A Rapid Prediction Model of Rainstorm Flood Targeting Power Grid Facilities

Abstract

Rainstorm floods constitute one of the major natural hazards threatening the safe and stable operation of power grid facilities. Constructing a rapid and accurate prediction model is of great significance in order to enhance the disaster prevention capacity of the power grid. This study proposes a rapid prediction model for urban rainstorm flood targeting power grid facilities based on deep learning. The model utilizes computational results of high-precision mechanism models as data-driven input and adopts a dual-branch prediction architecture of space and time: the spatial prediction module employs a multi-layer perceptron (MLP), and the temporal prediction module integrates convolutional neural network (CNN), long short-term memory network (LSTM), and attention mechanism (ATT). The constructed water dynamics model of the right bank of Liangshui River in Fengtai District of Beijing has been verified to be reliable in the simulation of the July 2023 (“23·7”) extreme rainstorm event in Beijing (the July 2023 event), which provides high-quality training and validation data for the deep learning-based surrogate model (SM model). Compared with traditional high-precision mechanism models, the SM model shows distinctive advantages: the R² value of the overall inundation water depth prediction of the spatial prediction module reaches 0.9939, and the average absolute error of water depth is 0.013 m; the R² values of temporal water depth processes prediction at all substations made by the temporal prediction module are all higher than 0.92. Only by inputting rainfall data can the water depth at power grid facilities be output within seconds, providing an effective tool for rapid assessment of flood risks to power grid facilities. In a word, the main contribution of this study lies in the proposal of the SM model driven by the high-precision mechanism model. This model, through a dual-branch module in both space and time, has achieved second-level high-precision prediction from rainfall input to water depth output in scenarios where the power grid is at risk of flooding for the first time, providing an expandable method for real-time simulation of complex physical processes.

1. Introduction

In recent years, driven by both climate change and human activities, the frequency and intensity of urban rainstorm floods in China have demonstrated a notable increasing trend [1]. Urban rainstorm floods have constantly triggered power grid facility failures and large-scale blackouts, causing severe economic losses and social impacts [2]. For example, the extreme rainstorm and flood disaster in Changsha in June 2017 caused approximately 2.32 million households to lose power. During the Zhengzhou “7·20” extreme rainstorm event in 2021, more than 1190 residential areas experienced prolonged power outages [3]. In the July 2023 event, power facilities in Beijing were severely damaged, with about 60,000 households in Fangshan District alone losing electricity. Thus, the increasing vulnerability of power grid facilities to urban rainstorm flooding necessitates the urgent development of rapid and accurate prediction tools for effective operational management and early warning.

China’s current early warning system for urban rainstorm floods, however, is primarily under the administration of multiple departments: meteorological departments are responsible for issuing weather warnings, and water resources departments issue warnings of river network floods and flash flooding [4]. Moreover, early warnings are generally issued at the administrative level of cities and counties or within natural river basins, and there are still concerns about the lack of precise warnings for specific locations of power grid facilities. In recent years, researchers have never ceased the attempt to develop urban rainstorm flood prediction technologies targeting power grid facilities. For instance, Liu Qiang et al. used the MIKE21 model to assess flood disasters at substations [5]. However, technical challenges still emerge in the application of such traditional hydrological and hydrodynamic model-based flood prediction technologies to the field of power grid facilities. On one hand, such traditional flood forecasting technologies rely on extensive basic geographic data and flood control engineering data, which results in longer construction cycles and higher operation and maintenance costs. On the other hand, power grid facilities are widely distributed and interconnected, which makes it difficult to widely apply traditional hydrological and hydrodynamic model-based methods.

Currently, rapid flood forecasting models based on deep learning technology are flourishing, gradually forming a dual-model paradigm comprising mechanism-based hydrological and hydrodynamic models and data-driven deep learning models. Urban rainstorm flood process samples are generated by using hydrological and hydrodynamic models, and subsequently, surrogate models of hydrological and hydrodynamic models are created by employing deep learning techniques to drive and train deep learning models. This approach significantly enhances the efficiency and applicability of hydrodynamic model-based urban rainstorm flood predictions. For example, Yang Yongchuan et al. [6] developed a surrogate model of spatiotemporal prediction for flash floods, enabling rapid prediction of the spatiotemporal processes of flash flood inundation over the following 6 h. Chen Jiarui [7] utilized a deep learning-based surrogate model, trained on rainfall data from six return periods (2-, 5-, 10-, 20-, 50-, and 100-year events) with a 120 min duration, to effectively identify and weight multi-source features (elevation, slope, aspect, soil moisture content, etc.) in high-risk urban water-logging points, significantly improving the prediction accuracy of maximum water depth and computational efficiency. Wang YiQuan et al. [8] employed a multi-layer perceptron (MLP) to predict the likelihood of flood disasters. Wang Yi et al. [9] found that the MLP model, as an artificial neural network, is capable of uncovering complex nonlinear relationships between flood occurrences and flood regulation factors. To predict the temporal water depth processes at water-logging points, Dai Xin et al. [10] utilized an LSTM neural network, based on six designed rainfall scenarios, to capture its dynamic variation characteristics, achieving satisfying prediction accuracy. Furthermore, the CNN-LSTM-ATT deep learning model proposed by Lin Kairong et al. [11], which was trained on rainfall data from eight return periods (2-, 3-, 5-, 10-, 20-, 30-, 50-, and 100-year events) with a 180 min duration, provides an effective method for predicting urban water logging and inundation. While these cited studies demonstrate the potential of deep learning surrogate models, their development and validation are typically constrained by a limited set of designed rainfall scenarios (typically six to eight return periods). This constraint can lead to increased predictive uncertainty, especially when models are used to predict extreme events beyond the range of their training conditions.

