# A MILP Optimization Method for Building Seasonal Energy Storage: A Case Study for a Reversible Solid Oxide Cell and Hydrogen Storage System

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Methods

#### 2.1. Optimization Method

_{i,j}is the energy storage load at the central point of time interval j in optimization step i. Step 1 also shows the storage load at the beginning, a, and the end, b, of the year. To guarantee a net zero annual storage balance, the storage loads at point a and b must be equal to one another. Otherwise, the energy storage cannot operate in a sustainable way for several years.

^{(n−1)}(where n is the number of the optimization step) new points will be added to the final solution from each optimization step, except for the first optimization step, where three points (center point, starting point and end point) are added to the final solution.

#### 2.2. Case Study: Reversible Solid Oxide Cell and Hydrogen Storage System for Seasonal Storage of Solar Energy

^{2}floor-area office building, VTT FutureHub, (Figure 2), located in Espoo, Finland. The building characteristics are presented in Table 1. The location of the building is 60°11′11.4″ N 24°48′49.0″ E and the Köppen climate classification of the site climate is Dfb [23].

#### 2.2.1. Calculated Cases

#### 2.2.2. Objective Function

#### 2.2.3. RSOC Functions

#### 2.2.4. Constraints

#### 2.2.5. Assumptions

^{2}and the PV generation profile is based on simulated data for a system where 50% of the solar panels are facing east and 50% are facing west [26].

## 3. Results

#### 3.1. Results of the Case Study

^{2}), with results presented in Figure A1, Figure A2, Figure A3, Figure A4, Figure A5, Figure A6, Figure A7, Figure A8 and Figure A9 in the Appendix A. All of the systems show the same trend; storing energy during summer and consuming the stored energy during winter. One surprising remark, which also holds true for all storage systems, is that the system uses large amounts of electricity from the grid to fill up the hydrogen storage, even during the summer, when there is a surplus of energy generated by the solar PV panels. This indicates that it is cost-effective to convert electricity from the grid to hydrogen gas and then use it for heating during the winter.

^{2}. For big PV installations, however, the significance of the hydrogen storage size is more crucial, especially when the OPEX of the systen drops below zero.

#### 3.2. Optimization Model Perfomance

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

a | The beginning of the year |

b | The end of the year |

${C}_{Pex,i}$ | Electricity export price at time step $i$ |

${C}_{Pim,i}$ | Electricity import price at time step $i$ |

${C}_{Qex,i}$ | Heat export price at time step $i$ |

${C}_{Qim,i}$ | Heat import price at time step $i$ |

CPU | Central processing unit |

${E}_{H,0}$ | Stored hydrogen gas at the beginning and end of the year |

${E}_{H,cap}$ | Hydrogen gas storage capacity |

${h}_{c,in}$ | Specific enthalpy of hydrogen gas at the inlet of the hydrogen compressor |

${h}_{c,out}$ | Specific enthalpy of hydrogen gas at the outlet of the hydrogen compressor |

$\mathrm{LHV}$ | Lower heating value of hydrogen gas |

m_{i,j} | Storage load at central point of time interval j in optimization step i |

MILP | Mixed integer linear programming |

n | The number of the optimization step |

N | Number of time steps |

OPEX | Operating expense |

PV | Photovoltaic |

${P}_{d,i}$ | Electrical power demand of the building at time step i |

${P}_{E,in,i}$ | Electrical power input to the RSOC at time step i |

${P}_{E,in,max}$ | Maximum electrical input of the RSOC |

${P}_{E,in,min}$ | Minimum electrical input of the RSOC |

${P}_{ex,i}$ | Exported electrical power at time step $i$ |

${P}_{F,out,i}$ | Electrical power output of the RSOC at time step $i$ |

${P}_{F,out,max}$ | Maximum electrical output of the RSOC |

${P}_{F,out,min}$ | Minimum electrical output of the RSOC |

${P}_{im,i}$ | Imported electrical power at time step $i$ |

${P}_{PV,i}$ | Solar PV electrical power output at rime step $i$ |

${\dot{Q}}_{b,i}$ | Heat generated by combustion of hydrogen gas at time step $i$ |

