# Real-Time Control for the EHU Stellarator

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data Acquisition System and Model Statement

_{Out}and current I signal measurements, while the input signal V

_{In}is also being monitored. Note that the main actuator of the system is composed of a coil system consisting of two external and two internal coils connected to a real-time fully controllable power source, as shown in Figure 1 and Figure 2. The resulting output voltage and currents are, therefore, the controlled variables, and are measured directly from the coils’ copper wires by means of physical resistance-based and Ohmic inductive sensors, respectively, providing the input for the control feedback loop through the corresponding conditioning circuits shown in Figure 3 and Figure 4 and the DAQ (Data acquisition) connected to the real-time target.

#### 2.1. Analytical Model

#### 2.2. Black Box Model

#### 2.3. Grey Box Model

#### 2.4. Model Validation via Experimentation

## 3. Control Design

#### 3.1. PID Controller Implementation

- Quick response. Since the system will have to handle the confinement of a rapidly varying plasma current, the control over the coil voltage will have to be as fast as possible.
- Low overshoot. Excessive overshoot translates into a more powerful magnetic field, which can lead to unexpected behavior and ultimately cause internal damage to the device, which must be avoided.
- Minimal steady-state error. A varying signal for voltage applied to coils creates magnetic fields that can change both in direction and in magnitude. This variable magnetic field will induce a leaking voltage into the conductive materials. The leaking currents can interfere with the plasma inside the device chamber and the instrumentation signals, and can also alter the temperature conditions.

#### 3.2. Model Predictive Control Design

- Both the dynamic and static behaviors are defined and foreseen by the model.
- Both the input and output variable constraints are considered by the control algorithm.
- The use of model prediction can foresee and anticipate potential problems.

## 4. Experimental Results

#### 4.1. PID Performance

#### 4.2. MPC Performance

#### 4.3. MPC and PID Performance Comparison

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- ITER Organization. What is ITER? 2019. Available online: https://www.iter.org/proj/inafewlines (accessed on 18 December 2019).
- Wikimedia Foundation, Inc. Nuclear Fusion-Wikipedia. 2019. Available online: https://en.wikipedia.org/wiki/Nuclear_fusion (accessed on 18 December 2019).
- ITER Organization. What is a TOKAMAK? 2019. Available online: https://www.iter.org/mach/Tokamak (accessed on 18 December 2019).
- Wikimedia Foundation, Inc. Stellarator-Wikipedia. 2019. Available online: https://en.wikipedia.org/wiki/Stellarator (accessed on 18 December 2019).
- De la Sen, M.; Garrido, A.J.; Soto, J.C.; Barambones, O.; Garrido, I. Suboptimal regulation of a class of bilinear interconnected systems with finite-time sliding planning horizons. Math. Probl. Eng.
**2008**, 817063. [Google Scholar] [CrossRef] - Kollár, I.; Pintelon, R.; Schouken, J. Frequency Domain System Identification Toolbox for MATLAB. IFAC Proc. Vol.
**1991**, 24, 1243–1247. [Google Scholar] [CrossRef] - Li, K.; Thompson, S. Fundamental grey-box modelling. In Proceedings of the 2001 European Control Conference (ECC), Porto, Portugal, 4–7 September 2001; pp. 3648–3653. [Google Scholar]
- Garrido, I.; Garrido, A.J.; Sevillano, M.G.; Romero, J.A.; Amundarain, M.; Alberdi, M. Tokamak state-space control modeling. In Proceedings of the Canadian Conference on Electrical and Computer Engineering, Niagara Falls, ON, Canada, 4–7 May 2008; Volume 1–4, pp. 840–847. [Google Scholar] [CrossRef]
- Garrido, A.J.; Otaola, E.; Garrido, I.; Lekube, J.; Maseda, F.J.; Liria, P.; Mader, J. Mathematical Modeling of Oscillating Water Columns Wave-Structure Interaction in Ocean Energy Plants. Math. Probl. Eng.
**2015**, 727982. [Google Scholar] [CrossRef] [Green Version] - Heong Ang, K.; Chong, G.; Li, Y. PID control system analysis, design, and technology. IEEE Trans. Control Syst. Technol.
**2005**, 13. [Google Scholar] [CrossRef] [Green Version] - De la Sen, M. On Cauchy’s Interlacing Theorem and the Stability of a Class of Linear Discrete Aggregation Models Under Eventual Linear Output Feedback Controls. Symmetry
**2019**, 11, 712. [Google Scholar] [CrossRef] [Green Version] - Babiarz, A.; Czornik, A.; Niezabitowski, M. Output controllability of the discrete-time linear switched systems. Nonlinear Anal. Hybrid Syst.
**2016**, 2016, 1–10. [Google Scholar] [CrossRef] - Garrido, A.J.; Garrido, I.; Barambones, O.; Alkorta, P.; Maseda, F.J. Simple Linear Models for Plasma Control in Tokamak Reactors. In Proceedings of the 2008 International Conference on Control, Automation and Systems, Seoul, Korea, 14–17 October 2008. [Google Scholar] [CrossRef]
- Padula, F.; Visioli, A. Tuning rules for optimal PID and fractional-order PID controllers. J. Process Control
**2011**, 21, 69–81. [Google Scholar] [CrossRef] - Vazquez, S.; Leon, J.I.; Franquelo, L.G.; Rodriguez, J.; Young, H.A.; Marquez, A.; Zanchetta, P. Model predictive control: A review of its applications in power electronics. IEEE Ind. Electron. Mag.
**2014**, 8, 16–31. [Google Scholar] [CrossRef] - Herrera-Cáceres, C.A.; Ibeas, A. Model predictive control of cash balance in a cash concentration and disbursements system. J. Frankl. Inst.
**2016**, 353, 4885–4923. [Google Scholar] [CrossRef] - Garrido, I.; Garrido, A.J.; Romero, J.A.; Carrascal, E.; Sevillano-Berasategui, M.G.; Barambones, O. Low Effort Li Nuclear Fusion Plasma Control Using Model Predictive Control Laws. Math. Probl. Eng.
**2015**, 527420. [Google Scholar] [CrossRef] [Green Version] - Sturzenegger, D.; Morari, M.; Semeraro, V.; Gyalistras, D.; Smith, R.S. BRCM Matlab Toolbox: Model generation for model predictive building control. In Proceedings of the American Control Conference (ACC), Portland, OR, USA, 4–6 June 2014. [Google Scholar]
- Garrido, A.J.; Garrido, I.; Alberdi, M.; Amundarain, M.; Barambones, O.; Romero, J.A. Robust Control of Oscillating Water Column (OWC) Devices: Power Generation Improvement. In Proceedings of the 2013 MTS/IEEE Oceans Conference-San Diego, San Diego, CA, USA, 23–27 September 2013. [Google Scholar]

