# Research on Temperature Control Index for High Concrete Dams Based on Information Entropy and Cloud Model from the View of Spatial Field

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## Abstract

**:**

## 1. Introduction

## 2. Temperature Field Characterization Using Information Entropy

#### 2.1. TFE Construction

#### 2.2. Weight Optimization

## 3. Temperature Control Index Construction Based on Cloud Model

#### 3.1. Cloud Model Algorithm

#### 3.2. Temperature Control Index

## 4. Case Study

#### 4.1. In Situ Observation Data

^{9}m

^{3}. The normal and checked water levels are 1418.00 m and 1421.07 m, respectively. The proposed methodology is adopted for the #6 dam section. Thirty-three observation points are installed to monitor temperature variations, as shown in Figure 4. Figure 5 gives the time curves of air temperature that show a periodic change pattern. Taking November 2011 as an example, the average daily temperature is 14.2 °C, the maximum temperature is 30.7 °C, and the minimum temperature is 4.7 °C. The average, maximum, and minimum temperature differences between day and night are 17.4 °C, 22.5 °C, and 7.4 °C, respectively.

#### 4.2. Weight Distribution

#### 4.3. Comparative Analysis

## 5. Conclusions and Suggestions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature and Abbreviations

Nomenclature | |

$i$ | an observation point |

$j$ | an observation date |

${\delta}_{ij}$ | the observation value of the $i\mathrm{th}$ point on the $j\mathrm{th}$ date |

${u}_{ij}$ | a parameter related to the order degree |

$p$ | a parameter related to ${u}_{ij}^{1}$ and ${u}_{ij}^{2}$ |

${u}_{ij}^{1}$ | the order degree of ${\delta}_{ij}$ |

${u}_{ij}^{2}$ | the disorder degree of ${\delta}_{ij}$ |

${f}_{i}\left(\delta \right)$ | the probability density function of ${\delta}_{ij}$ |

${F}_{i}\left(\delta \right)$ | the probability distribution function of ${\delta}_{ij}$ |

${S}_{ij}$ | the temperature entropy of ${\delta}_{ij}$ |

${\omega}_{i}$ | the weight of the $i\mathrm{th}$ point |

${S}_{j}$ | the TFE on the $j\mathrm{th}$ date |

$S$ | a matrix |

$m$ | the total number of observation dates |

$n$ | the total number of observation points |

$\mathit{P}$ | the unit projection direction |

$G\left(i\right)$ | the projection function |

${\mathit{P}}^{\ast}$ | the best projection direction |

$\overline{g}\left(i\right)$ | the mean value of $G\left(i\right)$ |

${S}_{G}$ | a parameter related to PPA |

${Q}_{G}$ | a parameter related to PPA |

${G}^{\ast}\left(i\right)$ | a parameter related to ${\mathit{P}}^{\ast}$ |

$R$ | a parameter related to PPA |

${r}_{ij}$ | the distance between $G\left(i\right)$ and $G\left(j\right)$ |

$f\left(t\right)$ | a unit step function |

$t$ | a parameter related to $f\left(t\right)$ |

$U$ | the domain |

$C$ | a qualitative concept |

$x$ | a variable |

$u$ | the certainty degree of $x$ |

${E}_{x}$ | the expectation of the cloud model |

${E}_{n}$ | the entropy of the cloud model |

${H}_{e}$ | The hyper entropy of the cloud model |

$\mathit{x}$ | a data set |

$\overline{x}$ | the mean value of $\mathit{x}$ |

${\sigma}_{x}$ | the variance of $\mathit{x}$ |

${\mathit{S}}^{\prime}$ | a data set |

$f\left(S\right)$ | the probability density function of ${\mathit{S}}^{\prime}$ |

$F\left(S\right)$ | the distribution function of ${\mathit{S}}^{\prime}$ |

$P$ | the abnormal probability |

$\overline{\mathit{S}}$ | the mean value of ${\mathit{S}}^{\prime}$ |

${\sigma}_{S}$ | the variance of ${\mathit{S}}^{\prime}$ |

$k$ | the size of the data sets $\mathit{x}$ and ${\mathit{S}}^{\prime}$ |

${y}^{u}$ | the upper bounds of cloud droplets |

${y}^{l}$ | the lower bounds of cloud droplets |

$\Delta U$ | a micro-interval in the domain $U$ |

$\Delta D$ | the contribution degree of $\Delta U$ |

${U}_{\Delta U}$ | the certainty degree of $\Delta U$ |

$\alpha $ | the confidence level |

${S}_{m,\alpha}$ | the temperature control index with a confidence level of $\alpha $ |