The above research demonstrates the broad application prospects of deep learning-based surrogate models in rapid flood forecasting. Due to the scarcity of urban rainstorm flood samples and the challenge posed by the technical complexity of mechanistic model construction, however, related studies are still in the preliminary exploration stage and far from meeting the application requirements for precise forecasting and early warning in the field of power grid facilities. High-precision mechanistic models remain the core foundation for driving intelligent surrogate models. This study uses the right bank of the Liangshui River in Fengtai District, Beijing, as a case study. First, a high-precision hydrological–hydrodynamic model is constructed, specifically targeting power grid facilities. Subsequently, this model is employed to simulate flood inundation processes under fifteen designed rainfall scenarios, generating a comprehensive and high-fidelity dataset. Based on this enhanced dataset, a high-precision deep learning surrogate model is trained and validated. Finally, the model’s performance is rigorously evaluated with multiple metrics, achieving accurate and rapid forecasting of urban rainstorm floods for targeted power grid facilities.

2. Materials and Methods

2.1. Research Area

The right bank of the Liangshui River Basin in Fengtai District, Beijing, was selected as the research area, covering a total area of 394.14 square kilometers at an elevation ranging from 14 to 218 m. The river network in this area extends 5.18 km in length and comprises 18 sluice gates and 5 rubber dams, as depicted in Figure 1. Additionally, owing to its low-lying topography and dense building concentration, this region is highly susceptible to urban rainstorm flood events, exemplified by the flood incident that occurred in Beijing’s Lianhua Bridge area on 21 July [12]. The area currently is equipped with 8 operational substations, labeled as D1 to D8 in Figure 1. These substations are utilized to ensure the power supply to critical functional zones and large communities, such as the Lize Financial Business District and the Capital Business New District. Flood prevention and safety in these areas are of paramount importance.

2.2. Technical Architecture

In this study, a rapid urban rainstorm flood prediction model targeting power grid facilities was constructed within the framework of a physically based hydrodynamic mechanism model integrating deep learning algorithms. The technical architecture of this model is illustrated in Figure 2. As in step 1, data about topographical features, historical monitoring data, and multi-return-period rainfall sequences are input to drive the hydrodynamic model to simulate physical mechanism, generating a training sample database with physical consistency (step 2). This sample database undergoes data cleaning, normalization processing, and sequence reconstruction before being fed into the deep learning model (step 3). The deep learning models are trained and validated (step 4) by adopting a spatiotemporal dual-branch architecture. In such an architecture, the spatial prediction branch employs an MLP to achieve deep fusion of multi-source features such as rainfall sequence characteristics and topographical features, outputting the spatial distribution of inundation depth; and the temporal prediction branch utilizes a CNN-LSTM-ATT hybrid model to learn the temporal dependencies of rainfall sequences and predict the temporal water depth processes at key water-logging points. Through this spatiotemporal dual-branch architecture, the model achieves the rapid prediction of both spatial distribution of inundation and the temporal water depth processes (step 5).

2.2.1. Urban Rainstorm Flood Model

In this study, a self-developed fully distributed model (DHMUrban) [13,14,15] was employed to construct the urban rainstorm flood mechanism model within the research area. This model consists of core modules, including a surface model, a pipe network model, and a river network model, which are fully coupled with one another (as shown in Figure 3). It enables simulation of the entire process with high precision and high efficiency: from rainfall generation and convergence at the ground surface, through pipe network confluence, and river network accumulation in complex urban environments [16,17,18].

2.2.2. Deep Learning Models

(1) Spatial Prediction Module

The spatial prediction module employs an MLP (multi-layer perceptron) model, which is a type of feed-forward artificial neural network architecture. In addition to an input layer and an output layer, this architecture can also incorporate multiple hidden layers between them [19]. Compared with a single-layer perceptron, this multi-layer architecture effectively addresses the modeling challenge of nonlinear separable problems by introducing nonlinear activation functions and a hierarchical feature extraction mechanism [20,21]. In this study, the MLP model was employed to learn the spatial characteristics of rainfall-induced inundation (Figure 2).

The architecture of each neural network is described using the notation “I–H₁–…–H_n–O”, where I denotes the input dimension, H_k represents the number of neurons in the kth hidden layer, and O is the output dimension. At the same time, this study assumes by default that all grid cells share consistent grid resolution and are spatially aligned through their unique grid IDs.

(1)

Rainfall pattern feature extraction: a three-layer fully connected network (24-64-64-251,450) is used to learn the complex nonlinear mapping from rainfall sequences to spatial inundation fields.

