Research on Downstream Safety Risk Warning Model for Small Reservoirs Based on Granger Probabilistic Radial Basis Function Neural Network

Abstract

Early warning of safety risks downstream of small reservoirs is directly related to the safety of people’s lives and property and the economic and social development of the region. The lack of data and low collaboration in downstream safety management of small reservoirs makes the existing safety risk warning methods for small reservoirs no longer fully applicable. The data from flood control and drought relief departments, small reservoir operation and management departments, etc., are used comprehensively. A machine learning model suitable for a large number of samples, a small amount of data, and the condition of incomplete information is applied and innovated, and from the holistic perspective of ‘upstream reservoir—downstream region’, the safety risk factors of the upstream reservoir are identified with the help of the Granger causality test. The risk losses of the disaster behavior are predicted with the three-dimensional $k~\epsilon$ two-equation model coupled with the VOF (Volume of fluid) method and the neural network model. The safety risk dynamics prediction, the prediction of the disaster-causing environment, and the prediction of the risk losses are integrated to construct the early warning method of the downstream safety risk of small reservoirs, and the simulation effect is verified with the example of the J Reservoir. The results show that the model can clarify the causal relationships and time lag dependencies between hydro-meteorological factors and the water level of small reservoirs, and calculate the inundation depth, inundation range, and flood velocity downstream of small reservoirs. The downstream safety warning model of small reservoirs constructed in this article can effectively integrate upstream and downstream information, further improve the timeliness and accuracy of warning, and provide a reference for downstream safety risk management of small reservoirs.

1. Introduction

The downstream safety of small reservoirs is affected by multiple factors such as reservoir engineering and scheduling, flood disasters, topography and geomorphology, and socio-economic activities, and the mechanism of safety risk formation is complicated. Small reservoirs are the primary focal points of disaster in mountainous areas prone to flooding within limited watershed areas, and have a significant impact on the downstream safety of the watershed [1]. Unlike large- and medium-sized reservoirs, small reservoirs in flash flood-prone areas often have different degrees of safety hazards due to special topography, construction standards and operation and maintenance, and a lack of corresponding operation and safety monitoring systems, which has a great impact on downstream safety, and makes it difficult to assess and conduct early warning. If the reservoirs are damaged, it will have a serious impact on the downstream safety [2]. Existing reservoir safety risk-related research methods are mostly for a single large-scale project based on a large number of monitoring equipment data, using big data analysis methods to carry out research on monitoring and early warning of safety risks in engineering. The downstream area of small reservoirs is wide, large, and with a relative lack of data, the downstream safety early warning research body is usually the relevant government departments, and the scope of the study involves a large number of small reservoirs in the region. Therefore, taking the upstream reservoir and downstream as the overall research object, exploring the downstream safety early warning method of small reservoirs is of utmost importance for safeguarding lives and protecting properties of the downstream people and reducing disaster losses.

Currently, scholars’ studies on downstream safety early warning of small reservoirs are divided into mechanism-based methods and data-driven early warning methods [3,4,5]. The former is mainly based on meteorological and hydrological hydrodynamic models to construct the mechanism model of flood formation and evolution. It can be subdivided into different natural environments, consideration of pre-conditions, and the introduction of new methods for critical rainfall research. Firstly, the study of critical rainfall in different natural environments mainly focuses on special natural geographic environments such as remote areas, data-poor areas, semi-arid areas, arid areas, and so on [6,7]. By studying the occurrence patterns of flash floods and determining the essential precipitation threshold for flash floods, this approach rests upon the statistical analysis of rainfall and flood data [8], based on which a critical rainfall calculation model and warning indicators for flash flood disasters are established [9,10]. Secondly, the critical rainfall measurement method considering the previous climate and soil water content can be based on the saturation of soil water content, the maximum rainfall 24 h before the occurrence of flash floods, to construct the critical dynamic rainfall warning model [11]. In this stage, some scholars introduce new methods and theories for critical rainfall estimation, such as the fuzzy identification method [12], distribution hydrological model [13], flood modulus [14], etc., to study the rainfall pattern with the calculation of flash flood early warning rainfall indicators. Therefore, more studies have coupled information technology with mechanistic models, introduced meteorological system data or constructed mechanistic warning models based on GIS systems to improve prediction accuracy [15,16].

The latter is mainly based on the principle of machine learning, the flow, water level, and other hydrological elements as a mathematical model of the training object, which can be used in view of the lack of meteorological observation data of the river cross-section, and needs to have a more sufficient water level or flow and other observation data. Once the parameters of the network have been determined, the model can be used to forecast the flooding process in real time and to issue warnings in accordance with the relevant standards [17]. The methodology allows the identification of the factors influencing flood hazards. Jodar [18] investigated the influence of land utilization patterns on the risk of mountain floods within the canyon basin of southeastern Spain, using GIS and SWAT models for simulation and comparison with the actual situation, and the results showed that the risk of flash floods in the study basin is increasing with the change in land use, especially with the increase in large cities. Satriagasa [19] predicted the future climate change for the Upper South River Basin in Northern Thailand using the last 30 years’ data and three GCMs and assessed the influence of climate change on future runoff and flooding in the area using SWAT and HEC-RAS models. Furthermore, machine learning and artificial intelligence methods are commonly utilized in data-driven flood warning approaches and used to study flood hazard distribution from a spatial scale. Costache [20] combined multi-attribute decision-making methods, machine learning, and multivariate statistical methods to construct flash flood-prone area identification methods. Panahi [21] used historical data of flooding in Golestan province, Iran, and geographic feature data based on CNN and RNN neural network models to construct a probabilistic prediction model and spatial distribution map of flood risk. Bui [22] combined PSO with an extreme learning machine to construct a spatial prediction model of flash flood risk based on 12 flash flood indicators and 654 floods as data in the northwestern region of Vietnam. Xu [23] constructed an XAJ model and a two-dimensional model based on remotely sensed hydrological data simulating flood events to support watershed flood management. Data-driven methods are also applied to peak flood prediction; e.g., Bahramian [24] used an event-based flood model to construct a short-term peak flood prediction model based on a soil moisture estimation method, and Al-Amri [25] simulated the reasonable model parameters by ternary lognormal distribution and estimated the peak flow and runoff better and quantified the uncertainty of the flood peak by Monte Carlo procedure. The uncertainty of flood peaks was quantified by a Monte Carlo procedure.

