# Adaline-Based Control Schemes for Non-Sinusoidal Multiphase Drives–Part I: Torque Optimization for Healthy Mode

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model 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 stator winding resistance of one phase; [

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

_{αβ}is the 7-dimensional vector of currents in α-β frames; (i

_{α}

_{1}, i

_{β}

_{1}), (i

_{α}

_{2}, i

_{β}

_{2}), and (i

_{α}

_{3}, i

_{β}

_{3}) are currents in two-dimensional decoupled frames (α

_{1}-β

_{1}), (α

_{2}-β

_{2}), and (α

_{3}-β

_{3}), respectively; i

_{z}is a current in zero-sequence frame z (one-dimensional); [

**T**] is the 7-by-7 Clarke transformation matrix expressed by the spatial angular displacement δ (δ = 2π/7) as follows:

_{1}-β

_{1}), (α

_{2}-β

_{2}), (α

_{3}-β

_{3}), and z, respectively [3]. A fictitious machine with its corresponding decoupled reference frame is associated with a group of harmonics as presented in Table 1.

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

_{dh}

_{1}, i

_{qh}

_{1}), (i

_{dh}

_{2}, i

_{qh}

_{2}), and (i

_{dh}

_{3}, i

_{qh}

_{3}) are currents in two-dimensional rotating frames (d

_{h}

_{1}-q

_{h}

_{1}), (d

_{h}

_{2}-q

_{h}

_{2}), and (d

_{h}

_{3}-q

_{h}

_{3}), respectively; h

_{1}, h

_{2}, and h

_{3}are respectively main NS-EMF harmonics presenting in FM1, FM2, and FM3 (see associated harmonics in Table 1); [

**P**] is the 7-by-7 Park transformation matrix with the electrical position θ as follows:

_{1}, h

_{2}, h

_{3}) in rotating frames depends on main harmonics (with highest amplitudes) presenting in fictitious machines of the considered NS-EMFs. In our case, the considered experimental seven-phase prototype has the 1st, 9th, and 3rd harmonics of NS-EMFs which account for the highest proportions in FM1, FM2, and FM3, respectively. Therefore, the corresponding rotating frames are (d

_{1}-q

_{1}), (d

_{9}-q

_{9}), and (d

_{3}-q

_{3}). If there is only one harmonic of NS-EMFs existing in one fictitious machine (e.g., only 1st in FM1, only 9th in FM2, and only 3rd in FM3), constant d-q currents (i

_{d}

_{1}, i

_{q}

_{1}, i

_{d}

_{9}, i

_{q}

_{9}, i

_{d}

_{3}, i

_{q}

_{3}) can be used to generate constant torque, facilitating PI controllers in the classical RFOC scheme.

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

_{n}

_{1}, E

_{n}

_{3}, E

_{n}

_{9}, E

_{n}

_{11}, E

_{n}

_{13}, E

_{n}

_{19}) and (φ

_{1}, φ

_{3}, φ

_{9}, φ

_{11}, φ

_{13}, φ

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

_{dq}can be a constant vector only if E

_{n}

_{11}, E

_{n}

_{13}, and E

_{n}

_{19}are zero. It means that NS-EMFs contain only 1st, 9th, and 3rd harmonics. In this study, the considered 13th, 19th, and 11th harmonics are called unwanted NS-EMF harmonics. It is noted that, harmonics 7k ($k\in \mathrm{N}$) associated with the zero-sequence machine ZM have no impact on torque generation due to the star connection (current i

_{z}= 0).

_{em}can be expressed by:

## 3. Classical Control of Multiphase Drives for Healthy Mode

_{em_ref}, seven reference phase currents i

_{ref}with minimum total copper losses can be calculated by using MTPA strategy [10,11] using the estimated NS-EMFs e

_{sn}. These currents are transformed into d-q frames (i

_{d}

_{1_ref}, i

_{q}

_{1_ref}, i

_{d}

_{9_ref}, i

_{q}

_{9_ref}, i

_{d}

_{3_ref}, i

_{q}

_{3_ref}, i

_{z}

_{_ref}). Current i

_{z}

_{_ref}is equal to zero due to the star connection. The electrical position θ obtained from an encoder is provided for the speed-normalized NS-EMF estimation and Park transformations.