(2)

Topographical features mapping: a parameter-sharing multi-layer perceptron (6-16-16-1) processes six static features (elevation, roughness coefficient, initial loss, initial infiltration rate, steady infiltration rate, and drainage capacity) to generate a topography-dominated flood-prone base field.

(3)

Feature fusion: a lightweight fusion network (2-16-1) is designed to perform pixel-wise intelligent weighting of the contributions from rainfall and topographical features, achieving a scenario-aware fusion mechanism.

(2) Temporal prediction module

The temporal prediction module is based on an LSTM (long short-term memory) neural network, a type of deep learning model primarily designed for processing and predicting time-series data. Through three logical structures, namely the forget gate, input gate, and output gate, the model filters and retains data, addressing the problems of vanishing or exploding gradients that traditional RNN models may encounter when handling long-range dependencies. This model is particularly well suited for capturing long-term dependencies in time series, leading to its widespread application across various fields such as stock market forecasting, weather forecasting, and hydrological prediction [22]. The initial standard LSTM, however, lacks inherent feature extraction capabilities and has limited ability to capture local patterns. In this study, several improvements were accomplished based on the traditional LSTM model by employing a CNN-LSTM-ATT architecture, as shown in Figure 2. The model uses rainfall sequences of shape (120, 1) as input and produces water depth predictions of shape (120,) as output.

First, local temporal features of the rainfall data are extracted using a CNN. On the first layer, 16 five-dimensional convolution kernels are employed to detect basic rainfall patterns such as the onset of downpour or sustained rainfall. On the second layer, 32 three-dimensional convolution kernels are utilized to combine lower-level features, identifying more complex composite rainfall patterns. Finally, self-adaptive pooling compresses the sequence to 32 time steps, enhancing computational efficiency while preserving key information.

Next, an LSTM layer is utilized to model the temporal dynamics of the features extracted by the CNN. By leveraging the coordinated operation of the forget, input, and output gates, the LSTM can learn the dynamic patterns of the rainfall-flood system, including processes such as delayed flood response, peak accumulation, and water recession. Its recurrent connection properties ensure effective capture of long-term dependencies. Subsequently, an attention (ATT) mechanism evaluates the importance of each time step in the LSTM, identifies critical periods by computing weight scores, and automatically focuses on crucial periods such as time of intense rainfall or rapid water rise. As result, a weighted context vector will be generated, which can improve prediction accuracy.

Finally, a fully connected layer transforms the above context vector into water-logging depth predictions. A two-layer neural network structure, combined with rectified linear unit (ReLU) activation and dropout regularization, ensures that the model captures complex relationships while avoiding overfitting. With the collaborative cooperation among the CNN, LSTM, and attention mechanism, this architecture achieves hierarchical processing from local feature extraction to temporal pattern learning and to key-focus weighting, thereby effectively reducing the complexity of rainfall-flood prediction.

2.3. Model Evaluation

In this study, the coefficient of determination (R²), mean absolute error (MAE), and root mean square error (RMSE) are selected as key indicators to evaluate the performance of the model. R² is used to assess the overall goodness of fit of the model, where a higher value indicates a better model fit. MAE is highly robust against outliers, while RMSE provides a more visualized reflection of the magnitude of prediction errors, with lower MAE and RMSE values representing smaller errors.

$$R^2=1-{\displaystyle \frac{{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}}{{\displaystyle \sum _{i=1}^n{(y_i-\overline{y})}^2}}}$$

(1)

$$MAE={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\left|y_i-{\widehat{y}}_i\right|}$$

(2)

$$RMSE=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{(y_i-{\widehat{y}}_i)}^2}}$$

(3)

(n denotes the sample size, ${\widehat{y}}_i$ represents the predicted value of sample i in the model, $y_i$ denotes its actual value, and $\overline{y}$ denotes the mean of the actual values of the samples.)

2.4. Construction of Urban Storm Flood Model

A hydrodynamic-based urban rainstorm flood model (HM model) was constructed utilizing high-precision digital elevation model (DEM) data with 2 m resolution in the research area, drainage network data, and detailed topography data of the substations and their surrounding areas. The area was subdivided into unstructured cells, with rivers and main roads serving as control lines for subdivision. Encryption processing was applied to local cells around the substations, resulting in a total of 251,450 cells. The land use types and corresponding parameter settings within the research area are illustrated in

Table 1. Parameters and area percentage of different land use types.

Land Use	Area Proportion (%)	Roughness Coefficient	Initial Loss (mm)	Initial Infiltration Rate (mm/h)	Stable Infiltration Rate (mm/h)	Drainage Capacity (mm/h)
Buildings	26.96	0.07	5	5	3	8
Grassland	24.35	0.065	30	20	5	4
Woodland	20.26	0.065	10	30	20	4
Roads	10.1	0.035	5	0	0	15
Hardened surfaces	8.8	0.035	5	0	0	15
Bare ground	6.93	0.04	10	30	20	4
Water bodies	2.38	0.035	0	0	0	0
Shrubs	0.22	0.065	10	30	20	4

.

