The proposed CTCTN structure consists of four key parts: DS-TC block, AAP, Transformer module, Multi-Head Attention Pooling (MHAP). These parts collectively facilitate hierarchical feature extraction and cross-scale fusion, enhancing the representation of degradation patterns.

The DS-TC block preserves the temporal modeling capabilities of conventional TCN while improving computational efficiency through the depthwise separable convolutions. Serving as the foundational unit of the DS-TC block, the dilated causal convolution (DCC) expands the receptive field without increasing the kernel size by adjusting the dilation rate. The receptive field comparison with standard convolution is shown in Fig. 2.

The causal convolution structure enables the network to learn historical information in the temporal data and avoid the effect of future information. Notably, this study incorporates depthwise and pointwise convolutions into the DCC, resulting in a depthwise separable convolution structure, as illustrated in Fig. 3.

Under identical input-output conditions, the computational cost of traditional convolution is given by Dc=DF2×DK2×M×N, whereas the cost of depthwise separable convolution is expressed as DDSC=DDepth+DPoint=DF2×DK2×M+DF2×M×N. The computational cost ratio between the two convolution types is 1/N+1/DK×DK, demonstrating that depthwise separable convolution effectively reduces the computational burden. Following this, batch normalization and activation functions are incorporated via residual skip connections to form a DS-TC block, as illustrated in Fig. 4. A depthwise separable convolution consists of a depthwise convolution followed by a pointwise convolution. In this study, the depthwise convolution is implemented as a DCC. By stacking i DS-TC blocks with dilation rates set to 2i in each block, multi-scale feature extraction is achieved.

A dimension mismatch exists between the output features of the DS-TC blocks and the input features of the cascaded Transformer modules, which is addressed by AAP. Specifically, the length of the input temporal sequences varies depending on the task, leading to a corresponding change in the output feature dimension of the DS-TC blocks. Additionally, the number of stacked layers in both the DS-TC blocks and Transformer modules, as well as the parameters within the network, contribute to the mismatch in feature dimensions. Considering that the output feature dimension of the DS-TC block is (B,C,T), where B denotes batch size, C represents the number of channels, and T stands the time steps, the AAP is used. It performs global average pooling across the T time steps for each channel, downsampling the temporal dimension to a fixed length DTransformer which matches the input dimension of the Transformer module. Therefore, the cascading mechanism in this paper is expressed as HTransformer=AAP(HDS-TC). The output of the DS TC block HDS-TC is connected to the input of the Transformer module HTransformer through AAP. Moreover, global average pooling not only preserves the key features of the channels but also suppresses noise interference, which is particularly advantageous for processing data from sensors exposed to space environments.

The Transformer module receives features from the AAP to capture global dependencies. The structure of Transformer module is shown in Fig. 5. The DS-TC block reduces the dimension of the temporal sequence, thereby reducing the computational cost of the Transformer module. Furthermore, AAP further aligns the feature dimensions, ensuring compatibility with the input requirements of the Transformer module. Initially, the features are added to the Transformer module through positional encoding. It encodes the positional information of the input sequence, enabling the network to learn how relative positions affect the structure, thus allowing it to extract temporal features. Global dependencies across different timesteps within the features are then captured through stacked Transformer encoders. The self-attention mechanism in Transformer enables each time step to attend to all other steps simultaneously, breaking the limitation of local receptive fields in traditional CNN or RNN. This is particularly beneficial for satellite data, where long-period events (e.g., orbital period) require modeling dependencies across many time steps. In multi-head attention setup, each head can learn to focus on different temporal scale, enabling the Transformer to process multi-scale features present in satellite failure scenarios. Notably, the activation function used in the Transformer encoders in this study is GELU which is preferred in attention-based Transformer architectures due to its beneficial properties, including nonlinearity, differentiability, and smoothness. These unique properties contribute significantly to training stability of the network. The output features of the stacked Transformer encoders in Fig. 5 form a tensor of shape (B,DTransformer,CTransformer), where B denotes the batch size, DTransformer the sequence length, and CTransformer the feature dimension determined by the Transformer’s hidden dimension. This representation preserves global contextual information and is subsequently fed into the MHAP module for further feature aggregation.