Upstream and downstream information is used to conduct systematic research on reservoirs and their downstream risks, and mostly uses meteorological, remote sensing, and other data to conduct research on reservoir safety, flood warning, and other aspects. Existing research has achieved relatively rich results in reservoir flood early warning, and the existing flash flood early warning methods are mostly based on meteorological and hydrological data used to carry out the construction of downstream safety early warning indicators, but the research on safety early warning methods integrating engineering, meteorological, hydrological, and downstream economic and social multi-source data are relatively small. In addition, coping with the non-uniform and non-linear characteristics of the water level when calculating based on the linear theoretical method is difficult, so the extrapolated results deviate from the actual value of the water level. And the projection method of water level has a large lag, which makes it difficult to realize early warning in the face of a sudden increase in the water level brought about by short-term heavy rainfall. On the other hand, the above methods mainly focus on the hydrological process when conducting forecasting and early warning, and less so consider the impact or risk of the reservoir project on the downstream safety, and it is difficult to comprehensively assess the downstream safety risk without considering the impact of the upstream reservoir. With the gradual improvement in monitoring data of small reservoirs and the continuous improvement in economic and social data, the data make a significant contribution to the scientific forecasting of the likelihood and scale of downstream disaster risk occurrences, as well as the extent of downstream disaster losses. For this reason, this study will be oriented to small reservoirs with certain inspection data, integrate multi-source data information to carry out downstream safety early warning model construction, and introduce upstream reservoir safety assessment information into downstream safety risk early warning by adding the dynamic prediction of reservoir safety risk in the early warning model, so as to realize upstream and downstream linkage early warning.

2. Research Ideas

Aiming at the comprehensive and synergistic early warning problem of the downstream safety risk of small reservoirs, the safety status information of small reservoirs is introduced into the flash flood early warning model, and the downstream safety risk early warning model is constructed by combining the environment factors affecting the safety of small reservoirs and the risk of downstream loss. The idea of model construction is shown in Figure 1.

(1) The Granger causality test is used to identify small reservoir warning factors. Accurate water level prediction data for small reservoirs is required for downstream safety risk warnings. The factors affecting the water level of small reservoirs are manifold, involving rainfall, evaporation, topography, subsurface, and meteorological conditions. When selecting the factors affecting the water level of small reservoirs, the characteristics of incoming floods should be fully considered, and data indicators that are directly related to the formation of floods and easily obtainable should be chosen simultaneously. The Granger causality test method [26] is applied to investigate the causal relationship between influencing factors and the water level of small-scale reservoirs and determine the time lag relationships between the water level and each effective influencing factor. Through the Granger causality test, the influencing factors are selected to determine the effective ones, which are imported into the prediction model as input parameters to further enhance the precision of the forecast for the water level and safety risk of small reservoirs.

(2) A BP neural network can be utilized to construct a small reservoir water level prediction model. The effective influencing factors obtained from screening are input into the BP neural network to achieve the prediction of the reservoir water level. The three-layer BP neural network utilizes the input layer to transmit influencing factor data by means of weight processing into the hidden layer, which, after processing by the hidden layer, is then transmitted to the output layer. During the neural network training phase, the error between the predicted water level for the training samples and the actual water level is calculated, and the error is propagated backward from the output layer through each hidden layer to the input layer, and the connection weights of each layer are gradually corrected. With this repeated error backward propagation correction, the final training of the small reservoir water level values will be constrained to the predefined error range. The neural network is capable of accurately representing the intricate nonlinear mapping between input and output values. It has been successfully employed in simulating the relationship between rainfall and runoff as well as predicting river water levels. The method will be used for water level prediction in small reservoirs to provide safety environment information for downstream safety warnings, drawing on related research.

(3) The $k~\epsilon$ two-equation model of the VOF method is used to predict the risk loss of the abnormal state of small reservoirs at a certain water level. Based on the predicted water level data of small reservoirs and using the coupled VOF $k~\epsilon$ two-equation model to simulate the evolution of the dam failure flood, the depth of inundation, the inundation range, and the flood flow rate of the downstream of the small reservoirs after dam failure at a certain water level are obtained. The $k~\epsilon$ two-equation model of the VOF method is a frequently employed approach for the analysis of small reservoirs, and it is a common approach for the analysis of small reservoirs, which is used for predicting the downstream safety loss and for the early warning model to provide prediction information about the risk loss.

(4) A probabilistic neural network model is used to construct a safety risk early warning model by integrating the predicted value of the downstream safety risk dynamics of small reservoirs, the forecasted value of the water level, as well as the forecasted value of the risk loss. Presently, back propagation (BP) and radial basis function (RBF) networks are extensively employed in the domain of robust classification prediction. Nevertheless, during network training, they often encounter susceptibility to local minimum values. The principal merits of probabilistic neural networks (PNN) lie in the fact that training necessitates only a single pass, and the decision surfaces are assured to converge towards the Bayes-optimal decision boundaries as the quantity of training samples increases. Li et al. [27] confirmed that probabilistic neural network models have significant advantages over vector quantized LVQ in terms of network structure learning speed and prediction accuracy, as well as self-organizing competing neural networks SOM with BP and Elman detection effects. The input layer of the probabilistic neural network completes the preprocessing of the risk level, water level, and risk loss of the input samples, and converts the input sample space into the data space; the pattern layer calculates the distance from the input vector to the weight vector; the superimposition layer obtains the output vector by applying a non-linear mapping with radial basis functions, which represents the predicted safety warning level; and the competitive output layer first obtains the weighted sum vector and then determines the network’s final output based on the maximum value in the obtained weighted sum vector, enabling the prediction of security risks downstream of small reservoirs.

