# Optimal Day-Ahead Scheduling of Microgrids with Battery Energy Storage System

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## Abstract

**:**

## 1. Introduction

- Presenting a practical methodology to compute the BESS availability cost in USD/kWh; modeling of BESS system considering battery, converter, and transformer useful values of efficiencies; modeling the BESS state of charge as a non-recursive constraint to the MILP problem;
- Modeling of scheduled intentional islanding in a microgrid with the possibility of curtailment and load shedding;
- Modeling of non-interruptible shiftable loads (continuous cycle) as washers and driers when participating in demand response programs;
- Presenting a practical methodology to compute the availability cost of PV systems.

## 2. Microgrid Energy Management System

## 3. Modeling Methodology

#### 3.1. Microgrid Modeling

#### 3.1.1. Objective Function

#### 3.1.2. Energy and Power Exchanges

#### 3.1.3. Power Balance

#### 3.1.4. Battery Energy Storage System

#### 3.1.5. Directly Controllable Loads and Load Shedding

#### 3.1.6. PV System

#### 3.1.7. Matrix Formulation

#### 3.2. Cost Estimation

#### 3.2.1. BESS Costs

#### 3.2.2. PV Costs

#### 3.2.3. Other Costs

## 4. Simulation Methodology

^{®}software. The optimizer function intlinprog() was used to solve the MILP problem from Equation (31), subject to the constraints from Table 7. Furthermore, it was used a time resolution $\Delta t$ of 1/4 h, and $N=96$ time slots.

#### 4.1. Normalized Energy Bill

#### 4.2. Simulation Cases

- Energy arbitrage as a result of the Brazilian White Tariff applied to a microgrid;
- MG scheduled intentional islanding using the resources of interruptible loads and PV output curtailment; two islanding periods: ${T}_{{isl}_{1}}=13$, from 02:00 to 05:15, and ${T}_{{isl}_{2}}=8$, from 10:30 to 12:30; $M{G}_{ss}=1$.
- MG scheduled intentional islanding using the resources of interruptible loads, PV output curtailment, load shedding, and shiftable loads; two islanding periods: ${T}_{{isl}_{1}}=16$, from 02:00 to 06:00, and ${T}_{{isl}_{2}}=8$, from 10:30 to 12:30; $M{G}_{ss}=1$.
- The impact of the microgrid self-sufficiency on the normalized energy bill; limit for energy import and export: $\phantom{\rule{0.166667em}{0ex}}{\overline{\phantom{\rule{-0.166667em}{0ex}}P}}_{pcc}=800$ kW; curves for ${E}_{r}=12$, 16, 20, and 30% of ${E}_{pv}$.

## 5. Results

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of the MG day-ahead EMS model addressed in the present work, showing its relationship with the input and output data and its interaction with the real-time EMS; CD, ED, FD, LD, PD, and TD stand for cost, estimated, forecasted, limit, price, and technical data, respectively.

**Figure 2.**Schematic diagram of the microgrid model used in the MILP problem of the present study. The model presents integer (binary) and real decision variables. The former ($s{t}^{\left(t\right)}$ type) work as virtual disconnectors and the latter (${P}^{\left(t\right)}$ type) represent the active powers of each element of the model. Type $\widehat{P}$ variables represent forecasted values.

**Figure 3.**Clarifying the nomenclature of BESS used throughout the present study. The definitions of SOC and DOD are following the IEEE Std 2030.2.1™ [34].

**Figure 4.**Exponential approximation curves for state of health, where ${SOH}_{tsh}=0.8$ and ${L}_{cyc}=6000$.

**Figure 6.**Results of simulation 1. (

**a**) Results of simulation 1 for the power balance curves; E

_{bill}= 249.36 USD; E

_{loss}represents energy losses throughout the day with BESS charging and discharging operations; T

_{pre,post}stands for the pre and post-peak period; T

_{peak}is the on-peak period; (

**b**) Results of simulation 1 for the BESS SOC.

**Figure 7.**Results of simulation 2. (

**a**) Results of simulation 2 for the power balance curves; [E

_{pvc}= 223.85, E

_{int}= 10.16] (kWh); E

_{bill}= 294.82 USD; (

**b**) Results of simulation 2 for the BESS SOC; (

**c**) Expanding the curve of interruptible loads during the first period of islanding.

**Figure 8.**Results of simulation 3. (

**a**) Results of simulation 3 for the power balance curves; [E

_{pvc}= 127.85, E

_{shd}= 61.98, E

_{int}= 14.92] (kWh); E

_{bill}= 285.92 USD; (

**b**) Results of simulation 3 for the BESS SOC.

**Figure 9.**Normalized energy bill for a sequence of simulations with increasing values of microgrid self-sufficiency; curves for BESS rated capacity (${E}_{r}$) from 12% to 30% of the average daily PV system capacity (${E}_{pv}$); $\phantom{\rule{0.166667em}{0ex}}{\overline{\phantom{\rule{-0.166667em}{0ex}}P}}_{pcc}=800$ kW; ${E}_{load}=2400$ kWh; ${E}_{pv}:=2400\phantom{\rule{0.166667em}{0ex}}{MG}_{ss}$ kWh.

