###### Abstract
All harmonic components can cause various problems in the electrical drive, including overheating the equipment, erratic behavior, and reduced efficiency. However, subharmonic currents and the heat generated by their losses can damage the electrical machine faster than other causes, such as voltage imbalance [21; 22]. Again, attempts have been made to solve this issue, and recent research works mitigate the situation by applying two or more switching vectors during the same sampling period. For example, zero and active voltage vector combinations are applied at each sampling time in [23]. The active voltage vector is selected in response to conventional MPC optimization problems, and the time of application is calculated online using a linear and reduced-order model that depends on the predicted, measured, and reference values of the controlled variable. Similar approaches can be found in [24], using appropriate PWM methods to ensure fixed switching frequencies. An alternative strategy is to use offline computation and definition of available voltage vectors. These voltage vectors are called virtual voltage vectors (VVVs), which are basically predefined voltage models that ensure low average voltages in the \(x_{1}\)-\(y_{1}\) plane [25]. The obtained results have demonstrated an improvement in controller efficiency by reducing the harmonic content in the \(x_{1}\)-\(y_{1}\) planes at the cost of higher switching frequencies and lower available voltage limits [26]. In this context, a predictive control action is defined in [27] where variable control times are utilized. The proposal is particularly relevant because it adds an extra degree of freedom to the control action with nonuniform control frequencies, which allows higher temporal resolutions and mitigates the computational cost of the predictive controller. The motivation of our work is to extend the idea in [27], introducing VVV to reduce the harmonic content of the controlled system and simplify the control algorithm, where the \(x\)-\(y\) plane is no longer considered. The rest of the paper is organized as follows. The next section details the system under study, which is based on a five-phase induction motor (5 pIM) drive. Then, the proposed controller, which combines the lead-pursuit and VVV concepts, is shown in Section 3. The proposed analysis and the results obtained are presented in Section 4, while the conclusions are finally indicated in Section 5. The Case Study: Five-Phase Distributed Winding Induction Motor Drive Using a Conventional Two-Level VSI  
The analyzed system is based on a 5 pIM, a five-phase two-level voltage source inverter (5p2L-VSI) constructed from two SEMIKRON SKS 22F modules, and a DC power supply that acts as a DC link (KDC 300-50 from California Instruments Corporation, San Diego, CA, USA). The core of the control unit is an MSK28335 board with a TMS320F28335 digital signal processor model from Texas Instruments (Dallas, TX, USA). The feedback of the mechanical speed signal is provided by the GHM510296R/2500 encoder, while four LH25-NP Hall effect sensors from LEM are used to measure the stator phase-current signals. Note that the 5 pIM has an isolated neutral point, a balanced load is assumed (\(v_{st}+v_{sb}+v_{sc}+v_{sd}+v_{sw}=0\) and \(i_{su}+i_{sb}+i_{sc}+i_{sd}+i_{se}=0\)), and a reduced number of current sensors (four instead of five) is used to obtain the stator phase currents. A DC motor is also used to generate load torque. The system is shown in Figures 1 and 2, where some pictures of the hardware and the schematic diagram of the 5 pIM drive are provided. The 5 pIM was designed using equally distributed windings, \(\vartheta=2\pi/5\), and the 5p2L-VSI generates 32 voltage vectors (30 active and 2 zero-voltage vectors), which is characterized by the switching vector [\(S_{a}\)\(S_{b}\)\(S_{c}\)\(S_{d}\)\(S_{e}\)] as follows:  
\[ \begin{bmatrix}v_{sa}\\ v_{sb}\\ v_{sc}\\ v_{sd}\\ v_{se}\end{bmatrix} =\frac{V_{dc}}{5}\cdot