# Design and Modeling of an Integrated Flywheel Magnetic Suspension for Kinetic Energy Storage Systems

^{*}

## Abstract

**:**

## 1. Introduction

- There are very limited winding losses because the steady-state suspension force is always applied by the PM flux;
- There are no flux variations in static conditions, resulting in negligible rotor losses and the possibility to adopt solid magnetic cores instead of laminated stacks;
- Possible eddy currents generated at dynamic condition provide a damping effect, improving the stability control;
- The flywheel itself provides a high surface available for the axial force application, without introducing additional rotating cores;
- There is independent control for the UCs and the LC;
- Simpler UCs control assessment as the force current characteristic is almost linear because of the DC bias due to the PM flux;
- There is an inherent self-centering effect in the radial direction, possibly controlled by both the coil currents.

## 2. System Description

- The rated air-gap length must be large enough to deal with high-rate disturbances, the compensation of which could require high electrical and thermal stresses to the AHMB coils;
- Temperature increase deteriorates the PM properties with possible loss of the stable condition;
- The lower the coercivity, the thicker the PM must be, therefore decreasing the useful steel surface; the consequent higher $\langle {B}_{g}\rangle $ can be obtained only with higher PM and core volumes.

## 3. AHMB Electromagnetic Design

^{®}2D, Ansys, Inc., Canonsburg, PA, USA). The purpose is to define the geometric configuration by parametric analyses, focused on the improvement of the force performances and the winding inductance values. The potential high number of geometric variables can be reduced by adopting some fair design assumptions. With reference to Figure 4, the following conditions are defined:

- (a)
- radial bounds defined by ${r}_{i}$ and ${r}_{o}$;
- (b)
- equal annular cross section for all the pole shoes, that are ${S}_{up}={S}_{up,h}$ (h = 1, …, 4) for the upper side and ${S}_{lp}={S}_{lp,j}$ (j = 1, 2) for the lower side; such a condition should ensure uniform flux density distribution, resulting in evenly distributed axial force among the pole shoes on the same side;
- (c)
- equal height for all the pole shoes (upper side ${h}_{up}$, lower side ${h}_{lp}$);
- (d)
- equal annular cross section of the core legs ${S}_{ul,h}={S}_{ul}={k}_{u}{S}_{up}$ (h = 1, …, 4) for the upper side and ${S}_{ll,j}={S}_{ll}={k}_{l}{S}_{lp}$ (j = 1,2) for the lower side; ${k}_{u}$ and ${k}_{l}$ are decreasing factors to enable enough room for the coil placement; their choice must consider local magnetic saturation, even with the coil mmf contribution;
- (e)
- equal cross section ${S}_{uc}={h}_{uc}^{\prime}{w}_{uc}^{\prime}={h}_{uc}^{\u2033}{w}_{uc}^{\u2033}$ of the upper coils considering that they should provide the same mmf with the same current density;
- (f)
- PM operation near the maximum energy point (little higher than ${B}_{r}/2$) at the rated air-gap length with unexcited coils;
- (g)
- fixed variation range for the PM height $\left\{{h}_{m,min},{h}_{m,max}\right\}$ and radius $\left\{{r}_{mu,min},{r}_{mu,max}\right\}$ for manufacturing and cost issues.

#### 3.1. Upper Side

^{2}). It is worth noting that strategy B provides a demagnetizing effect with a minimum for ${\varrho}_{u}\cong $ 8 A/mm

^{2}.

- the suspension force ${F}_{z0}^{*}$ at the air-gap ${g}^{*}$ due to the only PM;
- the suspension forces ${F}_{z0}^{\prime}$ and ${F}_{z}^{\prime}$ with augmented air-gap ${g}_{u,max},$ due to the PM only and in the presence of current excitation (current density ${\varrho}_{uM}$), respectively;
- the coil per-turn inductance ${L}_{uc}$ in the same condition as ${F}_{z}^{\prime}$ is calculated; because of the series connection and the chosen current polarity, it is ${L}_{uc}={L}_{uc}^{\prime}+{L}_{uc}^{\u2033}+2{M}_{uc}$, with ${L}_{uc}^{\prime},{L}_{uc}^{\u2033}$ being the inner and outer coil inductances, respectively, and ${M}_{uc}$ being the mutual inductance coefficient.

