# A Novel High Performance Discrete Flux Integrator for Control Algorithms of Fast Rotating AC Machines

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## Abstract

**:**

## 1. Introduction

## 2. Unified Electric Machine Model

- a: subscript that indicates where that value is measured, s for stator, r for rotor, m for mutual;
- b: subscript indicating if the value is referred to the d or q axis;
- c: superscript showing the reference frame that the value is given, s for stator, r for rotor reference frame.

- Input: voltage $\mathit{v}$;
- Output: current $\mathit{i}$;
- State: flux $\mathit{\phi}$

## 3. Solver Optimization

#### 3.1. Continuous Time

#### 3.2. Discretization

- $\left[M\right]$ does not depend on the position $\theta $, thus it is time invariant in case of constant switching frequency (${T}_{c}=const$).
- $\left[M\right]$ allows using infinite resistance coefficient for machines with no rotor currents because the matrix $\left[R\right]$ is always used as inverse. $\left[R\right]$ is diagonal and in case of a coefficient $r=\infty $ the corresponding element in ${\left[R\right]}^{-1}$ is simply 0.

#### 3.3. Sub-Interval Predictive Integration

## 4. Implementation

#### 4.1. CNT: Continuous Model

#### 4.2. SIMULINK-DSCR: Discrete Model with Simulink Integrator Block

#### 4.3. NAT-DSCR: Discrete Model with Sub-Interval Integration in Natural Reference Frame

#### 4.4. INV-DSCR: Discrete Model with Fast Matrix Inversion

#### 4.5. ROTOR-DSCR: Discrete Model with Sub-Interval Integration in Rotor Reference Frame

#### 4.6. FAST-DSCR: Discrete Model with Fast Rotation Transformation

#### 4.7. Testing Description

## 5. Results

#### 5.1. Model Execution Speed

#### 5.2. Model Precision

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

AC | Alternate Current |

AR-SM | Assistant Reluctance Synchronous Machine |

CNT | Continuous Model |

DSP | Digital Signal Processor |

FAST-DSR | Discrete model with fast rotation transformation |

FOC | Field Oriented Control |

IM | Induction Machine |

INV-DSCR | Discrete model with fast matrix inversion |

IPM-SM | Internal Permanent Magnets Synchronous Machine |

NAT-DSCR | Discrete model with sub-interval integration in natural reference frame |

PI | Proportional-Integral regulator |

PWM | Pulse Width Modulation |

R-SM | Reluctance Synchronous Machine |

ROTOR-DSCR | Discrete model with sub-interval integration in rotor reference frame |

SIM-DSCR | Discrete model with Simulink integration block |

SIMULINK-DSCR | Discrete model with Simulink integration block |

SM | Synchronous Machine |

SPM-SM | Surface Permanet Magnets Synchronous Machine |

## Appendix A

**Figure A1.**Discrete time model using the algorithm under study without optimization and PM flux: NAT-DSCR.

## References

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**Figure 10.**Execution time comparison increasing the discrete sub-interval integration number. CNT and SIMULINK-DSCR do not depend on the number of sub-intervals. Time is normalized to the CNT execution time.

**Figure 11.**Simulink flux machine model comparison example between the model presented: CNT, SIMULINK-DSCR, NAT-DSCR, INV-DSCR, ROT-DSCR and FAST-DSCR. (

**a**) overview, (

**b**) zoom to highlight differences.

**Figure 12.**Percentage difference between CNT and FAST-DSCR at low frequency increasing the integration sub-intervals for flux estimation in (from top to bottom) stator d-axis, stator q-axis, rotor d-axis, rotor q-axis. Stator field and rotor mechanical rotational speed equal to $6\phantom{\rule{3.33333pt}{0ex}}\frac{\mathrm{rad}}{\mathrm{s}}$.

**Figure 13.**Percentage difference between CNT and FAST-DSCR at high frequency increasing the integration sub-intervals for flux estimation in (from top to bottom) stator d-axis, stator q-axis, rotor d-axis, rotor q-axis. Stator field and rotor mechanical rotational speed have different values, 6200 and $5700\phantom{\rule{3.33333pt}{0ex}}\frac{\mathrm{rad}}{\mathrm{s}}$ respectively.

Description | Value | Unit |
---|---|---|

Solver | ode45 | - |

Max-Min Step Size | auto | - |

Simulation Time | 5 | s |

Discrete Sample Time | $0.125$ | ms |

Discrete Sample Frequency | 8000 | Hz |

Number of Sub-intervals | variable | - |

Nominal Voltage | 360 | V |

Nominal Current | 230 | A |

Max Speed | 15,000 | rpm |

Pole Pairs | 4 | - |

Stator Resistance | $3.4$ | m$\mathrm{\Omega}$ |

Rotor Resistance | $1.3$ | m$\mathrm{\Omega}$ |

Stator Inductance | $0.16$ | mH |

Rotor Inductance | $0.16$ | mH |

Mutual Inductance | $0.143$ | mH |

Magnet Flux | none | - |

**Table 2.**Mean error squared results of low and high frequency simulations. Base error is the one of the 1 sub-interval model.

