# Stochastic Optimal Strategy for Power Management in Interconnected Multi-Microgrid Systems

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## Abstract

**:**

## 1. Introduction

- The proposed UT-MJAYA algorithm can appropriately deal with multi-objective stochastic power management of MMG systems, while considering conflicting objectives;
- The suggested optimal power management strategy aims to minimize the total cost of day-ahead market transactions and fuel costs, while it is also intent on minimizing emission of the system;
- The uncertainties related to the forecasted values of the load demand, electricity price, and the available outputs of RESs are considered and investigated competently applying the presented UT-MJAYA algorithm;
- Since in the considered problem of optimal power management of MMGs the generations of renewable energies (i.e., wind and photovoltaic) as well as the load demand and electricity price are correlated, the correlation between these uncertain variables are considered; and
- By maximizing the utilization of RESs in the considered region, the suggested approach concentrates on the reduction of dependency of the MMG system on the main grid and the electricity market.

## 2. Problem Formulation

#### 2.1. Minimizing the Operational Cost

_{p}is the number of MGs in the MMG system, T is the horizon of operation which is considered equal to 24 h in this paper and $Cos{t}_{DG}^{n,t}$ is the hourly operational cost of dispatchable units in $. ${P}_{EX}^{n,t}$ is the shared power of n

^{th}MG with the electricity grid in MW and the positive values of ${P}_{EX}^{n,t}$ represent purchasing energy from the electricity grid while the negative values represent selling energy to the grid. ${B}_{EX}^{t}$ is the market price in electric market in $/MWh. Accordingly, ${P}_{EX}^{n,t}{B}_{EX}^{t}$ is representative of income (if ${P}_{EX}^{n,t}$ is negative) and cost (if ${P}_{EX}^{n,t}$ is positive) from power sharing with the electricity grid.

#### 2.2. Minimizing the Emission

#### 2.3. Technical Constraints

^{th}and m

^{th}MG in MW.

^{th}MG to the m

^{th}MG while the negative values represent the power sharing from the m

^{th}MG to the n

^{t}

^{h}MG. Moreover, all the generated and stored powers as well as the exchanged power with the electricity grid should satisfy the power limit constraints as follows:

^{th}and the m

^{th}MG in the MMG system in MW. ${P}_{EX,max}^{n}$ is the maximum exchangeable power with the electricity grid in MW. ${P}_{Batt,max}^{n}$ is the maximum hourly charging/discharging value of the battery of each MG in MW. $B{L}_{Batt,min}^{n}$ and $B{L}_{Batt,max}^{n}$ are, respectively, the minimum and the maximum stored energies in the battery of each MG in MWh.

## 3. Unscented Transformation (UT) Method

- calculating the set of sigma points;
- assigning weights to each sigma point;
- converting sigma points using the nonlinear function;
- calculating the Gaussian distribution of new states and their related assigned weights; and
- finding the covariance and mean value of the new Gaussian distribution.

- Step 1: generating the covariance matrix of uncertain variables: the elements of the main diagonal of the covariance matrix are the standard deviation of uncertain variables while the off-diagonal elements show the correlation among uncertain variables;
- Step 2: generating sigma points: sigma points should model the behavior of the uncertain variables so they should be generated based on covariance matrix. The following is used to generate the sigma points:$${X}^{k}=+{\left(\sqrt{m\times {P}_{xx}}\right)}_{k}k=1,2,\dots ,m$$$${X}^{k}=-{\left(\sqrt{m\times {P}_{xx}}\right)}_{k}k=1,2,\dots ,m$$
- Step 3: assigning weights to each sigma point: in this step, a weight is assigned to each sigma point based on the following:$${W}^{k}=\frac{1}{2m}k=1,2,\dots ,2m$$

- Step 4: output extraction for sigma points: for each sigma point the outputs of the problem are extracted based on the following:$${y}^{k}=f\left({X}^{k}\right)$$
- Step 5: achieving the mean value of the output: according to the extracted outputs and the assigned weights, the output of the problem is calculated as follows:$${\mu}_{y}={\displaystyle \sum}_{k=1}^{2m}{W}^{k}\times {Y}^{k}$$

## 4. System Structure

#### 4.1. Generation Units’ Model

^{2}, ${\mu}_{{R}^{t}}$ is the mean value of hourly irradiance in w/m

^{2}, $m$ is number of uncertain variables,${W}^{0}$ is the weighted value related to the scenario where the irradiance in all hours equals to its mean value. ${P}_{xx}$ is a square matrix the dimension of which is equal to the number of uncertain variables. The elements of the main diagonal equal to the standard deviation of irradiance in each hour. Afterward, the available hourly power of PV system is calculated based on (19):

^{2}. ${P}_{PV}^{rated}$ is the nominal power of PV units in MW, ${R}_{STD}$ represents the irradiance in standard conditions which equals to 1000 W/m

^{2}. ${R}_{C}$ is the constant solar irradiance which equals to 150 w/m

^{2}[39].

