begin{array}{c}\dot{\mathrm{X}}=\mathrm{A}(\Omega) \mathrm{X}+\mathrm{BU} \\ \mathrm{Y}=\mathrm{CX}\end{array}\right.$ (23)
where: $X=\left[i_{s d} i_{s q} \varphi_{r d} \varphi_{r q}\right]^{t}$: The state vector; $U=\left[V_{s d} V_{s q}\right]^{t}$: The input vector or command vector; $Y=\left[i_{s d} i_{s q}\right]^{t}$: The output vector.
$A=\left[\begin{array}{cc}{\left[\begin{array}{cc}a_{1} & 0 \\ 0 & a_{1}\end{array}\right]} & {\left[\begin{array}{cc}a_{2} & a_{3} \Omega \\ a_{3} \Omega & a_{2}\end{array}\right]} \\ {\left[\begin{array}{cc}a_{4} & 0 \\ 0 & a_{4}\end{array}\right]} & {\left[\begin{array}{cc}a_{5} & p \Omega \\ p \Omega & a_{5}\end{array}\right]}\end{array}\right]$
$B=\left[\begin{array}{cc}\frac{1}{\sigma L_{s}} & 0 \\ 0 & \frac{1}{\sigma L_{s}} \\ 0 & 0 \\ 0 & 0\end{array}\right] ; C=\left[\begin{array}{cccc}1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\end{array}\right]$
With:
$a_{1}=\frac{1}{\sigma}\left(\frac{1\sigma}{T_{r}}+\frac{1}{T_{s}}\right) ; a_{2}=\frac{M}{\sigma L_{s} L_{r} T_{r}}$
$a_{3}=p \frac{M}{\sigma L_{s} L_{r}} ; a_{4}=\frac{M}{T_{r}} ; a_{5}=\frac{1}{T_{r}} ; \omega_{m}=p \Omega$
The state matrix A(Ω) depends on the mechanical speed.
Figure 6. Representation of the bi phase model control of the SPIM
Two models were presented, the first considers the machine as two threephase machines, and applying the classic Park transformation for each star we obtain a doublephase model. This high order model does not simplify machine simulation and control. As a result, we established another sixphase model, applying a specific transformation matrix. This model considers the machine as a sixphase asynchronous machine. The model obtained is of a reduced order thus allowing its introduction into numerical simulation programs.
The representation of the SPIM as consisting of two threephase allows to control the sixphase inverter as two threephase inverters shown in Figure 7. Therefore, instead of using a sixphase SVM, it has been suggested to employ two SVM which simplifies the PWM control design, and exploited the research results developed in threephase multilevel inverter [25, 26].
Figure 7. Block diagram of the SPIM open loop
4.1 Transformation of the coordinate
The first step in the algorithm is to transform the vector of reference $\overrightarrow{\mathrm{V}}_{\text {ref}}$ in twodimensional plan. In order to obtain the simple and general threephase SVM algorithm for multilevel inverter, the threedimensional reference vectors $\overrightarrow{\mathrm{V}}_{\text {ref}}$ is transformed in twodimensional plan by
$\vec{g}\left(v_{a b}, v_{b c}, v_{c a}\right), \vec{h}\left(v_{a b}, v_{b c}, v_{c a}\right)=\left\{\left[\begin{array}{c}U \\ 0 \\ U\end{array}\right],\left[\begin{array}{c}0 \\ U \\ U\end{array}\right]\right\}$ (24)
For example, the threephase/twophase transformation (25) transforms the reference vector noted in the coordinate’s system between phase (24) in the coordinate’s system (g, h), and normalize, in parallel, the vector of reference with the length of the basic vector.
$\vec{V} \operatorname{ref}(g, h)=T \cdot \vec{V} \operatorname{ref}\left(v_{a b}, v_{b c}, v_{c a}\right)$ (25)
with:
$\left[\begin{array}{c}V_{a} \\ V_{b} \\ V_{c}\end{array}\right]=r \cdot\left[\begin{array}{c}\sin w t \\ \sin w t\frac{2 \pi}{3} \\ \sin w t+\frac{2 \pi}{3}\end{array}\right], T=\frac{1}{3} \frac{N1}{2} *\left[\begin{array}{ccc}2 & 1 & 1 \\ 1 & 2 & 1\end{array}\right]$ (26)
Figure 8 shows all of the vectors of commutation of the N levels converter, with the corresponding states of commutation in the new reference (g, h).
