# Design, Modeling and Control of Magnetic Bearings for a Ring-Type Flywheel Energy Storage System

^{*}

## Abstract

**:**

## 1. Introduction

- A low energy consumption ring-type FESS is proved to be feasible. The ring-type flywheel possesses higher moment of inertia and allows for higher energy storage.
- Magnetic force model of the ring-type HMB with Halbach array configuration is established.
- Magnetic force model of the ring-type PMB with Halbach array configuration is established.

## 2. System Modeling

#### 2.1. Description of Flywheel Energy Storage System

^{3}and specific energy of 4.74 kJ/kg. The system consists of axial PMB, radial HMB, and back-up ball bearings. Recall that this study focuses on the magnetic force modeling and static levitation of magnetic bearing, and hence the motor is not included here. The locations of PMB and HMB with Halbach array are also indicated in Figure 1. There are two sets of radial HMB, one at the top and the other at the bottom of the FESS, for the active control of four DOFs of the rotor. The schematic of the radial HMB is shown in Figure 2. On the rotor side of HMB, there are three layers of permanent magnet rings arranged in a Halbach array, and the corresponding actuator coils are mounted on the stator side of HMB. The rotor can be actively stabilized by the attraction and repulsion forces provided by the HMB, depending on positive or negative currents. Note that no vertical force will be generated with this configuration.

#### 2.2. Hybrid Magnetic Bearing

^{2}, and no iron core in the coil. For the Halbach array, each permanent magnet block is of 8 mm × 8 mm × 30 mm (a = 8 mm and b = 8 mm in Figure 2) with different polarity, and the material is Neodymium magnet (Nd-Fe-B).

^{2}value is 0.9997, indicating the excellent goodness of the fit. The results reveal different characteristic of the force model from the conventional electromagnet, where the force is proportional to the square of coil current and inversely proportianal to the square of air gap.

#### 2.3. Passive Magnetic Bearing

^{2}values for the fitted vertical and radial force functions are 0.9915 and 0.9819, respectively. Again, the results indicate the excellent goodness of the fit.

#### 2.4. Governing Equation

_{t}. From Equations (13)–(16), it is obvious that HMB requires no bias current theoretically if the rotor is stabilized at the center position (i.e., ${x}_{t}=0$, ${x}_{b}=0$, ${y}_{t}=0$, ${y}_{b}=0$).

## 3. Controller Design

#### 3.1. State Space Model

#### 3.2. Integral Sliding Mode Control

_{1}and b

_{2}are positive constants. If the system state is on the sliding manifold $\mathsf{\sigma}=0$ and can be maintained on the manifold ($\dot{\mathsf{\sigma}}=0$), the system dynamics will be like:

## 4. Experimental Results

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 5.**The magnetic force of a hybrid magnetic unit (N = 350): (

**a**) s = 1.5 mm; (

**b**) s = 2.0 mm; and (

**c**) s = 2.5 mm.

**Figure 10.**The moments caused by the vertical force ${f}_{pv}$ of 16 sets of passive magnetic units about X-axis and Y-axis.

**Figure 17.**Experimental results: (

**a**) rotor trajectory at the bottom HMB; and (

**b**) rotor trajectory at the top HMB.

**Figure 18.**Experimental results: (

**a**) displacement ${y}_{b}$ and ${y}_{b}$; and (

**b**) displacement ${x}_{t}$ and ${y}_{t}$.

**Figure 19.**Experimental results: (

**a**) control currents ${i}_{bx}$ and ${i}_{by}$; and (

**b**) control currents ${i}_{tx}$ and ${i}_{ty}$.

Symbol | Quantity | Value |
---|---|---|

m | mass of rotor | 20.76 kg |

J_{t} | cross-section radial mass moments of inertia | 0.1720 kg·m^{2} |

J_{p} | polar mass moments of inertia | 0.1793 kg·m^{2} |

l_{b} | nominal distances from mass center to the bottom HMB | 92.97 mm |

l_{t} | nominal distances from mass center to the top HMB | 93.03 mm |

l_{bp} | nominal distances from mass center to the bottom and top PMB | 42.12 mm |

l_{tp} | nominal distances from mass center to the bottom and top PMB | 41.88 mm |

Symbol | Quantity | Value |
---|---|---|

l_{0a} | nominal air gap of HMB | 2.0 mm |

l_{0p} | nominal air gap of PMB | 3.8 mm |

z_{0} | nominal offset of PMB | 3.4 mm |

N | number of coil turns | 350 |

a | width of permanent magnet for HMB | 8.00 mm |

b | height of permanent magnet for HMB | 8.00 mm |

c | width of permanent magnet for PMB | 5.00 mm |

d | height of permanent magnet for PMB | 5.00 mm |

Symbol | Quantity | Value |
---|---|---|

k_{1}, k_{2}, k_{3}, k_{4} | coefficients of AMB | 6.8467 N/A |

k_{5}, k_{7} | coefficients of PMB | 14,787 N/m |

k_{6}, k_{8} | coefficients of PMB | 5494 N/m |

k_{9}, k_{11} | coefficients of PMB | 5460 N/m |

k_{10}, k_{12} | coefficients of PMB | 14,819 N/m |

k_{mxb}, k_{myb} | coefficients of moments by PMB | 601.18 N |

k_{mxt}, k_{myt} | coefficients of moments by PMB | 599.25 N |

Symbol | Quantity | Value |
---|---|---|

b_{1} | parameter of equivalent control | 30.00 |

b_{2} | parameter of equivalent control | 55.00 |

k_{c} | parameter of switching control | 56.66 |

ε | parameter of switching control | 0.50 |

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**MDPI and ACS Style**

Toh, C.-S.; Chen, S.-L.
Design, Modeling and Control of Magnetic Bearings for a Ring-Type Flywheel Energy Storage System. *Energies* **2016**, *9*, 1051.
https://doi.org/10.3390/en9121051

**AMA Style**

Toh C-S, Chen S-L.
Design, Modeling and Control of Magnetic Bearings for a Ring-Type Flywheel Energy Storage System. *Energies*. 2016; 9(12):1051.
https://doi.org/10.3390/en9121051

**Chicago/Turabian Style**

Toh, Chow-Shing, and Shyh-Leh Chen.
2016. "Design, Modeling and Control of Magnetic Bearings for a Ring-Type Flywheel Energy Storage System" *Energies* 9, no. 12: 1051.
https://doi.org/10.3390/en9121051