MHAP extracts multi-dimensional features from the Transformer output and a following fully connected layer outputs predicted mean and variance of RUL. Unlike conventional average or max pooling, it introduces a learnable query vector to compute a weighted sum over the entire sequence, leveraging the multi-head attention mechanism for enhanced feature fusion. Specifically, the input sequence x and the expanded learnable query vector are linearly projected as follows. (6){Qh=query⋅WhQKh=x⋅WhKVh=x⋅WhVwhere WhQ, WhK, and WhV are trainable weight matrices that project the input into query, key, and value representations for each individual attention head. The attention scores for each head are normalized as follows. (7)Attentionh=softmax(QhKhT/dK)where dK donates the dimension of the key vectors, which is used to prevent gradient explosion. The score matrix encodes the similarity weights between the query and each position in the sequence. Notably, the attention distribution learned by each head differs, incorporating multi-dimensional degradation features. Subsequently, the value vectors are weighted and combined using the attention scores to generate the output vector for each head, as follows. (8)Outputh=Attentionh⋅Vhfinally, the concatenated outputs from all heads are integrated into the attention output through a linear layer, as follows. (9)Output=Concat(Output1,Output2,⋯,OutputH)⋅WOwhere WO is the output weight matrix. The output features are subsequently passed through a fully connected layer, yielding a two-dimensional vector representing the predicted mean and variance of RUL.

To quantify the uncertainty arising from multi-scale features, this paper proposes a method framework for quantifying both epistemic and aleatoric uncertainty, comprising two stages: network training and testing. During training, the outputs of the CTCTN are used as inputs to a heteroscedastic Huber loss function for network parameter optimization. During testing, epistemic and aleatoric uncertainties are then quantified through the MCD method to calculate confidence intervals.

In the training stage, the heteroscedastic Huber loss function is designed to optimize the the mean μ and variance σ2 output branches of the CTCTN. The network’s prediction is assumed to follow a Gaussian distribution: p(y|x)=N[μ(x),σ2(x)], where x denotes the input data and y corresponds to the predicted output [43,44]. To train this dual-output network, a heteroscedastic Huber loss function is given as follows. (10)Ltotal=Lhuber+αLuncertaintyLhuber=1/N∑i=1N{1/2(μi−yi)2,|μi−yi|<δδ(|μi−yi|−1/2δ2),|μi−yi|≥δLuncertainty=1/N∑i=1N{1/2[σi2+(μi−yi)2/σi2]+1/10ln⁡(1+σi2)}where Lhuber is the Huber loss and Luncertainty is a variance-based uncertainty constraint. The Huber loss [45], tailored for regression tasks, integrates the characteristics of both mean squared error and mean absolute error, thereby mitigating noise sensitivity and improving model robustness. The uncertainty constraint includes a variance calibration term 1/2[σi2+(μi−yi)2/σi2] and a variance regularization term 1/10ln⁡(1+σi2). The variance calibration term consists of the variance and the standardized residual (μi−yi)2/σi2. The predicted variance directly limits variance overestimation, while the residual term associates prediction errors with variance. The variance regularization term contains the natural logarithmic function to restrict excessive growth of variance and avoids instability when variance approaches zero. Network parameters are optimized using the AdamW algorithm.

In the testing stage, MCD is used to quantify predictive uncertainty. Originally introduced as a regularization method to avoid overfitting, dropout randomly deactivates neurons during training, thereby reducing reliance on specific network pathways. MCD extends this strategy by maintaining dropout activation in the testing stage. By executing multiple stochastic forward passes, it produces a distribution over predictions, which facilitates the estimation of uncertainty.

