# A New Approach for Assessing Heat Balance State along a Water Transfer Channel during Winter Periods

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## Abstract

**:**

## 1. Introduction

^{3}[3]. Since the start of the operation of this water transfer project in 2014, given the strong coupling of cascaded channels, it has been difficult to keep water levels at all gates along the route being stable. The cross-section of the main canal has a trapezoidal shape, with a bottom width range from 7.0 to 26.5 m. Along the water transfer route, there are no lakes or reservoirs for water detention, indicating the importance of safety operation of this project [4]. The main canal of this water transfer line is spanned from the place located at the north latitude of 33° to the north latitude of 40°. During winter periods, the temperature in northern China such as in Beijing is generally low. As a consequence, the formation of ice cover in the water transfer cannel creates a big ice problem [5].

## 2. Methodology

_{i}represents the extent that the cross-section is covered by ice. When an ice cover is present in a channel, the heat transfer of the water body is closely related to the heat loss of the ice body. The above equations characterize the physical mechanism of the heat transfer of the water body during a winter period [31].

**S**is calculated according to Equation (3):

**X**is the normalized sample matrix. By solving the characteristic equation, m non-negative features λ

_{k}(k = 1, 2, ⋯, m) of the covariance matrix

**S**are obtained and arranged in the order of λ

_{1}> λ

_{2}> ⋯ >λ

_{m}> 0. The corresponding orthogonal unit eigenvector

**µ**

_{k}is solved, and the formula of principal component calculation is as follows:

**Z**

_{k}represents the kth principal component (k ≤ m), and the contribution rate v

_{i}of the ith principal component is calculated using Equation (5):

**Z**= [

_{i}**Z**

_{i}

_{1};

**Z**

_{i}

_{2}], y

_{i}is the corresponding label of

**Z**. The heat gain of water body is labeled as “+1”, and the heat loss of water body is labeled as “−1”. The hyperplane formula for the sample classification is expressed in Equation (6), and the classification decision function is described as Equation (7) [33]:

_{i}**w**= (w

_{1}; w

_{2}; ⋯;w

_{m}) is the weight vector corresponding to

**Z**,

**b**is the displacement term which determines the distance between the hyperplane and the origin. The sign ( ) represents the sign function. The Lagrangian function for solving this problem can be described as Equation (8):

_{1}; α

_{2}; ⋯; α

_{p}) is a Lagrangian operator, α

_{i}> 0. By solving Equation (8) and substituting it into Equations (7) and (9) is obtained:

**Z**

_{i}) and η (

**Z**

_{h}) are the mapping transformation functions of the original space; and k (

**Z**

_{i},

**Z**

_{h}) is the kernel function. When meteorological data are relatively complete, based on the observed thermal-hydrodynamic data about winter ice-water regime of the MR-SNWTP, the heat budget process of water body and the mutual feedback relationship of the related parameters are identified.

## 3. Analysis of Heat Budget of Water Body

**Z**

_{i}and

**Z**

_{h}represent different input characteristics, and γ is the Gaussian kernel bandwidth parameter. By introducing the multiplier ${\alpha}_{i}^{*}$, and the penalty parameter C, a slack variable ε needs to be added to the threshold to reduce the error of the dual problem. The slack variable represents that the error term whose deviation is less than ε is not penalized, we can get:

_{w}

_{−h}represents the historical conditions of water temperature; T

_{a}

_{−s}represents air temperature conditions including daily maximum, minimum, average temperature and cumulative temperature during a period; T

_{t}represents the time-effect factor. Since the variation of water temperature approximately shows a periodic function and it fluctuates around the average annual water temperature, both sin (2πT

_{day}/T

_{year}) and cos (2πT

_{day}/T

_{year}) are selected to reflect the time-effect T

_{t}; where T

_{day}is the date count of the year (note: 1 January is 1, etc.); and T

_{year}is the number of annual dates; δv represents the component gradient of wind speed and flow velocity in the flow direction, which has an important influence on the heat transfer process; Fr is the flow Froude number; P is the average daily pressure; Ra is the solar radiation; Rel is the daily average relative humidity; Rp is the average daily precipitation; N is cloud amount; Hc is the cloud height. These parameters have either direct or indirect impact on the change of water temperature. Results of field observation showed that the changes of water temperature are compatible with the simulations of the collected data. If g > 0, it indicates that the water body absorbs heat and the water temperature rises; if g < 0, water body loses heat and water temperature drops. After standardizing parameters in Equation (10) based on data measured at the typical survey stations, the PCA was used to extract the first principal component and the second principal component as the input of the SVM. The cumulative contribution rate of each principal component output by using the PCA is shown in Table 2.

_{2}C and log

_{2}γare the logarithms of parameters C and γ, respectively. According to the calculation process of Equations (3) and (4), the composition of the principal components includes all the influencing factors, and the contribution of each factor is different. Due to the different features of various algorithms, the combination of parameters C and γ obtained by the optimization also fluctuates within a certain range. It can be seen from Figure 5 that the generated nonlinear partition hyperplane can better distinguish the heat budget of water body. In the prediction model, 63 out of the 89 classified data were predicted correctly, and the accuracy reached at 70.79%. From the number of iterations, the GS method has fewer iterations with a higher optimization efficiency. In view of the cross-validation rate, the GA algorithm performs well in the model training, and the number of the support vectors is less. By observing the execution time of the algorithm, with respect to the operation management of water transfer process, the calculation time consumed using these three algorithms is within the acceptable range, and the time need using the GA algorithm is slightly higher. Furthermore, by selecting the first five principal components in order to adequately reflect the influence of the original correlation factors, the SVM is used to predict the heat balance state of water body. The prediction results are summarized in Table 4.