3. Research Design

3.1. Analysis of Safety Risk Warning Factors Based on Granger Causality Test

3.1.1. Factor Identification Method Based on Granger Causality Test

It has been shown that there is a certain correlation between reservoir water level and meteorological and hydrological data [13], but it is difficult to determine whether the changes in water level can be attributed to these data. If data that are not causally related to water level changes are incorrectly imported into the prediction model, the accuracy of the prediction results will be reduced [28]. So a method is required to precisely gauge the alterations within a specific dataset as the reason for triggering the change in the reservoir water level. To this end, the Granger causality test was introduced.

The notion of an information set was introduced by Clive W.J. Granger in 1969 [29], with the aim of describing the temporal causal association amongst variables; i.e., if the historical data of one variable helps to predict the future value of another variable, then there is a Granger causal relationship between these two variables. This concept has been developed and refined over the years and has evolved into a crucial method for forecasting causal relationships. The core of the Granger test involves establishing a Vector Auto Regression (VAR) model. To conduct a two-variable Granger causality test, at least one of the following two conditions needs to be fulfilled: firstly, both variables are smooth series; secondly, the two variables are characterized by a cointegration relationship [30]. If both prerequisites are unmet, the Granger causality test’s conclusions become unreliable. Moreover, in the case of establishing that variable A acts as the Granger causal factor for variable B, two critical conditions must be satisfied. Firstly, A can enhance the predictability of B, meaning that in an autoregressive model of B based on its past values, the inclusion of past values of A as independent variables should substantially improve the model’s explanatory power. Secondly, B should not have a significant predictive effect on A. When variables A and B exhibit mutual predictability, there may exist one or more variables that serve as common causes for changes in both A and B. Hence, when applying the Granger test to identify factors influencing reservoir water levels, it is imperative to conduct a thorough analysis, including assessments of smoothness, cointegration, and verification of the two aforementioned conditions for each series.

3.1.2. Analysis of Early Warning Factors for Downstream Safety Risks in Small Reservoirs

In order to accurately predict the water level value in small reservoirs, firstly, a series of influencing factors affecting the water level in reservoirs are selected, and secondly, these influencing factors are screened and effective influencing factors are obtained by the Granger causality test, and the effective influencing factors are imported into the prediction model as the input parameters to achieve an accurate water level prediction. The influencing factors of incoming flood flow and water level are multifaceted, involving hydrometeorology, geomorphology, and other fields, including rainfall, evaporation, topography, and various subsurface and meteorological conditions [31]. The influencing factors affecting the water level of small reservoirs should be selected with full consideration of the characteristics of the incoming flood itself, and simultaneously, the data directly associated with flood formation and easy to obtain should be selected as the influencing factors for predicting the water level of small reservoirs.

During the study, complete data from 207 rainfall and meteorological stations near the small reservoirs are collected. Among them, there are 201 small reservoirs with rainfall stations within 10 km upstream and 157 small reservoirs with meteorological stations within a 10 km radius. Taking into consideration the above conditions and the information that the monitoring stations can actually provide, the rainfall $X\left(t\right)$, barometric pressure $H\left(t\right)$, and relative humidity $Z\left(t\right)$ from the stations over a specified period have been chosen as the variables affecting the time series prediction of the respective reservoir’s water level, and the effective influencing factors with prediction ability for the water level are screened out on this basis.

Assuming that $A$, $B$, and $C$ are all influential factors capable of effectively forecasting the water levels of small reservoirs $Y$, the following described relationship captures the non-linear functional connection between the water level prediction indicator $Y$ at time $t$ and the impactful factors $A$, $B$, and $C$:

$$Y(t)=f\left[A(t-p_a),B(t-p_b),C(t-p_c)\right]$$

(1)

In the equation, $p_a$, $p_b$, and $p_c$ are the lag times of each effective influencing factor on the water level, which can be obtained from the analysis results of the Granger test. The nonlinear function $Y\left(t\right)$ can be fitted using neural network training functions, and the predicted water level of small reservoirs can be achieved through the fitted functions [32].

3.2. Construction of Downstream Safety Early Warning Model for Small Reservoirs Based on Neural Network

3.2.1. Construction of BP Neural Network for Reservoir Level

The back propagation (BP) neural network represents a feedforward neural network with multiple layers [33]. Among them, the hidden layer is composed of non-linear transformation units, displaying robust capabilities in nonlinear mapping. This proficiency drives the application of a three-layer BP neural network in forecasting the water levels of small reservoirs in this article, which is then combined with the predicted value to analyze the downstream risk of small reservoirs.

The three-layer BP neural network includes an input layer, a hidden layer, and an output layer within its topological structure. Neurons in adjacent layers exhibit complete interconnections, while neurons within the same layer are devoid of connections. The input layer conveys the influencing elements related to small reservoir water levels, as derived in the preceding section, after being subjected to weight-based processing, the information is transmitted to the hidden layer, where it is further refined before being relayed to the output layer. The weights signify the interconnectedness of diverse layers. Throughout the training phase of the neural network, data from the training samples sequentially traverse the neurons in the input layer and progress towards the hidden layer and the output layer. In the output layer, data are fine-tuned by modifying the connection weights between layers with the aim of minimizing the discrepancy between the forecasted water level data for small reservoirs and the actual water level data derived from the training samples. Subsequently, this error is retrogressively disseminated from the output layer to the corresponding hidden layers, thereby step-by-step adjusting the linking weights for each layer. By repeating the error backpropagation correction, the final trained water level prediction value will be constrained within the established error range.

