# Application of the Polynomial Chaos Expansion to the Uncertainty Propagation in Fault Transients in Nuclear Fusion Reactors: DTT TF Fast Current Discharge

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

- Electrical (EL) simulation of the magnet power supply system;
- Electromagnetic (EM) modeling of the TF coil casing to evaluate the Joule power generated by eddy currents induced in it during the transient;
- Thermal–hydraulic (TH) analysis of the magnet to assess the effect of the Joule power deposition in the casing and AC losses in the superconducting (SC) cables on the coil performance.

#### 2.1. Electrical Model

#### 2.2. Electromagnetic Model

#### 2.3. Thermal–Hydraulic Model

#### 2.4. Uncertainty Propagation Analysis

#### 2.4.1. Polynomial Chaos Expansion

#### 2.4.2. Unscented Transform

## 3. Results

#### 3.1. Results of the Electrical Model

- $\tau \phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\to $ time required for the current to decrease from nominal value to $0.1\phantom{\rule{0.166667em}{0ex}}A$;
- ${I}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}t={\int}_{0}^{t}{I}^{2}\left({t}^{\prime}\right)d{t}^{\prime}$, considering its asymptotic value, which is proportional to the energy extracted from the TF coils during the FD;
- $\Delta {V}_{FDU}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\to $ peak voltage on the FDU during the discharge;
- $\Delta {V}_{TF}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\to $ peak voltage on the TF coil during the discharge.

**Figure 3.**Statistical distribution of (

**a**) $\tau $, (

**b**) ${I}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}t$, (

**c**) $\Delta {V}_{FDU}$, and (

**d**) $\Delta {V}_{TF}$ obtained with the MC method.

#### 3.2. Results of the Electromagnetic Model

- The peak of the deposited power ${P}_{peak}$;
- The overall deposited energy $E={\int}_{0}^{{t}_{end}}P\left(t\right)dt$.

#### 3.3. Identification of the Worst-Case Scenarios

#### 3.4. Results of the Thermal–Hydraulic Model

## 4. Conclusions and Perspective

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DTT | Divertor Tokamak Test |

EL | Electrical |

EM | Electro Magnetic |

FD | Fast Discharge |

FDU | Fast Discharge Unit |

HS | Hot Spot |

MC | Monte Carlo |

PCE | Polynomial Chaos Expansion |

RSD | Relative Standard Deviation |

SC | Superconductor/Superconductive |

STD | Standard Deviation |

TF | Toroidal Field |

TH | Thermal - Hydraulic |

UP | Uncertainty Propagation |

UT | Unscented Transform |

WCS | Worst-Case Scenario |

WP | Winding Pack |

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**Figure 1.**DTT TF coil geometry: view of the coil structure (

**left**) and cross sections where dimensions and pancake numbering are highlighted (

**right**).

**Figure 2.**Logical connections between the three aspect of physics (and sub-blocks) and related results.

**Figure 4.**Statistical distribution of (

**a**) $\tau $, (

**b**) ${I}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}t$, (

**c**) $\Delta {V}_{FDU}$, and (

**d**) $\Delta {V}_{TF}$ obtained with the PCE method. Comparisons between performances of different PCE quadrature orders are shown for each distribution.

**Figure 5.**Average evolution of (

**a**) current and (

**b**) ${I}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}t$ with respective 1–99% confidence range evaluated with PCE.

**Figure 6.**Statistical distribution of (

**a**) peak of the power deposition and of (

**b**) deposited energy, obtained using PCE with quadrature order 3.

**Figure 7.**Average evolution of the power deposited within the TF coil casing with its 1–99% confidence range evaluated with PCE.

**Figure 9.**Evolution of (

**a**) current and (

**b**) total power deposited in the coil casing during the two WCSs.

**Figure 10.**Voltage evolution computed in pancakes 5, 6, and 7 in both WCS1 and WCS2. The current evolution for WCS1 (solid) and WCS2 (dashed) is also plotted, to be read on the right axis.

**Figure 11.**Maximum hot spot temperature reached in each pancake during the transient for both WCS1 and WCS2.

**Table 1.**Comparison between the mean value and standard deviation of the monitored variables using the MC, PCE, and UT methods.

