Gradient-Based Predictive Pulse Pattern Control With Active Neutral Point Potential Balancing for Medium-Voltage Induction Motor Drives

This paper presents a control method for medium-voltage (MV) drive systems consisting of a three-level neutral-point-clamped (NPC) inverter and an induction machine. The proposed approach aims to achieve stator current reference tracking and balancing of the neutral point (NP) potential in one computational stage, without the help of additional control loops. To this end, a model predictive control (MPC) algorithm, designed as a multiple-input multiple-output (MIMO) controller, manipulates optimized pulse patterns (OPPs) in real time. As a result, minimal current harmonic distortions are produced, while the NP potential is kept balanced both during steady-state and transient operating conditions. The effectiveness of the proposed control approach is validated in a real-time system by simulating the MV drive of interest in a hardware-in-the-loop (HIL) setting.

semiconductor devices, such as gate turn-off thyristors (GTOs) and gate commutated thyristors (GCTs) [2].However, such low switching frequencies can lead to high current distortions, and thus adverse effects, such as increased losses in the machine.To address this issue, optimized pulse patterns (OPPs) can be employed as they produce the theoretical minimum current distortions [3], [4].OPPs are switching sequences calculated offline, designed to have optimal harmonic properties [5].The computation of the switch positions and switching angles of the OPPs involves solving a mathematical optimization problem that accounts for the total demand distortion (TDD) of the load current, e.g., the stator current of a machine [3], [4].
Control of OPPs, however, is a nontrivial task.This is due to the fact that OPPs do not have a fixed-length modulation interval, meaning that when sampling occurs, not only the fundamental component is sampled, but also the ripple.Moreover, the discontinuities in the switching angles with respect to the modulation index complicates the controller design.As a result, OPPs have been traditionally used with low-bandwidth controllers.Alas, such controllers cannot achieve satisfactory transient performance and disturbance rejection.
To address the above, high-bandwidth controllers that adopt the concept of trajectory tracking control have been proposed to manipulate OPPs in real time.For example, control methods based on deadbeat control principles were proposed in [5], [6], [7].While the control scheme in [5] is based on the concept of stator flux trajectory tracking, the control scheme in [6], [7] utilizes the stator current trajectory.A more evolved controller designed in the framework of model predictive control (MPC)known as model predictive pulse pattern control (MP 3 C)-was presented in [8].This method manipulates OPPs in real time and has been validated experimentally in industrial MV drive systems [9], [10].In particular, MPC with OPPs is an attractive option since it can take advantage of the excellent steady-state performance and low current harmonic distortions attributed to OPPs as well as the fast dynamic responses that can be achieved with MPC.In this direction, the control method named gradientbased predictive pulse pattern control (GP 3 C) was recently proposed [11] to achieve superior steady-state and dynamic performance for motor drive systems.The GP 3 C method tracks the optimal stator current reference by optimally modifying the switching time instants of the nominal OPP.Nevertheless, the above-mentioned control techniques must fulfill additional control objectives when multilevel converters are considered as their internal voltages need to be balanced during the whole operation of the system.For example, when neutral-point-clamped (NPC) inverters are of interest, the neutral point (NP) potential should be kept around zero to avoid deviations of the phase voltages from the expected voltage levels.Hence, even though the NPC inverter has an inherent natural balancing mechanism [12], active balancing techniques are commonly employed to prevent the NP potential from drifting away.
To this aim, a variety of control strategies have been used to tackle the problem of NP potential balancing.Most of these methods are based on the manipulation of the common-mode component of the output voltage [13], [14], [15].Therefore, control of the NP potential is achieved by using an outer loop to manipulate the common-mode component of the reference voltage that is fed to the modulator.Alternatively, control of the NP potential can be achieved by exploiting the redundant switching vectors of the NPC inverter [16].It should be noted, however, that the effectiveness of these methods diminishes as the phase difference between the inverter voltage and current approaches 90 • [15].
Addition of external loops, however, can further limit the controller bandwidth.For this reason, the developed closed-loop control methods that manipulate OPPs aim at incorporating the NP potential balancing mechanism into the inner control loop.In this direction, similar to [16], [17] adopts the concept of redundant vector manipulation.By selecting appropriate redundant sub-bridges during steady-state and transient operation, the control method eliminates the NP potential error at low modulation indices while operating the power converter at low switching frequencies.In [18], a push-pull configuration for a variablespeed drive is presented where a five-level OPP is mapped into two three-level OPPs for two NPC inverters.This gives rise to an additional degree of freedom which can be utilized to balance the NP potential.In [19], the MP 3 C method addresses the balancing of the NP potential by adding an extra term to the objective function to penalize the deviation of the dc component of the NP potential from its reference.Nevertheless, during the derivation of the control problem, a number of assumptions are made that can compromise the overall performance of the control scheme.As a result, the controller design becomes more complicated.
Motivated by the above, this paper refines the GP 3 C method [11] to tackle the problem of the NP potential balancing in a simple, yet effective, manner. 1 By exploiting the high design versatility of GP 3 C attributed to its modeling principle, i.e., the use of the gradient of the system output to predict its evolution, the current control and NP potential balancing problems are tackled in one computational stage.In doing so, unlike traditional balancing methods, the proposed method does not rely on manipulating the common-mode voltage of the inverter, but it rather directly regulates the (instantaneous) NP potential along its reference. 2This equips the GP 3 C method with a high bandwidth and a high degree of disturbance rejection.The efficacy of the proposed method is verified using real-time tests performed in a hardware-in-the-loop (HIL) environment based on a variable speed MV drive system.
The remainder of this paper is organized as follows.Section II presents the model of the MV VSD system adopted for the case study.Following, the proposed GP 3 C algorithm is presented in Section III along with the formulation of the optimal control problem as a quadratic program (QP).Section IV assesses the performance of the proposed control strategy during steady-state and transient operating conditions based on real-time HIL simulations.Finally, Section V concludes this article.