The model was verified by collecting and studying the measured rainfall process of the July 2023 event from 65 rainfall stations within and around the research area, and it measured water-logging depths in 5 monitoring points within the area. Figure 4 shows the measured rainfall process at the representative rainfall station in Fengtai. The downpour primarily happened between 29 July and 3 August 2023, with the most intense period occurring from 06:00 on 30 July to 00:00 on 1 August. This period was characterized by distinct short-duration rainstorms, with a maximum hourly rainfall of 44 mm.

2.5. Training of Surrogate Model

Based on the rainstorm intensity formula recommended in the Beijing Hydrological Manual, designed schemes of 15 24 h rainfall processes under different return periods were generated. Their rainfall characteristics are shown in

Table 2. Designed rainfall schemes and statistics on inundated areas.

Name of Rainfall Scenario	Rainfall Return Period (a)	Rainfall Intensity (mm/h)	Total Rainfall (mm)	Notes
P1	1	16.6	45
P2	1~3	28.2	72
P3	3	39.1	100
P4	3~5	42.1	120
P5	5	49.1	140
P6	5~10	55.5	170
P7	10	65.3	200	Validation samples
P8	10~20	67.3	225
P9	20	76.3	255
P10	20~50	83.6	297
P11	50	95.7	340
P12	50~100	102.8	370	Validation samples
P13	100	111.1	400	Test samples
P14	100~200	111.0	430
P15	200	118.7	460

. In this study, throughout the entire modeling area, each 24 h precipitation process was treated as uniformly distributed rainfall input data. These 15 designed rainfall schemes were individually input into the HM model for sequential simulation, aiming to obtain data on inundation spatial distribution, corresponding water depth, and temporal water depth processes. These data served as the training samples for the surrogate model.

After a comprehensive consideration of common, relatively severe, and extreme rainfall scenarios, 12 rainfall scenarios were chosen as training samples (P1–P6, P8–P11, P14, P15), 2 as validation samples (P7, P12), and 1 as a test sample (P13). The validation scenarios were selected based on their practical relevance: the 10-year return period represents common storm events critical for routine flood management, while the 50–100-year return period represents moderate-extreme events important for infrastructure resilience planning.

Before model training, spatial and temporal datasets required separate data preprocessing, which was conducted with strict anti-leakage protocols:

(1) Spatial Data Processing

The spatial dataset encompasses rainfall scenarios P1–P15 and their corresponding flood data, as well as six types of topographical feature data for the 251,450 cells. To prevent information leakage, each complete rainfall scenario (including its corresponding inundation data across all grid points) was treated as an indivisible data unit. Both rainfall and topographical data were normalized using Z-score standardization (with ε = 1 × 10⁻⁸ added to the denominator to prevent division by zero) to unify dimensions and accelerate model training. Critically, all normalization parameters (mean and standard deviation) were calculated exclusively from the training subset of scenarios (P1–P6, P8–P11, P14, P15), and the same parameters were then applied to the validation (P7, P12) and test (P13) subsets.

(2) Temporal Data Processing

The temporal dataset integrates the designed rainfall scenarios of P1–P15 and their corresponding temporal water depth processes at the substations, with the 24 h rainfall sequences extended to 120 h using the “last-value padding” method to ensure temporal consistency. Consistent with the spatial data protocol, the dataset was partitioned at the scenario level, guaranteeing that no data from the same rainfall scenario were mixed across training, validation, and test sets. After temporal alignment and min-max normalization (using training-set-derived parameters only), a rainfall-water depth mapping relationship was constructed for each substation as temporal prediction input.

(3) Overall Data Partitioning Strategy

The dataset partitioning follows a scenario-based approach where training, validation, and test sets contain mutually exclusive rainfall scenarios: training (12 scenarios: P1–P6, P8–P11, P14, P15), validation (2 scenarios: P7, P12), and test (1 scenario: P13). This ensures complete isolation between datasets and prevents any information leakage throughout the preprocessing pipeline.

This study adopts a lightweight architecture based on the PyTorch 2.9.1 framework which can run on a regular computer equipped with only 8 GB of memory and an Intel Core i7-11700-level processor. The model training and inference tasks can be efficiently completed. The model implementation relies on mainstream scientific computing libraries such as PyTorch 2.9.1, NumPy 2.3.5, Pandas 2.3.3, and Matplotlib 3.10.7. The entire training process does not require GPU acceleration, and the required storage space is controlled within 500 MB, with the training time within 20 min.

After multiple iterations and optimizations, the optimal parameter settings for the MLP and CNN-LSTM-ATT models were determined (

Table 3. Hyperparameter configuration.

Parameters		MLP	CNN-LSTM-ATT
Loss function		MSELoss	MSELoss
Optimizer		Adam	AdamW
Learning rate Schedule	Initial learning rate	0.001	0.005
	Patience	50	15
	Factor	0.5	0.5
	Min learning rate	1 × 10−6	1 × 10−6
Activation function		ReLu	ReLu, Sigmoid, Tanh, Softmax
Maximum epochs		1000	200
Early stopping criteria		100	30
Random seeds		514	514

).

Evaluation results of the models on the training and validation sets are presented in

Table 4. The evaluation results of models on the training set and validation set.

