Machines with more than three phases are called multiphase machines, which provide a better current distribution among phases, and lower current harmonic production in the power converter, than conventional threephase machines. However, multiphase drive applications require the development of complex controllers to regulate the torque (or speed) and flux of the machine. In this regard, direct torque controllers have appeared as a viable alternative due to their easy formulation and high flexibility to incorporate control objectives.
1. Introduction
Electric drives are the basis of locomotive traction, electric ship propulsion, electric aircraft with various auxiliary functions (e.g., fuel pumps, starter/generator solutions, etc.), and renewable energy production. Although conventional threephase drives represent the principal choice for industrial applications, multiphase ones have recently aroused the interest of practitioner engineers and researchers in the field. Any energy conversion system formed by a multiphase electric machine and converter and regulated by a certain control technique is called a multiphase drive. The first application of such a system, particularly for a fivephase drive, was used in the late 1960s ^{[1]}, showing the advantages of multiphase systems over conventional threephase ones. The main interest in the proposal was that the higher number of phases yields a torque ripple three times lower with respect to the equivalent threephase case due to a better power distribution per phase, this being one of the most reported problems in conventional drives at that time. However, it was not until the end of the 20th and the beginning of the 21st centuries that the interest of researchers in multiphase machines was renewed due to two main reasons. First, the development of highpower and highfrequency semiconductors and, consequently, the appearance of pulse width modulation (PWM) methods to control the ON and OFF states of these electronic devices, as well as the energy conversion process. Second, there is the development of microelectronic technology and the appearance of powerful electronic devices with the ability to implement control algorithms in real time, such as digital signal processors (DSPs) and fieldprogrammable gate arrays (FPGAs).
Notwithstanding the above, the crucial reason for the renewed interest in multiphase drives can be found in the intrinsic benefits that they provide versus the conventional threephase ones. These benefits are based on the extra degrees of freedom introduced by the higher number of phases and are principally the following:

The faulttolerant capability against a fault situation in the machine and/or the power converter, first presented in ^{[2]}. An nphase machine can operate after one or several fault occurrences without any external equipment, as long as the number of healthy phases remains greater than or equal to three (assuming a single isolated neutral connection). Consequently, the system reliability is enhanced at the expense of a reduction in the postfault electrical torque production.

The capability of increasing the power density in healthy operation by injecting specific current harmonics, exposed in ^{[3]}. This is possible in certain multiphase machine configurations based on concentrated windings, where the lower current harmonic components can be used to increase the torque production.
Although fieldoriented control (FOC) methods, based on decoupled control of the flux and electromagnetic torque and assisted by modulation stages, can be considered as the most popular control technique for conventional and multiphase drives ^{[4]}^{[5]}, direct control techniques have recently been presented as interesting competitors ^{[6]}^{[7]}^{[8]}. The essence of direct controllers is to eliminate any form of modulation, forcing the states of the power switches to rapidly track a reference value. Then, the meaning of ‘direct control’ techniques is related to control strategies without the intervention of a pulse width modulation or any other form of modulation, providing control commands that are applied directly to the power converter. As a main consequence, direct controllers, being direct torque controllers (DTC) are the most extended industrial alternative, can favor fast torque responses and control robustness with respect to the variation of the electrical parameters of the machine. In this regard, DTC appears to be a viable (from a commercial perspective) control alternative in conventional threephase drives due to an easy formulation and high flexibility to incorporate different control objectives. However, the use of DTC in multiphase drives is restricted in normal operation due to the impossibility of regulating more than two degrees of freedom (electrical torque and stator flux).
2. FivePhase Distributed Windings Induction Motor Drive Using a Conventional TwoLevel VSI
A graphical representation of the analyzed system is shown in Figure 1. It is based on a fivephase Induction Machine (IM) with a squirrelcage rotor and symmetrically distributed stator windings (spatial equal displacement between windings) fed by a DC power supply through a fivephase twolevel voltage source inverter (VSI).
Figure 1. Schematic diagram.
Figure 2 shows the twodimensional projections obtained for every vector, identified with the decimal number equivalent of their respective switching state [S_{a} S_{b} S_{c} S_{d} S_{e}]^{T} expressed in binary logic (1 or 0), being S_{a} and S_{e} the most and the least significant bits, respectively. These vectors uniformly divide the space that they occupy in 10 sectors with a separation of π/5 between them. Likewise, active voltage vectors can be classified according to their magnitude in long (0.647 V_{dc}), medium (0.4 V_{dc}), and short (0.247 V_{dc}) vectors. The switching states that generate long vectors in the α–β plane correspond to those that generate short vectors in the plane x–y and vice versa. The switching states corresponding to vectors of medium magnitude in the α–β plane, also generate medium vectors in the plane x–y. Null vectors are generated by the same switching states in both planes. This transformation allows for a detailed study of the harmonic components, since they are projected in certain planes. In particular, the fundamental frequency together with the harmonics of order 10 k ± 1 (k = 0, 1, 2, etc.) are mapped in the α–β plane, while the harmonics of order 10 k ± 3 are related to the plane x–y. The homopolar component and harmonics of order 5 k are projected on the zaxis.