The effective influencing factor is viewed as the input parameter for the neural network configuration, the water level value of each reservoir is taken as the output parameter, and the hidden layer neurons within the implemented BP neural network employ the S-type (Tan-Sigmoid) function as their activation function, characterized by the following functional expression:

$$f(x)=\frac{2}{1+{{\displaystyle e}}^{-2x}}-1$$

(2)

The output layer neuron employs a strictly linear function as its activation mechanism function, as demonstrated in Equation (3).

$$f(x)=x$$

(3)

Assuming that $h$ represents the quantity of neurons within the concealed layer, $m$ is the quantity of neurons within the input layer, $n$ indicates the quantity of neurons within the output layer, and $a$ is a consecutive ranging from 1 to 10. When working with substantial training data, an adjustment is required for the number of hidden layer neurons due to the application of a three-layer BP neural network. The trial calculations can be conducted based on Equation (4) to determine the number of hidden layer neurons.

$$h=\sqrt{m+n}+a$$

(4)

Assign ${{\displaystyle X}}_i(i=1,2,\cdots ,m)$ as the value for each neuron in the input layer, ${{\displaystyle Y}}_j(j=1,2,\cdots ,h)$ as the value for each neuron in the hidden layer, and ${{\displaystyle O}}_k(k=1,2,\cdots ,n)$ as the value for each neuron in the output layer. Designate ${{\displaystyle W}}_{ij}$ as the synaptic weight linking the input layer and the hidden layer, and ${{\displaystyle V}}_{jk}$ as the synaptic weight linking the input layer and the hidden layer. Set ${{\displaystyle \lambda }}_j$ and ${{\displaystyle \lambda }}_k$ as the thresholds for the hidden layer and the output layer, respectively. Consequently, the formula for computing the values of neurons in the hidden and output layers of the BP neural network can be derived as follows:

$${{\displaystyle Y}}_j=f({\displaystyle \sum _{i=1}^m{{\displaystyle W}}_{ij}{{\displaystyle X}}_i-{{\displaystyle \lambda }}_j})$$

(5)

$${{\displaystyle O}}_k=f({\displaystyle \sum _{j=1}^h{{\displaystyle V}}_{jk}-{{\displaystyle \lambda }}_k})$$

(6)

The model training and learning process is described as follows: firstly, as to the forward propagation process, the reservoir water level value sequence is input into the network to perform a weighted evaluation on the information X_i of each neuron in the input layer, and the provided value serves as input for the hidden layer neuron and is transmitted accordingly. Following transformation by the Tan-Sigmoid function, the derived output becomes the hidden layer neuron’s output. Then, the identical approach is used to pass the conversion to obtain the output layer results to compare the current real water level data with the predicted water level. For a given dataset of input neuron information,${{\displaystyle S}}_p(p=1,2,\cdots ,r)$ is the predicted value of water level for the $P$th group of input information, ${{\displaystyle O}}_p$ is the true value of water level for the $P$th group of input information, and if the input has a total of R groups of information, the root mean square deviation of the network is:

$$E=\frac{1}{2r}{\displaystyle \sum _{p=1}^r{({{\displaystyle S}}_p-{{\displaystyle Q}}_p)}^2}$$

(7)

If the error does not conform to the criteria, it is retrogressively propagated from the output layer to each hidden layer with the aim of minimizing the discrepancy between the network’s actual and intended outputs [34], thus gradually correcting ${{\displaystyle W}}_{ij}$ and ${{\displaystyle V}}_{jk}$. As the BP algorithm iterates, the discrepancy between the actual and target outputs diminishes progressively until it complies with the predefined accuracy standards. The three-layer BP neural network employed in this section sequentially adjusts the connection weights and thresholds of each layer according to Equations (8)–(11).

The input and implicit layer weights and bias adjustment equations are:

$${{\displaystyle W}}_{ij}(t+1)=-\eta \frac{\partial E(W,\lambda )}{\partial {{\displaystyle W}}_{ij}}+{{\displaystyle W}}_{ij}(t)$$

(8)

$${{\displaystyle \lambda }}_j(t+1)=-\eta \frac{\partial E(W,\lambda )}{\partial {{\displaystyle \lambda }}_j(t)}+{{\displaystyle \lambda }}_j(t)$$

(9)

The implicit and output layer weights and bias adjustment equations are:

$${{\displaystyle W}}_{jk}(t+1)=-\eta \frac{\partial E(W,\lambda )}{\partial {{\displaystyle W}}_{jk}}+{{\displaystyle W}}_{jk}(t)$$

(10)

$${{\displaystyle \lambda }}_k(t+1)=-\eta \frac{\partial E(W,\lambda )}{\partial {{\displaystyle \lambda }}_k(t)}+{{\displaystyle \lambda }}_k(t)$$

(11)

In these equations, $E$ stands for the accumulation of squared errors bridging the network’s output and the real output samples. The BP neural network systematically diminishes this error using the principle of learning through the descent of the negative gradient; $\eta$ is the learning step length of the training algorithm; the larger the value of $\eta$, the greater the change in the weights, and the faster the convergence, but an excessive $\eta$ size can easily cause oscillation; ${{\displaystyle W}}_{ij}(t)-t$ represents the input layer time of the $i$th neuron and the implied layer of the $j$th neuron connection weights; ${{\displaystyle V}}_{jk}(t)-t$ represents the implied layer of the $j$th neuron connection weights with the neuron in the output layer $k$th.

The analysis above illustrates that the BP neural network integrates both forward computation and backpropagation during the training phase. To derive the output value, forward propagation is employed by conducting network calculations. Backpropagation, on the other hand, is utilized to systematically propagate the error across layers and adjust the connection weights between neurons. In the case of making predictions with the BP neural network, solely forward computation is employed to obtain the predicted water level value for small reservoirs using the pre-trained network model from the prior stage. This value serves as an input for the subsequent modeling stage.

3.2.2. Construction of Three-Dimensional $k~\epsilon$ Two-Equation Model Using VOF Method for Risk Loss

In order to focus on the prediction method of downstream losses of small reservoirs under the risk of dam failure, the changes in the free surface of the dam failure flood are simulated by using the $k~\epsilon$ two-equation model coupled with the VOF method, so as to acquire critical data like the inundation depth, flood flow rate, and inundation extent downstream of the dam failure of a small reservoir. After the introduction of the VOF model, the basic differential equations of the dam failure flood evolution mathematical model are expressed as follows:

The continuity equation is:

$$\frac{\partial \rho }{\partial t}+\frac{\partial \rho u_i}{\partial x_i}=0$$

(12)

The Reynolds time-averaged momentum equation is:

$$\frac{\partial (\rho u_i)}{\partial t}+\frac{\partial (\rho u_i{u}_j)}{\partial x_i}=-\frac{\partial P}{\partial x_i}+\frac{\partial }{\partial x_i}[(\mu +{\mu }_i)(\frac{\partial u_i}{\partial x_j}+\frac{\partial u_j}{\partial x_i})]$$

(13)

In the equation, the coefficient of turbulent viscosity is ${\mu }_t=\rho c_{\mu }\frac{k^2}{\epsilon }$, where $c_{\mu }$ is an empirical constant, typically considered as $c_{\mu }=0.09$.

