# Adaline-Based Control Schemes for Non-Sinusoidal Multiphase Drives—Part II: Torque Optimization for Faulty Mode

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## Abstract

**:**

## 1. Introduction

## 2. Modeling of a Seven-Phase PMSM

_{sn}are the 7-dimensional vectors of phase voltages, phase currents, and speed-normalized NS-EMFs, respectively; R is the resistance of the stator winding of one phase; [

**L**] is the 7-by-7 stator inductance matrix; and Ω is the rotating speed of the rotor.

_{dq}is the 7-dimensional vector of currents in d-q frames; (i

_{d}

_{1}, i

_{q}

_{1}), (i

_{d}

_{9}, i

_{q}

_{9}), (i

_{d}

_{3}, i

_{q}

_{3}), and i

_{z}are currents in two-dimensional frames (d

_{1}-q

_{1}), (d

_{9}-q

_{9}), (d

_{3}-q

_{3}), and zero-sequence frame z, respectively; [

**T**] and [

**P**] are the 7-by-7 Clarke and Park transformation matrices, respectively (see part I of this study).

_{1}-q

_{1}), (d

_{9}-q

_{9}), (d

_{3}-q

_{3}), and z [4]. In healthy mode, a fictitious machine with its corresponding d-q frames is associated with a group of harmonics in natural frame as described in Table 1. Therefore, harmonic orders presenting in d-q frames of currents or NS-EMFs can be 0 (constant component), 14th, and 28th, as previously described in the first part of this study.

_{d}

_{1}, i

_{q}

_{1}, i

_{d}

_{9}, i

_{q}

_{9}, i

_{d}

_{3}, and i

_{q}

_{3}) are constant to generate constant torque, facilitating the use of PI controllers in the conventional RFOC scheme. When an open-circuit fault happens, the decoupled reference frames become coupled frames. A constant torque is only guaranteed when new complex reference currents are determined.

- The 1st and 13th are associated with FM1;
- The 9th and 19th are associated with FM2;
- The 3rd and 11th are associated with FM3.

_{z}= 0).

_{n}

_{1}, E

_{n}

_{13}, E

_{n}

_{9}, E

_{n}

_{19}, E

_{n}

_{3}, and E

_{n}

_{11}) and (φ

_{1}, φ

_{13}, φ

_{9}, φ

_{19}, φ

_{3}, and φ

_{11}) are the speed-normalized amplitudes and initial phase angles of back-EMF harmonics 1st, 13th, 9th, 19th, 3rd, and 11th, respectively; j is the phase number from 1 to 7 or from A to G; and θ is the electrical rotor position.

_{em}can be expressed as follows:

_{FM}

_{1}, T

_{FM}

_{2}, and T

_{FM}

_{3}are the torques of FM1, FM2, and FM3, respectively; and e

_{dq}is the 7-dimensional vector of NS-EMFs in d-q frames.

## 3. Conventional Control of Multiphase Machines for Faulty Mode

#### 3.1. Conventional RFOC Scheme for Faulty Mode

_{A}) is nullified. The conventional RFOC scheme used for healthy mode can be preserved in faulty mode as presented in Figure 1. However, new reference phase currents i

_{ref}need to be determined to obtain smooth torque. Conventionally, from a given reference torque T

_{em_ref}, MTPA strategy [9,10] or [11] can be used to calculate optimal currents in faulty mode. The reference phase currents are then transformed into d-q frames i

_{dq}

_{_ref}. Speed-normalized NS-EMFs and Park transformations are calculated with the rotor position θ from an encoder. Six PI controllers are used to control six reference d-q currents, generating voltage references. These voltage references are used to generate switching signals for a voltage source inverter (VSI) using the carrier-based pulse width modulation (PWM) strategy.

#### 3.2. Maximum Torque per Ampere for Faulty Mode (MTPA_Fault)

_{ref}for the six remaining healthy phases to generate a constant reference torque T

_{em_ref}are proposed in [10] as follows:

_{A}to zero; e

_{z}is a 7-dimensional vector to satisfy a null zero-sequence current in the star connection; $\Vert {\underset{\_}{e}}_{sn\_f}-{\underset{\_}{e}}_{z}\Vert $ is the norm of vector $\left({\underset{\_}{e}}_{sn\_f}-{\underset{\_}{e}}_{z}\right)$; and n is the number of phases (n = 7 in this study).

_{ref}) are transformed into reference d-q currents (i

_{dq}

_{_ref}) by using (2). These reference d-q currents are time-variant and controlled by six PI controllers of the conventional RFOC scheme as described in Figure 1.

