# Resilience-Driven Optimal Sizing of Energy Storage Systems in Remote Microgrids

^{*}

## Abstract

**:**

## 1. Introduction

- A novel formulation for the resilience-driven ESS sizing problem that provides the optimal capacity of energy storage devices to ensure the critical loads survivability for predefined time intervals. Simultaneously, the proposed algorithm optimizes the other MG resources scheduling in order to achieve an economically efficient operation.
- Validation of the proposed methodology effectiveness by conducting multiple simulations implying high and low values for the MG load and generation availability. To verify the advantages of using the proposed algorithm to enhance the MG resilience, the load restoration rates are investigated for both with and without ESS cases.
- Evaluation of the ESS impact on the MG resilience improvement by defining two new indices, precisely the restoration index and the resilience index.

## 2. Problem Formulation

#### 2.1. Objective Function

_{CG}[30], the annual cost of energy storage capacity investment defined by C

_{ESS}[31], the MG equipment maintenance cost C

_{O&M}, the emissions costs for the MT and diesel C

_{emissions}[28], and the load shedding penalties C

_{shedding}[32].

_{diesel}, fixed coefficients related to the diesel generator operation cost α

_{2}, α

_{1}, and α

_{0}, and the microturbine power production P

_{MT}and its exploitation costs defined by coefficients β

_{1}and β

_{0}. The ESS costs for the analysis duration are computed by dividing the annual investment to the number of days within a year of operation (N

_{yr}= 365 days) and multiplying by the number of operating days, N. In (3), P

_{ESS}and E

_{ESS}represent the power and energy capacity that resulted following the optimization calculation, ${\chi}_{P}^{ESS}$ and ${\chi}_{E}^{ESS}$ represent unit power and energy capacity cost coefficients, r is the discount rate, and L is the ESS lifespan in years. Based on the obtained power capacity, the operation and maintenance costs, and the installation costs are considered based on ${\chi}_{O\&M}^{ESS}$ and ${\chi}_{ins}^{ESS}$ coefficients. The maintenance costs for the diesel generation, microturbine, wind turbine, and photovoltaics are calculated in (4) for each time frame t using the cost coefficients ${\gamma}_{diesel}$, ${\gamma}_{MT}$, ${\gamma}_{WT}$, and ${\gamma}_{PV}$. Here, P

_{PV}and P

_{WT}are solar and wind generation during time frame t. The pollutants’ emissions costs are computed for the two types of conventional generators in (5) based on the price coefficient ψ and emission factor µ. In order to prioritize sensitive users, a shedding cost ${\tau}_{i}$ will be assigned to each load i in (6), while P

_{sh,i}defines the shedding power applied to each load i.

#### 2.2. Constraints

_{i}represents the demand of each of the three types of loads i considered in this study, while ${P}_{ch}^{ESS}$ and ${P}_{dis}^{ESS}$ define the ESS charging and discharging power during time interval t. The PV and WT production after curtailment measures must not exceed the maximum generation forecasting, $P{V}_{\mathrm{max}}$ and $W{T}_{\mathrm{max}}$.

_{ESS}is introduced to avoid the simultaneous charging and discharging of the system, where 1 will define the discharging mode while 0 will model the charging or standby mode. The energy storage system dynamics (state of charge—SOC) are modeled through a linear model as shown in (19), where ${\eta}_{ch}$ and ${\eta}_{dis}$ represent the charging/discharging efficiency of the ESS. Further, Equation (20) restricts the maximum allowable SOC of the ESS. In this paper, the time period length ∆t is of 1 h.

_{ESS}. To quantify the operation mode changes, the binary variable ω

_{ESS}will be 1 when λ

_{ESS}shifts its value during two consecutive time intervals, according to (22). Otherwise, ω

_{ESS}will be 0 due to the minimization nature of the problem. Finally, Equation (24) defines the state of charge at the beginning of the analysis based on a predefined coefficient, ξ

_{ESS}.