_{sn}contains additional unwanted harmonics (11th, 13th, and 19th as previously assumed). Therefore, to generate a constant torque, six reference d-q currents need to be time-variant and have main frequencies of 14θ, 28θ, and kn in the general case ($k\in \mathrm{N}$ and n is the phase number) [20]. Consequently, the use of six PI controllers with low bandwidth to control six varying reference d-q currents in the classical RFOC scheme may reduce torque quality at high speed.

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

#### 4.1. Simplified MTPA (SMTPA)

_{sn}in (1) and (6) by keeping only 1st, 9th, and 3rd harmonics.

_{em_}

_{139}is the total torque generated by SMTPA (with only 1st, 3rd, and 9th harmonics of currents); T

_{ave}is the average torque; T

_{14}

_{θ}and T

_{28}

_{θ}are harmonics 14θ and 28θ of the torque; E

_{n}

_{1}, E

_{n}

_{3}, E

_{n}

_{9}, E

_{n}

_{11}, E

_{n}

_{13}, and E

_{n}

_{19}are amplitudes of six NS-EMF harmonics as described in (6).

_{ave}and two harmonic components 14θ and 28θ. In general, torque harmonics of a n-phase machine under healthy condition are multiples of nθ, specifically 2knθ if n is odd and knθ if n is even ($k\in \mathrm{N}$) [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26]. Theoretically, to obtain a constant torque, torque ripples T

_{14θ}and T

_{28}

_{θ}in (12) can be used to calculate compensating currents. However, the estimated NS-EMFs are usually obtained from measured NS-EMFs in open-circuited stator windings. The mechanical load in the real-time operation of the drive may significantly affect the NS-EMF waveform, including amplitudes and initial phase angles of harmonic components. In addition, the dead-time voltages of VSI may impose harmonic components in machine phase voltages [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. Therefore, the compensating currents directly calculated from T

_{14θ}and T

_{28}

_{θ}cannot guarantee a constant torque.

#### 4.2. Structure of Proposed Control Scheme for Healthy Mode

_{em}of the seven-phase machine is calculated from measured phase currents and estimated NS-EMFs according to (8).

_{14θ}and T

_{28}

_{θ}in (12), a compensating torque T

_{em_com}will be adaptively determined from a specific Adaline. Thereafter, compensating currents are calculated from T

_{em_com}by using SMTPA as follows:

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

_{sn}in (1) and (6) by keeping only 1st, 9th, and 3rd harmonics.

_{com}is transformed into d-q frames as follows:

_{dq_com}is the 7-dimensional vector of d-q compensating currents; ${\underset{\_}{e}}_{dq}^{sim}$ is the constant vector as described in (10).

_{dq_ref}) from SMTPA in (11). The total reference d-q current vector (${\underset{\_}{i}}_{dq\_ref}^{\ast}$) is used for control as follows:

#### 4.3. Structure of Proposed Adaline for Healthy Mode

_{em_com}as well as compensating currents i

_{com}to make the total torque T

_{em}properly track its reference value T

_{em_ref}. It is assumed that the total torque with SMTPA in real time (experiments) ${T}_{em\_139}^{\ast}$ can be developed from (12) 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 expressed by five coefficients (${w}_{0}^{\ast}$, ${w}_{1}^{\ast}$, ${w}_{2}^{\ast}$, ${w}_{3}^{\ast}$, ${w}_{4}^{\ast}$). The torque ${T}_{em\_139}^{\ast}$ and its coefficients are not fixed and depend on the load and dead-time voltages of the drive [7,22].

_{error}(the error between reference torque T

_{em_ref}and the total torque T

_{em}) as presented in Figure 3.