## 4. Proposed Adaline-Based Control Scheme for Faulty Mode

#### 4.1. New Reference Currents with Equal Copper Losses (ECL)

_{e}is the amplitude ratio of the 3rd harmonic to 1st harmonic of NS-EMFs (k

_{e}= E

_{n}

_{3}/E

_{n}

_{1}); and φ

_{e}is the phase shift angle between the 3rd and 1st harmonics of NS-EMFs (φ

_{e}= φ

_{3}− φ

_{1}).

_{m}

_{1}is the amplitude of the 1st harmonic of the six remaining healthy phase currents; φ

_{B}, φ

_{C}, and φ

_{D}are initial phase angles of currents in phases B, C, and D, respectively.

_{em}is expressed by:

_{ave}is the average torque; and T

_{2θ}and T

_{4θ}are torque ripples with harmonics 2θ and 4θ, respectively.

_{ave}and harmonic components T

_{2θ}and T

_{4θ}. With S-EMFs, k

_{e}is zero, the total torque becomes constant when the first term of T

_{2θ}is nullified. In this case, the required values of φ

_{B}, φ

_{C}, and φ

_{D}are described as follows:

_{e}> 0) generates an additional term in T

_{2θ}and the entire harmonic T

_{4θ}. These harmonic components cannot be simultaneously eliminated with sinusoidal currents (no solution). In this case, an injection of the 3rd harmonic of currents can improve the torque quality. Indeed, new non-sinusoidal reference currents containing the 1st and 3rd harmonics for the remaining healthy phases are proposed as follows:

_{i}and φ

_{i}are the amplitude ratio and the phase shift angle of the 3rd harmonic to 1st harmonic of currents, respectively.

_{ave}(constant) and three harmonic components T

_{2θ}, T

_{4θ}, and T

_{6θ}. Due to the 3rd harmonic of currents with k

_{i}, the average torque T

_{ave}increases an amount proportional to (k

_{e}k

_{i}). Meanwhile, the torque harmonics T

_{2θ}, T

_{4θ}, and T

_{6θ}cannot be simultaneously eliminated because there are no solutions under the constraint on identical current waveforms in the six remaining healthy phases in (10). For the sake of simplicity, the initial phase angles in (9) can be used to eliminate the first term of T

_{2θ}which has the highest amplitude.

_{e}k

_{i}I

_{m}

_{1}E

_{n}

_{1}cos(φ

_{i}− φ

_{e})] when the 3rd harmonic of currents are injected. The amplitudes of torque harmonics (T

_{2θ}, T

_{4θ}, and T

_{6θ}) are proportional to k

_{e}and k

_{i}. The phase shift angle φ

_{i}must be equal to φ

_{e}to maximize the average torque T

_{ave}. The amplitude ratio k

_{i}is equal to k

_{e}to minimize torque ripples and the rest of torque ripples will be eliminated by the proposed Adaline in the next subsection.

_{ave}in (12), the amplitude of the 1st harmonic of reference phase currents can be expressed as:

_{ave}, reference non-sinusoidal phase currents for the remaining healthy phases can be determined by using (9), (10) and (13). When NS-EMFs contain only the 1st and 3rd harmonics, the torque ripples with sinusoidal currents include two harmonics 2θ and 4θ as described in (8). Meanwhile, with the injection of the 3rd harmonic of currents, there are three harmonics 2θ, 4θ, and 6θ as described in (11).

#### 4.2. Proposed Control Scheme for Faulty Mode

_{ave}equal to T

_{em_ref}. Then, compensating currents are adaptively added to identical waveform phase currents of ECL to eliminate torque ripples. Therefore, phase current waveforms are no longer identical but similar, leading to similar copper losses in the remaining healthy phases. These compensating currents are determined from the compensating torque T

_{em_com}that contains all harmonics or main harmonics of the torque in Table 2. This compensating torque is determined by an Adaline (“Adaline_Fault”) that will be described in the next subsection. The total torque T

_{em}is calculated from measured phase currents and estimated NS-EMFs according to (4).

_{em_com}, compensating currents can be calculated by using Simplified MTPA (SMTPA) that is discussed in the first part of this study as follows:

_{com}is the 7-dimensional vector of compensating currents in natural frame; ${\underset{\_}{e}}_{sn\_f}^{sim}$ is the simplified NS-EMF vector created from e

_{sn_f}in (5) by keeping only 1st, 9th, and 3rd harmonics. Then, currents i

_{com}in (14) are transformed into d-q frames i

_{dq_com}by using (2).