Low Frequency Example | High Frequency Example | |||
---|---|---|---|---|

$6\phantom{\rule{3.33333pt}{0ex}}\frac{\mathbf{rad}}{\mathbf{s}}$ | $6000\phantom{\rule{3.33333pt}{0ex}}\frac{\mathbf{rad}}{\mathbf{s}}$ | |||

Algorithm | Mean Err. Sq. | Variation | Mean Err. Sq. | Variation |

Stator d-axis - ${\phi}_{sd}^{n}$ | ||||

SIM-DSCR | $34.2\mathrm{e}-7$ | −37.1% | $120.0\mathrm{e}-5$ | $+113.6\%$ |

FAST 1 int. | $54.3\mathrm{e}-7$ | $0\%$ | $57.2\mathrm{e}-5$ | $0\%$ |

FAST 2 int. | $20.2\mathrm{e}-7$ | $-62.8\%$ | $26.3\mathrm{e}-5$ | $-53.9\%$ |

FAST 3 int. | $12.5\mathrm{e}-7$ | $-76.9\%$ | $18.7\mathrm{e}-5$ | $-67.3\%$ |

FAST 5 int. | $7.8\mathrm{e}-7$ | $-85.7\%$ | $13.5\mathrm{e}-5$ | $-76.4\%$ |

FAST 10 int. | $4.9\mathrm{e}-7$ | $-90.8\%$ | $10.2\mathrm{e}-5$ | $-82.2\%$ |

FAST 15 int. | $4.2\mathrm{e}-7$ | $-92.3\%$ | $9.2\mathrm{e}-5$ | $-84.0\%$ |

Stator q-axis - ${\phi}_{sq}^{n}$ | ||||

SIM-DSCR | $33.9\mathrm{e}-7$ | $-37.0\%$ | $150.0\mathrm{e}-5$ | $+113.3\%$ |

FAST 1 int. | $53.9\mathrm{e}-7$ | $0\%$ | $72.3\mathrm{e}-5$ | $0\%$ |

FAST 2 int. | $20.8\mathrm{e}-7$ | $-61.4\%$ | $33.6\mathrm{e}-5$ | $-53.5\%$ |

FAST 3 int. | $13.4\mathrm{e}-7$ | $-75.1\%$ | $24.0\mathrm{e}-5$ | $-66.8\%$ |

FAST 5 int. | $8.8\mathrm{e}-7$ | $-83.7\%$ | $17.5\mathrm{e}-5$ | $-75.8\%$ |

FAST 10 int. | $6.1\mathrm{e}-7$ | $-88.7\%$ | $13.4\mathrm{e}-5$ | $-81.5\%$ |

FAST 15 int. | $5.3\mathrm{e}-7$ | $-90.1\%$ | $12.1\mathrm{e}-5$ | $-83.3\%$ |

Rotor d-axis - ${\phi}_{rd}^{n}$ | ||||

SIM-DSCR | $14.9\mathrm{e}-7$ | $-87.7\%$ | $48.5\mathrm{e}-5$ | $-19.9\%$ |

FAST 1 int. | $121.8\mathrm{e}-7$ | $0\%$ | $60.6\mathrm{e}-5$ | $0\%$ |

FAST 2 int. | $29.2\mathrm{e}-7$ | $-76.0\%$ | $18.9\mathrm{e}-5$ | $-69.3\%$ |

FAST 3 int. | $12.6\mathrm{e}-7$ | $-89.7\%$ | $10.0\mathrm{e}-5$ | $-83.6\%$ |

FAST 5 int. | $4.4\mathrm{e}-7$ | $-96.4\%$ | $5.0\mathrm{e}-5$ | $-91.8\%$ |

FAST 10 int. | $1.3\mathrm{e}-7$ | $-98.9\%$ | $2.4\mathrm{e}-5$ | $-96.0\%$ |

FAST 15 int. | $0.8\mathrm{e}-7$ | $-99.3\%$ | $1.8\mathrm{e}-5$ | $-97.1\%$ |

Rotor q-axis - ${\phi}_{rq}^{n}$ | ||||

SIM-DSCR | $1.9\mathrm{e}-7$ | $-95.7\%$ | $42.3\mathrm{e}-5$ | $-22.8\%$ |

FAST 1 int. | $44.6\mathrm{e}-7$ | $0\%$ | $54.8\mathrm{e}-5$ | $0\%$ |

FAST 2 int. | $12.7\mathrm{e}-7$ | $-71.4\%$ | $16.8\mathrm{e}-5$ | $-69.3\%$ |

FAST 3 int. | $6.7\mathrm{e}-7$ | $-84.9\%$ | $9.0\mathrm{e}-5$ | $-83.6\%$ |

FAST 5 int. | $3.6\mathrm{e}-7$ | $-92.0\%$ | $4.5\mathrm{e}-5$ | $-91.8\%$ |

FAST 10 int. | $2.1\mathrm{e}-7$ | $-95.1\%$ | $2.1\mathrm{e}-5$ | $-96.1\%$ |

FAST 15 int. | $1.9\mathrm{e}-7$ | $-95.7\%$ | $1.6\mathrm{e}-5$ | $-97.2\%$ |

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**MDPI and ACS Style**

Rossi, C.; Pilati, A.; Bertoldi, M.
A Novel High Performance Discrete Flux Integrator for Control Algorithms of Fast Rotating AC Machines. *Appl. Sci.* **2021**, *11*, 2150.
https://doi.org/10.3390/app11052150

**AMA Style**

Rossi C, Pilati A, Bertoldi M.
A Novel High Performance Discrete Flux Integrator for Control Algorithms of Fast Rotating AC Machines. *Applied Sciences*. 2021; 11(5):2150.
https://doi.org/10.3390/app11052150

**Chicago/Turabian Style**

Rossi, Claudio, Alessio Pilati, and Marco Bertoldi.
2021. "A Novel High Performance Discrete Flux Integrator for Control Algorithms of Fast Rotating AC Machines" *Applied Sciences* 11, no. 5: 2150.
https://doi.org/10.3390/app11052150