#### 3.2. Lower side

- constant coil section ${S}_{lc}$ with variable aspect ratio ${k}_{lc}={h}_{lc}/{w}_{lc}$; it follows that ${w}_{lc}=\sqrt{{S}_{lc}/k\_lc}$ and ${h}_{lc}={k}_{lc}{w}_{lc}$;
- variable pole shoe height ${h}_{lp}={h}_{lp}^{\prime}$ + ${h}_{lp}^{\u2033}$ with fixed value for ${h}_{lp}^{\prime}$.

^{2}(total ampere-turns ${A}_{l}=$ 720 A). Moreover, the force characteristic is approximately linear, and the coil inductance is almost constant for ${\varrho}_{l}$ up to 8 A/mm

^{2}because of the limited magnetic saturation. Then, the net axial force $\Delta {F}_{zl}={F}_{zl}-{F}_{zl}^{*}$ and the per-turn coil inductance ${L}_{lc}$ as functions of ${k}_{lc}$ and ${a}_{1l}$ are examined, imposing that ${A}_{l}=$ 750 A and keeping unvaried the remaining parameters.

## 4. Control System

^{®}Simulink (The Mathworks, Inc., Natick, MA, USA) code. This code implements both the force balance equation governing the flywheel vertical motion and the electrical equation reproducing the coil supply condition. The solution of such equations requires, in turn, the definition of the electromagnetic models describing the axial forces and the coil inductances as functions of the air-gap length and of the coil ampere turns.

#### 4.1. Model of the Suspension Dynamics

#### 4.2. Electrical Model

#### 4.3. Overall Model

- an external force ${F}_{zd}$ added to the force balance; in practical cases, such contribution can derive from vertical attraction forces between the rotor and the stator of the electrical machine;
- an air-gap variation $\Delta {g}_{d}$ caused, for instance, by a sudden motion of the AHMB fixed parts;
- a superimposed vertical velocity $dz/dt$, simulating, for instance, the effect of an earthquake (instantaneous data of a seismic velocity).

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## List of symbols

${\alpha}_{1},{\alpha}_{2},{\alpha}_{3},{\beta}_{1}$ | weighting coefficients of the performance index P |

${a}_{1u},{a}_{2u},{a}_{3u}$ | per-unit variables used for the parametrization of the upper pole shoes geometry |

${a}_{1l}$ | per-unit variable used for the parametrization of the lower pole shoe geometry |

${A}_{u},{A}_{l}$ | upper and lower coil ampere-turns |

${b}_{1u},{b}_{3u}$ | slot openings of the upper cores |

${B}_{r},{H}_{c},{\alpha}_{T}$ | PM remanence, coercivity, and temperature coefficient |

${B}_{g}$ | mean air-gap flux density (axial component) |

${B}_{s},{\mu}_{r}$ | saturation flux density and magnetic permeability |

$\Delta {B}_{z}^{\prime}$, $\Delta {B}_{z}^{\u2033}$ | standard deviations of the axial flux density in the lower core legs |

$\Delta {F}_{zl}$ | net axial force with lower coil excitation |

$\Delta g,\Delta {g}_{M}$ | air-gap deviation with respect to ${g}^{*}$ and its maximum value |

$\Delta {g}_{d}$ | air-gap disturbance |

$\Delta {r}_{1u},\Delta {r}_{3u},\Delta {r}_{5u},\Delta {r}_{7u}$ | pole shoe widths of the inner and the outer cores (upper side) |

$\Delta {r}_{1l}$ | width of the inner pole shoe of the lower side core |

${F}_{z},{F}_{z0},{F}_{zu},{F}_{zl}$ | resultant, PM, upper side and lower side force contribution |

${F}_{z}^{*}$ | requested force for the total mass suspension |

${F}_{z0}^{*}$ | rated value of the PM suspension force |

${F}_{z0}^{\prime}$, ${F}_{z0}^{\u2033}$ | PM suspension forces at ${g}^{*}+\Delta {g}_{M}$ and ${g}^{*}-\Delta {g}_{M}$; |

${F}_{z}^{\prime}$ | upper side suspension forces at ${g}^{*}+\Delta {g}_{M}$ with current excitation; |

${F}_{zu}^{*},{F}_{zl}^{*}$ | PM force deviation at the maximum and minimum air-gap bounds; |