II. MODEL OF THE MV DRIVE SYSTEM
Consider the MV variable speed drive system in Fig. 1 consisting of a three-level NPC voltage source inverter and an induction machine (IM) with the rated values and parameters provided in Table I.The mathematical model of the system is derived in the stationary αβ-frame, where the transformation matrix is used to transform the three-phase quantities into the αβ-frame.Throughout this paper, the quantities are normalized and presented in the per unit (p.u.) system.The dc link of the inverter comprises two identical capacitors C dc with (inverse) reactance X dc ; the midpoint N is the so-called NP.The total (constant) dc-link voltage is where v dc,up and v dc,lo denote the upper and the lower dc-link capacitor voltages, respectively.Depending on the single-phase switch position u x ∈ {−1, 0, 1} in phase x ∈ {a, b, c} the inverter can produce three possible phase voltage levels, namely v dc,lo , 0, and v dc,up , respectively.Hence, the three-phase output voltage of the inverter in the αβ-frame is a function of the three-phase switch position u abc = [u a u b u c ] T , and it is given by where T is the component-wise absolute value of the three-phase switch position.Note that since the inverter is driving a machine, the output voltage of the inverter is equal to the stator voltage v s .
As can be seen in ( 3), the output inverter voltage fluctuates with the NP potential, defined as This potential evolves as a function of the current flowing through the NP [21], i.e., The NP current i n changes when a phase current i sx flows through the NP.This happens when the corresponding switch position u x is zero, meaning that i n is a function of u abc and the inverter (i.e., stator) current i s,abc = [i sa i sb i sc ] T according to where a star connection for the load is assumed, i.e., i sa + i sb + i sc = 0. Using ( 5) and ( 6), the evolution of the NP potential can be written as Regarding the dynamics of the squirrel cage IM in Fig. 1, these can be described by the differential equations of the stator current i s and the rotor flux ψ r ,3 i.e., [22] di In (8), R s and R r are the stator and rotor resistance, respectively, while X ls , X lr , and X m are the stator leakage, rotor leakage, and mutual reactance, respectively.Moreover, ) and τ r = X r /R r , are the transient stator and rotor time constants, respectively, while the constant m , with X s = X ls + X m and X r = X lr + X m .In addition, ω r is the angular rotor speed.Finally, I 2 is the two-dimensional identity matrix.
Let us consider the three-phase switch position u abc as input to the drive system, the stator current and NP potential as the system output, i.e., y = [i sα i sβ v n ] T ∈ R 3 , and the stator current, rotor flux and the NP potential as the system state, i.e., the state vector is 5 .By using (7) and (8), the continuous-time state-space model of the drive system is written as ) where the system F (t) ∈ R 5×5 , input G ∈ R 5×3 , and output C ∈ R 3×5 matrices are provided in Appendix A. It is important to point out that the system matrix F (t) contains nonlinear terms due to the nonlinear NP dynamics.Such a characteristic poses difficulties from a controller design perspective.Subsequently, ( 9) is discretized with the sampling interval T s by employing forward Euler discretization.In doing so, the discrete-time state-space model of the drive is computed as where A(k) = I 5 + F (t)T s and B = GT s are the discretetime matrices, and k ∈ N denotes the discrete time step.

