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

^{1}

^{2}

^{*}

## Abstract

**:**

## 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

- Eldeeb, H.; Hackl, C.M.; Horlbeck, L.; Kullick, J. A unified theory for optimal feedforward torque control of anisotropic synchronous machines. Int. J. Control
**2017**, 1–30. [Google Scholar] [CrossRef] [Green Version] - Schröder, D. Elektrische Antriebe—Regelung von Antriebssystemen, 4th ed.; Springer: Berlin/Heidelberg, Germany, 2015. [Google Scholar]
- Rang, G.; Lim, J.; Nam, K.; Ihm, H.B.; Kim, H.G. A MTPA Control Scheme for an IPM Synchronous Motor Considering Magnet Flux Variation Caused by Temperature. In Proceedings of the Nineteenth Annual IEEE Applied Power Electronics Conference and Exposition, 2004 (APEC ’04), Anaheim, CA, USA, 22–26 February 2004; Volume 3, pp. 1617–1621. [Google Scholar] [CrossRef]
- Jung, S.; Hong, J.; Nam, K. Current Minimizing Torque Control of the IPMSM Using Ferrari’s Method. IEEE Trans. Power Electron.
**2013**, 28, 5603–5617. [Google Scholar] [CrossRef] - Hoang, K.D.; Wang, J.; Cyriacks, M.; Melkonyan, A.; Kriegel, K. Feed-forward Torque Control of Interior Permanent Magnet Brushless AC Drive for Traction Applications. In Proceedings of the 2013 International Electric Machines Drives Conference, Chicago, IL, USA, 12–15 May 2013; pp. 152–159. [Google Scholar] [CrossRef]
- Mekri, F.; Elghali, S.B.; Benbouzid, M.E.H. Fault-Tolerant Control Performance Comparison of Three- and Five-Phase PMSG for Marine Current Turbine Applications. IEEE Trans. Sustain. Energy
**2013**, 4, 425–433. [Google Scholar] [CrossRef] [Green Version] - Mekri, F.; Charpentier, J.; Benelghali, S.; Kestelyn, X. High Order Sliding mode optimal current control of five phase permanent magnet motor under open circuited phase fault conditions. In Proceedings of the 2010 IEEE Vehicle Power and Propulsion Conference, Lille, France, 1–3 September 2010; pp. 1–6. [Google Scholar] [CrossRef]
- Kestelyn, X.; Semail, E. A Vectorial Approach for Generation of Optimal Current References for Multiphase Permanent-Magnet Synchronous Machines in Real Time. IEEE Trans. Ind. Electron.
**2011**, 58, 5057–5065. [Google Scholar] [CrossRef] [Green Version] - Bolognani, S.; Petrella, R.; Prearo, A.; Sgarbossa, L. Automatic Tracking of MTPA Trajectory in IPM Motor Drives Based on AC Current Injection. IEEE Trans. Ind. Appl.
**2011**, 47, 105–114. [Google Scholar] [CrossRef] - Kim, S.; Yoon, Y.; Sul, S.; Ide, K. Maximum Torque per Ampere (MTPA) Control of an IPM Machine Based on Signal Injection Considering Inductance Saturation. IEEE Trans. Power Electron.
**2013**, 28, 488–497. [Google Scholar] [CrossRef] - Sun, T.; Wang, J.; Chen, X. Maximum Torque Per Ampere (MTPA) Control for Interior Permanent Magnet Synchronous Machine Drives Based on Virtual Signal Injection. IEEE Trans. Power Electron.
**2015**, 30, 5036–5045. [Google Scholar] [CrossRef] - Ahmed, A.; Sozer, Y.; Hamdan, M. Maximum Torque per Ampere Control for Interior Permanent Magnet Motors using DC Link Power Measurement. In Proceedings of the 2014 IEEE Applied Power Electronics Conference and Exposition—APEC 2014, Fort Worth, TX, USA, 6–20 March 2014; pp. 826–832. [Google Scholar] [CrossRef]
- Armando, E.; Bojoi, R.I.; Guglielmi, P.; Pellegrino, G.; Pastorelli, M. Experimental Identification of the Magnetic Model of Synchronous Machines. IEEE Trans. Ind. Appl.