${F}_{zu,t},\Delta {F}_{zu,t}$ | contributions of upper side force model dealing with temperature variation |

${F}_{zd}$ | force and air-gap disturbances |

${g}^{*}$ | air-gap length at balanced condition with no coil supply |

${g}_{u},{g}_{l}$ | upper and lower air-gap length |

${h}_{up},{h}_{lp}$ | height of the upper and lower pole shoes |

${h}_{uc}^{\prime},{w}_{uc}^{\prime}$ | height and width of the inner coil (upper side) |

${h}_{uc}^{\u2033},{w}_{uc}^{\u2033}$ | height and width of the outer coil (upper side) |

${h}_{lc},{w}_{lc}$ | height and width of the lower coil |

${h}_{m},{l}_{m},{V}_{m}$ | PM height, length and volume |

$K,{W}_{F},{J}_{F}$ | flywheel shape factor, rated energy amount, and rotational inertia |

${k}_{0}$ | weighting coefficient to model the force with temperature variations |

${k}_{f},{S}_{uc},{S}_{lc}$ | coil filling factor, upper and lower coil cross sections |

$L,R,{\lambda}_{pm}$ | per-turn inductance, resistance, and PM flux |

${L}_{uc},{L}_{lc}$ | upper and lower coil inductance |

${M}_{F},{M}_{a}$ | flywheel and additional masses |

${n}_{max},s$ | maximum flywheel speed and speed ratio |

$P$ | performance index |

$\theta ,\Delta {\theta}_{uc},\Delta {\theta}_{uc}$ | operating temperature and temperature coil rise |

$\rho ,{\sigma}_{lim},{\sigma}_{f}$ | mass density, ultimate and low-cycle fatigue stresses |

${\rho}_{min}$, $\alpha $ | copper resistivity at ${\theta}_{min}$ and thermal coefficient |

${\varrho}_{u},{\varrho}_{l}$ | upper and lower coil current density |

${r}_{1u},\dots ,{r}_{6u}$ | radial coordinates of the upper pole shoes |

${r}_{1l},{r}_{2l}$ | radial coordinates of the lower pole shoes |

${r}_{mu}$ | mean PM radius |

${r}_{ml}$ | mean coil radius of the lower side coil |

${r}_{o},{r}_{i}$ | outer and inner radius of the flywheel rotor rim |

${r}_{uc}^{\prime},{r}_{uc}^{\u2033}$ | mean coil radius of the upper side coils |

${R}_{uc},{R}_{lc}$ | upper and lower coil ohmic resistance |

${S}_{r},{k}_{su}$ | rim surface and surface utilization factor |

${S}_{up,k},{S}_{lp,k}$ | annular cross section of the k-th pole shoe (up: upper side, lp: lower side) |

${S}_{ul,k},{S}_{ll,k}$ | annular cross section of the k-th core leg (ul: upper side, ll: lower side) |

$v,N$ | supply voltage and coil turns |

${\omega}_{max}$, ${\omega}_{min}$ | maximum and minimum flywheel angular speed |

${w}_{1u}^{\prime},{w}_{2u}^{\prime},{w}_{1u}^{\u2033},{w}_{2u}^{\u2033}$ | width of the inner and of the outer core legs (upper side) |

${w}_{l}^{\prime},{w}_{l}^{\u2033}$ | width of the lower side core legs |

${w}_{3u}^{\prime},{w}_{3u}^{\u2033}$ | thickness of the inner and outer yokes (upper side) |

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**Figure 1.**Configurations of an active magnetic bearing with the flux lines generated by the excited coils: (

**a**) radial heteropolar type; (

**b**) axial homopolar type acting on a disc integral with the shaft.

**Figure 2.**Configuration of the axial hybrid magnetic bearing integrated with the flywheel structure; (

**a**) 3D view with split sections; (

**b**) 2D section evidencing the axial hybrid magnetic bearing (AHMB) active parts.

**Figure 4.**Geometric parametrization of the AHMB; (

**a**,

**b**): radial cross section and bottom view (half cross section) of the upper side; (

**c**,

**d**): radial cross section and top view (half cross section) of the lower side; ${\ell}^{\prime}$, ${\ell}^{\u2033}$: lines for the sampling of the axial flux density.