Figure 8. Commutation Vectors of the N levels converter in the (g, h) plan
In the field of the operating control in a diphase reference, as is the case in most modern control for the different machines, this can be carried out using a transformation by basic change (d, q)→(g, h) [25]. The result of the transformation matrix is as follows:
$\vec{V} \operatorname{ref}(g, h)=T_{1} \cdot \vec{V} \operatorname{ref}(d, q)$ (27)
$T_{1}=\frac{3}{2} \frac{N1}{2} \cdot\left[\begin{array}{cc}1 & \frac{1}{\sqrt{3}} \\ 0 & \frac{2}{\sqrt{3}}\end{array}\right]$ (28)
4.2 Detection of the nearest three vectors (NTV)
The switching vectors have integer coordinates. This is advantageous because the four vectors nearest to the reference vector can be simply identified; these vectors whose coordinates are combinations of the rounded values greater and lower than the number of the reference vector are calculated as follows:
$\overrightarrow{V_{u l}}=\left[\begin{array}{l}{\left[V_{r e f g}\right]} \\ {\left[V_{r e f h}\right]}\end{array}\right], \overrightarrow{V_{l u}}=\left[\begin{array}{l}\left. V_{r e f g}\right\rfloor \\ {\left[V_{r e f h} \right.}\end{array}\right]$
$\overrightarrow{V_{u u}}=\left[\begin{array}{l}{\left[V_{r e f g}\right]} \\ {\left[V_{r e f h}\right]}\end{array}\right], \overrightarrow{V_{l l}}=\left[\begin{array}{l}\leftV_{r e f g}\right \\ \leftV_{r e f h}\right\end{array}\right]$
with:
⌈V_{ref}⌉: Indicates the upper rounded value of V_{ref};
⌊V_{ref}⌋: Indicates the lower rounded value of V_{ref}
The final points of the four nearest vectors form the equal parallelogram, which is divided into two equilateral triangles by the diagonal connecting the vectors V_{ul}. These are always two of the NTV. The third nearest vector is one of the two remaining vectors existing on the same side of the diagonal; it is taken as a reference. For that reason, the closest third vector can be found by evaluating the sign of the expression:
Ð=$V_{\text {refg}}+V_{\text {refh}}\left(V_{\text {ulg}}+V_{\text {ulh}}\right)$ (29)
If the variable Ð is positive, then the vector V_{uu} is the third nearest vector. That is, the vector V_{ll} is the nearest third vector. This concludes the identification of NTV for Nlevel inverters. Figure 9 explains how to obtain the closest third vector.
Figure 9. Localization of two different cases of the position of reference vector of the same four nearest vectors
4.3 Calculation of the switching times of the switches
The best way to synthesize the voltage reference vector is to use the three nearest vectors
$\vec{V} \operatorname{ref}=d_{1} \vec{V} 1+d_{2} \vec{V} 2+d_{3} \vec{V} 3$ (30)
With the following additional constraint on the conduction times:
$d_{1}+d_{2}+d_{3}=1$ (31)
Once the TVP are identified, the switching times of the switches can be found by solving (32) and (33), with:
$\left\{\begin{array}{l}\vec{V}_{1}=\vec{V}_{u l} \\ \vec{V}_{2}=\vec{V}_{l u} \\ \vec{V}_{3}=\vec{V}_{l l}\end{array}\right.$ (32)
$\left\{\begin{array}{l}\vec{V}_{1}=\vec{V}_{u l} \\ \vec{V}_{2}=\vec{V}_{l u} \\ \vec{V}_{3}=\vec{V}_{u u}\end{array}\right.$ (33)
Since all switching vectors always have integer coordinates, the solutions are essentially the partial parts of the coordinates (Figure 10).
If $\vec{V}_{3}=\vec{V}_{l l}$ Then $\left\{\begin{array}{l}d_{u l}=V_{r e f g}V_{l l g} \\ d_{l u}=V_{r e f h}V_{l l h} \\ d_{l l}=1d_{u l}d_{l u}\end{array}\right.$ (34)
If $\vec{V}_{3}=\vec{V}_{u u}$ Then $\left\{\begin{aligned} d_{u l} &=\left(V_{r e f h}V_{u u h}\right) \\ d_{l u} &=\left(V_{r e f g}V_{u u g}\right) \\ d_{l l} &=1d_{u l}d_{l u} \end{aligned}\right.$ (35)
4.4 Algorithm of threephase SVM, N level
The algorithm of threephases SVM of N levels inverters is summarized in Figure 10.
