# Experimental Identification of the Optimal Current Vectors for a Permanent-Magnet Synchronous Machine in Wave Energy Converters

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## Abstract

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## 1. Introduction

#### 1.1. Motivation

#### 1.2. Contribution of the Paper

#### 1.3. Outline of the Paper

## 2. Theory

#### 2.1. Model of a Synchronous Machine without Iron Losses

- The three phases of the electrical machine are star-connected (i.e., ${i}_{\mathrm{s}}^{\mathrm{a}}\left(t\right)+{i}_{\mathrm{s}}^{\mathrm{b}}\left(t\right)+{i}_{\mathrm{s}}^{\mathrm{c}}\left(t\right)=0$).
- The machine is symmetrical.
- Only the fundamentals are considered and spatial harmonics are neglected.
- Iron losses are neglected.

#### 2.2. Simplified Modelling of the Voltage Source Inverter

#### 2.3. Power Analysis

## 3. Methods for Efficiency Enhancement

#### 3.1. Maximum Torque Per Current for Linear Flux Linkages

#### 3.2. Maximum Torque Per Current for Nonlinear Flux Linkages

#### 3.3. Direct Measurement of the Current Vector with the Maximum Efficiency

## 4. Implementation and Measurements

#### 4.1. Setup of the Test Bench

#### 4.2. Measurement Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**One half-bridge of the voltage source inverter with (

**a**) the two MOSFETs, (

**b**) the equivalent circuit of the MOSFETs and (

**c**) the simplification of the half bridge.

**Figure 3.**Illustration of the torque contour lines ( ), the current limit ( ), and the numerically calculated Maximum Torque per Current (MTPC) hyperbola with linearised flux linkages ( ).

**Figure 4.**Illustration of the operation areas with the transient line ( ), the current limit ( ), and the voltage limit ( ) for two different speeds ${\omega}_{1}<{\omega}_{2}$.

**Figure 5.**Measured flux linkage maps. (

**a**) Flux linkage ${\psi}_{\mathrm{s}}^{d}$ with black contour lines of constant flux linkage; (

**b**) Flux linkage ${\psi}_{\mathrm{s}}^{d}$ in a 3D plot; (

**c**) Flux linkage ${\psi}_{\mathrm{s}}^{q}$ with black contour lines of constant flux linkage; and (

**d**) Flux linkage ${\psi}_{\mathrm{s}}^{q}$ in a 3D plot.

**Figure 7.**Measured efficiency over the direct current for a shaft speed of $200\phantom{\rule{3.33333pt}{0ex}}\mathrm{rpm}$. (

**a**) Roughly measured efficiency for different constant torques. (

**b**) Measured efficiency ( ) and fitted polynomial of second order ( ) over the direct current ${i}_{\mathrm{s}}^{d}$ for a constant torque of $-10\phantom{\rule{3.33333pt}{0ex}}\mathrm{Nm}$.

**Figure 8.**Illustration of the test bench. (

**a**) Schematics of the test bench; and (

**b**) Photograph of the test bench.

**Figure 9.**Comparison of the three presented methods. (

**a**) Measured torque contour lines ( ) and resulting trajectories for the two MTPC methods with linear approximated ( ) and nonlinear ( ) flux linkages and the measured trajectories with the maximum efficiency for different shaft speeds of $100\phantom{\rule{3.33333pt}{0ex}}\mathrm{rpm}$ ( ), $200\phantom{\rule{3.33333pt}{0ex}}\mathrm{rpm}$ ( ), and $300\phantom{\rule{3.33333pt}{0ex}}\mathrm{rpm}$ ( ) in the ($d,q$)-plane. (

**b**) Efficiency ( ) for a shaft speed of $300\phantom{\rule{3.33333pt}{0ex}}\mathrm{rpm}$ and a constant torque of $-15\phantom{\rule{3.33333pt}{0ex}}\mathrm{Nm}$ and the two points of MTPC for linear approximated ( ) and nonlinear ( ) flux linkages and the point of maximum efficiency ( ).

**Figure 10.**Steady-state electric model of the Permanent-Magnet Synchronous Machine (PMSM) with iron losses.

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

Number of pole pairs | ${n}_{\mathrm{p}}$ | 5 |

Gear ratio | ${g}_{\mathrm{r}}$ | 8 |

Stator resistance at $20{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{C}$ | ${R}_{\mathrm{s},20}$ | $396\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathsf{\Omega}$ |

Stator resistance at $120{\phantom{\rule{3.33333pt}{0ex}}}^{\circ}\mathrm{C}$ | ${R}_{\mathrm{s},120}$ | $570\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathsf{\Omega}$ |

Drain-source on resistance | ${R}_{\mathrm{DS},\mathrm{on}}$ | $60\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathsf{\Omega}$ |

Resistance of the cable | ${R}_{\mathrm{c}}$ | $12\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\mathsf{\Omega}$ |

Permanent-magnet flux linkage | ${\psi}_{\mathrm{pm}}$ | $75.79\phantom{\rule{3.33333pt}{0ex}}\mathrm{mVs}$ |

d-Inductance | ${L}_{\mathrm{s}}^{d}$ | $4.5\phantom{\rule{3.33333pt}{0ex}}\mathrm{mH}$ |

q-Inductance | ${L}_{\mathrm{s}}^{q}$ | $5.7\phantom{\rule{3.33333pt}{0ex}}\mathrm{mH}$ |

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**MDPI and ACS Style**

Krüner, S.; Hackl, C.M. Experimental Identification of the Optimal Current Vectors for a Permanent-Magnet Synchronous Machine in Wave Energy Converters. *Energies* **2019**, *12*, 862.
https://doi.org/10.3390/en12050862

**AMA Style**

Krüner S, Hackl CM. Experimental Identification of the Optimal Current Vectors for a Permanent-Magnet Synchronous Machine in Wave Energy Converters. *Energies*. 2019; 12(5):862.
https://doi.org/10.3390/en12050862

**Chicago/Turabian Style**

Krüner, Simon, and Christoph M. Hackl. 2019. "Experimental Identification of the Optimal Current Vectors for a Permanent-Magnet Synchronous Machine in Wave Energy Converters" *Energies* 12, no. 5: 862.
https://doi.org/10.3390/en12050862