III. GP 3 C WITH ACTIVE NP POTENTIAL BALANCING
The proposed GP 3 C control scheme exploits the inherent characteristics of OPPs [3], [4] and gradient-based direct MPC [23], [24], [25].In the sequel, the control problem and the working principles of the algorithm are presented.The block diagram of the presented control algorithm is given in Fig. 2.
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A. Preliminaries
OPPs are computed in an offline procedure by solving an optimization problem that typically accounts for the TDD of the stator current.This problem is solved for a given pulse number d and yields a set of optimal switching angles as a function of the modulation index m.Fig. 3(a) shows the optimal switching angles for the OPP p(d, m), with d = 5, over the whole range of modulation indices m ∈ [0, 4/π].For a given set of switching angles, the three-phase OPP can be constructed at a given modulation index, by assuming quarter-and half-wave symmetry as well as a balanced three-phase system.As an example, the three-phase OPP p(5, 1.046) is depicted in Fig. 3(b).Furthermore, based on a given OPP, the steady-state current trajectory (Fig. 3(c)) can be computed, as explained in [11].

B. Control Problem
Consider a prediction horizon T p of finite length, i.e., T p = N p T s , where N p ∈ N + is the number of prediction steps.Let z ∈ N be the number of switching time instants of the nominal OPP that fall within the horizon T p .Moreover, for controller design purposes, the following vectors are introduced where t ref ∈ R z is the vector of switching time instants of the nominal OPP that fall within T p , U ∈ U 3(z+1) is the vector of the corresponding OPP switch positions, 4 and t ∈ R z includes the to-be-computed modified switching time instants.
The discussed control algorithm aims to regulate the stator current i s along its optimal reference trajectory i s,ref by manipulating (i.e., modifying) the switching time instants of the nominal OPP t ref such that minimal current distortions are achieved in steady state.Moreover, the voltage over the upper and lower dc-link capacitors, v dc,up and v dc,lo , respectively, should be balanced by minimizing the deviation of the NP potential v n from its reference v n,ref .These goals are to be achieved while modifying the nominal OPP as little as possible.Finally during transients, the controller should exhibit high bandwidth to achieve fast dynamic performance.
To meet the above-mentioned objectives, the control scheme is formulated as a constrained optimization problem.Specifically, the objective function that captures the output tracking error and the changes in the switching time instants is5 with y ref being the output reference vector, i.e., being a diagonal positive-definite matrix, whose entries penalize the deviation of the output variables from their respective references, i.e., i s − i s,ref and v n − v n,ref . 6Moreover, Δt = t ref − t denotes the (to-be-applied) modifications on the nominal OPP.The weighting factor λ t ≥ 0 penalizes the deviation of the modified switching time instants with respect to the nominal OPP, and thus serves as a tuning parameter to prioritize between the output tracking and allowed modifications in the nominal OPP.
Function (12) needs to be minimized to obtain the vector of modified switching time instants t.To do so, the evolution of the output variables within the prediction horizon must be computed.As the OPP switch positions u abc that fall within T p are known, see the switching sequences U (11b), the evolution of the system output can be computed based on its gradient.), respectively, to provide the most accurate computation of the corresponding gradient.This is accomplished by employing the discrete-time system model (10).
Finally, based on the above expressions and by introducing some assumptions as outlined in [11], the objective function (12) is rewritten as where r is a vector that depends on the reference values and measurements of the outputs, and M is a matrix of the slopes with which the controlled variables evolve over the prediction horizon, see Appendix B.

