# Condition-Based Multi-State-System Maintenance Models for Smart Grid System with Stochastic Power Supply and Demand

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## Abstract

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## 1. Introduction

- Development of MSS reliability model for the generic smart grid system with HPS.
- Establishment of the power grid measures that simultaneously account for the stochastic behavior of the power supply and power demand.
- Formulation of PM models for the generic smart-grid system with HPS using the MSS reliability models.
- Design of a DOE scheme to simulate the normal/extreme settings related to climate changes, thereby evaluating its impact on the established power grid measurements.
- Although a generic smart grid with an HPS was considered, the ramifications of this study are adaptable and expandable to the specific smart grid structures with appropriate modifications, thereby extending the practicability.

## 2. Related Work

## 3. Materials and Methods

#### 3.1. Simulation of the Smart Grid

- Power efficiency-related measures

- 2.
- Total maintenance cost

#### 3.2. Construction and Optimization of Four PM Models

#### 3.2.1. Construction of Two Single-Objective PM Models

#### 3.2.2. Construction of Two Bi-Objective PM Models

#### 3.3. Single-Objective PM Optimization

#### 3.4. Multi-Objective PM Optimization

## 4. Results and Discussion

- As can be seen from Figure 21 and Figure 22, a high degradation rate combined with low repair rates, (corresponding to 1 and 4 in Table 6 for the experiments with the normal and climates, respectively, and both colored red) led to a lower measured performance in terms of the mean power system unavailability and total PM cost. This is in contrast to the case of the inverse combination, which corresponds to 3 and 6, both colored green. The comparison results were mathematically and practically appropriate, given that a high degradation rate with a low repair rate decreases the reliability of the smart grid, whereas a lacking workforce and limited maintenance resources contribute to inefficient repair. In comparison, the alternative, with a low degradation combined with high repair rates, provided a robust and stable system that demonstrated a superior power measurement performance with a low PM cost in a smart grid.
- As shown in Figure 23, the simulated normal climate (corresponding to 1, 2, and 3 in Table 6, as indicated by circle dots) outperformed the simulated extreme climate (corresponding to 4, 5, and 6, as indicated by stellular dots) in terms of the power system unavailability and PM cost, regardless of the different combinations of degradation rates and repair rates. For example, given the combination of degradation rate and repair rate at 0.5 and 2 under normal climates, the power system unavailability and PM cost are determined at 0.0324 and 1159, which is lower than that of 0.3023 and 1213 under extreme climates. This highlights the extent to which climate change influences the power grid performance, and in turn, poses a safety risk to humans amidst hurricanes, flash floods, droughts, and extreme high temperatures, among other natural disasters. Moreover, these disasters presently occur at an abnormally high frequency. Hence, research has been conducted to curb green gas emissions; thus preventing the planet temperature from rising to 1.5 °C above the preindustrial level, which leads to most catastrophic disasters.
- Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28 present the other power measures from different perspectives, namely, the power system availability, expected power deficiency, accumulated power deficiency, instantaneous power capacity, and accumulated power capacity. All present the same conclusion, as depicted in the second point, through differentiated diagrams reflecting the power-grid performances across various measures.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Block diagram of smart grid with HPS [33].

**Figure 9.**Transition diagram of the second single-objective optimal PM strategy for the apparatuses.

**Figure 10.**Power system availability with respect to time for the first and second single-objective PM models.

**Figure 11.**Expected power deficiency with respect to time for the first and second single-objective PM models.

**Figure 12.**Accumulated power deficiency with respect to time for the first and second single-objective PM models.

**Figure 13.**Instantaneous power capacity with respect to time for the first and second single-objective PM models.

**Figure 14.**Accumulated power capacity with respect to time for the first and second single-objective PM models.

The PM Models | Mean Power System Availability | Total Maintenance Cost |
---|---|---|

First single-objective PM model | 0.8951 | 2000 |

Second single-objective PM model | 0.9113 | 2173 |

**Table 2.**Three PM alternatives with the mean power system unavailability and total maintenance cost.

Non-Constrained PM | Constrained PM | |||
---|---|---|---|---|

$\overline{\mathit{U}\mathit{A}}$ | $\mathit{C}\mathit{s}$ | $\overline{\mathit{U}\mathit{A}}$ | $\mathit{C}\mathit{s}$ | |