Model	R2		MAE (m)		RMSE (m)
	Training Set	Validation Set	Training Set	Validation Set	Training Set	Validation Set
MLP	0.9881	0.9894	0.005	0.010	0.013	0.026
CNN-LSTM-ATT (D3)	0.8905	0.9923	0.0244	0.0171	0.0392	0.0333

. The MLP model maintained a higher goodness of fit (R² > 0.98) and low errors on both datasets, demonstrating stable performance. The CNN-LSTM-ATT model (using D3 as an example) shows moderate fitting on the training set (R² = 0.8905) but achieves an R² of 0.9923 on the validation set, with validation errors lower than training errors.

To properly interpret these results, it is important to note that the higher validation R² compared to training R² does not necessarily indicate superior generalization capability. This pattern can occur when (1) the validation subset (10-year and 50–100-year return periods) contains scenarios that are relatively less challenging or more homogeneous than the full training set, which encompasses a wider range of rainfall intensities; when (2) R² is sensitive to the variance of the target variable, and the validation set may have lower variance in water depth values; and when (3) the scenario-based split creates inherent differences in data distribution between training and validation sets. Thus, while the CNN-LSTM-ATT model performs well on the specific validation scenarios, its true generalization ability should be assessed through additional independent testing across diverse rainfall conditions.

To rigorously assess the generalization capability of the surrogate model and mitigate the risk of overfitting due to the limited number of rainfall scenarios, we implemented k-fold cross-validation on the combined training and validation sets (P1–P12, P14–P15, totaling 14 scenarios), while completely reserving P13 as an independent test set. Specifically, we divided these 14 rainfall scenarios into k = 5 folds, with each fold containing approximately 2–3 scenarios. In each iteration, k − 1 folds were used for training the neural network, while the remaining fold was reserved for validation. This process was repeated k times until each fold had been used exactly once as the validation set.

The evaluation results based on 5-fold cross-validation (

Table 5. The evaluation results of k-fold cross-validation.

Folds	MLP			CNN-LSTM-ATT (D3)
	R2	MAE (m)	RMSE (m)	R2	MAE (m)	RMSE (m)
1	0.9455	0.0194	0.0536	0.9712	0.0139	0.0287
2	0.9693	0.0244	0.0630	0.9877	0.0124	0.0228
3	0.9725	0.0110	0.0326	0.9900	0.0097	0.0173
4	0.9765	0.0126	0.0365	0.9892	0.0093	0.0168
5	0.9678	0.0161	0.0534	0.9898	0.0092	0.0178

) indicate that both MLP and CNN-LSTM-ATT (using D3 as an example) exhibit excellent predictive performance. Among them, MLP demonstrates a stable ability in explaining the spatial distribution of water depth, with an average R² reaching 0.9663, while CNN-LSTM-ATT excels in capturing the temporal process of water depth, with the average R² increasing to 0.9856, and the results show less fluctuation among different folds (standard deviation 0.0074 vs. 0.0109). Notably, this cross-validation performance substantially exceeds the training set R² observed in the fixed split (0.8905), suggesting that CNN-LSTM-ATT’s temporal prediction capability is more robust than indicated by the single-partition evaluation.

3. Results

3.1. Verification of Urban Storm Flood Model

The simulated water depth distribution for the July 2023 urban rainstorm flood is shown in Figure 5. A comparison between the simulated and measured maximum water depths at the 17 water-logging monitoring points is presented in

Table 6. Comparison and analysis of the measured maximum water depth and simulated maximum water depth at the water-logging monitoring point during the July 2023 event.

Number	Measured Maximum Water Depth (m)	Simulated Maximum Water Depth (m)	Difference (m)	R2	MAE (m)	RMSE (m)
R1	2.05	2.179	0.129	0.9913	0.038	0.061
R2	0.48	0.525	0.045
R3	0.63	0.706	0.076
R4	1.14	1.096	0.044
R5	0.47	0.402	0.068
R6	2.05	2.179	0.129
R7	0.48	0.525	0.045
R8	0.63	0.706	0.076
R9	1.14	1.096	0.044
R10	0.47	0.402	0.068
R11	1.01	1.107	−0.097
R12	0	0	0
R13	0	0.007	−0.007
R14	0	0	0
R15	0	0	0
R16	0	0	0
R17	0	0	0

. The results indicate that the water depth errors at the monitoring points range from 0 to 0.13 m. The coefficient of determination (R²) is close to 1 (R² = 0.9913), and both the mean absolute error (MAE) and root mean square error (RMSE) were small (MAE = 0.038 m, RMSE = 0.061 m).

Among the 11 points with water accumulation, the average absolute error between the simulated and measured values was 0.074 m, and the RMSE was 0.085 m. Among the six points without water accumulation, the model correctly predicted zero water accumulation at five points (accuracy rate 83%), with only a minor error (0.007 m) at point R13, demonstrating the high accuracy of the model.

3.2. Evaluation of Surrogate Mode

3.2.1. Spatial Prediction Module

Based on testing rainfall scenario P13, the performance of the spatial prediction module was evaluated by comparing its results with the high-precision urban rainstorm flood simulation outputs generated by the HM model. The prediction accuracy of the module was analyzed from two perspectives: overall accuracy and the precision of inundation water depth at substations.

