1. Introduction
In China, there are almost 98,000 dams with a combined storage capacity of 9.32 × 10
9 m
3 [
1]. To achieve goals like flood control, hydroelectric power, irrigation, and navigation, several large-scale cascade dam systems have been constructed in the Yangtze River, Jinsha, Yellow River, Yalong River, Lancang River, Wujiang, Red River, and Dadu River Basin [
2]. The statistical data reveal that more than 95% of these dams are embankment dams in China. From 1954 to 2013, approximately 3523 dam failure accidents occurred, resulting in fatalities and economic losses [
3,
4,
5]. Among the diverse natural hazards, flooding is the most important risk factor affecting dam breaking.
Flooding is the most disastrous natural hazard for the basin, and floods are transferred to the cascade dam system, like the domino effect [
6]. Therefore, flood risk analysis for cascade dam systems is important. Chen et al. developed a risk-based model for real-time flood control operation of dams under emergency and uncertain conditions [
7]. Due to the properties of the engineering system, Bayesian networks (BNs) are employed to quantify the complex relational dependencies using Bayes’ theorem. BNs are a type of probability graphical model that can accurately predict one event’s probability by combining historical data and expert experience [
8]. Due to the flexible structure and cause–effect inference engine, BNs are a promising tool for risk analysis in complex systems [
9]. BNs have been extensively applied for the analysis of the failure probability of gas pipelines [
10,
11], reliability estimation of system functioning [
12,
13], and risk assessment for reservoirs with respect to water quality and public health [
14].
Application of BNs for flood risk estimation of cascade dam systems is still in the preliminary phase. For instance, Li et al. used a BN and stochastic Monte Carlo to analyze the dam breaking risks of two reservoirs under floods and landslide surge [
15]. During the flood risk analysis, it is crucial to identify the risk factors and continuous breaking failure paths of cascade dam systems. On the basis of prior probability in the original BN model, sensitivity analysis can be used to rank the major risk factors for failure events of the system [
16,
17]. Considering the possible misperceptions of the expert experiment, the risk factors and failure path identified by the original prior probability and conditional probability tables (CPTs) cannot truly reflect the cascade dam system situation. Therefore, the original prior probability and CPTs need to be updated after the analysis of continuous breaking conditions.
During the simulation of the dam breaking process, the prediction accuracy of flood release is important for the breach of artificial or natural dams [
18]. In general, a combination of the hydraulic and the geological method is used to create the dam breaking analysis model; the flood release routing for dam breaking is often ensured by the broad crested weir flow formula [
19]. Several analytical models can be used to simulate the dam breaking process, such as Hydrologic Modeling System-River Analysis System (HEC-RAS) [
20,
21] and MIKE 11 [
22]. Wahl and Zhu et al. pointed out some deficiencies in the former publications, which mainly included the prediction uncertainties of the dam breaking process [
23,
24]. For instance, the empirical parametric models always overvalue the peak outflow. According to Liu et al. the simulated peak outflow is 4600 m
3/s, which is eight times as large as practical discharge [
25]. To solve this problem, Chen et al. developed a simple numerical algorithm to improve the existing dam breaking simulation method, which can avoid iteration increases each time [
26]. The proposed algorithm, DB-IWHR, can be run in Excel 2014. The improvements of the model are able to reduce the sensitivity of the input parameters for dam breaking analysis [
19]. The hydraulic details and lateral enlargement of dam breaking progress have been reported in the former two publications [
19,
26]. According to Zhou et al. and Zhang et al., DB-IWHR 2014 can be used to analyze the risks for Hongshiyan Barrier Lake and Dadu River cascade dam [
27,
28]. Based on the previous studies about the prediction of the dam breaking process, DB-IWHR 2014 was employed to simulate dam breaking progression in this study.