#### 2.3. Resilience Indices

## 3. Case Study

_{ESS}) is considered 0.4, based on authors’ previous work, for the improved operation performance of the microgrid [43]. The technology considered for the energy storage system is lithium-ion batteries.

## 4. Results and Discussion

#### 4.1. Scenario 1

#### 4.2. Scenario 2

#### 4.3. Scenario 3

#### 4.4. Overall Results

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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Parameter | ${\mathit{\alpha}}_{0}$ [€/kW] | ${\mathit{\alpha}}_{1}$ [€/kW] | ${\mathit{\alpha}}_{2}$ [€/kW] | ${\mathit{\gamma}}_{\mathit{d}\mathit{i}\mathit{e}\mathit{s}\mathit{e}\mathit{l}}$ [€/kWh] | ${\mathit{P}}_{\mathit{d}\mathit{i}\mathit{e}\mathit{s}\mathit{e}\mathit{l}}^{\mathbf{max}}$ [kW] | Ramp Up/Down Rate [kW/h] |
---|---|---|---|---|---|---|

Value | 0.4073 | 0.2193 | 0.0695 | 0.1433 | 125 | 50 |

Parameter | ${\mathit{\beta}}_{0}$ [€/kW] | ${\mathit{\beta}}_{1}$ [€/kW] | ${\mathit{\gamma}}_{\mathit{M}\mathit{T}}$ [€/kWh] | ${\mathit{P}}_{\mathit{M}\mathit{T}}^{\mathbf{max}}$ [kW] | Ramp Up/Down Rate [kW/h] |
---|---|---|---|---|---|

Value | 0.2368 | 0.0155 | 0.0419 | 125 | 50 |

Pollutant | ${\mathit{\mu}}_{\mathit{d}\mathit{i}\mathit{e}\mathit{s}\mathit{e}\mathit{l}}$ [kg/kWh] | ${\mathit{\mu}}_{\mathit{M}\mathit{T}}$ [kg/kWh] | $\mathit{\psi}$ [€/kg] |
---|---|---|---|

CO_{2} | 0.00131 | 0.000991 | 0.0309 |

SO_{2} | 0.00041 | 0.000005 | 2.1847 |

NOx | 0.02005 | 0.000027 | 9.2656 |

**Table 4.**Energy storage system economical parameters [29].

Parameter | ${\mathit{\chi}}_{\mathit{P}}^{\mathit{E}\mathit{S}\mathit{S}}$ [€/kW] | ${\mathit{\chi}}_{\mathit{E}}^{\mathit{E}\mathit{S}\mathit{S}}$ [€/kWh] | ${\mathit{\chi}}_{\mathit{O}\&\mathit{M}}^{\mathit{E}\mathit{S}\mathit{S}}$ [€/kW] | ${\mathit{\chi}}_{\mathit{i}\mathit{n}\mathit{s}}^{\mathit{E}\mathit{S}\mathit{S}}$ [kW] | r [%] | L [years] |
---|---|---|---|---|---|---|

Value | 250 | 450 | 60 | 150 | 5 | 10 |

**Table 5.**Energy storage system technical parameters [13].

Parameter | ${\mathit{P}}_{\mathit{E}\mathit{S}\mathit{S}}^{\mathbf{max}}$ [kW] | ${\mathit{\eta}}_{\mathit{c}\mathit{h}}$$/{\mathit{\eta}}_{\mathit{d}\mathit{i}\mathit{s}}$ [%] | $\mathit{S}\mathit{O}{\mathit{C}}_{\mathbf{min}}$$/\mathit{S}\mathit{O}{\mathit{C}}_{\mathbf{max}}$ [%] | ${\mathbf{\Omega}}_{\mathit{E}\mathit{S}\mathit{S}}$ [cycles/day] | ${\mathit{\xi}}_{\mathit{E}\mathit{S}\mathit{S}}$ [%] |
---|---|---|---|---|---|