In the comparison of overall simulation results, the spatial prediction module exhibits superior accuracy in predicting inundation water depths across the majority of cells areas, as depicted in Figure 6. Specifically, over 95% of the research area (cumulative proportion, 95.3%) exhibits an absolute prediction difference of below 0.05 m. Within this high-accuracy range, areas with extremely low differences (<0.01 m) constitute the dominant portion, covering up to 73.5% of the total area. Notably, within this high-precision zone, the module performed exceptionally well, with predicted values that closely align with simulated values (R² = 0.9939) and with extremely low error levels (MAE = 0.013 m, RMSE = 0.032 m).

Special accuracy assessment for high-risk inundation areas with water depths exceeding 0.5 m (Figure 7) indicates that the model still demonstrates excellent predictive performance: a total of 93.47% of the grid area has a prediction error of less than 0.1 m, and the coefficient of determination R² reaches 0.9936. This result not only confirms the model’s high accuracy at the overall scale but, more importantly, proves its highly reliable predictive capability in high-risk and high-water-depth scenarios, thereby meeting the practical demand for accurate prediction of key areas in urban flood warning and emergency dispatch.

Further analysis of the eight substations showed that the spatial prediction module demonstrated high accuracy and stability in predicting inundation water depths at these locations, as summarized in

Table 7. Comparative analysis of inundation water depth simulated by HM and predicted by SM model.

Number	HM Model Water Depth (m)	SM Model Water Depth (m)	Difference (m)	R2	MAE (m)	RMSE (m)
D1	0.654	0.668	0.014	0.991	0.024	0.030
D2	0.597	0.613	0.016
D3	0.972	1.003	0.031
D4	0.994	1.057	0.063
D5	0.601	0.610	0.009
D6	0.540	0.560	0.02
D7	0.781	0.797	0.002
D8	0.985	1.023	0.038

. The predicted values for D1–D8 closely matched the HM model simulations, with an R² of 0.991, indicating that the module can explain 99.1% of the variation in water depth. In terms of error indexes, the MAE and RMSE were 0.024 m and 0.030 m, respectively, which validates the reliability of the module’s prediction. Specifically, the absolute differences between predicted and simulated water depths at individual stations range from 0.002 m to 0.063 m. Station D7 exhibited the most accurate prediction, with a difference of only 0.002 m. Even for station D4, which shows the largest difference (0.063 m), the relative error remained under 6.3%. It is worth noting that the module maintained stable predictive performance even under conditions of relatively higher inundation depths (e.g., D3: 0.972 m vs. 1.003 m; D8: 0.985 m vs. 1.023 m), demonstrating its good generalization capability.

Overall, the spatial prediction module exhibits superior accuracy in predicting spatial inundation water depths as evidenced by the evaluation results depicted in Figure 6 and Table 7.

3.2.2. Temporal Prediction Module

Based on testing of scenario P13, the performance of the temporal prediction module was evaluated by comparing its results with the temporal water depth processes at the substations generated by the HM model. The prediction accuracy of the module was analyzed based on the trend and precision of the temporal water depth processes at the eight substations.

As depicted in Figure 8, the temporal prediction module demonstrates its excellence in trend capturing and fitting capabilities. The prediction curve from the module exhibits a high degree of synchronization with the actual value curve, reflected in its accurate prediction of key turning points and near-exact reproduction of peak values, which indicates that the module not only effectively tracks the overall trend but also accurately delineates the local dynamic characteristics of the sequence.

Specifically, the temporal prediction module demonstrates robust comprehensive predictive capability across the temporal water depth processes at D1–D8. The R² values of all substations exceed 0.92, with more than half of them (D1, D2, D5, D7) above 0.99, indicating an extremely high goodness of fit. Simultaneously, for the vast majority of substations, both the MAE and RMSE remain at notably low levels (MAE generally <0.03 m; RMSE generally <0.05 m), which reflects the module’s minimal prediction errors and high prediction accuracy. This module successfully integrates CNN’s spatial feature extraction, the temporal sequence modeling of LSTM, and the high key-information focusing capability of the ATT mechanism, thereby achieving high-quality predictions.

To assess the potential impact of the local topography on the model performance, the correlation between the prediction accuracy and the elevation of the substations was analyzed. As shown in Figure 9, the temporal module maintains a high level of accuracy (R² > 0.92) in all substations at an elevation ranging from 27 to 65 m, which suggests there is no significant correlation between the two parameters.

3.3. Computational Efficiency

In this study, the computational efficiency of the SM model and HM model was benchmarked using the 100-year return-period rainfall scenario as the test case. Runtime measurements followed a fully reproducible protocol: (1) for the HM model, the execution time reported by its built-in timing function in the executable (.exe) was used; (2) for the SM model, timing was implemented using Python 3.11’s time.perf_counter() to measure the complete prediction pipeline, including model loading, data I/O, and computation; (3) all values represent the mean of five independent runs.