Variable | $\overline{\mathit{V}}\pm 2\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}\mathit{\sigma}$ | $\overline{\mathit{V}}\pm 2\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}\mathit{\sigma}$ | $\overline{\mathit{V}}\pm 2\phantom{\rule{0.166667em}{0ex}}\mathit{\xb7}\phantom{\rule{0.166667em}{0ex}}\mathit{\sigma}$ |
---|---|---|---|

MC | PCE | UT | |

$\tau \phantom{\rule{0.166667em}{0ex}}\left[s\right]$ | $14.77\pm 4.70$ | $14.76\pm 4.69$ | $14.77\pm 4.69$ |

$\Delta {V}_{FDU}\phantom{\rule{0.166667em}{0ex}}\left[V\right]$ | $5065.85\pm 1708.03$ | $5067.26\pm 1708.40$ | $5067.27\pm 1698.46$ |

$\Delta {V}_{TF}\phantom{\rule{0.166667em}{0ex}}\left[V\right]$ | $843.98\pm 284.56$ | $844.21\pm 284.62$ | $844.22\pm 282.97$ |

${I}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}t\phantom{\rule{0.166667em}{0ex}}\left[G{A}^{2}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}s\right]$ | $4.67\pm 1.58$ | $4.67\pm 1.57$ | $4.67\pm 1.57$ |

**Table 2.**Comparison between the mean value and standard deviation of the peak power and the deposited energy evaluated with PCE and UT.

Variable | ||
---|---|---|

PCE | UT | |

${P}_{max}\phantom{\rule{0.166667em}{0ex}}\left[MW\right]$ | $1.01\pm 0.62$ | $1.00\pm 0.61$ |

$E\phantom{\rule{0.166667em}{0ex}}\left[MJ\right]$ | $4.42\pm 1.40$ | $4.42\pm 1.40$ |

**Table 3.**Comparison between the peak power obtained using the 3D-FOX and the PCE metamodel using the UT sigma points ($\sigma $) as input parameters.

${\mathit{P}}_{\mathit{max},3\mathit{D}-\mathit{FOX}}$ [MW] | ${\mathit{P}}_{\mathit{max},\mathit{PCE}}$ [MW] | Relative Difference—${\mathit{\epsilon}}_{{\mathit{P}}_{\mathit{peak}}}$ | ${\mathit{E}}_{3\mathit{D}-\mathit{FOX}}$ [MJ] | ${\mathit{E}}_{\mathit{PCE}}$ [MJ] | Relative Difference—${\mathit{\epsilon}}_{\mathit{E}}$ | |
---|---|---|---|---|---|---|

${\sigma}_{1}$ | $0.975$ | $0.977$ | $+0.28\%$ | $4.40$ | $4.40$ | $-0.03\%$ |

${\sigma}_{2}$ | $0.628$ | $0.625$ | $-0.45\%$ | $3.51$ | $3.51$ | $-0.11\%$ |

${\sigma}_{3}$ | $0.726$ | $0.728$ | $+0.31\%$ | $3.82$ | $3.82$ | $+0.01\%$ |

${\sigma}_{4}$ | $1.39$ | $1.39$ | $+0.28\%$ | $5.28$ | $5.28$ | $+0.02\%$ |

${\sigma}_{5}$ | $1.30$ | $1.30$ | $+0.05\%$ | $5.08$ | $5.07$ | $-0.06\%$ |

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**MDPI and ACS Style**

De Bastiani, M.; Aimetta, A.; Bonifetto, R.; Dulla, S.
Application of the Polynomial Chaos Expansion to the Uncertainty Propagation in Fault Transients in Nuclear Fusion Reactors: DTT TF Fast Current Discharge. *Appl. Sci.* **2024**, *14*, 1068.
https://doi.org/10.3390/app14031068

**AMA Style**

De Bastiani M, Aimetta A, Bonifetto R, Dulla S.
Application of the Polynomial Chaos Expansion to the Uncertainty Propagation in Fault Transients in Nuclear Fusion Reactors: DTT TF Fast Current Discharge. *Applied Sciences*. 2024; 14(3):1068.
https://doi.org/10.3390/app14031068

**Chicago/Turabian Style**

De Bastiani, Marco, Alex Aimetta, Roberto Bonifetto, and Sandra Dulla.
2024. "Application of the Polynomial Chaos Expansion to the Uncertainty Propagation in Fault Transients in Nuclear Fusion Reactors: DTT TF Fast Current Discharge" *Applied Sciences* 14, no. 3: 1068.
https://doi.org/10.3390/app14031068