^{2}, $/MW, and $. Moreover, the output power is limited to upper and lower bands:

#### 4.2. Load Model

## 5. Application of the Suggested Multi-Objective JAYA Algorithm (MJAYA)

^{th}iteration are updated based on the following:

^{th}variable in best solutions, and ${X}_{i,worst,iter}$ is the i

^{th}variable in worst solutions in iter

^{th}iteration. ${X}_{i,j,iter}^{\prime}$ is the modified version of ${X}_{i,j,iter}$ while ${r}_{1}{}_{i,iter}$ and ${r}_{2}{}_{i,iter}$ are two random values in range [0, 1]. The trend of a solution to approach to the best solution and to get far from the worst solution are, respectively, expressed by ${r}_{1}{}_{i,iter}({X}_{i,best,iter}-{X}_{i,j,iter})$ and ${r}_{2}{}_{i,iter}({X}_{i,worst,iter}-{X}_{i,j,iter})$. If ${X}_{i,j,iter}^{\prime}$ results in a better objective function value than that of ${X}_{i,j,iter}$, it will be accepted. Based on the values of objective functions, at the end of each iteration, best solutions are maintained and will be considered as the population for the next iteration. The best and worst solutions are determined in the recent maintained population for the next iteration. The process will pursue until the termination criterion is fulfilled [32,33,34].

## 6. Simulation Results

#### 6.1. Minimizing the Operational Cost

#### 6.2. Minimizing Emission

#### 6.3. Multi-Objective Optimal Operation of MMG

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 11.**Accumulated optimal operation of MMG elements for minimizing the cost (maximizing the profit) in deterministic conditions.

**Figure 12.**Accumulated optimal operation of MMG elements for minimizing the cost (maximizing the profit) in probabilistic conditions.

**Figure 13.**Accumulated optimal operation of MMG elements for minimizing emission in deterministic conditions.

**Figure 14.**Accumulated optimal operation of MMG elements for minimizing emission in probabilistic conditions.

**Figure 15.**Comparison of Pareto optimal fronts of environmental–economic optimal power management of MMG system in deterministic and probabilistic scenarios.

MG | ${\mathit{P}}_{\mathit{D}\mathit{G},\mathit{M}\mathit{i}\mathit{n}}^{}\text{}\left(\mathbf{MW}\right)$ | ${\mathit{P}}_{\mathit{D}\mathit{G},\mathit{M}\mathit{a}\mathit{x}}^{}\text{}\left(\mathbf{MW}\right)$ | α ($/MWh^{2}) | Β ($/MWh) | γ ($) |
---|---|---|---|---|---|

1, 2 | 0 | 1.285 | 0.0345 | 44.5 | 26.5 |

3, 4 | 0 | 2.47 | 0.0435 | 56 | 12.5 |

MG | ${\mathit{P}}_{\mathit{B}\mathit{a}\mathit{t}\mathit{t},\mathit{M}\mathit{i}\mathit{n}}^{}\text{}\left(\mathbf{MW}\right)$ | ${\mathit{P}}_{\mathit{B}\mathit{a}\mathit{t}\mathit{t},\mathit{M}\mathit{a}\mathit{x}}^{\mathit{n}}\text{}\left(\mathbf{MW}\right)$ | $\mathit{B}{\mathit{L}}_{\mathit{B}\mathit{a}\mathit{t}\mathit{t},\mathit{M}\mathit{i}\mathit{n}}^{}\text{}\left(\mathbf{MWh}\right)$ | $\mathit{B}{\mathit{L}}_{\mathit{B}\mathit{a}\mathit{t}\mathit{t},\mathit{M}\mathit{a}\mathit{x}}^{}\text{}\left(\mathbf{MWh}\right)$ | ${\mathit{E}}_{\mathbf{int},\mathit{B}\mathit{a}\mathit{t}\mathit{t}}^{}\text{}\left(\mathbf{MWh}\right)$ | ${\mathit{\eta}}_{\mathit{B}\mathit{a}\mathit{t}\mathit{t}}^{}$ |
---|---|---|---|---|---|---|