_{0}, w

_{1}, w

_{2}, w

_{3}, w

_{4}) are five Adaline weights corresponding to the five coefficients of the compensating torque in (17).

_{em_com}) is the weighted sum of the inputs as follows:

_{error}is the error between reference torque T

_{em_ref}and the total torque T

_{em}; T

_{em}is calculated from measured phase currents and estimated NS-EMFs in (8).

_{em_com}) such as amplitudes and phases of harmonics. An increase in η possibly results in faster convergence but it may lead to divergence. 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 minimized and the total torque T

_{em}is equal to the constant reference torque T

_{em_ref}. The proposed scheme using the Adaline can guarantee a smooth torque regardless of multi-harmonics existing in NS-EMFs.

_{ave}, these harmonics should be neglected in the Adaline inputs. Their corresponding weights are also removed. For example, the compensating torque in this study contains high-frequency components 14θ and 28θ. If absolute values of coefficients (${w}_{3}^{\ast}$, ${w}_{4}^{\ast}$) are very small compared to those of (${w}_{1}^{\ast}$, ${w}_{2}^{\ast}$, T

_{ave}), harmonic component 28θ of the inputs and its corresponding weights should be removed from the Adaline. The Adaline can be designed by a S-function builder block MATLAB/Simulink.

## 5. Numerical and Experimental Results

#### 5.1. Descriptions of Experimental Seven-Phase Test Bench

- FM1: The 1st (100%) and 13th (5% of the 1st);
- FM2: The 9th (12.5% of the 1st) and 19th (2% of the 1st);
- FM3: The 3rd (32.3% of the 1st) and 11th (10.3% of the 1st).

#### 5.2. Numerical Results

_{em_ref}is imposed at 33.5 N.m (rated torque) to obtain the rated RMS current of 5.1 A (Table 2).

_{em_max}and T

_{em_min}are maximum and minimum values of instantaneous torque T

_{em}, respectively; T

_{ave}is the average value of T

_{em}.

- Stage 1: The classical RFOC scheme (Figure 1) with SMTPA is used. Reference d-q currents are constant, but there are torque ripples as described in (12);
- Stage 2: The classical RFOC scheme (Figure 1) with MTPA is used. Reference d-q currents are no longer constant, but the torque ripples are theoretically eliminated;
- Stage 3: The proposed Adaline-based control scheme (Figure 2) is used. Three reference d-axis currents are zero while three reference q-axis currents are no longer constant. The torque ripples are theoretically eliminated. The compensating torque T
_{em_com}firstly contains both 14θ and 28θ (5 weights w_{0}, w_{1}, w_{2}, w_{3}, and w_{4}are used), then T_{em_com}contains only 14θ (3 weights w_{0}, w_{1}, and w_{2}are used).

_{3}and w

_{4}of harmonic 28θ are very small (<0.06) compared to those of w

_{1}(0.2), w

_{2}(1.49), and T

_{ave}(33.5 N.m). Therefore, using only w

_{0}, w

_{1}, and w

_{2}to eliminate harmonic 14θ still guarantees a high-quality torque. Indeed, from 1.2 s of Figure 6, the torque ripple with 3 weights at the three speeds is 0.1% or 0.2% slightly higher than the case with 5 weights (see stage 3 and zoom 4 in Figure 6).

_{error}that is the difference between the reference torque T

_{em_ref}and the total torque T

_{em}. It is noted that the main harmonic of torque ripples in stage 2 is T

_{14θ}due to 14 similar oscillations in one electrical period. When the output of the Adaline y converges to the compensating torque T

_{em_com}with harmonic 14θ, T

_{error}significantly reduces, and the torque ripple decreases from 6% to 2.8%. Torques generated by the three fictitious machines are presented in Figure 7b. In stages 2 and 3, the torque generated by FM2 (T

_{FM}

_{2}) is much smaller than those of FM1 (T

_{FM}

_{1}) and FM3 (T

_{FM}

_{3}) due to the lower amplitude of NS-EMFs in FM2 (9th harmonic). T

_{FM}

_{2}with MTPA has higher ripples than those of the Adaline-based scheme. Notably, in stage 3, harmonic components of T

_{FM}

_{1}and T

_{FM}

_{3}are opposite with similar amplitudes. Therefore, total torque ripples are almost eliminated in stage 3. Indeed, reference currents in the proposed scheme are adaptively generated to eliminate torque ripples by using the online-trained Adaline.