_{dq_ref}) from ECL strategy and the compensating currents i

_{dq_com}as follows:

#### 4.3. Structure of Adaline for Faulty Mode

_{em_com}as well as compensating currents i

_{com}, enabling the total torque T

_{em}to properly track reference torque T

_{em_ref}. It is assumed that the total torque ${T}_{em}^{\ast}$ generated by using only ECL strategy can be expressed from Table 2 as follows:

_{em_ref}, the compensating torque T

_{em_com}is given by:

_{em_ref}and ${T}_{ave}^{\ast}$. Therefore, the compensating torque T

_{em_com}is represented by (2h + 1) coefficients from ${w}_{0}^{\ast}$ to ${w}_{2h}^{\ast}$ and the electrical position θ.

_{j}is a weight corresponding to coefficient ${w}_{j}^{\ast}$ of the compensating torque T

_{em_com}in (17) with $j=\left\{0,1,2,\dots ,2h\right\}$.

_{error}is the error between reference torque T

_{em_ref}and the total torque T

_{em}; and T

_{em}is calculated from measured phase currents and estimated NS-EMFs in (4).

_{em_com}) such as amplitudes and phases of harmonic components. An increase in η possibly results in faster convergence but divergence may appear. On each iteration, the Adaline weights are updated to converge to the coefficients of the compensating torque in (17). Weight convergence is obtained after a given number of iterations as:

_{em_com}as described in (17).

_{error}is theoretically nullified, and the total torque T

_{em}well tracks the reference torque T

_{em_ref}. The proposed Adaline-based control scheme using ECL strategy can guarantee smooth torques in faulty mode regardless of multi-harmonics existing in NS-EMFs and impacts of the mechanical load.

_{ave}, this harmonic can be neglected in the Adaline inputs. Practically, a S-function builder block in MATLAB/Simulink can be used to implement the Adaline.

## 5. Numerical and Experimental Results

#### 5.1. Experimental Seven-Phase Test Bench Description

#### 5.2. Numerical Results

_{em_ref}is reduced from 33.5 N.m (rated torque) in healthy mode to 24.5 N.m in faulty mode to respect the rated RMS current (5.1 A), avoiding the overheating of machine windings. The four operating stages are described as follows:

- Stage 1: The conventional RFOC scheme with MTPA strategy is used in healthy condition (“MTPA_HM”) as explicitly described in the first part of this study;
- Stage 2: Phase A is opened without any reconfigurations;
- Stage 3: The RFOC scheme is preserved but MTPA_Fault is applied (Figure 1);
- Stage 4: The proposed control scheme using Adaline_Fault and ECL strategy (Figure 2), briefly called Adaline_Fault, is applied. Eleven even harmonics (h = 11) of torque ripples are used to generate compensating torque T
_{em_com}, leading to 23 weights (2h + 1).

_{em}) and min(T

_{em}) are maximum and minimum values of the electromagnetic torque T

_{em}, respectively; and T

_{ave}is the average value of T

_{em}.

_{error}, the difference between T

_{em_ref}and T

_{em}. When the Adaline output y converges to the compensating torque T

_{em_com}, T

_{error}is minimized to reduce the torque ripple from 9.8% to 4.3%. Torques generated by the three fictitious machines in stages 3 and 4 are presented in Figure 6b. Notably, torque harmonics in FM1, FM2, and FM3 eliminate each other in both stages. Torques with MTPA_Fault have more harmonic components than those of Adaline_Fault. The proposed explanation for this difference is that MTPA_Fault is a compensation with additions of harmonics to different d-q frames by using all harmonics of NS-EMFs ${\underset{\_}{e}}_{sn\_f}$ in (5). With Adaline_Fault, the compensation is directly calculated at the level of torque with its harmonic components. In addition, as Adaline_fault is online-trained, reference currents are adaptively generated to eliminate torque ripples.

_{DC}/2 = 100 V). Therefore, numerical voltage references and phase currents at 300 rpm are shown to compare with experimental results. Figure 7b presents phase currents (response signals) with the RFOC scheme using MTPA_Fault and the proposed scheme using Adaline_Fault. Notably, Adaline_Fault has a lower maximum peak current (8.7 A) compared with MTPA_Fault (10 A).

#### 5.3. Experimental Results

_{DC}/2 = 100 V). Therefore, from the numerical voltage references in Figure 7a, the rotating speed of the drive in experiments for faulty mode is limited to 300 rpm instead of the rated speed (750 rpm). Higher speeds could make the effectiveness of the proposed Adaline-based scheme more significant as presented in the numerical section (see Figure 5 and Table 4).