All simulations were conducted on an 11th-generation Intel (R) Core (TM) i7-11700@2.50 GHz CPU (with 16 GB memory). The results are shown in

Table 8. Comparison of model calculation speeds.

Model	CPU Execution Time	Speed Ratio (HM Model/SM Model)
HM model	35 ± 0.5 min	1:1860
SM model	1.15 ± 0.02 s

.

The SM model, which operates based on the collaborative work of its spatial and temporal prediction modules, can complete the prediction task for a single rainstorm water-logging event in 1.15 ± 0.02 s (mean ± SD, n = 5). Specifically, the spatial prediction module processes the inundation distribution for 251,450 grid points in 1.13 ± 0.02 s, while the temporal prediction module generates the water depth time-series process for a single substation point in 0.02 ± 0.002 s.

In contrast, the traditional HM model takes over 30 min to complete a simulation of equivalent scale, and the computation time rises as rainfall intensity increases. Measurements and computations indicate that the SM model achieves a computational speed approximately 1860 times faster than that of the HM model. This reduction transforms a simulation process that originally took tens of minutes to a second-level response meeting the practical needs for emergency flood prevention in power infrastructure in scenarios of extreme rainstorms.

4. Discussion

4.1. Advantages of the Surrogate Model

Traditional hydrological and hydrodynamic models can provide detailed delineation of physical processes of runoff generation and convergence with details, whereas their high computational costs severely restrict their application in early real-time warning and rapid multi-scenario evaluation. As pure data-driven models, although they have millisecond-level inference speed, they suffer from “black box” characteristics and unreliability in extrapolation scenarios, which restricts their use in critical decision-making. In this study, the hydrodynamic models are employed as “a physically consistent data generator,” ensuring that training data adheres to fundamental physical laws such as mass and momentum conservation. The maturely trained spatial and temporal prediction modules serve as “ultra-fast surrogate models” and can instantaneously generate inundation response under complex rainfall scenarios. As demonstrated in Figure 8 and Figure 9 and Table 7 and Table 8, with an R² value higher than 0.9, the surrogate model reduces the single simulation time from hours to seconds, laying a methodological foundation for real-time dynamic simulation and early warning systems of urban flooding.

4.2. Performance Analysis of Spatial and Temporal Prediction Modules

(1) MLP’s Capture Capability of Spatial Flooding Patterns

The performance of MLP exhibits distinct spatial heterogeneity, as shown in Figure 8. In flat topography areas, its prediction accuracy is extremely high (absolute difference below 0.05 m), whereas in areas near a river network and with rough topographical features, the prediction errors increase, which reveals the limitations of MLP. In essence, its fully connected structure learns an “averaged” response function, which indicates limited capacity to capture localized extremely variation features. Future improvements could be achieved by incorporating graph neural networks (GNNs) to better handle spatial heterogeneity and local interactions.

(2) Simulation Performance of the CNN-LSTM-ATT Model for temporal water depth processes

The CNN-LSTM-ATT model demonstrates remarkable robustness in predicting peak values across a range of tasks from D1 to D8. Nevertheless, distinct disparities exist in prediction accuracy of the model for the temporal process at different substations. For the high-accuracy station group (D1, D2, D5, D7), with R² > 0.99, the predicted and measured hydrographs virtually overlap. These stations are mainly located in areas with flat topography. The medium-accuracy station group (D3, D4, D8), with 0.969 < R² < 0.981, shows sound capture of peak values and trends but exhibits slight differences during the water rise or fall stage. The topography of the low-accuracy station (D6), with R² = 0.92, is significantly lower than those of the others. D6 is located in the downstream part of the research area, surrounded by low-lying topography areas and adjacent to a river confluence. For stations like D6, prediction results should be marked with higher uncertainty, and the utilization of physical models or human reviews is recommended.

4.3. Comparative Analysis with Existing Studies

The results of this study can be further specified by comparing our data, methods, and performance with those in the related research on rapid flood prediction.

First, in terms of dataset, many influential deep learning models are trained based on limited historical events [23] or a restricted set of design rainfall scenarios (e.g., six to eight return periods) [10,11]. In contrast, the training dataset of our surrogate model is generated by a high-precision hydrodynamic model covering 15 design scenarios. This approach ensures physical consistency and controllable diversity, directly addressing the common problem of data scarcity of extreme events.

Second, in terms of methodology, although purely data-driven models are fast, they are frequently criticized for their lack of physical interpretability and unreliable extrapolation, which is a widely recognized limitation in this field [24]. Our work adopts a hybrid paradigm, using mechanism models as physically constrained data generators and deep learning models as ultra-fast surrogate models.

Finally, in terms of performance and applicability, the research varies in target setting, ranging from regional flood inundation mapping to point-scale prediction [25]. Our model is specifically designed for critical infrastructure (substations) and is capable of achieving facility-level prediction with an R² value greater than 0.9 and a speed increase of approximately 1860 times that of high-precision models. However, this study has not yet conducted threshold-based evaluations for high-risk areas and power grid assets. This is a field that needs to be further explored in the future.

