1. Introduction
With the rapid development of power systems and the continuous growth in load demand, the operational reliability of power equipment has become a crucial factor affecting the safe and stable operation of the power grid [
1]. In modern power infrastructure, key components such as transformers and current transformers extensively use insulating oil for electrical insulation and thermal dissipation. However, during long-term operation, issues such as seal degradation, structural fatigue, and external corrosion can result in oil leakage over the surface of the equipment. Such leakage not only compromises the insulation performance but also introduces significant safety hazards, including short circuits, electrical arcing, and even equipment failure with fire. In addition, oil spills may lead to soil pollution, causing serious environmental repercussions [
2,
3,
4]. Therefore, the prompt and accurate detection of oil leakage in power equipment is essential to ensure the safe and reliable operation of the power system.
Currently, the detection of insulating oil leakage in power equipment primarily relies on three types of methods: manual inspection, sensor-based approaches, and image recognition techniques. Manual inspection entails visually examining the external surfaces of electrical equipment and surrounding areas for oil stains as indicators of leakage. However, this method is characterized by low efficiency, a high likelihood of missed or false detections, and limited accuracy in identifying subtle or irregular oil stain targets [
5,
6]. Sensor-based methods utilize a variety of devices, such as ultrasonic, infrared, fiber-optic, and gas-sensitive sensors, to monitor parameters like acoustic signals, temperature, gas composition, and pressure within the insulating oil in real time. These measurements are then analyzed to assess potential leakage events. For example, Qian et al. employed ultrasonic sensors to capture acoustic signals, which were used to evaluate the condition of transformer insulating oil and infer leakage indirectly [
7]. Similarly, Hashim et al. applied humidity and acoustic pattern sensors for detecting leaks in pipelines [
8]. Despite their theoretical advantages, sensor-based techniques are often susceptible to environmental fluctuations, which can degrade their performance and limit reliability in real-world applications.
With the widespread deployment of inspection robots and video surveillance systems, large volumes of images can now be collected without relying on manual inspections [
9,
10]. Image-based analysis enables the early detection of oil leakage, thereby contributing to the safe operation of power equipment. For instance, Xia et al. proposed an oil leakage detection method based on ultraviolet-induced hyperspectral imaging, combining fluorescence–reflectance spectral features with principal component analysis to identify and track various types of insulating oil leakage [
11]. Similarly, Lu et al. utilized the fluorescence emitted by insulating oil under ultraviolet illumination, applying the relationship between saturation and brightness in color space to detect leakage in images [
12]. However, these methods still largely depend on manual image interpretation. Prolonged visual analysis may lead to operator fatigue, ultimately compromising detection accuracy and efficiency.
With rapid advancements in computer vision, deep learning-based image recognition techniques have emerged as practical methods to enhance detection precision and visualization capabilities, offering new possibilities for intelligent oil leakage identification in power systems. Object detection methods in this domain are generally classified into two paradigms: two-stage detectors, such as Faster R-CNN (Faster Region-based Convolutional Neural Network), which separate region proposal and classification to achieve high accuracy [
13]; and one-stage detectors, such as YOLO (You Only Look Once) and SSD (Single Shot MultiBox Detector), which unify localization and classification in a single end-to-end architecture for real-time performance [
14,
15,
16,
17]. For example, Yang et al. enhanced the Faster R-CNN model using a ResNet-101 backbone to detect oil leakage defects in power equipment [
2,
18]. Ji et al. introduced a model based on the PFDAL-DETR (Progressive Feature Decoupling and Alignment Learning with DETR) framework, integrating DETR’s end-to-end detection capabilities with decoupling and alignment modules to improve performance under structurally complex scenarios [
19,
20,
21]. Although these models achieve high detection accuracy, their high computational complexity may increase inference latency and reduce their practicality in real-time power equipment inspection. In recent years, YOLO-based one-stage detection algorithms have gained popularity in practical applications due to their fast inference speed, low deployment cost, and strong real-time capabilities. Progressive improvements in versions such as YOLOv5, YOLOv8, and YOLOv11 have led to architectural refinements and performance enhancements—especially for small object detection [
22,
23,
24,
25]. Nonetheless, despite their strong performance in general object detection tasks, YOLO models face difficulties in detecting small-scale oil stains and maintaining robustness under complex visual conditions. Moreover, although YOLO architectures are relatively lightweight, further optimization is still required to reduce computational cost while maintaining detection accuracy. To address these issues, Luo et al. proposed LA-YOLOv8s, a lightweight model tailored for insulating oil leakage detection. This model redesigns the backbone and neck structures to significantly reduce parameters and computational overhead while maintaining detection performance [
26]. However, the dataset used in the study was relatively limited in scope, lacking comprehensive coverage of key scenarios such as equipment occlusion, complex backgrounds, small object detection, and multi-target detection. As a result, the model’s generalizability in real-world settings remains insufficiently validated. Given these limitations, improving the detection precision of lightweight models, particularly for small scale and multiple leakage targets in visually complex environments, has become a central focus of current research efforts. In contrast, YOLOv11s demonstrates significant advantages in terms of model compactness, lightweight design, and high inference speed, making it suitable for real-time monitoring applications in power systems, where low-latency image processing is required. Based on the above review,
Table 1 summarizes the main categories of existing oil leakage detection methods.