C. Control Algorithm
The proposed control method is designed in the discrete-time domain and works at equally spaced time instants kT s .The block diagram of the inner control loop is shown in Fig. 4. Furthermore, Algorithm 1 provides the pseudocode of the proposed control method.
Before the control algorithm is executed, the offline computed nominal OPP and harmonic current references are retrieved from the look-up tables (LUTs) where they are stored.With this, the three-phase OPP is computed and the stator current reference over the prediction horizon is constructed.Following, the current reference is aggregated into the output reference vector In a next step, according to (14), z unique output vector gradients are computed within the subintervals of the prediction horizon using the nominal OPP switching instants and the corresponding predicted output variables.This yields the gradient matrix M .Finally, the modified switching instants As per the receding horizon policy, the switch positions that fall within the first sampling interval T s are implemented at the corresponding time instants t * .7For a better understanding, the following example is provided.
Example 1: Consider the drive system in Fig. 1.As depicted in Fig. 5, , respectively, fall within the prediction horizon T p .These instants divide the horizon into five subintervals.Within each subinterval, it is assumed that the system output evolves with a constant gradient.Therefore, with the help of the discrete-time model (10) the evolution of the output can be predicted based on the corresponding gradients m(t ,ref ).For example, evolution of one of the controlled variables, i.e., i sα (dash-dotted, magenta line), is shown in Fig. 5 along with its sampled reference (dashed, black line).With the knowledge of the evolution of the system output within the horizon T p , the GP 3 C algorithm manipulates the OPP such that the error between the output and its reference is minimized, e.g., the error between i sα and i s,ref,α in Fig. 5.In doing so, the modified switching instants t 1 -t 4 are obtained that result in the stator current shown as solid (green) line.Finally, the (modified) pattern that falls within the first sampling interval T s -shown in red in Fig. 5-is applied to the inverter and the horizon is shifted by one T s .

IV. PERFORMANCE EVALUATION OF GP 3 C
This section assesses the effectiveness of the proposed GP 3 C scheme by means of real-time simulations based on a HIL setup.The block diagram of the HIL system is shown in Fig. 6.It consists of the PLECS RT-Box 1 real-time system which utilizes a Xilinx Z-7030 system-on-chip (SoC).The MV VSD system, shown in Fig. 1, is simulated on the processor of the PLECS RT-Box at a sampling interval of T s,HIL = 5 μs, while the field-programmable gate array (FPGA) is used for the data acquisition (DAQ).The measurements acquired from the VSD system are fed to the dSPACE control platform via an analog interface, while the switching signals for the three-level NPC inverter are received via digital inputs.For more details on the HIL setup, the reader is referred to [26].
The proposed control algorithm is implemented on the dSPACE SCALEXIO system which is based on a 2.8 GHz Intel i7 processor and a Xilinx Kintex-7 FPGA.The controller is implemented on the processor, and the data acquisition and the generation of the switching signals is done on the FPGA.
The sampling interval is T s = 50 μs and a 16-step prediction horizon (i.e., N p = 16) is chosen.The various parameters of the drive system are provided in Table I.Note that for the given parameters, a total leakage reactance X σ = 0.255 p.u. results.The dc-link voltage of the inverter is assumed to be constant.The weighting factors appearing in the objective function (15) are chosen as λ t = 1 • 10 6 and λ n = 5.The OPP in use has pulse number d = 5, while the modulation index is m = 1.046, unless otherwise stated.Finally, all results are shown in the p.u. system.

