# Effect of Flexible Operation on Residual Life of High-Temperature Components of Power Plants

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## Abstract

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## 1. Introduction

_{2}emissions have steadily increased over time, and Belbute [1] has suggested that the emissions will increase by 27.4% by 2030. Prior studies have addressed the adverse effects of fossil fuels on CO

_{2}emissions and the need to generate electricity using renewable energy sources (RES) for low-carbon growth [2,3]. Most RES have non-dispatchable characteristics that prevent them from adjusting the energy supply to meet the demand. Climatic or geographic conditions cause variability in electricity production and can render power plant systems more unstable. According to the average net electricity production per week in Germany in 2021, there was a power fluctuation at approximately 12:00 p.m. owing to the influence of solar power [4]. Τhe concern about power outages caused by over- and under-generation increases with the proportion of RES in the power generation system. As a result, it is critical to address the power uncertainty caused by the increased demand for RES [5]. Flexible operation is a method of changing the output level of a thermal power plant to maintain constant output power according to the change in the output power of the RES. This is the most economical way to handle the instability of RES energy generation [6].

_{2}emissions and that rapid load alternation reduces this efficiency. However, little attention has been paid to estimating residual life with fluctuating thermal loads by flexible operation. Therefore, a life evaluation study considering fatigue and creep life is required to investigate the reliability of high-temperature components because of thermal load fluctuations under flexible operating conditions.

## 2. Methodology

#### 2.1. Thermo-Structural Analysis for Evaluation of Strain Range during Flexible Operation

#### 2.2. Creep–Fatigue Damage Theory

#### 2.3. Machine Learning Techniques

#### 2.3.1. Feedforward Neural Network Model

#### 2.3.2. Hyperparameter Optimization Using Random Search

## 3. Numerical Examples

#### 3.1. Validation for Thermo-Structural FE Model

- (a)
- Radiant and convective heat transfer from combustion gas at the outer wall;
- (b)
- Conduction in the tube wall;
- (c)
- Convection at the inner wall of the working fluid.

#### 3.2. Estimation of Fatigue Life under Cyclic Thermal Loads

#### 3.3. Creep and Fatigue Life of the Header

#### 3.4. Response Surface Model

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Schematic diagram of the flexible operation cycle. (

**a**) General load operation section: the section where the power plant operates while maintaining the general load (${P}_{\mathrm{max}}$). (

**b**) Load-decreasing section: the flexible operating section adjusts power in response to an increasing proportion of RES. (

**c**) Minimum load operation section: the section that maintains the minimum load (${P}_{\mathrm{min}}$ ). (

**d**) Load increasing section: the section increasing the load from ${P}_{\mathrm{min}}$ to ${P}_{\mathrm{max}}$.

**Figure 6.**(

**a**) Schematic diagram of boundary conditions of the power plant. Thermal conditions for (

**b**) the header component, and (

**c**) tube component.

**Figure 7.**Validation of heat transfer analysis by comparing the analytical solutions with numerical solutions of (

**a**) wall temperature and (

**b**) heat flux in the power descending section.

**Figure 8.**Comparison of numerical solution with the analytic solution under (

**a**) thermal, (

**b**) structural, and (

**c**) thermo−structural condition.

**Figure 10.**Results of the thermo-structural analysis. The temperature of (

**a**) the reheater header and (

**b**) the superheater tube. The stress of (

**c**) the header and (

**d**) the tube.

**Figure 13.**Creep–fatigue envelop with damage accumulation after 3000 cycles. The red circle represents the cumulative damage of creep and fatigue that occurred after 3000 cycles for the plant header.

**Figure 14.**Performance of the ANN model. (

**a**) Epoch versus error graph, and (

**b**) regression results with coefficient of correlation R.

Elements | C | Mn | Si | Cr | Ni | N | Nb | P |
---|---|---|---|---|---|---|---|---|

Composition (wt%) | 0.07–0.13 | 1.00 | 0.010 | 17.0–19.0 | 7.5–10.5 | 0.05–0.12 | 0.30–0.60 | 0.040 |

Material Property | Value at 300 °C | Value at 500 °C | Value at 700 °C |
---|---|---|---|

Density, ρ $(kg/{m}^{3}$) | 7790 | 7700 | 7610 |

Thermal expansion coefficient, α $(\times {10}^{-6}/\xb0C)$ | 9.7 | 10.05 | 10.3 |

