**5. Results and Accuracy of the Proposed Model**

#### *5.1. A Better Mathematical Model for the Ageing Displacement of Concrete Dams*

To calculate the hydraulic displacement component by the FEM, a three-dimensional FEM model is established using ABAQUS, as shown in Figure 6. The instantaneous elastic modulus of the dam concrete is determined to be 38 GPa, which is an elastic inversion value conducted according to the measured dam displacement during the initial rising period of the upstream reservoir water level, rising from 1800.41 m to 1880 m [44]. The parameters of the foundation rocks are determined by their designed values. The instantaneous elastic modulus *E*<sup>1</sup> and delayed elastic modulus *E*<sup>2</sup> in the Burgers model are determined to be 46.5 GPa and 130.5 GPa. The corresponding viscosity coefficients are 376.2 GPa.d and 20,074.5 GPa.d, respectively. During the simulation period, there are a total of 32 incremental steps in the FEM simulation, with each step resulting ina5m change in water level. Figure 7 shows the relationship between the elastic FEM-calculated radial displacement and the upstream reservoir water depth for PL9-1.

**Figure 6.** FEM model of the Jinping-I super-high arch dam.

**Figure 7.** Relationship between the elastic FEM-calculated radial displacement and the upstream reservoir water depth for PL9-1.

#### *5.2. Calculation of Temperature Displacement by KPCA Method*

A large number of thermometers are embedded in the dam body. From the mechanical point of view, the thermometer measurement data of the concrete and rock foundation of the dam should be selected as the temperature factor. The principal component analysis (PCA) method was used in the previous studies [45,46] to analyse and separate the temperature measurement principal component. Its linear technique, on the other hand, has difficulty reducing large amounts of temperature variables to new uncorrelated variables while minimizing the loss of original information. In this paper, the principal components of thermometers are extracted by the KPCA method. As a result, the temperature components can be fitted more precisely. The advantages of using the measured temperatures of the concrete dam have been mentioned and testified by several articles [47–49]. In order to approve the KPCA's high performance on feature extraction and dimensionality reduction of input temperature dataset, the comparison process with PCA is necessarily proposed. The PCA method is based on the assumption that there was a linear hyperplane. The KPCA method is kernel-based, and the mapping performed by the KPCA method highly relies on the choice of the kernel function. Possible choices for the kernel function are the Linear kernel, Gaussian kernel, Polynomial kernel, Sigmoid kernel, and Laplacian kernel [50,51]. Table 1 shows the performance of the KPCA models based on different kernel functions. According to the results, the KPCA method with the Linear kernel, Polynomial kernel, and Sigmoid kernel achieves nearly the same accuracy rate and maximum contribution rate as the first kernel PC. In contrast, the Gaussian kernel and Laplacian kernel approach the low maximum contribution rate of the first kernel PC. As a result, the Polynomial kernel is chosen to replace the linear projection process. Figure 8 depicts the comparison of the two methods in the extraction result. It can conclude that after using the KPCA method with the polynomial kernel, two PCs are extracted. These two PCs can explain approximately 93.65% of the information in the original temperature dataset. They are considered to represent the thermal effect. The PC1, which has the maximum contribution proportion among all the principal components, explains 73.31% of the information in the original temperature dataset. In contrast, after using the PCA method, four PCs are extracted, and the PC1 explains only 46.87% of the information. As a result of using the KPCA method, a small number of principal components from the thermometers can be extracted and used for the temperature component as follows:

$$\delta\_T = \sum\_{i=1}^{2} b\_i T\_{P\bar{\mathbf{C}}i\prime} \tag{40}$$

where *TPCi* is the *i*th extracted temperature principal component; *bi* is the coefficient of the principal component.