$\beta $ | a parameter related to $\alpha $ |

Abbreviations | |

PPA | projection pursuit analysis |

TFE | temperature field entropy |

FE | finite element |

RCC | roller compacted concrete |

TE | temperature entropy |

WE | weight entropy |

K-S | Kolmogorov–Smirnov |

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**Figure 2.**Cloud droplet distributions with different mathematical features: (

**a**) ${E}_{x}=0$, ${E}_{n}=1$, and ${H}_{e}=0.05$; (

**b**) ${E}_{x}=1$, ${E}_{n}=1$, and ${H}_{e}=0.05$; (

**c**) ${E}_{x}=0$, ${E}_{n}=0.5$; and ${H}_{e}=0.05$; and (

**d**) ${E}_{x}=0$, ${E}_{n}=1$, and ${H}_{e}=0.1$.

**Figure 9.**The established FE models. (

**a**) The dam body and foundation, and (

**b**) the cooling water pipe.

**Figure 10.**Simulation results of temperature fields: (

**a**) pouring elevation at 1311.00 m; (

**b**) pouring elevation at 1360.00 m; (

**c**) pouring elevation at 1422.50 m; and (

**d**) 4 years after completion.

$\mathbf{Domain}\Delta \mathit{U}$ | $\mathbf{Contribution}\mathbf{Degree}\Delta \mathit{D}$ | $\mathbf{Value}\mathbf{of}\mathit{\beta}$ |
---|---|---|

$\left[{E}_{x}-0.67{E}_{k},{E}_{x}+0.67{E}_{k}\right]$ | 50.0% | 0.67 |

$\left[{E}_{x}-{E}_{k},{E}_{x}+{E}_{k}\right]$ | 68.3% | 1.00 |

$\left[{E}_{x}-1.96{E}_{k},{E}_{x}+1.96{E}_{k}\right]$ | 95.0% | 1.96 |

$\left[{E}_{x}-2{E}_{k},{E}_{x}+2{E}_{k}\right]$ | 95.5% | 2.00 |

$\left[{E}_{x}-2.58{E}_{k},{E}_{x}+2.58{E}_{k}\right]$ | 99.0% | 2.58 |

$\left[{E}_{x}-3{E}_{k},{E}_{x}+3{E}_{k}\right]$ | 99.7% | 3.00 |

Observation Point | Point Number | Maximum Value/ Date (mm-dd-yyyy) | Minimal Value/ Date (mm-dd-yyyy) | Maximum Annual Variation/Year (yyyy) |
---|---|---|---|---|