YOLOv11s is selected as the baseline model because it provides a favorable balance between detection accuracy and computational efficiency. Its one-stage detection framework enables fast end-to-end inference, which is important for real-time inspection in power equipment monitoring scenarios. In addition, the C3k2, SPPF, and C2PSA modules in YOLOv11s contribute to efficient feature extraction, multi-scale perception, and attention-guided representation learning. These characteristics make YOLOv11s suitable for detecting small, irregular, and low-contrast oil leakage targets in complex power equipment environments, while leaving sufficient room for further improvement through lightweight attention and localization optimization.
This study presents an improved method for the intelligent detection of oil leakage in power equipment, based on an enhanced YOLOv11s architecture. The proposed enhancements incorporate two key technical innovations. First, the original SimAM (Simple Attention Module) [
27,
28,
29] is extended by introducing a channel slicing strategy and localized normalized energy modeling, resulting in the development of SimAMWS (Simple Attention Module With Slicing). This module is embedded within the YOLOv11s backbone. By generating independent spatial attention maps for each channel, this content-aware attention mechanism enables fine-grained focus on different image regions, thereby improving the model’s ability to extract features from small targets. This enhancement is designed to address common challenges such as reduced detection accuracy in scenarios involving complex backgrounds or equipment occlusion. Second, the U-IoU (Unified Intersection over Union) loss function [
30] is integrated into the bounding box regression module. This loss function dynamically scales the predicted bounding boxes during training, guiding the model to prioritize low-quality boxes for faster convergence in the early training stages, while gradually shifting attention to high-quality boxes for enhanced localization accuracy in later stages. This design improves detection performance, particularly in scenarios involving small-scale and multiple leakage targets. To validate the effectiveness of the proposed approach, a dedicated image dataset was constructed, covering four typical operational conditions: equipment occlusion, complex background interference, small object instances, and multi-target scenarios. The dataset simulates the visual disturbances and target variability encountered in real-world power equipment environments, providing a solid foundation for model training and evaluation. Its development enhances the practical applicability of the proposed method and supports future application in power equipment inspection systems. To address these problems, this paper proposes SA-YOLOv11s, whose core idea is to improve local feature perception and bounding box localization within a lightweight detection framework. Specifically, a slicing-based attention mechanism is introduced to enhance the representation of small and irregular oil leakage regions, and a U-IoU loss is adopted to dynamically optimize bounding box regression. These designs enable the model to improve detection accuracy in complex backgrounds, occlusion, and multi-target scenarios while preserving real-time performance.
The main contributions are as follows:
1. A lightweight oil leakage detection model, SA-YOLOv11s, is proposed for power equipment inspection.
2. A SimAMWS slicing-attention module is designed to enhance local feature responses for small and irregular oil leakage targets.
3. A U-IoU loss function is introduced to improve bounding box regression and localization accuracy.
4. A power equipment oil leakage dataset is constructed, and extensive experiments verify the effectiveness and real-time performance of the proposed method.