The turbulent kinetic energy $k$ equation is:

$$\frac{\partial (\rho k)}{\partial t}+\frac{\partial }{\partial x_j}(\rho u_{i}k)=\frac{\partial }{\partial x_i}[(\mu +\frac{{\mu }_i}{{\sigma }_k})\frac{\partial k}{\partial x_i}]+G-\rho \epsilon$$

(14)

The turbulent kinetic energy dissipation rate $\epsilon$ equation is:

$$\frac{\partial (\rho \epsilon )}{\partial t}+\frac{\partial (\rho u_i\epsilon )}{\partial x_i}=\frac{\partial }{\partial x_i}\left[(\mu +\frac{{\mu }_i}{{\sigma }_{\epsilon }})\frac{\partial \epsilon }{\partial x_i}\right]+C_{1\epsilon }\frac{\epsilon }{k}G-C_{2\epsilon }\rho \frac{{\epsilon }^2}{k}$$

(15)

In the formula, $t$ is time (unit: s); ${{\displaystyle u}}_i$ and ${{\displaystyle u}}_j$ are velocity vectors (unit: m/s); ${{\displaystyle x}}_i$ and ${{\displaystyle x}}_j$ are spatial components (unit: m); $\rho$ is the fluid weight per unit volume (unit: kg/m³); $\mu$ is the molecular dynamic viscosity coefficient (unit: N·m/s); $k$ is the turbulence kinetic energy (unit: m²/s²); $\epsilon$ is the turbulence dissipation rate (unit: m²/s²); $P$ is the correction pressure (unit: Pa); ${\sigma }_k$ and ${\sigma }_{\epsilon }$ are the turbulence general values of $k$ and $\epsilon$, respectively. $C_{1\epsilon }$ and $C_{2\epsilon }$ are Lange numbers, where $C_{1\epsilon }=1.44$, and $C_{2\epsilon }=1.92$ are both empirical constants.

$G$ represents the term generated by turbulent kinetic energy $k$.

3.2.3. Construction of Probabilistic Radial Basis Neural Network for Risk Early Warning

(1) Model Construction

This section uses a probabilistic neural network to comprehensively predict the dynamic safety risks and small reservoir water level, and it uses risk loss prediction models to classify the downstream risk levels of small reservoirs.

The probabilistic neural network comprises the same number of hidden layer neurons as input layer vectors, and the quantity of output layer neurons corresponds to the predefined risk level within the training sample data. The architecture of the probabilistic neural network can be visualized, as shown in Figure 2.

The network output layer consists of the competition layer, and each output layer neuron corresponds to a data category. The activation function applied in the hidden layer uses the Gaussian function radbas, and the equation is:

$$f(x)={{\displaystyle e}}^{{{\displaystyle -x}}^2}$$

(16)

The input vector $n^1$ is equal to the product of the distance between the input vector p and the weight vector $W^1$ and the threshold $b^1$. $K$ is the number of neurons in the output layer; $C$ is the competition transfer function compete, employed to identify the peak value for each element within the input vector, and to output the corresponding category of neurons as 1 and the output of other categories of neurons as 0; $R$ is the quantity of elements within the input vector; $Q$ is the quantity of neurons in the hidden layer; $\parallel dist\parallel$ is used to calculate the Euclidean distance between the input vector $p$ and the weight vector $W^1$; operator $‘\times ’$ is the output vector corresponding to the threshold vector $b^1$, and $\parallel dist\parallel$ elements are multiplied; they are related as follows:

$$a_i{}^1=radbas(\parallel w_i{}^1-p\parallel b_i^1)$$

(17)

$$a^2=compet(W^2-a^1)$$

(18)

In the formula $a_i^1$ symbolizes the $i$th element of vector $a^1$, $b_i^1$ symbolizes the $i$th element of vector $b^1$, $W_i^1$ symbolizes the new vector composed of the $i$th row elements of vector $W^1$, and $y$ is the network output, that is, the safety risk warning level downstream of the reservoir.

(2) Early Warning Level Classification of Training Data

In order to train the model to obtain a radial basis neural network with high warning accuracy, a series of input samples and desired warning levels for the downstream safety warning of small reservoirs, such as ($X1,X2,X3 and Y$), are needed. Among them, $X1$ is the output of the downstream safety risk dynamic prediction model for small reservoirs, $X2$ is the output of the water level prediction model for small reservoirs, and $X3$ is the output of the risk loss prediction model, which have been provided by the model constructed in the previous section, and $Y$ is the safety warning level in accordance with the input information. In order to construct the downstream safety warning level of small reservoirs, the warning level division method adopted in the literature [35,36] is borrowed.

The evaluation system of downstream safety warning level of small reservoirs includes nine single indicators, which can reflect the main influencing factors of downstream safety risk of small reservoirs more comprehensively. Based on the validation of the literature [35,36], it can be obtained that the following nine indicators are important indicators in the flash flood database. Among them, the warning time R1 can be obtained from the optimal lag order of the Granger test results; the inundation depth R3 and the flood velocity R4 can be obtained from the numerical simulation of the VOF model, and the rest of the indicators are obtained from the downstream literature of small reservoirs [35,36], and the classification of the downstream safety risk of small reservoirs into early warning levels is shown in

Table 1. Early warning categorization of downstream safety risks of small reservoirs.

