# Design of a Neural Super-Twisting Controller to Emulate a Flywheel Energy Storage System

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Background

#### 2.1. FESS

#### 2.2. Nonlinear Systems

#### 2.3. RWFONN

#### 2.4. Filtered Error Training Algorithm

**Theorem**

**1.**

- 1
- ${\xi}_{j}^{i}$, ${\mathbf{w}}_{j}^{i}$∈${\mathcal{L}}_{\infty}$ ($i.e.,{\xi}_{j}^{i}$ and ${w}_{j}^{i}$ are uniformly bounded);
- 2
- ${lim}_{t\to \infty}{\xi}_{i}\left(t\right)=0$.

#### 2.5. Nonlinear Block Controllable Form

#### 2.6. Super-Twisting Control Algorithm

## 3. Proposed Methodology

- The charging procedure (storing kinetic energy in the flywheel). This procedure consists of the PMSM in the $dq$ coordinate frame [30] working in motor mode to the control the angular velocity through the RWFONN using the block control linearization technique. In this scenario, the kinetic energy is stored in the flywheel (dotted red block in Figure 1);
- Discharging procedure (releasing energy). In this stage, the stored energy is now transferred to the PMSM working in generator mode. This scenario is considered in the case when there exists an electrical failure in the utility grid or when the flywheel is required to compensate with active power to the utility grid to solve a problem of the power management office, and the velocity controller for the DC-motor emulates the deceleration of the flywheel through the use of Equation (2). The flywheel energy discharge is emulated by the DC-machine (dotted blue block in Figure 1).

#### 3.1. Flywheel Storage System

#### 3.2. Nonlinear Identification and NSTC Applied to PMSM

#### 3.2.1. PMSM Neural Identification

#### 3.2.2. NSTC Application to PMSM

#### 3.3. Nonlinear Identification and NSTC Applied to DC-Motor

#### 3.3.1. DC-Motor Neural Identification

#### 3.3.2. NSTC Application to DC-Motor

## 4. Simulation Results

#### 4.1. Neural Identification

#### 4.1.1. PMSM States Identification

#### 4.1.2. DC-motor States Identification

#### 4.2. PMSM and DC-motor Controller-Emulation of Flywheel

#### 4.3. Power Delivery

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AC | Alternating-Current |

DC | Direct-Current |

$dq$ | Direct-Quadrature |

FESS | Flywheel Energy Storage System |

HONN | High Order Neural Network |

NN | Neural Network |

NSTC | Neural Super-Twisting Control |

PMSM | Permanent Magnet Synchronous Machine |

RMS | Root Mean Square |

RHONN | Recurrent High Order Neural Network |

RSFONN | Recurrent Sigmoid First-Order Neural Network |

RWFONN | Recurrent Wavelet First-Order Neural Network |

STA | Super-Twisting Algorithm |

STC | Super-Twisting Control |

SVPWM | Space Vector Pulse Width Modulation |

UUB | Uniformly Ultimately Bounded |

## Appendix A. Identification Error Boundedness

**Theorem**

**A1.**

**Proof**

## Appendix B. Closed-Loop Stability Analysis

**Proof.**

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**Figure 10.**PMSM velocity controller performance: (

**a**) velocity error ${\epsilon}_{1}$, (

**b**) control signal ${v}_{q}$.

**Figure 11.**DC-motor velocity tracking performance: (

**a**) velocity error ${\zeta}_{1}$. (

**b**) control signal ${u}_{sdc}$ (10–30 and 52–70 s).

PMSM Parameters | Value |
---|---|

Stator Resistance R | $1.4$$\Omega $ |

Inductance ${L}_{d}$ | $6.6$ mH |

Inductance ${L}_{q}$ | $5.8$ mH |

Inertial Moment J | $0.00176$ kg·m${}^{2}$ |

Damping Coefficient B | $0.00038818$ N·m ·s/rad |

Flux Linkage ${\lambda}_{af}$ | $0.1546$ V·s/rad |

Pair Poles P | 3 |

DC-motor Nameplate Data and Parameters | Value |

Field Voltage | 120 V |

Field Current | $0.5$ A |

Armature Voltage | 120 V |

Armature Current | $3.0$ A |

Rotor Velocity | 1750 rpm |

Armature Resistance ${R}_{a}$ | $12.5$$\Omega $ |

Armature Inductance ${L}_{a}$ | $0.075$ H |

Motor Constant ${K}_{m}$ | $2.602$ |

Inertial Moment ${J}_{m}$ | $0.0036$ N·m·s${}^{2}$ |

Frictional Coefficient ${B}_{m}$ | $0.002$ N·m·s |

Neural Network Structure | ${\mathit{\omega}}_{\mathit{r}}$ | ${\mathit{i}}_{\mathit{d}}$ | ${\mathit{i}}_{\mathit{q}}$ |
---|---|---|---|

RWFONN | 0.1496 | 0.0311 | 0.0263 |

RSFONN | 0.0105 | 0.0015 | 0.0286 |

RHONN | 0.0386 | 0.0032 | 0.0128 |

Neural Network Structure | ${\mathit{\omega}}_{\mathit{m}}$ | ${\mathit{i}}_{\mathit{a}}$ |
---|---|---|

RWFONN | 0.0018 | 0.00004 |

RSFONN | 0.0004 | 0.00005 |

RHONN | 0.0206 | 0.0009 |

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**MDPI and ACS Style**

Magallón, D.A.; Castañeda, C.E.; Jurado, F.; Morfin, O.A.
Design of a Neural Super-Twisting Controller to Emulate a Flywheel Energy Storage System. *Energies* **2021**, *14*, 6416.
https://doi.org/10.3390/en14196416

**AMA Style**

Magallón DA, Castañeda CE, Jurado F, Morfin OA.
Design of a Neural Super-Twisting Controller to Emulate a Flywheel Energy Storage System. *Energies*. 2021; 14(19):6416.
https://doi.org/10.3390/en14196416

**Chicago/Turabian Style**

Magallón, Daniel A., Carlos E. Castañeda, Francisco Jurado, and Onofre A. Morfin.
2021. "Design of a Neural Super-Twisting Controller to Emulate a Flywheel Energy Storage System" *Energies* 14, no. 19: 6416.
https://doi.org/10.3390/en14196416