**Table 1.** The performance of the KPCA models based on different kernel functions.

**Figure 8.** The comparison of the PCA method and the KPCA method in the extraction result.

### *5.3. Fitting and Predicting Accuracy of the Proposed Hybrid Model*

To test the accuracy of the new hybrid model, the long monitoring data from 1 October 2013 to 31 December 2018 were divided into the training set, and the predicting set [52]. The first 1560 data in the dataset are used as the training set and the last 354 data are used as the predicting set. The coefficients of the Equation (39) are calculated and shown in Table 2. For the Jinping-I super high arch dam, the design and inversion values of the instantaneous elastic modulus of the dam concrete are 38.0 GPa and 38.5 GPa, respectively. As a result, the elastic hydraulic displacement calculated by FEM should be multiplied by the coefficient *X* of 0.98 (38.0/38.5).

**Table 2.** Coefficients of the new hybrid model for PL9-1.


The model results are tested and compared using several estimation criteria, including the mean absolute error MAE, the mean squared error MSE and the determination coefficient R2. The criteria mentioned above can be calculated as follows:

$$\text{MAE} = \frac{1}{N} \sum\_{i=1}^{N} |\mathcal{G}\_i - \mathcal{Y}\_i| \tag{41}$$

$$\text{RMSE} = \sqrt{\frac{1}{N} \sum\_{i=1}^{N} \left[ \hat{y}\_i - y\_i \right]^2} \tag{42}$$

$$\mathbf{R}^2 = \frac{\left[\sum\_{i=1}^N \left(\mathcal{Y}\_i - \overline{\mathcal{Y}}\right)(y\_i - \overline{y})\right]^2}{\sum\_{i=1}^N \left[\mathcal{Y}\_i - \overline{\mathcal{Y}}\right]^2 \sum\_{i=1}^N \left[y\_i - \overline{y}\right]^2} \tag{43}$$

where *N* is the number of observations; *y*ˆ is the predicted displacement value of the model; *y* is the measured displacement of the concrete dam; *y*ˆ and *y* are the mean value of predicted and measured displacement, respectively. The estimation criteria MAE, MSE, and R2 are displayed in Table 3, respectively for the HST model and the proposed hybrid model.


**Table 3.** The regression model performance comparison of the proposed model and the HST model.

As can be seen in Table 3, the two models realize the satisfactory performance that the correlation coefficients in the training set and predicting set are both higher than 0.95. The new hybrid model reaches deduced error and higher correlation coefficient than the HST model. It is reasonable to conclude that the model developed in this paper has both a clear physical meaning and a high fitting accuracy. As shown in Figure 9, the more significant error occurs when the measured value changes more violently, that is, when the water level changes more violently. However, the new hybrid model provides better stability in the fitting process compared to the HST model when the water level changes drastically.

**Figure 9.** Fitted process lines for measurement point PL9-1 in the NO.9 dam section: (**a**): The new hybrid model; (**b**) The HST hybrid model. Abbreviation: NH: new hybrid model; HST: hydraulicseasonal-time model; HC: hydraulic component; AC: ageing component; TC: temperature component.

The variation laws and values of the thermal displacement obtained by the new hybrid model and the HST model are displayed in Figure 10. The line temperature component (TC) of HST and the line TC of the new hybrid model (NH) are the ambient temperature of the dam and the internal temperature displacement of the dam, respectively, which both follow a periodic pattern. The internal temperature displacement of the dam has a slight lag relative to the ambient temperature displacement of the dam, in line with the law that the internal temperature of concrete has a phase difference relative to the ambient temperature. This phenomenon has the following possible explanations. The external ambient temperature at any given moment affects the change of its temperature field by means of heat conduction inside the dam concrete, and it takes some time for this heat conduction process to be finally completed. At the same time, the continuous change of external temperature causes the internal heat exchange of the dam body, i.e., when the heat input to the dam body in the current period has not yet been transferred to the deep concrete, the heat input to the dam body in the latter period has already begun to affect the layers of concrete, and so is the next. The slow and continuous heat transfer inside the dam body inevitably leads to the superposition of the influence of the external ambient temperature on the internal temperature field of the dam body at different times, thus causing a lag in the internal temperature field of the dam body relative to the external temperature.