	Level 1	Level 2	Level 3	Level 4	Level 5
R1 Alarm time/(h)	[0, 450]	[450, 525]	[525, 600]	[600, 675]	[675, 750]
R2 Average annual rainfall/(mm)	[0, 0.2]	[0.2, 0.8]	[0.8, 1.4]	[1.4, 2.0]	[2.0, 2.5]
R3 Depth of inundation/(m)	[0, 0.6]	[0.6, 1.2]	[1.2, 1.8]	[1.8, 2.4]	[2.4, 10]
R4 Flood velocity/(m3·s−1)	[0.25, 0.75]	[0.75, 1.75]	[1.75, 2.75]	[2.75, 3.75]	[3.75, 5.0]
R5 Population density at risk/(people·km−2)	[0, 100]	[100, 225]	[225, 350]	[350, 475]	[475, 600]
R6 Unit area GDP(/ten thousand yuan·km−2)	[0, 100]	[100, 300]	[300, 500]	[500, 700]	[700, 2000]
R7 Agricultural output intensity/(ten thousand yuan·km−2)	[0, 20]	[20, 80]	[80, 140]	[140, 200]	[200, 250]
R8 Density of commercial and industrial property in communes/(ten thousand yuan·km−2)	[0, 100]	[100, 500]	[500, 900]	[900, 1300]	[1300, 2000]
R9 Density of traffic arteries/(km·km−2)	[0, 0.2]	[0.2, 0.6]	[0.6, 1.0]	[1.0, 11.4]	[1.4, 2.0]

.

In this section, the downstream safety warning levels of small reservoirs are classified into five levels; i.e., the output of the probabilistic radial basis neural network has five warning levels. The criterion of each level can be regarded as the significance of the downstream safety warning level of small reservoirs. In this section, the ‘flood risk rating level’ is used to represent the overall situation of the regional safety status, and the table of safety warning levels downstream of small reservoirs is shown in

Table 2. Characteristics of downstream safety warning levels for small reservoirs.

Evaluation Level	Characteristics Status	Characteristics
Level 1	Slight risk	Flooding is less likely to occur, and the area subject to flooding is less economically developed than the surrounding area.
Level 2	Average Risk	The likelihood of flooding is low, and the economic development of the affected area is slightly below the general level of the local area.
Level 3	Medium Risk	The likelihood of flooding is moderate, and the economic development of the affected area is at a moderate level for the region.
Level 4	High Risk	The likelihood of flooding is high and the economic development of the affected area is above the local average.
Level 5	Heavy Risk	Higher likelihood of flooding and higher level of economic development in the affected area than in the surrounding area.

.

Based on the above criteria, the way to classify the warning level labels for the model input data is as follows: assuming that all the evaluation indicators of the downstream of a small reservoir belong to level n, the warning level of the unit is also level n; if all the evaluation indicators do not belong to the same level, the highest level is taken as the warning level of the downstream of the small reservoir in accordance with the principle of conservatism. When all the input sequences downstream of a small reservoir have completed the setting of the warning level, these input and output data can be used to train the safety risk warning model downstream of the reservoir.

4. Example Analyses

4.1. Case Background

(1) Overview of the J Reservoir Project

J Reservoir was built in February 1969 and completed in December 1976. The watershed area controlled above the dam site is 240 km²; the length of the main river above the dam site is 28 km, and the average slope drop of the main river is 5.5%, which is a mountainous river with good vegetation in the watershed; the reservoir watershed is in the low-latitude and -longitude area, and the average rainfall for many years is 1897.7 mm, and the distribution of rainfall is not uniform within the year. Reservoir hub buildings mainly include one main dam, one sub-dam, one spillway dam, two water conveyance culverts, power house, tailwater channel, etc. The upstream of J Reservoir is 41 km away from the urban area, with a population of more than 25,000 people in the town, and a total area of 265 km². There are local villagers living in the vicinity of the reservoir, and a number of towns and villages downstream of the J Reservoir have an impact on the population of more than 200,000 people, and the affected cultivated land covers an area of 150,000 acres.

(2) J Reservoir Risk Event

From 20:00 on 18 August 2013 to 8:00 on 19 August 2013, heavy rain fell in the reservoir area, and the automatic rainfall observatory around the reservoir measured 274.9 mm, while other rainfall observatories measured more than 170 mm, and the maximum value of rainfall at the rainfall observatory reached 477.4 mm. The water level of the reservoir rose sharply due to the torrential rain, and the water level of the reservoir was 97.65 m at 7:50 p.m. on 19 August, and the water level of the reservoir reached 102.20 m at 10:20. The reservoir water level reached 102.20 m, exceeding the flood limit level of 4.6 m, exceeding the calibration level of 0.88 m, and when the power plant has been flooded, there was grid power interruption. At 11:40, the reservoir water level reached 102.50 m, at this time 0.50 m away from the top of the dam, and in this emergency decisive order to break the sub-dam to increase the release of flood water, the flood discharge was up to 2250 m³/s at this time. At 19:00, the water level in the reservoir dropped to the flood limit level, due to the appropriate measures taken to ensure that J Reservoir’s main water level, with a flood limit of 478.4 mm, the maximum rainfall observed at the station, remained at 478.4 mm. Appropriate measures were adopted to ensure the safety of the main dam of J Reservoir, to avoid greater losses of life and property of the downstream groups. On 19 August 2013, the J Reservoir overflow flood inundation situation was as shown in Figure 3.

With the aim of studying the effectiveness of the probabilistic radial-based neural network model for risk early warning, this article validates the case of J Reservoir and compares the genuine circumstances of the safety events and the model prediction outcomes of J Reservoir.

4.2. J Reservoir Inflow Forecast

(1)

Observational Data Smoothness Test

Missing data points within the collected reservoir data series (denoted as J) were estimated through the application of the sliding average interpolation method. Subsequent to the data handling, ADF tests were conducted on the rainfall time series $X\left(t\right)$, barometric pressure time series $H\left(t\right)$, relative humidity time series $Z\left(t\right)$, and inlet flow time series $Y\left(t\right)$. This examination aimed to assess the data’s smoothness and stationarity.

Based on the test outcomes, presented in

Table 3. The results for variable $X$ from the unit root detection.

Variable	Null Hypothesis	t-Statistic	t Critical Values 1%	t Critical Values 5%	t Critical Values 10%	p Value	Conclusion
$X$	$X$ has a unit root	−30.31612	−2.566212	−1.940993	−1.616586	0.0002	stationary
$H$	$H$ has a unit root	−5.045221	−3.433736	−2.862920	−2.567552	0.0000	stationary
$Z$	$Z$ has a unit root	−3.671521	−3.433732	−2.862922	−2.567552	0.0001	stationary
$Y$	$Y$ has a unit root	−3.634508	−2.566215	−1.940994	−1.616583	0.0004	stationary

are the outcomes of unit root analysis for variables $X$, $H$, $Z$, and $Y$.