**Figure 5.**Flux density in the upper AHMB side with no excitation (${g}^{*}=$ 2 mm, $\Delta g=$ 0); (

**a**) map of the magnitude; (

**b**) axial component in the air-gap.

**Figure 6.**Possible upper coil (UC) supply strategies with concordant (

**A**,

**B**) or discordant (

**C**,

**D**) current polarities; the orange and black arrows indicate the permanent magnet (PM) and coil flux directions, respectively.

**Figure 7.**Force comparison between the coil supply strategies; A, B, C, D: excitation polarity represented in Figure 6; B1: excitation polarity as in B at a reduced air-gap length.

**Figure 8.**Results of the parametric analyses; (

**a**) per unit (p.u.) quantities; (

**b**) performance index.

**Figure 9.**PM axial force ${F}_{z0}$ as a function of the air-gap length ${g}_{u}$; ${F}_{zu}^{*}$, ${F}_{zl}^{*}$: force deviations at ${g}_{u,max}$ and ${g}_{u,min}$.

**Figure 10.**Net force and inductance of the AHMB lower side; (

**a**) dependence on the coil aspect ratio ${k}_{lc}$; (

**b**) dependence on the p.u. variable ${a}_{1l}$.

**Figure 12.**Force components developed by the AHMB upper side as functions of ${g}_{u}$ and ${A}_{u}$; (

**a**) force at the mean temperature; (

**b**) differential force.

**Figure 14.**Per-turn inductance as a function of ${g}_{u}$ for different ampere-turn values; (

**a**) upper winding; (

**b**) lower winding.

**Figure 16.**Dynamic progress of the air-gap and of the coil ampere-turns; (

**a**) trapezoidal time variation of the temperature between ${\theta}_{min}$ and ${\theta}_{max}$; (

**b**) step force disturbance ${F}_{zd}=-$ 150 N.

**Figure 17.**Simulation results of a seismic event; (

**a**) oscillation speed and upper air-gap variation; (

**b**) upper and lower coil ampere-turns (dotted box: zoomed interval on ampere-turn variation).

FESS | Materials | |||
---|---|---|---|---|

Rated energy ${W}_{F}$ | 2.2 kWh | AISI 4340 | Ultimate strength ${\sigma}_{lim}$ | 1790 MPa |

Maximum speed ${n}_{max}$ | 32000 rpm | Steel density $\rho $ | 7830 kg/m^{3} | |

Speed ratio $s$ | 3 | Magnetic permeability ${\mu}_{r}$ | 600 | |

Shape factor $K$ | 0.75 | Saturation flux density ${B}_{s}$ | 1.8 T | |

Rotational inertia ${J}_{F}$ | 1.59 kg m^{2} | NdFeB | Remanence ${B}_{r}$@20 °C | 1.1 T |

Flywheel mass ${M}_{F}$ | 74 kg | Coercivity ${H}_{c}$@20 °C | −838 kA/m | |

Outer radius ${r}_{o}$ | 200 mm | Temperature coefficient ${\alpha}_{T}$ | −0.12%/°C | |

Inner radius ${r}_{i}$ | 132 mm | AISI 1008 | Maximum permeability | 1250 |

Additional mass ${M}_{a}$ | 50 kg | Saturation flux density ${B}_{s}$ | 2.3 T |

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

Rated air-gap ${g}^{*}$ | 2 mm | Operating temperature ${\theta}^{*}$ | 80 °C |

Maximum air-gap deviation $\Delta {g}_{M}$ | 1 mm | Coil filling factor ${k}_{f}$ | 0.6 |

Average flux density ${B}_{g}$ | 0.24 T | Maximum core flux density ${B}_{cM}$ | 1.8 T |

Pole section ${S}_{up}$ | 131.4 mm^{2} | Core leg section ${S}_{ul}$ | 263 mm^{2} |

Maximum current density ${\varrho}_{uM}$ | 7 A/mm^{2} | Coil section ${S}_{uc}$ | 110 mm^{2} |

Pole shoe height ${h}_{up}$ | 4.5 mm | PM volume ${V}_{m}$ | 64 cm^{3} |

PM thickness ${l}_{m}$ | 10 mm | PM height ${h}_{m}$ | 6 mm |

Quantity | Reference Value | Optimal Value |
---|---|---|

$\left\{{a}_{1u},{a}_{2u},{a}_{3u}\right\}$ | $\left\{0.7,0.5,0.3\right\}$ | $\left\{0.7,0.7,0.3\right\}$ |