_{d}

_{9}and i

_{q}

_{9}with MTPA are no longer properly controlled at 750 rpm. Meanwhile, in the proposed Adaline-based scheme, three reference d-axis currents are null. Three q-axis reference currents oscillate with lower amplitudes compared to MTPA.

#### 5.3. Experimental Results

_{DC}= 200 V). Indeed, with the star connection and classical PWM strategy, the maximum voltage of each phase should not be greater than V

_{DC}/2 = 100 V. From Figure 9a, simulated voltage references at 400 rpm are about 100 V. If the speed could be higher (750 rpm in the numerical result section), the effectiveness of the proposed scheme will be clearer.

_{0}, w

_{1}, and w

_{2}are used to eliminate torque harmonic 14θ in experiments.

_{error}. It is noted that the main harmonic of torque ripples in stage 2 is T

_{14θ}due to 14 oscillations in one electrical period. When y converges to T

_{em_com}, T

_{error}significantly reduces, and the torque ripple decreases from 10.1% to 5.5%. Torques generated by the three fictitious machines are presented in Figure 11b. In general, T

_{FM}

_{1}and T

_{FM}

_{3}are opposite with similar amplitudes in both stages 2 and 3. However, ripples of these torques with the Adaline-based scheme are lower than those of MTPA, leading to the lower total ripple in stage 3.

_{1}-q

_{1}) frame at 100 rpm and 400 rpm is described in Figure 12. At 100 rpm, all currents in the three stages are well controlled. At 400 rpm, with constant d

_{1}-axis currents, SMTPA in stage 1 and the proposed scheme in stage 3 have better control quality than MTPA in stage 2. It is noted that the experimental control performance of q

_{1}-axis current in stage 3 is not much better than that of stage 2. Indeed, there are current harmonics existing in d-q frames that are caused by the unwanted harmonics of NS-EMFs and the nonlinearity of VSI (e.g., dead-time voltages) as previously presented in [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22]. However, the proposed scheme with Adaline can automatically generate a proper torque in each fictitious machine (Figure 11b) to obtain the total torque with minimum ripples as presented in Figure 10b.

_{0}due to weight oscillations.

## 6. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 2.**Torque control using the proposed Adaline-based control scheme of a seven-phase PMSM in healthy mode.

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

**Figure 4.**Experimental test bench of a seven-phase NS-EMF PMSM drive to validate the proposed Adaline-based control scheme.

**Figure 6.**(Numerical result) The torque performance in healthy mode at 100 rpm (green), 400 rpm (blue), and 750 rpm (red), in the three operating stages, and Adaline weights with η = 0.001 at 750 rpm.

**Figure 7.**(Numerical result) Switch from stage 2 to 3 in healthy mode at 750 rpm: (

**a**) elimination of T

_{error}with the proposed Adaline-based scheme (η = 0.001) in stage 3; (

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

_{FM}

_{1}, T

_{FM}

_{2}, T

_{FM}

_{3}) in one electrical period.

**Figure 8.**(Numerical result) Current control performance in one electrical period at 750 rpm in healthy mode using: (

**a**) RFOC scheme with SMTPA (Stage 1); (

**b**) RFOC scheme with MTPA (Stage 2); (

**c**) proposed Adaline-based scheme (η = 0.001) (Stage 3).

**Figure 9.**(Numerical result) Voltages and currents in healthy mode at 400 rpm in the three operating stages: (

**a**) phase voltage references; (

**b**) phase currents.

**Figure 10.**(Experimental result) The torque performance in healthy mode in the three operating stages, and Adaline weights with η = 0.01: (

**a**) at 100 rpm; (

**b**) at 400 rpm.

**Figure 11.**(Experimental result) Switch from stage 2 to 3 in healthy mode at 400 rpm: (

**a**) Elimination of T

_{error}with the proposed Adaline-based scheme (η = 0.01) in stage 3; (

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

_{FM}

_{1}, T

_{FM}

_{2}, T

_{FM}

_{3}) in one electrical period.