**Figure 1.**EHU (Euskal Herriko Unibertsitatea) stellerator located in the Automatic Control Group (ACG) at the Engineering Faculty of the Basque Country University (UPV/EHU).

**Figure 2.**Dimensions of the reaction chamber and the coils: the left the vacuum chamber and exterior coils; the right the inner coils, shown in a 90° setting for visualization purposes.

**Figure 7.**Comparison of the analytical model, black box model, and the grey box model responses in real time: comparison for voltage (

**top**) and current (

**bottom**).

**Figure 8.**Comparison between the real system and the theoretical model output signals: comparison with the black box model (

**left**) and grey box model (

**right**).

**Figure 11.**Comparison of the PID controlled voltage output of the real system and the grey box model.

**Figure 12.**Comparison of the MPC controlled voltage output of the real system and the grey box model.

Parameter | Physical Meaning | SI unit |
---|---|---|

V_{IN} | Input DC voltage | Volt (V) |

V_{OUT} | Measured DC voltage | Volt (V) |

R_{1} | Resistance of first outer and inner coil, plus second inner coil | Ohm (Ω) |

R_{2} | Resistance of second outer coil | Ohm (Ω) |

Le_{1} | Inductance of the first outer coil | Henry (H) |

Le_{2} | Inductance of the second outer coil | Henry (H) |

L_{i1} | Inductance of the first inner coil | Henry (H) |

L_{i2} | Inductance of the second inner coil | Henry (H) |

M | Mutual inductance (between inner coils) | Henry (H) |

Parameter | ${\mathit{R}}_{\mathbf{1}}$ | ${\mathit{R}}_{\mathbf{2}}$ | ${\mathit{L}}_{\mathit{e}\mathbf{1}}\left(={\mathit{L}}_{\mathit{e}\mathbf{2}}\right)$ | ${\mathit{L}}_{\mathit{i}\mathbf{1}}+{\mathit{L}}_{\mathit{i}\mathbf{2}}+\mathit{M}$ |
---|---|---|---|---|

Value | $426.870m\mathsf{\Omega}$ | $57.626m\mathsf{\Omega}$ | $509.990\mu H$ | $3.268\mu H$ |

Parameter | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{\mathit{d}}$ |
---|---|---|---|

Value | 14.7180 | 357.55 | 0.0018 |

Parameter | $\mathbf{Prediction}\text{}\mathbf{Horizon}{\mathit{N}}_{\mathit{p}}$ | $\mathbf{Control}\text{}\mathbf{Horizon}\text{}{\mathit{N}}_{\mathit{c}}$ | $\mathbf{Time}\text{}\mathbf{Step}\text{}{\mathit{T}}_{\mathit{s}}$ |
---|---|---|---|

Value | 25 | 13 | 250 µs |

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**MDPI and ACS Style**

Garrido, I.; Maseda, J.; Martija, I.; Garrido, A.J.
Real-Time Control for the EHU Stellarator. *Symmetry* **2020**, *12*, 11.
https://doi.org/10.3390/sym12010011

**AMA Style**

Garrido I, Maseda J, Martija I, Garrido AJ.
Real-Time Control for the EHU Stellarator. *Symmetry*. 2020; 12(1):11.
https://doi.org/10.3390/sym12010011

**Chicago/Turabian Style**

Garrido, Izaskun, Javier Maseda, Itziar Martija, and Aitor J. Garrido.
2020. "Real-Time Control for the EHU Stellarator" *Symmetry* 12, no. 1: 11.
https://doi.org/10.3390/sym12010011