Figure 2. Mapping of the phase stator voltages of the twolevel fivephase VSI in the α–β (left graph) and x–y (right graph) planes.
3. DTC in FivePhase Drives
Direct Torque Control is a wellknown strategy for threephase electrical drives. It was presented in the mid1980s by Takahashi ^{[9]} and Depenbrock ^{[10]}, showing fast flux and torque responses, as well as more robustness with respect to the variation of the electrical parameters of the machine and generating a hightorque/flux ripple and harmonic current content, compared to the more standard fieldoriented control technique. The operating principle is based on an offline lookup table, which is used to select the stator voltage to be applied to the machine. The selection is made taking into account the position of the flux vector and the stator flux and electromagnetic torque error signals, obtained from the difference between reference and estimated values and processed using hysteresis comparators. Another disadvantage of DTC that should be considered from the analysis of its operating principle is that it does not generate a constant switching frequency. In fact, this switching frequency is variable and depends on the operating point and the bandwidth of the hysteresis controllers. Note, however, that DTC schemes have been proposed to also be used with PI regulators and space vector PWM methods (see ^{[11]}), to compensate for the variable switching frequency and reduce the torque and flux ripple.
DTC has been commercialized ^{[12]} and extended to the case of multiphase drives in recent times, considering different types of machines ^{[13]}^{[14]}, machine neutral connections ^{[15]}^{[16]}, and drives without speed sensors ^{[17]}. In the case of multiphase drives, since the controller has only two freedom degrees (stator flux and electromagnetic torque), there is no chance of regulating the current and voltage components in the orthogonal α–β and x–y planes. In this sense, some DTC strategies have been developed that satisfy this additional requirement, controlling the current and voltage components in the α–β plane while reducing at the same time the current and voltage components in the x–y plane. For example, in ^{[18]}^{[19]}, a modification of the traditional control scheme is proposed, performing a twostep search to minimize the effect of loworder harmonics. Alternatively, the use of virtual vectors has been suggested to reduce current distortion ^{[17]}. Some criteria have also been included in the selection process within the lookup table to improve its performance in the lowspeed region and an optimization between the two zero vectors to minimize the average switching frequency obtained ^{[20]}. On the other hand, and based on the virtual vectors defined in ^{[17]}, different DTC schemes are presented defining new virtual vectors and avoiding the use of the zero vector to reduce the commonmode voltage generated by the VSI in ^{[21]}^{[22]}, to improve openphase fault operations in ^{[23]}, or to avoid any reconfiguration of the controller when openphase faults appear ^{[24]}.
3.1. SteadyState Operation
First, the performance of the system in steadystate operation at 500 rpm is analyzed in Figure 3 and Figure 4, where different load torques are applied (1 N·m in Figure 3 and 2.75 N·m in Figure 4). The reference and measured values are colored red and blue, respectively. The speed and electrical torque responses are shown in the upper rows, where it is appreciated that the controller works well and the mechanical speed is successfully maintained in the reference value. Note that the electrical torque is mathematically estimated using the machine model, which produces some estimation errors. The reference and estimated stator flux in the regulated α–β plane are then shown in the second row, where it can be observed that the estimated stator flux values coincide with their references. Lastly, the measured stator currents are depicted in the last two rows, where it is appreciated that the α–β stator current vector describes a circular trajectory, with nearly null x–y stator current components. Therefore, the control goals are met using the DTC controller in steadystate operation because the results obtained can be extended to different reference speed and load torques.
Figure 3. Experimental steadystate operation test where the reference speed is settled at 500 rpm and a load torque of 1 N·m is applied. Upper row: speed and torque responses. Second row: stator flux waveforms. Third row: current trajectories of the stator in the α–β and x–y planes. Last row: stator phase currents.
Figure 4. Experimental steadystate operation test where the reference speed is set at 500 rpm and a load torque of 2.75 N·m is applied. Upper row: speed and torque responses. Second row: stator flux waveforms. Third row: current trajectories of the stator in the α–β and x–y planes. Last row: stator phase currents.
3.2. Load Torque Rejection
Then load torque rejection tests were performed. The results obtained are summarized in Figure 5, where the reference speed is 500 rpm and the coupled DC machine imposes a heavy load torque within the system limits at t = 0.5 s. A drop in the speed is observed when the load is suddenly applied; although, the controller successfully manages this disturbance, upperleft plot. The estimated electrical torque is also regulated to be the referred one in steady and transient states, as shown in the upper right figure, while stator phase currents increase to manage the increment in the load (see bottomleft timing diagram). The estimated stator flux value is regulated in steady and transient states to coincide with the references, as can be appreciated in the bottom right figure. Then, these results, which summarize the ones obtained under different operating conditions, prove a controlled electrical torque in the multiphase drive. Note that flat lines are also observed in the x–y plane polar diagrams of the stator current, similar to the ones shown in Figure 3 and Figure 4.
Figure 5. Experimental response of the controlled system in a load torque rejection test. The reference speed is 500 rpm and a load torque is applied at 0.5 s. The upper row shows the speed and torque responses. The lower row shows the stator current waveform and modulus of the stator flux during the test.