Predictive uncertainty is quantified using the mean and variance of the outputs from MCD. This uncertainty consists of two components: epistemic and aleatoric uncertainty [46,47]. Epistemic uncertainty stems from the variability of network parameters (e.g., weights) and reflects the uncertainty of the network itself. It arises when limited training data prevent accurate estimation of optimal parameters. Aleatoric uncertainty is attributed to noise from the data acquisition process (e.g., sensor error, environmental disturbance) and is not related to the model parameters. Even when the network parameters are fixed, aleatoric uncertainty persists due to randomness in the outputs. The total predictive uncertainty is given by:(11)Var(y)=Var[E(y|x)]+E[Var(y|x)]where Var[E(y|x)] and E[Var(y|x)] represent epistemic and aleatoric uncertainty, respectively [48]. Epistemic uncertainty arises from variations in network parameters, which induce fluctuations in the expectation E(y|x). This variability is quantified by Var[E(y|x)], reflecting the intrinsic uncertainty within the network. In contrast, aleatoric uncertainty, expressed as E[Var(y|x)], directly captures the variance due to data noise. The 95% Confidence Interval (CI) [49] for predictions can be expressed as. (12)CI=E(y|x)±1.96⋅Var(y)

To address the challenge of multi-scale feature extraction in SAP degradation for LEO satellites, this paper proposes a CTCTN framework for satellite failure prognosis and uncertainty quantification, as illustrated in Fig. 6. The proposed framework consists of three main components: scenario configuration and data processing, prediction network construction, and uncertainty quantification.

In the first component, a LEO satellite simulation scenario is developed using a digital satellite development platform [50,51]. Subsequently, failures are injected into the satellite model during simulation to generate run-to-failure data, from which a satellite run-to-failure dataset is constructed under diverse operational conditions.

In the second component, the CTCTN architecture is proposed by combining temporal convolution with attention mechanisms to extract multi-scale degradation features from historical data. The network stacks DS-TC blocks to achieve local feature extraction and dimensionality reduction. An AAP then adaptively aligns feature dimensions, facilitating cascading of DS-TC blocks and Transformer modules. The Transformer module utilizes self-attention mechanisms to perform global contextual modeling of temporal features, establishing dependencies across different temporal scales and highlighting critical degradation patterns. By incorporating learnable query vectors with a multi-head attention, MHAP facilitates cross-scale feature interactions. Finally, fully connected layers are used to output the predictive mean and variance of the RUL.

In the third component, both epistemic and aleatoric uncertainties are quantified. During training, a heteroscedastic Huber loss function is employed to optimize the network parameters by jointly learning the predictive mean and variance of the RUL. The Huber loss function computes residuals using a piecewise structure. The uncertainty constraint processes predictive variance through a variance calibration term and a variance regularization term. During testing, the MCD method is applied to quantify both epistemic and aleatoric uncertainties. The proposed framework ultimately produces predictive intervals for RUL.

A LEO satellite model is developed to generate run-to-failure datasets under various operational conditions. Space debris poses a serious threat to satellite safety, and a damage to SAP can lead to significant degradation of power generation capacity [39,40]. Accurate RUL prediction for satellites SAP degradation is therefore critical [53]. Accordingly, this study develops a satellite model to simulate the power output degradation process of a damaged SAP. The satellite model integrates multiple subsystems, including attitude and orbit control, power, thermal control, propulsion, payload, and structure subsystem. For more model details, readers can refer to references [54,55]. The attitude and orbit control subsystem governs orbital and attitude maneuvers, while the power and thermal control subsystems manage energy and thermal balance, respectively. In the healthy state, the output power of the SAP is expressed as follows:(13)Pwing=CshadowS0AwingηXwingcos⁡(θSW)(βPΔTwing+1)where Cshadow identifies whether the solar wings are in the Earth’s shadow, S0 quantifies the solar irradiance, Awing represents the area of the photovoltaic array on the solar wings, η denotes the photoelectric conversion efficiency of the array, θSW describes the angle between the normal vector of the solar wings and the sunlight’s direction, βP is the power temperature coefficient of the solar wings, Xwing encompasses additional coefficients, and ΔTwing indicates the temperature differential from the standard temperature of the solar wings.

At a random time during the simulation, a failure is injected, after which the output power of the SAP begins to degrade. The injected parameter is the degradation duration tduration. Both the failure injection time and the degradation duration are randomized to generate diverse operating conditions for network training and to enhance its generalization capability. Following fault injection, the degraded output power of the SAP is expressed as Pwing,deg⁡radation=Pwing⋅e−λΔtdeg⁡radation, where λ=k/tduration, Δtdegradation represents the simulation time of the degradation process, and k denotes the degradation coefficient. To account the uncertainty in the degradation process, a multiplicative uniform noise of ±5% is introduced to the degradation coefficient.