Figure 10. Algorithm of threephase SVM, N level
In this part, an indirect field oriented control (IFOC) induction machine drive by a conventional PI practical to the SPIM is presented. The IFOC technique is principally a predictive approach in that it approximates the angular position of the rotor flux vector by exploiting the model of the SPIM [35,37]. A commonly used IFOC technique uses the following equations to satisfy the condition for proper orientation. Geometrically speaking, we will say that the stator variables projected on the plane (dq) are involved in the electromechanical conversion of energy. In contrast, the stator variables projected on the plane (xy) are not involved in the electromechanical conversion of energy. This decoupling will simplify the analysis and control of the SPIM. Orienting the axis system (dq) so that the axis d is in phase with the rotor flow (Figure 11), to obtain:
$\left\{\begin{array}{c}\varphi_{r d}=0 \\ \varphi_{r q}=\varphi_{r}\end{array}\right.$ (36)
Figure 11. The orientation of the rotor flow
The system of equations becomes:
$\left\{\begin{aligned} \varphi_{s d} &=L_{s}\left(1\frac{M^{2}}{L_{s} L_{r}}\right) i_{s d}+\frac{M}{L_{r}} \varphi_{r} \\ \varphi_{s q} &=L_{s}\left(1\frac{M^{2}}{L_{s} L_{r}}\right) i_{s q} \\ i_{r d} &=\frac{1}{L_{r}}\left(\varphi_{r}M i_{s d}\right) \\ i_{r q} &=\frac{M}{L_{r}} i_{s q} \end{aligned}\right.$ (37)
$\left\{\begin{array}{l}V_{s d}=R_{s} i_{s d}+\sigma L_{s} \frac{d i_{s d}}{d t}\omega_{s} \sigma L_{s} i_{s q} \\ V_{s q}=R_{s} i_{s q}+\sigma L_{s} \frac{d i_{s q}}{d t}+\omega_{s} \frac{M}{L_{r}} \varphi_{r}+\omega_{s} \sigma L_{s} i_{s d} \\ M i_{s d}=\varphi_{r d}+T_{r} \frac{d \varphi_{r d}}{d t} \\ \omega_{r}=\omega_{s}\omega=\dot{\theta}=p \Omega=\frac{M}{T_{r}} \frac{i_{s q}}{\varphi_{r}}\end{array}\right.$ (38)
Then using PI regulator of the obtained linearized systems (38). A comprehensive speed regulation diagram of the SPIM by the IFOC presented on the Figure 12.
Figure 12. The control strategy of speed closed loop on sixphase
In this section, the validation of the dual SVM threephase multilevel inverter to control an SPIM with IFOC is achieved. The SPIM parameters present in the Appendix. Figure 13 presents a speed response $\gamma=60^{\circ}$ represented with the application of a load torque, Cr=40N.m, at time t=0.4s, and reversing the direction of rotation at time 0.7s from 40rad/s to 60 rad/s, it can be observed that the machine speed follows the reference. When the load torque is applied, at the instant t = 0.4s, the speed is reduced, but it is reestablished again without static error, and the torque magnitude increases 0Nm to 40Nm showed in Figure 14 and remain constant at 40Nm as shown. When the reference speed is changed during simulation, the actual speed is observed to follow the change in speed as is seen from Figure 13. At the instant of speed change (0.7s), a spike is observed in the torque waveform Figure 14 which is attributed to the sudden change of speed. The actual speed is observed to track the reference. So, it highlights the good performance of the proposed structure, and we obtain the same results for $\gamma=30^{\circ}, 0^{\circ}$ in the speed and the torque electromagnetic. At the instant t = 0.4 s where the load torque is applied, the speed is reduced, but it is reestablished again without static error.
Figure 13. Simulation results of speed response of the SPIM with application of a load torque resistant, Cr=40N.m, at time t=0.4s
Figure 14. Electromagnetic torque for Cr=40N.m.
The stator current is shown in Figure 15 for $\gamma=60^{\circ}$. Initially, at the start, the current is observed to have a lot of ripples because of not using starting techniques. It can be seen that IFOC offered better dynamic performance for speed control and proved to be a better solution for variable speed drives.
Figure 15. Stator currents of the SPIM with application of a load torque resistant, Cr=40N.m
Figure 16 presents SPIM simulations results for displacement between stator winding ($\gamma=0^{\circ}, 30^{\circ}, 60^{\circ}$). The obtained sixstator currents similarly for $\gamma=0^{\circ}, 30^{\circ}, \text{and } 60^{\circ}$.