**Figure 10.** Radial displacement of hydraulic, thermal and ageing components separated by the HST hybrid model and the new hybrid model. Abbreviation: NH: new hybrid model; HST: hydraulicseasonal-time model; HC: hydraulic component; AC: ageing component; TC: temperature component.

The ageing displacement separated by the model is a combination of increasing and decreasing curves, containing reversible ageing displacement that varies with the main load of the dam in addition to irreversible trend ageing displacement, and the change in ageing displacement slightly lags behind the change in reservoir level at this stage, in line with the hysteresis effect of viscoelastic deformation. In comparison, the ageing component is expressed as *c*1*θ* + *c*2ln*θ* in the HST model, which only contains irreversible deformations. The instantaneous and hysteretic elastic hydraulic deformations are owned to the hydraulic component, which will result in larger hydraulic displacement in the HST model, as can be seen in Figure 10. While the hydraulic displacement, which is determined by the evolution of reservoir water level, has an annual evolution law comparable to the reversible ageing displacement in the new hybrid model. Thus, the development trend of the ageing displacement can be better reflected in the new hybrid model. It contains both reversible and irreversible components.

#### **6. Conclusions and Discussion**

In this paper, we proposed a new hybrid model for a concrete dam in which the temperature displacement and the ageing displacement of the traditional hybrid model are improved, and the validity and interpretability of the model accuracy of this paper can be confirmed through example validation according to the smaller predicting MSE (0.3404) and larger R2 (0.9902), whereas in the traditional HST hybrid model they are 2.2055 and 0.9898, respectively. The following conclusions are drawn:


the original hybrid model can be subsumed into the ageing displacement component, while the period factor was added to the ageing displacement to fit this component. Thus, a better mathematical model for the ageing displacement of concrete dams was established. It can fully reflect the recoverable creep component of the concrete and rock and accurately separate the ageing component from the total displacement.

However, some problems also should be pointed out. Firstly, the effectiveness of the proposed model is only validated in the experimental conditions. It needs to be applied in the actual engineering programs to verify the practicability. Moreover, the time range of the ageing deformation selected in this paper is only from 2013 to 2018, but the ageing deformation of the dam, especially in the stability period, will last for a long time. The change of water level component, temperature component, and ageing component for a longer time need to be discussed.

**Author Contributions:** Conceptualization, M.Y. and C.G.; methodology, X.C.; software, H.G.; validation, M.Y., X.C., H.G. and C.G.; formal analysis, M.Y.; investigation, X.C.; resources, H.G.; data curation, C.G.; writing—original draft preparation, M.Y. and X.C.; writing—review and editing, M.Y., X.C., H.G. and C.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the China Postdoctoral Science Foundation (Grant No. 2022M721668, 2021M701044); Central Public-Interest Scientific Institution Basal Research Fund, NHRI (Y423006, Y423004); the National Natural Science Foundation of China (51739008, U2243223, 51739003, 51779086 and 52079046); Science and technology projects managed by the headquarter of State Grid Corporation (5108-202218280A-2-417-XG); the Fundamental Research Funds for the Central Universities of Hohai (Grant No. B230201011); the Open Fund of Research Center on Levee Safety Disaster Prevention of Ministry of Water Resources under Grant (LSDP202204); Water Conservancy Science and Technology Project of Jiangsu (Grant No. 2022024); the National Natural Science Foundation for Young Scientists of China (Grant No. 51909173); Open fund of the National Dam Safety Research Center (Grant No. CX2020B02); The Fundamental Research Funds for the Central Universities (B210202017); the Jiangsu young science and technological talents support project (TJ-2022-076); the Open Fund of National Dam Safety Research Center (CX2020B02) and the Anhui Natural Sci-ence Foundation grant number (Grant No. 2208085US17).

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** The data presented in this study are available on request from the corresponding author. The data are not publicly available due to project confidentiality.

**Conflicts of Interest:** The authors declare that there are no conflict of interest.