**2013**, 49, 2116–2125. [Google Scholar] [CrossRef] [Green Version] - Mink, F.; Kubasiak, N.; Ritter, B.; Binder, A. Parametric Model and Identification of PMSM Considering the Influence of Magnetic Saturation. In Proceedings of the 2012 13th International Conference on Optimization of Electrical and Electronic Equipment (OPTIM), Brasov, Romania, 24–26 May 2012; pp. 444–452. [Google Scholar] [CrossRef]
- Kellner, S.L.; Seilmeier, M.; Piepenbreier, B. Impact of Iron Losses on Parameter Identification of Permanent Magnet Synchronous Machines. In Proceedings of the 2011 1st International Electric Drives Production Conference, Nuremberg, Germany, 28–29 September 2011; pp. 11–16. [Google Scholar] [CrossRef]
- Richter, J.; Dollinger, A.; Doppelbauer, M. Iron Loss and Parameter Measurement of Permanent Magnet Synchronous Machine. In Proceedings of the 2014 International Conference on Electrical Machines (ICEM), Berlin, Germany, 2–5 September 2014; pp. 1635–1641. [Google Scholar] [CrossRef]
- Morimoto, S.; Tong, Y.; Takeda, Y.; Hirasa, T. Loss Minimization Control of Permanent Magnet Synchronous Motor Drives. IEEE Trans. Ind. Electron.
**1994**, 41, 511–517. [Google Scholar] [CrossRef] - Uddin, M.N.; Zou, H.; Azevedo, F. Online Loss Minimization Based Adaptive Flux Observer for Direct Torque and Flux Control of PMSM Drive. In Proceedings of the 2014 IEEE Industry Application Society Annual Meeting, Vancouver, BC, Canada, 5–9 October 2014; pp. 1–7. [Google Scholar] [CrossRef]
- Barisa, T.; Sumina, D.; Kutija, M. Comparison of Maximum Torque per Ampere and Loss Minimization Control for the Interior Permanent Magnet Synchronous Generator. In Proceedings of the 2015 International Conference on Electrical Drives and Power Electronics (EDPE), Tatranska Lomnica, Slovakia, 21–23 September 2015; pp. 497–502. [Google Scholar]
- Pairo, H.; Shoulaie, A. Effective and simplified method in maximum efficiency control of interior permanent magnet synchronous motors. IET Electr. Power Appl.
**2017**, 11, 447–459. [Google Scholar] [CrossRef] - Dirscherl, C.; Hackl, C.; Schechner, K. Modellierung und Regelung von modernen Windkraftanlagen: Eine Einführung. In Elektrische Antriebe—Regelung von Antriebssystemen; Schröder, D., Ed.; Springer: Berlin/Heidelberg, Germany, 2015; Chapter 24; pp. 1540–1614. [Google Scholar] [Green Version]
- Richter, J.; Gemaßmer, T.; Doppelbauer, M. Predictive Current Control of Saturated Cross-Coupled Permanent Magnet Synchronous Machines. In Proceedings of the 2014 International Symposium on Power Electronics, Electrical Drives, Automation and Motion, Ischia, Italy, 18–20 June 2014; pp. 830–835. [Google Scholar] [CrossRef]
- Wittig, B. Verbesserung des Schalt- und Betriebsverhaltens von Leistungs-MOSFETs mit niedriger Spannungsfestigkeit und hoher Stromtragfähigkeit durch Optimierung der Treiberschaltung. Ph.D. Thesis, Christian-Albrechts-Universität zu Kiel, Kiel, Germany, 2012. [Google Scholar]
- Wintrich, A.; Nicolai, U.; Tursky, W.; Reimann, T. Applikationshandbuch Leistungshalbleiter, 2nd ed.; ISLE Verlag: Ilmenau, Germany, 2015. [Google Scholar]
- Hackl, C.M.; Kamper, M.J.; Kullick, J.; Mitchell, J. Current control of reluctance synchronous machines with online adjustment of the controller parameters. In Proceedings of the 2016 IEEE 25th International Symposium on Industrial Electronics (ISIE), Santa Clara, CA, USA, 8–10 June 2016; pp. 153–160. [Google Scholar] [CrossRef]
- Hackl, C.M. Non-Identifier Based Adaptive Control in Mechatronics; Springer International Publishing: Basel, Switzerland, 2017. [Google Scholar]
- Xu, L.; Yao, J. A Compensated Vector Control Scheme of a Synchronous Reluctance Motor Including Saturation and Iron Losses. IEEE Trans. Ind. Appl.
**1992**, 28, 1330–1338. [Google Scholar] [CrossRef]

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

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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