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

${st}_{{pur}_{g}}^{\left(t\right)}$, ${st}_{{pur}_{\mathsf{\mu}}}^{(j,t)}$ | represent the state of purchasing transaction at time t; index g stands for trading with the main grid and $\mathsf{\mu}$ trading with the j-th microgrid; |

${st}_{{sel}_{g}}^{\left(t\right)}$, ${st}_{{sel}_{\mathsf{\mu}}}^{(j,t)}$ | represent the status of energy sale transaction at time t; index g stands for trading with the main grid and $\mathsf{\mu}$ trading with the j-th microgrid; |

${st}_{chr}^{\left(t\right)}$, ${st}_{dch}^{\left(t\right)}$ | represent the state of charge and discharge of the battery, at time t, respectively; |

${st}_{int}^{\left(t\right)}$, ${st}_{sh\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}f}^{\left(t\right)}$ | represent the connection state of interruptible and shiftable loads, respectively. |

${st}_{shd}^{\left(t\right)}$ | represents the activation state of the load shedding resource. |

Power Variables (kW) | Description |
---|---|

${P}_{{pur}_{g}}^{\left(t\right)}$, ${P}_{{pur}_{\mathsf{\mu}}}^{(j,t)}$ | are the powers injected by the main grid into the microgrid ($pur$ index) through the $PCC$, at time t. The index g stands for trading with the main grid and $\mathsf{\mu}$ trading with the j-th microgrid; |

${P}_{{sel}_{g}}^{\left(t\right)}$, ${P}_{{sel}_{\mathsf{\mu}}}^{(j,t)}$ | are the powers injected by the microgrid into the main grid ($sel$ index) through the $PCC$, at time t. The index g stands for trading with the main grid and $\mathsf{\mu}$ trading with the j-th microgrid; |

${P}_{chr}^{\left(t\right)}$, ${P}_{dch}^{\left(t\right)}$ | are the battery charging and discharging power, respectively. |

${P}_{pvc}^{\left(t\right)}$, ${P}_{shd}^{\left(t\right)}$ | are the amount of PV output power curtailment and load shedding, respectively. |

${P}_{int}^{\left(t\right)}$, ${P}_{sh\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}f}^{\left(t\right)}$ | are the amount of interruptible and shiftable loads, respectively. |

Non-Decision Variables | Description |
---|---|

${\widehat{P}}_{load}^{\left(t\right)}$, ${\widehat{P}}_{pv}^{\left(t\right)}$ | are the day-ahead forecasted load and PV output power curves, respectively (kW). |

${st}_{con}^{\left(t\right)},{st}_{pv}$ | are the microgrid and PV system state of connection, respectively; state 0 means disconnected and 1 connected. |

State Parameter ($) | Description |
---|---|

${c}_{{sel}_{g}}$, ${c}_{{sel}_{\mathsf{\mu}}}$ | are fixed costs of service when selling energy to the grid or to the j-th microgrid. |

${c}_{{pur}_{g}}$, ${c}_{{pur}_{\mathsf{\mu}}}$ | are fixed costs of service when purchasing energy from the grid or from the j-th microgrid. |

${c}_{chr}$, ${c}_{dch}$ | represent a fixed cost for charging and discharging the battery, respectively; |

${c}_{int}$, ${c}_{sh\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}f}$ | represent a fixed cost for disconnecting interruptible loads, and for connecting shiftable loads, respectively; |

${c}_{shd}$, ${C}_{pv}$ | represent a fixed cost for disconnecting load (shedding), and the availability cost of the PV system, respectively; |

Power Parameter ($/kWh) | Description |
---|---|

${p}_{{sel}_{g}}^{\left(t\right)}$, ${p}_{{sel}_{\mathsf{\mu}}}^{(j,t)}$ | are the sale prices of energy, at time t, when trading with the grid (g index) or with the j-th microgrid ($\mathsf{\mu}$ index); |

${p}_{{pur}_{g}}^{\left(t\right)}$, ${p}_{{pur}_{\mathsf{\mu}}}^{(j,t)}$ | are the purchase prices of energy, at time t, when trading with the grid (g index) or with the j-th microgrid ($\mathsf{\mu}$ index); |

${p}_{chr}$, ${p}_{dch}$ | are the cost of each kWh for charging and discharging the battery, at time t, respectively. |