4. Experimental Results and Analysis
4.1. Sensitivity Analysis of SimAMWS Segmentation Configuration
To justify the choice of the 3 × 3 segmentation strategy in SimAMWS, a sensitivity analysis was conducted using different slicing configurations. In this experiment, only the segmentation configuration of SimAMWS was changed, while the network structure, training parameters, dataset split, and evaluation settings were kept the same. U-IoU was not included in this comparison, so that the influence of the slicing strategy could be evaluated independently. For an input image of 416 × 416 pixels, the 2 × 2, 3 × 3, and 4 × 4 configurations correspond to local regions of approximately 208.0 × 208.0, 138.7 × 138.7, and 104.0 × 104.0 pixels, respectively. The 2 × 2 setting preserves more spatial context, but each region still contains a relatively large amount of background information. The 4 × 4 setting provides finer local regions, but it may fragment small and irregular oil leakage areas and introduce additional computational cost. Therefore, the 3 × 3 setting was evaluated as a moderate configuration between local feature enhancement and spatial context preservation.
Table 4 shows that the 3 × 3 configuration provides the best overall performance. The 2 × 2 setting has lower accuracy and FPS than 3 × 3. This is mainly because each sub-region in 2 × 2 is relatively large and still contains considerable background information, which weakens the local attention response to small oil leakage regions. The 4 × 4 setting achieves the highest FPS, but its accuracy decreases because overly fine segmentation may fragment small and irregular leakage targets and reduce the surrounding context needed for discrimination. Therefore, 3 × 3 was adopted as a balanced configuration between local feature enhancement, spatial context preservation, localization capability, and computational efficiency.
4.2. Computational Cost and Comparison with Lightweight Attention Modules
To further evaluate the computational cost of the proposed SimAMWS module, GPU memory consumption and inference latency were measured on the same NVIDIA RTX 4060 platform. All models were tested with an input size of 416 × 416 under the same inference settings. Since the purpose of this comparison is to analyze the attention module itself, only YOLOv11s, YOLOv11s + SimAM, and YOLOv11s + SimAMWS are compared. The results are shown in
Table 5.
As shown in
Table 5, YOLOv11s has the lowest GPU memory consumption and latency. This is expected because it does not include an additional attention module. The original SimAM increases memory consumption to 746 MB and latency to 11.56 ms, indicating a higher computational burden. In contrast, SimAMWS reduces memory consumption to 681 MB and latency to 10.50 ms compared with SimAM. These results show that SimAMWS is more efficient than the original SimAM module. However, it still introduces a small additional cost compared with the baseline YOLOv11s due to feature-map slicing, regional attention calculation, and feature recombination. Therefore, SimAMWS provides a better accuracy–efficiency trade-off than SimAM, rather than completely eliminating the computational overhead of attention calculation.
4.3. Comparison with Mainstream Models
To comprehensively assess the proposed model’s computational efficiency and structural complexity, a comparative analysis was conducted against mainstream YOLO-based object detection models, including YOLOv5s, YOLOv8s, and YOLOv11s. The comparison focused on three key aspects: the number of parameters, the network depth (i.e., the number of layers), and the number of floating-point operations (GFLOPs) required for forward inference [
48].
The number of parameters refers to the total count of all learnable weights within the model. The number of layers represents the complete sequence of computational operations from input to output, primarily including convolutional layers, pooling layers, and fully connected layers. GFLOPs (Giga Floating Point Operations) quantify the total number of floating-point operations—specifically multiply–accumulate operations—required to process a single image, expressed in billions (10
9) of FLOPs. These metrics collectively reflect the model’s level of architectural efficiency and computational overhead, providing a quantitative basis for evaluating its suitability for real-world deployment. All measurements were obtained using the torchsummary tool within the PyTorch framework. The results are summarized in
Table 6.
As shown in
Table 6, YOLOv11s achieves a reduction in both the number of convolutional kernels and the overall channel width by incorporating lightweight modules in place of conventional convolutional structures, combined with an optimized feature fusion strategy. In the proposed model, although both the SimAMWS attention mechanism and the U-IoU loss function are integrated—each being simple and parameter-free—only a slight increase in network depth is observed, and the GFLOPs rises marginally to 21.8. This indicates that the architectural enhancements do not introduce model redundancy. Compared to YOLOv8s, which has a significantly higher parameter count, the proposed model reduces GFLOPs by more than 24%, demonstrating superior computational efficiency and greater suitability for real-time inspection in real-world applications.
To evaluate the convergence behavior and training stability of the proposed model, this study compares the loss function trends of several mainstream object detection algorithms during both the training and validation phases. The selected models include the classical SSD and Faster R-CNN, as well as the more recent and high-performing YOLOv5s, YOLOv8s, YOLOv11s, and the proposed improved model.