The initial parameters for the satellite model based on the real LEO satellite (referred to as SL) are listed in Table 1.

Fig. 7 presents a comparison between the telemetry data of SL and the simulation data generated by the satellite model. In Fig. 7, the solid line represents the telemetry data, while the marked line denotes the simulation data. It can be observed that the orbital and attitude of the satellite model are consistent with telemetry data. Moreover, the trends of component parameters—such as current, voltage, and temperature—also exhibit strong agreement. These results confirm that the operating condition of the satellite model are consistent with SL.

The multi-scale features in the degradation process are analyzed using the simulation data shown in Fig. 8, under both normal and degraded states. In Fig. 8, the blue line represents the normal operational data, while the red line corresponds to the run-to-failure data. Moreover, the tEOL at 32,100 s and the tFPT at 7000 s are both marked using red dashed lines. The orbital period, highlighted by a blue box, is 6000 s. A 2000-s window, highlighted by the green box, of SAP output power before the moment ti (marked by a yellow star) is used as a sample, with the RUL at ti serving as the label.

As shown in Fig. 8, the output power of SAP exhibits progressive degradation following fault injection. This simulation is based on the performance degradation process of a damaged SAP described in previous studies [5,7,52]. During the initial degradation stage, the limited extent of damage has only a minor impact on SAP power generation. The power subsystem is still able to maintain energy balance, and battery capacity remains largely unchanged. Subsequently, as the severity of degradation increases, the energy balance is disrupted, leading to an accelerated battery consumption. Due to the continuously changing relative positions of the satellite, Earth, and Sun, the satellite periodically enters Earth’s shadow. During these eclipse periods, the SAP cannot perform photoelectric conversion, causing its power output to drop to zero. The satellite also adjusts its attitude to an Earth-pointing mode during this time. It is noteworthy that the RUL cannot be inferred from the power data during eclipse, introducing uncertainty into the prediction. After exiting the eclipse, the satellite readjusts its attitude to a Sun-pointing mode, and the SAP power gradually recovers. Compared to the orbital period, the satellite’s attitude maneuver occurs rapidly, while SAP degradation is a slow process. These factors give rise to multi-scale features in the SAP power degradation process.

The satellite run-to-failure dataset comprises simulation data from 10 operational conditions, with the specific failure scenario parameters detailed in Table 2. The simulation model records data at a frequency of 1 Hz (once per second). Starting from the 2500-s into the simulation, the SAP output power over the preceding 2000 s is collected every 5 min as a sample. A total of 1022 samples are collected, with the number of samples per operating condition detailed in Table 2. The RUL at the time of sample collection, normalized between 0 and 1 according to Eq. (1), is assigned as the label. This data processing and labeling strategy is the same as logic of real satellite failure prognosis, where the RUL at a moment is inferred from recent historical data. Each operational condition is simulated once to generate the corresponding samples. To evaluate generalization capability, a leave-one-out strategy is adopted. Specifically, one operating condition is used as the test set, while the remaining conditions constitute the training set, and the average value of the evaluation metrics across all conditions are calculated. To mitigate overfitting, 20% of the training samples are reserved as a validation set.

To avoid the impact of hyperparameters on prediction performance, random search is adopted for hyperparameter optimization. The search ranges and corresponding results are listed in Table 3. To reduce the risk of overfitting, a dropout rate is set to 0.2. GELU is used as the activation function to provide a smooth, non-linear activation. The AdamW optimizer is employed to minimize the loss function and update network parameters. The learning rate is set to 0.001, the batch size is set to 128, and the number of epochs is 500. α and δ are set to 0.1 and 1, respectively, in proposed heteroscedastic Huber loss function. Finally, the number of Monte Carlo iterations is set to 100.

The predictive performance of the proposed CTCTN is evaluated using the satellite run-to-failure dataset. Fig. 9 illustrates the predicted RUL for operational conditions 9 and 10 under the leave-one-out testing. In the upper subplot, the blue line denotes the true RUL, the red line represents the predicted RUL, and the gray-shaded area indicates the 95% confidence interval. In the lower subplot, the blue points correspond to epistemic uncertainty, the red points to aleatoric uncertainty, and the black points to total uncertainty.