A. Steady-State Operation
To assess the steady-state behavior of the MV drive system, operation at nominal speed and rated torque is considered.This means that the desired electromagnetic torque is T e = 1 p.u., while the device switching frequency is 250 Hz since the chosen OPP has d = 5, and f 1 = 50 Hz.The corresponding results are presented in Fig. 7. 8 Starting with Fig. 7(a), which depicts the three-phase stator current over one fundamental period, it can be seen that the current reference tracking capability of the controller is excellent.Specifically, the deviations of the produced current from its optimal reference trajectory are only minute.As a result, the harmonic energy is very low, as indicated by the current TDD I TDD of 4.274%, which is very close to its theoretically minimum value.Furthermore, the harmonic energy is concentrated at frequencies that are odd, non-triplen integer multiples of the fundamental.This is thanks to the symmetry properties of the nominal OPP, which are preserved-to some extent-by the controller, as can be in Fig. 7(c) where the three-phase switching pattern generated by the controller is shown.As a result of the excellent tracking of the current reference trajectory, accurate tracking of the electromagnetic torque reference is also achieved, see Fig. 7(d).Finally, Fig. 7(e), shows that the controller successfully balances the NP potential around its reference.This is not realized at the expense of the current reference tracking performance, thus clearly demonstrating the multiple-input multiple-output (MIMO) feature and high versatility of the proposed control scheme.
Furthermore, for bench-marking purposes, the performance of the proposed GP 3 C scheme is compared with that of two established control strategies, namely, finite control set MPC (FCS-MPC) [27] and field oriented control (FOC) with space vector modulation (SVM) [10].This is done over the range of switching frequencies that are relevant for MV drives, specifically within the range of f sw ∈ [100, 500] Hz.For the presented analysis, and in line with the previous tests, operation at nominal speed and rated torque is considered.The associated trade-off curves between current TDD I TDD and switching frequency f sw are shown in Fig. 8.As evident from the results, the proposed control approach achieves significantly lower current TDD values than the other two control methods against which it is benchmarked.This reduction in harmonic current distortions becomes especially pronounced as the switching frequency decreases, where I TDD decreases by at least a factor of two.Thus, Fig.   clearly showcases the superior performance of the proposed GP 3 C during steady-state operating conditions as well as its suitability for operation at low switching frequencies.

B. Operation During Transients
To verify the effectiveness of the proposed GP 3 C scheme during transient operating conditions, its performance is tested for step changes in the desired torque.While operating at nominal speed, the torque reference is stepped down from T e,ref = 1 to 0 p.u. at t = 20 ms, see Fig. 9.Moreover, for reasons of completeness, a second case is examined where the torque reference is returned to its nominal value at t = 20 ms from no-load operating conditions, see Fig. 10.As observed in both of the aforementioned figures, the stator currents precisely follow their new reference values, leading to an excellent tracking of the torque reference, see Figs. 9(c) and 10(c).Despite the big changes in the torque reference, GP 3 C manages to quickly settle to the new operating points by significantly modifying the nominal OPP accordingly.This is clearly shown in Fig. 11, which zooms in on the time axis between t = 19 ms and 25 ms to show the control actions in detail.Additionally, the torque does not exhibit any overshoots during the step changes, while the settling times for the step-down and step-up tests are 3 ms and 4.7 ms, respectively.The former settling time indicates that high-bandwidth ability of the control scheme in discussion.The time required for the step-up transient, however, is somewhat longer due to physical limitations, i.e., the small voltage margin that is available in this scenario.
Notwithstanding the foregoing, the proposed controller modifies the nominal OPP during the torque reference step-up change such that the load angle changes as quickly as possible.For example, as can be seen in Fig. 11(e), this is done by significantly reducing the width of the pulses in all the phases and effectively removing the switching pulses in phases a and c.These results clearly show that the dynamic performance of GP 3 C is excellent and it is only limited by the available dc-link voltage.
Finally, Figs.9(d) and 10(d) show that the NP potential is kept balanced around zero, despite the large changes in the torque as well as operation at zero torque.With regards to the latter, as mentioned, balancing the NP potential at zero torque is challenging because the vectors (in the αβ-frame) of the applied voltage and stator current are perpendicular.Hence, this figure demonstrates the effectiveness of the active balancing mechanism of the proposed control method even when operating under such challenging conditions, i.e., zero power factor.