Figure 5 illustrates the variation in loss values on the training and validation sets as a function of the number of training iterations for each model.
Figure 5a and b respectively illustrate the loss function trends for each model on the training and validation sets. Overall, all models show a sharp decline in loss during the first 30 epochs, followed by gradual stabilization—indicating successful initial convergence. However, a more detailed comparison reveals that the proposed model achieves a significant reduction in loss within the first 10 epochs, demonstrating a notably faster convergence rate than the other models. This efficiency can be attributed to the dynamic weighting mechanism of the U-IoU loss function, which prioritizes the optimization of low-quality bounding boxes in the early training stages and progressively shifts focus toward high-quality boxes in later stages. This strategy enables more efficient gradient utilization and supports a gradually refined learning objective. Clear differences are also observed in the final steady-state loss values across models. The proposed model achieves a training loss stabilized around 0.55 and a validation loss near 0.58, both outperforming the baselines. In contrast, YOLOv5s and SSD stabilize at considerably higher values—approximately 0.9/1.0 and 2.0/2.2, respectively—demonstrating the superior convergence behavior and generalization capability of the proposed approach. Further analysis of the training phase between epochs 120 and 150 reveals that the amplitude of loss fluctuations reflects the model’s gradient stability post-convergence. Smaller fluctuations indicate smoother parameter updates and more complete convergence. Among all the compared models, the proposed model exhibits the smallest fluctuation amplitude, indicating superior training stability.
To further validate the overall performance advantages of the proposed model in the task of insulating oil leakage detection, a comparative evaluation was conducted against several mainstream object detection models, including SSD, Faster R-CNN, YOLOv5s, YOLOv8s, and YOLOv11s. The comparison was based on five key metrics: precision (Pre), recall (R), mean Average Precision at IoU threshold 0.5 (mAP@0.5), mean Average Precision across IoU thresholds from 0.5 to 0.95 (mAP@0.5:0.95), and frames per second (FPS). The detailed results are presented in
Table 7.
As shown in
Table 7, the proposed model achieves the best performance across all accuracy-related metrics. Specifically, it reaches an mAP@0.5 of 97.7% and an mAP@0.5:0.95 of 66.9%, representing improvements of 1.4% and 2.9%, respectively, over the baseline YOLOv11s model. Moreover, despite the integration of an attention mechanism and a refined loss function, the model achieves an inference speed of 96.4 FPS—surpassing both YOLOv5s and YOLOv8s—which demonstrates its strong capability for real-time detection. Overall, the results confirm that the proposed enhancements improve detection accuracy while maintaining real-time performance and training stability, making the model suitable for deployment in power equipment inspection tasks. To further evaluate the model’s robustness, five independent training experiments were conducted. The corresponding results are presented in
Figure 6 and
Figure 7.
As shown in the figures, the proposed model not only achieves the highest scores across four accuracy-related metrics—precision, recall, mAP@0.5, and mAP@0.5:0.95—but also exhibits significantly shorter error bars compared to the baseline models. This indicates that the model maintains stable performance under varying training conditions. In contrast, although SSD and Faster R-CNN demonstrate certain advantages in recall and FPS, they exhibit noticeably larger error ranges, reflecting limited robustness. Specifically, for the mAP@0.5:0.95 metric, the proposed model maintains an error margin within ±1.0%, whereas Faster R-CNN and YOLOv5s exceed ±1.5%, indicating that the proposed method achieves more accurate and consistent target localization. Furthermore, despite incorporating an attention mechanism and an optimized loss function, the proposed model sustains a high inference speed of 96.4 FPS, with variability confined to ±2.0 FPS. This demonstrates that performance gains were achieved without significantly compromising inference efficiency. Through multiple experimental repetitions and comprehensive error analysis, it is evident that the proposed model consistently outperforms other methods across all evaluation metrics, making it suitable for complex power equipment inspection tasks that demand both high accuracy and high reliability.
4.4. Ablation Study
To evaluate the individual contributions of the SimAMWS attention module and the U-IoU bounding box loss function to the overall detection performance, a series of systematic ablation experiments were conducted based on the YOLOv11s backbone. These experiments compared different combinations of the two components, and the resulting performance was analyzed across five key evaluation metrics. The detailed results are summarized in
Table 8.
