# Energy Management in Smart Building by a Multi-Objective Optimization Model and Pascoletti-Serafini Scalarization Approach

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## Abstract

**:**

## 1. Introduction

## 2. Multi-Objective Optimization Problem and Some Preliminaries

**Dominance:**Two feasible points $\widehat{\mathit{x}},\overline{\mathit{x}}\in \Omega $ for the MOOP (1) are considered. It is said that the solution $\widehat{\mathit{x}}$ dominates the solution $\overline{\mathit{x}}$, and it can be written that $\widehat{\mathit{x}}\prec \overline{\mathit{x}}$, if for all $i\in \{1,\cdots ,r\}$, we have ${f}_{i}\left(\widehat{\mathit{x}}\right)\le {f}_{i}\left(\overline{\mathit{x}}\right),$ and there is at least one $j\in \{1,\cdots ,r\}$ such that ${f}_{j}\left(\widehat{\mathit{x}}\right)<{f}_{j}\left(\overline{\mathit{x}}\right)$. Now, the Pareto optimality is defined by using the Dominance order.

**Pareto Minimizer(Pareto Point):**A feasible point ${\mathit{x}}^{*}\in \Omega $ is called a Pareto minimizer of MOOP (1) if there is no feasible points $\mathit{x}\in \Omega $ such that $\mathit{x}\prec {\mathit{x}}^{*}$. If the feasible point ${\mathit{x}}^{*}\in \Omega $ is a Pareto minimizer, then $\mathit{f}\left({\mathit{x}}^{*}\right)$ is called a non-dominated point.

**Pareto Front:**The image by $\mathit{f}$ of the set of all Pareto minimizers of (1) is called the Pareto front (or efficient set).

**Ideal Point:**For each $i=1,\cdots ,r$, let ${\mathit{x}}_{i}^{*}\in \Omega $ be an optimal solution of the following single objective optimization problem. Now, the vector ${\mathit{a}}^{*}=[{f}_{1}\left({\mathit{x}}_{1}^{*}\right),\cdots ,{f}_{r}\left({\mathit{x}}_{r}^{*}\right)]$ is called an ideal point of the MOOP (1).

## 3. Problem Description and Mathematical Model

- Solar photovoltaic production is considered for self-consumption but with the possibility of selling the surplus to the power grid;
- It is considered that each EV leaves and arrives to the building once a day. All EVs are connected to the electric network as soon as they get home.
- The arrival and departure times are known and for each EV the initial SoC is known at the arrival time. The EV battery can be charged/discharged between arrival and departure.

#### Formulation of MOOP

## 4. Case Study

#### Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**(

**a**) Distance of Pareto Points from the Ideal Point, (

**b**) Objective Function ${f}_{1}$ respect to $\u03f5$, (

**c**) Objective Function ${f}_{2}$ respect to $\u03f5$.

**Figure 3.**Trace of power among Grid, Building’s apartments, Photo Voltaics (PVs), Electric Vehicles (EVs) and Battery Energy Storage System (BESS) corresponding to $\u03f5=0.834$.

Parameter | Index | Description |
---|---|---|

I | Number of time-slots per Time-Study | |

$\tau $ | Time-slot duration (hour) | |

J | Number of apartments (or EVs) in the building | |

${T}_{\mathrm{EV}}^{\mathrm{in}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$, | The number of period-time in which j-th EV enters to the parking |

${T}_{\mathrm{EV}}^{\mathrm{out}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$, | The number of period-time in which j-th EV leaves the parking |

${S}_{\mathrm{EV}}^{\mathrm{max}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$ | Maximum allowable State of Charge(SoC) of j-th EV |

${S}_{\mathrm{EV}}^{\mathrm{initial}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$, | The initial SoC of j-th EV at the beginning departure in ${T}_{\mathrm{EV}}^{\mathrm{in}}\left(j\right)$ |

${S}_{\mathrm{EV}}^{\mathrm{min}\_\mathrm{out}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$, | The minimum allowable SoC for j-th EV at exit time |

${S}_{\mathrm{BE}}^{\mathrm{max}}$ | Maximum SoC for BESS | |

${S}_{\mathrm{BE}}^{\mathrm{initial}}$ | Initial SoC for BESS at the beginning of time-period | |

${P}_{\mathrm{SB}}\left(i\right)$ | $i\in \left\{1,\cdots ,I\right\}$ | Total power demand of SB at period i |

${P}_{\mathrm{PV}}\left(i\right)$ | $i\in \left\{1,\cdots ,I\right\}$ | Total generated power by PV at period i |

${P}_{\mathrm{G}}^{\mathrm{max}}\left(i\right)$ | $i\in \left\{1,\cdots ,I\right\}$ | Maximum power that can got from Grid at time-slot i |

${C}_{\mathrm{G}}^{\mathrm{buy}}\left(i\right)$ | $i\in \left\{1,\cdots ,I\right\}$ | Purchased electricity cost from grid in i-th time-slot |

${C}_{\mathrm{G}}^{\mathrm{sell}}\left(i\right)$ | $i\in \left\{1,\cdots ,I\right\}$ | Sell electricity cost to grid in i-th time-slot |

${P}_{\mathrm{EV}}^{\mathrm{ch}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$ | Active power related to the charging process of the j-th EV |

${P}_{\mathrm{EV}}^{\mathrm{diss}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$ | Active power related to the discharging process of the j-th EV |