C. Operation Across a Discontinuity in the Switching Angles
As shown in Fig. 3(a), the switching angles of the OPP exhibit multiple discontinuities.When operating at a modulation index in the vicinity of such discontinuities, small fluctuations in the dc-link voltage or speed, or big changes in the torque can lead to frequent changes in the modulation index.This, in turn, results Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.  in repeated jumps across the discontinuity, and thus significant changes in the pulse pattern.Such a phenomenon can have an adverse effect on the operation of the drive system as current and torque excursions may occur, while additional switching transitions may appear that could potentially lead to an overfrequency trip.Hence, operation close to these discontinuities is challenging, and tends to be a forbidding task for conventional control methods.
To validate the operation of the proposed control scheme around a discontinuity in the switching angles, its performance is tested under step changes in the desired torque. 9Specifically, the machine is initially operated at rated torque and 82% speed, which corresponds to modulation index m = 0.8677.This operating point appears to the right-side of a discontinuity in the switching angles, i.e., the OPP in use is p(5, 0.8677) = [16.876• 49.319 • 56.277 • 77.529 • 87.820 • ] T , see Fig. 3(a).While keeping the rotor speed fixed, the torque reference is stepped down to T e,ref = 0 p.u. at t = 20 ms, resulting in the modulation index m = 0.8584.This operating point corresponds to the angles on the left-side of the discontinuity, i.e., the new OPP is p(5, 0.8584) = [28.334• 33.187 • 44.015 • 50.186 • 56.741 • ] T .Moreover, the inverse scenario is examined where the torque reference is stepped up from T e,ref = 0 to 1 p.u.The performance of the proposed GP 3 C scheme across the discontinuity is depicted in Figs. 12 and 13 for the step-down an step-up cases, respectively.Moreover, Fig. 14 zooms in on the time axis to provide a more detailed view on the behavior of the controller during both examined scenarios.As can be observed, the controller operates smoothly across the discontinuity in the switching angles as Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.it changes seamlessly between the different OPPs.This is achieved while exhibiting favorable dynamic performance without excursions in the torque and/or stator current, thus ensuring the continuous and safe operation of the drive.

D. Evaluation of the Active NP Potential Balancing Mechanism
To further investigate the NP potential balancing ability of the proposed algorithm, the drive system is tested under two different torque scenarios, i.e., under rated torque and under zero torque10 conditions.As mentioned earlier, the latter operating point is acknowledged as a critical one for three-level NPC converters [15].
Starting with the first scenario, operation of the drive at nominal speed and rated torque is considered.The initial value of the NP potential is offset by 0.1 p.u..The weighting factor λ n is varied and the resulting performance is shown in Fig. 15.As can be inferred, the closed-loop natural balancing of the NP potential, i.e., when λ n = 0 (see Fig. 15(b)), is slower compared with the active NP balancing achieved with the proposed controller, i.e., when λ n > 0. Specifically, as the controller prioritizes the NP potential balancing, i.e., as λ n increases, the NP potential is balanced faster, see Fig.As the second test, this is performed at nominal speed and zero torque.An initial offset of 0.1 p.u. is again introduced to the NP potential.The open-loop response of the NP potential (see Fig. 16(a)) has an average decay rate of 0.018 p.u. per second and thus it takes around 15 s to remove the offset.On the contrary, the proposed GP 3 C scheme balances the NP potential almost thirty times faster, as it drives it to zero within around 0.5 s, see

E. Computational Burden
The main obstacle in implementing GP 3 C in real time is the need to solve the computationally intensive QP problem (16).
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To address this, the solver developed in [28] is employed.This solver is tailored to the structure of the optimization problem underlying GP 3 C as it fully exploits its geometry.Thus, thanks to this highly computationally efficient solver, the computation time involved remains modest.To provide a quantitative measure of this efficiency, Fig. 17 illustrates the probability distribution of the number of iterations n iter the QP solver needs to find the solution.As depicted, an average of 12 iterations is sufficient to achieve this.To further elucidate this point, Fig. 18 illustrates the probability distribution of the total turnaround time t tot for the complete GP 3 C scheme.As can be seen, in over 95% of instances, the total turnaround time t tot is under 33 μs for the selected controller settings.This highlights the capability of solving the entire GP 3 C algorithm within the chosen controller sampling interval T s .This is reflected by the total maximum turnaround time t tot,max = 39.786μs, which is smaller than the sampling interval T s = 50 μs, i.e., only about 80% of the available time is used to execute the controller.