The following are hence indicated by the test results:

The time series for rainfall $X\left(t\right)$, barometric pressure $H\left(t\right)$, relative humidity $Z\left(t\right)$, and inlet flow $Y\left(t\right)$ all exhibit smooth patterns.

(2)

Causality Tests for Variables

Because the series are smooth series, there is no need to carry out an additional cointegration test considering the series data can be directly applied to the Granger causality test; variable $H$, variable $X$, and variable Z were subjected to a causality test on variable Y; the original hypothesis is ‘not established causality’; and the results of the test are presented in

Table 4. Granger test for variable $H,X and Z$, respectively, for variable $Y$.

Excluded	Chi-sq	df	Prob.	Conclusion (t Critical Values 5%)
$H$	0.186633	2	0.3641	Accepted
$X$	6.252843	2	0.0424
$Z$	0.649981	2	0.9389	Accepted

.

The test outcomes reveal that the $p$-values for $H$ and $Z$ stand at 0.3641 and 0.9389, respectively. These values surpass the 5% critical threshold, leading to the acceptance of the null hypothesis. Consequently, it can be concluded that variables $H$ and $Z$ do not serve as Granger causes for variable $Y$, so the barometric pressure time series $H\left(t\right)$ and the relative humidity time series $Z\left(t\right)$ do not have the ability of predicting the inbound flow time series $Y\left(t\right)$.

(3)

Granger Test for the Variable

Using the Akaike Information Criterion to determine the maximum lag order of rainfall time series $X$ on the incoming flow time series $Y$ is 24, according to the adjusted lag order Granger test, and the results of the test are displayed in

Table 5. Granger test of variable $X$ against variable $Y$.

Null Hypothesis:	Obs	F-Statistic	Prob.
$Y$ does not Granger Cause $X$	1836	1.39385	0.0948
$X$ does not Granger Cause $Y$		1.74287	0.0163

.

The test findings reveal that with a maximum lag order of 24, the $p$-value for making a type 1 error in rejecting the null hypothesis ‘$Y$ does not Granger Cause $X$’ during the hypothesis test is 0.0948, placing it at the 5% significance level. Conversely, the $p$-value for making a type 1 error in rejecting the null hypothesis ‘$X$ does not Granger Cause $Y$’ during the hypothesis test is 0.0163, which is below the 5% significance threshold. Consequently, it can be concluded that variable $X$ serves as the Granger cause of variable $Y$, and the time series of rainfall $X\left(t\right)$ has the ability to predict the time series of the inlet flow $Y\left(t\right)$.

Through the Granger causality test, rainfall time series $X\left(t\right)$, barometric pressure time series $H\left(t\right)$, and relative humidity time series $Z\left(t\right)$ were screened, and the results show that only rainfall time series $X\left(t\right)$ was a valid influence factor on the inlet flow time series $Y\left(t\right)$.

The BP neural network is used to train and simulate the inlet flow time series $Y\left(t\right)$ using the effective influence factor rainfall time series $X\left(t\right)$.

(4)

Data Accuracy Check

The precision of the neural network data prediction is assessed using the recorded rainfall and inflow data from J Reservoir. Given the constraints of the available data, ensuring the neural network model’s effectiveness required the selection of data from a time period with comprehensive and abundant rainfall records. The data are further processed using the digital twin method proposed by Li et al. [37], to supplement and enhance the quality and availability of training data in deep learning methods. Accordingly, a dataset spanning 1836 h, from 31 May 2018 to 15 August 2018, is chosen for training, while a separate 24 h dataset is employed to evaluate the model’s predictive capabilities.

For predicting the inflow of J Reservoir, a three-layer BP network is employed. The specific number of neurons in the hidden layer is fine-tuned through practical curve-fitting assessments. The ultimate network topology adopted is 3-8-1. Model development is carried out using the MATLAB 2021 neural network toolbox, and the transfer function employed is purely, known for its linearity. The precision for the error term, denoted as E, is configured at 0.01. The maximum number of training cycles is capped at 30,000 iterations, with the network designed to output hourly inflow rates.

The forecasting accuracy of the results is assessed in compliance with the ‘Hydrological Information Forecasting Specifications (GB/T 22482-2008)’ [38].

(5)

Evaluation of the J Reservoir Inflow Prediction Results

The comparison between measured and predicted inflow flow values using multiple input parameters before data screening, and the comparison between measured and predicted inflow flow values with a single input parameter after data screening are shown in Figure 4. The evaluation of prediction result accuracy is presented in

Table 6. Table for evaluating prediction accuracy.

Prediction Stage	Predict Results before Data Filtering	Prediction Results after Data Filtering
Conformity rate/%	24.99	45.78
Average relative error level/%	39.11	21.50
Average error level/(m3/s)	629.55	387.41
Nash–Sutcliffe Efficiency	0.34	0.67

. The Nash–Sutcliffe efficiency (${{\displaystyle E}}_{ns}$) index offers the advantage of versatility, as it can be applied to various types of models. Therefore, the article uses this indicator for prediction accuracy evaluation. The equation is as follows:

$${{\displaystyle E}}_{ns}=1-\frac{{\displaystyle {\sum }_{i=1}^n{{\displaystyle ({{\displaystyle x}}_i-{{\displaystyle y}}_i)}}^2}}{{\displaystyle {\sum }_{i=1}^n{{\displaystyle ({{\displaystyle y}}_i-{{\displaystyle z}}_i)}}^2}}$$

(19)

where ${{\displaystyle x}}_i$ and ${{\displaystyle y}}_i$ represent the observed and computed flow values at time t, respectively, and ${{\displaystyle z}}_i$ represents the mean of the observed and computed flow values corresponding to different patterns.