As shown in Fig. 9, the proposed CTCTN effectively predicts the RUL of the satellite SAP. During the healthy stage, the predicted RUL closely matches the true RUL, and the associated uncertainty remains stable. In the degradation stage, both the predicted RUL and uncertainty exhibit periodic fluctuations, which are particularly pronounced under operational condition 10. These fluctuations correspond to eclipse periods, during which the SAP generates zero power, thereby affecting the RUL prediction. Despite this, the overall degradation trends of the true and predicted RUL remain consistent, and the true RUL is consistently encompassed within the predicted uncertainty interval. Furthermore, in the uncertainty quantification results, the aleatoric uncertainty is larger than the epistemic uncertainty, representing the main part of the total uncertainty. This indicates that the network has effectively learned the degradation patterns, while the multi-scale features of the satellite system primarily contribute to the aleatoric uncertainty in the predicted RUL. In summary, the proposed CTCTN framework effectively predicts the RUL of the satellite SAP and provides well uncertainty quantification.

To validate the effectiveness of the cascaded design in CTCTN, ablation experiments are conducted on modules and connection types. Experimental configurations are listed in Table 4, where Models A and B include only a single module, while Models C, D, and E employ different connection types.

The prediction results of each model are presented in Table 5, demonstrating the effectiveness of the proposed CTCTN structure. Firstly, compared to Model A, Model E achieves RMSE reductions of 50%, with corresponding MAE reductions of 49%. Similarly, compared to Model B, Model E reduces RMSE by 45%, and MAE by 42%. There results indicate that Models A and B, which incorporate only a single module, exhibit limited predictive capability for multi-scale degradation features, as they are restricted to processing either local or global features alone. Secondly, Model E reduces RMSE by 54% compared to Model C, with MAE reductions of 58%. Compared to Model D, Model E achieves RMSE reductions of 55%, and MAE reductions of 57%. These results demonstrate the impact of different connection types on predictive performance for the satellite SAP degradation process.

The impact of connection types on predictive performance is further examined. Model C shows no improvements over Model A and B. This suggests that, the parallel structure fails to fully exploit the advantages of the single module. In model D, the Transformer first transforms multi-scale features into a global representation, but the subsequent DS-TC is unable to sufficiently capture local patterns, resulting in a poor predictive performance. In contrast, the proposed cascaded design—where the DS-TC preceding the Transformer—achieves effective multi-scale feature fusion and superior prognostics. This arrangement allows the DS-TC to extract local features while performing dimensionality reduction, and simultaneously enhances the Transformer’s capacity to model long-range dependencies.

Furthermore, the capability for uncertainty quantification is evaluated using the PICP and MPIW metrics. Model E achieves the highest PICP values and the narrowest MPIW. This indicates that the proposed CTCTN structure not only provides the most reliable coverage of uncertainty intervals but also produces the tightest interval widths, demonstrating superior uncertainty quantification performance for satellite failure prognosis.

A comparative analysis is further conducted to evaluate performance differences among different modules integrated after the Cascade TC-Transformer. The results are summarized in Table 6. Compared with the Global Average Pooling (GAP), MHAP achieves a 63% reduction in RMSE and a 69% reduction in MAE. The inferior predictive performance of GAP can be attributed to its averaging across all time steps, which limits its ability to capture multiscale feature variations. In addition, compared with the Classification (CLS) token, MHAP yieldes a 46% reduction in both RMSE and MAE. By leveraging the multi-head attention mechanism, MHAP exhibits a stronger capability for extracting multi-scale features than the CLS token. Consequently, MHAP is better suited for the satellite fault prognosis task.

To further validate the performance of the proposed CTCTN for satellite failure prognosis, it is compared with current prognostic methods. The specific configurations of these comparative methods are detailed as follows, with hyperparameters kept consistent with those of the proposed method:

MCA-TCN [35]: The network integrates three TCN blocks with multi-channel attention for feature fusion. Each TCN block contains three dilated convolutional layers with 16 channels and a kernel size of seven. The fused features are processed by a GRU layer with 18 hidden units, followed by a fully connected layer for the RUL.