4.4. Uncertainties and Limitations of the Research

(1) Sample Representation Uncertainty

Although, in this study, the rainfall scenarios (P1–P15) have been designed to cover various return periods, the spatiotemporal patterns of real rainfall demonstrate virtually infinite diversity. The samples are inevitably insufficient to cover extremely rare events or rainfall with atypical spatial distribution patterns. In particular, the assumption of spatially uniform rainfall adopted in this study simplifies the representation of actual storm events; real rainfall exhibits significant spatial heterogeneity that can critically influence urban inundation patterns. This simplification may affect the transferability of the model to real storms with non-uniform rainfall distributions.

The currently used hydrological sequences are based on historical observation data and the assumption of robustness, both of which have inherent limitations. First, historical rainfall records with limited duration may not fully capture the true statistical characteristics of extreme climatic events, leading to biases in estimation of low-frequency and high-intensity extreme rainfall events. Second, under the background of global climate change, hydro-meteorological systems exhibit distinct non-stationary characteristics, which challenges the concept of “return period” based on historical patterns and may result in a systematic underestimation of the risks of future extreme scenarios.

To address the limitations related to spatial uniformity, future work should include at least one non-uniform rainfall case in the training or validation set or implement a concrete validation plan that explicitly tests model performance under spatially heterogeneous rainfall conditions. Such a plan could involve the following: (i) using radar-based rainfall fields from historical storm events as input, (ii) comparing the predicted inundation patterns with those simulated by a hydrodynamic model under the same spatially variable rainfall, and (iii) evaluating the model’s sensitivity to key spatial rainfall features (e.g., rainfall gradient, location of peak intensity).

It is suggested that future research incorporate rainfall prediction sequences under multiple climate models to assess the impact of climate change on flood risks in substations [26] and simultaneously extend the scenario library to include spatially non-uniform rainfall patterns to enhance the model’s robustness and real-world applicability.

(2) Impact of Hydrodynamic Model Uncertainties and Their Propagation

The HM model, which serves as an alternative to the foundation for training the SM model, was validated primarily based on a single extreme rainfall event (the July 2023 event) and a limited number of monitoring points. Although this event is representative, and the validation results are satisfactory, validation based on a single event is insufficient to fully confirm the model’s generalizability across all potential hydrological conditions.

A key limitation is that simulation errors caused by HM model parameter uncertainties (the variability of empirical parameters such as Manning’s roughness coefficient [27]) enter the SM model training set directly as “noisy labels.” For instance, if the HM model systematically overestimates water depth due to an underestimated roughness parameter, the SM model will learn this biased mapping, leading to a similar overestimation tendency in all its predictions. This error-propagation mechanism effectively caps the upper bound of the SM model’s prediction accuracy at the validation reliability of the HM model.

Future work should aim to collect observed data from more historical flood events of various types and magnitudes to enable more extensive calibration and validation of the mechanistic model. This would further solidify its robustness as a reliable “physical data generator,” thereby enhancing the reliability and extrapolation capability of the overall deep learning surrogate modeling framework.

(3) Uncertainty in Surrogate Models

Although the surrogate model architectures are powerful, they may not perfectly capture certain complex hydrological physical processes. Specifically, the temporal prediction module exhibits lower accuracy when simulating flood inundation under smaller return-period rainfall events (e.g., 1- to 10-year scenarios), which suggests that the module’s capability to capture low-intensity rainfall-water depth responses requires further enhancement. Even given an optimal architecture, limited training data and optimization algorithms can lead to learned weight parameters that are not the “true” optimal solution but rather a single estimation of the posterior distribution of the parameters. Future work should be dedicated to developing more physics-aware neural network architectures capable of handling heterogeneous relationships and, meanwhile, incorporating domain-specific knowledge to design targeted features, thus aiming to approximate or even surpass the ability of physical models to simulate complex system behaviors while maintaining computational efficiency [28].

5. Conclusions

In this study, high-precision cell data simulated by the self-developed fully distributed model named DHMUrban model are utilized as the driving data to construct a rapid prediction model for urban rainstorm floods targeting power grid facilities based on deep learning methods. The main conclusions are as follows:

(1)

The constructed HM model, validated based on measured data from the July 2023 event, demonstrates strong agreement between simulation results and measured data, which indicates that the model possesses high reliability and applicability within the research area.

(2)

Compared to the simulation results of the HM model, the SM model exhibits superior performance in prediction accuracy. The spatial prediction module achieved an R² of 0.9939 in predicting overall inundation water depth distribution, with a mean absolute error of only 0.013 m. The temporal prediction module achieved R² values above 0.92 at all substations in predicting temporal water depth processes, with more than half of them (D1, D2, D5, D7) exceeding 0.99, demonstrating an extremely high fitting quality.

(3)

Based on deep learning, the SM model provides rapid and accurate prediction capabilities targeting power grid facilities under extreme rainfall scenarios similar to those in its training set. Upon inputting rainfall data, the model can dynamically simulate water-logging distribution and water depth at power grid facilities within seconds, representing a significant efficiency gain over traditional numerical simulations. Consequently, the SM model enables scenario-specific rapid assessment of flood risk levels for critical infrastructure such as substations, thereby contributing to a shift from passive response towards more proactive and precise defense.