First PM alternative | 0.0918 | 2066 | 0.1411 | 1880 |

Second PM alternative | 0.1244 | 1865 | 0.1499 | 1653 |

Third PM alternative | 0.1248 | 1852 | 0.1595 | 1542 |

Transition Rates | Coal | Gas | PV | ||
---|---|---|---|---|---|

Apparatus | Degradation rates | ${\lambda}_{4,3}$ | 0.0144 | 0.0234 | 0.0108 |

${\lambda}_{4,2}$ | 0.0081 | 0.0045 | 0.0081 | ||

${\lambda}_{4,1}$ | 0.0045 | 0.0063 | 0.0072 | ||

${\lambda}_{3,2}$ | 0.0405 | 0.0036 | 0.0216 | ||

${\lambda}_{3,1}$ | 0.0032 | 0.0018 | 0.0063 | ||

${\lambda}_{2,1}$ | 0.0045 | 0.0045 | 0.0081 | ||

Minimum repair rates | 0.2628 | 0.1116 | 0.0504 | ||

${\mu}_{1,3}$ | 0.2196 | 0.3474 | 0.045 | ||

${\mu}_{1,4}$ | 0.0108 | 0.0162 | 0.009 | ||

${\mu}_{2,3}$ | 0.144 | 0.0684 | 0.0216 | ||

${\mu}_{2,4}$ | 0.162 | 0.153 | 0.0684 | ||

${\mu}_{3,4}$ | 0.144 | 0.0594 | 0.0576 | ||

Power demand | Low–High rate | ${b}_{1,2}$ | 1.52 | ||

High–Low rate | ${b}_{2,1}$ | 2.43 |

Apparatus States | Coal | Gas | PV |
---|---|---|---|

1 | 0 | 0 | 0 |

2 | 204 | 136 | 112 |

3 | 282 | 194 | 173 |

4 | 360 | 228 | 228 |

Apparatuses Costs | Coal | Gas | PV | |
---|---|---|---|---|

Minimum repair costs | ${C}_{1,2}$ | 250 | 230 | 240 |

${C}_{1,3}$ | 600 | 450 | 530 | |

${C}_{1,4}$ | 650 | 600 | 630 | |

${C}_{2,3}$ | 400 | 300 | 350 | |

${C}_{2,4}$ | 500 | 400 | 450 | |

${C}_{3,4}$ | 300 | 200 | 250 |

Climate Statuses | Rates | $\overline{\mathit{U}\mathit{A}}(\mathit{t})$ | $\overline{\mathit{A}}(\mathit{t})$ | $\mathit{C}\mathit{s}$ | ||
---|---|---|---|---|---|---|

$\lambda $ | $\mu $ | |||||

1 | Normal climate | 2 | 0.5 | 0.5782 | 0.4218 | 1665 |

2 | 1 | 1 | 0.1411 | 0.8589 | 1880 | |

3 | 0.5 | 2 | 0.0324 | 0.9676 | 1159 | |

4 | Extreme climate | 2 | 0.5 | 0.7885 | 0.2115 | 2592 |

5 | 1 | 1 | 0.6508 | 0.3492 | 1657 | |

6 | 0.5 | 2 | 0.3023 | 0.6977 | 1213 |

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**MDPI and ACS Style**

Wang, C.-H.; Huang, C.-H.; You, D.-G.
Condition-Based Multi-State-System Maintenance Models for Smart Grid System with Stochastic Power Supply and Demand. *Sustainability* **2022**, *14*, 7848.
https://doi.org/10.3390/su14137848

**AMA Style**

Wang C-H, Huang C-H, You D-G.
Condition-Based Multi-State-System Maintenance Models for Smart Grid System with Stochastic Power Supply and Demand. *Sustainability*. 2022; 14(13):7848.
https://doi.org/10.3390/su14137848

**Chicago/Turabian Style**

Wang, Chun-Ho, Chao-Hui Huang, and Deng-Guei You.
2022. "Condition-Based Multi-State-System Maintenance Models for Smart Grid System with Stochastic Power Supply and Demand" *Sustainability* 14, no. 13: 7848.
https://doi.org/10.3390/su14137848