# Mixed Logic Dynamic Models for MPC Control of Wind Farm Hydrogen-Based Storage Systems

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## Abstract

**:**

## 1. Introduction

#### Nomenclature

## 2. System Description and Modeling

#### 2.1. Power Demand Reference Model

#### 2.2. Electrolyzer and Fuel Cell Models

#### 2.3. Hydrogen Storage Model

#### 2.4. Feasibility and Operating Constraints

#### 2.5. Power Balance Constraint

## 3. Implementation of the Proposed MPC Controller

#### 3.1. Electrolyzer and Fuel cell Cost Functions

#### 3.2. Load Tracking Cost Function

#### 3.3. MPC Formulation

- protection of the hydrogen storage tank from excessive discharging and overcharging;
- limitation of the power rate of the fuel cell and of the electrolyzer to protect them;
- tracking of the power reference request according to the forecasted conditions;
- in case an expected event occurs, the fuel cell is employed as a contingent energy storage system to satisfy the power demand.

## 4. Case Study

## 5. Simulations and Numerical Results

#### 5.1. Example 1

#### 5.2. Example 2

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A.

#### Appendix A.1. Constraints Formulation of the Logical States

#### Appendix A.2. Mathematical Model and Constraints Formulation of the State Transitions

## References

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**Figure 1.**Block diagram of the microgrid under investigation. ${P}_{w}$, ${P}_{e}^{\mathrm{in}}$, ${P}_{f}^{\mathrm{out}}$, ${P}_{\mathrm{ref}}$ and ${P}_{\mathrm{dump}}$ are the power by wind generation, the power inward the electrolyzer, the power outward the fuel cell, the power demand reference and the dumped power, respectively.

**Figure 2.**Automata of the electrolyzer ($i=e$) and of the fuel cell ($i=f$). Each node represents a particular state (i.e., operational mode), while the edges represent the state transition, for each $i\in \{e,f\}$.

**Figure 4.**Numerical results for Example 1. (

**a**) Electrical Reference Demand. (

**b**) Forecasted Wind Power. (

**c**) Power of the Electrolyzer. (

**d**) Power of the Fuel Cell. (

**e**) Level of Hydrogen. (

**f**) Dump Load. (

**g**) Electrolyzer Switching States. (

**h**) Fuel cell Switching States.

**Figure 5.**Numerical results for Example 2. (

**a**) Electrical Reference Demand. (

**b**) Forecasted Wind Power. (

**c**) Power of the Electrolyzer. (

**d**) Power of the Fuel Cell. (

**e**) Level of Hydrogen. (

**f**) Electrolyzer Switching States. (

**g**) Fuel cell Switching States.

Parameters | Description |
---|---|

${H}^{\mathrm{tank}}$ | Hydrogen level in the storage unit [Nm^{3}] |

${H}^{max}$ | Maximum level of the hydrogen storage unit [Nm^{3}] |

${H}^{min}$ | Minimum level of the hydrogen storage unit [Nm^{3}] |

${P}_{e}^{max}$ | Maximum power level of the electrolyzer [$\mathrm{k}$$\mathrm{W}$] |

${P}_{e}^{\mathrm{STB}}$ | Standby power of the electrolyzer [$\mathrm{k}$$\mathrm{W}$] |

${P}_{e}^{min}$ | Minimum power level of the electrolyzer [$\mathrm{k}$$\mathrm{W}$] |

${P}_{f}^{max}$ | Maximum power level of the fuel cell [$\mathrm{k}$$\mathrm{W}$] |

${P}_{f}^{min}$ | Minimum power level of the fuel cell [$\mathrm{k}$$\mathrm{W}$] |

${P}_{f}^{\mathrm{STB}}$ | Standby power of the fuel cell [$\mathrm{k}$$\mathrm{W}$] |

${\mathrm{NH}}_{e}$ | Number of life hours of the electrolyzer |

${\mathrm{NH}}_{f}$ | Number of life hours of the fuel cell |

${\mathrm{HY}}_{e}$ | Number of per year life hours of the electrolyzer |

${\mathrm{HY}}_{f}$ | Number of per year life hours of the fuel cell |

${d}_{e}$ | Degradation rate of the electrolyzer at maximum input power and over the number of yearly life hours |

${d}_{f}$ | Degradation rate of the fuel cell at maximum output power and over the number of yearly life hours |

${S}_{\mathrm{rep},\mathrm{e}}$ | Electrolyzer stack replacement cost [€/$\mathrm{k}$$\mathrm{W}$] |

${S}_{\mathrm{rep},\mathrm{f}}$ | Fuel cell stack replacement cost [€/$\mathrm{k}$$\mathrm{W}$] |

${T}_{s}$ | Sampling period [$\mathrm{h}$] |

T | Simulation horizon [$\mathrm{h}$] |

Forecasts | Description |
---|---|

${P}_{w}$ | Wind power production [$\mathrm{k}$$\mathrm{W}$] |

${P}_{\mathrm{ref}}$ | Electrical load demand [$\mathrm{k}$$\mathrm{W}$] |

Variables | Description |
---|---|

${\delta}_{e}^{\mathrm{ON}}$ | On state of the electrolyzer |

${\delta}_{e}^{\mathrm{OFF}}$ | Off state of the electrolyzer |

${\delta}_{e}^{\mathrm{STB}}$ | Standby state of the electrolyzer |

${\delta}_{f}^{\mathrm{ON}}$ | On state of the fuel cell |

${\delta}_{f}^{\mathrm{OFF}}$ | Off state of the fuel cell |

${\delta}_{f}^{\mathrm{STB}}$ | Standby state of the fuel cell |

${P}_{e}$ | Electrical power of the electrolyzer [$\mathrm{k}$$\mathrm{W}$] |

${P}_{f}$ | Electrical power of the fuel cell [$\mathrm{k}$$\mathrm{W}$] |

${P}_{\mathrm{avl}}$ | Available system electrical power [$\mathrm{k}$$\mathrm{W}$] |