${\dot{Q}}_{d,i}$ | Heating demand of the building at time step $i$ |

${\dot{Q}}_{ex,i}$ | Exported heat rate at time step $i$ |

${\dot{Q}}_{im,i}$ | Imported heat rate at time step $i$ |

RSOC | Reversible solid oxide cell |

RSOCHS | Reversible solid oxide cell and hydrogen gas storage |

SA | Simulated annealing |

SOEC | Solid oxide electrolysis cell |

SOFC | Solid oxide fuel cell |

$t$ | Time step size |

${\delta}_{i}$ | binary operation mode decision variable for the time step $i$ |

${\eta}_{c}$ | Efficiency of the hydrogen gas compressor |

${\eta}_{is}$ | Isentropic efficiency of the hydrogen gas compressor |

${\eta}_{m}$ | Electrical motor efficiency of the hydrogen gas compressor |

## Appendix A

## References

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**Figure 1.**Graphical description of the first and second optimization step in the interval halving optimization method.

**Figure 5.**Nord Pool electricity spot prices 2017 [27].

**Figure 7.**Optimal annual operating expense (OPEX) (for the optimal hydrogen storage size) as a function of installed solar photovoltaic (PV) panel area for the three RSOCHS systems and the reference system.

**Figure 8.**Annual imported energy (electricity and heat) (for the optimal hydrogen storage size) as a function of installed solar PV panel area for the three RSOCHS systems and the reference system.

**Figure 9.**Annual imported electricity (for the optimal hydrogen storage size) as a function of installed solar PV panel area for the three RSOCHS systems and the reference system.

**Figure 10.**Annual imported heat (for the optimal hydrogen storage size) as a function of installed solar PV panel area for the three RSOCHS systems and the reference system.

**Figure 11.**Optimal hydrogen storage size as a function of installed solar PV panel area for the three RSOCHS systems and the reference system.

**Figure 12.**Optimal OPEX of the 50/200 kW RSOC system as a function of the hydrogen storage capacity for different solar PV installations.

Floor Area | 7874 m^{2} |
---|---|

Volume | 29,483 m^{3} |

Roof area | 1163 m^{2} |

Construction year | 2020 |

**Table 2.**Statistics of the optimization problem as well as the central processing unit (CPU) time required to solve the problem.

Number of Binary Variables | 1248 |
---|---|

Number of non-integer variables | 9985 |

Number of constraints | 6240 |

CPU time | 15 s |

Case Parameter | Tested Values |
---|---|

RSOC sizes (kW) | 20/80, 50/200, 100/400 |

Solar PV areas (m^{2}) | 500, 1000, 2000, 3000, 4000, 5000, 6000, 7000, 8000, 9000, 10.000 |

Hydrogen storage (kWh) | 10, 20, …, 600 |

Export (c/kWh) | Import (c/kWh) | |
---|---|---|

Electric energy | Nord Pool spot price Average: 3.30 | Nord Pool spot price Average: 3.30 |

Daytime distribution tariff, winter ^{1,2} | - | 2.42 |

Other time distribution tariff ^{1} | - | 1.15 |

Transfer fee | −0.24 | - |

Grid tax | - | 2.25 |

Total average price | 3.06 | 9.12 |

^{1}Electricity distribution tariff for a 400 V connection;

^{2}Daytime distribution tariff, winter: Mon–Sat 7am–10pm, Nov–Mar.

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**MDPI and ACS Style**

Lindholm, O.; Weiss, R.; Hasan, A.; Pettersson, F.; Shemeikka, J.
A MILP Optimization Method for Building Seasonal Energy Storage: A Case Study for a Reversible Solid Oxide Cell and Hydrogen Storage System. *Buildings* **2020**, *10*, 123.
https://doi.org/10.3390/buildings10070123

**AMA Style**

Lindholm O, Weiss R, Hasan A, Pettersson F, Shemeikka J.
A MILP Optimization Method for Building Seasonal Energy Storage: A Case Study for a Reversible Solid Oxide Cell and Hydrogen Storage System. *Buildings*. 2020; 10(7):123.
https://doi.org/10.3390/buildings10070123

**Chicago/Turabian Style**

Lindholm, Oscar, Robert Weiss, Ala Hasan, Frank Pettersson, and Jari Shemeikka.
2020. "A MILP Optimization Method for Building Seasonal Energy Storage: A Case Study for a Reversible Solid Oxide Cell and Hydrogen Storage System" *Buildings* 10, no. 7: 123.
https://doi.org/10.3390/buildings10070123