${p}_{pvc}^{\left(t\right)}$, ${p}_{shd}^{\left(t\right)}$ | are the cost of each kWh of PV output power curtailment and load disconnected (shedding), at time t, respectively. |

${p}_{int}^{\left(t\right)}$, ${p}_{sh\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}f}^{\left(t\right)}$ | are the cost of each kWh to disconnect interruptible loads and to connect shiftable loads, at time t, respectively. |

Battery Technology | Coupling Transformer | Efficiency | |||
---|---|---|---|---|---|

${\mathbf{\eta}}_{\mathit{t}\mathit{r}\mathit{f}}$ | ${\mathbf{\eta}}_{\mathit{c}\mathit{o}\mathit{n}\mathit{v}}$ | ${\mathbf{\eta}}_{\mathit{b}\mathit{a}\mathit{t}}$ | ${\mathbf{\eta}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{s}}$ | ||

Lead-acid | no | − | >$0.97$ | $0.9220$ | $0.8943$ |

yes | >$0.97$ | >$0.97$ | $0.9220$ | $0.8675$ | |

Lithium-ion | no | − | >$0.97$ | $0.9487$ | $0.9202$ |

yes | >$0.97$ | >$0.97$ | $0.9487$ | $0.8926$ | |

LiFePO${}_{4}$, and Lithium-ion of higher technology | no | − | >$0.97$ | $0.9747$ | $0.9454$ |

yes | >$0.97$ | >$0.97$ | $0.9747$ | $0.9170$ |

**Table 7.**Summary of the equations, organized by features, required to build the matrices and vectors of constraints of the MILP problem; $\mathbf{0}$ is a $N\times 1$ vector of zeros.

Features | Equations for A and b | Equations for ${\mathit{A}}_{\mathbf{eq}}$ and ${\mathit{b}}_{\mathbf{eq}}$ | Equations for ${\mathit{l}}_{\mathit{b}}$ and ${\mathit{u}}_{\mathit{b}}$ |
---|---|---|---|

State variables in general | $\mathbf{0}\le {\mathbf{x}}_{st}\le \mathbf{1}$ | ||

Energy exchange and islanding | (4) | (4) | |

Power exchange | (5), (6) | (5) | |

Power balance | (8) | ||

BESS, state variables | (17) | (17) | |

BESS, power variables | (15), (18) | (16) | (18) |

Load shedding, state variable | (20) | ||

Load shedding, power variable | (19) | (19) | |

Interruptible load, state variable | (22), (21) | ||

Interruptible load, power variable | (22) | (22) | |

Shiftable load, state variable | (26) | (24), (25) | |

Shiftable load, power variable | (24) | (23), (24) | |

PV output power curtailment | (27) | (27) |

**Table 8.**The Brazilian White Tariff; average time blocks were computed from 104 companies data; average prices were calculated from the 11 major energy distributors in Brazil, weighted by the number of customers of each company.

Tariff | Average Time Blocks | Average Prices | Percentage Prices (%) | |||
---|---|---|---|---|---|---|

Start Time | End Time | BRL/kWh | USD/kWh | |||

Conventional tariff | − | − | 0.558 | 0.130 | $100\phantom{\rule{14.22636pt}{0ex}}$ | |

ToU, off-peak | 00:00 | 17:00 | 0.466 | 0.109 | $84\phantom{\rule{14.22636pt}{0ex}}$ | |

ToU, pre-peak | 17:00 | 18:00 | 0.679 | 0.158 | $122\phantom{\rule{14.22636pt}{0ex}}$ | |

ToU, on-peak | 18:00 | 21:00 | 1.059 | 0.247 | $190\phantom{\rule{14.22636pt}{0ex}}$ | |

ToU, post-peak | 21:00 | 22:00 | 0.679 | 0.158 | $122\phantom{\rule{14.22636pt}{0ex}}$ | |

ToU, off-peak | 22:00 | 24:00 | 0.466 | 0.109 | $84\phantom{\rule{14.22636pt}{0ex}}$ |

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**MDPI and ACS Style**

Silva, V.A.; Aoki, A.R.; Lambert-Torres, G.
Optimal Day-Ahead Scheduling of Microgrids with Battery Energy Storage System. *Energies* **2020**, *13*, 5188.
https://doi.org/10.3390/en13195188

**AMA Style**

Silva VA, Aoki AR, Lambert-Torres G.
Optimal Day-Ahead Scheduling of Microgrids with Battery Energy Storage System. *Energies*. 2020; 13(19):5188.
https://doi.org/10.3390/en13195188

**Chicago/Turabian Style**

Silva, Vanderlei Aparecido, Alexandre Rasi Aoki, and Germano Lambert-Torres.
2020. "Optimal Day-Ahead Scheduling of Microgrids with Battery Energy Storage System" *Energies* 13, no. 19: 5188.
https://doi.org/10.3390/en13195188