1, 2, 3,4 | −0.4 | 0.4 | 0.240 | 1.2 | 0.375 | 0.75 |

MG | ${\mathit{P}}_{\mathit{P}\mathit{V}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}\left(\mathbf{MW}\right)$ | Rc (W/m^{2}) | Rstd (W/m^{2}) |
---|---|---|---|

1, 3 | 0.3 | 150 | 1000 |

Plant | ${\mathit{P}}_{\mathit{W}\mathit{T}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}\left(\mathbf{MW}\right)$ | ${\mathit{\nu}}_{\mathit{i}\mathit{n}}^{\mathit{C}}(\mathbf{m}/\mathbf{s})$ | ${\mathit{\nu}}^{\mathit{r}\mathit{a}\mathit{t}\mathit{e}\mathit{d}}(\mathbf{m}/\mathbf{s})$ | ${\mathit{\nu}}_{\mathit{o}\mathit{u}\mathit{t}}^{\mathit{C}}(\mathbf{m}/\mathbf{s})$ |
---|---|---|---|---|

2, 4 | 0.45 | 3.5 | 13.5 | 25 |

MG1 ${\mathit{P}}_{\mathit{E}\mathit{X},\mathit{M}\mathit{a}\mathit{x}}^{1}\left(\mathbf{MW}\right)$ | MG2 ${\mathit{P}}_{\mathit{E}\mathit{X},\mathit{M}\mathit{a}\mathit{x}}^{2}\left(\mathbf{MW}\right)$ | MG3 ${\mathit{P}}_{\mathit{E}\mathit{X},\mathit{M}\mathit{a}\mathit{x}}^{3}\left(\mathbf{MW}\right)$ | MG4 ${\mathit{P}}_{\mathit{E}\mathit{X},\mathit{M}\mathit{a}\mathit{x}}^{4}\left(\mathbf{MW}\right)$ |
---|---|---|---|

2.5 | 3.5 | 4 | 7 |

E_{DG} (Kg/MWh) | E_{Grid} (Kg/MWh) |
---|---|

725 | 927 |

**Table 7.**The values of the achieved profit of all MGs and the total profit of the system in both deterministic and probabilistic analysis.

MG1 | MG2 | MG3 | MG4 | Multi MG System | Run Time (s) | |
---|---|---|---|---|---|---|

Expected Profit ($) | −3631 | 1985 | 2232 | 7425 | 8011 | 262 |

Profit ($) | −2422 | 947 | 3625 | 6382 | 8559 | 1902 |

**Table 8.**The values of the emission of all MGs and the total emission of the system in both deterministic and probabilistic analysis.

MG1 | MG2 | MG3 | MG4 | Multi MG System | Run Time (s) | |
---|---|---|---|---|---|---|

Expected Emission (kg) | 21,822 | 13,939 | 17,708 | 7816 | 61,285 | 262 |

Emission (kg) | 26,555 | 10,515 | 18,244 | 4725 | 60,039 | 1715 |

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**MDPI and ACS Style**

Javidsharifi, M.; Pourroshanfekr Arabani, H.; Kerekes, T.; Sera, D.; Guerrero, J.M.
Stochastic Optimal Strategy for Power Management in Interconnected Multi-Microgrid Systems. *Electronics* **2022**, *11*, 1424.
https://doi.org/10.3390/electronics11091424

**AMA Style**

Javidsharifi M, Pourroshanfekr Arabani H, Kerekes T, Sera D, Guerrero JM.
Stochastic Optimal Strategy for Power Management in Interconnected Multi-Microgrid Systems. *Electronics*. 2022; 11(9):1424.
https://doi.org/10.3390/electronics11091424

**Chicago/Turabian Style**

Javidsharifi, Mahshid, Hamoun Pourroshanfekr Arabani, Tamas Kerekes, Dezso Sera, and Josep M. Guerrero.
2022. "Stochastic Optimal Strategy for Power Management in Interconnected Multi-Microgrid Systems" *Electronics* 11, no. 9: 1424.
https://doi.org/10.3390/electronics11091424