Fig. 10 presents the prediction results of all methods under operating conditions 9 and 10, where the lines represent the true RUL and the predicted RUL of each method and the scatter plots depict the overall uncertainty. It is evident that the proposed CTCTN achieves superior predictive performance. Specifically, during the healthy state, the RUL predicted by CTCTN remains consistently close to the true RUL. In contrast, methods including CNN-Transformer and MCA-TCN yield lower RUL predictions under condition 9. Furthermore, during the degradation state, the predicted RUL from CTCTN more accurately tracks the true RUL, whereas TCLSTM, Attention-BiLSTM, Transformer-LSTM, and CNN-Transformer exhibit larger fluctuations.

The statistic prediction results of all methods are summarized in Fig. 11 and Table 7. To avoid the impact of randomness, 10 repetitions of the experiment are conducted and the mean and variance are reported. The findings demonstrate that the proposed CTCTN method achieves the best overall performance in terms of both prediction accuracy and uncertainty quantification. TCLSTM exploits the strengths of TCN in extracting temporal features from historical data for RUL prediction. However, their feature extraction capabilities are limited in the satellite SAP degradation scenario, which involves multi-scale features. Although TCLSTM incorporates an attention mechanism, its single-head attention cannot simultaneously capture degradation patters across multiple time scales. A similar limitation is also observed in the attention-based MCA-TCN. The predictive performance of Transformer-LSTM is constrained by its preceding Transformer module, which is consistent with the findings of the ablation study. Although the architecture of CNN-Transformer is similar to that of the proposed CTCTN, CNNs have limited capacity in processing temporal data. Attention-BiLSTM, owing to its residual connections, effectively mitigates the vanishing gradient problem and consequently achieves relatively low RMSE. Informer and PatchTST are recent strong baselines. Compared with Informer, the proposed CTCTN reduces RMSE and MAE by 10.60% and 12.65%, respectively. Furthermore, while achieving a PICP of 98.48%, the proposed CTCTN also yields a relatively low MPIW. In summary, the proposed CTCTN effectively captures multi-scale features in satellite SAP degradation data, leading to state-of-the-art predictive performance and superior uncertainty quantification.

For resource-constrained satellite applications, computational complexityof the prognostic method is critical. Table 8 presents the parameter counts, FLOP counts, and average inference time per sample for each comparative method. It can be observed that the proposed CTCTN achieves a computational efficiency that satisfies the requirements for real-time processing. Specifically, compared with Informer and PatchTST, the proposed CTCTN achieves better predictive performance while having lower inference time and fewer parameters.

Based on the experiments presented above, the proposed CTCTN achieves superior prognostic performance compared with existing methods in the satellite SAP degradation scenario. This advantage arises from the alignment between the network architecture and the multi-scale characteristics of the degradation process. Traditional TCN-based methods, although capable of enlarging the receptive field through stacked DCCs, still process sequences using local sliding windows, which inherently limits their ability to model global dependencies. Attention-based methods excel at capturing long-range dependencies but exhibit limited capability in modeling local temporal features. CTCTN addresses these limitations through a cascaded design that leverages complementary strengths. The front-end DS-TC block extracts local features while simultaneously reducing sequence length. The subsequent Transformer module then performs multi-head self-attention within the compressed feature space, enabling each time step to capture long-term dependencies across the entire degradation process. Fig. 12 shows the heat maps of the attention weights of the 4 heads in the Transformer module. The x-axis and y-axis represent the time steps being focused on and the current time step, respectively. From the diagonal distribution to the global distribution, the different distribution patterns of the attention weights indicate that features of the different time scales have been learned by the Transformer module. This progressive fusion strategy makes CTCTN superior to both single-module architectures and traditional hybrid structures. This cascaded design constitutes the core innovation of this paper.

While CTCTN demonstrates robust prognostic capability in simulated scenarios, it presents several challenges against transferring CTCTN to other prognostics domains, which also indicate promising directions for future research. The distribution differences between the source domain and the target domain limit the generalizability of the prediction methods. Therefore, cross-domain research represents a potential direction.