Table 8 reports the ablation results of the proposed modules. When only U-IoU is introduced, Precision, Recall, mAP@0.5, and mAP@0.5:0.95 increase from 91.6%, 93.0%, 93.3%, and 64.0% to 93.0%, 94.0%, 94.7%, and 65.4%, respectively. Since the network architecture is unchanged, this improvement is mainly attributed to better bounding box regression during training. The FPS remains close to that of the baseline, which is expected because U-IoU is not involved in the inference stage. The standard SimAM module improves mAP@0.5 to 95.2%, but the FPS drops to 86.5. This indicates that its attention calculation brings additional computational cost. In comparison, SimAMWS achieves 96.8% mAP@0.5 and 65.5% mAP@0.5:0.95 while maintaining 95.2 FPS. The result suggests that the slicing strategy improves local feature representation for leakage regions with a smaller impact on inference speed. The best performance is obtained when SimAMWS and U-IoU are used together. The final SA-YOLOv11s model achieves 96.4% Precision, 95.8% Recall, 97.7% mAP@0.5, and 66.9% mAP@0.5:0.95. Compared with the SimAMWS-only model, mAP@0.5 and mAP@0.5:0.95 are further improved by 0.9 and 1.4 percentage points, respectively. The slight FPS difference between SimAMWS and SimAMWS + U-IoU is within the normal fluctuation of inference-time measurement, rather than a change caused by the loss function. Overall, compared with the baseline YOLOv11s, SA-YOLOv11s improves mAP@0.5 by 4.4 percentage points and mAP@0.5:0.95 by 2.9 percentage points, with only a 2.2 FPS decrease. This indicates a practical trade-off between detection accuracy and real-time performance.
To reduce the influence of a single train-validation-test split, a repeated random split experiment was conducted for SA-YOLOv11s. The dataset was randomly divided five times using the same 8:1:1 ratio, and the model was trained and evaluated under identical parameter settings. Since the purpose of this experiment is to verify the stability of the reported performance of the proposed model, only SA-YOLOv11s was evaluated in this repeated-split setting. The repeated random split results for SA-YOLOv11s are summarized in
Table 9.
4.5. Detection Result Visualization
To further validate the practicality and robustness of the proposed model under diverse and complex conditions, four representative working scenarios for insulating oil leakage detection in power equipment were selected for visual comparison experiments. These include: (1) small object detection, (2) complex background, (3) multi-object detection, and (4) equipment occlusion. These scenarios closely reflect the typical challenges encountered in actual substation environments, making them both representative and realistic. To comprehensively assess the model’s generalization ability, five mainstream object detection models were selected as baselines. Visual comparisons were conducted on the same set of images, contrasting their detection outputs with those of the proposed model. The results are presented in
Figure 8.
In the visualizations, the confidence score reflects the model’s certainty in its predictions—higher values indicate greater confidence that the predicted region contains a true target. As observed, the proposed model performs particularly well in small object detection tasks. Even for extremely small leakage regions located at the image corners—occupying less than 2% of the total image area—the model is able to accurately localize the target with a high confidence score of 0.86, whereas other models either assign low confidence scores or fail to detect the target altogether. In scenarios involving complex background interference, such as stone-paved grounds where the color and texture closely resemble those of oil stains, most models are susceptible to background noise, leading to false positives or significant drops in confidence. In contrast, the proposed model maintains a high confidence level of 0.97, demonstrating strong robustness against background complexity.
In multi-target detection scenarios, where multiple leakage regions are densely distributed and some targets are located in close proximity, conventional models such as SSD tend to generate overlapping bounding boxes, leading to reduced detection clarity. In contrast, the proposed model consistently detects all targets with clear separation and maintains an average confidence score above 0.93, demonstrating strong capability in handling densely packed scenes. In equipment occlusion scenarios—where portions of the leakage areas are obscured by structural components or affected by strong reflective surfaces—Faster R-CNN fails to detect certain targets, while YOLOv11s exhibits noticeable misalignment of bounding boxes. The proposed model, however, accurately localizes even partially occluded leakage regions, with confidence scores consistently around 0.92, underscoring its robustness in complex industrial visual environments.