In this study, the failure probability for the cascade dam system was estimated through a BN combined with DB-IWHR 2014. First, the BN model was created to study the logical dependency relationship of the cascade dam system. The prior probability and CPTs of each node were obtained from historical data, computational formulas, and expert experience. In the BNs model, the sensitivity was analyzed to determine the original continuous break failure path. Next, DB-IWHR 2014 was applied to analyze the reasonability of the original failure path, which can avoid the use of subjective factors to some extent. Posteriori continuous break failure paths can be certified by DB-IWHR 2014 under the different sates of each node. After the posteriori paths are determined, the original CPTs are sequentially refined. Finally, based on the redefined CPTs and new evidence, a new BN model was created to calculate the system failure probability to assess the effects of each dam on the system safety. The proposed method was applied to the Bala–Busigou–Shuangjiangkou (BL–BSG–SJK) cascade dam system, which is located in the Dadu River Basin in China. On the basis of the application, three reasonable continuous breaking failure paths were identified, and the cascade dam system failure probabilities of the determined continuous breaking failure paths were calculated by the refined BNs model.
2. Study Area
Dadu River, the largest tributary of the Ming River, is located in Sichuan Province, China, as shown in
Figure 1. The total length of its main stream is 1062 km and its catchment area is 77,400 km
2. Dadu River is geographically positioned between 99°42′ and 103°48′ E and 28°15′ and 33°33′ N within the transitional zone from the southwest area of the Tibetan plateau to the Sichuan catchment. Above the SJK dam, the typical alpine valley region and fast-flowing river have an average bed slope of approximately 6.2‰.
Dadu River has rich water resources. According to the Dadu River hydropower program (Power China Chengdu engineering corporation, 2013), there are 22 dams constructed or under construction in the Dadu River Basin, with a total installed capacity of 2.3 × 107 KW. For a controlled engineering project in Dadu River Basin, the SJK dam is not only in charge of flood control for the upstream Dadu River Basin, but also provides flood control pressure relief for the Three Gorges Dam (TGD) in the south of China. Hence, it is necessary to analyze the safety of SJK. We selected two upstream dams adjacent to SJK so the three dams included the BL–BSG–SJK cascade dam system.
In this paper, the BL–BSG–SJK cascade dam system was selected as a research object. The checking standard of BL–BSG is a 5000-year return period, and the checking standard of SJK is the probable maximum flood (PMF). The main parameters of the three dams are listed in
Table 1. BL, BSG, and SJK are all embankment dams, according to total capacity and maximum dam height. The three dams are classified as Type I according to the rank of the water and hydropower project and flood protection criteria (SL252-2017) in China [
29].
5. Conclusions
Considering the characteristics of a cascade dam system, BNs and DB-IWHR 2014 were combined in this study to analyze the flood risks of a cascade dam system. The method was applied to the BL–BSG–SJK cascade dam system in the Dadu River basin in China.
On the basis of the inference and the information-transmitting functions of BNs, the original sequential breaking path of the cascade dam system was confirmed using sensitivity analysis. However, some errors in the results occur since the original BN model is heavily decided by historical data and experts’ experience. To overcome this limitation, DB-IWHR 2014 and the flood regulation method were applied to simulate the breaking process. After analyzing the breaking simulations, the more reasonable continuous breaking failure paths were determined, such as the first: BL Flood = overtopping, BSG Flood = check flood, and SJK Flood = check flood. BL, BSG, and SJK sequentially break under this situation. The second path is BL Flood = overtopping, BSG Flood = check flood, and SJK Flood = normal flood; BL and BSG break, while SJK is safe under this situation. The third path is BL Flood = overtopping, BSG Flood = normal flood, and SJK Flood = normal/check flood. Here, BL and BSG break, while SJK is safe, which is the same as for path two. Through analyzing the three paths, we found that continuous breaking events occur in both overtopping flood and in check/normal flood of the cascade dam system. For this reason, the CPTs of the original BNs model should be updated by adding the new information. Finally, a new BN model was established to quantify the failure probabilities of the cascade system under the three continuous breaking failure paths.
The subjectivity can be reduced and the reliability of flood risk analysis of cascade dam systems can be enhanced with the combination of BNs and DB-IWHR 2014. The results can provide some suggestions to decision-makers. If the BL dam overtopping happened, according to the return period of natural floods, proper flood control measurements should be provided to avoid a cascade of continuous dam breaks. This method can be easily adapted to other cascade dam systems.