${F}_{z0}$ [N] | 1210 | 1212 |

${F}_{z}^{\prime}$ [N] | 1211 | 1205 |

${l}_{coil}$ | 1.21 | 1.11 |

${F}_{ob}$ | 1 | 0.979 |

$\left\{\Delta {r}_{1u},\Delta {r}_{3u},\Delta {r}_{5u}\right\}$ [mm] | $\left\{13.2,12.5,10.6\right\}$ | $\left\{13.9,12.3,10.6\right\}$ |

$\left\{{b}_{1u},{b}_{3u}\right\}$ [mm] | $\left\{3.2,3.2\right\}$ | $\left\{4.5,2\right\}$ |

$\left\{{w}_{uc}^{\prime},{w}_{uc}^{\u2033}\right\}$ [mm] | $\left\{5,6.1\right\}$ | |

$\left\{{w}_{1u}^{\prime},{w}_{2u}^{\prime},{w}_{3u}^{\prime}\right\}$ [mm] | $\left\{5.6,4.7,8\right\}$ | |

$\left\{{w}_{1u}^{\u2033},{w}_{2u}^{\u2033},{w}_{3u}^{\u2033}\right\}$ [mm] | $\left\{4.4,4.2,5.9\right\}$ |

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

Coil section ${S}_{lc}$ | 200 mm^{2} | Coil aspect ratio ${k}_{lc}$ | 4.25 |

Pole shoe height ${h}_{lp}^{\prime}$ | 4 mm | Pole shoe width $\Delta {r}_{1l}$ | 29 mm |

Core leg reduction factor ${k}_{l}$ | 0.4 | Current density ${\varrho}_{l}$ | 6.25 A/mm^{2} |

Coil mean radius ${r}_{ml}$ | 164 mm | Pole shoe tapered height ${h}_{lp}^{\u2033}$ | 8 mm |

Standard deviation (inner leg) $\Delta {B}_{z}^{\prime}$ | 0.364 T | Coil per-turn inductance ${L}_{lc}$ | 11.17 μH |

Standard deviation (outer leg) $\Delta {B}_{z}^{\u2033}$ | 0.249 T | Net axial force $\Delta {F}_{zl}$ | −59.1 N |

Parameters | Upper Side | Lower Side |
---|---|---|

Design current density $\varrho $ | 4 A/mm^{2} | 6 A/mm^{2} |

Winding design voltage $v$ | 60 V | 60 V |

Maximum converter voltage ${v}_{max}$ | 200 V | 200 V |

Turns/coil $N$ | 33 | 46 |

Ohmic resistance $R$@20 °C | 21.2 mΩ | 6.1 mΩ |

Current regulator $\left\{{K}_{P},{K}_{I}\right\}$ | $\left\{0.4,0.1\right\}$ | $\left\{0.12,0.1\right\}$ |

Anti-windup constant ${K}_{Ia}$ | 10^{−4} | 5∙10^{−5} |

Air-gap regulator $\left\{{K}_{P},{K}_{I},{K}_{D},{K}_{N}\right\}$ | $\left\{25000,24000,10000,100\right\}$ | |

Temperature range $\left\{{\theta}_{min},{\theta}_{max}\right\}$ | {20 °C, 80 °C} |

© 2020 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**

Andriollo, M.; Benato, R.; Tortella, A.
Design and Modeling of an Integrated Flywheel Magnetic Suspension for Kinetic Energy Storage Systems. *Energies* **2020**, *13*, 847.
https://doi.org/10.3390/en13040847

**AMA Style**

Andriollo M, Benato R, Tortella A.
Design and Modeling of an Integrated Flywheel Magnetic Suspension for Kinetic Energy Storage Systems. *Energies*. 2020; 13(4):847.
https://doi.org/10.3390/en13040847

**Chicago/Turabian Style**

Andriollo, Mauro, Roberto Benato, and Andrea Tortella.
2020. "Design and Modeling of an Integrated Flywheel Magnetic Suspension for Kinetic Energy Storage Systems" *Energies* 13, no. 4: 847.
https://doi.org/10.3390/en13040847