${P}_{\mathrm{dump}}$ | Dumped electrical power [$\mathrm{k}$$\mathrm{W}$] |

z | Electric power formulated as mixed logic dynamic (MLD) variables for the electrolyzer and the fuel cell [$\mathrm{W}$] |

$\sigma $ | Logical variables ON/OFF/STB states for the electrolyzer and the fuel cell |

${\zeta}_{e}$ | Electrolyzer degradation rate [Nm^{3}/$\mathrm{h}$$\mathrm{W}$] |

${\zeta}_{f}$ | Fuel cell degradation rate [$\mathrm{h}$$\mathrm{W}$/Nm^{3}] |

PEM Electrolyzer Parameters | |
---|---|

${\mathrm{Cost}}_{\mathit{e}}^{\mathrm{STB}}=0.0042\phantom{\rule{0.277778em}{0ex}}$€ | ${\mathrm{NH}}_{\mathit{e}}=\mathrm{40,000}\phantom{\rule{0.277778em}{0ex}}\mathrm{h}$ |

${\mathrm{Cost}}_{\mathit{e}}^{\mathrm{ON}}=0.123\phantom{\rule{0.277778em}{0ex}}$€ | ${\mathrm{Cost}}_{\mathit{e}}^{\mathrm{OFF}}=0.0062\phantom{\rule{0.277778em}{0ex}}$€ |

${\mathrm{Cost}}_{[\mathrm{deg},\mathit{e}]}=\phantom{\rule{0.277778em}{0ex}}$$0.05$ € | ${\mathrm{Capex}}_{\mathit{e}}=$$1.55$ €/$\mathrm{k}$$\mathrm{W}$ |

${\mathrm{Cost}}_{\mathit{e}}^{\mathrm{OM}}=\phantom{\rule{0.277778em}{0ex}}$$0.00210$ €/$\mathrm{h}$ | ${P}_{\mathit{e}}^{max}=3000\mathrm{k}\mathrm{W}$ |

${P}_{\mathit{e}}^{min}=300\mathrm{k}\mathrm{W}$ | ${P}_{\mathit{e}}^{\mathrm{STB}}=1\mathrm{k}\mathrm{W}$ |

${\mathrm{NY}}_{\mathit{e}}$ = 8000 h | |

PEM Fuel cell Parameters | |

${\mathrm{Cost}}_{\mathit{f}}^{\mathrm{STB}}=0.003\phantom{\rule{0.277778em}{0ex}}$ € | ${\mathrm{NH}}_{f}=\mathrm{40,000}$$\phantom{\rule{0.277778em}{0ex}}\mathrm{h}$ |

${\mathrm{Cost}}_{\mathit{f}}^{\mathrm{ON}}=0.01\phantom{\rule{0.277778em}{0ex}}$€ | ${\mathrm{Cost}}_{f}^{\mathrm{OFF}}=\phantom{\rule{0.277778em}{0ex}}$$0.005$ € |

${\mathrm{Cost}}_{[\mathrm{deg},\mathit{f}]}=0.01\phantom{\rule{0.277778em}{0ex}}$€ | ${\mathrm{Capex}}_{\mathit{f}}=\phantom{\rule{0.277778em}{0ex}}$$1.55$ €/$\mathrm{k}$$\mathrm{W}$ |

${\mathrm{Cost}}_{\mathrm{OM}}^{\mathit{f}}=\phantom{\rule{0.277778em}{0ex}}$$0.01$ €/$\mathrm{h}$ | ${P}_{\mathit{f}}^{max}=132\mathrm{k}\mathrm{W}$ |

${P}_{f}^{min}=12\mathrm{k}\mathrm{W}$ | ${P}_{\mathit{f}}^{\mathrm{STB}}=1\mathrm{k}\mathrm{W}$ |

${\mathrm{NY}}_{\mathit{f}}$ = 8000 h | |

Hydrogen Tank Parameters | |

Volume = $20\phantom{\rule{0.277778em}{0ex}}{\mathrm{m}}^{3}$ | Pressure $=\phantom{\rule{0.277778em}{0ex}}$ 30 bar |

© 2019 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**

Shehzad, M.F.; Abdelghany, M.B.; Liuzza, D.; Mariani, V.; Glielmo, L. Mixed Logic Dynamic Models for MPC Control of Wind Farm Hydrogen-Based Storage Systems. *Inventions* **2019**, *4*, 57.
https://doi.org/10.3390/inventions4040057

**AMA Style**

Shehzad MF, Abdelghany MB, Liuzza D, Mariani V, Glielmo L. Mixed Logic Dynamic Models for MPC Control of Wind Farm Hydrogen-Based Storage Systems. *Inventions*. 2019; 4(4):57.
https://doi.org/10.3390/inventions4040057

**Chicago/Turabian Style**

Shehzad, Muhammad Faisal, Muhammad Bakr Abdelghany, Davide Liuzza, Valerio Mariani, and Luigi Glielmo. 2019. "Mixed Logic Dynamic Models for MPC Control of Wind Farm Hydrogen-Based Storage Systems" *Inventions* 4, no. 4: 57.
https://doi.org/10.3390/inventions4040057