A6-T-06 | 1 | 31.1/09-11-2007 | 17.0/08-30-2007 | 14.2/2007 |

A6-T-07 | 2 | 31.9/10-22-2008 | 20.0/01-30-2012 | 10.8/2008 |

A6-T-08 | 3 | 34.0/09-05-2008 | 23.0/02-01-2010 | 10.9/2008 |

A6-T-09 | 4 | 33.0/09-10-2008 | 17.9/03-11-2008 | 15.1/2008 |

A6-T-10 | 5 | 30.8/05-12-2008 | 17.4/03-11-2008 | 13.5/2008 |

A6-T-11 | 6 | 34.6/10-07-2008 | 19.9/02-22-2012 | 11.3/2008 |

A6-T-12 | 7 | 40.6/05-19-2008 | 22.2/03-22-2008 | 18.4/2008 |

A6-T-13 | 8 | 33.8/09-02-2008 | 18.1/02-01-2010 | 15.0/2008 |

A6-T-14 | 9 | 30.4/06-06-2008 | 18.1/08-19-2010 | 10.3/2008 |

A6-T-15 | 10 | 30.5/07-04-2008 | 17.4/01-03-2011 | 10.3/2008 |

A6-T-16 | 11 | 32.3/05-23-2008 | 19.9/08-11-2011 | 11.8/2008 |

A6-T-17 | 12 | 31.7/05-16-2008 | 18.5/03-18-2011 | 11.8/2008 |

A6-T-18 | 13 | 31.5/05-16-2008 | 20.7/12-28-2011 | 10.9/2008 |

A6-T-19 | 14 | 31.7/05-16-2008 | 19.6/04-19-2008 | 12.2/2008 |

A6-T-20 | 15 | 37.7/05-17-2008 | 14.7/12-28-2011 | 18.9/2008 |

A6-T-21 | 16 | 37.1/08-15-2008 | 17.9/05-05-2008 | 19.2/2008 |

A6-T-22 | 17 | 35.5/08-29-2008 | 18.1/05-05-2008 | 17.4/2008 |

A6-T-23 | 18 | 32.1/06-06-2008 | 18.6/05-05-2008 | 13.5/2008 |

A6-T-24 | 19 | 38.0/07-24-2008 | 19.0/07-03-2008 | 19.0/2008 |

A6-T-25 | 20 | 34.2/09-05-2008 | 19.0/07-03-2008- | 15.3/2008 |

A6-T-26 | 21 | 30.5/02-11-2010 | 20.0/07-03-2008 | 10.5/2008 |

A6-T-27 | 22 | 31.7/01-14-2009 | 12.4/03-10-2011 | 10.6/2008 |

A6-T-28 | 23 | 28.5/12-17-2010 | 18.3/12-03-2008 | 5.9/2008 |

A6-T-29 | 24 | 28.4/08-11-2010 | 19.0/12-03-2008 | 4.9/2008 |

A6-T-30 | 25 | 40.2/11-05-2009 | 18.1/12-03-2008 | 14.3/2009 |

A6-T-31 | 26 | 32.0/12-17-2010 | 18.6/12-03-2008 | 5.8/2010 |

A6-T-32 | 27 | 31.8/12-17-2010 | 18.0/12-03-2008 | 6.3/2008 |

A6-T-33 | 28 | 27.9/12-17-2010 | 17.8/12-03-2008 | 6.9/2008 |

A6-T-34 | 29 | 30.5/08-26-2010 | 14.0/12-04-2008 | 7.8/2010 |

A6-T-35 | 30 | 49.4/05-28-2009 | 17.3/07-22-2011 | 21.2/2009 |

A6-T-36 | 31 | 39.6/05-28-2009 | 17.5/01-20-2011 | 20.6/2009 |

A6-T-37 | 32 | 48.5/10-04-2009 | 19.0/04-06-2011 | 22.0/2009 |

A6-T-38 | 33 | 45.3/09-28-2009 | 22.4/02-10-2012 | 16.5/2009 |

Year | 2008 | 2009 | 2010 | 2011 | 2012 | 2013 |

Minimum value of TFEs | 0.723 | 0.968 | 1.126 | 1.034 | 1.170 | 1.160 |

Probability Distribution | Confidence Level |
---|---|

Lognormal distribution | 0.23 |

Normal distribution | 0.15 |

Uniform distribution | 0.64 |

Triangular distribution | 0.33 |

Exponential distribution | 0.41 |

$\gamma $ distribution | 0.87 |

$\beta $ distribution | 0.78 |

Method | Temperature Control Index | |
---|---|---|

Typical small probability method | $\alpha =$5% | $\alpha =$1% |

1.0431 | 1.0031 | |

Cloud model | $\alpha =$5% | $\alpha =$1% |

1.0418 | 1.0107 |

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## Share and Cite

**MDPI and ACS Style**

Yang, G.; Sun, J.; Zhang, J.; Niu, J.; Luan, B.; Huang, Z.; Zhao, A.
Research on Temperature Control Index for High Concrete Dams Based on Information Entropy and Cloud Model from the View of Spatial Field. *Water* **2023**, *15*, 4023.
https://doi.org/10.3390/w15224023

**AMA Style**

Yang G, Sun J, Zhang J, Niu J, Luan B, Huang Z, Zhao A.
Research on Temperature Control Index for High Concrete Dams Based on Information Entropy and Cloud Model from the View of Spatial Field. *Water*. 2023; 15(22):4023.
https://doi.org/10.3390/w15224023

**Chicago/Turabian Style**

Yang, Guang, Jin Sun, Jianwei Zhang, Jingtai Niu, Bowen Luan, Zhendong Huang, and Ahui Zhao.
2023. "Research on Temperature Control Index for High Concrete Dams Based on Information Entropy and Cloud Model from the View of Spatial Field" *Water* 15, no. 22: 4023.
https://doi.org/10.3390/w15224023