V. CONCLUSION
This paper refined the GP 3 C algorithm introduced in [11] to incorporate the balancing of the NP potential of a three-level NPC inverter into the control problem.The mathematical model adopted within the framework of the proposed controller equips GP 3 C with high versatility and enables it to simultaneously address multiple control objectives.To exploit these favorable characteristics, the discussed method was applied to an MV drive system, and its performance was assessed in a real-time environment by employing a HIL system.As highlighted by the presented results, thanks to the combination of optimal constrained control and optimal modulation, the proposed strategy not only exhibits superior performance during steady state, i.e., minimal current TDD for a given switching frequency, but it is characterized by high bandwidth as demonstrated by the short settling times during transients and its ability to quickly balance the NP potential over the whole operating regime.Finally, the presented comparisons with established control methods, such as FOC with SVM and FCS-MPC, further showcased the superior performance of the proposed control method.

APPENDIX B OBJECTIVE FUNCTION MATRICES
The vector r and matrix M in (15)

Fig. 2 .
Fig. 2. Block diagram of the outer control loops of the GP 3 C scheme.

Fig. 3 .
Fig. 3. (a) OPP p(d, m) for a three-level converter with d = 5 switching angles per quarter of the fundamental period.The optimal switching angles for the modulation index m = 1.046 are indicated by (black) circles.(b) Three-phase OPP for m = 1.046.(c) The current reference trajectory (solid blue line) for the given three-phase OPP is a combination of the fundamental component i s1,ref (red dash-dotted line) and the harmonic component i sh,ref .

Fig. 5 .
Fig.5.Example of trajectory tracking of GP3 C for one of the controlled variables (e.g., stator current i sα ) within a four-step (T p = 4T s ) prediction horizon.The nominal OPP instants t i,ref (dotted line) and the modified switching instants t i (solid line) are shown, where i ∈ {1, 2, 3, 4}.The modifications introduced by the controller are highlighted in yellow.In the bottom figure, the dash-dotted (magenta) line represents the current trajectory when applying the nominal OPP, while the solid (green) line shows the current trajectory based on the modified pulse pattern.The dashed (black) line is the current reference sampled at the nominal OPP time instants.

Fig. 7 .
Fig. 7. HIL results of the proposed GP 3 C algorithm at steady-state operation, nominal speed and rated torque.The modulation index is m = 1.046, the pulse number d = 5, and the switching frequency is 250 Hz.

Fig. 8 .
Fig.8.Trade-off between current TDD I TDD and switching frequency f sw for the proposed GP 3 C scheme, one-step FCS-MPC and FOC with SVM.

Fig. 9 .
Fig.9.HIL results during a torque reference step-down change (rated load to no load) when operating at nominal speed.

Fig. 10 .
Fig.10.HIL results during a torque reference step-up change (no load to rated load) when operating at nominal speed.

Fig. 11 .
Fig. 11.Transient performance of GP 3 C at nominal speed during a torque reference (a)-(c) step-down change, and (d)-(f) step-up change (zoomed in).In (b) and (e), the (black) dash-dotted lines indicate the switching sequence of the unmodified, nominal OPP, whereas the solid lines refer to the modified switching sequence as computed by GP 3 C.

Fig. 12 .
Fig.12.HIL results during a torque reference step-down change (rated load to no load) when operating at 82% speed.

Fig. 13 .
Fig.13.HIL results during a torque reference step-up change (no load to rated load) when operating at 82% speed.

Fig. 15 .
Fig. 15.Comparison of natural balancing and NP potential balancing for different values of λ n when operating at rated torque.The initial offset of the NP potential is 0.1 p.u.

15
(c) and (d).It is also worth noting that the closed-loop natural balancing of GP 3 C is significantly faster than the open-loop natural balancing of the nominal OPPs, see Fig. 15(b) and (a), respectively.

Fig. 16 .
Fig. 16.Comparison of natural balancing and NP potential balancing when operating at zero torque.The initial offset of the NP potential is 0.1 p.u.

Fig. 17 .
Fig. 17.Probability distribution of the number of iteration steps required by the QP solver to find a solution.The average number of iterations is indicated by the solid vertical line.The 95 and 99 percentiles are shown as dashed and dotted vertical lines, respectively.

Fig. 18 .
Fig. 18.Probability distribution of the total turnaround time t tot .The average turnaround time is indicated by the solid vertical line.The 95 and 99 percentiles are shown as dashed and dotted vertical lines, respectively.

Fig. 16 (
Fig.16(b).These results clearly highlight the inherent ability of the proposed control method to actively and effectively balance the NP potential.