# Model Predictive Operation Control of Islanded Microgrids under Nonlinear Conversion Losses of Storage Units

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Literature Review

#### 1.2. Statement of Contributions

- (i)
- We derive the model of an islanded MG with uncertain renewable generation and loads with a very high share of RES. This model, motivated by [22], considers a possible limitation of renewable infeed, while limitations on transmission lines are approximately accounted for using DC power flow approximations.
- (ii)
- We model storage devices as grid-forming units, and, to make the MG model more realistic, we consider the conversion losses of the storage units, the losses of the power electronic devices when converting Alternating Current (AC) to Direct Current (DC) (and vice versa), as well as ohmic losses in the batteries as the quadratic functions in the dynamics of storage units.
- (iii)
- We take into account power-sharing with the active conventional units. Therefore, the load fluctuations and renewable units influence all units’ power and the storage units’ state of charge. In this way, the model can also work where only RES and storage units are active, and no conventional unit is required.
- (iv)
- We propose a novel MPC approach for the optimal operation of an islanded MG with a very high share of renewable energy sources. To solve the optimization problem and mitigate the effect of errors in the storage units’ state of charge prediction, we reformulate the conversion loss functions as the mixed-integer linear inequality functions and include them in the proposed scheme.
- (v)
- We confirm the properties of the proposed operation control scheme via realistic simulations with a high renewable share.

#### 1.3. Paper Organization

#### 1.4. Mathematical Notation

## 2. Microgrid Model

**Assumption**

**1**

**Assumption**

**2**

**Assumption**

**3**

**Assumption**

**4**

## 3. Certainty Equivalence Model Predictive Control

#### 3.1. Plant Model Interface

#### 3.2. Power of Units

#### 3.2.1. Active Power at RES Units

#### 3.2.2. Active Power at Conventional Units

#### 3.2.3. Active Power at Storage Units

#### 3.3. Power Sharing of Grid-Forming Units

#### 3.4. Dynamics of Storage Units

**Remark**

**1.**

#### 3.5. Transmission Network

#### 3.6. Overall Model

## 4. Operating Costs

**Remark**

**2.**

## 5. Case Study

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Certainty equivalence MPC scheme for operation of islanded MG at time instant k and $\forall j=0,\dots ,J-1$.

**Figure 4.**The prediction error of the state of charge (Up) with the dynamic storage without piecewise affine loss model (26)–(29d) in the controller; (Down) with the dynamic storage with piecewise affine loss model (10)–(25) in the controller.

Symbol | Explanation | Unit | Size |
---|---|---|---|

x | Energy of storage units (state) | $\mathrm{pu}\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$ | S |

${u}_{\mathrm{t}}$ | Setpoint inputs of conventional units | $\mathrm{pu}$ | T |

${u}_{\mathrm{s}}$ | Setpoint inputs of storage units | $\mathrm{pu}$ | S |

${u}_{\mathrm{r}}$ | Setpoint inputs of renewable units | $\mathrm{pu}$ | R |

u | Setpoint inputs of all units | $\mathrm{pu}$ | U |

${\delta}_{\mathrm{t}}$ | Boolean inputs of conventional units | - | T |

v | Vector of all control inputs | - | Q |

${w}_{\mathrm{r}}$ | Uncertain available renewable power | $\mathrm{pu}$ | R |

${w}_{\mathrm{d}}$ | Uncertain load | $\mathrm{pu}$ | D |

w | Vector of all uncertain inputs | $\mathrm{pu}$ | W |

${p}_{\mathrm{t}}$ | Active power of conventional units | $\mathrm{pu}$ | T |

${p}_{\mathrm{s}}$ | Active power of storage units | $\mathrm{pu}$ | S |

${p}_{\mathrm{r}}$ | Active power of renewable units | $\mathrm{pu}$ | R |

p | Active power of all units | $\mathrm{pu}$ | U |

${p}_{\mathrm{e}}$ | Power over transmission lines | $\mathrm{pu}$ | E |

${\delta}_{\mathrm{r}}$ | Boolean auxiliary variables | - | R |

$\rho $ | Real-valued auxiliary variable | - | 1 |

$\overline{q}$ | Vector of all auxiliary variables | - | Q |

Parameter | Value | Weight | Value |
---|---|---|---|

$\left[{p}_{t}^{min},{p}_{r}^{min},{p}_{s}^{min}\right]$ | $\left[\begin{array}{c}0.4,0,-1\end{array}\right]$${}_{\mathrm{pu}}$ | ${c}_{\mathrm{t}}$ | 0.1178 |

$\left[\begin{array}{c}{p}_{t}^{max},{p}_{r}^{max},{p}_{s}^{max}\end{array}\right]$ | $\left[\begin{array}{c}1,2,1\end{array}\right]$${}_{\mathrm{pu}}$ | ${\widehat{c}}_{\mathrm{t}}$ | $0.0048$${}_{1/\mathrm{pu}}$ |

$\left[\begin{array}{c}{x}^{min},{x}^{max}\end{array}\right]$ | $\left[\begin{array}{c}0,7\end{array}\right]$${}_{\mathrm{pu}}$${}_{\mathrm{h}}$ | ${\tilde{c}}_{\mathrm{t}}$ | $0.751$${}_{1/\mathrm{pu}}$ |

$\left[\begin{array}{c}{\tilde{x}}^{min},{\tilde{x}}^{max}\end{array}\right]$ | $\left[\begin{array}{c}0.5,6.5\end{array}\right]$${}_{\mathrm{pu}}$${}_{\mathrm{h}}$ | ${\tilde{c}}_{\mathrm{r}}$ | 0.0001 |

${x}^{0}$ | 3 ${}_{\mathrm{pu}}$ ${}_{\mathrm{h}}$ | ${\widehat{c}}_{\mathrm{r}}$ | 1 ${}_{1/\mathrm{pu}}$ |

$\left[\begin{array}{c}{K}_{t},{K}_{s}\end{array}\right]$ | $\left[\begin{array}{c}1,1\end{array}\right]$ | ${\tilde{c}}_{\mathrm{s}}$ | 0.09 |

${\tilde{M}}_{i}$ | 0.1 | ${\widehat{c}}_{\mathrm{s}}$ | 0.01 |

${\tilde{m}}_{i}$ | −0.17 | ${c}_{\mathrm{t}}^{\mathrm{sw}}$ | 0.1 |

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## Share and Cite

**MDPI and ACS Style**

Gholami, M.; Pisano, A.
Model Predictive Operation Control of Islanded Microgrids under Nonlinear Conversion Losses of Storage Units. *Electricity* **2022**, *3*, 33-50.
https://doi.org/10.3390/electricity3010003

**AMA Style**

Gholami M, Pisano A.
Model Predictive Operation Control of Islanded Microgrids under Nonlinear Conversion Losses of Storage Units. *Electricity*. 2022; 3(1):33-50.
https://doi.org/10.3390/electricity3010003

**Chicago/Turabian Style**

Gholami, Milad, and Alessandro Pisano.
2022. "Model Predictive Operation Control of Islanded Microgrids under Nonlinear Conversion Losses of Storage Units" *Electricity* 3, no. 1: 33-50.
https://doi.org/10.3390/electricity3010003