_{error}. It is noted that the main harmonic of torque ripples in stage 3 is T

_{2θ}due to 2 peaks in one electrical period. When y converges to T

_{em_com}, T

_{error}is almost eliminated, making the torque ripple decrease from 30.3% to 4%. Torques generated by the three fictitious machines are presented in Figure 10b. In general, T

_{FM}

_{1}, T

_{FM}

_{3}, and T

_{FM}

_{3}compensate each other. As numerical results (see Figure 6), torques with the proposed Adaline-based scheme contain less harmonics than those of the RFOC scheme using MTPA_Fault do.

_{3}- and q

_{9}-axis currents.

_{0}, w

_{1}, and w

_{2}related to the main harmonics of torque ripples (T

_{2θ}) are zoomed in. The rotating speed variations between 200 rpm and 300 rpm have effects on torque quality as shown in Figure 13a with torque overshoots in transient states, especially with small values of learning rate. In steady states, the torque ripple is about 4%. When the learning rate increases to 0.02, the overshoots slightly decrease. Thus, weight variations can reduce the effect of the speed variation on torque. However, an increase in learning rate results in more oscillations of weights, leading to slightly longer learning time. When the reference torque varies between 10 N.m and 20 N.m at 300 rpm as shown in Figure 13b, the torque ripple increases from 5.2% to 10.3%. However, the torque quality is guaranteed because this ripple increase is due to the decrease in the average torque. Adaline weights are updated in each torque variation and remain almost constant at steady state. An increase in learning rate results in shorter learning time but higher overshoots appear.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Torque control with the conventional RFOC scheme using MTPA_Fault for a seven-phase PMSM when phase A is opened.

**Figure 2.**Torque control with the proposed control scheme using Adaline_Fault and ECL strategy for a seven-phase PMSM when phase A is opened.

**Figure 3.**Adaline structure with single layer for the proposed control scheme of a seven-phase PMSM in faulty condition.

**Figure 4.**Experimental test bench and the multi-harmonic NS-EMF spectrum of the seven-phase PMSM to validate the proposed control scheme using Adaline_Fault and ECL strategy in faulty mode.

**Figure 5.**(Numerical result) The torque at 100 rpm (green), 300 rpm (blue), and 750 (red) rpm in the four operating stages, and 23 Adaline weights at 750 rpm with learning rate η = 0.0003 in stage 4.

**Figure 6.**(Numerical result) Switch from stage 3 to 4 when phase A is opened at 750 rpm: (

**a**) elimination of T

_{error}with the proposed Adaline-based scheme (η = 0.0003) in stage 4; and (

**b**) torques of three fictitious machines T

_{FM}

_{1}, T

_{FM}

_{2}, and T

_{FM}

_{3}in one electrical period.

**Figure 7.**(Numerical result) Voltages and currents at 300 rpm in stages 3 and 4 when phase A is opened: (

**a**) phase voltage references; and (

**b**) phase currents.

**Figure 8.**(Numerical result) Harmonic spectrum of phase-B current with the RFOC scheme using MTPA_Fault (Stage 3), and the proposed scheme using Adaline_Fault (Stage 4), when phase A is opened at 300 rpm.

**Figure 9.**(Experimental result) The torque performance and 23 Adaline weights in the four operating stages in which stage 4 applies the proposed scheme using Adaline_Fault (η = 0.01) and ECL strategy: (

**a**) at 100 rpm; and (

**b**) at 300 rpm.

**Figure 10.**(Experimental result) Switch from stage 3 to 4 when phase A is opened at 300 rpm: (

**a**) elimination of T

_{error}with the proposed Adaline-based scheme (η = 0.01) in stage 4; and (

**b**) torques of fictitious machines (T

_{FM}

_{1}, T

_{FM}

_{2}, T

_{FM}

_{3}) in one electrical period.

**Figure 11.**(Experimental result) The q-axis current control performance in one electrical period when phase A is opened at 300 rpm with: (

**a**) conventional RFOC scheme using MTPA_Fault in stage 3; and (

**b**) proposed scheme using Adaline_Fault (η = 0.01) and ECL strategy in stage 4.

**Figure 12.**(Experimental result) Phase current analyses when phase A is opened at 300 rpm: (

**a**) measured phase currents in stages 3 and 4; and (

**b**) harmonic spectrum of phase-B current.

**Figure 13.**(Experimental result) Dynamic performance of the proposed control scheme using Adaline_Fault and ECL strategy with four values of learning rate η when phase A is opened with: (

**a**) rotating speed variations; and (

**b**) reference torque variations.

**Table 1.**Fictitious machines, reference frames, and associated harmonics in natural frame of a seven-phase machine (only odd harmonics are considered).