In Table 6, the prediction results’ accuracy assessment is presented, with the following comparative analysis:

(1) The application of the Granger causality test method for data screening led to substantial improvements in the quality of the BP neural network predictions. This resulted in an 83.23% increase in prediction accuracy, a 45.05% reduction in average relative error, a 38.46% decrease in average error, a 97.06% enhancement in the Nash–Sutcliffe efficiency, and a significant improvement in the alignment between predicted and actual values. Furthermore, employing filtered data for generalization testing within the prediction process substantially lowered the incidence of fitting failures in the results.

(2) In this case, the success rate of J Reservoir inflow predictions falls below 60%. This outcome can be ascribed to the limited quantity of network training samples and the lower data accuracy. The number of training samples will be increased in subsequent research. And improving the accuracy of sample data can enhance the precision of predictions.

4.3. J Prediction of Risk Level Downstream of J Reservoir

4.3.1. Analysis of Risk Level Prediction Process Downstream of J Reservoir

The probabilistic neural network is used to classify risk levels downstream of J Reservoir. Given the limited data available, the selection of representative data associated with reservoir risk events is crucial to ensure the effectiveness of the neural network model’s training. A total of 104 sets of data on risk events downstream of the reservoir with complete records and certain typical characteristics of J Reservoir are selected: 80 sets of data are selected for training, and the data of the remaining 24 sets of risk events are selected to verify the accuracy of model prediction. Modeling is conducted using the MATLAB neural network toolbox, and the Spread parameter is set to 1.5 after actual debugging.

The input values of the PNN network are the dynamic prediction level of the reservoir safety status corresponding to the J Reservoir risk event, the predicted value of the inflow flow, the average 1 h rainfall, the average 6 h rainfall, the average 24 h rainfall, the annual rainfall, the annual maximum 24 h rainfall, risk population density, GDP per unit area, agricultural output value density, township industrial and commercial property density, and township industrial and commercial property density downstream of the reservoir, combined with the dynamic security risk prediction status above as input indicators, and the PNN’s output value is the risk level downstream of J Reservoir. According to the requirements in the ‘J Reservoir Flood Control and Rescue Emergency Plan’, five levels of risk are set according to the working status of the dam, the degree of flood discharge hazards, and the development trend of the danger. Among them, level 4, level 3, level 2, and level 1 risk levels are from low to high, respectively. The high order corresponds to the five-level emergency response in the emergency plan, and level 5 risk is set as the normal state of the dam. The fitting situation of the PNN network training samples is shown in Figure 5.

The graph shows the PNN network’s training sample fitting results for different risk event numbers downstream of a reservoir. The primary conclusion that can be drawn is that the PNN model has varying degrees of success in predicting the correct risk level. There are instances where the predicted risk level (dashed lines) matches the actual risk level (solid lines) closely, indicating accurate predictions. However, there are also several mismatches, where the predictions deviate from the actual values. These deviations suggest areas where the model may require further tuning or additional training data to improve its predictive performance.

The fitting error of the PNN network training sample is shown in Figure 6.

The graph represents the fitting error of PNN model predictions for various risk events downstream of a reservoir. Most of the plotted points lie close to the zero line, indicating minor discrepancies between the predicted and actual risk levels. This suggests the model has a generally good fit. However, there are noticeable spikes where the errors are significant, which might point to specific instances where the model’s predictive accuracy is poor. These outliers could be due to anomalies in the data or areas where the model struggles to capture the underlying pattern.

4.3.2. Analysis of the Prediction Results for Risk Levels of Downstream of J Reservoir

Figure 7 displays the prediction results of the PNN network model.

According to the prediction results of the risk level downstream of the reservoir by the PNN network, only 4 of the 24 test samples had deviations in the risk level prediction. The overall prediction accuracy of the model was 83.3%, and the false alarm rate was 16.7%. In the results, only the risk level predictions of the two samples were lower than the actual risk levels. One sample in the test sample had false positives and false negatives. The false negative rate and false negative rate were both 4.2%. In addition, all Level 1 emergency response events downstream of the reservoir are accurately predicted, which shows that the reservoir downstream risk prediction model constructed in this article has certain applicability.

5. Conclusions

This article studies and constructs a downstream safety risk early warning model for small reservoirs based on a neural network, which integrates reservoir safety and comprehensive risk prediction of flash flood disasters. Based on the existing critical rainfall model and data-driven model, the Granger causality test method is employed for the examination of rainfall. Based on the time-series logic relationship with the inflow and water level of the upstream small reservoir, a neural network model for small reservoir water level prediction is constructed. The three-dimensional $k~\epsilon$ equation of the VOF method is employed to create a prediction model for downstream risk loss, which integrates the dynamic prediction of safety risks downstream of the reservoir and the reservoir environment. Dynamic prediction of safety risks downstream of reservoirs, prediction of reservoir environment, and prediction of risk losses are combined to build a probabilistic radial basis neural network model for risk early warning. The model is trained and corrected through test data to continuously improve the prediction and early warning accuracy of the model. The method proposed in this article integrates the reservoir’s own information, hydrological and meteorological information, downstream information, etc., thereby effectively improving the accuracy and predictability of safety warnings downstream of the reservoir, and providing a reference for downstream safety risk management and control of small reservoirs.

Currently, the number of small reservoirs continues to increase globally. The safe operation of small reservoirs is mostly affected by flash floods, and compared to large reservoirs, it is very difficult to carry out safety condition monitoring and fault diagnosis based on operational monitoring data, using statistical analysis, machine learning, and other methods. Therefore, the research proposed in this paper broadens the research perspective of risk early warning methods for small reservoirs, which can be applied to risk early warning under the conditions of incomplete and inaccurate data. The problem of the downstream safety risk assessment of small reservoirs involves many aspects, and the safety risk indicator system was constructed from the perspective of the vulnerability of small reservoirs and analyzed and explored only on the basis of the available data. With the improvement in the theoretical system and the enhancement of data completeness, subsequent studies can construct a safety risk indicator system for small reservoirs from different perspectives or different datasets. In addition, this study only conducted early warning research on safety risks downstream of small reservoirs in a narrow sense. The next step in this research can be expanded to the study of safety risks downstream of reservoirs in a broad sense, including other indirect risks such as environmental and ecological risks caused by natural disasters.