## 4. Conclusions

- (1)
- By analyzing the factors that influence water heat balance during winter periods, the correlation coefficient matrix of thermal and hydrodynamic characteristic data is studied. By using the PCA method to extract principal components as model input, the correlation between multiple variables was described by a few variables. By inputting data using the machine learning model, it can effectively reduce data dimension and eliminate redundancy, and thus improve the computational efficiency. With the increases in the number of principal components extracted, the prediction accuracy of the model increases accordingly.
- (2)
- Regarding the selection of the SVM parameters, the GS algorithm, the PSO algorithm and the GA algorithm are selected to optimize the parameters, which are applied to identify heat loss or heat gain of water body in channels during winter periods. After the optimization of parameters, the recognition rate of model prediction algorithm is better. By comparing the differences between the algorithms, the GA algorithm is more suitable for the SVM method used for the assessment of water heat balance state.
- (3)
- Aiming at the problem of heat budget of water body in the study channel in winter, considering water temperature change by means of environmental variables, a new approach is proposed to assess the heat balance state by analyzing the observed data in the field. In view of the nonlinear change of water temperature, the RBF kernel function with better performance in classification is selected. Regarding the sample learning, the SVM can quickly and accurately conduct classification. Thus, the SVM can used to effectively solve the identification problem of water heat exchange.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 5.**(

**a**) the PCA-GS-SVM classification; (

**b**) parameter optimization using the GS method; (

**c**) the PCA-PSO-SVM classification; (

**d**) the fitness curve of the PSO algorithm; (

**e**) the PCA-GA-SVM classification; (

**f**) the fitness curve of the GA algorithm.

Parameter | Interpretation | Parameter | Interpretation |
---|---|---|---|

ρ_{w} | mass density of water | c_{p} | specific heat capacity of water |

T_{w} | water temperature | Q_{w} | flow discharge of water |

A_{w} | flow cross-sectional area of the channel | E_{x} | diffusion coefficient |

B_{w} | width of water surface channel cross-section | φ_{wa} | water—air heat flux |

N_{i} | the ratio of the length of the water surface of the cross-section covered by ice cover to the width of the water surface of the same cross-section | φ_{wi} | water—ice cover heat flux |

Component | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|

cumulative contribution rate | 33% | 52% | 68% | 77% | 83% | 89% | 93% |

Parameter | PCA-GS-SVM | PCA-PSO-SVM | PCA-GA-SVM |
---|---|---|---|

Iteration times | 276 | 286 | 291 |

C | 0.11 | 0.10 | 0.13 |

γ | 48.50 | 54.34 | 53.20 |

Execution time (ms) | 83,139 | 83,085 | 86,598 |

Cross-validation rate | 71.83% | 72.02% | 71.46% |

Number of boundary support vectors | 363 | 380 | 349 |

Number of support vectors | 385 | 398 | 374 |

Number of classification predictions | 89 | 89 | 89 |

Correct classification number | 63 | 63 | 63 |

Classification accuracy | 70.79% | 70.79% | 70.79% |

Parameter | PCA-GS-SVM | PCA-PSO-SVM | PCA-GA-SVM |
---|---|---|---|

Iteration times | 3893 | 2146 | 950 |

C | 48.50 | 19.64 | 7.82 |

γ | 0.57 | 1.61 | 1.71 |

Execution time (ms) | 87,288 | 96,072 | 87,401 |

Cross-validation rate | 72.02% | 71.08% | 72.02% |

Number of boundary support vectors | 260 | 244 | 257 |

Number of support vectors | 293 | 286 | 291 |

Number of classification predictions | 89 | 89 | 89 |

Correct classification number | 64 | 65 | 65 |

Classification accuracy | 71.91% | 73.03% | 73.03% |

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**MDPI and ACS Style**

Cheng, T.; Wang, J.; Sui, J.; Zhao, H.; Hao, Z.; Huang, M.; Li, Z.
A New Approach for Assessing Heat Balance State along a Water Transfer Channel during Winter Periods. *Water* **2022**, *14*, 3269.
https://doi.org/10.3390/w14203269

**AMA Style**

Cheng T, Wang J, Sui J, Zhao H, Hao Z, Huang M, Li Z.
A New Approach for Assessing Heat Balance State along a Water Transfer Channel during Winter Periods. *Water*. 2022; 14(20):3269.
https://doi.org/10.3390/w14203269

**Chicago/Turabian Style**

Cheng, Tiejie, Jun Wang, Jueyi Sui, Haijing Zhao, Zejia Hao, Minghai Huang, and Zhicong Li.
2022. "A New Approach for Assessing Heat Balance State along a Water Transfer Channel during Winter Periods" *Water* 14, no. 20: 3269.
https://doi.org/10.3390/w14203269