 Figure 16 (A) presents the current superposed two threephase IM $\gamma=0^{\circ}$;
 Figure 16 (B) presents the current of asymmetrical sixphase IM for $\gamma=30^{\circ}$;
 Figure 16 (C) presents the current of symmetrical sixphase IM for $\gamma=60^{\circ}$.
Figure 16. Phase currents for (A) dual threephase, for (B) asymmetrical sixphase IM, and (C) symmetrical sixphase IM
The results obtained using three configurations of the SPIM are presented in Figure 17
 Figure 17 (A) shows the output of the three levels inverter. The simulation results show the three levels of the output voltage. The THD is obtained as 10.38 % by FFT analysis in simulation and is shown in the figure 18 (A);
 Figure 17 (B) shows the output of the five levels inverter. The simulation results show the five levels of the output voltage. The THD is obtained as 7.12 % by FFT analysis in simulation and is shown in the figure 18 (B);
 Figure 17 (C) shows the output of the seven levels inverter. The simulation results show the seven levels of the output voltage. The THD is obtained as 5.86 % by FFT analysis in simulation and is shown in the figure 18 (C).
Figure 17. Phase voltage for (A) threelevel, for (B)fivelevel, and (C) sevenlevel
We find that when increasing the number of levels, the tensions will be closer to the sinusoid
Figure 18. Percentage of the THD according to the angle $\gamma$ with different levels (3L, 5L, and 7L)
The results obtained using three configurations of the IM are presented in Figure 19 with $\gamma=30^{\circ}$
 Figure 19 (A) presents current for three levels;
 Figure 19 (B) presents current for five levels;
 Figure 19 (C) presents current for seven levels.
Figure 19. Phase currents for (A) threelevel, for (B) fivelevel, and (C) sevenlevel for $\gamma=30^{\circ}$
Clearly, when the level number increases, the stator's currents become smooth and better. We find that when the number of levels increases, the stator currents become smoother. The following Table I summarizes the percentage of the THD according to the angle $\gamma$ with different levels (3L, 5L, and 7L). In fact, it is clearly observable in table 1 the THD% decreases gradually and progressively as the number of levels of the inverter increases. According to the results found there that when the level of the inverter voltage is N=3, N=5 and N =7 the output voltage approaches more and more perfect sinusoidal form. Best THD for $\gamma=30^{\circ}$ phase shift angle and level 7.
Table 1. Percentage of the THD according to angle $\gamma$ with different levels for SPIM

For $\gamma=0^{\circ}$ 
For $\gamma=30^{\circ}$ 
For $\gamma=60^{\circ}$ 

THD % 
3l 
5l 
7l 
3l 
5l 
7l 
3l 
5l 
7l 
10.38 
7.12 
5.86 
10.35 
7.10 
5.82 
10.36 
7.11 
5.83 
This paper presents a new dual SVM algorithm for the sixphase multilevel inverterSPIM. Where the six phase multilevel inverter has controlled as two threephase multilevel inverter separately by the Nlevel SVM of the threephase SDCS inverter. The two threephase multilevel inverter diphase by γ=0°, γ=30° and γ=60°, according to the type of the six phases machine with placed windings; two threephase induction machine, asymmetric six phase, symmetrical sixphase IM respectively. This control strategy of the Nlevel inverters SDCS is general, applicable to any number of levels converter, where the number of steps involved in computation is always the same despite the area of the reference voltage vector. In addition to that, the efficiency and the ease of implementation of this algorithm makes it well suited for simulation on digital computers, and can become a very useful device in assist investigation of the properties of multilevel power converters. The main advantage of phase shift γ is to show that the dual SVM is general regardless of the phase shift γ. It has been shown by simulation that the stator current harmonic loss and output torque ripple are reduced. The studied SPIM multilevel inverter is suited for applications requiring high power synchronous motor drives.
Machine parameters: Rated Power: Pn=5,5 kW, Rated voltage: Vn=220 V, Rated current: In=6 A, Rated speed: Nn=1000 rpm, Number of poles: P=6, Stator resistance: Rs=2.03 Ω, Rotor resistance: Rr=3 Ω, Stator inductance: Ls=0.2147 H, Rotor inductance: Lr=0.2147 H, Mutual inductance: M=0.2 H, Moment of inertia: J=0.06 kg.m^{2}.
Acronyms
SVM: Space Vector Modulation
SPIM: Six Phase Induction Machine
IFOC: Indirect Field Oriented Control
VSI: VoltageSource Inverter
PWM: Pulse Width Modulation
SDCS: Separate Direct Current Source
NTV: Nearest Three Vectors
VSD: Vector Space Decomposition
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