Value | 1000 | 90/90 | 20/100 | 2 | 0.4 |

**Table 6.**Cost of short-duration electricity outages [32].

User | Load 1 | Load 2 | Load 2 |
---|---|---|---|

Shedding cost [€/kWh] | 37.52 | 23.41 | 0.53 |

Scenario | Case | ${\mathit{C}}_{\mathit{C}\mathit{G}}$ [€] | ${\mathit{C}}_{\mathit{E}\mathit{m}\mathit{i}\mathit{s}\mathit{s}\mathit{i}\mathit{o}\mathit{n}\mathit{s}}$ [€] | ${\mathit{C}}_{\mathit{E}\mathit{S}\mathit{S}}$ [€] | ${\mathit{C}}_{\mathit{O}\&\mathit{M}}$ [€] | ${\mathit{C}}_{\mathit{s}\mathit{h}\mathit{e}\mathit{d}\mathit{d}\mathit{i}\mathit{n}\mathit{g}}$ [€] | MG Total Operation Costs [€] |
---|---|---|---|---|---|---|---|

Scenario 1 | Without ESS | 2424.35 | 113.21 | 0 | 396.98 | 826.12 | 3760.66 |

With ESS | 17.423 | 0.2839 | 760.05 | 510.3561 | 0 | 1288.11 | |

Scenario 2 | Without ESS | 15,704.58 | 719.86 | 0 | 913.14 | 15,933 | 33,270.58 |

With ESS | 18,674.6 | 839.73 | 2876.6 | 1058.7 | 0 | 23,449.63 | |

Scenario 3 | Without ESS | 27,258.12 | 1266.65 | 0 | 2400.8 | 16,822 | 47,747.57 |

With ESS | 6760.1 | 333.06 | 5167.3 | 2723.9 | 2886.1 | 17,870.49 |

Scenario | Case | Load 1 [%] | Load 2 [%] | Load 3 [%] |
---|---|---|---|---|

Scenario 1 | Without ESS | 0 | 0 | 5.74 |

With ESS | 0 | 0 | 0 | |

Scenario 2 | Without ESS | 0 | 1.68 | 25.84 |

With ESS | 0 | 0 | 0 | |

Scenario 3 | Without ESS | 0 | 0.61 | 13.28 |

With ESS | 0 | 0 | 2.73 |

Scenario | Case | ${\mathit{I}}_{\mathit{r}\mathit{e}\mathit{s}\mathit{t}\mathit{o}\mathit{r}\mathit{a}\mathit{t}\mathit{i}\mathit{o}\mathit{n}}$ | ${\mathit{I}}_{\mathit{r}\mathit{e}\mathit{s}\mathit{i}\mathit{l}\mathit{i}\mathit{e}\mathit{n}\mathit{c}\mathit{e}}$ |
---|---|---|---|

Scenario 1 | Without ESS | 0.9872 | 0.9997 |

With ESS | 1 | 1 | |

Scenario 2 | Without ESS | 0.9257 | 0.9943 |

With ESS | 1 | 1 | |

Scenario 3 | Without ESS | 0.9654 | 0.9978 |

With ESS | 0.9932 | 0.9999 |

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## Share and Cite

**MDPI and ACS Style**

Picioroaga, I.; Luca, M.; Tudose, A.; Sidea, D.; Eremia, M.; Bulac, C.
Resilience-Driven Optimal Sizing of Energy Storage Systems in Remote Microgrids. *Sustainability* **2023**, *15*, 16002.
https://doi.org/10.3390/su152216002

**AMA Style**

Picioroaga I, Luca M, Tudose A, Sidea D, Eremia M, Bulac C.
Resilience-Driven Optimal Sizing of Energy Storage Systems in Remote Microgrids. *Sustainability*. 2023; 15(22):16002.
https://doi.org/10.3390/su152216002

**Chicago/Turabian Style**

Picioroaga, Irina, Madalina Luca, Andrei Tudose, Dorian Sidea, Mircea Eremia, and Constantin Bulac.
2023. "Resilience-Driven Optimal Sizing of Energy Storage Systems in Remote Microgrids" *Sustainability* 15, no. 22: 16002.
https://doi.org/10.3390/su152216002