${E}_{\mathrm{EV}}^{\mathrm{ch}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$ | The charge efficiency of j-th EV |

${E}_{\mathrm{EV}}^{\mathrm{diss}}\left(j\right)$ | $j\in \left\{1,\cdots ,J\right\}$ | The discharge efficiency of j-th EV |

${P}_{\mathrm{BE}}^{\mathrm{ch}}\left(i\right)$ | $i\in \left\{1,\cdots ,I\right\}$ | Active power of the charging process of the BESS in period i |

${P}_{\mathrm{BE}}^{\mathrm{diss}}\left(i\right)$ | $i\in \left\{1,\cdots ,I\right\}$ | Active power of the discharging process of BESS in period i |

Variable | Type | Index | Description |
---|---|---|---|

${\alpha}_{\mathrm{EV}}(i,j)$ | $\{0,\phantom{\rule{-1.5pt}{0ex}}1\}$ | $i\in \left\{1,\cdots ,I\right\},j\in \left\{1,\cdots ,J\right\}$ | j-th EV charging process in period i |

${\beta}_{\mathrm{EV}}(i,j)$ | $\{0,\phantom{\rule{-1.5pt}{0ex}}1\}$ | $i\in \left\{1,\cdots ,I\right\},j\in \left\{1,\cdots ,J\right\}$ | j-th EV discharging process in period i |

${\alpha}_{\mathrm{BE}}\left(i\right)$ | $\{0,\phantom{\rule{-1.5pt}{0ex}}1\}$ | $i\in \left\{1,\cdots ,I\right\}$ | BESS charging process in period i |

${\beta}_{\mathrm{BE}}\left(i\right)$ | $\{0,\phantom{\rule{-1.5pt}{0ex}}1\}$ | $i\in \left\{1,\cdots ,I\right\}$ | BESS discharging process in period i |

${S}_{\mathrm{EV}}(i,j)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\},j\in \left\{1,\cdots ,J\right\}$ | SoC of the j-th EV at the start of period i |

${S}_{\mathrm{BE}}\left(i\right)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\}$ | SoC of the BESS at the start of period i |

${P}_{\mathrm{G}}\left(i\right)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\}$ | Active power extracted from the grid in period i |

${P}_{\mathrm{G}\phantom{\rule{-0.7pt}{0ex}}\to \phantom{\rule{-0.7pt}{0ex}}\mathrm{BE}}\left(i\right)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\}$ | Active power of charging the BESS by grid in period i |

${P}_{\mathrm{G}\phantom{\rule{-0.7pt}{0ex}}\to \phantom{\rule{-0.7pt}{0ex}}\mathrm{EV}}(i,j)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\},j\in \left\{1,\cdots ,J\right\}$ | Active power of charging the j-th EV by grid in period i |

${P}_{\mathrm{EV}\phantom{\rule{-1.0pt}{0ex}}\to \phantom{\rule{-0.7pt}{0ex}}\mathrm{B}}(i,j)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\},j\in \left\{1,\cdots ,J\right\}$ | Active power of discharging of j-th EV to SB in period i. |

${P}_{\mathrm{PV}\phantom{\rule{-1.0pt}{0ex}}\to \phantom{\rule{-0.7pt}{0ex}}\mathrm{B}}\left(i\right)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\}$ | Active power from PV to SB in period i |

${P}_{\mathrm{PV}\phantom{\rule{-1.0pt}{0ex}}\to \phantom{\rule{-0.7pt}{0ex}}\mathrm{BE}}\left(i\right)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\}$ | Active power from PV to BESS in period i |

${P}_{\mathrm{PV}\phantom{\rule{-1.0pt}{0ex}}\to \phantom{\rule{-0.7pt}{0ex}}\mathrm{G}}\left(i\right)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\}$ | Active power from PV to grid in period i |

${P}_{\mathrm{BE}\phantom{\rule{-0.7pt}{0ex}}\to \phantom{\rule{-0.7pt}{0ex}}\mathrm{G}}\left(i\right)$ | ${\mathbb{R}}_{0}^{+}$ | $i\in \left\{1,\cdots ,I\right\}$ | Active power from BESS to grid in period i |

**Table 3.**Comparison of Proposed Model (MOMBLP) with Reference Case Study for 1 day from external Supplier.

Total Cost $\left(\mathbf{EUR}\right)$ | Peak Load (kW h) | |
---|---|---|

MOMBLP Model | $39.2349$ | $4.9134$ |

Reference Case Study [12] | $60.8838$ | $9.0190$ |

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Foroozandeh, Z.; Ramos, S.; Soares, J.; Vale, Z.
Energy Management in Smart Building by a Multi-Objective Optimization Model and Pascoletti-Serafini Scalarization Approach. *Processes* **2021**, *9*, 257.
https://doi.org/10.3390/pr9020257

**AMA Style**

Foroozandeh Z, Ramos S, Soares J, Vale Z.
Energy Management in Smart Building by a Multi-Objective Optimization Model and Pascoletti-Serafini Scalarization Approach. *Processes*. 2021; 9(2):257.
https://doi.org/10.3390/pr9020257

**Chicago/Turabian Style**

Foroozandeh, Zahra, Sérgio Ramos, João Soares, and Zita Vale.
2021. "Energy Management in Smart Building by a Multi-Objective Optimization Model and Pascoletti-Serafini Scalarization Approach" *Processes* 9, no. 2: 257.
https://doi.org/10.3390/pr9020257