Comprehensive analysis indicates that the proposed model achieves accurate detection of small, densely distributed, and partially occluded targets across all four representative scenarios, consistently outperforming mainstream object detection algorithms. These results validate the effectiveness of the proposed improvements in enhancing detection precision under complex and challenging conditions. To more intuitively demonstrate the feature attention capabilities of different models under various detection scenarios, this study introduces Grad-CAM visualization for comparative analysis [
49]. The resulting attention heatmaps reveal how each model responds to representative scenes, as shown in the figure. The selected scenarios are consistent with the previous visualizations and cover four typical conditions: small objects, complex backgrounds, multiple targets, and equipment occlusion. The comparison includes the proposed improved model alongside five mainstream object detection models: SSD, Faster R-CNN, YOLOv11s, YOLOv8s, and YOLOv5s.
As shown in
Figure 9, SA-YOLOv11s produces more concentrated activation regions around oil leakage areas than the comparison models. In small-target and complex-background scenes, the heatmaps of SSD and YOLOv5s are more dispersed, while the proposed model shows stronger responses near the annotated leakage regions. In multi-target scenes, SA-YOLOv11s separates adjacent leakage regions more clearly. In occlusion scenes, the activation regions are mainly distributed around the visible parts of the leakage area. These results suggest that SimAMWS helps the network focus on leakage-related local features under challenging visual conditions. Nevertheless, missed detections and false detections may still occur in some difficult scenes. For example, extremely small or heavily occluded oil stains may not be recognized, and low-contrast leakage regions may only be partially detected. In addition, rust, dust, shadows, reflections, or water-like marks may be confused with oil leakage. These cases show that SA-YOLOv11s still needs to be improved under complex field conditions.
5. Conclusions
This study proposes SA-YOLOv11s for insulating oil leakage detection in power equipment. The method is designed to address common detection difficulties in field images, including small leakage regions, dense oil stains, equipment occlusion, and complex background interference. Based on YOLOv11s, two targeted improvements are introduced: the SimAMWS attention module and the U-IoU loss function.
(1) The main contribution of SimAMWS is to improve local feature representation. By dividing the feature map into local regions and calculating attention responses within each region, SimAMWS helps the model focus on small and irregular leakage areas while reducing the influence of large background regions. Compared with the original SimAM module, the slicing strategy reduces part of the attention-related computational cost.
(2) U-IoU further improves the bounding box regression process. Its dynamic scaling strategy provides a coarse-to-fine optimization mechanism during training. In the early stage, the expanded predicted boxes improve the tolerance of low-quality predictions and help stabilize training. In the later stage, the contracted boxes impose stricter localization constraints, encouraging more accurate boundary alignment. This is useful for oil leakage targets with small size, irregular shape, and partial occlusion.
(3) Experimental results show that SA-YOLOv11s achieves 97.7% mAP@0.5 and 66.9% mAP@0.5:0.95, outperforming the baseline YOLOv11s and other compared detection models. The model contains 9.46 million parameters and requires 21.8 GFLOPs, while maintaining an inference speed of 96.4 FPS. Although the FPS is slightly lower than that of the original YOLOv11s, the decrease is small compared with the improvement in detection accuracy. These results indicate that the proposed method provides a practical balance between accuracy and real-time performance for power equipment inspection.
(4) The visualization results and Grad-CAM heatmaps show that SA-YOLOv11s produces more concentrated responses around leakage-related regions under small-target, complex-background, and occlusion conditions. This supports the effectiveness of combining local attention enhancement with improved bounding box regression.
Several limitations should be noted.
(1) This study focuses only on insulating oil leakage in power equipment. Other liquids, such as rainwater, condensed water, cleaning liquid, or lubricant stains, were not used as independent detection categories. Therefore, the proposed model should be regarded as an oil leakage detector rather than a general liquid-stain classifier. In real inspection scenes, visually similar regions such as water marks, rust, dust, shadows, and reflections may still lead to false detections.
(2) The dataset size is still limited compared with large-scale object detection datasets. Although the repeated random split experiment reduces the influence of a single train-test split, it cannot fully replace validation on larger independent field datasets. Moreover, the proposed model has not yet been fully evaluated on external datasets or completely unseen substations. Future work will collect more independent field samples from additional substations, equipment types, weather conditions, and imaging devices to evaluate the real-world generalization ability of the proposed model.