**Figure 12.**(Experimental result) Current control performance in one electrical period of the three operating stages in healthy mode: (

**a**) At 100 rpm; (

**b**) At 400 rpm.

**Figure 13.**(Experimental result) Measured phase currents in healthy mode at 400 rpm with the RFOC scheme using SMTPA (Stage 1), MTPA (Stage 2), and the proposed Adaline-based control scheme with η = 0.01 (Stage 3).

**Figure 14.**(Experimental result) Dynamic performance of the proposed Adaline-based control scheme for healthy mode using four different weight learning rates η (0.001, 0.005, 0.01, 0.05) with: (

**a**) rotating speed variations; (

**b**) reference torque variations.

**Table 1.**Fictitious machines, reference frames, and associated harmonics of a seven-phase machine (Only odd harmonics).

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

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

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

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

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

_{0}.

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

Resistance of one phase 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 |

Amplitude of 1st harmonic of NS-EMF E_{n}_{1} | V/rad/s | 1.27 |

Number of pole pairs p | 3 | |

Rated RMS current | A | 5.1 |

Rated torque | N.m | 33.5 |

Rated speed | rpm | 750 |

Rated power | kW | 2.5 |

Rated voltage | V | 120 |

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

PWM frequency | kHz | 10 |

**Table 3.**Comparisons between the three operating stages (strategies) in terms of torque ripples at different speeds.

Speed Ω (rpm) | ∆T with SMTPA (%) | ∆T with MTPA (%) | ∆T with Adaline_HM (%) | |||
---|---|---|---|---|---|---|

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

100 | 11 | 11.8 | 1.3 | 5 | 1.5 | 3.9 |

400 | 11.5 | 9.5 | 3 | 10.1 | 2.3 | 5.5 |

750 | 12 | - | 6 | - | 2.8 | - |

^{1}simulation (numerical) result;

^{2}experimental result.

**Table 4.**Comparisons between the three operating stages (strategies) at 400 rpm in terms of RMS and peak currents, and peak voltage references.

Strategies | I_{RMS} (A) | I_{peak} (A) | V_{peak} (V) | |||
---|---|---|---|---|---|---|

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

SMTPA | 5.06 | 5.09 | 9.1 | 10 | 80 | 93.5 |

MTPA | 5.1 | 5.08 | 10.5 | 10.1 | 127 | 98 |

Adaline_HM | 5.07 | 5.14 | 9.4 | 10.1 | 94 | 95 |

^{1}simulation (numerical) result;

^{2}experimental result.

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**MDPI and ACS Style**

Vu, D.T.; Nguyen, N.K.; Semail, E.; Wu, H.
Adaline-Based Control Schemes for Non-Sinusoidal Multiphase Drives–Part I: Torque Optimization for Healthy Mode. *Energies* **2021**, *14*, 8302.
https://doi.org/10.3390/en14248302

**AMA Style**

Vu DT, Nguyen NK, Semail E, Wu H.
Adaline-Based Control Schemes for Non-Sinusoidal Multiphase Drives–Part I: Torque Optimization for Healthy Mode. *Energies*. 2021; 14(24):8302.
https://doi.org/10.3390/en14248302

**Chicago/Turabian Style**

Vu, Duc Tan, Ngac Ky Nguyen, Eric Semail, and Hailong Wu.
2021. "Adaline-Based Control Schemes for Non-Sinusoidal Multiphase Drives–Part I: Torque Optimization for Healthy Mode" *Energies* 14, no. 24: 8302.
https://doi.org/10.3390/en14248302