Elastic modulus E $\left(GPa\right)$ | 164.78 | 148.93 | 132.38 |

Poisson’s ratio ν | 0.2874 | 0.2946 | 0.3018 |

Thermal conductivity, k $(W/m\xb0C)$ | 21.461 | 24.923 | 30.98 |

Specific heat, c $(J/kg\xb0C)$ | 542.62 | 579.71 | 616.81 |

Boundary Condition | Under 100% Condition | Under 30% Condition | ||
---|---|---|---|---|

Tube | $\mathrm{Steam}\mathrm{temperature},{T}_{\infty ,\hspace{0.17em}in}$ $(\xb0C)$ | Inlet | 501.71 | 474.22 |

Outlet | 502.29 | 475.32 | ||

$\mathrm{Flue}\mathrm{gas}\mathrm{temperature},{T}_{\infty ,\hspace{0.17em}ex}$ $(\xb0C)$ | Inlet | 1057.89 | 1066.17 | |

Outlet | 843.19 | 857.56 | ||

Internal pressure p $\left(MPa\right)$ | Inlet | 25.303 | 9.787 | |

Outlet | 25.298 | 9.786 | ||

$\mathrm{Convective}\mathrm{film}\mathrm{coefficient},{h}_{conv}$ ($W/{m}^{2}$) | $\mathrm{Flue}\mathrm{gas},{h}_{conv,\hspace{0.17em}ex}$ | 8.582 | 8.216 | |

$\mathrm{Steam},{h}_{conv,\hspace{0.17em}in}$ | 5436.24 | 1620.95 | ||

Emissivity | $\mathrm{Tube},{\epsilon}_{tube}$ | 0.8 | ||

$\mathrm{Gas},{\epsilon}_{gas}$ | 0.281 | |||

Header | $\mathrm{Steam}\mathrm{temperature},{T}_{\infty ,\hspace{0.17em}in}$$(\xb0C)$ | 596 | 572 | |

Convective film coefficient, ($W/{m}^{2}$) | 2403.8 | 980.54 | ||

Internal pressure, p ($MPa$) | 4.599 | 1.483 |

Part | Num. of Nodes | Num. of Elements |
---|---|---|

Tube | 31,648 | 24,021 |

Header | 24,324 | 17,527 |

**Table 5.**Analysis results at minimum load (${P}_{\mathrm{min}}$ ) of 20% and ramp rate of $3\hspace{0.17em}\%/\mathrm{min}$.

Value | At Header Component | At Tube Component |
---|---|---|

Maximum temperature over time (°C) | 595.52 ° | 527.70 ° |

Minimum temperature over time (°C) | 566.20 ° | 513.34 ° |

Maximum von Mises stress over time (MPa) | 228.09 | 113.61 |

Minimum von Mises stress over time (MPa) | 13.169 | 74.761 |

Maximum strain range | $14.648\times {10}^{-4}$ | $5.175\times {10}^{-4}$ |

Value | ${\mathit{n}}_{\mathit{l}\mathit{a}\mathit{y}\mathit{e}\mathit{r}}$ | ${\mathit{n}}_{\mathit{n}\mathit{o}\mathit{d}\mathit{e}\mathit{s}}$ | ${\mathit{\zeta}}_{\mathit{i}\mathit{n}\mathit{i}\mathit{t}\mathit{a}\mathit{l}}$ |
---|---|---|---|

Max | 2 | 50 | 0.1 |

Min | 1 | 5 | 0.000001 |

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**MDPI and ACS Style**

Heo, J.; Park, M.; Kim, J.-M.; Jang, D.-W.; Han, J.-H.
Effect of Flexible Operation on Residual Life of High-Temperature Components of Power Plants. *Processes* **2023**, *11*, 1679.
https://doi.org/10.3390/pr11061679

**AMA Style**

Heo J, Park M, Kim J-M, Jang D-W, Han J-H.
Effect of Flexible Operation on Residual Life of High-Temperature Components of Power Plants. *Processes*. 2023; 11(6):1679.
https://doi.org/10.3390/pr11061679

**Chicago/Turabian Style**

Heo, Jun, Mingyu Park, Jeong-Myun Kim, Dong-Won Jang, and Ji-Hoon Han.
2023. "Effect of Flexible Operation on Residual Life of High-Temperature Components of Power Plants" *Processes* 11, no. 6: 1679.
https://doi.org/10.3390/pr11061679