Fictitious Machine | Reference Frame | Associated Harmonic * |
---|---|---|

1st fictitious machine (FM1) | d_{1}-q_{1} | 1, 13, 15, …, 7k ± 1 |

2nd fictitious machine (FM2) | d_{9}-q_{9} | 5, 9, 19, …, 7k ± 2 |

3rd fictitious machine (FM3) | d_{3}-q_{3} | 3, 11, 17 …, 7k ± 3 |

Zero-sequence machine (ZM) | z | 7, 21, …, 7k |

**Table 2.**Possible harmonic components of torque generated by NS-EMF harmonics and current harmonics of the considered seven-phase machine in faulty mode.

NS-EMF Harmonics | 1st | 3rd | 9th | 11th | 13th | 19th | |
---|---|---|---|---|---|---|---|

Current Harmonics | |||||||

1st | 2θ | 2θ, 4θ | 8θ, 10θ | 10θ, 12θ | 12θ, 14θ | 18θ, 20θ | |

3rd | 2θ, 4θ | 6θ | 6θ, 12θ | 8θ, 14θ | 10θ, 16θ | 16θ, 22θ |

Parameter | Unit | Value |
---|---|---|

Stator resistance R | Ω | 1.4 |

Self-inductance L | mH | 14.7 |

Mutual inductance M_{1} | mH | 3.5 |

Mutual inductance M_{2} | mH | −0.9 |

Mutual inductance M_{3} | mH | −6.1 |

The 1st harmonic of speed-normalized NS-EMFs E_{n}_{1} | V/rad/s | 1.27 |

Number of pole pairs p | 3 | |

Rated RMS current | A | 5.1 |

Rated voltage | V | 120 |

Rated torque | N.m | 33.5 |

Rated power | kW | 2.5 |

Rated speed | rpm | 750 |

PWM frequency | kHz | 10 |

Maximum DC-bus voltage V_{DC} | V | 200 |

**Table 4.**Comparisons between stages 3 (MTPA_Fault) and 4 (Adaline_Fault) in terms of torque ripples at different speeds when phase A is opened.

Speed Ω (rpm) | ∆T (%) | |||
---|---|---|---|---|

Stage 3 (MTPA_Fault) | Stage 4 (Adaline_Fault) | |||

sim^{1} | exp^{2} | sim^{1} | exp^{2} | |

100 | 2.1 | 16.1 | 2.5 | 3.5 |

300 | 4.2 | 30.3 | 3.2 | 4 |

750 | 9.8 | - | 4.3 | - |

^{1}simulation (numerical) result;

^{2}experimental result.

**Table 5.**Comparisons between stage 3 (MTPA_Fault) and stage 4 (Adaline_Fault) at 300 rpm in terms of copper losses when phase A is opened.

Phase | P_{loss} (pu) * | |||
---|---|---|---|---|

Stage 3 (MTPA_Fault) | Stage 4 (Adaline_Fault) | |||

sim^{1} | exp^{2} | sim^{1} | exp^{2} | |

B | 1.89 | 1.82 | 1.71 | 1.79 |

C | 1.37 | 1.35 | 1.62 | 1.54 |

D | 1.26 | 1.22 | 1.65 | 1.57 |

E | 1.27 | 1.22 | 1.67 | 1.53 |

F | 1.17 | 1.21 | 1.60 | 1.51 |

G | 1.61 | 1.62 | 1.62 | 1.64 |

Total | 1.23 | 1.22 | 1.41 | 1.38 |

^{1}simulation (numerical) result;

^{2}experimental result.

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## Share and Cite

**MDPI and ACS Style**

Vu, D.T.; Nguyen, N.K.; Semail, E.; Wu, H.
Adaline-Based Control Schemes for Non-Sinusoidal Multiphase Drives—Part II: Torque Optimization for Faulty Mode. *Energies* **2022**, *15*, 249.
https://doi.org/10.3390/en15010249

**AMA Style**

Vu DT, Nguyen NK, Semail E, Wu H.
Adaline-Based Control Schemes for Non-Sinusoidal Multiphase Drives—Part II: Torque Optimization for Faulty Mode. *Energies*. 2022; 15(1):249.
https://doi.org/10.3390/en15010249

**Chicago/Turabian Style**

Vu, Duc Tan, Ngac Ky Nguyen, Eric Semail, and Hailong Wu.
2022. "Adaline-Based Control Schemes for Non-Sinusoidal Multiphase Drives—Part II: Torque Optimization for Faulty Mode" *Energies* 15, no. 1: 249.
https://doi.org/10.3390/en15010249