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Voltage H_{∞} Control of a Vanadium Redox Flow Battery

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## Abstract

**:**

## 1. Introduction

## 2. System Modeling

#### 2.1. Operation of a VRFB

#### 2.2. VRFB Electrochemical Model

## 3. Equilibrium Points Analysis

## 4. Controller Design

#### 4.1. Introduction

#### 4.2. Model Simplification

- Vanadium concentrations are the same on both sides of the half cell. Therefore, it is possible to have the following correspondence:$${c}_{2}^{cell}={c}_{5}^{cell}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{c}_{3}^{cell}={c}_{4}^{cell}$$
- The tanks concentration can be expressed in terms of SOC and total vanadium concentration ${c}_{v}$:$${c}_{2}^{tank}={c}_{v}SOC\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{c}_{3}^{tank}={c}_{v}\xb7(1-SOC)$$

#### 4.3. System Linearization

#### 4.4. Uncertainty Modeling

- Determine a frequency range where to model the uncertainty. In our case the, range $[{10}^{-4.5},{10}^{0.5}]$ $\mathrm{rad}/\mathrm{s}$ has been selected. As can be seen in Figure 6, out of this frequency range there is almost no variability in the response. 500 points logarithmically distributed have defined in this range.
- Sample the set of plants, obtained in step 1 and the range of frequencies selected in step 2.
- Compute the error (distance) between each point with respect the nominal plant.
- Obtain stable and minimum phase rational function of polynomials that bounds all of the points obtained in step 4. The rational function order must be selected making a trade-off between the order of the function and the goodness of the fit.

#### 4.5. Controller Design

#### 4.6. Integral Controller Design

#### 4.7. Materials and Methods

## 5. Results and Discussion

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

ESS | Energy storage systems |

NP | Nominal performance |

OCV | Open circuit voltage |

RES | Renowable energy sources |

RFB | Redox flow battery |

RP | Robust performance |

RS | Robust stability |

SOC | State of charge |

VRFB | Vanadium redox flow battery |

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**Figure 9.**Modulus of the error between ${G}_{n}\left(s\right)$ and the analysed plants (in blue), fitting obtaining with a second order ${W}_{u}^{a}\left(s\right)$ (in orange).

**Figure 15.**Aproximated robust performance analysis for the reduced order controller, $\overline{C}\left(s\right)$.

**Figure 16.**Formal robust performance analysis for the reduced order controller, $\overline{C}\left(s\right)$.

**Figure 17.**Comparison between the two weighing function, $1/{W}_{e}\left(s\right)$ and $1/{W}_{e}^{\prime}\left(s\right)$ frequency responses.

**Figure 18.**Approximated robust performance check for the closed-loop system step response with ${\overline{C}}_{i}\left(s\right)$ and ${W}_{e}^{\prime}\left(s\right)$.

**Figure 19.**Formal robust performance check for the closed-loop system step response with ${\overline{C}}_{i}\left(s\right)$ and ${W}_{e}^{\prime}\left(s\right)$.

Parameter | Meaning | Unit—Value |
---|---|---|

${c}_{i}^{cell}$ | Concentration of specie i inside the cell | $\mathrm{mol}\xb7{\mathrm{m}}^{-3}$ |

E | Electrode potential | V |

${E}^{\theta}$ | Standard potential | 1.259 V |

T | Temperature of the cell | K |

F | Faraday’s constant | 96,485 $\mathrm{C}\xb7{\mathrm{mol}}^{-1}$ |

R | Gas constant | 8.314 $\mathrm{J}\xb7{\mathrm{K}}^{-1}\xb7{\mathrm{mol}}^{-1}$ |

**Table 2.**Parameters of the electrochemical model [18].

Parameter | Meaning | Unit—Value |
---|---|---|

${V}_{cell}$ | Volume of the cell | $9.5\xb7{10}^{-4}$${\mathrm{m}}^{3}$ |

${V}_{tank}$ | Volume of each tank | $0.4$${\mathrm{m}}^{3}$ |

Q | Flow rate | ${\mathrm{m}}^{3}\xb7{\mathrm{s}}^{-1}$ |

I | Current | $\mathrm{A}$ |

S | Surface area of the electrode | 0.15 ${\mathrm{m}}^{2}$ |

d | Membrane thickness | 1.27·10${}^{-4}$ $\mathrm{m}$ |

${k}_{2}$ | Diffusion coefficient of ${V}^{2+}$ | $8.768\xb7{10}^{-12}$$\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

${k}_{3}$ | Diffusion coefficient of ${V}^{3+}$ | $3.221\xb7{10}^{-12}$$\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

${k}_{4}$ | Diffusion coefficient of ${V}^{4+}$ | $6.825\xb7{10}^{-12}$$\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

${k}_{5}$ | Diffusion coefficient of ${V}^{5+}$ | $5.896\xb7{10}^{-12}$$\mathrm{m}\xb7{\mathrm{s}}^{-1}$ |

N | Number of cells of the stack | 19 |

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**MDPI and ACS Style**

Clemente, A.; Ramos, G.A.; Costa-Castelló, R.
Voltage *H*_{∞} Control of a Vanadium Redox Flow Battery. *Electronics* **2020**, *9*, 1567.
https://doi.org/10.3390/electronics9101567

**AMA Style**

Clemente A, Ramos GA, Costa-Castelló R.
Voltage *H*_{∞} Control of a Vanadium Redox Flow Battery. *Electronics*. 2020; 9(10):1567.
https://doi.org/10.3390/electronics9101567

**Chicago/Turabian Style**

Clemente, Alejandro, Germán Andrés Ramos, and Ramon Costa-Castelló.
2020. "Voltage *H*_{∞} Control of a Vanadium Redox Flow Battery" *Electronics* 9, no. 10: 1567.
https://doi.org/10.3390/electronics9101567