**1. Introduction**

Since the increasing requirement and utilization of water resources in China, a large amount of concrete dams have been constructed and contributed significantly to economic development [1,2]. The safety issues and the risks of dam failure are always unavoidable due to uncertainties in geology, hydrology, design, construction, and operational management. The dam failure, which resulted in major casualties, is a tragic lesson worldwide. Hence, dam safety has always been a high priority for governments and relevant authorities and has played an important role in preventing dam failures.

Displacement, stress, strain, and cracks [3–9] et al. are essential for dam health monitoring and operational instruction. As the most intuitive reflection of the integrity of concrete dams, displacement is frequently adopted for predicting the behaviour of the dam. Among the models that were used to fit and predict dam displacements, there are two main categories. The first one is based on deterministic functions, physical extrapolation methods, and multiple linear regression. For instance, statistical, deterministic, and hybrid models are the commonly used monitoring models in engineering. Chen et al. [10] adopted a semi-parametric statistical model, which has more imitative effect and explanatory than the parametric statistical model. Shang [11] analysed the key dam section of the roller compacted concrete (RCC) gravity dam with statistical modeling methods. Hu et al. [12] proposed a statistical hydrostatic-thermal-crack-time model to deal with the influential horizontal cracks in concrete arch dams. Another type is based on machine learning models.

**Citation:** Gu, C.; Cui, X.; Gu, H.; Yang, M. A New Hybrid Monitoring Model for Displacement of the Concrete Dam. *Sustainability* **2023**, *15*, 9609. https://doi.org/10.3390/ su15129609

Academic Editor: José Ignacio Alvarez

Received: 15 December 2022 Revised: 9 January 2023 Accepted: 27 March 2023 Published: 15 June 2023

**Copyright:** © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Kang et al. [13] presented a dam health monitoring model based on kernel extreme learning machines. Chen et al. [14] combined the advantages of extreme learning machines and elastic networks for predicting dam deformation. Su et al. [15] used rough set theory and a support vector machine to build the early-warning models of dam safety. The monitoring models based on machine learning techniques; however, do not take the structural characteristics of the concrete dams into account. Furthermore, they lack direct mathematical expressions, causing them to only be useful for making predictions rather than providing a causal interpretation of dam deformation, such as statistical models [16].

The causes of ageing components in dams are complex, including cracking [17], alkali-aggregate reactions (AAR) [18], dissolution [19], etc. It can also be interpreted by the compression deformation of the rock foundation structure, as well as self-generated volume deformation and irreversible displacement due to cracks in the dam. Therefore, the ageing displacement is relatively difficult to give the mathematical expression directly according to the actual situation. Among a variety of HST-based models, the ageing component is expressed as the exponential function or the combination of the linear and logarithmic functions. However, the formation mechanism of the ageing displacement of the concrete dam caused by the theological properties of the dam material is extremely complicated. Therefore, the HST model based on a simple exponential function or a logarithmic function may not appropriately represent the relationships between the influencing factors and the dam displacement. The development law of the ageing component is studied and expressed as some function of time, which can then be separated in some way based on the relationship between the ageing displacement and the dam load to optimize the traditional HST model. For instance, Li et al. presented a hydrostatic-seasonal-state (HSS) model for rationally obtaining and accurately interpreting ageing displacement from total displacement monitoring data [20]. Hu and Wu [12] presented a statistical hydrostaticthermal-crack-time model and applied it to assess the impact of the large-scale horizontal crack on the Chencun dam's downstream face. To explain the abnormal displacement behaviour of the Jinping-I arch dam, Wang et al. [21] introduced a hysteretic hydraulic component into the HST model, transforming the traditional HST model into an improved HHST model. The ageing component reflects a combination of reversible and irreversible deformation of the dam body and rock foundation. According to the changing pattern of the displacement and the periodic regulation of the reservoir water level for the concrete dams, Gu and Wu [22] concluded that a combination of the exponential and periodic harmonic functions, which are used for the irreversible and reversible ageing displacements [23], should be used to represent the ageing displacement according to the periodic regulation of the reservoir water level for most dams.

The temperature displacement component is the displacement due to temperature changes in the dam body and rock foundation of the dam; thermometer measurements of the dam body and rock foundation of the dam should be selected as a factor. The traditional hybrid model expressed the temperature component in the statistical mode and then was optimally fitted to the measured values. In recent years, the thermal components have been extracted or considered in dam displacement prediction by various methods [24]. Mata et al. conducted the Short Time Fourier Transform analysis of the residuals to determine the impact of the daily air temperature variation on the displacement of concrete dams [25]. Kang et al. considered that the temperature varies over time and substituted harmonic sinusoidal functions to the long-term air temperature for thermal effect simulation [26]. Chen et al. [27] adopted kernel principle component analysis (KPCA) to extract the temperature variables. Among a significant number of the thermometers buried in the dam sections, principal component extraction is an important process. Additionally, the KPCA method used in the paper (20) validates that it assists to reduce the dimensionality of the thermal measurement results by minimizing the loss of the original information.

The main idea of this paper is to propose a new hybrid model. For the extraction of the ageing component, a better mathematical model for the ageing displacement of concrete dams is proposed. For the temperature component, the KPCA method is used to extract the inherent characteristics of the thermometer measurement data to substitute the periodic harmonic factors. Then, a multiple linear regression (MLR) model is developed to fit the measured displacement to validate the reasonableness of the extracted ageing component and temperature component. In this paper, the MLR model is established based on Matlab. The accuracy of the new hybrid model and the extracted components are compared with the components in the HST model. The displacement components of the concrete dam can be more easily understood with the help of the suggested model.

The research significance is for dam safety based on the monitoring model of displacement messages, which can fully reflect the recoverable creep component of the concrete and rock and accurately separate the ageing component from the total displacement.

#### **2. Model Establishment**

The factors influencing dam displacement can be broadly summarized as water level, temperature, and time. The traditional HST hybrid model can be generally interpreted as the following equation:

$$\delta = \delta\_H + \delta\_T + \delta\_\theta = X\delta\_H' + \sum\_{i=1}^2 \left[ b\_{1i}\sin\frac{2\pi it}{365} + b\_{2i}\cos\frac{2\pi it}{365} \right] + c\_1\theta + c\_2\ln\theta,\tag{1}$$

where *δH*, *δT,* and *δθ* represent the hydraulic displacement component, temperature displacement component, and ageing displacement component, respectively; *δ <sup>H</sup>* is the FEMcalculated hydraulic component; *X* is the adjustment coefficient; *H* is the water depth of the upstream reservoir on the same monitoring day; *t* is the number of cumulative days since the initial monitoring day; *θ* is equal to *t*/100.
