# Piecewise Affine Magnetic Modeling of Permanent-Magnet Synchronous Machines for Virtual-Flux Control

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## Abstract

**:**

## 1. Introduction

## 2. PMSM Model

## 3. Piecewise Affine Magnetic Model

#### 3.1. Choosing ${\mathcal{I}}_{P}$

#### 3.2. Subdomains and the Delaunay Triangulation

#### 3.3. Subdomain Coefficients

#### 3.4. Assign Coefficients to Simplices

## 4. Magnetic Model Optimization

## 5. Experimental Results

#### Flux Error Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DSP | Digital signal processor |

FEA | Finite element analysis |

IPMSM | Interior permanent-magnet synchronous machine |

MCU | Microcontroller |

MM | Magnetic model |

MPC | Model predictive control |

MTPA | Maximum torque per ampere |

PM | Permanent magnet |

PMSM | Permanent-magnet synchronous machine |

PWA | Piecewise affine |

SPMSM | Surface-mounted permanent-magnet synchronous machine |

THD | Total harmonic distortion |

pu | Per unit |

VF-MPC | Virtual-flux model predictive control |

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**Figure 1.**Example motor drive system (gray) and virtual flux controller architecture (white) demonstrating the use of the PWA MM flux linkage map.

**Figure 2.**(

**top**) (

**left**) The d-axis flux linkage map, (

**middle**) q-axis flux linkage map, and (

**right**) torque function. (

**bottom**) (

**left**) The d-axis iso-flux lines, (

**middle**) q-axis iso-flux lines, and (

**right**) iso-torque lines.

**Figure 4.**(

**left**) Mesh of simplices ${\mathcal{I}}_{j}\in {\mathcal{I}}_{M}$ created using a 13-by-13 regularly gridded ${\mathcal{I}}_{P}$ in current space. (

**right**) Corresponding mesh of simplices ${\Lambda}_{j}\in {\Lambda}_{M}$ in flux space.

**Figure 5.**Voronoi diagram (

**left**) of a seven-point irregular current grid and the corresponding eight-simplex Delaunay triangulation (

**right**).

**Figure 6.**Visualization of ${f}_{\mathrm{pwa}}$: (

**left**) current simplex (triangle) $\overline{{\mathcal{I}}_{j}}$ shifted by ${i}_{j0}$ vector and (

**right**) corresponding flux simplex (triangle) ${\overline{\Lambda}}_{j}$ shifted by ${\lambda}_{j0}$ vector.

**Figure 7.**PWA MM functions at various resolutions using regularly gridded ${\mathcal{I}}_{P}$: (left) d-axis flux ${\lambda}_{d}$ from ${f}_{\mathrm{PWA}}$ and (right) q-axis flux ${\lambda}_{d}$.

**Figure 8.**Two-norm flux error distribution between ${f}_{\mathrm{PWA}}\left(i\right)$ and ${f}_{S}\left(i\right)$, ${f}_{\mathrm{PWA}}\left(i\right)$ constructed using regularly gridded $13\times 13$${\mathcal{I}}_{P}$, where ${f}_{S}\left(i\right)$ uses high-fidelity spline-interpolated FEA data points.

**Figure 9.**PWA MM optimization algorithm to minimize the maximum two-norm flux error in ${f}_{\mathrm{PWA}}$ given an allotted N number of of points to use ($|{\mathcal{I}}_{P}|=N$).

**Figure 10.**Visualization of three regions of current to optimize ${f}_{\mathrm{PWA}}$: $\mathcal{I}$, ${\mathcal{I}}_{\mathrm{MTPA}}$, and ${\mathcal{I}}_{\mathrm{derated}}$.

**Figure 11.**(

**left**) Average and (

**right**) maximum two-norm flux error vs. N points in various magnetic models (MMs): PWA ${f}_{\mathrm{PWA}\left(i\right)}$ (targeting error reduction in specific regions), linearized inductance model ${f}_{l}\left(i\right)$, and spline model ${f}_{s}\left(i\right)$ (targeting error reduction in MTPA region).

**Figure 12.**(

**top**row) Simplical mesh ${\mathcal{I}}_{M}$ comprising all simplices ${\mathcal{I}}_{j}$. (

**middle**row) Placement of points in ${\mathcal{I}}_{P}$ by the optimization algorithm. (

**bottom**row) Two-norm flux error distribution. (first column) ${f}_{\mathrm{PWA}}$ using regularly gridded current points covering $\mathcal{I}$. (second column) ${f}_{\mathrm{PWA}}$ using irregularly gridded current points optimizing $\mathcal{I}$. (third column) ${f}_{\mathrm{PWA}}$ using irregularly gridded current points optimizing ${\mathcal{I}}_{\mathrm{MTPA}}$. (fourth column) ${f}_{\mathrm{PWA}}$ using irregularly gridded current points optimizing ${\mathcal{I}}_{\mathrm{derated}}$.

Type of Link | Lin. | Bidir. | Cont. | Diff. | Sat. | Cross-Sat. ${}^{1}$ |
---|---|---|---|---|---|---|

Linearized Inductance [22,23] | • | • | • | • | ||

Linearly Interpolated LUT | • | • | • | |||

Spline-Interpolated LUT [15,17,18] | • | • | • | • | ||

Polynomial Functions [13,19] | • | • | • | • | ||

Piecewise Nonlinear Function [14] | • | • | • | • | ||

Other Nonlinear Function [12,20] | • | • | • | • | ||

Piecewise Linear (PWA) [21] | • | • | • | • | • |

^{1}Linear, bidirectional, continuous, differentiable, saturation, and cross-saturation, respectively.

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

Type | Interior PM synchronous machine (IPMSM) |

Rated current ${i}_{\mathrm{rated}}$ | 10 A |

Rated torque ${T}_{\mathrm{rated}}$ | 8.0 Nm |

Rated flux ${\lambda}_{\mathrm{rated}}$ | 142.5 mWb |

Inductance (d-axis) ${L}_{d}$ | 9.1 mH |

Inductance (q-axis) ${L}_{q}$ | 14.6 mH |

Stator Resistance ${R}_{s}$ (@ 20C) | 636 m$\Omega $ |

PM rotor flux $\psi $ | 88.3 mWb |

Pole pair p | 5 |

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**MDPI and ACS Style**

Steyaert, B.; Swint, E.; Pennington, W.W.; Preindl, M.
Piecewise Affine Magnetic Modeling of Permanent-Magnet Synchronous Machines for Virtual-Flux Control. *Energies* **2022**, *15*, 7259.
https://doi.org/10.3390/en15197259

**AMA Style**

Steyaert B, Swint E, Pennington WW, Preindl M.
Piecewise Affine Magnetic Modeling of Permanent-Magnet Synchronous Machines for Virtual-Flux Control. *Energies*. 2022; 15(19):7259.
https://doi.org/10.3390/en15197259

**Chicago/Turabian Style**

Steyaert, Bernard, Ethan Swint, W. Wesley Pennington, and Matthias Preindl.
2022. "Piecewise Affine Magnetic Modeling of Permanent-Magnet Synchronous Machines for Virtual-Flux Control" *Energies* 15, no. 19: 7259.
https://doi.org/10.3390/en15197259